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A994 6661

AIAA 99-0812 Oblique Modes in Attachment-Line Boundary Layer

R.-S. Lin and M. R. Malik High Technology Corporation Hampton, VA

h AIAA Aerospace Sciences Meeting and Exhibit

January 11-I 4,1999 / Reno, NV

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AIAA 99-0812

OBLIQUE MODES IN ATTACHMENT-LINE BOUNDARY LAYER

Ray-Sing Lin* and Mujeeb R. Malik** High Technology Corporation

Hampton, VA 23666, USA

ABSTRACT

In this paper the linear stability of attachment- line boundary layer is considered using the two- dimensional eigenvalue approach developed earlier, with modifications to include the effects of compressibility and to improve computational efficiency. Re- sults for hfW=3.5 and 1.6 are presented. In both cases, it was found that the instability is dominated by travelling waves that propagate at oblique angles with respect to the attachment line. For the M,=l.6 case, effects of surface heating on the increase of instability are investigated. Results agree very well with available experimental observations. Additionally, a revisit to the incompressible swept Hiemenz flow is made, and the results reveal that three-dimensional disturbances are more unstable than the two-dimensional disturbance which was previously thought to be the most unstable perturbation in this particular three-dimensional boundary-layer flow. It is therefore concluded that in an attachment-line flow three-dimensional disturbances are the dominant disturbances at all Mach numbers.

1. INTRODUCTION

A generalized linear stability theory for the attachment-line boundary-layer flow on a swept body was developed and applied to the swept Hiemenz flow earlier by Lin & Malik’. That study was the first in which no assumption about the particular form of the disturbance was made. Their approach results in an eigenvalue problem associated with a set of two-dimensional partial differential equations. Later, the effect of leading- edge curvature on the stability of an incompressible attachment-line boundary layer was also investigated, where higher-order boundary-layer ef-

* Senior Research Scientist, Member AI&. **Chief Scientist, Associate fellow AIAA. Copyright 0 1999 by High Technology Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

fects (specifically, displacement and surface curvature effects) of the mean flow were considered using second-order boundary-layer theory; see Lin & Malik2. It was shown that the surface-curvature effect can be significant as the leading-edge Reynolds number decreases. In both previous studies, analysis was limited to the incompressible attachment- line flow at finite Reynolds numbers.

Here, the method developed earlier is extended to compressible attachment-line boundary layers where the linear stability problem cannot be studied with a similar convenient approach as that previously used for incompressible flows by Hall et al3 Though the most unstable mode in an incompressible attachment-line boundary layer has been thought to be two-dimensional in nature; yet, for high speed compressible flows, it is generally expected that the instability will be dominated by three-dimensional disturbances. In order to prop- erly resolve the eigenfunction of oblique waves, the original two-dimensional eigenvalue method would have required a prohibitively fine spatial resolution in the chordwise direction; see $2 for details. To overcome this difficulty, in this paper a trigonometric factorization has been introduced into the previous stability approach in order to improve convergence and to provide a means for studying disturbances travelling at a particular wave angle. Additionally, the present code uses an advanced iterative method to effectively locate selected unstable eigenmodes of the resulting large algebraic eigenvalue problem. Details of these implementations are given in $2 & 3.

In the supersonic regime, up to date there have been very few experimental studie@ which were designed and conducted specifically for improving our understanding about the physics of attachment- line stability and transition. In the hypersonic regime, recently a few experimental work7-lo had been performed due to the realization that laminar- to-turbulent transition can have a big impact on the design of thermal protection system of a hypersonic vehicle; such as the one in the X-33 program. In this paper, we limit our interest to cases where the edge

(c)l999 American Institute of Aeronautics & Astronautics 4

.

Mach number is below hypersonic velocity.

One of the cases studied in the present paper is the compressible attachment-line transition experiment conducted by Creel et a1.4,5 in the Lang- ley Mach 3.5 Pilot Quiet Tunnel during mid 80’s. The models used were two circular cylinders with sweep angles of 45” and 60”. The Langley experiment showed that the natural transition on the attachment line, 12.7cm from the tip, occurred at freestream Reynolds numbers of about R,,~=7.5 x lo5 in fair agreement with R,,D-ciiteria of Bushnell & Huffmanll and the range of R,,D values for the criteria of Po11i2. Subsequently, Malik & Beckwith13 made a theoretical investigation for the stability of this supersonic attachment-line boundary layer by using a 1D quasi-parallel linear stability theory and found that the critical RO is about 230. Also the effect of wall cooling was studied and found to be stabi- lizing. An attempt to use the traditional eN method (e.g. N=lO) for correlating the transition data was also made for the condition of R,,~=8.6xlO~. Later, Wie & Collieri re-examined the problem, except the computation of the basic flow was done by using a different compressible boundary-layer code. They obtained a slightly lower critical Re which was about 205 (fi x 573). Using the quasi-parallel theory, Wie & Collieri also computed the variation of the critical Reynolds number with Mach number.

More recently, Coleman et al6 measured instabilities on a 76” swept cylinder in a Mach 1.6 freestream. The large sweep angle was chosen to maximize the achievable /? for the NASA Ames wind tunnel operating conditions. The condition of the boundary layer was determined by using a 2.5pm copper plated tungsten hot wire anemometer, which was also used to measure the frequencies of the amplified disturbances. In their experiment, the surface temperature of model was increased by electri- cal heating to enhance the attachment-line instability. The frequencies of the most amplified disturbances were determined over a range of temperature settings. A controlled data set is available for msta- bilities in the heated attachment line at R = 800.

In this paper, we review our results of 2D stability computations for the two cases described above. In both configurations, it was found that the instability is dominated by oblique travelling waves that propagate at a non-zero angle with respect to the attachment line. In the second case, we focused on the effect of surface heating on the increase of instability and found that the computational results agree very well with available experimental observations. This comparison provides a solid validation for the

2D stability approach and puts the current method on a firm ground. In this paper, we also paid a revisit to the incompressible swept Hiemenz flow and found that three-dimensional disturbances are more unstable than the famous two-dimensional similarity disturbance which was previously thought to be the most unstable perturbation in an incompressible att&-nnent-line flow. It is therefore concluded that in an attachment-line flow three-dimensional disturbances are the dominant unstable disturbances at all Mach numbers.

-2. COMPRESSIBLE 2D -;STABILITY THEORY

The stability results presented in this paper have been obtained using a modified version of the 2D linear stability approach developed earlier by au- thors. The modifications include: (1) an extension to incorporate compressible effects, and (2) an introduction of a~ trigonometric factorization to facilitate the study of oblique waves. Since the compressible 2D stability theory itself is not radically different from its incompressible counter part which has been discussed with considerable details in Refs. 1 & 2, hence only a brief review is necessary here. In this section, more attention will be paid to the second part of the modifications.

The coordinate system used here is illustrated in Figure 1. As in the incompressible case, we express the instantaneous flow variables in the form

Q(I, Y, -7 t) = &(G Y) + &, Y, ,‘I t) (1) _;

where q=(u,u,w,p,T), while barred and primed quan- tities denote basic flow and perturbation, respec- tively. Substituting the above expression into the unsteady compressible Navier-Stokes (N-S) equations, subtracting the basic state, and linearizing with respect to the small perturbations results in a set of 3D linear partial differential equations which governs the stability characteristics of small perturbations propagating in compressible attachment- line flow. (These equations have been given in the Appendix of Ref. 15.) Coefficients of these equations are all variable functions depending on the basic state. Owing to the infinite-span assumption the basic attachment-line flow, a( Z, y), is uniform in the spanwise direction. Consequently, the solutions of the linear stability equations are separable in the variables z and t, and can be further expressed in a normal-mode form

Q’(E, y, z, t) = $(z, y)eq[iP(= - ct,] f C.C. (2) -7

2

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where d is the spanwise wavenumber, and c is the phase speed. In this paper, we limited ourVstu&es to the scope of temporal theory. Hence, 9 is taken to be real, and the real part of the complex phase velocity represents the propagating speed in the spanwise direction of a travelling disturbance and the imag- inary part is proportional to the temporal growth rateyi (ul,=,J ci). Note that the introduction of normal mode reduces the 3D linear stability equations into a set of 2D partial-differential equations in the x-y domain. This set of 2D partial-differential equations forms the core of our compressible 2D stability theory.

Oblique modes have been known as the most unstable disturbances in a compressible flat plate boundary layer when the edge Mach number is greater than 1. Naturally, we expected that oblique travelling waves would also be relevant in an attachment-line Aow of high edge Mach number. In general, an oblique traveling wave in an attachment-line flow can be expressed in the following form

a’E)dF+Pr-tit 4’ = G-t(t, Y)e 1 + C.C.

= &(z, y)e~[~z-utl + C.C. (3)

where qr+(zr y) q 9,+(x, y)e is “e’dE. Above expression represents a travelling wave propagating in both positive x and z-direction

To study this type of disturbance using 2D stability theory, we are confronted by three major dif- ficulties. First, what would be the proper boundary conditions, specifically in the chordwise direction? Clearly, for a single oblique wave as expressed by Eq.(3), suitable boundary conditions can hardly be found a priori unless the solution has been fully determined. The second difficulty is regarding the grid resolution, particularly in the chordwise direction. It is noted that, in the context of 2D stability theory, the effect of chordwise wavenumber a(x) is com- pletely absorbed in the eigenfunction, e.g. &+( 2, y). For a wave oscillating rapidly in the x-direciton, a fine mesh in this direction will be necessary in order to resolve the eigenfunction. However, our previous experience indicates that the 2D stability computation can be too expensive to be useful for cases demanding a large number of grid point. Ad- ditionally, direct numerical solutions of large matrix eigenvalue problems can be contaminated by round-off error due to increasing stiffness of associated matrix problem. The third difficulty has to

do wgfth how to interpret the solutions? It is ap- carent that, for a fixed grid, solutions of each 2D eigenvalue computation will consist of both grid converged eigenvalues (e.g. solutions associated with low (Y’S) and non-converged spurious solutions (e.g. solutions of high (Y’S). To separate grid converged solutions from others would require multiple calculations with different levels of spatial resolution. Fur- thermore, among grid converged solutions tedious analyses of the eigenfunction, which is itself expensive to be obtained, will be needed in order to determine the direction of propagation. As demonstrated in this paper, we have success in addressing the first two issues and have also largely resolved the third one. However, current approach still requires multiple (at least two) calculations in order to identify grid converged solutions. Our innovation to these issues are explained in the following.

It turns out that the chordwise boundary conditions can be readily constructed. Owing to the symmetric nature of the attachment-line flow with respect to z=O, it is important to recognize that in addition to the oblique wave depicted by Eq.(3), a disturbance propagating toward negative x (still in the positive z) direction should also be possible. The linear combination of these two waves can be expressed in the form

q’ = i5t(2, y),&P”-wtl + &-(z, y)e”[P”-wtl + C.C.

= 9(x, y)eQ-J~l + C.C.

(*)

where G(z, y) = &+(z, y) + Qz-(L, y). Note that the summation of these two oblique waves does not alter the eigenvalue which remains to be w for a given 8. With that, proper boundary conditions in the chordwise direction for the composite disturbance are,

for v’, w’, $3 T’ (5)

i(r,y) = -d-t1 Y>, for IL'

As usual, boundary conditions in the wall-normal (y) direction are

~=~=~;I~rO, y=o,m (6) _

which assign zero-disturbance amplitudes at the solid surface (y = 0) and at the far field ( y - ~0).

Again we would like to emphasize that the solution of 2D stability equations with boundary conditions (5) & (6) represents the sum of a pair of oblique travelling waves propagating in opposite X- directions. Considering the slow variation of meanflow near attachment line, the leading-order behav-

3


Y

ior of the solution in the chordwise direction will re- semble that of a stationary or standing wave, which is characterized by the spatial modulation of perturbation amplitude in some trigonometric function form. It is therefore possible to develop a solution method for resolving oblique waves without an ex- cessive grid requirement and still render good in- terpretation of the results. In the present paper, a trigonometric factorization is thereupon introduced.

p’( I, y, z, t) = q( I, y)ei[P’-utl + C.C.

= T,(a, x)<(z, y)ei[~“-Wtl + C.C. (7)

where I!+(a,d=sin(crx) for q’=u’ and T~(~~,x)=cos(c~x) for q’=(v’, w’,p’,F). After applying this trigonometric factorization, for a fixed fi and a non-zero cy, the new eigenfunction of the problem, i( r , y), is expected to be a slowly varying function in the chordwise direction. Advantages of this trigonometric factorization are two folds: (a) the 2D eigenvalue problem can be solved with a much reduced grid in chordwise direction, and (b) it provides a means for studying disturbances travelling in a specific direction without tedious examination of eigenfunction. Here, we loosely define the wave angle of perturbations as $=tan-‘(a I p). With this trigonometric factorization, the size of current numerical eigenvalue problems fits well within the capability of most of modem workstation.

To validate the code, we consider the stability of an attachment-line boundary-layer flow on an yawed cylinder at the conditions of A&=0.7, T,=410”R, and I?=800. In this paper, the Reynolds number l? is defined as that used in incompressible flow

03)

where

\i

VF rl = (dUe/dr),=o

(9)

To simplify the task without losing generality, we consider a parallel version of the attachment-line flow, i.e. a(~, Y) = (0, CW(Y), P(Y), T(Y)). With this assumption, the stability problem can be correctly and conveniently studied by using conven- tional 1D quasi-parallel linear stability theory as that used by Malik & Beckwith13. In Figure 2, we compare the stability curves obtained by 1D theory with that obtained bv the-current 2D stability approach. Results show excellent agreement. In the 2D calculation, a grid of xc=4 (only 2 grid points on each side of the attachment line) and ny=71 is used. This clearly demonstrates that oblique

waves of a prescribed wave angle can be correctly captured by the 2D stability approach described here. In Figure 3, we show the result of another sample calculation for a fully nonparallel case, i.e. a(~: y) = (L’(i> y)> I’(r, y), I\‘(x~ y), . ..). at the conditions of M,=O.7, T,=410”R, and k=600. Notice how grid converged solutions are accompanied by spurious solutions. The numerical approach used here was unable to filter these spurious modes forcing us to adopt a rather brute force procedure. For each case, we performed (at least) two calculations, each with different chordwise grid resolution. This al- lows us to separate spurious modes from desired solutions, at the cost of extra computation.

-: 3. COMPUTATIONAL METHOD

In this study, we discretized the 2D stability equations by using Chebyshev collocation method in the chordwise (x-) direction and a Gth-order finite difference scheme in the wall normal b-) direction. The resulting discrete problem defines the disper- sion relation, w = w( Q, @), and characterizes the stability nature of compressible attachment-line flow. The eigenvalues were obtained by either a classi- cal direct solver which yields all the eigenvalues, or an iterative method which produces a few desired eigenvalues. For the later approach, we use Amoldi’s method16 with a shift strategy.

The Amoldi method is based on similarity transformation which maps the original problem onto Krylov subspaces, i.e. subspaces spanned by the iterates of the simple power method. In general, the degree of freedom of the subspace is far smaller than that of the original problem. Arnoldi’s method probably is one of the most effective methods for extracting eigenvalues of large matrices. Recently, Lin et all7 applied this method in their study of streamwise comer-layer stability and demonstrated that the wo-rkin computing a few selected eigenvalues can be far less (at least 10 times) than that of a direct method.

4. RESULTS AND DISCUSSION

4.1 &, = 3.5, 60’ Swept Cylinder Since in this case the freestream component

normal to the leading edge is highly supersonic, i.e. l&=2.387, Wie et all4 had chosen to prescribe the inviscid pressure distribution near the leading edge by using the modified Newtonian theory:

P - = cos28 + g (1 - cos%) p3

(10) ~: s

4

(c)l999 American Institute of Aeronautics& Astronautics

where 6’ is the azimuthal angle measured fro-m the attachment line, and P, is the pressure obtained by stagnating the normal Mach number behind the shock. Recently, by comparing with numerical Navier-Stokes solution, Lin & Maliki5 demonstrated that the modified Newtonian distribution provides a very reasonable approximation to the pressure distribution near the leading edge, the difference between these two approaches is utterly insignifi- cant in the region near leading edge. In this paper, we used the numerical solution obtained by Lin & Malik15 to describe the leading-edge pressure distribution, and the attachment-line flow is obtained by solving 3D boundary-layer equations with an adiabatic wall. Though, we had used a different method to compute the mean flow compared to that of Wie et a1.14, however, the large difference in stability results, as will be shown later, should be recog- nized as due to the fact that our 2D stability theory has taken the nonparallel effects of attachment-line flow into consideration while the 1D quasi-parallel theory used by Wie et al. has not.

In Figure 4, we plot the stability curves for the case of R =lOOO and Te=210.4”R. Temporal growth rates for travelling waves with wave angles 11 = 40”) 50”, and 60” with respect to the attachment line are calculated by using both 1D and 2D stability theories. The growth rates are highest for waves propagating at near 60”. The frequency of the most unstable mode is about 15OKHz at this Reynolds number. The large difference in the growth rates between 1D and 2D theories clearly indicates that the terms omitted in the 1D parallel theory have a strong destabilizing effect and, therefore, should not be neglected. This destabilizing effect due to the non-parallel terms has also been observed in our previous incompressible studyl, however, the effect was weaker.

Figure 5 shows the stability curves computed by the 2D stability theory at R=SOO. Note that the wave angle of the most unstable travelling wave shifts from 60’ to 50’ while Reynolds number decreases. Further studies indicate that the critical Reynolds number is given by an oblique wave with a wave angle near 50’ (accurate within 5”) and the phase velocities in z-direction of unstable perturbations are in the range of 0.5 to 0.6 (normalized with boundary-layer edge velocity). In addition, the frequencies of unstable modes are of the order of 1OOKHz. Theoretically, a hot wire placed within the boundary layer should be able to observe oscilla- tions in the unstable frequency range predicted by the theory. However, to determine the wave angle can be very challenging for such a measurement

would require placing multiple hot wires in a boundary layer less than 0.2mm thickness.

Figure 6 shows the neutral curve for the 50” ohlique travelling mode computed by 2D stability theory and compared with the results of Wie et a1.14 based on 1D quasi-parallel theory. Also indicated in Figure 6 is the Reynolds number R,,o at which transition on the attachment line was detected by a thermocouple located 12.7cm from the tip of the swept cylinder in Langley’s experiment (Creel et a14s5). For the 50” oblique wave, the critical Reynolds number predicted by the 2D stability theory is about l? = 349, which is substantially lower than R = 573 predicted earlier by the 1D parallel theory. This again demonstrates the strong destabilizing effect of the non-parallel terms in compressible attachment-line flow. Additionally, note that 1D and 2D theory each predicts a different critical freestream Reynolds number, R,,D, which are all substantially lower than the transition criteria given by Bushnell & Hufl?nanl’ and Po1112. Particularly, the one given by the current 2D theory is almost an order of magnitude lower. Probably, the relatively short model used in the experiment did not provide enough distance for instability to grow to large amplitudes, therefore, no transition can be detected at lower value of R,,D. This perhaps is the reason for this discrepancy. -

4.2 Heated Mm = 1.6, 76’ Swept Cylinder

In this case, measured chordwise Cp distribution at z/D=3.6 was used to compute the boundary- layer flow near attachment line; where D is the di- ameter of the cylinder. Experimental data taken from this location was considered to be a represen- tative example of how the flow along the attachment line would behave with surface heating. In the experiment, the surface temperature was increased from O’?? (near adiabatic wall condition) to 138’F. The boundary-layer and stability calculations had employed an infinite swept assumption. This assumption was a fair approximation away from the tip of the model since properties are changing signif- icantly faster in the chordwise direction than in the spanwise direction. Though the model was placed at a 76” sweep with respect to the freestream, however, due to finite-length effects, specifically, a shock emitting from the tip of the model, we found that at a local effective sweep of 72.8” the measured Cp at attachment line matches better with the theoretical value based on an infinite swept assumption. Hence, the boundary-layer solutions are computed based on this effective sweep.

5


In air, it is typically expected that surface heating will destabilize boundary layers. In Figure 7(a- d), the effects of heating on the boundary-layer profiles are illustrated. Since heating reduces the den- sity of the fluid near the surface, as a result, the inertial force can no longer balance with the favor- able pressure gradient in the chordwise direction. Hence, as shown in Figure 7(b), the overshoot in the chordwise velocity profile is enhanced as surface temperature increases. Also, since the kinetic vis- cosity of air increases as temperature increases, the boundary-layer thickness increases as indicated in the spanwise velocity profiles shown in Figure 7(c). In addition, the distribution of $,- (p%) , a quantity defines the general inflection point in compressible flow, is given in Figure 7(d). It is observed that heating pushes the generalized inflection point upwards. Over all, as shown in Table I, surface heating causes a reduce in the Reynolds number, Ro, based on the momentum thickness at attachment line.

Table I. Variation of Ro as function of surface temperature for l?=800, MeG1.495, Te=331”R.

Surface T (“F) Re 0 305.7

46 301.5

92 297.6

138 293.7

Figure 8(a) shows the computed evolution of the most amplified disturbances with increasing surface temperature for R=806, Te=331”R, and a local Mach number equals 1.495; which are the conditions used in the experiment. Note that the free-stream Reynolds number R,,D is about 4.8~10~ for this experiment. As indicated in the figure, the results of, 2D stability computations show that the ampli- fication rate, frequency and wave angle all increase with increasing surface temperature. At the adiabatic temperature (i.e. O’F), the frequency of the most unstable disturbance was found to be about 65kHz with a 30” wave angle. When surface temperature reaches 138”F, it increases to 72kHz and a 40’ wave angle. The measured power spectra from hot wire anemometer (provided by Dr. Coleman) are shown in Figure 8(b). As the level of heating increases, a “bulge” appeared. The bulge signals the existence of unstable waves in attachment line and the instability associated with these modes was enhanced by heating. Comparing Figure 8(a) with

8(b), the most striking thing is the close agreement between theoretical prediction and experimental results. Both show the same behavior, not only in the way amplitude and frequency increase with heating, but also in the range of frequency of the amplified disturbance. In experiment an attempt had been made to confirm the wave angle by using two hot wires simultaneously, however, due to mutual in- terference the aspect about wave angle has yet to be verified. Nevertheless, the experiment of Coleman et a1.14 provided the first experimental verification of the current 2D stability theory.

4.3 Swept Hiemenz Flow Revisited

We now turn our attention back again to the oblique waves in the swept Hiemenz flow which had been examined before based on an earlier version of 2D stability approach without phase factorization. The former conclusion was that in this particular flow the two-dimensional disturbance of the form considered by Hall et a1.3 is the most amplified mode, and the critical Reynolds number given by this mode is R=583.1 and compared well with natural-transition experiments. For an incompressible two-dimensional boundary layer, the Squire transformation warrants that the most amplified perturbation is a two-dimensional mode. However, for a three-dimensional boundary layer such as attachment line, no such transformation exists. Re- sults of our revisit to this flow (given below) shows that disturbances with a small oblique angle can be slightly more unstable than the 2D disturbance.

In Figure 9, the temporal growth rates of travelling disturbances with wave angles I/J = 60, 5’, lo”, and 15” are plotted as functions of phase speed for the swept Hiemenz flow at fi=583.1. At this particular Rey-nold number, the most unstable 2D wave is known to be neutrally stable as shown in the figure. However, note that at this Reynolds number some oblique waves with small wave angles re- main mildly unstable. Further analyses show that the most unstable disturbance in the incompressible swept Hiemenz flow at all Reynolds number is the oblique wave with a wave angle near 10” (accurate up to 5O). This is contradictory to the former belief that 2D disturbances are the dominant mode in an incompressible attachment line. Further note that the critical Reynolds number, based on the 10’ oblique waves, drops slightly down to about 560 and remains in the good range of experimental results.

In Table- II we highlight the effect of compress- mdity on the critical Reynolds numbers which will be relevant to “natural transition”. The M,=O.7

6

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analysis was performed on the Langley cylinder based on the modified Newtonian Cp distribution. To facilitate meanflow calculations, an effective sweep of 20.77” was used. The results show a strong decrease in critical Reynolds number as the edge Mach number increases. In general, this trend agrees with recent experimental study of Benard et al’s who found that at about M,=6.5 the critical Reynolds number ii* (following Poll’s notation), based on a modified temperature, went down to about 200 for the occurrence of natural transition. This may be partly due to the destabilizing effect of wall cooling on the second mode present in hypersonic boundary layers, but no stability information about their flow are currently available.

Table II. Variation of critical Reynolds numbers.

5. CONCLUSIONS The 2D eigenvalue approach developed earlier

has been extended to include effects of compressibility. A phase factorization method has been introduced to facilitate the study of oblique waves. This also relaxes the gird requirement in the chordwise direction. In addition, an iterative Amoldi method has been applied to effectively locate a few desired eigenvalues of the large matrix problem associated with 2D stability equations.

Results for swept cylinders in &f-=3.5 and 1.6 are presented. In both cases, it was found that the instability is dominated by oblique travelling waves. In the ,21,=3.5 case, we found that the critical Reynolds number based on the current 2D stability theory is substantially lower than that predicted by 1D quasi-parallel theory, moreover, is one order of magnitude lower than the eldsting transition criteria. Probably, the applicability of the existing transition criteria may be limited to small models. When instabilities are relatively weak, transition can not be seen in a short model. In the M,=I.6 case, it is found that surface heating enhances the instabilities which are also oblique waves in nature. In the second case, computational results were in close agreement with experimental data. This comparison puts the current method on a firm ground. The

revisit to Swept Hiemenz flow concludes that three- dimensional perturbations in attachment-line flow constitute the dominate instability at all Mach numbers, and the critical Reynolds number decreases rapidly as the edge Mach number increases.

References

‘Lin, R.-S., and Malik, M. R., 1996. “On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow” J. Fluid Mech.,311, pp. 239-255.

‘Lin, R.-S., and Malik, M. R., 1997. “On the stability of attachment-line boundary layers. Part 2. The effect of leading-edge curvature” J. Fluid Mech. ,333, pp. 125-137.

3Hall, P., Malik, M. R., and Poll, D.I.A., 1984. “On the stability of an infinite swept attachment line boundary layer,” Proc. R. Sot. Lond., A395, pp. 229-245.

4Creel, T. R., Beckwith, I. E., and Chen, F.-J., 1986. “Effects of wind-tunnel noise on swept cylinder transition at Mach 3.5” AIAA Paper 86-1085.

5Creel, T. R., Beckwith, I. E., and Chen, F.-J., 1987. “Transition on swept leading edges at Mach 3.5” Journal of Aircraft, 25, pp. 710-717.

6Coleman, C. P., Poll, D.I.A., and Lin, R.-S., “Ex- perimental and computational investigation of leading Edge Transition at Mach 1.6”, AIAA Paper No. 97-1776, 1997.

7Amal, D., Vignau, F., and Laburthe, F., 1991. “Re- cent supersonic transition studies with emphasis on the swept cylinder case.” Boundary Layer Transition and Control Symposium, Cambridge, UK.

sHolden, M. and Kolly, J., 1995. “Attachment line transition studies on swept leading edges at Mach Numbers from 10 to 12” AIAA Paper 95-2279.

gMurakami, A., Stanewsky, E., and Krogmann, P., 1995. “Boundary Layer Transition on Swept Cylin- ders at Hypersonic Speeds,” AIAA Paper 95-2276.

‘OBenard, E., Gaillard, L., and Alziary de Roquefort, T., 1997. “Influence of roughness on attachment line boundary layer transition in hypersonic flow.” Experiments in Fluids, 22, pp. 286-291.

llBushnell, D. M., and Huf&an, J. K., 1967. “In- vestigation of Heat Transfer to a Leading Edge of a 76’ Swept Fin With and Without Chordwise Slots

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and Correlations of Swept-Leading-Edge Transition Data For Mach 2 to 8,” NASA TMX-1475.

12Poll, D. I. A., 1983. “The Development of Inter- mittent Turbulence on the Swept Attachment Line Including The Effects of Compressibility,” The Aero- nautical Quarterly, Vol. 34, pp l-23.

13Malik, M. R., and Beckwith, I. E., 1988. “Stabil- ity of a Supersonic Boundary Layer Along a Swept Leading Edge,” AGARD Conference Proceedings, No. 438.

14Wie, Y.-S. and Collier, F. S., Jr, 1993. “Instabil- ity of a swept leading-edge attachment-line boundary layer.” Transitional and Turbulent compressible Flows - 1993, FED-Vol. 151, pp. 111-116, ASME.

i5Lin, R.-S. and Malik, M. R., 1995. “Stabil- ity and transition in compressible attachment-line boundary-layer flow.” SAE Paper 95-2041.

i%aad, Y., 1992. Numerical Methods for Large Eigenvalue Problems. Manchester U. Press.

17Lin, R.-S., Wang, W.-P., and Malik, M. R., 1996. “Linear stability of incompressible viscous flow along a comer.” FED-Vol. 237, Vol. 2, pp. 633-638, ASME.

isBenard, E. et al. at Queen’s U. of Belfast. Private communication in 1998. ~

Figure 1. Coordinate system for leading-edge attachment line boundary layer.

0 15

0.10

“0

z-

0.05

0.00

-0 05 i 0

0.15

010

"0 I 5

0 05

0.00

Figure 2. Comparison between the 2D stability result and then-ID solution for a parallel attachment- line flow of T,=410°R, M,=O.7, and R=SOO.

gnd convergad

0.15~ 0 20 0.25 0.30 0

Figure 3. Numerical solutions of 2D stability equations for a nonparallel attachment-line flow of T,=410°R, M,=O.7, and R=600. Symbols: (nz,ny)=(12, 71); Lines: (nx,ny)=(20, 71) with spurious modes manually removed.

8


0.20- _ R=lQQQ y1=60" -20 Theory

----1DTheory

0.15 - “0

0.05 -

0.00 - ,

0 50 100 150 200 250 300 356

Frequency (KHz)

Figure 4. Comparison of temporal growth rates computed by different methods, Solid line : 2D stability (nonparallel) theory; Dashed line : 1D parallel the- QW.

0.20

t

i=6QQ

0.15

2D Stability Theo.

“0

T; 0.10

Ei

0.05

0.00

0 50 100 150 200 250 300 350 Frequency (KHz)

Figure 5. Temporal growth rates for the l? = 600 attachment-line boundary layer (2D stability theory).

i2D StabilityTheo.

- - - 1 D Parallel Theo.

3.5 -

0.0; 200 ’ 400 6QQ- 900 1000 1200 1400 R

Figure 6. Comparison of neutral curves computed by 1D parallel theory and by 2D stability approach.

/ I I

I

i I

I ! , I I

I1 i ;

-3 I ( 0.90 0.95 1.00 1.05 I ,__

I

I

-

0.0 0.2 0.4 0.6 0.8 1.0 / ;

Figure 7(a-d). Effect of heating on attachment-line boundary-layer profiles.

9


30 10 -50 60 70 60 90 100

frequency/kHz

(a)

lncreasmg Surface Temperature

0 0004

Figure 9. Temporal growth rates for swept Hiemenz flow at R=583.1.

0 10 20 30 40 50 60 70 80 90 100 hqurncy/kHz

(b)

Figure 8. (a) Theoretical predictions; (b) hot-wire measurement for the heated attachment line at z/D=3.6, R=800 and local Mach number = 1.495.

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