A Coupled Runge Kutta Discontinuous Galerkin-Direct GhostFluid (RKDG-DGF) Method to Near-field Early-time Underwater

Explosion (UNDEX) Simulations

Jinwon Park

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Aerospace Engineering

Dr. Alan J. Brown, Chair

Dr. Owen F. Hughes

Dr. Rakesh K. Kapania

Dr. Leigh S. McCue-Weil

Dr. Danesh K. Tafti

August 28, 2008

Blacksburg, Virginia

Keywords: Underwater Explosion (UNDEX), Runge Kutta Discontinuous Galerkin (RKDG),

Ghost Fluid Method (GFM), Bubble, Cavitation, Level Set Method (LSM), Multi-fluid

Copyright 2008, Jinwon Park

A Coupled Runge Kutta Discontinuous Galerkin-Direct GhostFluid

(RKDG-DGF) Method to Near-field Early-time Underwater Explosion

(UNDEX) Simulations

Jinwon Park

(ABSTRACT)

A coupled solution approach is presented for numerically simulating a near-field underwater ex-

plosion (UNDEX). An UNDEX consists of a complicated sequence of events over a wide range

of time scales. Due to the complex physics, separate simulations for near/far-field and early/late-

time are common in practice. This work focuses on near-field early-time UNDEX simulations.

Using the assumption of compressible, inviscid and adiabatic flow, the fluid flow is governed by

a set of Euler fluid equations. In practical simulations, we often encounter computational diffi-

culties that include large displacements, shocks, multi-fluid flows with cavitation, spurious waves

reflecting from boundaries and fluid-structure coupling. Existing methods and codes are not able

to simultaneously consider all of these characteristics.

A robust numerical method that is capable of treating large displacements, capturing shocks, han-

dling two-fluid flows with cavitation, imposing non-reflecting boundary conditions (NRBC) and

allowing the movement of fluid grids is required. This methodis developed by combining nu-

merical techniques that include a high-order accurate numerical method with a shock capturing

scheme, a multi-fluid method to handle explosive gas-water flows and cavitating flows, and an

Arbitrary Lagrangian Eulerian (ALE) deformable fluid mesh.These combined approaches are

unique for numerically simulating various near-field UNDEXphenomena within a robust single

framework. A review of the literature indicates that a fullycoupled methodology with all of these

characteristics for near-field UNDEX phenomena has not yet been developed.

A set of governing equations in the ALE description is discretized by a Runge Kutta Discontinuous

Galerkin (RKDG) method. For multi-fluid flows, a Direct GhostFluid (DGF) Method coupled with

the Level Set (LS) interface method is incorporated in the RKDG framework. The combination of

RKDG and DGF methods (RKDG-DGF) is the main contribution of this work which improves the

quality and stability of near-field UNDEX flow simulations. Unlike other methods, this method is

simpler to apply for various UNDEX applications and easier to extend to multi-dimensions.

Dedication

To Yejoo, my wife, Jesung (Jason), my son and my parents

iii

Acknowledgements

I would like to express my deep and sincere gratitude to my advisor, Dr. Alan J. Brown for

introducing me to this wonderful research field. I would alsolike to acknowledge and thank him

for his guidance and support during the past four years of my doctoral program. Dr. Owen F.

Hughes, Dr. Rakesh K. Kapania, Dr. Leigh S. McCue-Weil and Dr. Danesh K. Tafti were the other

professors in the research committee and I am grateful for their valuable suggestions. I am indebted

to all the past and present members of Virginia Tech Ship Survivability Research group, especially

Ashley Nisewonger, Bradley Klenow, Keith Webster, Jason Cordell, and Dr. John Sajdak for their

constructive comments and suggestions during the researchmeetings and general discussions. I

would like to thank Republic of Korea Navy (ROKN) for providing this opportunity to pursue

a doctorate degree and for the financial support during the past four years. I also wish to thank

Professor Sukyoon Hong of Seoul National University, Koreafor providing the first experience in

graduate research which has given me the confidence to succeed in my doctoral program.

Personally, there are a lot of people who made my stay in Blacksburg very memorable and enjoy-

able. A special thanks to Sungsub Lee (ROKAF commander), Dr.Jonghoon Nam at University of

Wisconsin-Madison, Dr. Feng Zhou at Intergraph Corporation, Korean Graduate Students in Me-

chanics (KGSM) at Virginia Tech and Blacksburg United Methodist church Preschool (BUMP).

Finally, I would like to thank my parents, sisters and brothers for their life long love, encourage-

ment and support.

iv

Contents

1 Introduction 11.1 Near-field underwater explosions. . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Analytical/empirical methods. . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Experimental methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Numerical methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3.1 Acoustic wave approaches. . . . . . . . . . . . . . . . . . . . . 71.2.3.2 Cavitating fluid approach. . . . . . . . . . . . . . . . . . . . . 81.2.3.3 Computational fluid dynamics. . . . . . . . . . . . . . . . . . 91.2.3.4 Hydrocodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Recent VT ship survivability research. . . . . . . . . . . . . . . . . . . . . . . . 141.3.1 Deep spherical TNT explosion simulation. . . . . . . . . . . . . . . . . . 141.3.2 U.S Navy blast test simulation. . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Objective and contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Continuum mechanics approach to the UNDEX problem 212.1 Basics of continuum mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Fundamental definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.3 Strains and stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Governing equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 Conservation of mass: continuity equation. . . . . . . . . . . . . . . . . . 332.2.2 Conservation of momentum: momentum equation. . . . . . . . . . . . . 352.2.3 Conservation of energy: energy equation. . . . . . . . . . . . . . . . . . 372.2.4 Additional equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.4.1 Constitutive relations. . . . . . . . . . . . . . . . . . . . . . . 412.2.4.2 Equations of state. . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Discretizations 533.1 Brief introduction to the RKDG method. . . . . . . . . . . . . . . . . . . . . . . 543.2 Spatial discretizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 One-dimensional weak form. . . . . . . . . . . . . . . . . . . . . . . . . 55

v

3.2.2 Multi-dimensional weak form. . . . . . . . . . . . . . . . . . . . . . . . 573.2.3 Basis function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.4 Numerical flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Temporal discretizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4 Generalized slope limiter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Solution methods 774.1 Two-fluid method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.1 Interface methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1.1.1 Interface presentation. . . . . . . . . . . . . . . . . . . . . . . 814.1.1.2 Interface evolution. . . . . . . . . . . . . . . . . . . . . . . . . 834.1.1.3 Interface reinitialization. . . . . . . . . . . . . . . . . . . . . . 84

4.1.2 Two fluid methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1.2.1 Ghost fluid method. . . . . . . . . . . . . . . . . . . . . . . . 884.1.2.2 Direct ghost fluid method. . . . . . . . . . . . . . . . . . . . . 94

4.2 Fluid-structure interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.1 Solution method for the structure. . . . . . . . . . . . . . . . . . . . . . 994.2.2 Interface conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2.3 Mesh Coupling algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2.4 Cavitation in FSI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Assessment 1085.1 One-dimensional assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1.1 Cartesian 1-D cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.1.2 Symmetric 1-D cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Multi-dimensional assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.1 1D Spherical Bubble Pulses. . . . . . . . . . . . . . . . . . . . . . . . . 1305.3.2 Cavitating flows in multi-dimensions. . . . . . . . . . . . . . . . . . . . 135

6 Conclusions 148

Bibliography 151

A Mathematical theorems and rule 162

B FE-FCT algorithm 163

C The slope limiting procedure 166

D 2D Geometric quantities 168

E Communication between the fluid and the structure at the interface 169

F The implementation of Boundary Conditions 170

vi

List of Figures

1.1 Classification of UNDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Process of a near-field UNDEX close to a target [65]. . . . . . . . . . . . . . . . 41.3 Experimental UNDEX studies [63]. . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 The pyramid-shaped fluid model for the deep spherical TNTexplosion problem. . 151.5 Bubble radius vs. time for the 1D deep spherical bubble problem . . . . . . . . . 161.6 US Navy blast tests and LS-DYNA simulations. . . . . . . . . . . . . . . . . . . 17

2.1 The motion of a material pointO [143] . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Maps between different configurations [143]. . . . . . . . . . . . . . . . . . . . . 232.3 Example of the Lagrangian mesh and the Eulerian mesh [143] . . . . . . . . . . . 242.4 Remapping of Eulerian and ALE descriptions [99, 105]. . . . . . . . . . . . . . . 252.5 A schematic of the deformation gradientF [21] . . . . . . . . . . . . . . . . . . . 292.6 An illustration of the initial condition and flow fields attime t = t . . . . . . . . . 45

3.1 1D presentation of DG approximations. . . . . . . . . . . . . . . . . . . . . . . 563.2 2D representation of the flux computation between element Ωj and it neighbors. . 583.3 Canonical elements in one and two dimensions [18]. . . . . . . . . . . . . . . . . 593.4 Illustration of Legendre polynomials, p=0,1,2...,5. . . . . . . . . . . . . . . . . . 603.5 The density profiles and percentage errors (%). . . . . . . . . . . . . . . . . . . 623.6 The convergence of DG approximations (p=0, p=1 and p=2). . . . . . . . . . . . 633.7 1D mapping between element(xj− 1

2, xj+ 1

2) and canonical element (-1, 1). . . . . 64

3.8 Description of the FVM and the DGM. . . . . . . . . . . . . . . . . . . . . . . . 653.9 The initial condition of the Fedkiw’s strong shock tube problem [135] . . . . . . . 703.10 Numerical results of the Fedkiw’s strong shock tube problem . . . . . . . . . . . . 713.11 The illustration of a general minmod slope limiter. . . . . . . . . . . . . . . . . 743.12 The performance of the generalized slope limiter for the Fedkiw’s problem . . . . 76

4.1 Comparison of a multi-fluid and a multi-phase flow [128]. . . . . . . . . . . . . 784.2 Description of interface methods in two-dimensions [76] . . . . . . . . . . . . . . 794.3 Graphical representations of the interface [140]. . . . . . . . . . . . . . . . . . . 81

vii

4.4 Contours of the LS functionφ with a centered circle. . . . . . . . . . . . . . . . . 824.5 Contours of the LS functionφ with two circles. . . . . . . . . . . . . . . . . . . . 824.6 Numerical diffusion generated after single Eulerian step. . . . . . . . . . . . . . . 864.7 Identification of the interface location in one-dimension . . . . . . . . . . . . . . . 884.8 Description of the GFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.9 Analytic solutions of the Sod’s shock tube problem in[0, 1] at t=0.2 second. . . . 904.10 1D presentation of the GFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.11 1D presentation of the GFM coupled with the isobaric fix technique . . . . . . . . 914.12 Initial conditions of an explosive gas-water flow example . . . . . . . . . . . . . . 924.13 Density profile with a overshoot at the interface. . . . . . . . . . . . . . . . . . . 934.14 The description of the DGFM for explosive gas-water flows . . . . . . . . . . . . . 944.15 RKDG-DGF results for a 1-D explosive gas-water flow. . . . . . . . . . . . . . . 954.16 Description of a two-dimensional DGFM procedure. . . . . . . . . . . . . . . . . 964.17 A feedback of FSI simulations[71] . . . . . . . . . . . . . . . . . . . . . . . . . . 984.18 Flow chart of the two-step Newmarkβ time scheme [143] . . . . . . . . . . . . . 994.19 A simple axial-bar example using the two-step Newmarkβ time scheme. . . . . . 1004.20 FSI mesh coupling algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.21 An example of mesh couplings: red (fluid) and blue (structure) . . . . . . . . . . . 1024.22 A 2D configuration of FSI in near-field UNDEX. . . . . . . . . . . . . . . . . . . 1034.23 Description of an averaging mesh- smoothing scheme in two-dimensions [20]. . . 1034.24 The traditional view of cavitation mechanism [6]. . . . . . . . . . . . . . . . . . 1044.25 The flow-chart of FSI simulations using the non-matching scheme . . . . . . . . . 106

5.1 Initial conditions for Case 5.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2 RKDG-DGF results for Case 5.1.1 at 0.16E-3 seconds. . . . . . . . . . . . . . . 1115.3 Initial conditions for Case 5.1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4 RKDG-DGF results for Case 5.1.2 at 1.55921E-04 seconds. . . . . . . . . . . . 1125.5 Initial conditions for Case 5.1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.6 RKDG-DGF results for Case 5.1.3 at 0.1E-3 seconds. . . . . . . . . . . . . . . . 1145.7 Initial conditions for Case 5.1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.8 RKDG-DGF results for Case 5.1.4 at 0.5E-3 seconds. . . . . . . . . . . . . . . . 1155.9 Initial conditions for Case 5.1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.10 RKDG-DGF results for Case 5.1.5 at 0.25 seconds. . . . . . . . . . . . . . . . . 1185.11 Initial conditions for Case 5.1.6. . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.12 RKDG-DGF results for Case 5.1.6 at 0.455E-3 seconds. . . . . . . . . . . . . . 1205.13 Initial conditions for Case 5.1.7. . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.14 RKDG-DGF results for Case 5.1.7. . . . . . . . . . . . . . . . . . . . . . . . . . 122

viii

5.15 Initial conditions for Case 5.2.1 at 0.1E-3 seconds. . . . . . . . . . . . . . . . . 1245.16 RKDG-DGF results for Case 5.2.1 at 0.1E-3 seconds. . . . . . . . . . . . . . . . 1255.17 Density contours for Case 5.2.1 at 0.1E-3 seconds. . . . . . . . . . . . . . . . . 1255.18 Pressure contours for Case 5.2.1 at 0.1E-3 seconds. . . . . . . . . . . . . . . . . 1265.19 Initial conditions for Case 5.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.20 RKDG-DGF results for Case 5.2.2 at 0.3E-3 seconds. . . . . . . . . . . . . . . . 1285.21 Density contours for Case 5.2.2 at 0.3E-3 seconds. . . . . . . . . . . . . . . . . 1285.22 Pressure contours for Case 5.2.2 at 0.3E-3 seconds. . . . . . . . . . . . . . . . . 1295.23 UNDEX gas bubble dynamics [116]. . . . . . . . . . . . . . . . . . . . . . . . . 1315.24 Initial conditions for Case 5.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.25 RKDG-DGF results for Case 5.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . 1325.26 Initial conditions for Case 5.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.27 RKDG-DGF results for Case 5.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . 1345.28 Initial conditions and the fluid mesh for Case 5.3.3. . . . . . . . . . . . . . . . . 1365.29 RKDG-DGF results on the rigid wall for Case 5.3.3. . . . . . . . . . . . . . . . 1375.30 Pressure contour of Case 5.3.3 at 0.0036 seconds. . . . . . . . . . . . . . . . . . 1385.31 Pressure contour of Case 5.3.3 at 0.0051 seconds. . . . . . . . . . . . . . . . . . 1385.32 Experimental arrangement and initial conditions for Case 5.3.4. . . . . . . . . . . 1405.33 RKDG-DGF results for Case 5.3.4 with the rigid wall. . . . . . . . . . . . . . . 1415.34 Pressure contours for Case 5.3.4 with the rigid wall. . . . . . . . . . . . . . . . . 1425.35 The initial FSI meshes for Case 5.3.4. . . . . . . . . . . . . . . . . . . . . . . . 1435.36 RKDG-DGF results for Case 5.3.4 with the elastic wall. . . . . . . . . . . . . . . 1445.37 Pressure contours for Case 5.3.4 with the elastic wall. . . . . . . . . . . . . . . . 145

B.1 The schematic of FCT procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.2 The initial conditions for the SOD’s shock tube problem. . . . . . . . . . . . . . 165B.3 Comparison of FE-FCT results and RKDG results at 0.2 seconds . . . . . . . . . . 165

D.1 Numbering in the counter-clockwise direction for quadrilateral element . . . . . . 168

E.1 Communication between the fluid and the structure. . . . . . . . . . . . . . . . . 169

F.1 Cell numbering and concept of dummy cells. . . . . . . . . . . . . . . . . . . . 170F.2 Description of sponge layer concept. . . . . . . . . . . . . . . . . . . . . . . . . 173F.3 The initial mesh with the sponge layer for Case 5.3.3. . . . . . . . . . . . . . . . 174F.4 Pressure contours for Case 5.3.3 with the sponge layer. . . . . . . . . . . . . . . 175

ix

List of Tables

1.1 Important UNDEX simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Prominent mesh-based methods and mesh-free methods [54] . . . . . . . . . . . . 11

1.3 Comparison of prominent hydrocodes. . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Transformation between stresses [143]. . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Summary of universal conservation laws in the non-conservative form . . . . . . . 40

2.3 Summary of universal conservation laws in the conservative form . . . . . . . . . . 40

2.4 Lagrangian governing equation for the structure. . . . . . . . . . . . . . . . . . . 40

2.5 ALE governing equations for the fluid. . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 Use of JWL EOS and ideal gas law for explosive gas. . . . . . . . . . . . . . . . 46

2.7 Coefficients in the JWL equation of state. . . . . . . . . . . . . . . . . . . . . . 46

2.8 Use of equations of state for water. . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.9 Coefficients in the Gruneisen equation of state for water. . . . . . . . . . . . . . 49

2.10 Material properties and coefficients in the stiffened gas EOS . . . . . . . . . . . . 50

2.11 Material property and coefficients for waters in the Tait EOS . . . . . . . . . . . . 51

3.1 ALE - Euler fluid governing equations. . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 TheL1 errors and the orders of accuracy for the densityρ . . . . . . . . . . . . . . 63

4.1 Use of interface methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Compressible multi-fluid methods in the literature. . . . . . . . . . . . . . . . . . 87

5.1 Comparisons with the reference results. . . . . . . . . . . . . . . . . . . . . . . 133

5.2 Comparisons with the reference results. . . . . . . . . . . . . . . . . . . . . . . 135

x

Acronyms

ALE Arbitrary Lagrangian Eulerian

AMR Adaptive Mesh Refinement

AWA Acoustic Wave Approaches

BC Boundary Condition

BE(M) Boundary Element (Method)

BI(M) Boundary Integral (Method)

CASE/FE Cavitation Acoustic Spectral Element/Finite Element

CEL Coupled Eulerian Lagrangian

CFA Cavitating Fluid Approaches

CFD Computational Fluid Dynamics

DGF(M) Direct Ghost Fluid (Method)

DG(M) Discontinuous Galerkin (Method)

EOS Equation of State

FCT Flux Corrected Transport

FD(M) Finite Difference (Method)

FE(M) Finite Element (Method)

FORCE First Order Center

FSI Fluid Structure Interaction

FV(M) Finite Volume (Method)

GCL Geometric Conservation Law

xi

GF(M) Ghost Fluid (Method)

HE High Explosive

JWL Jones-Wilkins-Lee

LHS Left Hand Side

LLF Local Lax-Friedrich

LS(M) Level Set (Method)

MGFM Modified Ghost Fluid Method

NPG Naval Postgraduate School

NRBC Non Reflecting Boundary Condition

NSWC Naval Surface Warface Center

ODE Ordinary Differential Equation

OOP Objected-Oriented Programming

PC Prediction-Correction

PDE Partial Differential Equation

PETN Pentaerythritol tetranitrate

PK2 Piola Kirchhoff

RCM Random Choice Method

RHS Right Hand Side

RK(DG) Runge Kutta (Discontinuous Galerkin)

SF Simple Fix

SPS Sandwich Panel System

TNT Trinitrotoluene

TVD Total Variation Diminishing

UNDEX Underwater Explosion

USA Underwater Shock Analysis

VOF Volume of Fluid

VT Virginia Tech

xii

Chapter 1

Introduction

An underwater explosion (UNDEX) consists of a complicated sequence of the events that include:

detonation waves, shock waves, fluid-fluid interactions of detonation-produced explosive gas and

water, fluid-structure interactions and cavitations [8, 131, 155]. These events occur over a wide

range of time scales. Figure1.1 represents the classification of UNDEX based on hydrodynamic

phenomena and structural behaviors [8, 99, 131].

Figure 1.1. Classification of UNDEX

1

To accurately model all these aspects of UNDEX, extensive method testing and development,

fine-tuning of numerical methods and large computing resources are required [54, 83, 99]. Due

to these difficulties and complex physics, separate simulations for near/far-field and early/late-

time are common in practice [9, 14, 25, 51, 55, 73, 83, 92, 152, etc]. This work focuses on the

near-field early-time UNDEX simulations that include shock-bubble interactions, bubble-structure

interactions and cavitation.

1.1 Near-field underwater explosions

As shown in Figure1.1, UNDEX may be defined by the response of a structure subjectedto

a charge detonation under the surface of water. If a sufficiently large detonation occurs close

to the structure, severe plastic deformation may occur as well as structural failure [8]. As the

distance between the charge and the structure, or the detonation distance increases, a point is

reached where the structure no longer ruptures, but the plastic deformation still occurs [8, 155]. By

increasing the detonation distance ever further, a point isreached where only elastic deformation

occurs [8, 131, 155].

A far-field UNDEX event often involves the global response ofthe structure such as whipping (i.e.

ship hull girder vibration) [8, 131]. However, it can also cause shock damage inside a ship that is of

interest in shock-resistance design [63]. Using the assumption of inviscid and incompressible flow,

the fluid in far-field UNDEX is commonly assumed to be an acoustic medium [14, 71, 111, 152,

163]. Due to many necessary assumptions, the far-field UNDEX approaches [25, 26, 87, 129, 152,

153] are not valid for the near-field UNDEX simulations in which strong shock-bubble interaction

may occur, and the motions of both the fluid and the structure are large [54, 71, 131].

Unlike far-field UNDEX, shock waves, charge fragmentationsand bubble effects may cause con-

siderable plastic deformation and structural failure [8]. For example, the peak magnitude of pri-

2

mary shock wave is on the order of109 Pa for a small-radius TNT charge [7, 17]. The structural

responses involve many complex phenomena over a wide range of time scales from microseconds

for the response to shock waves to milliseconds for the response to bubble effects [8, 131, 155].

In late-time, the structure undergoes complex responses due to bubble effects such as bubble-

structure interaction, bubble jetting and so forth [8, 65, 131, 155]. Since the fluid state inside the

bubble is almost uniform (i.e., nearly constant spatial variation of density [142]), the fluid flow is

often modeled to be incompressible [9, 22, 23, 39, 41, 101, 121, 139]. Using the assumption of

incompressible, inviscid and irrotational flow (i.e. potential flow), Q. X. Wang [121], A.M. Zang

[9], E. Klaseboer [41] and C. Wang [22, 23] used the Boundary Element (BE)/Boundary Integral

(BI) method to discretize the resulting Laplace equation for simulating bubble behaviors.

In early-time, the intensity of the shock wave determines ifa structure undergoes elastic deforma-

tion, yielding, plastic deformation or rupture [8]. Initially, there is an upward acceleration of the

structure, due to the transient intensive pressure loading, which continues until the structure begins

to move faster than the surrounding water [14, 131]. At this moment, water near the structural

interface is exposed to tension, resulting in local cavitation (i.e. sufficiently low pressure,≪1bar).

Water cannot sustain the tension. Once the structure reaches its maximum upward velocity, or

kickoff velocity, the structural motion begins to slow downand the local cavitation region closes

[8, 131]. This closure causes the structure to be reloaded. The reloading pressure is not as strong as

that of the primary shock but may still be significant [14, 131]. The shock wave impact is dominant

so that the fluid flow is best characterized as a compressible inviscid flow [7, 30, 45, 85, 92].

Because of the large difference in terms of required numerical schemes and physical aspects, a

separate formulation is preferred to characterize solutions at different times [45, 71, 81, 86, 91,

132]. The numerical results of an early-time UNDEX simulation can be used as initial conditions

for the late-time UNDEX simulation including bubble effects [103]. This is discussed in Example

5.1.7 in Chapter 5. Figure1.2 illustrates the sequence of a near-field UNDEX. The key features

are shock wave, local cavitation, reloading and bubble jetting.

3

Figure 1.2. Process of a near-field UNDEX close to a target [65]

Near-field early-time UNDEX simulations must simultaneously consider various characteristics

such as large displacement, shock wave, multi-fluid flow, cavitation formation and collapse, spuri-

ous wave reflecting at boundaries, and fluid-structure coupling. Large displacement caused by the

extreme fluid motion results in large mesh distortion [143]. When the mesh is extremely distorted,

the method may suffer from stability problems which degradethe quality of numerical results or

lead to the sudden breakdown of computation [54]. Lagrangian-based methods are not generally

recommended in the flow computation [91]. Shock-involved flows may encounter spurious pres-

sure oscillations near the shock front during the computation. These oscillations may interfere

with the fluid field. Without eliminating or minimizing thesespurious pressure oscillations, the

fluid computation is not able to continue [92]. The UNDEX fluid flow involves multiple fluids

with different physical and thermodynamic properties across a material interface [140]. Since an

equation of state (EOS) for each fluid is used to describe the fluid field, different EOSs are applied

to each side across the material interface during the computation. This discontinuous employment

of EOSs can also be a source of nonphysical behaviors in the multi-fluid flow computations [92].

Classical single-fluid methods cannot solve this problem since the source of these oscillations dif-

fers from that of a shock-capturing mechanism [92]. J.P. Cocchi indicates that in multi-fluid flows,

general numerical diffusion created by classical methods causes nonphysically diffused density

resulting in spurious pressure oscillations [92]. The finite size of the computational domain is

usually determined by the size of the physical domain which we wish to solve. Computational

constraints and other considerations force a smaller sub-domain with arbitrary limits instead of a

4

larger physical domain [98, 132]. However, when using such a sub-domain, fluid waves reach the

arbitrary limits and the waves reflect from these limits backinto the flow [98, 132]. To minimize

their influence, a non-reflecting boundary condition (NRBC)is usually required in external explo-

sion simulations [147]. When a nearby structure is exposed to intensive UNDEX pressure loading,

it may undergo significant plastic deformation. During the computation, the permanent contact

between the structure and the fluid must be enforced [71]. Since Lagrangian methods are generally

employed to analyze the structural behavior [91], a deformable fluid mesh is required to adjust the

fluid mesh against the deformation of the structural interface.

1.2 Literature survey

UNDEX simulations have been the subject of analytical/empirical, experimental and numerical

research since the 1950s. Table1.1 overviews lists of methods that have been used in important

UNDEX studies.

Analytical/empirical Experimental Numerical

R. H. Cole[131], G. I.Taylor[52], I. S.

Sandler[67], R. P.Godwin[137]

R. R. Rajendran[126], K.Ramajeyahtilagam[97],R. H. Cole[131], A. H.

Keil[8]

Acoustic approximations:T. L. Geer[152],

C. A. Felippa[25],R. D.Mandlin[129],

C. C. Liang[28]

Cavitating fluid approximations:R. E. Newton[130],

B. Klenow[14],M. A. Sparague[111]

CFD methods:T. G. Liu[147], J. P. Cochhi[92], C. H.

Cooke[31], J. Flores[73], R. R.Nourgaliev[138], J. Qiu[78], A. P.

Pishevar[10], H. Luo[57]

Hydrocodes:Pangilnan[51], J. J. Dike[88], J. E.

Chisum[83], P. Ding[116], K.Webster[99]

Table 1.1. Important UNDEX simulations

5

The ability to predict structural damage subjected to UNDEXpressure loading is of great interest

in ship survivability design. Success at this task requiresthe accurate prediction of fluid flows as

well as the dynamic coupling between the structure and the fluid [7, 71].

1.2.1 Analytical/empirical methods

The similitude methods, developed in the 1940s, offer a quick and simple way to approximate the

characteristics of fluid flows in UNDEX events where the presence of a structure has no influence

[14, 131]. Using experimental parameters, generic equations and numerous assumptions, the meth-

ods provide a good understanding of fluid flows in UNDEX eventsthat include peak pressures, gas

bubbles and secondary pressure pulses [14, 131]. The Taylor plate model analyzes the response

of an infinite air-backed flat plate subjected to a plane shockwave [52, 131]. The Bleich-Sandler

model considers local cavitation near a structure [67]. The Rayleigh-Plesset equation focuses on

the behavior of gas bubbles with the assumption of potentialflow [55, 56].

Since parameters and equations in analytical/empirical methods are for the most part experimen-

tally determined, these methods are limited to simple casesand range of variables [85, 131].

1.2.2 Experimental methods

Experimental studies have been conducted to investigate local structural deformation [97, 126],

bubble pulses [131] and ship response [8, 63] to a designed charge detonation. Experimental

methods provide a good understanding of specific UNDEX events, but are costly and have poten-

tially negative impact on the environment [63]. Although experiments with total ships and ship

components are often conducted in naval ship acquisitions ,naval engineers wish to substitute re-

liable numerical simulations for ship trials/tests to reduce related costs, environmental impact and

safety concerns, and to evaluate a wider range of explosion scenarios and structural designs.

6

Figure 1.3. Experimental UNDEX studies [63]

Experimental methods are useful for predicting specific UNDEX cases, but these methods are not

generally applicable to realistic encounter situations.

1.2.3 Numerical methods

Various numerical approaches have been applied to assess the dynamic response of the fluid and

the structure in UNDEX simulations. To accurately predict the response of a structure subjected to

an UNDEX loading, both the accurate prediction of fluid flows and a reasonable dynamic coupling

of the structure to the fluid are necessary components [71]. Numerical methods used are divided

into four categories: Acoustic Wave Approaches (AWA), Cavitating Fluid Approaches (CFA),

Computational Fluid Dynamics (CFD) methods and Hydrocodes.

1.2.3.1 Acoustic wave approaches

In far-field UNDEX events, one can assume that the fluid has sufficiently small fluid perturbations,

negligible density change from its initial valueρ0, and small particle velocities [14, 131, 163].

With these assumptions, the fluid flow is often treated as an acoustic medium which allows the

7

use of acoustic wave approach (AWA) that include the plane wave approximation, the virtual mass

approximation, the doubly asymptotic approximation, etc [25, 26, 87, 129, 152, 153]. The AWA

estimates the acoustic wave pressure on the interface surface in the fluid-structure interaction. The

approximate wet surface pressure is employed as input load in computing the structural response.

Displacements output from the structure are used to force the fluid [14]. Since the AWA simply es-

timates the fluid pressure acting on the structure, it is typically decoupled from the structural code

to assess the response of the structure subjected to far-field UNDEX, greatly reducing the compu-

tational effort for far-field UNDEX simulations [14]. Due to such simplifications, classical AWA

does not account for explosive gas bubbles, cavitation and multi-fluid flows which are dominant in

near-field UNDEX [152].

1.2.3.2 Cavitating fluid approach

The Underwater Shock Analysis (USA) program is often used inconjunction with commercial FE

Codes such as the LS-DYNA [14, 105]. B. Klenow [14] at Virgina Tech used the LS-DYNA/USA

program to simulate a boxbarge-like structure subjected tofar-field UNDEX. LS-DYNA was used

to model the fluid domain with acoustic elements, and also analyze the response of a boxbarge-like

structure subjected to acoustic wave pressure. USA was usedto model the rest of the fluid domain

by acting as a radiation boundary on the fluid domain boundary. This method allowed the use of

a smaller sub-domain which greatly reduces computational efforts and time. Unlike the AWA, the

cavitating fluid approach (CFA) allows cavitation to be modeled in far-field UNDEX.

To model shock-induced cavitation, R.E. Newton developed aone-fluid cutoff homogeneous model

for the acoustic FEM [130]. The irrotational formulation of Newton is based on two scalar vari-

ables: displacement potentialΨ( ~X, t) and dynamic condensations( ~X, t) = 2Ψ. Recently, the

Cavitation Acoustic Spectral-Element/Finite Element (CASE/FE) method has been used to analyze

cavitating flows in far-field UNDEX simulations [106, 111]. Since far-field UNDEX simulations

only consider a one-fluid acoustic model with many assumptions [14, 25, 26, 87, 106, 111, 129,

130, 152, 153], these methods are not valid for near-field UNDEX simulations that include multi-

fluid flows.

8

1.2.3.3 Computational fluid dynamics

CFD codes have been used in conjunction with structural codes to assess the response of the struc-

ture subjected to a fluid loading [51]. For fluid-structure interaction simulations, the fluid pressure

obtained from a CFD code can be mapped to the input load for a structural code. Classical CFD

techniques [30, 31, 92, 102, 147] have been applied to near-field UNDEX flow simulations which

are of interest in this research.

J. Flores [73] first applied Glimm’s method which is an accurate method in 1-D simulations [27, 45,

132], to a spherical 1-D underwater explosion flow. Later, J.P. Cocchi [92] and R. R. Nourgaliev

[138] employed a Godunov-type prediction-correction method using local Riemann solutions to

correct affected values near the material interface. C.H. Cooke [30] employed a front tracking

method to solve the same problem [92, 73]. These early methods captured the interface sharply, and

provided reasonable fluid results without spurious pressure oscillations. However, these methods

were limited to simple 1-D cases since the extension to multi-dimensions is not trivial and a local

analytic solution in multi-dimensions is not available. Since the local analytic solutions require the

use of an ideal-form EOS such as the ideal gas law, a general EOS such as the JWL model is not

applicable in the framework of these methods.

More recently, R. P. Fedkiw developed the Ghost Fluid Method(GFM) for various multi-fluid

flows [135, 140]. Unlike earlier methods, this method is robust and easier to extend to multi-

dimensions with no geometric complexities [134, 135, 136, 140]. However, it is not usable in the

case where the interface interacts with a strong shock wave as in near-field UNDEX [148]. To

overcome the inability of the original GFM at predicting strong shock-involved flows, T.G. Liu

presented a modified GFM (MGFM) using local shock values nearthe interface [147, 148, 157].

The MGFM improves the quality of multi-fluid flow results, butloses the advantages of the original

GFM method which are easy extension to multi-dimensions, and straightforward implementation

of interface conditions. Furthermore, it is uncertain thatthe general equations of state for the fluid

are usable with this method. The methods mentioned earlier are mostly Eulerian methods which

consider mixed elements where the fraction of fluids coexists. As a Lagrangian counterpart, A. P.

9

Pishevar [10] and H. Luo [57] employed an ALE interface method to simulate two-fluid flowsin

near-field UNDEX. The material interface is explicitly tracked as a Lagrangian surface. The use of

these methods is limited to simple cases where the interfaceis not subjected to large displacement

[10, 57].

Although CFD techniques have been used in near-field UNDEX simulations, these methods are

limited to simple cases where the interface motion is not extreme and/or the explosive gas is gov-

erned by a simple ideal EOS rather than a real EOS. Since the majority of these methods are

based on an Eulerian mesh fixed in space, these methods were not able to adjust the fluid mesh to

the deformation of the structural interface. All of these reasons make previous CFD approaches

inadequate for numerically simulating complex phenomena in near-field UNDEX events.

1.2.3.4 Hydrocodes

Hydrocodes are computational tools for numerically simulating multi-material, compressible and

transient continuum mechanics [64]. Hydrocodes simulate both the fluid and the structure under

highly dynamic conditions such as shock propagation [64, 65]. Pangilnan [51] used the hydro-

dynamic code CTH to obtain the pressure-time history in a PETN detonation in water. CTH is a

multi-dimensional finite volume hydrocode developed by Sandia National Laboratory [74]. J. J.

Dike [88] coupled CTH with the finite element structural code DYNA3D to predict UNDEX dam-

age. J.E. Chisum [83] used the multi-dimensional finite volume code MSC/DYTRAN to model

a spherical TNT explosion ignited at 178.6m below the surface of water. P. Ding [116] used

MSC/DYTRAN to simulate a cylinder submerged under water subjected to an explosion. K. Web-

ster [99] at Virginia Tech used the general-purpose commercial FE code LS-DYNA, to simulate a

deep spherical TNT explosion and blast test cases.

Since most hydrocodes are general-purpose commercial codes, the user cannot access the detail

of the codes to modify the codes, and a thorough understanding of code theories is often difficult

due to poor documentation. As in previous works [83, 99, 116], fine-tuning of numerical methods

relating to meshing, boundary and initial conditions is almost always required. These significant

10

drawbacks of hydrocodes often motivate researchers to makein-house codes for research and ex-

periment.

Terminologies used to describe the numerical aspects of computational frames and mesh descrip-

tions are as follows:

Computational frames To numerically solve a set of governing equations, the geometry of a

computational domain must first be divided into discrete components. According to the kind of

discrete components, numerical methods can be classified into mesh-based approaches and mesh-

free approaches as listed in Table1.2.

Mesh-based approaches Mesh-free approaches

Finite Element Method(FEM) Smoothed Particle Hydrodynamics(SPH)

Finite Difference Method(FDM) Element free Galerkin(EFG) method

Finite Volume Method(FVM) Meshless Local Petrov-Galerkin(MLPG) method

Table 1.2. Prominent mesh-based methods and mesh-free methods [54]

Mesh-based approaches, including the FEM, the FDM and the FVM, are distinguished by the com-

putational frame and the form of governing equations to be discretized. The computational frame

is traditionally a set of grid points or nodes, which approximate the geometry of a computational

domain and also represent the locations at which field variables are evaluated [54]. The FEM dis-

cretizes a problem domain by a set of finite elements, the FDM uses a set of finite nodes and the

FVM uses a set of finite cells. Both the FEM and the FDM use the differential form of governing

equations to approximate the domain, while the FVM uses the integral form of governing equa-

tions [151]. The FVM is most often used in the simulations of fluid flows which strictly require the

conservation of fluid variables [132]. The direct discretization of the integral form of conservation

laws satisfies conservative discretization automatically[19]. Piece-wise discontinuous features

make FVM methods useful for handling the shock-involved flows in UNDEX simulations.

11

To improve intrinsic difficulties (e.g. mesh distortion) inclassical mesh-based approaches, the

mesh-free approaches have recently been developed and applied to various problems including the

UNDEX simulations [54]. Although mesh-free approaches have successfully been applied to some

problems, they are not mature enough to replace mesh-based approaches due to many complexities

in UNDEX simulations [65].

Mesh descriptions There are four types of mesh descriptions that depend on meshand material

motions: Lagrangian, Eulerian, Coupled Eulerian-Lagrangian (CEL) and Arbitrary Lagrangian

Eulerian (ALE). A Lagrangian mesh remains fixed on the material in time, therefore the mesh

moves with material motion. Since the mass within each cell or element remains fixed, there is

no mass flux across inter-cell or element boundaries. Lagrangian methods are very popular in

solving solid mechanics applications [115, 143], however, these methods are limited to relatively

low deformation problems [10, 57, 64, 65].

An Eulerian mesh remains fixed in space during the computation. The nodes or cells do not change

with time, while field variables flow through the mesh. Thus, Eulerian methods do not produce the

same types of problems associated with mesh distortion as inLagrangian methods; however, it is

relatively difficult to determine the free surface, the deformable boundary and the moving material

interface necessary to simulate UNDEX events accurately. Numerical results obtained by Eulerian

methods are usually more diffusive than those from Lagrangian methods [132, 143].

These difficulties led to the development of coupled methodssuch as Coupled Eulerian Lagrangian

(CEL) and Arbitrary Lagrangian Eulerian (ALE). The methodsusing CEL coupling use both Eu-

lerian and Lagrangian meshes in separate regions of the domain. The structure in which material

strength plays a dominant role is discretized with the Lagrangian mesh and the fluid with the Eule-

rian mesh. Both regions continuously interact with each other during the computation. A coupling

algorithm is necessary for the interface between the fluid and the structure. Computational infor-

mation is exchanged by a complex mapping between the two meshes. Such complexities can cause

serious errors if the two regions overlap [63, 151].

12

ALE methods are dependent on mesh-smoothing or mesh-remapping techniques for the Lagrangian

mesh or the Eulerian mesh [54]. By allowing the mesh to move independent of material motion,

mesh distortion can be minimized. Thus, these methods are effectively applicable to large defor-

mation problems. Unlike CEL methods, ALE methods couple thefluid region with the structure

region directly without the complex exchange of computational information. Since the mesh mo-

tion is independent of material motion, it is free to choose arbitrary mesh motions. The following

advantages of ALE methods make them very suitable for simulating fluid-structure interactions in

UNDEX events:

X Fluid mesh is easily adjustable to the deformation of the structural interface

X Mesh-smoothing can overcome problems with mesh-distortion under highly dynamic con-

ditions

The ALE mesh option is included in many recent hydrocodes. Table1.3lists prominent hydrocodes

and available solution methods for each hydrocode.

codes Lagrangian Eulerian CEL SALE MALE

ALE3D o o o o

ALEGRA o o o o

AUTODYN-3D o o

CTH o

CTH-EPIC o o o

DYNA3D o

DYSMAS/ELC o o o

EPIC o

LS-DYNA o o o o o

HULL o o o

MSC/DYTRAN o o o o o

PRONTO3D o

Table 1.3. Comparison of prominent hydrocodes; SALE(single-material ALE), MALE(multi-material ALE)[65]

13

It is concluded that for near-field UNDEX simulations a multi-fluid method based on compressible

inviscid Euler equations in the ALE description is best suitable not only to handle hydrodynamic

phenomena such as shock waves, cavitation formation and collapse and shock-bubble interactions,

but also to couple the dynamic system of the fluid and the structure .

1.3 Recent VT ship survivability research

The VT Ship Survivability Research Group has investigated various UNDEX simulation tech-

niques for both far-field UNDEX and near-field UNDEX [14, 99]. Due to the complex physics,

far-field simulations and near-field simulations have been considered separately. This section de-

scribes a subset of these simulations that provide insight into alternative methods for simulating

near-field early-time UNDEX events.

1.3.1 Deep spherical TNT explosion simulation

The VT research group [99] first chose LS-DYNA 970 which is an explicit finite element code to

assess the use of a multi-fluid ALE method for near-field UNDEXsimulations. LS-DYNA, devel-

oped by LSTC, is a general-purpose finite element hydrocode for analyzing nonlinear behaviors

of structural components and fluid flows with fluid-structureinteractions. Material behaviors are

described with many constitutive models for the structure and equation of state models for the fluid

in LS-DYNA.

A deep spherical TNT explosion at a depth of 178.6 meters was simulated and compared with the

results obtained using MSC/DYTRAN at Naval Postgraduate (NPG) School [83, 162]. A quasi-1D

fluid model shown in Fig1.4 was used in [83, 99]. The fluid model (i.e., seawater with an initial

density 1025 kg/m3) expands 178 meters from the center of the TNT charge to the free surface.

14

Figure 1.4. The pyramid-shaped fluid model for the deep spherical TNT explosion problem

The non-reflecting boundary condition (NRBC) in LS-DYNA wasplaced at the far right segment

to prevent nonphysical wave reflections from boundaries back into the fluid field. The program ran

with a final time of 0.05 seconds. To assess the performance ofLS-DYNA, we obtained the bubble

radius-time history from the post-processor, and comparedthe results with the reference results

in [83]. LS-DYNA demonstrated its ability to simulate the axi-symmetric motion of the bubble;

however, the simulation results were not reproduced accurately as shown in Figure1.5. The ability

to reproduce results depended on many factors which includemeshing-techniques, mesh-densities,

material models, boundary conditions such as the non-reflecting condition and radial constraints

all of which were investigated [99]. K. Webster [99] said that the variation between [99] and [83]

may also be due to the difference between the finite element LS-DYNA and the finite volume

MSC/DYTRAN.

Since the FVM is usually superior to the FEM in simulating shock-involved fluid flows [94, 132],

MSC/DYTRAN could produce better results of the behavior of the bubble. However, a further in-

vestigation of MSC/DYTRAN in the areas of fine-tuning parameters, grid-dependency and bound-

ary conditions is also required to be certain.

15

Figure 1.5. Bubble radius vs. time for the 1D deep spherical bubble problem

1.3.2 U.S Navy blast test simulation

A U.S Navy blast test conducted on 1/2 inch thick air-backed circular plates was simulated us-

ing LS-DYNA [99]. Under similar blast loads, the test explored the structural behaviors of an

equivalent weight steel plate, and Intelligent Engineering’s Sandwich Panel System (SPS) plate

which is a steel-elastomer-steel composite structure laminate [141]. The test investigated the ef-

fectiveness of the SPS plate compared to the steel plate. Although detailed information of the test

was unavailable, the predicted plate deformations were similar to the test as shown in Figure1.6.

The experimental results and LS-DYNA results indicate thatthe elastomer core plays an impact

resistant barrier in absorbing load impacts and preventingthe plate tearing [99, 141].

In the simulation, the effect of local cavitation and bubblejetting was not observed because com-

putational constraints, the size of the fluid domain and the simulation duration were limited. Res-

olution was limited by the mesh-matching requirement in LS-DYNA (i.e. the meshing technique

using a single mesh density). The fluid mesh determined by a rough structural mesh density was

not sufficient to capture detailed near-field UNDEX characteristics in the simulation. As the struc-

tural mesh density increases, the computing cost is greatlyincreased. Prediction of the immediate

response to the incoming shock wave which dominated this test was good as shown in Figure1.6.

16

(a) US Navy blast test results [141]

(b) LS-DYNA simulation results [99]

Figure 1.6. US Navy blast tests and LS-DYNA simulations

17

For multi-fluid flow simulations, LS-DYNA uses a volume of fluid (VOF) method in classical CFD

simulations [105, 132, 140]. This method initially assigns values between 1.0 and 0.0 based on the

fraction of fluids within each cell [79, 140]. Updating the fraction of cells on the Eulerian mesh

reflects the progress of the material interface [79]. Although the VOF method is useful in certain

situations, this method has several drawbacks in simulating complex fluid flows: a large number

of cells are required to obtain accurate interface information and a complex front-reconstruction is

frequently required during the computation [79, 140].

The fluid flows in near-field UNDEX have variable discontinuities (e.g. pressure, density, etc)

initially set by discontinuous fluid fields. Such discontinuities are not easy to treat in continuous

interpolation-based FEM because of its smearing nature [18]. Moreover, the hyperbolic nature

of compressible flows spontaneously leads to discontinuities even if initial disturbances are suffi-

ciently smooth [132].

Without remedying the complexities and limitations discussed above, satisfactory numerical results

cannot be produced. Thus, it was concluded that LS-DYNA which uses the continuous FEM,

the inefficient mesh-matching requirement, and the VOF method is not adequate for simulating

complex fluid flows in near-field UNDEX events although it may adequately capture the initial

shock response [99].

1.4 Objective and contribution

The primary objective of this work is to present a coupled solution approach for numerically sim-

ulating near-field early-time UNDEX events. After the review of computational difficulties which

may be encountered in practice, it was concluded that current methods and tools are not adequate

for simulating near-field UNDEX events. No existing code hasall of the necessary characteristics.

The VT ship survivability research group initially used LS-DYNA to simulate some near-field ap-

plications, however, it was noted that although LS-DYNA maybe a good nonlinear FE code for

18

the structure, LS-DYNA is not adequate for the fluid. The literature survey and the VT research

group’s prior works provide the evidence that a coupled solution approach is required to overcome

the computational difficulties in existing methodologies.This approach is accomplished by com-

bining solution methods that include an accurate numericalmethod with a shock capturing scheme

to reduce numerical diffusion and remove spurious pressureoscillations, a multi-fluid method to

handle explosive gas-water flows and cavitation, and a deformable mesh technique to adjust the

fluid mesh to the deformation of the structural interface. These make our approach unique and

effective to simulate various near-field UNDEX phenomena within a single framework.

Based on the continuum assumption, a set of ALE governing equations is first derived. These

governing equations are discretized by the Runge Kutta Discontinuous Galerkin (RKDG) method

in time and space, which provides an attractive framework toobtain high orders of accuracy, rel-

atively easy implementation, and piecewise discontinuouspolynomial approximations useful for

near-field UNDEX fluid flows [18]. The ALE technique plays an important role in modeling the

arbitrary movement of fluid grids adjacent to the structure interface in the fluid-structure interac-

tion [10, 11, 71, 143]. A problem-independent limiting mechanism for shock-capturing is included

in the implementation [12, 18, 132]. For multi-fluid flows, a Direct Ghost Fluid (DGF) Method

is incorporated in the RKDG framework. To minimize the influence of spurious wave reflections

from boundaries, a sponge-layer NRBC is also implemented [81, 98].

The novel combination of RKDG and DGF (RKDG-DGF) methods is the main contribution of

this work which improves the quality and the stability of explosive gas-water flow simulations.

Unlike previous works, this work has wider application for various near-field UNDEX events and

is easier to extend to multi-dimensions. The literature survey indicates that an efficient and accurate

multi-fluid ALE method for near-field early-time UNDEX simulations has not previously been

developed. For research and experiment, the suggested approach is implemented in an in-house

code using FORTRAN 90 instead of a general-purpose commercial code to avoid the complexities

and limitations discussed for commercial codes.

19

1.5 Outline

Chapter 1 reviews the general characteristics of underwater explosions, the literature survey, the

computational difficulties, and the recent VT ship survivability research. It is concluded that the

ability to accurately simulate near-field UNDEX events may best be accomplished with an effi-

cient and accurate multi-fluid ALE approach. The review of recent VT ship survivability research

motivates the assessment of a coupled solution approach forsimulating near-field UNDEX events.

The objective and contribution of this work is addressed. Chapter 2 discusses the basics of theo-

retical concepts and conservation laws underlying continuum mechanics as applied to this work.

We review the description of conservation laws, material models for the structure and equation of

state models for the fluid in the context of continuum mechanics. Chapter 3 provides the details

of discretization techniques for the fluid governing equations. The governing equations derived

in Chapter 2 are discretized using a RKDG method in space and time. A problem-independent

limiting process ensuring Total Variation Diminishing (TVD) properties is described. Chapter 4

provides the details of a two-fluid method, cavitation and anALE technique for fluid-structure

interaction simulations. A grid-stretching sponge-layerNRBC is presented to model the physical

domain as a reduced computational domain, thereby greatly reducing computational cost. A Direct

Ghost Fluid (DGF) Method coupled with the Level Set (LS) interface method is presented. Chapter

5 performs several one and two dimensional test cases to assess the effectiveness of our approach.

The numerical results are compared with the analytic solution, experimental data and results from

previous work. To assess the movement of fluid grids, a simplefluid-structure interaction test using

an elastic structure is also performed. All of these demonstrate the potential of our approach to

near-field early-time UNDEX simulations. Finally, Chapter6 includes the conclusions and future

works recommended for the VT ship survivability research group.

20

Chapter 2

Continuum mechanics approach to the

UNDEX problem

Analysis of physical phenomena often requires a thorough understanding of microscopic, macro-

scopic, mechanical, thermal and other properties of materials [104]. Continuum mechanics is an

effective and powerful approach for analyzing various physical phenomena without knowledge of

these complexities [18, 104]. Continuum mechanics is applicable to a wide range of engineer-

ing problems that include fluid mechanics, solid mechanics and thermal analysis. Any physical

system must always meet the fundamental laws in continuum mechanics: conservation of mass,

conservation of momentum and conservation of energy [100, 143]. This work considers continuum

concepts for numerically solving physical phenomena in near-field UNDEX. This chapter provides

a theoretical background for subsequent chapters.

Section2.1 describes basic continuum mechanics applicable to the UNDEX problem that in-

cludes fundamental definitions, kinematics and stress-strain relations used in this work. Sec-

tion 2.2 presents a brief derivation of the governing equations in Lagrangian, Eulerian and Ar-

bitrary Lagrangian Eulerian descriptions. For the detailsof continuum mechanics, References

[66, 70, 100, 104, 143] are recommended. This chapter is based on the terminologies and nota-

tions in [70, 143].

21

2.1 Basics of continuum mechanics

2.1.1 Fundamental definitions

Some definitions in general use are provided in this subsection. To distinguish the reference frame,

three configurations and coordinates are defined. Accordingto the mesh motion, we classify the

descriptions into Lagrangian, Eulerian and Arbitrary Lagrangian Eulerian (ALE).

Configurations

The motion of a material point is usually specified with regard to three configurations: the initial

configuration (or undeformed configuration)Ω0 is the domain of a body in the initial state. The

current configuration (or deformed configuration)Ω is the domain of a body at any moment of

time. The reference configurationΩ represents the domain of a body to which governing equations

are referred [70, 143]. For example, either the initial configuration, the current configuration or an

arbitrary configuration can be chosen as the reference configuration. Figure2.1shows the motion

of a material pointO through the initial state inΩ0 and the current state inΩ.

Figure 2.1. The motion of a material pointO [143]

22

Coordinates

In Figure2.1, the position vector of a material pointO in Ω0, X, is called the material coordinate

(or Lagrangian coordinate). The position vector of a material pointO in Ω, x, is called the spatial

coordinate (current or Eulerian coordinate). Similarly, the position vector of a material pointO in

Ω, χ, is called the reference coordinate (or ALE coordinate), remaining coincident with the mesh

during the computation [143]. Unless otherwise specified, material coordinates are identical to

spatial coordinates inΩ0 [143]

X = x(X, t = 0)

The motion of a body is defined as

x = φ(X, t) (2.1)

which maps the initial configurationΩ0 to the current configurationΩ in terms of material coordi-

nateX and timet. The motion of a body can also be expressed as

x = φ(χ, t) (2.2)

which maps the reference configurationΩ to the current configurationΩ [143]. The motion of a

mesh point is expressed as

χ = Ψ(X, t) (2.3)

where is a function of material coordinateX and timet [71, 143].

Figure2.2 represents the one-to-one relationship between configurations via mapping functions.

Note that one point in one configuration corresponds to one point in other configurations [70, 143].

Figure 2.2. Maps between different configurations [143]

23

Descriptions

The nodes in the Lagrangian mesh remain coincident with the material points during the com-

putation which means that the mesh motion is identical to thematerial motion. Since boundary

nodes remain on the boundary throughout the computation, the imposition of boundary conditions

is straightforward, but the mesh can be distorted if it is subjected to large deformations [143]. On

the other hand, the nodes in the Eulerian mesh remain fixed to spatial points during the computa-

tion. Therefore, the mesh motion is different from the material motion. Since boundary nodes do

not remain coincident with the boundary throughout the computation, the imposition of boundary

conditions is relatively difficult, but the mesh is free fromdistortion [143].

Figure 2.3. Example of the Lagrangian mesh and the Eulerian mesh [143]

The ALE description is an advantageous combination of both descriptions lying somewhere be-

tween the Eulerian description and the Lagrangian description. Since nodes are neither attached to

material points nor fixed to spatial points [143], the ALE description does not suffer from the dif-

ficulties of each description. Since the 1970s, the ALE description has received much attention in

a wide range of engineering applications such as fluid mechanics, solid mechanics and fluid-solid

interactions [71].

Both the Eulerian and the ALE descriptions require a remapping procedure throughout the com-

putation because of the difference between the material motion and the mesh motion.

24

Figure2.4 illustrates the remapping process in three descriptions. Note that the Lagrangian mesh

does not require the remapping of variables since the mesh motion is identical to the material

motion.

Figure 2.4. Remapping of Eulerian and ALE descriptions [99, 105]

The ALE description allows the freedom to choose arbitrary mesh motions for specific purposes.

It can be used to reduce the error associated with mesh-distortion in the Lagrangian description

[42, 57, 59]. A second crucial purpose is to adapt the fluid mesh against the deformation of struc-

tural interface in the fluid-structure interaction [72, 144, 160]. Due to their limitations, both the

Lagrangian and the Eulerian descriptions are relatively difficult to use in fluid-structure interac-

tion simulations. Chapter 4 discusses the role of the ALE description in fluid-structure interaction

simulations.

This subsection reviewed fundamental definitions which aremajor ingredients in continuum me-

chanics approach to near-field UNDEX problem. These definitions are useful for understanding

kinematics in the following subsection.

25

2.1.2 Kinematics

In continuum mechanics, the governing equations are in the form of partial differential equations

(PDEs), and the independent variables depend on the description. The understanding of material

time derivatives for a functionf is helpful in the derivation of governing equations in Section 2.2.

Lagrangian description

The material coordinateX and timet are chosen to describe the material motion. As shown in

Figure2.1, the displacement of a material point is the difference between current coordinatex and

initial coordinateX

u(X, t) = φ(X, t) − φ(X, 0) = x − X (2.4)

whereφ(X,t) maps material coordinate in the initial configuration tospatial coordinate in the cur-

rent configuration [100, 143]. At time t = 0, the spatial coordinate,x = φ(X, t = 0) is usually

equal to material coordinateX [143]. Using Equation (2.4), velocity corresponding to the time

derivative of displacement is given by

v(X, t) =∂u(X, t)

∂t=

∂φ(X, t)

∂t=

∂x∂t

≡ u (2.5)

and accelerationa(X, t) corresponding to the time derivative of velocity is given by

a(X, t) =∂v(X, t)

∂t=

∂2u(X, t)

∂t2≡ v ≡ u (2.6)

Equations (2.5-2.6) represent the material time derivatives in the Lagrangiandescription resulting

in the ordinary time derivatives [143]. Similarly, the material time derivative for a functionf(X, t)

can be expressed as

Df(X, t)

Dt=

∂f(X, t)

∂t(2.7)

26

Eulerian description

Instead of the material coordinateX, the spatial coordinatex is employed in the Eulerian descrip-

tion. By the chain rule, the accelerationa(x, t) is expressed as

a(x, t) =Dv(x, t)

Dt=

∂vi(x, t)

∂t+

∂vi(x, t)

∂xj

∂xj

∂t(2.8)

and in vector notation,

Dv(x, t)

Dt=

∂v(x, t)

∂t+ v · v (2.9)

where the second derivative part represents the convectiveterm or transport term [143]. Equation

(2.9) shows how to define the material time derivative for a functionf(x, t) as

Df(x, t)

Dt=

∂f(x, t)

∂t+ vi

∂f(x, t)

∂xi=

∂f

∂t+ v · f (2.10)

where convection velocityv effects the transport of a functionf(x, t) from the current configuration

to the initial configuration.

Arbitrary Lagrangian Eulerian description

The ALE description focuses more on the mesh motion than the material motion. The material

motion, which is a function of reference coordinatesχ and timet, is defined asx = φ(χ, t).

Similar to Equation (2.4), the displacement of a mesh point is expressed as [143]

u(χ, t) = φ(χ, t) − φ(χ, 0) = x − χ (2.11)

For a functionf(χ, t), the material time derivative becomes

Df(χ, t)

Dt=

∂f(χ, t)

∂t+

∂f(χ, t)

∂χi

∂χi

∂t=

∂f(χ, t)

∂t+

∂f(χ, t)

∂χi

wi (2.12)

wherewi is the reference particle velocity [143]. The material time derivative (2.12) is more

complex than those in the other descriptions due to the presence of the convective term∂f(χ,t)∂χi

. For

a simpler form of Equation (2.12), we consider the relationship between the material, mesh and

reference velocities. First, we define the material time derivative of displacement with Equation

(2.2) as

27

vj =∂φj(X, t)

∂t=

∂φj(χ, t)

∂t+

∂φj(χ, t)

∂χi

∂χi

∂t= vj +

∂xj

∂χiwj (2.13)

whereφ(χ, t) maps the reference configuration to the current configuration. Equation (2.13) gives

the ALE convection velocitym [143] which is the difference between the material velocitiesv and

the mesh velocitiesv

mj = vj − vj =∂φj(χ, t)

∂χi

∂χi

∂t=

∂xj

∂χi

wj (2.14)

The second RHS term in Equation (2.12) can be recast via the chain rule as

∂f(χ, t)

∂χi

=∂f(χ, t)

∂xj

∂xj

∂χi

(2.15)

Substituting Equations (2.14-2.15) into Equation (2.12) provides the material time derivative for a

functionf(χ, t) as

Df(χ, t)

Dt=

∂f(χ, t)

∂t+

∂f(χ, t)

∂xj

∂xj

∂χi

wj (2.16)

=∂f(χ, t)

∂t+ mj

∂f(χ, t)

∂xj

which replaces the derivative to the reference coordinatesχj by the derivative to the spatial coor-

dinatexj with the ALE convection velocitymj. In vector notation,

Df(χ, t)

Dt= f,t[χ] + mj · gradf,j (2.17)

The material time derivatives for a functionf are summarized by

Df

Dt= f,t[X] Lagrangian description in terms of(X, t)

= f,t[x] + f,ivi Eulerian description in terms of(x, t) (2.18)

= f,t[χ] + f,imi ALE description in terms of(χ, t)

Depending on the independent variables and convection velocity, these various descriptions may

be identical. For example, if we replaceχ = X andm = 0, then the material time derivative in the

28

ALE description reduces to that in the Lagrangian description. If we replaceχ = x andm = v, the

material time derivative in the ALE description becomes identical to that in the Eulerian description

[70, 143]. As mentioned previously, no convection occurs in the Lagrangian description because

the spatial coordinatex and material coordinateX remain coincident during the computation. In the

Eulerian description, the convection is due to the difference between the current configuration and

the initial configuration with flow velocityv. In the ALE description, the convection is due to the

difference between the current configuration and the reference configuration with ALE convection

velocity m. For more details, References [70, 143] are recommended. Material time derivatives

(2.18) will play a crucial role in deriving governing equations inSection2.2.

2.1.3 Strains and stresses

The behavior of a material point is described by its stress and strain [66, 70, 104]. This subsection

reviews some strain and stress definitions often used in nonlinear continuum mechanics. The de-

formation gradient and its determinant may help in the understanding of the strain-stress relations

used in this work. As shown in Figure2.5, the deformation gradientF is the transformation of an

initial differential elementdX to a deformed differential elementdx as [21, 66, 104]

dx = F · dX or Fij =dxi

dXj

(2.19)

and it determinant J is defined as

J = det(F) (2.20)

Figure 2.5. A schematic of the deformation gradientF [21]

29

Strains

The Green-Lagrange strainE measures the difference of the square of the length of an infinitesimal

segment [66, 143] as

dxdx − dXdX = 2dX · E · dX (2.21)

The first term of Equation (2.21) with Equation (2.19) can be rewritten as

dxdx = (F · dX) · (F · dX) = dX · (FT · F) · dX (2.22)

Equation (2.22) and the expressiondX · dX = dX · I · dX give

dX ·(

(FT · F) − I − 2E)

· dX = 0 (2.23)

whereI is the identity matrix with ones on main diagonal and zeros elsewhere [66, 70, 143]. The

Green-Lagrange strainE is then rearranged as

E =1

2(FT · F − I) =

1

2

(

∂ui

∂Xj+

∂uj

∂Xi+

∂uk

∂Xi

∂uk

∂Xj

)

(2.24)

where

FT · F = FkiFkj =

(

∂ui

∂Xj+

∂uj

∂Xi+

∂uk

∂Xi

∂uk

∂Xj+ δij

)

There are other strains to consider including the following:

X the Euler-Almansi straine

e =1

2(I − (FFT )−1) =

1

2

(

∂ui

∂xj+

∂uj

∂xi−

∂uk

∂xi

∂uk

∂xj

)

(2.25)

is similar to the Green-Lagrange strain except the derivatives of the displacements with respect to

the spatial coordinatesx [66].

30

X the Cauchy infinitesimal strainǫ for small deformation

ǫ =1

2

(

∂ui

∂xj

+∂uj

∂xi

)

(2.26)

is a reduced version of the Euler-Almansi straine. Under the assumption of small deformation

[66], the last term in Equation (2.25) disappears.

X the velocity strainD or the rate-of-deformation

Dij =1

2

(

∂vi

∂xj

+∂vj

∂xi

)

(2.27)

is often recommended in fluid mechanics [115].

Stresses

There are three popular stresses; the Cauchy stressσ, the nominal stressP and the second Piola-

Kirchhoff (or PK2 for brevity) stressS [70, 143]. The Cauchy stress is the ratio of the force applied

and the area in the current configuration as

σ =F

A(2.28)

Since the Cauchy stress follows the symmetry asσ = σT [70, 100, 143], the conservation of

angular momentum can be decoupled from the system of equations [70, 71, 143] and the Cauchy

stress is most commonly used in continuum mechanics [66, 70, 115, 143].

The nominal stress is defined as

P =F

A0(2.29)

whereA0 is the reference area which can be chosen as one in the initialconfiguration. The nominal

stress, or engineering stress is especially useful for measuring the stress in experiments where the

current areaA is not easy to obtain accurately [70, 143].

31

The PK2 stress is the ratio of the force and the area in the reference configuration as

S = F−1

(

F

A0

)

(2.30)

whereF is the deformation gradient (Equation (2.19)). Since the PK2 stress is constant in pure

rotations (or called rigid rotations), the PK2 stress is often used in describing the effects of large

deformation due to rotations [143].

The different stresses are inter-related by transformations in Table2.1with the deformation gradi-

entF and the Jacobian J.

Cauchy stress (σ) Nominal stress (P) PK2 stress (S)

σ - J−1F · P J−1F · S · FT

P JF−1 · σ - S · FT

S JF−1 · σ · F−T P · F−T -

Table 2.1. Transformation between stresses [143]

This section outlined the basics of continuum mechanics that include the fundamental definitions,

kinematic concepts and stress-strain relations which willfrequently be mentioned in this work.

The material time derivative in the Lagrangian descriptionis in the form of ordinary differential

equations (ODE) while those in the Eulerian and the ALE descriptions are in the form of partial

differential equations (PDE) with convection terms. The ALE convection velocitym was defined

to distinguish the ALE description from the others. Note that the convection effect is due to the

difference between material motion and mesh motion.

With these concepts, Section2.2 provides a brief derivation of the fundamental equations which

govern physical phenomena in near-field UNDEX. These equations play a major part in the nu-

merical methodologies, discussed in subsequent chapters.

32

2.2 Governing equations

The fundamental equations which govern the motion of continuums are identical both in a fluid

and a solid, and even applicable to any particular medium. This is a very remarkable property.

The fundamental equations are based on following universallaws of conservation: conservation of

mass, conservation of momentum and conservation of energy.The resulting equations are called

the continuity equation, the momentum equations and the energy equation, respectively.

This section provides a brief derivation of the governing equations using the basic concepts given

in Section2.1. The numerical discretizations of these equations are discussed in the subsequent

chapters. The Reynolds transport theorem, the product rule, and the Gauss’ theorem are useful in

manipulating the mathematical expressions throughout this section [70, 71, 143]. See AppendixA.

The equations are grouped into the Lagrangian expression and the advection expression depending

on the presence of the convection term.

2.2.1 Conservation of mass: continuity equation

The principal of mass conservation requires that the mass within an infinitesimal domain remains

constant; the mass within the domain cannot be created nor bedestructed [27], even if it is rear-

ranged in the domain. The mass within the domain can be transported only by convection. Hence,

there is no diffusion [19].

Lagrangian expression

Let us define massm as [143]

m =

∫

Ω

ρ(X, t)dΩ (2.31)

whereρ(X, t) is density in the domainΩ. For the conservation of mass, the material time derivative

of massm must always vanish as [27, 96, 143]

33

Dm

Dt=

D

Dt

∫

Ω

ρdΩ = 0 (2.32)

which is the integral expression of the continuity equationin the Lagrangian description. As an

alternative, the algebraic form of the continuity equationis often used in the Lagrangian description

as

ρJ = ρ0 (2.33)

where J andρ0 are the determinant of the deformation gradient, det(F) and initial density, respec-

tively [70, 143].

Advection expressions

By applying the Reynolds transport theorem to Equation (2.32), the continuity equation is obtained

as∫

Ω

(

Dρ

Dt+ ρ ·v

)

dΩ = 0 (2.34)

and in the differential form

Dρ

Dt+ ρ ·v = 0 (2.35)

By replacing DDt

in Equation (2.35) by Equation (2.18), we obtain

∂ρ

∂t+ v · ρ + ρ ·v = 0 in the Eulerian description (2.36)

∂ρ

∂t+ m · ρ + ρ ·v = 0 in the ALE description (2.37)

wherev is material velocity andm is the ALE convection velocity which implies there is convec-

tion effect in both descriptions. Equations (2.36-2.37) are in the quasi-linear or non-conservative

form. Applying the product rule to the above equations provides the continuity equations in con-

servative form as

34

∂ρ

∂t+ · (ρv) = 0 in the Eulerian description (2.38)

∂ρ

∂t+ · (ρm) = 0 in the ALE description (2.39)

where the first term is the rate of change of density and the second term is the rate of mass flux

passing out of domain surface [27, 132]. In the steady-state,∂∂t

= 0, Equation (2.38) is simplified

to

·(ρv) = 0 (2.40)

and for incompressible flows

·v = 0 (2.41)

where the densityρ is constant in time.

2.2.2 Conservation of momentum: momentum equation

The principle of momentum conservation requires that the time rate of change of momentum of a

body be equal to the net force exerted on a body [27, 96, 143]

D

Dt(mv) = F (2.42)

wherem, v andF are mass, velocities and net force. Momentum is defined as a vector quantity,mv

and forceF is the resultant of all the forces acting on a body. For constant massm, the momentum

equation is the general expression of Newton’s second law (F = ma). Net forceF consists of body

forceb and surface tractiont.

Lagrangian expression

The momentum equation within the domainΩ is expressed as

D

Dt

∫

Ω

ρvdΩ =

∫

Ω

ρbdΩ +

∫

Γ

tdΓ (2.43)

35

whereρv is linear momentum per unit mass,ρb is body force exerted on a body per unit mass,

andt is surface traction per unit area [143].

The surface integral in Equation (2.43) can be transformed into the volume integral by using the

Cauchy’s relation and the Gauss’ theorem

∫

Γ

tdΓ =

∫

Γ

n · σdΓ =

∫

Ω

· σdΩ (2.44)

whereσ is the Cauchy stress tensor [143]. By replacing the second RHS term of Equation (2.43)

by Equation (2.44), we obtain

D

Dt

∫

Ω

ρvdΩ =

∫

Ω

(ρb + · σ) dΩ (2.45)

where · σ representsσij,i in tensor notation. By applying the Reynolds transport theorem and

the product rule to the LHS of Equation (2.45), it is rewritten as

D

Dt

∫

Ω

ρvdΩ =

∫

Ω

(

D

Dtρv + (ρv) ·v

)

dΩ

=

∫

Ω

(

ρDvDt

+ v(

Dρ

Dt+ ρ ·v

))

dΩ

=

∫

Ω

ρDvDt

dΩ (2.46)

The term (DρDt

+ρ·v) was deleted by the definition of the continuity equation (2.35). Substituting

Equation (2.46) into Equation (2.45), and moving all terms to the left-hand side gives∫

Ω

(

ρDvDt

− ρb − · σ

)

dΩ (2.47)

and in differential form

ρDvDt

= ρb + · σ (2.48)

36

Advection expressions

By applying material time derivatives to Equation (2.48), the momentum equation in the Eulerian

description is obtained as

ρ

(

∂v∂t

+ v · v)

= ρb + · σ (2.49)

and

ρ

(

∂v∂t

+ m · v)

= ρb + · σ (2.50)

in the ALE description. By applying the product rule to the above equations, we obtain the con-

servative form of the momentum equations in terms of linear momentumρv as

∂(ρv)

∂t+ · (ρvv) = ρb + · σ (2.51)

in the Eulerian description and

∂(ρv)

∂t+ · (ρvm) = ρb + · σ (2.52)

in the ALE description.

2.2.3 Conservation of energy: energy equation

The principle of energy conservation requires that the rateof change of total energy within a

domain be equal to the sum of the work done by external forces,and heat energy supplied by heat

sources and heat flux [27, 96, 143]. The conservation of energy is also known as the first law of

thermodynamics:

wtotal = wext + wheat (2.53)

wherewtotal is the rate of change of total energy,wext is the work done by external forces andwheat

is the heat energy added by heat sources and heat flux [27, 96, 143].

37

Lagrangian expression

The rate of change of total energy within the domainΩ is expressed as

wtotal =D

Dt

∫

Ω

(

ρeint +1

2ρv · v

)

dΩ (2.54)

whereρeint is internal energy per unit volume and12ρv · v is kinetic energy per unit volume. The

work done by external forcewext is the sum of the rate of the work done by the body forceb and

the surface tractiont as

wext =D

Dt

∫

Ω

v · ρbdΩ +D

Dt

∫

Γ

v · tdΓ (2.55)

The heat energywheat is the sum of the rate of the work done by heat sources and heat fluxq as

wheat =D

Dt

∫

Ω

ρsdΩ −D

Dt

∫

Γ

n · qdΓ (2.56)

where the sign of heat flux term is negative due to physical reason [143]. For the sake of simplicity,

suppose that no heat sources exist in a control volume,s = 0 and no heat flux flows in and out

through the surface of a control volume,q = 0. These assumptions mean that the system is

thermally isolated [71]. Based on these assumptions, we can decouple the contribution of the

heat energy from the energy equation. Substituting Equations (2.54-2.55) into Equation (2.53), we

obtain the integral form of a simplified energy equation as

D

Dt

∫

Ω

(

ρeint +1

2ρv · v

)

dΩ =

∫

Ω

v · ρbdΩ +

∫

Γ

v · tdΓ (2.57)

By applying the Cauchy’s law and the Gauss’s theorem to the surface integral of Equation (2.57),

the surface integral can be converted into the volume integral as [71]∫

Γ

v · tdΓ =

∫

Γ

v · (σ · n)dΓ =

∫

Ω

· (σ · v)dΩ (2.58)

where · (σ · v) represents(viσij)j in tensor notation. For detail derivations, References [70, 71,

38

143] are recommended. Rearranging all terms to the left-hand side gives

∫

Ω

[

D

Dt

(

ρeint +1

2ρv · v

)

− v · ρb − · (σ · v)

]

dΩ = 0 (2.59)

and the differential form of the energy equation becomes

ρDe

Dt= v · ρb + · (σ · v) (2.60)

wheree is the total energy per unit mass.

Advection expressions

By applying the material time derivatives to Equation (2.60), the energy equations is obtained as

ρ

(

∂e

∂t+ v · e

)

= v · ρb + · (σ · v) (2.61)

in the Eulerian description and

ρ

(

∂e

∂t+ m · e

)

= v · ρb + · (σ · v) (2.62)

in the ALE description. By applying the product rule and the continuity equation, we obtain the

conservative form of the energy equations in terms of conservative variableρe (i.e. total energy

per unit volume)

∂(ρe)

∂t+ · (ρev) = v · ρb + · (σ · v) (2.63)

in the Eulerian description and

∂(ρe)

∂t+ · (ρem) = v · ρb + · (σ · v) (2.64)

in the ALE description.

The universal conservation laws are summarized in Tables2.2and2.3. On the basis of the principle

of continuum mechanics, these equations are applicable to any particular medium. According

to the mesh motion we wish to use, we can choose the Lagrangiangoverning equations or the

advection governing equations.

39

Table 2.2. Summary of universal conservation laws in the non-conservative form

Table 2.3. Summary of universal conservation laws in the conservative form

For the structure, the Lagrangian form is most commonly selected in the literature [70, 91, 100,

143]. If the structure undergoes severe deformation, the ALE governing equations can be chosen

as an alternative to relax severe mesh-distortion associated with large displacements [71]. It is

usually assumed that the densityρ is constant and the entire process is adiabatic, thus allowing

the continuity equation and the energy equation to be eliminated from the system of equations

[91, 100, 143].

Table 2.4. Lagrangian governing equation for the structure

40

For the fluid, the conservative form of the governing equations is used. In Chapter 1, we assumed

that the fluid flow induced by near-field UNDEX is compressible[54]. The non-conservative form

does not provide physically meaningful results in shock-involved flow simulations. It cannot treat

properly the conservation of variables, and also capture the discontinuity of flow variables which

is important in compressible flow simulations [71, 143]. Accordingly, the conservative form of

governing equations has received widespread use in compressible flow applications [7, 18, 71,

143]. For the fluid computation, the following set of ALE governing equations is considered.

Table 2.5. ALE governing equations for the fluid

2.2.4 Additional equation

As shown in Table2.5, there are four unknowns with three equations when we solve 1-D fluid

governing equations: density, velocity, energy and stress. For the structure, there are two unknowns

with the single equation given in Table2.4: velocity and stress. Since a unique solution requires

an equal number of equations and unknowns, one more equationis necessary. The constitutive

relation is the additional equation used to complete the setof governing equations.

2.2.4.1 Constitutive relations

The material’s behaviors are distinguished by their way of resistance against deformation [18]. The

structural behaviors are often defined by

σij = Cijklǫkl (2.65)

41

whereCijkl is the fourth-order tensor of material constants andǫij is the linear strain. Some strains

and stresses were previously described in Subsection2.1.3. Equation (2.65) is a strain-stress re-

lationship for linear elastic materials, also referred to as Hooke’s law [70, 143]. The Kirchhoff

material model which is an extension of the Hooke’s law to large deformations [70, 143], is simply

achieved by replacing the stressσij by the PK2 stressSij , and the linear-strainǫij by the Green-

Lagrange strainEij [70, 143] as

Sij = CijklEkl (2.66)

By the transformation of stresses in Table (2.1), we can convert the PK2 stress to the Cauchy stress

commonly used

σ = J−1F · S · FT (2.67)

where F and J are the deformation gradient and its determinant [143]. For nonlinear plastic mate-

rial models, more complex constitutive relations are required, but these models are not considered

in this work. For the details of such models, References [66, 70, 100, 143] are recommended.

The constitutive relation for the fluid is usually

σij = −pδij + 2µ

(

ǫij − δijǫkk

3

)

(2.68)

whereǫ is the velocity strain andµ is the dynamic (shear) viscosity [115]. Since the displacement

of fluid particles is much larger than those of solid particles, the velocity strain is preferred to

model the motion of fluid particles [71, 115]. For inviscid flows, i.e.µ = 0, Equation (2.68) is

reduced to

σij = −pδij (2.69)

Equation (2.69) is a function of pressurep only. An expression for pressure in Equations (2.68,

2.69) is still required. It is called the equation of state which is a relationship between flow vari-

ables, e.g.p = p(ρ, e, . . .).

42

2.2.4.2 Equations of state

Thermodynamic concepts play an important role in the understanding of compressible flow phe-

nomena that include microscopic properties such as density, pressure and internal energy. This

section provides a brief discussion of thermodynamic concepts and some equations of state for

UNDEX fluid flow simulations.

Thermodynamic relations

For a perfect gas which does not involve intermolecular forces [27, 82], a relation between state

variables, first synthesized by Boyle in the 1600s [82], is defined as

p = ρRT or pv = RT (2.70)

wherep, ρ = 1/v, R andT are pressure, density per unit mass, gas constant and temperature,

respectively. A useful state variable, enthalpy is defined as

h = eint + pv (2.71)

which is a function of internal energyeint, pressurep and specific volumev.

For a calorically perfect gas with constant specific heatscv andcp [82], internal energy per unit

masseint and enthalpyh are given as

eint = cvT (2.72)

h = cpT

By manipulating Equations (2.70-2.72), we obtain the Carnot’s law [53] and other expressions as

cp − cv = R, cp =γR

γ − 1and cv =

R

γ − 1(2.73)

whereγ = cp/cv is the ratio of specific heats. Equations (2.71-2.73) yield a relationship between

pressure, density per unit mass and internal energy per unitvolume as

43

ρeint =p

(γ − 1)(2.74)

which is referred to as the equation of state for perfect gas or the ideal-gas law [27, 53, 82].

Any number of other equations of state can be obtained in a similar way, either theoretically or

experimentally.

Entropy, which indicates how much of the internal energy is available for useful work [27, 82], is

defined as

s = cv log

(

p

ργ

)

+ const. (2.75)

For the isentropic process, which is a reversible and adiabatic process [27, 53], pressure is ex-

pressed as

p = sργ (2.76)

In isentropic flow, the entropy is constant over all the flows excepts at shocks [27, 53, 82]. This

assumption permits a simple treatment of compressible flow complexities which has often been

used in compressible flow simulations [27, 29, 30, 31, 40, 135].

The speed of sound is another useful state variable in thermodynamics. For the relationp =

p(ρ, eint), the speed of sounda is determined by

a =

√

p

ρ2peint + pρ (2.77)

wherepeint andpρ represent partial derivatives of pressurep with respect toeint andρ [27, 135].

For example, the speed of sounda for a perfect gas obeying Equation (2.74) is

a =

√

p

ρ2(γ − 1)ρ + (γ − 1)eint =

√

γp

ρ(2.78)

Since no interaction of particles between microscopic and macroscopic levels exists in incompress-

ible flows [27], incompressible flow simulations do not require the equation of state.

44

Equations of state for explosive gas

The explosion process is largely divided into the detonation phase through a high explosive mass at

rapid constant velocity, and the expansion phase into a surrounding media, e.g. water in UNDEX

events [54, 120]. Currently, this work focuses on the simulation of the expansion phase. Since

the detonation phase can be canceled instantaneously because of its infinite detonation velocity

(e.g. 6930m/s for TNT [17]), the original explosive mass is immediately transformedinto a high

pressure homogeneous gas bubble, and resulting mediums including explosive gas and water are

considered to be compressible and inviscid [1, 54]. For the details of detonation simulations,

References [1, 17, 32] are recommended.

The initial state of UNDEX simulations is usually defined as the moment when a high-pressure

gas bubble is formed after the completion of the detonation process [5, 7, 10, 40, 57, 83, 88, 92,

161]. The initial gas bubble has the same volume and internal energy as the original explosive

in water. See Figure2.6. Both the gas bubble and shock wave initially expand outward[27, 45,

120, 132]. A strong shock wave propagating into the water suddenly raises the flow velocity from

zero, giving rise to the decaying of pressure caused by the rapid decrease of density [120, 131].

As time continues, the shock wave propagates outward fasterthan the interface. The shock wave

propagates at a speed determined by the sum of flow velocity and speed of sound (v + c) in the

water, whereas the interface travels only by flow velocity (v). Across the gas-water interface, two

different equations of state must be applied to describe thestate of each fluid.

Figure 2.6. An illustration of the initial condition and flowfields at timet = t

45

For explosive gas, the JWL EOS and the ideal-gas law have beenextensively employed in the

literature as summarized in Table2.6.

JWL EOS Ideal gas law

A. B. Wardlaw, Jr[7, 5], R. P.Fedkiw[135, 136], J. E.Chisum[83, 162], J. J. Dike[88], HongLuo[57], G. R. Liu[54], K. Webster[99],H. J. Schittke[60], A. Alia[1]

R. R. Nourgaliev[138],A. Pishevar[11], Keh-MingShyue[102], X.Y. Hu[161], C. H.Cooke[30], F. H. Harlow[46], T. G.Liu[147, 148], W. F. Xie[157], J. P.Cocchi[92], M. A. Jamnia[110], R.Saurel[127], Peiran. Ding[116], B.Koren[15], J. Qiu[78], C. Wang[24]

Table 2.6. Use of JWL EOS and ideal gas law for explosive gas

X Jones-Wilkins-Lee (JWL) EOS

p = A

(

1 −ωη

R1

)

e−R1η + B

(

1 −ωη

R2

)

e−R2η + ωρeint (2.79)

where coefficientsA, B, R1, R2 andω are experimentally determined,ρ is density andeint is

internal energy per unit mass,η = ρρ0

is the ratio of current density and initial density [7, 54]. For

typical trinitrotoluene (TNT) and pentaerythritol tetranitrate (PETN), the material properties and

coefficients are listed in Table2.7. These coefficients were determined from the experimental data.

TNT [83] PETN [58]

Initial density,ρ 1630kg/m3 1770kg/m3

Fitting coefficient,A 3.712 × 1011Pa 6.1327 × 1011 Pa

Fitting coefficient,B 0.0321 × 1011 Pa 0.15069× 1011 Pa

Fitting coefficient,R1 4.15 4.4

Fitting coefficient,R2 0.95 1.2

Fitting coefficient,ω 0.30 0.25

Initial internal energy,eint0 4.29 × 106J/m3 10.1 × 109J/m3

Table 2.7. Coefficients in the JWL equation of state

46

X Ideal-gas law

Based on the assumption that the reaction products obey the ideal gas principle (i.e. no intermolec-

ular force exists), the ideal-gas law can be used to model theexpansion phase of the explosive gas

by

p = (γ − 1)ρeint (2.80)

whereρ is the density,eint is the internal energy per unit mass andγ is the ratio of specific heats.

P. Ding [116] employed this model for a compressed hot gas bubble in a fluid-structure interaction

application in MSC/Dytran. In [147, 157, 158], W. F. Xie et al. performed several UNDEX

applications using this model withγ = 2.0. This equation can also model helium withγ = 1.67,

hydrogen, oxygen, nitrogen and air withγ = 1.4, and carbon dioxide withγ = 1.3 [46]. The ratio

of specific heatsγ is based on statistical mechanics [46].

The JWL EOS can model both the detonation and the expansion phase with parameters experimen-

tally determined [32], while the ideal-gas law can model only the expansion phase. Even though

both models have different characteristics, they have often been utilized in the simulation of near-

field UNDEX because of its asymptotic nature. A. Alia [1] found that for the expansion state

(V → ∞), the third term in JWL EOS becomes dominant so that the reduced form asymptotically

matches the ideal gas law as

p ≈ ρωeint (2.81)

whereω representsγ − 1. See Figure 2 in Reference [1]. G.R. Liu found that the ideal gas

law produces close results with the JWL EOS beyond the regionof 8-10 times the charge radius

[54]. Both EOSs were used in a 1-D TNT slab detonation problem to assess their applicability

to near-field UNDEX simulations by comparing with experimental data. The JWL EOS produced

a stronger impulse than that made by ideal-gas law within theregion of 8-10 times the charge

radius; however, beyond 8-10 times the charge radius the numerical results were similar [54]. The

47

JWL EOS approximated peak pressure close to experiments. Itwas concluded that the JWL EOS

produces more reliable results in near-field early-time UNDEX simulations where the domain size

is smaller than a region 8-10 times the charge radius [54].

This work uses both EOSs to describe the state of explosion-produced gas even though the JWL

EOS requires more computational effort. The ideal gas law isvery useful in developing an analytic

solution because of its common use in gas dynamics [27, 31, 45, 73, 85, 92, 132].

Equations of state for water

To describe the state of water, a number of EOSs such as the Gruneisen EOS, the stiffened-gas

EOS, the modified-gas EOS and the Tait EOS have been used. Table 2.8 summarizes the use of

equations of state to model water behavior in the literature.

Gruneisen EOS Stiffened gas Modified gas Tait EOS

Shin [83] A.R.Pisher[10],J.P.Cocchi[92],F.H.Harlow[46], R.Saurel[127],R.Kuszla[119], P.Glaister[117], K.M.Shyue[102], EricJohnsen[40], R.R.Nourgaliev[138], C.H.Chang[29], N.Andrianov[113], B.Koren[15], B.Lombard[16], C.Wang[24], A. Chertock[2],R. Abgrall[122]

M.A.Jamnia[110]

C.C.Cooke[30], H.Luo[57], J. Flores[73],A.B. Wardlaw[7], S.Y.Kadioglu[142], P.Ding[116], J. Qiu[78],T.G. Liu[147]

Table 2.8. Use of equations of state for water

48

X Gruneisen EOS

The Grunesien EOS is a general EOS which can describe the state of a large class of materials such

as gases, liquids and solids [46]. Shin [83] employed a modified Gruneisen equation of state with

specific coefficients obtained by fitting the experimental data in MSC/DYTRAN. Two expressions

depending on the state of the material are provided. In the compression state, water is governed by

p =ρ0C

2[

1 +(

1 − ρ0

2

)

µ − a2µ2

]

[

1 − (S1 − 1)µ − S2µ2

µ+1− S3

µ3

(µ+1)2

]2 + (γ0 + aµ)e (2.82)

and in the expansion state,

p = ρ0C20µ + (ρ0 + aµ)e (2.83)

whereρ0 is initial density,µ = ρ0

ρ− 1 is a switch,C andC0 are the speeds of sound. Other

coefficients are listed in Table2.9.

Coefficient Value [83]

Gruneisen coefficient,γ0 0.5

Volume correction coefficient,a 0.0

Fitting coefficient,S1 2.56

Fitting coefficient,S2 1.986

Fitting coefficient,S3 1.2268

Table 2.9. Coefficients in the Gruneisen equation of state for water

X Stiffened-gas EOS

F.H. Harlow stated that “when the density change of a medium under high-pressure conditions is

slight, the Grunesien EOS can be simplified to a stiffened-gas law which is a reasonable approxi-

mation for a large class of materials” [46].

p = (γB − 1)ρe − γBB (2.84)

49

where coefficientsγB andB are obtained using the experimental data [92]. When we use this EOS

to model a perfect gas withγB = 1.4 andB = 0.0, Equation (2.84) becomes equal to Equation

(2.80) which is the ideal-gas law. Material properties and coefficients for various materials are

listed in Table2.10.

[113, 127] Air Explosive gas Sea Water Copper Zinc

Density (kg/m3), ρ 1 1630.0 1025(1000) 8924 7139

Coefficient,γB 1.4 1.4 or 2.0 5.5 (4.4) 4.0 4.17

Coefficient (Pa),B 0 0 4.921e8 (6e8) 3.41e10 1.57e10

Table 2.10. Material properties and coefficients in the stiffened gas EOS; ( ) is for fresh water

X Modified-gas EOS

This EOS uses the ideal-gas law with a modifiedγ to describe the state of water. In [110], Jammnia

calculatedγm as

γm =2c0

u+

(γ − 1)

2(2.85)

wherec0 is the speed of sound1448m/s, u is the flow velocity behind shocks4.798m/s andγ is

the ratio of specific heats1.10 for water. Equation (2.85) givesγm = 603. Jammnia employed

this model for numerically solving a shock reflection simulation to obtain a reasonable reflection

ratio from the rigid wall [110] . However, this model is restricted to the case where Mach number

(M = u/a) is smaller than1 [110].

X Tait EOS

p = B

[

(

ρ

ρ0

)N

− 1

]

+ A (2.86)

which coefficientsA, B andN are experimentally determined, andρ0 is the initial density. Due to

its barotropic property, the Tait EOS has often been used in fluid dynamics. This equation can also

be rearranged as

50

p = (N − 1)ρeint − γB (2.87)

whereB is B − A andρeint is the internal energy per unit volume [15, 135, 148, 161]. Note that

Equation (2.87) is identical to the stiffened-gas law (2.84). For water,N , A andB are set equal

to 7.15,105 Pa and3.31 × 108 Pa, respectively. Like the stiffened-gas EOS, this model permits a

single formulation for gas-water computations since this equation becomes identical to the ideal-

gas law when we setB = 0.0 andN = 1.4. Material properties and coefficients are listed in Table

2.11[7, 30].

Sea water Fresh water

Density (kg/m3), ρ 1025.0 1000.0

Coefficient (Pa),A 1 × 105 0.0

Coefficient (Pa),B 3.31 × 108 2.322 × 108

Coefficient,N 7.15 7.415

Table 2.11. Material property and coefficients for waters inthe Tait EOS

The near-field UNDEX simulations require two equations of state, one for explosive gas and one

for water. Four combinations of equations of state used in UNDEX simulations are investigated as

follows:

1. JWL EOS for explosive gas and Gruneisen EOS for water [83]

2. JWL EOS for explosive gas and Tait EOS for water [5, 7, 57]

3. Ideal gas EOS for explosive gas and Tait EOS for water [5, 30, 73, 142, 147, 150, 93]

4. Ideal gas EOS for explosive gas and stiffened gas EOS for water [15, 24, 29, 46, 92, 102,

113, 117, 119, 127]

Although the ideal-gas law was unable to accurately predicta real explosion phase as demonstrated

by G.R. Liu [54], it still has been widely used due to its simplicity. For thepurpose of this work

51

and effectiveness, we consider both the JWL EOS and the ideal-gas law for explosive gas, and the

stiffened-gas law and the Tait EOS for water. With the ideal-gas law for explosive gas, the use

of stiffened-gas law/Tait EOS permits a single formulationof gas-water flows which is useful for

obtaining the analytic solution based on gas dynamics.

This chapter outlined necessary theoretical background that includes the basis of continuum me-

chanics, conservation laws and constitutive relations. Conservation laws were provided for the

Lagrangian, the Eulerian and the ALE descriptions: the continuity equation, the momentum equa-

tion and the energy equation. Even though Section2.2listed the differential form of the governing

equations, we can readily return to the integral form for thefinite volume methods. To complete

the system of governing equations, constitutive relationsfor the solid and the fluid were addressed.

For the fluid, equations of state commonly used were summarized. For UNDEX problems, we

consider the JWL EOS and the ideal-gas law for explosive gas,and the stiffened-gas law and the

Tait EOS for water, respectively. Subsequent chapters dealwith numerical methodologies and the

assessment of our approach for near-field UNDEX simulations.

52

Chapter 3

Discretizations

The basic concepts of continuum mechanics, the governing equations and the constitutive relations

were provided in the previous chapter. In general, the differential form of the governing equations

is not directly applied to the numerical simulation involving flow discontinuities [71, 132]. The

weak formulation, which is based on the weighted-residual method, as opposed to the strong for-

mulation, which is based on the differential form of governing equations, is usually advised in the

discretization of the governing equations [18, 49, 71, 132, 143, 151]. We consider the method of

lines, also known as the semi-discretized scheme, which uses a separate discretization in space and

time. Since this method allows a different order of accuracyin space and time discretizations, it

provides a much greater flexibility compared to combined space and time discretizations such as

Lax-Wendroff scheme [69].

This chapter discusses the Runge-Kutta Discontinuous Galerkin (RKDG) method for numerically

solving the governing equations of near-field UNDEX fluid flows. Based on the DG weak formu-

lation, a solution algorithm coupled with a time integratorand a slope limiter for removing variable

oscillations is presented.

53

3.1 Brief introduction to the RKDG method

W.H. Reed, et al. in 1973 first developed the discontinuous Galerkin (DG) method, which may be

considered a method between the Finite Volume Method (FVM) and the Finite Element Method

(FEM), to solve a neutron transport problem [159]. Since then, the method has received much

attention in simulating compressible fluid flows [12, 13, 59, 76, 77, 78]. Comparable features to

classical methods are high-order accuracy, a simple treatment of complex geometries, and the weak

connectivity between elements [12, 18, 94]. High-order accurate approximations are achieved by

increasing the order of basis functions. Since all the termsexcept the boundary integral are treated

as in the standard Galerkin method, the DG method simply treats complex geometries as in the

FEM. A local weak form provides weak connectivity between elements through the boundary in-

tegral only [12, 18, 94]. It allows a simple extension of numerical flux functions proven successful

in the FVM. These features make the DG method especially attractive for compressible flow sim-

ulations where variable discontinuities are included.

The governing equations in Chapter 2 require a time integration scheme. Since the DG formulation

offers fully decoupled time derivatives∂∂t

between elements, the DG method can use an efficient

time marching algorithm. A commonly used time marching algorithm for the DG method is an

explicit Runge-Kutta (RK) method [12, 18]. B. Cockburn first presented a Total Variation Dimin-

ishing (TVD)-RKDG method for hyperbolic conservation equations [12]. It consists of the DG

method based on the local weak form in space discretization,the explicit RK method in temporal

discretization and a generalized slope limiter to ensure the TVD properties [12, 13, 18]. Compared

to some FE and FV methods, the RKDG method is an accurate and efficient solution method with

no additional effort associated with inter-element continuity constraints, complex geometries and

stability issues [12, 13, 18, 94, 95].

Subsequent sections provide a description of the RKDG method: the spatial discretization, the

temporal discretization and the slope limiter. For detailsof the RKDG method, general References

[12, 13, 18, 59, 76, 77, 78] are recommended.

54

3.2 Spatial discretizations

In Chapter 1, the near-field UNDEX fluid flows were assumed to becompressible and inviscid be-

cause of extremely rapid fluid motion. The set of ALE Navier-Stokes fluid equations is transformed

into a set of ALE Euler fluid equations by substituting the constitutive relation (2.69) as

Table 3.1. ALE - Euler fluid governing equations

3.2.1 One-dimensional weak form

A one-dimensional analysis is first considered to provide some insight for the general RKDG

method. The set of 1-D Euler fluid equations is expressed as

U(x, t),t + F(U(x, t))x = 0 (3.1)

where U = [ρ, ρu, ρe]T and F(U) = [ρm, ρum + p, ρem + pu]T . Note that the domainΩ is

subdivided by elementsdxj = [xj− 12, xj+ 1

2] with j = 1, 2, ..., N andN is the number of elements

or cells. Multiplying Equation (3.1) by a test functionω(x) and integrating it over the elementj

gives

∫ xj+1

2

xj− 1

2

[U,t(x, t) + F(U(x, t)),x] ω(x)dx = 0 (3.2)

55

Integrating by parts on the spatial differential term, we obtain a local weak form: [12, 18]

∫ xj+1

2

xj− 1

2

[

U,t(x, t)ω(x) − F(U(x, t))ω(x),x

]

dx + F(U(x, t))ω(x)|j+ 1

2

j− 12

= 0 (3.3)

where

F(U(x, t))ω(x)|j+ 1

2

j− 12

= F(U(xj+ 12, t))ω(xj+ 1

2) − F(U(xj− 1

2, t))ω(xj− 1

2)

Note Equation (3.3) is similar to that in the standard Galerkin method, but computationally differ-

ent. Unlike the FEM, a jump appears as the flux through inter-element boundaries F(U(x, t))ω(x)|j+ 1

2

j− 12

,

and unlike the FV method, the flux integral∫ x

j+ 12

xj− 1

2

F(U(x, t))ω(x),xdx is added. A 1-D presentation

of DG approximations is illustrated in Figure3.1.

Figure 3.1. 1D presentation of DG approximations; | marks the inter-element boundary, and dots for the centerof cells [18]

Since the solution U(x, t) in shock-involved flows must be discontinuous at inter-element bound-

aries, we replace the nonlinear fluxes F(U(xj± 12, t)) by appropriate numerical fluxes using two

values of U at inter-elemental boundaries as

F(U(xj− 12, t)) = F(U−

j− 12

, U+j− 1

2

) and (3.4)

F(U(xj+ 12, t)) = F(U−

j+ 12

, U+j+ 1

2

) (3.5)

where U−j− 1

2

= limx↓x−

j− 12

U(x, t) and U+j− 1

2

= limx↓x+

j− 12

U(x, t) [18]. The interior information is

denoted by a superscript “-” and the exterior by a superscript “+”, respectively.

56

As opposed to the inter-elemental continuity requirement in the standard Galerkin method, the DG

method relaxes this requirement by allowing a jump of numerical fluxes at inter-element bound-

aries [12, 18]. The approximate solutionU is then obtained by

∫ xj+ 1

2

xj− 1

2

(

U,t(x, t)ω(x) − F(U(x, t))ω(x),x

)

dx +[

F(U(x, t))]

j= 0 (3.6)

where

[

F(U(x, t))]

j= F(U

−

j+ 12, U

+

j+ 12)ω(xj+ 1

2) − F(U

−

j− 12, U

+

j− 12)ω(xj− 1

2)

Equation (3.6), which is the semi-discretized equation, requires the temporal discretization because

it is still continuous in time.

3.2.2 Multi-dimensional weak form

The procedure of the 1-D weak formulation is easily extendedto multi-dimensions. The three-

dimensional Euler equations are written as

Ut + · F(U) = 0 (3.7)

where

U =

ρ

ρu

ρv

ρw

ρe

, F(U) = [f(U), g(U), h(U)] =

ρmu ρmv ρmw

(ρu)mu + p (ρv)mu (ρw)mu

(ρu)mv (ρv)mv + p (ρw)mv

(ρu)mw (ρv)mw (ρw)mw + p

(ρe)mu + pu (ρe)mv + pv (ρe)mu + pu

and termsmu, mv andmw are the ALE convection velocities in each direction.

57

By multiplying Equation (3.7) by a test functionω(x), integrating overΩj and applying the diver-

gence theorem to the flux term, we obtain

∫

Ωj

ω(x)UtdΩ +

∫

Γj

ω(x)F(U) · ndΓ (3.8)

−

∫

Ωj

(

ω(x),xf(U) + ω(x),yg(U) + ω(x),zh(U))

dΩ = 0

where the vectorn = [n1, n2, n3]T is the local unit outward normal vector toΓj. For the calcu-

lation of the unit normal vector, see AppendixD. The 2D numerical flux functions are obtained in

the same way as the one-dimensional flux functions using two values on each side of four edges

Γje,jw,jn,js in Figure3.2[12, 84].

Figure 3.2. 2D representation of the flux computation between elementΩj and it neighbors

To evaluate the integral terms in the weak formulation, we use the Gauss quadrature rule with an

appropriate number of Gauss points [91, 100]. For the line integral, we use the three point Gauss

quadrature rule as

58

∫ 1

−1

f(x)dx ≈∑

i

Wif(ξi) =5

9

[

f(−

√

3

5) + f(

√

3

5)

]

+8

9f(0) (3.9)

whereWi are weighting coefficients andξi are Gauss points [100]. For the area integral, we use

the3 × 3 point Gauss quadrature rule as

∫ 1

−1

∫ 1

−1

f(x, y)dxdy ≈∑

i

∑

j

WiWjf(ξi, ηj)

=25

81

[

f(−

√

3

5,−

√

3

5) + f(

√

3

5,−

√

3

5) + f(−

√

3

5,

√

3

5) + f(

√

3

5,

√

3

5)

]

+40

81

[

f(0,−

√

3

5) + f(−

√

3

5, 0) + f(

√

3

5, 0) + f(0,

√

3

5)

]

(3.10)

+64

81f(0, 0)

whereWi andWj are weighting coefficients, and(ξi, ηj) are Gauss points [100].

Figure 3.3. Canonical elements in one and two dimensions [18]

59

3.2.3 Basis function

Since the DG formulation relaxes the element-by-element continuity requirement, several options

are available to choose a basis function. For this work, a function based on the Legendre polynomi-

als is chosen [12, 18]. This basis function diagonalizes the mass matrix withoutlumping so that it

is most commonly used in practical DG applications [12, 13, 76, 77, 78]. We can also choose other

basis functions as in the standard Galerkin method with appropriate mass lumping [18, 91, 95].

The first three Legendre polynomials commonly used are [12, 18, 112]

P0(ξ) = 1, P1(ξ) = ξ andP2(ξ) =3ξ2 − 1

2(3.11)

and have following properties

|Pj(ξ)| ≤ 1 for − 1 ≥ ξ ≥ 1 , Pj(±1) = (±1)j,

dPj(±1)

dξ= (±)j j(j + 1)

2and

∫ 1

−1

Pj(ξ)2dξ =

2

2j + 1(3.12)

These polynomials are shown in Figure3.4.

Figure 3.4. Illustration of Legendre polynomials, p=0,1,2...,5

60

Below is a sample MATLAB code to generate the representationof P=0~5 Legendre polynomials.

Algorithm 1 MATLAB code to plot the representation of Legendre polynomials

function mainxi=-1.0:0.01:1.0;for iorder=0:5

temp=Legendre(iorder,xi)plot(xi,temp,’linewidth’,2); hold on

end

% Definition of Legendre polynomials(P=0~5)function LP=Legendre(iorder,xi)if(iorder == 0) LP=ones(size(xi));

elseif(iorder == 1) LP=xi;elseif(iorder == 2) LP=(3.0*xi.^2-1.0)/2.0;elseif(iorder == 3) LP=(5.0*xi.^3-3.0*xi)/2.0;elseif(iorder == 4) LP=(35.0*xi.^4-30.0*xi.^2+3)/8.0;elseif(iorder == 5) LP=(63.0*xi.^5-70.0*xi.^3+15.0*xi)/8.0;

end

Normalized Legendre polynomials are often used [12, 77] :

P0(ξ) = 1, P1(ξ) = ξ andP2(ξ) = ξ2 −1

3(3.13)

where the original Legendre polynomials are divided by its L2 norm since the basis function with

the original Legendre polynomials is not orthonormal [77].

A simple numerical experiment with the smoothed initial condition (ρ(x, 0) = 1.0+0.2 sin(πx) v(x, 0) =

1.0 andp(x, 0) = 1.0 in [77]) is performed to assess the performance of the RKDG method de-

pending on the order of the Legendre polynomials. The domain[0, 2] is discretized with several

grids (6, 12, 24, 48 and 96) to measure theL1 norm for the error estimation of conservation laws

[132]. Both boundaries are treated using a periodic condition and the 1D RKDG code is run to

a final time of 1 second. According to the RKDG CFL condition (3.35), the CFL numbers for

61

the casesP 0, P 1 andP 2 are set equal to 0.5, 0.3 and 0.2, respectively. The exact solutions are

ρ(x, t) = 1.0 + 0.2 sin(π(x − t)), v(x, t) = 1.0 andp(x, t) = 1.0. Figure3.5 represents the cal-

culated densitiesρ compared to the exact density, and the percentage errors (i.e., |ρ(x,t)−ρh(x,t)||ρ(x,t)|

×100

whereρ(x, t) andρh(x, t) are the exact density and the calculated densities ). For ease of distinc-

tion, the domain is discretized with only 20 elements.

Figure 3.5. The density profiles and percentage errors (%)

TheP 0 approximation generates the largest diffusive result. TheP 0 DG approximation performs

similar to a first-order FV method. Neither theP 1 nor theP 2 DG approximations present signifi-

cant differences in the quality of the numerical results. Compared to theP 1 DG approximation, the

P 2 DG approximation generates slightly less diffusion in someregions. This experiment demon-

strates that the DG method, based on higher-order polynomials aboveP 0, may produce satisfactory

numerical results. As a good choice for efficiency and accuracy, this work chooses the basis func-

tion based on theP 2 Legendre polynomials (3.13). TheL1 errors and the orders of accuracy for

the densityρ are summarized in Table3.2. Using theL1 errors versus the number of elements,

the convergence of DG approximations is also illustrated inFigure3.6. The designed orders of

accuracy for these cases are achieved.

62

p=0 p=1 p=2

N L1 error order L1 error order L1 error order

6 4.4017E-02 - 2.7852E-02 - 3.9263E-03 -

12 1.9962E-02 1.1408 6.5837E-03 2.0808 4.8650E-04 3.0127

24 9.2603E-03 1.1081 1.5524E-03 2.0844 6.6057E-05 3.0037

48 4.2986E-03 1.1072 3.7529E-04 2.0485 7.7568E-06 3.0010

96 2.0452E-03 1.0716 9.2389E-05 2.0222 9.4682E-07 3.0004

Table 3.2. TheL1 errors and the orders of accuracy for the densityρ

Figure 3.6. The convergence of DG approximations (p=0, p=1 and p=2)using theL1 errors versus the number of elements

63

To apply the Gauss quadrature rule for the integrals in the weak formulation requires the trans-

formation of the global coordinatesx to the local coordinatesξ by a linear transformation. An

element(xj− 12, xj+ 1

2) is mapped to a canonical element (-1, 1) via a linear transformation as [91]

x(ξ) =(1 − ξ)

2xj− 1

2+

(1 + ξ)

2xj+ 1

2and ξ(x) =

2x − xj+ 12− xj− 1

2

xj+ 12− xj− 1

2

(3.14)

where

x(−1.0) = xj− 12, x(1.0) = xj+ 1

2and

dx

dξ=

xj+ 12− xj− 1

2

2=

∆xj

2

The mapping between elements is demonstrated in Figure3.7.

Figure 3.7. 1D mapping between element(xj− 12, xj+ 1

2) and canonical element (-1, 1)

On canonical elements, the weak form (3.6) can be rewritten as

∆xj

2

∫ 1

−1

(

U,t(ξ, t)ω(ξ)− F(U(ξ, t))ω(ξ),ξ

)

dξ +[

F(U(ξ, t))]

j= 0 (3.15)

and using a modal expansion, the solutionU(ξ, t) and the test function are approximated by

U(ξ, t) =

p∑

k=0

Uk(t)Pk(ξ) and ω(ξ) =

p∑

k=0

ωPk(ξ) (3.16)

whereω is a constant. Both expansions take the same basis function (i.e. the Bubnov-Galerkin

method which is the most widely used [100]). For theP 2 Legendre polynomial, the approximate

solutionU(ξ, t) within the element[xj− 12, xj+ 1

2] is expressed by [12]

64

U(ξ, t) = U(t) + Ux(t)P1(ξ) + Uxx(t)P2(ξ) (3.17)

= U(t) + Ux(t)ξ + Uxx(t)(ξ2 −

1

3)

whereU, Ux and Uxx represent the constant value, the slope and the quadratic ofthe approximate

solution. The FVM and the DGM are compared in Figure3.8. The FVM uses the piecewise-

constant approximation which means the values are constantwithin the element, while the DGM

employs the piecewise-polynomial approximation which means the values vary within the element.

As mentioned earlier, the accuracy in spatial discretization can be improved simply by increasing

the order of the basis function.

(a) Piecewise-constant approximations in FVM

(b) Piecewise-polynomial approximations in DGM

Figure 3.8. Description of the FVM and the DGM

65

By substituting Equation (3.16) into Equation (3.15), we obtain

Uk(t),t = M−1

L

∫ 1

−1

dPk(ξ)

dξF(U(ξ, t))dξ −

[

Pk(1)F(U(1, t))j − Pk(−1)F(U(−1, t))j

]

for k = 0, 1, 2, ...p (3.18)

wherep is the highest order of Legendre polynomial. The local mass matrix ML can be inverted at

very little cost. Each element of the local mass matrix for theP 2 Legendre polynomial is

mij =∆x

2

∫ 1

−1

Pi(ξ)Pj(ξ)δijdξ (3.19)

such that [12]

ML = diag[

m00 m11 m22

]

= ∆xdiag[

1 1/3 4/45

]

(3.20)

The numerical fluxes are computed by

F(U(1, t))j ≈ F(U(1, t)j, U(−1, t)j+1) (3.21)

F(U(−1, t))j ≈ F(U(−1, t)j, U(1, t)j−1) (3.22)

where each depends on the values on either side of inter-elemental boundaries. The communication

between elementj and its neighbors occurs only by the flux computation across inter-elemental

boundaries [12, 18]. See Subsection3.2.4 for the numerical flux computation. Algorithm 2 is

the pseudo-code to treat the integral in the DG approximate Equation (3.18). For the detail of the

Gauss quadrature rule, refer to Equations (3.9-3.10).

66

Algorithm 2 Pseudo-code for the integral part of the scalar equation, U,t + (0.5U2),x = 0

! 1. Define Gauss points

ξ−1 = −√

35; ξ0 = 0; ξ1 =

√

35;

DO i=1,nx ! nx=the number of elements in x-direction

! 2. Reconstructs variables at Gauss points

U−1 = Ui(t) + Ui,x(t)ξ−1 + Ui,xx(t)(ξ2−1 −

13) ! approximate solutionU(ξ−1, t)i

U0 = Ui(t) + Ui,x(t)ξ0 + Ui,xx(t)(ξ20 −

13) ! approximate solutionU(ξ0, t)i

U1 = Ui(t) + Ui,x(t)ξ1 + Ui,xx(t)(ξ21 −

13) ! approximate solutionU(ξ1, t)i

! 3. Do Gauss quadrature integration, INT=∫ 1

−1dPk(ξ)

dξF(U(ξ, t))dξ

INT(i, 0) = 0 ! dP0(ξ)dξ

= 0

INT(i, 1) = 59(0.5U2

−1) + 89(0.5U2

0) + 59(0.5U2

1) ! dP1(ξ)dξ

= 1

INT(i, 2) = 2

59(0.5U2

−1ξ−1) + 89(0.5U2

0ξ0) + 59(0.5U2

1ξ1)

! dP2(ξ)dξ

= 2ξ

ENDDO

For two-dimensional problems, the solutionUij(ξ, η, t) is similarly approximated by [12, 94]

Uij(ξ, η, t) =

p∑

m=0

p∑

n=0

Uijmn(t)Pm(ξ)Pn(η) (3.23)

wherep is the highest order of Legendre polynomial chosen. The mapping between a 2D element

and a canonical element is given in Figure3.3. For theP 2 Legendre polynomial, the approximate

solutionU within an element[xi− 12, xi+ 1

2] × [yj− 1

2, yj+ 1

2] becomes [12]

U(ξ, η, t) = U(t) + Ux(t)P1(ξ) + Uy(t)P1(η) + Uxy(t)P1(ξ)P1(η) + Uxx(t)P2(ξ)

+Uyy(t)P2(η)

= U(t) + Ux(t)ξ + Uy(t)η + Uxy(t)ξη + Uxx(t)(ξ2 −

1

3) (3.24)

+Uyy(t)(η2 −

1

3)

67

Each element of the local mass matrix in two-dimensions is defined as

mi(ξ)j(η)=

∆x

2

∆y

2

∫ 1

−1

∫ 1

−1

Pi(ξ)Pj(η)dξdη (3.25)

such that [12, 95]

ML = ∆x∆ydiag[

m0(ξ)0(η)m1(ξ)0(η)

m0(ξ)1(η)m1(ξ)1(η)

m2(ξ)0(η)m0(ξ)2(η)

]

(3.26)

= ∆x∆ydiag[

1 1/3 1/3 1/9 4/45 4/45

]

By substituting Equation (3.23) into Equation (3.8), we obtain the two-dimensional DG approxi-

mate equation. Since only the normal components of the fluxesare involved in the surface integral

of Equation (3.8), the multi-dimensional numerical fluxes are computed in the same way as the

one-dimensional numerical fluxes using values on each side at inter-elemental boundaries [12].

The discussion in this subsection indicates that the numerical flux functions play a major role in

the communication of an element and its neighbors in the framework of the DG approximation.

3.2.4 Numerical flux

Numerical flux functions typically use the same approaches as proven successful in FV methods

for hyperbolic wave problems. B. Cockburn presented some useful numerical fluxes such as the

Godunov flux FG, the Engquist-Osher flux FEO and the local Lax-Friedrichs (LLF) flux FLLF

[12]. He found that as the order of the basis function increases,the choice of the flux function

has no significant influence on the quality of the numerical results [12]. J. Qiu investigated the

performance of various flux functions in great detail [77]. He found that the RKDG method with

the first-order center (FORCE) flux FFORCE costs less CPU time and the numerical errors are also

less. Both the LLF and the FORCE functions are relatively simple to apply for nonlinear hyperbolic

problems and performs similarly to the Godunov flux function[12, 18, 45, 77]. The Godunov flux

68

function requires the solution of the local Riemann problemto construct flux functions, while

the LLF and the FORCE flux functions do not require this complexity [12, 18, 45]. Due to its

simplicity, the LLF function has been most commonly used in the literature, but introduces more

diffusion than is actually required [12, 18, 76, 77]. The FORCE function produces less diffusive

solutions than those of the LLF function [77]. Considering the computational cost and the quality

of discontinuous solutions, our work uses the first-order FORCE flux function FFORCE averaged

with the LLF function and the second-order Richtmyer flux function FR [45, 77]. For the details

of flux functions, References [45, 77, 132, 151] are recommended.

The LLF function FLLF is defined as

FLLF (U(1, t)j, U(−1, t)j+1) =1

2

F(U(1, t)j) + F(U(−1, t)j+1)

−λmax(U(1, t)j − U(−1, t)j+1)

(3.27)

whereλmax is the maximum absolute eigenvalue of the Jacobian∂F(U)/∂U over elementj and

j + 1 [12, 18, 77]. For scalar problems,λmax is taken as an upper bound between adjacent two

valuesU(t)j andU(t)j+1 [12, 77]. Equation (3.17) gives

U(1, t)j = Uj(t) + Uj,x(t) +2

3Uj,xx(t) (3.28)

U(−1, t)j+1 = Uj+1(t) − Uj+1,x(t) +2

3Uj+1,xx(t)

and fluxes FLLF are assembled with interface variables inherent. In the second-order Richtmyer

function, the variables are first reconstructed by

U∗

=1

2

U(1, t)j + U(−1, t)j+1 −∆t

∆xj

(

F(U(1, t)j) − F(U(−1, t)j+1))

(3.29)

then fluxes FR are assembled by using the stared variablesU∗as

FR(U(1, t)j, U(−1, t)j+1) = F(U∗) (3.30)

69

where∆t is the size of time step and∆xj is the increment of elementj in x-direction [77].

The FORCE flux function FR is defined as [77]

FFORCE(U(1, t)j, U(−1, t)j+1) =1

2

FLLF (U(1, t)j, U(−1, t)j+1) (3.31)

+FR(U(1, t)j, U(−1, t)j+1)

A simple numerical experiment was performed to compare the performance of the LLF function

and the FORCE function. The initial conditions of a one-dimensional Riemann problem are given

in Figure 3.9. The domain[0, 4]m was discretized with 100 elements. Both boundaries were

treated as a rigid wall, and the code ran to a final time of 0.002seconds.

Figure 3.9. The initial condition of the Fedkiw’s strong shock tube problem [135]

Figure3.10shows the calculated density and pressure profiles versus the analytic solution at time

t = 0.002 seconds, and the percentage errors. The profiles indicate that both flux functions behave

similarly but the FORCE function gives less diffusive solutions particularly near the shock front.

Both the FORCE function FFORCE and LLF function FLLF required a compact stencil size which

means that the flux function F(U)j depends on the two valuesUj andUj+1 at the previous time

step. This dependency provides a highly parallelizable structure for large-scale problems. For

multi-dimensions, the numerical fluxes are applied using the same procedure in other directions.

70

(a) Density and pressure profiles versus the analytic solution

(b) Percentage errors

Figure 3.10. Numerical results of the Fedkiw’s strong shocktube problem

71

Algorithm 3 represents the pseudo-code to assemble the numerical fluxes for a scalar equation. For

the system, we evaluate the numerical flux based on the flux expressions in Equations (3.1-3.7).

Algorithm 3 Pseudo-code for numerical fluxes of the scalar equation, U,t + (0.5U2),x = 0

DO i=2,nx-1 ! For internal elements only.

! 1. Reconstructs variables at inter-elemental interfaces, x+j− 1

2

andx−j− 1

2

Uminus = Ui(t) + Ui,x(t) + 23Ui,xx(t) ! approximate solutionU(1, t)i

Uplus = Ui+1(t) − Ui+1,x(t) + 23Ui+1,xx(t) ! approximate solutionU(−1, t)i+1

! 2. Find an upper bound between Ui and Ui+1

λmax = MAX(Ui, Ui+1)

! 3. Assemble the fluxes using reconstructed variables,[

F(U(ξ, t))]

i

! 3.1 Define fluxes for the given scalar equationFplus = 0.5U2

plus; Fminus = 0.5U2minus

! 3.2 Compute Local Lax-Friedriche flux functions FLLF

FLLF (i) = 12Fplus + Fminus − λmax (Uplus−Uminus)

! 3.3 Compute Richtmyer flux functions FR

FR(i) = 12

Uplus+Uminus −∆t∆x

(Fplus − Fminus)

! 3.4 Compute FORCE flux functions FFORCE

FFORCE(i) = 12

FLLF (i) + FR(i)

ENDDO

3.3 Temporal discretizations

To treat the semi-discretized governing equations in time,an explicit Runge-Kutta (RK) time inte-

grator is used throughout our work. B.Q.Li found that to conduct an accurate prediction, the order

of the time integrator should at least be compatible with theorder of the spatial approximation [18].

B. Cockburn also recommended that for a polynomial of degreek, the order of a RK integrator

must bek + 1 or higher [12]. Based on these experiences, we chose the third-step RK integrator

for P 2 spatial approximation [12, 18, 94]

72

U(1) = U(0) + ∆tR(U(0))

U(2) =3

4U(0) +

1

4

(

U(1) + ∆tR(U(1)))

(3.32)

U(3) =1

3U(0) +

2

3

(

U(2) + ∆tR(U(2)))

where R(Uk) is the RHS vector in Equation (3.18) as

RHS= M−1L

∫ 1

−1

dPk(ξ)

dξF(U

k(ξ, t))dξ −

[

Pk(1)F(Uk(1, t))j − Pk(−1)F(U

k(−1, t))j

]

(3.33)

The first integral term is obtained by the algorithm 2 and the second flux terms are taken from the

algorithm 3. Then, these terms are multiplied by the inverseof the local mass matrix (3.20). The

values of the polynomialPk(±1) before the fluxes are given as

P0(1) = 1 and P0(−1) = 1

P1(1) = 1 and P1(−1) = −1 (3.34)

P2(1) =2

3and P2(−1) =

2

3

where the Legendre polynomials areP0(ξ) = 1, P1(ξ) = ξ andP2(ξ) = ξ2 − 13.

The size of time step in theP k RKDG approximation is determined by following the Courant-

Friedrichs-Levy (CFL) condition

|c|dt

dx≤

1

2k + 1(3.35)

where|c| is the absolute value of the speed of sound [12, 18].

3.4 Generalized slope limiter

A Total Variation Diminishing (TVD) method is one where the total variation of the solution does

not increase as

TV (Un+1) ≥ TV (Un

) where TV (Un) =

∑

j

|Uj+1 − Uj| (3.36)

73

where variablesU refer to the values at the center of elements [18, 27, 71, 132, 151]. The TVD

property is desirable to obtain stable results in most numerical methods [27, 151]. To ensure the

TVD property, nonlinear limiters have often been included in numerical computations even for

linear fluid flow problems [18, 132, 151]. There are several types of limiters, such as flux-based

limiters and slope-based limiters. Flux limiters are applied directly to the fluxes rather than fluid

flows or characteristic variables while slope limiters introduce systematic dissipation arising from

nonlinearities of fluid flows [19].

Flux limiters are based on the idea of combining a high-orderscheme and a low-order scheme (e.g.,

FE-FCT algorithms) [124, 132]. An accurate high-order scheme is applied to smooth flow regions

while a diffusive low-order scheme is applied near discontinuities. For the details of FE-FCT

algorithms, refer to AppendixB.

This work uses a generalized slope limiter to obtain stable results. For the details of slope limiters,

References [18, 132, 151] are recommended. A minmod slope limiter, which operates onthe slope

Uj,x using the slope itself and the differences of constant values between two neighboring elements

and elementj, is chosen as [12, 132]

Uj,x = minmod(Uj,x, Uj+1 − Uj , Uj − Uj−1) (3.37)

where

minmod(a, b, c) =

sgn(a)minmod(|a| , |b| , |c|) if sign(a) = sign(b) = sign(c)

0 otherwise

The schematic of this slope limiter is shown in Figure3.11.

Figure 3.11. The illustration of a general minmod slope limiter; ux means the slope of solutions and () represents the sign of theslope[84]

74

The limiter compares the slope and its neighbors, then sets the slope to the smaller value in case

(a) while in case (b) or (c) the slope sets equal to zero [84]. Once the slope Uj,x is limited, the

quadratic value Uj,xx is set equal to zero to avoid nonphysical errors. For stable results with no

spurious pressure oscillations, this generalized slope limiter is always applied before computing

the right-hand side of the equations at each level of a singletime step. Below is the pseudo-code

of the slope limiter for a scalar equation.

Algorithm 4 Pseudo-code of the modified minmod slope limiter for the scalar equations

DO i=2,nx-1 ! for internal elements only

! 1. Define minmod argumentsUS=U(i,1) ! slope of element iU1=U(i+1,0)-U(i,0) ! the difference of constant values betweeni + 1 andiU2=U(i,0)-U(i-1,0) ! the difference of constant values betweeni andi − 1

! 2. Apply the minmod limiterUS1=MINMOD(US,U1,U2)IF(US1 .NE. US) THEN ! Check whether the slope is limited or not

U(i,1)=US1; U(i,2)=0ENDIF

ENDDO

FUNCTION MINMOD(US,U1,U2)

IF(US > 0 .AND. U1 > 0 .AND. U2 > 0) THEN ! All signs are positiveMINMOD=MIN(US,U1,U2)

ELSEIF(US< 0 .AND. U1 < 0 .AND. U2 < 0) THEN ! All signs are negativeMINMOD=MAX(US,U1,U2)

ELSEMINMOD=0

ENDIF

ENDFUNCTION

We performed a numerical experiment to assess the performance of this slope limiter by solving

the Fedkiw’s strong shock tube problem given in Subsection3.2.4. Figure3.12 represents the

potential of the slope limiter for suppressing the oscillations near discontinuities. The RKDG

method without the limiting had several kinks near discontinuities which may spoil the numerical

results in many cases.

75

Figure 3.12. The performance of the generalized slope limiter for the Fedkiw’s problem

The slope limiting was applied to local characteristic variables rather than system variables which

are interrelated. The limiting procedure is summarized in AppendixC.

This chapter provided a description of the spatial discretization, the temporal discretization and the

generalized slope limiter in the framework of the RKDG method for compressible inviscid fluid

flows. Section3.1briefly introduced the RKDG method. Section3.2provided a detail of the spatial

discretization based on the DG weak formulation. The DG weakform provides the discontinuous

formulation which is effective for treating shock-involved compressible flows [12, 18]. By relaxing

the inter-element continuity in the discontinuous formulation, the communication between an ele-

ment and its neighbors is conducted only by numerical flux functions. Subsection3.2.4discussed

the first-order FORCE flux function which is better than otherflux functions in accuracy, efficiency

and CPU time [77]. Section3.3provided a description of the temporal discretization based on the

third-step Runge-Kutta time integrator. Section3.4 presented the generalized slope limiter based

on the slope and two neighboring values. Several pseudo-codes for the scalar equation were pro-

vided to explain the implementation of the RKDG method. Thiscompletes the discussion of a

single-fluid flow solution method based on the RKDG method.

This single-fluid method cannot be directly applied to the simulation of two-fluid flows in which

explosive gas and water are presented together since there is no consideration of the discontinuous

adoption of the equations of state across the interface separating fluids. The next chapter discusses

a two-fluid flow solution method incorporated in the RKDG framework.

76

Chapter 4

Solution methods

The previous chapter described the RKDG method consisting of the RK scheme in time, the DG

scheme in space, and the slope limiter in suppressing spurious oscillations. A set of the governing

equations in the ALE description was provided. The numerical approach in Chapter 3 cannot

directly treat two-fluid flows with material discontinuities and handle a deformable fluid mesh

required in near-field UNDEX simulations.

This chapter introduces the solution methods for two-fluid flows in which explosive gas and water

coexists, cavitation effects, and an ALE-based deformablefluid mesh. The solution methods are

integrated in the framework of the RKDG method discussed previously. This provides an ALE-

RKDG compressible two-fluid approach which is a unique coupled method usable for numerically

simulating complex two-fluid flows and fluid-structure interactions in near-field UNDEX events.

Section4.1 provides a description of interface methods and two-fluid methods. A reliable two-

fluid method is suggested for numerically treating explosive gas-water flows. Section4.2presents

a fluid-structure interaction (FSI) algorithm based on a grid moving strategy. Cavitation is dis-

cussed with FSI which may effect the state of the surroundingfluid and the pressure loading on

the structure. The cavitation mechanism consists of cavitation cutoff, its subsequent collapse and

reloading the target. Through several examples in Chapter 5, the effectiveness and robustness of

these approaches are assessed.

77

4.1 Two-fluid method

Multi-fluid/multi-phase compressible flows occur in many industrial situations. There are different

physical and thermodynamic properties across the interface separating fluids. R. Saurel defined a

multi-fluid flow as homogeneous phases separated by well-defined interfaces, and a multi-phase

flow with a number of the interfaces such as a bubbly flow [128]. See Figure4.1.

Figure 4.1. Comparison of a multi-fluid and a multi-phase flow[128]

In Chapter 1, we observed that near-field UNDEX fluid flows consist of homogeneous explosive

gas and water separated by a well-defined interface. Therefore, we restrict our concern to the

two-fluid flows in which immiscible explosive gas and water coexist.

Once the explosion is initiated, the high pressure gas bubble produces a shock wave followed by

the material interface, and a rarefaction wave. At the initial state, both the shock wave and the

material interface propagate radially outward, while the rarefaction wave travels toward the origin

[45, 132]. This distinguishable wave structure provides a good way to identify the location of

the material interface throughout the computation and therefore we can readily distinguish one

fluid from the other across the material interface. Different equations of state (EOS) are then

applied to each fluid. The discontinuous EOS may create spurious pressure oscillations near the

material interface, perhaps resulting from nonphysicallydiffused system variables [31, 92, 135].

This section describes a two-fluid method coupled with an interface method suitable for near-

field UNDEX flow simulations. The first subsection provides a description of interface methods

commonly used.

78

4.1.1 Interface methods

Interface methods can be largely divided into two main classes; Lagrangian approaches and Eule-

rian approaches. Lagrangian approaches can be further subdivided into fitting methods and track-

ing methods. Figure4.2represents the description of interface methods in two-dimensions.

Figure 4.2. Description of interface methods in two-dimensions [76]

The Interface-fitting method (a) employs a set of grids aligned along the interface which is trans-

ported in Lagrangian fashion [10, 57, 79]. This method provides an explicit, accurate interface

location, but for large interface motion the interface-aligned grids become extremely distorted

[57, 145]. In the Interface-tracking method (b), the flow computation is carried out on a spatially-

fixed Eulerian mesh with a set of mass-less markers representing the interface [79]. This method

may enhance the mesh quality in large interface motion, but the motion of markers may still be-

come complicated when the interface motion is extreme [140]. In Lagrangian approaches, the

interface is transported by

dxdt

= v (4.1)

where x = [x, y, z] and v = [u, v, w] are the position and the velocity vectors at every point

[79, 140]. The use of Lagrangian approaches is often limited to the case where interface motion

is not extreme [10, 57, 79, 140]. In case (c), the interface is embedded on a spatially-fixedmesh

and transported by flow velocities in Eulerian fashion [79, 132, 140]. This method may prevent

topological problems as in the Lagrangian approaches when the interface motion is extreme. A

tracer function,φ(x, t), labels every point in the domain as either one fluid or the other, and is

79

transported by

dφ

dt+ v · φ = 0 (4.2)

where v·φ representsuφx + vφy + wφz [79, 132, 140]. Equation (4.2) implies that the interface

defined initially is transported by flow velocities v.

Eulerian methods include theγ-based method, the mass fraction or volume of fluid (VOF) method

and the level set (LS) method [2, 15, 75, 132, 140]. Recently, the LS method has been used in many

disciplines such as graphics, geometry optimization and computational fluid dynamics [79, 140].

The method handles the time-varying shape by solving Equation (4.2). The tracer functionφ(x, t)

becomes the signed distance functionφ = ±d, or the so-called LS function [79, 140]. This method

naturally provides topological information such as curvatures and normal vectors, and is easily

applicable to multi-dimensions with no extra cost associated with parametrizing curves or surfaces

[79, 140]. These characteristics make the LS method a good mathematical tool for modeling the

time-varying nature of complex two-fluid flows. Table4.1 lists the use of interface methods in the

literature.

Interface-fitting Interface-tracking Interface capturing methodmethod method VOF LSM γ-basedWardlaw Jr.[5],A. Alia [1], A.R.Pishevar[10],G.R. Liu[54]

A. Chertock[2],J.P. Cocchi[92],C.H. Cooke[30]

M.Schaffar[108],D. Benson[44]

J. Naber[76],G.F. Duivesteijn[50],B.Koren[15], R.P.Fedkiw[135], S.Y.Kadioglu[142], T.G.Liu[147], R.R.Nourgaliev[138], C.Wang[24]

H.S. Tang[62],P. Jenny[118],K.M.Shyue[102], R.Saurel[127], P.Kuszla[119]

Table 4.1. Use of interface methods

This work uses the LS method to identify the interface location in explosive gas-water flows.

The following sub-subsections detail the mathematical expressions of the interface, the evolution

and reinitialization of the LS function to ensure that it is areal distance function throughout the

computation.

80

4.1.1.1 Interface presentation

S. Osher states that the interface can be expressed mathematically as ann − 1 dimensional com-

ponent inn spatial dimension [140]. Therefore, the interface is expressed as points on a line in

one-dimension, a curve on a surface in two-dimensions and a closed surface on a volume in three-

dimensions. Figure4.3gives respective graphical representations of the interface in one-dimension

and two-dimensions.

Figure 4.3. Graphical representations of the interface [140]; external region is denoted byΩ+, internal regionΩ− and interface∂Ω

The LS functionφ(x, t) is the signed distance function±d(x), being negative in one fluid and

positive in the other. The LS function implies thatφ = 0.0 is for points on∂Ω, φ = −d(x) for

points∈ Ω− andφ = d(x) for points∈ Ω+ [140]. A distance functiond labels every point in a

domain with the value of the shortest normal distance to the interface∂Ω [140]

d(x) = min(|x − xi|) where xi ∈ ∂Ω (4.3)

For example, the initial LS functionφ0 is set as

φ0 =

|x| − 1 in 1D√

x2 + y2 − 1 for unit -circle in 2D√

x2 + y2 + z2 − 1 for unit-sphere in 3D

(4.4)

then the LS functionφ is transported by Equation (4.2).

81

Consider a gas bubble in domain[−1, 1] × [−1, 1] as a centered unit-diameter circle. Figure4.4

represents the contours of the initial LS functionφ0 (a), and the LS functionφ (b) transported by a

uniform outward velocity v.

Figure 4.4. Contours of the LS functionφ with a centered circle

For multi-objects, the LS functionφ is expressed as the intersection of two sub-LS functionsφ1

andφ2 [140]

φ = min(φ1, φ2) (4.5)

To illustrate, consider two circles initially embedded in domain[−1, 1]×[−1, 1]. Figure4.5demon-

strates the potential of the LS method showing it is capable of treating the merging of interfaces.

Figure 4.5. Contours of the LS functionφ with two circles

82

This simple initialization and evolution of the LS functionφ makes the LS method a good math-

ematical tool for modeling the time-varying nature of a gas bubble in explosive gas-water flows.

Below is the sample MATLAB code to initialize a unit-diameter circle centered in domain[−1, 1]×

[−1, 1].

Algorithm 5 MATLAB code of the initialization of 2-D LS functionφ for a centered unit-circle[164]

! 1. Define the number of nodes N, the size of elements h and the size of matricesN=40; h=2/N; x=zeros(N,N); y=zeros(N,N); phi0=zeros(N,N); phx=phi0; phy=phi0;

! 2. Set the mesh grid and initial LS functionφ0 by Equation (4.4)[x,y]=meshgrid(-1:h:1,-1:h:1); phi0=SQRT(x.^2+y.^2)-0.5;

! 3. Compute normal vector of internal grids n= φ0

|φ0|by central differential scheme

for i=2,N-1for j=2,N-1phx(i,j)=(phi0(i+1,j)-phi0(i-1,j))/(2*h); phy(i,j)=(phi0(i,j+1)-phi0(i,j-1))/(2*h)

endendnx=phx/sqrt(phx.^2+phy.^2+eps); ny=phy/sqrt(phx.^2+phy.^2+eps);

! 4. Plot the contoursfigure(1); surfc(x,y,phi0); hold on; contour(x,y,phi0,zeros(size(phi0)),’k’); axis(’square’);figure(2); quiver(x,y,nx,ny); hold on; contour(x,y,phi0,zeros(size(phi0)),’k’); axis(’square’);

4.1.1.2 Interface evolution

Because of the presence of flow velocity v in the LS Equation (4.6), the system of equations and the

LS equation must be solved simultaneously. Since the LS function φ is a scalar function, Equation

(4.6) can be decoupled from the system of equations. The interface (i.e.,φ = 0) is transported by

dφ

dt+ v · φ = 0 (4.6)

where v is the flow velocity given by the system of governing equations. Note that both the fluid

flow and the LS function do not necessarily have to use the samenumerical approach. For the spa-

tial discretization of Equation (4.6), upwind differencing, Hamilton-Jacobi ENO, Hamilton-Jacobi

83

(HJ) WENO and other schemes have been used [76, 140, 164]. In this work, a FV approximation

is employed to evolve the LS function. A DG approximation forthe LS method is not addressed

here since it would require a separate method development.

The standard FV discretization to Equation (4.6) gives

φn+1i,j = φn

i,j −

[

ui,j

∆x(φi+ 1

2,j − φi− 1

2,j) +

vi,j

∆y(φi,j− 1

2− φi,j− 1

2)

]

(4.7)

whereφi,j is piecewise constant LS values,φi± 12,j andφi,j± 1

2are cell-face LS values. A third-order

flux limiter by Koren [76, 132] is employed to evaluate cell-face LS values as

If ui,j ≥ 0

φi+ 12,j = φi,j + 1

2Φ(ri+ 1

2,j)(φi,j − φi−1,j)

φi− 12,j = φi−1,j + 1

2Φ(ri− 1

2,j)(φi−1,j − φi−2,j)

then

else

φi+ 12,j = φi,j −

12Φ(ri+ 1

2,j)(φi+1,j − φi+2,j)

φi− 12,j = φi,j −

12Φ(ri− 1

2,j)(φi,j − φi+1,j)

(4.8)

where

ri+ 12,j =

φi+1,j − φi,j

φi,j − φi−1,j

and Φ(r) = max

(

min

(

2r, min(1

3+

2

3r, 2)

))

(4.9)

Other limiters in [132] can also be used in evaluating the cell-face values. For they-direction, the

same scheme (4.8) with velocityvi,j is applied to the cell-face valuesφi,j± 12.

The FV spatial scheme is then coupled with third-step Runge-Kutta (RK) temporal scheme which

was discussed in Chapter 3.

4.1.1.3 Interface reinitialization

The LS functionφ in Equation (4.6) is numerically transported by real non-uniform flow velocity

v. Non-uniform velocity generates noisy features that are not appropriate in the LS computation

because the numerical results with distorted LS function become unreasonable at subsequent time

steps [79, 140]. The frequent reinitialization of the distorted LS functionφ is recommended in the

84

literature to correct this problem [15, 79, 122, 135, 140, 147]. The commonly used method is the

reinitialization equation developed by Sussman, Smereka and Osher [79, 140]

∂φ

∂τ+ S(φ0)(| φ| − 1) = 0 (4.10)

whereS(φ0) = 2H(φ0) − 1, τ andH(φ0) are a sign function, fictitious time within the reinitial-

ization equation and a Heaviside function, respectively. The key to this correction is to repeatedly

solve Equation (4.10) until steady state is reached as

∂φ

∂τ= 0 (4.11)

such that

| φ| = 1 (4.12)

At steady state, the distorted LS function can be converted back to real signed distance function

[79, 140]. The absolute gradient of the LS function in Equation (4.10) is determined by a second-

order approximation in [76] as

|φ|i,j =

√

(

∆xφ

∆x

)2

i,j

+

(

∆yφ

∆y

)2

i,j

(4.13)

where

∆xφ =

(∆Lxφ)i,j if S(φi,j)(∆

Lxφ)i,j > 0 and S(φi,j)(∆

Lxφ)i,j + S(φi,j)(∆

Rx φ)i,j > 0

(∆Rx φ)i,j if S(φi,j)(∆

Rx φ)i,j < 0 and S(φi,j)(∆

Lxφ)i,j + S(φi,j)(∆

Rx φ)i,j < 0

0 if S(φi,j)(∆Lxφ)i,j < 0 and S(φi,j)(∆

Rx φ)i,j > 0

(4.14)

, (∆Lxφ)i,j = φi,j − φi−1,j +

1

2min(φi+1,j − 2φi,j + φi−1,j, φi,j − 2φi−1,j + φi−2,j)

and (∆Rx φ)i,j = φi+1,j − φi,j +

1

2min(φi+1,j − 2φi,j + φi−1,j, φi+2,j − 2φi+1,j + φi−,j)

The same scheme is applied to y-directional∆yφ.

In summary, the LS method may play an important role in any two-fluid method due to its ap-

pealing features. The method provides a simple way to capture a time-varying shape regardless

85

of its complexities, and to extend to multi-dimensions withno additional cost. However, Eulerian

approaches, including the LS method, always require the modeling of mixed elements in which

fractions of fluids coexist [79, 140]. Since physical and thermodynamic properties are discontinu-

ous across the material interface, a specific treatment nearthe interface is necessary in developing

any two-fluid method. Using the LS interface method, the following subsection proposes a reliable

two-fluid method suitable for near-field UNDEX fluid simulations.

4.1.2 Two fluid methods

Classical Eulerian approaches including the RKDG method generally produce numerical diffusion

which is acceptable in some single-fluid flows. However, in multi-fluid flows, numerical diffusion

may result in nonphysical density at the interface that diffuses from one fluid to the other [92].

Since pressure is calculated by a thermodynamic relationp = p(ρ, e, · · · ), numerical results with

diffused density may produce spurious pressure oscillations in some situations as shown in Figure

4.6.

Figure 4.6. Numerical diffusion generated after single Eulerian step

A sharp interface method (i.e., a zero-point interface capturing method) is often recommended

to eliminate or minimize numerical diffusion which is a major source of non-physical results in

multi-fluid flow simulations [30, 92, 135].

86

A large number of compressible multi-fluid methods have beendeveloped as summarized in Ta-

ble 4.2. The approaches described here can be divided into prediction-correction (PC) methods

which use the solution of the local Riemann problem to correct affected values near the interface,

simple fix (SF) methods which allow a little numerical diffusion in gaseous flows, ALE-based

methods which do not allow mixed elements by adjusting the interface in Lagrangian manner, and

ghost fluid (GF) methods which deal with two separate single fluid flows with a special interface

treatment.

PC methods SF methods ALE-basedmethods

GF methods

J. P. Cocchi[92],D. Schwendeman[38],R. R. Nourgaliev[138],

A. Chertock[2],P. Jenny[118],

N. Andrianov[113],T. J. Chen[150],

J. Naber[76],R. Abgrall[122],

K. M.Shyue[102],

R. Saurel[127],

A. R. Pishevar[10],A. Alia [1],

A.B. Wardlaw[7],H. Luo[57]

G. F. Duivesteijn[50],B. Koren[15],

R. P. Fedkiw[135],T. G. Liu[147],

B. Lombard[16],J. Qiu[78],

D. A. Bailey[35],D. Tam[34]

Table 4.2. Compressible multi-fluid methods in the literature

PC methods directly correct affected values near the interface so that they can capture the interface

sharply. However, the extension to multi-dimensions is notsimple since the multi-dimensional

correction of affected elements is fairly difficult and a multi-dimensional Riemann solution is un-

available [45, 92, 138]. SF methods have been successfully applied to compressible multi-gaseous

flows. However, it is unclear if they are always able to remedythe errors in liquid-involved flows

where the density jump is relatively large [122, 127, 128]. ALE-based methods have been re-

strictively employed to simple 1-D cases where the interface motion is not large [1, 7, 10, 57].

In comparison with other approaches, the extension of GF methods to multi-dimensions is fairly

straightforward since no geometric complexities are involved in the computation. The solution

procedure is relatively easy to implement, and applicable to a variety of multi-fluid flows that

include gas-water flows. These characteristics make GF methods an excellent tool for treating

87

two-fluid flows in near-field UNDEX with no additional cost associated with geometric complex-

ities and complex physics. Subsequent sections detail the solution procedure for a GF method in

conjunction with the LS method.

4.1.2.1 Ghost fluid method

In 1998, R.F. Fedkiw first developed the Ghost Fluid Method (GFM) to simulate various multi-fluid

flows [135]. The positive qualities that make it applicable are easy extension to multi-dimensions,

straightforward implementation of interface conditions and simple substitution into multi-level

time integrators [78, 135, 147]. Making the interface invisible during the GFM computation keeps

a sharp density profile from smearing out [135]. The GFM is usually composed of two steps: step

one solves two respective single fluid flows and step two treats the interface separating fluids with

a problem-dependent algorithm [135]. The LS function,φ(x, t), plays a crucial role in defining the

interface location in the domain. The interface location can be identified by a sign change of the

LS functionφ as

φj × φj+1 < 0 (4.15)

which means that the interface is located somewhere betweenxj andxj+1. Figure4.7presents an

example of the identification of the interface location in one-dimension.

Figure 4.7. Identification of the interface location in one-dimension

Every element in a domain has mass, momentum and energy from the real fluid or from the other

fluid (i.e., ghost fluid) that does not really exist [135]. Once the ghost values are set, the RKDG

method updates a single-fluid flow with no special concern about the interface and gives flow

88

velocities v as one output. The LS functionφ is then transported using flow velocities v (i.e., one

fluid velocity in the region whereφ > 0, and the other fluid velocity in the region whereφ < 0)

for next time step. Figure4.8represents a flow-chart and graphical description of the GFM.

Figure 4.8. Description of the GFM

The key of the GFM is to define variables in the ghost region of each fluid. In one-dimension,

three variables are usually considered [135, 140, 147]. Based on physical reasons observable

in numerical experiments, pressure and velocity are chosenas two of the three variables [135,

140]. This can be demonstrated in the one-dimensional Sod’s shock tube problem with the initial

conditions given in FigureB.2. The domain[0, 1] m was discretized with 100 elements and the

code ran to 0.2 seconds. The analytic solutions are given in Figure4.9.

89

Figure 4.9. Analytic solutions of the Sod’s shock tube problem in[0, 1] at t=0.2 second

Primitive variables at the interface satisfy the followingjump conditions

[ρ] 6= 0, [u] = 0, [p] = 0 and[e] 6= 0 (4.16)

where jump[f ] for any functionf means|fi+1 − fi|. Since pressure and velocity are continuous at

the interface, we can directly copy real pressure and velocity into ghost values by a simple node-by-

node manner [135]. This simple copy procedure naturally imposes interface conditions for pressure

and velocity without explicitly identifying complex jump conditions. Discontinuous entropy at the

interface is most commonly used for the final variable [135]. To remove or minimize numerical

diffusion, the GFM often uses the one-sided constant extrapolation of entropy across the interface

which transforms discontinuous density at the interface tocontinuous density [135]. Based on the

isentropic assumption, densityρi+1 at elementi + 1 is modified by

Si =pi

ργi

and ρi+1 =

(

pi+1

Si

)1γ

(4.17)

90

whereS andγ are the entropy at elementi and the ratio of specific heats, respectively. EntropySi

is directly copied to elementi + 1 by constant extrapolation and densityρi+1 is modified by the

isentropic relation (4.17). Once density is modified, conservative variables[ρ, ρv, ρe] in the ghost

region must be re-assembled. For ease of reference, this procedure is called the indirect variable

extrapolation. Figure4.10depicts a 1D presentation of the GFM [135, 140].

Figure 4.10. 1D presentation of the GFM

An isobaric fix technique is often coupled with the GFM to minimize nonphysical temperature

and density when a compressible flow interacts with a stiff medium [134, 140]. Entropy Si−1

is extrapolated to elementi in the real region as well as those in the ghost region. Figure4.11

represents a 1D presentation of the GFM coupled with the isobaric fix technique [135, 140].

Figure 4.11. 1D presentation of the GFM coupled with the isobaric fix technique

91

R. Fedkiw points out that a variable extrapolation is most likely problem dependent [140]. Instead

of entropy, we can use any degenerate variable such as density or temperature.

Equations (4.15-4.17) are the key of the original GFM [135, 140]. To update two respective single-

fluid flows, the RKDG method addressed in the previous chapteris used. The feature of high

accuracy in the RKDG method produces less-diffusive numerical results compared to low-order

approximate methods. See FigureB.3 in Appendix B. This method helps minimize inherent

numerical diffusion which is a source of error in Eulerian-based multi-fluid methods. Classical

Eulerian simulations coupled with the original GFM have been successfully applied to various

multi-fluid flow simulations including gas-gas, gas-liquidand liquid-gas flows [133, 135, 136].

In order to assess the performance of the RKDG-original GFM,a 1D explosive gas-water flow

simulation [78] is performed. The initial conditions are shown in Figure4.12. The domain[0, 1]

m is discretized with 200 elements and the 1D RKDG-original GFM code is run to a final time of

1E-3 second.

Figure 4.12. Initial conditions of an explosive gas-water flow example

In this case, the coupling failed. The erroneous density profile versus the analytic solution at very

early time is provided in Figure4.13. A strong shock wave was generated by the large pressure

jump across the interface. Since a small density error in stiff water generated spurious pressure

oscillations, the density treatment in water is probably a source of numerical instabilities and the

sudden breakdown of the computation.

92

Figure 4.13. Density profile with a overshoot at the interface

At the first time step, density showed a large overshoot at theinterface as observed in Figure4.13

ultimately leading to the sudden end of the computation at a subsequent time step. The density

treatment by the original GFM was not applicable to the explosive gas-water flows we wish to

solve.

T.G. Liu pointed out that the incorrect interface conditions set by the original GFM may cause

nonphysical pressure oscillations [147]. He presented a modified GFM (MGFM), which is sim-

ilar to PC methods using local shock values for correct interface conditions [147]. The MGFM

solve local Riemann problems to set ghost values across the material interface. Compared to the

original GFM which failed at the case shown in Figure4.12, the MGFM has wider application for

various compressible fluid flows [147], but also has several disadvantages: its extension to multi-

dimensions does not seem trivial because of geometric complexities as much as in PC methods,

and it may not be applicable directly to explosive gas-waterflows modeled by a general EOS such

as the JWL EOS. Thus, a GFM based on the original GFM is better suited for explosive gas-water

flow simulations, which may still possess appealing features of the original GFM and does not re-

quire local Riemann solutions. This need required extensive fine-tuning of a variable extrapolation

to obtain reasonable outputs in explosive gas-water flow simulations.

93

4.1.2.2 Direct ghost fluid method

For the water, we directly extrapolate density from the realwater to the ghost water (i.e., in real

gas region) rather than using the indirect variable extrapolation (4.17). For the gas, we directly

extrapolate density from the real gas to the ghost gas (i.e.,in real water region) with the isobaric

fix technique minimizing the “overheating error” [135, 140]. To distinguish it from the indirect

variable extrapolation mentioned above, the current treatment is called the Direct Ghost Fluid

Method (DGFM) which is a variant GFM suitable for explosive gas-water flow simulations. The

DGFM helps decrease the density jump across the interface, thus minimizing spurious pressure

oscillations near the interface. The procedure is described in Figure4.14.

Figure 4.14. The description of the DGFM for explosive gas-water flows

94

Since this approach does not need the entropy calculation asin Equation (4.17) including thermo-

dynamic coefficients, there are no difficulties when a general equation of state such as the JWL

EOS is employed in the computation.

The RKDG-DGF method using the direct extrapolation performs well in the previous 1D explosive

gas-water flow test case. Figure4.15shows that the gas-water interface is fairly well captured at

the correct location, and RKDG-DGF results agree well with the analytic solution. Some density

diffusion occurs in the region near the interface, but no spurious pressure oscillations are observed

in the domain. These results demonstrate the potential of this approach for explosive gas-water

flows common in near-field UNDEX.

Figure 4.15. RKDG-DGF results for a 1-D explosive gas-waterflow

For multi-dimensions, we extend the variables into the ghost region along the normal to the inter-

face as in the one-dimensional manner. To extend discontinuous variables across the interface, we

solve an advection equation only with having to do a few iterations

I,τ ± N · I = 0 (4.18)

95

whereI is a discontinuous variable such as density and tangential velocity andτ is an artificial

time scale [135, 140]. The normal vectors at each point,N = [nx, ny, nz] are provided by the LS

method. To apply the isobaric fix technique, we populate a thin bandǫ = 1.0 ∼ 1.5∆x of real

elements bordering the interface. The positive sign+ is used to extend the variablesI in the real

regionφ < 0 to the ghost regionφ > 0, and the negative sign− is used to extend the variablesI

in the real regionφ > 0 to the ghost regionφ < 0 [135]. Velocities (u, v) in the ghost region are

recomputed by

Vn = unx + vny , Vt = −uny + vnx

u = Vnnx − Vtny and v = Vnny + Vtnx (4.19)

whereVn andVt are the normal velocity directly copied from the real fluid, and the tangential

velocity extrapolated from the other fluid.

Figure 4.16. Description of a two-dimensional DGFM procedure

The DGFM using the direct variable extrapolation performedwell in the 1-D explosive gas-water

flow simulation which failed using the original GFM [135, 140]. Compared to the MGFM, which

uses the local analytic solutions at the interface [147], the DGFM is simpler to extend to multi-

dimensions without solving multi-fluid Riemann problems.

96

Algorithm 6 Pseudo code of the DGFM using the direct variable extrapolation

1. Define two sets of fluidU for water andV for gas

2. Evolve the LS functionφ with flow velocitiesφt + uφx = 0 : Gas denoted by negativeφ and water by positiveφ

For water flow3.1 Set the ghost water located to the left of the interface with densityρR at the right of the interfaceDO i=1, the number of nodes

IF(φi< 0.0 ) THENρi = ρR : copy the densityρR to the density at node imi = ρiui : reassemble momentum flux using modified density at node iEi = ρiei : reassemble energy flux using modified density at node i

END IFEND DO

3.2 Solve the system of equations for waterUt + Fx(U) = 0

For gas flow4.1 Set the ghost gas located to the right of the interface with densityρL nearφ = −1.5∆xDO i=1, the number of nodes

IF(φi > −1.5∆x ) THENρi = ρL : copy the density at node L to the density at node imi = ρiui : reassemble momentum flux using modified density at node iEi = ρiei : reassemble energy flux using modified density at node i

END IFEND DO

4.2 Solve the system of equations for gasVt + Fx(V) = 0

We are ultimately concerned about the effects of underwaterweapons on naval ships. In close-

proximity UNDEX, the hull structure is usually subjected toan extremely high intensity transient

fluid loading perhaps resulting in structural deformation.The structural deformation may cause

a modification of the fluid flow near the structure by a feedbackloop. Therefore, without cou-

pling between the fluid and the structure, we cannot conduct an accurate UNDEX simulation that

97

includes cavitation formation and closure, shock-bubble interaction, and bubble-structure interac-

tion. The following section discusses the fluid-structure interaction with a coupling algorithm, and

a cavitation mechanism including its effect and numerical approach

4.2 Fluid-structure interaction

The fluid-structure interaction (FSI) is commonly considered a feedback loop shown in Figure

4.17[42, 71, 72, 160]. The interface pressures at timen − 1 acting on the structure are provided

by the fluid computation at timen − 1. The structure computation at timen provides interface

displacements to accommodate the fluid mesh against the deformation of the structural interface.

Figure 4.17. A feedback of FSI simulations[71]

In practical FSI simulations, boundary nodes behave in Lagrangian fashion while internal nodes

move independently of flow motion. Accordingly, flow motion and mesh motion are not equal so

the flow variables must be convected to an updated fluid mesh. For the details of this convection ef-

fect, refer to Chapter 2. Since the Eulerian mesh is spatially fixed and the Lagrangian mesh suffers

from excessive mesh distortion, we use the fluid governing equations in the ALE description.

This section presents the solution method of a Lagrangian governing equation for the structure, its

coupling with the fluid , an ALE-based moving mesh algorithm and cavitation associated with FSI.

98

4.2.1 Solution method for the structure

Consider the Lagrangian momentum equation for the structure given in Chapter 2:

ρDvDt

= · σ (4.20)

where the Cauchy stressσ is determined by a constitutive relation. By applying the classical

Galerkin method to the weak form of Equation (4.20), we obtain the matrix form of the momentum

equation

Mdvdt

+ fint = fext (4.21)

where M, fint and fext are mass matrix, internal force vector and external force vector [143]:

M =

∫

Ω

ρNT NdΩ, fint =

∫

Ω

dNT

dxσ and fext =

∫

∂Ω

NT td∂Ω

Now, we temporally discretize Equation (4.21) due to the presence of a time-dependent term. Since

oscillatory structure solutions can contaminate the fluid flow near the structure, a stable solution

method is required. This work uses a two-step Newmarkβ time scheme commonly used in solid

mechanics [91, 100, 143]. Figure4.18represents the flow chart of the two-step Newmarkβ time

scheme used when solving Equation (4.21). Parametersγ andβ play a major role in controlling

the stability of structural solutions. Asγ increases, more artificial viscosity is added [143].

Figure 4.18. Flow chart of the two-step Newmarkβ time scheme [143]

99

To see the effect of artificial viscosity introduced by the Newmarkβ time scheme, a simple axial-

bar example loaded by a distributed external load at the leftend was tested. In Figure4.19, the

largerγ, at a fixedβ, the more artificial viscosity.

Figure 4.19. A simple axial-bar example using the two-step Newmarkβ time schemewith different ratiosγ/β

4.2.2 Interface conditions

The fluid-structure interactions require kinematic and dynamic conditions along the normal to the

interface [71]. No particles may cross the interface and stresses must be continuous across the

interface [71]. To accomplish these requirements, all interface nodes must remain permanently

aligned during the computation by prescribing that the gridvelocities of fluid nodes vg are equal

to the material velocities v of structural nodes on the interface [71].

100

Using permanently aligned interface nodes, we implement the kinematic interface condition. For

inviscid flow, the displacements along the normal to the interface must be continuous as

nf · uf = ns · us (4.22)

wheren andu are normal vector and displacements, and subscriptsf ands represent the fluid and

the structure. This procedure insures that the fluid and structure never detach or overlap during the

computation [71].

The dynamic condition requires that stresses along the normal to the interface be continuous [71].

For inviscid flow, it is achieved by

nfp = nsts (4.23)

wherep andts are fluid pressure and surface traction acting on the structure.

4.2.3 Mesh Coupling algorithm

The two types of FSI mesh coupling schemes, a matching and a non-matching schemes, are shown

in Figure4.20. In a matching scheme, both the fluid and structure meshes arediscretized in a same

mesh density along the interface. A non-matching scheme allows for different mesh densities.

Figure 4.20. FSI mesh coupling algorithms

101

The matching scheme has permanently aligned nodes along theinterface so that the imposition

of interface conditions (4.22-4.23) is straightforward. Since the size of the time step,∆t, may be

determined by the structural mesh density, the computationbecomes expensive.

If a valid data exchange between the fluid and the structure with different mesh densities can be

carried out, computational efficiency can be improved with alarger time step,∆t, compared to the

matching scheme. This method is called the non-matching scheme. The communication is carried

out via a set of virtual interface nodes. Although this scheme requires careful dimensional grid

matching along the interface, it can be considered an alternative coupling method for the matching

method which requires intensive computing cost. This work uses the non-matching scheme with

a coarser structure mesh to increase the size of time step,∆t, which is usually determined by the

structure mesh density. For the details of the grid matchingbetween the fluid and structure meshes,

see AppendixE. Figure4.21depicts an example of the different mesh densities in both methods.

Figure 4.21. An example of mesh couplings: red (fluid) and blue (structure)

Next, we consider a 2-D nodal configuration for near-field UNDEX simulations shown in Figure

4.22with a high pressure circular gas bubble surrounded by water. The high intensity transient

fluid loading (i.e. shock) impinges against the structure, causing the structure to deform. To ac-

commodate this resulting interface deformation, the fluid mesh is adjusted during the computation.

This adjustment is accomplished by allowing the motion of interface nodes in Lagrangian fashion.

102

Figure 4.22. A 2D configuration of FSI in near-field UNDEX

After updating the fluid interface nodes, the remapping of fluid internal nodes is needed to maintain

the quality of the fluid mesh. This is accomplished using an averaging mesh-smoothing scheme

shown in Figure4.23. For other mesh-smoothing schemes, References [109, 143] are recom-

mended.

Figure 4.23. Description of an averaging mesh- smoothing scheme in two-dimensions [20]

Using the updated nodal positionsxn+1, the grid velocity at timen + 1/2 is calculated by

vn+1/2g =

xn+1 − xn

∆t(4.24)

The ALE convection velocitym is defined as

103

mn+1 = vn+1 − vn+1/2g (4.25)

which represents the difference between flow velocity and grid velocity.

The flow computation using any ALE scheme must satisfy the Geometric Conservation Law (GCL)

which states that the numerical scheme preserve the state ofa uniform flow independently of the

grid deformation [69]. M. Lesoinne [107] provided the discussion of a GCL in aeroelastic problems

and its impact on fluid-structure interactions. He stated that the uniform flow can be recovered if

the flow computation is evaluated at the mid-point configuration, and the grid velocity is assumed

to be constant over the time step [107]. The GCL is satisfied by evaluating the flux and integral

terms using the mid-point configurationxn+1/2 and the grid velocityvn+1/2g .

4.2.4 Cavitation in FSI

FSI with cavitation effects is depicted in Figure4.24.

Figure 4.24. The traditional view of cavitation mechanism [6]

The transient intensive pressure loading gives the structure a downward acceleration which contin-

ues until the structure begins to move faster than the surrounding fluid [14, 131]. At this point, the

fluid next to the structure is exposed to tension. Since watercannot sustain tension, a local cavita-

tion region develops near the interface [8, 131]. After reaching its maximum downward velocity,

the structure begins to slow and the local cavitation regioncloses [8, 14, 131]. This closure causes

104

the structure to be reloaded [8, 14, 131]. Due to the presence of cavitating water, it is necessary to

include the cavitating flows to simulate real fluid-structure interactions.

J. M. Brett, et al. [90] performed small-scale experiments showing that cavitation reloading and

bubble effects in near-field UNDEX are as important as the primary shock impact. The duration

and magnitude of the cavitation effects were not negligiblein close-in explosions. Some numerical

studies were also conducted by A.B. Wardlaw [5, 6] and W.F. Xie [157, 158]. To assess cavitation

formation, collapse and reloading, A.B. Wardlaw used a pressure-cutoff model which requires that

cavitation occurs when the water pressure is lower than a pressure limit [6]. The low limit is usually

0.05 bars [6]. He found that the low pressure limit has little impact on the solution as long as it is

positive and much less than 1 bar [6]. When the water pressure drops below the low pressure limit,

the pressure is set to a given low limit as

p = MAX(pcalculated, plimit) (4.26)

wherepcalculated andplmit are the calculated pressure and the given low limit, respectively. The

comparison with experimental data in [6] showed that this simple model performs well in FSI sim-

ulations with cavitation effects. T.G. Liu [149] recently proposed an isentropic one-fluid model

that assumes the cavitating flow is a homogeneous mixture comprising isentropic vapor and liquid

components. This approach calculates the sound speed and pressure of a mixture, separately. W.F.

Xie conducted some numerical experiments using this model that analyze the bulk cavitation near

the free water surface and an internal explosion in a water-filled cylinder [157, 158]. This model

also performed well, but is more complicated than the cutoffmodel. In [149], the comparison

between these two common models shows that the choice of models is not significant to the solu-

tion. For ease of implementation, we use the pressure cutoffmodel, Equation (4.26). In Chapter

5, we examine some cases which include FSI with cavitation effects. For other cavitation models,

Reference [149] is recommended.

The flow-chart of FSI simulations is summarized in Figure4.25. The interface information is

exchanged via a set of virtual interface nodes.

105

Figure 4.25. The flow-chart of FSI simulations using the non-matching scheme

Section4.1 discussed a two-fluid model (i.e. explosive gas-water) based on the LS interface

method. Since the LS method is able to capture any time-varying shape, it has been applied to

a wide range of engineering problems [140]. However, the mixed elements in which fractions of

explosive gas and water coexist make the use of LS method alone inadequate for two-fluid flow

simulations. Physical and thermodynamic properties are discontinuous across the interface so the

classical methods may suffer from nonphysical behaviors. The DGFM using direct variable ex-

trapolation to minimize or remove the spurious pressure oscillations near the interface is used.

This method performed well in the 1D explosive gas-water flowsimulation in which the original

GFM failed. Compared to existing methodologies, this method was simpler to implement and to

extend to multi-dimensions. Section4.2 discussed the procedures for fluid-structure interaction

simulations. Without coupling between the fluid and the structure, we cannot carry out a valid

UNDEX simulation. The structural solution method with the ALE-based moving mesh algorithm

106

was addressed. The fluid mesh which is usually fixed in space must be adjusted to the deformation

of the structural interface.

Chapters 1 to 4 concluded that near-field UNDEX simulations require the treatment of two-fluid

compressible flows and a deformable fluid mesh for fluid-structure interaction simulation. This

required the ALE-based RKDG compressible two-fluid method:the RKDG method, the DGFM

and the ALE methodology. In Chapter 5, we examine 1D and 2D cases to assess the performance

of the suggested solution methods.

107

Chapter 5

Assessment

This chapter describes assessment of RKDG-DGF approach outlined in the previous chapters.

Section 5.1 examines several 1D Cartesian cases (Cases 5.1.1~5.1.4) and 1D symmetric cases

(Cases 5.1.5~5.1.7). RKDG-DGF results are compared with analytic solutions and other reference

results. For the 1D Cartesian cases, the analytic solution is obtained using a Riemann problem so-

lution FORTRAN code provided in [45]. For symmetric cases, an analytic solution is unavailable

[27, 45] so reference results from previous work are used. Section 5.2 examines the extension of

the 1D RKDG-DGF approach to multi-dimensions. 1D symmetricRKDG-DGF results on a fine

mesh are used to assess RKDG-DGF results in multi-dimensions. Section 5.3 extends the RKDG-

DGF approach to practical UNDEX applications that include bubble behavior (Cases 5.3.1 and

5.3.2), shock-bubble interaction and cavitation with fluidstructure interaction (FSI) (Cases 5.3.3

and 5.3.4). Experimental data and reference results are used to assess the RKDG-DGF approach

in these cases. Since the gas bubble caused by UNDEX has spherical symmetry, the 1D spherical

symmetric RKDG-DGF approach can be used to simulate bubble behavior (e.g., peak pressure,

bubble radius-time history and maximum bubble radius). Cases 5.3.3 and 5.3.4 consider an ex-

ternal explosion near a rigid wall, and an internal explosion within a water-filled aluminum tube.

The pressure cutoff model discussed in Section 4.2 is used topredict the formation and collapse of

108

cavitation which is an important phenomenon in these cases.The differences between a rigid-wall

case and a deformable-wall case are discussed to describe the shock-bubble interaction and wall

deformation effects with FSI. The cavitation mechanisms with FSI are described. An ALE-based

deformable fluid mesh is used and explored in the fluid-structure coupling. These cases show the

potential of our approach for near-field UNDEX applicationsand identify the deficiencies we need

to correct.

In all test cases, fluid computations use the RKDG-DGF methodwith CFL=0.2. Our 1D and 2D

RKDG-DGF codes use FORTRAN 90. For the details of the boundary conditions used, see Ap-

pendixF. Unless stated otherwise, we use a uniform mesh of 200 elements along each direction,

a rigid wall boundary condition, and the c-g-s unit system (i.e., density (g/cm3), velocity (cm/s),

pressure (dyne/cm2), length (cm) and time (second)), respectively. The interface is initially lo-

cated at the center of the domain. Variablesρ andp are chosen to show the numerical results.

5.1 One-dimensional assessment

Several 1-D cartesian and 1D symmetric cases are consideredin this section.

5.1.1 Cartesian 1-D cases

This subsection explores the applicability of RKDG-DGF approach for simple high pressure gas-

water flows which are qualitatively similar to explosion-produced gas and water flows in near-field

UNDEX. Assessment of RKDG-DGF approach is carried out by comparing with the Riemann

analytic problem solution [27, 45, 92]. In gas dynamics, the ideal equations of state are often

employed to easily obtain the analytic solution [45, 92]. For consistency, this approach is followed

in Cases 5.1.1~5.1.3. Unless stated otherwise, the Tait equation of state (EOS) withγ = 7.15,

B = 3.31E9 dyne/cm2 andρ0 = 1.0 g/cm3, and the ideal gas law withγ = 1.4 are used to model

the water and gas. Other EOS model are also used in other test cases.

109

Case 5.1.1: This case considers a high pressure gas-water shock tube problem previously modeled

by J. Qiu [78]. The initial conditions are given in Figure5.1. J. Qiu solved this problem using a 1D

two-fluid RKDG method which treats the moving interfaces conservatively [78]. J. Qiu’s method

obtains the solution of the cells near the interface by solving multi-medium Riemann problems

defined at the interface [78]. The solution procedure in J. Qiu’s method [78] is similar to that of

Godunov-type Prediction Correction method which requireslocal Riemann solutions to correct

affected cell values in the region near the interface (See Section 4.2). The 1D RKDG-DGF code is

run to a final time of 0.16E-3 seconds. RKDG-DGF results versus the Riemann problem analytic

solution and J.Qiu’s results are compared in Figure5.2.

Figure 5.1. Initial conditions for Case 5.1.1

Figure5.2 shows that at 0.16E-3 seconds, the shock front and material interface predicted by the

RKDG-DGF method, which are initially located at the center of the domain, are observed at about

86cm and 55cm. Since the shock travels at a speed by the sum of the speed of sound and flow

velocity, the shock front propagates outward faster than the material interface [45]. The material

interface is transported by flow velocity only. The RKDG-DGFmethod for this case resolves

110

Figure 5.2. RKDG-DGF results for Case 5.1.1 at 0.16E-3 seconds

the shock front with only 3 elements. The head and tail of rarefaction wave are also very well

resolved. Small density oscillations are observed in the region near the interface, but no pressure

oscillations occur. Compared to J. Qiu’s results taken from[78], the RKDG-DGF method produces

a less-diffusive and oscillatory density profile.

Case 5.1.2: This case considers a strong shock tube problem previouslyinvestigated by H. Luo

[57]. H. Luo assessed the application of various numerical fluxes on an ALE moving grid. [57]. He

states that the ALE approach is attractive when the interface is not subjected to large deformation

[57]. Since large interface deformation occurs in real UNDEX problems and may lead to significant

distortion of the mesh, this is a major concern which is avoided in the RKDG-DGF method. To

compare the performance of the RKDG-DGF method and H. Luo’s ALE-based method, the same

initial conditions shown in Figure5.3are used. The 1D RKDG-DGF code is run to a final time of

1.55921E-04 seconds. The domain [0, 100] cm is discretized with 100 elements as in [57]. RKDG-

DGF results versus the Riemann problem analytic solution and H. Luo’s results are compared in

Figure5.4.

111

Figure 5.3. Initial conditions for Case 5.1.2

Figure 5.4. RKDG-DGF results for Case 5.1.2 at 1.55921E-04 seconds

112

Compared to the analytic solution, H. Luo’s results are verydiffusive. To improve the resolution of

results, the number of elements must be increased. The RKDG-DGF method using the same num-

ber of elements produces less-diffusive profiles for the density and the pressure compared to those

of H. Luo’s ALE-based interface method without the requirement for interface mesh adjustment.

Case 5.1.3: This case considers a TNT-produced equivalent gas and water flow problem previously

modeled by J. Qiu [78]. J. Qiu considered this problem to assess the application of a 1D RKDG

two-fluid method for an explosion flow simulation. To comparethe performance of the two dif-

ferent RKDG approaches (i.e., J. Qiu’s two-fluid RKDG methodand RKDG-DGF method), the

same initial conditions are considered as shown in Figure5.5. Both initial density and pressure in

the gaseous medium are equivalent to those of TNT-produced gas from [5]. The 1D RKDG-DGF

code is run to a final time of 0.1E-3 seconds. RKDG-DGF resultsversus the analytic solution and

J.Qiu’s results are compared in Figure5.6.

Figure 5.5. Initial conditions for Case 5.1.3

113

Figure 5.6. RKDG-DGF results for Case 5.1.3 at 0.1E-3 seconds

With both methods, the shock front and material interface are sharply captured at the correct loca-

tions and magnitudes. Only 2~3 elements are used to resolve the shock front. As in Case 5.1.1, our

RKDG-DGF approach provides a less-diffusive and oscillatory density profile in the region near

the interface without the requirement of multi-medium Riemann solutions at the interface.

Case 5.1.4: This case considers a TNT-produced gas and water flow problem modeled by A.B.

Wardlaw using an ALE-based interface method [5]. The ALE-based interface method treats the

interface in a Lagrangian manner, and adjusts fluid nodes to the interface deformation [5]. As

discussed in Case 5.1.2, unlike in the RKDG-DGF method, special concern for mesh adjustment is

required. In this case, RKDG-DGF results are compared only to the results from [5]. An analytic

solution is unavailable. The JWL EOS is used to model the explosive gas. Material properties and

coefficients for the JWL EOS are summarized in Table2.7. The initial conditions are shown in

Figure5.7. The 1D RKDG-DGF code is run to a final time of 0.5E-3 seconds. RKDG-DGF results

versus the reference results from [5] are compared in Figure5.8.

114

Figure 5.7. Initial conditions for Case 5.1.4

Figure 5.8. RKDG-DGF results for Case 5.1.4 at 0.5E-3 seconds

115

Both methods provide sharp density and stable pressure profiles. The shock front in the RKDG-

DGF results is sharply captured with only 2 elements. The results show that the RKDG-DGF

method is applicable for numerically simulating explosivegas flows modeled using the JWL EOS.

Unlike A.B. Wardlaw’s ALE approach, our approach does not require special concern for mesh

adjustment.

Cases 5.1.1~5.1.4 were selected to assess the application of the RKDG-DGF method for simple

high pressure gas-water flows which are very similar to TNT-produced gas-water flows in near-

field UNDEX applications. The high pressure gaseous medium produces a strong shock travelling

into the water and simultaneously a rarefaction wave movingtoward the origin. Compared to

the analytic solutions and reference results from previousworks, the RKDG-DGF method sharply

captures both the shock front and interface at the correct locations and magnitudes. The head

and tail of rarefaction are also very well resolved. Some density diffusion occurs in the region

near the interface, but it is less than with other methods andno spurious pressure oscillations

are visible in the domain. Compared to the original GFM and the MGFM methods which add

difficulty and complexity in multi-fluid using the JWL EOS andmulti-dimension implementation,

the RKDG-DGF approach did not increase difficulty and complexity even in the implementation

and computation of the JWL EOS. Since the RKDG-DGF method in the fluid is based on an

Eulerian description, no special concern for mesh adjustment is required as in the ALE-based

methods for interface tracking.

To assess the applicable range of the RKDG-DGF method, initial pressures in the gaseous medium

were increased by 8E+8Pa to 7.81E+9Pa. The RKDG-DGF approach produced no severe error

through Cases 5.1.1~5.1.4 when compared to the analytic solution. Since the initial pressure pro-

duced by typical TNT explosives is around 7.81E+9Pa [5, 17, 32], the RKDG-DGF method is very

applicable to real near-field UNDEX simulations without magnitude limitations.

116

5.1.2 Symmetric 1-D cases

1D symmetric solutions can be used to assess multi-dimensional solution in method development

[45, 102], and to analyze symmetric UNDEX fluid flows [5, 7, 10, 92]. A reliable 1-D symmetric

solution on a very fine mesh can provide very accurate numerical results. Section 5.1.1 presented

a series of 1D assessment for the RKDG-DGF method. This Cartesian approach can be extend to

a 1D symmetric approach as follows. The governing equationsare rewritten as

Ut + F (U)r = S(U) (5.1)

where

U =

ρ

ρu

ρe

, F =

ρu

ρu2 + p

u(ρe + p)

and S = −α

r

ρu

ρu2

u(ρe + p)

wherer is the stream-wise distance andu is the stream-wise velocity. The vectorS(U) with α

is the geometric source vector reflecting the symmetric effect [30, 45, 73, 92]. For cylindrical

symmetry,α is 1.0. For spherical symmetry,α is 2.0. Whenα is zero, Equation (5.1) becomes

equal to the 1-D cartesian conservation laws.

Equation (5.1) can be solved either by a fully coupled method or a splittingmethod [45, 132]. For

ease of implementation, this work uses the coupled method. For splitting methods, References [45,

132] are recommended. Cases 5.1.5 to 5.1.7 explore this symmetrical 1D RKDG-DGF approach

for cylindrical and spherical symmetry.

Case 5.1.5: This case considers a single-gas cylindrical explosion problem previously modeled

by E.F. Toro using a FVM [45]. The non-dimensionalized initial conditions are shown inFigure

5.9. E.F. Toro’s 2D results were assessed by comparing with 1D cylindrical results on a very fine

mesh [45]. To reflect cylindrically symmetry, we setα = 1 in Equation (5.1), and run the 1D

cylindrical RKDG-DGF code to a final time of 0.25 seconds. In Figure5.10, RKDG-DGF results

are compared with E.F. Toro’s 1D cylindrical results taken from [45]. The results show good

agreement.

117

Figure 5.9. Initial conditions for Case 5.1.5

Figure 5.10. RKDG-DGF results for Case 5.1.5 at 0.25 seconds

118

Case 5.1.6: This case considers a spherical UNDEX flow which has been used extensively in

method developments [30, 73, 92, 102, 146]. It was first numerically studied by J. Flores using the

Random Choice Method (RCM) [73]. C.H. Cooke employed the subcell resolution method with

the interface tracking scheme for this problem [30]. The subcell resolution method produced very

good numerical results compared with those reported in [73]. This case requiresα = 2 to reflect

spherical symmetric flow. The initial conditions are shown in Figure5.11. The domain[0, 30.48]

cm is discretized with 200 elements and the 1D spherical RKDG-DGF code is run to a final time

of 0.455E-3 seconds. The interface is initially located 10.16 cm from the origin. In Figure5.12,

RKDG-DGF results are compared to the results of the subcell resolution method in [30]. As in

[30, 73, 92, 102, 146], density, pressure and radial distance on the plots are non-dimensionalized

by ρ′ = ρ/1.007, p′ = p/3.331231E9 andr′ = r/30.48 [30]. The Tait EOS withγ = 7.0 and

B = 3.311301E9 dyne/cm2, and the ideal-gas law withγ = 1.4 are used to model the water and

explosive gas, respectively.

Figure 5.11. Initial conditions for Case 5.1.6

119

Figure 5.12. RKDG-DGF results for Case 5.1.6 at 0.455E-3 seconds

Once the explosion is initiated, a spherical outward shock wave travels into the water and a spheri-

cal inward rarefaction wave propagates toward the origin, simultaneously. Both the location of the

interface and shock front compare very well to those reported in [30]. The interface is sharply cap-

tured, and no spurious pressure oscillations occur. Eulerian methods used in [30, 73, 92, 102, 146]

require local Riemann solutions to correct affected variables across the material interface, but the

RKDG-DGF method does not require this complexity.

Case 5.1.7: This case considers a spherical UNDEX shock problem taken from A.B. Wardlaw’s

test cases [5]. As in Case 5.1.4, A.B. Wardlaw used the ALE-based interface method. The JWL

EOS is used to model the TNT-produced explosive gas. Similarly, Case 5.3.1 considers this prob-

lem to approximate the characteristics of a spherical UNDEXbubble pulse. However, Case 5.3.1

requires a larger mesh and longer time duration for bubble pulses. The initial conditions are shown

in Figure5.13. For initial conditions, we assume that a TNT-produced gas bubble with radius

r0 = 16 cm is surrounded by water at rest. We setα = 2 in Equation (5.1) for spherically symme-

try. The domain[0, 250] cm is discretized with 200 non-uniform elements, and the 1D spherical

RKDG-DGF code is run to 0.5E-3 seconds and 1.0E-3 seconds, respectively.

120

Figure 5.13. Initial conditions for Case 5.1.7

RKDG-DGF results are compared to those from [5]. Figure5.14shows that both methods, which

one method is A.B. Wardlaw’s 1D ALE interface method and the other method is our 1D RKDG-

DGF method, predict very similar pressure profiles at two times. In [5], reference density profiles

are unavailable. At 0.5E-3 seconds, the locations of the material interface and shock front are about

41 cm and 120 cm. A secondary outward shock is created, and themagnitude of peak pressure

is about 1.9Edyne/cm2. At 1.0E-3 seconds, the locations of the material interfaceand shock front

are about 52 cm and 250 cm. A reflected inward shock and a transmitted shock near the interface

are observed, and the magnitude of peak pressure is about 0.75Edyne/cm2. In Figure5.14, the

locations of material interfaces and shock fronts, and the magnitudes of peak pressures compare

very well to those from [5].

The shock fronts propagate outward faster than the materialinterfaces during time interval. The

shock front travels about 130 cm, but the material interfaces travels about only 11 cm. It is mainly

due to the difference of the characteristic speeds as discussed in Case 5.1.1. The magnitudes of

peak pressure are also decreased.

121

(a) Density distributions

(b) Pressure distributions

Figure 5.14. RKDG-DGF results for Case 5.1.7

122

Note in Figure (5.14) (a) that the fluid flow appears incompressible or weakly compressible at 1E-3

seconds. Spatial variation of the density is negligible. Based on this finding in mid-time UNDEX

events, incompressible approaches are often used in the simulations of a gas bubble [121, 9, 41, 23].

This result provides some insight as to why a compressible approach may suffer from a loss of

efficiency and accuracy in mid-to-late time multi-dimensional UNDEX simulations and may not

be necessary. Since the Mach numberM = uc

for incompressible/weakly compressible flows

ranges from 0.0 to 0.3 [142], the usual CFL condition (e.g.∆t = ∆xmax(u+c)

wherec is the sound

speed) strictly requires an extremely small time step size to avoid stability problems at mid-to-late

time simulations. Separate simulations for early-time andmid-to-late time are potentially more

effective in both cost and accuracy. This work focuses on early-time UNDEX simulations. For

low-Mach number flows, References [33, 69, 142] are recommended.

The ALE-based interface method is often used to prevent mixed elements caused by diffused den-

sity, which usually occurs in Eulerian methods. However, its extension to multi-dimensional appli-

cations where the interface motion is large, may not be effective [10, 57]. Early Eulerian methods

and J. Qiu’s two-fluid method [78] require multi-medium Riemann solutions to correct affectval-

ues in the region near the interface. The RKDG-DGF method does not have geometric limitations,

and does not require multi-medium Riemann solutions to treat explosive gas-water interfaces.

Section5.1provides reasonable test cases for 1D Cartesian and 1D symmetric RKDG-DGF meth-

ods. RKDG-DGF results are compared with the Riemann problemanalytic solution for 1D Carte-

sian cases, and the results from previous works for 1D symmetric cases. In these cases, the

shock front and material interface which are important hyperbolic characteristics in compressible

flow simulations, are sharply captured at correct locationsand magnitudes. No spurious pressure

oscillation and excessive density diffusion are observed.Now, to assess the multi-dimensional

applicability of our RKDG-DGF approach, Section 5.2 extends the 1D RKDG-DGF method to

multi-dimensions. Cases 5.2.1 (a 2D cylindrical case) and 5.2.2 (a 3D axi-symmetric case) are

considered.

123

5.2 Multi-dimensional assessment

An alternative way to assess the RKDG-DGF results in multi-dimensions is required because an

analytic solution is unavailable. To accomplish this, our RKDG-DGF results in multi-dimensions

are compared with our equivalent 1D symmetric RKDG-DGF results. The 1D domains are dis-

cretized with 1000 uniform elements to insure well-defined results.

Case 5.2.1: This case considers a 2D cylindrical explosive gas-water flow. Since the analytic

solution is unavailable, 2D RKDG-DGF results are compared with 1D cylindrical results on a fine

mesh. The initial conditions are shown in Figure5.15. The 2D domain[−100, 100]× [−100, 100]

cm is discretized with200 × 200 uniform elements. An initial gas bubble with radiusr0 = 50 cm

is located at the origin. The 2D RKDG-DGF code is run to a final time of 0.1E-3 seconds.

Figure 5.15. Initial conditions for Case 5.2.1 at 0.1E-3 seconds

2D RKDG-DGF results versus 1-D cylindrical results (α = 1.0 in Equation (5.1)) are compared in

Figure5.16. The half profiles of 2D results taken along the horizontal center are compared with the

1D results. The contour plots of density and pressure are shown in Figures5.17and5.18. Results

are very consistent.

124

Figure 5.16. RKDG-DGF results for Case 5.2.1 at 0.1E-3 seconds

Figure 5.17. Density contours for Case 5.2.1 at 0.1E-3 seconds

125

Figure 5.18. Pressure contours for Case 5.2.1 at 0.1E-3 seconds

Case 5.2.2: This case considers a 3D axi-symmetric flow that exhibits symmetry about an axis of

rotation. The fluid flow is described in cylindrical coordinates (r, θ andz). We assume that the

flow has symmetry around the axial coordinatez and the radial coordinater is defined as the dis-

tance from the axisz. Since the flow variables are a function of radial and axial coordinates(r, z)

only, the solutions are independent of the circumferentialcoordinateθ. Under these conditions,

the governing equations can be reduced from 3D to 2D with geometric sourceSsym(U) as

Ut + F (U)r + G(U)z = Ssym(U) (5.2)

where

U =

ρ

ρu

ρv

ρe

, F =

ρu

ρu2 + p

ρuv

u(ρe + p)

, G =

ρv

ρuv

ρv2 + p

v(ρe + p)

and Ssym = −1

r

ρu

ρu2

ρuv

u(ρe + p)

126

For the details of axi-symmetric equations, References [45, 123, 132] are recommended. Equation

(5.2) is often applied to axi-symmetric UNDEX flow applications (See Case 5.3.4) [158]. The

initial conditions are shown in Figure5.19. The initial gas bubble with radiusr0 = 100 cm is

located at the origin of the domain. The domain[0, 200] × [−200, 200] cm is discretized with

100×200 uniform elements. A symmetric boundary condition is applied along the axis z. The 2D

RKDG-DGF code is run to a final time of 0.3E-3 seconds.

Figure 5.19. Initial conditions for Case 5.2.2

Figure5.20compares the 3D axi-symmetric RKDG-DGF results and 1D reference results (i.e.,

α = 2.0 in Equation (5.1)). Figures5.21 and 5.22 represent the contour plots of density and

pressure at 0.3E-3 seconds. The shock front is sharply captured with only 3 elements and the

second outgoing shock wave reflected from the origin is very well resolved. Results are again very

consistent.

127

Figure 5.20. RKDG-DGF results for Case 5.2.2 at 0.3E-3 seconds

Figure 5.21. Density contours for Case 5.2.2 at 0.3E-3 seconds

128

Figure 5.22. Pressure contours for Case 5.2.2 at 0.3E-3 seconds

Cases 5.2.1 and 5.2.2 demonstrate UNDEX fluid-only results in multi-dimensions. Case 5.2.1

shows results at the time before the inward rarefaction wavereaches the origin. Case 5.2.2 shows

results at the time when the low-pressure region near the origin is created immediately after re-

flecting the inward rarefaction wave. The locations of both the interface and shock front are very

similar to those of the 1-D reference results. No spurious pressure oscillations and contour distor-

tions occur.

Section5.2 explored two test cases (i.e. a 2D cylindrical flow and a 3D axi-symmetric flow) to

assess a multi-dimensional RKDG-DGF method for modeling near-field UNDEX events in the

fluid only. In Cases 5.2.1 and 5.2.2, the multi-dimensional extension of the RKDG-DGF method

was examined by comparison of multi-dimensional results to1D symmetric RKDG-DGF results.

The RKDG-DGF results in multi-dimensions compared very well with the 1D reference results.

No contour distortion occurred in the domain. These resultsshow the potential of the RKDG-

DGF method for multi-dimensional UNDEX applications. The RKDG-DGF method in multi-

dimensions is not susceptible to error caused by distortionof the mesh as in ALE interface methods,

and the JWL EOS which provides better early-time/near-fieldaccuracy in the explosive gas model

as discussed in [54], does not require additional effort in implementation or computation.

129

5.3 Applications

Sections 5.1 and 5.2 examined simple fluid-only cases where the analytic solution or reference

results are available. These demonstrated the potential ofthe RKDG-DGF method for real UNDEX

simulations. This section extends the RKDG-DGF method to practical UNDEX applications which

include bubble behavior (Cases 5.3.1 and 5.3.2), shock-bubble interaction and cavitation in FSI

(Cases 5.3.3 and 5.3.4).

5.3.1 1D Spherical Bubble Pulses

Cases 5.3.1 and 5.3.2 consider 1D spherical bubble pulses which are important phenomenon in

near-field UNDEX. Assuming spherical symmetry, the fluid flowcan be modeled using the 1D

spherical RKDG-DGF method based on Equation (5.1) with α = 2.0. Experimental data or ref-

erence results from previous work are used to assess the RKDG-DGF results. In these cases, the

JWL and stiffened-gas EOSs are used to model the TNT-produced gas and water. The inner and

outer boundaries are treated using a rigid wall condition and a outflow condition, respectively.

The outflow condition helps decrease wave distributions at the boundary, even though it may be

troublesome in oblique shock impinging cases [98, 132].

Typical UNDEX gas bubble dynamics is described in Figure5.23. Following the explosion, the

pressure rises instantaneously to its peak value and then decreases exponentially. The high pressure

inside the gas bubble causes the bubble to expand, and pushesout the surrounding water. As the

gas pressure drops to the hydrostatic level, this outward expansion slows down until the expansion

stops. The high-pressure surrounding water pushes in the bubble causing it to be contracted inward.

This process is known as bubble collapse. If the bubble reaches a minimum radius, the bubble

begins to be re-compressed, producing another pulse. Several other cycles may follow at reduced

shock strength. The pressure-time history shows that the magnitudes of pressure are inversely

proportional to the bubble volumes.

130

Figure 5.23. UNDEX gas bubble dynamics [116]

Case 5.3.1: This case considers a 1D spherical bubble collapse previously modeled by A. B. Ward-

law [5], and introduced in Case 5.1.7. This problem has often been used in UNDEX method de-

velopments [57, 89, 103, 142]. A.B. Wardlaw conducted this problem using a 1D ALE interface

method code with highly resolved meshes [5]. It models the UNDEX of a 28kg TNT explosive

surrounded by water at a depth of 178m. In our model, the TNT charge is replaced by a gas bubble

with equivalent initial volume and internal energy for the original High Explosive (HE) material.

The bubble radiusr is initialized usingr0 =(

3V0

4π

)1/3with the approximate initial volumeV0 for the

HE material [17]. The initial conditions are shown in Figure5.24. The 1D domain[0, 10000] cm

is discretized with 2000 non-uniform elements. The center of a gas bubble withr0 = 16 cm is

located at the origin. The 1D spherical RKDG-DGF code is run to a final time of 0.15 seconds.

Figure5.25shows the bubble radius-time history, and the magnitudes ofpeak pressures taken at

121cm from the origin.

131

Figure 5.24. Initial conditions for Case 5.3.1

Figure 5.25. RKDG-DGF results for Case 5.3.1

132

The bubble radius is estimated using the LS functionφ(r) (see Subsection 4.1.2). The maximum

bubble radius is approximately 220 cm at about 0.07 seconds,and the minimum bubble radius is

approximately 30 cm at about 0.13 seconds. In Table5.1, these results are compared with those

from A.B. Wardlaw [5]. The peak pressure at 5E-4 seconds is approximately1.9 E9 dyne/cm2 as

observed in Case 5.1.7. There exists a small difference in bubble radius. This is mainly due to the

fact that A.B. Wardlaw identified the interface location by aLagrangian fashion, but we use the

sign change of the LS function (See Section 4.2).

Reference results Numerical results Differences (%)

Max. bubble radius (cm) 220 221 +0.45

Bubble period (sec.) 0.132 0.130 -1.50

Peak pressure (dyne/cm2) 1.9E9 1.9E9 0.0

Table 5.1. Comparisons with the reference results

Case 5.3.2: This case considers an experiment conducted by Swift and Decius in 1950 [43]. Swift

and Decius presented measurements on the oscillations and maximum radii of gas bubbles from

underwater explosions [43]. The experiment provides a useful graph of the bubble radius-time

history for a 300g charge detonated at a depth of 91.46m. J. M.Brett investigated this problem

using a nonlinear explicit FE-DYNA2D code [89]. The 1D domain[0, 9146] cm is discretized

with 2000 non-uniform elements. The initial conditions areshown in Figure5.26. Compared to

Case 5.3.1, we use a smaller-radius gas bubble withr0 = 3.529 cm which corresponds to a 300g

TNT charge. The 1D spherical RKDG-DGF code is run to a final time of 0.045 seconds. The

RKDG-DGF results are compared with the experimental data inFigure5.27and Table5.2.

Both the RKDG-DGF and DYNA2D approaches perform well for thefirst bubble pulse, but devi-

ate from the experimental data during the second pulse. DYNA2D predicts a very different second

pulse. This deviation may result from the fact that the real bubble behavior includes other mech-

anisms such as turbulence, buoyancy producing, asymmetricflows and heat loss from hot gases

[83, 89, 131, 99].

133

Figure 5.26. Initial conditions for Case 5.3.2

Figure 5.27. RKDG-DGF results for Case 5.3.2

134

Experiment data Numerical results Differences (%)

Max. bubble radius (cm) 48.1 48.0 +1.0

Bubble period (sec.) 0.298 0.291 -1.7

Table 5.2. Comparisons with the reference results

The RKDG-DGF approach is particularly useful for estimating the pressure loading associated

with the primary shock wave and the first bubble pulse. However, as discussed in Section5.1,

this compressible approach may lose efficiency and accuracyin simulating mid-to-late time multi-

dimensional UNDEX flows where the fluid flow becomes incompressible/weakly compressible.

The 1D spherical results can effectively be used to a fine initial condition for mid-to-late time

UNDEX simulations [103].

Subsection5.3.1modeled two 1D spherical gas bubble collapses which one is a numerical bench-

mark and the other is an experiment. RKDG-DGF results were compared to those of the results

from References [5, 89] and experimental data [43]. Bubble radius-time histories compared very

well. Tables5.1 and5.2 list maximum bubble radii, bubble periods and peak pressures. Cases

5.3.1 and 5.3.2 show the applicability of the 1D spherical RKDG-DGF method for modeling the

first bubble pulse, and the deficiency surrounding the inaccuracy and inefficiency of a compress-

ible approach to the second pulse. Subsection 5.3.2 extendsthe RKDG-DGF method to cavitating

flows in multi-dimensions.

5.3.2 Cavitating flows in multi-dimensions

An underwater explosion occurring near a structure generates a low pressure zone in the water

next to the structure, that has substantial influence on the response of a structure. Since water

cannot sustain tension, the water subjected to a sufficiently low pressure may be cavitated [6,

111]. Prediction of the formation and collapse of cavitation isan important component in UNDEX

135

simulations. Reference [6] provides a detail description of cavitation mechanisms inUNDEX

applications, and References [149, 156] overview various cavitation prediction methods applicable

to UNDEX. This work uses the pressure cutoff cavitation method described in Section 4.2. The

low pressure limit is taken as 0.05 bar.

To explore the applicability of our RKDG-DGF approach coupled with the pressure cutoff cavi-

tation method, the RKDG-DGF results from Cases 5.3.3 (a 2D cylindrical case) and 5.3.4 (a 3D

axi-symmetric case) are compared with those from previous works.

Case 5.3.3: This case considers a 2D cylindrical external explosion near a rigid wall as previously

modeled by W.F. Xie [156]. W.F. Xie obtained the pressure-time history in this cavitating flow

using the Modified Ghost Fluid Method (MGFM) with a modified Schmidt cavitation method (see

References [149, 156]). The modified Schmidt method models the cavitation as a homogeneous

mixture of vapor and liquid [156]. The initial conditions and the fluid mesh are shown in Figure

5.28. The domain[−600, 600] × [300,−600] cm is discretized with100 × 100 non-uniform ele-

ments. The initial gas bubble with radiusr0 = 100 cm is located at the origin. Our 2D RKDG-DGF

code is run to a final time of 0.01 seconds. The ideal-gas law with γ = 2.0 and the Tait EOS with

γ = 7.15 andB = 3.31E9 dyne/cm2 are used to model the explosive gas and water, respectively.

Figure 5.28. Initial conditions and the fluid mesh for Case 5.3.3

136

The rigid wall is located at the top boundary, and other boundaries are treated using a sponge-layer

non-reflecting boundary condition (NRBC) to minimize nonphysical wave reflections at bound-

aries. For the details of the sponge-layer NRBC, see Appendix F and References [80, 98]. Using

the pressure cut-off cavitation method, the pressure-timehistory at the center of the upper rigid

wall is shown in Figure5.29. Pressure contours at 0.0036 seconds and 0.0051 seconds when the

formation and collapse of cavitation are observed, are provided in Figures5.30and5.31.

RKDG-DGF results are compared with the results from [156]. Figure5.29shows that the RKDG-

DGF method provides similar peak pressure, cavitation cutoff and collapse times, and reloading

in the pressure-time history. Figures5.30and5.31show the cavitation mechanism in FSI with a

rigid wall. At 0.0036 seconds, the cavitation forms in the region below the upper wall as shown in

Figure5.30. At 5.1E-3 seconds, the cavitation collapses from the high-pressure surrounding water,

causing the upper rigid wall to be reloaded. The collapsed cavitations are observed at both sides of

the collapse region. The cavitation closes within a few milliseconds.

Figure 5.29. RKDG-DGF results on the rigid wall for Case 5.3.3

137

Figure 5.30. Pressure contour of Case 5.3.3 at 0.0036 seconds

Figure 5.31. Pressure contour of Case 5.3.3 at 0.0051 seconds

138

There is a small pressure deviation after the cavitation collapse at 5.1E-3 seconds. This may be

due to the difference of methods. The sponge-layer NRBC performs well in preventing significant

interference from reflected waves at boundaries. No contourdistortions and nonphysical waves

occur. For the treatment of interface cells, the MGFM used byW.F. Xie [156] requires local shock

values to be calculated at the gas-water interface, but the RKDG-DGF method does not require

these complexities in the implementation and computation.Compared to the modified Schmidt

method, the pressure cut-off method is simpler and faster. The comparison between both methods

shows that the choice of cavitation methods has no significant impact on the solution of cavitating

flows in FSI. Although the approaches used different cavitation models, the formation and collapse

of cavitation is very similar.

Case 5.3.4: This case considers a 3D axi-symmetric internal explosionmodeled by H. Sandusky

[58], G. Chambers [47], T.G. Liu [149], W.F. Xie [156] and A.B. Wardlaw [6] in separate work.

The experiment by H. Sandusky [58] provides the plastic deformation history of a water-filled

aluminum tube subjected to a 3g PETN explosion. The 2.8g PETNcharge plus 0.2g detonator

were located at the center of a water-filled aluminum tube [58]. The tube bottom was sealed by a

thin plastic sheet and the tube top left open [6, 47, 58]. The pressure-time history is measured at

the inner center of the tube wall. For the details of the experiment, see References [6, 47, 58].

Using experimental data for this case, H. Sandusky [58] assessed the applicability of a coupled

Eulerian-Lagrangian DYSMAS code developed by the Naval Surface Warface Center (NSWC)

Indian Head Division. G. Chambers [47] and A.B. Wardlaw [6] used DYSMAS and GEMINI-

DYNA_N codes (i.e., GEMINI for the fluid and the Navy version of DYNA (DYNA_N) for the

structure) for the same problem. Both used the pressure-cutoff cavitation method withPlimit =

0.05 bar to predict the formation and collapse of cavitation withFSI. More recently, W.F. Xie

and T.G. Liu [149, 156] used the MGFM coupled with the modified Schmidt cavitation method.

In [149, 156], the charge was replaced by an ideal gas bubble. A.B. Wardlaw using DYSMAS

provides a good description of the cavitation mechanisms with UNDEX FSI [6]. The differences

139

between a rigid-wall case and a deformable-wall case are discussed to assess the shock-bubble

interaction and wall deformation effects with FSI [6]. Wardlaw [6] provides useful results for

exploring the RKDG-DGF results.

Based on the experimental arrangement for our model, the initial conditions are shown in Figure

5.32. Since the fluid flow has axi-symmetric flow along the axis z, the fluid computation using

Equation (5.2) treats only the right half of the domain[0, 4.415]× [−8.9, 8.9] cm discretized with

40 × 160 uniform elements. The JWL EOS with PETN parameters in Table2.7, and the Tait EOS

are used to model the explosive gas and water, respectively.

Figure 5.32. Experimental arrangement and initial conditions for Case 5.3.4

The internal explosion case with a rigid-wall tube is considered, first. Along the axisz, a symmetry

condition is applied. Both the top and bottom are treated using a outflow condition to minimize the

interference from reflection waves at boundaries. The 3D axi-symmetric RKDG-DGF code is run

to a final time of 1.5E-4 seconds. The pressure-time history at the inner center of the outer wall is

shown in Figure5.33.

140

The peak pressures at the shock arrival and reloading moments (i.e. at 1.5E-5 seconds and 1.2E-4

seconds) are predicted to be 6.6E9 dyne/cm2 and 3.2E9 dyne/cm2 which are very close to the refer-

ence results taken from [6]. A pressure jump at around 1.4E-4 seconds occurs. This is likely due to

a different boundary condition from Wardlaw [6] used for both top and bottom where we assumed

a simple outflow condition. The condition used in [6] is unknown. Otherwise, the pressure-time

histories show excellent agreement.

Figure 5.33. RKDG-DGF results for Case 5.3.4 with the rigid wall

The pressure contours at various times are shown in Figure5.34. Before the cavitation cutoff at 4E-

5 seconds, interactions between the primary shock, expansion wave and gas bubble are dominant.

After the initial shock reflects off from the rigid wall, the reflected wave interacts with the gas

bubble at 3E-5 seconds. The interaction between the reflected wave and gas bubble creates an

expansion wave traveling back toward the wall. The expansion wave reduces the surface pressure

on the wall. This low pressure next to the wall causes the surrounding water to be cavitated.

141

Figure 5.34. Pressure contours for Case 5.3.4 with the rigidwall; the red line denotes the material interface

142

As in Case 5.3.3, the local cavitation originates from the center of the rigid wall around 4E-5

seconds. The high-pressure surrounding water rushes into the low-pressure zone, and the low-

pressure zone collapses at 1.2E-4 seconds. The cavitation collapse causes the rigid wall to be

reloaded [6, 8, 131]. These results show that the cavitation mechanism in a rigid wall case is

mainly influenced by the shock-bubble interaction.

Next the rigid tube wall is replaced by a deformable wall (i.e., Al5086 cylinder). The wall behav-

ior is described by the Kirchhoff material model (See Subsection 2.2.4.1). Typical properties of

Al5086 from [154] are used in the calculation. The “three-to-one” non-matching mesh technique

described in Section 4.2 and Appendix E is used to match the fluid and structural meshes along the

interface. The initial FSI meshes are shown in Figure5.35. The structural computation is based on

the method described in Section 4.2. Both the top and bottom structural nodes are fixed in r and z-

directions during the computation. The tube wall is treatedusing a moving wall condition, and the

domain[0, 4.415] × [−8.9, 8.9] cm is discretized with60 × 240 uniform elements. Unless stated

otherwise, the same fluid computation used in the rigid-wallcase is applied to this FSI application.

Figure 5.35. The initial FSI meshes for Case 5.3.4

143

Figure 5.36. RKDG-DGF results for Case 5.3.4 with the elastic wall

In Figure5.36, the pressure-time history from the RKDG-DGF calculation is similar to experi-

mental data [58] and results from A.B. Wardlaw’s work [6]. The predicted peak pressure 6.05E9

dyne/cm2 is somewhat lower than the experimental value 6.25E9 dyne/cm2. In the experimental

results, the first cavitation occurs at 3E-5 seconds with reloading and a second cavitation at 6E-5

seconds. References [6, 58] conclude that at about 9 E-5 seconds, final cavitation collapse and

reloading occur. The RKDG-DGF results show pressure reloadings at about 4.7E-5 seconds and

8.5E-5 seconds which are earlier than those of the experiment. Earlier time appearances are likely

due to the elastic model used in our study. Pressure contoursat various times are provided in

Figure5.37. The first cavitation occurs at about 2.8E-5 seconds in the water next to the tube wall

which is earlier than in the rigid wall case. This is mainly due to the wall deformation from the

initial shock impact. The deformed wall generates an extra volume required to be filled by the

surrounding water [6].

144

Figure 5.37. Pressure contours for Case 5.3.4 with the elastic wall; the red line denotes the material interface

145

This volume increase makes the initial shock strength lowerthan that of the rigid wall case after the

peak pressure (Compare Figures5.33and5.36) and a low pressure cavitation region on the wall

is created at about 2.8E-5 seconds. At around 3E-5 seconds, another cavitation occurs midway

between the tube and bubble. This is due to the interaction between the reflected wave from the wall

and the gas bubble. This interaction generates an expansionwave traveling back toward the tube

wall. These observations lead to the conclusion that the first cavitation is related to the lowered wall

pressure and second cavitation is associated with the shock-bubble interaction. The first cavitation

may not occur in the rigid wall case, since it is mainly related to the wall deformation. Two

cavitations merge at about 4E-5 seconds and collapses later[6]. Unlike the DYSMAS calculation,

a small cavitation and collapse at about 4.7E-5~5.8E-5 seconds is reflected in the RKDG-DGF

calculation. The tube wall is re-loaded at about 8.5E-5 seconds.

The RKDG-DGF method using the elastic material model provided a reasonable description of

the significant cavitation mechanisms (one cavitation related to the wall deformation and the other

cavitation due to the shock-bubble interaction). Unlike the rigid wall case, the presence of the

deformable wall in the FSI simulation is very important.

To connect the fluid and structure meshes in FSI simulations,the non-matching mesh technique

described in Ch. 4 is used. To keep the quality of fluid mesh during the calculation, the fluid

nodes are continuously adjusted by the ALE mesh-smoothing scheme. A single FSI framework

based on the ALE scheme is assessed by this case. Compared to the previous NSWC work using

a Coupled Eulerian Lagrangian (CEL) scheme, the ALE-based FSI framework is more attractive

for UNDEX FSI simulation; There is no error caused by mesh-overlapping, no complex mesh

information exchange required, and no geometric complexities.

Sections5.1 and5.2 provided several 1D and 2D test cases to assess the RKDG-DGF method in

the fluid by comparing with analytic solutions and referenceresults. Both the shock front and

material interface were sharply captured at correct locations and magnitudes. No spurious oscil-

lation, excessive diffusion or contour distortion occurred. Compared to existing methods such as

146

the original GFM, the ALE interface method and the MGFM, the RKDG-DGF approach has fewer

difficulties in the implementation and computation: The RKDG-DGF method has no geometric

complexities for multi-dimensional applications where ALE interface method may suffer from er-

rors caused by distortion of the mesh in large interface motion. The RKDG-DGF method also has

no limitations on the use of the JWL EOS and shock strengths. Section5.3extended RKDG-DGF

approach for practical UNDEX applications that include bubble behavior, shock-bubble interac-

tion and cavitation with FSI. Cases 5.3.1 and 5.3.2 investigated the behavior of a spherical gas

bubble produced by TNT explosions. The RKDG-DGF results including maximum bubble radius,

bubble pulse and peak pressures showed good agreement with reference results. Compared to a 3D

flow simulation, this 1D symmetric approach provided a simple and fast way to obtain the charac-

teristics of UNDEX gas bubbles. This 1D symmetric approach is also useful for providing initial

profiles in mid-to-late time UNDEX simulations as in [103]. Case 5.3.3 considered the 2D external

UNDEX application near a rigid wall. The pressure cut-off model was used to detect the formation

and collapse of cavitation on the wall. The performance of the sponge-layer NRBC was shown

in the pressure contours. No nonphysical wave reflections occurred during the computation. Case

5.3.4 considered internal explosions in a water-filled tube. Due to axi-symmetric flow, Equation

(5.2) with the pressure cutoff cavitation model was applied to the half of the tube. The cavitation

mechanisms were explored.

These assessments support the conclusion that the RKDG-DGFmethod has wider applications for

near-field UNDEX applications and is easier to extend to multi-dimensions. Compared to other

multi-fluid methods, the JWL EOS does not increase difficulties and complexities in the imple-

mentation and computation of our RKDG-DGF approach. Compared to the early CFD approaches

which require the solution of local Riemann problems at the interface, the RKDG-DGF approach

does not require these additional computations. Compared to the ALE-based interface method,

error associated with distortion of the mesh is managed evenwhen the interface motion is large in

multi-dimensions.

147

Chapter 6

Conclusions

A coupled solution approach was presented for numerically simulating a near-field early-time UN-

DEX. The approach consists of the Runge Kutta DiscontinuousGalerkin (RKDG) method to dis-

cretize the Euler fluid equations, the Direct Ghost Fluid (DGF) method to treat explosive gas-water

flows and the ALE deformable fluid mesh to adjust grid points tothe structural interface deforma-

tion. The combination of RKDG and DGF (RKDG-DGF) methods forexplosive gas-water flows

is the main contribution of this work. Compared to existing two-fluid methods, the RKDG-DGF

method has wider application for various near-field UNDEX simulations, and is easier to extend

to multi-dimensions.

Several test cases (e.g., 1D Cartesian, 1D symmetric, 2D cylindrical and 3D axi-symmetric cases)

were examined and assessed by comparing the RKDG-DGF results with analytic solutions, ex-

perimental data, results from previous work and results from equivalent 1D symmetric simulation.

These comparisons showed excellent agreement. Both the shock front and material interface were

sharply captured at the correct locations and magnitudes. The comparison between RKDG-DGF

results and results from previous work showed that the RKDG-DGF method produces less diffu-

sive and oscillatory fluid results, allows easier extensionto multi-dimension and simply models

explosive gas governed by the JWL EOS.

148

To assess the applicability of the RKDG-DGF method for near-field UNDEX, practical UNDEX

applications that include bubble pulses, shock-bubble interactions and cavitations with FSI, were

also studied. In spherically symmetric UNDEX flows, the motion of a gas bubble can effectively

be simulated by the 1D spherically symmetric RKDG-DGF approach. An investigation in the

simulation of a small-radius gas bubble showed that the approach predicts very well for the first

bubble pulsation and reasonably well for subsequent pulses. A 2D underwater explosion near a

rigid wall was studied to obtain the pressure-time history in a cavitating fluid flow. The pressure-

time history obtained from the 2D RKDG-DGF method coupled with the pressure-cutoff cavitation

model and the sponge-layer NRBC, agreed well with results from W.F. Xie’s previous work [156].

Although the two approaches used different solution methods, the comparison of peak pressure,

cavitation formation and collapse times showed excellent agreement. It was found that the choice

of a cavitation model does not have significant impact on the pressure-time history. Compared

to W.F. Xie’s one-fluid homogeneous cavitation model, insertion of the pressure cut-off model

was simpler and easier. The sponge layer allowed the shock wave to propagate outward without

any disturbance at boundaries. Compared to other NRBC approaches which require a modifica-

tion of governing equations, the implementation of the sponge layer NRBC was straightforward.

A 3D spherical internal explosion within a water-filled tube, which has received much attention

from many researchers, was studied. Cavitation mechanismswere discussed. The rigid wall case

showed excellent agreement with results from DYSMAS. The cavitation was mainly dominated by

the shock-bubble interaction. When using a deformable walltube, the pressure-time history from

the RKDG-DGF approach was similar to experimental data and results from DYSMAS. A series

of cavitation effects was observed: cavitation formation,cavitation collapse and reloading. This

case provided significant insight into the cavitation mechanisms distinguishable between the rigid

wall and the deformable wall. The ALE deformable fluid mesh was used to adjust fluid nodes to

the structural interface deformation. Compared to previous NSWC work using a CEL DYSMAS

code, the ALE-based deformable fluid mesh has less geometriccomplexity in the communication

between the structure and the fluid.

149

To extend the application of RKDG-DGF method to wider near-field UNDEX applications, the

following future works are recommended:

X Add plastic/nonlinear material models

– In current code

– Linking with a commercial structural code

X Extend applicability to other UNDEX mechanisms

– Contact explosion

– Bubble jetting

150

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Appendix A

Mathematical theorems and rule

Gauss theorem: For a differentiable functionf(x),∫

Ω

∂f(x)

∂xi

dΩ =

∫

Γ

nif(x)dΓ (A.1)

Reynolds transport theorem: The material time derivative in the Lagrangian description is

D

Dt

∫

Ω

f(x, t)dΩ =

∫

Ω

(

Df(x, t)

Dt+ f

∂vi

∂xi

)

dΩ (A.2)

Substituting the material time derivatives (2.16) into Equation (A.2) gives

D

Dt

∫

Ω

f(x, t)dΩ =

∫

Ω

(

∂f(x, t)

∂t+

∂ (vif)

∂xi

)

dΩ (A.3)

Applying Equation (A.1) to Equation (A.3) gives

D

Dt

∫

Ω

f(x, t)dΩ =

∫

Ω

∂f(x, t)

∂tdΩ +

∫

Γ

fvinidΓ (A.4)

Product rule: For differentiable functionf(x) andg(x),

d(f g)

dx= f

dg

dx+ g

df

dx(A.5)

162

Appendix B

FE-FCT algorithm

This work first chose the Finite Element Method-Flux Corrected Transport (FEM-FCT) scheme for

UNDEX flow simulations. Compared to other shock capturing schemes [132, 151], the FEM-FCT

method provides a simple and easy limiting procedure in the implementation and computation

[125]. The key is the combination of a high-order accurate schemewhich is accurate but oscil-

latory, and a low-order accurate scheme which is diffusive but monotonic. In an implicit way, a

high-order accurate scheme is applied to smooth flow region and a low-order accurate scheme

is applied to the region near the discontinuity. For the details of FCT procedure, References

[48, 68, 71, 124, 125] are recommended. The schematic of FCT procedure is shown inFigure

B.1.

Figure B.1. The schematic of FCT procedure

163

A high-order accurate scheme advances the solutions from timetn to timetn+1 as

un+1 = un + ∆uH (B.1)

where∆uH = un+1H − un is the solution increment obtained from a high-order accurate scheme.

Equation (B.1) can be rewritten as

un+1 = un + ∆uL + (∆uH − ∆uL) = un+1L + (∆uH − ∆uL) (B.2)

where∆uL = un+1L −un is the solution increment obtained from a low-order accurate scheme, the

termun+1L is diffusive low-order solution and the term(∆uH −∆uL) is anti-diffusion. Merely sub-

tracting both increments does always not provide stable solutions since anti-diffusion is globally

applied.

A nonlinear limiter to anti-diffusion is often used as

un+1 = un+1L + limiter · (∆uH − ∆uL) where 1 ≥ limiter ≥ 0 (B.3)

wherelimiter is a local modulation coefficient to suppress oscillations.Whenlimiter is equal to

0.0, Equation (B.3) reduces to a low-order scheme solution as

un+1 = un+1L (B.4)

Whenlimiter is equal to 1.0, Equation (B.3) reduces to a high-order scheme solution as

un+1 = un+1L + (un+1

H − un − un+1L + un) = un+1

H (B.5)

The SOD’s shock tube problem was used to assess the performance of the FE-FCT method com-

parable to the RKDG method we use. The 1D domain [0, 1]m is discretized with 1,000 uniform

elements. FE-FCT results and RKDG results are compared in FigureB.3.

164

Figure B.2. The initial conditions for the SOD’s shock tube problem

Figure B.3. Comparison of FE-FCT results and RKDG results at0.2 seconds

Compared to RKDG results using the same conditions, the FE-FCT scheme produces more diffu-

sive profiles. J. Cocchi states that these numerical diffusion can be a source of nonphysical pressure

oscillations in multi-fluid flows [92]. For the details of numerical diffusion effect, refer to Section

4.2. This finding motivated this research to use a piece-wisediscontinuous solution method such as

Discontinuous Galerkin Method (DGM). Piece-wise discontinuous feature helps decrease variable

diffusion in the region near the discontinuity by allowing avariable jump.

165

Appendix C

The slope limiting procedure

For the system of 1D Euler equations, the limiting procedureis as follows [12, 132]

1. Find eigenvectors of the jacobian matrix A= ∂F/∂U

(a) convert conservative form equation to quasi-linear form with a jacobian matrix A1

U,t + F,x = 0 ⇒ U,t + AU,x = 0 where Ai,j =∂fi(U)

∂uj(C.1)

(b) Compute matrices R and R−1 of eigenvectors2 as

R−1 ∂F∂U

R = R−1AR = Λ so A = RΛR−1 (C.2)

where

Λ = diag[λ1, λ2, λ3] = diag[u − a, u, u + a]

2. Transform system variablesU to local variablesW by multiplying variables by R−1

W = R−1U (C.3)

where inter-related system variables are converted into decoupled local variables.

1Since the system is hyperbolic, the jacobian matrixA can be diagonalized and has distinct real eigenvalues.2Variablesu anda are flow velocity and the speed of sound in a medium.

166

3. Apply the minmod slope limiter to local variablesW.

Wj,k = minmod(Wj,k, Wj+1 − Wj, Wj − Wj−1) (C.4)

4. Transform local variables to original system variables by multiplying local variables by R

on the left.

U = RW (C.5)

For the system of 2D Euler equations,

U,t + F(U),x + G(U),y = 0 or U,t + AU,x + BU,y = 0 (C.6)

where A and B are Jacobian matrices,∂F/∂U and∂G/∂U. The slope limiting is applied by the 1D

manner (i.e., C.1-5) to each direction.

For multi-dimensions, R.Fedkiw [140] provides the right eigenvector matrixR and its inverseR−1

as

R =

1 1 0 1

u − Ac u B u + Ac

v − Bc v −A v + Bc

H − (Au + Bv)c H − ρc2

peint

−v H + (Au + Bv)c

(C.7)

and

R−1 = 1/2

(

peint

ρc2((u2 + v2) − H) + 1 + (Au+Bv)

c

)

−(

peint

ρc2u + A

c

)

−(

peint

ρc2v + B

c

)

peint

ρc2

−2p

eint

ρc2((u2 + v2) − H) 2

peint

ρc2u 2

peint

ρc2v −2

peint

ρc2

Av − Bu B −A 0(

peint

ρc2((u2 + v2) − H) + 1 − (Au+Bv)

c

)

−(

peint

ρc2u − A

c

)

−(

peint

ρc2v − B

c

)

peint

ρc2

(C.8)

whereu, v, c, p, ρ, H are velocities in x and y-directions, the speed of sound, pressure, density and

enthalpy. Thepeint is the derivative of the pressure with respect to internal energyeint. For Jacobian

matrix A, we setA = 1 andB = 0. For Jacobian matrixB, we setA = 0 andB = 1. For 1D, we

setA = 1, B = 0 andv = 0. For the details of eigenvectors, References [18, 19, 27, 45, 49, 135]

are recommended.

167

Appendix D

2D Geometric quantities

Figure D.1. Numbering in the counter-clockwise direction for quadrilateral element

Face vectors are

~L1 =

[

y2 − y1

x1 − x2

]

, ~L2 =

[

y3 − y2

x2 − x3

]

, ~L3 =

[

y4 − y3

x3 − x4

]

and ~L4 =

[

y1 − y4

x4 − x1

]

(D.1)

Area and unit normal vector at face i are

∆Ai =∣

∣

∣

~Li

∣

∣

∣=

√

L2x,i + L2

y,i and ~ni =~Li

∆Ai(D.2)

We assume that these quantities are constant along the face.For triangular element and 3D element,

Reference [69] is recommended.

168

Appendix E

Communication between the fluid and the

structure at the interface

There are three data exchanges between the fluid and the structure: fluid pressure to the structure,

structural displacement to the fluid and structural velocity to the fluid. This work uses the “three-

to-one” non-matching mesh algorithm. Three fluid elements communicate with one structural

element.

Figure E.1. Communication between the fluid and the structure

Fluid pressures are directly copied to quadrature points along structural interface. Both cell-vertex

displacements and cell-centered velocities in the fluid aretaken by the linear interpolation of struc-

tural values. For the review of coupling algorithms, see reference [3].

169

Appendix F

The implementation of Boundary

Conditions

RKDG method evaluates piecewise value at celli by using values at neighboringi + 1 andi − 1

(see Subsection3.2.4). The first and last cells have no a neighboring value so that special care is

required [45, 69, 132]. The imposition of BC is conducted by extending the domain with additional

cells which are called dummy cells [69].

Figure F.1. Cell numbering and concept of dummy cells

170

One dummy cell is added along each side. A high-order numerical scheme may require more

dummy cells. At the beginning of each time step, the values atdummy cells are treated in some

manner[132]. For the details of BC, References[19, 27, 45, 69, 132] are recommended.

The following BC are often considered:

X rigid wall (also called solid wall or reflective wall) condition

X moving wall condition

X symmetry condition

X outflow (also called transmissive or open-end) condition

X non-reflecting ( also called far-field or infinite or absorbing ) condition

The method of image(also called mirror) is often used to implement some BC. The reflection is

accomplished by reflecting scalar quantities, and vector quantities with a sign change onto the

fictitious region[27, 45].

F.1 Rigid wall condition

The rigid wall condition requires no flow normal to the surface as

~v · ~n = 0 (F.1)

For example, along the left and top boundaries, we treat

u0, j = −u1, j and vi, ny+1 = −vi, ny (F.2)

whereu andv are velocities in x and y-directions.

Other quantities along surfaces (0, j) and (i, ny + 1) are taken from values of the nearest interior

cell as

Q0, j = Q1, j and Qi, ny+1 = Qi, ny (F.3)

171

F.2 Moving wall condition

This condition is often considered in FSI simulations. Whenthe right boundary of a fluid domain

interacts with an impermeable wall, velocityunx+1, j is defined as

unx+1, j = −unx, j + 2uwall, j (F.4)

whereuwall, j is moving wall velocity from structure computation [45]. Other quantities are treated

as in the rigid wall condition.

F.3 Symmetric condition

This condition requires no flow normal to the symmetric surface. Thus, the implementation is the

same as in the rigid wall condition F.1 [61].

F.3 Outflow condition

This condition requires no wave reflections from boundariesback into the domain

∂Q

∂~x= 0 (F.5)

whereQ denote conservative quantities in the fluid equations. Thiscondition allows wave to

propagate outward without any disturbance at boundaries. The simplest way is the zero-order

extrapolation (or zero-gradient) of interior values into fictitious values as [37, 45, 132]

Q0,j = Qi,ny+1 and Qi,ny+1 = Qi,ny (F.6)

Depending on the angle of outgoing wave, this condition is often inadequate for preventing non-

physical wave reflections; Since the zero-order extrapolation may lose information of the oblique

wave, an incorrect representation of the outgoing oblique wave may occur [37, 132]. See Figures

15.1-15.4 in Reference [98], volume 2.

172

F.4 Non reflecting condition

To remedy the deficiency of outflow condition mentioned above, a non-reflection condition is in-

stead considered. Here, the sponge layer concept (also called the buffer layer) which adds artificial

layer [98], is considered. In FigureF.2, colored regions represent the sponger layer: blue regions

have only x-directional contributions, red regions have y-directional contributions and green re-

gions have the sum of both directional contributions.

Figure F.2. Description of sponge layer concept

The sponge layer absorbs or dissipates the wave to prevent nonphysical wave reflections back into

the domain. Without modification of governing equations, this is simply accomplished by adding

a sourceSBC(U) to governing equations as

Ut + F (U)x + G(U)g = SBC(U) (F.7)

where

U =

ρ

ρu

ρv

ρe

, F =

ρu

ρu2 + p

ρuv

u(ρe + p)

, G =

ρv

ρuv

ρv2 + p

v(ρe + p)

and SBC = −σ(x)

ρ

ρu

ρv

ρe

173

The termσ(x) is an artificial damping function which controls wave absorption. For bottom left

corner,

σx = a

[

x − xL1

WL

]n

for xL2 ≤ x ≤ xL1 and σy = a

[

y − yB1

WB

]n

for yB2 ≤ y ≤ yB1 (F.8)

such that

σ = σx + σy (F.9)

For middle bottom side,

σ = σy = a

[

y − yB1

WB

]n

for yB2 ≤ y ≤ yB1 (F.10)

where constantsa andn control amplitude and distribution of the damping coefficient. K.A. Hoff-

mann [98] investigated the performance of the sponge-layer with a parameter study. Practically,

a, n and the number of additional cells are set to 0.05, 4 and 20 [98]. For other regions, func-

tion σ is obtained from the same manner. Functionσ within interior region is set to 0.0. The

sponge-layer approach is considered in Case 5.3.3 to prevent wave reflections as shown in Figure

F.3. The performance comparison of outflow condition and Spongelayer NRBC at 4E-5 seconds

is demonstrated in FigureF.4.

Figure F.3. The initial mesh with the sponge layer for Case 5.3.3

174

Figure F.4. Pressure contours for Case 5.3.3 with the spongelayer

The sponge layer allows shock wave to propagate outward without any disturbance at the bound-

ary. Since the sponge layer approach does not require a modification of governing equations,

the implementation is straightforward. For the details of the sponge layer concept, References

[4, 36, 80, 98, 114] are recommended.

175