Analysis of Parabolized Stability Equations for Perturbed Flows

Indian Institute of Technology, Gandhinagar

Summer Research Fellowship Programme

Analysis of PSE for Perturbed Flows

By:Nishant KumarB.Tech IVMechanical EngineeringS.V. National Institute ofTechnology, Surat

Supervisor:Dr. Vinod NarayananDepartment of MechanicalEngineeringIndian Institute of Technology,Gandhinagar

May-July, 2010


Abstract

The parabolized stability equations (PSE) are a new approach to analyse the streamwiseevolution of single or interacting Fourier modes in weakly nonparallel flows such as boundarylayers. The concept rests on the decomposition of every mode into a slowly varying amplitudefunction and a wave function with slowly varying wave number. The neglect of the smallsecond derivatives of the slowly varying functions with respect to the streamwise variableleads to an initial boundary-value problem that is solved by numerical marching procedures.The PSE approach is valid in convectively unstable flows. The PSE codes have developedinto a convenient tool to analyse basic mechanisms in boundary-layer flow. In this report, theemphasis has been laid on the analysis of PSE concept in natural variables, along with theunderlying background and concepts. The implementation of the equations using FORTRANcodes was done to analyse the solutions of the equations.

1 Background

The theoretical background required for understanding the concept of hydrodynamic stability andanalysis of fluid flows is presented here.

1.1 Navier Stokes Equations

The NSE, along with the continuity equation and well formulated boundary conditions, is usedto determine the flow field. The NSE is basically a statement of conservation of momentum. Forincompressible flow of Newtonian fluids, the NSE in vector form may be represented as

ρ

(∂v

∂t+ v.5 v

)= −5 p+ µ52 v (1)

The work deals basically with the axisymmetric flows and hence it is beneficial to representeqn. (1) in cylindrical coordinates systems in nondimensional form. A change in variables of thecartesian form of the equation will yield the desired form in cylindrical coordinates r, θ, z

∂u

∂t+ v

∂u

∂r+w

r

∂u

∂θ+ u

∂u

∂u= −∂p

∂z+

1

Re

[1

r

∂

∂r

(r∂u

∂r

)+

1

r2

∂2u

∂θ2+∂2u

∂z2

](2)

∂v

∂t+ v

∂v

∂r+w

r

∂v

∂θ+ u

∂v

∂z− w2

2=∂p

∂r+

1

Re

[1

r

∂

∂r

(r∂v

∂r

)+

1

r2

∂2v

∂θ2+∂2v

∂z2− v

r2− 2

r2

∂w

∂θ

](3)

∂w

∂t+ v

∂w

∂r+w

r

∂w

∂θ+ u

∂w

∂z+v.w

r= −1

r

∂p

∂θ+

1

Re

[1

r

∂

∂r

(r∂w

∂r

)+

1

r2

∂2w

∂θ2+∂2w

∂z2+

2

r2

∂v

∂θ− w

r2

](4)

where u, v, w represent the velocity components in z, r, θ directions respectively. We have used thefollowing vector transformations for the derivations

dr

dθ= θ and

dθ

dθ= −r (5)

1.2 Orr Sommerfeld Equation (OSE)

Amplification of disturbances: The flow may consist of disturbances caused due to free streamcondition or surface roughness. Disturbances of a certain band of frequencies get damped by thefluid viscosity and some frequencies get amplified, which may cause turbulence downstream of theflow.

1


The OSE is an eigenvalue equation describing the linear two-dimensional modes of disturbanceto a viscous parallel flow. The perturbed velocity field is

u = (U(z) + u′(x, z, t), 0, w′(x, z, t)) (6)

where perturbed velocity component is determined from

ψ(x, y, t) = φ(y)eiα(x−ct) (7)

In non-dimensional form, OSE can be obtained from the linearised version of NSE. The final formof OSE can be represented as

(U − c)(φ′′ − α2φ) = − 1

αRe

(φ′′′′ − 2α2φ′′ + α4φ

)(8)

Solution of OSE : For the OSE to give non-trivial solution, we must have

f(α, c,Re) = 0 (9)

It is assumed that the disturbances are temporally growing. Thus, depending on the value of c,the α−Re plane can be divided into following domains• c < 0, flow is stable (i.e., disturbances decay),• c = 0, flow is neutrally stable, and• c > 0, flow is unstable (i.e., disturbances grow).

Squire’s Theorem: To each unstable three dimensional disturbance there corresponds a moreunstable two-dimensional one.

1.3 Boundary Layer Flow

Order of magnitude analysis: The inviscid solution gives u = U =constant everywhere, ∂p/∂x = 0and the governing equations become

u∂u

∂x+ v

∂u

∂y=

1

Re

∂2u

∂x2(10)

∂u

∂x+∂v

∂y= 0 (11)

For similarity solution of the above equations, we consider

u

U= f ′(η) (12)

where

η =y

δ(x)= y

√U

νx(13)

Blasius profile: This equation is used to analyse flow over a flat plate at zero angle of attack. Weconvert the NSE which is a PDE into an ODE which can be solved easily using numerical methods.

f ′′′ +1

2ff ′′ = 0, (14)

f(0) = f ′(0) = 0, f(∞) = 1 (15)

where the dash represents differentiation with respect to η.

2


Falkner-Skan equation: Here, we generalize the Blasius boundary layer by considering a wedge atan angle of attack β from some uniform velocity field

ue(x) = U0

( xL

)m(16)

Thus,

f ′′′ + ff ′′ + β(1− f ′2

)= 0 (17)

where,

β =2m

m+ 1(18)

2 Solution of Orr-Sommerfeld equation in cylindrical co-ordinates

For axisymmetric flows, to investigate the stability of the basic flow, we study the evolutions of theperturbations u, v and p requiring that the total flow parameters satisfy the equations of motionand that they are constrained by the same flow-forcing boundary conditions as the basic flow.

Here, we consider the perturbed flow to be given as the combination of the unperturbed meanflow V,P and the perturbations v,p

uz = u = U + u, ur = v = V + v, uθ = w = W + w, p = P + p (19)

where ¯ denotes the total flow parameter.For the mean flow, we apply the following boundary conditions

W = 0 and U, V 6= f(θ)

The disturbances are also assumed to be parallel flows. Then the most general form for the 1-Ddisturbance q is that of a travelling wave whose amplitude varies with r and which moves alongthe boundary at an angle θ with respect to the z axis. In complex notation, the disturbances maybe represented as

q = q(r)exp[i(αx+ nθ − βt)] (20)

All disturbances have wave number α and frequency β. They are referred to as Tollmein-Schlichtingwaves which are indications of laminar-flow instability.

The basic flow satisfies the momentum equation. In vector form, the momentum equation maybe represented as

D ~V

D t+ (~V .~∇)~V = −1

ρ~∇P + ν.~∇2~V (21)

By suitable transformations, the N-S equations in non-dimensional form for the basic flow canbe written in the cylindrical coordinate system for r,θ,z directions.

r momentum:

∂V

∂t+ (~V .~∇)V − 1

rW 2 = −∂P

∂r+

1

Re

(~∇2V − V

r2− 2

r2

∂W

∂θ

)(22)

θ momentum:

∂W

∂t+ (~V .~∇)W +

V.W

r= −1

r

∂P

∂θ+

1

Re

(~∇2W − W

r2+

2

r2

∂V

∂θ

)(23)

3


z momentum:

∂U

∂t+ (~V .~∇)U = −∂P

∂z+

1

Re~∇2U (24)

Now, we substitute the perturbed velocity components given by eqn. (19) into the N-S equation

and subtract out the basic flow ~V and P equalities, as given by eqs. (22-24), and neglect higherpowers and product of p. The linearized disturbance equations may then be written as

∂u

∂t+ u

∂U

∂z+ U

∂u

∂z+ V

∂u

∂r+ v

∂U

∂r= −∂P

∂z+

1

Re

(∂2u

∂r2+

1

r

∂u

∂r+

1

r2

∂2u

∂θ2+∂2u

∂z2

)(25)

∂u

∂t+ u

∂V

∂z+ U

∂v

∂z+ V

∂v

∂r+ v

∂V

∂r= −∂P

∂r+

1

Re

(∂2v

∂r2+

1

r

∂v

∂r+

1

r2

∂2v

∂θ2+∂2u

∂z2− v

r2− 2

r2

∂w

∂θ

)(26)

∂w

∂t+ U

∂w

∂z+ V

∂w

∂r+

1

rV.w = −1

r

∂P

∂θ+

1

Re

(∂2w

∂z2+

1

r

∂w

∂r+

1

r2

∂2w

∂θ2+∂2w

∂z2− w

r2+

2

r2

∂v

∂θ

)(27)

To eliminate the pressure terms appearing in the eqs. (25-27), at first, we subtract the r-derivative of eqn. (25) from the z-derivative of eqn. (26)

∂2u

∂r∂t− ∂2v

∂z∂t+

(∂V

∂r

∂u

∂r+ V

∂2u

∂r2+∂v

∂r

∂U

∂r+ v

∂2U

∂r2+∂U

∂r

∂u

∂z+∂u

∂r

∂U

∂z+ u

∂2U

∂r∂z

)−

(∂U

∂z

∂v

∂r+∂u

∂z

∂V

∂r+ u

∂2V

∂z∂r+ 2

∂V

∂z

∂v

∂z+ V

∂2v

∂z2− 2w

r

∂w

∂z

)=

1

Re

(− 1

r2

∂u

∂r− 1

r

∂2u

∂r2+∂3u

∂r3− 2

r3

∂2u

∂θ2+

1

r2

∂3u

∂r∂θ2+

∂3u

∂r∂z2

)− 1

Re

(∂3v

∂z∂r2+

1

r2

∂3v

∂z∂θ2+∂3v

∂z3− 1

r2

∂v

∂z− 2

r2

∂2w

∂z∂θ

)Similarly, from the θ and z direction equations, the equation after eliminating pressure terms gives

∂2w

∂z∂t− 1

r

∂2u

∂θ∂t+

(∂V

∂z

∂w

∂r+ V

∂2w

∂z∂r+∂V

∂z

∂w

∂z+ U

∂2w

∂z2+V

r

∂w

∂z− w

r

∂V

∂z

)−

(1

r

∂V

∂θ

∂u

∂r+v

r

∂2u

∂θ∂r+

1

r

∂U

∂θ

∂u

∂z+U

r

∂2u

∂θ∂z+

1

r

∂v

∂θ

∂U

∂r+v

r

∂2U

∂θ∂r+

1

r

∂u

∂θ

∂U

∂z+u

r

∂2U

∂θ∂z

)=

1

Re

(∂2w

∂z∂r+

1

r2

∂3w

∂z∂θ2+∂3w

∂z3+

2

r2

∂2v

∂z∂θ− 1

r2

∂w

∂z

)− 1

Re

(1

r

∂3u

∂θ∂r2+

1

r3

∂3u

∂θ3+

∂3u

∂θ∂z2

)We eliminate the axial velocity term from the above equations by using the continuity equation

and by incorporating the exponential disturbance equation from eqn. (20), we get

∂u

∂z+∂v

∂r+v

r+

1

r

∂w

∂θ= 0 (28)

⇒ u = − 1

iα

(v′ +

v

r+in

rw

)(29)

where ′ represents derivative in r directionConsidering the flow to be invariable along θ, the above substitution leaves us with a linear

equation and its boundary conditions which are both homogeneous, it follows that the result-ing equation is an eigenvalue problem. The problem can be represented in matrix form as (forreadability, we drop the cap on the perturbed velocity components)

4


[A11 A12

A21 A22

] [vw

]= β

[B11 B12

B21 B22

] [vw

](30)

The values of the elements of the matrix are given as(here D → ∂

∂r

)A11 = − 1

α

∂U

∂z

(D2 − 1

r2+D

r

)+i

α

∂2U

∂r∂z

(D +

1

r

)− ∂U

∂r

(D +

1

r

)− U

(D2 − 1

r2+D

r

)+

i

α

∂V

∂r

(D2 − 1

r2+D

r

)+i

αV

(D3 +

2

r3− 2

r2D +

D2

r

)+∂V

∂rD +

∂2U

∂r2− iα∂U

∂z

+ Uα2 +∂V

∂z

(D +

1

r

)− ∂2V

∂z2

(D +

1

r

)− 1

α

∂V

∂z

(D2 − 1

r2+D

r

)− iαV D − iα∂V

∂r

− ∂2V

∂z∂r+iα

Re

(D2 − 1

r2+D

r

)− 2n2

Re.r3

i

α

(D +

1

r

)+

n2

Re.r2

i

α

(D2 − 1

r2+D

r

)− 1

Re

i

α

(D4 − 6

r4+

6D

r3− 3D2

r2+D3

r

)− 1

Re.r

i

α

(D3 +

2

r3− 2

r2D +

D2

r

)+

1

Re.r2

i

α

(D2 − 1

r2+D

r

)+

1

ReiαD2 +

iα

Re.rD − iαn2

Re.r2− 1

Reiα3 − iα

Re.r2

A12 = − 1

α2

∂U

∂z

(− inr2

+in

rD

)+i

α

∂2U

∂r∂z

(in

r

)− ∂U

∂r

(in

r

)− U

(− inr2

+in

rD

)+

i

α

∂V

∂r

(− inr2

+in

rD

)+i

αV

(2in

r3− 2in

r2+in

rD2

)+∂V

∂z

(in

r

)− ∂2V

∂z2

(in

r

)− 1

α

∂V

∂z

(− inr2

+in

rD

)+iα

Re

(− inr2

+in

rD

)− in2

α

2

Re.r3

(in

r

)+

in2

α

1

Re.r2

(− inr2

+in

rD

)− i

α

(−6in

r4+

6in

r3D − 3in

r2D2 +

in

rD3

)− 1

Re.r

i

α

(2in

r3− 2in

r2+in

rD2

)+

1

Re.r2

i

α

(− inr2

+in

rD

)− 2iα

r

(in

r

)

B11 =β

α

(D2 − 1

r2+D

r

)− αβ

B12 =β

α

(− inr2

+in

rD

)

A21 = − n

r ∗ α

(D +

1

r

)+

i

rα

∂2U

∂θ∂z

(D +

1

r

)− 1

r

∂U

∂θ

(D +

1

r

)− in

rU

(D +

1

r

)+

1

rα

∂V

∂θ

(D2 − 1

r2+D

r

)− V n

rα

(D2 − 1

r2+D

r

)+in

r

∂U

∂r+

1

r

∂2U

∂θ∂r

− nα

Re.r

(D +

1

r

)− n3

Re.r3α

(D +

1

r

)− n

Re.rα

(D3 +

2

r3− 2

r2D +

D2

r

)− n

Re.r2α

(D2 − 1

r2+D

r

)− 2αn

Re.r2

5


A22 = − n

rα

∂U

∂z

(in

r

)+

i

rα

∂2U

∂θ∂z

(in

r

)− 1

r

∂U

∂θ

(in

r

)+Un2

r2+

i

rα

∂V

∂θ

(− inr2

+in

rD

)− V n

rα

(− inr2

+in

rD

)− iα∂U

∂z+ Uα2 − ∂V

∂zD − iαV D − iαV

r− 1

r

∂V

∂z− nα

Re.r

(in

r

)− n3

Re.r3α

(in

r

)+

n

Re.rα

(2in

r3− 2in

r2+in

rD2

)+

n

Re.r2α

(− inr2

+in

rD

)+

iα

ReD2 +

iα

Re.rD − iαn2

Re.r2− iα3

Re− iα

Re.r2

B21 =iβn

αr

(D +

1

r

)

B22 =iβn

αr

(1n

r

)− αβ

The eqn. (30) has three parameters: α,c and ν. For a given profile, only a certain continuous butlimited sequence of these parameters the eigenvalues will satisfy the relation. The mathematicalproblem is to find this sequence, which has a different functional form for spatial versus temporalgrowth of disturbances.

A FORTRAN program that takes the velocity profile as input and evaluates the correspondingeigenvalues from the relation given by eqn. (30) was used to analyse the flow for stability.

3 Parabolized Stability Equations (PSE) for Fluid Flow

3.1 Development

The laminar-turbulent transition of boundary layer flows led to the linear stability theory, assumingparallel flow. The resulting eigenvalue problem, expressed as Orr-Sommerfeld (OSE) and Squireequations, when used for boundary layer transition, showed deviations from the experimentalresults for the neutral curve. These discrepancies were attributed to the local assumption ofparallel flow and the consideration of no upstream development of boundary layer. Consequently,this led to a more general analysis that takes into account both the streamwise development andupstream data of the instability. This analysis further led to the PSEs, applicable to 2D-flows (2D-PSE). These equations are valid for convective instabilities and relax the assumption of parallelflow.

3.2 Introduction

The PSEs are used to analyze the streamwise evolution of Fourier modes in weakly non-parallelflows. It relies on the decomposition of each mode into a slowly varying amplitude function anda wave function with slowly varying wave number. Analogous to the OSE, a BiGlobal stabilityanalysis generalises the flow assumptions in 2D plane. This concept is inadequate in prediction ofbreakdown as it neglects the axial gradients that may influence the vortex. This necessitates ananalysis technique that can account for velocity and pressure gradients. Thus, the 3D-PSE conceptoriginated which has an advantage of being able to solve for arbitrarily complicated basic statesin the x, y plane.

6


3.3 2D-PSE for Incompressible Flow

The continuity and non-linear momentum equations for an incompressible fluid with constantdensity and viscosity are given by

∇~v = 0 (31)

∂~v

∂t+ (~v.∇)~v = −1

ρ∇p+ ν∇2~v (32)

In Cartesian coordinates (x, y, z) we use velocity components v = (u, v, w) and

∇ =∂

∂xi+

∂

∂yj +

∂

∂zk (33)

We consider the stability of 2D steady laminar boundary-layer flows neglecting the mean flowand derivatives in the spanwise z direction. For stability analysis, we decompose the total flowfield v, p into the steady laminar basic flow V, P and the disturbance v, p.

v = V + v, p = P + p (34)

We introduce a term ε such that ξ = εx, where ε is of order O(Re−1). This accounts for the slowvariation of variables with respect to the streamwise direction. Owing to uniformity in z, the steadybasic flow takes the form V = [U(ξ, y), εV (ξ, y), 0]. Here, Reynolds number is Rex = Urx

ν with aproper reference velocity Ur. Under boundary-layer approximation, the basic flow is governed by

Ux + Vy = 0 (35)

UUx + V Uy = −Px +1

ReUyy (36)

0 = −Py (37)

UWx + VWy =1

ReWyy (38)

For disturbances v, p we obtain the stability equations

ux + vy + wz = 0 (39)

vt + Uux + uUx + V uy + vUy +Wuz + px −1

Re(uxx + uyy + uzz) = − (uux + vuy + wuz) (40)

vt + Uvx + V vy + vVy +Wvx + py −1

Re(vxx + vyy + vzz) = − (uvx + vvy + wvz) (41)

wt + Uwx + uWx + V wy + vWy +Wwz + pz −1

Re(wxx + wyy + wzz) = − (uwx + vWy + wwz)

(42)

where we have neglected the small term u′Vx in the y-equation. For a given basic flow, this systemis equivalent to the Navier-Stokes equations and hence of elliptic type.

For the instability study, we apply the following homogeneous boundary conditions

u = 0, v = 0, w = 0 at u = 0 (43)

u −→ 0, v −→ 0, w −→ 0 as y −→∞ (44)

as the inhomogeneous conditions are satisfied by the basic flow. In non-linear problems, however,the fluctuating components of the disturbance also create a distortion of the steady mean flow

7


(u, v, w). Since the mean flow and its distortion are governed by the boundary-layer equations, wemust admit v 6= 0 as y −→∞ with

vy −→ 0 as y −→∞ (45)

and px = 0. In general, the nonlinear stability equations can be solved under inhomogeneousboundary conditions to account for disturbances at the wall or in the free stream.

3.4 PSE Concept in Natural Variables

We maintain the decomposition of the disturbance q into an amplitude function q′ and a wavefunction χ and write the disturbance in the form

q(x, y, z, t) = q′(x, y)χ(x, z, t) (46)

χ = exp(iθ(x) + iβz − iΩt), θx = α(x) (47)

We note that

qx = (iαq′ + q′x)χ (48)

qxx =(−α2q′ + 2iαq′x + q′xx+ iαxq

′)χ (49)

and by substituting the components of eqns. (46 − 47) into the linearized eqns. (39 − 42), weobtain in nondimensional form (Henceforth, the ′ from the amplitude function is dropped and ′

terms will represent the disturbances)

L1u+ L2v +M1∂u

∂x+M2

∂v

∂x+dα

dxN1u+

dα

dxN2v = 0 (50)

where the operators L,M and N act only in y and are given as follows

L1 = −iΩD + iαUy + iαUD + UxD + Uyx + VyD + V D2 + iβWy + iβWD

+α2

ReD − 1

ReD3 − β2

ReD − iαΩ

βD

+iα2

βUy +

iα2

βUD −WxD −Wxy +

α

βVyD

+α

βV D2 + iαWy +

α3

βReD − α

βReD3 +

αβ

ReD

L2 = UyD + Uyy − αΩ− αUx + α2U − VxD − iαV D − iαVy − iβWx

+ αβW − iα3

Re+iα

ReD2 − Vxy −

iαβ2

re

+ βΩ− αβU + iβV D + iβVy − β2W +iα2β

Re

− iβ

ReD2 − iV

βD3 −WyD −Wyy +WyD −

iα2

βReD2

M1 = Uy + UD − 2iα

ReD − Ω

βD +

2α

βUy

+2Uα

βD − iV

βD2 +Wy −

iVyβD − 3iα2

βReD

+i

βReD3 − iβ

ReD

8


M2 = iΩ− Ux − 2iαU − V D − Vy − iβW −3α2

Re+

1

ReD2

− β2

Re+ iβU +

2αβ

Re− iUy

βD − iU

βD2 − 2α

βReD2

N1 = − i

ReD +

Uyβ− 3iα

βReD +

U

βD

N2 = iU − 3α

Re− β

Re− 1

βReD2

where now D = ∂/∂yThe derivation of eqn. (60) from eqns. (31−32) first involves the elimination of the pressure by

taking the curl of the momentum equations which provides the vorticity transport equations. TheOrr-Sommerfeld equation is obtained by subtracting the z derivative of the x-vorticity equationfrom the x derivative of the z-vorticity equation and eliminating w, via continuity. The Squireequation is the z derivative of the y-vorticity equation.

For a proper function α(x), the streamwise variation of q′ is governed by the wave functionχ while the derivatives qx and αx are small. The PSE approximation assumes the variation of qand α as sufficiently small to neglect qxx, αxx, αxqx and their higher derivatives with respect tox. The eqn. (50) is obtained after application of this approximation. Within PSE approximation,the streamwise derivatives of q′ take the form

∂kq′

∂xk= ik

[αkq− ikαk−1 ∂q

∂x− ik(k − 1)

2αk−2 dα

dxq

]χ (51)

and hence, the operators may be related as,

M = −i∂L∂α

,N =1

2

∂M

∂α(52)

The operators defined above have been found to comply with these relations.

3.5 Closure and Norm

Solving eqn. (50) as an initial boundary value problem is difficult due to the appearance of dα/dxwhich affects the operators. In absence of any information regarding the streamwise changes, wecan impose the restriction

αx = 0 (53)

and absorb all streamwise changes into the amplitude function. This choice is most convenientfor the first step of the numerical integration of eqn. (50) as application of the equation for all xwould maintain the error preventing convergence to the correct solution.

Alternatively, a streamwise independent norm can be imposed on q. Such a norm can beapplied at a fixed position given by the maximum streamwise rms fluctuation of the function |u|.The deformation of this function may lead to jumps in value of location. We, thus, prefer anintegral norm.

From eqn. (49), we obtain

−i (lnq′)x = α− iqxq

(54)

where the last term causes a dependence of the local wave number on y, except if qx = 0. Toremove this dependency, the equation is multiplied with the weight |q|2, integrated over the domainΘ in y, and divided by the integral of |q|2. This yields

−i∫

Θ|q|2(lnq′)xdy∫

Θ|q|2dy

= α− i∫

Θq†qxdy∫

Θ|q|2dy

(55)

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where the dagger denotes the complex conjugate. We now choose∫Θ

q†qxdy = 0 (56)

to normalize q and consider eqn. (55) as the definition of α(x).Eqn. (56) minimises the streamwise change ∂q/∂x in a weighted sense across the y domain.

The definition of α(x) obtained from eqn. (55) by deleting pressure and replacing q by the vectorof velocity amplitudes v is of particular importance. The denominator then is the kinetic energyintegral, and the norm is ∫

Θ

v†vxdy = 0 (57)

Equation (50) along with one of the conditions (56) and (57), or any suitable norm are theclosed system of parabolized stability eqquations (PSE). This system permits the simultaneouscalculation of α(x) and q(x, y) in a streamwise marching procedure. Different norms lead todifferent partitions (46) of the solution q′. The physical solution q′, however, is the same to withinsmall effects of the norm on the PSE approximation.

3.6 Solution Procedure

Before solving eqns. (50) and (57), we need to specify initial and boundary conditions. We specifyRe, β,Ω and V, provide the same estimates α0 for the eigenvalue and q0 for the eigenfunction asinitial conditions at xs, and apply the homogeneous boundary conditions (43− 44).

To solve the PSE, the differential operators (52) and boundary conditions are converted intoalgebraic form by any method suitable for the Orr-Sommerfeld problem. We use a spectral collo-cation method with Chebyshev polynomials, together with a domain transformation.

The streamwise derivative is in the simplest case approximated by the two-point finite difference

qx ≈1

4xj(qj+1 − qj

)(58)

where j ≥ 0 is the step index and qj = qj(y) = q(xj , y). Since α appears in the differential opera-tors, the system is nonlinear, and a predictor-corrector approach is employed to find the solution atxj+1. the explicit forward difference formula leads to numerical instability. Although second-orderaccurate, the central difference formula has not been found advantageous in comparison with thefirst-order accurate backward difference

(qx)j+1 ≈1

4xj(qj+1 − qj

)(59)

Eqn. (50) then assumes the form[4xjLnj+1 +Mn

j+1 + (αx)nj+1Nnj+1

]qnj+1 = Mn

j+1qj (60)

where the subscripts 1, 2 have been dropped from the operators L,M,N and (αx)nj+1 = αnj+1−αj ,n being the iteration count. Starting with αnj+1 = αj according to eqn. (53), we obtain qnj+1 andexploit eqn. (55) to obtain an updated wavenumber

αn+1j+1 = αnj+1 −

2i

4xj

∫Θ

(qnj+1)†(qnj+1 − qj)dy∫Θ|qnj+1|2dy

(61)

The integration-update cycle is continued until eqn. (56) is satisfied within a given error limit andthe solution has converged to αj+1,qj+1(y). We then proceed to the next step in x where a betterestimate for α and αx is known. The marching procedure terminates if the convergence criterionis not satisfied within a given number of iterations.

The use of marching technique for solving the PSE is permitted only if the stability problemis governed by downstream propagating information. The PSE approach is valid for convectivelyunstable flows which consist of pipe and channel flows; entrance flows; and boundary layers, mixinglayers, and far wakes.

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Conclusion

The parabolized stability equations have developed into a mature approach to analyse the receptiv-ity, linear, and nonlinear stability of convectively unstable flows into the late stages of transition.The applications of the PSE approach for transition analysis in aerodynamic flows aim towardsdisturbance environment and location of the transition point based on the nonlinear changes inskin friction. The methods discussed here in the report will be useful in better documentation ofthe disturbance environment.

References

[1] Michael S. Broadhurst and Spencer J. Sherwin. The parabolised stability equations for 3d-flows: Implementation and numerical stability. Applied Numerical Mathematics, 58:10171029,2008.

[2] Thorwald Herbert. Parabolized stability equations. Special Course on Progress in TransitionModelling, AGARD REPORT 793:4/1–34, 1994.

[3] Thorwald Herbert. Parabolized stability equations. Annual Review of Fluid Mechanics, 29:245–283, 1997.

[4] Fei Li and Mujeeb R. Malik. Spectral analysis of parabolized stability equations. Computers& Fluids, Vol. 26:pp. 279–297, 1997.

[5] S.S. Motsa and P. Sibanda. On the chebyshev spectral collocation method in channel and jetflows. Journal of Pure Applied Mathematics, 1(1):36–47, 2001.

[6] Helen L. Reed. Parabolized stability equations: 2-d flows. NATO, RTO-EN-AVT-151:4/1–26.

[7] Vassilios Theofilis. Advances in global linear instability analysis of nonparallel and three-dimensional flows. Progress in Aerospace Sciences, 39:249315, 2003.

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Analysis of Parabolized Stability Equations for Perturbed Flows

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