[american institute of aeronautics and astronautics 37th aiaa fluid dynamics conference and exhibit...

Biorthogonal Eigenfunction System for Supersonic

Inviscid Flow Past a Flat Plate

Carlos Chiquete∗ and Anatoli Tumin†

The University of Arizona, Tucson, Arizona 85721

In order to illustrate the application of the biorthogonal eigenfunction system to recep-

tivity problems and to multimode decomposition, a study case is chosen so that all steps of

the method are accompanied by analytical solutions. The receptivity of an inviscid super-

sonic flow past a flat plate to localized periodic-in-time perturbations emanating from the

wall is revisited within the scope of the method of the biorthogonal eigenfunction system

for linearized Euler’s equations. In addition, application of the biorthogonal eigenfunction

system to projection of computational results onto modes of continuous spectra is shown.

Nomenclature

ρ density perturbationπ pressure perturbationv normal velocityu streamwise velocity perturbationθ temperature perturbationM Mach numberA perturbation function vectorE,H1,H2 4 × 4 matricess Laplace transform variableαv Fourier transform variableω frequencyAα eigenfunction of the direct problemBα eigenfunction of the adjoint problemγ specific heats ratiox streamwise coordinatey coordinate normal to the freestreamt time

Subscript

∞ freestream parameters

I. Introduction

The progress being made in computational fluid dynamics provides an opportunity for reliable simulationof such complex phenomena as laminar-turbulent transition, where dynamics of flow transition depends onthe instability of small perturbations excited by external sources. Computational results provide completeinformation about the flow field that would be impossible to measure in real experiments. However, validationof these results might be a challenging problem. Sometimes, numerical simulations of small perturbations inboundary layers are accompanied by comparisons with results obtained within the scope of the linear stability

∗Graduate Student, Program in Applied Mathematics, AIAA Member.†Professor, Aerospace and Mechanical Engineering, AIAA Associate Fellow.

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American Institute of Aeronautics and Astronautics

37th AIAA Fluid Dynamics Conference and Exhibit25 - 28 June 2007, Miami, FL

AIAA 2007-3982

Copyright © 2007 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

theory. In principle, this is possible in the case of a flow possessing an unstable mode. Far downstream fromthe actuator, the perturbations might be dominated by the unstable mode, and one may compare thecomputational results for the velocity and temperature perturbation profiles and their growth rates with thelinear stability theory. However, this analysis does not work when the amplitude of the unstable mode iscomparable to that of other modes, or when one needs to evaluate the amplitude of a decaying mode.

Recently, a method of normal mode decomposition was developed for two- and three-dimensional per-turbations in compressible and incompressible boundary layers.1–3 The method is based on the expansionof solutions of linearized Navier–Stokes equations for perturbations of prescribed frequency into the normalmodes of discrete and continuous spectra. The instability modes belong to the discrete spectrum, whereasthe continuous spectrum is associated with vorticity, entropy, and acoustic modes. Because the problemof perturbations within the scope of the linearized Navier–Stokes equations is not self-adjoint, the eigen-functions representing the normal modes are not orthogonal. Therefore, the eigenfunctions of the adjointproblem are involved in the computation of the normal modes’ weights. Eigenfunctions of the direct andadjoint problems comprise the biorthogonal eigenfunction system (BES).

Originally, the method based on the expansion into the normal modes was used for analysis of discretemodes (Tollmien–Schlichting–like modes) only.4 After clarification of uncertainties associated with the con-tinuous spectra,1 the method was also applied to the analysis of roughness-induced perturbations.5–8 Inorder to find the amplitude of a normal mode, one needs profiles of the velocity, temperature, and pressureperturbations, together with some of their streamwise derivatives, given at only one station downstreamfrom the disturbance source. Because computational results can provide all the necessary information aboutthe perturbation field, application of the multimode decomposition is straightforward. Analysis of the com-putational results by Wang and Zhong9 in Ref. 10 demonstrated how the method can be used for validationof computations and for a detailed description of the perturbation field.

The objective of the present work is to illustrate the main ideas of the multimode decomposition usinga study case where all steps of the method are accompanied by analytical solutions. The problem of per-turbations in inviscid uniform flow past a flat plate provides the opportunity to formulate the biorthogonaleigenfunction system in analytical form.

The structure of the paper is as follows: In §II, we formulate the spatial Cauchy problem for the linearizedEuler equations with periodic-in-time initial conditions and present the formal solution. Section III introducesthe biorthogonal eigenfunction system, and in §IV, we prove that the formal solution developed in §II can berecast as the eigenfunction expansion. Finally, two examples of the use of the BES technique are presentedin §V. Discussion and summary follow in §VI.

II. Spatial Cauchy Problem

The first step in the method is to demonstrate that periodic-in-time solutions of linearized Euler’s equa-tions can be presented as an expansion into normal modes. This problem is analogous to analysis of pertur-bations in boundary layers in Refs. 1, 3, and 11.

Consider two-dimensional perturbations in an inviscid uniform supersonic flow past a flat plate. Axis x ischosen in the flow direction; coordinate y stands for the distance from the plate. The governing equations forthe perturbations are the linearized Euler equations, which can be written in dimensionless form as follows:

∂ρ

∂t+

∂ρ

∂x+

∂u

∂x+

∂v

∂y= 0

∂u

∂t+

∂u

∂x= −

∂π

∂x∂v

∂t+

∂v

∂x= −

∂π

∂y

∂θ

∂t+

∂θ

∂x= (γ − 1)M2

(

∂π

∂t+

∂π

∂x

)

(1)

where u, v, π, ρ, and θ are perturbations of the x and y velocities, pressure, density and temperature,respectively. The free-stream velocity U∞, density ρ∞, and temperature T∞ are chosen as the characteristicscales in (1). The pressure is scaled with the help of ρ∞U2

∞. The coordinates x and y are scaled with alength scale L, whereas the time scale is L/U∞. M and γ in (1) are the free-stream Mach number and the

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specific heats ratio, respectively. In addition, one can find from the linearized equation of state that

ρ = γM2π − θ (2)

In the present work, we consider periodic-in-time solutions of the governing equations in the complex formq(x, y, t) = q(x, y) exp (−iωt). As a result, the equations (1) are recast with the help of the equation of state(2) (we omit ‘ ˆ ’):

− iω(

γM2π − θ)

+∂

∂x

(

γM2π − θ)

+∂u

∂x+

∂v

∂y= 0

− iωu +∂u

∂x= −

∂π

∂x

− iωv +∂v

∂x= −

∂π

∂y

− iωθ +∂θ

∂x= (γ − 1)M2

(

−iωπ +∂π

∂x

)

(3)

The system of four equations can be written in the matrix-vector form with the help of the vector functionA(x, y) = (u, v, θ, π)T , where T refers to the matrix transpose.

E∂A

∂y= H1A + H2

∂A

∂x

E =

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 1

, H1 =

iω 0 0 0

0 0 −iω iωγM2

0 0 iω −iω(γ − 1)M2

0 iω 0 0

H2 =

−1 0 0 −1

−1 0 1 −γM2

0 0 −1 (γ − 1)M2

0 −1 0 0

(4)

Solution of (4) is subject to the following boundary conditions on the wall (y = 0) and at y → ∞:

y = 0 : A2 = 0 (5)

y → ∞ : |Aj | → 0, (j = 1, ..., 4) (6)

At x = 0, the initial data for the amplitude functions are provided:

x = 0 : A = A0(y) = (u0(y), v0(y), θ0(y), π0(y))T (7)

where the initial data vector A0(y) is assumed to decay at y → ∞. This defines the spatial Cauchy problem.Solution of the problem, (4) - (7), can be found with the use of the Laplace transform with respect to x:

As(y) =

∫ ∞

0

A(x, y)e−sxds (8)

The transformation leads to the following boundary-value problem for inhomogeneous ordinary differentialequations:

EdAs

dy− H1As − sH2As = F

y = 0 : As,2 = 0

y → ∞ : |As,j | → 0

(9)

where F = −H2A0. A fundamental solution of the corresponding homogeneous system of (9) can be foundas ∝ exp(−λy), where λ is a root of the characteristic polynomial

−(s − iω)2(−λ2 + M2(s − iω)2 − s2) = 0 (10)

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There are two distinct roots

λ1,2(s) = ±µ(s) (11)

µ(s) =√

M2(s − iω)2 − s2

where the root branch is chosen to have Re(µ) > 0. This defines two fundamental solutions

z1 =

(

−s

s − iω,

µ

s − iω,M2(γ − 1), 1

)T

(12)

and

z2 =

(

−s

s − iω,−

µ

s − iω,M2(γ − 1), 1

)T

(13)

The non-homogeneous system given by (9) has a solution expressed in the form

As(y) = MQ(y) + G(y)

G ≡ (F1/(s − iω), 0, F3/(s − iω), 0)T

(14)

where M = [[z1 exp(−µy), z2 exp(µy)]] is the matrix of fundamental solutions, and the vector of coefficients

Q(y) = (Q1(y), Q2(y))T

has to be found. Applying (14) to (9), we arrive at the following reduced 2 × 2system:

mdQ

dy= f(y),

f =

(

F2 + F3 −s

s − iωF1, F4

)T (15)

where m is a 2× 2 matrix: One can solve the algebraic equations (15) and write down the solution of (9) asfollows

As(y) =

(∫ y

0

C1(y′; s)dy′ − c1(s)

)

z1e−µy +

(∫ y

∞

C2(y′; s)dy′

)

z2eµy + G(y) (16)

where

c1(s) =

∫ ∞

0

C2(y′; s)dy′

C1(y; s) =eµy

2µ

[

µv0 − iωu0 +(

[M2(s − iω) − s)

π0

]

C2(y; s) =e−µy

2µ

[

µv0 + iωu0 −(

[M2(s − iω) − s)

π0

]

G(y) =

(

π0 + u0

s − iω, 0,−

M2(γ − 1)π0 − θ0

s − iω, 0

)T

(17)

Finally, the solution of the spatial Cauchy problem can be written as the inverse Laplace transform

A(x, y) =1

2πi

∫

Γ

As(y)esxds (18)

where Γ indicates a vertical path in the complex plane of s to the right of any singularities of the integrand.There is a pole located at s = iω and two branch points, s = iωM/(M ± 1). In order to have solution (16)decaying at y → ∞ everywhere in the complex plane s, we choose two vertical branch cuts (defined by theequation Re(µ) = 0) as shown in Fig. 1.

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Figure 1. A sketch of the complex plane of s.

The path of integration, Γ, can also be closed, as in Fig. 1. By Cauchy’s residue theorem, the contourintegral over the closed path is equivalent to the residue of the integrand at s = iω, i.e.,

1

2πi

(∫

Γ

· · · +

∫

γ−

· · · +

∫

γ+

· · · +

∫

C∞

. . .

)

= Res(Asesx)s=iω (19)

Because the integral vanishes along the contour C∞, we can represent the contour integral over Γ as a sumof residues derived from the poles of the integrand, and two contour integrals along the branch cuts γ+ andγ−.

One can find explicitly the residue of As(y) exp(sx) at s = iω:

Res(Asesx)s=iω = eiωx

−iω e−ωy[f1(y) + f2(0)] + eωyf2(y) + π0 + u0

ω e−ωy[f1(y) + f2(0)] − eωyf2(y)

θ0 − M2(γ − 1)π0

0

(20)

where

f1(y) =

∫ y

0

eωy′

2[v0 − i(u0 + π0)] dy′ (21a)

f2(y) =

∫ y

∞

e−ωy′

2[v0 + i(u0 + π0)] dy′ (21b)

Then solution of the spatial Cauchy problem is given by

A(x, y) = Res(Asesx) −

1

2πi

(∫

γ−

Asesxds +

∫

γ+

Asesxds

)

(22)

This defines the formal solution to the initial value problem. However, the branch cut integral can be simpli-fied by parameterizing the complex variable s along the branch cut contours. This procedure will facilitatethe proof of the equivalence of the formal solution and the expansion in the biorthogonal eigenfunctionsystem in §IV.

In the following the integrals in (22) along either path (γ+ or γ−) are considered at once, and denotedas γ±. Then writing the explicit path along the branch cuts gives

−1

2πi

∫

γ±

. . . ds = −1

2πi

(

∫ ±i∞

s∗±

As(y)esxds +

∫ s∗±

±i∞

As(y)esxds

)

(23)

where s∗+ = iωM/(M −1) is the upper branch point of the function µ(s) and s∗− = iωM/(M +1) is the lowerbranch point. The vector function As(y) is discontinuous across the branch cut, i.e., if we parameterize

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µ(s) = ±ik for k > 0 on the right-hand side of the branch cut, then µ(s) = ∓ik on the left-hand side. Oneach branch cut, we have s = sm(k) (m = 1, 2), where we solve

µ(s) =√

M2(s − iω)2 − s2 = ±ik

for s. Two solutions for s can be found and are denoted s1,2(k),

s1(k) = i

(

M2ω +√

(M2 − 1)k2 + M2ω2

M2 − 1

)

, s2(k) = i

(

M2ω −√

(M2 − 1)k2 + M2ω2

M2 − 1

)

(24)

Figure 2 illustrates the path of integration and values of µ(k) on the sides of the branch cuts.

Figure 2. The change of variable scheme.

Integrals along the sides of the branch cuts in (23) are then recast as integrals with respect to theparameter k via the transformation s = sm(k) and so ds = (dsm/dk)dk:

−1

2πi

∫

γ±

. . . ds = −1

2πi

(∫ ∞

0

[As(y)]µ=−iks=sm(k) esm(k)x dsm

dkdk . . .

+

∫ 0

∞

[As(y)]µ=iks=sm(k) esm(k)x dsm

dkdk

)

=1

2πi

∫ ∞

0

(

[As(y)]µ=iks=sm(k) − [As(y)]

µ=−iks=sm(k)

)

esm(k)x dsm

dkdk

Writing the full solution,

A(x, y) = Res(Asesx) +

1

2πi

∑

m=1,2

∫ ∞

0

(


µ=−iks=sm(k)

)

esm(k)x dsm

dkdk

Explicitly evaluating the vector valued factor in the integrand,(


µ=−iks=sm(k)

)

= Dm(k)[

eikyz1(s = sm(k), µ = ik) + e−ikyz2(s = sm(k), µ = ik)]

(25a)

where Dm(k) =

∫ ∞

0

[

iv0 sin ky′ −

(

ωu0

k+ i

(M2(sm(k) − iω) − sm(k))

kπ0

)

cos ky′

]

dy′.

(25b)

The final form of the formal solution is therefore as follows:

A(x, y) = Res(Asesx)+

1

2πi

∑

m=1,2

∫ ∞

0

Dm(k)

(

eikyz1(s = sm(k), µ = ik) + e−ikyz2(s = sm(k), µ = ik)

)

esm(k)x dsm

dkdk

(26)

It will be shown in section IV that this formal solution can be recast as an eigenfunction expansion.

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III. Biorthogonal Eigenfunction System

The idea that any disturbance governed by the linearized Navier-Stokes equations can be considered as asuperposition of vorticity, entropy, and acoustic modes was expressed a long time ago (see, for example, Ref.12, pp. 519 and 524). A proof for an ideal gas having Prandtl number 3/4 was given by Wu in Ref. 13. Inthe case of an inviscid gas without heat conduction, the acoustic modes have non-zero velocity, pressure, andtemperature perturbations, whereas there are no vorticity and entropy perturbations. The entropy modeshave non-zero entropy and temperature perturbations, whereas the velocity and pressure perturbations areabsent. The vorticity modes have non-zero velocity and vorticity perturbations, and there are no pressure,entropy, and temperature perturbations.

In the present section, we introduce a biorthogonal eigenfunction system that can serve as a tool forprojection of the perturbation field onto the normal modes, i.e., for decomposition into vorticity, entropy,and acoustic modes.

We define the following biorthogonal eigenfunction system Aα,Bα

EdAα

dy− H1Aα − iαH2Aα = 0

y = 0 : Aα,2 = 0

y → ∞ : |Aα,j | < ∞

(27)

−EdBα

dy− HH

1 Bα + iαHH2 Bα = 0

y = 0 : Bα,4 = 0

y → ∞ : |Bα,j | < ∞

(28)

The superscript H in (28) stands for the Hermitian transpose.Fundamental solutions of the equations (27) can be found in the form ∼ exp(λy), which yields the

following characteristic equation for λ:

−(α − ω)2(λ2 + (M2 − 1)α2 − 2M2ωα + M2ω2) = 0 (29)

leads to four roots of α at a given λ. In order to ensure the boundary conditions at y = 0 and y → ∞ aresatisfied, we impose λ = ±ik (where k is a positive real number), and construct the eigenfunction as a sumof two fundamental solutions ∝ exp(±iky). Therefore, one can find from (29)

α1,2 =M2ω ±

√

(M2 − 1)k2 + M2ω2

M2 − 1(30a)

α3,4 = ω (30b)

Each root α(k) corresponds to the following four distinct eigenfunctions representing the normal modes.The first of these modes we define as the slow acoustic mode, i.e.,

Aac1(y; k) =

α1

k

1M2(γ−1)(ω−α1)

kω−α1

k

eiky −

−α1

k

1

−M2(γ−1)(ω−α1)k

−ω−α1

k

e−iky (31)

The phase speed of the slow acoustic mode at k = 0 is equal to c = 1 − 1/M .The fast acoustic mode is found similarly,

Aac2(y; k) =

α2

k

1M2(γ−1)(ω−α2)

kω−α2

k

eiky −

−α2

k

1

−M2(γ−1)(ω−α2)k

−ω−α2

k

e−iky (32)

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The phase speed of the fast acoustic mode at k = 0 is equal to c = 1 + 1/M .We define the following as the vorticity mode,

Av(y; k) =

1

−ωk

0

0

eiky +

1ωk

0

0

e−iky (33)

where αv = ω. This mode is denoted as the vorticity mode since it produces a non-zero vorticity.Finally, the entropy mode is given by

Ae(y; k) =

0

0

1

0

eiky ±

0

0

1

0

e−iky (34)

where αe = ω. Because the boundary condition on the wall is satisfied for each fundamental solutioncorresponding to the entropy mode, both of them can be considered as independent entropy modes. Instead ofdealing with the fundamental solutions as modes, we construct symmetric and antisymmetric eigenfunctionsby choosing + or − in (34).

The adjoint solution corresponding to the slow acoustic mode is given by

Bac1(y; k) =

−α1

k

−ω−α1

k

−ω−α1

k

1

e−iky −

α1

kω−α1

kω−α1

k

1

eiky (35)

Similarly, the adjoint solution corresponding to the fast acoustic mode is

Bac2(y; k) =

−α2

k

−ω−α2

k

−ω−α2

k

1

e−iky −

α2

kω−α2

kω−α2

k

1

eiky (36)

There is an associated adjoint vorticity mode,

Bv(y; k) =

kω

0

0

1

e−iky −

− kω

0

0

1

eiky (37)

Finally, the adjoint solution associated with the entropy mode is of the form

Be(y; k) =

0

0

1

0

e−iky ±

0

0

1

0

eiky (38)

To derive vectors (35) - (38), we used the theorem14 about the relationship between fundamental solutionsof the adjoint system (28) and the matrix of fundamental solutions of the direct problem (27). The j-thfundamental solution of Eq. (28) can be obtained as a vector comprised of cofactors of the j-th column ofthe matrix of fundamental solutions for the direct problem, Eq. (27).

One can establish the following orthogonality relation for the modes of the continuous spectra:1

〈H2Aα′(y; k′),Bα(y; k)〉 = Qαδ(k − k′) (39)

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where δ(k − k′) is the Dirac delta function and Qα is a constant, which depends on the normalization of theeigenfunctions (Qα = 0 if α and α′ belong to different normal modes or belong to different symmetries inthe case of the entropy modes).

For the particular normalizations of Aα(y; k) and Bα(y; k) used above, one can find

Qac1 = −4πω − α1

k

[

dα1

dk

]−1

Qac2 = −4πω − α2

k

[

dα2

dk

]−1

Qv = −2π

(

ω

k+

k

ω

)

Qe = −2π

(40)

IV. The Coefficients of the Expansion

In the present section, we are going to show that the solution (22) can be written as an expansion intothe eigenfunctions so that

A(x, y) ≡∑

α′

∫ ∞

0

Cα′(k′)Aα′(y; k′)eiα′xdk′ (41)

One can recognize that the input from the residue value in (22) is associated with the entropy and vorticitymodes, whereas the integrals along the branch cuts lead to the fast and slow acoustic modes in (41).

Using the orthogonality condition in (39) and that A0(y) is a given vector function in (7), it follows that

Cα(k) =〈H2A0(y),Bα(y; k)〉

Qα(42)

In order to confirm that the eigenfunction system representation (41) together with (42) is equivalent to(22), we proceed with the discussion of the modes separately.

A. Vorticity Modes

The vorticity mode has two non-zero components (u and v) and therefore its contribution to the solutioninvolves only these two components. Using the definition for the vorticity coefficient defined in (42), we willshow that the contribution of the vorticity mode is equivalent to the residue value of these two componentsin (20). The coefficient of the expansion for the vorticity mode is given by Eq. (42), therefore,

Cv(k) =k2

π(k2 + ω2)

∫ ∞

0

(u0 + π0) cos ky′dy′ +iωk

π(k2 + ω2)

∫ ∞

0

v0 sin ky′dy′ (43)

The contribution to the solution from the vorticity mode can be evaluated as the following integral:

Av(x, y) =

∫ ∞

0

Cv(k)Av(y; k)eiαvxdk = eiωx

∫ ∞

0

Cv(k)Av(y; k)dk (44)

The first component of Av(x, y) is exp(iωx)uv(y) where uv(y) is determined by the following integral:

uv(y) = 2

∫ ∞

0

Cv(k) cos kydk

=2

π

∫ ∞

0

(∫ ∞

0

k2(u0 + π0)

k2 + ω2cos ky′dy′

)

cos kydk +2

π

∫ ∞

0

(∫ ∞

0

iωkv0

k2 + ω2sin ky′dy′

)

cos kydk

(45)

Since the integrand of both double integrals is a bounded function of k and y′, we can exchange the order ofintegration,

uv(y) =2

π

∫ ∞

0

(u0 + π0)

(∫ ∞

0

k2

ω2 + k2cos ky cos ky′dk

)

dy′ +2

π

∫ ∞

0

iv0

(∫ ∞

0

ωk

ω2 + k2cos ky sin ky′dk

)

dy′

(46)

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The integrals inside the parentheses can be evaluated explicitly:15

∫ ∞

0

k2

ω2 + k2cos ky cos ky′dk =

π

2δ(y − y′) −

ωπ

2

e−ωy cosh ωy′, y′ < y

e−ωy′

cosh ωy, y′ > y

∫ ∞

0

ωk

ω2 + k2cos ky sin ky′dk =

ωπ

2

−e−ωy sinh ωy′, y′ < y

e−ωy′

coshωy, y′ > y

(47)

Therefore, one can obtain from (46)

uv(x, y) = (π0 + u0) − ωe−ωy

∫ y

0

[(u0 + π0) cosh ωy′ + iv0 sinhωy′] dy′ . . .

− ω coshwy

∫ ∞

y

e−ωy′

[−iv0 + (u0 + π0)] dy′

(48)

Expanding each of the hyperbolic sine and cosine functions and regrouping terms lead to the following result:

uv(x, y) = (π0 + u0) − iωe−ωy

∫ y

0

[v0 sinh ωy′ − i(u0 + π0) cosh ωy′] dy′ . . .

+ iω coshwy

∫ ∞

y

e−ωy′

[v0 + i(u0 + π0)] dy′

= (u0 + π0) − iω

(

e−ωy

[

∫ y

0

eωy′

2[v0 − i(π0 + u0)]dy′ . . .

−

∫ ∞

0

e−ωy′

2[v0 + i(u0 + π0)]dy′

]

− ewy

∫ ∞

y

e−ωy′

2[v0 + i(u0 + π0)]dy′

)

(49)

Finally, Eq. (49) can be recast as

uv(x, y) = eiωx(u0 + π0) − iω[

e−ωy[f1(y) + f2(0)] + eωyf2(y)]

(50)

where f1(y) and f2(y) are defined in (21a) and (21b), respectively.The same procedure is used to derive the second component of Av(x, y), which is defined as exp(iωx)vv(y).

In order to determine vv(y), we write that

vv(y) = −2i

∫ ∞

0

ω

kCv(k) sin ky′dy′

=2

π

∫ ∞

0

v0

(∫ ∞

0

ω2

k2 + ω2sin ky sin ky′dk

)

dy′ −2

π

∫ ∞

0

i(π0 + u0)

(∫ ∞

0

ωk

k2 + ω2sin ky cos ky′dk

)

dy′

The integrals in parentheses can be found explicitely,15 i.e.,

∫ ∞

0

ω2

k2 + ω2sin ky sin ky′dk =

ωπ

2

e−ωy sinhωy′, y′ < y

e−ωy′

sinh ωy, y′ > y

∫ ∞

0

ωk

k2 + ω2sin ky cos ky′dk =

ωπ

2

e−ωy coshωy′, y′ < y

−e−ωy′

sinh ωy, y′ > y

(51)

Expanding of the hyperbolic functions and regrouping the terms lead to

vv(y) = ω

[

e−ωy

∫ y

0

eωy′

2[v0 − i(u0 + π0)]dy′ . . .

− e−ωy

∫ y

0

e−ωy′

2[v0 + i(u0 + π0)]dy′ + eωy

∫ ∞

0

e−ωy′

2[v0 + i(u0 + π0)]dy′

] (52)

10 of 18


Finally, Eq. (52) can be recast as

vv(x, y) = ωeiωx[

e−ωy[f1(y) + f2(0)] + eωyf2(y)]

There are no other non-zero terms in Av, therefore the contribution to the solution from the vorticity modeis

Av(x, y) = eiωx

(u0 + π0) − iω [e−ωy[f1(y) + f2(0)] + eωyf2(y)]

ω [e−ωy[f1(y) + f2(0)] + eωyf2(y)]

0

0

(53)

If we refer to Eq. (20), the vorticity contribution is exactly equivalent to the first two components of theresidue value of As(y) exp(sx) at s = iω. In the following we shall show that the third component of theresidue value is represented by the entropy mode.

B. Entropy Modes

For brevity of the discussion, we assume that the initial data allow expansion only into the symmetric entropymode. The entropy mode coefficient is calculated using Eq. (42), and so we obtain

Ce(k) =1

π

∫ ∞

0

(θ0 − M2γπ0) cos ky′dy′ (54)

The entropy mode has only one non-zero component, i.e., Ae3(y; k) = 2 cos ky. Its contribution, exp(iωx)θe(y),can be evaluated as the following integral:

θe(y) = 2

∫ ∞

0

Ce(k) cos kydk

=2

π

∫ ∞

0

(∫ ∞

0

[θ0 − M2(γ − 1)π0] cos ky′dy′

)

cos kydy

=2

π

∫ ∞

0

[θ0 − M2(γ − 1)π0]

(∫ ∞

0

cos ky cos ky′dk

)

dy′

(55)

The integral in parentheses can be found to be proportional to the Dirac delta function, specifically,

∫ ∞

0

cos ky cos ky′dk =π

2δ(y − y′) (56)

Consequently,θe(y) = θ0 − M2(γ − 1)π0

Now it is clear that the contribution from the entropy and vorticity modes to the solution A(x, y) is thesame as the contribution from the residue value at s = iω in the formal solution, Eq. (22),

Res(As(y)esx) =

∫ ∞

0

Cv(k)Av(y; k)eiαvxdk +

∫ ∞

0

Ce(k)Ae(y; k)eiαexdk

=

−iω e−ωy[f1(y) + f2(0)] + eωyf2(y) + (π0 + u0)

ω e−ωy[f1(y) + f2(0)] − eωyf2(y)

θ0 − M2(γ − 1)π0

0

(57)

C. Fast and Slow Acoustic Modes

It will be shown the integrals along the branch cut in Fig. 1 that have been parameterized by k > 0 in (26),can be represented via the fast and slow acoustic modes. The equivalence will follow from first expressing theexpansion into the acoustic modes, and subsequently showing the acoustic mode expansion to be equivalent

11 of 18


to the integrals that appear in the formal solution in (26). First, the expansion into the acoustic modesusing the coefficients defined in (42) for the slow and fast modes is given by

Aac(x, y) =∑

m=1,2

∫ ∞

0

Cacm(k)Aacm(y; k)eiαmxdk (58)

where the coefficients are found from (42) as

Cacm(k) =1

2π

k

ω − αm

dαm

dkCm(k) (59a)

Cm(k) =

∫ ∞

0

iv0 sin kydy −

∫ ∞

0

(

ω

ku0 +

M2(ω − αm) + αm

kπ0

)

cos kydy (59b)

The wave numbers αm(k) (m = 1, 2) in (59) are defined by (30a).From (24) it is clear that in fact sm(k) = iαm(k), and the acoustic mode coefficients is recast as:

Cacm(k) =1

2π

k

iω − sm

dsm

dkCm(k) (60a)

Cm(k) =

∫ ∞

0

iv0 sin kydy −

∫ ∞

0

(

ω

ku0 +

M2(sm − iω) − sm

kπ0

)

cos kydy (60b)

Also, we can show that the acoustic mode vector can be represented in terms of the fundamental solutionsof the Laplace transform solution in (26),

Aacm(y, k) =iω − sm(k)

k

(

1

i

[

eikyz1(s = sm(k), µ = ik) + e−ikyz2(s = sm(k), µ = ik)]

)

And therefore the contribution of the acoustic modes is equivalent to

Aac(x, y) =∑

m=1,2

∫ ∞

0

Cacm(k)Aacm(y; k)eiαmxdk =

1

2πi

∑

m=1,2

∫ ∞

0

Cm(k)[

eikyz1(s = sm(k), µ = ik) + e−ikyz2(s = sm(k), µ = ik)]dsm

dkesm(k)dk

Given that Dm(k) = Cm(k) where Dm(k) is defined in (25), the acoustic mode component is exactly equalto the integrals that appear in the formal solution in (26). Therefore, it is concluded that integrals over γ+

and γ− can be represented by the fast and slow acoustic modes, i.e.,

−1

2πi

(∫

γ−

. . . ds +

∫

γ+

. . . ds

)

=∑

m=1,2

∫ ∞

0

Cacm(k)Aacm(y; k)eiαmxdk (61)

with the coefficients defined from Eq. (59). Analysis of the branches corresponding to the slow and fastacoustic modes ends the proof of the equivalence of the Cauchy problem formal solution (22) and eigenfunc-tion expansion (41).

V. Two Examples of BES Application

A. Projection of Computational Results onto the Normal Modes

The biorthogonal eigenfunction system can be used to obtain insight into computational results by decompo-sition of the perturbations into the normal modes. For the purpose of illustration, we analyze computationalresults for the receptivity problem involving a periodic-in-time actuator placed on a flat plate in uniformsupersonic inviscid flow. This part of the work is similar to the analysis of acoustic perturbations in Ref. 10.

The actuator is emulated by the inhomogeneous boundary condition on the wall:

v(x, y = 0, t) = vw(x, t) = εg(x) sin ωt (62)

12 of 18


For the numerical example the following dimensionless parameters are chosen: freestream Mach numberM = 6.62; angular frequency of the perturbations ω = 0.1, the actuator’s length w = 100, and the specificheats ratio γ = 1.4. These parameters are close to the local parameters in the direct numerical simulationof perturbations emanating from the wall of a wedge in supersonic flow in Ref. 9.

The actuator shape function g(x) is given by

g(x) =

20.25l5 − 35.4375l4 + 15.1875l2, (l ≤ 1)

−20.25(2.0 − l)5 + 35.4375(2.0 − l)4 − 15.1875(2.0 − l)2 (l ≥ 1)(63)

where l(x) = 2x/w. A plot of the shape function is shown in Fig. 3. The actuator amplitude ε is chosen as10−4.

x

g(x

)

100806040200

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

Figure 3. The shape function g(x).

The nonlinear Euler equations are solved with the help of a code based on the Conservation Ele-ment/Solution Element (CE/SE) Euler method introduced by Chang et al.16

For the problem at hand, we use the following computational domain: 0 ≤ x ≤ 200 and 0 ≤ y ≤ 35. Twogrids, Nx × Ny, having Nx and Ny intervals in the x and y directions, respectively, were used in order tocheck the numerical convergence:

• Coarse Grid: 250 × 200 ,

• Fine Grid: 500 × 400.

The time steps were τ = 0.1 and 0.05 for the coarse and fine grids, respectively.

ΩΩ

Figure 4. The alternating grids Ω1 and Ω2.

The CE/SE method uses two alternating grids, Ω1 and Ω2. These are shown in Fig. 4. The methodbegins with grid Ω2 at t = 0, intermediately finds a solution on grid Ω1 at t = τ/2, and finishes on grid

13 of 18


Ω2 at t = τ , and so forth. We notice that at certain x-coordinates, the grid uses points slightly above andbelow the wall. The actuator boundary condition (62) is applied at these points in spite of their shift fromy = 0 in order to simplify the algorithm. We can estimate the effect of the simplified boundary conditionsby comparing the fine and coarse grid solutions in the vicinity of the wall. Downstream of the actuator,the no-penetration boundary condition is imposed at the wall, y = 0, following Ref. 16. The boundaryconditions at the right and the upper sides of the domain are the non-reflecting boundary conditions usedin Ref. 16. On the left-hand side of the domain, the no-perturbation boundary condition was used.

The CE/SE scheme includes three additional parameters: α and β (both related to the control of os-cillations resulting from discontinuities such as shock waves) and ε0 (serving to control effects of numericalviscosity). In this case, α = β = 0 and ε0 = 0.3. The value for ε0 was chosen after experiments with theparameter and was found to mostly affect the layer close to the wall.

(a)(a)

x

y

200150100500

30

25

20

15

10

5

0

(b)(b)

x

y

200150100500

30

25

20

15

10

5

0

Figure 5. Coarse grid result for the (a) pressure perturbation (increment of .5 × 10−5 between the contours)and (b) streamwise velocity perturbation (increment 1 × 10−5)

The numerical results for the pressure field, π, and streamwise velocity field, u, at t = 200, are shownin Fig. 5. The dashed contours represent negative contours while the solid lines represent the positivecontours. Each contour represents an increase or decrease of the indicated increment. For example, theoutermost positive contour in Fig. 5(a) has value 0.5 × 10−5, and the innermost positive contour has value3.0 × 10−5. These results were produced using the coarse grid.

Comparison of the numerical results with ε = 2 × 10−4 in (62) demonstrated that ε = 10−4 is smallenough that the perturbations can be interpreted as linear.

Figure 6 shows the pressure perturbation and streamwise velocity as function of y at x = 160 for bothcoarse and fine grids. The results show nearly perfect agreement. In the case of the streamwise velocity,

(a)(a)

y

π

2520151050

3e-05

1.5e-05

0

-1.5e-05

(b)(b)

y

u

2520151050

5e-05

4e-05

3e-05

2e-05

1e-05

0

Figure 6. Coarse (crosses) and fine (solid-line) grid comparison for x = 160 and t = 200 for the (a) pressureperturbation and (b) streamwise velocity perturbation

there is a relatively larger difference between the coarse and fine grid solutions in the vicinity of the wall,which we attribute to the inaccuracy in the application of the actuator boundary condition at the staggeredpoints in the vicinity of the wall.

14 of 18


As an example of the BSE application, one can use the computational results together with (42) in orderto find a projection onto the normal modes at a prescribed coordinate x. Figure 7 shows the amplitudesof the slow and fast acoustic modes as functions of the parameter k. Because there is no dissipation, theamplitude distributions shown in Fig. 7 are independent of x. One can see that the fast acoustic mode

(a)(a)

k

|Cac1|

21.81.61.41.210.80.60.40.20

7e-05

6e-05

5e-05

4e-05

3e-05

2e-05

1e-05

0

(b)(b)

k

|Cac2|

21.81.61.41.210.80.60.40.20

0.0003

0.00025

0.0002

0.00015

0.0001

5e-05

0

Figure 7. Projection result for the (a) slow acoustic mode (Cac1) and (b) fast acoustic mode (Cac2)

amplitude is a magnitude larger than the slow acoustic mode. The BES technique reveals this insight, whichotherwise would remain hidden from view if one knew the numerical solution only.

It is worthwhile to compare the computational results and their projection onto the fast and slow acousticmodes presented in this section with the results of Ref. 10. The results are very close qualitatively. Thelatter means that the main features of the numerical results for acoustic modes observed in Ref. 10 have aninviscid character.

B. Projection of the Analytical Result onto the Normal Modes: the Receptivity Problem

Solution

The receptivity problem that was solved numerically in the preceding section has an analytical solution.17

It can be found as a solution of the linearized velocity potential equation with inhomogeneous boundarycondition on the wall. The solution provides the perturbation flow field, and it does not reveal anythingabout the amplitudes of the fast and slow acoustic modes. Using the BES, one can decompose the analyticalsolution as was done for the numerical result. Instead of the decomposition of the available analytical result,we are going to show how the BES can be used to find the receptivity problem solution as an expansion intothe normal modes directly. (This part of the work is similar to the receptivity problem solution in Ref. 7,within the scope of the linearized Navier-Stokes equations.) After that, one can prove that the solution isequivalent to the result of Ref. 17.

Assuming that the solutions of the linearized Euler equations are proportional to exp(−iωt), the governingequations are identical to (4) except for differing boundary conditions on the wall.

E∂A

∂y= H1A + H2

∂A

∂x

y = 0 : A2(x, 0) = vw(x)

y → ∞ : |Aj | → 0

(64)

where vw(x) = εg(x) as defined in (63).To find the formal solution of the problem, we begin with the Fourier transform with respect to x,

Af (y) =

∫ ∞

−∞

A(x, y)e−iαf xds (65)

As a result, the problem (64) is transformed to the following boundary-value problem for ordinary differential

15 of 18


equations:

EdAf

dy− H1Af − iαfH2Af = 0

y = 0 : Af,2 = ρ(αf )

y → ∞ : |Af,j | → 0

(66)

where

ρ(αf ) =

∫ ∞

−∞

vw(x)e−iαf xdx (67)

One can find the solution of the problem (66) as follows:

Af (y) = ρ(αf )

(

−iαf

µ, 1,

iM2(γ − 1)(αf − ω)

µ,i(αf − ω)

µ

)T

e−µy (68)

where

µ(αf ) =√

α2f − M2(αf − ω)2 (69)

and the positive branch of the square root, Re(µ) > 0 is chosen.Finally, one can write the formal solution of problem (64)

A(x, y) =1

2π

∫ +∞

−∞

Af (y)eiαf xdαf (70)

The formal solution can be presented as a sum of the normal modes with coefficients

Cα(k) =〈H2A(y),Bα(y; k)〉

Qα(71)

=

∫ +∞

−∞〈H2Af ,Bα〉e

i(αf−α)xdαf

2πQα

Using the dot product of Eq. (64) and Bα, together with integration by parts, one can derive

〈H2Af ,Bα〉 = −ρ(αf )Bα2(y = 0; k)

i(αf − α(k))

Therefore,

Cα(k) = −Bα2(y = 0; k)

Qα(k)

(

1

2πi

∫

L

ρ(αf )ei(αf−α(k))x

αf − α(k)dαf

)

(72)

The integral in (72) can be evaluated explicitly as the residue value at αf = α(k) by completing the path inthe upper half-plane αf . As a result, we arrive at

Cα(k) = −ρ(α(k))

Qα(k)Bα2(y = 0; k) (73)

If we refer to the equations defining the adjoint modes, it is clear from Eqs. (37) and (38) that the entropyand vorticity mode coefficients are zero since the second component of both modes is zero, i.e., Be2 = 0 andBv2 = 0. This result explains the observation in Ref. 7 that amplitudes of the entropy and vorticity modesare very small in the case of a periodic-in-time actuator placed at the bottom of the boundary layer.

For the slow and fast acoustic modes, the second component at y = 0 is non-zero for both modes, i.e.,Bacm2 = −2(ω − αm)/k, and one can find explicitly amplitudes of the slow, Cac1(k), and fast, Cac2(k),modes as follows:

Cac1(k) = −kρ(α1(k))

2π√

(M2 − 1)k2 + M2ω2(74)

16 of 18


Cac2(k) =kρ(α2(k))

2π√

(M2 − 1)k2 + M2ω2(75)

The Fourier transform ρ(α) for the particular case of vw(x) was found the help of Mathematica,18 yielding

ρ(α) =2iεe−iαw/2

w5α6(8wα(48a2 + a3w

2α2) coswα

2− 3840a1 sin

wα

2+

384(5a1 − a2)wα − 8(10a1 − 6a2 + a3)w3α3 + (a1 − a2 + a3)w

5α5)

with a1 = 20.25, a2 = −35.4375, and a3 = 15.1875.We are now able to compare the amplitudes, (74) and (75), with the amplitudes derived from the

projection of the numerical results for the slow and fast acoustic modes (see Fig. 8). The close agreement of

(a)(a)(a)(a)

k

|Cac1|

21.81.61.41.210.80.60.40.20

7e-05

6e-05

5e-05

4e-05

3e-05

2e-05

1e-05

0

(b)(b)

k

|Cac2|

21.81.61.41.210.80.60.40.20

0.0003

0.00025

0.0002

0.00015

0.0001

5e-05

0

Figure 8. Comparison for the projection (crosses) and theoretical result (solid line) for the (a) slow acousticmode (Cac1) and (b) fast acoustic mode (Cac2).

the results serves as validation for the numerical method that was employed in solving the Euler equations forthe receptivity problem. Figure 9 is a comparison of the theoretical and computational results for pressureand streamwise velocity perturbations at x = 160, t = 200.

(a)(a)(a)(a)

y

π

2422201816141210

3e-05

1.5e-05

0

-1.5e-05

(b)(b)

y

u

252015105

5e-054.5e-05

4e-053.5e-05

3e-052.5e-05

2e-051.5e-05

1e-055e-06

0-5e-06

Figure 9. Comparison of the theory (solid-line) with the coarse (circle) and fine (cross) results for the (a)pressure and (b) streamwise velocity

VI. Conclusions

The practical use of the BES technique with respect to analysis of computational data was illustrated byconsidering the receptivity problem of a periodic-in-time actuator on the wall in uniform flow. A numericalmethod derived from Ref. 16 was used to solve the problem numerically. The computational data wereprojected onto the normal modes of the continuous spectra (slow and fast acoustic modes). As a result, it

17 of 18


was found that the slow and fast acoustic modes compose the perturbation field found through the numericalmethod. Thus, The BES system helps to reveal the physical structure of the flow.

The receptivity problem was solved analytically, as well. With the help of the BES technique, it wasshown that the actuator does not excite vorticity or entropy modes. We compared the theoretical andnumerical amplitudes of the fast and slow acoustic modes, finding a close agreement between the predictionsderived from our theory and the projection of the computational results. This type of analysis can be usedfor validation of numerical methods in other problems.

Acknowledgments

This work was sponsored by the Air Force Office of Scientific Research, USAF under grant No. FA9550-05-101 monitored by Dr. J. D. Schmisseur. The views and conclusions contained herein are those of theauthor and should not be interpreted as necessarily representing the official polices or endorsements, eitherexpressed or implied, of the Air Force Office of Scientific Research or the U. S. Government.

References

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Fluids, Vol. 15, 2003, pp. 2525–2540.2Guydos, P. and Tumin, A., “Multimode decomposition in compressible boundary layers,” AIAA J., Vol. 42, 2004,

pp. 1115–1121.3Tumin, A., “Three-dimensional spatial normal modes in compressible boundary layers,” AIAA Paper 2006–1109, 2006.4Zhigulev, V. N. and Tumin, A. M., Origin of Turbulence, Nauka, Novosibirsk, 1987, (in Russian) [NASA TT-20340,

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AIAA Paper 2005–5025, 2005.10Tumin, A., Wang, X., and Zhong, X., “Direct numerical simulation and the theory of receptivity in a hypersonic boundary

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of Mathematics and Physics, Vol. 35, 2005, pp. 13–27.14Kamke, E., Differentialgleichungen. Losungsmethoden und Losungen, Akademische Verlagsgesellschaft Geest & Portig,

Leipzig, 1959.15Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, Academic Press, Inc., San Diego, CA, 5th

ed., 1994, p. 453.16Chang, S.-C., Wang, X.-Y., and Chow, C.-Y., “The Space-Time Conservation Element and Solution Element Method:

A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws,” Journal of Computational

Physics, Vol. 156, 1999, pp. 89–136.17Ashley, H. and Landahl, M., Aerodynamics of Wings and Bodies, Addison-Wesley Publishing Company, Inc., Reading,

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