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1 American Institute of Aeronautics and Astronautics

MULTIMODE DECOMPOSITION IN COMPRESSIBLE BOUNDARY LAYERS

Paul Gaydos* and Anatoli Tumin† The University of Arizona, Tucson, Arizona 85721

Abstract Two-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier-Stokes equations. Because a spatially growing solution can be expanded into a biorthogonal eigenfunction system, the latter can be utilized for decomposition of flow fields derived from computational studies when pressure, temperature, and all velocity components, together with their derivatives, are available. The method can be used also in a case where partial data are available when a priori information leads to consideration of a finite number of modes.

Nomenclature ( , , )x y tA = vector-function of 9 components

jA = j-th component of vector A D = d dy 2 / 3e = ratio of the bulk viscosity to the dynamic

viscosity , ij ijH L = ij-matrix element

m = 2( 1) / 3e − M = local Mach number p = Laplace variable Pr = Prandtl number r = 2( 2) / 3e + R = Reynolds number

sT = mean flow temperature

sU = mean flow velocity u = streamwise velocity disturbance v = normal velocity disturbance x = streamwise coordinate y = coordinate normal to the wall α = streamwise wave number γ = specific heat ratio µ = viscosity disturbance

sµ = mean flow viscosity

sµ ′ = d dTs sµ ____________ *Research Assistant, Department of Aerospace and Mechanical Engi- neering. †Assistant Professor, Department of Aerospace and Mechanical Engi-neering ([email protected]). Senior Member AIAA.

π = pressure disturbance θ = temperature disturbance ρ = density disturbance

sρ = mean flow density Superscripts T = transposed Subscripts e = edge of the boundary layer s = mean flow

Introduction The progress being made in computational fluid dynamics (CFD) provides an opportunity for reliable simulation of such complex phenomenon as laminar-turbulent transition. The dynamics of flow transition depends on the instability of small perturbations excited by external sources. CFD provides complete informa-tion about the flow field, which would be impossible to measure in real experiments. However, this increase in available information does not furnish a physical in-sight of the transition because the leading mechanisms still remain hidden behind a messy disturbance field. Sometimes, a flow possesses a few instability modes that are equally significant in the transition process, and it might be desirable to distinguish the dynamics of each mode in the complex non-steady flow field. Figure 1 illustrates the dependence of the real parts of wave numbers of the first and the second discrete modes on the local Reynolds number, /e eR Uxρ µ= , in a hypersonic boundary layer over a sharp cone, where , e eρ µ , and eU are density, viscosity, and mean velocity at the edge of the boundary layer. The coordinate x is measured along the surface of the cone, and the wave numbers are dimensionless with the help of the length scale /e e ex Uµ ρ . The frequency parameter is 6150 10F −= × , and the Mach number is equal to 6. One can see that there is a synchronism of these two modes in the vicinity of R = 1600. The synchronism is not absolute because the imaginary parts of the wave numbers are different. Nevertheless,

33rd AIAA Fluid Dynamics Conference and Exhibit23-26 June 2003, Orlando, Florida

AIAA 2003-3724

Copyright © 2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


200 400 600 800 1000 1200 1400 1600 1800 2000

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Mode 1

Mode 2

Cone. Me = 5.6. Adiabatic wall. F = 150×10-6

α r

R

Fig. 1 Wave numbers of two discrete modes. the complex wave numbers are very close, and Fig. 2 demonstrates that the mode shapes are very close as well. In simple words, one has to recognize these modes in the presence of acoustic, vorticity, and entropy modes and separate them from one another. Consequently, the problem of decomposing the flow fields into normal modes arises. Because CFD provides complete information about the flow field, one can expect that the decomposition may be formulated as a rigorous procedure. We shall discuss the problem of two-dimensional perturbations in two-dimensional compressible bound-ary layers. The linearized Navier-Stokes equations are considered as the governing equations for the pertur-bations, and they provide the basis for the decomposi-tion. Gustavsson1 solved a temporal initial-value prob-lem in a parallel boundary layer by using Fourier-Laplace transforms. The inverse Laplace transform was presented as a sum of integrals along a branch-cut (continuous spectrum) and the residue values at poles (discrete spectrum). Salwen and Grosch2 formulated a biorthogonal eigenfunction system within the scope of a temporal stability theory and derived an orthogonality condition between the eigenfunctions of the direct and adjoint linearized equations. They also showed that the results in Ref. 1 can be recast as an expansion of the solution into the eigenfunctions of continuous and dis-crete spectra, with coefficients depending on the initial perturbation. The results in Ref. 2 can be used for decomposition of, for example, wave packets in bound-ary layers. Later on, Fedorov and Tumin3 extended the analysis of the temporal initial-value problem to com-pressible boundary-layer flows. Another class of problems is associated with spatially growing perturbations in boundary-layer flows. The signaling problem may serve as an exam-

0 5 10 15 20 25 300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

|u|

y

mode 1 mode 2

Cone. Me = 5.6. Adiabatic wall. R = 1600. F = 150×10-6

Fig. 2 Profiles (normalized to 1) of the discrete modes at R = 1600.

ple.4,5 This problem is related to a disturbance generator of prescribed frequency starting oscillations at initial time 0t = . After a transient period of time, the solution of the linearized Navier-Stokes equations may be ex-pressed as a sum of integrals along branch-cuts (con-tinuous spectrum) and the residue values at poles (dis-crete spectrum) in the complex wave number plane. The corresponding biorthogonal eigenfunction system for spatially growing disturbances was introduced independently by Zhigulev et al.6 and by Salwen and Grosch2 for two-dimensional perturbations in two-dimensional incompressible boundary layers. In addition to discrete modes (Tollmin-Schlichting-like modes), there are four branches of continuous spectra. The modes of two branches have arbitrary large growth rates in the downstream direction. This indicates that a spatial initial-value problem is ill-posed, and Hadamard’s example7 illustrates the ill-posedness of the problem. The ill-posedness was a stumbling block in the decomposition of spatially growing disturbances into modes because the method in Refs. 1 and 2 could not be applied. However, the ill-posedness was recog-nized by Tumin and Fedorov8 (see also Ref. 9), and a suggestion was made to consider spatial initial-value problems when the solution has a finite growth rate in the downstream direction. An analysis of compressible8 and incompressible boundary layers9 using a Laplace transform with respect to the streaming coordinate, x, demonstrated completeness of the biorthogonal eigen-function system, similar to the temporal theory.1,2 The results in Ref. 9 could be used for decom-position of CFD data when pressure, temperature, and disturbance perturbations, velocity disturbance profiles, and their streamwise derivatives are known at one station, x. In experiments, one usually deals with partial


information, i.e., only one velocity component is measured. If for some reason, one can expect that only a few modes are responsible for the total signal measured in the experiment, this a priori information also can be helpful to decompose the experimental flow field. The first application of multimode decomposition was carried out for an analysis of experimental data in a laminar wall jet when only one velocity component was measured.10 The main feature of the laminar wall jet is that two unstable discrete modes can coexist in the flow. This allowed the assumption that these two modes provide the main input into the measured signal some-where far downstream from a vibrating ribbon. There-fore, the decomposition becomes possible with the help of one velocity component measured at a downstream station. Comparison of recovered and measured data illustrated good agreement. Later on, the method was applied to an analysis of numerical results for forced transitional wall jets.11 The objective of the present work is to discuss/ illustrate the possible application of the biorthogonal eigenfunction system to the multimode decomposition of DNS and experimental data.

Governing Equations and Formal Solution to the Spatial Cauchy Problem

We analyze two-dimensional perturbations in a two-dimensional compressible boundary layer, and the boundary layer is considered in the parallel flow approximation. After Fourier transform with respect to time, the linearized Navier-Stokes equations are recast in the following matrix form:

0 1 1 2L L H Hy y y x

∂ ∂ ∂ ∂+ = + ∂ ∂ ∂ ∂

A A AA (1)

where 0 1 1, , L L H , and 2H are 9 9× matrices. Their non-zero elements are presented in Appendix A. The vector-function, ( , )x yA , is defined with the help of the streamwise and normal velocity perturbations, u and v, respectively; pressure perturbation,π ; and temperature perturbation,θ .

(

), / , , , , / ,

/ , / , /T

u u y v y

u x v x x

π θ θ

θ

= ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

A (2)

The boundary conditions are a no-slip condition on the wall and decaying perturbations outside the boundary layer. For the purpose of clarity, we briefly recapitulate the main ideas of Ref. [8]. The Cauchy problem for (1) with initial data at x = 0 is ill-posed. The latter is

associated with possible upstream influence of the downstream boundary of the domain. However, if there is no evidence of the upstream impact, one can assume that the solution has a finite growth rate along the coordinate x. Assuming that the solution has a finite growth rate, we apply the Laplace transform with respect to x:

( ) ( )0

, ppxy x y e dx

∞ −= ∫A A (3)

As a result, we arrive at the following inhomogeneous system of ordinary differential equations:

0 1 1 2

2 0

p pp p

d ddL L H pH

dy dy dy

H

+ − − =

= −

A AA A F

F A

(4)

where vector-function 0A stands for the initial data at x = 0. A homogeneous system of equations correspond-ing to (4) can be recast in the form

0d

Hdy

=zz (5)

where 0H is a 6 6× matrix (see Appendix A) and z is a vector-function comprised of the first six elements of vector-function pA . The system of equations (5) has six fundamental solutions, 1( )yz , 2 ( )yz , ... , 6 ( )yz . Because their properties will be used in further analysis, we shall discuss them in detail. Classification of the fundamental solutions stems from their behavior outside the boundary layer, where ( ) 1sU y ≡ , ( ) 1sT y ≡ , and derivatives of the mean flow profiles are equal to zero. Solutions of (5) outside the boundary layer have an exponential form as 0 exp( )j j yλz . For y → ∞ , the characteristic equation

0det 0H λ− =I (6)

can be written in the explicit form

( ) ( ) ( )2 2 211 22 33 23 32

21 42 24 43 34 46 6411 0 22 0 0 0 0 0 0

42 25 43 35 46 6523 0 0 0 0 0 0

64 6532 0 33 0

0

,

,

b b b b b

b H b H H H H H H

b H H H H H H

b H b H

λ λ λ − × − − − =

= = + +

= + +

= =

(7)


where elements of matrix 0H are evaluated at y → ∞ . The roots of equation (7) are

( )

( )

( )

2 21,2 11

22 22 333,4 22 33 23 32

22 22 335,6 22 33 23 32

14

2 2

14

2 2

b iR

b bb b b b

b bb b b b

λ α α ω

λ

λ

= = + −

+= − − +

+= + − +

(8)

Solution of the inhomogeneous system of differential equations (4) can be expressed in terms of the funda-mental solutions 1( )yz , 2 ( )yz , ... , 6 ( )yz with coeffi-cients depending on the initial data 0A . The inverse Laplace transform could be presented as a sum of residue values at poles corresponding to the modes of discrete spectrum and a sum of integrals along branch-cuts in the complex plane p. These branch-cuts are associated with the continuous spectrum, and they can be found from equations

2 2j kλ = − , ( )1,...,6j = (9)

where k is a positive parameter. In the case of the spatially growing disturbances, there are seven branches of the continuous spectrum. This number stems from the fact that the characteristic equation represents a polynomial of seventh order with respect to the wave number, ipα = (the continuity equation has the first derivative with respect to x, the two momentum equations and the energy equation have second deriva-tives with respect to x). Figure 3 demonstrates branches of the continuous spectrum at local Mach number eM = 0.5 in the complex plane ipα = . Figure 4 shows the branches in the case of a supersonic flow ( eM = 2). In both examples the dimensionless frequency was chosen equal to 1. One can see that there are three modes having negative imaginary parts of α . They are asso-ciated with upstream perturbations. The other four modes correspond to the downstream modes. One of the branches has a limiting point as parameter k tends to infinity. Although the limiting point exists at signifi-cantly high positive values of iα that are usually not of interest, we demonstrate the structure of the branches for completeness of the illustration. The branches can be interpreted as acoustic, entropy, and vorticity modes in accordance with their properties outside the bound-

-15 -10 -5 0 5 10 15-30

-20

-10

0

10

20

30

40

50

60

70

-15 -10 -5 0 5 10 15-30

-20

-10

0

10

20

30

40

50

60

70

-15 -10 -5 0 5 10 15-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-10 -5 0 5 10-15

-10

-5

0

5

10

15

20

25

30

Acoustic mode Entropy mode Vorticity mode Acoustic mode

b)R = 20

α iα

r

c) R = 25

α i

αr

d)R = 28

Me = 0.5; ω = 1

α i

a)R = 10

Fig. 3 Branches of the continuous spectrum in the complex planeα .

-5 0 5-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

Acoustic mode Entropy mode Acoustic mode Vorticity mode

α i

αr

Me = 2; ω = 1; R = 40

Fig. 4 Branches of the continuous spectrum in the com-plex plane α . ary layer. Note that the branch having a limiting point in the upper half-plane was calculated incorrectly in Ref. [8]. The correct behavior of this branch was found by D. E. Ashpis (private communication, 1993). How-ever, the error was associated with strongly decaying modes, and it did not affect the main result and conclu-sions of Ref. [8].


Biorthogonal Eigenfunction System It was shown in Ref. [8] that the formal solution of the spatial initial-value problem could be represented as a sum of the continuous and discrete spectra. The direct and the adjoint problems corresponding to these modes ( αA and αB , respectively) are the following:

1 1 20

51 30 : 0

: j

d ddL L H i H

dy dy dy

y A A A

y A

α αα α

αα α

α

α

+ = +

= = = = → ∞ < ∞

A AA A

(10)

0 1 1 2

2 4 60 : 0

:

T T T T

j

d ddL L H i H

dy dy dy

y B B B

y B

α αα α

α α α

α

α

− = +

= = = = → ∞ < ∞

B BB B

(11)

In the definition of the adjoint problem, we do not introduce the complex conjugate due to the conven-ience associated with a numerical realization. The subscript α indicates that the mode is associated with the wave number. The system of equations (10) can be recast in the form of Eq. (5), whereas Eq. (11) can be recast as

0Td

Hdy

− =YY (12)

We have found relationships between components of vectors B and Y with the help of Mathematica12 (see Appendix B). The following orthogonality condition exists:

( )9

2 2 ,1 0, jjj

H H B dy Qα αα α α α∞

′ ′ ′=≡ = ∆∑ ∫A B A (13)

where ,α α ′∆ is the Kronecker symbol if α or α ′ belongs to the discrete spectrum, and ,α α ′∆ =

( )δ α α ′− is the delta function if both α and α ′ belong to the continuous spectrum. The coefficient Q on the right-hand side of (13) depends on normalization of ( )yαA and ( )yαΒ . If we have a vector function ( )0 yA (for example, from a computational study), we can find the amplitude of a mode as follows: 2 0 2, ,C H Hα α α α= A B A B (14)

An example of two discrete modes is given in Fig. 2. Examples of the acoustic modes in the boundary layer over a cone 1600R = and 6150 10F −= × are shown in Fig. 5. The parameter k in these examples is equal to 1 and the normalization of the modes was defined by the equation /u y∂ ∂ = 1 on the wall.

0 5 10 15 20 25 30

-1.0

-0.5

0.0

0.5

1.0

1.50 5 10 15 20 25 30

-2

-1

0

1

2

3

y

ur

ui

Fast Acoustic Mode

Me = 5.6; R = 1600; F = 150×10-6; k = 1

ur

ui

Slow Acoustic Mode

Fig. 5 Streamwise velocity component of acoustic modes.

Examples of the Multimode Decomposition We consider two examples. The first one is an emulation of the analysis of a flow field obtained by means of a computational method, i.e. all parameters of the flow are assumed to be known at a prescribed station x. The second example will be an emulation of the decomposition of “experimental” data. Details of the numerical methods are presented in Appendix C. “CFD” Flow Field Figure 6 gives an example of a flow field, ( )0 yA , comprised of two discrete modes (Fig. 2) with ampli-tudes C1 = 1 and C2 = -1, corresponding to acoustic and vorticity modes, respectively. The sum of the discrete modes only is shown in Fig. 6. It is very difficult to imagine presence of the discrete modes in the total flow field (Fig. 7). The decomposition procedure, Eq. (14), leads to the coefficients C1 = 1.0018+i1.58e-03 and C2


0 5 10 15 20 25 30 35 40

-40

-20

0

20

40


ur

ui

u

y

R = 1600

Fig. 6 Real and imaginary parts of the combined flow field (two discrete modes, vorticity and acoustic). = -0.9997-i1.59e-04. The negligible discrepancy with the actual values is attributed to the accuracy of the numerical evaluation of the integrals. Evaluation of the coefficient Q in Eq. (12) in the case of the discrete spectrum is straightforward: numer-ical integration on the interval max[0, ]y and analytical calculation on the interval max[ , )y ∞ . In the case of the continuous spectra, the coefficient Q can be found with the help of the asymptotic solutions outside of the boundary layer. An outline of the evaluation is given in Appendix D. “Experimental” Flow Field In the previous example, all components of the vector 0 ( )yA were known. If, for some reason, one can expect that only a few modes are responsible for the total signal measured in the experiment, this a priori information also can be helpful in decomposing the experimental flow field. The assumption about the main input from specific modes actually allows recovery of the other components in the vector 0 ( )yA with un-known coefficients. For example, we have “experi-mental” data for the x-velocity component as in Figs. 8 and 9 (amplitude and phase of the streamwise velocity perturbation corresponding to Fig. 7). If there is a priori information that the data are comprised of two discrete modes associated with wave numbers 1α and 2α , one can represent the vector 0 ( )yA as follows:

0 5 10 15 20 25 30-0.2

-0.1

0.0

0.1

0.2

0.3


ur

ui

u

y

R = 1600

Fig. 7 Real and imaginary parts of the combined flow field (two discrete modes only). Compare magnitudes of the functions in Figs. 6 and 7!

(

)

1 2 1

2 1 2 1

2 1 2 1

2 1 2 1

2 1 2

0 1 2 1 1 2

2 2 1 3 2 3 1 4

2 4 1 5 2 5 1 6

2 6 1 7 2 7 1 8

2 8 1 9 2 9

( ) , ,

, ,

, ,

, ,

,

A y u C A C A C A

C A C A C A C A

C A C A C A C A

C A C A C A C A

C A C A C A

α α α

α α α α

α α α α

α α α α

α α α

= +

+ +

+ +

+ +

+ +

(15)

where

j mAα stands for the m-th component of the vector

jαA . At the same time, the a priori information suggests that the vector ( )0 yA can be represented also

as a sum of these two modes

( ) ( ) ( )1 20 1 2

A y C A y C A yα α= + (16)

From the orthogonality condition, one can obtain

1 1 1

2 2 2

2 0 1 2

2 0 2 2

, ,

, ,

H C H

H C H

α α α

α α α

=

=

A B A B

A B A B (17)

Substitution of 0 ( )yA from Eq. (15) into Eq. (17) leads to a system of algebraic equations for 1C and 2C . This decomposition procedure has been applied to data in Figs. 8 and 9 and the coefficients C1 and C2 were found as 1.00002+i1.16e-04 and -0.999996+i1.65e-06, respectively.


0 2 4 6 8 10 12

0.00

0.05

0.10

0.15

0.20

0.25Cone. M

e = 5.6. Adiabatic wall. F = 150×10-6

|u|

y

R = 1600

Fig. 8 “Experimental” streamwise velocity disturbance amplitude distribution across the boundary layer.

0 2 4 6 8 10 12-350

-300

-250

-200

-150

-100

-50

0

50

100

150Cone. M

e = 5.6. Adiabatic wall. F = 150×10-6

Pha

se, d

eg

y

R = 1600

Fig. 9 “Experimental” streamwise velocity disturbance phase distribution across the boundary layer.

Conclusion The formulated method of multimode decom-position might be very helpful for the analysis of computational results related to laminar-turbulent transition in compressible boundary layers. Although the present formulation deals with two-dimensional disturbances and two-dimensional boundary layers, the orthogonality relationship can be easily extended to a general case, when the Fourier transform is employed in the spanwise direction. The method might be useful also in the analysis of real experimental data when a priori information about the modes is available.

Acknowledgment This work is supported by the U.S. Air Force Office of Scientific Research.

References 1. Gustavsson, L. H., “Initial-Value Problem for

Boundary Layer Flows,” Physics of Fluids, Vol. 22, No. 9, 1979, pp. 1602-1605.

2. Salwen, H., and Grosch, C. E., “The Continuous Spectrum of the Orr-Sommerfeld Equation. Part 2. Eigenfunction Expansion,” Journal of Fluid Mechanics, Vol. 104, 1981, pp. 445-465.

3. Fedorov, A., and Tumin, A. “Initial-Value Problem for Hypersonic Boundary-Layer Flows,” AIAA Journal, Vol. 41, No. 3, 2003, pp. 379-389.

4. Ashpis, D., and Reshotko, E., “The Vibrating Ribbon Problem Revisited,” Journal of Fluid Mechanics, Vol. 213, 1990, pp. 513-547.

5. Huerre P., and Rossi M., “Hydrodynamic Instabilities in Open Flows,” Hydrodynamics and Nonlinear Instabilities, edited by C. Godreche and P. Manneville, Cambridge University Press, Cam-bridge, 1998, pp. 81-294.

6. Zhigulev, V. N., Sidorenko, N. V., and Tumin, A. M., “Generation of Instability Waves in a Bounda-ry Layer by External Turbulence,” Journal of Applied Mechanics and Technical Physics, Vol. 21, No. 6, 1980, pp. 28-34.

7. Hadamard, J., Le Problème de Cauchy et les Éuations aux Derives Partielles Linéares Hyprboliques, Hermann, Paris, 1932.

8. Tumin, A. M., and Fedorov, A. V., “Spatial Growth of Disturbances in a Compressible Bound-ary,” Journal of Applied Mechanics and Technical Physics, Vol. 24, No. 4, 1983, pp. 548-554.

9. Zhigulev, V. N., and Tumin, A. M., Origin of Turbulence, Nauka, Novosibirsk, 1987 (in Rus-sian); NASA TT-20340, October 1988 (translated).

10. Tumin, A., Amitay, M., Cohen, J., and Zhou, M. “A Normal Multi-Mode Decomposition Method for Stability Experiments,” Physics of Fluids, Vol. 8, No. 10, 1996, pp. 2777-2779.

11. Wernz, S., and Fasel, H. F., “Numerical Investi-gation of Forced Transitional Wall Jets,” AIAA Paper, No. 97-2022, June 1997.

12. Wolfram, S., The Mathematica Book, 4th ed., Wolfram Media and Cambridge University Press, New York, 1999.

13. Ryzhik, I. M., and Gradshteyn, I. S., Tables of Integrals, Series and Products, 6th ed., edited by A. Jeffrey and D. Zwillinger, Academic Press, New York, 2000, pp. 1114-1119.

Appendix A: The Non-zero Elements of the Matrices in Eqs. (1) and (5)

The velocity, temperature, and viscosity are scaled using their values at the upper boundary-layer


edge; the pressure is scaled using 2e eUρ . Non-zero

elements of the matrices in Eq. (1) are

430

srL

R

µ= − , ( )1 1 , = 1,...,6ijL i j=

281 ( 1)L m= + , 12

1 1H =

211

s s

i RH

T

ωµ

= − , ( )221 ln sH D µ= −

231 s

s s

RH DU

Tµ= ,

( )251

s s

s

D DUH

µµ′

= −

261

ss

sH DU

µµ

′= − , 33

1s

s

DTH

T=

34 21 eH i Mωγ= , 35

1s

iH

T

ω= −

431

s

iH

T

ω= , 561 1H =

( )63 21 2 Pr 1s eH DU Mγ= − −

( )64 21

Pr1 e

s

RH i Mω γ

µ= −

( ) ( )

( )

2265

1

Pr 1

Pr

es s

s

s s

s s s

MH DU

D DT i R

T

γµ

µµ ωµ µ

−′= −

′− −

661

2 s s

s

DTH

µµ′

= −

77 88 991 1 11, 1, 1H H H= = =

212

s

s s

RUH

Tµ= , ( )23

2 = ln sH D µ−

242

s

RH

µ= , 27

2H r= −

48 692 2, 1sH H

R

µ= = −

312 1H = − , 34 2

2 e sH M Uγ= − , 352

s

s

UH

T=

412

smDH

R

µ= , ( )42

2 1 sH mR

µ= +

432 s

s

UH

T= − , 45

2s sDU

HR

µ′=

( )62 22 2Pr 1s eH DU Mγ= − −

( )64 22

Pr1 e s

s

RH M Uγ

µ= − − , 65

2Pr

ss s

RH U

T µ=

71 83 952 2 21, 1, 1H H H= − = − = −

12 560 0 1H H= =

( )21 20 /s s sH i U R Tα α ω µ= + −

220 /s sH Dµ µ= −

( )230 1 / / /s s s s s s sH i m DT T i D RDU Tα α µ µ µ= − + − +

( ) ( )24 20 / 1s e sH i R m M Uα µ γ α α ω= − + −

( )( ) ( )250 1 / /s s s s sH m U T D DUα α ω µ µ′= + − −

260 /s s sH DUµ µ′= −

31 330 0, /s sH i H DT Tα= − =

( )34 20 e sH i M Uγ α ω= − −

( )350 /s sH i U Tα ω= −

( )1

2e s

s

Rir M Uχ γ α ω

µ

−

= + −

( )410 / 2 /s s s sH i rDT T Dα χ µ µ= − +

420 H iα χ= −

( )43 20

2

/

/ /

s s s

s s s s s s

H i U R T

rD T T rD DT T

χ α α ω µ

µ µ

= − − −+ +

( ) ( )

44 20

+ / /

e s

s s s s s

H i r M DU

U DT T D

χ γ α

α ω µ µ

= − − +

( )

450 / /

/

s s s s s

s s s s

H i r DU T DU

r U D T

χ α α µ µ

α ω µ µ

′= ++ −

( )460 /s sH ir U Tχ α ω= −

( )62 20 2 1 Pre sH M DUγ= − −

( )63 20 2 1 Pr Pr /e s s s sH i M DU R DT Tα γ µ= − − +

( ) ( )64 20 Pr 1 /e s sH iR M Uγ α ω µ= − − −

( )( ) ( )

65 20

22 2

Pr /

1 Pr / /

s s s

e s s s s

H iR U T

M DU Ds

α α ω µ

γ µ µ µ µ

= + −

′− − −


660 2 /s sH Dµ µ= −

Appendix B: Correspondence Between

Solutions of the Adjoint Problems, Eqs. (11) and (12)

( )4

1 12e s

s

i rYB Y

Rir M U

α

γ α ωµ

= −

+ −

2 2B Y=

( )

( )

( )

3 2 3

4

2

4

2

1

1

s

s

s

e ss

se s

rDTB i m Y Y

T

rY d

R dyRir M U

Y

ri M U

R

α

µ

γ α ωµ

µ γ α ω

= − + + +

× −

+ − × + −

( )4

421 se s

YB

ri M U

R

µ γ α ω= + −

( )

( )4

5 52

s

se s

s

ir U YB Y

T Rir M U

α ω

γ α ωµ

−= +

+ −

6 6B Y=

7 2B i rBα=

( ) 28 2 41

dBB m i H B

dyα= − + −

9 6B i Bα=

Appendix C: Numerical Method

The numerical method used in the present work is the same as in Ref. [3]. The only difference is that in Ref. [3] the temporal stability analysis was employed, whereas in the present work we consider spatially growing perturbations of prescribed frequency. The conventional form of Eqs. (5) and (12) for the direct

and adjoint problems were used in the numerical evalu- ations. Afterwards, the adjoint eigenfunction αB was found with the help of the relationships in Appendix B. The numerical procedure incorporates the integration of Eqs. (5) and (12) for three (discrete spectrum) or four (continuous spectrum) fundamental solutions. The inte-gration was carried out from outside the boundary layer ( max 14y ≈ ) toward the wall employing the Gram-Schmidt orthonormalization procedure. The integration was fulfilled with a fourth-order Runge-Kutta scheme with a constant step (301 points on the interval). In the analysis of the discrete spectrum, the com-plex wave number, α, was found using Newton’s itera-tion procedure. The iterations were considered con-verged if the temperature disturbance on the wall was less than 10-5 while u and v were kept equal to zero. In consideration of the continuous spectra, k was used as a parameter, and the wave number, α, was found from the equation 2 2

j kλ = − . The inner product 2 ,H α α⟨ ⟩A B in (13) was calculated numerically on the interval max[0, ]y and analytically on the interval max[ , )y ∞ . The iteration procedure in the analysis of the discrete spectrum depends on the initial approach to the eigenvalue α. This dependency might be especially strong in the vicinity of the synchronisms described above. In order to find the initial approach and to validate the results obtained with the code based on the Runge-Kutta scheme, we developed a code using the two-domain Chebyshev spectral collocation method as it was in Ref. [3]. Appendix D: Derivation of Coefficient in Orthogo-

nality Condition (13) in the Case of a Continuous Spectrum

For the purpose of numerical implementation, it is necessary to find the coefficient Q in Eq. (12) at an arbitrary normalization of the eigenfunctions. In the case of a discrete mode, the evaluation of the integral in (12) is straightforward and it can be easily implemented numerically. To derive the coefficient in the case when α and α ′ belong to the continuous spectrum, we follow Ref. [2] and consider a narrow wave packet of the continuous modes with ( , )k k kε ε∈ − + , where ε is a small parameter ( 0)ε → . This leads to integrals of the following type:

( ) ( ) ( )0

, ,j

j

k

kk y C k k y dk dy

ε

εψ φ

+∞

−′ ′ ′∫ ∫ (D1)


where functions ψ and φ have asymptotics

exp( )iky±∼ and exp( )ik y′±∼ , respectively, outside the boundary layer. The integral with respect to y can be recast as a sum:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

0

0

0

0

, ,

, ,

, ,

, ,

j

j

jout

j

jout

j

j

j

k

k

k

k

k

asymp asympk

k

asymp asympk

k y C k k y dk dy

k y C k k y dk dy

k y C k k y dk dy

k y C k k y dk dy

ε

ε

εδ

ε

εδ

ε

ε

ε

ψ φ

ψ φ

ψ φ

ψ φ

+∞

−

+

−

+

−

+∞

−

′ ′ ′∫ ∫

′ ′ ′= ∫ ∫

′ ′ ′− ∫ ∫

′ ′ ′+ ∫ ∫

(D2)

where outδ stands for a coordinate outside the bound-ary layer and the subscript “asymp” indicates asymp-totic representations of the function as exp( )iky± and exp( )ik y′± . The first and the second terms on the right-hand side of (D2) tend to zero at 0ε → . The third term leads to integrals of the following type:

( )

( )

0 0

0

0

sin2

cos sin 2

sin sin 2

j

j

k i k k yjiky ik y

k

j

j

ye e dk dy e dy

y

k k y ydy

y

k k y yi dy

y

ε

ε

ε

ε

ε

+∞ ∞ −′−

−

∞

∞

′ =∫ ∫ ∫

− = ∫

− + ∫

(D3)

One can find the latter integrals in Ref. [13] ,

( ) ( )( )

( )( )( )( )

0

0

sin sin 1log

2

, 2

cos sin,

4

0,

j j

j

j

j

j

j

k k y y k kdy

y k k

k k

k k y ydy k k

y

k k

ε ε

ε

π εε π ε

ε

∞

∞

− + − =∫− −

− < − = − =∫

− >

(D4)

Therefore, integrals (D1) can be easily evaluated at 0ε → by consideration of asymptotics of functions ψ

and φ , and representing each term exp( ( ) )i k k y′± − after integration with respect to y as ( )k kπδ ′− . The corresponding analytical and numerical implementa-tions become straightforward if asymptotics of vector functions ( )kαA and ( )kαB are known.

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