Linear stability analysis in fluid–structure interaction with transpiration. Part II: Numerical analysis and applications

Comput. Methods Appl. Mech. Engrg. 192 (2003) 4837–4873

www.elsevier.com/locate/cma

Linear stability analysis in fluid–structureinteraction with transpiration.

Part II: Numerical analysis and applications

Miguel �AAngel Fern�aandez a,*, Patrick Le Tallec b

a INRIA, Projet MACS, F-78153 Le Chesnay Cedex, Franceb �EEcole Polytechnique, DGAE, F-91128 Palaiseau Cedex, France

Received 27 September 2002; received in revised form 12 July 2003; accepted 18 July 2003

Abstract

This paper constitutes the numerical counterpart of the mathematical framework introduced in Part I. We address

the problem of flutter analysis of a coupled fluid–structure system involving an incompressible Newtonian fluid and a

reduced structure. We use the Linearization Principle approach developed in Part I, particularly suited for fluid–

structure problems involving moving boundaries. Thus, the stability analysis is reduced to the computation of the

leftmost eigenvalues of a coupled eigenproblem of minimal complexity. This eigenproblem involves the linearized in-

compressible Navier–Stokes equations and those of a reduced linear structure. The coupling is realized through specific

transpiration interface conditions. The eigenproblem is discretized using a finite element approximation and its smallest

real part eigenvalues are computed by combining a generalized Cayley transform and an implicit restarted Arnoldi

method. Finally, we report three numerical experiments: a structure immersed in a fluid at rest, a cantilever pipe

conveying a fluid flow and a rectangular bridge deck profile under wind effects. The numerical results are compared to

former approaches and experimental data. The quality of these numerical results is very satisfactory and promising.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Fluid–structure interaction; Transpiration; Flutter; Finite elements; Sparse generalized eigenproblems; Cayley transform;

Arnoldi method

1. Introduction

A body immersed in a fluid flow undergoes vibrations which can modify its geometry. Indeed, if a fluid–

structure equilibrium is subjected to an initial small disturbance, the generated oscillations will either decay

or diverge, depending on whether the flow energy transmitted to the structure is less than or surpass

the energy dissipated by the damping of the system. If the fluid is at rest, any oscillation caused by the

* Corresponding author. Address: �EEcole Polytechnique F�eed�eerale de Lausanne, IACS, CH-1015 Lausanne, Switzerland.

E-mail address: [email protected] (M.�AA. Fern�aandez).

0045-7825/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.08.001

mail to: [email protected]

4838 M.�AA. Fern�aandez, P. Le Tallec / Comput. Methods Appl. Mech. Engrg. 192 (2003) 4837–4873

disturbance will be damped (for instance, by the fluid viscosity). When the velocity of the flow is augmentedgradually the damping of the oscillations increases. However, with further increase in the flow velocity, a

point is reached from where the system is no longer subject to damping. The oscillation just maintain its

amplitude at the point where the damping vanishes. Above this point, any small disturbance generates

oscillations of large amplitude. This is a flutter instability; the point where the damping reduces to zero is

referred to flutter boundary.

Flutter instabilities can take place in a number of civil engineering processes: heat exchanger tubes in

axial flow, flexible pipes with internal flow, wind effects on long span bridges, aircraft wings and so forth.

Such flow-induced vibrations can damage the structure under study. For example, it may happen that thefluid forces feed energy into the vibrating structure progressively increasing the amplitude of the motion

until the structure collapses. The analysis of flutter instabilities is a major concern in the design of civil

engineering systems involving a fluid–structure coupling. Hence, a great number of experimental [33,40,

46,47] and numerical [4,23,30,32,34,36] works have been carried out on this subject.

In this work we treat the flutter problem of a coupled fluid–structure system involving an incompressible

Newtonian fluid and a reduced structure. This is carried out by a linear stability approach. We use the

Linearization Principle formulation developed in Part I [15], particularly suited for fluid–structure problems

involving moving boundaries. Thus, the analysis of the above flow-induced vibrations reduces to thecomputation of the leftmost eigenvalues of a coupled spectral problem, see [15, Section 3]. This coupled

eigenproblem involves the linearized incompressible Navier–Stokes equations (written in a fixed domain)

and those of a reduced linear structure. The coupling is realized through specific transpiration interface

conditions (see [13,24,32,35,36]).

We must notice that the use of a linear model for flutter analysis was already addressed (see for instance

[4,23,30]). However, the originality of our approach lies in the linearization–transpiration formulation

developed in Part I [15] which provides a coupled eigenproblem of minimal complexity, involving tran-

spiration interface conditions. In this sense, the present paper constitutes the numerical counterpart of thementioned formulation. The eigenproblem is discretized using a finite element approximation, and its

smallest real part eigenvalues are approximated by combining a generalized Cayley transform [18,19] and

an implicit restarted Arnoldi method (IRAM) [29,41]. Finally, several numerical experiments will outline,

on the one hand, the performance of our approximation scheme and, on the other hand, the robustness of

our linearization–transpiration formulation for flutter instabilities detection.

The organization of this paper is as follows. In Section 2 we introduce the spectral problem arising from

the linearization–transpiration formulation developed in Part I [15]. In Section 3 this problem is approxi-

mated using a stabilized finite element method. The velocity and pressure are both approximated usingcontinuous functions with P1 interpolation per element. The discrete formulation leads to a sparse gen-

eralized eigenvalues problem. In Section 4 we deal with the leftmost eigenvalues computation of this

generalized eigenproblem. We briefly discuss the IRAM method and the generalized Cayley transform. We

summarize the main steps of the Cayley transform Arnoldi algorithm (introduced in [27]) and we provide

some implementation techniques for the matrix–vector operations. Finally, the numerical experiments are

reported in Section 5. We consider three situations: a structure immersed in a fluid at rest, a cantilever pipe

conveying a fluid flow and a rectangular bridge deck profile under wind effects. The numerical results are

compared to former approaches [6,36] and to experimental data [46,47].

2. Linear stability: spectral problem

We consider a steady fluid–structure system in equilibrium. In this configuration, the solid is located in a

domain Xs � R3 with boundary c. As in many problems of aeroelasticity at low Mach numbers, it is

surrounded by a fluid in R3. We introduce a control volume X � R3 containing the solid. Hence, the fluid

Fig. 1. The computational domain X, defined by the system in its equilibrium configuration.

M.�AA. Fern�aandez, P. Le Tallec / Comput. Methods Appl. Mech. Engrg. 192 (2003) 4837–4873 4839

evolution is restricted to the domain Xf ¼ X� Xs. In the sequel we set C ¼ oX with C ¼ Cin [ Cout,

Cin \ Cout ¼ ;. Here, Cin stands for the inlet boundary and Cout for the outlet boundary, see Fig. 1.

We assume the fluid to be Newtonian viscous, homogeneous and incompressible. Its behavior is de-

scribed by its velocity u0 and pressure p0. At equilibrium, these fields satisfy the following incompressible

Navier–Stokes equations written in eulerian conservative formulation in Xf ,

div u0 � u0 �1

qrðu0; p0Þ

� �¼ 0; in Xf ;

divu0 ¼ 0; in Xf ;u0 ¼ uCin

; on Cin;rðu0; p0Þn ¼ 0; on Cout;u0 ¼ 0; on c;

ð1Þ

with

eðu0Þ ¼ 12½ru0 þ ðru0ÞT�; rðu0; p0Þ ¼ �p0I þ 2leðu0Þ;

q > 0 stands for the volume fluid density, l for the kinetic viscosity of the fluid, n for the unit normal vector

on c pointing inside the solid Xs, and uCinfor the fixed incoming velocity on Cin. Since, at equilibrium the

structure is at rest, the interface condition (1)5 expresses the continuity of the velocity field at the fluid–

structure interface c. In the sequel we will suppose that u0 and p0 are smooth functions.

The elastic solid under large displacements is described by its velocity and its stress tensor. As in [15], inthis paper we will suppose that the displacement of the structure, around the equilibrium configuration Xs,

is given by a linear combination of a finite number of vibration modes ui : Xs ! R3, 16 i6 ns, in such a

way that the motion of the structure can be written as IXs þ Us in Xs, with s 2 Rns and U ¼ ½u1ju2j � � � juns �is a 3� ns matrix standing for the reduced modal basis. In this way, the structural behavior is driven by

given mass and stiffness operators, M and K respectively. Thus, at equilibrium the structural equations

reduce to

K0 ¼ �ZcUTrðu0; p0Þnda; ð2Þ

where K0 stands for the residual structural stress equilibrating the fluid load at the interface c.

Remark 1. In the particular case where the configuration at equilibrium is known, problems (1) and (2) are

uncoupled (see numerical experiments below). However, in general, the equilibrium configuration X de-

pends on the structural displacement, and then problems (1) and (2) are strongly coupled (see [8,22]).

In this paper we focus on the numerical solution of the following quadratic eigenvalue problem, char-

acterizing the stability of the equilibrium (1) and (2): find the pulsation k 2 C, the perturbation velocity

u : Xf ! C3, pressure p : Xf ! C and displacement parametrization s 2 Cns , with ðu; p; sÞ 6¼ 0, such that


ru0uþruu0 � 2mdiv eðuÞ þ 1

qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on Cin;

rðu; pÞn ¼ 0; on Cout;

u ¼ �kUs� ku0Us; on c;

k2Msþ ðK þ B0Þs ¼ �ZcUTrðu; pÞnda;

ð3Þ

where M and K denote, respectively, the tangential mass and stiffness matrices of the structure and B0 is a

ns � ns real geometric stiffness matrix, given by the following expression:

B0ij ¼

Zcfrrðu0; p0Þujnþ rðu0; p0Þ½I divuj � ðrujÞ

T�ng � ui da; ð4Þ

for 16 i, j6 ns.This eigenproblem was derived in Part I [15, Section 3.3] (see also [14, Chapter 4]) by combining the

‘‘Linearization Principle’’ approach (see [21]) with the recent linearization method developed in [13,14],

particularly suited for problems involving moving boundaries. In this way, the above steady equilibriumstate will be considered as linearly asymptotically stable, if all the eigenvalues of (3) have positive real parts.

However, this steady state will be termed asymptotically unstable if there exists, at least, one eigenvalue

with negative real part.

Remark 2. The transpiration boundary condition (3)5 and the above ‘‘added stiffness’’ matrix B0 come from

the geometric interaction between the fluid and the structure [13–15], and take into account the possible

motion of the interface.

3. Numerical approximation

We address, in this section, the finite element discretization of the eigenvalue problem (3). We first re-write the ‘‘fluid part’’ of (3) in variational form and then we perform a finite element approximation. Fi-

nally, we provide the corresponding matrix formulation, which leads to a sparse generalized eigenvalue

problem.

3.1. Variational formulation

Introducing the variable z ¼ �ks, eigenproblem (3) takes the following traditional form (see also Part I

[15, Section 4.2.1]): find k 2 C, u : Xf ! C3, p : Xf ! C and s; z 2 Cns , with ðu; p; s; zÞ 6¼ 0, such that

qðru0uþruu0Þ � 2ldiv eðuÞ þ rp ¼ kqu; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on Cin;

rðu; pÞn ¼ 0; on Cout;

u ¼ Uz�ru0Us; on c;

�z ¼ ks;

ðK þ B0ÞsþZcUTrðu; pÞnda ¼ kMz:

ð5Þ


We consider the following complex Sobolev spaces (see [9]):

H 1CinðXfÞ ¼ fv 2 H 1ðXfÞjv ¼ 0; on Cing;

H 1Cin[CðX

fÞ ¼ fv 2 H 1ðXfÞjv ¼ 0; on Cin [ cg;

and a linear continuous lift operator

R : H 1=2ðcÞ3 ! H 1CinðXfÞ3: ð6Þ

By multiplying Eq. (5)1 by v 2 H 1Cin[cðX

fÞ3, integrating by parts and taking into account the boundary

conditions, we get that k 2 C and ðu; pÞ 2 H 1ðXfÞ3 � L2ðXfÞ satisfy

u� RðUz�ru0UsÞ 2 H 1Cin[cðX

fÞ3;aðu; vÞ þ bðp; vÞ þ bðq; uÞ ¼ kdðu; vÞ 8ðv; qÞ 2 H 1

Cin[cðXfÞ3 � L2ðXfÞ;

ð7Þ

with notation

aðu; vÞ ¼ qðru0uþruu0;�vvÞ0;Xf þ 2lðeðuÞ; eð�vvÞÞ0;Xf ;

dðu; vÞ ¼ qðu;�vvÞ0;Xf ;

bðp; vÞ ¼ �ðp; div�vvÞ0;Xf :

On the other hand, by multiplying (5)1 by RðuiÞ, integrating by parts and taking into account the

boundary conditions, we getZcðrðu; pÞnÞ � ui da ¼ aðu;RðuiÞÞ þ bðp;RðuiÞÞ � kdðu;RðuiÞÞ: ð8Þ

Taking into account (7) and (8), we can write (5), under variational form, in the following way: find

k 2 C and ðu; p; s; zÞ 6¼ 0 in H 1ðXfÞ3 � L2ðXfÞ � Cns � Cns such that

u� RðUz�ru0UsÞ 2 H 1Cin[cðX

fÞ3;aðu; vÞ þ bðp; vÞ þ bðq; uÞ ¼ kdðu; vÞ 8ðv; qÞ 2 H 1

Cin[cðXfÞ3 � L2ðXfÞ;

� z ¼ ks;

ðK þ B0Þsþ F aðuÞ þ F bðpÞ ¼ kðMzþ F dðuÞÞ;

ð9Þ

with F aðuÞ, F bðpÞ, F dðuÞ 2 Cns given, from (8), by the following expressions:

½F aðuÞ�i ¼ aðu;RðuiÞÞ; ½F bðpÞ�i ¼ bðp;RðuiÞÞ; ½F dðuÞ�i ¼ dðu;RðuiÞÞ;for i ¼ 1; . . . ; ns.

3.2. Finite element discretization

In the sequel, we will assume that Xf � R3 is a polygonal domain with which we associate a regular

family of triangulations fThgh>o (see [3]), such that

Xf ¼

[K2Th

K 8h > 0;

where h is defined by h ¼ maxK2Th hk, with hk the diameter of K.In order to approximate the continuous spaces H 1ðXfÞ and L2ðXfÞ, we introduce the finite dimensional

space Vh defined by


Vh ¼ fvh 2 C0ðXfÞjvhjK 2 P1ðKÞ 8K 2Thg;

where P1ðKÞ stands for the space of polynomials on K of degree less or equal to 1. Thus, we define the

following discrete spaces:

Xh ¼ Vh \ H 1CinðXfÞ; Qh ¼ Vh; Vh;0 ¼ Vh \ H 1

Cin[cðXfÞ: ð10Þ

In a discrete framework, the operator R in (6) is replaced by a discrete lift operator

Rh : TrðVhÞ3jc ! X 3h :

In the same way, we introduce a P1 Lagrange-piecewise interpolation operator Ph on c

Ph : C0ðcÞ3 ! TrðVhÞ3jc;

defined as the restriction on c of the classical P1 Lagrange-piecewise interpolation operator Ph in Xf .

We approximate problem (9) using the following stabilized finite element formulation: find k 2 C and

ðu; p; s; zÞ 6¼ 0 in V 3h � Qh � Cns � Cns such that

u� RhðUz�ru0UsÞ 2 V 3h;0;

aðu; vÞ þ bðp; vÞ þ bðq; uÞ þ asðu; vÞ þ bsðp; vÞ þ btsðq; uÞ þ csðp; qÞ¼ kðdðu; vÞ þ dsðu; vÞ þ esðq; uÞÞ 8ðv; qÞ 2 V 3

h;0 � Qh;

� z ¼ ks;

ðK þ B0Þsþ F ah ðuÞ þ F b

h ðpÞ ¼ kðMzþ F dh ðuÞÞ;

ð11Þ

with

asðu; vÞ ¼XK2Th

ðqðru0uþruu0Þ; sKqr�vvu0Þ0;K ;

bsðp; vÞ ¼XK2Th

ðrp; sKqr�vvu0Þ0;K ;

btsðu; qÞ ¼ �XK2Th

ðqðru0uþruu0Þ; sKr�qqÞ0;K ;

csðp; qÞ ¼ �XK2Th

ðrp; sKr�qqÞ0;K ;

dsðu; vÞ ¼XK2Th

ðqu;sKqr�vvu0Þ0;K ;

esðu; qÞ ¼ �XK2Th

ðqu; sKr�qqÞ0;K ;

½F ah ðuÞ�i ¼ aðu;RhPhðuiÞÞ þ

XK2Th

ðqðru0uþruu0Þ; sKqrðRhPhðuiÞÞu0Þ0;K ;

½F bh ðpÞ�i ¼ bðp;RhPhðuiÞÞ þ

XK2Th

ðrp; sKqrðRhPhðuiÞÞu0Þ0;K ;

½F dh ðuÞ�i ¼ dðu;RhPhðuiÞÞ þ

XK2Th

ðqu; sKqrðRhPhðuiÞÞu0Þ0;K ;


for i ¼ 1; . . . ; ns, the stabilization parameter sK being given by

sK ¼hK

2qku0k2nðRheKÞ; RheK ¼

ku0k2hK12m

; nðxÞ ¼ x si 06 x < 1;1 si xP 1:

�

Remark 3. It is straightforward to verify that (11) is consistent with the solutions of (5). Indeed, (11) is

obtained by adding the strong form of the differential equation (5)1 to the variational form (9).

As we have already pointed out in Part I [15, Section 4.2.1], and as it will be confirmed below (in Section4.4), the numerical solution of problem (11), involves a P1=P1 finite element approximation of linearized

Navier–Stokes problems with right-hand side and zero-order reaction term. Unfortunately, to our present

knowledge, there is no a stabilized finite element method for the discretization of a general problem of this

type, see [44]. Scheme (11) is a direct extension of those introduced in [43] and [45] for the Stokes equations

with convection. In fact, we have kept the choice of sK for this type of equations as proposed in [45]. We

have employed the same triangulation Th in the computation of u0 and u (it is not mandatory, but

practical). Thus, an accurate computation of the permanent flow ðu0; p0Þ and of ðru0Þjc, requires a refined

grid in the vicinity of c. As pointed out in [16], this reduces the complications associated when dealing withdominating reaction terms. The numerical experiments reported in Section 5 will point out the performance

of scheme (11).

3.3. Matrix formulation

In this paragraph, problem (11) is reformulated in a matricial form. This will allow us to explicitly

compute its solutions. Let nf ¼ nfðhÞ be the number of vertices of the triangulation Th on the fluid domain.

We introduce the finite element real basis f/ig3nf

i¼1 and fwignf

i¼1 of V3h and Qh respectively. Thus, each element

ðu; pÞ 2 V 3h � Qh can be written as

u ¼X3nfj¼1

uj/j; p ¼Xnfj¼1

pjwj; ð12Þ

with uj, pj 2 C. We introduce also the following subsets of I ¼ f1; . . . ; 3nfg � N:

IXf ¼ fi 2 I jd:o:f : i is not on C [ cg; ICout ¼ fi 2 I jd:o:f : i is on Coutg;

ICin ¼ fi 2 I jd:o:f : i is on Cing; IC ¼ fi 2 I jd:o:f : i is on cg;

and then we denote

nXf ¼ cardðIXf Þ; nCout ¼ cardðICoutÞ; nCin ¼ cardðICinÞ; nc ¼ cardðIcÞ:

By substituting (12) in (11)2 we get

X3nfj¼1

ujað/j; vÞ þXnfj¼1

pjbðwj; vÞ þX3nfj¼1

ujbðq;/jÞ þX3nfj¼1

ujasð/j; vÞ

þXnfj¼1

pjbsðwj; vÞ þX3nfj¼1

ujbtsð/j; qÞ þXnfj¼1

pjcsðwj; qÞ

¼ kX3nfj¼1

ujdð/j; vÞ

þX3nfj¼1

ujdsð/j; vÞ þX3nfj¼1

ujesð/j; qÞ!8ðv; qÞ 2 V 3

h;0 � Qh: ð13Þ


We denote by uXf 2 CnX

f

, uCout 2 CnCout , uCin 2 CnCin and uc 2 Cnc the degrees of freedom of u corre-sponding, respectively, to shape functions in IX

f

, ICout , ICin and Ic. In the sequel we will assume that the shape

functions f/ig3nf

i¼1 are ordered in such a way that the first degrees of freedom correspond to uXf

, next to uCout ,

next to uCin and finally to uc.By taking in (13) v ¼ /i, with i 2 IX

f [ ICout , and q ¼ wi, with i ¼ 1; . . . ; nf , we obtain a

ðnXf þ nCout þ nfÞ � ð4nf þ 2nsÞ matrix expression of type

AXf

1 ACout

1 ACin

1 Ac1 B1 0 0

AXf

2 ACout

2 ACin

2 Ac2 B2 0 0

BXf

BCout BCin Bc C 0 0

264

375

uXf

uCout

uCin

uc

pzs

2666666664

3777777775¼ k

DXf

1 DCout

1 DCin

1 Dc1 0 0 0

DXf

2 DCout

2 DCin

2 Dc2 0 0 0

EXf

ECout ECin Ec 0 0 0

264

375

uXf

uCout

uCin

uc

pzs

2666666664

3777777775: ð14Þ

In the same way, by substituting (12) in (11)4 we get

ðK þ B0ÞsþX3nfj¼1

ujF ah ð/jÞ þ

Xnfj¼1

pjF bh ðwjÞ ¼ k Mz

þX3nfj¼1

ujF dh ð/jÞ

!;

which leads to the following matrix expression of size ns � ð4nf þ 2nsÞ:

F Xf

a F Couta F Cin

a F ca Fb 0 K þ B0

� �uX

f

uCout

uCin

uc

pzs

2666666664

3777777775¼ k F Xf

d F Cout

d F Cin

d F cd 0 M 0

� �uX

f

uCout

uCin

uc

pzs

2666666664

3777777775:

ð15Þ

The transpiration interface condition (11)1 is taken explicitly on each interface vertex, xi, of the trian-

gulation. Therefore, we obtain the following matrix expression of size ðnCin þ ncÞ � ð4nf þ 2nsÞ:

0 0 I 0 0 0 0

0 0 0 I 0 �G0 G1

� �uX

f

uCout

uCin

uc

pzs

2666666664

3777777775¼ k

0 0 0 0 0 0 0

0 0 0 0 0 0 0

� �uX

f

uCout

uCin

uc

pzs

2666666664

3777777775; ð16Þ

with the following notation:

½G0�ij ¼ ½uj�ncðiÞðxnvðiÞÞ; i 2 Ic; j ¼ 1; . . . ; ns;

½G1�ij ¼ ½ru0uj�ncðiÞðxnvðiÞÞ; i 2 Ic; j ¼ 1; . . . ; ns:

Here, ncðiÞ 2 f1; 2; 3g is the component of the velocity corresponding to the velocity degree of freedom i,and nvðiÞ 2 f1; . . . ; nfg the label of the vertex where the velocity degree of freedom i lies, for i ¼ 1; . . . ; 3nf .


In short, taking into account (11)3 with (14)–(16) we obtain that the discrete problem (11) is equivalent tothe following generalized eigenvalue problem of size n ¼ 4nf þ 2ns: find k 2 C and 0 6¼ x 2 Cn such that

AXf

1 ACout

1 ACin

1 Ac1 B1 0 0

AXf

2 ACout

2 ACin

2 Ac2 B2 0 0

0 0 I 0 0 0 0

0 0 0 I 0 �G0 G1

BXf

BCout BCin Bc C 0 0

0 0 0 0 0 �I 0

F Xf

a F Couta F Cin

a F ca Fb 0 K þ B0

2666666666664

3777777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A

uXf

uCout

uCin

uc

p

z

s

2666666666664

3777777777775

|fflfflfflffl{zfflfflfflffl}x

¼ k

DXf

1 DCout

1 DCin

1 Dc1 0 0 0

DXf

2 DCout

2 DCin

2 Dc2 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

EXf

ECout ECin Ec 0 0 0

0 0 0 0 0 0 I

F Xf

a F Couta F Cin

a F ca 0 M 0

2666666666664

3777777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

uXf

uCout

uCin

uc

p

z

s

2666666666664

3777777777775

|fflfflfflffl{zfflfflfflffl}x

: ð17Þ

The matrices A and B are real, sparse, non-symmetric and, in most of the applications, of large size.

Following the ‘‘Linearization Principle’’ approach developed in Part I [15, Section 3], in a linear stability

analysis the goal is to find eigenvalues with negative real part in order to detect instabilities. Typically,

almost all eigenvalues of (17) have positive real part and only a small number cross the imaginary axis, seeCorollary 23 of Part I [15, Section 4.3]. Therefore, to detect a stability change, the interest lies in computing

the few eigenvalues with smallest real part.

In the next section, we will deal with the numerical approximation of a small number (compared to n) ofsolutions of the generalized eigenproblem (17). To fully exploit the sparse character of the matrices, it will

be crucial to use methods which only involve operations of type matrix–vector product (i.e. which do not

destroy the sparsity). For instance, subspace iteration methods or Arnoldi methods [1,38] are very ap-

propriate.

4. Eigenvalues computation

In this section we will follow the ideas of Garrat [18] and Lehoucq and Scott [27]. In this sense, the

leftmost eigenvalues of (17) will be approximated using an iterative algorithm combining a generalized

Cayley transform and an IRAM method. This algorithm is fully developed in [27]. This choice is justified,

on the one hand, by the proven performance of the IRAM method, implemented in the ARPACK library

[29], and on the other hand, by the results of Lehoucq and Scott [27] that point out the efficiency of theiralgorithm.

In the following paragraphs we will briefly describe the IRAM method and the generalized Cayely

transform. We will summarize the main steps of the Cayley transform Arnoldi algorithm [27] together with

some implementation techniques used in the numerical experiments reported in Section 5.


4.1. Generalized Cayley transform

A complication arising in problem (17) is the singularity of matrix B. This implies that (17) has less than

n eigenvalues [5,18]. The missing eigenvalues, called ‘‘infinite’’ eigenvalues [18,42], are defined as the zero

eigenvalues of the inverse problem

Bx ¼ xA x; x ¼ 1

k:

Each zero value of x corresponds to an infinite eigenvalue of (17).

Since matrix B in (17) has one block of columns and two blocks of rows filled with zeros, eigenproblem

(17) has three kinds of infinite eigenvalues. The pure pressure modes correspond to nf columns of zeros. The

two other groups of infinite eigenvalues, nCin þ nc, come from the explicit discrete treatment of the boundary

conditions on Cin and c, which are not implicitly incorporated in the discrete space V 3h .

Although the infinite eigenvalues are not true eigenvalues of (17), in practice, they can introduce nu-

merical difficulties [5,18,27,31]. When working in finite arithmetic, the matrix B is often perturbed so that it

may become ‘‘not-singular’’. Eigenvalues with a very large module (and perhaps with negative real part)may appear which are, of course, irrelevant for the stability analysis.

Iterative methods for non-symmetric problems, such as subspace iteration and Arnoldi, cannot be ap-

plied directly to the generalized eigenproblem (17). Before we must transform problem (17) in a standard

problem of the form

Tx ¼ hx: ð18Þ

The difficulty now is to choose T . It is well known that iterative methods quickly provide good ap-proximations to well-separated eigenvalues. In general, these eigenvalues do not match with those of

smallest real part. Thus T must satisfy some a priori properties:

• matrix–vector products, Ty, should be carried out efficiently;

• there is a known transformation between the solutions of (17) and (18);

• the eigenvalues with smallest real part of (17) are mapped to the eigenvalues of (18) which are easily ap-

proximated by the iterative method performed on (18).

The complication associated to the singularity of the matrix B also restricts the choice of the transfor-

mation T . We must use rational transformations, namely, it is necessary to invert problem (17). In this

framework, standard choices when computing leftmost eigenvalues of problems like (17) are shift-invert

and generalized Cayley transformation, see [11,12,26,27].

In the sequel we will assume that (17) has m eigenvalues fkigmi¼1, which we suppose ordered with in-

creasing order of their real parts,

Reðk1Þ6Reðk2Þ6 � � � 6 ReðkmÞ:

Let a 2 C, with a 6¼ ki, for i ¼ 1; . . . ;m. By subtracting aBx from both sides of (17) we get

ðA� aBÞx ¼ ðk� aÞBx:

By multiplying this identity by ðA� aBÞ�1, we obtain

ðA� aBÞ�1B|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}TSIðaÞ

x ¼ 1

k� a

� �|fflfflfflfflfflffl{zfflfflfflfflfflffl}

h

x: ð19Þ


The matrix TSIðaÞ ¼ ðA� aBÞ�1B, is called a shift-invert transformation. The eigenvectors of (17) andTSIðaÞ are identical. Hence, the eigenvalues k of (17) and h of TSIðaÞ, are related by the following expression:

k ¼ aþ 1

h; h ¼ 1

k� a: ð20Þ

By selecting the shift (or pole) a near the imaginary axis, the left-most eigenvalues of (17) can be mapped

onto those of TSIðaÞ with largest magnitude. However, since A and B are real matrices, and in order to keep

the computation in real arithmetic, we will not consider the use of complex shifts.Let a1; a2 2 R be such that

a1 < a2; a1 6¼ ki; i ¼ 1; . . . ;m: ð21ÞBy subtracting a2Bx from both sides of (17) we obtain

ðA� a2BÞx ¼ ðk� a2ÞBx:Let us suppose that x is an eigenvector and k an eigenvalue of (17). By multiplying by ðA� a1BÞ�1 we get

ðA� a1BÞ�1ðA� a2BÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}TCða1;a2Þ

x ¼ ðk� a2Þ ðA� a1BÞ�1B|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}TSIða1Þ

x ¼ k� a2k� a1

� �|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

h¼cðkÞ

x: ð22Þ

The matrix

TCða1; a2Þ ¼ ðA� a1BÞ�1ðA� a2BÞ; ð23Þis termed generalized Cayley transform, see [18,19]. The scalars a1 and a2 are, respectively, referred to as

pole and zero of the transformation. Transformation (23) is a generalization of the standard Cayley

transform where a1 ¼ �a2, see [2,17].

From (22) we can easily derive the following lemma, giving the relationship between the finite eigen-

values of (17) and those of TCða1; a2Þ, see [18].

Lemma 4. Let a1; a2 2 R satisfying (21). The pair ðk; xÞ is an eigensolution of (17) if and only if ðh; xÞ is aneigensolution of TCða1; a2Þ, where

h ¼ cðkÞ ¼ k� a2k� a1

; k ¼ c�1ðhÞ ¼ a1h� a2h� 1

: ð24Þ

Here, c is a bijection of C� fa1g in C� f1g.

The main interest of the generalized Cayley transform lies in the properties of the bijection c and in the

role of parameters a1 and a2 in this mapping. In particular, how do they affect the way the eigenvalues of (17)

are mapped to those of TCða1; a2Þ. Garrat, in [18], provides a complete study of this mapping (see also [20]).

The main properties of the generalized Cayley transform can be summarized in the following theorem(see [18]):

Theorem 5. Let a1, a2 2 R with a1 < a2, and h ¼ cðkÞ with k 2 C� fa1g. ThenReðkÞ < 1

2ða1 þ a2Þ if and only if jhj > 1;

ReðkÞP 12ða1 þ a2Þ if and only if jhj6 1:

The above result implies that the eigenvalues of (17) lying on the left of the straight line ReðkÞ ¼ða1 þ a2Þ=2 in the complex plan, are mapped to extreme eigenvalues of TCða1; a2Þ.


More precisely, we have the following corollary (see [18]):

Corollary 6. Let k1; k2; . . . ; km to be the eigenvalues of (17) ordered with increasing real parts. Take an index16 k < m such that ReðkkÞ < Reðkkþ1Þ.

Let a1, a2 2 R satisfying (21) with

12ða1 þ a2Þ ¼ Reðkkþ1Þ;

then

hi ¼ cðkiÞ 62 Bð0; 1Þ; i ¼ 1; . . . ; k;

hi ¼ cðkiÞ 2 Bð0; 1Þ; i ¼ k þ 1; . . . ;m:

The above corollary suggests a strategy for choosing parameters a1 and a2 suitable for accelerating the

convergence to h1 ¼ cðk1Þ. Indeed, a1 and a2 may be chosen in such a way that the first unwanted eigen-

value, hkþ1 (with kP 1), is located on the unit circle Bð0; 1Þ and that the distance of the dominant eigenvalueh1 ¼ cðk1Þ from Bð0; 1Þ is maximal. In other words, under the hypothesis of Corollary 6, we have to

maximize jh1j with respect to a1 and a2 subjected to the constraints

a1 þ a22

¼ Reðkkþ1Þ; a1 < a2: ð25Þ

In the case where k is real, the following lemma holds (see [18]):

Lemma 7. Suppose k1 2 R. Let 16 k < m be such that ReðkkÞ < Reðkkþ1Þ and j > 1 a given real number.If we take

a1 ¼ Reðkkþ1Þ �jþ 1

j� 1ðReðkkþ1Þ � k1Þ;

a2 ¼ Reðkkþ1Þ þjþ 1

j� 1ðReðkkþ1Þ � k1Þ;

then

j ¼ jh1jP jh2jP � � � P jhkj > 1 ¼ jhkþ1jP jhij; i ¼ k þ 2; . . . ;m:

In this case, h1 may be made as large as wanted by increasing j. However, when k1 is complex the theoryis less satisfactory, we just have the following result (see [18,19]):

Lemma 8. Let us suppose that k1 ¼ x1 þ iy1 with x1 < Reðkkþ1Þ. If

a1 ¼ Reðkkþ1Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðReðkkþ1Þ � x1Þ2 þ y21

q;

a2 ¼ Reðkkþ1Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðReðkkþ1Þ � x1Þ2 þ y21

q;

then the maximum of jh1j subjected to the constraints (25) is reached. In addition,

jh1j ¼ jh2j ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 þ 1p

� n> 1; with n ¼ Reðkkþ1Þ � x1

jy1j

and jhij6 1 for i ¼ k þ 1; . . . ;m.


The above lemma implies that the optimal size of h1 is directly related to the ratio

n ¼ Reðkkþ1Þ �Reðk1ÞImðk1Þ

: ð26Þ

The approximation of h1 becomes delicate in situations where:

(1) jReðkkþ1Þ �Reðk1Þj is very small;

(2) jImðk1Þj is very large.

In both cases, the eigenvalue h1 is very close to the unit circle, which complicates its computation. The

main difference compared to the case where k1 is real comes from the fact that Lemma 8 does not ensure

that h1 is a large eigenvalue of TC. We emphasize that this is a direct result of using a rational transfor-

mation h ¼ cðkÞ, which is necessary because B is singular.

Remark 9. Let us notice that the eigenfrequency localization provided in Part I (see Remark 23 in [15,

Section 4.3]) guarantees that (at least at continuous level) the ratio n, defined in (26), has a positive lower

bound. In other words, h1 can be computed by increasing sufficiently the number k of requested eigenvalues(see Section 4.3.1).

4.2. Arnoldi’s method with implicit restart

The IRAM, developed in [41], provides a efficient and numerically stable way for implementing restart,

without explicitly computing a new Arnoldi factorization. This method combines the Arnoldi�s method [38]

with the implicitly shifted QR algorithm [42].

Let us consider a m-step (m < n) Arnoldi factorization of a real n� n non-symmetric matrix A, i.e.

AV ¼ VH þ reTm; ð27Þin such a way that V 2 Rn�m has m orthonormal columns, r 2 Rn with V Hr ¼ 0, and H 2 Rm�m is an upper

Hessemberg matrix with non-negative lower subdiagonal.

A QR type algorithm with explicit shift (see [42]) can be applied to (27). Let b 2 R be a given shift. We

factorize H � bI ¼ QR, with R upper triangular and Q upper Hessemberg orthogonal. Then, from (27),

we get

ðA� bIÞV � V ðH � bIÞ ¼ reTm;

ðA� bIÞV � VQR ¼ reTm;

ðA� bIÞðVQÞ � ðVQÞðRQÞ ¼ reTmQ;

AðVQÞ � ðVQÞðRQþ bIÞ ¼ reTmQ;

ð28Þ

in such a way that setting Vþ ¼ VQ and Hþ ¼ RQþ bI ¼ QTHQ, we obtain

AVþ ¼ VþHþ þ reTmQ:

The matrix Vþ has m orthonormal columns (it is the product of V by an orthogonal matrix Q) and Hþ is aupper Hessember matrix (it is a classical property of the QR factorization, see [42]). In addition, the firstm� 2 entries of eTmQ are zero. Thus, a new m� 1 Arnoldi factorization can be obtained by condensing,

namely, by equating the first m� 1 columns of each side:

AV þm�1 ¼ V þm�1Hþm�1 þ rþm�1e

Tm�1: ð29Þ


Therefore, shifting by b does not disturb the structure of the Arnoldi�s factorization. The main result ofthis operations is that the first column of Vþ has the direction of ðA� bIÞv1, where v1 is the first column of

V . Indeed, by multiplying (28)2 by the canonical vector e1, we get

ðA� bIÞv1 ¼ VþRe1 ¼ ðeT1Re1ÞVþe1:We can iterate this process by extending the new (m� 1)-step factorization (29) to a m-step factorization,

applying a shift and condensing. The payoff is that each iteration applies to v1 a linear polynomial in A, andwhere the root of this polynomial is the applied shift. These ideas can be generalized to the application of pimplicit shifts. Starting from a k-step Arnoldi factorization

AVk ¼ VkHk þ rkeTk ;

we extend this factorization to a (k þ p)-step Arnoldi factorization

AVkþp ¼ VkþpHkþp þ rkþpeTkþp: ð30Þ

Then, p implicit shifts fbjgpj¼1 may be applied to the factorization, resulting in a new factorization

AV þkþp ¼ V þkþpHþkþp þ rkþpeTkþpQ;

with V þkþp ¼ VkþpQ, Hþkþp ¼ QTHkþpQ and Q ¼ Q1;Q2; . . . ;Qp, with Qj the upper Hessemberg orthonormal

matrix resulting from the factorization of Hhþp � bjI for j ¼ 1; . . . ; p. A new k-step Arnoldi factorization

can be obtained by equating the first k columns on each side (see [41]):

AV þk ¼ V þk Hþk þ rþk eTk : ð31Þ

We can iterate by applying p steps of the Arnoldi�s algorithm to extend the factorization (31) to a new

(k þ p)-factorization (30), then applying shifts and condensing.

As mentioned above, each implicit application of a shift bj replaces the initial vector, v1, by a vector in

the direction of ðA� bjIÞv1. Thus, after application of p implicit shifts we get v1 wðAÞv1, where w is a pthdegree polynomial with roots fbjg

pj ¼ 1. The choice of the shifts, and hence the construction of the poly-

nomial, is motivated by the fact that if we choose as shifts the eigenvalues that are ‘‘unwanted’’, we can

effectively filter the starting vector v1 so that it will be rich in the direction of the ‘‘wanted’’ eigenvectors.

One possible shift selection strategy is the so-called ‘‘exact shift strategy’’ (see [41]), where the ðk þ pÞ eigen-values of Hkþp are partitioned into a set of k wanted and a set of p unwanted elements. The p unwanted

eigenvalues are used as the shifts in the restarting. This is equivalent to restarting the Arnoldi factorization

with a linear combination of the approximate eigenvectors associated with the wanted eigenvalues. We no-

tice that the use of implicits shifts of zero (see [27,31]) is equivalent to performing subspace iteration on Vk.

4.3. Generalized Cayley transform IRAM algorithm

In this paragraph we summarize the algorithm we use for the approximation of the leftmost eigenvaluesof (17). This algorithm, fully developed by Lehoucq and Scott in [27], combines shift-invert and generalized

Cayley transformations with an IRAM method. Shift-invert is used to get a first approximation of the

spectrum and is also used to purify the starting vector and the computed eigenvectors, see [31].

In the sequel k stands for the number of sought-after eigenvalues ðk � nÞ and k þ p ðp > 0Þ for the

number of computed Arnoldi vectors (i.e. the dimension of the Krylov�s subspace). We denote by h and xthe approximate eigenvalues and eigenvectors of TSI and TC. Hence, k ¼ c�1ðhÞ stands for the correspondingapproximate eigenvalue of the original problem (17).

Algorithm 1 outlines the Lehoucq and Scott�s Generalized Cayley transform IRAM algorithm. In thisalgorithm, symbol } indicates the steps provided by the ARPACK library (see [27,29]).


Algorithm 1. Cayley transform IRAM algorithm.

begin

} v1 2 Rn random unit vector

purify: v1 TSIð0Þv1} compute h1; . . . ; hkþp from a ðk þ pÞ-step Arnoldi factorization of TSI

ki 1

hi; i ¼ 1; . . . ; k þ p

order ki with increasing real parts

} v1 2 Rn random unit vector

v1 TSIð0Þv1repeat

compute a1 and a2 from k1 and kkþ1 (Lemmas 7 and 8)

} compute a ðk þ pÞ-step Arnoldi factorization of TC} update v1 by implicit application of p shifts of 0 (see Section 4.3.3)

} compute h1; . . . ; hkþp by extension to a ðk þ pÞ-step factorization

ki c�1ðhiÞ, i ¼ 1; . . . ; k þ porder ki with increasing real parts

(a) convergence test on hi largest in magnitudeuntil k converged eigenvalues

} compute eigenvectors xi corresponding to the converged eigenvalues

obtain eigenvectors of ðA;BÞ by purifying: xi TSIð0Þxiend

4.3.1. Missing eigenvalues

Once the dominant eigenvalues of TCða1; a2Þ are converged, it is necessary to be cautious to avoid eigen-

values which are not of smallest real part. This kind of uncertainty comes from the fact that, unfortunately,the largest eigenvalue of TCða1; a2Þ in magnitude does not necessarily coincide with the leftmost eigenvalues

of (17). It is important to notice that there is no theory available to verify whether the left-most eigenvalues

have been computed, see [18,26,27].

In an attempt to overcome this drawback, in [27], Lehoucq and Scott introduce an additional test on the

computed eigenvalues, after step (a) in Algorithm 1. Once the dominant eigenvalues (largest in magnitude)

of TCða1; a2Þ have acceptable approximations, we order the k þ p eigenvalues in decreasing order of their

moduli. Then we compute the corresponding eigenvalue approximations of (17) and order them in de-

creasing order of their real parts. If there is an eigenvalue in the package of the p smallest in magnitudewhich corresponds to a eigenvalue in the package of the k with smallest real part, we increase the number kof requested eigenvalues. The numerical experiments carried out in [27] demonstrates the effectiveness of

this strategy for the computation of the leftmost eigenvalues of the discretized linearized Navier–Stokes

equations. Such success is directly related to the fact that, in this situation, the ratio (26) is lower bounded

(see [21,39]). This is also the case for eigenproblem (3) (see Remark 9).

4.3.2. Spurious eigenvalues

From Lemma 4 the generalized Cayley transform maps the infinite eigenvalues of (17) on to +1. Theseeigenvalues are not relevant for the stability analysis but, in practice, are likely to be computed. Indeed,

from (24) and since a1 < a2, if h ¼ 1þ d with d > 0 small, then

k ¼ c�1ðhÞ ¼ 1

dða1ð1þ dÞ � a2Þ;


become large in magnitude with negative real part. Thus, this eigenvalue could be taken into account in the

update of a1 and a2. Lehoucq and Scott, in [27], propose that on each iteration all real hi which are close to

+1 must be excluded. They also exclude such spurious eigenvalues when searching for possible missing

eigenvalues.

4.3.3. Implicit shifts

We have noticed that the main difficulty in our eigenproblem comes from the existence of infinite eigen-

values. We could try to purify v1 with the application of implicit shifts of +1. Unfortunately, the experi-ments performed in [27] show that this technique can give bad results. Lehoucq and Spence, in [28], pointed

out that the application of shifts corresponding to very good approximations of the eigenvalue can in-

troduce instabilities due to the rounding errors. Lehoucq and Scott in [27] have shown that the application

of p shifts of 0 give consistently good results. This is equivalent to perform subspace iteration with TCða1; a2Þon Vk. With this choice the effects of the spurious eigenvalues are strongly attenuated.

4.3.4. Starting Cayley iterations

The information contained in the starting vector is crucial for the performance of the Arnoldi�s algo-rithm. At each Cayley iteration we can improve v1 by taking into account the information provided by the

last iteration. More precisely, we can take v1 in the invariant subspace associated to the converged eigen-

values. Therefore, see [37], after each Cayley iteration we set

v1 Xli¼1

wi; v1 v1kv1k

;

where fwigli¼1 is a Schur basis of the invariant subspace associated to the eigenvalues which have converged

in the preceding iteration.

4.4. Matrix–vector product computation

In Algorithm 1, each shift-invert purification or Arnoldi step requires, respectively, a matrix–vectorproduct y ¼ TSIða1Þx or y ¼ TCða1; a2Þx, depending on the transformation in progress. At this point, it is

important to provide the following identity

TCða1; a2Þ ¼ ðA� a1BÞ�1ðA� a2BÞ ¼ ðA� a1BÞ�1½ðA� a1BÞ þ ða1 � a2ÞB�

¼ I þ ða1 � a2ÞðA� a1BÞ�1B ¼ I þ ða1 � a2ÞTSIða1Þ:Therefore, on each Arnoldi step or purification step, we have to perform the matrix–vector product

y ¼ TSIða1Þx. In a equivalent way, we have to solve the following linear system:

ðA� a1BÞy ¼ Bx: ð32ÞIn the sequel we will interpret and define a method for the resolution of this system based on the

mathematical analysis developed in Part I [15], which can be viewed as a generalized added mass procedure.

Let us set

x ¼

f Xf

f Cout

f Cin

f c

qhg

2666666664

37777777752 Rn; y ¼

uXf

uCout

uCin

uc

pzs

2666666664

37777777752 Rn:


After elimination of z by the fifth line of (32), z ¼ �g � a1s, (32) reduces to

AXf

1 � a1DXf

1 ACout

1 � a1DCout

1 ACin

1 � a1DCin

1 f Cin Ac1 � a1D

c1 B1 0

AXf

2 � a1DXf

2 ACout

2 � a1DCout

2 ACin

2 � a1DCin

2 f Cin Ac2 � a1D

c2 B2 0

0 0 I 0 0 0

0 0 0 I 0 a1G0 þ G1

BXf � a1EXf

BCout � a1ECout BCin � a1ECin Bc � a1Ec C 0

F Xf

a � a1F Xf

d F Couta � a1F

Cout

d F Cina � a1F

Cin

d F ca � a1F

cd Fb K þ B0 þ a21M

2666666664

3777777775

uXf

uCout

uCin

uc

ps

2666666664

3777777775

¼

DXf

1 f Xf þ DCout

1 f Cout þ DCin

1 f Cin þ Dc1f

c

DXf

2 f Xf þ DCout

2 f Cout þ DCin

2 f Cin þ Dc2f

c

0

�G0gEXf

f Xf þ ECoutf Cout þ ECinf Cin þ Ecf c

F Xf

d f Xf þ F Cout

d f Cout þ F Cin

d f Cin þ F cd f

c þMðh� a1gÞ

2666666664

3777777775: ð33Þ

As in Section 4.2 of Part I [15] (problem (26) of Part I is the PDE equivalent of (33)) we decompose the

solution of the preceding problem as ðu; pÞ ¼ ðu1; p1Þ þ ðu2; p2Þ. By block Gaussian elimination of the fluidsubproblem, ðu1; p1Þ is solution of the first four lines of (33) used with s ¼ 0:

AXf

1 � a1DXf

1 ACout

1 � a1DCout

1 ACin

1 � a1DCin

1 f Cin Ac1 � a1D

c1 B1

AXf

2 � a1DXf

2 ACout

2 � a1DCout

2 ACin

2 � a1DCin

2 f Cin Ac2 � a1D

c2 B2

0 0 I 0 0

0 0 0 I 0

BXf � a1EXf

BCout � a1ECout BCin � a1ECin Bc � a1Ec C

26666664

37777775

uXf

1

uCout

1

uCin

1

uc1p1

2666664

3777775

¼

DXf

1 f Xf þ DCout

1 f Cout þ DCin

1 f Cin þ Dc1f

c

DXf

2 f Xf þ DCout

2 f Cout þ DCin

2 f Cin þ Dc2f

c

0

�G0gEXf

f Xf þ ECoutf Cout þ ECinf Cin þ Ecf c

26666664

37777775: ð34Þ

Similarly, we introduce elementary solutions of the first four lines of (33) (fluid subproblem) with unitright-hand side

AXf

1 � a1DXf

1 ACout

1 � a1DCout

1 ACin

1 � a1DCin

1 f Cin Ac1 � a1D

c1 B1

AXf

2 � a1DXf

2 ACout

2 � a1DCout

2 ACin

2 � a1DCin

2 f Cin Ac2 � a1D

c2 B2

0 0 I 0 0

0 0 0 I 0

BXf � a1EXf

BCout � a1ECout BCin � a1ECin Bc � a1Ec C

26666664

37777775

wXf

j

wCoutj

wCinj

wcj

qj

26666664

37777775 ¼

0

0

0

�ða1G0 þ G1Þej0

2666664

3777775ð35Þ

with ej the jth canonical vector of Rns .

From the above considerations we propose the following modular resolution for the linear system (32).

For a1 given, we start by factorizing the matrix of (34) or of (35). The solution of the elementary problems

(35) and the residual computations


Fijða1Þ ¼ ½ F Xf

a � a1F Xf

d F Couta � a1F

Cout

d F Cina � a1F

Cin

d F ca � a1F

cd Fb �

wXf

j

wCoutj

wCinj

wcj

qj

26666664

37777775

26666664

37777775

i

;

with i; j ¼ 1; . . . ; ns, allow us to compute the ns columns of the influence matrix F ða1Þ, describing the ret-

roaction of the fluid on the structure when the fluid is set into motion through the structural displacement s.Thus, we can factorize the matrix of the condensed problem, obtained after elimination of ðu2; p2Þ in the last

line of (33):

K þ B0 þ F ða1Þ þ a21M :

At this level, the computation of ðu; pÞ is immediate. On the one hand, ðu1; p1Þ can be obtained by solving

(34). Thus, from the residual

r ¼ F Xf

a

h� a1F Xf

d F Couta � a1F

Cout

d F Cina � a1F

Cin

d F ca � a1F

cd Fbi

uXf

1

uCout

1

uCin

1

uc1

p1

2666666664

3777777775

� F Xf

d f Xfh

þ F Cout

d þ f Cout þ F Cin

d f Cin þ F cd f

ci;

we obtain the fluid load at the interface. By solving the condensed problem

ðK þ B0 þ F ða1Þ þ a21MÞs ¼ �r þMðh� a1gÞ; ð36Þ

we get s. Once s is computed, we obtain ðu2; p2Þ from

ðu2; p2Þ ¼Xnsj¼1

sjðwj; pjÞ

and then with ðu1; p1Þ we finally compute ðu; pÞ.Clearly, we could separately build the matrices A and B and then factorize A� a1B. However, this would

imply, on the one hand, a larger cost of factorization, and on the other hand, a loss in the modular

character of the solution. It is well known that the modularity of an algorithm implies reliability and ro-

bustness. In the numerical experiments reported in Section 5, the numerical resolution of the linear systems

resulting from the fluid subproblems is performed using the UMFPACK package (available for download

on www.netlib.org/linalg). This package implements an hybrid algorithm unifrontal/multifrontalfor the direct resolution of sparse linear systems, see [10]. The resolution of the condensed problem (36), of

small size, is realized using a standard LU factorization, as implement in the LAPACK library (available

for download on www.netlib.org/lapack).

In the next section we will point out the performance of Algorithm 1 for the computation of the leftmost

eigenvalues of (17), when performing the matrix–vector multiplication as described above.

http://www.netlib.org/lapack


5. Numerical experiments

In this section we report the numerical results obtained in three significant two-dimensional test cases. Of

course, three-dimensional experiments can be addressed with our approach, by providing the corre-

sponding linearized fluid solver.

These numerical experiments will allow us to illustrate, on the one hand, the performance of the pro-

posed discretization scheme (11) and, on the other hand, the efficiency of the linearization–transpiration

approach developed in Part I [15]. In the following paragraphs we compare our numerical results to formerapproaches and to experimental data coming from wind-tunnel experiments.

5.1. Structure placed in a viscous flow at rest

In this paragraph we will consider the stability problem of a rigid tube immersed in a incompressible

viscous fluid at rest. The fluid is contained in a three-dimensional cavity containing the tube. It is assumed

that the tube is elastically mounted (with stiffness k > 0) in such a way that it can only vibrate in a

transverse plane perpendicular to the tube. Moreover, axial effects are not considered. Then, the problem isstudied in two dimensions restricting it to any of the sections, X, of the cavity that are perpendicular to the

tube, see Fig. 2. In the sequel Xf stands for the domain section occupied by the fluid, Xs for the transverse

section of the tube, c for the section of the fluid–structure interface, and c ¼ oX for the section of the cavity

boundary.

We assume that the fluid and the structure are at equilibrium in this configuration. Since the fluid is at

rest, we have u0 ¼ 0 and p0 ¼ 0. Therefore, eigenproblem (3) reduces to (see also [15, Section 4.1.1]): find

k 2 C, u : Xf ! C2, p : Xf ! C and s 2 Cns , withRXf pdx ¼ 0 and ðu; p; sÞ 6¼ 0, such that

�2mdiv eðuÞ þ 1

qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on c;u ¼ �ks; on c;

k2msþ ks ¼ �Zcrðu; pÞnda:

ð37Þ

Here, m > 0 is the mass of the tube.

Fig. 2. The section X.


This model problem, coupling the Stokes equations with those of a rigid body in translation, was alreadyproposed and studied in [7], for the determination of the vibration frequencies of a tube rack immersed in a

viscous fluid at rest, see also [8]:

Theorem 10. The spectrum of (37) consists of a countable infinite quantity of complex numbers which convergeto infinity. Moreover, the eigenvalues have the following properties:

(1) ReðkiÞ > 0 8iP 1;(2) There exists at most four eigenvalues with non-zero imaginary part;(3) If ImðkÞ 6¼ 0, then

jkj6ffiffiffiffikm

r:

A numerical study of problem (37) was carried out in [6]. The problem was approximated with a P2=P1

finite element discretization of the Stokes equations. The numerical experiments obtained then corroborate

the results of Theorem 10.

Theorem 10 ensures that the eigenvalues of (37) have positive real part. Therefore, the steady equilibrium

in which the fluid and structure are at rest is linearly stable, namely, the vibrations of the tube are damped

by the fluid viscosity.

In order to validate the discretization scheme (11) and the Algorithm 1, we have taken again the nu-

merical experiments reported by Conca and Dur�aan in [6]. Our numerical experiments were performedkeeping the same geometry: a closed cavity X ¼ ð�3; 3Þ � ð�3; 3Þ with a perforation Xs ¼ ð�1; 1Þ � ð�1; 1Þ,and using P1=P1 finite elements.

To discretize the fluid domain Xf ¼ X� Xs, we have considered four triangulations represented in Fig. 3.

As in [6], the density q, the kinematic viscosity m and the mass of the tube m were fixed to 1. For each

triangulation, the left-most eigenvalues of (17) were approximated for several values of k: 0.01, 0.1, 1, 10and 100. The number of requested eigenvalues was fixed to 10 and the dimension of the Krylov subspace to

30. Finally, the tolerance for the Ritz estimates was 10�6 (see [27]). On the finer grid, Fig. 3(d), the com-

putations took in average two iterations of Cayley and 4 min of CPU time on a HP Visualize 8200workstation. In addition, the relative residuals kAx� kBxk, with kxk ¼ 1, were between 10�7 and 10�15.

It is important to notice that, in [6], the authors have computed all the eigenvalues of the discretized

problem, that was done with the payoff of using very coarse meshes. In Figs. 4–8 we report, for each value of

k, the real part of the computed leftmost eigenvalue as a function of the mesh step h. The dashed line

represents the more accurate value provided in [6], it corresponds to a 128 triangles mesh. We can notice (for

small h) a linear convergence of the approximations (except for k ¼ 100) and a good agreement of both

numerical results.

In Table 1 we have reported the 10 left-most eigenvalues corresponding to our finer grid (Fig. 3(d)). Onthe other hand, Table 2 provides the eigenvalues obtained in [6] with a 128 triangles mesh.

We can immediately point out the good agreement of these results. In both cases the number of complex

eigenvalues is not higher than 4, that was predicted by Theorem 10. In addition, both spectra show a similar

behavior when k varies. In particular, for small values of the parameter k the spectra are made up only of

real numbers, and when k increases complex eigenvalues appear.

In Figs. 9–13 we present the velocity field and the isobaric lines corresponding to eigenvectors associated

with some eigenvalues of Table 1. These figures agree perfectly with those provided in [6]. A noticeable

feature, already observed in [6], lies in the fact that the simple eigenvalues of large magnitude are insensitiveto variations of k. In other words, they are purely fluid eigenmotions. This phenomenon can be easily noted

in Figs. 10, 12 and 13. Here, the coupled eigenmotions do not correspond to a displacement on the tube. We

recall that u ¼ ks on the interface c.

Fig. 3. Triangulations of the fluid domain Xf . (a) 96 triangles, (b) 320 triangles, (c) 1152 triangles and (d) 4608 triangles.

Fig. 4. Approximations of Re(k1) for k ¼ 0:01.


Fig. 5. Approximations of Re(k1) for k ¼ 0:1.

Fig. 6. Approximations of Re(k1) for k ¼ 1.


5.2. Cantilever pipe conveying a fluid

In this paragraph we address the problem of the linear stability of a cantilever pipe conveying a fluid, see

Fig. 14. It is well known, see [33,36], that this fluid–structure system loses stability by flutter for high flow

velocities.

Pa€ııdoussis, in [33], derives a simplified linear model for the motion of the pipe by adding fluid forces to

the pipe structural equations, the pipe being treated as a beam, modeled as (see [25])

EIo4wox4þ m€ww ¼ f in ð0; LÞ;

w ¼ owox¼ 0; on x ¼ 0;

o2wox2¼ o3w

ox3¼ 0; on x ¼ L;

ð38Þ



Table 1

The 10 left-most eigenvalues with the grid 3(d)

k ¼ 0:01 k ¼ 0:1 k ¼ 1 k ¼ 10 k ¼ 100

1.810· 10�4 1.811· 10�3 1.818· 10�2 1.894· 10�1 2:061þ 1:696i1.810· 10�4 1.811· 10�3 1.818· 10�2 1.894· 10�1 2:061� 1:696i2.651 2.651 2.651 2.651 2:061þ 1:696i3.870 3.869 3.855 3.707 2:061� 1:696i3.870 3.869 3.855 3.707 2.651

8.152 8.152 8.152 8.152 8.152

8.238 8.238 8.238 8.238 8.238

8.382 8.382 8.381 8.379 8.361

8.382 8.382 8.381 8.379 8.361

9.218 9.218 9.218 9.218 9.218


Fig. 9. Eigenvalue k1 ¼ 1:818� 10�2. (a) Velocity field and (b) isobaric lines.

Fig. 10. Eigenvalue k3 ¼ 2:651. (a) Velocity field and (b) isobaric lines.

Table 2

The 10 left-most eigenvalues reported in [6]

k ¼ 0:01 k ¼ 0:1 k ¼ 1 k ¼ 10 k ¼ 100

1.868· 10�4 1.869· 10�3 1.876· 10�2 1.960· 10�1 2:067þ 1:752i1.868· 10�4 1.869· 10�3 1.876· 10�2 1.960· 10�1 2:067� 1:752i2.625 2.625 2.625 2.625 2:067þ 1:752i3.863 3.862 3.847 3.696 2:067� 1:752i3.863 3.862 3.847 3.696 2.625

7.909 7.909 7.909 7.909 7.909

7.934 7.934 7.934 7.934 7.934

8.167 8.167 8.167 8.164 8.138

8.167 8.167 8.167 8.164 8.138

9.223 9.223 9.223 9.223 9.223



Fig. 12. Eigenvalue k6 ¼ 8:152: (a) Velocity field and (b) isobaric lines.



Fig. 14. Cantilever pipe conveying a fluid.


where w stands for the flexural displacement, E for the Young�s modulus, I for the inertia momentum, m for

the mass per unit length, L for the pipe length and f for the transverse load applied on the pipe per unit

length. In [33], Pa€ııdoussis provides the following expression for f

f ¼ �M €ww� 2MUo _wwox�MU 2 o

2wox2

:

Here, M represents the mass per unit length of the fluid and U the mean axial velocity of the internal

flow. Hence, substituting the above expression in (38), we get that the flexural motion of the pipe under the

flow effects can be modeled by the following linear equation:

EIo4wox4þ 2MU

o _wwoxþMU 2 o

2wox2þ ðM þ mÞ€ww ¼ 0; in ð0; LÞ; ð39Þ

provided with the boundary conditions (38)2;3. Pa€ııdoussis analyzes the stability of the coupled system by

computing the harmonic solutions of (39). More precisely, setting wðx; tÞ ¼ wðxÞe�kt with k 2 C, we obtain,

from (39), the following quadratic eigenproblem:

EIo4wox4� 2kMU

owoxþMU 2 o

2wox2þ k2ðM þ mÞw ¼ 0; in ð0; LÞ:

In [33], the solutions of the above eigenproblem are approximated by computing the eigenvalues of the

algebraic system resulting from the projection of Eq. (39) on the first eigenmodes of the beam.

In [36], a similar approach was used for the stability analysis of the coupled system. Again, it is assumed

that the pipe walls vibrate with beam modes. Then, by projecting Eq. (38)1 on the first ns beam eigenmodes

we obtain

M€ssþ Ks ¼ fg; ð40Þwith M and K the ns � ns matrices of mass and stiffness and s; fg 2 Rns the vectors of generalized dis-

placements and fluid loads on the interface, respectively. The numerical computations reported in [36] are

based on the assumption that these fluid loads at the fluid–structure interface, fg, can be expressed in terms

of added mass, added damping and added stiffness, i.e.

fg ¼ �Ma€ss� Ca _ss� Kas; ð41Þwhere Ma, Ca and Ka stand, respectively, for the ns � ns full matrices of added mass, damping, and stiffness.The above temporal decomposition of the fluid loads is not really exact. Indeed, even for small oscillations,

the non-linear character of the fluid introduces a dependence of the added mass and damping matrices with

respect to the frequency of the motion. The added matrices Ma, Ca and Ka are classically obtained using the

forced oscillation method, see [36,46,47]. The structure is subjected to a forced sinusoidal oscillation and the


above added matrices are computed by measuring the resulting fluid interface load over a fixed number of

periods. In [36] this is done numerically from the solution of the incompressible Navier–Stokes equations,

where the interface motion is taken into account via transpiration boundary conditions (see [13,36]). It is

noticed, in [36], that these fluid computations become very expensive when dealing with high frequency

oscillations (130 min of computing time for three oscillation periods).

Once the above added matrices have been computed, Eq. (40) can be rewritten as

ðM þMaÞ€ssþ Ca _ssþ ðK þ KaÞs ¼ 0: ð42Þ

Thus, the stability analysis can be carried out by computing its harmonic solutions, sðtÞ ¼ se�kt withs 2 Cns and k 2 C. Equivalently, we can compute the roots of the following characteristic polynomial:

detðk2ðM þMaÞ � kCa þ ðK þ KaÞÞ ¼ 0:

In this paragraph we intend to reproduce the main stability predictions obtained by Pa€ııdoussis and

Renou in [33,36], using the linearization–transpiration approach developed in Part I [15]. For the sake of

simplicity, and following [36], we will couple the flow into the pipe with the first three modes of a cantilever

beam modeling the pipe walls.In the sequel we will deal with a cantilever pipe of length L and width l, conveying an homogeneous

incompressible viscous fluid of density q and kinematic viscosity m, see Fig. 14. Let us consider the followingparabolic velocity profile on the inlet boundary Cin:

uCinðyÞ ¼ 6U

yl

1� y

l

�; y 2 ½0; l�;

where U stands for the mean velocity of the flow. The coupled fluid–structure system exhibits a steady

equilibrium state ðu0; p0; IXÞ satisfying Eqs. (1) and (2), i.e.

ru0u0 � 2mdiv eðu0Þ þ1

qrp0 ¼ 0; in Xf ;

divu0 ¼ 0; in Xf ;

u0 ¼ uCin; on Cin;

rðu0; p0Þn ¼ 0; on Cout;

u0 ¼ 0; on ct [ cb;

K0 ¼ �Zct[cb

UTrðu0; p0Þnda:

ð43Þ

Remark 11. For a symmetric velocity profile (with respect to the pipe axis) the fluid load is opposite on the

walls. Therefore, the resultantZct[cb

UTrðu0; p0Þnda ¼ 0

and then the fluid domain Xf at equilibrium is known.

The stability analysis of this equilibrium state can be carried out using the linearization–transpiration

framework introduced in Part I [15]. Hence, the study of the linear stability of ðu0; p0; IXÞ reduces to the

approximation of the leftmost eigenvalues of problem (3), i.e., find k 2 C, u : Xf ! C2, p : Xf ! C and

s 2 Cns , with ðu; p; sÞ 6¼ 0, such that


ru0uþruu0 � 2mdiv eðuÞ þ 1

qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on Cin;

rðu; pÞn ¼ �rrðu0; p0ÞUsn; on Cout;

u ¼ �kUs�ru0Us; on ct [ cb;

k2Msþ ðK þ B0Þs ¼ �Zct[cb

UTrðu; pÞnda:

ð44Þ

Remark 12. In this specific case, the outlet boundary Cout is mobile. Therefore, we must impose a ‘‘tran-

spiration’’ boundary condition which takes into account the transport termrrðu0; p0ÞUsn, coming from the

linearization of the ‘‘do nothing’’ boundary condition in the perturbed problem (see Part I [15]).

Using the discretization scheme (11) the above eigenproblem becomes a generalized eigenvalue problemof type (17). Its left-most eigenvalues can be approximated using Algorithm 1. In our numerical compu-

tations we have chosen the following data (reported in [36]):

L ¼ 1 m; m ¼ 160 kg=m2;

E ¼ 1:5� 109 Pa; q ¼ 1000 kg=m3;

l ¼ 0:04 m; M ¼ 40 kg=m2;

I ¼ 1:0053� 10�8 m4; m ¼ 5� 10�5 m2=s:

As in many studies of Pa€ııdoussis [33], the above mass values correspond to a mass ratio b ¼ M=ðM þ mÞof 0.2. In addition, the mean velocity U (in m/s) takes the following values: 0, 0.614, 1.228, 1.842, 2.456,

3.07 and 3.684.

The fluid domain inside the pipe was discretized using a 4000 triangles mesh, see Fig. 15. For each value

of U we compute the fluid state at equilibrium ðu0; p0Þ using a P2=P1 Navier–Stokes solver (see Fig. 16).

Remark 13. Since the fluid computation are carried out using a P2=P1 discretization of (43), we can define a

first explicit approximation of the added stiffness matrix B0 defined in (4).

In Algorithm 1 the number of requested eigenvalues was fixed to 6, the dimension of the Krylov sub-

space to 20 and the Ritz estimates tolerance to 10�6. For each value of U , the mean computational time was

3 min on a HP Visualize 8200 workstation with three Cayley iterations.

In Tables 3–9 we have reported, for each value of U , the first smallest real part eigenvalues and thecorresponding residuals. This results point out the sensitivity of the approximations of the imaginary part.

This phenomenon was already announced in Section 4.1, see also [18,27]. Indeed, the eigenvalues of (17)

Fig. 15. Mesh of the fluid domain Xf at equilibrium.

Fig. 16. Velocity magnitude at equilibrium for U ¼ 3:684 m/s.

Table 3

Eigenvalues and residuals for U ¼ 0 m/s

k kAx� kBxk1:371� 10�3 þ 9:662� 10�1i 3.560· 10�151:371� 10�3 � 9:662� 10�1i 3.560· 10�157:945� 10�3 þ 6:069i 5.063· 10�117:945� 10�3 � 6:069i 5.063· 10�111:966� 10�2 þ 17:01i 6.284· 10�71:966� 10�2 � 17:01i 6.284· 10�7

Table 4

Eigenvalues and residuals for U ¼ 0:614 m/s

k kAx� kBxk1:907� 10�1 þ 16:93i 1.241· 10�81:907� 10�1 � 16:93i 1.241· 10�82:060� 10�1 þ 5:988i 1.164· 10�102:060� 10�1 � 5:988i 1.164· 10�102:392� 10�1 þ 9:718� 10�1i 2.435· 10�122:392� 10�1 � 9:718� 10�1i 2.435· 10�12

Table 5


k kAx� kBxk3:713� 10�1 þ 16:67i 1.377· 10�53:713� 10�1 � 16:67i 1.377· 10�53:854� 10�1 þ 5:698i 4.279· 10�93:854� 10�1 � 5:698i 4.279· 10�95:067� 10�1 þ 9:871� 10�1i 3.942· 10�115:067� 10�1 � 9:871� 10�1i 3.942· 10�11

Table 6


k kAx� kBxk4:829� 10�1 þ 5:191i 5.889· 10�134:829� 10�1 � 5:191i 5.889· 10�135:304� 10�1 þ 16:17i 2.097· 10�65:304� 10�1 � 16:17i 2.097· 10�68:627� 10�1 þ 1:067i 1.631· 10�138:627� 10�1 � 1:067i 1.631· 10�13

Table 7


k kAx� kBxk3:220� 10�1 þ 4:476i 6.652· 10�143:220� 10�1 � 4:476i 6.652· 10�146:577� 10�1 þ 15:43i 4.131· 10�86:577� 10�1 � 15:43i 4.131· 10�81:495þ 1:288i 1.643· 10�141:495� 1:288i 1.643· 10�14


Table 8


k kAx� kBxk�3:477� 10�1 þ 4:000i 2.709· 10�14�3:477� 10�1 � 4:000i 2.709· 10�147:243� 10�1 þ 14:39i 1.731· 10�107:243� 10�1 � 14:39i 1.731· 10�102:675þ 1:295i 2.205· 10�142:675� 1:295i 2.205· 10�14

Table 9


k kAx� kBxk�1:026þ 4:181i 6.751· 10�15�1:026� 4:181i 6.751· 10�156:633� 10�1 þ 12:93i 1.270· 10�126:633� 10�1 � 12:93i 1.270· 10�123:930þ 6:927� 10�1i 1.636· 10�143:930� 6:927� 10�1i 1.636· 10�14


with large imaginary parts are mapped to eigenvalues of TC close to the unit circle, which are difficult toapproximate.

The left-most eigenvalues, namely, the smallest damping provided by our numerical computations are

compared to those obtained by Pa€ııdoussis and Renou [33,36] in Fig. 17. We can notice the good agreement

of our computations and those of Renou. These values are a little lower than those of the Pa€ııdoussis model.

However, all curves give comparable stability results. A flutter instability is predicted in the range of flow

velocities U 2 ½2:5; 3:5�. Fig. 17 shows a typical situation of flutter instability. Indeed, if the velocity of the

flow is augmented gradually the damping of the oscillations increases. However, with further increase in the

flow velocity, a point is reached from where the damping of the system decreases again, down to negative(unstable) values.

Fig. 17. Real part of the left-most eigenvalue: damping.

Fig. 18. Real part of the displacement in the unstable mode for U ¼ 3:070.

Fig. 19. Imaginary part of the displacement in the unstable mode for U ¼ 3:070.


In Figs. 18 and 19 we have reported, respectively, the real and imaginary part of the unstable mode (for

U ¼ 3:070). Finally, in Fig. 20, we have reported a comparison between the frequencies of this unstable

mode for each value of U . We can immediately point out the good agreement of these results.

5.3. Wind effects on a simplified bridge deck profile

In this paragraph we deal with the aeroelastic flutter instability analysis of bluff-bodies in a steady wing.

More precisely, we address the problem of flutter instabilities of prismatic cylinders with rectangular sec-

tion, and chord c to thickness d ratio equal to 4, i.e. c=d ¼ 4, see Fig. 21. This simple section has been

chosen because several experimental data are available [46,47].

As mentioned above, flutter is a self-excited oscillatory instability in which the fluid aerodynamic forces

put energy into the structure and progressively increase the amplitude of the motion. It corresponds to anegative damping and occurs at any velocity above the flutter boundary. This clearly distinguishes it from

vortex shedding instabilities, which are associated with a flow instability and arise when the frequency of

the shed vortex in the wake coincides with the natural frequency of the structure. Moreover, vortex-excited

Fig. 21. Fluid–structure configuration at equilibrium.

Fig. 20. Frequency of the mode that becomes unstable for U ¼ 3:070.


oscillation never produce divergent amplitudes. For a more detailed description of these aeroelastic phe-

nomena the reader can refer to [40].

In [46,47], Washizu et al. investigate experimentally the aeroelastic instabilities of several prismatic

cylinders with rectangular section in a heaving or in a torsional mode. We will consider here their wind

tunnel free oscillation experiments. In the free oscillation method the velocity of the uniform flow U isincreased step by step and, at each step, the structure is given a small initial heaving or rotating dis-

placement and let to go. The goal here is to obtain the mean limit cyclic amplitude of the free oscillation as

a function of the flow velocity.

In Fig. 22 we report the free oscillation results obtained by Washizu et al. in [46,47], for a rectangular

cylinder of ratio 4 in a heaving and a rotating mode. In these figures, the ordinate is the amplitude of the

limit cycle and the abscissa the dimensionless velocities Ufyd

and Ufhd, where fy and fh stand, respectively, for

the natural frequency of the structure in the heaving and torsional mode. In both figures the symbol }indicates the resonance speed, namely, the uniform flow velocity where the frequency of the shedding vortexin the wake coincides with the natural frequency of the structure in the corresponding mode.

Fig. 22. Experimental amplitude of the limit cycle, reported in [46,47]. (a) Heaving mode and (b) torsional mode.


Fig. 22(a) shows two regions of instability of which one is in the vicinity of the resonance speed. Washizuet al., in [46], pointed out that if the structural damping becomes large these two regions disappear. Thus,

they concluded that these two unstable regions correspond to vortex-shedding instabilities, and that no

transverse flutter occurs. On the other hand, see [47], Fig. 22(b) indicates that vortex-excited oscillation was

not observed in the free oscillation experiments, and torsional flutter can develop from a initial oscillation

of small amplitude. In short, a prismatic bar of ratio 4 cannot suffer flutter in a heaving mode but this can

take place in a torsional motion.

In this paragraph we wish to reproduce the above experimental results using the linearization–tran-

spiration method developed in Part I [15] and the numerical analysis introduced in Sections 3 and 4 of thepresent paper: namely, the finite element approximation (11) combined with Algorithm 1 for the left-most

eigenvalue computation.

We consider an elastically mounted rigid prismatic cylinder with rectangular section immersed in a

transverse incompressible flow. We assume that the structure can only vibrate on a transverse plane per-

pendicular to the prism and axial effects are not considered. Hence, the problem is studied in two di-

mensions. As in [46,47], we will confine our study to vertical and torsional body motions. In the sequel, Xf

stands for the two-dimensional domain occupied by the fluid and Xs for the rectangular section of the prism

with ratio c=d ¼ 4, and c denotes the fluid–structure interface.Let us consider a uniform flow velocity uCin

¼ ðU ; 0ÞT on the inlet boundary Cin, see Fig. 21. For low

Reynolds regimes, the coupled fluid–structure system exhibits a steady equilibrium state satisfying Eqs. (1)

and (2). Here the modal basis U is given by

U ¼ 0

1

� �0 �11 0

� �� xy

� �� ;

that generates the vertical displacement and the rotation around the center of mass of the structure.

Remark 14. Since the uniform flow is symmetric, the resultant of the fluid loads at the interface

K0 ¼ �ZcUTrðu0; p0Þnda ¼ 0;

and then the fluid domain Xf , at equilibrium is known.


The stability analysis of this equilibrium state can be carried out using the linearization–transpirationframework introduced in Part I [15] leading to the spectral problem (3). Here s ¼ ðy; hÞT 2 R2, with y the

vertical displacement and h the rotation. Using the discretization scheme (11) the above eigenproblem

becomes a generalized eigenvalues problem of type (17). Thus, its left-most eigenvalues will be approxi-

mated using Algorithm 1.

In our numerical computations we have used the following parameter values: q ¼ 1:293 kg/m3,

m ¼ 5� 10�5 m2/s, c ¼ 4 m, d ¼ 1 m,

M ¼ m 0

0 Ih

� �; K ¼ ky 0

0 kh

� �; ky ¼ mw2

y ; kh ¼ mw2h;

with m ¼ 100 kg, Ih ¼ 100 kgm2 and fy ¼ fh ¼ 10�4 Hz. All flow regimes are at low speed, the range ofReynolds number Ud=m being ½0; 80�, so that, as in [46,47], we cover a range of dimensionless velocities

between 0 and 40.

The fluid domain Xf around the deck section was discretized using an unstructured mesh of 6648 tri-

angles, see Fig. 23. For each value of U the steady flow ðu0; p0Þ was computed using a P2=P1 Navier–Stokes

solver (Fig. 24) and P1=P1 elements for the stability analysis. In Algorithm 1 the number of requested

eigenvalues was fixed to 10, the dimension of the Krylov subspace to 30 and the Ritz estimates tolerance to

Fig. 23. Triangulation of the fluid domain at equilibrium.

Fig. 24. Iso-values of the flow x-velocity at equilibrium for U ¼ 4 m/s.


10�6. For each value of U , the mean computational time was 13 min on a HP Visualize 8200 workstationwith 2 Cayley iterations.

First, we consider problem (3) with the rigid body oscillating in its heaving mode. In Fig. 25 we have

plotted the damping of the system, namely, the real part of the left-most eigenvalue, as a function of the

dimensionless velocity. Since Re(k1) only takes positive values we conclude that the reference equilibrium is

stable, i.e. not flutter occurs in the range of non-dimensional velocities ½0; 45�, as predicted by Washizu et al.

in [46]. As mentioned above, the vortex-shedding instability observed in [46] are associated with flow

separation around the body, namely, with a flow instability. This is an unsteady phenomena that cannot

take place in our low speed framework. Indeed, the fluid flow is stable without interface motion (see Fig.24). Therefore, vortex-shedding instabilities cannot be recovered with our approach.

In Fig. 26 we have reported the stability results for the torsional motion. These numerical results are in

very good agreement with experiments in [47], see Fig. 22. When the velocity of the uniform flow U

Fig. 25. Real part of the left-most eigenvalue (heaving mode).

Fig. 26. Real part of the left-most eigenvalue (torsional mode).


becomes large, the real part, Re(k1), of the left-most eigenvalue takes negatives values. Thus, Fig. 26 in-dicates that torsional flutter can occur in the range of dimensionless velocities [15,45].

6. Conclusion

In this work we addressed the flutter problem of a coupled fluid–structure system involving an in-

compressible Newtonian fluid and a reduced structure. This was done by a linear stability approach. We

used the linearization–transpiration formulation developed in Part I [15]. This allows us to reduce thestability analysis to the computation of the left-most eigenvalues of a coupled spectral problem. This

coupled eigenproblem involves the linearized incompressible Navier–Stokes equations (written in a fixed

domain) and those of a reduced linear structure. The coupling is realized through specific transpiration

interface conditions.

We proposed a stabilized finite element method for the discretization of the fluid–structure eigenprob-

lem. Although the numerical results pointed out the good behavior of this scheme, no error analysis was

carried out in this paper. It will be a forthcoming topic of our work. The discrete formulation leads to a

sparse generalized eigenvalues problem with ‘‘infinite’’ eigenvalues. The smallest real part eigenvalues,which are relevant for the stability analysis, were approximated by means of an iterative algorithm which

combines a generalized Cayley transform and an IRAM method.

Finally, we have reported three significant numerical experiments: a structure immersed in a fluid at rest,

a cantilever pipe conveying a fluid flow and a rectangular bridge deck profile under wind effects. The

corresponding numerical results were compared to former approaches and experimental data. The com-

parisons have indicated that our numerical results are very satisfactory and promising.

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Documents

Linear stability analysis in fluid–structure interaction with transpiration. Part II: Numerical analysis and applications