Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 5.1.3 - Stabilization of wall shear beneath vortical disturbance using a thin film
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Title: 5.1.3 - Stabilization of wall shear beneath vortical disturbance using a thin film Instabilities
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Saha, S.
Zaki, T.A.
Jung, S.Y.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: boundary layers
linear stability
DNS
two-phase flow
 Notes
Abstract: Free-stream vortical disturbances promote bypass transition to turbulence in boundary layers. The initial stages of bypass-transition are dominated by the amplification of streaks, which are large amplitude fluctuations of the streamwise velocity. These streaks are generated by the vorticity-tilting mechanism, where the wall-normal component for the disturbance tilts the mean vorticity and, as a result, produces a wall-normal vorticity perturbation. In this study we demonstrate that the efficacy of the vorticity tilting mechanism to generate streaks is reduced when a low viscosity wall-film is introduced. The temporal evolution of a linear perturbation is studied using an initial value problem. The linear analysis also reveals another mechanism for energy amplification, which is due to a neutral interface mode. This neutral mode has the appearance of a corrugated sheet, and generates velocity fluctuation by virtue of the discontinuity of mean shear across the interface. The results from temporal analysis are verified using direct numerical simulations (DNS) of the equivalent spatial problem. The DNS tracks the amplification of an inflow vortical disturbance in two-fluid boundary layers with different viscosity ratios. The downstream evolution is compared to the prediction of linear theory.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Stabilization of wall shear beneath vortical disturbances using a thin film


SANDEEP SAHA, SEO Y. JUNG, and TAMER A. ZAKI*

Mechanical Engineering, Imperial College, London, SW7 2AZ, UK
t.zaki@imperial.ac.uk
Keywords: Boundary layers, linear stability, direct numerical simulations, two-phase flow




Abstract

Free-stream vortical disturbances promote bypass transition to turbulence in boundary layers. The initial stages of
bypass-transition are dominated by the amplification of streaks, which are large amplitude fluctuations of the stream-
wise velocity. These streaks are generated by the vorticity-tilting mechanism, where the wall-normal component for
the disturbance tilts the mean vorticity and, as a result, produces a wall-normal vorticity perturbation. In this study we
demonstrate that the efficacy of the vorticity tilting mechanism to generate streaks is reduced when a low viscosity
wall-film is introduced. The temporal evolution of a linear perturbation is studied using an initial value problem. The
linear analysis also reveals another mechanism for energy amplification, which is due to a neutral interface mode. This
neutral mode has the appearance of a corrugated sheet, and generates velocity fluctuation by virtue of the discontinuity
of mean shear across the interface. The results from temporal analysis are verified using direct numerical simulations
(DNS) of the equivalent spatial problem. The DNS tracks the amplification of an inflow vortical disturbance in
two-fluid boundary layers with different viscosity ratios. The downstream evolution is compared to the prediction of
linear theory.


1. Introduction

Sheared wall-films are ubiquitous in engineering flows,
including cooling applications and coating flows. In
these flows, the wall-film is sheared by an external
stream with different density and viscosity. As a result,
the flow becomes prone to both shear and interfacial in-
stabilities. In addition, the presence of background vor-
tical disturbances can lead to transient instabilities and
bypass breakdown of the two-fluid boundary layer (see
figure 1 for a schematic of the flow setup). This paper
demonstrates the influence of a lower viscosity wall-film
on transient growth due to free-stream disturbances. In
particular, this work seeks to identify the optimal vis-
cosity ratio in order to minimize transient growth due to
free-stream vortical forcing.
The free-stream turbulent fluctuations can be regarded
as a superposition of a spectrum of modes which pene-
trate the boundary layer to various extents. The low-
frequency component is most effective at permeating
the boundary layer shear, while the high-frequencies
are sheltered (Hunt & Durbin 1999; Jacobs & Durbin
1998; Zaki & Saha 2009). This selective receptivity ex-
plains the elongated appearance of disturbances inside
the boundary layer. The amplification of the streaks
downstream has been explained by rapid distortion the-


ory (Philips 1969), and the lift-up mechanism (Landahl
1980). An alternative explanation is providedby the Orr-
Sommerfeld and Squire equations for linear perturba-
tions: A three-dimensional, wall-normal v-disturbance
tilts the mean vorticity, and thus generates a strong nor-
mal vorticity disturbance (Zaki & Durbin 2005, 2006).
At low frequencies, the generation of wall-normal vor-
ticity is equivalent to strong streamwise velocity distur-
bance, or streaks. The amplification of the streaks occurs
on a time-scale much shorter than that of the well-known
Tollmien-Schlichting wave.
In the current investigation, the physical mechanisms
of disturbance amplification in two-fluid boundary lay-
ers are explained. The tilting of mean vorticity is the
dominant mechanism for the transient growth of per-
turbations in single-fluid boundary layers; however the
presence of a thin wall-film alters the dominant mecha-
nism at low viscosity ratios. As the viscosity of the thin
film is reduced, the short time transient amplification is
inhibited and a neutrally stable interfacial mode gradu-
ally becomes dominant.
Since the streaks dominate the initial stages of bypass-
transition, researchers have studied extensively the tem-
poral amplification of a linear perturbations which can
lead to the formation of streaks (Ellingsen & Palm 1975;
Gustavsson 1979; Landahl 1980; Salwen & Grosch











1981; Butler & Farrell 1992; Reddy & Henningson
1993; Olsson & Henningson 1995; Schmid & Henning-
son 2001; Yecko & Zaleski 2005; Malik & Hooper 2007;
Schmid 2007). Investigation of the viscous initial value
problem for bounded flows demonstrated the possibil-
ity of short time transient growth of perturbations due
to resonant or near-resonant interactions Gustavsson &
Hultgren (1980); Gustavsson (1981); Benney & Gus-
tavsson (1981). The initial value problem is viewed as
a forced response problem where the normal vorticity
is driven by normal velocity component of the pertur-
bation. Therefore near-resonance between the discrete
Orr-Sommerfeld and Squire eigenvalues results in short
time growth of normal vorticity (Gustavsson & Hultgren
1980). Eventually at large time, t > 1/c, the viscous
dissipation overcomes the algebraic growth and there-
after the behavior is governed solely by viscous decay.
In bounded flows, the occurrence of direct resonance
between Orr-Sommerfeld and Squire eigenvalues is re-
stricted to particular wavenumbers and Reynolds num-
bers, and hence the notion of near-resonance was intro-
duced.
For semi-bounded and unbounded flows the continu-
ous spectra of the Orr-Sommerfeld and Squire equations
overlap, and therefore infinitely many instances of reso-
nance can occur. Zaki & Durbin (2005, 2006) studied
the temporal evolution of normal vorticity forced by a
continuous Orr-Sommerfeld mode in a laminar bound-
ary layer. Their solution explicitly demonstrated that the
resonant Squire mode grows as teh-t. The extent of
amplification is determined by the forcing term in Squire
equation, which is the coupling between the normal ve-
locity perturbation and mean shear, (Ov'/dz)(dU/oy).
The magnitude of this coupling is governed by the pene-
tration depth of the continuous Orr-Sommerfeld mode
into the boundary layer, and the mean shear distribu-
tion. The effect of viscosity stratification on penetra-
tion depth was evaluated by (Zaki & Saha 2009). The
authors showed that there exists an optimum viscosity
ratio, tPBT = PB/IT < 1, for which penetration of
continuous modes is maximized. For viscosity ratios
lower than the optimum value, penetration of continu-
ous modes is inhibited due to the shear-sheltering mech-
anism. The mean shear distribution is also affected by
introduction of a thin film. Films with viscosity lower
than the outer fluid absorb mean shear due to the tangen-
tial stress continuity at the interface. As a result the shear
in the top fluid is weaker. The net effect of a lower vis-
cosity wall-film on amplification of streaks is therefore
not clear. On the one hand, penetration of free-stream
disturbances can be enhanced, and on the other hand the
mean shear is weaker in the outer fluid.
Despite the importance of vorticity tilting mechanism
in the amplification of perturbations in a single-fluid


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010









--I I


ix


Figure 1: Schematic of a two-fluid boundary layer under
the influence of a continuous mode and an in-
terface mode.


boundary layer, it has not been investigated in the con-
text of two-fluid boundary layers. Furthermore, the pres-
ence of the interface in a two-fluid boundary layer can
lead to interfacial instability mechanisms (Yih 1967;
Hooper & Boyd 1983; Hinch 1984; Hooper & Boyd
1987). In this investigation we take advantage of the
thin layer effect which stabilizes the interfacial mode,
(Renardy 1987; Charm & Hinch 2000) when the film
viscosity is lower than the outer stream. The stabiliza-
tion of the interface mode, however, explains only the
large time asymptotic behavior of the system. In studies
of transient amplification, even a stable interface mode
must be taken into account in evaluating the short time
energy amplification. Attempts to compute optimal dis-
turbances which give rise to maximal transient ampli-
fication have been made for two-phase mixing layers
(Yecko & Zaleski 2005) and two-layer Poiseuille flow
(Malik & Hooper 2007). These studies support the view
that short-time energy growth can play a significant role
in two-fluid boundary layers.
In the current paper the amplification of streaks forced
by free-stream vortical disturbances is investigated using
linear theory and direct numerical simulations of two-
fluid boundary layers. The solution to an initial value
problem which governs the boundary layer response is
obtained using the eigen-modes of the Orr-Sommerfeld
and Squire equations. Thereafter the boundary layer re-
sponse to low-frequency vortical forcing is evaluated for
different viscosities of the wall film. The influence of the
interface mode is explained using asymptotic techniques
under the long wavelength approximation. In addition,
the optimal initial disturbance, which causes maximal
energy amplification in time, is computed in order to es-
tablish that transient growth is reduced by lowering of
PBT. The temporal, linear results demonstrate that the
streaks can be weakened by appropriate choice of the
wall film. Further evidence of streak attenuation, in the











context of the spatial problem, is obtained from direct
numerical simulations (DNS) of a two-fluid boundary
layer. The mean and root-mean-square perturbations are
extracted from the simulations, and compared to the pre-
diction of the linear analyses.


2. Linear theory

The base flow

The stability of a parallel, two-fluid base flow is exam-
ined by studying the evolution of a infinitesimal pertur-
bations to the base state (Uj(y), 0, 0),. The subscript
j T, B denotes the top and bottom fluids respectively,
and the streamwise, wall-normal and spanwise coordi-
nates are denoted x, y and z. The base flow of inter-
est in the current study is the two-fluid boundary layer
model described by Nelson et al. (1995). The same base
state was used by Zaki & Saha (2009) for examining the
structure of the continuous Orr-Sommerfeld modes in
a two-fluid boundary layer. The base flow satisfies the
following governing equation,


d2F+ d3Fj
dF2 dF3


where U = dF and F y The wall (y -0)
and free-stream (y 0) boundary conditions on F are,

FB (0) = 0; dFB(0) 0; F( ) .
dF dF(y U.

In addition, at the interface location, Ff = y 2, ,
the following conditions must be satisfied,


FB FT; dFB dFT. 2
dF dF 1 ""B df2


d2 FT
T dF2


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


normal velocity and normal vorticity perturbations, re-
spectively,


( kU) (d k2)r + kxd' Ujf
+(iRej) (d k 2)2j=
(W kxUj)ij + (iRej) (d2 k 2) =


0 (2)
kzdUq (3)


where k + k and dy In addition, the
interfacial displacement, f, is governedby the kinematic
condition,


iwf + ikxUT(yf)f


OT(Yf) (4)


The following boundary conditions are imposed on the
eigenfunctions p and 2 at the wall and in the free stream,


OB(O) 0; 1 ., () o0;


B(O)= 0


T(Y- 00) = 0; .'., (y 0oo) = 0; XT(Y 00)

In addition, the following conditions of velocity and
stress continuity at the interface must be enforced on 9
at y yf,

4T = 4B
S. 1. ikdyUrTf = .'. ikdyUBf
PT(d2 + k2)T I .1 r Tf
PB (d + k)B I .1?UBf

P 1'T Re7 d 3 3-k 2



p[- ...: ReT d- )33k

+ik(U, .T UB) +Wel k4f.

The interfacial conditions on 2 are as follows,


The eigenvalue problem

The linearized disturbance equations for a small pertur-
bation around the base state admit solutions of the form,


XT XB
., T .'


-ikz(dyUT dyUB)f
ikz(,,, .12rjT ._ .. 1 rB)f


v f
/iS


()
'j j(Y) )


where v, r and f are the normal velocity, n
ticity and interfacial displacement. The streak
spanwise wavenumber are denoted by kx and
tively, and w is the frequency of the perturb
above normal-mode assumption yields the v
Orr-Sommerfeld (2) and Squire (3) equati


The Orr-Sommerfeld and Squire equations, along
t), (1) with the boundary and interface conditions, constitute
an eigenvalue problem,

normal vor- )
-w 0 1 0 f =
mwise and i 0 1 )
rken- r o \A1e/


ation. The
cell known
ons for the


S3
15J


0
Yf) _J
0


0 /
&7Wy)/










The operators, j, 1, 'j,, _y are defined as,


Y 1 2r (d 2- k2)2
S -iku(yf)


1 (d 2

1 .1 Ijf


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


solution is expressed in terms of the eigenfunctions of
the Orr-Sommerfeld and interface equations, for a given
wavenumber vector k (ku, kz).


ff( t))
f{ W )


kI2) ikUj


The numerical discretization of equation 5 is carried
out in terms of C hb6 silc\ polynomial expansion on a
mapped Gauss-Lobatto grid (Orszag 1981). The domain
was truncated at a large but finite y-location, and the
C lb6 shic\ polynomials were mapped onto the physi-
cal grid using a linear mapping function. The domain
height was chosen such that the results were unaffected
upon further extension.

The initial value problem
For semi-bounded flows, the eigenspectra of the Orr-
Sommerfeld and Squire equations consist of a finite
number of discrete modes and a continuous spectrum.
The corresponding eigenfunctions form a complete ba-
sis (Salwen & Grosch 1981). The completeness of the
Orr-Sommerfeld and Squire eigenfunctions can be used
in order to describe the evolution of a general distur-
bance. Here, the focus is on the amplification of wall-
normal vorticity perturbation due to an initial vertical
velocity disturbance. Since the system of equations is
linear, it is sufficient to investigate the effect of a par-
ticular Orr-Sommerfeld mode. In the general case, the
Squire response to a superposition of Orr-Sommerfeld
eigenfunctions can be found by superposing the solution
for each forcing mode. The system of equations which
governs the evolution of v, f, and r is,


V2v


(
-fo 3y
fo0 6(y
Jo ^v


0 (f

7 W


Introducing the Fourier transform in the streamwise and
the spanwise directions, the dependent variables can be
expressed as,


, (y, t)

fM(t)

l,k (Y,t)


Jj Jj f(x,z,t)e -ik:cx ikz dxdz
JJ-f ;q00 ; t tk- OOdc

r f(o % e iYe ikx dzdxdz.
-00 J -


Nos /

n-l

+ f i (t)
,k Y


In order to derive the amplitude functions, ,,
in equation 6, the adjoint problem is invoked. The ad-
joint Orr-Sommerfeld eigenfunctions, yt, satisfy the
following bi-orthogonality condition (see appendix A
for the derivation of adjoint and the definition of the bi-
orthogonality condition 25),

[,mV27. 6m; [ V2 6(k k,).

The time dependent coefficients of the eigenfunction ex-
pansion can therefore be obtained according to (Salwen
& Grosch 1981),


,, V ; v 2(Y' 0)] e t

t VV(y, 0) eC_
k ky I

The initial condition on normal velocity must be spec-
ified before the forced Squire response can be solved. A
normal velocity disturbance in general consists of a su-
perposition of Orr-Sommerfeld modes. Here, we focus
on the normal vorticity due to a particular Orr-Sommer-
feld mode. The general case of Squire response to a
spectrum of Orr-Sommerfeld eigenfunction can then be
found by superposing all the individual solutions. There-
fore, for the initial condition, a particular Orr-Sommer-
feld mode, cw, is prescribed,


S(0) 0


where, A is the initial amplitude of the normal velocity
perturbation. Therefore the full time evolution of the
vertical velocity and interface height can be expressed
as,


V, (Y, t)

fk(t)


AYjSky (y)e
Af, ,c


The normal velocity and the interface equations form
an autonomous sub-system, independent of the normal
vorticity. Therefore, this subsystem is solved first. The


Zaki & Durbin (2005) proposed the solution to the
forced Squire equation for a single-fluid boundary layer,
in terms of the eigenfunctions of the Squire modes.


t (y))
k (Y))


ik U (d2 k2)


(0) = A6(ky ky)











The Squire eigenfunctions, 2, satisfy the homogeneous
Squire equation,


(-iw + ikU)(d k2) - (d' k 2)

and homogeneous interfacial conditions,

YT -Y B 0
I IA 0.


For a two-fluid boundary layer, the homogeneous
Squire eigenfunctions remain continuous across the in-
terface. This introduces a difficulty since the normal
vorticity we wish to represent is discontinuous at the in-
terface, and must satisfy the following interfacial jump
conditions,


t) ".' ty ) ,tT)( '/ t)
TIT, TI'B,(Yf, t)=


-i k (dUT(yf) IdUB(yf)) ff(t)


-ik. (,, 19 [IT yf() ;, ... (Y)W .

In order to satisfy the above jump conditions implicitly,
a change of variables is introduced. The new variable 4I
is continuous and defined as,

IjF,(y, t) rJ,(Yy, t) + 7 .1 jffg(t)g(y). (10)
The function g(y) must be continuous and differentiable
over the interval [0, oo) for I to be continuous and
satisfy the interfacial jump conditions. Moreover g(y)
should equal unity at the interface and zero at the wall.
The choice of g(y) does not affect our results and any
g(y) which satisfies the above conditions can serve the
purpose of solving the initial value problem. In this par-
ticular case, we use g(y) Yey-. Thus the interfa-
#f
cial boundary conditions on I are identical to those on
the eigenfunctions 2,

T,(yfY,It) B,I(Yf, t) 0=
IT ) BOyTl ,- (y I .".,t) = O-
Substituting the definition 10 into the Squire equation
yields the evolution equation for I,


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Following Zaki & Durbin (2005), we expand I in
terms of the homogeneous Squire eigenfunctions,


0 (8)


,F(y,t)


Nsqh
S bk(t)Y(y)

+ ()Y (y;,
y


(12)


The amplitude functions, bkm(t) and 1 (t) can be
derived using the bi-orthogonality condition satisfied by
the adjoint Squire eigenfunctions, t, (see Appendix B
for the derivation of the adjoint Squire eigenfunctions
and the bi-orthogonality condition 30),

,(2 ,min) -8mn ( ; ,) 6(k k'), (13)

where (X, Y) =PB fo XBYBddy + pr J XTYTdy.
The inner product of the adjoint eigenfunctions with ex-
pansion 12 yields,


i', (t)


(KY k k,' ))


(14)


The solution to the initial value problem is obtained
by substituting the expansion 12 into the governing
equation 11 and applying the orthogonality conditions
13. This yields the evolution equations for the spectral
coefficients,


db, (t)


km' k(yt)) ()


,k(t, _, (y, Y )) + k)}e


The expansion 12 is again applied in place of I, which
yields,


d
dt ,b,,m (t)
d
- (t)
dt


- (t)+ ( )
kt ( k Ky }c


Equation 15 is a first order, ordinary differential equation
and can be solved exactly for the amplitude functions,


a
at- 4jp, (y, t)


bkm(t) = bkm(0)ei, ,e

Ktm


e krt


where the forcing term j is,


dy j (y)


j,k,k y (yf)g(y))


+Rejl (d k ) (dUg(y) f~,


Since the continuous spectrum of the Squire and the
Orr-Sommerfeld equation overlap, there exists a contin-
uous Squire mode which resonates with the forcing Orr-
Sommerfeld mode. Therefore, for the resonant mode,


t (15)


S-ikA [


-x (y, t)







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


ky = ky, the amplitude is given by


1' (t)


'. (O)e

KXkY&Jt[


and for non-resonant Squire modes k ,

' (t) = (0)

-> [k.


t] (16)


S t


Finally, the normal vorticity r is recovered from I using
equation 10.

3. Results from linear theory

Although the solution to the initial value problem is gen-
eral, our objective is to evaluate the effect of viscosity
stratification on a two-fluid boundary layer. Therefore,
we ignore surface tension and assume unit density ra-
tio. We consider the temporal evolution of a streamwise
oriented vortex in a two-fluid boundary layer.
A streamwise oriented vortex has no normal vortic-
ity. However, any initial interface deformation requires
that there exists a jump in the normal vorticity perturba-
tion across the interface. Therefore, in order to prescribe
zero initial vorticity, I (y, 0) = 0, the interface defor-
mation, f&, must be set to zero at the initial time. Each
continuous mode has a non-trivial f' , associated with
it. An interface mode is added to the initial perturbation
field with an amplitude such that the interface displace-
ment due to the interfacial and the continuous modes ex-
actly cancel. The solution to the initial value problem
was described for forcing by a continuous Orr-Sommer-
feld mode. However it can be applied to the interfacial
mode as well. The individual solutions due to both forc-
ing modes are superimposed in order to obtain the full
boundary layer response.
The continuous Orr-Sommerfeld mode resonantly
forces Squire equation, and the solution 16 explic-
itly demonstrates initial algebraic growth with time for
the resonant Squire eigenmode. The inner product,
I ('t fg I is shown in figure 2, for forcing by (i) the
continuous Orr-Sommerfeld mode and (ii) the interface
mode. The peak in the projection is at the resonant
Squire mode, and hence algebraic transient growth is ex-
pected in the response.
The normal velocity profile is shown in figure 3(a) at
various times. It is oscillatory in the free-stream and
decays inside the boundary layer similar to the single-
fluid case. The entire normal velocity field decays with
time owing to viscous dissipation. At large times there is


100 res ant mode




10-2



10-4
102 100


Figure 2: Projectionofforcing = ( ) on the
homogeneous Squire eigenfunctions.
interfacial mode; - -, continuous Orr-
Sommerfeld mode. k, 0.001, ky,cont =
i, wuint 0.000373, k, = i, Re
800, pBT = 0.3, d 0.1.


no vertical velocity present because the continuous Orr-
Sommerfeld mode is exponentially stable, and there is
no vertical velocity perturbation associated with the in-
terface mode.
The time evolution of the interface displacement is
shown in figure 3(b), and can be explained by examining
equation 17,


f,(t) = A k, f kec
v ,k ,k


+ A ,intfkinte (17)


The two amplitudes Akky and A ,in, in equation 17
were chosen such that the interface displacement is zero
at t 0 However this cancellation is no longer main-
tained at later times since the continuous Orr-Sommer-
feld mode decays much faster than the interfacial mode.
The net interface displacement, which is the difference
between the individual interfacial displacements of the
two modes, also increases. At large times, only the inter-
face mode is present and therefore interfacial displace-
ment should decay as e ki",n '. An detailed discussion
of the asymptotic behavior of the interface mode in the
limit k, -- 0 is presented in the subsequent subsec-
tion 3. The asymptotic analysis reveals that the interface
mode is neutrally stable to leading order and therefore
the interface displacement tends to the asymptotic be-
havior as shown in figure 3(b).
Figure 4(a) shows the normal vorticity response as a
result of the forcing by the interfacial and continuous
Orr-Sommerfeld mode. The free-stream solution is triv-
ial at all instants of time. Normal vorticity is generated
by the vorticity tilting mechanism, whereby the normal
velocity component of the perturbation tilts the mean
shear and generates perturbationvorticity. Since there is
no mean shear in the free-stream, no normal vorticity is


















10 102


Figure 3: (a) Normal velocity, (b) Interface displace-
ment at different instants of time.- t 0;
t = 44; t 314. k,
0.001,ky,cont = 7wint = 0.000373,k, =
7, Re = 800, PBT = 0.3, d = 0.1.


generated. However inside the boundary layer, the nor-
mal vorticity perturbation amplifies due to the vorticity
tilting mechanism.
Viscous decay sets in on a time-scale, tvscous
O(Re/(k2 + k2 + k )). At larger times, the normal
velocity perturbation vanishes, and the normal vorticity
equation is driven purely by the displacement of the in-
terface. In the long-time limit, a jump in normal vor-
ticity is observed across the interface and the vorticity
perturbation decays away from the interface, as shown
in figure 4(b). The large-time normal vorticity field is
identical to the particular vorticity associated with the
interface mode. The interface displacement increases
in magnitude with time and, as a result, the vorticity
jump across the interface intensifies. While the nor-
mal vorticity generated by the continuous Orr-Sommer-
feld mode has vanished due to viscosity, the neutrally
stable interface mode has generated normal vorticity
by virtue of the jump in mean shear across the inter-
face. The jump in normal vorticity is proportional to
(1 yBT).7 IT(Yyf)(t) and hence a large interface de-
formation leads to large normal vorticity field. We refer
to the vorticity generated by the interface mode as the
neutral interfacial wave mechanism.
The total kinetic energy of the system, E(t), is shown
in figure 5,


E E(0) [PB f | I'dy + k |1| + |jl| )d y +
E(t) f (l N2 k2


pr (.-1 + k 212 + i 2)@ .
Yv

The contribution from normal velocity,

E,, = 2o) PB Jof 2dy + PT fyf I\|1
and normal vorticity,

2k = pE) PB |p M 2ldy + PT fJy I2dy
are also shown in the same figure. Only the energy
present inside the boundary layer is computed by choos-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010



(a) (b) 1

y/8 / 99


0 20 40 4,,

Figure 4: (a) Normal vorticity response at different in-
stants in time, -- t 0; - -, t
44; - t 314. A zoomed in
view of the boundary layer is shown in pane
(b). The forcing mode parameters are k,
0.001, kyon t = wit = 0.000373, k,
7t, Re 800, ItBT 0.3, d = 0.1.


60

50

40
E 30

20

10

101 100 101 102 103
t

Figure 5: Variation in energy with time. E(t);
S -, E,, E;. k
0.001, ky, ont ,= 7 i, t = 0.000373, k
7t, Re 800, PBT = 0.3, d = 0.1.


ing the upper limit of integration for E(t) to be 899.
The perturbation energy in the free-stream is solely due
to normal velocity, which is due to the initial condition
and decays in time. The disturbance amplification is re-
stricted to the boundary layer, where normal vorticity is
generated.
Figure 5 reveals that initially at t ~ 0(1), the Orr-
Sommerfeld mode contains all the energy and E z E,.
As time progresses, the normal vorticity contribution E,
amplifies and is approximately equal to the total energy
of the system for t > 0(10). The first peak in E, is
a result of the mean-vorticity tilting mechanism and the
second peak is caused by the neutral interfacial wave, as
explained by figure 4.


Asymptotic behavior of interface mode

The energy at large time is due to the interface mode.
Therefore, in this subsection, the asymptotic behavior of







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


the interface mode is examined in the limit k, -- 0. The
leading order solution presented herein demonstrates
that the energy associated with the interface mode in-
creases as the viscosity of the thin film is reduced.
The frequency, wit, of the interface mode is deter-
mined by assuming an asymptotic expansion with k, as
the small parameter, w wo + kw1 + .... Substituting
the expansion in equation 5 and the interface equation
leads to,


0 1" fo(


fj (y -


0U (y foj(y )
ikz(yf)) ( fo )


where, 2% = j(d k~). To leading order, the in-
terface conditions on the normal velocity perturbation
reduce to,

eOT OB = 0,
0 ,
I .*,,,. 0 .7,.... O,
(d + k2)oT B PT(d + k )o = 0,
(d 2- .1,)OTr PBT(d 2I.i,)OB = 0.

The interfacial boundary conditions on Yo are inde-
pendent of fo, but the interface equation depends on
eo(yf). This implies that there exists only one way
coupling between 4o and fo, and that the eigenvalues
of Orr-Sommerfeld equation and the interface equation
are independent of each other, to leading order. There-
fore the interface mode does not have an associated nor-
mal velocity perturbation. The frequency of the inter-
face mode, wit ~ O(k/) 0, essentially represents a
standing wave with a spanwise corrugation.
The normal vorticity eigenfunction associated with
the interface mode is found by solving the reduced
Squire equation,

(d k2)xoj 0. (18)

The solution to the equation 18 is Xoj = Celj +
C2j e-ky. The constants of integration can be obtained
by imposing boundary conditions on Xo which are,

xoT(y 00) = 0,
IoB(O) 0.
with interface conditions at y = yf,


XOT XOB
dyXOT i. ^, OB


-ik (dyUr
-ikz(d UT


dyUB)fo,
i,. B)fc


Therefore the normal vorticity eigenfunction is,

XOT = CB(1e- )
IOB = CB (e6-y e-ky).


9 E 20

10

0 2 4 6 0 05 10

Figure 6: (a) Normal vorticity eigenfunction Xo, PBT
0.3 (b) Energy, E, of the interface mode as
a function of the viscosity ratio.kz d
0.1.


where,

S(d2UT i, .. UB) + kz(dUT dUB)
S ekY -e ky + BT (ekyf + e-ky)
o [-PTk(dyUT dyUB)(ck, S + e-, y)
e2T[ e-yT + PBT(eky f + e-ky)

(d2UT -' ..' rJ)(eCky ,yf) 1
elkyfj e-k Y + PBT(eky + e-,y)]

The normal vorticity eigenfunction associated with the
interface mode, io, is shown in figure 6(a). The mode
corresponds to a unit interface displacement and is sim-
ilar in shape to the large time remnant vorticity field in
figure 4(a). The jump in normal vorticity across the in-
terface is proportional to the term dy UT dy U. There-
fore as the viscosity ratio, PBT, is reduced t,he jump in
normal vorticity increases due a larger discontinuity in
the mean flow gradient. As a result the energy E, due to
the interface mode increases for smaller PBT as shown
in figure 6(b).

Viscosity stratification effects
A reduction in viscosity ratio, PBT, increases the en-
ergy of the interface mode and hence the neutral inter-
facial wave generates large amplitude normal vorticity
for small viscosity ratios. The effect of viscosity strati-
fication on the vorticity tilting mechanism must also be
evaluated. Figure 7 shows the Squire response at the
time of maximum amplification in E(t). The peak in
normal vorticity response associated with the vorticity
tilting mechanism reduces as the viscosity ratio PBT is
reduced. The forcing term, which is shown in figure
8(a-b) is also attenuated. Therefore, it can be concluded
that the temporal amplification due to the vorticity tilting
mechanism is reduced by lowering of PB T
'. The energy amplification curves are examined in fig-
ure 9(a-b) for two different film thicknesses, over a range
of viscosity ratios. As PBT is reduced, the magnitude of
the first peak in E(t) is reduced whereas that of the sec-
ond peak increases.
























4z;=! zw
0 10 20 30 40 50 60


Figure 7: Squire response in a two-fluid boundary layer
at the time of maximum amplification in en-
ergy. d = 0.1 PBT = 1.0; - tBT
0.5; -- --, BT 0.3; - -, PBT -0.2.
k = 0.001, ky =, kz = -7, Re = 800




3 1
(a) (b)
y/399 y/
999


'. 0 00 I |dyU0 05 1 0

Figure 8: (a)Tilting term, '.1 r4| for a two-fluid bound-
ary layer (b) Tilting term, '.1 rT1\ inside the
boundary layer. d 0.1 PBT 1.0;
...., PBT 0.5 ; -, BT 0.3;
- -, BT 0.2. k = 0.001, ky = 7, kz
t, Re = 800.




100 100
(a) (b)
E 50 E
50 50


0 100 102 10 102
t t

Figure 9: Energy amplification for various viscosity ra-
tios (a) d = 0.1 BT = 1.0; .. .
PIBT 0.5 ; BT 0.3; - -
PBT = 0.2. (b) d 0.05 PBT 1.0;
.... PBT 0.3 ; BT 0.2;
- -, PBT = 0.1. k = 0.001, ky = k
t, Re = 800.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


The origin of the two energy peaks has been ad-
dressed; the early energy maximum is due to the tilting
of mean vorticity and the second is associated with the
neutral interfacial wave. The first peak is therefore de-
pendent on the strength of the forcing term, { (Zaki &
Durbin 2005). This term is affected by the mean shear
distribution and the variation in p with viscosity stratifi-
cation. As tBT is reduced, the shear in the bottom fluid
increases and the shear in the top fluid is reduced in order
to maintain the mean shear stress continuity at the inter-
face. The shape of p in a two-fluid boundary layer was
discussed in detail in the work of Zaki & Saha (2009).
It was reported that the penetration of normal velocity
perturbation into the boundary layer is inhibited for low
viscosity ratios due to the presence of stronger shear in
the bottom fluid. Therefore the overall forcing term {
is reduced in strength as the viscosity ratio is reduced
and hence the reduction in the first peak in figure 9. The

coupling coefficient 0 = proposed by Zaki &
Durbin (2005) was computed, and is shown in figures
10(a-c). As the viscosity ratio is reduced, the optimum
coupling coefficient is significantly reduced compared to
the single-fluid value.



The energy amplification figures 9(a-b) show that
a critical viscosity ratio exists for each film-thickness
where the energy due to the neutral interfacial wave ex-
ceeds the mean-vorticity tilting mechanism. For a given
viscosity ratio a thicker film absorbs the mean shear to a
greater extent and therefore the coupling between mean
shear and normal velocity is weaker. This implies that
the vorticity tilting mechanism is expected to be weaker
for a thicker film. This prediction is substantiated by fig-
ure 11 which shows the coupling coefficient optimized
over all k,, ky and k,. The optimal 0 is larger for the
thinner film.



The solution to the initial value problem demonstrated
that the temporal amplification of streaks, due to a par-
ticular Orr-Sommerfeld mode, can be reduced by intro-
ducing a thin film. However, a stronger statement re-
garding the efficacy of the wall-film is sought. To this
end, we evaluate the maximal possible temporal growth,
optimized for all possible initial disturbances with unit
energy. A reduction in the energy amplification of the
optimal initial perturbation would confirm the efficacy
of lower PBT to reduce streak amplitude. The optimal
transient growth is evaluated according to the method
proposed by Reddy & Henningson (1993). In addition,
in order to account for the presence of the interface,
we adopt the --norm introduced by Malik & Hooper







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


10-3 10-2 101 100
k.


0.2 0.4 0.6 0.8 1.0


Figure 11: Optimum coupling coefficient,
fluid boundary layer. -, d
d =0.1. Re= 800


00pt fora two-
S0.05; - -,


(2007),


10-2 100
k


(c) 'v I


100


10-2


Figure 10: Coupling coefficient contours, 0 for a
fluid boundary layer, (a) BT = 1.0
PBT 0.5. (c) BT 0.1. Re
Contour levels correspond to 0 < 0 <
an increment of 5.


"8 + 2 12 + |2)dy +

(i.1 i + k21Vl2 + M2)y +
/YO


(19)


The maximum transient amplification is defined as,


G(t) max q(t)112 l
q(0) o |q(0) |2


two-
S(b)
800.
70 at


(20)


where L is the matrix on the right hand side of equa-
tion 5. The disturbance energy, q, can be expressed in
terms of the eigenfunctions of equation 5. Figure 12
shows the envelope of maximum transient amplification
achievable at any instant of time for various viscosity
ratios. The single-fluid boundary layer shows a single
peak whereas the two-fluid cases have two peaks. Mul-
tiple peaks for G(t) in two-fluid flows have been re-
ported by other researchers as well (Yecko & Zaleski
2005; Olsson & Henningson 1995). The first peak in
G(t) decreases for lower viscosity ratio PBT, whereas
the second peak increases. This observation is in accord
with the fact that the vorticity tilting mechanism is weak-
ened, and the neutral interfacial wave is strengthened by
reduction in PBT.
The solution to the initial value problem, and the re-
sults of optimal transient growth, indicate that the tem-
poral amplification of streaks is reduced in presence of
low-viscisoty wall film. The equivalent spatial problem
is now studied using direct numerical simulations. In
the next section, the numerical method used in our di-
rect numerical simulations is introduced, followed by a
discussion of the results from our direct computations.


8k2_(1 r) f I/I2]







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


100


I ,
l''.''\ .'--
I '

I -
,I/~ t% ~ ~
tt I


Figure 12: Variation in G(t) with JBT. k,
0.001,k 2, Re 800,d 0.1.- ,
IBT=I; ..., -IBT = 0.5 .-
PBT = 0.4; - -, PBT = 0.3 = 0.01


Figure 13: Vertical profiles of streamwise mean veloci-
ties at various x-stations. tBT = 1.0;
.... BT = 0.5; -, BT = 0.3;
- -, BT = 0.2.


4. Direct numerical simulations


Numerical method

Direct numerical simulations (DNS) account for non-
parallel effects and non-linearity in two-fluid boundary
layers which were not captured in the above linear anal-
yses. Both non-parallel effects and non-linearity are im-
portant due to the low-frequency of the inflow distur-
bance and the large amplitude of the boundary layer
response, respectively. In this section, direct numeri-
cal simulations of two-fluid boundary layers with small-
amplitude inflow perturbations are carried out in order to
complement the results of linear theory. Thus, the em-
phasis is placed on the linear limit, rather than the inves-
tigation of non-linear aspects. A single low-frequency
continuous Orr-Sommerfeld mode is prescribed at the
inflow of our computational domain, and its spatial
evolution is computed for six different viscosity ratios
([PBT 0.2, 0.3,0.5,0.7,0.8, 1.0). The film-thickness
is set to 1I' of the boundary layer thickness at the inlet.
A fractional step method is utilized to solve the
time-dependent, incompressible Navier-Stokes equa-
tions. The computational algorithm is based on a stag-
gered grid with a local volume flux formulation in curvi-
linear coordinates (Rosenfeld et al. 1991). Adams-
Bashforth scheme is implemented for the explicit time
advancement of convective terms. Pressure and diffu-
sion terms are treated by implicit Euler method, and by
Crank-Nicolson method, respectively. This code is par-
allelized using Message Passing Interface (MPI).
The inflow is composed of a two-phase boundary
layer profile, and a single mode perturbation at the
inlet Reynolds number, Re6o = 800, based on the
inlet boundary-layer thickness and free-stream veloc-
ity. The computational domain is (L,/6o, Ly,/o,


L,/6o)=(300, 20, 8). The number of grid points used
is (N,, Ny, N,)=(257, 129, 65). The grid is uniformly
spaced in the streamwise x and spanwise z directions,
and is clustered near the interface in the wall-normlal y
direction. The height of the computational domain is se-
lected to be sufficiently large in order to ensure that the
boundary layer is unaffected by the far field boundary
conditions. The convective outflow condition is applied
at the exit plain. A no-slip boundary condition is im-
posed at the solid wall. At the free stream boundary,
u Uo and 9 = 0. Periodic boundary con-
ditions are employed in the spanwise direction. For the
single-mode perturbation, the inflow mode is c = 0.01,
ky = k = -, and the amplitude of the inflow distur-
bance is normalized such that v,,r = 0.01% in the free
stream.

In the present DNS, a level-set method is employed
for capturing the interface between two-fluids. In the
level-set approach, the interface is represented implic-
itly by an iso-surface of a smooth function. Advan-
tages of the level-set method include automatic handling
of topological changes, efficient parallelization as well
as easy representation of the interface curvature. The
conventional level-set technique relies on representing
the interface as the zero level set of a signed distance
function. However, one of the main limitations of this
method is poor mass conservation. The present study
exploits the conservative level-set method in conjunction
with a ghost fluid approach (Desjardins et al. 2008). The
conservative level set method ensures the accurate and
robust interface transport by using a hyperbolic tangent
level set function. Moreover, the sharp discontinuity of
the interface can be captured by the ghost fluid method.


100







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Figure 14: Vertical profiles of u,,, at various x-stations.
BT 1.0; ... PBT 0.5;
-- -- BT = 0.3; -, PBT = 0.2.


DNS results

The influence of the viscosity ratio on the mean veloci-
ties is shown in figure 13. Here, the y-axis is normalized
by the local boundary layer thickness and four different
viscosity ratios are plotted. It is clearly seen that the wall
film absorbs the mean shear and this tendency is more
pronounced with decreasing viscosity of the film. This
implies that the mean-vorticity mechanism is expected
to be weaker for a less viscous film by reducing mean
shear of the top fluid as shown in figures 8 and 9. In
order to observe the boundary layer response to the in-
flow perturbations, the root-mean-square (rms) stream-
wise velocity fluctuations at different x-locations are
displayed in figure 14. The wall-normal axis is shown
in the range 0.3 < y/gS9 < 0.7 are provided in or-
der to demonstrate the attenuation of the streak clearly.
The streamwise velocity fluctuations for PBT 1.0 are
increasing from the inlet to x/6o z 45 and then de-
cay as they convect further downstream. This trend is
also observed for other viscosity ratios. The boundary
layer streaks are most amplified at x/60o 45, which
is very close to the time of the first peak in figure 9.
Furthermore, the streak amplitude becomes weaker with
decreasing viscosity ratio. This is consistent with the
linear prediction that lower viscosity wall films attenu-
ate the energy amplification inside the boundary layer.
It is interesting to note, however, that the decay rate of
the velocity fluctuations becomes slower with decreas-
ing viscosity ratio for x/6o > 45. It is attributed to the
reduction of the viscous dissipation by introducing the
less viscous wall-film.
In order to evaluate the energy amplification of the
inlet perturbation and the interfacial instability, the max-
imum u,,, is extracted at every downstream location
and is shown in figure 15. In figure 15(a), the maximum


Figure 15: Maximum u,,,, versus downstream distance
from the inlet plane for (a) the outer fluid and
(b) for the entire y-domain. -- PBT 1;
.... BT 0.8; --- IBT 0.7;- -
IBT = 0.5; -B- PBT 0.3; V--
PBT = 0.2.


is computed from the disturbance field in the outer fluid,
and therefore reflects the strength of the streaks in the
outer stream due to tilting of mean vorticity. vorticity-
tilting. the streak strength by the altered vorticity-tilting
mechanism solely. The maximum values of the u,,,
for the streaks are located at x/6o z 40 which is similar
to the first peak location of the energy amplification in
9. Furthermore, the peak value decreases at lower vis-
cosity ratios. The reduction rate by lowering viscosity
ratio is from 0.5'. to 1I'. It should be noted in figure
15(b), however, that the maximum values of u,,, for
PIBT 0.2 and 0.3 are significantly higher than those
of other viscosity ratios and the maxima show slow de-
cay rates for x/6o > 100. This indicates that the inflow
disturbances decay much faster than the interfacial in-
stability as explained by linear theory. Plan views of the
instantaneous streamwise velocity fluctuations are pro-
vided in figure 16. This also demonstrates the reduction
of the streak strengthby a lower-viscosity wall-film. The
solution to the initial value problem presented earlier
neglects both non-parallel and nonlinear effects. Here,
the former are fully represented, while the amplitude of
the disturbance was kept small in order to preserve lin-
ear behavior. The agreement between the DNS and lin-
ear theory therefore complements the analyses presented
herein.


5. Conclusions

In this paper, the amplification of streaks was examined
in two-fluid boundary layers. The temporal amplifica-
tion of normal vorticity due to forcing by a continuous
and an interface mode was solved. Our analysis revealed


0.7 .
if I" f I1 l I
ii ": i: i Ii it; 11

1 I .. .. I i I
II ii i 1ii ii II
I I I 'I 'I
1. I3 I I 'I
i li I I i I I I


0.3,-


51 25 35


60 /7



























Figure 16: Instantaneous contours of streamwise veloc-
ity fluctuations evaluated at y/So 0.7. (a)
PBT 0.7, (b) BT = 0.5, (c) BT = 0.3.
Contour levels correspond to -0.002 < u <
0.002 at an increment of 0.0004.


two disturbance generation mechanisms: (a) the vortic-
ity tilting and (b) the neutral interfacial wave. The extent
of vorticity generationby the two mechanisms varies de-
pending on the coupling coefficient between normal ve-
locity and normal vorticity, as well as the mean shear
discontinuity across the interface.
The two amplification mechanisms generate normal
vorticity simultaneously and their individual effects
must be distinguished. The interface eigenmode was
chosen to demonstrate the role of the interface in gen-
eration of normal vorticity. The neutral interfacial wave
results in the amplification of normal vorticity despite
the absence of an associated normal velocity perturba-
tion. Therefore, at large times, the interface displace-
ment is finite while the normal velocity vanishes as can
be seen in figure 3. The large-time normal vorticity gen-
erated by the interface is clearly visible in figure 4.
If, however, two continuous modes were chosen, the in-
terface deformation would have initially increased, but
at large time the fk(t) would tend to zero because both
modes are decaying. In the absence of any interface de-
formation, there would be no normal vorticity present.
In this case, it would be difficult to distinguish the nor-
mal vorticity generated by the normal velocity (tilting of
mean vorticity) and from interface deformation.
We computed the coupling coefficient in order to
demonstrate that the mechanism of streak generation by
tilting of mean vorticity is weakened by a reduction in
viscosity ratio. An asymptotic analysis was also per-
formed in the long wavelength limit in order to exam-
ine the effect of viscosity ratio on the interface mode.
The analysis revealed that this mechanism is strength-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


ened as the viscosity ratio is reduced. Optimal transient
growth for single- and two-fluid boundary layers was
also compared, and the results corroborated the conclu-
sions based on the initial value problem.
The spatial evolution of small amplitude perturba-
tions was also studied using direct numerical simula-
tions. The mean flow profile at various downstream
locations showed that a low viscosity film absorbs the
mean shear and, as a result, curtails the coupling be-
tween the mean shear and the normal velocity perturba-
tion. Thus the vorticity tilting mechanism is not as effec-
tive at low viscosity ratios and that the streak amplitude
decreases as the viscosity of the thin film is reduced.

6. Acknowledgements

This work was supported by the UK Engineering and
Physical Sciences Research Council (EP/F034997).

A. Adjoint Orr-Sommerfeld operator

The adjoint Orr-Sommerfeld equation is derived based
on the following inner product,


{}t*,YZos } ={t* =} 0


(21)


where, {X, Y} pPB f XBYBdy + pr fJ XTYTdy
and t denotes the adjoint. The definition of the adjoint
can be further expanded into,

PB /j f os,B B dy + pT J pTYos,T T dy
af
PB J sBPB)os B B dy

+PT (s,T ) *T dy (22)

where os,j = ij-)(d' k2). Integration-by-parts
is carried out in order to derive the adjoint Orr-Sommer-
feld equation,


Yst t
o j 0


where,

os,j


(23)


i(w kU)(d -k 2)-
ik1 dyU Re (d k2)2.


The boundary terms which arise from the above proce-
dure must be set to zero in order to satisfy the definition
of the adjoint in equation 22. This yields the following
boundary conditions on the adjoint equation,

^;(0) 0 .(0) 0
(y o0) -0 ; ., (y oo) 0.











and the adjoint interfacial conditions at y = yf,

t t
9B 9T


PB(d + k2)qt


I ?t

PT(d + k 2)'t


PB i(wt* kUB,' + Re -(d 3k dy) +


BkiUB wt*) Jj9B

Re1 ( d i(kmU wt*) T
-PT [i(wt* kmUTU. + Re(d 3k2 d) +

R ( .(kU UT( --k

k4We- 4
= o0. (24)
i(kUT wt*)
The bi-orthogonality condition is derived starting from
the product of two eigenfunctions,

{,'.,kAos= } {$2 ,,, o

where subscripts m, n denote the the ," adjoint and
Forward Orr-Sommerfeld eigenmodes. Using the
definition of the forward and adjoint operators, and the
boundary conditions 24, the following form of the bi-
orthogonality condition is obtained,


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010




YJ
P Bt*-O' dy pT t* T dy

PB t t ,)*B dy

+PT (-T,)* XT dy
J f
Using integration-by-parts, the adjoint Squire operator
is given by,


(27)


The boundary terms lead to the following boundary con-
ditions on the adjoint equation,

S(0) 0 ; (y-0o) 0

and at y yf,

t t tt Bt
XB T definition of the adoint,

Based on the definition of the adjoint,


K*'X^,Xn}


(28)


for any two eigenfunctions of It and xn. Using the def-
inition of the forward and adjoint eigenvalue problems
yields,

iWm(Xn Xn) iwn{X, Xn)


(29)


S{q,, (d k2)q}

+ 5int,mn 6mn.


where,

* .int,mn [(PT PB-.

ik~B(dyUT dyUB)(d + k2)
(ik UT iwn)(ikuUT iwm)
We- k4 + ( ,. [ UT-- rUt
(ikuUT iw,)(ikuUT iwu)


nB] t*
anB |,, .


B. Adjoint Squire operator

Consider the homogeneous Squire equation for two-
fluid boundary layer flow where the interface is located
at y yf,

jXj iWXj = 0. (26)

The adjoint operator jt is defined according to,

(xY, x=) =( xtt, X)


Equation 29 implies that unless cm w= the inner
(25) product of the (xt, xn) must vanish. Therefore, the or-
thogonality condition is given according to,


(Qx, xn)


(30)


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7th International Conference on Multiphase Flow,
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