[american institute of aeronautics and astronautics 15th aiaa computational fluid dynamics...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

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AIAA 2001-2781Initial-Value Problem for HypersonicBoundary Layer FlowsA. FedorovMoscow Institute of Physics and Technology

A. TuminUniversity of Arizona, Tucson, AZ

31st AIAA Fluid DynamicsConference & Exhibit

11-14 June 2001 / Anaheim, CA

For permission to copy or repubiish, contact the copyright owner named on the first page. For AIAA-heid copy-right, write to AIAA, Permissions Department, 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344.


AIAA-2001-2781

INITIAL-VALUE PROBLEM FOR HYPERSONIC BOUNDARY LAYER FLOWS

Alexander Fedorov*Moscow Institute of Physics and Technology, Moscow region, Zhukovski, 140180, Russia

Anatoli Tuminf

The University of Arizona, Tucson, Arizona 85721

AbstractAn initial-value problem is analyzed for a two-di-

mensional wave packet induced by a local two-dimen-sional disturbance in a hypersonic boundary layer. Theproblem is solved using Fourier transform with respectto the streamwise coordinate and Laplace transformwith respect to time. It is shown that the solution can bepresented as an expansion in the biorthogonal eigen-function system. This provides a compact and robustformalism for theoretical and numerical studies of exci-tation and evolution of wave packets generated by localsources. The temporal continuous spectrum is revisited,and the uncertainty associated with the overlapping ofcontinuous-spectrum branches is resolved. It is shownthat the behavior of the discrete spectrum's dispersionrelationship is non-analytic due to its relationship withthe synchronization of the first or second mode with thevorticity/entropy waves of the continuous spectrum.The characteristics of the wave packet are numericallycalculated using an expansion to the biorthogonaleigenfunction system, which comprises modes of dis-crete and continuous spectra. It is shown that the hyper-sonic boundary layer is highly receptive to vorticity/entropy disturbances in the synchronism region. Thefeasibility of experimental verification of this recep-tivity mechanism is discussed.

AJD

HiJ9I?

Nomenclature= vector-function of six components=yth component of vector A= d/dy= ratio of the second viscosity to the first= ij* matrix element

* Associate Professor, Faculty of Aeromechanics andFlight Techniques, Membert Assistant Professor, Department of Aerospace andMechanical Engineering ([email protected]), onleave from Tel-Aviv University, Senior Member'Copyright © 2001 The American Institute of Aeronau-tics and Astronautics, Inc. All rights reserved.

mMPPrrRtTUuV

X

ya

2(e-\)/3Mach NumberLaplace variablePrandtl number

Reynolds numbertimemean flow temperaturemean flow velocitystreamwise velocity disturbancenormal velocity disturbancestreamwise coordinatecoordinate normal to the wallstreamwise wave numberspecific heat ratioviscosity disturbancemean flow viscosity

n = pressure disturbance^ = temperature disturbanceP = density disturbancePS = mean flow densitySuperscriptsT = transposedSubscripts and Other Symbolsp = Laplace transforms = mean flowa = Fourier transform

I. IntroductionStudies of laminar-turbulent transition in hyper-

sonic boundary-layer flows have a long history. Never-theless, understanding of this phenomenon is still verypoor compared to the low-speed case.1 There are sev-eral reasons for this gap. For example, experimentalconditions are severe in hypersonic wind tunnels.Because of very high levels of freestream noise, it isdifficult to perform experiments with controlled dis-turbances. It is also difficult to design perturbers pro-viding high-frequency artificial disturbances of flow

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with well-controlled characteristics. Usually experi-mentalists deal with wave trains or wave packets inter-nalized in the boundary layer. Furthermore, interpre-tation of experimental data is not straightforwardbecause of the complexity of receptivity and stabilityprocesses at hypersonic regimes. This issue leads to anecessity of close coordination between theoreticalmodeling and experimental design and testing.

A high Reynolds-number, Mach-6, quiet Ludwiegtube has being developed at Purdue University.^3

Experiments with controlled disturbances in thisfacility could provide a breakthrough in experimentalmodeling and detailed examination of governing mech-anisms associated with hypersonic laminar-turbulenttransition. Its test section fits well to experiments withbodies of revolution, and a sharp cone can be con-sidered as a good candidate for transition studiesbecause of its relatively simple mean flow. This windtunnel provides high Mach numbers in combinationwith high Reynolds numbers. Under these conditions,the second instability mode becomes dominant in thetransition process.4 The existence of the second-modeinstability was established by the experiments ofKendall,5 Demitriadis,6 Stetson et al.,7 and Stetson andKimmel.8 The Mach-8 stability and transition experi-ments7 for the boundary layer on a sharp cone indicatedthat the unstable high-frequency second mode plays amajor role in the conical boundary-layer transition.These data are consistent with the second-mode stabil-ity calculations,4'9'10 which show that two-dimensionalwaves of the second mode have a growth rate higherthan that of three-dimensional waves. To address thetwo-dimensional nature of the second mode, one shouldlook for an experimental setup providing excitation ofpredominantly two-dimensional disturbances.

Several methods for excitation of artificial disturb-ances in a hypersonic boundary layer are available. Aglow discharge technique is used in the design of two-dimensional11'12 and three-dimensional13'14 perturbersthat are able to generate disturbances at the boundary-layer bottom. They may also be used for excitation ofhigh-frequency acoustic disturbances.12 In Ref. 15, alaser is used to induce local disturbances in a freestream. A relatively small region of heated air isgenerated in front of the cone tip. The temperature spotpropagates downstream and translates to a ringembracing the cone surface. The laser perturber canprovide both wave trains of fixed frequency and localwave packets of a broad frequency band. This methodmay serve for experimental studies of interactionbetween two-dimensional freestream disturbances andthe boundary-layer flow. Such experiments should beaccompanied by theoretical modeling in all phases,

from the design of the experimental setup to theanalysis and interpretation of data.

To meet the experimental constraints discussedabove, we need to focus on theoretical modeling ofexcitation and development of two-dimensional wavetrains and/or wave packets in a conical hypersonicboundary layer. Another motivation originates from thetheoretical results of Gushchin and Fedorov16 andFedorov and Khokhlov.17'18 They found that the second-mode instability is associated with synchronization ofthe first mode with the second mode. In the synchro-nism region, the eigenvalue spectrum splits into twobranches. The asymptotic analyses17'18 of the spectrumbranching show a strong inter-modal exchange due tononparallel effects. In accordance with the inter-modalexchange rule17'18 the first (stable) mode effectivelyexcites the second (unstable) mode in the synchronismregion. To predict the initial amplitude of the second-mode instability, one should account for excitation ofboth the first and second modes, despite the fact that thefirst mode may decay downstream.

If we exclude high-frequency vibrations of thewall, then a major source of the instability excitation isassociated with the freestream disturbances. Theoreticalstudies17'19 showed that the first and second modes aresynchronized with acoustic waves near the leading edgeof a flat plate (or near the sharp cone tip). That maylead to a strong excitation of the discrete modes byfreestream acoustic noise. Fedorov and Khokhlov17'19

developed a theoretical model of this receptivitymechanism. Corresponding experimental studies wereperformed by Maslov et al.I2 A similar experiment withacoustic disturbances radiating the boundary layer overa sharp cone would be more complicated, as the inter-action of the acoustic field with the shock wave wouldgenerate all types of modes from the continuous spec-trum,20 and interpretation of the data would be ques-tionable. Another option is associated with synchroni-zation of a discrete mode with vorticity/entropy wavesof the continuous spectra. Fedorov and Khokhlov18

showed that the first mode (for adiabatic and weaklycooled walls) and the second mode (for strongly cooledwalls) are synchronized with these freestream disturb-ances. The synchronization provides favorable condi-tions for hypersonic boundary-layer receptivity tofreestream turbulence and temperature spottiness. Thesefindings along with the possibility of generating two-dimensional spottiness near a sharp cone model,12

motivated the present work.Our objective is to solve the initial-value problem

for a wave packet generated by a local source, whichcan be a temperature spot, a vortex, or a combination ofthe two. Because the second-mode instability is maxi-mal for two-dimensional (2D) waves, we focus on the



analysis of 2D disturbances propagating in a boundarylayer on a sharp cone at a zero angle of attack. As thefirst step of this research, we consider the locallyparallel approximation.

2. The Problem FormulationWe consider a locally parallel boundary-layer flow

of a calorically perfect gas. At the initial time moment/ = 0, a localized (with respect to the streaming coordi-nate) 2D disturbance of a small amplitude isinternalized into the mean flow. The problem is todescribe the downstream evolution of this disturbance,which can be treated as a wave packet. The linearizedNavier-Stokes equations are written as a system ofpartial differential equations for the disturbance vector-function

d—dy

d A ]——d y )

dA——dy dt

d A—cbc

(la)

(Ib)

dxdy

where I0, //,0, //n, //2, //3, and H4 are 6x6matrices. Their non-zero elements are presented inAppendix A. At / = 0, the disturbance vector isspecified as

(2)

The disturbance vector satisfies the boundary condi-tions

y ->

(3a)

(3b)

Specifically, we will consider the initial boundary-valueproblem, (l)-(3), with the initial vector, A 0 , of the non-zero temperature disturbance

AQ . =0 (j^5,6) (4a)

AQ5=0Q(x,y) (4b)

3. Formal Solution of the Initiai-Value ProblemAlthough the compressibility adds complexity to

the problem, our approach is similar to the case of anincompressible boundary layer considered byGustavsson.21 The problem, (l)-(3), is solved using

Fourier transform with respect to * and Laplace trans-form with respect to /,

V2;r 0(5)

Applying the transformations (5), we obtain from(1) the non-homogeneous system of ordinary differ-ential equations

(6)

where A0ff (7) is the Fourier transform of the initialdisturbance field A0 (x,y). The solution of (6) satisfiesthe boundary conditions

(7a)

(7b)

Further analysis is similar to the case of a harmonicdisturbance propagating in space.22 We begin with con-sideration of the homogeneous part of Equation (6).The latter can be recast as a standard stability system ofordinary differential equations for 2D compressibleboundary layers,

dy (8)

where H0 is a 6x6 matrix; its non-zero elements arepresented in Appendix A. There are six fundamentalsolutions of the homogeneous system of equations:z,,...,z6. Each vector-function has an exponentialasymptotic, expf/l - y ] , outside the boundary layer. Fory -» oo, the characteristic equation

det =0

can be written in the explicit form

)x -A

= 0

=//21

(9)

(lOa)

(lOb)



622 = //042 //024 + // J3 H204 + H*6 Hf ( 1 Oc)

42H25 + //43//035

,65

(lOd)

(10e,f)

where elements of matrix //„ are evaluated at y -* oo.The roots of Equation (10) are

A2, = b.. = a2 +iR(a-ip) (lla)

( l ib)

26=^l+U^2.^)2+4^^ (Uc)

For the sake of defmiteness, we choose the rootbranches as Re(^,A J ,>i 5)<0, and introduce thematrix of fundamental solutions

Z = zp...,z6 (12)

A solution of the non-homogeneous system (6) isexpressed in the form

A nsy =

where the vector of coefficients, Q , is determined bythe method of variation of the parameters

21 i L Z0 °dy dy dy2 dy dy

Accounting for

we obtain the system

(14a)

(14b)

(15)

3Z^ =3 dy

(16)

Let us consider the fourth equation of system (16)

(17)«V "^0 -^J

+ Zd • ————— + ———^— Z-> • ————— — FA

J dy dy y dy

One can derive the following relations,

dy

dFdy dy

= 0;

2 T-(r'o dy

*= 0

(18a)

(18b)

(18c,d)

Accounting for the non-zero elements of H\J only, wecan write (17) as

(19)dy

(13) A similar analysis of the second equation,

23̂ -

with

^ dy ''I

leads to the form

(20)

(21)

(22)

Thus, we can rewrite the non-homogeneous system (14)as

(23)

where



(H A }(n\Q*Qa)i

1

dy

^5 =0

A formal solution of Equation (21) is expressed in theform

(24a)

(24b)

where constants Q- and y • are obtained from theboundary conditions at y = 0 and oo. The function

is given by the equation

(25)

where Jf(oo) can be evaluated with known asymp-totics of the fundamental solutions outside the boundarylayer.

Summarizing, we obtain the following solution,

I y Da} + J — dy I z,

0 W '

y D,- d y z j +\ a^ + \-^-dy z,

oo W 2 i 3 0 W ' 3

y D*f — - d y z . + \ a* 4- f —-dyi W 4 I 5 o W

W

(26a)

oo D,Cj ^l——J o W

;.,ijk = det

Z Z5i Z5j Z5

(26b)

(26c)

(26d)

(26e)

(26f)

where zi} stands for the /* component of they411 vector.The inverse Laplace transform of (24) is defined by thepoles (relevant to the discrete spectrum) and branchcuts (relevant to the continuous spectrum) in the com-plex plane/?.

4. Biorthogonal System of EigenfufictionsA solution of the initial boundary-value problem,

(l)-(3), can also be presented as an expansion in thebiorthogonal eigenfunctions system {A^B^}, wherethe vector A^ is a solution of the direct problem

I0 "(27a)

>>->oo:

and the vector

dy *

~'A<»}=Aa>5 (27b)

,j\«» <27c)

is a solution of the adjoint problem

io

dy

(28a)

(28b)

(28c)

The asterisk in (28) denotes a Hermitian adjoint matrixand the "overbar" stands for a complex conjugate. Thedirect problem, (27), can be expressed in the standardform, (15). The adjoint problem, (28), can also beexpressed in a similar form as

-^ = tf*Y (29a)

\Yj «x>

(29b)

(29c)

>>=o

This has been done with the help of Mathematicasoftware,23 and the results are presented in Appendix B.

The solutions of problems (27) and (28) belong todiscrete and continuous spectra. The discrete modescorrespond to the case when all roots (11) have nonzeroreal parts. Therefore, only three fundamental solutionscan be used to satisfy the boundary conditions outsidethe boundary layer. The discrete eigenmodes are asso-



elated with zeros of £135 (p) = 0 in (26). Solutions ofthe continuous spectrum correspond to the case when a

is pure imaginary, i.e.,characteristic number A.-

For the system {Afgonality relation is valid,

\, the following ortho-

), (30)

where A^, is the Kronecker symbol if CD or co'belongs to the discrete spectrum; AWi<B, = S(co-co'} isdelta-function if both co and co' belong to the contin-uous spectrum. We assume that solutions of the adjointproblem (28) are properly normalized to get the coeffi-cient on the right side of (30) equal to unity.

The temporal analysis of the continuous spectrum( a is a real parameter and co is a complex eigenvalue)was carried out by Grosch and Saiwen24 and Ashpis andErlebacher25 for incompressible and compressibleboundary layers, respectively. The spatial analysis ( cois a real parameter and a is a complex eigenvalue) wasperformed by Grosch and Saiwen26 and Zhigulev et al.27

for an incompressible boundary layer, and by Tuminand Fedorov22 (see also Zhigulev and Tumin28) andBalakumar and Malik29 for a compressible (supersonic)boundary layer. We shall recapitulate main results ofthe analysis reported in Ref. 25.

The first pure oscillatory solution corresponds toA2, = -k2 in (1 1) and leads to the relation

P\=' R(31)

This solution is interpreted as a vorticity branch. Theequation

02)

is a third-order polynomial with respect to p . It hasthree roots at A2 = -k2 . Numerical evaluation of theseroots was performed with the help of Mathematicaland the results are shown in Figure 1 for Me = 5.585 ,a = 1 , 7 = 1 .4 , Pr=0.7 , and R=1000 . One branch hasa finite limiting point, and two others extend to infinity(see Ref. 25). The horizontal branch is associated withentropy freestream disturbances (we do not showbranch (31) as it is overlapping with the entropybranch). The upper and lower branches are relevant toacoustic waves. They start from the branch points— ia(l T 1 / Me ) , which correspond to plane acousticwaves of the phase speed c = 1 + 1 / Me .

Solutions for acoustic waves include four funda-mental vector-functions. Two of them decay outside theboundary layer, and two others oscillate as e±l^ . Simi-

larly, the vorticity disturbance (outside its overlappingwith the entropy waves) includes two decaying and twooscillating fundamental vectors. An uncertainty existsat the overlapping points, where we have, simultane-ously, four oscillating and one decaying fundamentalsolutions; i.e., the number of fundamental solutions ismore than necessary to satisfy the boundary conditionsin (27). This issue has not been discussed elsewhere. Itwill be resolved in Section 5 using the analysis of theformal solution (26).

Figures 2-5 show examples of the eigenmodes.Hereafter, we consider the boundary layer over anadiabatic sharp cone at a zero angle of attack. Thelength scale is defined as H = ̂ pex/ peUe . Allnumerical results are obtained for the local Machnumber Mp =5.6 , the Reynolds number R =

ue = 1219.5, and the stagnation temperaturero=470K.

5, Inverse Laplace TransformUsing the biorthogonal eigenfunction system, (27)-

(28), we can present a solution of linearized Navier-Stokes equations as a decomposition of the discrete andcontinuous modes. This approach provides a compactand robust formalism for theoretical and numericalstudies of receptivity and stability problems. Grosh andSaiwen24 analyzed the solution21 of the initial- valueproblem for incompressible flow and proved that theeigenfunction system is complete. Tumin and Fedorov22

analyzed the spatially developing disturbances in acompressible boundary layer. They showed that thecorresponding eigenfunction system is also complete.The completeness of the system (27)-(28) can beproved in a way similar to Refs. 22 and 24. The inverseLaplace transform is expressed as a sum of integralsalong the sides of the branch cuts and residues associ-ated with poles in the complex /?-plane (Figure 6).Then, it can be shown that the poles represent the inputfrom the discrete spectrum, and the integrals along thebranch cuts represent summation over the continuousspectrum. However, there is an obstacle associated withoverlapping of two branches of the continuous spec-trum. At each overlapping point /?, we have one decay-ing solution and two pairs of oscillating solutions, e±iky

and e±lk>y , where k and k\ are positive and differentThis leads to uncertainty in solving the boundary- valueproblem, because the number of fundamental solutionsis larger than the number of boundary conditions.

To resolve this uncertainty, we consider the inte-grals along both sides of the branch cut. In the over-lapping zone, we denote one side of the branch cut as"+" and the other as "-". We choose the notation offundamental vectors in accordance with the followingasymptotic behavior,



z4" ~ e^,z~ - e~l^,z+ ~ e~1^

z2 ~ e , z3 ~ e , z3 ~ e

z^ ~ e 4 , z^ ~ e 4 , z^" ~ e 'rty + -/*y /*y

Z5 ~ e ' Z6 ~ e ' Z6 ~ e

Then, the vectors are related as

Zj = Z2 , Z2 = Zj , Z3 = Z3

Z4 = Z4'Z5 = Z6'Z6 = Z5

and we have

iif ' . _ r/^ . r~\ _ T\* 7~) _ f~)

3 3'4 4 ' 5 o

6 5

The integrals along the branch-cut sidesas one integral of the difference

^+||U)^f£4

( y D$ \ y D?5 n W* 5 i W+

\ J( y D{ ^ ^ D2

( - y D3 } - y D4

( y D~ } y D~/7 i f 5 rhi \ 7 f 6 /A°5 ' J HV Z5 J ^0

Since the underlined terms are canceled,

/ +^\ z+

One can derive the following relations,+ rr+ . r+ pr+ . + c+

+ C2 ^235 + C4 ^435 + C6 ^635

£,+35— + . — 4- . — -1-

(33a)

(33b)

(33c)

(33d)

(34a)

(34b)

(35a)

(35b)

(35c)

can be written

—— dy Z-,

' Z4

; x,

(36)' Z

' Z

o

we obtain

(37)

(38a)

(38b-d)

£-35=£5+36; £f35=£i6 (38g,h)

+ + c{ E135 +c2 E235 +c4 £435 +c6 E635a{ +cj + (j8i)£135

-^ + C2E236+C\E\36+C4E436+C5E536 ,^a{ +c2 ~ + (j8j)^236

^5 - + (j8k)

C2 = cl+ ' C4 - C4 » C6 = C5 (381-n)

^32 = ^231' 134 = 234 (J^^P)

£f36=£2+35; £r35=£2+36 (38V)

0+-f.c+ = ——————————————————— (3gs)^135

a- +c+ _ C6 ^236 +C1 ̂ 23 1 +C4 E234 +C5 E235 „ ̂E236

a+ - °2 125 °4 145 C6 165

135 (38u)c+ £+ +c+ r+

+c+ r^1 216 4 246 5 256

+

After substitution of (38) into (37), we obtain (thesuperscripted "+" is omitted)

^ ^ _c.£135+c2£235+c4£435+^6£635 x

C2 E236+C\E\36+ C4 E436 + C5 £536 ,E 2^•236

- ( £,.,

cE +c E +c E (39)

£236

+ E Z$

- ——————————————————— — z6£236



We have found, with the help of Mathematical thatthe latter result might be presented in the compact form

f** F F^135^235

r 7 -£ z + £L z -u£. z 1^235 1 135 2 ^25 3 ' ^32 5 J

1^-231 + C4^234 +C5^235 +C6^236

(40)

^235 ̂ 236

[£35X 3 5 6 Z 2 - +£236Z5 ~

The first term of (40) represents the vorticity wave withz5 ~ e l> , and the second term represents the entropywave with z2 - e~l ' . Equation (40) specifies weightsof vorticity and entropy disturbances in the fundamentalsolution relevant to the overlapping zone.

Following Refs. 22 and 24, one can analyze thecoefficients in the formal solution and show that theinverse Laplace transform can be expressed in the formof expansion into the biorthogonal eigenftmctionsvstem

(41)

Here, the summations Iv and I, stand for the sum overthe discrete and continuous spectra, respectively. InEquation (41), we use the frequency cj instead of theLaplace variable p: these parameters are coupled ascj = ip. The coefficients cv and Cj can be found fromthe Fourier transform of the initial data, A^ (y), withhelp of the orthogonality relation (30),

6. Synchronism of the ModesFigure 7 shows numerical results for eigenvalues

cj of the first and second discrete modes. It is seen thatthere is a synchronism between the first mode and theentropy and vorticity modes of the phase velocity c= 1.This coalescing occurs at a wave number, a, of about0,2. The streamwisc velocity profile of the disturbanceat a = 0.2 is shown in Figure 8. In the synchronismvicinity, the eigenfunction decays very slowly outsidethe boundary layer and oscillates similar to vorticity/entropy waves. As the discrete mode coalesces with thecontinuous spectrum from one side of the branch cutshown in Figure 1, it re-appears on its other side atanother point. This topology leads to a jump of theimaginary part <vt shown in Figure 7. Contours of o)iin the complex a -plane (see Figure 9) indicate that

jumps of (oi are observed along an almost vertical linestarting from the point a «0.19-/0.02. To demon-strate that the discontinuity of G)l is associated with thecoalescing with the entropy/vorticity mode, we plotcontours of Re A t in the complex a -plane (seeFigure 10).

Note that a similar topology of the disturbancespectrum was reported by Fedorov and Khokhlov's18

analysis of the spatial eigenvalues a. They assumedthat in the synchronism region, the vorticity andentropy waves may effectively generate the first mode.Using discretization of the entropy/vorticity continuousspectrum Fedorov and Khokhlov18 showed that thevorticity/entropy wave translates to the first mode in thevicinity of the synchronism point. In a similar way, thefirst mode translates to the vorticity/entropy wave.

Figure 9 also shows that there is another irregu-larity at the point a « 0.21 - /O.016. To illuminate moredetails of the discontinuity, we plot coi versus ai inFigure 11. This irregularity is associated with branchingof the first and second discrete modes. The branchingtopology is consistent with the results presented inRefs. 16-18. Fedorov and Khokhlov17'18 investigated aninteraction between the first and second modes in thebranch-point vicinity due to nonparallel effects. Theyrevealed a strong inter-modal exchange leading toeffective excitation of the second mode instability.

The coalescing of the discrete mode with thecontinuous spectrum needs to be studied in more detail,with emphasis on the analytic properties of the solution(41) in the complex a-plane. The non-analyticity mayimpact the integration path in the inverse Fouriertransform and affect both excitation and downstreamevolution of the wave packet. This issue was brieflydiscussed by Reutov and Rybushkina30 in connectionwith the wave packet of vortical disturbances propa-gating in incompressible flow over a flat plate.

7. Receptivity to Temperature SpotsAs an example, we consider the initial temperature

spot localized at a distance yo from the wall, i.e., at/ = 0, 6(x, y) = S(y - yQ )S(x). As discussed in §6, thesynchronization between modes of discrete and contin-uous spectra can lead to effective penetration of free-stream disturbances into the boundary layer. Figure 12shows the maximum streamwise velocity amplitude,"max = max[w(>>)], of the first mode generated by a -components of the temperature spot localized at various>>0. These data can be treated as a distribution of thereceptivity coefficient with respect to a and yQ. Asexpected, the extreme receptivity is observed in thevicinity of the synchronism point a « 0.19. Figure 13shows the distributions of Hmax(>>o) for various wavenumbers of the temperature spottiness. The receptivity

8American Institute of Aeronautics and Astronautics


maximum is observed near the upper boundary-layeredge.

ConclusionsIn this paper, we analyzed the initial-value problem

for two-dimensional disturbances propagating in ahypersonic boundary layer over a sharp cone at a zeroangle of attack. It is shown that its solution can bepresented as an expansion in the biorthogonal eigen-fiinction system. This approach provides a compact androbust formalism for theoretical studies of hypersonicboundary-layer receptivity to freestream disturbancessuch as temperature spots and vortical perturbations.The analytical and numerical results lead to the follow-ing conclusions:1. The spectrum topology of a hypersonic boundary

layer is substantially different from the subsoniccase. The discrete spectrum has a non-analyticbehavior of its dispersion relationship due to syn-chronization of the first mode (for adiabatic wall)or the second mode (for strongly cooled wall) withthe vorticity/entropy waves of continuous spec-trum.

2. In the synchronism region, the boundary layer ishighly receptive to temperature spottiness and/orfreestream turbulence. The receptivity maximum islocated near the upper boundary-layer edge, wherevortical and temperature disturbances are alsomaximal.

3. This receptivity mechanism may play an importantrole in the initial phase of transition for the case offree flights and "quiet" wind tunnels when thefreestream acoustic field is relatively weak. It maybe competitive with the leading-edge receptivity,especially for conical configurations with relativelysmall leading-edge areas.

Our theoretical findings can be validated by experi-ments in the high Reynolds number, Mach-6, quietLudwieg tube of Purdue University.2^ The laser-perturber method15 developed at Purdue University canbe used to generate controlled temperature spots in afree stream and to conduct detailed measurements ofreceptivity characteristics. The theoretical model dis-cussed in this paper can help to design such an experi-ment and to interpret data.

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12. Maslov, A. A., Shiplyuk, A. N., Sidorenko, A.A., and Arnal, D., "Leading-Edge Receptivity ofa Hypersonic Boundary Layer on a Flat Plate,"Journal of Fluid Mechanics, Vol. 426, 2001, pp.73-94.

13. Kosinov, A. D., Maslov, A. A., and Shevelkov,S. G., "Experiments on the Stability of Super-sonic Laminar Boundary Layers," Journal ofFluid Mechanics, Vol. 219, 1990, pp. 621-633.

14. Ladoon, D. W., and Schneider, S. P., "Measure-ments of Controlled Wave Packets at Mach 4 ona Cone at Angle of Attack," AIAA Paper 98-0436, 1998.

15. Schmisseur, J. D., Schneider, S. P., "Receptivityof the Mach-4 Boundary-Layer on an EllipticCone to Laser-Generated Localized Free-StreamPerturbations," AIAA Paper 98-0532, 1998.

16. Gushchin, V. R., and Fedorov, A. V., "Excitationand Development of Unstable Disturbances in aSupersonic Boundary Layer," Fluid Dynamics,Vol.25, 1990, pp. 344-352.



17. Fedorov, A. V., and Khokhlov, A. P., "Excitationand Evolution of Unstable Disturbances inSupersonic Boundary Layer," Proceedings of1993 AS ME Fluid Engineering Conference,FED-Vol. 151, Transitional and TurbulentCompressible Flows, ASME, New York, 1993,pp. 1-13.

18. Fedorov, A. V., and Khokhlov, A. P., "Pre-History of Instability in a Hypersonic BoundaryLayer," Theoretical and Computational FluidDynamics (to be published in 2001).

19. Fedorov, A. V., and Khokhlov, A. P., "Excitationof Unstable Modes in a Supersonic BoundaryLayer by Acoustic Waves," Fluid Dynamics, Vol.26, 1991, pp. 531-537.

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Appendix A: The Non-zero Elements of theMatrices in Equations (I) and (8)

The velocity, temperature, and viscosity are scaledusing their values at the upper boundary-layer edge; thepressure is scaled using plj]. Non-zero elements ofthe matrices in Equations (1) and (8) are

RH21 -"10 ~

//35- l"\o ~7~2s

rr65H\Q ~

HU -1-fill -1'

H%=-YMt

H43_ 1^10 ~~^T

64 / ?P r ,

^11 =~~^~DUs

\. rr25)' ^U = ~'

H*,-?LDUs. H f f = l

r/33 _ TS . r /63 _/ / i , — —— , / / i | —11 ' "

H =.

66

- l A / 2 'DT

Hf2=-2DUsPr(r-\)M;

/ / -=-

43

R //242=!

45

R

R

= -2PrD[/5(^-l)A/<



=J

'

rt*s

rr56 _H ~"0 0

#21 =

//024 = iaR I ns - (m + 1) Y M2ea (aUs - ip)

-ip)ITs -(£DUS)I

Hi '=-/«; H*=DTSITS

Hf=i(aUs-ip}lTs

Z= — + iryMg (aUs - ip)

H4Q

6=irZ(aUs-ip)/Ts

Hf = -2/a (y - 1) A/2 Pr DUS +RPr DTS

//O64 = -iR Pr(r-l)M2 (aUs - ip] I fis

Hf =a2 + iR Pr (aUs - ip) I /JSTS

,66

Appendix B: Correspondence Between Solutionsof the Adjoint Problems , (28) and (29)

iarYA

B2 =Y2

R

ys dR dy

H^ =-iXryM2e[aDUs

+ (aUs -ip)(DTs/Ts+ D

//045 = iX[raDUs/Ts +a ^DUS

B5=Y5-ir(aUs-co]



-0.6

-0.8

-1

-1.2

-1.4

-0.08 -0.06 -0.04-0.02

Figure 1. Three branches of the continuous spectrum in the complex plane/? (with/without zoom)Me =5.585; a = 1; 7 = 1.4; Pr=0.7; R=1000.

0.60.50.40.30.20.10.0

-0.1-0.2-0.3-0.4-0.5-0.6

10 15

y/H20 25 30

Figure 2. Stream wise velocity disturbance of the acoustic modes. Me = 5.6; /?=1219.5; a :

0.215;*= l .a)C<l-l /A/ e ;b)C> HI/A/,.



20-

-20-

25 30

Figure 3. Streamwise velocity disturbance of the Figure 4. Streamwise velocity disturbance of thevorticity mode. Me = 5.6; R=\2\9.5\ a = 0.215; k = 1. entropy mode. Me = 5.6; R=\2\9.5i a = 0.215; k = 1.

3 5 10 15

y/H20 25 30

Figure 5. Streamwise velocity disturbance of the discrete(first) mode. Me = 5.6; ^1219.5; a = 0.215.

Figure 6. Path of integration for the inverseLaplace transform.



0.005

0.000

-0.005

'-0.010

-0.015

-0.020

0.35

a)^-^^

^\

-1stmod€

^<<c~2nd r

k

"\lode !

i

0.050.10 0.30

Figure 7. Numerical evaluation of eigenvaluesfor the first and the second discrete modes.Cone. Me = 5.6; R= 1219.5. T0 = 470 K.

1.0-

0.5-

0.0-

-0.5-

-1.00 5 10 15

y/H

Figure 8. Streamwise velocity disturbance of thediscrete (first) mode. Me = 5.6; ̂ 1219.5; a = 0.2.

-0.02 -

-0.04

-0.02

-0.04

0.1 0.15 0.2 0.25 0.3

Figure 9. Contours of COl in the complex plane a . Figure 10. Contours of Re |/l, in the complex plane a



-0.008

-0.010-

-0.012-

-0.014-

-0.016

8--0.018-

-0.020-

-0.022-

-0.024 -

-0.026

a=-0.014-6.015-0.016-0.017

0.10 0.15 0.20

a

0.25 0.30

0.3-

0.25-

0.2-

0.15 10

Figure 11. G). in the vicinity of a{ - -0.0155. Figure 12. Amplitude of the streamwise velocitycomponent of the first discrete mode generated by OLcomponent of the temperature spot.

0.030-T

0.025-

0.020-

0.015-

0.010-

0.005-

0.00010 15

Figure 13. Amplitude of the streamwise velocity component of the first discrete modegenerated by a component of the temperature spot.


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