Algo for Modelling (Pre)2005

8/2/2019 Algo for Modelling (Pre)2005

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An algorithm for modeling the interaction of a flexible rod with a

two-dimensional high-speed flow

D. Tam1, R. Radovitzky1, and R. Samtaney2

1 Department of Aeronautics and Astronautics, Massachusetts Institue of Technology, Cambridge, MA,

U.S.A.

2 Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ, U.S.A.

SUMMARY

We present an algorithm for modeling coupled dynamic interactions between very thin flexible

structures immersed in a high-speed flow. The modeling approach is based on combining an Eulerian

finite volume formulation for the fluid flow and a Lagrangian large-deformation formulation for the

dynamic response of the structure. The coupling between the fluid and the solid response is achieved via

an approach based on extrapolation and velocity reconstruction inspired in the Ghost Fluid Method.

The algorithm presented does not assume the existence of a region exterior to the fluid domain

as it was previously proposed and, thus, enables the consideration of very thin open boundaries and

structures where the flow may be relevant on both sides of the interface. We demonstrate the accuracy

of the method and its ability to describe disparate flow conditions across a fixed thin rigid interface

without pollution of the flow field accross the solid interface by comparing with analytical solutions

of compressible flows. We also demonstrate the versatility and robustness of the method in a complex

Correspondence to: Department of Aeronautics and Astronautics, Massachusetts Institue of Technology,

Cambridge, MA, 02139, U.S.A.


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fluid-structure interaction problem corresponding to the transient supersonic flow past a transverse,

highly flexible structure.

Copyright c 2004 John Wiley & Sons, Ltd.

key words: fluid-solid interaction, compressible flows, flexible structures

1. Introduction

Current and future interplanetary exploration missions demand the availability of numerical

tools for the design of light structures such as gossamer spacecraft and parachutes, [ 1, 2, 3]. In

many situations of interest, an adequate description of the continuum fields in both the fluid

flow and the solid structure dynamic deformations, as well as of their coupled interactions, is

necessary. In this work we propose a computational strategy for modeling the coupled response

of a thin structure immersed in a supersonic flow.

In general, the dynamic deformation of solid structures is most adequately described in a

Lagrangian framework, especially in the case of large deformations. The main advantage of the

Lagrangian approach lies in its natural ability to track the evolution of properties at material

points in materials with history, as well as in the treatment of boundary conditions at material

surfaces such as free boundaries or fluid-solid interfaces. In contrast to Eulerian approaches,

boundary conditions are enforced at material surfaces ab initio and therefore require no special

attention. In this work, we propose a Lagrangian formulation for describing the large dynamic

deformations of two-dimensional thin structures (rods) having both bending and membranal

stiffness.

By contrast, Lagrangian formulations are inadequate in the case of high-speed flows or

Copyright c 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 00:00

Prepared using nmeauth.cls


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flows involving significant vorticity due to the unavoidable mesh distortion incurred during

deformation which reduces the stable time step and the overall accuracy of the simulation

and, eventually, breaks the numerical method. This problem can be partially remedied by the

use of remeshing [4]. However, remeshing increases the complexity of the algorithm and of its

implementation and suffers from robustness problems in the three-dimensional case. Eulerian

approaches, in which the field equations are formulated in terms of spatial variables and fixed

or adaptivealbeit not distortingmeshes, are more adequate for most fluid flows. We concern

ourselves with flows where the viscous time scales far exceed the convection time scales, i.e.

we model the fluid flow with the compressible Euler equations. In this work, the supersonic,

unsteady flow conditions are modeled by recourse to a finite volume formulation of the Euler

equations of compressible flow following Samtaney et al [5, 6].

A number of different strategies for coupling fixed-grid Eulerian fluid dynamics formulations

with Lagrangian solid mechanics formulations have been proposed. For incompressible viscous

flows, the immersed boundary method of Peskin and McQueen [7] has received significant

attention, especially owing to its success in modeling the complex conditions of blood flow in

the heart. A recent review of the method may be found in Peskin [ 8]. Several extensions of

this method have been recently proposed by Liu [9].

Our work is concerned with problems involving high-speed compressible flows. For this type

of problems, the so-called Cartesian boundary method [10, 11] and the Embedded boundary

approach of Colella et al [12] have recently gained significant popularity. In these approaches,

the computational domain is discretized by rectangular finite volume cells and the geometry

is represented by intersections with the underlying Cartesian grid. This leads to cut-cells

in those grid locations where the boundary intersects the grid. A detailed description of this




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approach, along with some issues related to the unavoidable appearance of small cells, is given

by Colella [12] and references therein. An alternative approach that explicitly avoids these

issues from the outset by replacing the imposition of boundary conditions with an approach

based on field extrapolation into exterior ghost cells has been proposed [13, 14, 15, 16, 17]. This

class of methods is inspired in the Ghost Fluid Method of Fedkiw et al [ 18]. The convergence

properties of this Eulerian-Lagrangian coupling approach have been carefully studied by

Arienti et al [17]. A similar treatment of irregular boundaries in cartesian grid approaches

including second order accurate formulation of boundary conditions has been recently given

by Sussman [19, 20].

The algorithms presented in the references above are adequate for flows interacting with

bulk solids [13, 14] or thin shells [21, 22]. However, in the case of thin shells they are limited

to situations in which the shell is closed and flow takes place only on one side of it. This

restriction is imposed by the assumption that the fluid domain has a well-defined interior and

exterior, which is the basis of the coupling algorithm based on level sets.

In this work, we extend this approach to the case of thin structures that at the same time are

open and in which the flow on both sides of the structure may be relevant. These situations

arise in important applications such as the deployment of parachutes used as decelaration

devices during planet entry in space exploration missions, [1, 2, 3], The extended approach

retains the basic concepts of the original algorithm, but allows an unbiased consideration of the

flow conditions on both sides of the immersed structure as well as an adequate treatment of the

boundary conditions on both sides of the boundary. Among the advantages of the approach one

finds its simplicity, robustness and ease of implementation especially considering the minimal

modifications required in each solver. Another advantage of this class of fluid-solid coupling




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methods is their suitability for parallel implementation. In [14, 22], the three-dimensional

parallel implementation of this class of fluid-solid coupling algorithms was demonstrated,

including scalability properties on up to 1856 processors. For simplicity, we restrict our

attention to the two dimensional problem.

In the following sections we first present the formulation and numerical approach for

describing large dynamic deformations of a thin rod structure. This is followed by a review of

the numerical method adopted for the fluid. Subsequently, we describe the fluid-solid coupling

algorithm and the proposed extension to thin open immersed structures. The last section of the

paper is devoted to establishing the feasibility and properties of the method. We first present

verification simulations and a convergence study corresponding to the supersonic flow past a

very thin flat rigid boundary at different angles of attack. These simulations demonstrate the

ability and accuracy of the proposed approach to describe the flow on both sides of a very thin

structure. We finally present a fully coupled simulation of a supersonic flow initially normal to

a flexible structure which demonstrates the versatility and robustness of the overall method in

simulating complex fluid-structure interaction problems.

2. Large-displacement rod dynamics model

In this section, we briefly summarize the model adopted for describing the dynamics of slender

rods. For more general beam or slender rod models, the vast literature on the subject, which

traces its origins to the work of Euler and Bernoulli [23], may be consulted, see for example

[24, 25, 26] and references therein. A representative rod element is shown in Figure 1. The rod

element is allowed to undergo a motion consisting of a finite rotation, a finite uniform stretch

and a small bending distortion. With the conventions shown in this schematic, the deformation




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Figure 1. Schematic describing the conventions and kinematics of the rod model proposed

mapping of the rod element follows as:

x1 l

LX1 (X1)X2 (1)

x2 X2 (2)

and the stretch of the longitudinal fibers of the rod element follows as

=x1X1

l

L (X1)X2 (3)

The small bending distortion measured from the rotated and stretched configuration (axes

x1,x2) is assumed to follow the classic Euler-Bernoulli hypothesis:

w

x1(4)




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A strain energy density per unit undeformed area of the rod of the form:

W() = E(1 + log) (5)

is assumed, where E is the Youngs modulus. This energy density gives a linear relation between

the nominal stress and the logarithmic or true strain.

The strain energy of the rod is obtained by integrating equation (5) over the volume of the

undeformed rod after inserting the assumed rod kinematics, equation (3), with the result

U EA [L l + l log(l/L)] +EI

2 l

0

2

(x1)dx1 (6)

where A and I are the area and moment of inertia with respect to the axis normal to the

bending plane of the undeformed cross section of the rod. In evaluating the integral along

the undeformed axis of the rod, a change of variables to the deformed configuration has been

conveniently taken advantage of.

Linear momentum balance is enforced weakly by recourse to Hamiltons principle for

continuous media, i.e. by finding the paths between two arbitrary times t1 and t2 for which

the action integral is stationary:

t1t0

Ldt = 0 (7)

where L is the associated Lagrangian defined as L = K , K is the kinetic energy of the

system, = U+ ext is the potential energy of the system and ext is the potential of the

external forces. Equation (7) must hold for any variationally admissible virtual displacement.

A complete derivation of Hamiltons principle for continuous systems may be found in standard

references, see for example [27].

We take Hamiltons Principle, Equation (7), as the basis for finite element discretization.

The explicit derivation of the first variation of the action integral, Equation ( 7), leading to




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the explicit expression of the continuous variational form is omitted for conciseness, since it

is not necessary for the numerical formulation. This derivation for different linearized beam

models may be found in standard textbooks, see for example Reddy [ 28]. We use hermitian

cubic interpolation to represent w and its derivatives as a function ofx1. This automatically

satisfies the requirement ofC1 interelement continuity ofw. In the finite element formulation

proposed, the unknowns represent the physical displacements and rotation at extremity (node)

a = 1, 2 of the rod element. Explicit expressions of the strain and kinetic energy of the rod

element in terms of the degrees of freedom are derived in [29]. Upon spatial discretization, the

stationarity condition (7) leads to the semi-discrete system of nonlinear ordinary differential

equations:

Mhxh +Finth (xh) = F

exth (t) (8)

In these expressions, Mh is the mass matrix, xh the array of nodal accelerations and

Finth (xh) =U

xh(9)

Fexth = extxh(10)

are the arrays of internal forces and time-varying external forces, respectively. In the fluid

structure interaction problems of interest in this work, the array Fexth in equation (10)

represents the external nodal forces equivalent to the traction boundary conditions imposed

by the flow on the structure. The computation of these forces is discussed in section 4.

The equations of motion (8) are integrated in time using Newmarks family of algorithms:

xn+1 = xn + txn + t212 xn + xn+1

xn+1 = xn + t

(1 )xn + xn+1

Mxn+1 +Fint(xn+1) = F

ext(tn+1)

(11)




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where and are the Newmark algorithm parameters. For = 0, a conventional implicit

predictor-corrector algorithm [30] is adopted to solve the system of equations (11), leading to

the incremental nonlinear algebraic system:

MU

t2+ Fint(xn+1 +U) = Fextn+1 (12)

where U = t2xn+1. A consistent linearization of this nonlinear algebraic equation about

the predictor configuration leads to the computation of the tangent stiffness matrix:

K =Fintx

bxn+1

(13)

which enables a quadratic convergence of the Newton Raphson algorithm used to obtain

dynamic equilibrium at t = tn+1. Explicit expressions for the mass matrix, array of internal

forces and consistent tangent moduli in our slender rod model are derived in [ 29].

3. Eulerian compressible fluid solver

In this section we summarize the formulation of the fluid solver. Further details may be found

in the original references. [31, 5, 6]. The flow is modeled as compressibe and inviscid, leading

to the governing Euler equations of compressible flow. These equations may be expressed in

the following strong conservative form:

U,t + F,x(U) + G,y(U) = 0 (14)

where

U = ,u,v,ET

F(U) =u, u2 +p, uv, (E+

p

)uT

G(U) =v,uv,v2 +p, (E+

p

)vT




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where is the density, u and v are the Cartesian components of the velocity vector, p is the

pressure, E is the specific total energy, U is the vector of conservative variables, F(U) and

G(U) are the components of the flux vector. An additional equation of state closes the system

of equations. In this work, the equation of state of perfect gases:

p = ( 1)e (15)

is adopted, where is the specific heat ratio and e is the specific internal energy with

E= e + 12u2.

A finite volume formulation is adopted as the numerical approximation of these equations.

The discretized equations may be written as:

Ui,jt

=Fi 1

2,j Fi+ 1

2,j

h+Gi,j1

2 Gi,j+ 1

2

h(16)

where Ui,j is an average value ofU over the (i, j)th cell, and Fi 1

2,j , Fi+ 1

2,j , Gi,j 1

2and Gi,j+ 1

2

are the fluxes at the cell interfaces. This formulation is numerically conservative and thus, the

variation ofU over one cell of the mesh is equal to the inward flux. For stability reasons, flows

with strong compressibility effects leading to the formations of shocks are best modeled by a

conservative formulation [32, 33].

The fluxes at the cell interfaces may be calculated either by the Equilibrium Flux Method

(EFM) (a kinetic flux vector splitting scheme) [31], or the Godunov [34] or Roe method [35] (a

flux difference splitting scheme). These discretization schemes are first order in space and can

be taken as a starting point for the formulation of higher order schemes. In our case, second

order accuracy is achieved via linear reconstruction with Van Leer type slope limiting applied

to projections in characteristic state space. This method is often referred to as the MUSCL

approach (Monotone Upstream-centered Schemes for Conservation Laws) [36, 33, 37, 38].

Equations (16) are integrated explicitly in time using the second-order Runge-Kutta algorithm:




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First step:

Un+ 1

2

i,j = Uni,j +

t2h

[Fi1/2,j(Un) Fi+1/2,j(U

n)

+ Gi,j1/2(Un) Gi,j+1/2(U

n)] (17)

Second step:

Un+1i,j = Uni,j +

t

h[Fi1/2,j(U

n+ 12 ) Fi+1/2,j(U

n+ 12 )

+ Gi,j1/2(Un+ 1

2 ) Gi,j+1/2(Un+ 1

2 )] (18)

The fluid solver imposes restrictions on the stable time step given by the Courant-Friedrichs-

Levy (CFL) stability condition [32, 38, 33]. The resulting fluid model is second order in time

and space. Details of the parallel implementation of this algorithm, including adaptive mesh

refinement capability may be found in Ref. [39].

4. Eulerian-Lagrangian coupling algorithm

Our objective in this work is to develop a fluid-structure coupling algorithm with the ability

to describe situations in which the details of the flow on both sides of a very thin structure

are of equal importance. Such situations arise, for example, when the structure is a manifold

with boundary (e.g. an open shell) or, if the manifold is closed, when there is fluid in the

shell interior as well as in the exterior. This objective is relatively easy to achieve with an

unstructured mesh, finite element, Arbitrary Lagrangian Eulerian (ALE) formulation [ 40] or

with other alternative mesh moving techniques[41] in the case that the thin structure is fixed,

i.e., in the case of a thin rigid boundary, or when the structure deformations are relatively

small. However, when the deformations are large, these methods usually suffer from stability

[42] and excessive mesh distortion problems [43].




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In previous work, [16], we have focused on flow geometries with closed boundaries and a well

defined exterior region. In this work we extend this algorithm in a way that this restriction

can be eliminated. For simplicity, we restrict our attention to the two-dimensional case.

The Eulerian fluid solver and the Lagrangian solid solver are weakly coupled by applying

appropriate boundary conditions at the fluid-solid interface at the beginning of each time step.

Other possible implicit and staggering schemes in coupled systems have been proposed and

studied in detail in [44, 45]. In the case of inviscid flows considered, these boundary conditions

correspond to continuity of the normal component of the velocity field:

[ v n ] = 0, on the fluid-solid boundary (19)

where [ . ] represents field jumps, and continuity of the normal component of the traction across

the fluid-solid interface:

[ t n] = [ ijninj ] = [ n ] = 0, on the fluid-solid boundary (20)

which enforce conservation of mass and linear momentum, respectively. For simplicity, heat

transfer across the fluid-solid interface is neglected.

The formulation of the algorithmic steps to enforce these conditions is described in the

following. In the model proposed, we consider that the only aerodynamic force acting on the

structure is due to the fluid pressure. Equation (20) is enforced weakly by directly applying

the pressure exerted by the fluid on the structure at time tn as traction boundary conditions

for time step tn+1 in a variationally consistent manner. This results in the following expression

for the external force array Fexth in Equation (10):

Fextia

n+1=

So2

pnNanids (21)




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where i is the degree of freedom, Na the shape function of node a and pn is the local value of

the pressure, which is interpolated bilinearly from the computed flow field at time tn.

On the fluid, mass, momentum and energy conservation at the boundary are enforced via

extrapolation and a flow reconstruction step. First velocity, pressure and density cell averages

from the physical fluid domain are extrapolated to a narrow band of ghost cells across the

boundary. This extrapolation is done by advection in pseudo-time :

q

+ n q = 0 (22)

where q = (,p,u,v) is the array of extrapolated quantities and n is the normal to the interface

computed from the level set function. We employ simple upwinding along the normal n to

march forward in in the extrapolation step above. When the steady state (q/ = 0)

is reached, the non-physical extrapolated velocity field in the ghost cells is reconstructed

according to the expression:

vF = (2vS vF) n

n + (vF t)t (23)

where vF is the fluid velocity extrapolated from the active fluid cells and vS is the velocity of

the solid interface. The resulting ghost velocities correspond to a reflection of the normal

component of the fluid velocity relative to the moving boundary, whereas the tangential

component is left unchanged. The fluxes thus computed from real and ghost values at the

boundary, see section 3, in effect, enforce Equation (19). In the case of flows with a well-

defined exterior domain, ghost and real flow values can be supported on the same grid. By

contrast, in the case of open boundaries a separate data structure is required to store the

extrapolated ghost values, as ghost and real fluid regions overlap. To this end, two arrays are

used: one storing the real values of the conserved variables on the whole domain on which




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the main computation takes place and another one storing the values of the extrapolated

variables in the ghost fluid cells next to the interface, as shown in Figure 2, which is used for

the application of boundary conditions in the fluid. It bears emphasis that significant storage

savings may be achieved by adopting specialized sparse arrays [46] to store ghost values, which

effectively reduce the dimensionality of the required storage size. It should be noted that the

use of an additional array to store and access ghost data imposes some additionalalbeit

straightforwardmodifications to the fluid solver, as compared with conventional ghost fluid

approaches one of whose attractive features is the minor solver modifications required.

Figure 2. Real fluid and ghost fluid arrays

The location of the boundary as well as the boundary normal required to apply

these boundary conditions, as described above, need to be computed efficiently to avoid

computational bottlenecks. To this end, the level set function (x) which gives the minimum

distance to the fluid-solid interface at each grid point of the fluid domain is used. The normal

n to the interface is also interpolated directly from the level set function:

n =

(24)




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The boundary is located where (x) = 0. The computation of the distance function at

each time step is accomplished with an optimal algorithm developed by Mauch [47], whose

complexity is O(m + n), where m is the number of elements in the shell mesh and n is the

number of grid points in the subset of the fluid grid where the level set is required. In the

case of flows with a well-defined exterior, as assumed in [13, 21], a sign is assigned to the level

set function. (x) is taken as negative (positive) in the interior (exterior) of the physical fluid

domain. This facilitates the immediate identification of real and ghost fluid cells. The main

limitation of this approach is that it precludes the possibility of flows coexisting on both sides

of a thin boundary.

In the case of open boundaries, by contrast, interior and exterior cannot be defined. However,

a key observation is that the boundary remains an orientable manifold, i.e., it has two

unequivocally identifiablealbeit arbitrarily chosenpositive and negative sides which can

be conveniently assigned to the adjacent fluid domain. Both in the two and three dimensional

case, the manifold may be endowed with an orientation by a suitable choice of a specific

parametrization, from which the surface tangent vector(s) and a positive normal may be

defined. Based on this observation, the side of the boundary is identified on the fluid grid

by endowing the distance function i,j with a sign next to the interface. This sign is obtained

from thedimension independentformula:

sign(i,j) =

+1 if di,j n 0

1 otherwise

(25)

where di,j is the distance vector from grid point i, j to the boundary and n is the local normal

to the boundary. Thus, cells lying in the half space pointed to by the surface normal are

assigned a positive sign and all other a negative one, as shown in Figure 3. Therefore, this




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pseudo-signed level set function offers a straightforward manner to determine if two grid

points lie on the same side of the boundary or not.

Figure 3. Pseudo sign defined by the orientation of the interface

Once the pseudo-signed distance function is computed, values from each side of fluid-solid

interface are symmetrically extrapolated to the corresponding other side and stored on the

auxiliary ghost array. The velocity fields are then reconstructed in the entire ghost region to

impose non-penetration boundary condition, as described earlier.

After the explicit application of boundary conditions by extrapolation in the fluid, Equations

(22) and (23), and in the structure, Equation (21), at the beginning of the time step, time

integration proceeds independently in each solver as described in sections 2 and 3. The

computation of the fluxes in the fluid, Equation (16), for cells next to the fluid-solid boundary,




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i.e. where i,j changes sign, needs to be done using both real and ghost values. The latter are

obtained from the auxiliary array described above. Apart from this consideration, the solution

is computed on the whole fluid grid, without the need of any additional special treatment of

the boundary.

One of the main advantages of this explicit coupling approach is its suitability for parallel

implementation. The scalability properties of this coupling scheme on upwards of 1800

processors has been reported in [14].

It is important to remark that in order to avoid singularities stemming from the violation of

the Sobolev cone condition at the ends of vanishing-thickness boundaries, structures embedded

in the flow are endowed with a finite thickness, as real structures are expected to have.

The staggering method adopted remains stable if the time step is chosen as:

t min(tCFL, tb) (26)

where tCFL is the stable time step for the fluid and:

tb =min (x,y)

vS(27)

which prevents the solid boundary from crossing more than one fluid cell per time step. As

explained in section 2, the structure does not impose additional time step restrictions related to

stability, as the integration is done implicitly. The stability of different weak coupling schemes

in coupled systems has been studied in detail in [45].

The resulting fluid-structure coupling algorithm is summarized in Algorithm 1.




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Algorithm 1. Fluid-Solid Coupling

1. - Apply boundary conditions using solution from previous time-step:

(a) For the fluid solver:

i. Compute pseudo-signed distance function from updated location of solid boundary

ii. Extrapolate flow field values to the ghost region using Equation (22) and store

extrapolated fields in auxiliary ghost array.

iii. Reconstruct velocity field in the ghost region using Equation (23)

(b) For the solid solver:

i. Interpolate pressure field at the interface directly from fluid grid

ii. Apply pressure as external loading on the structure using Equation (21)

2. - Compute stable time step:

(a) Compute stable time step for the fluid solver tf according to CFL condition

(b) Compute time step restrictions at the interface tb, Equation (27)

(c) Adopt stable time step as: t = min(tf,tb), Equation (26)

3. Integrate solution in time:

(a) Integrate in time the fluid solution Second order Runge-Kutta integration, Equations (17),

(18). Next to the boundary, access ghost values from auxiliary array to compute fluxes in

Equation (16).

(b) Integrate in time the solid solution using Newmarks algorithm, Equations (11).

(c) Increment time-step in both fluid and solid: t tn + t

(d) Update location of the interface, i.e., the reference configuration of the solid.




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Figure 4. Schematic solution of the supersonic flow past a thin rigid plate

5. Numerical examples

5.1. Supersonic flow past a fixed thin plate

The first example is intended to assess the ability of the extended ghost fluid method to

describe supersonic flows past open thin boundaries. To this end, the flow past a fixed thin

plate immersed in a high-speed flow at different angles of attack , see schematic in Figure 4,

is computed using the numerical method described in the foregoing. This problem is amenable

to analytical treatment [48, 49] and, therefore, provides a convenient means of assessing the

accuracy and convergence properties of the extended ghost fluid method. The plate profile

induces a weak shock attached to its leading edge on the side where the cross section of the

flow decreases and an expansion wave on the opposite side. The pressure behind the shock and




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the expansion wave is uniform and can be computed analytically as a function of using the

Rankine-Hugoniot relations and, respectively, the Prandtl-Meyer function [49].

Figure 5. Computed solution of the supersonic flow past a thin rigid plate. In the case shown, the Mach

number is M = 1.8 and the angle of attack is = 15 degrees. Contours show pressure normalized

with upstream value

The Mach number adopted in these calculations is M = 1.8 and the initial properties of

the gas are: p = 1.0atm, = 1.293 kgm3 , and = 1.4. The fluid domain is discretized with a

400 680-cell fluid grid. In order to avoid singularities in the solution, the rigid boundary is




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(a) y=-0.245 (b) y=0.095

(c) y=0.275

Figure 6. Comparison of numerical and analytical horizontal pressure profiles for different values of

the vertical coordinate y. Analytical values are shown in thick gray lines. Numerical results are shown

in thin black lines and + symbols. Values shown are normalized with upstream pressure.




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given a finite thickness, as real structures are expected to have. The numerically-computed

flow field for the case of = 15 degrees is shown in Figure 5. The contours indicate the value

of the pressure normalized with its upstream value. Figures 6 (a)-(c) show comparisons of the

normalized horizontal pressure profiles against the analytical solution for different values of

the vertical coordinate. Figure 6 (b) corresponds to a cross section through the center of the

plate (y = 0.095) and shows that the pressure values behind the shock and in the expansion

region behind the plate are accurately computed up to the interface. It bears emphasis that

there is no pollution of the solution from one side of the boundary to the other, which usually

constitutes a challenge in methods based on extrapolation. Figures 6-(a) and (c) correspond

to cross sections one grid cell away from the bottom (y = 0.245) and top (y = 0.275) tips of

the rigid boundary, respectively. As it can be seen in these figures, the quality of the numerical

solution is very good both far and near the rod ends.

Several simulations were conducted for angles of attack ranging from = 5 to 18 degrees.

Above 18 degrees and for an upstream Mach number of M = 1.8 the shock at the leading

edge of the profile detaches. Figure 7 shows the variation of the pressure behind the shock

as a function of the angle of attack. As expected, the pressure behind the shock increases

with . The corresponding dependence of the pressure in the expansion region behind the

plate on the angle of attack is shown in Figure 8. As is increased, the pressure behind the

expansion wave decreases, as expected. The numerically computed values are plotted on the

same Figures 7 and 8. In both cases, a very good agreement between the exact and numerical

results is obtained.

A critical aspect of fluid-structure interaction models is the ability to compute the

aerodynamic loads on the structure with sufficient accuracy, as these loads determine the




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Figure 7. Comparison of analytical and numerical values of the pressure behind the shock vs. angle of

attack

structural response. In the following, we study the convergence of the pressure load on the

structure in the supersonic flow past a fixed plate problem. Other aspects of the convergence

of the coupling approach based on the ghost fluid method were previously reported by Arienti

et al [17].

A series of simulations corresponding to the case of upstream Mach number M = 1.8 and

angle of attack = 18 degrees is conducted for grid resolutions starting at 85 50. In each

subsequent simulation, the resolution is increased by a factor of 2 in each direction. The

finest grid resolution is 680 400. The exact nondimensional value of the aerodynamic lift

(= 1.5759) for M = 1.8 and = 18 degrees is readily obtained from the difference between

the analytical pressure values in the windward ( p1p = 2.5515) and leeward (p2p

= 0.3453)

sides, where p =1

= 0.7143, and the normalized length of the flat plate (=1

cos=1.051).




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Figure 8. Comparison of analytical and numerical values of the pressure in the expansion region behind

the plate vs. angle of attack

The error in the computed lift, defined as the absolute value of the difference between the

numerical and the analytical values normalized by the analytical value, as well as the rate of

convergence, is reported in Table 5.1. The first order convergence rate obtained is attributed

Grid resolution Computed value Error Error in % Convergence Rate

85 x 50 1.4864950 0.08946629 5.6772 0.92

170x100 1.5285353 0.04734034 3.0041 1.29

340x200 1.5564916 0.01938427 1.2300 1.31

680x400 1.5680707 0.00780513 0.4952 -

Table I. Convergence analysis of the aerodynamic lift on the solid structure.




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to the first order description of the geometry, and the first order interpolation of the fluid

pressures on the boundary. Second order schemes for similar treatment of irregular boundaries

in cartesian grids have recently been proposed [19, 20], but are more expensive in terms of

CPU and memory. It can be concluded from these results that the algorithm proposed applies

boundary conditions on both sides of the thin profile in a consistent manner and results in

convergent pressure distributions caused by the flow on the solid boundary. It can therefore be

expected that this, in turn, will result in correct traction boundary conditions on the structure

in coupled simulations.

5.2. Supersonic flow past a highly-flexible structure

In this section, we demonstrate the versatility of the overall computational methodology in

describing complex fluid solid interactions. The simulation corresponds to a supersonic flow

transverse to an initially-flat structure made of an elastic fabric with a Youngs Modulus

E = 6.0 109Pa and mass density = 1000.0 kgm3 . The length of the structure is 1.m and is

discretized with 50 elements as described in section 2, its thickness is 3.0 103m, its cross

sectional area A = 1.0 103m2 and its moment of inertia I = 2.25 109m4. A schematic of

the simulation set up is provided in Figure 9.

The initial properties of the gas are: upstream pressure p = 1.0atm, mass density

= 1.293kgm3 , and = 1.4. The flows Mach number is M = 2.0. The size of the

computational fluid domain is 5.20m9.60m and the grid resolution adopted in this calculation

is 260 480 fluid cells.

At first, the structure is held fixed and the steady-state flow around the flat structure is

computed. A strong shock develops upstream of the structure. The highly flexible structure is




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Figure 9. Schematic of simulation of supersonic flow past a flexible structure tranverse to the flow

then released, except at its tips which are restrained horizontally, and starts inflating under the

pressure of the flow, inducing complex interactions between the flow and the thin structure,

see Figure 10.

Figure 10 (a) shows the initial steady-state flow past the fixed structure. A strong shock

forms in front of the structure which causes a very high (low) pressure pp 5.0 (p

p 1.0) on

the windward (leeward) side of the structure. When released, the structure starts to accelerate

rigidly except at the extremities where the horizontal supports create a flexural wave which

propagates towards the center, Figure 10 (b). The forward motion of the structure releases

an expansion wave in the flow, which travels upstream towards the strong shock lowering the

upstream pressure, Figures 10 (b)-(c). The flexural waves in the structure converge at its center

at t = 2.80ms. As the structure deforms until it reaches a maximum deflection, the flow in




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(a) Step 0, t = 0.00ms (b) Step 150, t = 1.20ms

(c) Step 350, t = 2.80ms (d) Step 450, t = 3.60ms

Figure 10. Simulation of supersonic flow past a flexible structure




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(a) Step 650, t = 5.20ms (b) Step 850, t = 6.80ms

(c) Step 1050, t = 8.40ms (d) Step 3050, t = 24.40ms

Figure 11. Simulation of supersonic flow past a flexible structure (continued)




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the windward side stagnates and the pressure increases again, as shown in Figure 10 (c). At

t = 3.60ms, Figure 10 (d), the expansion wave propagating upstream reaches the shock front,

affecting its shape and causing it to move downstream, thus following with some delay the

initial forward motion of the rod. In the same Figure, it can be observed that, at this point,

the elastic strain energy stored in the structure starts causing it to recoil. As the elastic energy

is released, Figure 11 (a), the structure pushes the upstream flow generating a compression

wave. This emphasizes the ability of the method to describe aspects of the flow caused by the

dynamic deformation of the structure. Figure 11 (b) clearly shows the compression wave shed

by the structure propagating upstream towards the shock front. At t = 8.40ms, Figure 11

(c), the compression wave reaches the shock front, affecting its shape and causing it to move

backwards, following again with some delay the motion of the structure.

This process in which the structure first inflates storing elastic energy and then recoils

restoring the energy to the flow continues ostensibly unchanged, as no physical dissipation

mechanisms are taken into account in the model. In order to reach a steady-state inflated

configuration, a small numerical dissipation is added to the structure by a convenient choice

of the parameters of Newmarks algorithm, equation (11), as = 0.5 and = 1. Figure 11 (d)

shows the steady-state configuration reached at time t = 24.40ms after the structure was first

released.

This example illustrates the robustness of the coupling algorithm in describing complex

fluid-solid interactions in which the structure undergoes large nonlinear elastic deformations

which, in turn, affect the flow in a non-trivial manner. The ability of the model to describe

these interactions across a very thin structure without cross pollution of the flow across the

interface is particularly noteworthy.




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Summary and conclusions

We have proposed a computational strategy for the coupling of high-speed flows interacting

with the large, dynamic deformations of very thin open structures. Among the necessary

components of the overall computational framework, a formulation is presented for the large

dynamic deformations of thin rod structures including the bending and membrane response.

The coupling algorithm constitutes an extension of the ghost fluid method without the

restrictions of thick solid structures and closed boundaries in which a well-defined exterior

to the fluid domain exists. The new algorithm was verified against the analytical solution of

the supersonic flow past a flat rigid plate at different angles of attack. The numerical solution

is shown to converge to the analytical solution on both the shocked and rarefied regions on the

windward and leeward side of the plate without pollution of the solution across the infinitely

thin boundary. A convergence analysis of the lift load on the structure confirms the theoretical

first order accuracy of the coupling approach. As an example of a coupled application, a

simulation of the transient supersonic flow normal to a highly-flexible structure is presented.

The simulation shows that a complex pattern of highly unsteady coupled interactions are set

in motion between the flow and the structure, leading to the large oscillations of the structure

until a steady-state is reached in its final inflated configuration.

ACKNOWLEDGEMENTS

The support of the U.S. Department of Energy through the ASC Center for the Simulation of the

Dynamic Response of Materials (DOE W-7405-ENG-48, B523297) is gratefully acknowledged.

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