linear instability of orthogonal compressible leading-edge...

Linear instability of orthogonal compressible

leading-edge boundary layer flow

E. M. Gennaro1,∗, D. Rodrıguez2,

†, M. A. F. Medeiros1 and V. Theofilis3,

‡

1 Sao Carlos School of Engineering, University of Sao Paulo

Av. Trabalhador Sao Carlense, 400, CEP-13566-590 Sao Carlos/SP, BRASIL2 Division of Engineering and Applied Science, California Institute of Technology, Pasadena CA, 91125 USA

3School of Aeronautics, Universidad Politecnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, SPAIN

Instability analysis of compressible orthogonal swept leading-edge boundary layer flowwas performed in the context of BiGlobal linear theory.1,2 An algorithm was developedexploiting the sparsity characteristics of the matrix discretizing the PDE-based eigenvalueproblem. This allowed use of the MUMPS sparse linear algebra package3 to obtain adirect solution of the linear systems associated with the Arnoldi iteration. The developedalgorithm was then applied to efficiently analyze the effect of compressibility on the stabilityof the swept leading-edge boundary layer and obtain neutral curves of this flow as a functionof the Mach number in the range 0 ≤ Ma ≤ 1. The present numerical results fully confirmedthe asymptotic theory results of Theofilis et al.4 Up to the maximum Mach number valuestudied, it was found that an increase of this parameter reduces the critical Reynoldsnumber and the range of the unstable spanwise wavenumbers.

I. Introduction

Investigation of linear instability mechanisms is essential for understanding the process of transitionfrom laminar to turbulent flow. Many studies over several decades have reported results in simple one-dimensional steady laminar basic flows, such as those governing boundary- and shear layers. However, mostflows of practical engineering significance remain unexplored. The reason is that the underlying basic stateof most practical flows depends on an inhomogeneous manner on more than one spatial direction and thecost of performing a complete parametric instability analysis can be formidable.

In any situation, when a basic flow is established, the Navier-Stokes equations can be written in terms ofdisturbance variables and linearized to study small-amplitude disturbances developing around this basic flow.If the latter depends in an inhomogeneous manner on two spatial directions, the disturbance can be consideredperiodic along the third homogeneous spatial direction, along which Fourier modes can be introduced. Anexponential dependence on time is assumed for the disturbances, and consequently the problem becomes atwo-dimensional, partial-derivative-based eigenvalue problem (EVP) where the eigenvalues are the complexfrequencies, and the eigenfunctions depend on the two non-homogeneous spatial directions. This approach isreferred to as BiGlobal instability analysis,1 in order to avoid confusion with the use of this term by Chomazet al.5 BiGlobal instability analysis has been shown to provide useful insight in flows over or through complexgeometries and is a promising path in devising theoretically founded flow control strategies; see Theofilis2

for a recent review.In principle, all these assumptions lead to a problem easier to solve than the direct numerical simulation

(DNS), but the large size of the discretized matrices makes the numerical solution challenging. The mosteffective techniques available for the direct solution of the generalized eigenproblem are based on subspace

∗Research Assistant, AIAA Member, Correspondence to: [email protected]†Postdoctoral Scholar, AIAA Member‡Research Professor, Associate Fellow AIAA

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American Institute of Aeronautics and Astronautics

6th AIAA Theoretical Fluid Mechanics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3751

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.

projection-iterative methods such as the Arnoldi iteration, which is one member of Krylov-subspace tech-niques.6 The Arnoldi method delivers a window of the eigenspectrum, but it favors the eigenvalues with thelargest modulus, thus inversion of the matrix is required in order to introduce an eigenvalue shift towardsthe most interesting part of eigenspectrum. One solution of this problem, within the context of matrixformation and direct inversion, has been discussed by Rodrıguez & Theofilis.7 The use of dense linear alge-bra algorithms in the latter work underlined the need for alternative matrix inversion methodologies. Thepresent contribution discusses one such approach, which exploits the sparsity of the matrix resulting fromdiscretization of the compressible BiGlobal EVP using spectral collocation. Alternative methodologies arepresented and discussed in depth elsewhere in this conference.8,9

From a physical point of view, instability of the flow near the leading edge of swept wings has a greatengineering significance. This kind of instability promotes the growth of disturbances which will be convecteddownstream, having direct influence on the transitional process from laminar to turbulent flow on the wingsurface and, in turn, will affect aerodynamic performance. Study of instability mechanisms in this flow canprovide a useful insight in the aerodynamic design of wings. Hall et al.10 were the first to study the linearstability of the incompressible swept attachment-line boundary layer, adopting the swept Hiemenz11 basicflow and the Gortler-Hammerlin12,13 similarity model for the perturbations. Lin and Malik14 used a two-dimensional representation of the perturbations around the Hiemenz flow. They went on to include curvatureand concluded that its effect is stabilizing.15 Theofilis et al.16 performed temporal BiGlobal analysis of theincompressible swept Hiemenz flow and proposed a polynomial model to describe the chordwise dependence ofthe amplitude functions, reducing the cost of performing global analysis without loss of physical informationin the linear regime. Study of compressibility effects on the leading edge boundary layer was first introducedin a global analysis context by Theofilis et al.4 who solved a dense BiGlobal eigenvalue problem and presentedan asymptotic theory along the lines of their earlier incompressible work. Instability in the vicinity of aswept cylinder in compressible flow has recently been addressed by Collis and Lele17 and Schmid and co-workers.18,19 Unlike these works, in which large-scale computation has been used to establish the basicstate and study the subsequent instability development, the present contribution retains the compressibleleading-edge boundary layer basic flow model and, exploiting the newly-developed efficient computationalapproach, completes the parametric study initiated by Theofilis et al.4 Section II introduces the basic flowanalyzed and the theoretical concept employed. Section III discusses some details and presents validations ofthe newly-developed algorithm, while section IV contains the main results obtained up to the present time.

II. Theory

II.A. Basic Flow

The flow in the vicinity of the leading-edge of a swept wing is approximated in this work as a stagnation-lineflow, with constant, non-zero free-stream velocity along the attachment line. Compressible flow is consideredhere (as was done in Ref. 4), as opposed to previous works10,14,16 in which the flow was assumed to beincompressible. Let x and y be the tangential (chordwise) and wall normal coordinates, z the (spanwise)coordinate along the stagnation line and U, V and W the velocity components along the three directions. Asimilarity solution was proposed by Cohen and Reshotko20 (see also Mack21), for which the flow in the x− yplane is obtained in the form of a two-dimensional stream-function. If ν is the kinematic viscosity and S alocal strain rate defined as S = (dUe/dx)x=0 (where the subscript e refers to the boundary-layer edge), alength scale can be defined as ∆ = (ν/S)1/2. The similarity variables ξ and η are then defined as x/∆ andy/∆, respectively. The stream-function for the similarity solution takes the form

ψ(ξ, η) = νξf(η). (1)

It is assumed that the spanwise velocity component depends solely on the normal direction η. The localMach number and Reynolds number are defined using the inviscid spanwise velocity We, the sound speedae, kinematic viscosity νe

a at the far-field and the strain rate S:

Ma = We/ae, Re = We∆/νe, (2)

and the Prandtl number is taken to be equal to 0.72.aA dependence of the viscosity with temperature is permitted

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The compressible, Reynolds-independent laminar boundary layer equations can be obtained by using theIllingworth-Stewartson transformation defined by:

X =∫ x

0

λpecep0c0

dx , Y =ae

a0

∫ y

0

λρ

ρ0dy , (3)

where p is the pressure and subscript o indicates free-stream stagnation values.Taking into account equations (1) and (3), the velocity and temperature profiles can be obtained as

solutions of the following set of ordinary differential equations:20,21

V ′ = −U + VT ′

T(4)

U ′′ =1µ

(U2 + V U ′

T− 1− ∂µ

∂TT ′U ′

)(5)

W ′′ =1µ

(VW ′

T− ∂µ

∂TT ′W ′

)(6)

T ′′ =Pr

µ

(− ∂µ∂T

T ′2

Pr+T ′V

T− (γ − 1)Ma2µW ′2

)(7)

(8)

complemented with the boundary conditions:

U(0) = W (0) = 0, V (0) = 0 (9)U(∞) = W (∞) = T (∞) = 1. (10)

The system (4-10) is solved using a shooting method and the velocity and temperature profiles may beseen in Figure 1. Table 1 presents the shear-stresses and temperature at the wall for different Mach numbers.

Table 1. Dependence of basic flow shear-stress and wall-temperature on Ma

Ma U0′(0) W0

′(0) T0(0)10−5 1.232588 0.570465 1.0000000.25 1.228483 0.566095 1.0107460.50 1.216689 0.553511 1.0429900.75 1.198610 0.534132 1.0967411.00 1.176141 0.509881 1.1720162.00 1.076630 0.399360 1.6886293.00 1.000514 0.308888 2.550336

II.B. BiGlobal linear stability analysis

The basic flow described in the previous subsection is dependent on two spatial directions (x and y) andhomogeneous in the third, spanwise direction z. In this case, the BiGlobal linear stability analysis is adequatein order to study the stability of the flow which impinges a solid surface. The flow field is decomposedaccording to

q(x, y, z, t) = Q(x, y) + εq(x, y, z, t), ε� 1, (11)

into a basic flow Q = (U , V , W , T , P )T , and disturbance flow q = (u, v, w, θ, p)T . The basic flow is determinedby the solution of the system (4-10), through U = xU(y), V = V (y), W = W (y) and T = T (y).

Introducing the decomposition (11) into the compressible continuity, Navier-Stokes and energy equationsand linearizing about Q, we obtain a system of partial-differential equations for the temporal evolution ofthe disturbances. The eigenmodes are introduced as

q(x, y, z, t) = q(x, y) exp(iβz − ωt) + c.c., (12)

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where c.c. denotes complex conjugate, ω is a complex frequency and β is a real wavenumber related to aspanwise periodicity length defined by Lz = 2π/β. A positive value of the amplification rate ωi indicatesgrowth and the mode is temporally unstable, while ωi < 0 indicates damping, and the mode is stable. Theproblem can be rewritten as a two-dimensional, partial-derivative-based eigenvalue problem, that written inmatrix notation takes the form:

Lq = ωRq. (13)

The complete description of L and R are shown in detail in Theofilis and Colonius22 and Gennaro etal.23 The above eigenvalue problem is complemented here with the following set of boundary conditions:

• no-slip boundary conditions are imposed to the velocity components at the wall, along with ∂p∂y = 0;

• a fast decay is assumed for the disturbances in the normal direction and Dirichlet condition is imposedto all the quantities at the far field;

• linear extrapolation from the interior of the computational domain was used for all the fluid variablesat x = ±Lx.

III. Methodology

The matrix eigenvalue problem (13) is discretized using Chebyshev-Gauss-Lobatto points in both thex and y directions. In order to better resolve the boundary layer, an appropriate mesh stretching is usedfor the y direction. The generalized matrix eigenvalue problem resulting from the discretization of thelinear operators L and R is solved using a shift-and-invert implementation of the Arnoldi algorithm. Thegeneralized problem is transformed into the standard eigenvalue problem

Lq = µq, (14)

where L = (L − σR)−1 R, µ = (ω−σ)−1 and σ is a shift parameter. The most computationally demandingpart of the algorithm is storing and performing a LU decomposition of the matrix L − σR.

The matrices resulting from the discretization of the operators L and R contain a large number ofzero elements (see figure 2), that suggest the use of sparse techniques in order to reduce both the memoryrequirements and the computational time required in the LU decomposition. Consequently, two differentimplementations of the algorithm23 were tested: a dense algebra version using the LAPACK routine ZGETRFfor the LU decomposition of a general matrix, and the sparse algebra library MUMPS (MUltifrontal MassivelyParallel Solver3).

In order to obtain the optimal implementation of the sparse version, the manner in which MUMPS storesand works with sparse matrices must be understood. The linear system is solved by MUMPS on three steps:analysis, factorization and solution. The original matrix is mapped during the analysis step that returnsestimates of the memory needed for the matrix storing the LU form. This estimate considers the operationsrequired to perform a LU decomposition

a(i, j) := a(i, j)− (a(i, k) ∗ a(k, j))/a(k, k), (15)

so that if a(i, k), a(k, j) and a(k, k) are nonzero, the output a(i, j) will be nonzero as well even if it was equalto zero in the original matrix. The number of these “fill-in coefficients” depends on many aspects, but hasa strong influence of the order in which the pivots are eliminated by the ordering method. For this reason,at the end of the analysis phase, MUMPS returns an estimate of the memory needed. If a large numberof fill-in coefficients appears during the LU factorization, the benefits of using sparse techniques could beremarkably reduced. Fill-in reducing ordering for sparse matrices within MUMPS is provided in the presentimplementation by the package METIS (a set of serial programs for partitioning graphs, partitioning finiteelement meshes, and producing fill reducing orderings for sparse matrices). Using the BiGlobal stabilityeigenvalue problem as an example for the reduction of the memory reduction associated with using sparsetechniques, for a problem of resolution 60×60 the number of nonzero entries is 3.2×106, a number well belowthe estimated number of non-zero entries by MUMPS, which was 2.5× 107. The numerical factorization isthen performed on a sequence of dense factorizations and the factor matrices are stored to be used during the

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solution step. The codes were run on a desktop computer featuring a 3.06GHZ Intel Core 2 Duo processor8Gb 1067 MHZ DDR3 of RAM.

Table 2 compares the CPU time and memory requirements for the solution of the BiGlobal eigenvalueproblem when using the dense (LAPACK) and sparse (MUMPS) implementations, for different problem sizes.A important reduction in the required memory is observed, that increments with the resolution employed inthe discretization of the problem so that the problem with the higher resolution considered, 60× 60, cannotbe solved in the present computer using dense linear algebra. In addition, a significative reduction in theCPU-time is achieved for the sparse algebra implementation.

Table 2. Comparison of the CPU time and memory required for the LU factorization, using MUMPS (using METISordering) and ZGETRF.

LU decomposition MemoryResolution ZGETRF (s) MUMPS (s) ZGETRF (Gb) MUMPS (Gb)

10× 10 0.1 0.1 0.01 0.005820× 20 2.3 1.2 0.17 0.09330× 30 23.6 13.1 0.73 0.4140× 40 124.6 75.6 2.22 1.350× 50 456.3 262.1 5.04 3.360× 60 1342.8 777.1 10.32 6.7270× 70 - 2092.7 18.92 13.6

IV. Results

IV.A. Incompressible swept attachment-line

The linear stability of the swept attachment-line in the incompressible limit is addressed here in orderto validate the present algorithm. The Gortler-Hammerlin (GH) model of linear perturbations was firstadopted by Hall et al.,10 and then recovered in the context of a global stability analysis by Lin and Malik14

and Theofilis et al.16 In order to reproduce their results with the present compressible code, the very lowMach number Ma = 0.02 was used. Table 3 shows the convergence history of the leading GH mode andthe first antisymmetric eigenmode (A1) corresponding to the present computations, and compared with thereference values in the literature. Note that the scaled eigenvalues c = ω/β are displayed in the table, insteadof ω. The Reynolds number and spanwise wavenumber considered are, respectively, Re = 800 and β = 0.255.

IV.B. Compressible swept attachment-line

The global stability analysis of a compressible swept attachment-line flow presented in Theofilis et al.4 isrevisited here. Theofilis et al. performed a BiGlobal stability analysis and an asymptotic expansion of thelocal stability equations for two Mach numbers, Ma = 0.5 and 0.9, Reynolds number Re = 800 and arange of spanwise wavenumbers β. In their computations, a consistent disagreement existed between theresults delivered by the two approaches. The same combination of parameters is considered here. Tables 4and 5 summarize the grid refinement and domain size convergence histories. Figure 3 show the amplitudefunctions corresponding to the leading eigenmode at Ma = 0.9, Re = 800 and β = 0.19, used in theconvergence analyses. The spatial structure is identical with the first of the polynomial modes recoveredin incompressible flow. The simple linear variation of the eigenfunctions with the chordwise direction x,along with the linear extrapolation boundary conditions imposed to the eigenvalue problem, explains theconvergence of the eigenvalue for a small number of discretization points. More interestingly, it also explainswhy the eigenvalues recovered do not experience substantial changes when the domain length is increased,in contrast with other BiGlobal stability analysis of unbounded flows.

On account of the results of the convergence checks, a domain length of [−100, 100] and a resolution of60 × 60 collocation points were used in the subsequent parametric study varying the spanwise wavenum-ber β. The amplification rates are plotted in figure 4, superimposed to the reference4 results. A perfectagreement between the predictions of the asymptotic analyses and the present BiGlobal stability analyses

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Table 3. Grid refinement history in the numerical solution for incompressible attachment-line boundary flow at Re = 800and β = 0.255.

Gortler-Hammerlin mode A1 modeResolution cr ci cr ci

08× 48 0.35844146 0.0058564 0.3573547 0.004354916× 48 0.35844146 0.0058640 0.3580007 0.004299724× 48 0.35844146 0.0058564 0.3458880 0.004391232× 48 0.35844146 0.0058564 0.3580521 0.004146148× 48 0.35844146 0.0058564 0.3579257 0.004095660× 48 0.35844151 0.0058565 0.3579401 0.004041260× 60 0.35844147 0.0058555 0.3579401 0.0040403

Lin and Malik14 and Theofilis et al.16 0.35840982 0.0058325 0.3579170 0.0040988

Table 4. Grid refinement history in the numerical solution for compressible attachment-line boundary flow at Re = 800,Ma = 0.90 and β = 0.19 for Gortler-Hammerlin mode.

Resolution ωr ωi

20× 20 0.07648758 0.0012266630× 30 0.07647268 0.0011875040× 40 0.07647143 0.0011902750× 50 0.07647193 0.0011894760× 60 0.07647181 0.0011894670× 70 0.07647184 0.00118944

Table 5. Independence with respect to the chosen domain at Re = 800, Ma = 0.90, β = 0.19 for resolution 60× 60.

Domain ωr ωi

[−25, 25] 0.07652226 0.00119659[−50, 50] 0.07644808 0.00115308

[−100, 100] 0.07647181 0.00118946[−150, 150] 0.07647105 0.00118941

is attained, indicating that the previous numerical solutions of Theofilis et al. were under-resolved, due tomemory limitations. The agreement with the analytical results further underlines the quality of the presentimplementation in terms of numerical accuracy, as well as in terms of computational efficiency.

The combined effect of Mach and Reynolds numbers on the stability properties of the swept attachment-line flow is studied now. Figure 5 presents the temporal amplification rate obtained at three differentReynolds numbers (Re = 585, 800 and 1500) as a function of β, for the range of subsonic Mach numbersMa = [0.02 − 0.9]. For unstable Reynolds numbers (figures 5a and 5b) the compressibility stabilizes theleading eigenmode and reduces the dominant wavenumber β, so that it becomes less amplified and with asmaller spanwise characteristic length. This stabilization is more pronounced with higher Reynolds numbers.This is the usual behavior on linear stability associated with compressibility. However, in the close vicinity ofthe critical Reynolds number, the effect of compressibility is apparently inverted. Figure 5c, shows that forRe = 585 the increase of Mach number destabilizes the leading eigenmode. The effect of the compressibilityon the critical instability conditions will be discussed further below.

Figure 6 shows the neutral stability curves for the range of Mach numbers considered. Again, it can beseen that increasing the Mach number reduces the range of unstable spanwise wavenumbers for all Reynoldsnumbers. The neutral curves also show how the critical conditions change on account of the Mach number.The Reynolds number and wavenumber at the critical conditions for each Mach number, denoted by Recrit

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Table 6. Critical parameters

Ma 0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Recrit 583.1 582.7 581.6 579.8 577.5 574.8 571.9 569.0 566.4 564.5βcrit 0.2878 0.2868 0.2836 0.2784 0.2713 0.2625 0.2524 0.2411 0.2290 0.2164

and βcrit respectively, are shown in table 6 and figure 7. A significative result is that increasing the Machnumber reduces the critical Reynolds number from Rec ≈ 583 at Ma = 0.02 to Rec ≈ 565 at Ma = 0.9.

V. Conclusions

An algorithm for the numerical solution of the compressible BiGlobal linear eigenvalue problem wasdeveloped. It features matrix storage and inversion in sparse serial format and was shown to provide O(35%)savings in both CPU time and memory required, when compared with dense matrix inversion as, for example,used in earlier work of the same vein.7

The code developed was utilized to revisit and complete the parametric analysis of global instability incompressible swept Hiemenz flow that where initiated in Theofilis et al.4 The results agree well with of theasymptotic theory addressed from Theofilis et al.4 The results showed that the effects of the compressibilityon the leading-edge boundary layer was to reduce the critical Reynolds number and the range of the unstablespanwise wavenumbers. Further improvements in performance may be obtained by parallelization, which ispresently being considered.

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Acknowledgments

The authors acknowledge the financial support of the National Council of Scientific and Technolog-ical Development - CNPq/Brazil, CAPES and the Spanish Ministry of Science and Innovation throughGrant MICINN-TRA2009-13648: “Metodologias computacionales para la prediccion de inestabilidades glob-ales hidrodinamicas y aeroacusticas de flujos complejos”. Computations were performed on the FinisTerraesupercomputer of the Centro de Supercomputacion de Galicia.

References

1Theofilis, V., “Advances in global linear instability of nonparallel and three-dimensional flows,” Prog. Aero. Sciences,Vol. 39 (4), 2003, pp. 249–315.

2Theofilis, V., “Global linear instability,” Annu. Rev. Fluid Mech., Vol. 43, 2011, pp. 319–352.3P. R. Amestoy, I. S. Duff, J. K. and L’Excellent, J. Y., “A fully asynchronous multifrontal solver using distributed

dynamic scheduling,” SIAM Journal of Matrix Analysis and Applications., Vol. 1, 2001, pp. 15–41.4Theofilis, V., Fedorov, A. V., and Collis, S. S., “Leading-edge boundary layer flow - Prandtl’s vision, current developments

and future perspectives,” IUTAM Symposium on One Hundred Years of Boundary Layer Research, Vol. 129, 2004.5Chomaz, J. M., Huerre, P., and Redekopp, L. G., “Bifurcation to local and global modes in spatially developing flows,”

Phys. Re. Lett., Vol. 60, 1988, pp. 25–28.6Saad, Y., Iterative methods for sparse linear systems, SIAM, 2000.7Rodrıguez, D. and Theofilis, V., “Massively parallel numerical solution of the BiGlobal linear instability eigenvalue

problem using dense linear algebra,” AIAA J., Vol. 47, No. 10, 2009, pp. 2449–2459.8Paredes, P., Theofilis, V., Rodrıguez, D., and Tendero, J. A., “The PSE-3D instability analysis methodology for flows

depending strongly on two and weakly on the third spatial dimension,” 6th Theoretical Fluid Mechanics Conference, AmericanInstitute of Aeronautics and Astronautics, Reston, VA (submitted for publication), 2011.

9Theofilis, V. and Clainche, S. L., “Global linear instability at the dawn of its 4th decade: a list of challenges. A practicalguide on how to contain the euphoria and avoid the oversell,” 6th Theoretical Fluid Mechanics Conference, American Instituteof Aeronautics and Astronautics, Reston, VA (submitted for publication), 2011.

10Hall, P., Malik, M. R., and Poll, D. I. A., “On the stability of an infinitive swept attachment line boundary layer,” Proc.R. Soc. Lond. A, Vol. 395, 1984, pp. 229–245.

11Hiemenz, K., “Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder,”Dingl. Polytechn. J., Vol. 326, 1911, pp. 321–324, Thesis, Gottingen.

12Gortler, H., “Dreidimensionale Instabilitat der ebenen Staupunktstromung gegenuber wirbelartigen Storungen,” 50 JahreGrenzschichtforschung, edited by H. Gortler and W. Tollmien, Vieweg und Sohn, 1955, pp. 304–314.

13Hammerlin, H., “Zur Instabilitatstheorie der ebenen Staupunktstromung, 50 Jahre Grenzschichtforschung,” Vieweg undSohn, 1955, pp. 315–327.

14Lin, R. S. and Malik, M. R., “On the stability of attachment-line boundary layers. Part 1. The incompressible sweptHiemenz flow,” J. Fluid Mech., Vol. 311, 1996, pp. 239–255.

15Lin, R. S. and Malik, M. R., “On the stability of attachment-line boundary layers. Part 2. The effect of leading-edgecurvature,” J. Fluid Mech., Vol. 333, 1997, pp. 125–137.

16Theofilis, V., Fedorov, A., Obrist, D., and Dallmann, U. C., “The extended Gortler-Hammerlin model for linear instabilityof three-dimensional incompressible swept attachment-line boundary layer flow,” J. Fluid Mech., Vol. 487, 2003, pp. 271–313.

17Collis, S. S. and Lele, S. K., “Receptivity to surface roughness near a swept leading edge,” J. Fluid Mech., Vol. 380, 1999,pp. 141–168.

18Mack, C. J., Schmid, P. J., and Sesterhenn, J. L., “Global stability of swept flow around a parabolic body: connectingattachment-line and crossflow modes,” J. Fluid Mech., Vol. 611, 2008, pp. 205 – 214.

19Mack, C. J. and Schmid, P. J., “A preconditioned Krylov technique for global hydrodynamic stability analysis of large-scale compressible flows,” J. Comp. Phys., Vol. 229, No. 3, 2010, pp. 541 – 560.

20Cohen, C. B. and Reshotko, E., “Similar solutions for the compressible laminar boundary layer with heat transfer andpressure gradient,” NACA. Report , Vol. 1293, 1956.

21Mack, L. M., “Boundary layer linear stability theory,” AGARD–R–709 , 1984, pp. 3.1–3.81.22Theofilis, V. and Colonius, T., “Three-dimensional Instabilities of Compressible Flow Over Open Cavities: Direct Solution

of the BiGlobal Eigenvalue Problem,” AIAA Paper , , No. 2004-2544, 2004.23Gennaro, E. M., Perez, J. M., Theofilis, V., and Medeiros, M. A. F., “Global instability of compressible leading-edge flow

by sparse matrix techniques and parallelization,” Int. J. Numer. Meth. Fluids, Vol. (submitted for publication), 2011.

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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6

η

(a)

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0 1 2 3 4 5 6

T0

η

Ma = 0.02

Ma = 0.10

Ma = 0.20

Ma = 0.30

Ma = 0.40

Ma = 0.50

Ma = 0.60

Ma = 0.70

Ma = 0.80

Ma = 0.90

(b)

Figure 1. The base solutions to compressible flow: (a) the streamwise (solid lines) and spanwise (dashed lines) velocitycomponents at Mach number Ma = 0.02 (black color) and Ma = 0.90 (red color). (b) temperature profiles at differentMach numbers.

(a) (b)

Figure 2. The matrix forms L and R of the compressible BiGlobal EVP (13) for Nx = 12 and Ny = 12.

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−100

−50

0

50

100

0

5

10

15−1

−0.5

0

0.5

1

xy

(a) Real(u)

−100

−50

0

50

100

0

10

20

30−0.1

0

0.1

0.2

0.3

0.4

xy

(b) Real(v)

−100

−50

0

50

100

0

10

20

30−4

−3

−2

−1

0

1

xy

(c) Real(w)

−100

−50

0

50

100

0

10

20

30−1

−0.8

−0.6

−0.4

−0.2

0

xy

(d) Real(p)

Figure 3. The real part of the eigenfunctions at Ma = 0.9, Re = 800 and β = 0.19 corresponding to mode ω =0.07647181 + i0.00118946.

c i

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32-0.002

0

0.002

0.004

0.006

0.008

(a) Ma = 0.50

c i

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3-0.0075

-0.005

-0.0025

0

0.0025

0.005

0.0075

0.01

(b) Ma = 0.90

Figure 4. Dependence of ci on β for mode GH at (a) Ma = 0.5 and (b) Ma = 0.9, Re = 800. Solid line obtained byasymptotic analysis and square symbols by BiGlobal analysis from (Theofilis et al.4). Circle symbols are results of thispresent paper.

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0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.05 0.1 0.15 0.2 0.25 0.3 0.35

ωι

β

Ma=0.02

Ma=0.10

Ma=0.20

Ma=0.30

Ma=0.40

Ma=0.50

Ma=0.60

Ma=0.70

Ma=0.80

Ma=0.90

(a)

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.05 0.1 0.15 0.2 0.25 0.3 0.35

ωι

β

Ma=0.02

Ma=0.10

Ma=0.20

Ma=0.30

Ma=0.40

Ma=0.50

Ma=0.60

Ma=0.70

Ma=0.80

Ma=0.90

(b)

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0.05 0.1 0.15 0.2 0.25 0.3 0.35

ωι

β

Ma=0.02

Ma=0.10

Ma=0.20

Ma=0.30

Ma=0.40

Ma=0.50

Ma=0.60

Ma=0.70

Ma=0.80

Ma=0.90

(c)

Figure 5. Temporal amplification rate as a function of the β at (a) Re = 1500, (b) Re = 800 and at Re = 585.

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0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

500 600 700 800 900 1000 1100 1200 1300 1400 1500

β

Re

Ma=0.02Ma=0.10Ma=0.20Ma=0.30Ma=0.40Ma=0.50Ma=0.60Ma=0.70Ma=0.80Ma=0.90

Figure 6. Neutral curves of orthogonal compressible leading-edge boundary layer flow obtained by BiGlobal analysis.

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

Ma

(a)

564

566

568

570

572

574

576

578

580

582

584

0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Re

Ma

(b)

Figure 7. Spanwise wavenumber β (a) and Reynolds number Re (b) at the critical conditions for different Mach numbers,extracted from the neutral curve.

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linear instability of orthogonal compressible leading-edge...

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