SECONDARY INSTABILITY OF HIGH-SPEED FLOWS ANDTHE INFLUENCE OF WALL COOLING AND SUCTION

Nabil M. El-Hady tAnalytical Services and Materials, Inc.

Hampton, Virginia 23665

ABS1RACT

The periodic streamwise modulation of the supersonic and hypersonic boundary layersby a two-dimensional first-mode or second-mode wave makes the resulting base flow suscepti­ble. to a broad-band spanwise-periodic three-dimensional type of instability. The principalparametric resonance of this instability (subhannonic) has been analyzed using Floquet theory.The effect of Mach number and the effeetiveness of wall cooling or wall suction in controllingthe onset, the growth rate, and the vortical structure of the subharmonic secondary instabilityare assessed for both a first-mode and a second-mode primary wave.

1. INTRODUCTION

Interest in boundary-layer transluon of highMach number flows has increased, and boundary­layer transition has become an important factorinfluencing the design of the hypersonic vehicle - itsconfiguration, thermal protection system, and itsengine requirements. Prediction and control of tran.­sition at high Mach numbers is extremely beneficialsince skin friction and aerodynamic heating are con­siderably lower for laminar Hows at high speeds.

For compressible and high-speed boundary­layer flows, the stability problem is more complex,and the direct relationship between instability andtransition is unknown. A thennal boundary layerwith mean density variations develops in addition tothe velocity boundary layer leading to changes in thedistribution of the angular momentum through theboundary layer. Due to that, the generalizedinflection point ( the location in the boundary layerwhere the gradient of the product of density and vor­ticity is zero) moves out toward the outer edge ofthe boundary layer as Mach number increases.

Compressibility is known to have a stabilizingeffect on the primary wave due to a change in thenature of the instability as Mach number increases.In order to describe the physical me{;hanism leadingto the transition process in high-speed Hows, the

'I' Senior Scientist. AlAA Senior Member.

secondary instability approach is being used. Thisapproach enables us to select the proper wavesamongst a spectrum of amplified three-dimensional(3D) waves to model the nonlinear interaction.

Recent progress in the early nonlinear stage oftransition where strong three dimensionality takesplace has identified a major link in the transitionprocess between the linear and fully nonlinear stagesfor incompressible boundary layers. It showed thatthere is a well defined transition from laminar two­dimensional (2D) to laminar 3D waves through asecondary-instability mechanism. This mechanismhas been reviewed for incompressible flows by Her­bert [1] and Bayly et a1. [2]. The extension of thesecondary-instability theory in a spatial form tocompressible and high Mach number boundary-layerflows was examined by El-Hady [3-5], and Masadand Nayfeh [6], while Ng and Erlebacher [7] exam­ined the temporal theory. The linear growth of a pri­mary wave may parametrically excite a secondarygrowth with 3D character. This seeondary instabilitymay not lead to transition by itself, but as it grows,it interacts with both the mean flow and the primarywave leading rapidly to transition.

At high speeds, the effect of cooling or suctionin some of the early stages to transition needs to befully assessed as means of transition control. This

can serve the goal of optimizing laminar now con­trol systems for supersonic and hypersonic vehicles.The effect of cooling or suction on the primary waveat high-speed flows depends on the type of instabilityin the boundary layer. This effect is well establishedin the literature, and is briefly reviewed here. In thelinear stage, and in the absence of cooling or suc­tion, there exists one generalized inflection point inthe compressible boundary layer. For subsonic andlow supersonic boundary layers, the instability is ofviscous type and the most amplified disturbance isoblique, and is called the first mode. As the Machnumber increases, the generalized inflection pointmoves toward the outer edge of the boundary layeruntil it disappears. With that, the viscous instabilitybecomes weaker and the inviscid instability picks up.Consequently, the primary first-mode disturbance,which is dependent upon the presence of the general­ized inflection point, is stabilized due to the changein the nature of the instability. Also as Machnumber increases, multiple eigenvalues of amplifiedand damped modes (Mack modes) result as solutionto the compressible stability equations. The first ofthe Mack modes is called the second mode and isthe most unstable of all the modes as a 2D distur­bance. This mode is a high frequency, acoustical­type disturbance.

As the boundary layer is cooled or sucked, asecond inflection point appears. Both inflectionpoints move away from the wall and then disappearas the cooling or suction increases. When this hap­pens, the primary first-mode disturbance is com­pletely stabilized. The effect of wall cooling on theprimary stability of boundary layers was one of thesignificant findings of Mack's calculations [8]. Aprimary first mode is strongly stabilized and its mostunstable frequency is decreased by cooling. In con­trast, primary higher modes are destabilized by wallcooling, and the unstable frequency band shifts tohigher values [9]. This means that a wave withfixed frequency may be stabilized by sufficient cool­ing, but the growth rate of the most amplified pri­mary second mode increases quite rapidly with cool­ing. Malik's calculations [10] for transition predic­tion on sharp cones using eN method, showed that itis the first oblique mode and not the second modethat is responsible for transition up to about Machnumber 7 ( the first produces higher value of N­factor than the second). This is despite the fact thatgrowth rates of the second mode are much higher,however, the streamwise extent of their region ofinstability is much shorter (producing lower value ofN-factor). With wall cooling, the primary first modeis stabilized, while the primary second mode

- 2 -

becomes more unstable with larger streamwise extentfor a given second mode frequency. Consequently, aswaH cooiing increases, the second mode dominatesthe transition process, and its role becomes increas­ingly important at lower Mach numbers.

Suction is more effective in· stabilizing theviscous instability, and hence it is more effective atlow Mach numbers [10]. As Mach numberincreases, the minimum amount of suction needed toeliminate the generalized inflection point increases.The first mode is strongly stabilized by suction, onthe other hand, suction losses its effectiveness in sta­bilizing the second mode at high Mach numbers[11]. With the increase in suction, the frequency ofthe most unstable second mode shifts to a highervalue. Although a second mode with a fixed fre­quency may be destabilized by suction, the growthrate of the most unstable frequencies decrease quiterapidly.

The nonlinear evolution and breakdown insupersonic and hypersonic boundary layers as well asthe structure of the flow near transition are still unk­nowIl. None of the high-speed stability experimentswere designed LO study these phenomena. However,a direct numerical simulation of parallel compressi­ble boundary layers performed by Erlebacher andHussaini [12] was able to unveil a secondary insta­bility at Mach number 4.5 triggered by the interac­tion between a finite amplitude 2D wave with a 3D(first mode) disturbance. This instability was foundto be weaker than those found in incompressibleflow but qualitatively similar to the k-type break­down. On the other hand, with respect to thetheoretical developments, the secondary instabilitytheory was extended by the author [3-5] and by oth­ers [6,7] for compressible boundary layers.

In this paper, we study the linear secondary3D instability of supersonic and high-speed flows.We investigate the effect of a small but finite ampli­tude 2D or oblique compressible Tollmien­Schlichting (TS) wave on the growth of 3D perturba­tions in supersonic boundary layers, and the effect ofsmall but finite amplitude 2D second mode on thegrowth of 3D perturbations in hypersonic boundarylayers. The study focuses on the growth of thesecondary subharmonics due LO their importance in alow disturbance environment. The influence of wallcooling and suction on the onset, growth rate, andthe vortical structure of the secondary instability ofthe boundary layer is also investigated to assess theireffectiveness as a means of transition control at highspeeds.

cc denotes a complex conjugate, and A o is an initialamplitude. For the spatial stability analysis, a is thecomplex wavenumber of the primary wave definedas a = ar+i aj, and (0 is the frequency which is real.We shall consider the variation of the primary ampli­tude A (x) in Eq (8) to be weak. This variation willbe neglected, and A is assumed to be locally con­stant. Also, we shall neglect terms 0 (A 2) in theanalysis, and assume that the 2D compressible flowis modulated only by a periodic component of thelinear stability problem. Justification of theseassumptions was considered in [5].

In Eq (7), qo stands for boundary-layer flowquantities uo, Po, Elo, Po, and I!o, while ql stands forthe eigensolutions of the primary wave, they areu I> v I> P I> Ell ,PI> and I!I' These quantities representvelocities, pressure, temperature, density, and viscos­ity, respectively. The state equation relates theboundary-layer temperature and density profiles by

PoElo= 1 (9)

and also relates the primary density disturbance tothe temperature and pressure disturbances by

A 2'YM~ PI = plElo+ PoElI + o (A 2) (10)

The boundary-layer viscosity is related to the tem­perature through Sutherland's law, while the primaryviscosity disturbance is assumed to be linearlyrelated to the temperature disturbance by

dl!oI!I = dEl

oEll + 0 (A 2) (11)

The eigenfunctions ql in Eq (1) are normalized suchthat the amplitude A of the primary wave measuresdirectly the maximum root mean square value of themass-How disturbance in the flow direction, that is

max 2fm I (y)f = 1 (12)05:y5:~

where In I is the mass-flow disturbance given by

Uo A 2 Ell UIml=-('YM~PI--)+- (13)

Elo Elo Elo

EI-Hady [5] used the streamwise velocity perturba­tion to nornlalize the eigenfunctions q I' That choicewas adequate for the low range of Mach numbers.As Mach number increases, the temperature distur­bance increases rapidly, and the mass-flow distur-

In section 2 the analysis is developed. Section3 discusses the numerical procedures. ResulL<; anddiscussions arc given in section 4. Then we endwith concluding remarks in section 5.

2. ANALYSIS

The flow field is described by the laminarcompressible 3D Navier-Stokes and energy equa­tions. Lengths, velocities, and time are made dimen­sionless using a reference length L=(vQooX/UO~)1I2,

the free-stream velocity UO~, and LJu~, respectively.Here x is the distance from the leading edge of theflat plate and v~ is the kinematic viscositycoefficient evaluated at the free stream. The pres­sure is made dimensionless using P~ u6.0. The tem­perature, density, specific heats, viscosity, and ther­mal conductivity of air (treated as perfect gas) aremade dimensionless using their corresponding free­stream values. In terms of these dimensionlessquantities and in a vectorial form, the governingequations read

i7+V.(PV) = 0 (1)

a(aPV) +V.(pVV) =-Vp+-lV.'t (2)t R

P(~~ +V.vEl) = <Y-l)M~[~: +V.Vp

1 1+IicI>]+ RP V.(I!VEl) (3)

r

with the state equation

yM~p = pEl (4)

In the above equations 't is the dimensionless viscousstress tensor, and cI> is the dimensionless dissipationfunction. They are defined as

't = l![vv+(VVl]+Av.vI (5)

cI> = 't:VV (6)

Also, yis the ratio of specific heats, M~ is the free­stream Mach number, R=po~ Uo~ LJI!o~ is Reynoldsnumber, Pr=cp J!/k is Prandtl number, I! and A arethe first and second coefficients of viscosity, respec­tively, I is a unit tensor, and l' denotes a transpose.

2.1. The Uasic Flow

The basic flow consists of a 2D compressiblelocally parallel boundary layer modulated by a smallbut finite-amplitude compressible TS or obliquewave, here called primary wave. The basic flow isassumed to be a solution of the equations of motion,it takes the form

- 3 -

where

q(x ,y ,t) = q o(y )+A [q I (y )e j (arx--{OI)

+cc. J+O (A 2)

-Ja.dxA == A (x) = A o e I

(7)

(8)

- 4 -

n=-oa

(17)

~

q2(X ,y ,z,1) = eYX+<J/ eipz L 1ll1l (y )eill lX(:HOlIlX)

and no attempt was made to study the effect of thaton the solution. When the secondary instability istuned with the basic state, then Ill-I =<iiI' Using Eq(18), the governing equations for the 3D subhar­monic instability ( governing l\>l ) can be written inthe form

(18)

-21

i(ax-<.o/)q2ex,y,z ,1) = e yx eiPz [Ill1(y)e

_1- i (ax-<.o/)

+ Ill-l(y)e 2 ]

periodic coefficients can be applied to give a solu­tion to these equations in the form

q2(X ,y ,z ,I) = eYX+<J/ ei Pz (j>(x ,y ,1) (16)

where ~ is a real spanwise wavenumber of thesecondary disturbance. It is a measure of the angleof divergence of the direction of propagation of thesecondary disturbance from the primary-wave vector.r=¥r+iYi and a=ar+iai are two complex charac­teristic exponents, and (j>(x,y ,1) is a periodic functionof (X-Wl fa) , the same as the period of the basic/low. We express (j>(x,y,1) in terms of Fourier seriesto obtain the following expression for q 2(X ,y ,z ,I)

Equation (17) represent a general Floquet formfor the eigenmodes of a periodic basic flow. Thesubharmonic and fundamental modes are specialcases of this form. Given two values of the four realexponents Yr' Yi, ar , andai in Eq (17), the solutionof the resulting eigenvalue problem determines theother two values. For the purpose of our study of thespatial instability of subharmonic modes, we let Yrrepresent the growth rate of the secondary distur­bance, ar=O ( no temporal growth), ai=-(f)f2 for apure subharmonic mode, and let Yi represent the shiftin the streamwise wavenumber of the secondarywave with respect to the primary one (detunedmodes). A value of Yi = -al2 means that the secon­dary disturbance is tuned with the basic state.

In the spatial numerical treatment, the lowestpossible truncation of Fourier series for subharmonicmodes is used , that is,

(0 D2+b D+c) Illl =A(d D 2+e D+f) <iiI

+ o (A 2) (19)

with a similar set of equations for l\>-I> where,

l\>l =' (U2' V2, W2' P2' ez)T, (20)

D =d Idy, and a, b, c, d, e, f are 5x5 matrices that

bance seems more appropriate for this normalization.

The eigenfunctions of the prilnary \vave aregoverned by a six- order system of equations for 2Dprimary wave, or by eight-order system of equationsfor 3D primary wave.

2.2. The Secondary Instability

To study the linear 3D instability of the basicnow given by (7), we superpose a small unsteadydisturbance on each velocity, thermodynamic andtransport quantity of the basic flow, that is

q (x ,y ,z ,1) = q(x ,y ,1) + Bq 2(X ,y ,z ,1 ) (14)

where q 2 is a secondary disturbance eigenfunctionthat represents velocities U2, v2, w2, pressure P2,temperature e2, density P2, and viscosity ~2; they arenormalized such that the amplitude B of the secon­dary disturbance measures the maximum root meansquare value of the secondary mass-flow disturbancem2, which is given by

_ Uo A 2 e2 U2m2- - (yM~ P2- -)+-eo 80 eo

+ A (P1U2 + U lPz) (15)

where the overbar indicates a complex conjugate. Ina linear analysis, the amplitude B of the secondarydisturbance is assumed small compared to the ampli­tude A of the primary wave in such a way that theprimary will in/luence the modulaLion of the secon­dary but not vise versa.

The basic flow given by (7) neglects the non­linear distortion of the eigenfunction q I at a finiteamplitude of the primary wave. This has beenjustified in the incompressible secondary instabilitytheory [1] on the basis that the 3D secondary insta­bility occurs at small amplitudes of the primarywave where the nonlinear distortion is weak. It wasalso noticed that the vortical nature of the 3D secon­dary instability is not affected by the nonlinearity.These justifications are still valid for the compressi­ble secondary instability theory, if we accept thenotion that compressibility will probably retard andattenuate any nonline~lr distortion.

EquaLion (14) is substituted into Eqs (1-6), thebasic flow is subtracted, and the resulting equaLionsare linearized with respect to the secondary distur­bance (j2' We end up with five coupled partialdifferential equations for the secondary 3D instabil­ity. The coefficients of these stability equations arefunctions of the basic flow and its derivatives, theyare independent of the coordinate z, and periodic inx and I. Hence, the z -variation can be separated,and Floquet theory of differential equations with

- 5 -

for given boundary-layer velocity and temperatureprofiles uoCY) and OoCY), respectively.

The system of Eqs (19) governs the spatialsecondary 3D subharmonic instability of compressi­ble 2D flows. They are five coupled ordinarydifferential equations for Uz, Vz, Wz, pz and Oz.When supplemented with homogeneous boundaryconditions, they constitute an eigenvalue problem inthe form

are dependent upon !he basic flow.

The density secondary disturbance pz is relatedto !he pressure s~condary disturbance pz by !he stateequation

yM;, Pz = PoOz + 00Pz + A (p192+0 jP2) + 0 (A z) (2])

while the viscosity secondary disturbance Ilz isrelated to the temperature secondary disturbance Ozthrough a Taylor's expansion of the total viscositythat yields

d Ilo dZllo - ZIlz = -Oz + A--z OJ Oz + 0 (A) (22)

dOo dOo

and gn/> and hnl ,n=1,2,..8 are !he elements of 8x8variable coefficient matrices. They are given inAppendix B. While gill are function of the mean/low, frequency, and wavenumber of !he subhar­monic, hnl are function of the basic flow parameters(including the primary wave).

We assume !hat the amplitude of the primaryvanishes far in the free stream at y:2:Ye, e denotesthe edge of the boundary layer. Then the system(24) will have constant coefficients, and can besolved analytically producing four linearly indepen­dent, exponentially decaying solutions to conformwith the boundary condition (26). With the free­stream solution as initial condition, Eqs (24) areintegrated from Y=Ye to y=O at the wall, using avariable step-size algorithm [13], based on theRunge-Kutla -Fehlburg fifth-order formulas. Thesolution is orthonormalized at a preselected set ofpoints using a modified Gram-Schmidt procedure. ANewton-Raphson technique is used to iterate on theeigenvalue to satisfy the last wall boundary conditionwi!hin a specified accuracy of 0 (10-5

).

(23)Y= rca, p, R; A)

wi!h the boundary conditions,

Zt =Z3=Z5=Z7=O at y=O (25)

Z 1> Z3, Z4, Z7 ~o as Y~oo (26)

3. NUMERICAL PROCEDURES

The mean flow equations (Appendix A) arenumerically integrated by using a combination of ashooting technique and Runge-Kutta and Adam­Moulton integrator. The thermodynamic and tran­sport properties of the perfect gas are computed ateach integration step, as they vary with the tempera­ture.

The primary instability which modulate the 2Dcompressible boundary layer is governed by sixfirst-order set of ordinary differential equations.They are numerically integrated as initial value prob­lem using a free-stream solution as initial condition.

In this study we limit our concern to thesubharmonic instability that is tuned with the basic/low. The system of Eqs (18) may be written, byneglecting terms 0 (A /R), as eight first-order com­plex equations in the form

8 8

DZn - L gn/ Z/ = ALhn/2;.1=1 1=1

n = 1,2,..,8 (24)

4. RESULTS AND DISCUSSION

Experiments indicate that the subharmonic ins­tability mechanism leading to breakdown is favoredwhen !he amplitude of the primary wave is low ormoderate, whereas the fundamental breakdownoccurs for higher primary amplitudes. Because flightapplications are mostly characterized by low distur­bance background, the subharmonic 3D instabilityappears to be more realistic and more dangerous insuch applications.

In a supersonic boundary layer, 3D primarywaves become dominant, having an oblique angleabout 40-60 deg for the most unstable wave. Theinfluence of the primary wave angle on the secon­dary growth rate was examined by Ng andErlebacher [7] at Mach number 1.6. Their investiga­tion was temporal and local. They concluded thatthe strongest subharmonic modes occur when the pri­mary wave propagates in the mean-flow direction,and that they arc tuned wi!h the basic state (havingthe same phase velocity as the 2D primary wave).They also concluded that wi!h a primary-wave angleof 45 deg, !he secondary subharmonic growth rate ismuch less than the 2D case, but the first five Fouriermodes, see Eq (18), are required to capture thesecondary growth rate correctly.

where

For the previous reasons, we focus our studyon the possibility of the formation of strong threedimensionality of the subharmonic type from purelycompressible 2D basic flow. Results reported hereshow that a parametrical excitation by the finiteamplitude primary wave will produce strong growthof secondary 3D subhannonics along a broad bandof spanwise wavelengths. All reported results use anondimensional frequency F defined as F = 106ro1Rand a spanwise wavenumber parameter b defined asb = 103PIR. They represent a wave traveling down­stream with a fixed physical frequency, and a fixedphysical spanwise wavenumber.

4.1. Effect of Mach Number

Figure 1 compares the growth rates of thesecondary subharmonic mode at Mach numbers 1.6,2.2, 3.0, and 4.5. Calculations are performed at afree-stream stagnation temperature as = 3UoK, andR = 800. The primary wave is 2D first mode with afrequency F =60, and amplitude Am = 0.06 ( thesubscript m refers to mass flow). The figure hasbasically the same features of subharmonic growthrates in incompressible [1,14] and low Mach number[5] flows, the large values of the growth rates, thebroad band of unstable subharmonics, and the sharpcutoff at low spanwise wavenumbers. The stronggrowth of the secondary subharmonics compared tothat for the primary waves [ which is 0 (10-3

) or less] specially in the region of interest where they aremostly unstable, justifies the assumption of neglect­ing the variation of the primary amplitude and con­sidering it as locally constant, see Eq (8).

At these conditions, Fig 1 indicates thatcompressibility has a stabilizing effect on the secon­dary subharmonic instability with a considerabledecrease in the' growth rates and reduction in theunstable band of the spanwise wavenumbers as Machnumber increases. It should be noted that at a fixedfrequency F, an increase in the Reynolds numberand/or the amplitude of the primary wave will subse­quently increase the growth rates as well as theunstable band of the secondary subharmonics for allMach numbers. At the local conditions of Fig 1, itseems that the mechanism of the secondary subhar­monic instability, resulted from the parametric exci­tation by the finite amplitude first mode primarywave, is weakened as Mach number increases.

At different Mach numbers, Fig 2 shows thevariation across the boundary layer of the root meansquare values of the secondary subharmonic distur­bance normalized with respect to mz. Thecorresponding variations for the primary first-mode

- 6 -

disturbances normalized with respect to ml areshown in the same figure for comparison. These cal­culations are performed at R =800, F=60,Am = 0.06, and b=.18,

The location of the critical layer, indicated inFig 2 by y", moves away from the wall as Machnumber increases, followed by the peak amplitude ofall the primary and secondary disturbance quantities.At high Mach numbers, fluctuations take place nearthe edge of the boundary layer. In comparison withother secondary disturbance quantities, Vz is verysmall, while the primary v1 disturbance has a consid­erable value. As Mach number increases, both theprimary and secondary temperature disturbancesincrease rapidly , with a second peak developingnear the wall for the primary disturbance. Boththe streamwise and spanwise velocity components (Uz and Wz ) of the secondary disturbance, whichhave considerable amplitudes at low Mach numbers[5], diminish as Mach number increases. The figurealso shows that the rate of decay of the secondarydisturbances in the free stream is much faster thanthe rate of decay of the primary waves.

With the onset of the secondary instability,three-dimensionality is induced in the flow field andthe initial 2D vorticity of the base flow is deformedproducing a vortical structure. Figure 3a shows thespanwise vorticity contours of the total flow in thex-y plane at z=O for different Mach numbers. Thefigure is plotted over four primary wavelengths. Thevortices are inclined to the mean-flow direction at anangle. As Mach number increases, these vorticesextend to and concentrate around the critical layer(the tick mark on the left side of each graph indi­cates the position of the critical layer). The span­wise velocity variations produce a streamwise vorti­city shown in Fig 3b and plotted over two spanwisewavelengthes in the z-y plane at x=O, for differentMach numbers. It shows counter-rotating vorticesextending away from the wall toward the criticallayer as Mach number increases. The interactionbetween the streamwise vorticity and the deformedspanwise component is the main drive to flow break­down.

At high Mach numbers, a primary 2D secondmode dominates the primary stage, and a parametri­cal excitation of its finite amplitude produces agrowth of a secondary 3D subharmonic along abroad band of spanwise wavelengths. Figure 4shows the growth rates of these secondary subhar­monies at Mach numbers 4.5 and 7.0. These calcu­lations are performed at a free-stream temperaturel20oK, R = 1000 and the primary wave is a 2Dsecond mode with amplitude Am = 0.06, and fre-

quency F = 200 for M ~ = 4.5, and F = 180 forM~ = 7.0.

The normalized variation across the boundarylayer of both the secondary subharmonic and the pri­mary wave are shown in Fig 5 for Mach numbers4.5 and 7.0. Calculations are performed at the mostunstable subharmonic spanwise wavenumberscorresponding to Fig 4. Figure 5 shows that the crit­ical layer (Yc) moves toward the edge of the boun­dary layer as Mach number increases. Both the pri­mary and the secondary temperature and mass flowdisturbances have sharp peaks compared to theircounterpart first mode in Fig 2. The Vz disturbance isalmost vanishing, while the induced spanwise velo­city w z disturbance is very small.

In spite of the week growth of the subhar­monic secondary instability at these hypersonicMach numbers, the flow field is completely altereddue to the vortical structure produced by the secon­dary instability. This structure is concentratedaround the critical layer as shown in Fig 6 (the tickmark on the left side). The figure shows com­ponents of the angular momentum ( the vorticitymodulated by the mean density). The spanwiseangular momentum is inclined to the mean-flowdirection, while the streamwise angular momentumshows counter rotating vortices.

It is noted that, the free-stream temperatureaffects the growth rate of the secondary instability.At M ~ =1.6, Fig 7 shows that the growth rate of thesubharmonic increases with the increase in the free­stream temperature ( the primary wave is a 2D firstmode). Whereas at M ~ = 4.5, Fig 8 shows theopposite, the growth rate of the secondary subhar­monic decreases with the increase of the free-streamtemperature ( the primary wave is a 2D secondmode).

4.2. Effect of Wall Cooling

The effect of wall cooling on the subharmonicsecondary instability is investigated at Machnumbers 0.8 and 1.6 using a 2D first-mode primarywave, and at Mach number 4.5 using a 2D second­mode primary wave. The wall cooling parameterOw/Oad is used for this purpose, where the sUffi~e.s wand ad refer to wall and adiabatic wall condluonsrespectively. Cooling the wall influences both themean flow and the amplitude of the primary wave.The net outcome depends largely on identifying themajor disturbance in the flow field. First-mode andsecond-mode primary disturbances give oppositeeffeCL'l.

- 7 -

The local effect of cooling ( both Reynoldsnumber and the amplitude of the primary are fixed)is demonstrated at different Mach numbers in Figs9-11. The figures show that whether the primarywave is a first mode ( Figs 9 and 10 ) , or a secondmode ( Fig 11 ) cooling can be stabilizing or desta­bilizing depending on the spanwise wavenumber ofthe secondary subharmonic. A similar conclusionwas reached by El-Hady [5] concerning the localeffect of compressibility, which also can be stabiliz­ing or destabilizing depending on the amplitude ofthe primary wave and the spanwise wavenumber ofthe secondary subharmonic. In the low Machnumber figures, we notice that cooling shifts themost unstable spanwise wavenumber to a highervalue and stabilizes it. At high Mach numbers, thesame shift occurs, but the most unstable spanwisewavenumber is not stabilized. The local effect ofwall healing is shown to be stabilizing at low Machnumbers in contradiction wilh its known destabiliz­ing effect in air. Wall heating effect is not investi­gated at higher Mach numbers since it is not practi­cal, the adiabatic wall temperature is already highfor the metal to sustain.

Figures 12 and 13 show a comparison of theeigenfunctions of the secondary disturbance atdifferent wall cooling levels at Mach numbers 1.6and 4.5, respectively. In both figures, the Vz com­ponent is very small (not shown) and is slightlyaffected by wall cooling. Primary-wave eigenfunc­tions are also shown in both figures for comparison.Wall cooling tends to move the critical layer closerto the wall (the opposite occurs for wall heating) andthe maximum amplitudes of both the primary andthe secondary waves follow that. This is truewhether the primary is a first-mode wave (its growthrate decreases by cooling), or the primary is asecond-mode wave (its growth rate increases bycooling). So, the stabilizing or destabilizingmechanism due lO wall cooling does not depend onthis fact. While the value of the maximum ampli­tude of the velocity components (uz,wz) is slightlyaffected by wall cooling (or heating), the tempera­ture anlplitude changes drastically. At M ~ =4.5, wenotice an opposite effect due to wall cooling on thelWo temperature-peaks of the 2D second-mode pri­mary wave. The wall peak is destabilized by cool­ing, while the outer peak is stabilized.

The vortical structure that appears in the flowfield due to the secondary instability is affected bywall cooling or wall heating. Figures 14 and 15show that effect on the contours of the spanwise andstreamwise angular momentum at Mach numbers 1.6and 4.5, respcctively. Notice that these figures show

a local cooling effect where both R and A are fixed.It represents the influence of the mean-flowmodification due to walI cooling on the componentsof the vortical structure.. Regions of concentratedspanwise angular momentum are convected down­stream confining themselves near the wall as walIcooling level increases. The concentrated angularmomentum follows the critical layer as wall coolinglevel changes. Figure 14 shows that wall heating atM .. = 1.6 has an opposite effect, where the vorticalstructure stretches away from the wall.

The local effect of wall cooling or heating hasno practical value. To evaluate the overall effect onthe onset and growth rate of the secondary 3Dsubharmonic, .we should combine the effect of thechange in the amplitude A of the primary wave, aswell as the increase in R as the disturbance developdownstream. This is shown in Figs 16-18 atdifferent Mach numbers. In these calculations, theinitial amplitude of the primary wave is assumed atits first neutral point, and the spanwise wavenumberparameter b is held fixed. At Mach numbers 0.8and 1.6, Figs 16 and 17, respectively use a 2D firstmode as a primary wave. They show that the totaleffect of wall cooling is to delay the onset of thesecondary instability, and to decrease significantly itsgrowth rate. Heating has the opposite effect. Noticein these figures the explosive growth rates of the 3Dsecondary subharmonic disturbances compared to theprimary wave as Reynolds number increases. Also,with a cooling parameter 0.85 at M .. = 0.8, and withinitial primary amplitude A Om = 0.0023, the primarygrowth is almost zero, while a considerable 3Dgrowth takes place (not shown in the figure). AtM .. = 1.6, and with initial primary amplitudeA Om = 0.01, the cooling parameter 0.8 overstabilizethe primary wave, yet large growth of the 3D secon­dary subharmonic takes place.

At M~ = 4.5, and for a second-mode primarywave, Fig 18 shows the effect of cooling on the pri­mary wave as well as the overall effect of coolingon the onset and the growth rate of the 3D secondarysubharmonic. The figure emphasizes the known factthat the primary wave ( second mode) is destabilizedby cooling. The figure shows that in spite of thedelay in the onset of the 3D secondary subharmonic,its growth rate increases rapidly due to wall cooling.Notice that at this Mach number, the 3D growth isnot explosive, it is almost the same order as the pri­mary wave when the initial amplitude of the primaryA om = 0.01. The 3D growth increases as A omincreases ( shown in the figure for A Om = 0.03 ) withthe total effect of cooling still destabilizing. Figure18 shows that although the growth rates of the 3D

- 8 -

subharmonic increases with wall cooling, the stream­wise extent of the unstable region is reduced. Thisindicates that a secondary amplification factor [ wecall it S-factor, and define it as S = In (B IB 0), withBoas the initial amplitude of the secondary distur­bance ], may not significantly be affected by wallcooling.

4.3. Effect of Suction

The similarity suction parameter 'Yo, defined inAppendix A is used to investigate the effect of suc­tion control on the secondary instability at Machnumbers 0.8, 1.6 and 4.5. Moderate suction of theboundary layer stabilizes the mean flow, andreduces the amplitude of the primary wave.

Figures 19-21 demonstrate the local effect ofsuction (both Reynolds number and the amplitude ofthe primary wave are fixed) at different Machnumbers. They represent the influence of the meanflow modification by suction on the secondarysubharmonic growth rates. Contrary to the findingsin the previous section of the mixed effect of wallcooling, these figures show that suction even locallyreduces the growth rate of the whole unstable bandof the secondary subharmonic and limit it to fewerspanwise wavenumbers. These results are in har­mony with previous results by El-Hady [14] on theeffect of suction in controlling the secondary insta­bility for incompressible boundary layers. We noticethat while the most unstable spanwise wavenumberis not influenced by suction at Mach numbers 0.8and 1.6 ( same as the incompressible results in [14]),the most unstable spanwise wavenumber becomeshigher as suction level increases at M ~ = 4.5, wherea second-mode primary wave is used.

Figure 22 shows the effect of suction on theeigenfunctions of both the primary and the secondarysubharmonic at M .. = 1.6, where the primary is a 2Dfirst mode. Figure 23 shows the effect of suction ona second-mode primary wave and the resultingsecondary subharmonic. Suction, like wall cooling,tends to move the critical layer closer to the wallfollowed by the peak amplitudes of both the primaryand the secondary waves.

Figure 24 shows the effect of suction on thevortical structure due to secondary instability atM ~ = 1.6. The figure shows contours of the span­wise component of the angular momentum of theflow field before and after the onset of the secondaryinstability. With suction, the concentrated spanwiseangular momentum is confined nearer to the wallfollowing the critical layer.

The overall effect of suction, like wall cooling,on the onset and growth rate of the secondary insta­bility needs to incorporate the changes in both Rand,A as the disturbance develops downstream. Becausethe local effect of suction proved to be always stabil­izing for different Reynolds numbers and primaryamplitudes, we expect that the overall effect of suc­tion is always stabilizing. Figures similar to 16 and17 are expected at M~ =: 0.8 and M~ =: 1.6, but withsuction instead of wall cooling.

At M ~ =: 4.5, and for a second mode primarywave, Fig 25 shows the effect of suction on the pri­mary wave as well as the overall effect of suction onthe the onset and growth rate of the 3D secondarysubharmonic. The figure emphasizes the known factthat the primary wave ( second mode) is stabilizedby suction. The figure shows that not only the onsetof the secondary subharmonic is delayed by suction,but also the maximum growth rate is decreased andthe streamwise extent of the subharmonic instabilityis reduced. Again, as we discussed before at thisMach number, the 3D growth is not explosive, it isalmost the same order as the primary wave when theinitial amplitude of the primary A Om =: 0.01. The 3Dgrowth increases as A Om increases ( shown in thefigure for AOm =: 0.03 ) with the total effect of suc­tion still stabilizing. This indicates that a secondaryamplification factor ( S-factor ), will significantlydecrease by suction.

5. CONCLUDING REMARKS

We investigated the principal parametric reso­nance of the spatial three-dimensional instability ofhigh-speed boundary layers due to small but finiteamplitude two-dimensional compressible Tollmien­Schlichting wave at high speeds. Control of theseearly transition instabilities by wall cooling or suc­tion is studied.

Numerical results performed for a primaryfirst-mode wave with frequency F=:60 show that themechanism of secondary subharmonic instability issubstantially weakened as Mach number increases.

The normal component V2 of the secondarydisturbance is almost vanishing as Mach numberincreases. Both the streamwise and the spanwisevelocity components of the secondary disturbance,which have a considerable amplitude at M ~ =: 0,decrease rapidly as Mach number increases, affectingthe production of the spanwise and the streamwisecomponents of the angular momentum, respectively,and hence may be slowing the process of break­down.

- 9 -

The free-stream temperature influences' thegrowth of the secondary subharmonic. At M ~ =: 1.6,and the primary wave is a 2D first mode theinfluence is directly proportional. At M ~ =: 4.5, andthe primary wave is 2D second mode, the influenceis inversely proportional.

In spite of the week growth of the secondaryinstability at high Mach numbers, the flow field iscompletely altered by a vortical structure that is con­centrated around the critical layer.

At the investigated Mach numbers, the localeffect of wall cooling can be stabilizing or destabil­izing whether the primary wave is first or secondmode. Also, the spanwise wavenumber for the mostunstable secondary subharmonic increases with cool­ing. On the contrary, the local effect of suction isalways stabilizing, the growth rates of the secondarysubharmonics are reduced and the band of unstablespanwise wavenumbers is narrowed. The mostunstable subharmonic mode is not influenced by suc­tion at low Mach numbers, but becomes higher assuction level increases at M ~ =: 4.5 where the pri­mary wave is a second mode.

Practically, the overall effect of compressibil­ity, wall cooling, or suction includes the changes inReynold number and the primary amplitude as anyinstability wave develops downstream. When thesechanges are incorporated in the calculations,compressibility is shown to be stabilize the secon­dary subharmonic instability [5J. Also, suction isshown here to stabilize the secondary instability dueto a first or a second mode primary wave. The onsetof the instability is delayed, the growth rates arereduced, and the streamwise extent of the instabilityis narrowed.

The overall effect of cooling is also shown tostabilize the secondary instability the same way suc­tion does at low Mach numbers, where the primarywave is a first mode. At higher Mach numbers,where the primary wave is a second mode, wallcooling does delay the onset of the secondary insta­bility, and narrows the streamwise extent of thesecondary instability, but the growth rates of thesecondary subharmonic increases rapidly.

ACKNOWLEGEMENT

This work was supported by the TheoreticalFlow Physics Branch of NASA Langley ResearchCenter under contract NASl-19320.

APPENDIX A: The Compressible Mean Flow

The 2D compressible boundary",layer equationsfor zero pressure gradient and with suction and heattransfer are reduced to the following set of ODE's.

(p~uY + gu' - You' =0

(p~O'IPr)' + gO' + 2p~u '2 - YoO' = 0

, 1g - "2Pu = 0

u = 0, g = 0, e' = 0 or e = ew atTl = 0

U .--7 1, 0.--7 0 as 'J1.--7'J1 e

by using the transformation

(Rx

)1/2Y

TI =--Jp dyx 0

as well as a stream function to satisfy the continuityequation. Here Rx is the free-stream x- Reynoldsnumber, P is the density, x is the distance along theplate and y is the normal to it, and

h-h P vo= e Yo = ~(Rx )112hoe-he p",U '"

where h and ho are the fluid enthalpy and the stag­nation enthalpy, respectively, and e and w denotesconditions at the edge of the boundary layer. and atthe wall, respectively, and Yo is the similarity suctionparameter.

The variation of of the viscosity ~ and thermalconductivity k of the perfect gas with temperature 0are given in [4]. For the variation of the enthalpyand Prandtl number with temperature, the NBS per­fect gas tables are used. The specific heat cp iscomputed from the definition of Prandtl numberusing the calculated values of ~ and k and the tabu­lated values of Pr'

APPENDIX B: Nonzero Elements of Matrices gand h in Eq (24)

g 12 = 1,

2 2 'g21=RV/~000)-A +~, g22=-~o/~,

g23 = RU~/(~eo) - A(~~~o + mIO~eO)'

g24 = A(R I~ + mtyM~V),

g25 = -(l1ou~)'/J.I{) - mlAV/eo• g26 = -~u~/~,, A 2

g3l = -A, g33 = eo 190, g34 = -yM", V.

g35 = V/eo, g37 =-~

g4l =XA[m~~+m2~00~00-~)]IR,

g42 = XA~o(m 1+m2)IR

- 10 -

g45 = X[Au ~~ + m2(~V+~ou~A)/Oo]IR

g46 = xm~V/(Reo)

g47 = x[m ~~~ - m2~(~ + ~O~Oo)]IR

g48 = -X~WR

g56 = 1,A 2 '

g62 = -2(y--I)M:Pr Uo,I A 2 '

g63 = RPrOol(~Oo) - 2(y--l)M:PrAuo

g 64 = -R cr-1)M:Pr V/~o,A 2 '2

g65 = RPr V/~Oo) - (y--l)M:Pr~UO I~

2 2 "+ '2- A + ~ - (~OO + ~O 00 )/~o

g66 = -2~~0

gn = 1

g 83 = ~(~~O + m 1O~eO)A 2

g84 = ~(m1'YM '" V - R I~), g85 = ~mlV/Oo

h2l = R (Aul/eo+ VPl)/~o, h22 = RVl/(~oeo)

h 23 =R(u;/Oo+ U~Pl)l~OA 2' I

h24 =-R yM", [I «(J)-UrUO)Ul - UOVl]l(J..tOOo),A 2

h25 =-h 24/(yM", 00)

• "/ A 2-h3l = -Pli Ureo, h33 = -Oo(Pl + PIeo eo) + yM", Vv l/eo

A 2 - I I

h 34 = yM", (OOPI V + eovl/Oo + vel/Oo - AUI - vd- '2 2 'h35=-PlV - 200vl/Oo + (PI - 0l/Oo)V + (AUI + vd/Oo

h36 = v1/00

h41 = -X[iUr - (Yr- ~ iur)]vl/eo

1 . ,-, 2h 43 = -X[(Yr-"2t<Xr )U l /Oo+ vl/Oo + VPl + 0ovl/Oo]

h44 = XyM 2[i «(0 - Uruo) + V]vl/Oo

h 45 =-h44/(yM2eo), h47 = X~rvI/OOA 2

h61 = RPriur[el/eo - (y-l)M,:;pI]~

, I A 2'-h63 = RPrlel/eo + PleO - (y-l)M,:;pl - VVl/eo]/~

A 2 1h64 = -R (y-l)M :Pr(Yr-"2i ur)ul/~o

+ yM ZRPr H «(J)-urUo)el + e~v l]/(~eo)

1 -h65 = RPr [(Yr-"2iUr)U 1/00 + PlV

/'. - PP " If" A \"06 ~.f.U r" l' \f-"-OVOI

1 . -h87 = R [(Yr-2l cxr)uI/eo+ Vpd/J.lo

h 88 = RvI/(J.loeo)

Where m = 2(e-l)/3 is the ratio of the secondviscosity coefficient Ao to the first viscositycoefficient Ilo, e=O corresponds to the Stokeshypothesis, it is taken 0.8 in this analysis. Also,mI=m+l, mz=m+2, !1o=dJ.lo/deo, Ilt=dJ1oldeo, and,

AI.

=Yr + 2lCXr

V = Auo + (Ij - ~ i CJ)

X = 11(1 + m2 lloyM2VIR)A 2

PI = (yM ~ PI - eI/eo)/eo

REFERENCES

1. Herbert, Th., Secondary Instability of BoundaryLayers, Ann. Rev. Fluid Mech., vol. 20, 1988, pp.487-526.

2. Bayly, B. J., Orszag, S. A. and Herbert, Th., Insta­bility Mechanisms in Shear-Flow Transition. Ann.Rev. Fluid Mech., vol. 20, 1988, pp. 359-391.

3. El-Hady, N. M., Secondary Three-DimensionalInstability in Compressible Boundary Layers. Tran­sonic Symposium: Theory, Application, and Experi­ment, NASA CP-3020, vol. I, part 2, 1988, pp.691-704.

4. El-Hady, N. M., Secondary Instability ofCompressible Boundary Layer to SubharmonicThree-Dimensional Disturbances. AIAA Paper No.89-0035, 1989, also NASA CR-4251, September1989.

5. El-Hady, N. M., Spatial Three-DimensionalSecondary Instability of Compressible Boundary­Layer Flows. AIAA J. vol. 29 (5), May 1991, pp.688-696.

6. Masad, J., A., and Nayfeh, A H., SubharmonicInstability of Compressible Boundary Layers. Phys.Fluids A, vol. 2, 1990.

7. Ng, L. L., and Erlebacher, G., Secondary Instabili­ties in Compressible Boundary Layers. ICASEReport No.90-58, submitted to Phys. Fluids.

8. Mack, L. M., Review of Linear Compressible Sta­bility Theory. In Stability for Time Dependent andSpatially Varying Flows, ed. D. L. Dwoyer and M.

- 11 -

Y. Hussaini, 1985, pp. 164-187, Springer-Verlag,New York.

9. Lysenko, V. I., and Maslov, A A, The Effect ofCooling on Supersonic Boundary-Layer Stability. J.Fluid Mech., vol. 147, 1984, P 39.

10. Malik, M. R., Prediction and Control of Transi­tion in Hypersonic Boundary Layers. AIAA Paper87-1414, 1987.

11. AI-MaaitalI, A A., and Nayfeh, A H., Effect ofSuction on the Stability of Supersonic and Hyper­sonic Boundary Layers. Proc. Int. Conf. Fluid Mech.3, Cairo, Egypt, January 2-4, 1990, vol. 2, p 677.

12. Erlebacher, G. and Hussaini, M. Y., Stability andTransition in Supersonic Boundary Layer. AIAAPaper 87-1416, 1987.

13. Scott. M. R. and Watts, H. A, ComputationalSolution of Linear Two- Point Boundary-Value Prob­lems Via Orthonormalization. SIAM J. Numer.Anal., vol. 14, 1977, ppAO-70.

14. El-Hady, N. M., Effect of Suction on Controllingthe Secondary Instability of Boundary Layers.NASA-CR 4306, July 1990, also Phys. Fluids A,vol. 3, March 1991.

- 12 -

2.52.01.51.0

primary

0.5

.11.=4.5

.If.=3.0

.If.=2.2

"'~'~~_~~ ~ ~,:~:..~1.6

~:~;~~~;~/.

0.5 .0

Fig 1 Effed of Mach Number on the growth rate and'the unstable band of the spanwise wavenumbers ofthe secondary subharmonics at R =800, F =60,Am =0.06, and as =311oK. The primary wave is a2D first mode.

20 II'I'," ~"I'I I

15 I lI:, : Ye\ i10

\ !;::, i,. ! ..:, I I.

5

0.8

M",,=4.5

M",,=3.0

M",,=2.2

M",,=1.6

0.6

/ ..,.-.......... .., ....

.,. .................,"-................ ............, ....... ..........

0.2 0.4

spanwise wavenumber parameter

0.0000.0

0.005

0.010

0.015

n r\t'"\1"\U.UIGU

2.52.01.51.00.5

seconday .11.=4.5

U.=3.0

" U.=2.2-'----~--- ~ ~,_"~.=1.6

,-->;-;;;:;<

0.5 .00.5 .01.0 .00.5

~

1.0 .00.5

a t<-.::::.'--L-'--L..-I~--l----"-...l._L---'----l._i~---l._[---'----l.---'---L.---'----l..--'---l...-'-.-J

0.0

Fig 2 Variation across the boundarylayer of the primary 2D first modeand the secondary subharmonic eigen­functions at different Mach numbers,and the conditions of Fig 1.b = 0.18.

(a) Moo = 1.6 Moo = 2.2 Moo= 3.0 Moo = 4.5

y

y

(b) z 2t..z 0 z z

Fig 3 Vorticity contours of the total 3D flow at different Mach numbers for the conditions of Fig 1. (a) spanwisecontours in the x-y plane at z=O, (b) streamwise contours in the z-y plane at x=O. The primary wave is a 2Dfirst mode.

- 13 ~

M.=7.0M.=4.5

un M.=7.0_ M.=4.5

seconday

primary

1.5 2.0 2.5 3.0

0.5.0 0.5 1.0 1.5 2.0 2.5 3.0

"'=:----~i'"

0.5 .0

a iL....L-l...~.L..I~-L.._JL.........L_L.:'"'--....,.=:::L.............L-..........L-'--L............J

0.0 0.5 1.0.0 0.5.0 0.5.0 0.5 1.0

"'t '"t vI

5

10

25

20 ..::::::::::::..:.~:>;::, 15

a L.......---I..--'-..L-.L-.........J_L--'--l.-.L-........JL-!~......L.""'---'--.........J--'---I..-'-...L...........J

0.0 0.5 1.0.0 0.5.0

~ ~

10

5

25

20 :.::::.~::::=::::>;::, 15

0.40.30.20.1o0.0

2

3

1

xl0-3

4,---,.--,.--,--,--,....----,,....----,,.----,

spanwise wavenumber parameter

Fig 4 Effect of Mach number on the growth rate andthe unstable band of the spanwise wavenumbers ofthe secondary subharmonic due to a 2D second modeprimary wave at R = 1000, F = 200 for M ~ = 4.5,F = 180 for M~ = 7.0, Am = 0.06, and e~ = l20oK.

2cdH

o

Moo =4.5

x

Fig 5 Variation across the boundarylayer at hypersonic speeds of theprimary 2D second mode and thesecondary subharmonic eigenfunctionsat the corresponding most unstablespanwise wavenumber and the condi­tions of Fig 4.

y

Fig 6 Contours of the angularmomentum components of thetotal 3D flow at hypersonicspeeds and the conditions ofFig 5.(a) spanwise contours in thex-y plane at z=O,(b) streamwise contours in thez-y plane at x=O.The primary wave is a 2D second mode.

- 14 -

0.020

:5 0.015~o~

tlD

~ 0.010Ul

?-,~

ro] 0.005ouQ)Ul

e",=1200K

e",=2000K

e",=280oK 3

~o~

tlD

.-d 2

.gUl

1

e",=600K

e",=120oK

................................,." .. ., '.: ~., ".

I •I ••••

0.0000.0 0.2 0.4 0.6 0.8

o0.1 0.2 0.3 0.4

spanwise wavenumber parameter spanwise wavenumber parameter

Fig 7 Effect of the free-stream temperature on thegrowth rate and the unstable band of spanwisewavenumbers of the secondary subharmonics atM", = 1.6. R = 800, F =60, and Am = 0.06. Theprimary wave is 2D first mode.

Fig 8 Effect of the free-stream temperature on thegrowth rate and the unstable band of spanwisewavenumbers of the secondary subharmonics atM~ = 4.5. R = 956, F = 227, and Am = 0.023. The·primary wave is 2D second mode.

0.80.6

ew/ead=·55ew/ead=·70ew/ead=1.0

ew/ead=1.2

0.40.2

....•-e:::: ./ ~.:.::~ .

. ....."" ..:::-- .........~ ..........., ......

......................~:.......................

...............

0.02

Q)...,ro~

..c1...,~0~

tlD

.-d 0.01.gUl

?-,~

ro'd~0uQ)

Ul

0.000.00.80.60.40.2

0.000.0

2ro~

:5~ 0.02~

tlD

.-d

.gUl

i::' 0.01ro'd~ouQ)

Ul

0.03

Spanwise wavenumber parameter Spanwise wavenumber parameter

Fig 9 The local effect of wall cooling and heating atM~ = 0.8 on the growth rate and the unstable bandof spanwise wavenumbers of the secondary subhar­monies. R = 950, F = 60, Am = 0.02, ande~ = 2750K. The primary wave is a 2D first mode.

Fig 10 The local effect of wall cooling and heatingat M ~ = 1.6 on the growth rate and the unstableband of spanwise wavenumbers of the secondarysubharmonics. R = 750, F = 60, Am = 0.02, ande~ =2050K. The primary wave is a 2D first mode.

15 -

2.0

2.01.5

1.0 1.5

.- 8J8a4=.55- 8J8a4=1.0- 8J9a4=1.4

secondary

1.0

primary

0.5

0.50.5 .0

1.0.0 0.0

0.5 .0

u,.

0.5

Uz

1.0 .0

m"

1.0 .0

0.5a 1..o:l:~"=:l::::'-'--..L...-L._e:::::I-_L........_.1-._L.Oo::;..._.L...-"""--L_'--...L...-'---'

0.0

3

_ 9J8a4=.55_ 8J8a4=1.0_ 9J8a4=1.4

0.5

l7Iz12 .......~--.--,.......-r-- ......,.,,.......-r--r-.,.....,--.--,.,,.....,.-,...;..........--r-...-..,....--.--,

!IiI'19 I

3

9

;:" 60.4

Bw/Bad=0.7Bw/Bad=0.9

Bw/Bad=LO

0.30.2

spanwise wavenumber parameter

3

2 .""/

o0.1

Fig 11 The local effect of wall cooling at M ~ = 4.5on the growth rate and the unstable band of spanwisewavfmumbers of the secondary subharmonics.R = 956, F = 227, Am = 0.023, and e~ =62oK.The primary wave is a 2D second mode.

15

;:" 10

5

ii

-"'T-'~l~-;-T--r-r~-'T

lJu/6ad=0.7

6w/6ad=0.9

6w/6ad=1.0

secondary

Fig 12 Effect of wall cooling andheating on the variation across theboundary layer of the primary 2Dfirst mode and the secondary subharmoniceigenfunctions at M ~ = 1.6.R = 750, F = 60, Am =0.02, b = 0.2,and e~ = 2050K.

a L,~--,--.L_L--,--.L-.'-o.....J_[-'----L--'---'---'--l-..J--L-'--'---'---'

0.0 0.5 1.0.0 0.5.0 0.5.0 0.5 1.0 1.5 2.0 2.5 3.0

15

0.5 .0

6,jlJad=0.7

6w/lJ ad=0.9

6w/6ad=1.0

, .............._ _._._..... .~-

primary

0.5.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig 13 Effect of wall coolingon the variation across theboundary layer of the primary2D second mode and thesecondary subharmoniceigenfunctions at M ~ = 4.5.R = 956, F = 227, Am = 0.023,b = 0.25, and e~ = 62oK.

16 -

Fig 14 Effect of wall coolingand wall heating on thecontours of the 2D-spanwisecomponent and the 3D-spanwisecomponent of the angularmomentum at M ~ = 1.6,and the conditions of Fig 12.B = 1, the x-y plane is at z=O, andthe primary wave is a 2D first mode.

x

3D-spanwise2D-spanwise

x

::::::=:::-:::=:::=::::::=:::= e e d=1.4

111111111111111(1 a ~+~~~\-----------------

o

y

y

y

3D-streamwise 3D-spanwise

Fig 15 Effect of wall coolingon the contours of the streamwiseand spanwise components of theangular momentum of the total 3Dflow field at M ~ = 4.5,and the conditions of Fig 13.B = 1, the x-y plane is at z=O,the z-y plane is at x=O, andthe primary wave is a 2D second mode.

y

y

o-, --

z~-----~==--==~_-l

x

- 17 -

secondary

ew/ead=·80

ew/ead=·90eW/ead=1.0

ew/ead=1.2

0.012

0.008

0.004

primary

0.000 1----.....L.+..:.,....,.=o...,~~~..L---:.--~

Reynolds Number

Fig 17 The overall effect of wall cooling and heatingon the growth rates of the 2D first mode primarywave, and the secondary subharmonic at M~ = 1.6.Marching downstream includes the influence ofincreasing both A and R. F = 60, b = 0.2, and theinitial amplitude of the primary wave AOm = 0.01.

-0.0040.0 0.2 0.4 0.6 0.8 1.0 1.2

I

secondary

ew/ead=0.85

ew/ead=0.9

ew/e ad=1.0

ew/ead=1. 2

l·'· .. •.... ..,." '\

/ \../ \

I ,

primary ,/ ;..... \

~\......::......-..... ...... ' \.

0.2 0.4 0.6 0.8 1.0 1.2

Reynolds Number

Fig 16 The overall effect of wall cooling and heatingon the growth rates of the 2D first mode primarywave, and the secondary subharmonic at M ~ = 0.8.Marshing downstream includes the influence ofincreasing both A and R. F = 60, b = 0.2, and theinitial amplitude of the primary wave A Om = 0.0023.

0.05

0.04

(]) 0.03-+JroH

..c: 0.02-+J

~0H

t.'J 0.01

0.00

-0.010.0

1.21.0

secondary

0.81.2 0.61.0

ew/ead=0.7ew/ead=1.0

x10-3

10

8

(I) 6-+-'<0~

...c: 4-+-'

~0 2l-<00

0

-2

-40.6 0.8

Reynolds Number

Fig 18 The overall effect of wall cooling on the growth rate of the 2D second mode primary wave, and on theonset and growth rate of the secondary subharmonic at M ~ = 4.5. Marching downstream includes the influenceof increasing both A and R. F =227, b =0.21. The initial amplitudes of the primary wave are A Om = 0.01 andA Om = 0.03.

- 18 -

0.8

%=-.2

1:=-.1"1=0.0"

0.60.4

0.02

II)......to~

~~0~

00

..d 0.01.gUl

:>,~

cd'"d1:10()II)Ul

0.000.0 0.20.80.60.4

"10=-.2

-ro=-.1

%=0.0

0.2

/~~:~:~~ .., ,.' , ..

, ", ...... .,............ .. .." ..

......... ...." '.

0.000.0

0.03

2cd~

~o 0.02~

..d.gUl

t:' 0.01cd

'"d1:1o()II)Ul

spanwise wavenumber parameter spanwise wavenumber parameter

Fig 19 The local effect of wall suction at M ~ = 0.8on the growth rate and the unstable band of thespanwise wavenumbers of the secondary subharmon­ics. R = 950, F =60, Am =0.02. and e~ = 2750K.The primary wave is a 2D first mode.

Fig 20 The local effect of wall suction at M~ = 1.6on the growth rate and the unstable band of thespanwise wavenumbers of the secondary subharmon­ics. R =750, F =60. Am =0.02, and e~ =2050K.The primary wave is a 2D first mode.

~ 3o~

00

..d 2.gUl

~'"d 11:1o()II)(/)

"10=-.10

1'.=-.07o

'Y,=-.05()

1'.=0.0o

Fig 21 The local effect of wallsuction at M ~ = 4.5 on thegrowth rate and the unstableband of the spanwise wavenumbersof the secondary subharmonics.R = 956. F = 227.Am = 0.023, and e~ = 62oK.The primary wave is a 2D second mode.

0.40.30.2

01.-.--'-----'---.1..---'----'-----'

0.1

spanwise wavenumber parameter

- 19 -

12 ,..---.-..,--.----.--r---,.-....--.--,--r---.-..,.-r--..-....-...---.--.-----,

9

- "1'0=-.4-- "1'0=-·2- "1'0=0.0

second.a.ry

- 20 -

2D-spanwise 3D-spanwise

1'0=-0.4

y ------------------

lUUI!IJI!llll!ll

y

o x

Fig 24 Effect of suctionon the contours of the2D-spanwise componentand the 3D-spanwisecomponent of the angularmomentum at M ~ = 1.6,and the conditions of Fig 22.B = I, the x-y plane is at z=O,and the primary wave is a 2Dfirst mode.

xlO-3

8

'Yo=-O.l

6'Yo=O.O Aom=0.03

Q)

4-+-'l1:Sl-< AOm=O.Ol

..c:-+-'

~ 20l-<QO

0

secondary

-20.6 0.8 1.0 1.2 .6 0.8 1.0 1.2 1.4 x103

Reynolds number

Fig 25 The overall effect of suction on the growth rate of the 2D second mode primary wave, and on the onsetand growth rate of the secondary subharmonic at M ~ = 4.5. Marching downstream includes the influence ofincreasing both A and R. F = 227, b = 0.21. The initial amplitudes of the primary wave are A Om = 0.01 andA Om = 0.03.