numerical simulation of a multi-store separation phenomenon: a fictitious domain approach

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Comput. Methods Appl. Mech. Engrg. 195 (2006) 5566–5581

Numerical simulation of a multi-store separation phenomenon:A fictitious domain approach

Roland Glowinski a,b, Tsorng-Whay Pan a,*, Jacques Periaux c,d

a Department of Mathematics, University of Houston, Houston, TX 77204, USAb Universite P. et M. Curie, Paris, France

c CIMNE, University Polytechnic of Cataluna, 08034 Barcelona, Spaind Pole Scientifique Dassault Aviation/UPMC, 92552 Saint Cloud, France

Received 20 July 2005; received in revised form 9 September 2005; accepted 20 September 2005

To the memory of J.H. Argyris (1913–2004)

Abstract

In this article we discuss the numerical simulation of multi-store separation phenomena taking place in incompressible Newtonianviscous fluids. We assume that these phenomena are modeled by the Navier–Stokes equations coupled to the Euler–Newton equationsdescribing solid rigid body motions. The numerical methodology relies on the combination of a finite element approximation with time-discretization by operator-splitting and a Lagrange multiplier based fictitious domain method allowing the flow calculations to take placein a fixed simple shape space region. The results of numerical experiments are presented; they concern the separation and free fall ofelongated ellipsoids under an airfoil shaped cylinder.� 2005 Elsevier B.V. All rights reserved.

Keywords: Store separation; Fictitious domain method; Distributed Lagrange multipliers; Operator-splitting methods; Finite element methods

1. Introduction

The two senior authors of this article (R.G. and J.P.) had the privilege of being participants to the conference held at theUniversity of Stuttgart in 1978, and honoring J.H. Argyris and W. Prager on the occasion of their 65th and 75th birthdays,respectively. Glowinski, Periaux and collaborators contributed to the proceedings of this most remarkable event by an arti-cle [1] which appeared also in Computer Methods in Applied Mechanics and Engineering, a journal founded by Argyris andPrager. The present article has a much more sad motivation, being written in memory of Argyris, who left the scientificcommunity he contributed so much in 2004. Argyris is better known for his many contributions to the application of finiteelement methods to the solution of problems from Structural Mechanics; however he has also many contributions in Com-putational Fluid Mechanics, including [2], an article which had a profound influence on the authors of the present article.Indeed, his belief that finite element methods can significantly contribute to Fluid Mechanics has been verified later, inmany situations, particularly in Aeronautics.

The present article is a tribute to Argyris vision; it shows that finite element methods, coupled to other computationaltechniques, can address the solution of complex problems in Fluid Mechanics, involving time varying geometries. Theproblem discussed here concerns the numerical simulation of three-dimensional multi-store separation phenomena. To be

0045-7825/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2005.09.018

* Corresponding author. Tel.: +1 713 743 3448; fax: +1 713 743 3505.E-mail address: [email protected] (T.-W. Pan).

mailto:[email protected]

R. Glowinski et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5566–5581 5567

more precise we are going to simulate the release and free fall under the effect of gravity of elongated ellipsoids, initially, toan airfoil cylinder. This problem has multiple applications (Aeronautics, . . .). The content of this article is as follows:

In Section 2, we provide a mathematical model for incompressible viscous fluid/rigid body interaction, including a dis-tributed Lagrange multiplier based fictitious domain variational formulation. In Section 3, we discuss the time discretiza-tion by operator-splitting, starting from the above fictitious domain formulation, and then the finite elementapproximation of the problem. The results of numerical experiments are presented and discussed in Section 4.

2. A model problem and its fictitious domain formulation

To perform the direct numerical simulation of the interaction between rigid bodies and fluid, we have developed a meth-odology which combines a distributed Lagrange multiplier based fictitious domain (also called domain embedding) methodwith operator splitting methods [6,7,9–11]. This approach (or closely related ones derived from it) has been used by otherinvestigators (e.g., [8,12–14]). We already applied this methodology to simulate a two-dimensional multi-store separationphenomenon [7]. In this article we extend this methodology to simulate the release and free fall under the effect of gravityof elongated ellipsoids attached, initially, to a fixed airfoil-shaped cylinder. We are going first to recall the ideas at the basisof the above methodology by considering the motion of a single elongated three-dimensional rigid body B in a Newtonianviscous incompressible fluid (of density qf and viscosity lf) separating from a fixed airfoil-shaped cylinder A under the effectof gravity. For the situation depicted in Fig. 1, the flow is modeled by the Navier–Stokes equations, namely, (with obviousnotation)

qf

ou

otþ ðu � rÞu

� �� lfDuþrp ¼ qfg in fðx; tÞ j x 2 X n ðBðtÞ [ AÞ; t 2 ð0; T Þg; ð1Þ

r � uðtÞ ¼ 0 in X n ðBðtÞ [ AÞ; 0 < t < T ; ð2Þuð0Þ ¼ u0ðxÞ ðwith r � u0 ¼ 0Þ; ð3Þ

u ¼ g0 on C� ð0; T Þ with

ZC

g0 � ndC ¼ 0

� �; u ¼ 0 on oA; ð4Þ

where C is the union of the top, bottom, left and right boundaries of X as in Fig. 1, g denotes gravity and n is the unitnormal vector pointing outward to the flow region. We assume a no-slip condition on c(=oB) and on oA and that the flowis periodic in the x2 direction. The motion of the rigid body B satisfies the Euler–Newton’s equations, namely

vðx; tÞ ¼ VðtÞ þ x!ðtÞ �GðtÞx��!

; 8x 2 BðtÞ; 8t 2 ð0; T Þ; ð5ÞdG

dt¼ V; ð6Þ

MpdV

dt¼ Mpgþ FH þ Fr; ð7Þ

dðIpxÞdt

¼ TH þGxr��!� Fr ð8Þ

with the resultant and torque of the hydrodynamical forces given by, respectively,

FH ¼ �Z

crndc; TH ¼ �

Zc

Gx�!� rndc ð9Þ

Fig. 1. The flow region with an elongated ellipsoid and a fixed airfoil-shaped cylinder.

5568 R. Glowinski et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5566–5581

with r = lf($u + $ut) � pI. Relations (1)–(9) are completed by the following initial conditions

Gð0Þ ¼ G0; Vð0Þ ¼ V0; xð0Þ ¼ x0; Bð0Þ ¼ B0. ð10ÞAbove, Mp, Ip, G, V and x are the mass, inertia, center of mass, velocity of the center of mass and angular velocity of therigid body B, respectively. In (8) we found preferable to deal with the kinematic angular momentum Ipx making the formu-lation more conservative. In order to avoid the overlapping of the rigid bodies and rigid body-wall penetration which canhappen in the numerical simulation, we have introduced a artificial short-range repulsion force Fr in (7), which becomesactive when the shortest distance between two (convex) rigid bodies or between (convex) rigid body and wall is less thana pre-chosen distance (for more details, see, e.g., [6,7]; see also [15] for another approach) and then a torque in (8) acting onthe point xr where Fr applies on B. For non-convex particles, we can apply similar approach to activate the short-rangerepulsion force Fr.

To solve system (1)–(10) we can use, for example, arbitrary Lagrange–Euler (ALE) methods as in [3–5], or fictitious

domain methods, which allow the flow calculation on a fixed grid, as in [6,7,9–11]. The fictitious domain methods thatwe advocate have some common features with the immersed boundary method of Peskin (see, e.g., Refs. [16–18]) but alsosome significant differences in the sense that we take systematically advantage of distributed Lagrange multipliers to forcethe rigid body motion inside the particle. As with the methods in [16–18], our approach takes advantage of the fact that theflow can be computed on a grid which does not have to vary in time, a substantial simplification indeed.

The principle of fictitious domain methods is simple. It consists of

• Filling the rigid bodies with a fluid having the same density and viscosity as the surrounding one.• Compensating the above step by introducing, in some sense, an anti-body of mass ð�1ÞMp

qf

qsand inertia ð�1ÞIp

qf

qs, taking

into account the fact that any rigid body motion v(x, t) verifies $ Æ v = 0 and D(v) = 0 (qs: rigid body density).• Finally, imposing the rigid body velocity on BðtÞ, namely

vðx; tÞ ¼ VðtÞ þ xðtÞ��!�GðtÞx��!

; 8x 2 BðtÞ; 8t 2 ð0; T Þ; ð11Þvia a Lagrange multiplier k supported by BðtÞ. Vector k forces rigidity in B(t) in the same way that $p forces $ Æ v = 0 forincompressible fluids.

We obtain then an equivalent formulation of (1)–(10) defined on the whole domain X, namelyFor a.e. t > 0, find u(t) 2Wg0

(t), pðtÞ 2 L20ðXÞ, V(t) 2 R3, G(t) 2 R3, x(t) 2 R3, k(t) 2 K(t), kA 2 KA such that

qf

ZX

ou

otþ ðu � rÞu

� �� vdx�

ZX

pr � vdxþ lf

ZXru : rv dx� hk; v� Y� h�GxiKðtÞ � hkA; viKA

þ 1� qf

qs

� �Mp

dV

dt� Yþ dðIpxÞ

dt� h

� �� Fr � Y�Gxr

��!� Fr � h

¼ 1� qf

qs

� �Mpg � Yþ qf

ZX

g � v dx; 8v 2W0; 8Y 2 R3; 8h 2 R3;

8>>>>>>>><>>>>>>>>:

ð12Þ

ZX

qr � uðtÞdx ¼ 0; 8q 2 L2ðXÞ; ð13Þ

hl; uðtÞ � VðtÞ � xðtÞ �GðtÞxiKðtÞ ¼ 0; 8l 2 KðtÞ; ð14Þ

hlA; uðtÞiKA¼ 0; 8lA 2 KA; ð15Þ

dG

dt¼ V; ð16Þ

Vð0Þ ¼ V0; xð0Þ ¼ x0; Gð0Þ ¼ G0; Bð0Þ ¼ B0; ð17Þ

uðx; 0Þ ¼ ~u0ðxÞ ¼u0ðxÞ; 8x 2 X n ðBð0Þ [ AÞ;V0 þ x0 �G0x; 8x 2 Bð0Þ;0; 8x 2 A

8><>: ð18Þ

with the following functional spaces

W ¼ fv j v 2 ðH 1ðXÞÞ3; v is periodic in the x2 directiong;Wg0ðtÞ ¼ fv j v 2W; v ¼ g0ðtÞ on Cg;

W0 ¼ fv j v 2W; v ¼ 0 on Cg;


L20ðXÞ ¼ q j q 2 L2ðXÞ;

ZX

qdx ¼ 0

� �;

KðtÞ ¼ ðH 1ðBðtÞÞÞ3;

KA ¼ fl j l 2 ðH 1ðAÞÞ3; l is periodic in the x2 directiong.In (12), (14) and (15), h�; �iKðtÞ and h�; �iKA

are inner product on K(t) and KA, respectively. Various examples for are given in[7,19, Chapter 8]. The zero velocity field inside A is enforced in (12) and (15) via the Lagrange multiplier kA supported by A.

Remark 1. The second gravity term in the right-hand-side of Eq. (12) can be combined with the pressure. Hence in thefollowing, we will not use this term anymore.

Remark 2. The hydrodynamical forces and the torques from the hydrodynamical forces are not shown in the (12)–(18)since they are built-in implicitly in the formulations. This is the advantage of the distributed Lagrange multiplier basedfictitious domain method in the sense that we do not need to compute the hydrodynamical forces and their torques explic-itly in order to move the rigid bodies.

In (12)–(18), only the center of mass, the translation velocity of the center of mass and the angular velocity of the particleare considered. Knowing these two velocities and the center of mass of the particle, one is able to translate and rotate theparticle in space by tracking two extra points x1 and x2 in each particle, which follow the rigid body motion

dxi

dt¼ VðtÞ þ x!ðtÞ �GðtÞxi

��!; xið0Þ ¼ xi;0; i ¼ 1; 2. ð19Þ

In practice we shall track two orthogonal normalized vectors rigidly attached to the body B and originating from the centerof mass G.

3. Time and space discretization

3.1. Lie’s scheme: a first order operator-splitting scheme

Many operator-splitting schemes can be applied to problem (12)–(19). One of the advantage of operator-splittingschemes is that we can decouple difficulties such as (i) the incompressibility condition, (ii) the nonlinear advection term,and (iii) the rigid body motion, so that each one of them can be handled separately, and in principle optimally. Let Dt

be a time discretization step and tn+s = (n + s)Dt. The Lie’s scheme is a first order operator-splitting scheme [20], which,when applied to problem (12)–(19), yields:

u0 ¼ ~u0; G0 ¼ G0; V0 ¼ V0; x0 ¼ x0; x01 ¼ x1;0; x0

2 ¼ x2;0 given ð20Þ

for n P 0, un(’u(tn)), Gn, Vn, xn, xn1 and xn

2 being known, we first compute unþ16, pnþ1

6 via the solution of

qf

ZX

ou

ot� v dx�

ZX

pr � vdx ¼ 0; 8v 2W0; a.e. on ðtn; tnþ1Þ;ZX

qr � u dx ¼ 0; 8q 2 L2ðXÞ;

uðtnÞ ¼ un;

uðtÞ 2Wg0ðtnþ1Þ; pðtÞ 2 L2

0ðXÞ;

8>>>>>>><>>>>>>>:and set unþ1

6 ¼ uðtnþ1Þ; pnþ16 ¼ pðtnþ1Þ.

ð21Þ

Next, compute unþ26 via the solution ofZ

X

ou

ot� vdxþ

ZXðunþ1

6 � rÞu � vdx ¼ 0; 8v 2Wnþ1;�0 ; a.e. on ðtn; tnþ1Þ;

uðtnÞ ¼ unþ16;

uðtÞ 2W; uðtÞ ¼ gðtnþ1Þ on Cnþ1� � ðtn; tnþ1Þ;

8>>><>>>:and set unþ2

6 ¼ uðtnþ1Þ

ð22Þ

with Cnþ1� ¼ fx j x 2 C; g0ðtnþ1ÞðxÞ � nðxÞ < 0g and Wnþ1;�

0 ¼ fv j v 2W; v ¼ 0 on Cnþ1� g. Then, compute unþ3

6 via the solu-tion of


qf

ZX

ou

ot� vdxþ alf

ZXru : rvdx ¼ hkA; viKA

; 8v 2W0; a.e. on ðtn; tnþ1Þ;

hlA; uiKA¼ 0; 8lA 2 KA;

uðtnÞ ¼ unþ26;

uðtÞ 2Wg0ðtnþ1Þ;

8>>>>>>><>>>>>>>:and set unþ3

6 ¼ uðtnþ1Þ.

ð23Þ

Now predict the motion of the center of mass and the angular velocity of the particle via

dG

dt¼ VðtÞ=2; ð24Þ

1� qf

qs

� �Mp

dV

dt¼ Fr=2; ð25Þ

1� qf

qs

� �dðIpxÞ

dt¼ Gxr��!� Fr=2; ð26Þ

dxi


��!; for i ¼ 1; 2; ð27Þ

GðtnÞ ¼ Gn; VðtnÞ ¼ Vn; ðIpxÞðtnÞ ¼ ðIpxÞn; x1ðtnÞ ¼ xn1; x2ðtnÞ ¼ xn

2 ð28Þ

for tn < t < tn+1. Then set Gnþ46 ¼ Gðtnþ1Þ, Vnþ4

6 ¼ Vðtnþ1Þ, ðIpxÞnþ46 ¼ ðIpxÞðtnþ1Þ, x

nþ46

1 ¼ x1ðtnþ1Þ, and xnþ4

62 ¼ x2ðtnþ1Þ.

Using Gnþ46, x

nþ46

1 and xnþ4

62 obtained in the above step, we enforce the rigid body motion in the region Bnþ4

6 occupied by theparticle

qf

ZX

ou

ot� vdxþ ð1� aÞlf

ZXru : rvdxþ 1� qf

qs

� �Mp

dV

dt� Yþ 1� qf

qs

� �dðIpxÞ

dt� h

¼ 1� qf

qs

� �Mpg � Yþ hk; v� Y� h�Gnþ4

6x��!

iKnþ4

6; 8v 2W0; Y 2 R3; h 2 R3; a.e. on ðtn; tnþ1Þ;

uðtnÞ ¼ unþ36; VðtnÞ ¼ Vnþ4

6; ðIpxÞðtnÞ ¼ ðIpxÞnþ46;

uðtÞ 2Wg0ðtnþ1Þ; kðtÞ 2 Knþ4

6; VðtÞ 2 R3; ðIpxÞðtÞ 2 R3;

8>>>>>>>>><>>>>>>>>>:

ð29Þ

hl; u� V� x�Gnþ46x

��!iKnþ4

6¼ 0; 8l 2 Knþ4

6 ð30Þ

and set unþ1 ¼ uðtnþ1Þ; Vnþ56 ¼ Vðtnþ1Þ; ðIpxÞnþ

56 ¼ ðIpxÞðtnþ1Þ.

(In (29) and (30), we have Knþ46 ¼ ðH 1ðBnþ4

6ÞÞ3 where Bnþ46 is the region occupied by the particle B according to

Gnþ46, x

nþ46

1 and xnþ4

62 .)

Correct the motion of the center of mass and the angular velocity of the particle via

dG


1� qf

qs

� �Mp

dV

dt¼ Fr=2; ð32Þ

1� qf

qs

� �dðIpxÞ


dxi


��!; for i ¼ 1; 2; ð34Þ

GðtnÞ ¼ Gnþ46; VðtnÞ ¼ Vnþ5


x1ðtnÞ ¼ xnþ4

61 ; x2ðtnÞ ¼ x

nþ46

2 ð35Þ

for tn < t < tn+1. Then set Gn+1 = G(tn+1), Vn+1 = V(tn+1), (Ipx)n+1 = (Ipx)(tn+1), xnþ11 ¼ x1ðtnþ1Þ, and xnþ1

2 ¼ x2ðtnþ1Þ.


3.2. Space discretization

We assume that X � R3 and is a rectangular parallelepiped. Concerning the finite element approximation of problem(12)–(19) we have

Wh ¼ fvhjvh 2 ðC0ðXÞÞ3; vhjT 2 ðP 1Þ3; 8T 2Th; vh is periodic in the x2 directiong; ð36ÞW0h ¼ fvhjvh 2Wh; vh ¼ 0 on Cg; ð37ÞL2

h ¼ fqhjqh 2 C0ðXÞ; qhjT 2 P 1; 8T 2T2hg; ð38Þ

L20h ¼ qhjqh 2 L2

h;

ZX

qh dx ¼ 0; qh is periodic in the x2 direction

� �; ð39Þ

where Th is a tetrahedrization of X, T2h is twice coarser than Th, and P1 is the space of the polynomials in three variablesof degree 61. (In practice, we first divide the computational domain of the shape of rectangular parallelepiped into a col-lection of cubes with edges of length 2h and then divide each cube into six tetrahedra to obtain T2h. By similar way, weobtain Th.) A finite dimensional space approximating K(t) is as follows: let fnig

Ni¼1 be a set of points from BðtÞ which cover

BðtÞ (uniformly, for example); we define then

KhðtÞ ¼ lhjlh ¼XN

i¼1

lidðx� niÞ; li 2 R3; 8i ¼ 1; . . . ;N

( ); ð40Þ

where d(Æ) is the Dirac measure at x = 0. Then we shall use h�; �iKhðtÞ defined by

hlh; vhiKhðtÞ ¼XN

i¼1

li � vhðniÞ; 8lh 2 KhðtÞ; vh 2Wh. ð41Þ

A typical choice of points for defining (40) is to take the grid points of the velocity mesh internal to the particle B andwhose distance to the boundary of B is greater than, e.g., 3h/4 (used in the simulation), and to complete with selected pointsfrom the boundary of B(t) (e.g., see Fig. 2 for an example of selected points on the boundary of B(t)). As we did for Kh(t)and h�; �iKhðtÞ

, we define the finite dimensional space KAh and the inner product h�; �iKAhvia a set of points of the velocity mesh

internal to the region A and whose distance to the surface of A is greater than, e.g., h/2, and a set of the points chosen fromthe surface of the airfoil shaped cylinder A, e.g., points chosen from each cross-section cut by the plane x2 = ihv (hv is themesh size for the velocity field).

Remark 3. The inner product like bracket h�; �iKhðtÞ in (41) makes little sense for the continuous problem, but it ismeaningful for the discrete problem; it amounts to forcing the rigid body motion of B(t) via a collocation method. A similartechnique has been used to enforce Dirichlet boundary conditions by Bertrand et al. in [21].

Using the above finite dimensional spaces and the backward Euler’s method for most of the subproblems in scheme(20)–(35), we obtain the following scheme after dropping some of the subscripts h (similar ones are discussed in [6,7,9–11]):

u0 ¼ ~u0; G0 ¼ G0; V0 ¼ V0; x0 ¼ x0; x01 ¼ x1;0; x0

2 ¼ x2;0 given ð42Þ

Fig. 2. An example of selected points on the boundary of the rigid body.


for n P 0, un(’u(tn)), Gn, Vn, xn, xn1 and xn

2 being known, we compute unþ16, pnþ1

6 via the solution of

qf

ZX

unþ16 � un

Dt� vdx�

ZX

pnþ16r � v dx ¼ 0; 8v 2W0h;Z

Xqr � unþ1

6 dx ¼ 0; 8q 2 L2h;

unþ16 2Wh; unþ1

6 ¼ gnþ10h on C; pnþ1

6 2 L20h.

8>>>>><>>>>>:

ð43Þ

Next, compute unþ26 via the solution ofZ

X

ou

ot� vdxþ

ZXðunþ1

6 � rÞu � vdx ¼ 0; 8v 2Wnþ1;�0h ; a.e. on ðtn; tnþ1Þ;

uðtnÞ ¼ unþ16;

uðtÞ 2Wh; uðtÞ ¼ gnþ10h on Cnþ1

� � ðtn; tnþ1Þ

8>>><>>>:and set unþ2

6 ¼ uðtnþ1Þ.

ð44Þ

Then, compute unþ36 and k

nþ36

Ahvia the solution of

qf

ZX

unþ36 � unþ2

6

Dt� vdxþ alf

ZXrunþ3

6 : rv dx ¼ hknþ36

Ah; viKAh

; 8v 2W0h;

hlA; unþ3

6iKAh¼ 0; 8lA 2 KAh ;

unþ36 2Wh; unþ3

6 ¼ gnþ10h on C; k

nþ36

Ah2 KAh .

8>>>>><>>>>>:

ð45Þ

Now predict the motion of the center of mass and the angular velocity of the particle via

dG


1� qf

qs

� �Mp

dV

dt¼ Fr=2; ð47Þ

1� qf

qs

� �dðIpxÞ


dxi


��!; for i ¼ 1; 2; ð49Þ

GðtnÞ ¼ Gn; VðtnÞ ¼ Vn; ðIpxÞðtnÞ ¼ ðIpxÞn; x1ðtnÞ ¼ xn1; x2ðtnÞ ¼ xn

2 ð50Þ

for tn < t < tn+1.

Then set Gnþ46 ¼ Gðtnþ1Þ; V nþ4

6 ¼ Vðtnþ1Þ; ðIpxÞnþ46 ¼ ðIpxÞðtnþ1Þ, x

nþ46

1 ¼ x1ðtnþ1Þ, and xnþ4

62 ¼ x2ðtnþ1Þ. With the center Gnþ4

6,

xnþ4

61 and x

nþ46

2 obtained at the above step, we enforce the rigid body motion in the region Bðtnþ46Þ occupied by the particle

qf

ZX

unþ1 � unþ46

Dt� vdxþ ð1� aÞlf

ZXrunþ1 :rvdxþ 1� qf

qs

� �Mp

Vnþ56 �Vnþ4

6

Dt�Yþ 1� qf

qs

� �ðIpxÞnþ

56 � ðIpxÞnþ

46

Dt� h

¼ 1� qf

qs

� �Mpg �Yþ hknþ4

6; v�Y� h�Gnþ46x

��!iK

nþ46

h

; 8v 2W0h; Y 2 R3; h 2 R3;

unþ1 2Wh; unþ1 ¼ gnþ10h on C; knþ4

6 2 Knþ4

6h ; Vnþ5

6 2 R3; xnþ56 2 R3;

8>>>>>>><>>>>>>>:

ð51Þ

hl; unþ1 � Vnþ56 � xnþ5

6 �Gnþ4

6j x��!

iK

nþ46

h

¼ 0; 8l 2 Knþ4

6h . ð52Þ

Correct the motion of the center of mass and the angular velocity of the particle via

dG


1� qf

qs

� �Mp

dV

dt¼ Fr=2; ð54Þ


1� qf

qs

� �dðIpxÞ


dxi


��!; for i ¼ 1; 2; ð56Þ

GðtnÞ ¼ Gnþ46; VðtnÞ ¼ Vnþ5


x1ðtnÞ ¼ xnþ4

61 ; x2ðtnÞ ¼ x

nþ46

2 ð57Þ

for tn < t < tn+1. Then set Gn+1 = G(tn+1), Vn+1 = V(tn+1), (Ipx)n+1 = (Ipx)(tn+1), xnþ11 ¼ x1ðtnþ1Þ, and xnþ1

2 ¼ x2ðtnþ1Þ.In (42)–(57), we have Cnþ1

� ¼ fx j x 2 C; gnþ10h ðxÞ � nðxÞ < 0g and Wnþ1;�

0h ¼ fv j v 2Wh; v ¼ 0 on Cnþ1� g, Knþs

h ¼ KhðtnþsÞ,gnþ1

0h is an approximation of gnþ10 belonging to

cWh ¼ fzhjzh 2 ðC0ðCÞÞ3; zh ¼ ~zhjC with ~zh 2Whgand verifying

RC gnþ1

0h � ndC ¼ 0. In the numerical simulation, we usually choose a = 1.

Remark 4. In above (42)–(57), we do not combine problem (45) with problem (51) and (52) since we have taken theadvantage of the fixed cylinder. When solving (45) with Lagrange multipliers obtained from previous time step, the numberof iterations can be reduced to less than 10 quickly. But when solving the problem (51) and (52), the same strategy does notwork since the rigid bodies keep moving.

3.3. On the solution of subproblems (43), (44), (45), (46)–(50), and (51)–(52)

The degenerated quasi-Stokes problem (43) is solved by an Uzawa/preconditioned conjugate gradient algorithm as in[19,22], where the discrete elliptic problems used for preconditioning are solved by a matrix-free fast solver from FISHPAKdue to Adams et al. in [23]. The stopping criterion for the preconditioned conjugate gradient algorithm is krkk/kr0k 6 �where rk is the residue at the kth iteration. It typically takes about 10 iterations in the simulation with � = 10�5. The advec-tion problem (44) for the velocity field is solved by a wave-like equation method as in [19,24,25].

Systems (46)–(50) and (53)–(57) are systems of ordinary differential equations thanks to operator splitting. For its solu-tion one can choose a time step smaller than Dt, (i.e., we can divide Dt into smaller steps) to predict the translation velocityof the center of mass, the angular velocity of the particle, the position of the center of mass and the regions occupied byeach particle so that the repulsion forces can be effective to prevent particle–particle and particle–wall overlapping. At eachsub-cycling time step, keeping the distance constant between points x1 and x2 in each particle is important since we aredealing with rigid particles. To satisfy the above constraint we have applied the following approach:

• Translate x1 and x2 according to the new position of the mass center at each sub-cycling time step.• Rotate Gx1 and Gx2, the relative positions of x1 and x2 to the center of mass G, by the following Crank–Nicolson

scheme (a Runge–Kutta scheme of order 2, in fact):

Gxnewi �Gxold

i

s¼ x�Gxnew

i þGxoldi

2ð58Þ

for i = 1, 2 with s as a sub-cycling time step. By (58), we have jGxnewi j

2 ¼ jGxoldi j

2 for i = 1, 2 and jGxnew2 �Gxnew

1 j2 ¼

jGxold2 �Gxold

1 j2 (i.e., scheme (58) is distance and in fact shape preserving).

Remark 5. In order to activate the short range repulsion force, we have to find the shortest distance between twoellipsoids. Unlike the cases for spheres, it is not trivial to locate the point from each surface of the ellipsoid where thedistance is the shortest between two ellipsoids. There is no explicit formula for such distance. In practice, we first choose aset of points from the surface of each ellipsoid. Then we find the point among the chosen points from each surface at whichthe distance is the shortest. We repeat this (kind of relaxation) process in the neighborhood of the newly located point oneach surface of ellipsoid until convergence, usually obtained in very few iterations.

For the shortest distance between the wall and ellipsoid, there exists an explicit formula. To check whether two ellipsoidsoverlap each other, there exists an algorithm used by people working on computer graphics and in robotics (e.g., see, [26]).

The rigid body motion is enforced in Bðtnþ46Þ, via Eq. (52). At the same time those hydrodynamical forces and gravity

acting on the particles are also taken into account in order to update the translation and angular velocities of the particles.To solve (51) and (52), we use a conjugate gradient algorithm as discussed in [6]. Since we take b = 0 in (51) for the sim-ulation, we actually do not need to solve any non-trivial linear systems for the velocity field; this saves a lot of computingtime. To get the angular velocity xn+1, computed via


xnþ1 ¼ ðInþ46

p Þ�1ðIpxÞnþ1

; ð59Þ

we need to have Inþ46

p , the inertia of the particle Bðtnþ46Þ. We first compute the inertia I0 in the coordinate system attached to

the particle. Then via the center of mass Gnþ46 and points x

nþ46

1 and xnþ4

62 , we have the rotation transformation Q

(QQT = QTQ = Id, detQ = 1) which transforms vectors expressed in the particle frame to vectors in the flow domain

coordinate system and Inþsp ¼ QI0QT. Actually in order to update matrix Q we can also use quaternion techniques, as

shown, in the review paper [27].Problem (45) is a classical saddle-point problem which can be solved by conjugate gradient algorithms. Actually, problem

(45) is a particular case of

aZ

Xu � vdxþ l

ZXru : rvdx ¼

ZX

f � vdxþ hk; vi; 8v 2W0h;

hl; ui ¼ 0; 8l 2 K; u 2Wh; u ¼ g on C; k 2 K.

8<: ð60Þ

A conjugate gradient method for the solution of problem (60) reads as follows:

k0 2 K is given; ð61Þ

solve

aZ

Xu0 � vdxþ l

ZXru0 : rvdx ¼

ZX

f � vdxþ hk0; vi;

8v 2W0h; u0 2Wh; u ¼ g on C;

8<: ð62Þ

then solve

hg0; li ¼ hl; u0i; 8l 2 K; g0 2 K; ð63Þ

and set

w0 ¼ g0. ð64ÞFor m P 0, assuming that km, um, wm, gm are known, compute km+1, um+1, wm+1, gm+1 as follows:

Solve

aZ

Xum � vdxþ l

ZXrum : rvdx ¼ hwm; vi;

8v 2W0h; um 2W0h;

8<: ð65Þ

and set

hgm; li ¼ hl; umi; 8l 2 K; gm 2 K. ð66Þ

Then compute

qm ¼ hgm; gmi=hgm;wmi; ð67Þ

and set

kmþ1 ¼ km � qmwm; umþ1 ¼ um � qmum; gmþ1 ¼ gm � qmgm. ð68Þ

If hgm+1,gm+1i/hg0,g0i 6 �, then take u = um+1. If not, compute

cm ¼ hgmþ1; gmþ1i=hgm; gmi; ð69Þand set

wmþ1 ¼ gmþ1 þ cmwm. ð70ÞDo m = m + 1 and go back to (65).

Remark 6. The above conjugate gradient algorithm is similar to the one discussed in [28,29]; here a distributed Lagrangemultiplier has been used instead of the boundary Lagrange multiplier used in [28,29]. It takes about 7 (resp., 8) iterationson the average to solve subproblem (45) for the case of separation tested in Section 4.1 (resp., Section 4.2).

Fig. 3. The position of an elongated ellipsoid at t = 1.2, 1.5, 1.6, and 1.75 (from left to right and from top to bottom) for {hv,Dt} = {1/30,0.001}.

Fig. 4. Portion of the velocity field at t = 1.0 projected onto the x1x3-plane (top left), x2x3-plane (top right) and x1x2-plane (bottom) passing through themass center of the elongated ellipsoid.



4. Numerical experiments

4.1. Separation of an elongated ellipsoid initially placed under an airfoil shaped cylinder

In the first test case, we consider the simulation of the motion of an elongated ellipsoid separating from a fixed airfoil-shaped cylinder. The computational domain is X = (0, 12) · (0, 3) · (0,5). The fluid density is qf = 1 and the fluid viscosityis lf = 0.01. The flow field initial condition is u = 0. We have chosen the boundary condition of the flow field as

g0ðx; tÞ ¼ ð1� e�20tÞ1

0

0

0B@

1CA if x 2 C

for t P 0 to have a smooth transition. The fixed airfoil-shaped cylinder has a cross-section with the shape of a NACA0012airfoil with characteristic length 2.25 and zero degree angle of attack. The axis passing the mass center of each cross-sectionis x1 = 3 and x3 = 4. Hence the Reynolds number is 225 with respect to the characteristic length of the NACA0012 and themaximal in-flow speed. The density of the ellipsoid is qs = 1.05. The three semi-axes of the ellipsoid are 0.15, 0.15 and 0.75.Initially its longest axis is parallel to the x1 direction. The initial position of the mass center is at (4, 3.5,1.5) (the relativeposition between the elongated ellipsoid and the fixed airfoil-shaped cylinder is shown in Fig. 1). The initial translation andangular velocities of the ellipsoid are 0. In the simulation the elongated ellipsoid is completely fixed without possiblemotion up to t = 1. After t = 1 we allow the ellipsoid to move freely. The snapshots of the elongated ellipsoid positionfor {hv,Dt} = {1/30,0.001} at different times are shown in Fig. 3. We can see the ellipsoid not only falling but also turningitself against the streamlines as expected. The projections of the velocity field, obtained with {hv,Dt} = {1/38,0.001},onto three planes parallel to the coordinate planes, respectively, are shown in Figs. 4–6. For hv = 1/38 and Dt = 0.001,

1 1.1 1.2 1.3 1.4 1.5 1.6 1.73

3.5

4

4.5

5

5.5

6

1 1.1 1.2 1.3 1.4 1.5 1.6 1.70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7–5

–4

–3

–2

–1

0

1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7–5

–4

–3

–2

–1

0

1

Fig. 7. Histories of the x1-coordinate of the mass center (upper left), the x3-coordinate of the mass center (upper right), the angular velocity with respect tothe x2-axis (lower left), and the angular velocity with respect to the x3-axis (lower right) (hv = 1/30 and Dt = 0.001, dashed lines; hv = 1/30 and Dt = 0.0005,dashed-dotted lines; hv = 1/38 and Dt = 0.001, solid lines).


the averaged speed of the ellipsoid for 1.5 6 t 6 1.75 is about 3.40 (the maximal speed is about 3.71) so the averaged par-ticle Reynolds number with the longest axis as characteristic length is 510.

To check the convergence, we have chosen the following three pairs for the mesh size of the velocity field and the timestep: {hv,Dt} = {1/30,0.001}, {1/30,0.0005} and {1/38,0.001}. The mesh size of the pressure is always hp = 2hv. The his-tories of the x1-coordinate and x3-coordinate of the mass center and the angular velocity with respect to the x2-axis and x3-axis are shown in Fig. 7. We have obtained good convergence results. The number of grid points for the velocity field is4,960,501 (resp., 10,038,005) for hv = 1/30 (resp., 1/38). The memory used in the simulation is about 611 MB (resp.,1.2 GB) for hv = 1/30 (resp., 1/38). For each time step, it takes about 108 s (resp., 343 s) for hv = 1/30 (resp., 1/38) on aLinux based computer with AMD 2.4 GHz Opteron CPU.

Fig. 8. The position of two elongated ellipsoids at t = 1, 1.25, 1.5, and 1.65 (from left to right and from top to bottom) for {hv,Dt} = {1/30,0.001}.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7–4

–3.5

–3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7–4

–3.5

–3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1

Fig. 9. Histories of the angular velocities with respect to the x2-axis (left), and the angular velocities with respect to the x3-axis (right).

Fig. 11. Portion of the velocity field at t = 1.5 projected onto the x1x3-plane (top left), x2x3-plane (top right) and x1x2-plane (bottom) passing through themass center of the right elongated ellipsoid.





4.2. Separation of two elongated ellipsoids initially placed under an airfoil shaped cylinder

In this test case all the parameters are the same as in Section 4.1 except those mentioned in the following. The three semi-axes of the ellipsoids are 0.14, 0.1 and 0.7. The initial position of the mass centers are at (4.2,3.5,1.5) and (2.5,3.5,1.5). Thefluid viscosity is lf = 0.005. Hence the Reynolds number is 450 with respect to the characteristic length of the NACA0012and the maximal in-flow speed. Both elongated ellipsoids are initially lined up under the airfoil-shaped cylinder along a lineparallel to the x1 axis as shown in Fig. 8. In the simulation two elongated ellipsoids are completely fixed without possiblemotion up to t = 1. After t = 1 we allow the two ellipsoids to move freely. The snapshots of the elongated ellipsoids posi-tion for {hv,Dt} = {1/30,0.001} at different times are shown in Fig. 8. We can see that the right ellipsoid is turning fasteragainst the streamlines, probably due to the wakes created by the leading ellipsoid. The histories of the angular velocitieswith respect to the x2-axis and the angular velocities with respect to the x3-axis are shown in Fig. 9. The projections ofvelocity field, obtained with {hv,Dt} = {1/30,0.001}, onto three planes parallel coordinate planes, respectively, are shownin Figs. 10–12. For hv = 1/30 and Dt = 0.001, the averaged speed of the two ellipsoids for 1.45 6 t 6 1.65 is about 3.45 (themaximal speed is about 3.68) so the averaged particle Reynolds number with the longest axis as characteristic length is 966.

Remark 7. In the simulation, actually the short range repulsion force has not been activated due the fact that the shortestdistance between the elongated ellipsoids and the fixed cylinder and that between two elongated ellipsoids are greater thanthe safe zone of the size 2hv. Hence it has no effect on the simulations discussed in this section. But for the cases involvedmany solid bodies moving freely in fluids, it is an issue to check the effect of different short range forces, which has beendone in, e.g., [30,31].

5. Conclusion

In this article we have presented the extension the methodology in [7] to simulate the multi-store separation phenomenataking place in incompressible Newtonian viscous fluids. To move the non-spherical rigid body in fluid, we track two unitvectors rigidly attached to the rigid body (the third one then can be obtained via the cross-product of these two). Then wehave applied it to simulate the store separation of elongated ellipsoids under the cylinder of the cross-section with the shapeof a NACA0012 airfoil. We observed the ellipsoids turn the broadsides against the main direction of streamlines in thesimulation results as expected.


Acknowledgments

We acknowledge the helpful comments and suggestions of R. Bai, E.J. Dean, J. He, H.H. Hu, P.Y. Huang, G.P. Galdi,D.D. Joseph, Y. Kuznetsov, V. Paulsen, and reviewers. We acknowledge also the support of Dassault Aviation, NSF(grants ECS-9527123, CTS-9873236, DMS-9973318, CCR-9902035, DMS-0209066, DMS-0443826) and DOE/LASCI(grant R71700K-292-000-99).

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numerical simulation of a multi-store separation phenomenon: a fictitious domain approach

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