2011 AIM ED 03 Theofilis Notes

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ADVANCED INSTABILITY METHODS (AIM)- EDUCATION 1sp


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ADVANCED INSTABILITY METHODS(AIM) - EDUCATION

International Graduate School on Stability,Transition to Turbulence and Flow Control

Vassilis TheofilisSchool of Aerospace Engineering, Universidad Politecnica de Madrid, Pza. CardenalCisneros 3, E-28040 Madrid, Spain


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CONTENTS

1 Numerical Methods for Linear Instability Analysis 1

1.1 Introduction - The full picture 1

1.2 Local theory: the LNSE in 1D 4

1.2.1 The Orr-Sommerfeld equation 5

1.2.2 The Rayleigh equation 5

1.2.3 Physical interpretation 5

1.3 Numerical solution of the 1D EVP 7

1.3.1 The need for accuracy 7

1.4 The PSE 8

1.5 Global theory: the LNSE in 2D 9

1.5.1 The (direct) BiGlobal EVP 9

1.5.2 The adjoint BiGlobal EVP 10

1.5.3 The 2D Helmholtz EVP 11

1.6 Numerical solution of the 2D EVP 111.6.1 Matrix-forming or Time-stepping 11

1.6.2 The Kronecker product 11

1.6.3 On the sparsity patterns in global instability analysis 12

1.6.4 On iterative solutions of the EVP - the Arnoldi algorithm 13

1.6.5 The linear algebra work 13

1.7 The PSE-3D 13

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vi CONTENTS

1.8 Global theory: the LNSE in 3D 14

1.8.1 The TriGlobal EVP 14

1.8.2 The 3D Helmholtz EVP 14

2 Numerical Solution Methods 17

2.1 Spatial discretization 17

2.1.1 Collocation solution of linear differential operators 17

2.1.2 Spectral Collocation 18

2.1.3 The FD-q high-order finite-difference method [22] 25

3 Results 29

3.1 LNSE in 1D: The Plane Poiseuille Flow 29

3.1.1 Tutorial Exercise I 30

3.2 LNSE in 2D: Preliminaries - the 2D Helmholtz EVP 303.2.1 Tutorial Exercise II 30

3.3 LNSE in 2D: The BiGlobal eigenvalue problem 30

3.4 The PSE-3D 31

3.5 LNSE in 3D: Preliminaries - the 3D Helmholtz EVP 31

References 37


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CHAPTER 1

NUMERICAL METHODS FOR LINEARINSTABILITY ANALYSIS

1.1 INTRODUCTION - THE FULL PICTURE

A short reminder of theoretical concepts underlying global linear instability analysis isnecessary in order to set the scene for the discussion that follows. Our concern is with the

development in time and space of small-amplitude perturbations superposed upon a given

laminar flow. This can be described exactly by the linearized Navier-Stokes, continuity

and energy equations, without the need to invoke the parallel (or weakly-non-parallel) flow

assumption: the flow analyzed with respect to its global stability may be any laminar two-

or three-dimensional solution of the equations of motion, as well as flows with strong

dependence on two inhomogeneous- and weakly varying along the third spatial direction.

The respective theoretical concepts are referred to as BiGlobal, TriGlobal and PSE-3D

analyses. Table 1.1. classifies and refines the different kinds of linear stability theory,

demarcating the boundaries between local analysis based on variants of the Orr-Sommerfeld

equation (OSE), non-local analysis based on the standard Parabolized Stability Equations

(PSE) [21] andthe three aforementioned versionsofglobal linear theory; symbols appearingto be defined shortly.

Linearization of the equations of motion may be performed around steady or unsteady

laminar basic flows, q = (, u, v, w, T)T. This is to be contrasted against some of thecurrent global instability literature which concerns time-averaged turbulent flows, which

will not be dealt with in the present article, although some comments on the applicability

of the theory, supported by the still scarce evidence in the literature, will be made in what

follows.

AIM-Education. By XXX

ISBN A-BBB-CCCCC-D c2011 The Authors

1


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2 NUMERICAL METHODS FOR LINEAR INSTABILITY ANALYSIS

Basic flows in complex geometries are typically provided by two- or three-dimensional

direct numerical simulations, potentially exploitingspatial invariance. Steady laminar flows

exist only at low Reynolds numbers, but numerical procedures exist for the recovery of basic

flows also at conditions where linear global instability would be expected, e.g. based on

mirroring the solution computed under the imposition of symmetries, continuation [28]or selective frequency damping [1]. In using the term small-amplitude perturbations, the

decomposition

q = q+ q + c.c., 1, (1.1)

is assumed, and solutions to the initial-value-problem

B(Re,Ma, q)dq

dt= A(Re,Ma, q)q, (1.2)

are sought. Complex conjugation is introduced in (1.1) since both the basic state, q and the

total field, q, are real while the perturbations, q, are in general complex. Unless otherwise

stated, the c.c. terms will be omitted in what follows, in order to simplify presentation, buttheir existence in the full expansion should be borne in mind.

Specific comments on the dependence of these quantities on the spatial coordinates, x,and time, t, will be made in what follows. The operators A and Bare associated with thespatial discretization of the linearized equations of motion and comprise the basic state,

q(x, t) and its spatial derivatives. In case of steady basic flows, the separability betweentime- and space coordinates in (1.2) permits introducing a Fourier decomposition in time,

q = qei, (1.3)

with a phase function, leading to the generalized matrix eigenvalue problem

Aq = Bq. (1.4)

Here matrices A and B discretize the operators A and B, respectively, incorporating theboundary conditions; q(x; t) = (, u, v, w, T)T is the vector comprising the amplitude

functions of linear density, velocity-component and temperature or pressure perturbations.The eigenvalue problem adjoint to (1.4) may also be derived, after suitable definition of

an inner product, typically associated with perturbation energy in incompressible [48] and

compressible[39] flow, and enforcement of the bi-linear concommitant to zero [38]. Both

the direct EVP (1.2) and its associated adjoint describe a modal global linear instability

scenario, applicable to the dynamic behavior of the linearized equations of motion at the

asymptotic limit t .On the other hand, re-writing (1.2) as

dq

dt= Cq, (1.5)

withC = B1A, the autonomous system (1.5) has the explicit solution

q(t) = eCt

q(0) (t)q(0). (1.6)Here q(0) q(t = 0) and the matrix exponential,(t) eCt, is known as thepropagatoroperator[14]. A solution of the initial value problem (1.5) distinguishes between the limits

t 0 and t ; while the latter limit may be described by the eigenvalue problem (1.4),growth, , of an initial linear perturbation, q(0), may be computed at all times via

2 =

eC

teCtq(0), q(0)

(q(0), q(0))=

((t)(t)q(0), q(0))

(q(0), q(0)). (1.7)


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INTRODUCTION - THE FULL PICTURE 3

Implicit here is the definition of an inner product, (, ), and the associated adjoints, andC

, of the operator and matrix C, respectively [38]. The discussion is completed by

introducing the singular value decomposition (SVD) of the propagator operator

(t) e

Ct

= UV

. (1.8)Here the unitary matricesV andU respectively comprise (as their column vectors) initial

and final states, as transformed by the action of the propagator operator, while is diagonal

and contains the growth associated with each initial state as the corresponding singularvalue. Given that (1.8) may be formed at all times t, without the need to introduce modalperturbations in this coordinate, the framework in which global linear analysis is performed

in this context is referred to as non-modal, solution of (1.7) permitting study of both modal-

and non-modal perturbation growth. Note also that the operator appearing in (1.7)

is symmetric, which has important consequences both for its computation and in relation

with the interpretation of its results.

In case the basic flow has an arbitrary time-dependence, the propagator operator, (t),may also be defined [15] and (1.6) may be generalized as

q(t0 + ) = () q(t0). (1.9)

The propagatormay be understoodas the operator evolving the small-amplitudeperturbation

from its state at time t0 to a new state at time t0 + . If the time-dependence of the basicstate is periodic, t0 : q(x; t0 +T) = q(x; t0), the propagator is denoted as the monodromyoperatorand it isalso Tperiodic,(t0 +T) = (t0); the monodromy operator is definedby [27]

= exp

t0+T

t0

C (q(x; t)) dt

. (1.10)

Solutions to the instability problem, indicating the development of small-amplitude pertur-

bations during one period of evolution, are obtained through Floquet theory, which seeks

the eigenvalues of the monodromy operator, also known as Floquet multipliers [6], . Tothis end, the monodromy operator is evaluated at a time T and the eigenvalue problem

(T)q = q (1.11)

is solved. The Floquet multipliers can also be expressed in terms of the Floquet exponents,

, as = eT, which identifies || = 1 as a bifurcation point and indicates that

|| < 1 : periodic flow stability, (1.12)

|| > 1 : periodic flow instability. (1.13)

Barkley et al. [5] provide an up-to-date description of global instability analysis of time-

periodic flows, with special focus on non-modal analysis and recovery of optimal (global)

disturbances arising from localized regions of classic linear (local) instability. The dis-

cussion is completed by reference to the dimensionality of the basic state and the ensuing

global instability analysis methodologies.The preceding discussion is a reasonably complete account of linear stability theory, as

is presently understood in terms of both local and global analysis, as well as in terms of

both modal and non-modal theory. Further discussion of this exciting research area may

be found in [52] and references therein. For the purposes of the course the present lecture

will mainly focus on numerical solution of the one-dimensional eigenvalue problem and

elements of the algorithms necessary to extend the analysis to multiple spatial dimensions

using a matrix-forming approach.


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Table 1.1. Classification of global linear theory approaches

Denomination Basic State Assumptions Phase Function

GlobalTriGlobal - q(x1, x2, x3) tPSE-3D 1q 2q, 3q q(x

1, x2, x3)

(x1)dx

1 t

BiGlobal 1q = 0 q(x2, x3) x1 t

Non-Local PSE 1q 2q; 3q = 0 q(x1, x2)

(x1)dx

1 + x2 t

Local OSE 1q = 3q = 0 q(x2) x1 + x3 t

1.2 LOCAL THEORY: THE LNSE IN 1D

We now focus on some of the concepts shown in Table 1.1., starting from the situation

pertinent to a basic flow which may be described by an one- or two-component, one-

dimensional velocity vector, (u(y), 0, w(y))T; an one-dimensional basic pressure, p(y)completes the definition of the basic flow vector q. Note that in order to simplify notation,

the coordinates (x1, x2, x3) have been substituted by (x,y,z). Under these assumptions,the Ansatz (1.1) becomes

q(x,y,z,t) = q(y) + qei(x+zt) + c.c. (1.14)

Inserting (1.14) into the full Navier-Stokes and continuity equations and linearizing,

three systems are obtained, at O(1), O() and O(2), respectively. The system at O(1)defines the basic state and is satisfied either exactly, e.g. in the case of the plane Poiseuille

flow, or approximately, e.g. in the case of boundary-layer flows. The system at O(2) isneglected, on account of the smallness of the expansion parameter . The system at O()are the linearized Navier-Stokes equations (LNSE) which, after introduction of the modal

Ansatz (1.3), become a generalized eigenvalue problem (1.4); in incompressible flow thisEVP reads

iu + vy + iw = 0, (1.15)

L1du uyv ip + iu = 0, (1.16)

L1dv py + iv = 0, (1.17)

wy v + L1dw ip + iw = 0. (1.18)

where L1d 1Re

d2

dy2

2 + 2

iu iw. The analogous expansion and

LNSE for compressible flows in which the local theory assumptions are satisfied may be

found in Mack [36]. Instability analysis based on (1.15-1.18) has been used extensively

to describe instability of free-shear and boundary-layer flows. It is reiterated here that thekey assumption made in the context of local theory is that of a parallel-flow, according

to which the basic flow velocity component v, along the spatial coordinate normal to themean flow direction, y, is neglected. Consequently, the basic state may comprise up to twovelocity components, namely the streamwise u(y) and the spanwise w(y), both of whichare functions of one spatial coordinate alone.

The boundary conditions completing the description of linear instability in the context

of local theory are the viscous boundary conditions u = v = w = 0 at solid walls or in the


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LOCAL THEORY: THE LNSE IN 1D 5

"far-field" (where the basic flow velocity vector assumes a constant value), alongside with

the boundary condition vy = 0, derived from the continuity equation.

1.2.1 The Orr-Sommerfeld equationThe LNSE in 1D may be re-written in the form of a single ordinary differential equation, in

which disturbance pressure is eliminated, as no boundary conditions are directly available

for this quantity. Eliminating pressure perturbation by cross-differentiation of (1.16) and

(1.18), using continuity (1.15) and substituting the result into (1.17) results in the Orr-

Sommerfeld equation for the disturbance velocity component v(y),

i

Re

v(4) 22v

+ 4v

+ (u )

v

2v

u

v = 0. (1.19)

Primes denote differentiation with respect to y andSquires theorem [34] hasbeen used asjustification of focusing on two-dimensional perturbations, for which = 0. The boundaryconditions v = v

= 0 at both ends of the integration domain complete the description of

the problem.

1.2.2 The Rayleigh equation

Theinstabilityof basic flows having inflectional profiles, e.g. theshearlayeror theboundary

layer profile inside a laminar separation bubble, may be analyzed by solution of the Rayleigh

equation

(u )

v

2v

u

v = 0, (1.20)

which may be obtained from the Orr-Sommerfeld equation in the limit Re .The no-penetration boundary condition v = 0 completes the description of the instabilityproblem. From a physical point of view, the implicit assumption in this class of analysis is

that viscosity does not affect stability. Precisely the vanishing of viscosity has the potentialfor introduction of singularities which need to be circumvented in the numerical solution.

1.2.3 Physical interpretation

The phase function in local stability analysis is 1D = x + z t. This harmonicAnsatz satisfied by the linear perturbations along the streamwise and the spanwise direc-

tions implies that these perturbations assume the form of planar or oblique waves. The

parameters describing these waves have a different physical interpretation, depending on

whether instability analysis is performed in the so-called temporal or spatial context.

1.2.3.1 Temporal stability In temporal linear stability analysis, the quantities and

appearing in (1.15-1.18), (1.19) or (1.20) are real wavenumber parameters, related withperiodicity wavelengths, Lx and Lz , along the x and z directions, respectively, by =

2Lx

and = 2Lz . The eigenvalue sought is and is in general complex, since the matrixcoefficients are complex. The real part of is interpreted as the instability wave frequency,while its imaginary part is the (temporal) amplification or damping rate of the wave. The

equivalent formulation of linear two-dimensional waves

q1d = q1dei(xct) + c.c. = q1de

i[x(cr+ici)t] + c.c. (1.21)


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introduces the complex eigenvalue c, the real part of which may be interpreted as beingrelated to the phase speed of the wave while its imaginary part is again the amplification or

damping rate of the perturbation.

Instability may initially be thought of as a linear superposition of monochromatic waves

which develop in streamwise (and spanwise, if = 0) periodic boxes of length Lx = 2/(and width Lz = 2/). Each wave travels inside the basic flow at its own constantphase speed cr and is growing or being attenuated at its own rate ecit. As time grows, theexponential nature of amplification/damping results in just one of these waves being the

most significant from the point of view of determination of the flow dynamics; this is the

leading eigenmode, calculation of which is the central part of the instability analysis.

The complete instability analysis involves varying the Reynolds and wavenumber pa-

rameters and identifying the frequency and amplification/damping rate of the leading eigen-

mode. Plotting the i = i(Re,) surface, two curves of interest may be defined: theneutral curve, which is the intersection of the above surface with the plane i = 0 and de-fines the boundary of stability/instability of the flow, and the maximum amplification curve,

which is the projection of the maximum values ofi on the plane i = 0. Both curves may

be utilized to identify linear instability characteristics in experiment or simulation.

1.2.3.2 Spatial stability The assumption of a constant streamwise wavenumber maybe relaxed and solutions of (1.15-1.18), (1.19) or (1.20) may be sought for a constant real

monochromatic frequency , treating as the eigenvalue. Under this point of view, linearperturbations are written as

q1d = q1dei[(r+ii)x+zt] + c.c. (1.22)

i being the (spatial) amplification or damping rate of the wave, while its wavenumber isr . This instabilityanalysis framework is preferred by experimentalists, since identificationof perturbation characteristics is more straightforward when dealing with monochromatic

waves. From a theoretical/numerical point of view the eigenvalue problem to be solved is

now nonlinear.

In a manner analogous to temporal theory, the most significant instability wave is that

having the largest amplification rate i, determination of which becomes the key objectiveof the theory. Besides calculating the (same as that yielded by temporal analysis) neutral

curve, spatial instability analysis results may also be used for laminar-turbulent transition

location prediction, using the eN method discussed in detail by Mack [36].

1.2.3.3 Spatio-/temporal stability The dichotomy of the assumptions of temporaland spatial stability is overcome by considering the most general case of both the and parameters being simultaneously complex. Monitoring results of numerical solutionof the ensuing problem, details of which are given elsewhere in this course, it becomes

straightforward to differentiate between two situations which may predominate in a laminar

flow after a perturbationhas been introduced to it. Aflow will be calledconvectively unstableor stable if the perturbation is convected away from the point of its introduction while it

is being amplified or damped, respectively, while if the perturbation remains at the point

where it was introduced in the flow and grows in time, it is called absolutely unstable.

The review of Huerre & Monkewitz [25] is essential reading in this context and discusses

several parallel flows to which this concept is applicable. The extension of the absolute-

/convective or spatio-/temporal instability analysis concept to quasi-parallel flows led to the

introduction of one of the three different contexts of the terminology global instability that


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NUMERICAL SOLUTION OF THE 1D EVP 7

are still in use in the literature, although their theoretical bases are distinct; the reviews of

Theofilis [51, 52] discuss details.

1.3 NUMERICAL SOLUTION OF THE 1D EVP

The solution of the eigenvalue problems introduced implies two steps, spatial discretization

which converts the linear operators associated with the temporal or spatial EVPs to linear

or nonlinear matrix eigenvalue problems, respectively, and solution of the latter using an

appropriate algorithm.

Spatial discretization may be performed using spectral methods, e.g. collocation based

on the Chebyshev Gauss-Lobatto points, finite-difference methods, ranging from standard

second-order centered to compact [31], Dispersion Relation Preserving (DRP) [49] or stable

very high-order finite differences of order q(FD-q) [22], low-order finite-element or high-order spectral-element [27] methods.

As far as the solution of the matrix eigenvalue problem is concerned, the QZ algorithm

[17] can be used for this one-dimensional eigenvalue problem. Although the memory and

runtime requirements of this algorithm scale with the square and the cube of the matrix

leading dimension, respectively, modern hardware can easily deliver converged solutions

of at least the most interesting leading eigenmodes. When the leading dimension becomes

rather large, e.g. in case of secondary instability analysis in which several Floquet coef-

ficients need be calculated in a coupled manner, iterative algorithms for the recovery of

the leading eigenvalues come into consideration. Such algorithms will be outlined in the

context of global stability analysis.

In practical terms, the linear algebra work is undertaken using software packages such

as the Open Source LAPACK library, or proprietary packages which are equivalent in terms

of their functionality as far as the solution of the problems at hand is concerned, such as

the cross-platform NAG R library or the platform-specific Intel MKL R or the IBM ESSLR linear algebra software. Finally, MATLAB R can be used in order to obtain both the

entire eigenspectrum, using the direct approach, or the most interesting part it, employingiterative means.

1.3.1 The need for accuracy

A comment regarding the importance of accuracy in stability calculations closes the present

section and provides a fitting introduction to the next. For the sake of argument, attention

is focused on the temporal EVP. Expanding (1.14) for any of the disturbance vector com-

ponents, say for the streamwise perturbation velocity u, and focusing on temporal analysisof two-dimensional traveling waves, the following expression results

u(x,y,t) = u(y) + {ur(y) cos[(x crt)] ui(y)sin[(x crt)]} ecit, (1.23)

where subscripts r and i denote real and imaginary parts of the respective quantities.The time and space development of the observable instability, namely the l.h.s. of equation

(1.23), is fully described by the r.h.s. of this equation. The ingredients of the latter are

the basic flow, u, which is taken to be known to arbitrary accuracy, the space and timecoordinates, x and t, which are also known at each measurement point and time and theresult of the solution of the eigenvalue problem, c = cr + ici or, equivalently, = r +ii. Precise theoretical knowledge of the sought eigenvalue completes the description


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of the perturbations and permits comparison with frequency results extracted from the

observed signal by Fourier transform and amplification rates computed from the slope of

the logarithmic time-derivative of (1.23) in a simulation, or from that of measured quantities

at different time instances. Errors in the calculation of either quantity amplify exponentially

and may lead to erroneous conclusions regarding the validity of the analysis, e.g. in termsof its capacity to predict transition location using empirical means such as the eN method.

1.4 THE PSE

Efforts to remedy the inconsistencies arising from the parallel-flow approximation, as well

as the desire to substitute empirical transition prediction methods by theoretically-founded

approaches, have resulted in improved descriptions of the instability problem for boundary-

layer and shear-layer flows. The most successful approach has been based on the utilization

of a multiple-scales approach for the description of instability. In this context both the

basic state and the perturbations are taken to depend, besides y, on the streamwise spatialcoordinate, x, as well. However, the key assumption of multiple-scales approaches is thattwo scales exist along x, one on which the basic state depends and a different one, onwhich the perturbations are taken to develop. The first is taken to be the coordinate x itself,while the second is x, with the small parameter = O(Re1). This multiple-scalesapproximation (also called WKBJ approach, after the initials of the authors who introduced

it) has been used prior to the introduction of the PSE in conjunction with solutions of the

Orr-Sommerfeld equation in order to improve prediction of the critical conditions of Blasius

boundary layer flow. However, the missing ingredient in early attempts along this path was

the idea of parabolizing the stability equations; this was independently introduced by Hall

[18] for the study of stability of Gortler vortices on a curved surface and by Herbert for the

study of the stability of the Blasius boundary layer [7].

Formally one proceeds using a consistent expansion of all flow quantities in terms of a

small parameter, typically the inverse of the Reynolds number. The particular form of the

Ansatz (1.1) in the case of PSE is

q(x,y,z,t) = q(x, y) + q(x, y) ei[Rx()d+zt] + c.c. (1.24)

Note that q(x, y) = (u, v, w, p)T(, y), which permits the introduction of the small butessential non-zero wall-normal velocity component v into the instability analysis. Secondderivatives of the perturbations with respect to the streamwise flow direction are neglected.

Analysis of the resulting equations [32, 33] shows that the associated ellipticity of the

LNSE is eliminated (albeit not fully - see original references for details) and the stability

problem may be parabolized, marching the perturbation equations along the streamwise

flow direction. This is the origin of the Parabolized Stability Equations (PSE) nomenclature

being utilized to differentiate this stability analysis approach from earlier efforts within the

same WKBJ class. The eigenvalue problem nature of the stability equations is lost in a PSE

context, although it is straightforward to relate partial derivatives of the perturbation flowquantities with the wavenumber and amplification rates obtained from solution of EVPs at

successive downstream flow locations. Note also that introduction of nonlinearity into the

instability analysis is now possible in a self-consistent manner. One of the most interesting

results of the classic paper by Bertolotti et al. [7] has been the excellent agreement of

nonlinear PSE and spatial DNS results for transition on the flat plate boundary layer, results

of the nonlinear PSE being orders-of-magnitude less expensive to obtain compared with

those delivered by DNS. An equally important result was obtained by Haynes& Reed [19] in


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GLOBAL THEORY: THE LNSE IN 2D 9

their theoretical prediction of the experimentally discovered crossflow vortex stabilization

by Distributed Roughness Elements [46, 47].

Numerical solution of the PSE will not be discussed in detail here. However, it is a

straightforward task to derive the equations and set up an algorithm for their solution, based

on solution of an Ordinary Differential Equation at each downstream location, combinedwith a simple Euler scheme along the marching spatial direction. As discussed by Li &

Malik [32] care should be taken in order to avoid the residual ellipticity in the PSE equations

by choosing marching steps that are not too short for this ellipticity to destroy the solution

procedure.

1.5 GLOBAL THEORY: THE LNSE IN 2D

1.5.1 The (direct) BiGlobal EVP

When the basic flow the stability of which is to be studied depends on two out of the

three spatial directions and is independent of the third, q = [u, v, w, p]T(x, y), the analysis

proceeds following the Ansatz

q(x,y,z,t) = q(y, z) + q(y, z)ei[xt] + c.c. (1.25)

The disturbances are three-dimensional, but a sinusoidal dependence along the homoge-

neous x-direction is assumed, with the periodicity length Lz = 2/. Upon substitutionof (1.25) into the LNSE the system of equations results

iu + vy + wz = 0, (1.26)

L2du + uyv + uzw + ip iu = 0, (1.27)

(L2d + vy)v + vzw + py iv = 0, (1.28)

(L2d + wz) w + wyv + pz iw = 0, (1.29)

where L2d = 1Re (yy + zz

2) + iu + vy + wz.

Note that, if{.}z i{.} is considered, the operators L2d and L1d become identical

and the PDE-based system (1.26-1.29) reduces to the ODE-based system (1.15-1.18). The

elliptic eigenvalue problem (1.26-1.29)is complemented withadequate boundary conditions

for the disturbance variables, typically no-slip on solid walls for the velocity components

and the associated compatibility conditions for pressure, if the latter variable is maintained

in the vector sought. Alternatively, pressure may be eliminated to yield the two-dimensional

analogue of the Orr-Sommerfeld equation, as originally done by Tatsumi & Yoshimura [50],

or the two-dimensional analogue of the Rayleigh equation, first presented and solved by

Henningson [20].

The primitive-variable formulation (1.26-1.29) can be written as a two-dimensional

partial-derivative-based generalized eigenvalue problem, which can be recast in matrixform as the (linear) temporal BiGlobal eigenvalue problem,

A q = Bq (1.30)

or the (nonlinear) spatial BiGlobal eigenvalue problem,

A q =2

k=1

kBk q. (1.31)


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The physical interpretation of the parameters and depends on whether the temporal orthe spatial framework has been chosen for the analysis, much like in the classic 1D stability

theory, while the operators A and Bdefining (1.30) may be found explicitly in the literature,e.g. in Paredes et al.[42].

1.5.2 The adjoint BiGlobal EVP

Hill [23, 24] has introduced the utility of adjoints in identifying flow receptivity and sensi-

tivity in both one and two-dimensional flows, while more recently Luchini and co-workers

have put forward the idea of structural sensitivity of global modal perturbations, which

may be identified through a direct-adjoint BiGlobal eigenmode product, and successfully

applied the concept to the lid-driven cavity flow [35] and the cylinder wake [16].

The derivation of the complex BiGlobal eigenvalue problem governing adjoint pertur-

bations is constructed using the Euler-Lagrange identity [38, 13, 8, 43, 35, 16],

q

t+Nq+ p q

+ qp +q

q

t+Nq + p

+ p q

=

t(q q) + j(q, q), (1.32)

as applied to the linearizedincompressible Navier-Stokes and continuityequations. Here

the operator N(q) results from linearization of the convective and viscous terms in thedirect and adjoint Navier-Stokes equations and is explicitly stated elsewhere (e.g. [13]).

The quantities q = (u, v, w)T and p denote adjoint disturbance velocity componentsand adjoint disturbance pressure, andj(q, q) is the bilinear concomitant. Vanishing of theRHS term in the Euler-Lagrange identity (1.32)defines the adjoint linearizedincompressible

Navier-Stokes and continuity equations

q

t+Nq + p = 0, (1.33)

q = 0, (1.34)

Assuming modal perturbations and homogeneity in the spatial direction x eigenmodesare introduced into (1.33-1.34) according to

q = q(y, z)ei(xt). (1.35)

Note the opposite signs of the spatial direction x and time in (1.25) and (1.35) denotingpropagation ofq in the opposite directions compared with the respective one for q. Sub-

stitution of (1.35) into the adjoint linearized Navier-Stokes equations (1.33-1.34) results

in the complex adjointBiGlobal EVP, details of which are omitted here, but will be made

explicit in the next Lecture. For completeness, it is mentioned that an EVP analogous instructure to the direct BiGlobal EVP is obtained, whose linear operator is

L =1

Re

2

y 2+

2

z 2 2

iu + v

y+ w

z. (1.36)

Boundary conditions for the partial-derivative based adjoint EVP may be devised fol-

lowing the general procedure of expanding the bilinear concomitant in order to capture

traveling disturbances [13].


17/45

NUMERICAL SOLUTION OF THE 2D EVP 11

1.5.3 The 2D Helmholtz EVP

Closer inspection of (1.26-1.29) and (1.36) reveals that the two-dimensional Laplace oper-

ator is at the heart of the operator L2d defining the BiGlobal direct and adjoint eigenvalue

problems. When setting up a numerical procedure for the solution of the BiGlobal EVPit is advisable to take advantage of partial validation cases, one of which is offered by the

well-known (multi-dimensional) Helmholtz equation. The eigenvalue problem2

y 2+

2

z 2

+ 2 = 0 (1.37)

in the rectangular membrane domain y [0, Ly] z [0, Lz] is known to have ananalytical solution [38] for the eigenvalues

2ny,nz = 2

nyLy

2+

nzLz

2; ny, nz = 1, 2, 3, (1.38)

and associated eigenfunctions

ny,nz (y, z) = sin(nyy

Ly)sin(

nzz

Lz). (1.39)

The eigenvalue problem (1.37-1.39) is thus useful in assessing the accuracy of the

proposed spatial discretization method, especially in the recovery of the higher eigenval-

ues/eigenfunctions, ny, nz 1.

1.6 NUMERICAL SOLUTION OF THE 2D EVP

The solution of a multi-dimensional eigenvalue problem also implies two steps, spatial

discretization which converts the linear operator into a linear or nonlinear matrix eigenvalue

problem, and solution of the latter using an appropriate (typically iterative) algorithm.

Thesame spatial discretization methodused forlocal analysis mayalso be used for the2D

(or 3D) eigenvalue problem. Spectral collocation based on the Chebyshev Gauss-Lobatto

points, finite-difference, finite-volume, finite-element and spectral-element methods have

all been employed in this context.

1.6.1 Matrix-forming or Time-stepping

A dichotomy exists in terms of solving the multidimensional eigenvalue problem by ex-

tending the techniques used in local stability analysis and forming a multi-dimensional

matrix ("matrix-forming") or by downsizing and expanding a direct numerical simulation

algorithm to obtain solutions to the multi-dimensional (direct or adjoint) EVP by "time-

stepping" techniques. The former approach will be discussed in this presentation, while the

latter is the topic of the following lecture.

1.6.2 The Kronecker product

A key element in the matrix-forming approach in two spatial dimensions is the Kronecker

product, M A B, where matrices A Cmn and B Crs represent discretizationin either of the two spatial directions. Then M Cmrns discretizes in both dimensionsin a coupled manner and is defined as


18/45


M A B =

a11B a12B a1nBa21B a22B a2nB

..

.

..

.

. . ...

.am1B am2B amnB

. (1.40)The extension to three spatial dimensions is straightforward. However, note that the

dimensions of the two-dimensional matrix M are now mr ns, which all but precludesusing directmethods based on theQZ algorithmfor therecovery of itseigenvalues. Themost

usedalternative for the computation of an interestingsubset of eigenvaluesin multiple spatial

dimensions is based on subspace iteration, notably the Arnoldi algorithm. However, before

introducing this algorithm, some thoughts on the properties of matrix M are appropriate.

1.6.3 On the sparsity patterns in global instability analysis

The nature of the global eigenvalue problem to be solved determines the choice of both

spatial discretization and EVP solution approach to be followed. In this respect, it is in-structive to examine the sparsity pattern of relevant discretized linear operators. Figure

1.3. shows these patterns (obtained using the "spy"function of MATLAB R) for the Lapla-cian, the incompressible BiGlobal eigenvalue problem in a two-component basic flow (the

lid-driven cavity[53]) and the compressible BiGlobal eigenvalue problem pertinent to a

three-component basic flow (the swept leading edge boundary layer [55]). All three prob-

lems have been discretized using the spectral collocation method based on the Chebyshev

Gauss-Lobatto grid, which will be outlined in the next section, and have been plotted using

the same scale, in order for relative differences to be appreciated.

The first point concerns the symmetric nature of the Laplacian, as opposed to the non-

symmetric nature of both of the other two problems, which is immediately obvious in figure

1.3.. This is a key consideration when choosing an algorithm for the solution of the EVP:

what will work nicely for the Laplace/Poisson/Helmholtz EVP may not converge as rapidly

or not work at all for the global stability problem. Interestingly, if the SVD of the stability

matrix is sought, then the symmetry property may be recovered and methods for the Laplace

equation may also be employed.

The second point to be made is that the sparsity pattern of the global EVP is affected

by two factors, whether incompressible or compressible flow instability is monitored, and

whether (two-dimensional) one-, two- or three-component flow instability is addressed.

While a rather large number of zero elements may be seen in the analysis of a single basic

flow velocity component incompressible flow (e.g. the rectangular duct [50]) the non-zero

elements predominate when global instability analysis of compressible flow is performed, in

which all three velocity components are present (e.g. the swept leading edge boundary layer

[55]). Consequently, sparse linear algebra software which performs nicely for the former,

may become uncompetitive for the latter class of problems, for which the computing cost

of the solution approaches that of a dense approach.This leads naturally to the last and probably most significant point that can be deduced

from theresults presented: if thematrix-forming approach is to be made competitive to other

methodologies for the solution of the global EVP, the sparsity pattern needs to be increased

substantially. This could be achieved by substituting the spectral collocation approach by

a low-order finite-difference method. However, the loss of accuracy of the finite-difference

method would not compensate for the increase of efficiency. What is really needed is a

finite-difference method which is (very) high-order accurate (and stable) and at the same


19/45

THE PSE-3D 13

time requires much shorter stencils than those used in the spectral collocation approach.

Such a method will be introduced in the next section.

1.6.4 On iterative solutions of the EVP - the Arnoldi algorithm

By far the widest-employed and most successful approach for numerical solution of large-

scale eigenvalue problems is the Arnoldi [4, 45] member of the Krylov subspace iteration

methods. On of the variants of the Arnoldi algorithm [51] converts the generalized EVP

(1.4) into the standard eigenvalue problem

Aq = q, A = (A B)1B, =1

(1.41)

and constructs a Krylov subspace, a Hessenberg matrix Hij and a sequence of Ritz

vectors in an iterative manner, as schematically shown in figure 1.4.. The eigenvaluesof the (much smaller, compared with the original problem dimension) Hessenberg matrix

may be calculated by direct means using the QZ algorithm and are increasingly better

approximations of thoseof the original problem as the Krylov subspace dimensionincreases.

1.6.5 The linear algebra work

Finally, the linear algebra workfor large matrix leading dimensionsmay be undertaken either

in serial manner, if practically possible, or as is more common, in parallel. If operations are

performedin a dense linearalgebra context theOpen SourceScaLAPACK or thecommercial

equivalents offered by the NAG R, Intel MKL R or IBM PESSL R libraries may be used.If sparse linear algebra operations are envisaged, the MUMPS [2, 3] package offers an

alternative.

1.7 THE PSE-3D

The derivation of the parabolized equations in two fully resolved and one slowly-varying

spatial direction, an analysis concept abbreviated as PSE-3D [10], is again based upon the

introduction of a small parameter = O(Re1) and the associated slowly varying spatialcoordinate = x, as was done in the classic PSE approach. By contrast to the PSE, thebasic flow is now q = [u, v, w, p]T( , y , z), i.e. it corresponds to a fully three-dimensionalflow which is strongly dependent on two and weakly on the third spatial direction. A system

of trailing vortices behind a finite-span lifting surface is a flow which would typically be

analyzed by the PSE-3D approach. The Ansatz pertinent to PSE-3D is

q(x,y,z,t) = q( , y , z) + q( , y , z)ei[Rx()dt] + c.c. (1.42)

Substituting (1.42) into the linearized Navier-Stokes equations and neglecting terms of

order 2 the incompressible linear PSE-3D equations are obtained [10, 11, 42]


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(L2d i + ux)u + uyv + uzw + ip +

u

2i

Re

ux + px

i

Re

d

dxu = 0,(1.43)

(L2d i + vy)v + vxu + vzw + py +u 2i

Re vx i

Reddx

v = 0,(1.44)

(L2d i + wz) w + wxu + wyv + pz +

u

2i

Re

wx

i

Re

d

dxw = 0,(1.45)

ux + iu + vy + wz = 0.(1.46)

Hence the PSE-3D can be written in a compact form as

(L2d i) q+ Mq

x=

d

dxNq

(1.47)

where operators L2d, M and N define PDEs dependent on y and z. The term in squarebrackets can be neglected becauseN arises from the viscous terms and is ofO(Re1) as is

the derivative d/dx. Technical details of this class of analysis, which is an active researchtopic, are provided by Paredes et al. [42].

1.8 GLOBAL THEORY: THE LNSE IN 3D

1.8.1 The TriGlobal EVP

Finally, when the base state depends essentially on all three spatial coordinates, q =[u, v, w, p]T(x,y,z), instability analysis proceeds along with the Ansatz

q(x,y,z,t) = q(x,y,z) + q(x,y,z)eit + c.c. (1.48)

Linearization follows and the the eigenvalue and the three-dimensional eigenfunctionsq = [u, v, w, p]T(x,y,z) are determined through solution of the TriGlobal EVP

ux + vy + wz = 0, (1.49)

[L3d ux = i] u uy v uzw + px = 0, (1.50)

vxu + [L3d vy + i] v vzw = 0, (1.51)

wxu wyv + [L3d wz + i] w = 0, (1.52)

with L3d 1Re (xx + yy + zz)uxvywz . TriGlobal analysis is the present-

day state-of-the-art in terms of the challenging nature of the underlying EVP solution; see

Theofilis [52] for a review. In the closing section of these notes brief mention will be made

regarding the ongoing efforts in the authors group to generate an efficient and accurate

matrix-forming approach for the numerical solution of the TriGlobal EVP.

1.8.2 The 3D Helmholtz EVP

As with the analogous two-dimensional eigenvalue problem, the Helmholtz eigenvalue

problem in three spatial dimensions and regular geometries offers the possibility of vali-

dating the Laplace operator part of the TriGlobal EVP solution algorithm. In three spatial

dimensions the Helmholtz eigenvalue problem


21/45

GLOBAL THEORY: THE LNSE IN 3D 15

Figure 1.1. Matrix-forming vs. time-stepping for linear stability in multiple dimensions

2

x2+

2

y 2+

2

z 2

+ 2 = 0 (1.53)

in the domain x [0, Lx] y [0, Ly] z [0, Lz] is known to have an analyticalsolution [38] for the eigenvalues

2nx,ny,nz = 2

nxLx

2+

nyLy

2+

nzLz

2; nx, ny, nz = 1, 2, 3, (1.54)

and associated eigenfunctions

nx,ny,nz (x,y,z) = sin(nxx

Lx)sin(

nyy

Ly)sin(

nzz

Lz). (1.55)

Much like the analogous two-dimensional case, the eigenvalue problem (1.53-1.55) is

thus useful in assessing the accuracy of any given spatial discretization method, especially

in the recovery of the higher eigenvalues/eigenfunctions, nx, ny, nz 1.


22/45


Figure 1.2. Spatial discretization methods for linear stability in multiple dimensions

Figure 1.3. Sparsity patterns of the Laplace operator (left) incompressible BiGlobal EVP (middle)

and compressible BiGlobal EVP (right), discretized using the spectral collocation method.

Compute #A A il And overwrite #A by its LU-decomposition

Compute the entries hi;j of the Hessenberg Matrix %Hm

Initialize#r0 1; 1;y; 1

T; r0 : #r0=jj#r0 jj2;h0;0 jj#r0 jj2; %V0 r0

For j 0;y;m 1 do

Set rj : B; rjUse #A to solve (55)

For i 0;y;j dohi;j : %Vi; rjrj : rj hi;j %Vi

Form hj1;j jjrjjj2

and %Vj1 rj=hj1;j Compute the eigenvalues of Hm %Hmlast row using the QZ

Select an interesting eigenvalue and calculate its Ritz vector q %V;y

Figure 1.4. The Arnoldi algorithm (from [51])


23/45

CHAPTER 2

NUMERICAL SOLUTION METHODS

2.1 SPATIAL DISCRETIZATION

2.1.1 Collocation solution of linear differential operators

The one-dimensional linear eigenvalue problems exposed in the previous section may be

solved by constructing an operator which discretizes an n-th order linear ODE in the form

L(x) = an(x)D(n) + ... + a1(x)D

(1) + a0(x) (2.1)

where D(m) is the mth derivative with respect to the independent variable x anda0, a1,...an are functions of x. A collocation approach requires a discrete grid, xi, andassociated differentiation matrix, both of which will be defined in subsequent sections. For

now, we keep the symbolic form

Lij = an(xi)D(n)ij + ... + a1(xi)D

(1)ij + a0(xi) (2.2)

with D(m)ij the mth derivative matrix corresponding to the collocation points xi and

a0, a1,...an evaluated at xi. Here Lij denotes the contribution of grid point xj to theoperator L at xi.

The Ordinary Differential Equation

Lu(x) = f(x) (2.3)

may then be transformed into



17


24/45

18 NUMERICAL SOLUTION METHODS

j

Liju(xj) = f(xi), (2.4)

that is, a matrix equation

AX = B, (2.5)

withX the vector of unknowns u(xj), the coefficient matrixA constructed with the aidofLij and the RHS vectorB comprising the (known) function values f(xi).

2.1.1.1 Boundary ConditionsBefore (2.5) may be solved the m boundary conditions associated with (2.3) have to be

imposed. One of the ways to accomplish this is by dropping m rows in (2.5) and replac-ing them by rows derived from the boundary conditions. Specifically, a Robin boundary

condition

Dy(x) + y(x) = 0, at x = x0 (2.6)

may be imposed by replacing the appropriate row of (2.5) by the spectral analogue of

(2.6) at x = x0. For the sake of demonstration of the ideas presented, consider the followingproblems.

2.1.2 Spectral Collocation

In this section we discuss construction of collocation differentiation matrices and associated

grids from first principles, and introduce the Chebyshev-Gauss-Lobatto spectral collocation

method to accomplish the task of solving linear ODEs.

Fundamental to a spectral representation of a function on an as yet unspecified grid is

the concept of the basis functions and, associated with it, the expansion coefficients of thefunction. The function value on a gridpoint may be reconstructed through the contribution

ofall expansion coefficients (defined on all gridpoints), weighted accordingly by the basis

function value on the corresponding gridpoint. It may already be anticipated that spectral

methods inherit their accuracy (and their potential miscarriages!) through this "global"

character. We may call "the function" either the set of numbers values-of-the-function on

this grid ("collocation") or, equivalently, the set of numbers values-of-the-coefficients (the

"spectrum") on the same grid. Symbolically,

Function Values Spectrum.

The importance of this equivalence cannot be overemphasized. When the solution proce-

dure needs to be extended to include nonlinear terms, as the case is in the PSE and PSE-3D

approaches introduced earlier, It is through successive sweeps between the so-called "phys-ical" (="real") space, where function values are defined and "spectral" space, where the

coefficients live, that nonlinear terms arising in equations are evaluated.

Central to the expansion of a function is, of course, the grid on which it is expanded.

This issue should not be taken light-heartedly since it potentially holds the balance between

success or failure of a spectral method. Grids associated with spectral expansions are

discussed in depth by Boyd[9]. Appropriate grid choices are also at the heart of the FD-q

finite-difference methods which will be introduced in what follows.


25/45

SPATIAL DISCRETIZATION 19

2.1.2.1 Interpolation with cardinal polynomialsStarting from first principles, we assume that a set of(N+ 1) different points xk [a, b]

is given. This set is called the

grid : (xk), k = 0, 1, ..., N, a xk b (2.7)

The polynomial of degree (N+1) which vanishes at the grid is

(x) = A

Nk=0

(x xk), A = 0 (2.8)

The cardinal polynomial Kj is the unique polynomial with the properties:

(i) degree(Kj) = N

(ii) Kj(xi) = ij =

0, i = j

1, i = j(2.9)

The cardinal polynomials may be used to represent the polynomial (INf) of degree N, which interpolates a (general) function f at the grid, viz.

(INf)(x) =Nj=0

Kj(x)fj , fj = f(xj) (2.10)

The Lagrange interpolation formula, including the error, is

f(x) = (INf)(x) +fN+1()

(N + 1)!

(x)

A, = (x) [a, b] (2.11)

2.1.2.2 DifferentiationThe derivative of the interpolating polynomial (INf), by reference to (2.10), is given by

d

dx(INf)(x) =

Nj=0

K

j(x)fj (2.12)

Evaluated at the x = xi grid point this becomes

d

dx(INf)(xi) =

Nj=0

K

j(xi)fj i = 0, 1,...,N (2.13)

Hence,


26/45


(INf)

= Df (2.14)

where D denotes the (N + 1) (N + 1) matrix with elements

D = (dij), dij = K

j(xi), 0 i, j N (2.15)

and

f = (f0, f1,...,fN)T

(2.16)

and similarly for (INf)

. The matrix D is called the collocation derivative matrix. Dis fully characterized by the equation

u

= Du, u PN (2.17)

where PN denotes the set of all polynomials with degree N.

2.1.2.3 The second and higher derivativesThe second collocation derivative matrix D(2) can be defined in the same way as the first

derivative D. It will be clear that this leads to

D(2) =

d(2)ij

=

K

j (xi)

, 0 i, j N (2.18)

Again, this matrix is completely characterized by

u

= D(2)u, u PN (2.19)

Now, ifu PN then also u

PN and it follows that

u

= Du

= D (Du) = D2u, u PN (2.20)

Hence,

D(2) D2 (2.21)

In other words, the second derivative matrix may be obtained by squaring the first

derivative matrix. This may be generalized to construct the general mth collocationderivative matrix, simply by

D(m) Dm (2.22)


27/45


Two points should be remarked in this context. First, it may be argued that repeated

(numerical) operations may result in loss of significant digits in the higher matrices. This is

especially true of machines with limited precision arithmetic. For a given system of linearequations which may be recast in the form of an equivalent single equation in which higher-

order derivatives appear, e.g. the LNSE in 1D vs. Orr-Sommerfeld equation, a trade-off

exists regarding the cost at which a given level of precision is expected: one may choose

to work with the original system in which lower-order differentiation is needed, or with the

single equation, which needs a smaller-size matrix for its discretization at the price of loss

of significant digits due to the discrete representation of higher-order derivatives. Second,

a further advantage of the collocation approach as compared to standard finite-differences

is that no special formulae are required for the calculation of derivatives at the endpoints of

the domain.

2.1.2.4 Formulae and properties of Chebyshev polynomialsUp to now, the grid (xk) hasnot been specified, the only demand being that the gridpoints

are contained in some interval [a, b] and are mutually different. In this section we will use[a, b] = [1, 1] and choose grids related with the Chebyshev polynomials Tn(x).

There exists a grid which minimizes the maximum of

(x)A = N

k=0

|x xk| (2.23)

which appears in the interpolation error, eq. (2.11). If the gridpoints xk are the zeroesofTN+1(x), then it can be shown that

max1x1

N

k=0

|x xk| = 2N (2.24)

while any other choice of the grid points in [1, 1] results in a larger maximum. In theparticularly simple and attractive choice of equidistant gridpoints, the maximum is much

larger than the value given above. The minmax property given above is an important and

useful property of Chebyshev polynomials, e.q. in approximation theory.

In order to explicitly derive collocation derivative matrices for the grids based on Cheby-

shev polynomials we need a number of relations for these polynomials and this will now

be considered. The Chebyshev polynomials may be defined by the recursion

T0(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) Tn1(x), n = 1, 2,... (2.25)

Note the odd/even character; note also that the highest power of x in Tn+1(x) has acoefficient 2n. From the trigonometric identity

cos(n + 1) = 2 cos() cos(n) cos(n 1) (2.26)


28/45


it can be shown that

Tn(cos()) = cos(n), x = cos() (2.27)

This expression is called the trigonometric representation of Tn+1(x). An immediateconsequence is

||Tn|| = max1x1

|Tn(x)| = 1 (2.28)

and

TN+1(x) = 0, at x = xk = cos

k + 1

2

N + 1

, k = 0, 1,...,N (2.29)

2.1.2.5 The Gauss-Lobatto gridFrom the trigonometric representation ofTn(x) it is clear that the extrema of TN(x) are

1 and these are attained at the Gauss-Lobatto grid points

xk = cos(k/N) , k = 0, 1,...,N (2.30)

depicted in Figure 2.1.. These points are the zeroes of the polynomial

(x) = (x2 1)T

N(x) =N

2

(TN+1(x) TN1(x)) (2.31)

which follows from (1.43).

The collocation derivative matrix D for the Gauss-Lobatto grid is denoted by

D(1)GL (dij), 0 i, j N (2.32)

and the elements d as obtained from (1.17), (1.18), (1.65) and (1.68) are

d00 = dNN = (2N2 + 1)/6djj =

xj2(x2j1)

, 1 j N 1

dij = (1)ijci/c

jxixj , i = j

with cj defined by (1.66).

2.1.2.6 Chebyshev approximation of a non-periodic functionThe Gauss-Lobatto grid is taken to be

xj = cosj

N, j = 0,...,N (2.33)


29/45


the extrema of the Nth order Chebyshev polynomials1

TN(x) = cos(Ncos1x) (2.34)

and is seen to be such that xj [1, 1]. We again appeal to mappings, to be introducedlater, which may take virtually any grid of interest onto [1, 1] and, hence, remove theapparent restriction imposed by use of the grid (2.4). As with a periodic function, we need

a sum to construct the (real space) interpolating polynomial INf(x) to f(x). This is

(INf) (x) =

Nn=0

fnTn(x) (2.35)

where the expansion coefficients (in spectral space) are given by

fn =2

N

1

cn

N

j=01

cjf(xj)Tn(xj) (2.36)

with

cj =

2, j = 0 or j = N1, 1 j N 1

2.1.2.7 Differentiation: The collocation derivativeFrom the outset it should be born in mind that differentiation, when using a spectral

method, maybe performedin either real or spectral space. This goesto underline (again) the

equivalence (and sometimes necessity) of working with either of the discrete representations

of a function. There are several ways to calculate derivatives within spectral expansions.

Here we focus on representing derivatives with the aid of collocation derivative matrices.

2.1.2.8 Non-periodic functions The collocation derivative matrix for Chebyshev ex-pansions, as defined in 2.1.2 and the Gauss-Lobatto points xj is (see Chapter 1)

(D(1)GL)i,j =

2N2+16

, i = j = 0

xj2(1x2j )

, i = j = 0, N

cicj

(1)j+i

xixj, i = j

2N2+16

, i = j = N

(2.37)

The mth derivative may be calculated through

dm

dxmINf(xk) =

Nj=0

(D(m)GL )k,jf(xj) (2.38)

when the Gauss-Lobatto points are used.

1It is not difficult to see why Boyd[9] calls these functions Fourier cosines in disguise!


30/45


2.1.2.9 MappingsRarely will the case be that the domain on which a function we are interested in coincides

with the standard domains on which the spectral grids are defined2

If a mapping function

j = (xj)

is used to map the domain of interest onto the standard collocation domain xj the collo-cation derivative matrices are modified by use of the chain rule as

bD(1)k,j

() =dx

dD(1)

k,j(x) (2.39)

bD(2)k,j() =dx

d

2D

(2)k,j(x) +

d2x

d2D

(1)k,j(x) (2.40)

bD(3)k,j() =dx

d

3D

(3)k,j(x) + 3

d2x

d2dx

dD

(2)k,j(x) +

d3x

d3D

(1)k,j(x) (2.41)

bD(4)k,j() = dx

d4

D(4)

k,j

(x) + 6d2x

d2dx

d2

D(3)

k,j

(x)

+3

d2x

d2

2D

(2)k,j(x) + 4

d3x

d3dx

dD

(2)k,j(x) +

d4x

d4D

(1)k,j(x) (2.42)

The mapping function to be utilized should be chosen according to the criteria (1) obvi-

ously, it should be transforming the domain of interest onto one of the collocation domains

presented earlier; whether the problem is periodic or not should decide the spectral expan-

sion to be used; (2) more subtly, significant regions in the flow field, such as walls, internal

sharp gradients or wave-like solutions, should be given sufficient points to be resolved prop-

erly. Various authors have suggested mappings appropriate to specific problems. Boyd[9]

and Canuto et al.[12] discuss some alternatives. A widely used mapping for semi-infinite

domains, such as that in the normal to a flat plate, is

j = l1 xj

1 + s + xj(2.43)

with l a parameter depending on the extent of the domain and s = 2l

2.1.2.10 Direct solution of Lu = f We can now make the above discussion com-plete by solving the linear system Lu = f in a direct (non-iterative) manner. For the sakeof simplicity of exposition of the ideas involved, take L in (2.3) to be

L(x) = D(2) + (x)D + (x)

and the boundary conditions associated with the problem both Dirichlet and Neumann

conditions at one endpoint of the integration domain. We proceed by setting up the equation

at these collocation points. Given that in this example ODE no special symmetries exist, thesafe choice of basis functions for expansion of u(x) is Chebyshev polynomials.3 Choosingthe Gauss-Lobatto as the collocation grid xi [1, 1] on which the ODE is to be solved,4

the spectral linear differential operator may be constructed as

2although Poiseuille flow in a straight channel is an exception, since we may normalize the normal-to-the-wallsdirectionby the channel half-widthso as to makethe calculation gridcoincidewith the standardChebyshev [1,1]3see Boyd[9] for a rational discussion of the procedure of selecting basis functions guided by the ODE.4by virtue of a linear transformation any integration domain of interest may be mapped onto this standard grid


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Lij = (D(2)GL)i,j + (xi)(D

(1)GL)i,j + (xi), i = 2,...,N, j = 0,...,N (2.44)

The two first rows ofLij have been reserved for the boundary conditions to be imposed,namely

L0,j = 0, j = 1,...,N L00 = 1,

L1,j = (D(1)GL)0,j, j = 0,...,N.

Correspondingly, the right-hand-side vectorB in equation (2.5) comprises the boundary

conditions as its first two entries and the function f(xi) in the rest.

2.1.3 The FD-q high-order finite-difference method [22]

Consider the generic initial boundary value problem for u(x, t) comprised by the partialdifferential equation

u

t= L

x,t,u,

u

x,

2u

x2

(2.45)

and the following boundary and initial conditions:

B1(u,u/x) = 0, x = 1,

B2(u,u/x) = 0, x = +1,(2.46)

u(x, 0) = f(x), x [1, +1]. (2.47)

A numerical approximation to the above formulated differential problem is sought on

a discrete set of grid nodes {xi, i = 0, . . . , N } located in the domain [1, +1] by em-ploying finite difference approximations to the first and second order space derivatives.

The approach followed for the derivation of the finite difference approximations is briefly

presented in the following subsections. See the work by Hermanns and Hernandez [22]

for in-depth details of the presented method as well as its application to time evolution

problems.

2.1.3.1 Piecewise polynomial interpolation In order to derive the finite differenceapproximations to the spatial derivatives, a piecewise polynomial interpolant is constructed

that matches the discrete values ui(t) of the function u(x, t) at the grid nodes xi, and whosederivatives are then computed to obtain the sought finite difference formulas. Fig. 2.2.

represents such a piecewise polynomial interpolant formed out of individual polynomial

interpolants Ii(x) which are only valid in their respective domains of validity i. Each of

these domains i includes the corresponding grid node xi and their union is equal to thewhole domain [1, +1] of the problem.

Given a set of grid nodes, the expressions for Ii(x) can readily be obtained through theLagrange interpolation formulae:

Ii(x) =

si+qj=si

ij(x)uj , ij(x) =

qm=0

si+m=j

x xsi+mxj xsi+m

. (2.48)


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where qis the polynomial degree andthe seed si istheindexoftheleftmostnode xi involvedin the construction of the interpolant Ii(x). For the case of even polynomial degrees, whichwill be the choice from now on, the following selection of values for si is made:

{si} = {0, . . . , 0 q/2 times

, 0, 1, . . . , N q centered FD

, N q , . . . , N q q/2 times

}. (2.49)

The piecewise polynomial interpolant represented in Fig. 2.2. corresponds to the case of

q = 6 and N = 10. As can be seen from the represented stencils of the individualinterpolants, sufficiently far away from the boundaries, the above selection of seeds leads to

centered finite difference formulas, whereas close to the boundaries the stencils are biased

towards the center of the domain in order to only make use of existing grid nodes.

It should be noted, that in virtue of the uniqueness of the interpolating polynomials, the

finite differences formulas obtained from the differentiation of the piecewise polynomial

interpolant introduced above coincide with the ones obtained by classical means. Thus,

no differences compared to conventional finite difference methods on arbitrary grids exist,

only the way in which they are formulated and derived, but no in the end result.

2.1.3.2 Uniform interpolation errors In the above definition of the piecewise poly-nomialinterpolant thechoice of thegrid nodes xi hasbeen left open so far. However, by theirproper selection it is possible to make the interpolation error of the piecewise polynomial

interpolant to be uniform across the interval [1, +1]. The result is a non-equispaced gridthat is unique for each pair of values ofqand N. This same idea underlies the Chebyshevpolynomials, where the condition that the interpolation error is uniform across the interval

[1, +1] is also imposed, but on a single polynomial interpolant instead [26, 9]. The resultis also a non-equispaced grid, known as the Chebyshev roots or Chebyshev-Gauss quadra-

ture points, that is unique for each value ofN. Both approaches achieve the same result,namely the suppression of the Runge phenomenon that spoils the accuracy of polynomial

interpolations close to the ends of the interpolation interval.

In Fig. 2.3. the resulting grid spacings xi

= xi+1

xi

for the piecewise polynomial

interpolant and for the Chebyshev interpolant are shown for different cases, both of them

normalized with the equispaced grid spacing xi,Eq = 2/N. The details of the algorithmfor the former one can be found in [22], while the derivation of the Chebyshev grids can

be found in any classical textbook on spectral methods or interpolation theory [26, 9]. The

case q = 6 and N = 10 from Fig. 2.2. is shown in Fig. 2.3.(a), where it can be seen thatthe proposed non-uniform grid for the piecewise polynomial interpolant lies in between the

equispaced grid and the Chebyshev grid.

Very enlightening are the following limiting cases: (i) q Nand(ii) q = N. In the firstcase, only a few points O(q) close to the boundaries need to be clustered in order to controlthe interpolation error, while far from the boundaries the grid points are equally spaced, as

seen in Fig. 2.3.(b), where the case of a piecewise polynomial interpolant for q = 6 andN = 30 is shown. In the second case, when q = N, only one interpolating polynomial can

be constructed out of the N+ 1 grid nodes, thus I0(x) = I1(x) = . . . = IN(x). Due to theuniqueness of the interpolating polynomials and the fact that the same error uniformization

strategy is used for the piecewise polynomial interpolant than for the derivation of the

Chebyshev polynomials, both approaches are identical. Thus, in the limit q = N, theproposed piecewise polynomial interpolant with the proposed non-equispaced grid presents

all the properties of spectral collocation methods, especially their spectral accuracy [12, 9].

When q < N, most of the nodes are affected by the presence of the boundaries andthe resulting grid point distributions are in between the two limiting cases. This can be


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-1 0 1

//T

Figure 2.1. The Chebyshev Gauss-Lobatto grid

.

seen in Figure 2.3.(c), where the grid spacing of the proposed non-equispaced grids fordifferent values of qand N = 50 are shown. As the degree of the interpolation increases,the node distribution approaches the Chebyshev grid, while for small values of qit is moreclose to the equispaced grid. Figure 2.3.(d) shows that the minimum grid spacing xminpresent in the proposed non-equispaced grids is always greater than the minimum grid

spacing xmin,Ch of the Chebyshev grid. Moreover, from the figure in can be inferred thatxmin = O (xmin,ChN/q) = O

(qN)1

.

Figure 2.2. Stencils, seeds si, and domains of validity i of the individual polynomial interpolantsIi(x) of a piecewise polynomial interpolation of degree q= 6 on 11 nodes (N = 10). The dashedbox separates the centered stencils from those affected by the presence of the boundaries.


34/45


i

x

i

/x

i,Eq

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

i

x

i

/x

i,Eq

0 5 10 15 20 250

0.5

1

1.5

(a) (b)

q

i

x

i

/x

i,Eq

0 5 10 15 20 25 30 35 40 450

0.5

1

1.5

q /N

q

x

min,C

h/xmin

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

(c) (d)

Figure 2.3. Grid spacing xi = xi+1xi of the non-equispacedgrid for the piecewise polynomialinterpolation (solid line), for the Chebyshev interpolation (dotted line), and for the equispaced grid

(dashed line) normalized with the equispaced grid spacing xi,Eq = 2/Nfor (a) q= 6 andN = 10,(b) q = 6 and N = 30, and (c) q = 10, 20, 30, 40 and N = 50. (d) Variation ofxmin,Ch/xminwith qof the non-equispaced grid for the piecewise polynomial interpolation with N = 50.


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CHAPTER 3

RESULTS

3.1 LNSE IN 1D: THE PLANE POISEUILLE FLOW

The Plane Poiseuille flow, instability of which was first solved by spectral methods in theclassic paper of Orszag [40], serves as the testbed for validations of the numerical methods

discussed herein. This section presents results which may be obtained by the code provided

for the Tutorial work. The linearized Navier-Stokes equations have been discretized using

Chebyshev Gauss-Lobatto spectral collocation and the LAPACK implementation of the QZ

algorithm for the recovery of the entire eigenspectrum. The eigenvalue problem is solved

either in its original generalized form (1.15-1.18) or as the standard EVP which results after

matrix A has been inverted and premultiplied with matrix B (both options are implementedin the code).

Table 3.1. shows the convergence history of the leading eigenmode as a function of

the number of collocation points at Re = 104, = 1. The highly-accurate spectral-element results of Kirchner [29] are also presented for comparison. Figure 3.1. presents the

streamwise perturbation velocity component of the eigenvector, u(y) [12], which can alsobe obtained by course participants using the MATLAB R routines provided as part of theintroductory Lecture.



29


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30 RESULTS

3.1.1 Tutorial Exercise I

It is expected that during the Tutorial participants of this course will:

familiarize themselves with the codes provided,

modify the codes to solve the Rayleigh equation for the hyperbolic tangent profile,

u(y) = tanh(y), (3.1)

for which it is necessary to implement a mapping into the differentiation matrices,along the lines presented in (2.39-2.42). Parameters and comparison results are

offered by the classic work of Michalke [37]; this reference is provided to course

participants.

3.2 LNSE IN 2D: PRELIMINARIES - THE 2D HELMHOLTZ EVP

Figure 3.2. shows a result of the numerical solution of the two-dimensional Helmholtz EVP

(1.37) in terms of the convergence history of a high (ny , nz 1) eigenvalue, (/)2 = 34.Besides the CGL spectral collocation method used up to the present point in the exposition,

several high-order finite-difference methods have also been employed, including the FD-q

method discussed in section 2.1.3. Results such as this, and analogous obtained in the

Blasius boundary layer, have established the FD-q method as a viable alternative to the

CGL for the formation and efficient solution of multi-dimensional eigenvalue and singular

value problems [54, 30, 41].

For completeness, figure 3.3. displays the eigenvector corresponding to the eigenvalue

(/)2 = 34. The spatial structure of the corresponding eigenfunction results in the needto use a relatively high number of nodes for its accurate description. This is in contrast with

the first few eigenvalues, which are recovered using ny = nz 10.

3.2.1 Tutorial Exercise II

It is expected that participants of this course will also:

understand the extension to two spatial dimensions of the one-dimensional discretiza-tion scheme provided in the codes, making use of the Kronecker product (1.40)

write their own routines to implement spatial discretization of the Helmholtz eigen-value problem (1.37) in a Cartesianrectangular domain, and use LAPACK subroutines

for the solution of the resulting EVP

compare their results with (1.38) and (1.39).

3.3 LNSE IN 2D: THE BIGLOBAL EIGENVALUE PROBLEM

Implementation of the two-dimensional Laplace operator is an essential first step for the so-

lution of the incompressible BiGlobal eigenvalue problem (1.26-1.29) and its compressible

counterpart.

Some results of a flow problem, the basic flow of which may itself be obtained by

solution of the Poisson equation, are included herein in order to assist course participants


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THE PSE-3D 31

in their development of their own codes. The flow in question is that in a rectangular duct

of cross-sectional aspect ratio A, driven by a constant pressure gradient along the axial(unbounded) direction; the large-aspect-ratio (A ) limit of this configuration is PPF.The global linear instability of the rectangular duct flow was originally studied by Tatsumi

& Yoshimura [50].Consider therectangularduct definedin thedomain = {z [A, A]}{y [1, 1]},

where A is the aspect ratio. A constant pressure gradient in th unbounded axial, x, directiondrives a steady laminar flow which is independent of x and possesses a velocity vectorq = [u(y, z), 0, 0]T; the single velocity satisfies the Poisson equation

22du(y, z) = P/x = c, (3.2)

where 22d = 2/y2 + 2/z 2 and viscous conditions are imposed on all four duct walls.

Taking c = 2 in 3.2 and scaling the result with the value of U at the midpoint of theintegration domain, the Poisson problem may be solved in closed series form [44], which

assists comparison with numerically obtained solutions.

The instability analysis problem is solved by forming the matrix, spatial derivatives

being calculated either by the spectral collocation method based on the Chebyshev Gauss-Lobatto grid (CGL) or the FD-q method at different orders, q. In order to exploit thesparsity of the matrix, the MUMPS package [2, 3] has been used for the serial direct

inversion of the matrix. Subsequently, the Arnoldi algorithm has been employed to extract

the most interesting eigenvalues. Table 3.2. presents convergence history results using the

two methods at a subcritical Reynolds number, Re = 1000, and wavenumber parameter = . In addition, Richardson-extrapolation results are also shown.

3.4 THE PSE-3D

Results of work in progress (P. Paredes) close this lecture. As discussed in the introduc-

tory section, the PSE-3D instability analysis concept is appropriate for flows with strongdependence on two and weak on the third spatial direction. Unlike typically used models

of systems of counter-rotating vortices in which no axial development is permitted and

subsequently either BiGlobal EVPs or DNS in which the axial direction is taken to be ho-

mogeneous, DNS work has demonstrated [11] that different instability behavior is to be

expected when axial vortex-system development is permitted. Motivated by this finding

PSE-3D analysis of an axially evolvingpair of counter-rotating vorticeshas been undertaken

[42].

Figure 3.4. shows linear superposition of such a basic flow and its leading eigenmode.

The main observations made are the non-axisymmetric nature of this perturbation, clearly

seenat large streamwisecoordinate values, as wellas the interaction between the instabilities

on the individual vortices. Further details of this work may be found in the original reference

[42].

3.5 LNSE IN 3D: PRELIMINARIES - THE 3D HELMHOLTZ EVP

Finally, the FD-q method for spatial discretization, in conjunction with the serial version of

the MUMPS [2, 3] package for the direct linear algebra operations has been used to solve

(3.5.) in three spatial dimensions, on a standard workstation (P. Paredes, unpublished).

Figure 3.5. shows one of the eigenvectors corresponding to a large eigenvalue, (/)2 =


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32 RESULTS

Table 3.1. Convergence history of instability analysis of PPF at Re = 104 and = 1, = 0.

N r i

20 0.2343850714231792 0.0013789375577136

40 0.2375263827503545 0.0037385876638924

60 0.2375264888136988 0.0037396705884981

80 0.2375264888205456 0.0037396706231850

100 0.2375264888204468 0.0037396706229840

120 0.2375264888204677 0.0037396706229719

140 0.2375264888204739 0.0037396706229729

Kirchner[29] 0.2375264888204682 0.0037396706229799

-1

-0.5

0

0.5

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

u-velocity

y

Re = 10000, = 1

Re

Im

Figure 3.1. Streamwise perturbation velocity component u(y) of plane Poiseuille flow (PPF) atRe = 104, = 1, = 0 [40].

43, such that a non-trivial structure in terms of gradients be obtained. As with its two-dimensional analogue, spatial discretization of the three-dimensional Poisson operator has

been at the heart of the solver for the TriGlobal eigenvalue problem currently used in the

authors research group, first validations results of which have been presented elsewhere

[41].


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LNSE IN 3D: PRELIMINARIES - THE 3D HELMHOLTZ EVP 33

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

10 100

N

CGL

FD-q04

FD-q08

FD-q12

FD-q16

FD-q20

Figure 3.2. Convergence history of the solution of the Helmholtz eigenvalue problem (1.38) for the

eigenvalue (/)2 = 34, obtained using the Chebyshev Gauss Lobatto and a suite of FD-q methodsof orders 4(4)16. The number of discretization nodes used is the same in both spatial directions and

is denoted by N [54, 30, 41].

Table 3.2. Convergence history of BiGlobal instability analysis of square duct flow at

A = 1, Re = 1000 and = .

CGL FD-q16

N2 r i N2 r i

302 2.9027647730 -0.103535353975 302 2.9027679758 -0.103527154668402 2.9027654432 -0.103524928082 502 2.9027654409 -0.103524924455502 2.9027654495 -0.103524926162 702 2.9027654496 -0.103524924216602 2.9027654518 -0.103524926085 902 2.9027654520 -0.103524925116702 2.9027654528 -0.103524926094 1102 2.9027654529 -0.103524925545

2.9027654556 -0.103524926119 2.9027654547 -0.103524926414


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34 RESULTS

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Figure 3.3. The eigenvector corresponding to (/)2 = 34 in the solution of (1.38). Shown arefive isolines between zero and maximum (solid line) and between minimum and zero (dashed line).

Figure 3.4. The leading linear perturbation u( ,y ,z ) obtained by PSE-3D analysis, superposed atlinearly small amplitudes upon a basic flow u,y,zcomposed of two counter-rotating vortices withslow axial dependence [42].


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LNSE IN 3D: PRELIMINARIES - THE 3D HELMHOLTZ EVP 35

Figure 3.5. The eigenvector of the 3D Helmholtz equation (1.53) pertaining to the eigenvalue

(/)2 = 43.


42/45


43/45

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