amr_066_02_024804

Matthew P. JuniperEngineering Department,

Cambridge University,

Trumpington Street,

Cambridge CB2 1PZ, UK

e-mail: [email protected]

Ardeshir HanifiSwedish Defence Research Agency,

FOI, SE-164 90 Stockholm, Sweden;

Linne FLOW Centre,

KTH Royal Institute of Technology,

Stockholm SE-100 44, Sweden


Vassilios TheofilisSchool of Aeronautics,

Universidad Politecnica de Madrid,

Plaza Cardenal Cisneros 3,

Madrid E-28040, Spain


Modal Stability TheoryLecture notes from the FLOW-NORDITA Summer Schoolon Advanced Instability Methods for Complex Flows,Stockholm, Sweden, 2013

This article contains a review of modal stability theory. It covers local stability analysisof parallel flows including temporal stability, spatial stability, phase velocity, groupvelocity, spatio-temporal stability, the linearized NavierStokes equations, the OrrSommerfeld equation, the Rayleigh equation, the BriggsBers criterion, Poiseuille flow,free shear flows, and secondary modal instability. It also covers the parabolized stabilityequation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigen-value problems, spectral collocation methods, and other numerical solution methods.Computer codes are provided for tutorials described in the article. These tutorials coverthe main topics of the article and can be adapted to form the basis of research codes.[DOI: 10.1115/1.4026604]

1 Introduction

Traditionally, the instability of flows to small amplitude pertur-bations has been analyzed using the modal approach. Given an op-erator that describes the evolution of small perturbations, thisapproach considers the temporal or spatial development of indi-vidual eigenmodes of that operator. The resulting linear stabilitytheory (LST) relies on the decomposition of any flow quantity qinto a steady part q, which is usually called the basic state or baseflow,1 and an unsteady part ~q

qx; t qx e~qx; t (1)

where x is the space coordinate vector, t is time, and e 1 is asmall amplitude. Substituting this ansatz into the NavierStokesequations, subtracting the equations for the steady flow, and drop-ping the terms in e2 yields the linearized perturbation equations,referred to as the linearized NavierStokes equationsLNSE. Inincompressible flow, the LNSE take the form

@~u

@t ~u ru u r~u r~p 1

Rer2~u (2)

r ~u 0 (3)

Once a basic state q has been provided, the LNSE may be solvedas an initial-boundary-value problem and are valid for any smallamplitude perturbation, both modal (to be defined shortly) andnonmodal, discussed in the article by Schmid and Brandt [1] inthis volume.

It should be mentioned that the choice/computation of the basicstate q around which the linearization is performed is not alwaysan easy task. This is especially cumbersome for flows that areglobally/absolutely unstable. The choices are usually either thesteady solution of the NavierStokes equations or the timeaverage of the flow field. A classical case is the flow behind acylinder at moderate to high Reynolds numbers, for which resultsbased on the steady solution have the wrong frequency of unstablemodes, while results based on the time-averaged flow have similarfrequencies to those observed experimentally [2]. The main

problem in such cases is that it is difficult to predict the character-istics of instability waves because the time average of the flowcan be obtained only when the flow field is measured/computedwith perturbations being present. For examples of such flow casessee Sipp and Lebedev [3]. A numerical approach for computationof the steady state for strongly unstable flows, when using time-stepping solvers, is so-called selective frequency damping(Akervik et al. [4]). Here, the amplitude of the solution corre-sponding to the strong instability is damped through applicationof an appropriate forcing. Alternatively, Newton methods may beused to obtain the steady state solution.

In order to assess whether a given basic state is stable or unsta-ble to an arbitrary small-amplitude perturbation, one can eithersolve the LNSE or perform a direct numerical simulation of thefull nonlinear equations of motion. Both of these are very compu-tationally expensive, however, so this article describes somecheaper alternatives, which involve LST. In the absence ofexperimental results to compare with those from LST, numericalsolutions of the LNSE or a direct numerical simulations (DNS)of small-amplitude perturbations superposed upon a givenbasic state should be obtained [57]. This checks the LSTresults and, in the case of DNS, also shows how nonlinear pertur-bations develop. Further discussion of this point may be found inSec. 4.4.

Mathematical expediency and the lack of sufficient computingpower during the last century dictated drastic simplifications tothe form of perturbations, ~qx; t in Eq. (1) that were studied byLST. These limited LST to flows that are homogeneous in two outof three spatial directions. Classic examples include free and con-fined shear flows, such as mixing layer flows and planar channelflows. In these cases, the perturbations are expanded in terms oftheir Laplace transform in time and Fourier transform in one ortwo spatial dimensions. The stability is determined separately foreach mode, giving rise to the term modal LST. Explicitly, the per-turbation vector ~q in Eq. (1) takes the form

~q q^eiH (4)

where q^ and H are the amplitude and phase functions of the linearperturbations, respectively.

In recent decades, efficient algorithms for the solution of multi-dimensional eigenvalue problems, as well as the wide availabilityof appropriate computing hardware, have allowed modal stability

Manuscript received August 11, 2013; final manuscript received January 20,2014; published online March 25, 2014. Assoc. Editor: Gianluca Iaccarino.

1Not to be confused with the term mean flow/state, which is reserved for timeand/or space-averaged turbulent flow.

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analysis to be performed on any laminar steady flow. When start-ing a new flow stability problem, the first consideration is thenumber of homogeneous directions in the basic state. Table 1shows the underlying assumptions about the homogeneity of thebasic state q, the dimensionality of the perturbations ~q, and theresulting framework and associated nomenclature for LST.

At the strongest level of approximation, the basic state is con-sidered homogeneous along two spatial directions and then theparallel flow approximation is made. In this approximation, thewall-normal velocity component and the streamwise derivativesof the mean flow are assumed to be negligible. One can thenassume that [810]

~qx; t q^x2 expfiax1 bx3 xtg (5)

Inserting the above ansatz into the perturbation Eqs. (2) and (3)leads to a one-dimensional generalized eigenvalue problem.Details are given in Sec. 2.5. In the past century, LST was per-formed mainly on flows that satisfied this ansatz, either exactly(e.g., planeor HagenPoiseuille flow) or approximately (e.g., theflat plate boundary layer).

When a flow varies slowly in one spatial direction, say x1, aWKBJ type of approach can be considered. The ansatz (5) can bereformulated as

~qx; t q^x1; x2 exp ix1

x0

ax0dx0 bx3 xt

(6)

where streamwise variations of q^x1; x2 and ax scale withRe1. This approximation results in a set of local equations suchas Eq. (5), which can either be solved as 1D eigenvalue problemsin a local stability analysis, Sec. 2.5, or as a set of PSE [11], whichwill be discussed in Sec. 4. The PSE is typically used to studyinstability in the important class of the (weakly nonparallel) flatplate boundary layer flow, and can also account for curvatureeffects and nonlinear perturbation development.

When the basic flow is strongly dependent on two spatial direc-tions but homogeneous in the third, the following anstaz isappropriate:

~qx; t q^x2; x3 expfiax1 xtg (7)

substitution of which into the LNSE results in a two-dimensional(so-called BiGlobal) eigenvalue problem; this will be discussed inSec. 5. BiGlobal LST can be used to analyze the stability of flowssuch as that over a cylinder, a step, or an open cavity, all of whichmay be homogeneous along the span. It can also be used to studyaxially homogeneous flows in ducts of arbitrary cross section, iso-lated model vortices, and systems of infinitely long model vortices.

When the base state depends strongly on two spatial directionsand weakly on the third, an extension of the classic PSE concept,labeled PSE-3D, may be used following the ansatz:

~qx; t q^x1; x2; x3 exp ix1

x0

ax0dx0 xt

(8)

By contrast to the PSE, which advances a perturbation with aone-dimensional perturbation depending on x2 along the stream-wise spatial direction x1, PSE-3D advances a two-dimensionalperturbation defined on the plane (x2, x3) along the direction nor-mal to the plane x1.

Finally, when no homogeneity assumption is permissible on thebasic flow, one has to resort to addressing the three-dimensionaleigenvalue problem resulting from substituting

~qx; t q^x1; x2; x3 expfixtg (9)

into the LNSE. The related context is referred to as TriGlobalLST. The areas of BiGlobal, PSE-3D, and TriGlobal modal linearstability theories, collectively referred to as global LST, arerapidly expanding; see Ref. [12] for a recent review. The presentarticle presents a common theoretical foundation for all of thelocal, nonlocal, and global frameworks shown in Table 1.

Section 2 introduces local stability analysis and terminologysuch as temporal instability, spatial instability, convective insta-bility, and absolute instability. The first tutorials in this sectionfocus on flows with piecewise-linear and top hat velocity pro-files, whose governing equations can be solved without numeri-cal methods. Later tutorials focus on parabolic and tanh velocityprofiles, whose stability requires the use of simple numericalmethods, which are detailed in the Appendix. Section 3 introdu-ces secondary instabilities, which occur on top of the primaryinstabilities investigated in Sec. 2. Section 4 introduces the para-bolized stability equations, which allow nonlocal and nonlineareffects to be included in slowly developing flows. Section 5introduces two-dimensional problems in linear stability. Section6 describes numerical solution methods for all types of linearstability problems.

2 Local Stability Analysis

Local stability analysis concerns either parallel flows (5) or, viathe WKBJ approach, flows that vary slowly in one spatial direc-tion (6). Many of the concepts in local stability analysis, such asconvective and absolute instability, are used to provide physicalinsight into the behavior found in global stability analysis. Inglobally unstable flows, local analysis is, therefore, increasinglyused as a diagnostic tool, rather than as a predictive tool. In glob-ally stable flows, however, local stability analysis can be moreuseful than global stability analysis, as described in Sec. 5. Thebasic concepts of local stability analysis are introduced in this sec-tion. These are then applied to a simple class of unstable flows inthe MATLAB tutorials.

2.1 Temporal Local Stability. In temporal linear stabilityanalysis, the quantities a and b appearing in Eqs. (5)(7) arereal wavenumber parameters, related to periodicity wavelengths,Lx1 and Lx3 , along the x1 and x3 directions, respectively, bya 2p=Lx1 and b 2p=Lx3 . The eigenvalue sought is x, which isin general complex. This approach has been used for well over acentury and can be found in many classic texts [9,13]. The equiva-lent formulation of linear two-dimensional waves

Table 1 Classification of modal linear stability theories

Denomination Basic state assumption Amplitude function Phase function H

Global TriGlobal qx1; x2; x3 q^x1; x2; x3 xtPSE-3D @1q @2q; @3q qx1; x2; x3 q^x1; x2; x3

x1ax0dx0 xt

BiGlobal @1q 0 qx2; x3 q^x2; x3 ax1xtNonlocal PSE @1q @2q; @3q 0 qx1; x2 q^x1; x2

x1ax0dx0 bx3 xt

Local OSE @1q @3q 0 qx2 q^x2 ax1bx3xt

024804-2 / Vol. 66, MARCH 2014 Transactions of the ASME


~q q^x2eiax1ct c:c: q^x2eiax1cricit c:c: (10)

introduces the complex eigenvalue c, the real part of which maybe interpreted as the phase speed, while the imaginary part is theamplification or the damping rate of the perturbation. (The com-plex conjugate is labeled c.c.)

A perturbation may initially be thought of as a linear superposi-tion of monochromatic waves that develop in streamwise (andspanwise, if b 6 0) periodic boxes of length Lx1 2p=a (andwidth Lx3 2p=b). Each wave travels on the base flow at its ownconstant phase speed cr and grows or attenuates at its own rateeacit. As time passes, just one of these waves dominates. This isthe leading eigenmode and its calculation is the central part ofinstability analysis.

A complete instability analysis involves varying the Reynoldsand wavenumber parameters and identifying the frequency andamplification/damping rate of the leading eigenmode. There aretwo particularly interesting curves in the (Re,a) plane. The first isthe neutral curve, on which xi 0, which defines the boundary ofstability/instability. The second is the maximum values of xi atfixed Re or fixed a. Both curves may be used to identify linearinstability characteristics in experiments or simulations.

2.2 Spatial Local Stability. The assumption of a constantstreamwise wavenumber may be relaxed and solutions of theLNSE may be sought for a constant real monochromatic fre-quency x, treating a as the eigenvalue. From this point of view,linear perturbations are written as

~q q^x2eiariaix1bx3xt c:c: (11)

ai being the (spatial) amplification or damping rate of the wave,while its wavenumber is ar. This instability analysis framework ispreferred by experimentalists because identification of perturba-tion characteristics is more straightforward when dealing withmonochromatic waves. From a theoretical/numerical point ofview, the eigenvalue problem to be solved is now nonlinear.

By integrating the spatial growth rate of convective perturba-tions, we can obtain amplification of their amplitude along thestreamwise direction. This quantity is usually called the N-factorand is defined as

N ln AA0

x1x1;0

aixdx (12)

where A is the amplitude of the perturbation and subscript 0refers to the location where the perturbation first starts to grow.Smith and Gamberoni [14] and van Ingen [15] independentlydeveloped an empirical transition-prediction method using theN-factor values. The method assumes that transition occurs whenrelative amplification of discrete-frequency disturbances firstreaches a preset ad hoc transition level of eNtr . The critical valueNtr is found through correlation between computed N-factors andmeasured transition location. An extensive description of thismethod is given by Arnal [16]. Of course, transition locationdepends on the spectrum and the amplitude of perturbations pres-ent in the flow. Because the eN method does not account for theinitial amplitude of perturbations, the value of Ntr strongly

depends on the perturbation environment. In order to get a reliableprediction of the transition location, the method should include areceptivity analysis, which provides the initial amplitude of theperturbation. Readers interested in the subject are referred to Refs.[1723].

2.3 Phase Velocity and Group Velocity. In expressions (5)and (6), setting b 0 for convenience, the perturbations can alsobe written as ~qx; t q^x2 expfiax1 ctg where c x=a.For a given a, the value of ~q is, therefore, constant along a particu-lar ray with (x1 ct) constant. For real a, c is, therefore, thevelocity of the wave crests, which is the phase velocity. If (andonly if) x is directly proportional to a, then every wavenumberhas the same phase velocity and the medium is known asnondispersive.

In general, the perturbation consists of a superposition of manywaves, which interfere with each other constructively and destruc-tively. The patterns of constructive and destructive interferenceusually cause the perturbation to have an identifiable envelope, asshown in Fig. 1. If the wavecrests of all wavenumbers move at thesame phase velocity, then this interference pattern, and thereforethis envelope, also move at the phase velocity and they do notchange shape over time. If, however, the wavecrests of differentwavenumbers move at different phase velocities, then this inter-ference pattern changes shape over time and the envelope movesat a different velocity to the wavecrests. The velocity of the enve-lope is known as the group velocity and it is equal to dx=da, asshown in Ref. [13191]. In this case, the medium is known asdispersive.

2.4 Spatio-Temporal Stability. In a spatio-temporal stabilityanalysis, both a and x can be complex. It is then necessary to dif-ferentiate between three types of flow by considering the flowsresponse to an impulse. A flow is stable if the impulse responsedecays in every directioni.e., along every ray emanating fromthe point of impulse. A flow is convectively unstable if theimpulse response grows along some rays that move away from thepoint of impulse but decays at the point of impulse itself. A flowis absolutely unstable if the impulse response grows at the pointof impulse. The review of Huerre and Monkewitz [24] is essentialreading in this context and discusses several parallel flows towhich this concept is applicable. Further details can be found inRefs. [25,26]. The extension of this analysis to quasi-parallelflows led to one of the three contexts for the terminology globalinstability, which is used regularly in the literature. The theoreti-cal bases of these contexts are distinct [12,27].

2.5 The LNSE in 1D. After introduction of the modal ansatz(5) into Eqs. (2) and (3), the LNSE become a generalized eigen-value problem. For an incompressible flow, and with (x1, x2,x3) (x, y, z), these are

iau^ v^y ibw^ 0 (13)

L1du^ uyv^ iap^ ixu^ 0 (14)

L1dv^ p^y ixv^ 0 (15)

Fig. 1 Waves often travel in packets and have a well-defined envelope. The phase velocity isthe velocity at which the wave crests travel. The group velocity is the velocity at which theenvelope travels.

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wyv^ L1dw^ ibp^ ixw^ 0 (16)

where L1d 1=Re d2=dy2 a2 b2 iau ibw, and the

subscript y denotes d/dy. The analogous expansion and LNSE forcompressible flows in which the local theory assumptions are sat-isfied may be found in Mack [28]. Instability analysis based onEqs. (13)(16) has been used extensively to describe instability offree-shear and boundary-layer flows. The key assumption made inthe context of local theory is that of a parallel-flow, according towhich the base flow velocity component v is neglected. Conse-quently, the basic state may comprise up to two velocity compo-nents, namely the streamwise uy and the spanwise wy, both ofwhich are functions of one spatial coordinate alone.

The boundary conditions completing the description of linearinstability in the context of local theory are the viscous boundaryconditions u^ v^ w^ 0 at solid walls or in the far-field (wherethe basic flow velocity vector assumes a constant value), along-side the boundary condition v^y 0, derived from the continuityequation. For flows that are unbounded in the normal direction,for example, boundary-layer flows, the boundary conditions forperturbations can be relaxed and just require that u^; v^; w^ isbounded as y ! 1. This allows for solutions that propagate intoor out from the boundary layer. For details see Refs. [28,29].

In compressible flow, use of the ansatz (5) and linearizationleads to an analogous system for the determination of perturbationvelocity components and disturbances to two of the three thermo-dynamic quantities: density, pressure, and temperature. The sys-tem is closed with an appropriate equation of state, which can bethat of ideal gases for the Mach number range 0

varicose perturbations of an unconfined planar jet/wake

D S1 K x=k2 cothn 1 K x=k2 nR 0 (21)

sinuous perturbations of an unconfined planar jet/wake

D S1 K x=k2 tanhn 1 K x=k2 nR 0 (22)

perturbations of an unconfined single shear layer

D S1 K x=k2 1 K x=k2 n 0 (23)

(R 1 for an unconfined single shear layer because the surfacetension defines the reference lengthscale h1).

The variable n is required in order to keep branch points off thekr axis, as described in Refs. [33] and [35]. It is given byn k2 k2y1=2 as ky ! 0. Here, however, we shall examineonly k 0 and, therefore, n can be replaced by k.

Surface tension is required in order to prevent x from becominginfinite as k becomes infinite. Surface tension has a strong effecton absolute instability, however, as described in Ref. [34]. Analternative strategy is to introduce a finite shear layer thickness.This prevents x from becoming infinite and has only a mild effecton long wavelength instabilities. Analytical dispersion relationsexist for the analogous flows [33].

2.8.2 Coding the Dispersion Relations. It is quite easy (butrather tedious) to use the quadratic formula to derive cx/k as ananalytic function of k. It is quicker to use MATLAB functions. Thefunction fun_eval_c_single.m uses the quadratic formulato evaluate the two values of c for a given value of k. (In order toproduce smooth plots, it also swaps MATLABs default ordering ofthe two solutions at certain values of k. This does not becomeimportant until we consider the spatio-temporal analysis inSec. 2.8.7)

The function has been constructed so that entire arrays of c1and c2 can be calculated simultaneously. This requires operatorssuch as (.*) to be used instead of (*). The parameters K and S areheld in a MATLAB structure as param.L and param.S. This nota-tion makes it easy for a single script to call a variety of functions.

(1) Write MATLAB functions to calculate x/k for the remainingdispersion relations. Name themfun_eval_c_unconf_jetwake_sin.mfun_eval_c_unconf_jetwake_var.mfun_eval_c_conf_jetwake_sin.mfun_eval_c_conf_jetwake_var.m

(2) Try calling these functions with the generic functionsfun_eval_c.m and fun_eval_w.m in order to becomefamiliar with the program structure.

(3) With the exception of the single shear layer, iterative proce-dures are required in order to find k as a function of x.These involve setting the function D to zero by changing kat given x. Write MATLAB functions to calculate D for eachdispersion relation, given x and k. Name themfun_eval_D_single.mfun_eval_D_unconf_jetwake_sin.mfun_eval_D_unconf_jetwake_var.mfun_eval_D_conf_jetwake_sin.mfun_eval_D_conf_jetwake_var.m

(4) Try calling these functions with the generic functionfun_eval_D.m.

(5) Write a script that calculates x using the first set of func-tions and then checks that this (x, k) pair satisfies D 0 bycalling the second set of functions.

These functions will be the building blocks for the rest of thistutorial.

2.8.3 Temporal Stability Analysis. In a temporal stabilityanalysis, we constrain k to be a real number. Physically, thismeans that we consider only waves that do not grow or decay inthe streamwise direction. We then ask whether such waves growor decay in time. An example is shown in Fig. 3 (left) for a singleshear layer with K 0.9 and S 1.

(1) Write a script called script_temporal_001.m to gen-erate Fig. 3 (left).

(2) There are two regimes, either side of k 1.6. How do thewaves behave in these two regimes?

(3) Vary K and S to see how this model behaves.(4) Write a script called script_temporal_002.m to per-

form the temporal analysis on all the dispersion relationsand to plot their growth rates as a function of positive k onthe same plot, as shown in Fig. 3 (right).

(5) What do you observe? In particular, how does confinementaffect the temporal stability of the flows?

2.8.4 Group Velocity. The phase velocity cx/k is the veloc-ity of wavecrests. The group velocity cg dx=dk is the velocityof wavepackets. In a dispersive medium, c 6 cg.

(1) Write a function called fun_eval_dwdk.m to evaluatethe group velocity as a function of k. A first order finite dif-ference method will be good enough.

(2) Write a script called script_groupvel_001.m tocompare the phase and group velocities as shown in Fig. 4.

(3) Investigate the different dispersion relations at differentparameter values. What do you observe? Look particularly

Fig. 2 The base flows in this tutorial are all planar jets/wakes with top hat velocityprofiles. These flows have analytical dispersion relations, which can be calculatedwithout numerical methods.



at unconfined low density jets (K 1, S 1) at small val-ues of R.

2.8.5 Spatial Stability Analysis. In a spatial stability analysis,we constrain x to be a real number. Physically, this means that weforce sinusoidally at a particular point in space and see whetherthe resulting perturbations grow (negative ki) or decay (positiveki) in space. This is known as the signalling problem.

Algorithm fun_eval_k.m uses an iterative technique to findk, given x and an initial guess for k. It uses MATLABs functionfsolve.m, which tries to set the output, D, of fun_eval_-D_loc to zero by varying its argument k. This is where the func-tions fun_eval_D_*.m become useful.

(1) Write a script called script_spatial_001.m to per-form a spatial stability analysis on a single shear layer withK 0.5 and S 1, as shown in Fig. 5.

(2) At small x, the iterative procedure finds a second solution.Try changing the initial guess for k so that it always findsthe first solution.

(3) What is the most amplified frequency?(4) What happens to the most amplified frequency as K

increases?

Spatial stability analyses are more difficult to perform than tem-poral stability analyses and, as we shall see later, have no physicalrelevance when the flow is absolutely unstable because the tran-sient behavior overwhelms the signal if the wave with zero groupvelocity has positive growth rate.

2.8.6 Gasters Transformation. The relationship betweentemporal and spatial stability is given in Ref. [36]. The angular

frequency and wavenumber pairs from the temporal analysis arelabeled x(T) and k(T), respectively. The angular frequency andwavenumber pairs from the spatial analysis are labeled x(S) andk(S), respectively. These are approximately related by

krS krT (24)xrS xrT (25)xiTkiS

@xr@kr

(26)

The final term in Eq. (26) can be written Redx=dk and is sim-ply the real component of the group velocity.

(1) Write a script called script_spatial_002.m to com-pare ki(xr) calculated with script_spatial_002.mwith that calculated from Gasters transformation, as shownin Fig. 5 (right). (You can use Gasters transformation toimprove the initial guess of k for the spatial analysis.)

(2) Vary S and K to discover when Gasters transformationworks well and when it works badly.

Gasters transformation begs the question of what happenswhen the real part of the group velocity is zero. Is the spatialgrowth rate infinite? Rather than ponder this conundrum now, it isbetter to progress to a spatio-temporal stability analysis, whichwill make everything clear.

2.8.7 Spatio-Temporal Stability Analysis. In a spatio-temporal analysis, k and x are both allowed to be complex num-bers. Physically, this means that we consider waves that grow or

Fig. 3 Left: The imaginary and real components of x(k) calculated with a temporal stabilityanalysis, in which k is constrained to be real. Right: The imaginary components of x(k) for thefive dispersion relations.

Fig. 4 The phase velocity (solid lines) and group velocity (dashed lines) as a func-tion of wavenumber k for a single shear layer with surface tension. This is a disper-sive medium, which means that waves at different wavenumbers travel at differentspeeds and that the group velocity does not, in general, equal the phase velocity.



decay in the streamwise direction and that grow or decay in time.A useful way to picture this is to imagine exciting every wavesimultaneously with an impulse at (x, t) (0, 0) and then observ-ing the evolution of the resulting wavepacket. This is known as animpulse response or, equivalently, the Greens function. Theresponse to any type of forcing can be found from this by convo-luting the impulse response with the forcing signal.

Only some waves are permitted (these are the ones that satisfythe dispersion relation). They propagate away from the point ofimpulse at different velocities, given by their group velocitydx=dk, and grow or decay at different rates, given by their growthrate, xi.

If every wave decays in time then the flow is stable. If somewaves grow in time then the flow is unstable and a further distinc-tion is necessary. After a long time, the only wave to remain at thepoint of impulse is the wave that has zero group velocity. If thiswave decays in time, then the impulse response decays to zero atthe point of impulse and the flow is called convectively unstable.If the wave grows in time, then the impulse response grows to

infinity at the point of impulse and the flow is called absolutelyunstable.

In Sec. 2.8.8, we will examine this in more detail. For now,however, we will find the wave with zero group velocity byplotting contours of x in the complex k-plane.

(1) For the single shear layer, write a script called script_-spatemp_001.m to calculate x1 and x2 on a grid ofcomplex k with kr [0, 2] and ki [1, 1]. Plot contoursof xi and xr on the same plot, as in Fig. 6. (The stencilbecomes useful here.) Plot the contour xi 0 with doublethickness.

(2) What do you notice about the relationship between the con-tours of xi and the contours of xr?

(3) Identify by eye the position where the group velocity iszero. Does it have positive or negative growth rate?

(4) Write a function called fun_eval_dwdk0.m that usesfsolve to converge to this point, given an initial guess,and then displays (x, k) at this point.

Fig. 5 The real and imaginary components of k, calculated with a spatial stabilityanalysis, in which x is constrained to be real. Left: The exact spatial analysis, per-formed via iteration from a guessed value of k. Right: The approximate spatial anal-ysis from Gasters transformation, compared with the exact spatial analysis.

Fig. 6 Contours of xi(k) (grayscale) and xr(k) (gray lines) for a single shear layerwith K5 0.9, S51. This is a spatio-temporal stability analysis in which both x andk are complex. The points with dx/dk5 0 can be identified by eye (there is one ineach frame). The thick black line is xi5 0.



(5) Write a script called script_click.m that asks the userto click near a point where dx=dk 0, then convergesaccurately to that point, then determines whether the flow isabsolutely or convectively unstable.

The contours of xi and xr are always at right angles to eachother. Furthermore, the two components of curvature on the sur-face xi(k) always have opposite sign. This means that pointswhere dx=dk 0 are saddle points of x(k). In the long time limit,the impulse response at the point of impulse is dominated by thebehavior of these saddle points. At a saddle point, the values of xand k are called the absolute frequency and wavenumber and aregiven the symbols x0 and k0:

(1) Compare Fig. 3 with Fig. 6 to check that the temporal anal-ysis corresponds to a slice through x(k) along the kr axis.

(2) What does the spatial analysis correspond to in Fig. 6?(3) Explain why a spatial analysis does not work for absolutely

unstable flows.

2.8.8 The BriggsBers Criterion. The aim of this section is topresent a more rigorous calculation of the impulse response and toshow that some saddle points do not contribute to it. We want toevaluate

ux; t 11

11

u^k;xeikxxt dx dk (27)

To evaluate the response at the point of impulse, see Ref. [37],particularly Eq. (9), and [26], which gives more steps in the calcu-lation. To evaluate the response at every point in space, see Ref.[35], particularly Fig. 11, and Ref. [33], one result from which isshown in Fig. 7.

Figure 7 shows contours of the growth rate as a function ofgroup velocity, (x/t, z/t), for all the permitted waves in the impulseresponse. (It is for varicose perturbations of an unconfined lowdensity jet with K 1/1.1 and S 0.1.) The impulse responsepropagates and grows at the point of impulse, (x/t, z/t) 0, so thisflow is absolutely unstable. It grows fastest, however, in the down-stream direction, x/t> 0. The maximum growth rate of theimpulse response (8.5) is given by the growth rate of the temporalstability analysis because, at this point in the impulse response,ki 0.

The integral (27) can be evaluated by integrating either over xand then k, or over k and then x. For these dispersion relations, itis easiest to integrate over x first because, for each k, there areonly two permitted values of x. The integral becomes

ux; t 11

u^k;xeikxx1kt dk 11

u^k;xeikxx2kt dk(28)

Initially, the integration path lies on the real k-axis but it can beshifted into the complex k-plane without changing the integral, aslong as it is not shifted through branch points or poles (i.e., pointswhere x61). The integral is easiest to evaluate if the integra-tion path is shifted onto lines of constant xr because then thephase of expfikx xtg remains constant and the integrand doesnot oscillate as k changes. (Remember that x 0 because we arelooking at the point of impulse, so the only term that can oscillateas k varies is expixrkt.)

In order to pass from k1 to k1, the integration pathmust cross some (but not all) of the saddle points in the k-planeand, for each of these saddle points, there is only one possiblevalue of xr. This means that the integration path must cross thesesaddle points at this value of xr and then change to another valueof xr at points where xi is strongly negative, so that the oscillat-ing contribution to the integral there is negligible.

This is best explained by showing an example. Figure 8 showscontours of xi(k) for varicose perturbations of a confined jet flowwith K 1, S 1, h 1.3, R 1. The integration path needs to passover two saddle points s1 and s3 in order to pass from k 0 to k1,but it misses out saddles s2a and s2b. As the surface tension reduces,which is shown in Fig. 9, saddle s2a comes onto the integration con-tour. As it reduces further, s2b etc. come onto the contour too.

In the long time limit, the dominant contribution to the integralcomes from the point on the integration path with highest xi,which is always a saddle point. At these saddle points, contours ofconstant xr extend into the top half k-plane on one side and intothe bottom half k-plane on the other side. This is the BriggsBerscriterion. It is another way of stating that the saddle point must lieon the integration path. Alternative but equivalent explanationscan be found in Refs. [26,37]:

(1) Write a script called script_spatemp_002.m to plotx1(k) for varicose perturbations of a confined jet/wake flowwith K 1, S 1, h 1.3, R 1. Compare it with Fig. 9.

(2) Investigate how the dominant saddle points change as Sdecreases, R decreases, and h increases. Look particularlyat the s2a saddle.

(3) Write a script called script_spatemp_003.m to repeatthis for varicose perturbations of an unconfined jet/wakeflow with K 1, S 1, R 0. This corresponds to anunconfined jet. Find the s2a saddle point exactly withscript_click.m

The main point of this section is that only some saddle pointscontribute to the impulse response and that, in the case of a lowdensity jet with vanishing surface tension, the s2a saddle is one ofthese.

2.9 Application to Wall-Bounded Shear-Flow: PlanePoiseuille Flow. (See the Supplemental Data tab for this paperon the ASME Digital Collection.) One of the most studied two-dimensional flows is the flow between two parallel walls (planePoiseuille flowPPF). This flow, which is invariant in the stream-wise direction, can be described as

u Uy; 0; 0; Uy 1 y2; y 2 1; 1 (29)

Plane Poiseuille flow, the instability of which was first solved byspectral methods in the classic paper of Orszag [38], serves as thetestbed for implementation and validation of a stability analysison a realistic flow.

This section presents results that may be obtained with the codeprovided for the Tutorial work. The stability problem is solvedeither in its original generalized form Eqs. (13)(16) or consider-ing the OrrSommerfeld Eq. (17). To solve the stability equations,

Fig. 7 The impulse response for varicose perturbations of anunconfined low density jet with K5 1/1.1, S5 0.1, and finitethickness shear layers. The impulse is at (x,z)5 (0,0) and the re-sultant waves disperse at their individual group velocities inthe x- and z-directions. The chart plots the growth rate, xi, ofthe wave that dominates along each ray (x/t,z/t), i.e., the wavethat has group velocity (x/t,z/t). We see that most of the wave-packet propagates and grows downstream (to the right) but thatsome waves propagate and grow upstream (to the left). Thegrowth rate at the point of impulse (x/t,z/t)5 (0,0) is positive,which means that this is an absolutely unstable flow [33].



we discretize the equations using the spectral collocation methodusing Chebyshev polynomials, as shown in the Appendix. Thisyields a generalized matrix eigenvalue problem of the form

Aq^ xBq^ (30)

where q^ u^; v^; w^; p^T for discretized Eqs. (13)(16) or q^ v^ forthe OrrSommerfeld equation.

In Fig. 10, the spectrum of the eigenvalues for Re 104, a 1,and b 0 is plotted. Figure 11 presents the velocity and pressureperturbations corresponding to the most unstable eigenvalue.

The following tutorial exercises are recommended:

(1) Familiarize yourself with the codes provided. Scriptrun_LNS.m computes the eigenvalue and eigenfunctionscorresponding to Eqs. (13)(16) and script run_OS.m toEq. (17).

(2) Run the scripts for Re 104, a 1, and b 0 by changingthe values of parameters Re, alpha and beta in filerun_OS.m (and run_LNS.m).

(3) Change the number of Chebyshev polynomials and investi-gate convergence of the most unstable mode (the one withthe largest xi) as well as the whole spectrum. Compare

Fig. 8 Contours of xi(k) for varicose perturbations of a confined jet flow with sur-face tension, showing the integration path from k52 to k51. The integrationpath passes over saddle points s1 and s3, which means that they contribute to theintegral. It does not pass over saddle points s2a or s2b, however, which means thatthey do not contribute to the integral. This is another way to visualize theBriggsBers criterion, which states that a saddle point is only valid if it is pinchedbetween a k1 and a k2 hill [33].



your results with those of Kirchner [39] x 0.2375264888204682 0.0037396706229799i.

(4) Compare the results of the two scripts run_LNS.m andrun_OS.m.

(5) Plot the eigenfunctions of different modes to see the char-acteristic shape of different branches of the spectrum.

2.10 Free Shear-Flow: The tanh Profile. (See theSupplemental Data tab for this paper on the ASME DigitalCollection.) Chebyshev GaussLobatto-spaced points are notalways conveniently spaced. In this tutorial, the Rayleigh equationis solved for perturbations to a tanh velocity profile, which hashigh gradients around y 0 and has infinite extent in they-direction. In order to solve this, a mapping function needs to beintroduced. The following tutorial exercises are recommended:

(1) Familiarize yourselves with the codes provided.(2) Change the base flow and its derivatives in script run_-

OS.m to those for the hyperbolic tangent profile

uy tanhy (31)

(3) Implement the following mapping

y l y^=1 s y^2

p(32)

where l 10; s1 20; s l=s12, into the differentiationmatrices, along the lines presented in Eqs. (A10)(A13).

(4) Modify the function OS_temp.m to solve the Rayleighequation.

(5) Run the code for parameters in the classic work ofMichalke [40] and compare the results. The data formMichalke are stored in file Michalke.dat and plotted inFig. 12. Note that the real part of phase velocity for thisflow is independent of wavenumber a and is equal to 1.

(6) Compare results for the OrrSommerfeld and Rayleighequations.

3 Secondary Linear Modal Instability Theory

The linear instability of wall-bounded shear flows has beenstudied for over a century. The latter stage of the linear instabilityprocess in a boundary layer is of particular interest, motivated bythe turbulent breakdown of laminar linearly unstable flow. Thekey point is to discover the scenario that follows linear amplifica-tion of small-amplitude perturbations in a boundary layer. How-ever small such perturbations may be initially, the exponentialnature of modal growth suggests that these perturbations mustreach amplitudes at which the linearity assumption is invalidated.There exists a theoretical vacuum between the arbitrarily scalableresults of linear local theory and the essentially nonlinear natureof the phenomenon of transition to turbulence.

A breakthrough was provided by the almost simultaneous inde-pendent theoretical [4143], experimental [44,45], and DNS[46,47] efforts in the area of secondary (linear modal) instabilitytheory. The analysis proceeds by considering a steady laminarone-dimensional basic flow q which has been modified by the lin-ear growth of its most unstable perturbation q^. In order for the

Fig. 9 Contours of xi(k) for the same flow as Fig. 8, as the sur-face tension is reduced. This reduction in R causes the s1 sad-dle to move to lower xi and, therefore, causes the integrationpath to pass over the s2a saddle point. When the surface ten-sion tends to zero (not shown here) all the s2 saddle pointsmove onto the integration path [33].

Fig. 10 Eigenvalue spectrum for PPF at Re5 104, a5 1, b50. So-lution of LNS () and the OrrSommerfeld () equations. Noticethat the vorticity modes are not represented by OrrSommerfeldequation. Results produced by run_LNS.m and run_OS.m.

Fig. 11 Velocity and pressure perturbations corresponding tothe most unstable eigenvalue of PPF at Re5 104, a5 1, b5 0.Data produced by run_LNS.m for the most unstable mode inFig. 10. All components are normalized with max(|u^|).



secondary instability analysis to remain linear, the amplitude, A,at which the (so-called primary) eigenmode needs to be super-posed upon the basic state needs to be small initially, but is subse-quently allowed to become finite and modify the steady laminarbasic state in a periodic manner, while linear theory is still valid.In a temporal framework the ensuing new two-dimensional basicstate

q2dx; y; t q2dx 2p=a; y; t qy A q^yeiaxxrixit c:c:

(33)

is periodic in a frame of reference moving with the phase speed ofthe leading eigenmode (the TollmienSchlichtingTS wave)with a periodicity length Lx 2p=a, where a is the wavenumberof the most amplified TS mode. Consequently, Floquet theory isused to analyze the modified base state, three-dimensional pertur-bations assuming the form [43]

~qx; y; t ertecxeibzq^x; y (34)

Two interesting classes of linear secondary perturbations havebeen identified in experiments [44,48] and are modeled in second-ary linear theory, fundamental perturbations, for which

q^f x; y X

m even

q0f yeima2x (35)

and subharmonic, for which

q^sx; y Xm odd

q0f yeima2x (36)

respectively, satisfying

q^f x; y; t q^f x Lx; y; t (37)

and

q^sx; y q^sx 2Lx; y (38)

The ansatz (33) alongside either of Eq. (35) or Eq. (36) is intro-duced into the LNSE, like-order terms are grouped together andan eigenvalue problem is obtained for the determination of thesecondary growth rate r as a function of the flow parameters, Re,the streamwise and spanwise wavenumbers, a and b, respectively,given an (arbitrarily chosen) amplitude parameter A; Fig. 13shows results for subharmonic resonance in the Blasius boundarylayer.

From a physical point of view, secondary instabilities representlambda vortices formed during the late stages of transition. Funda-mental instabilities, also known as K-type instabilities, are alignedwith each other, while subharmonic instabilities, also known asH-type instabilities, form a staggered pattern of vortices. Both areclearly visible in experiments on both flat plate and axisymmetricboundary layers.

Excellent agreement is obtained between theoretical results ofsecondary instability theory and DNS in the plane Poiseuille flow[46,49]. In the celebrated experiments of Kachanov andLevchenko [44,45] both three-dimensional spatial distribution oflambda vortices as well as secondary amplification rates r weremeasured during transition in the Blasius boundary layer and theresult were found to be in excellent agreement with the predic-tions of the subharmonic H-type scenario. Although the amplitudeparameter A in Eq. (33) can be chosen to match experimentallymeasured amplitudes of the primary instability, an alternativeapproach that altogether abandons the modal concept for thedescription of boundary layer stability, namely the PSE [11], hasovershadowed secondary stability theory, on account of the poten-tial of PSE to predict both linear and nonlinear disturbance growthin a self-consistent manner. The PSE is discussed next.

4 Nonlocal Nonparallel TheoryThe PSE

Traditional stability theory, based on the quasi-parallel flowassumption, does not account for growth of the boundary layer.The local character of this theory exclude any history effectsassociated with the initial condition or the varying properties ofthe basic flow. Early work on the effects of a growing boundarylayer on the characteristics of the disturbances was based on themultiple-scales method (Refs. [5054]). Recent work by Zuccherand Luchini [23] gives a good review of the application of themultiple-scales method for receptivity analysis.

The idea of solving parabolic evolution equations for disturban-ces in the boundary layer was first introduced by Hall [55] for

Fig. 13 Subharmonic secondary growth rate r as a function ofthe spanwise wave number b for F5xr /Re5 1.243 10

24 atBranch II, Re5 606 [43]. The parameter A indicates the ampli-tude of the primary disturbance as a percentage of the maxi-mum value of the streamwise basic flow velocity component.Fig. 12 Growth rate as a function of wave number for the tanh

profile. Data from Michalke [40].



steady Gortler vortices. Itoh [56] derived a parabolic equationfor small-amplitude TollmienSchlichting waves. Herber andBertolotti [57] and Bertolottiet et al. [58] developed this methodfurther and derived the nonlinear PSE. Simen [59] independentlydeveloped a similar theory. Their contribution was to consistentlyand in a general way model convectively amplified waves withdivergent and/or curved wave-rays and wave-fronts propagatingin nonuniform flow. Due to the fact that the disturbance character-istics predicted by such a method are influenced by local andupstream flow conditions, this theory is called nonlocal.

It should be mentioned that the parabolic stability equationsconstitute an initial value problem and in principle can be used tocompute the evolution of any perturbation. However, the auxiliaryconditions (see Sec. 4.2) usually cause the solution to converge tothe most unstable (or least stable) mode. Starting the integrationwith the dominant mode given by local theory, it always con-verges to the nonlocal solution for that mode. The linearizedboundary-layer equations, which are similar to PSE but witha 0, have been used by Andersson et al. [60], and Luchini [61]to compute the optimal transient growth in the Blasius boundarylayer. Recently, Tempelmann et al. [62,63] modified PSE to com-pute the optimal perturbation in three-dimensional boundarylayers.

Next, we discuss the basics of the nonlocal method. A detaileddiscussion on this subject can be found in Bertolotti [64] and Her-bert [65].

4.1 Assumptions. Here, we restrict ourselves to quasi-three-dimensional flows (basic flows that are independent of the z coor-dinate). We start the derivation of the nonlocal stability equationsby introducing the following assumptions:

(1) The first assumption is of WKB type, in which the depend-ent variables are divided into an amplitude function and anoscillating part, i.e.,

~qx; t q^x; y exp ix

x0

ax0dx0 bz xt

(39)

Here, as in the spatial local theory, a is a complex number.Note that in nonlocal theory, in contrast to the local theory,both the shape function q^ and the phase function H areallowed to vary in the streamwise direction.

(2) As a second assumption, a scale separation 1/Re is intro-duced between the weak variation in the x-direction andthe strong variation in the y-direction, analogous to themultiple-scales method. Here, the Reynolds number is basedon a length scale proportional to the boundary-layer thick-ness (d x??=U?ep ). Important implications of thisassumption are that the x-dependence of the basic and dis-turbance flow, and the normal basic flow velocity componentV are also assumed to scale with the small parameter 1/Re.

@

@x;V ORe1; @

@y;U;W O1 (40)

With the above scaling and ansatz (39) introduced into the line-arized NavierStokes equations, considering only terms that scalewith powers of Reynolds number up to 1/Re, we obtain the nonlo-cal, linear stability equations that are nearly parabolic. Theydescribe the kinematics, nonuniform propagation, and amplifica-tion of wave-type disturbances with divergent or curved waverays in a nonuniform basic flow. These equations can be writtenas

Aq^ B @q^@y C @

2q^

@y2D @q^

@x 0 (41)

The entries of these matrices for an incompressible boundary layerare

A

ia 0 ib 0

f @U@x

@U

@y0 ia

0 f @V@y

0 0

@W

@x

@W

@yf ib

26666666664

37777777775; B

0 1 0 0

V 0 0 0

0 V 0 1

0 0 V 0

26664

37775;

C

0 0 0 0

1Re

0 0 0

0 1Re

0 0

0 0 1Re

0

266666664

377777775; D

1 0 0 0

U 0 0 1

0 U 0 0

0 0 U 0

26664

37775

where

f ix iaU ibW 1Rea2 b2

The corresponding terms for compressible flows given in a curvi-linear coordinate system can be found in Refs. [66,67].

The disturbances are subjected to the following boundaryconditions

u^ 0 on y 0 and u^ ! 0 as y !1 (42)

where u^ is the perturbation velocity. The system of Eq. (41) isintegrated in the downstream direction with a given initial condi-tion at x x0, e.g. the solution of the local stability theory.

4.2 Auxiliary Condition. As both the amplitude and phasefunction depend on the x-direction, we follow Refs. [64,65,68]and remove this ambiguity by specifying various forms ofauxiliary conditions. These conditions can be based either onthe amplitude of one of the disturbance quantities q^k of Eq. (41) atsome fixed y ym, i.e.,

@q^k@x

yym

0 (43)

or on the integral quantity10

q^@q^

@xdy 0 (44)

where the superscript refers to the complex conjugate. The aux-illary conditions enforce the variation of the shape function q^ toremain small enough to justify the (1/Re) scaling of @q^=@x. Ateach streamwise position, the value of a is iterated such that auxil-iary condition is satisfied.

4.3 Definition of Growth. In the frame of the nonlocaltheory, the physical growth rate r of an arbitrary disturbancequantity n can be defined as

r ai Re 1n@n@x

(45)

where the first RHS term is the contribution of the exponentialpart of the disturbance. The second term is the correction due tothe changes of the amplitude function. Usually, n is taken to be acomponent of q^, measured either at some fixed y or at the locationwhere it reaches its maximum value. Moreover, the disturbancekinetic energy E, here defined as



E 1

0

ju^j2 jw^j2 jv^j2

dy

can be used as a measure for the disturbance growth. Then,Eq. (45) becomes

r ai @@x

lnE

p

It should be noticed that the growth rate based on Eq. (45) can bea function of the normal coordinate y.

The corresponding physical streamwise wavenumber is givenby

a ar Im 1n^

@n^@x

( )(46)

The direction of wave propagation is given by the physicalwaveangle

w tan1 ba

: (47)

The waveangle w also describes wave-front distortion or wave-raycurvature because w can vary both in the x- and y-directions.

4.4 Comparison With Direct Numerical Simulations. Themost accurate stability results are those from DNS. A drawback ofthese methods is the long central processing unit (CPU) timeneeded in such calculations, and the limitation to simple geome-tries. The comparisons of the DNS and nonlocal results over awide range of Mach numbers have shown great potential of thenonlocal method for prediction of the stability characteristics aswell as transition location.

Berlin et al. [69] calculated the neutral stability curves for two-dimensional disturbances in an incompressible flat plate boundarylayer. Their results are given in Fig. 14. As can be seen there, aclose agreement between the results from the DNS and nonlocalcalculations is found. These results are also in good agreementwith experimental data of Klingmann et al. [70]. The CPU time ofthe DNS calculations were about one hundred times larger thanthat of the PSE calculations.

4.5 Mathematical Character of the PSE. Although no sec-ond x-derivatives of q^ are present in Eq. (41), there is still some ellip-ticity in the stability equations. This is demonstrated by the rapidoscillations of the solutions when the marching step size becomestoo small. This is illustrated in Fig. 15 for a TollmienSchlichtingwave in a flat-plate boundary layer. The residual ellipticity is due tothe upstream propagation through the pressure terms. The ill-posedness of the nonlocal stability equations has been studied by,e.g., Herbert [65], Haj-Hariri [71], Li and Malik [72], Airiau [73]. Liand Malik showed that the dropping of @p^=@x relaxes the step sizerestriction. They also gave the limit for the smallest stable step sizeto be 1=jarj. Andersson et al. [74] suggested the addition of a stabi-lizing term which is of the same order as the numerical errors. Thenthe modified equations are written as

@q^

@x Lq^ sL @q^

@x(48)

This solution is derived for a first-order accurate backward Euler dis-cretization of the PSE. In Fig. 16 the efficiency of this approach isdemonstrated for instability waves in a flat-plate boundary layer.

4.6 Tutorial on the PSE. (See the Supplemental Data tabfor this paper on the ASME Digital Collection.) The followingexercises aim to provide an understanding of different aspects ofPSE computations.

(1) Familiarize yourself with the MATLAB codes provided. ThePSE are discretized using a Chebyshev spectral discretiza-tion for wall-normal derivatives and a first- or second-orderaccurate backward Euler scheme for streamwise deriva-tives. The main program is called pse.m.

(2) Choose a perturbation with a reduced frequency ofF 2pf ?=U?2e x=Re 104 and b 0 and the com-putational domain Re 2 400 1000 for the followingcomputations. This should be done in the main codepse.m. The parameters to be set are F, beta, Re, andRe_end. These parameters correspond to results presentedin Figs. 15 and 16.

(3) Perform calculations for different step size Dx usingstabfalse and find the smallest stable solution andcompare with the analytical value of 1=jarj. The step sizecan be changed by modifying the value of Nstation.

Fig. 14 Neutral stability curves based on the maximum of u^and v^ . The nonlocal results are compared to the DNS calcula-tions (Berlin et al. [69]) and experimental data (Klingmann et al.[70]).

Fig. 15 Effect of streamwise step size on the growth rates of2D waves in a Blasius boundary layer flow, F5 1.03 1024. Com-puted by pse.m with stab5false.



(4) Perform calculations with a stable step size Dx using auxil-iary conditions (43) and (44). This is done by changingvalue of iaux from 1 (for integral norm) to 0. Verify thatvalues of the physical growth rates and streamwise wave-number a are the same in both cases.

(5) Study the function integratePSE.m and implementa-tion of the stabilization term given in Eq. (48). Run thecode for different critical step sizes using stabtrue.This implementation is correct for first-order backwardEuler scheme (FDorderrst).

(6) Put a loop around the main program and run the code fordifferent values of the reduced frequency to generate theneutral curves given in Fig. 14. The data plotted in Fig. 14are found in the file neu.mat.

5 BiGlobal Linear Theory: The LNSE in 2D

When the base flow varies in two out of the three spatial direc-tions and is independent of the third, q u; v; w; pTy; z, theanalysis proceeds following the ansatz:

qx; y; z; t qy; z eq^y; zei axxt c:c (49)

The disturbances are three-dimensional, but a harmonic depend-ence along the homogeneous x-direction is assumed, with the peri-odicity length Lx 2p/a. Upon substitution of Eq. (49) into theLNSE the system of equations results

iau^ v^y w^z 0 (50)L2du^ uyv^ uzw^ iap^ ixu^ 0 (51)L2d vyv^ vzw^ p^y ixv^ 0 (52)L2d wzw^ wyv^ p^z ixw^ 0 (53)

where L2d @yy @zz a2=Re iau v@y [email protected] that, if f:gz ibf:g is considered, the operators L2d andL1d become identical and the PDE-based system (partial differen-

tial equation) (50)(53) reduces to the ODE-based system (ordi-nary differential equation) (13)(16). The elliptic eigenvalueproblem (50)(53) is complemented with adequate boundary con-ditions for the disturbance variables, typically no-slip on solid

walls for the velocity components and the associated compatibilityconditions for pressure, if the latter variable is maintained in thevector sought. Alternatively, pressure may be eliminated to yieldthe two-dimensional analogue of the OrrSommerfeld equation,as originally done by Tatsumi and Yoshimura [75], or the two-dimensional analogue of the Rayleigh equation, first presentedand solved by Henningson [76].

Since the LNSE in two spatial dimensions can be reduced tothe system of equations arising when only one inhomogeneousspatial direction exists, it is natural to pose two questions: (i) canEqs. (50)(53) be used to predict parallel flow instability? (ii)when the assumptions underlying parallel flow theory are relaxed,can solution of Eqs. (50)(53) provide information inaccessible tosolution of the classic OrrSommerfeld equation or absolute/con-vective instability analysis as applied to parallel or weakly non-parallel flows? Both questions were posed first in the context ofadverse-pressure-gradient-generated laminar separation bubble ina flat plate boundary layer by Theofilis et al. [77], while in Blasiusflow the same issues were addressed first by Akervik et al. [78]and Rodrguez and Theofilis [79] and by several authors since.

Answers depended on the type of flow analyzed. In the sepa-rated flow, the discovery of the three-dimensional stationaryamplified global mode of the laminar separation bubble [77] wasonly possible by application of global linear theory. On the otherhand, in Blasius flow, imposition in a single BiGlobal computa-tion of a given streamwise periodicity length and elimination ofthe wall-normal basic flow velocity component led to recovery ofresults identical to those delivered by multiple solutions of theOrrSommerfeld equation, each for a single streamwise wave-number parameter. In other words, no information additional tothat already contained in solutions of the classic one-dimensionallinear instability analysis problem could be provided by applica-tion of global linear theory. However, relaxation of streamwiseperiodicity in the convectively unstable Blasius boundary layer inthe context of a BiGlobal eigenvalue problem (EVP) solutionposes the question of inflow/outflow boundary conditions appro-priate for open flows; several choices are discussed by Rodrguezand Theofilis [79]. The branch(es) recovered in both related works[78,79] correspond to stable BiGlobal eigenmodes, while the loca-tion of the eigenspectra in (xr, xi)space is strongly affected bythe imposed inflow/outflow boundary conditions [79].

In the opinion of the authors, use of global linear theory is onlyjustified when local modal analysis suggests that the flow willcontain regions of local absolute instability. In that case, globallinear theory can provide definitive answers which are free fromthe assumptions of local theory. However, when the instability inquestion is convective, classic linear instability analysis toolsbased on solution of the OrrSommerfeld equation not only areadequate from a physical point of view but also orders-of-magni-tude more efficient than global linear theory.

5.1 Temporal BiGlobal Stability. Temporal linear stabilityanalysis may also be performed in the context of BiGlobal LST.The significance of the parameters a and x appearing in Eq. (49)is analogous to that in local LST: a is a real wavenumber parame-ter, associated with a periodicity length along the (single) homo-geneous spatial direction, x1, through L 2p=a and is an inputparameter for the analysis, as shown in Eq. (54). The box consid-ered here is periodic only along the x1 spatial direction, whilethe limiting case a 0 refers to two-dimensional perturbationssuperposed upon the two-dimensional basic state. The parameterx is the sought complex eigenvalue, the real part of which,xr 0, and the damping rate if =fxg < 0.

~q q^y; zeiaxxt c:c: q^y; zeiaxcricit c:c (54)The primitive-variable formulation (50)(53) can be written as atwo-dimensional partial-derivative-based generalized eigenvalue

Fig. 16 Effect of stabilizing term on the growth rates of 2Dwaves in a Blasius boundary layer flow, F51.031024. Com-puted by pse.m with stab5true.



problem, which can be recast in matrix form as the (linear) tempo-ral BiGlobal eigenvalue problem

A q^ xB q^ (55)

Identification of a neutral stability curve is also the objective ofBiGlobal linear theory and is also accomplished by projecting theamplification rate surface on the xi 0 plane. Common with thelocal theory, the flow Reynolds number (and Mach number, ifcompressible) is a flow parameter that needs to be varied. By con-trast to local LST, here only one wavenumber parameter, a, exists.Also, the two-dimensional nature of the temporal eigenvalueproblem to be solved makes it order(s) of magnitude more expen-sive than that governing local analysis. Alternative numericalmethods need to be utilized for the solution of the BiGlobal EVP,which will be discussed next.

5.2 Spatial BiGlobal Stability. In a global instability analy-sis context, most of the presently available literature has been per-formed in the temporal BiGlobal framework. However, the recentemergence of the PSE-3D concept gave rise to the associatednecessity to initialize such computations using solutions of thespatial BiGlobal eigenvalue problem [80]. In this context, the sig-nificance of the parameter x appearing in Eq. (7) is that of a realfrequency, while the sought eigenvalue is now a, the real part ofwhich is again associated with a periodicity length L 2p=ar ,while the imaginary part of this quantity ai is the sought amplifi-cation/damping rate.

~q q^y; zeiaraixxt c:c: (56)

The primitive-variable formulation (50)(53) can also be writtenas a two-dimensional partial-derivative-based generalized eigen-value problem, which can be recast in matrix form as the (nonlin-ear) spatial BiGlobal eigenvalue problem,

A q^ X2k1

akBk q^ (57)

The operators A and B defining Eq. (55) may be found explicitly inthe literature, e.g., in Paredes et al. [80]. The nonlinearity (in a) ofthe eigenvalue problem (55) makes the size of the matrix discretizingthe linear operator four times as large compared with that governingtemporal BiGlobal instability when the companion matrix approach[81,82] is used. A schematic of the size of the matrices governinglocal and global temporal and spatial instability analysis is shown inFig. 17. The efficient numerical solution of global eigenvalue prob-lems in which the matrix is formed and stored is the subject of cur-rent research [84]. The neutral curve, which is identical to thatobtained in a temporal BiGlobal context, is now obtained by projec-ting the amplification surface onto the ai 0 plane.

5.3 2D EVP Tutorials

5.3.1 The Helmholtz Equation. (See the Supplemental Datatab for this paper on the ASME Digital Collection.) Closer inspec-tion of Eqs. (50)(53) reveals that the two-dimensional Laplaceoperator is at the heart of the operator L2d defining the BiGlobaldirect and adjoint eigenvalue problems. When setting up a numer-ical procedure for the solution of the BiGlobal EVP it is advisableto take advantage of partial validation cases, one of which isoffered by the well-known (multidimensional) Helmholtz equa-tion. The eigenvalue problem

@2

@y2 @

2

@z2

/ k2/ 0 (58)

in the rectangular membrane domain y 2 0;Ly z 2 0;Lz isknown to have an analytical solution [85] for the eigenvalues

k2ny;nz p2nyLy

2 nz

Lz

2" #; ny; nz 1; 2; 3; (59)

and associated eigenfunctions

/ny;nzy; z sinpnyyLy

sin

pnzzLz

(60)

The eigenvalue problem (58)(60) is therefore useful in assessingthe accuracy of the proposed spatial discretization method, espe-cially in the recovery of the higher eigenvalues/eigenfunctions, ny,nz 1.

Generating a solver for the two- and three-dimensional eigen-value problems in a matrix-forming context requires the followingsteps:

(1) Choose a spatial discretization method for each spatialdirection.(a) The spatial discretization method either can be different

in each resolved spatial direction or can be the same,for example spectral collocation based on Chebyshevpolynomials on the GaussLobatto nodes (CGL). Atypical situation in which different spatial discretiza-tions should be used is the coupled resolution of wall-bounded and periodic directions, such as would occurwhen resolving Gortler or crossflow vortices on a flat-plate boundary layer. Here we will use the CGL methodto set up the Laplacian operator.

(2) Use of the Kronecker product to create the two- or three-dimensional matrix.(a) The use of the Kronecker product is explained in

Sec. 6.3.1.(3) Impose boundary conditions.

(a) The imposition of boundary conditions can be made ina weak or a strong manner, respectively, using basisfunctions which automatically satisfy boundary condi-tions (e.g., Fourier spectral collocation for periodicproblems) or explicitly using rows of the matrix toimpose Dirichlet or Neumann boundary conditions.Here, we use strong imposition of boundary conditions.

Fig. 17 Relative size of the discretized operators describingmodal stability, when the same number of collocation nodes isused in one (local, PSE) and two (global, PSE-3D) resolved spa-tial directions (from de Tullio et al. [83]).



(4) Choose a direct or iterative eigenvalue solution procedure(a) MATLAB provides access to both methods, respectively,

through use of the eig and eigs routines.


(1) Familiarize yourself with aspects of the codes provided,such as Chebyshev collocation derivative matrices and theKronecker product.set_y_grid.m, set_chebyshev_derivative_-

matrices.m: set the Chebyshev GaussLobatto collo-cation grid; form 1st and 2nd collocation derivativematrices.

Helmholtz2D.m: set up and solve the 2D Helmholtz EVPform_Helmholtz_2D.m: form the 2D Helmholtz EVP

(58) using the collocation derivative matrices in the twospatial directions and the Kronecker product.

impose_Helmholtz_BCs.m: impose homogeneousDirichlet boundary conditions on the discretized 2DHelmholtz operator.

plot_Helmholtz2D.m: plot selected eigenvalues andeigenfunctions of the 2D Helmholtz EVP

(2) Run the script Helmholtz2D.m to obtain the spectrum ofeigenvalue problem (58).

(3) Plot the eigenfunctions of different modes to see their char-acteristic shape. Compare the eigenfunction correspondingto k/2p2 34 with that provided in Fig. 18.

(4) Change the order of Chebyshev polynomials and plot con-vergence of the leading members of the eigenspectrum.

5.3.2 Plane Poiseuille Flow. (See the Supplemental Data tabfor this paper on the ASME Digital Collection.) This tutorial intro-duces numerical solution of the BiGlobal eigenvalue problem by amatrix-forming tensor-product-based method. For simplicity, theequations solved are those to which Eqs. (50)(53) reduce whenthe two-dimensional perturbation limit is considered. Denotingthe resolved spatial directions by x (streamwise) and y (wall-normal), the linearized NavierStokes equations can be written as a3 3 matrix eigenvalue problem Aq^ xBq^. Explicitly

L Uy @x0 L @y@x @y 0

24

35 u^v^

p^

2435 x I 0 00 I 0

0 0 0

24

35 u^v^

p^

2435 (61)

where L 1=Re @xx @yy U@x. In order for comparisons of

results of the BiGlobal EVP with those of the OrrSommerfeldequation to be possible, the assumptions underlying local analysisare made, namely that flow is parallel and has a single streamwisevelocity component U(y). This permits taking the streamwise spa-tial direction x to be homogeneous and resolve it using Fouriercollocation [86].

A convenient geometry in which these ideas can be demonstratedis the plane channel, the eigenspectrum of which is known from Sec.2.9 and Fig. 10. By contrast to local theory, here streamwise perio-dicity is imposed implicitly through the choice of a streamwise perio-dicity length Lx, which corresponds to a wavenumber a 2p/Lx andits harmonics. Spectral collocation based on the ChebyshevGaussLobatto nodes is used to discretize the wall-normal spatialdirection. The Kronecker product discussed in Sec. 6.3.1 is used toform the two-dimensional matrices A and B.


(1) Familiarize yourself with Fourier spectral discretization inthe code provided set_x_grid.m, set_fourier_-derivative_matrices.m: set the Fourier collocationgrid; form first and second collocation derivative matrices.map_x.m: linearly transform the Fourier system.calculate_ddx.m: compute Fourier collocation deriva-

tive of an 1D periodic functionEVP2d.m: set up and solve the 2D BiGlobal EVPform_EVP2d.m: form the 2D EVP using the collocation

derivative matrices in the two spatial directions, usingFourier spectral collocation along x, Chebyshev colloca-tion based on the GaussLobatto points along y and theKronecker product.

impose_BCs_EVP2d.m: impose boundary conditions(along y) on the discretized LNSE operator.

plot_EVP2d.m: plot selected eigenvalues and eigenfunc-tions of the 2D BiGlobal EVP

(2) Run the code EVP2d.m to obtain the eigenspectrum ofEq. (61). Verify that it is identical to that shown in Fig. 10.The amplitude functions corresponding to the unstableTollmienSchlichting (TS) mode and those of the leastdamped member of the P family are shown in Fig. 19. Theeigenfunctions of the TS mode correspond exactly to thoseobtained by local analysis and are shown as 1D amplitudefunctions in Fig. 11.

(3) Using the code provided, perform calculations for differentvalues of the Reynolds number, Re, using a streamwise pe-riodicity length, Lx 2p. Try Re 5772, and Re 7500and compare your results with those obtained by solvingthe OrrSommerfeld equation in which a 2p/Lx is explic-itly imposed [38,86].

(4) Modify the code provided to solve the 2D EVP for theBlasius boundary layer. The 3 3 system is still solved,but you need to modify the wall-normal domain, includinga mapping function, as well as the base flow (and itsderivatives!) to include both a streamwise, Ux; y, and awall-normal, Vx; y, velocity component. Use the bound-ary layer displacement thickness as the lengthscale toreproduce the near-critical conditions of Mack [87]:Re 998, a 0.309; at those parameters x 0.36412129 0.00796250i [88]. Compare the eigenfunctions with thoseshown by Akervik et al. [89].

(5) Extend the code provided to solve the full (4 4) system(50)(53), including spanwise flow perturbations. Use therectangular duct to test your code and references [75,90] tocompare your results.

6 Numerical Solution Methods

6.1 One-Dimensional Eigenvalue Problems. (See theSupplemental Data tab for this paper on the ASME DigitalCollection.) The solution of the eigenvalue problems (58) and(61) introduced in Secs. 5.3.1 and 5.3.2, respectively, implies two

Fig. 18 The eigenvector corresponding to (k/p)25 34 in the so-lution of Eq. (59). Shown are five isolines between zero andmaximum (solid line) and between minimum and zero (dashedline).



steps: spatial discretization, which converts the linear operatorsassociated with the temporal or spatial EVPs to linear or nonlinearmatrix eigenvalue problems, respectively, and solution of the lat-ter using an appropriate algorithm.

Spatial discretization may be performed using spectral meth-ods, e.g., collocation based on the Chebyshev GaussLobattopoints, finite-difference methods, ranging from standardsecond-order centered to compact [91], dispersion relation

Fig. 19 Amplitude functions corresponding to the unstable TollmienSchlichting mode (left column) andthe least stable eigenmode member of the P family (right column) in PPF flow at Re5 104, a5 1. Upper tolower rows: u^(x,y), v^ (x,y), and p^ (x,y).



preserving (DRP) [92] or stable very high-order finite differen-ces of order q (FD-q) [84,93], low-order finite-element orhigh-order spectral-element [94] methods; see Ref. [12] forfurther details.

As far as the solution of the matrix eigenvalue problem isconcerned, the QZ algorithm [95] can be used for this one-dimensional eigenvalue problem. Although the memory and run-time requirements of this algorithm scale with the square and thecube of the matrix leading dimension, respectively, modern hard-ware can easily deliver converged solutions of at least the mostinteresting leading eigenmodes. When the leading dimensionbecomes rather large, e.g., in case of secondary instability analysisin which several Floquet coefficients need be calculated in acoupled manner, iterative algorithms for the recovery of the lead-ing eigenvalues come into consideration. Such algorithms will beoutlined in the context of global stability analysis.

In practical terms, the linear algebra work is undertaken usingsoftware packages such as the Open Source LAPACK library,or proprietary packages which are equivalent in terms of theirfunctionality as far as the solution of the problems at hand is con-cerned, such as the cross-platform NAG

VRlibrary or the platform-

specific Intel MKLVR

or the IBM ESSLVR

linear algebra software.Finally, MATLAB can be used in order to obtain both the entire

eigenspectrum, using the direct approach, or the most interestingpart of it, employing iterative means.

Figure 20 shows a comparison between results for the relativeerror in the leading eigenvalue of PPF at Re 104, a 1, obtainedusing the classic spectral method of Orszag [38] and a suite of theabove mentioned high-order finite difference methods. It can beseen that, at the same order of accuracy, the FD-q method [84]outperforms all of its peers: a preset level of accuracy, denoted asex in this figure, is arrived at earlier in terms of number of discre-tization nodes N when use of the FD-q method is made as opposedto any of the standard high-order finite-difference methods, Pade,SBP, or DRP methods. Conversely, if a given level of discretiza-tion is available, indicated by a constant value of the number ofdiscretization nodes in the figure, the relative error in the FD-qapproximation is substantially smaller than that in any of its peers.When the order q increases, the results of the FD-q methodconverge toward those of the spectral method, albeit using

banded matrix discretization of the operator. This offers the poten-tial to use sparse linear algebra operations without sacrificingaccuracy, which makes the FD-q method an excellent candidate todiscretize the linearized NavierStokes operator in higher spatialdimensions.

6.2 Two- and Three-Dimensional Eigenvalue Problems.The solution of a multidimensional eigenvalue problem alsoimplies two steps: spatial discretization, which converts the linearoperator into a linear or nonlinear matrix eigenvalue problem andsolution of the latter using an appropriate (typically iterative)algorithm.

The same spatial discretization method used for local analysismay also be used for the 2D (or 3D) eigenvalue problem. Spectralcollocation based on the Chebyshev GaussLobatto points, finite-difference, finite-volume, finite-element, and spectral-elementmethods have all been employed in this context.

6.3 Matrix-Forming Versus Time Stepping. A dichotomyexists in terms of solving the multidimensional eigenvalue prob-lem by extending the techniques used in local stability analysisand forming a multidimensional matrix (matrix-forming) or bydownsizing and expanding ag direct numerical simulation algo-rithm to obtain solutions to the multidimensional (direct oradjoint) EVP by time-stepping techniques. A short critical dis-cussion of advantages and disadvantages of either methodologyfollows.

The main advantages of the matrix-forming approach rely inprogramming simplicity and flexibility. Given a 1D differentiationmatrix D Cnn it is straightforward to form any of the linearone-dimensional operators discussed in this paper. In two or threespatial dimensions the Kronecker product of matrices, to be intro-duced shortly, can be used. An additional key advantage of amatrix-forming approach is its flexibility: incompressible or com-pressible, viscous or inviscid operators may be programmed in astraightforward manner into the same code, thus minimizing theamount of validation and verification work necessary when a newproblem is to be tackled. The main disadvantage of the matrix-forming approach is the curse of dimensionality associated withmatrix storage and inversion. If 1m 5 is the number of equa-tions that need be solved, memory and CPU time requirements fordirect matrix operations scale as shown in Table 2.

The theoretical scalings shown in Table 2 are encountered innumerical solutions of eigenvalue problems described by densematrices. Treating the matrix as dense is the only option whenspectral discretization of the spatial LNSE operator is employed.Use of alternative spatial discretization methods, which result inbanded or sparse matrices, permits reducing the high cost of directinversion shown in Table 2 through the use of sparse matrix linearalgebra. As an example, the FD-q methods discussed earlier resultin the following scalings:

Memsparse q 1N 1

a

Memdense (62)

and

Timesparse q 1N 1

b

Timedense (63)

Fig. 20 Relative error in the computation of the leading eigen-mode in PPF at Re5 104, a5 1, using spectral collocation (circlesymbol), a suite of common high-order finite-difference meth-ods, and the recently proposed FD-q finite-difference method[84].

Table 2 Memory and CPU time requirements for dense-matrixoperations in LST

Memory CPU time

1D: O((mn)2) O((mn)3)2D: O((mn)4) O((mn)6)3D: O((mn)6) O((mn)9)



with a 2.7 and 3 b 4 for 8 q 24, determined empirically[84]. This gives the high-order finite-difference method a strongcompetitive advantage over the spectral spatial discretization ofthe multidimensional EVP.

When use of matrix formation is prohibitively expensive, thealternative time-stepping approach may be used. No memoryavailability or memory management issues are present in thetime-stepping methodology, since operations are performed inthe same manner as those of the underlying DNS code used.Relatively minor code modifications are required; steady andtime-periodic flows may be analyzed with one code which maytime-advance either the LNSE or the full NavierStokes equa-tions. On the other hand, the flexibility of the matrix-formingapproach is lost and every time one needs to solve a different sta-bility problem, an appropriate (validated) DNS code needs to beavailable. Incompressible and compressible flow instability prob-lems need to be treated separately. More subtly, the length oftime-integration in a time-stepping algorithm is a priori unknown,while the efficiency of the method is progressively lost withincreasing mKryl, the dimension of the subspace in which theeigenvalue problem is projected. Only the matrix-formingapproach will be discussed in this article; an extensive discussionof both approaches may be found in Ref. [12]. Guidelines for thecoupling of a time-stepping approach with a direct simulationcode are offered in the recent work of Gomez et al. [100].

6.3.1 The Kronecker Product. A key element in the matrix-forming approach in two spatial dimensions is the Kroneckerproduct, MA B, where matrices A 2 Cmn and B 2 Crs rep-resent discretization in either of the two spatial directions. ThenM 2 Cmrns discretizes in both dimensions in a coupled mannerand is defined as

M A B a11B a12B a1nBa21B a22B a2nB... ..

. . .. ..

.

am1B am2B amnB

0BBB@

1CCCA (64)

The extension to three spatial dimensions is straightforward. How-ever, note that the dimensions of the two-dimensional matrix Mare now mr ns, which all but precludes using direct methodsbased on the QZ algorithm for the recovery of its eigenvalues.The most used alternative for the computation of an interestingsubset of eigenvalues in multiple spatial dimensions is based onsubspace iteration, notably the Arnoldi algorithm. However,before introducing this algorithm, some thoughts on the propertiesof matrix M are appropriate.

6.3.2 Sparsity Patterns in Global Instability Analysis. Thenature of the global eigenvalue problem to be solved determines the

choice of both spatial discretization and EVP solution approach to befollowed. In this respect, it is instructive to examine the sparsity pat-tern of relevant discretized linear operators. Figure 21 shows thesepatterns (obtained using the spy function of MATLAB) for the Lapla-cian, the incompressible BiGlobal eigenvalue problem in a two-component basic flow (the lid-driven cavity [96]) and the compressi-ble BiGlobal eigenvalue problem pertinent to a three-component ba-sic flow (the swept leading edge boundary layer [97]). All threeproblems have been discretized using the spectral collocation methodbased on the Chebyshev GaussLobatto grid, which will be outlinedin the Appendix and have been plotted using the same scale, in orderfor relative differences to be appreciated.

The first point concerns the symmetric nature of the Laplacian,as opposed to the nonsymmetric nature of both of the other twoproblems, which is immediately obvious in Fig. 21. This is a keyconsideration when choosing an algorithm for the solution of theEVP: what will work nicely for the Laplace/Poisson/HelmholtzEVP may not converge as rapidly or not work at all for the globalstability problem. Interestingly, if the SVD of the stability matrixis sought, then the symmetry property may be recovered andmethods for the Laplace equation may also be employed.

The second point to be made is that the sparsity pattern of theglobal EVP is affected by two factors, whether incompressible orcompressible flow instability is monitored, and whether (two-dimen-sional) one-, two-, or three-component flow instability is addressed.While a rather large number of zero elements may be seen in theanalysis of a single basic flow velocity component in incompressibleflow (e.g., the rectangular duct [75]), the nonzero elements predomi-nate when global instability analysis of compressible flow is per-formed, in which all three velocity components are present (e.g., theswept leading edge boundary layer [97]). Consequently, sparse linearalgebra software that performs nicely for the former may becomeuncompetitive for the latter class of problems, for which the comput-ing cost of the solution approaches that of a dense approach.

This leads naturally to the last and probably most significantpoint that can be deduced from the results presented: if thematrix-forming approach is to be made competitive to other meth-odologies for the solution of the global EVP, the sparsity patternneeds to be increased substantially. This could be achieved bysubstituting the spectral collocation approach with a low-order fi-nite-difference method. However, the loss of accuracy of thefinite-difference method would not compensate for the increase ofefficiency. What is really needed is a finite-difference method thatis (very) high-order accurate (and stable) and at the same timerequi

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