L

.

AIAA 95-0778 NATURAL TRANSITION IN TRANSONIC FLOWS USING AN EFFICIENT TEMP ORAL/SPATIAL FORMULATION Christian Masson Chaire J.-A. Bombardier Ecole Polytechnique de Montrkal Montrkal, CANADA Robert Martinuzzi Department of Mechanical Engineering the University of Western Ontario London, CANADA Mohsen Mirshams Chaire J.-A. Bombardier Ecole Polytechnique de Montrkal Montrkal, CANADA Ion Paraschivoiu Chaire J.-A. Bombardier Ecole Polytechnique de Montr6al Montrhal, CANADA

33rd Aerospace Sciences Meeting and Exhibit

January 9-1 2,1995 / Reno, NV For permission to copy o r republish, contact the American Institute of Aeronaullcs a n d Astronautics 370 L'Enfant Promenade , S.W., Washington, D.C. 20024

AIM-95-0778

4 NATURAL TRANSITION IN TRANSONIC FLOWS USING AN EFFICIENT TEMPORAL/SPATIAL FORMULATION

Ecole Polytechnique de Montrbal Montrbal, CANADA

Mohsen Mirshamst Chaire J.-A. Bombardier

Ecole Polytechnique de Montrbal Montrbal, CANADA

Christian Masson' Robert Martinuzzit Chaire J.-A. Bombardier Department of Mechanical Engineering

the University of Western Ontario London, CANADA

Ion Paraschivoius Chaire J.-A. Bombardier

Ecole Polytechnique de Montrbal Montrbal. CANADA

Abstract

Transition prediction capabilities are critical to wing design for minimizing skin friction while retain- ing the desired lift characteristics. A new stability analyzer is presented which combines beneficial as- pects of both temporal and spatial formulations, to result in an efficient stability analyzer. This tech- nique was validated by comparison with the recog- nized ONERA/CERT method and observed transi- tion locations. I t is demonstrated, through numeri- cal examples, that the temporal frequency tracking procedure is equivalent to its spatial counterpart.

-'

Nomenclature Non-dimensional unless otherwise specified. A = amplitude C = chord length, m ( M i ] , [Pa], [P,] f = frequency, Hz M = Mach number N n = amplification factor P = mean pressure P = instantaneous pressure Q = mean flow quantity ¶ = instantaneous flow quantity Re t = time

= eigenvalue problem matrices

= number of grid points

= Reynolds number, = %

*Research Scientist 'Assistant Professor 'Currently at Flight Systems Dept., CAE Electronics Ltd. SAcronautical Chair Professor, AIAA Member

Copyright 0 1 9 9 5 by the American Institute of Aero- nautics and Astronautics, Inc. All rights reserved.

W

1

U , V u , v , w 6 = group velocity +, y, z cz, p = wavenumbers 7i

of propagation 5, = boundary-layer thickness, m 6' = boundary-layer displacement

5 = generalized eigenvector * = wave orientation V I = kinematic viscosity T = instantaneous temperature rl W = complex frequency superscripts:

subscripts: e

P, i

= mean velocity components = instantaneous velocity components

= local Cartesian coordinates

= spatial growth rate along the path

thickness, m

= transformed normal coordinate

- = perturbation quantities

= dimensional quantities a t the edge of the boundary layer

= real and imaginary parts

1 Introduction The potential for reduced frictional losses repre-

sented by sustained laminar flow over the airfoil has long been recognized. In this context, Natu- ral Laminar Flow (NLF) airfoil technology finds a promising field of application. However, drag pre- diction for N L F wings presents one of the most de- manding problems posed in applied aerodynamics mainly because it hinges on obt,aining an accurate

estimate of the location of laminar/turbulent tran- sition of the boundary layer. In the industrial en- vironment, the development of NLF airfoils requires a cost-effective design tool capable of delivering a good approximation to the location of transition. The linear stability theory, along with the empiri- cal e" method, provides an appropriate framework for transition predictions over wings in the cruise regime. Mack [l], Malik and Orszag [2], and C e beci and Stewartson [3], amongst others, have de- veloped numerical methods for the solution of the linear stability equations. Mack [l] used an Initial Value Method (IVM) for the solution of the spa- tial stability equations. Malik and Orszag [2] con- sidered IVM's computationally slow and proposed a more efficient solver for the solution of the tempo- ral stability equations based on a Boundary Value Method (BVM). Cebeci and Stewartson [3] and Ar- nal and Juillen [4] proposed BVM's for the solution of the spatial stability equations. The group of the J.-A. Bombardier Chair has recently developed a lin- ear stability analyzer (SCOLIC) based on the tem- poral formulation [j]. Pressure distributions were taken from available experimental data. The bound- ary layer was predicted using the MAIN computer code [6].

This paper presents an extension of SCOLIC: an hybrid method was developed [7] to exploit the strengths and advantages of both the temporal and spatial formulations, resulting in a very efficient sta- bility analyzer. The resulting stability analyzer has been validated through comparisons with available numerical solutions and experimental data.

2 Linear Stability Theory 2.1 General Formulation The customary notation is adopted for a gen- eral three-dimensional, compressible boundary layer. For a local Cartesian coordinate system, the flow variables are assumed to be of the form:

(1) ei(~ztp.--Y1) Y, 2, t ) = Q(Y) + i(y)

where Q(y) is the mean laminar profile and the tilde denotes the perturbation quantities for any of the velocity components, pressure or temperature. The wavenumbers, a and 4, and the frequency, w are generally complex.

The problem is formulated using the parallel flow assumption (V(y) = 0) which implies that the pres- sure in the boundary layer is constant in the normal direction ( g = 0). The definitions of Eq. (1) are substituted into the momentum, state and energy Y

equations. The non-homogeneous terms are elimi- nated by subtracting the laminar mean flow solution

higher-order terms yields a homogeneous system of five linear, second-order ordinary differential equa- tions. This mathematical model is, in general, not amenable to analytical solution techniques. Con- sequently, the solution procedure proposed here is based on numerical techniques.

There are several discretizations available for the solution of the linear stability equations. In our ap- proach, a finite-difference method with a staggered mesh for pressure is employed. I t is convenient to map the physical domain 0 < y 5 6, into a trans- formed domain 0 < 11 < N - 1 as proposed by Malik and Orszag [2]. This transformation was proposed to allow a fine resolution in the critical layer of the physical domain while using a uniform grid spacing in the transformed domain. The finite-difference ex- pressions for the first and second derivatives in the transformed domain are then second-order accurate. The discretized system in the transformed domain can be expressed as:

~ [ ~ ~ I + ~ [ ~ Z ] + ~ ~ [ M ~ ] + P [ M ~ I + ~ ~ [ ~ ~ ] + [ M ~ D ( ~ ) = o (2)

from the equations. Neglecting the non-linear and u

W where 6 is the perturbation vector: ... o = ( ( I? i+pG, i j ,&? ,ac -pc ) (3)

{$} is a (5N - 9) coefficient vector representing the discrete eigenvectors, [M;] are (5N - 9) x (5N - 9) coefficient matrices.

The disturbances vanish at the wall and in the freestream, except for the pressure fluctuations. The staggering obviates the need for special treatment on pressure at the boundaries. Hence, in the framework of the linear stability theory, the boundary-layer sta- bility analysis reduces to a homogeneous eigenvalue problem for which non-trivial solutions exist only for certain combinations of the wavenumbers, 0 and p, and the frequency, w . In the general formulation, a, p and w are complex numbers, corresponding to six real parameters. Solution of the eigenvalue problem provides a relation for only two of these so that the resulting system of equations is not mathematically closed. It is thus customary and necessary to make some basic assumptions about the nature of a, /3 and w.

2.2 Temporal Stability Theory

In the temporal stability theory, (I and 4 are as- L sumed real. For given values of a and p, the system

of equations is amenable to a linear eigenvalue prob- lem in w : - [PlltG} = w[M1]{6} (4)

where:

[PI]= - ( ~ [ M z ] +a2 [&I +P[M41 +P2 [ M ~ ] + [MsD (5)

The real part of the temporal eigenvalue, w,, is the frequency and the imaginary part, w,, the temporal growth rate.

The temporal stability theory yields an eigenvalue problem which can be solved using classical numer- ical algorithms. These methods can be divided into two categories: global and local methods. Global methods are used to obtain the complete eigenvalue spectrum. These are quite expensive in terms of computational resources but they do not need any initial guess for the eigenvalue. Local methods are more efficient and generally more accurate but re- quire an initial guess for the eigenvalue.

The SCOLIC implementation of the temporal sta- bility theory calls for a coarse grid solution to a global calculation, using the QZ algorithm [E], per- formed to yield a spectrum of (5N - 9) eigenvalues. A suitable candidate is then selected as initial guess for a local method applied to a finer grid for the purpose of improving the solution estimate. Local calculations use an inverse Rayleigh iteration proce- dure [Z].

- 2.3 Spatial Stability Theory In the spatial stability theory, w is assumed real and given, n and P are complex. For given values of 0, the resulting system of equations can thus be ex- pressed as a nonlinear eigenvalue problem in a:

[PSI{@ = ((*[W] + n2[M3]) ( 6 )

where:

[pa] = - ( 4 4 1 1 + P[M41+ P2[M5] + [M6]) (7)

The real part of the spatial eigenvalue, ar, yields the wavenumber and the imaginary part, ai, the spa- tial growth rate.

In the spatial stability theory, the resulting eigen- value problem is nonlinear in a. It can be solved using the companion matrix method [9], for exam- ple. This standard approach uses a simple transfor- mation, first proposed by Bridges and Morris 191, to linearize all terms. The resulting form of the eigen- value DrOblem is suited to the standard QZ algo-

as large as the order of that corresponding to the temporal formulation, which makes the global cal- culations in the context of the spatial formulation much longer to perform.

Such a global method was considered inefficient for engineering calculations. Therefore, the inverse Rayleigh iteration method was preferred and applied to the following linearized problem:

[P81t39 = a([Mz] +"g"e.r[M31) {a (8)

where aguea$ represents the needed initial guess for the eigenvalue of interest. The solution is ob- tained iteratively.

In a traditional spatial method, the initial guess is provided by a global calculation using an appropri- ate solution procedure such as the companion matrix method. In this paper, a more eRcient procedure, presented in Section 3.3, is proposed for the calcula- tion of the initial guess.

3 Transition Prediction The main assumption regarding transition predic-

tion is that there exists a critical amplitude of the disturbances at the transition location. The arnpli- tude ratio A/Aa can he calculated assuming that the growth of the disturbance from its initial amplitude, An, t o the critical value, A,,it, can be predicted by the linear stability characteristics. The amplitude ratio AJAa or the more commonly used amplifica- tion factor, n = In (AIA,) , is therefore calculated by integrating the spatial growth rate along the path of propagation, yi, wh$h is parallel to the real part of the group velocity, Vgp:

(9)

where s is a point of the path of the disturbance and so is the point of inception of the perturbation (yi = 0). The real part of the group velocity is given by:

For low background turbulence levels in twc- dimensional flows, it has been observed experimen- tally that n is approximately 9 at transition. This correlation of the amplification factor with transition location is the basis of the e" method. The n-factor calculation is usually done under the constraint of constant dimensional frequency: -

rithm for computing the entire eigenvalue spectrum. v, W I f = -- 6' 2a The order of this eigenvalue problem is almost twice

3

A finite number of frequencies are selected, and a n factor is calculated for each of them. The fre- quency that reaches the critical n factor first is con- sidered the most relevant frequency for the predic- tion of transition location.

3.1 Temporal Frequency Tracking In the temporal stability theory, there are four inde- pendent parameters: a, p, wr, and wi . The temporal eigenvalue problem provides two real relations. In the calculation of the n factor, the frequency wr is given (based on an appropriate frequency selection criterion). The additional relation (or constraint) needed to close the problem is provided by the max- imization of the temporal growth rate, w i . These four real relations are the basis for the frequency- trackina Drocedure imdemented in SCOLIC.

I I

- . The spatial amplification factor in the direction of

the real part of the group velocity is obtained using Figure 1: Algorithm for the Temporal Frequency Gaster's relation [lo]: Tracking.

yi = -- (12)

The evaluation of yi involves the knowledge of the = &c

group velocity is given by: trated in Fig. 1. u

Wi

V g r l &L

(18) BO group velocity. In the temporal formulation, where

only is a complex number, the real part of the The algorithm for the frequency tracking is illus-

v,, .. = (2,s) The basis of the temporal frequency tracking is the maximization of wi for a given w r . First, the desired frequency, wSPEC is found, and then wi is maxi- mized along this constant frequency contour. In or- der to find the desired frequency, Am and AD are given by:

where

Aw, = wSPEC - wv (16)

Once the specified frequency is reached, w; is max- imized along the constant frequency contour. The appropriate expressions for Aa and Ap are obtained from the complex dispersion relation:

3.2 Spatial Frequency Tracking

In the spatial stability theory, there are five indepen- dent parameters: CY?, ai, &, p;, and w. T h e spatial eigenvalue problem, the given frequency w , and the optimization of the spatial growth rate provide four real relations. The fifth real relation is obtained from the constraint that aa a0 - E Real

This relation was derived by Cebeci & Stewartson [3] and Nayfeh [ll]. These five real relations can he used to derive a spatial frequency tracking method. The spatial amplification factor in the x direction, ai, is a direct result of the nonlinear eigenvalue problem of the spatial stability theory. The spatial amplifi- cation factor in the direction of the real part of the group velocity, needed for the calculation of the n factor, is given by:

L In the spatial formulation, Eq. (13) is not valid since a and p a r e complex. Therefore, the real part of the group velocity is given by Eq. (10).

4

3.3 Proposed Hybrid Method The advantage of the hybrid method is that the tem- poral formulation is used to obtain a first guess for the spatial growth rate. The linearity of the tem- poral system results in time saving with respect to the non-linear spatial system. In the proposed hy- brid method, global calculations are performed us- ing the temporal formulation: for given real a and 0, the temporal eigenvalues wI. and w, can he obtained. The temporal eigenvalues obtained with the global calculation are then transformed into spatial eigen- values using appropriate relations that exist between the temporal and the spatial formulations [lo, 111. This provides the initial guess for the local solution of the linearized spatial eigenvalue problem, Eq. (8).

In the case of an infinite swept wing, the approx- imation Pi = 0 can be used, and assuming that the imaginary part of the group velocity is much smaller than the real part , the relation between the temporal and spatial formulations is given by:

Eq. (21) is the relation used in the hybrid method to calculate the initial guesses for the spatial eigen- values from the results of the global calculation per- formed in the context of the temporal formulation.

In the hybrid method, the instability tracking is done in the context of the temporal formulation. Once the appropriate disturbance is found, a lo- cal calculation based on the spatial formulation is achieved to obtain directly the spatial growth rate needed in the calculation of the n factor. The algo- rithm used by the hybrid method for the calculation of the n factor is illustrated in Fig. 2. This algo- rithm assumes tha t the group velocity based on the temporal formulation is a good approximation of the spatial group velocity.

The main differences between the temporal and spatial frequency tracking procedures are: (i) max- imization is applied on the temporal growth rate in the former while in the latter the spatial growth rate is optimized; (ii) an additional constraint, Eq. (19), is applied in the spatial frequency tracking.

It can he shown that the frequency trackings used in the hybrid and spatial methods are equivalent. According to Nayfeh [I l l the additional constraint, Eq. (19), can be formulated as:

This equation stipulates that for a physical problem, the ratio of the group velocity components must he

I . I I

Figure 2: Overall Algorithm for the Hybrid Method.

real. Eq. (22) can he recast as:

In the context of the temporal formulation,

This condition corresponds to the maximization of wi for a given wr. Therefore, the maximization of wi for a constant wp of the temporal frequency tracking procedure is equivalent t o the "saddlepoint" condi- tion used in the spatial frequency tracking. How- ever, the maximization of w; for a constant wr does not ensure that the spatial growth rate in the direc- tion of the real part of the group velocity is maxi- mized. This question is of physical relevance in some cases where multiple local maxima of w, exist for a given frequency w,. Multiple local maxima of wi can exist in regions of the flow where both crossflow and Tollmien-Schlichting instabilities are present. In principle, in the context of the temporal formula- tion which is used in the hybrid method, it is possi- ble to select the local maximum of w, corresponding to the maximum value of the spatial growth rate 7i. At a such maximum, the two instability tracking procedures should be equivalent so that the results obtained with the proposed hybrid method should be the same within numerical accuracy as those ob- tained with a traditional spatial formulation. Fur- thermore, the proposed hybrid method is believed to be more efficient than the traditional spatial for-

5

mulation. The main advantages of the proposed hybrid

method are: (i) the initial guesses for the spatial eigenvalues are obtained using a global calculation with the temporal formulation which requires less CPU time and memory than the corresponding spa- tial problem; (ii) Gaster's relation is not needed for the n-factor calculations: the spatial growth rate is a direct result of the linearized spatial eigenvalue problem.

4 Results To demonstrate the capability of the proposed hy-

brid method, stability/transition calculations over the suction side of a 15-degree swept tapered wing with an AS409 cross-section were undertaken. Ex- perimental pressure distributions and transition lo cations were obtained in the T 2 transonic tunnel of ONERAICERT [12, 13, 141. Cebeci, Chen, and Ar- nal (131 and Niethammer [14] have also presented stability/transition calculations over this wing using the spatial theory. In order to avoid fully three- dimensional stability/transition analysis, their cal- culations were performed under the approximation of an infinite swept wing having the mean sweep an- gle of the actual swept tapered wing. The results presented in this paper correspond to the experi- mental runs # 42, 60, and 79 [12]. The experimental Mach number distributions are given in Fig. 3. Some physical irregularities are present on the test model as can be seen from the Mach number distributions near x/c = 0.3 and 0.5.

The results for run #42 are presented in Fig. 4, where the predictions of the hybrid method are com- pared with the solutions obtained using the ON- ERA/CERT method (13, 141. Run #42 was for a stagnation temperature and pressure of 145 K and 2 bars, respectively, and a chord Reynolds nurn- ber of 1 2 . 8 ~ 1 0 ~ . Fig. 4a shows the orientation of the most amplified disturbance as a function of the distance from the attachment line, s/c, for two se- lected frequencies. For the purpose of comparison, the frequencies used in this study are the ones se- lected in Refs. [13] and [14]. The results of the proposed hybrid method and the ONERA/CERT method are in very good agreement. The orientation of the most amplified low-frequency disturbance (f = 4923 Hz) is almost constant along the wing with a value near 88" indicating a pure crossflow insta- bility. In this case, the temporal frequency tracking procedure used in the hybrid method produces re- sults identical to the spatial procedure of the ON-

1.00 @ 00

ow 020 0.40 IIc 0.80 0.00

(b) Run #60.

t i o r

o,zo.. 0.00 ow o m 0.40 yc 0 . w 0 . U 1.00

(c) Run #79.

Figure 3: Experimental Mach number Distributions. u

6

v

i-

r . . . n . . . a . . . t . . . n . . . r . . . L . . . I ow 0 1 0 o m 0 3 0 0.0 OB0 o w e,@

"C

,) Orientation of the &tost Amplified Disturbance

"I 30770

(b) n-Factor Calculations.

Figure 4: Run #42.

ERA/CERT method [13, 141. In the case of the higher frequency (f = 30770 Hz), large variations of the most amplified disturbance orientation are noted along the wing. Near the attachment line, the dis- turbance is mainly in the direction of the crossflow instabilities. The large variations in the most am- plified disturbance orientation are the result of the locally adverse pressure gradients induced by the presence of small hollows in the model [14]. The hybrid method predicts these large variations in rel- atively good agreement with the solution obtained using the ONERA/CERT method. These successful comparisons of the orientation of the most ampli- fied disturbance strongly suggest that the frequency tracking procedure of the temporal formulation is equivalent to its spatial counterpart. n-factor cal- culations are presented in Fig. 4b. Here again the - present results are very close to the ones produced by the ONERA/CERT method.

I . . . , . . . , . . . , . . . n . . : I . . . 8 . . . 1 o w 010 020 090 W O O W 080 070

UC

a) Orientation of the Most Amplified Disturbancc

28652

(b) n-Factor Calculations.

Figure 5: Run #60.

The results concerning run #SO are illustrated in Fig. 5. The stagnation temperature and pres- sure, and the Reynolds number for this run are 134 K, 2 bars, and 14.4x106, respectively. The orientation of the most amplified disturbance pre- dicted by the proposed hybrid method and the ON- ERA/CERT method (see Fig. 5a) are again in very good agreement, supporting the validity of the fre- quency tracking procedure implemented in the pro- posed hybrid method. The hybrid method consis- tently produces n factors lower than the solution of the ONERA/CERT method. For a given frequency, the difference is relatively constant along the wing with an absolute value near 0.2 for the high fre- quency (f = 28653 Hz) and 0.4 for the low frequency (f = 5731 Hs). This nearly constant difference indi- cates that the point ofinception predicted by the hy- brid method is situated downstream of the one pro- duced by the ONERA/CERT method. I t is however

7

30770

Run 1 70

M- - 0.74

Re.- 13.4.40'

0.w 0.lO *.go @.,O 0.40 050 0.w 0.70 "C

(b) n-Factor Calculations.

Figure 6: Run #79.

expected that a finer marching step near the leading edge during the laminar boundary layer calculations would result in a closer agreement of the point of inception.

The influence of larger stagnation temperature and pressure are presented in run #79 where the stagnation temperature and pressure, and the Reynolds number are 164 K, 2.5 bars, and 13.4x106, respectively. The comparisons, presented in Fig. 6, show again good agreement adding confidence in the validity of the proposed hybrid method.

The prediction of the critical n factor for runs #42 and 79 has been conducted using the frequency se- lection strategy proposed in Ref. 1151. The results are presented in Fig. 7. The experimental transition location for run #42 is around x/c = 0.47. Based on the results of Fig. 7a, the critical n factor is around 5.6. Experimental transition location is not available for run #79, but since its drag coefficient is similar

(a) Run #42.

I .0 - . Run 1 7 0 . M- - 0.74 I I

0.w 0.10 0.20 0.30 o m 010 .Io

(a) Run #79.

Figure 7: Critical %Factor.

L

U

to the one of run #42, it may be assumed that tran- sition occurs near x/c = 0.47 [13, 141. Based on the results of Fig. 7b, and using this assumed transi- tion location, the critical n factor for run #79 is 6.5. The proposed hybrid method is certainly adequate for transition predictions, considering that the crit- ical n factor for the T2 transonic tunnel is between 7 and 8 [13, 141. The critical n factor for run #60 is not predicted since the transition for this run was triggered by ice crystals, a phenomenon which is not taken into account in this analysis.

5 Conclusion

The major contribution of this work is the imple- mentation of the spatial formulation and its original

The main idea in the proposed hybrid method is to inclusion in SCOLIC yielding a new hybrid method. b

R

join the advantages and strengths of the two for- mulations resulting in an effective method: (i) the global calculations are pelformed on the linear eigen- value problem of the temporal formulation since the order of the matrices involved is almost half the or- der of those involved during the global solution of the nonlinear eigenvalue problem associated to the spatial formulation; (ii) the selection of the most critical frequency is done using the temporal formu- lation since the frequency is part of the solution in this formulation, making the prediction of the fre- quency of the most amplified disturbance an easier task than in the context of the spatial formulation; (iii) the tracking of the instability during the n- factor calculations is conducted using the temporal formulation since it involves four parameters instead of the five parameters associated to the spatial for- mulation; ( i u ) the spatial growth rate used in the calculation of the n factor is obtained by a local so- lution of the linearized spatial eigenvalue problem eliminating reliance on Gaster’s relation.

The proposed hybrid method has been validated through comparisons with available numerical solu- tions and observed transition locations.

L-‘

Acknowledgement - ’ The support of the Natural Sciences and Engineering Research Council of Canada (NSERC), in the form of a R & D Grant with Bombardier Inc./Canadair is gratefully acknowledged. The authors would like to thank Dr. D. Arnal of ONERA/CERT who kindly provided the experimental pressure distribu- tions over the AS409.

References [l] Mack, L.M., “On the Stability of the Bound-

ary Layer on a Transonic Swept Wing”, A I A A Paper 79-0264, 1979.

[2] Malik, M.R. and Orszag, S.A., “Efficient Com- putation of the Stability of Three-Dimensional Compressible Boundary Layers”, A I A A Paper 81-1277, 1981.

[3] Cebeci, T., and Stewartson, K., “Stability and Transition in Three-Dimensional Flows”, A I A A Journal, Vol. 18, pp. 398-405, 1980.

[4] Arnal, D. and Juillen, J .C., “Three-Dimen- sional Transition Studies at ONERA/CERT”, A I A A Paper 87-1335, 1987.

v [5] Martinuzzi, R., Lamarre, F., and Paraschivoiu, I., “Natural Laminar Flow Airfoils for Swept

Wings in the Transonic Regime”, Final Report: Canadair N/D: C132, 1992.

[6] Kaups, K . and Cebeci, T. , “Compressible Lam- inar Boundary Layers with Suction on Swept and Tapered Wings”, Journal of Aircrnfl , Vol. 14, pp. 661-667, 1977.

[7] Mohsen, M., “Transition de la couche limite sur une aile en flkche en rdgime transsonique”, M.A.Sc. Thesis, Ecole Polytechnique de Mont- rGal, Montrhal, Canada, 1994.

(81 Moler, C.B., Stewart, W., “An Algorithm for Generalized Matrix Eigenvalue Problems”, S I A M Journal for Numerical Anolysis , Vol. 10, No. 2, p. 241, 1973.

[9] Bridges, T.J., and Morris, P.J., “Differential Eigenvalue Problems in which the Parameter Appears Non-Linearly”, Journol of Compula- lional Physics, Vol. 55, pp. 437-460, 1984.

[lo] Gaster, M., “A Note on the Relation be- tween Temporally-Increasing and Spatially- Increasing Disturbances in Hydrodynamic Sta- bility”, Journal of Fluid Mechanics, Vol. 14, pp. 222-224, 1962.

[ll] Nayfeh, A.H., “Stability of Three-Dimensional Boundary Layers”, A I A A Journal, Vol. 18, NO. 4, pp. 406-416, 1980.

[12] Archambaud, J.P., Payry, M.J., Seraudie, A., “Etude exphrimentale de la laminaritd sur I’aile AS409 jusqu’i des nombres de Reynolds de I’ordre de 14 millions dans la soufflerie T2”, Rapport technique de synthkse 33/5006-19, 1989.

[13] Cebeci, T., Chen, H.H., and Arnal, D., “Natu- ral Transition in Compressible Flows on Wings: Spatial Theory and Experiment”, A I A A Paper 94-0824, 1994.

1141 Niethammer, R., “Boundary-Layer Stability Computations Related to Laminar Flow Ex- periments at Low Temperatures”, Projet de fin d’itudes, Dhpartement d’htudes et de recherches en acirothermodynamique, Toulouse, July 1991.

(151 Masson, C., Martinuzzi, R., Langlois, M., Paraschivoiu, I., Tezok, F., “Transition Pre- diction Capabilities for Conical Wings in the Transonic Regime”, submitted to the Canadian Aeronautics and Space Journal.

9