A Preconditioned Navier-Stokes Method for Two-Phase Flows ... · Prtk,Prtε turbulent Prandtl...

AIAA-99-3329

A Preconditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Prediction

Robert F. Kunz*, David A. Boger, David R. Stinebring*, Thomas S. Chyczewski, Howard J. GibelingApplied Research Laboratory

The Pennsylvania State University, University Park, PA 16802

Sankaran Venkateswaran*, T.R. Govindan*NASA Ames Research Center

Moffett Field, CA 94035

AbstractAn implicit algorithm for the computation of vis-

cous two-phase flows is presented. The baseline differen-tial equation system is the multi-phase Navier-Stokes equations, comprised of the mixture volume, mixture momentum and constituent volume fraction equations. Though further generalization is straightforward, a three-species formulation is pursued here, which separately accounts for the liquid and vapor (which exchange mass) as well as a non-condensable gas field. The implicit method developed here employs a dual-time, precondi-tioned, three-dimensional algorithm, with multi-block and parallel execution capabilities. Time-derivative precondi-tioning is employed to ensure well-conditioned eigenval-ues, which is important for the computational efficiency of the method. Special care is taken to ensure that the result-ing eigensystem is independent of the density ratio and the local volume fraction, which renders the scheme well-suited to high density ratio, phase-separated two-fluid flows characteristic of many cavitating and boiling sys-tems. To demonstrate the capabilities of the scheme, sev-eral two-dimensional and three-dimensional examples are presented.

NomenclatureSymbols

Aj flux JacobiansCµ, C1, C2 turbulence model constantsCdest, Cprod mass transfer model constantsCi pseudo-sound speedCP pressure coefficientCD drag coefficientD source Jacobiand body diameterdm bubble diameterE, F, G flux vectorsgi gravity vectorH source vectorI identity matrixJ metric JacobianKj transform matrixk turbulent kinetic energyL bubble lengthMj similarity transform matrices

, mass transfer ratesP turbulent kinetic energy productionPrtk,Prtε turbulent Prandtl numbers for k and εp pressureQ transport variable vector

m·-

m·+

*

1American Institute of Aeron

Copyright 1999 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

Member, AIAA.

Re Reynolds numbers arc length along configuration

time coordinate, mean flow time scale ( )Uj velocity magnitude, contravariant velocity componentsui, u, v, w Cartesian velocity componentsxi Cartesian coordinatesα volume fraction, angle-of-attackβ preconditioning parameterΓ, Γe time derivative preconditioning and transform matricesε turbulence dissipation rate, numerical Jacobian parameterκ, φ MUSCL parametersΛj, λj eigenvaluesµ molecular viscosityρ densityσ cavitation number ( )τ pseudo-time coordinateν dissipation sensorξj curvilinear coordinates

Subscripts, Superscripts

1φ single-phase valuei, j coordinate indicesk constituent index, pseudo-time-step indexL, R dependent variable values on left and right of facel liquidm mixtureng non-condensable gast turbulentv condensable vapor, viscous

free stream valuetransformed to curvilinear coordinates

+/- production/destruction, right/left running

IntroductionMulti-phase flows have received growing research atten-

tion among CFD practitioners due in large measure to the evolv-ing maturity of single-phase algorithms that have been adapted to the increased complexity of multi-component systems. How-ever, there remain a number of numerical and physical modeling challenges that arise in multi-phase CFD analysis beyond those present in single-phase methods. Principal among these are large constituent density ratios, the presence of discrete interfaces, sig-nificant mass transfer rates, non-equilibrium interfacial dynam-ics, the presence of multiple constituents (viz. more than two) and void wave propagation. These naturally deserve special attention when a numerical method is constructed or adapted for multi-phase flows.

The authors’ interest here is in the analysis of sheet- and super-cavitation, wherein significant regions of the flow are occupied by gas phase. Depending on the configuration, such “cavities” are composed of vapor and/or injected non-condens-able gas.

t t∞, d/U∞

p∞ pv–( )/ 1/2ρlU∞2

∞

autics and Astronautics

Early efforts to model large cavities relied on potential flow methods applied to the liquid flow, while the bubble shape and closure conditions were specified. Recently, more general CFD approaches have been devel-oped to analyze these flows. In one class of methods, a single continuity equation is considered with the density varying abruptly between vapor and liquid densities (Song and He [1998], for example). Although these methods can directly model viscous effects, they are inherently unable to distinguish between condensable vapor and non-con-densable gas, a requirement of our current application.

By solving separate continuity equations for liquid and gas phase fields, one can account for and model the separate dynamics and thermodynamics of the liquid, con-densable vapor, and non-condensable gas fields. Merkle et al. [1998] have employed a two-species formulation for the analysis of natural sheet cavitation. This is the level of modeling employed here, though a three-species formula-tion is used to account for two gaseous fields.

Full two-fluid modeling, wherein separate momen-tum (and in principle energy) equations are employed for the liquid and vapor constituents, have also been utilized for natural cavitation (Grogger and Alajbegovic [1998]). However, in sheet-cavity flows, the gas-liquid interface is known to be nearly in dynamic equilibrium; for this rea-son, we do not pursue a full two-fluid level of modeling.

Sheet- and super-cavitating flows are character-ized by large density ratios (ρl/ρv > 104 is observed in near-atmospheric water applications), discrete cavity-free stream interfaces and multiple gas phase constituents (with differing liquid interfacial thermodynamics). Therefore, the CFD method must accommodate these physics effec-tively. Also, many relevant applications exhibit large scale unsteadiness associated with re-entrant jets, periodic ejec-tion of non-condensable gas, and cavity “pulsations”. Accordingly, we employ a time-accurate formulation.

The purpose of this paper is to present the numeri-cal method and physical models used in this research. Par-ticular emphasis is placed on unique aspects of the numerics including the preconditioning strategy, resultant eigensystem characteristics, and flux evaluation and limit-ing strategies associated with the resolution of interfaces. To date, the CFD method has been validated against a number of two-dimensional and axisymmetric configura-tions. Sample results of this capability are presented. In addition, we present two sets of three-dimensional results for which validation and application are underway.

The paper is organized as follows: The theoretical formulation of the method is summarized, including the baseline differential model, inviscid eigensystem, physical models, and key elements of the numerical method. This is followed by two sets of results. The first set of results includes axisymmetric steady-state and transient analyses of natural cavitation about several configurations. The sec-ond set of results includes three-dimensional analyses of flows about an axisymmetric ogive at angle-of-attack and a control surface interacting with a phase-separated gas-liquid stream. Available experimental measurements and photographs are used for comparison with the CFD results.

Theoretical FormulationGoverning Equations

The governing differential system employed is cast in Cartesian coordinates as

(1)where αl and αng represent the liquid phase and non-con-densable gas volume fractions, and mixture density and mixture turbulent viscosity are defined as

(2)

In the present work, the density of each constituent is taken as constant. The mass transfer rates from vapor to liquid and from liquid to vapor are denoted and , respectively. Mass transfer terms appear in the mixture continuity equation because this equation is a statement of mixture volume conservation. Also, note that each of the equations contains two sets of time-derivatives - those written in terms of the variable “t” correspond to physical time terms, while those written in terms of “τ” correspond to pseudo-time terms that are employed in the time-itera-tive solution procedure. The forms of the pseudo-time terms will be discussed presently.

In the development of the baseline differential sys-tem presented above, a number of physical, numerical, and practical issues were considered. First, a mixture volume continuity equation is employed rather than a mixture mass equation. This choice was made based on the authors’ experience that the nonlinear performance of seg-regated pressure based algorithms (Kunz et al. [1998]) is improved by doing so for high density ratio multi-phase systems. Because of this choice, neither a physical time derivative nor mixture density appears in the continuity equation, although the mixture density can vary in space and time. To render the system hyperbolic and to facilitate the use of time-marching procedures, we then introduce a pseudo-time derivative term (signified by “τ”) in the mix-ture continuity equation, a strategy that derives from the work of Chorin [1967] and others.

1

ρmβ2-------------

τ∂∂p

+xj∂

∂uj m·+

+m·-

( ) 1ρl----

1ρv-----–

=

t∂∂ ρmui( )+

τ∂∂ ρmui( )+

xj∂∂ ρmuiuj

( ) =

- xi∂

∂pxj∂∂

+ µm,t xj∂∂ui +

xi∂∂uj( ) ρmgi+

t∂∂α l+

α l

ρmβ2-------------

τ∂∂p

+τ∂

∂α l+xj∂∂ α luj

( ) m·+

+m·-

( ) 1ρl----

=

t∂∂αng+

αng

ρmβ2-------------

τ∂∂p

+τ∂

∂αng+xj∂∂ αngu

j( ) 0 ,=

ρm ρlα l ρvαv ρngαng+ +≡

µm t,ρmCµk

2

ε--------------------=

m·+

m·-

2American Institute of Aeronautics and Astronautics

Second, corresponding artificial time-derivative terms are also introduced in the component phasic conti-nuity equations, which ensures that the proper differential equation (in non-conservative form) is satisfied. That is, combining equations 1a and 1c,

(3)

Again, inclusion of such “phasic continuity enforcing” terms has a favorable impact on the nonlinear performance of multi-phase algorithms when mass transfer is present (Siebert et al. [1995]).

Third, we desired an eigensystem that is indepen-dent of density ratio and volume fractions so that the per-formance of the algorithm would be commensurate with that of single-phase for a wide range of multi-phase condi-tions. These considerations give rise to the preconditioned system in equation 1.

In generalized coordinates, equations 1 can be written in vector form as

, (4)

where the primitive solution variable, flux, and source vectors are written

(5)

and J is the metric Jacobian, .Matrix Γe is defined by

, (6)

and the preconditioning matrix, Γ, takes the form

(7)

where and . The compati-bility condition, , is incorporated implicitly in equations 6 and 7.

Eigensystem

Of interest in the construction and analysis of a scheme to discretize and solve equation 4 is its inviscid eigensystem. In particular, the eigenvalues and eigenvec-tors of matrix are required, where

, . (8)

, , and can be computed straightfor-wardly, and expressions for these are provided in the Appendix. The eigenvalues and eigenvectors of can be found by first considering the reduction of equation 4 to a single-phase system. With αl = 1, ρm = ρl = ρv = ρng (= constant), = 0, equation 4 collapses to the widely used single-phase “pseudo-compressibility” scheme, which can be written for inviscid flow as

. (9)

Flux Jacobian matrix has a well known form (Rogers et al. [1989], for example) and is given in the Appendix.

Comparing the expressions for and , one can write

t∂∂α l α l

ρmβ2-------------

τ∂∂p

τ∂∂α l

xj∂∂ α luj

( )+ + + =

t∂∂α l α l

ρmβ2-------------

τ∂∂p

τ∂∂α l α l

1–

ρmβ2-------------

τ∂∂p

uj xj∂∂α l+ + + + =

t∂∂α l

τ∂∂α l uj xj∂

∂α l+ + 0≡

Γe t∂∂

Q Γτ∂

∂Q

ξ j∂∂Ej

ξ j∂∂Ej

v

– H–+ + 0=

Q JQ J p ui α l αng, , ,( )T= =

Ej J Uj ρmuiUj ξ j i, p+ α lUj αngUj,,,( )T

=

Ejv

J 0 µ, m,t ∇ξ j ∇ξ j•( )∂ui

∂ξ j------- ξ j i,

∂uk

∂ξ j--------ξ j k,+ 0 0, ,

T

=

H J m·+

+m·-

( ) 1ρl----

1ρv-----–

ρmgi m·+

+m·-

( ) 1ρl----

0, , , T

,=

J ∂ x y z, ,( )/∂ ξ η ζ, ,( )≡

Γe

0 0 0 0 0 0

0 ρm 0 0 u∆ρ1 u∆ρ2

0 0 ρm 0 v∆ρ1 v∆ρ2

0 0 0 ρm w∆ρ1 w∆ρ2

0 0 0 0 1 0

0 0 0 0 0 1

=

Γ

1

ρmβ2-------------

0 0 0 0 0

0 ρm 0 0 u∆ρ1 u∆ρ2

0 0 ρm 0 v∆ρ1 v∆ρ2

0 0 0 ρm w∆ρ1 w∆ρ2

α l

ρmβ2-------------

0 0 0 1 0

αng

ρmβ2-------------

0 0 0 0 1

=

∆ρ1 ρl ρv–≡ ∆ρ2 ρng ρv–≡α l αv αng+ + 1=

Aj

Aj Γ -1Aj≡ Aj

Q∂

∂Ej≡

Aj Γ -1Aj

Aj

m·+/-

t∂∂Q

1φ

+ ξ j∂∂Ej

1φ

0=

Q1φ

J p ui,( )T=

Ej1φ

Uj uiUj ξ j i, p+,( )T=

Aj1φ

Q1φ

∂

∂Ej1φ

≡

Aj1φ

Aj1φ

Aj


. (10)

Diagonalizing ,

. (11)

The elements of the diagonal matrix are the eigenvalues of . Similarity transform matrices and

contain the right and left eigenvectors of . Forms for these matrices are sought. Using equation 10, we can write equation 11 as

. (12)

This brief analysis illustrates that the inviscid eigenvalues of the present preconditioned multi-phase sys-tem are the same as the standard single-phase pseudo-compressibility system with two additional eigenvalues introduced, Uj, Uj. That is,

. (13)

Equation 12 also illustrates that a complete set of linearly independent eigenvectors exists for the present three-component system. These results generalize for an arbitrary number of constituents. The eigenvalues are seen to be independent of the volume fractions and density ratio. This is not the case for other choices of precondi-tioning matrix, Γ, or if a mixture mass conservation equa-tion is chosen instead of the mixture volume equation.

The local time-steps and matrix dissipation opera-tors presented below are derived from the inviscid multi-phase eigensystem above, which has been shown to be closely related to the known single-phase eigensystem. This has had the practical advantage of making the single-phase predecessor code easier to adapt to the multi-phase system.

Physical Modeling

Mass Transfer

For transformation of liquid to vapor, is mod-eled as being proportional to the liquid volume fraction and the amount by which the pressure is below the vapor

pressure. This model is similar to that used by Merkle et al. [1998] for both evaporation and condensation. For transformation of vapor to liquid, , a simplified form of the Ginzburg-Landau potential is employed:

. (14)

In this work, Cdest and Cprod are empirical con-stants (here Cdest = 100., Cprod = 100.). αng appears in the production term to enforce that as . Both mass transfer rates are non-dimensionalized with respect to a mean flow time scale. Further details of the mass transfer modeling can be found in Kunz et al. [1999].

The mass transfer model presented in equation 14 retains the physically observed characteristic that cavity sizes, and thereby the dynamics of the two-fluid motion, are nearly independent of liquid-vapor density ratio. One outcome of the eigensystem characteristics summarized above is that the numerical behavior of the code parallels this physical behavior. This is illustrated in Figure 1a. There, the convergence histories are provided for six sim-ulations of cavitating flow over a hemispherical forebody with cylindrical afterbody (1/2 caliber ogive, σ = 0.3, from series of results presented below). As the density ratio is increased from 1 to 10, the convergence history is modi-fied somewhat, but beyond ρl/ρv = 10, the performance of the solver is virtually independent of density ratio up to ρl/ρv = 105. This behavior is consistent with the modeled physics of this problem as shown in Figure 1b. There, drag coefficient and number of predicted vaporous cells are seen to reach a nearly constant value at ρl/ρv > 10.

Aj˜

K1–

Aj1φ{ } K 0 0

0 Uj 0

0 0 Uj

K1

ρm------- 0

0 I

≡,=

Aj˜

Aj˜ MjΛjMj

1–=

Λ jAj˜ Mj

Mj1–

Aj˜

K1–

Aj1φ

K 0 0

0 Uj 0

0 0 Uj

K1–

Mj1φ

0 0

0 1 0

0 0 1

× =

Λj1φ

0 0

0 Uj 0

0 0 Uj

Mj1φ

K 0 0

0 1 0

0 0 1

Λj Uj Uj Uj Cj+ Uj Cj– Uj Uj, , , , ,( )T=

Cj Uj2 β2 ξ j,iξ j,i( )+=

m·-

m·+

m·- Cdestρvα

lMIN 0 p pv–,[ ]

1/2ρlU∞2( )t∞

---------------------------------------------------------------=

m·+ Cprodρv α l αng–( )2

1 α l– αng–( )t∞

---------------------------------------------------------------------------------=

m·+

0→ αv 0→

0 500 1000 1500 2000Iteration

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

log 10

RM

S[∆

u]

ρl/ρv = 1ρl/ρv = 10ρl/ρv = 100ρl/ρv = 1000ρl/ρv = 10000ρl/ρv = 100000

Figure 1a. Comparison of convergence histories with den-sity ratio for hemispherical forebody simulation.


Turbulence Closure

A high Reynolds number form k-ε model (Jones and Launder [1974]) with standard wall functions is implemented to provide turbulence closure:

(15)

As with velocity, the turbulence scalars are interpreted as being mixture quantities.

Numerical Method

The baseline numerical method is evolved from the UNCLE code of Taylor and his co-workers at Mississippi State University (Taylor et al. [1995], for example). UNCLE is based on a single-phase, pseudo-compressibil-ity formulation. Roe-based flux difference splitting is uti-lized for convection term discretization. An implicit procedure is adopted with inviscid and viscous flux Jaco-bians approximated numerically. A block-symmetric Gauss-Seidel iteration is used to solve the approximate Newton system at each time-step.

The multi-phase extension of the code retains these underlying numerics but also incorporates two vol-ume fraction transport equations, mass transfer, non-diag-onal preconditioning, flux limiting, dual-time-stepping, and two-equation turbulence modeling.

Discretization

The transformed system of governing equations is discretized using a cell centered finite volume procedure. Flux derivatives are computed as

, (16)

with similar expressions for , , and the cor-responding viscous fluxes. The inviscid numerical fluxes are evaluated using a flux difference splitting procedure (Whitfield and Taylor [1994]),

(17)where, with the non-diagonal preconditioner used here, the matrix dissipation operator is defined by =

, with defined from .The extrapolated Riemann variables, and are obtained using a MUSCL procedure (Anderson

et al. [1986], for example):

(18)For first order accuracy, φ = 0. The choice φ = 1,

κ = 1/3, yields the third order accurate upwind bias scheme used for the results presented in this paper.

The flows of interest here typically contain regions with sharp interfaces between liquid and gas phases. Accordingly, higher order discretization practices are required to retain adequate interface fidelity in the simula-tions. This is particularly important in three-dimensional super-cavitating vehicle or control surface computations such as those presented below. There, predicted lift and drag can be severely over-predicted if liquid phase (and its much higher inertia) diffuses numerically into low-lift gas-eous regions of the lifting surface.

Attendant to the third order upwind bias scheme employed are overshoots in solution variables at these interfaces. These can be highly destabilizing, particularly for the volume fraction equations, if sufficient mass trans-fer or non-condensable vapor is present to yield locally. To ameliorate this difficulty, the flux evaluation is rendered locally first order in the presence of large gradi-ents in αl or αng. This is affected through the use of a “dis-sipation sensor” in the spirit of Jameson et al. [1981]. Specifically, a sensor is formulated for each coordinate direction as

. (19)

This parameter is very small except in the immediate vicinity of liquid-gas interfaces. In this work, the higher order component of the numerical flux in equation 18, i.e. term φ, is multiplied by (1-νi).

To illustrate the foregoing discretization issues, consider adjacent parallel streams of two constituents. In the absence of shear, if a flow-aligned Cartesian mesh is employed, the mixing layer interface will be perfectly pre-served using the present modeling (no mass diffusion).

100 101 102

ρl/ρcv

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

CD/C

D, 1φ

0

50

100

150

200

250

300

350

400

# α l <

1 c

ells

Figure 1b. Comparison of predicted drag coefficient and number of vaporous cells with density ratio for

hemispherical forebody simulation.

t∂∂ ρmk( )

xj∂∂ ρmku

j( )+

xj∂∂ µm t,

Prtk-----------

xj∂∂k

P ρε–+=

t∂∂ ρmε( )

xj∂∂ ρmεu

j( )+

xj∂∂ µm t,

Prtε-----------

xj∂∂ε

C1P C2ρε–[ ]+

εk---

=

∂E/∂ξ Ei+1/2 Ei-1/2–( )=

∂F/∂η ∂G/∂ζ

Ei+1/2 E Qi+1/2L( ) A

-Qi+1/2

RQi+1/2

L,( ) Qi+1/2R

Qi+1/2L

–( )•+=

A-

Γ Γ -1A[ ]

-≡

Γ MΛ-M

1–( ) Λ- Λ- Λ Λ–( )/2≡Qi+1/2

R

Qi+1/2L

Qi+1/2R

Qi+1φ4--- 1-κ( ) Qi+2 Qi+1–( ) 1+κ( ) Qi+1 Qi–( )+[ ]–=

Qi+1/2L

Qi φ4--- 1-κ( ) Qi Qi-1 –( ) 1+κ( ) Qi+1 Qi–( )+[ ]+=

α l 0→

νi

α i+1 2α i– α i-1+

α i+1 2α i α i-1+ +-------------------------------------------≡


This is true independent of the density ratio of the streams and whether first or higher order discretization is employed. However, if the interface encounters a region of significant grid non-orthogonality, the interface will be smeared.

We consider a two-dimensional grid slice extracted from the three-dimensional fin computation presented below, as illustrated in Figure 2a. Two-dimensional invis-cid computations of a high density ratio mixing stream were computed on this grid. Gas and liquid (ρl/ρg = 1000) were injected axially, at the same velocity, above and below an inlet location seen in Figure 2b. No solid bound-aries are specified, so ideally, the interface would be per-fectly preserved through the domain. However, the initially sharp mixing layer interface encounters severe grid non-orthogonality, as it does in full scale three-dimen-sional computations, even well away from solid bound-aries, due to grid topology. Though this case is particularly pathological, regions where local grid quality suffers due to the geometry of the problem are inescapable in struc-tured multiblock analyses of complex aerodynamic con-figurations.

Downstream of the grid “stripe” associated with the fin leading edge, Figures 2b and c show that the inter-face is badly smeared when first order discretization is employed (0.05 < αl < 0.95 for 11-12 nodes). This is largely remedied when the 3rd order flux difference with dissipation sensor is employed (0.05 < αl < 0.95 for 3-4 nodes).

The dissipation sensor used does not completely eliminate overshoots, as illustrated in Figure 2c. If the vol-ume fractions are “clipped” at 1.0 after each pseudo-time-step the impact on the solution accuracy is minimal, as illustrated in Figure 2c. But, clipping causes the non-linear convergence to flat-line, so in general, we accept the mod-est overshoots in volume fraction.

Second order accurate backward differencing is used to discretize the physical transient term as

, (20)

where index n designates the physical time-step. Second order accurate central differencing is utilized for the viscous flux terms.

Implicit Solution Procedure

Adopting Euler implicit differencing for the pseudo-transient term, equation 4 can be written in ∆-form as

(21)

where , with index k designating the pseudo-time-step. Terms , and are the invis-cid flux, viscous flux, and source Jacobians. The source Jacobian is evaluated analytically as illustrated below. The inviscid and viscous flux Jacobians are evaluated numeri-cally as

(22)

where ε is taken as the square root of a floating point num-ber close to the smallest resolvable by the hardware.

The pseudo-time-step is defined based on the spec-tral radii of Γ-1Aj as

. (23)

For steady state computations, the physical time-step, ∆t, is set to infinity, and a CFL number of 5 is typi-cally used.

In the present work, we seek to resolve transient features characterized by the quasi-periodic shedding of vorticity and gas from the cavity. Strouhal numbers on the

Γe∂Q∂t------- Γe

3Qn+1,k

4Qn

– Qn-1

+( )2∆t

---------------------------------------------------------→

Γe3

2∆t---------

Γ 1∆τ------

∂∂ξ j-------Aj

∂∂ξ j-------Aj

v– D

-–+ +

∆Q =

Γe3Q

n+1,k4Q

n– Q

n-1+

2∆t----------------------------------------------------

–∂

∂ξ j-------Ej–

∂∂ξ j-------

Ejv

D ,+ +

∆Q Qn+1,k+1

Qn+1,k

–≡Aj Aj

vD

-

vapor

liquid

flow direction

0 0.1 0.2 0.3 0.4 0.5 0.6y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

α l

3rd order3rd order, clipping1st order3rd order, cartesian mesh1st order, cartesian mesh

Figure 2. a) Two-dimensional grid slice extracted from three-dimensional control fin model. b) Predicted liquid volume fraction contours using present third order dis-cretization. c) Predicted liquid volume fraction profiles

for various grids and discretizations.

c)

Aj∆Q( )i+1/2∂E

∂Qi

---------∆Qi=∂E

∂Qi+1

--------------∆Qi+1+ =

E Qi ε Qi+1,+( ) E Qiˆ Qi+1,( )–

ε------------------------------------------------------------------------ ∆Qi +

E Qi Qi+1, ε+( ) E Qiˆ Qi+1,( )–

ε------------------------------------------------------------------------ ∆Qi+1 ,

∆τ CFL

Uj Cj+j∑--------------------------=

a) b)


order of 0.1 are encountered in practice. Accordingly, non-dimensional physical time-steps on the order of .005 - .01 are required. A CFL number of 5 is typically used for the inner iterates in transient computations. This gives rise to a dual-time scheme that provides a 1 to 3 order-of-magni-tude drop in residuals in 5-10 pseudo-time-steps.

Optimum non-linear convergence is obtained using a pseudo-compressibility parameter, β2/U2

∞ ≅ 10. Upon application of the discretization and numeri-

cal linearization strategies defined, equation 21 represents an algebraic system of equations for ∆Q. This block (6x6 blocks) septadiagonal system is solved iteratively using a block symmetric Gauss-Seidel method. Five sweeps of the BSGS scheme are applied at each pseudo-time-step.

Source Terms

Following the strategy of Venkateswaran et al. [1995] for the numerical treatment of mass transfer source terms in reacting flow computations, we identify a source and sink component of the mass transfer model and treat the sink term implicitly and the source term explicitly. Specifically, with reference to equation 1 we have

. (24)

With from equation 14 we have

. (25)

It can easily be shown that the non-zero eigenvalue of D- is

(26)

which is less than zero for ρv < ρl. Hence, the identifica-tion of as a sink is numerically valid. Implicit treat-ment of this term provides that αl approaches zero exponentially, so that cases like those considered below, where significant mass transfer results in extremely low liquid volume fractions, remain stable.

The production mass transfer term is treated explic-itly. A relaxation factor of 0.1 is applied at each pseudo-time-step to keep this term from destabilizing the code in early iterations.

Turbulence Model Implementation

The turbulence transport equations are solved sub-sequent to the mean flow equations at each pseudo-time-step. A first order accurate flux difference splitting proce-dure similar to that outlined above for the mean flow equa-tions is utilized for convection term discretization. The k and ε equations are solved implicitly using conventional implicit source term treatments and a 2x2 block symmetric Gauss-Seidel procedure.

Boundary Conditions

Velocity components, volume fractions, turbulence intensity, and length scale are specified at inflow bound-aries and extrapolated at outflow boundaries. Pressure dis-tribution is specified at outflow boundaries (p=0 for single-phase or non-buoyant multi-phase computations) and extrapolated at inflow boundaries. At walls, pressure

and volume fractions are extrapolated, and velocity com-ponents and turbulence quantities are enforced using con-ventional wall functions. Boundary conditions are imposed in a purely explicit fashion by loading “dummy” cells with appropriate values at each pseudo-time-step.

Parallel Implementation

The multiblock code is instrumented with MPI for parallel execution based on domain decomposition. Inter-block communication is affected at the non-linear level through boundary condition updates and at the linear solver level by loading ∆Q from adjacent blocks into “dummy” cells at each SGS sweep. This is not as implicit as solving an optimally ordered linear system for the entire domain at each SGS sweep, but this potential shortcoming will not deteriorate the non-linear performance of the scheme if the linear solver residuals are reduced ade-quately at each pseudo-time-step.

ResultsTwo sets of results are presented here. The first set

includes axisymmetric steady-state and transient analyses of natural cavitation about several configurations. These solutions are compared to experimental measurements to demonstrate the capability of the modeling employed. The second set of results includes three-dimensional analyses of flows about an axisymmetric ogive at angle-of-attack and a control surface interacting with a phase-separated gas-liquid stream. More results and physical interpretation are provided in Kunz et al. [1999].

Two-Dimensional Results

Steady State and Transient Natural Cavitation on a Series of Axisymmetric Forebodies

Rouse and McNown [1948] carried out a series of experiments wherein cavitation induced by convex curva-ture aft of various axisymmetric forebodies with cylindri-cal afterbodies was investigated. They took photographs and pressure measurements along and aft of the forebodies from which bubble size and approximate shape could be deduced.

A series of 2-, 1-, and 1/4-caliber ogive forebodies with cylindrical afterbodies was analyzed at several cavi-tation numbers and compared with these measurements. Grid sizes ranging from 193 x 65 (axially x radially) to 705 x 65 were used depending on bubble size (larger grids for lower σ, larger bubbles). A grid study reported in Kunz et al. [1999] confirms the adequacy of these mesh sizes. Figures 3 through 7 show sample results for these axisym-metric computations. Figure 3 shows predicted and mea-sured surface pressure distributions at several cavitation numbers for a 1-caliber ogive forebody with cylindrical afterbody. The code is seen to accurately capture the bub-ble size as manifested by the decrease in magnitude and axial lengthening of the suction peak with decreasing cavi-tation number. Also captured is the overshoot in pressure recovery associated with the local stagnation due to bubble closure. Figure 4 illustrates the qualitative physics as cap-tured by the model. There, surface pressure contours, field liquid volume fraction contours, selected streamlines and the grid used are shown for the 1-caliber case at σ = 0.15. Here the cavitation bubble is quite long (L/d > 3). As with all large cavitation bubbles, the closure region is charac-

H+/-

m·+/- 1

ρl----

1ρv-----–

0 0 0 m·+/- 1

ρl----

0,, , , ,T

=

m·-

∆H-

D-∆Q with D

- ∂H-/∂Q≡,≡

λ D-( ) Cdestρv

1ρl----

1ρv-----–

α l1ρl----

p pv–( )+=

m·-


terized by an unsteady “re-entrant” jet. The significant flow recirculation and associated shedding of vorticity and vapor in these flows require that a transient simulation be carried out. The solution depicted in Figure 4 represents a snapshot in time of an unsteady simulation.

The ability of the code to predict important hydro-dynamic performance parameters has been verified by application to numerous cavitator shapes across a wide range of cavitation numbers. Figure 5 assembles the present results with those published previously (Kunz et al. [1999]). The non-dimensional quantity, is plotted vs. σ for a number of forebody or “cavitator” shapes. A number of experimental cases assembled by May [1975] are plotted along with 31 simulations carried out with the present method for six different cavitator shapes. Lower cavitation number simulations were tran-sient and are designated with shaded symbols in the plot. Clearly, the code captures the correct relationship between bubble length, drag and cavitation number except for large values of σ (say > 0.4), where the rather small bubble lengths are somewhat underpredicted.

For relatively bluff forebody shapes at moderate cavitation numbers, it has been widely observed that the entire cavity can be highly unsteady, with “re-entrant” liq-uid issuing quasi-periodically from the aft end of the bub-ble and traveling all the way to the front of the bubble. Figure 6 illustrates the ability of the method to capture these physics. There, a time-sequence of predicted vapor volume fractions are reproduced for a 1/4-caliber ogive simulation at a cavitation number of 0.3. A 193 x 65 mesh and a non-dimensional physical time-step of 0.007 was used for this computation. Clearly captured is the transport of a region of liquid towards the front of the cavity. There, the liquid interacts with the bubble leading edge, the top of this liquid region being sheared aftward while the bulk of the fluid proceeds upstream “pinching off” the bubble near the leading edge. These computed physics correspond well to film footage of blunt cavitators at intermediate cavita-tion numbers.

Figure 7 shows two other elements of this particu-lar transient simulation. The inner- or pseudo-time conver-gence history for four successive physical time steps is shown in Figure 7a. Nearly a three order-of-magnitude drop in the axial velocity residuals is obtained at each physical time step using 10 pseudo-time-steps. Figure 7b shows a time history of predicted drag coefficient for this case.

Three-Dimensional Results

As demonstrated above, and more extensively in another recent publication (Kunz et al. [1999]), the two-dimensional capability of the method has been established for steady and unsteady flows by application to a number of configurations. The three-dimensional capability of the method is maturing. Two sample calculations that repre-sent our current capability are presented here, though it is recognized that significantly more development, testing, and validation are required in three dimensions, especially for transients.

Steady Natural Cavitation on a 1-Caliber Ogive at Angle-of-Attack.

Figures 8 and 9 show the results of a three-dimen-sional simulation of cavitating flow over a 1-caliber ogive forebody with cylindrical afterbody at a 10o angle-of-attack. A cavitation number of 0.32 was specified. A 97 x 33 x 65 mesh was utilized consistent with grid studies pre-sented in Kunz et al. [1999]. The domain was decomposed into eight subdomains azimuthally and run on eight pro-cessors.

L/dCD1/2

s/d

Cp

0.0 1.0 2.0 3.0 4.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1-φanalysis2-φanalysis, σ=0.462-φanalysis, σ=0.402-φanalysis, σ=0.322-φanalysis, σ=0.24Data, 1-φData, σ=0.46Data, σ=0.40Data, σ=0.32Data, σ=0.24

Figure 3. Comparison of predicted and measured surface pressure distributions at several cavitation numbers for a

1-caliber ogive forebody.

Figure 4. Predicted liquid volume fraction and surface pressure contours, selected streamlines and computational

grid for a 1-caliber ogive forebody, σ = 0.15. Solution shown at a given time-step in transient analysis.


At this cavitation number a steady state solution could be obtained. Figure 8 illustrates the predicted bubble shape and streamline pattern for this simulation. Several interesting features are observed in the prediction. In par-ticular, the cavitation bubble shape (as identified with an isocontour of αl = 0.99) is seen to be highly three-dimen-sional in nature. A significant recirculation zone aft of the bubble diminishes the pressure recovery associated with the bubble-induced blockage and this in turn leads to a local collapse of the bubble on the top of the body. Indeed, the bubble is seen to have its greatest axial extent off of the symmetry plane of the geometry. Also, the predicted cavi-tation bubble closes outboard of the symmetry plane on the pressure side of the body, and significant azimuthally ori-ented vortical structures are observed.

Figure 9 shows the convergence history for this case. A three order-of-magnitude drop in the axial velocity residual is achieved in 800 pseudo-time-steps. It is observed that the 8-block/8-processor simulation exhibits nearly identical convergence behavior to a 1-block/1-pro-cessor run. The single processor case took 6.60 times as long to run as the 8 processor case on the SGI O2 cluster at the DoD ARL-MRSC, for a parallel efficiency of 82.5%.

Interaction of a Control Surface with Phase-Sepa-rated Non-condensable Gas and Liquid Impinge-ment Streams

The second three-dimensional simulation presented is that of a wedge shaped control surface interacting with an incoming stream of phase-separated water and air. This configuration was tested by the third author in the 12” water tunnel at the Penn State Applied Research Labora-tory (unpublished). The test was run with co-directed air

10-2 10-1 100

log10σ

100

101

102

log 10

[L/(

d*C

D1/2 )]

XXX

XXX

XXX

XX

X

Data, SphereData, Stagnation CupData, ConeComputed, 1/4-CaliberComputed, 1-CaliberComputed, 2-CaliberComputed, HemisphereComputed, ConeComputed, Blunt Head

X

Figure 5. Comparison of vs. σ for numerous cavitator shapes. Experimental data from May [1975].

Open symbols represent steady computations, filled sym-bols represent transient computations.

L/ dCD1/2( )

t=0.05

t=0.10

t=0.15

t=0.20

t=0.25

t=0.30

t=0.35

t=0.40

t=0.45

t=0.50

Figure 6. Time sequence of predicted vapor volume frac-tion for flow over a 1/4 caliber ogive with cylindrical

afterbody, σ = 0.3

0 10 20 30 40Pseudo-time-step

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

log 10

RM

S[∆

u]

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Physical Timestep

0.590

0.595

0.600

0.605

0.610

0.615

0.620

CD

Figure 7. a) Pseudo-time convergence history at four suc-cessive physical time steps for transient flow over a 1/4

caliber ogive with cylindrical afterbody, σ = 0.3. b) Time history of predicted drag coefficient for this case.


and water streams. Air was injected along the bottom of the tunnel such that the unperturbed gas-liquid interface impinged upon the sharp leading edge at about 25% span from the base of the fin. Air and water velocities were approximately the same. The configuration was tested at a range of angles-of-attack.

A 189,546 vertex grid was used for the results pre-sented here. The simulation was run on 16 processors with a 10o angle-of-attack and a cavitation number of 0.15. Using this relatively coarse mesh and the k-ε model, a steady-state solution was obtained, despite the presence of a blunt trailing edge and its associated recirculation zone along the span. As we refine our analyses of this class of application, time accurate simulations using upwards of 106 nodes will be required.

These flows are characterized by several physical features of academic and practical interest. At angle-of-attack, the liquid and gas streams are turned through approximately the same angle, but because the density ratio is high (≅ 1000 here), significant spanwise pressure gradients arise along the lifting surfaces. Accordingly, on the pressure side, the gas-liquid interface is deflected downward, giving rise to deceleration and acceleration of the liquid and gas streams respectively. On the suction sur-face the lower pressure in the liquid gives rise to an upward deflection of the gas interface. Attendant to these interface deflections is a loss in lift.

Adjacent to the sharp leading edge on the suction side, the local static pressure becomes low due to leading edge separation. Local natural cavitation occurs, and this vapor merges with the swept up non-condensable interface so that the entire suction side is enveloped in gas phase. Aft of the blunt trailing edge, the flow is recirculating and therefore the local static pressure is also low. This gives rise to a sweeping up of the pressure side cavity and some natural cavitation so that the entire wake region is princi-pally gas phase.

Figure 8. Predicted bubble shape (designated by αl = 0.99 isosurface) and streamlines for a 1-caliber ogive at a

10o angle-of-attack.

Iteration

log 10

[RM

S(∆U

)]

0 300 600 900-3

-2

-1

0

1

2

single blockmultiblock

Figure 9. Comparison of 1-block/1-processor and 8-block/8-processor convergence histories for a 1-caliber

ogive at a 10o angle-of-attack.

pressure side suction side

natural cavitation adjacent toleading edge

interface

2-phase wake

liquid

non-condensable gas

Figure 10. Front/top view of CFD simulation of super-cavitating fin configuration at 10o angle-of-attack.

Isocontours of αng = 0.9, αv = 0.9 designate non-con-densable gas and vapor regions respectively.


Figure 10 shows an oblique front/top view of the predicted flow field. Each of the flow features discussed above is clearly observed in the simulation. Of particular note is that the three-species formulation enables the sepa-rate prediction and identification of vaporous and non-condensable regions of the flow. Figure 11a shows a suc-tion side video frame of the tested flow configuration at this angle-of-attack. The enveloping of the suction surface in gas phase and the primarily gas phase wake are observed. Figure 11b shows a pressure side photograph. A similar view of the simulation is presented in Figure 11c which shows that the pressure surface interface deflection and gas-vapor wake region are qualitatively consistent with experimental observation.

ConclusionsA multi-phase CFD method has been presented and

applied to a number of high density ratio sheet- and super-cavitating flows. Several aspects of the method were out-lined and demonstrated that enable convergent, accurate and efficient simulations of these flows. These include a differential model and preconditioning strategy with favorable eigensystem characteristics, a block implicit dual-time solution strategy, a three species formulation that separately accounts for condensable and non-condens-able gases, higher order flux differencing with limiters and the embedding of this scheme in a parallel multi-block Navier-Stokes platform.

The two-dimensional simulations presented verify the ability of the tool to accurately analyze steady-state and transient sheet- and super-cavity flows. The three-dimensional capability of the code was demonstrated as well.

As the authors proceed with this research, we are focusing on several areas including: 1) extension of the method to compressible constituent fields, 2) improved physical models for mass transfer and turbulence, 3) extended application and validation for steady and tran-sient three-dimensional flows and 4) improved error damping through preconditioning and pseudo-time-step-ping formulations that locally adapt to problem parame-ters.

AcknowledgmentsThis work is supported by the Office of Naval

Research, contract # N00014-98-1-0143, with Mr. James Fein and Dr. Kam Ng as contract monitors. The authors acknowledge Brett Siebert, Charles Merkle and Phil Bue-low with whom several conversations benefited the present work. This work was supported in part by a grant of HPC resources from the Arctic Region Supercomputing Center and in part by a grant of SGI Origin 2000 HPC time from the DoD HPC Center, Army Research Labora-tory Major Shared Resource Center.

References1) Anderson, W.K., Thomas, J.L., Van Leer, B. (1986) “Comparison of Finite Volume Flux Vector Splittings for the Euler Equations,” AIAA Journal, Vol. 24, No. 9, pp.1453-1460.2) Chorin, A.J. (1967) “A Numerical Method for Solving Incompressible Viscous Flow Problems,” Journal of Com-

suction side enveloped in gas phase

interface

a)

b)

pressure side

non-condensable cavity

2-phase wake

Figure 11. a) Suction side video frame of super-cavitating fin at 10o angle-of-attack. b) Pressure side photograph.

c) Pressure side view of CFD simulation.

c)


putational Physics, Vol. 2, pp. 12-26.3) Grogger, H.A., Alajbegovic, A., 1998, “Calculation of the Cavitating Flow in Venturi Geometries Using Two Fluid Model,” ASME Paper FEDSM 98-5295.4) Jameson, A., Schmidt, W., Turkel, E. (1981) “Numeri-cal Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” AIAA Paper 81-1259.5) Jones, W.P., Launder, B.E. (1972) “The Prediction of Laminarization with a Two-Equation Model of Turbu-lence,” International Journal of Heat and Mass Transfer, Vol. 15, pp. 301-314.6) Kunz, R.F., Cope, W.K., Venkateswaran, S. (1997) “Stability Analysis of Implicit Multi-Fluid Schemes,” AIAA Paper 97-2080, to appear Journal of Computational Physics.7) Kunz, R.F., Siebert, B.W., Cope, W.K., Foster, N.F., Antal, S.P., Ettorre, S.M (1998) “A Coupled Phasic Exchange Algorithm for Three-Dimensional Multi-Field Analysis of Heated Flows with Mass Transfer,” Comput-ers and Fluids, Vol. 27, No. 7, pp. 741-768.8) Kunz, R.F., Boger, D.A., Chyczewski, T.S., Stinebring, D.R., Gibeling, H.J., Govindan, T.R., (1999) "Multi-Phase CFD analysis of Natural and Ventilated Cavitation About Submerged Bodies," ASME Paper FEDSM99-7364, Pro-ceedings of 3rd ASME/JSME Joint Fluids Engineering Conference.9) May, A. (1975) “Water Entry and the Cavity-Running Behaviour of Missiles,” Naval Sea Systems Command Hydroballistics Advisory Committee TR-75-2.10) Merkle, C.L., Feng, J., Buelow, P.E.O. (1998) “Com-putational Modeling of the Dynamics of Sheet Cavita-tion,” 3rd International Symposium on Cavitation, Grenoble, France.11) Rogers, S. E., Chang, J. L. C., Kwak, D. (1997) “A Diagonal Algorithm for the Method of Pseudocompress-ibility,” Journal of Computational Physics, Vol. 73, pp. 364-379.12) Rouse, H., McNown, J. S. (1948) “Cavitation and Pressure Distribution, Head Forms at Zero Angle of Yaw,” Studies in Engineering, Bulletin 32, State University of Iowa.13) Siebert, B.W., Maneri, C. C., Kunz, R.F., Edwards, D. P. (1995) “A Four-Field Model and CFD Implementation for Multi-Dimensional, Heated Two-Phase Flows,” 2nd International Conference on Multiphase Flows, Kyoto, Japan.14) Song, C., He, J., 1998, “Numerical Simulation of Cav-itating Flows by Single-Phase Flow Approach,” 3rd Inter-national Symposium on Cavitation, Grenoble, France.15) Taylor, L. K., Arabshahi, A., Whitfield, D. L. (1995) “Unsteady Three-Dimensional Incompressible Navier-Stokes Computations for a Prolate Spheroid Undergoing Time-Dependent Maneuvers,” AIAA Paper 95-0313.

16) Venkateswaran, S., Merkle, C. L. (1995) “Dual Time Stepping and Preconditioning for Unsteady Computa-tions,” AIAA Paper 95-0078.17) Venkateswaran, S., Deshpande, M., Merkle, C.L. (1995) “The Application of Preconditioning to Reacting Flow Computations,” AIAA Paper 95-1673, from Pro-ceedings of the 12th AIAA Computational Fluid Dynam-ics Conference.18) Whitfield, D. L., Taylor, L. K. (1994) “Numerical Solution of the Two-Dimensional Time-Dependent Incom-pressible Euler Equations,” Mississippi State University CFD Laboratory Report MSSU-EIRS-ERC-93-14.



Appendix

(A.1)

(A.2)

(A.3)

(A.4)

AjQ∂

∂Ej≡

0 ξ j x, ξ j y, ξ j z, 0 0

ξ j x, ρm Uj ξ+j x, u( ) ρmξ j y, u ρmξ j z, u uUj∆ρ1 uUj∆ρ2

ξ j y, ρmξ j x, v ρm Uj ξ+j y, v( ) ρmξ j z, v vUj∆ρ1 vUj∆ρ2

ξ j z, ρmξ j x, w ρmξ j y, w ρm Uj ξ+j z, w( ) wUj∆ρ1 wUj∆ρ2

0 ξ j x, α l ξ j y, α l ξ j z, α l Uj 0

0 ξ j x, αng ξ j y, αng ξ j z, αng 0 Uj

=

Γ 1–

ρm 0 0 0 0 0

α l∆ρ1 αng∆ρ2+( )u

ρm-------------------------------------------------

1ρm------- 0 0

u– ∆ρ1

ρm------------------

u– ∆ρ2

ρm------------------

α l∆ρ1 αng∆ρ2+( )v

ρm------------------------------------------------- 0

1ρm------- 0

v– ∆ρ1

ρm------------------

v– ∆ρ2

ρm------------------

α l∆ρ1 αng∆ρ2+( )w

ρm--------------------------------------------------- 0 0

1ρm-------

w– ∆ρ1

ρm-------------------

w– ∆ρ2

ρm-------------------

α l– 0 0 0 1 0

αng– 0 0 0 0 1

=

Aj˜ Γ 1–

Aj( )

0 β2ρmξj x, β2ρmξ

j y, β2ρmξj z, 0 0

ξ j x,ρm--------- Uj ξ+

j x, u( ) ξ j y, u ξ j z, u 0 0

ξ j y,ρm--------- ξ j x, v Uj ξ+

j y, v( ) ξ j z, v 0 0

ξ j z,ρm--------- ξ j x, w ξ j y, w Uj ξ+

j z, w( ) 0 0

0 0 0 0 Uj 0

0 0 0 0 0 Uj

= =

Aj1φ

Q1φ

∂

∂Ej1φ

≡

0 β2ξ j x, β2ξ j y, β2ξ j z,

ξ j x, Uj ξ+j x, u( ) ξ j y, u ξ j z, u

ξ j y, ξ j x, v Uj ξ+j y, v( ) ξ j z, v

ξ j z, ξ j x, w ξ j y, w Uj ξ+j z, w( )

=

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