efficient method to compute full eigenspectrum of ...efficient method to compute full eigenspectrum...

© 2015 The Korean Society of Rheology and Springer 177

Korea-Australia Rheology Journal, 27(3), 177-188 (August 2015)DOI: 10.1007/s13367-015-0018-8

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Efficient method to compute full eigenspectrum of incompressible viscous flows:

Application on two-layer rectinear flow

Jaewook Nam1,* and Marcio S. Carvalho

2

1School of Chemical Engineering, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 440-746, Republic of Korea

2Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marques de Sâo Vicente 225, Gávea, Rio de Janerio, RJ 22453-900, Brazil

(Received March 3, 2015; final revision received May 1, 2015; accepted May 4, 2015)

An efficient algorithm based on the matrix transformation method (Valério et al., 2007) is presented forsolving the generalized eigenvalue problem (GEVP) derived from linear stability analysis of incompressibleviscous flow. The proposed method uses the formulation based on primitive variables, i.e. velocity and pres-sure, instead of streamfunction used by typical Orr-Sommerfeld equation. A series of matrix operationsremoves non-physical eigenvalues at infinity and leads to a non-singular smaller size eigenvalue problem(EVP), which contains full eigenspectrum, than the original GEVP. Two different solution strategies for thetransformed EVP are proposed, and their accuracies are discussed. The proposed procedure is used to solvethe stability of two layer rectilinear flow. The computed eigenspectrum are compared to previously reportedvalues.

Keywords: eigenvalues, two-layer channel flow, interlayer, stability analysis, interfacial mode, finite ele-

ment method

1. Introduction

Linear stability analysis of viscous flow is an important

tool for a complete study of different applications. In par-

ticular, rectilinear two-layer flow is an important funda-

mental model system in many engineering applications,

such as macroscopic oil transportation (Hu and Joseph,

1989) and microscopic coating process (Severtson and

Aidun, 1996, Lee et al., 2010). The nature of instabilities

that occur in this model flow can be effectively studied by

considering Couette flow, Poiseuille flow, or combination

of both. Linear stability of a given base flow is determined

by tracking the evolution of an infinitesimal disturbance in

the flow. Since the flow has an interlayer between the two

fluids, it has additional instabilities that are not observed

in single phase flow. For example, viscosity and density

difference between the liquid layers may lead to growing

disturbances in the interlayer, so called the interfacial

mode (Yih, 1967).

In many cases, stability of the two-layer parallel flow is

described by two coupled by Orr-Sommerfeld equations.

When the perturbation is not restricted to short-wave-

length or long-wavelength limits, it is difficult to get an

analytical solution using conventional perturbation scheme.

Therefore, the equation needs to be solved numerically.

However, the stability formulation is well-known for suf-

fering form parasitic growth problem or stiff eigenvalue

problem, and this is a driving force for development of

various numerical methods (Drazin and Reid, 2004).

Finite element method (Li and Kot, 1981; Yiantsios and

Higgins, 1986), compound matrix method (Yiantsios and

Higgins, 1988a), and spectral Chebyshev tau method (Su

and Khomami, 1992; Severtson and Aidun, 1996) have

been proposed to overcome these problems. While the

compound matrix method can only track a single eigen-

mode with a proper initial guess, the finite element method

and the spectral method can be used to compute the whole

eigenspectrum at once without an initial guess.

The discretization of the system of linear differential

equations that describes the evolution of infinitesimal small

amplitude perturbation of a viscous incompressible flow

leads to a non-Hermitian, generalized eigenvalue problem

(GEVP), Jc = ωMc, where J, M, ω and c are Jacobian,

mass matrix, generalized eigenvalue and corresponding

eigenvector, respectively. However, finding eigensolutions

of a GEVP is a challenging task. In general, accurate

eigenvalues requires high degree of discretization that

leads to large matrices. The size of the full eigenproblem

rules out any algorithm that computes the entire eigen-

spectrum; rather, only a portion of the spectrum − typi-

cally, eigenvalues with largest real parts or leading modes

− is calculated. As discussed by Saad (Saad, 1989), var-

ious numerical techniques that are based on iterative

methods, such as subspace iteration methods and projec-

tion methods are used to approximate the original eigen-

problem by a smaller problem. Typically, these methods

are coupled with preconditioning, for example, exponen-

tial preconditioner (Christodoulou and Scriven, 1988) or*Corresponding author; E-mail: [email protected]

Jaewook Nam and Marcio S. Carvalho

178 Korea-Australia Rheology J., 27(3), 2015

Chebyshev acceleration technique (Saad, 1984) to amplify

the leading modes.

Moreover, because the mass matrix is singular, the

GEVP that describes the growth of perturbations presents

obviously non-physical eigenvalues with unrealistic large

moduli, or called eigenvalues at infinity, which come from

inevitable perturbations of original GEVP problem during

numerical computation. A simple way to overcome this

difficulty is mapping eigenvalues at infinity to specified

values, e.g. shift-and-invert and mapping techniques (Gous-

sis and Perlstein, 1988), or evade direct computations of

them, e.g. QZ algorithm (Gloub and Van Loan, 1996) and

LZ algorithm (Kaufman, 1975). Because of the usual large

dimension of the problem, these methods are combined

usually with subspace or projection method to handle a

large-scale problem.

To compute full eigenspectrum efficiently, one has to

decrease the dimension of matrices. For a single layer rec-

tilinear flow, Orszag (Orszag, 1971) used Chebyshev spec-

tral method for the stability analysis and solved the resulting

eigenvalue problem using QR algorithm (Gloub and Van

Loan, 1996). There is an attempt to reduce the size of

eigenvalue problem using finite difference scheme (Gary

and Helgason, 1970). Recently, Valério et al. showed that

the GEVP resulting from linear stability analysis of vis-

cous flows in Galerkin finite element framework can be

reduced to a smaller non-singular eigenvalue problem

(Valério et al., 2007). By exploiting the structure of the

mass and Jacobian matrices, eigenvalues at infinity are fil-

tered by an algebraic procedure. This approach was later

extended to viscoelastic flow (Valério et al., 2009).

Here, we show that the approach proposed by Valério et

al. (2007) can be generalized to more complicated flows.

The proposed method is used to compute the spectrum of

two-layer rectilinear flow. The linear stability analysis is

formulated in terms of the primitive variables, i.e. velocity

and pressure. Unlike the forth-order Orr-Sommerfeld equa-

tion that requires Hermite cubic basis functions, in the

mixed formulation used here, the primitive variables can

be expanded by quadratic and linear discontinuous basis

functions. Extending and generalizing Valério et al. (2009)

approach, the eigenvalues at infinity are removed, leading

to a non-singular smaller size simple eigenvalue problem,

by a series of matrix operations. The resulting reduced

eigenvalue problem is solved by three diffeent methods,

discussed in Sec. 3.2. The computed eigenspectrum are

compared to previously reported values and the perfor-

mance of the different methods are compared.

2. Derivation of generalized eigenvalue problem(GEVP)

We use the two-layer parallel channel flow as shown in

Fig. 1, as the model problem to evaluate the solution

approach proposed in this work. Galerkin/finite element

method is used to discretize the set of differential equa-

tions that described the amplitude of the perturbations.

In this particular formulation, the disturbances are

approximated by quadratic basis functions for velocity φj

and linear discontinuous basis function for pressure ψj:

, (1)

where N is the number of elements used to discretize the

domain, Uj = iUj + kWj and Pj are the coefficients for veloc-

ity vector and pressure, respectively.

The velocity field disturbance has a jump across the

interlayer. To describe this jump, we use two degrees of

freedom for the interfacial velocity. The pressure jump

across the interlayer is naturally handled by the discon-

tinuous basis function. Detail formulations and discretiza-

tion by Galerkin’s method and finite element basis functions

are discussed elsewhere (Nam and Carvalho, 2010).

The growth rate of the perturbation and their amplitude

are described by the generalized eigenvalue problem (GEVP)

Jc = ωMc (2)

where ω is the eigenvalue (growth rate) and c is the eigen-

vector that contains all the finite element coefficients for

all unknown fields. The Jacobian and mass matrix entries

are summarized in Tables 1 and 2. The scheme used to

number each equation and degree of freedom of the sys-

tem is presented in Fig. 2. This numbering scheme is

important to define the matrices sub-blocks that are used

in the procedure to eliminate the infinite eigenvalues pre-

sented next.

The choice of dimensionless parameters and variables

depends on the type of flow − either (nearly) Couette flow

or (nearly) Poiseuille flow. There are three ratios, which

are independent of the flow type: viscosity ratio m = μ1/μ2,

density ratio r = ρ1/ρ2, and thickness ratio n = H1/H2.

ul = j 1=

2N 2+

∑ Ujφj Pl = j 1=

2N

∑ Pjψj

Fig. 1. Base flow configuration for the two-layer channel flow.

Pd and Pu are the downstream and upstream pressure, ΔP = Pd

− Pu is the pressure difference across the flow domain, and L is

the channel. Note that the flow is driven by both pressure gra-

dient (Pd − Pu/L) and moving wall with velocity Uw. The system

consist of two fluids with the interlayer, moving substrate and sta-

tionary wall. With the pressure gradient across the system, the

base flow profile is the combination of Couette and Poisuille flow.

Efficient method to compute full eigenspectrum of incompressible viscous flows: Application on two-layer rectinear flow

Korea-Australia Rheology J., 27(3), 2015 179

Reynolds number represents the ratio between viscous and

inertia forces based on the fluid property of the second

layer and a characteristic velocity:

for (nearly) Couette flow,

for (nearly) Poiseuille flow (3)

where Uw is the moving wall velocity and (z = 0) is the

velocity at the interface. The interfacial tension number

represents the ratio between interfacial tension and vis-

cous stress for (nearly) Couette flow or inertia for (nearly)

Poiseuille flow:


for (nearly) Poiseuille flow. (4)

The ratio of a characteristic velocity to a gravitational

wave velocity is represented as the Froude number:



NRe C, = ρ2UWH2

μ2

-------------------

NRe P, = ρ2U2

bz = 0( )H2

μ2

-----------------------------------

U2

b

NT C, = σ

μ2Uw

------------

NT P, = σ

ρ2H2 U2

bz = 0( )[ ]

2-----------------------------------------

NF C, = ρ2 r 1–( )gH2

2

μ2Uw

-----------------------------

NF P, = ρ2 r 1–( )gH2

2

U2

bz = 0( )[ ]

2-----------------------------

Table 1. Mass and Jacobian matrix entries for the two-layer par-

allel flow system. l = 1 is for the first layer and l = 2 for the sec-

ond layer. Ω1 and Ω2 span and , respec-

tively. One can find the residual equations ,

and in Nam and Carvalho (2010).

Name Equations

Mass

matrix

Jacobian

matrix

H1– z 0≤ ≤ 0 z H2≤ ≤Rt,mx

j, Rt,mz

j, Rs,mx

j, Rt,mx

j

Rc

j

∂Rt,mx

j

∂Uk

-------------– = − Ωl

∫ ρlφjφkdz

∂Rt,mz

j

∂Wk

-------------– = − Ωl

∫ ρlφjφkdz

∂Rs,mx

j

∂Uk

-------------- = − Ωl

∫ ρliαUl

bφjφk + μl

dφj

dz-------

dφk

dz-------- + μlα

2φjφk dz

∂Rs,mx

j

∂Wk

-------------- = − Ωl

∫ ρl

dUl

b

dz---------φjφk dz

∂Rs,mx

j

∂Pk

-------------- = Ωl

∫ iαφjψk[ ]dz

∂Rs,mx

j

∂Wk

-------------- = Ωl

∫ ρliαUl

bφjφk + μl

dφj

dz-------

dφk

dz-------- + μlα

2φjφk dz

∂Rs,mx

j

∂Pk

-------------- = Ωl

∫dφj

dz-------– ψk dz

∂Rc

j

∂Uk

--------- = Ωl

∫ iαψjφk[ ]dz

∂Rc

j

∂Wk

--------- = Ωl

∫ ψj

dφk

dz-------- dz

Table 2. Mass and Jacobian matrix entries for interfacial and

boundary conditions in the two-layer parallel flow system. Inter-

facial node for velocity is Ni and pressure node near the interlayer

is Np,i and Np,i+1. Note that the position where interfacial condition

are evaluated is z = 0. One can find the residual equations ,

, and in Nam and Carvalho (2010).

Name Equations

Shear stress

continuity

Normal stress

balance

Velocity

continuity

Kinematic

condition

No-slip

condition

Rs mx,

j

Rs mx,

jRc

j

∂Rs mx,

Ni

∂Uk

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

μ1

dφk

dz-------- k = Ni 1– ,Ni 2–( )

μ1

dφk

dz-------- −

dφk

dz-------- k = Ni( )

μ2

dφk

dz--------– k = Ni 1+ , Ni 2+ , Ni 3+( )

⎩⎪⎪⎪⎨⎪⎪⎪⎧

∂Rs mx,

Ni

∂Wk

-------------- = iμiα k = Ni( )

iμ2α– k = Ni 1+( )⎩⎨⎧

∂Rs mx,

∂h-------------- = μ1

d2U1

dz2

----------- − μ2

d2U2

dz2

-----------

∂Rs mx,

Ni

∂Wk

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

2μ1

dφk

dz-------- k = Ni, Ni 1– ,Ni 2–( )

2μ2

dφk

dz--------– k = Ni 1+ , Ni 2+ , Ni 3+( )

⎩⎪⎪⎨⎪⎪⎧

∂Rs mz,

Ni

∂Pk

-------------- = ψk– k = Np i, 1– ,Np i,( )

ψk k = Np i, 1+ ,Np i 2+,( )⎩⎨⎧

∂Rs mz,

Ni

∂h-------------- = α

2σI

∂Rs mx,

Ni

∂Uk

-------------- = 1 k = Ni( )

1– k = Ni 1+( )⎩⎨⎧

∂Rs mx,

Ni

∂h-------------- =

dU1

dz---------

z 0=

− dU2

dz---------

z 0=

∂Rs mz,

Ni

∂Wk

-------------- = 1 k = Ni( )

1– k = Ni 1+( )⎩⎨⎧

∂RtK

∂h----------– = −1

∂RsK

∂Wk

----------- = −1 k = Ni( )

∂RsK

∂h----------- = iαU1

b z = 0( )

∂Rs mx,

k

∂Uk

-------------- = 1 k = 1, Nv1 Nv2+( )

∂Rs mz,

k

∂Wk

-------------- = 1 k = 1, Nv1 Nv2+( )



The choice of unit for the pressure value is different for

each flow type. Therefore the dimensionless pressure gra-

dient is defined as



3. Filtering eigenvalues at infinity with matrixtransformation

The mass matrix M is singular, because the continuity

equation for incompressible fluid, the no-slip boundary

conditions and the interfacial conditions (except kinematic

condition) have no time derivative. Thus, the number of

finite eigenvalues of the generalized eigenvalue problem

(2) is smaller than the dimension of the problem 6N + 5.

The missing 4N + 3 eigenvalues are commonly referred to

as eigenvalues at infinity, because if the mass matrix is

slightly perturbed to remove the singularity, e.g. M* = M

+ εI, large eigenvalues appear in the spectrum, and they

grow unbounded as (Valério et al., 2007). During

the computation of the eigenspectrum, truncation errors

and round-off errors may cause perturbations of the mass

matrix and lead to unrealistically large eigenvalues.

The eigenvalues ω of Eq. (2) are the roots of the char-

acteristic polynomial p(ω) = det(J − ωM). The eigenval-

ues are the value of ω for which the homogeneous system

has non-trivial solution c. A different matrix

pair and may define a new GEVP

with non-trivial solution and the same eigenvalues ω.

and can be constructed by multiplying ω-independent,

full-ranked matrices X and Y to left and right side of both

J and M, such that and . The matrices

X and Y arise while solving with a two-sided

Gaussian elimination, in the sense that row and column

operations are allowed. The non-trivial solutions of the

original system and of the transformed sys-

tem are related by . Therefore one

can recover the solution of the original system only by

simple matrix multiplications to the solution of the trans-

formed system.

3.1. Matrix transformations

To proceed with the appropriate matrix operations to fil-

ter the infinite eigenvalues, it is convenient to assemble

the Jacobian and mass matrices following the equation

and variable-numbering scheme shown in Fig. 2. The

resulting matrices can be partitioned into 5 × 5 block

structure with square blocks along the diagonal. The val-

ues on the right and underneath the matrices indicate the

dimension of each block

NG C, = dP/dx( )H2

2

μ2Uw

-------------------------

NG P, = dP/dx( )H2

ρ2 U2

bz = 0( )[ ]

2----------------------------------

ε 0→

J ωM–( )c = 0

J M J ωM–( )c = 0

c J

M

J = XJY M = XMY

Jc = ωMc

J ωM–( )c = 0

J ωM–( )c = 0 Yc = c

Fig. 2. The sequence of entries in weighted residual equations inside residual vector and unknown coefficient for finite element expan-

sion inside solution vector.



. (7)

Note that most of the sub-blocks are zero, especially in

the mass matrix, and the last block of the Jacobian matrix

I[4] is the identity matrix of dimension four. Algebraic

equations associated with Dirichlet boundary conditions −

no-slip conditions − do not have a time derivative, and the

perturbed velocity field vanishes at these boundaries. In

the equation and variable numbering scheme used, they

correspond to the last 4 equations of the system. Conse-

quently, in the matrix configuration of Eq. (7), they belong

to the last blocks of J and M, which are the identity matrix

and zero matrix, respectively. Therefore they can be elim-

inated from the system without affecting the eigenspec-

trum.

After elimination of the rows and columns related to no-

slip boundary conditions, the upper 4 × 4 sub-blocks of

the mass matrix Mb and the Jacobian matrix Jb remains. It

is convenient to redefine the block structure of B = Mb −

ωMb as :

. (8)

Note that B44 is always invertible when Jb is not singu-

lar. Therefore, a transformation matrix can be defined

to eliminate such block matrices B41 and B42. The resulting

transformed matrix is defined as :

(9)

where

. (10)

Matrix A is defined as the first 3 × 3 blocks of :

. (11)

Since the matrix is a lower triangular matrix with

diagonal entries equals to one, its determinant is equal to

one. The characteristic polynomial of is equal to the

characteristic polynomial of the original matrix B,

. (12)

Furthermore, the characteristic polynomial of is related

to that of A

. (13)

Because B44 is independent of ω, i.e. its determinant is just

a number, the roots of the characteristic polynomial of

are the same roots of det A = 0, i.e. the transformation

does not affect the eigenspectrum of B. Note that the rank

of , B, and Jb are the same, because B44 is invertible.

The eigenspectrum of B is equal the eigenspectrum of A.

Now, the matrix structure is similar to the rectilinear sin-

gle-layered flow analyzed in Valério et al. (2007). The

sub-blocks and B13 are invertible, when matrix is

nonsingular. Therefore, one can construct matrices

and to eliminate and B23 from A. The transformed

matrix is defined as:

(14)

where

,

. (15)

As in Eq. (12), the multiplication by and does

not change the spectrum of A:

M =

M11 M12 0 M14 M15

M21 M22 0 M24 M25

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ 2N

2N 2–

2N

3

4

2N 2N 2– 2N 3 4

J =

J11 J12 J13 J14 J15

J21 J22 J23 J24 J25

J31 J32 0 J34 J35

J41 J42 0 J44 J45

0 0 0 0 I 4[ ]⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ 2N

2N 2–

2N

3

4

B =

B11 ω( ) B12 ω( ) B13 B14 ω( )

B21 ω( ) B22 ω( ) B23 B24 ω( )

B31 B32 0 B34

B41 B42 0 B44⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

2N

2N 2–

2N

3

2N 2N 2– 2N 3

Tr

1( )

B

B =

B11 ω( ) B12 ω( ) B13 B14 ω( )

B21 ω( ) B22 ω( ) B23 B24 ω( )

B31 B32 0 B34

0 0 0 B44⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

= B Tr

1( )

Tr

1( ) =

I 2N[ ] 0 0 0

0 I 2N 2–[ ] 0 0

0 0 I 2N[ ] 0

B44

1–– B41 B44

1–– B42 0 I 3[ ]⎝ ⎠

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

B

A =

B11 ω( ) B12 ω( ) B13

B21 ω( ) B22 ω( ) B23

B31 B32 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

Tr

1( )

B

pB ω( ) = det B = det B det Tr

1( ) = det B = pB ω( )

B

pB ω( ) = det B = det B44 det A = det B44pA ω( )

B

B

B31 B

Tl

2( )

Tr

2( )B32

A

A =

B11 ω( ) A12 ω( ) B13

A21 ω( ) A22 ω( ) 0

B31 0 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

= Tl

2( )A Tr

2( )

Tl

2( ) =

I 2N[ ] 0 0

B23B13

1–– I 2N 2–[ ] 0

0 0 I 2N[ ]⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

Tr

2( ) =

I 2N[ ] B31

1–B32– 0

0 I 2N 2–[ ] 0

0 0 I 2N[ ]⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

Tl

2( )Tr

2( )



. (16)

Furthermore, one can rewrite the characteristic polynomial

of in terms of block matrices:

. (17)

Again, and B13 are independent of ω, so the eigen-

spectrum of is the same as .

From the relationship between the characteristic poly-

nomials of the transformed matrix, Eq. (12) and (16), the

final relationship between the characteristic polynomial of

the original matrix and that of the final transformed matrix

becomes

. (18)

Therefore the eigenspectrum of is the

same as that of the original GEVP, when the original Jaco-

bian matrix is nonsingular.

The reduced eigenvalue problem can be written as

(19)

where the matrices and can be

evaluated in terms of the blocks of the original GEVP

matrices:

,

, (20)

and matrices with bar are computed as

,

,

,

,

,

,

,

,

,

. (21)

It is important to note that the dimension of the simple

eigenproblem is (2N − 2), less than 1/3 of the original gen-

eralized eigenproblem.

3.2. Solution of the reduced eigenvalue problemAs discussed by Valério et al. (2007), both and

are invertible, when the original Jacobian J is non-singu-

lar. Therefore the eigenvalue ω can be obtained by either

, (22)

or

. (23)

In most of our computations, is usually well-con-

ditioned, i.e. it has smaller condition number, comparing

with . One may attempt to use Eq. (22) instead of Eq.

(23), because the error from the inversion of may be

less than that from the inversion of . However, the

accuracy of eigenvalue depends on the conditioning of

eigenvalue.

If the original matrix A is perturbed with an arbitrary

small parameter t − − then the eigenvalue

problem becomes , where is the

perturbed eigenvalue. In numerical computations using

floating point operations, it is reasonable to approximate

the norm of the perturbation matrix in terms of

the norm of the original matrix , i.e. ≈ ,

where is a machine epsilon (Anderson et al., 1999).

If t is small enough, the perturbed eigenvalue can be

approximated as . According to the

eigenvalue perturbation theory (Gloub and Van Loan, 1996),

the sensitivity of the eigenvalue is bounded by

(24)

where cl and cr are the left and right eigenvector of A, and

κ(λ) is the condition number for eigenvalue λ.

From above, one can rewrite estimated error bound of

the computed eigenvalue as below. When Eq. (22) is con-

sidered, the error bound becomes

. (25)

For Eq. (23), the error bound is roughly approximated as

. (26)

Both Eqs. (25) and (26) share the same left and right

eigenvectors, i.e. their condition numbers are the same.

Hence, there is the possibility that Eq. (23) can lead to

more accurate results than Eq. (22), especially when the

matrix norm is small and the modulus of the perturbed

eigenvalue ω(t) is large. At some conditions, we found

spurious wiggles of the most dangerous growth rate versus

pAω( ) = det A = det Tl

2( )det A det Tr

2( )= det A = ± pA ω( )

A

pAω( ) = det A = det B31 det A22 ω( ) det B13

= det B31 det B13pA22ω( )

B31

A A22

pA22ω( ) = ±

pA ω( )

det B31 det B13

---------------------------------- = ±pB ω( )

det B44det B31 det B13

--------------------------------------------------

A22 J22= σM22–

J22c2 = ωM22c2

2N 2–( ) 2N 2–( )× M22 J22

J22 = J23J13

1–J22–( ) J31

1–– J32( ) + J23J13

1–J22–( )J12 + J22

M22 = J23J13

1–M22–( ) J31

1–– J32( )

+ J23J13

1–M22–( )M12 + M22

J11 = J14– J44

1–J41 + J11

J12 = J14– J44

1–J42 + J12

J21 = J14– J44

1–J41 + J21

J22 = J14– J44

1–J42 + J22

J31 = J34– J44

1–J41 + J31

J32 = J34– J44

1–J42 + J32

M11 = J14– J44

1–J41 + M11

M12 = J14– J44

1–J42 + M12

M21 = J14– J44

1–J41 + M21

M22 = J14– J44

1–J42 + M22

J22 M22

M22

1–J22c2 = ω c2

J22

1–M22c2 =

1

ω----c2

M22

J22

M22

J22

A t( ) = A + tE

A t( )c t( ) = λ t( )c t( ) λ t( )

L2 tE 2

L2 A 2 tE 2 εm A 2

εm

λ t( ) = λ + dλ/dt( )t

dλ/dt

dλ

dt--------

λ t( ) λ–

t------------------- E 2≤≈

cl 2 cr 2

cl cr⋅-------------------

κ λ( )

⎧ ⎨ ⎩

ω t( ) ω– εm≤ M22

1–J22 2κ ω( )

1

ω t( )----------

1

ω----– εm J22

1–M22 2κ

1

ω------

⎝ ⎠⎛ ⎞≤

ω t( ) ω– εm≤J22

1–M22 2

ω t( ) 2---------------------κ ω( )⇒



wavenumber plot computed from Eq. (22) in the small

wavenumber regime. In most cases, however, the resulting

values from both equations do not show any significant

difference.

Most of numerical methods for eigenvalue computations

are based on an iterative scheme, like QR iteration using

Schur decomposition. The convergence property and

accuracy of computed eigenvalues depend largely on the

separation between eigenvalues (Gloub and Van Loan,

1996). In other words, an eigenvalue ωi having a large

modulus ratio to the closet one ωi+1, i.e. a large value of

, will shows a better accuracy and convergence

rate. The separation between eigenvalues can be adjusted

easily by the shift-and-invert method:

(27)

where δ is shift factor. This enable to get a large separa-

tion of eigenvalues near δ. Because the size of the reduced

eigenvalue problem is smaller than the original one, the

shift-and-invert method applied on the reduced problem

can save significant amount of computational time. Note

that Eq. (23) is a special case of Eq. (27).

In this study, LAPACK subroutines (Anderson et al.,

1999) are used to solve the corresponding eigenvalue pro-

blems. A generalized eigenvalue solver for square com-

plex nonsymmetric matrices using QZ algorithm, ZGGEV

subroutine, is used to handle the original and reduced

GEVP, Eq. (2) and Eq. (19). For different forms of the

simple eigenvalue problem, Eqs. (22), (23), and (27),

eigenvalues are computed by QR algorithm for square

complex nonsymmetric matrix, ZGEEV subroutine.

3.3. Recovering the original generalized eigenvectorsThe generalized eigenvectors c of the original GEVP,

Eq. (2) can be recovered from the eigenvector c2 in Eq.

(19). The procedure is simply reversing the transformations.

The original eigenvalue problem without the boundary

conditions can be expressed in terms

of the block matrices used in Sec. 3.1:

,

,

,

,

,

(28)

where . Therefore, the original

problem is equivalent to a new problem ,

and the eigenvectors for the original problem can be

retrieved by

. (29)

In terms of block matrices used in Sec. 3.1 and

, Eq. (28) can be written as

(30)

where . Accord-

ing to Sec. 3.1, when the Jacobian Jb is invertible, B44 and

is non-singular. Hence, c1 = c4 = 0 from the third and

the fourth equations of Eq. (30). This leads the second

equation of Eq. (30) to the reduced GEVP, Eq. (19), i.e.

. Note that c3 can be obtained

from the first equation: . The solution of

Bcb = 0 in terms of c2 is

. (31)

Therefore the original generalized eigenvector cb can be

directly recovered from the reduced eigenvector c2.

4. Applications of the proposed method

4.1. An example: two-layer rectilinear flow with fourelements

As an illustration, we consider a mesh with four ele-

ments, e.g. N = 4, two for the first layer and two for the

second layer as shown in Fig. 3. The number of unknown

coefficients for velocity, pressure, and interfacial height

are 2(2N + 2) = 20, 2N = 8, and 1, respectively. After

ω i /ω i 1+

J22 δ M22–( )1–

M22c2 = 1

ω δ–------------ c2

Jb

ωMb

–( )cbBc

b= = 0

Bcb = 0

BTr

1( )Tr

1( )1–

cb = 0

A B

0 B44⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

Tr

1( )1–

cb = 0

Tl

2( )1–

ATr

2( )1–

B

0 B44⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

Tr

1( )1–

cb = 0

Tl

2( )1–

0

0 I⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

A B

0 B44⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

Tr

2( )1–

0

0 I⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

Tr

2( )1–

cb = 0

Tl

2( )1– A B

0 B44⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=A

Tr

2( )1–

Tl

1( )1–

cb

=c

= 0

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

⎧ ⎪ ⎨ ⎪ ⎩

B = B14 ω( ), B24 ω( ), B34[ ]T

Bcb = 0 Ac = 0

cb = Tr

1( )Tr

2( )c

c = c1, c2, c3, c4[ ]T

B11 ω( )c1 + A12 ω( )c2 + B13c3 + B14 ω( )c4 = 0,

A21 ω( )c1 + A22 ω( )c2 + A24 ω( )c4 = 0,

B31c1 + B34c4 = 0,

B44c4 = 0.⎩⎪⎪⎪⎨⎪⎪⎪⎧

A12 ω( ) = − J11 ωM11–( )J31

1–J32 + J12 ωM12–( )

B31

A22 ω( )c2 = J22 ωM22–( )c2 = 0

c3 = −B13

1–A12 ω( )c2

cb = Tr

1( )Tr

2( )

0

c2

−B13

1–A12 ω( )c2

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

−B31

1–B32c2

c2

−B13

1–A12 ω( )c2

B44

1–B41B31

1–B32c2 − B44

1–B42c2⎝ ⎠

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞



elimination of no-slip conditions, the resulting matrix

structure is shown in Fig. 4a, the solution vector order and

equation numbers are shown in Fig. 3. The next step is to

eliminate the blocks and using the transformation

matrix defined in Eq. (10). In practice, the inverse oper-

ation does not occur explicitly for evaluating and

Instead, Gauss elimination was performed on the

rectangular matrices and in order to

decrease number of operations for minimizing round-off

error as discussed in Valério et al. (2007). The trans-

formed matrix structure is shown in Fig. 4b. The structure

of the top 3 × 3 block of , defined as matrix A, is shown

in Fig. 5a. Similarly, row and column permutation-based

Gauss elimination is used to compute and

efficiently and eliminate the matrix blocks A32 and A23.

The structure of the transformed matrix is shown in

Fig. 5b. Now all the information of the eigenspectrum

boiled down to the central block in the

transformed matrix . In this example, the dimen-

sion of the central block matrix is 6, as shown in Fig. 5b.

4.2. Solution of the reduced eigenvalue problemHere, we compare the computed spectrum of the reduced

eigenvalue problem and the original generalized eigen-

B41 B42

B44

1–B41

B44

1–B42

B41, B44[ ]T B42, B44[ ]

B

B23B13

1–B32

1–B32

A

2N 2–( ) 2N 2–( )×A22 ω( )

Fig. 3. Numbering scheme for 4 elements, 10 nodes and 29 degrees of freedom: two elements for both layers, 10 for the x-velocity

U, 10 for the z-velocity W, 6 for the pressure P, and 1 for the interfacial height h. The 29 related coefficients C1,..., C29 are inserted

in the vector following the order described in Fig. 2.

Fig. 4. (Color online) Non-zero entries after the eliminating no-slip conditions (a) and after the first matrix transformation Eq. (10) (b).



value problem. For the original problem, QZ method was

chosen to solve Eq. (2). From now on, FullQZ stands for

the solution of the full GEVP by QZ method. For the

reduced eigenvalue problem, three different methods are

used:

1. QZ method to solve the reduced GEVP Eq. (19) −

MTwQZ,

2. QR method to solve the simple reduced EVP Eq. (22)

− MTwQR(M),

3. QR method to solve the simple reduced EVP Eq. (23)

− MTwQR(J).

Total number of elements used in this test is 200, 100

elements for each layer, and the corresponding number

degrees of freedom is 1207. Flow is mainly driven by the

moving substrate and distorted by relatively small pressure

gradient: dimensionless pressure gradient NG,P is 0.196.

Reynolds number NRe,C viscosity ratio m, density ratio d

and thickness ratio n are 2.61, 0.5, 1, 0, and 1.5, respec-

tively. There is no gravity in this system, i.e. Froude num-

ber NF,C is zero.

Table 3 shows the computing time, the real part of the

leading eigenvalue , the condition number κ of the

eigenvalue and the estimated error bound from Eq. (24)

for the four solution strategies tested. Predictions were

obtained for wavenumber α = 0.1 and α = 10.

As expected, the CPU time of the reduced simple eigen-

value problem is approximately one order of magnitude

smaller than the solution of the full GEVP. For the low

wave number case (α = 0.1), the most dangerous mode

predicted by the MTwQZ method (solution of Eq. (19))

does not match that of the original problem. The other two

ωMD R,

εb

Fig. 5. (Color online) Non-zero entries for before (a) and after (b) the second matrix transformation, Eq. (15). Note that the mid block

become full matrix after finishing matrix transformation.

Table 3. Comparison between full QZ method and matrix transformation method. means the most dangerous growth rate or real

part of the eigenvalue, stands for condition number of the eigenvalue, and is estimated error bound from Eq. (24)

with approximating . Total number of element is 200. Note that condition number for eigenvalue are computed from

LAPACK expert driver subroutine Anderson et al. (1999) : ZGEEVX for QR iteration and ZGGEVX for QZ iteration.

MTwQZ MTwQT(M) MTwQR(J) Full QZ

CPU time (sec) 10.8 7.3 7.2 80.2

α = 0.1 case

ωMD,R(sec−1

) −1.0661 × 10−1

6.0551 × 10−3

6.0527 × 10−3

6.0544 × 10−3

κ(ωMD) 1.759 × 101

8.478 × 101

8.475 × 101

8.467 × 104

εb(ωMD) 1.6719 × 101

6.5698 × 10−3

2.5129 × 10−13

3.207 × 10−8

α = 10 case

ωMD,R(sec−1

) 4.0196 × 101

4.0195 × 101

4.0191 × 101

4.0193 × 101

κ(ωMD) 1.862 × 101

4.161 × 103

4.163 × 103

1.629 × 102

εb(ωMD) 1.770 × 10−3

2.892 × 10−2

5.4627 × 10−12

1.632 × 10−10

ωMD R,

κ ωMD( ) εb ωMD( )tE 2 εm A 2≈



solution strategies used to solve the reduced simple eigen-

problem led to the correct most dangerous mode. This

inaccuracy can be explained by the relatively large value

of the estimated error bound . For the large wave num-

ber case (α = 10), the most dangerous mode predicted by

the solution methods used to solve the reduced eigenprob-

lem were approximately that of the original GEVP. For

both case, the smallest error bound was for the solution of

Eq. (23), e.g. MTwQR(J).

The most dangerous growth rate as a function of wav-

enumber is presented in Fig. 6 for the four solution meth-

ods. The results show that the matrix transformation

method with QZ iteration used in the reduced problem

fails to get the most dangerous eigenvalue for small wav-

enumbers. At high wavenumber, the three methods yield

similar growth rates. The ten most dangerous eigenvalues

in the eigenspectrum, shown in Fig. 7, also reveal similar

trend: growth rates computed by MTwQZ deviate from

others. The trend clearly supports that solving Eq. (22) or

(23) is better than solving Eq. (19). Therefore one can

conclude that either MTwQR(J) or MTwQR(M) computes

accurate growth rates at much smaller computational cost.

The amplitude of the disturbed fields associated with the

leading eigenvalue, = 6.0527−1− i 4.8055 × 102 sec−1,

i.e. the associated eigenvector, for α = 1 mm−1 is shown in

for Fig. 8. The largest disturbance amplitudes are all

located at the interface, located at z = 0 in the plots. The

most dangerous mode is clearly the interfacial mode. It is

important to note that the eigenvector obtained from the

reduced problem recovers the eigenvector of the original

GEVP.

εb

ωMD

Fig. 6. Comparison between three eigenvalue solving methods:

MTwQZ, MTwQR, and FullQZ. Here MTwQR stands for solv-

ing Eq. (23): Because MTwQR(J) and MTwQR(M) show virtu-

ally the same results, the plot shows only one of them. Note that

MTwQZ shows strange patterns at low wavenumber.

Fig. 7. Ten most dangerous leading eigenvalues for α = 0.1 com-

puted from different methods: MTwQZ, MTwQR(J), MTwQR(M),

and FullQZ. Except the results from MTwQZ, rest of them close

to the origianl generalized eigenvalue problem.

Fig. 8. Modulus of the velocity component and pressure across

the flow direction z related to the most dangerous growth

rate ωMD = 6.0527 × 10−1 − i4.8055 × 102 sec−1. “Original” and

“Reduced” stand for the eigenvector from the original general-

ized eigenproblem and the reduced eigenproblem. Especially, the

reduced eigenvector computed by solving Eq. (23). Number of

element is 100, and wavenumber is 1 mm−1. Interlayer is located

at z = 0. The jump of velocity component and pressure across the

interlayer are shown in (a) and (d).



4.3. Comparison with previously reported resultsAs mentioned before, stability of two layer rectilinear

flow has been studied in the past. Different solution meth-

ods have been proposed to evaluate the growth factor of

perturbations. Yiantsios and Higgins (1988a) extended the

compound matrix method proposed by Ng and Reid (1979).

for parallel flow. However, the compound matrix method

can only track a single mode and requires a good initial

guess. To overcome this limitation, Yiantsios and Higgins

(1988a) proposed using finite element method to discret-

ize the governing equations and to evaluate the complete

eigenspectrum and thus the calculation of a desired mode

is refined using the compound matrix method. However, it

is important to note that the finite element formulation

proposed here is based on the primitive variables and that

used by Yiantsios and Higgins was based on streamfunc-

tion, e.g. they solved two coupled Orr-Sommerfeld equa-

tions. The results in Table 4 shows that the dimensionless

wavespeed c* computed from the proposed method matched

up to four significant digits in real part and three digits in

imaginary part comparing with that from the compound

matrix method. The number of nodes for the proposed

method is 402, and they are distributed by the stretching

function (Vinokur, 1983) to concentrate near the interface.

Note that they are considerably smaller than the number of

steps used in the compound matrix method, 2000.

We also compare the results from our method with those

from spectral Chebyshev tau method. The method was

developed by Orszag (1971) to evaluate the linear stability

for a plane Poiseuille flow, and Su and Khomami (1992)

expanded it for a two-layer plane flow. Table 5 shows the

results from both methods for a plane Couette two-layer

flow. The analysis was done at NRe = 10.0, r = 1, n = 1,

NF = 0 and NT = 0 with different m and α. The growth rate

are converted to the dimensionless wavespeed for the pur-

pose of comparison. Similar to the previous Poiseuille

flow case, wavespeed computed from the proposed method

matched up to about four digits in real part and about three

digits in complex part comparing with that from the spec-

tral method.

5. Final remarks

A procedure to extend the method proposed by Valério

et al. (2007) to remove non-physical infinite eigenvalues

Fig. 9. Ten most dangerous leading eigenvalues computed from

different number of elements: 50, 100, 200, and 300. Eigenvalues

are computed by solving Eq. (23), and wavenumber is α = 10

mm−1.

Table 4. Comparison between previously reported results with

different mesh scheme for dimensionless complex wave speed

c*(NRe,P = 1, m = 5, n = 1, r = 1). The results from the compound

matrix (CM) excerpted from Yiantsios and Higgins (1988b), and

this shooting-based technique is done with step size 0.001, which

is essentially discretize the domain into 2000 steps. Number of

elements used in the proposed method proposed in this study is

100, which discretize the domain into 201 nodes, but nodes are

concentrated near the interlayer using the stretching function

(Vinokur, 1983).

NT,P α*

c* (CM method) c

* (Proposed method)

1 10 0.99998 − i0.008199 0.999968 − i0.008201

1 20 0.99956 − i0.004145 0.999983 − i0.004146

2 10 0.99996 − i0.016537 0.000050 − i0.016440

2 20 0.99907 − i0.008312 0.999980 − i0.008314

Table 5. Comparison between numerical results by spectral tau

method and numerical model for dimensionless complex wave

speed c*. Numerical results by spectral tau method are excerpted

from Su and Khomami (1992). The mesh configurations for the

proposed method is the same as Table 4.

Viscosity

ratio mc (Spectral method) c (Proposed method)

100 2.71932 + i2.05300 × 10−5 2.71942 + i2.05346 × 10−5

60 2.56766 + i8.26909 × 10−6 2.56727 + i8.27006 × 10−6

20 2.06021 + i1.58952 × 10−6 2.06033 + i1.59030 × 10−6

10 1.67220 + i1.24810 × 10−6 1.67232 + i1.24896 × 10−6

5 1.33333 + i7.52439 × 10−7 1.33333 + i7.53490 × 10−7

α = 1.0 × 10−5, NRe,C = 10.0, r = 1, n = 1, NF,C = 0, NT,C = 0

Viscosity

ratio mc (Spectral method) c (Proposed method)

100 2.71925 + i2.05258 × 10−2 2.71861 + i2.05263 × 10−2

60 2.56747 + i8.26842 × 10−3 2.56673 + i8.26950 × 10−3

20 2.06008 + i1.58908 × 10−3 2.06037 + i1.59049 × 10−3

10 1.67213 + i1.24773 × 10−3 1.67219 + i1.24820 × 10−3

5 1.33333 + i7.52910 × 10−4 1.33333 + i7.53208 × 10−4

α = 1.0 × 10−2, NRe,C = 10.0, r = 1, n = 1, NF,C = 0, NT,C = 0



that arise from linear stability analysis of incompressible

viscous flows is presented. It is based on a series of matrix

transformations that leads to a simple eigenvalue problem

such that its spectrum is exactly the finite eigenvalues of

the original generalized eigenvalue problem. Since the di-

mension of the transformed problem is much smaller than

the original one, the computational cost is greatly reduced.

We used the proposed procedure to study the stability of

two-layer rectilinear flow. For this problem, the compu-

tation cost was reduced approximately a factor of 10. These

different methods were used to solve the resulting simple

EVP. The results show that QR methods lead to accurate

solution of the entire spectrum.

Acknowledgments

This research was supported by Basic Science Research

Program through the National Research Foundation of

Korea(NRF) funded by the Ministry of Science, ICT &

Future Planning (Grant No. NRF-2013R1A1A1004986)

Nam thanks Prof. J.V. Valério for discussions during his

stay at PUC-Rio, Brazil.

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