Akademisk avhandling som med tillstνand av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknisk doktorsexamen freda

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1 Stability results for viscous shock waves and plane Couette flow Mattias Liefvendahl Stockholm 2001 Doctoral Dissertation Royal Institute of Technology Department of Numerical Analysis and Computer Science

2 Akademisk avhandling som med tillstνand av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknisk doktorsexamen fredagen den 2 november 2001 kl 9.15 i Kollegiesalen, Administrationsbyggnaden, Kungl Tekniska Högskolan, Valhallavägen 79, Stockholm. ISBN TRITA-NA-0132 ISSN ISRN KTH/NA/R--01/32--SE cfl Mattias Liefvendahl, September 2001 Universitetsservice US AB, Stockholm 2001

3 Abstract Stability of stationary solutions is a basic question in the theory of nonlinear partial differential equations. Understanding the dynamics of the system in the vicinity of a stationary state is of obvious importance from the point of view of applications, it is also of intrinsic mathematical interest. In this thesis we study viscous shock wave solutions to systems of conservation laws in one space dimension. Sufficient conditions for stability are given. This analysis applies to strong shock waves of Lax type, but is restricted to perturbations with zero mass. We also numerically investigate an example of an unstable viscous shock wave for a perturbed version of the cubic model, which isa2 2 system of conservation laws. Another part of the study concerns plane Couette flow for the incompressible Navier-Stokes equations. We give analytical and numerical results for the resolvent operator corresponding to this flow. The computations are done using a spectral method based on Chebyshev polynomials, we present the details of this method. Experimenting with different weighted norms we find one such norm more suitable for the stability question than the standard energy norm. Using this norm we prove nonlinear stability for finite amplitude perturbations of plane Couette flow. This result gives a lower bound, including Reynolds number dependence, of the threshold amplitude below which all perturbations are stable. The bound is an improvement of an existing result in the literature. A unifying theme in this thesis is the approach to stability problems via a study of the resolvent equation, and also the combination of analytical and numerical methods for this purpose. Keywords: nonlinear stability, shock waves, Couette flow, Chebyshev spectral method, resolvent estimates, threshold amplitudes. ISBN ffl TRITA-NA-0132 ffl ISSN ffl ISRN KTH/NA/R--01/32--SE iii

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5 Preface This thesis contains five papers concerning stability results for viscous shock waves and plane Couette flow. There is also included an overview of introductory character giving background information to the research articles. The papers are presented in the order they were written. This order also groups the papers according to the problems I have worked on during my graduate studies. The work presented in this thesis was carried out at the Department of Numerical Analysis and Computer Science, Royal Institute of Technology during the period February 1997 to October I would like to express my gratitude to my advisor Prof. Gunilla Kreiss for her excellent guidance on this work. I would like to thank my colleagues at the department for providing a pleasant and relaxed atmosphere to work in, and the department for the financial support needed to complete this thesis. Finally, I want to thank my family for their constant support over the years, and also extend a big and special thanks to Maria. Mattias Liefvendahl Stockholm, October 2001 v

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7 Contents 1 Introduction Remarks on stability definitions Conservation laws Basic theory Shock waves Stability of strong viscous shock waves The Navier-Stokes equations Basic theory Stability of shear flows Computational problems related to stability Solution of the full nonlinear problem Calculation of eigenvalues and eigenvectors Viscous shock waves Plane Couette flow Computation of pseudospectra Summary of the papers Paper 1: Stability of viscous shock waves for problems with nonsymmetric viscosity matrices Paper 2: Numerical investigation of examples of unstable viscous shock waves Paper 3: A Chebyshev tau spectral method for the calculation of eigenvalues and pseudospectra Paper 4: Analytical and numerical investigation of the resolvent for plane Couette flow Paper 5: Bounds of the threshold amplitude for plane Couette flow 23 A Definitions for theorem 2 25 RESEARCH PAPERS (From page 31 onwards) vii

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9 Chapter 1 Introduction Yet not every solution of the equations of motion, even if it is exact, can actually occur in Nature. The flows that occur in Nature must not only obey the equations of fluid dynamics, but also be stable. - L.D. Landau & E.M. Lifshitz (1959) The material presented in this thesis concerns the stability of stationary solutions of nonlinear partial differential equations (PDEs). Two such situations are considered. The first is, viscous shock waves for systems of conservation laws. This is the subject of paper 1 and 2. Papers 3-5 are related to the second subject, plane Couette flow for viscous incompressible flow (the Navier- Stokes equations). The purpose of these introductory chapters is to provide background information to the research papers. In the next section we comment on some possible variations when stating a precise definition of stability. In the PDE case the inequivalence of norms makes the situation more complicated than one might first expect. We illustrate this by two examples. The second chapter is devoted to systems of conservation laws. First we present basic material on the PDE, then the discussion becomes gradually more specialized leading to the question of nonlinear stability of strong viscous shock waves. We also, of course, give references to related work. In chapter three the treatment is similarly organized, starting with the formulation of the Navier-Stokes equations and leading to the precise problems investigated in paper 4 and 5. Both for shock waves and plane Couette flow we apply numerical methods to compute e.g. eigenvalues or the norm of the resolvent operator. An overview of the computational problems is given in chapter four where we discuss direct numerical simulation and computation of the spectrum and pseudospectrum. Chapter five contains a short summary of each of the five research papers. The summary contains much of the same information as the abstracts but also more on the interrelation of the papers. Here we also give complete references concerning the publication of the articles. 1

10 2 Chapter 1. Introduction 1.1 Remarks on stability definitions It is appropriate to discuss here how to define the intuitively clear concept of stability. When this is to be done rigorously one finds that there are some possible variations of the concept. In this section we work in a rather general setting. In the research papers we will of course formulate definitions adapted to those specific problems. For ordinary differential equations (ODEs), dx dt = f(x); x 2 RN ; (1.1) it is customary to distinguish between the Lyapunov stability and asymptotic stability which we now define. Definition 1 A stationary solution x 0 of equation (1.1) is said to be stable in Lyapunov's sense if given ffl>0, there exists a ffi>0 such that every solution x(t) with jx(0) x 0 j <ffisatisfies jx(t) x 0 j <fflfor all t>0. Definition 2 A stationary solution x 0 of equation (1.1) is said to be asymptotically stable if it is stable in Lyapunov's sense and if lim t!1 x(t) =x 0 for every solution x(t) with initial data in a sufficiently small neighborhood of x 0. REMARK: By jxj above we intend the Euclidean norm of the vector. For ODEs however both stability concepts are topological properties (of the vector field defining the ODE), [3]. It is immaterial which metric is used in the definition. For PDEs the distinction between Lyapunov and asymptotic stability is also used. The situation is however more complicated because norms need not be equivalent. This implies that the same problem may be stable according to a definition using a certain norm and unstable according to another definition which uses another norm. Below we give two examples of this phenomena. The guiding principle" in the research papers of this thesis is to consider asymptotic stability in the norm of the space L 1. Example 1. Consider + = u=2; x 2 R;t >0

11 1.1. Remarks on stability definitions 3 The analytical solution is u(x; 0) = u (0) (x): u(x; t) =e t=2 u (0) xe t : The time dependence of the L p -norms of the solution is easily calculated, ku( ;t)k L p = e 2 p 2p t ku (0) k L p: From this expression we see that the zero solution is asymptotically stable in L p for p>2, it is Lyapunov stable for p 2 and it is unstable for p<2. Example 2. This example, which istaken from [52], is more interesting in connection with paper 3-5. Consider the following problem in the strip jyj < 1in the (x; + = 0 ψ(x; y; 0) = ψ (0) (x; y) ψ(x; ±1; t) = 0 Here denotes the Laplacian. This problem is obtained as a result of linearization at plane Couette flow of the two-dimensional equations of motion of an ideal fluid. Introducing the vorticity! by we find for the vorticity the expression ψ =!; ψ (0) =! (0) ;!(x; y; t) =! (0) (x yt; y): For the vorticity there is Lyapunov stability inc (the space of continuous functions with maximum-norm) max x;y j!(x; y; t)j = max j!(0) (x; y)j: In C 1 there is, on the other hand, no stability since max x;y x;y fi fi fi fi fififi fi (x; y) = maxfi x;y fi fi fi fi fififi fi (x; y) maxfi x;y t max x;y (x yt; (x; y) fi fififi (x yt; y) fi fififi Here the right hand side tends to infinity (if! (0) depends on x) ast!1.

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13 Chapter 2 Conservation laws Our present analytical methods seem unsuitable for the solution of the important problems arising in connection with partial differential equations and, in fact, with virtually all types of nonlinear problems in pure mathematics. The truth of this statement is particularly striking in the field of fluid dynamics. Only the most elementary problems have been solved analytically in this field. - J. von Neumann, 1946 The conservation laws considered here 1 are those of physics and mechanics where the state of the system is described by a set of space and time dependent functions which are conserved. By this we mean that the rate of change of each quantity in any volume of space is given by the flux of the quantity across the surface of the volume, i.e. nothing is produced or annihilated. A mathematical formalization of this situation leads to the partial differential equation (2.1) below. This equation has been extensively studied, we refer to the comprehensive works [29], [37], [45], [17] and [44] of which [44] contains the present state of the art. The book [17] is concerned both with the theory and the numerical approximation and contains a bibliography of 40 pages which, even if not claimed exhaustive, provides a good starting point for exploration of the literature up to The study of conservation laws started with specific equations from physics of which the mechanics of a compressible fluid (The Euler equation) must be considered the most important. For this application, the book [10] is the classical reference. Other physical models leading to conservation laws include magnetohydrodynamics, multi phase flow and electromagnetics. The generalization from specific equations to the study of (2.1) took place around We do not consider the Noether theorem, see [4] p.88, of classical mechanics which connects invariances with conservation of energy, momentum etc. Of course, these are related concepts but when we refer to conservation laws we have the precise meaning of equation (2.1) in mind. 5

14 6 Chapter 2. Conservation laws The phenomenon of shock waves occur for conservation laws. These are discontinuous solutions of the equation. In the case of gas dynamics shock waves occur e.g. around supersonic aircraft which thus induce a discontinuity in the state variables, including the pressure. The theory of shock waves go back to Riemann and the 19th century, we content ourselves with the references above and the references therein. Questions studied include the existence and stability of shock waves. As we elaborate in the introduction to paper 1, in the context of viscous shock waves in one space dimension, we distinguish two aspects of the stability question. The first is most extensively studied and concerns weak shock waves (it is assumed that the discontinuity is sufficiently small for the argument to hold), see [36], [19] and [47]. The second, more recent, approach changes perspective and treats shock waves of arbitrary strength (size of discontinuity). To be able to derive results in this situation one relies on spectral properties of the linearized problem. This approach is represented in [23], [53] and [34]. The spectral conditions for stability may or may not hold in specific cases. In [30] itisshown to hold for a shock wave for the Euler equations with artificial viscosity. In [14] and [24] an unstable example is investigated. 2.1 Basic theory The Cauchy problem for a hyperbolic system of p conservation laws in d space dimensions (1» d» 3) is given k f(u) = 0; x 2 R d ;t>0 (2.1) u(x; 0) = u (0) (x); where u 2 R p. We denote the Jacobian matrix of f k by A k. The system is called hyperbolic if the matrix A(!; u) = dx k=1 A k (u)! k has real eigenvalues 1 (!; u)» 2 (!; u)» :::» p (!; u) and the corresponding eigenvectors fr l (!; u)g p l=1 are linearly independent. The above should hold for all! and u. If the eigenvalues are distinct the problem is called strictly hyperbolic. This is the hypothesis in paper 1. If there exists a positive definite matrix A 0 (u) such that the matrices A 0 (u)a k (u); 1» k» d; are symmetric then the problem is called symmetrizable. For symmetrizable systems we have local existence of a classical solution. This important result is easy to

15 2.2. Shock waves 7 formulate precisely so we state it in the following theorem which is adapted from [44] where it is contributed to Gνarding and Leray. Theorem 1 Let s>d=2+1 and let U be abounded subset of R p. If 2 u (0) 2 H s (R d ) and u (0) takes values in U then there exists a time T > 0 such that the symmetrizable system of conservation laws has a solution u 2 C 1 ([0;T] R d ). If the classical solution breaks down at time T then we have, [44] p.115, lim max t!t jffj»1 fl fl fl fl flflflc(r ff ( ;t) p ) = 1: Thus either the function becomes infinite or some of its first derivatives become infinite. The second possibility is the case when shock waves develop. 2.2 Shock waves In this section we restrict the discussion to one space dimension (d = 1), which is the case studied in paper 1 and 2. We denote f 1 by f. To allow discontinuous solution one cannot use the differential equation but must generalize the solution concept. This generalization is accomplished using smooth test functions much in the same way as in distribution theory. However, we do not extend the solution concept to include distributions but are still restricted to functions (the appropriate class of functions depend on the nonlinearity f). We call u a generalized solution if it satisfies Z 1 0 Z R (ffi t u + ffi x f(u)) dxdt =0; 8ffi 2 C 1 0 (R [0; 1)): (2.2) The class of test functions consists of vector functions where each component is smooth and has compact support in an open set containing R [0; 1). It is easily verified that classical solutions are generalized solutions. We mention the global existence theorem of Glimm [16] which roughly states that if the initial data has sufficiently small total variation and L 1 -norm, then there is a global weak solution. The theorem thus applies to initial data which is almost constant and for these initial data the theorem guarantees a solution which also is almost constant for all times. Refinements and extensions of this theorem has been the topic of many research papers. 2 We use standard notation for the Sobolev norm [1] containing s derivatives of vector functions defined on R d with values in R p.

16 8 Chapter 2. Conservation laws Now we turn to the study of discontinuous solutions which consists of two constant states separated by a discontinuity which moves with velocity s. u(x; t) = ρ ul u R x<st x > st: (2.3) Inserting this ansatz into (2.2) we easily derive the following equation, called the Rankine-Hugoniot condition, connecting the left and right states and the velocity. s(u R u L )=f(u R ) f(u L ) The rest of this section is dedicated to a motivation why the so called Lax shock, defined below, is the most basic and non-degenerate" solution of the form (2.3). We assume that the problem is strictly hyperbolic and that there is an index 3 k such that k 1 (u L ) < s < k (u L ) (2.4) k (u R ) < s < k+1 (u R ): (2.5) The above inequalities are the defining properties of Lax shocks. The inequalities (2.4) means that n k +1characteristics enter the shock from the left and (2.5) means that k characteristics enter the shock from the right. The remaining characteristics leave the shock. Now we will investigate a small perturbation of the solution (2.3). The solution is determined for all characteristic variables which enter the shock. The question is whether we can determine the outgoing characteristic variables using the Rankine- Hugoniot conditions. Before we answer this we first introduce some notation u L = u (0) L + u L u R = u (0) R + u R (2.6) s = s (0) + s: The variables with (0)-superscript denotes the unperturbed states and the variables with over-bar" denotes the small perturbations. It is sufficient to consider a perturbation which is constant to the left and right of the shock. We also need notation for the matrix of eigenvectors of the matrices A(u (0) L ) and A(u(0) R ). We use block matrices according to A(u (0) L )(SI L SL II )=(SL I SL II Λ I ) L 0 0 Λ II L and a corresponding partitioning of A(u (0) R ). Here we have Λ I L = diag( 1(u (0) L );:::; k 1(u (0) L )) ΛI R = diag( 1(u (0) R );:::; k(u (0) R )) Λ II L = diag( k(u (0) L );:::; p(u (0) L )) ΛII R = diag( k+1(u (0) R );:::; p(u (0) R )): 3 We use the convention that 0 = 1 and p+1 = 1

17 2.3. Stability of strong viscous shock waves 9 We see that the columns of SL I are the eigenvectors corresponding to the the eigenvalues on the diagonal of Λ I L. With the notation above we can also introduce the characteristic variables u L =(SL I SL II w I ) L w II ; u R =(SR I SR II w I ) R L w II : R As mentioned above w II L and w I R, which correspond to incoming characteristics, are known from the initial data. It thus remain to determine the p unknowns s, w I L and w II R. This will be done through the linearized Rankine-Hugoniot equations. We need f(u L ) ß f(u (0) L )+A(u(0) L )u L; and the corresponding linearization at u (0) R. Now we insert the ansatz into the Rankine-Hugoniot condition, use the linearization and simplify the equations to u (0) R u (0) L ; SI L(s (0) I Λ I L); S II R (s (0) I Λ II R ) s w II L w I R 1 A = b: (2.7) Here b contains two terms, the first of which depend linearly on the characteristic variables entering the shock from the left and the second term depending linearly on the characteristic variables entering the shock from the right side. We donot give the expression for b. The coefficient matrix in (2.7) is square (size p p). In the non-degenerate case, when the matrix is nonsingular, we can solve the problem and thereby determine all the unknowns. We note that the coefficient matrix is similar to the matrix D = u (0) R u (0) L ; SI L; SR II The other factors in the expression in (2.7) only scale the column vectors (with non-zero numbers because of (2.4) and (2.5)). Summarizing, we observe that the Lax shock leads to a square matrix D and is to be considered the basic shock wave case motivated by the fact that it is then possible to determine the characteristic variables leaving the shock. The nonsingularity of D is crucial in investigations of Lax shock waves, [13]. In paper 1 it is designated as assumption 4 and is thus part of the sufficient condition for stability. In paper 2 a Lax shock for a system depending on a parameter is investigated. For one value of the parameter the matrix D is singular, for more information we refer to paper 2. : 2.3 Stability of strong viscous shock waves Without repeating too much of the content of paper 1 and 2 we will make some remarks concerning the stability of strong viscous shock waves. Both from a mathematical and a physical point of view it is natural and fruitful to consider the

18 10 Chapter 2. Conservation laws addition of a second derivative, viscous, term to the conservation law. u t + f x (u) =Bu xx This modification gives a parabolic problem if the eigenvalues of B has positive real part. The physical motivation is that often (e.g. in fluid mechanics) a small viscous" effect is present which have been neglected in the hyperbolic case. From the mathematical point of view the modification gives a regularization of the problem. The solutions of the parabolic problem are (often) very similar to those of the hyperbolic problem but they are more regular (differentiable). Discontinuous shock waves of the hyperbolic problem correspond to smooth shock profiles for the parabolic problem. Stability of weak viscous shock waves has been studied in many works as we mentioned in the beginning of the chapter. Result for strong shock waves on the other hand has only appeared in the last five years. The exception being the paper [38], where a stability result is proven for strong shock waves of a model system for compressible viscous gas. Apart from this work we distinguish two approaches for the problem. Both rely on spectral information and a study of the Laplacetransformed linearized equation s^u +(A(U)^u) x = B^u xx ; x 2 R;s 2 C : (2.8) Here ^u denote the Laplace-transform of u and U denotes the shock profile. The first approach of Kevin Zumbrun and collaborators is based on a study of the Green's function for (2.8). Estimates of this are, by the inverse transform (contour integrals), converted to pointwise bounds on the parabolic" Green's function which in turn are used to prove stability, we refer to [15] and [53]. The second approach which is used in paper 1 relies on a reduction to a finite x-interval of the boundary value problem (2.8). This is possible by a detailed study of the growth or decay at x = ±1 of the solutions of (2.8). Many of the techniques of paper 1 where developed in [23] where stability in the case B = I (identity matrix) was proven.

19 Chapter 3 The Navier-Stokes equations... hydrodynamics with its spectacular empirical laws remains a challenge for mathematicians. For instance, the phenomena of turbulence has not yet acquired a rigorous mathematical theory. Furthermore, the existence problems for the smooth solutions of hydrodynamic equations of a three-dimensional fluid are still open. - V.I. Arnold & B. Khesin 1998 In contrast to conservation laws, which is a mathematical class of differential equations, the Navier-Stokes equations (3.1) (below) constitute a specific physical model describing the dynamics of an incompressible viscous fluid. The equations were derived in the first half of the 19th century and are of immense importance for technological applications since they are an appropriate model for a wide range of fluids, e.g. air, water, oil etc. The literature is of course vast. For the physical point of view we refer to [28] and for the mathematical theory [26], [27] and [48]. Hydrodynamical stability has been recognized as one of the central problems of fluid mechanics for a century. It is concerned with if and how laminar flow breaks down, its subsequent development and possible transition to turbulence. The comprehensive treatises [35], [9], [11] and [43] deal exclusively with stability questions. In the first section of the chapter we formulate the initial boundary value problem for the Navier-Stokes equations. In the same section we give a precise stability theorem, taken from [52], to illustrate the relation between spectral information and nonlinear stability. The formulation of the theorem also illustrates the many technical points when formulating stability conditions. In section 3.2 we discuss the stability of shear flows and important concepts including subcritical instability and pseudospectra (defined in section 4.3). 11

20 12 Chapter 3. The Navier-Stokes equations 3.1 Basic theory The initial boundary value problem for the Navier-Stokes equations in a domain Ω of three dimensional space is given + u k + grad p = 1 u; x 2 Ω;t>0 k R div u = 0; x 2 Ω;t>0 u = 0; x u(x; 0) = u (0) (x): Here u is the velocity field, p is the pressure and R is the Reynolds number. In two space dimensions the problem is well-posed, see e.g. [26]. In three dimensions on the other hand the well-posedness is an open question. We refer to [27] for a short time (local) existence theorem. For initial data in the vicinity of a linearly stable stationary solution, however, precise information on the solution is available, see the theorem below. The situation is somewhat analogous to the Glimm result for conservation laws in the case of initial data near (the stable) constant state. Because of its relation to papers 4 and 5 and because it is not so easily found in the literature, we give such a well-posedness/stability theorem. The theorem is adapted from [52], the notation needed for its formulation is collected in appendix A. Before we state the theorem we must formulate the linearized equations for a given stationary flow U as the following differential equation on the Banach space S p (defined in appendix A). Here A is the operator defined by Au = 1 R Π u +Π 3X dv dt = Av (3.2) v(0) = v U k + u k for any solenoidal vector u 2 W 2;p (Ω) vanishing Theorem 2 Suppose the spectrum of the operator A is situated in the left half plane. ff(a) ρfs 2 C : Re s< s 0 < 0g (3.3) Then the steady flow U is asymptotically stable in S p for p>3. Moreover, if v (0) has sufficiently small norm in S p (p>3) then the solution v of (3.2) satisfies e s0t kv( ;t)k Sp + kvk Sp;r;t;s 0 + 3X fl fl fl k fl flflflsp;r;t;s 0» Ckv (0) k Sp :

21 3.2. Stability of shear flows 13 Here r>1 and the constant C does not depend on t. To apply this theorem it is thus sufficient to show (3.3) for the spectrum. In the case of plane Couette flow (3.3) was established in [41]. We also remark that the theorem contains no information on the physically interesting energy (L 2 ) norm since the result only holds for p > 3. The paper 5 uses a different approach to nonlinear stability which includes an unbounded Ω and the L 2 norm but the stability condition also require for the initial data that the norm of space derivatives is small, see theorem 1 of paper Stability of shear flows In this section we make some remarks on the stability of shear flows and plane Couette flow in particular. There will be some overlap in content with paper 4 and 5. We mention pipe flow, (cylindrical) Couette flow and boundary layers as other important situations but restrict the discussion here to Poiseuille flow and plane Couette flow. The Poiseuille and plane Couette flows are stationary flows in the domain Ω=fx 2 R 3 : jx 2 j < 1g: The flows are parallel and depend only on the normal coordinate, the expression is U = U 1(x 2 ) 0 0 with an appropriately chosen coordinate system. The function U 1 is the following in the cases of Poiseuille and Couette flows respectively. 1 A U P 1 (x 2 ) = 1 x 2 2 U C 1 (x 2 ) = x 2 Using these examples we will now discuss some important concepts. Poiseuille flow is stable for R < 5772:22 (and unstable for larger R) as was discovered in [49] and [42]. The accurate value of the critical Reynolds number was determined in [39] using a numerical method very similar to that applied in paper 3 and 4. We use the term critical Reynolds number in the precise sense that below it the flow is (linearly) stable, and above it there are unstable eigenvalues. For all flows it is possible to derive a Reynolds number below which the energy of the perturbation decays monotonically. Such a result is obtained by energy methods (partial integration) leading to the Reynolds-Orr equation, see e.g. [43]. The monotonic decay requirement, however, typically gives a Reynolds number very much below the critical Reynolds number.

22 14 Chapter 3. The Navier-Stokes equations In contrast to the Poiseuille case plane Couette flow is stable for all Reynolds numbers. The result proven in [41] is that all eigenvalues satisfies» ffi R ; for some ffi > 0 independent of R. The bound for the spectrum thus approaches the imaginary axis for increasing R. In experiments however, plane Couette flow always becomes turbulent for large (larger than 5000 say) Reynolds numbers. The reason for the turbulent behavior is that there is a threshold amplitude for the perturbations. Below this threshold the perturbation decays to zero and above the threshold amplitude there are perturbations which lead to turbulence. This phenomenon is referred to as subcritical stability. The existence of a threshold is a nonlinear effect, for the linearized problem the asymptotic time dependence is determined by the spectrum. There may however be a transient amplification of the initial perturbation. This linear phenomena is of critical importance for the threshold value since the transient growth is found to be large (it increases with R) for Poiseuille flow and plane Couette flow. The spectrum gives no information on transient growth since the growth depends on non-orthogonality of eigenvectors. This is the reason for the, rather recent, shift in focus of the research from the spectrum to the resolvent (or the pseudospectra [50]) which contains more information. For completeness we now define the resolvent set and operator in the context of closed (possibly unbounded) operators in a Banach space, following [22]. Let X be a Banach space and T : X! X a closed linear operator. Then T si is also a closed linear operator for all complex numbers s. If T si is invertible with R(s) =(T si) 1 2B(X); (Bounded operators in X) s is said to belong to the resolvent set of T. The one parameter family R(s) thus defined on the resolvent set is called the resolvent oft. Finally we give some references. For a discussion of hydrodynamical stability from the point of view of the resolvent/pseudospectra, in contrast to eigenvalues, see [50], which also contain many references on the topic. The non-normality of eigenvectors was pointed out in [40] and the fact that it can cause transient growth of factors of thousands was found in [21] and [7]. The recent book [43] is, among other things, concerned with these topics and we refer to it and the references therein.

23 Chapter 4 Computational problems related to stability Linearization of a system of conservation laws or the Navier-Stokes equations typically lead to a non-self-adjoint problem. This is the case in paper 1-5. The investigation of such problems is difficult and for the majority of questions it is necessary to have recourse to numerical methods. In this chapter we discuss different computational problems which are directly related to stability questions. Section 4.1 contains some remarks on simulation of the full non-linear problem with initial data near the stationary solution. Section 4.2 presents the traditionally most important problem of calculating the spectrum of the linearized problem. As mentioned in section 3.2 it is sometimes more interesting to determine the pseudospectra (norm of the resolvent), in particular this is important for investigations of the threshold amplitude. In section 4.3 we present this problem which is closely related to the material in paper Solution of the full nonlinear problem We consider the solution of the full nonlinear problem (or direct numerical simulation) because of its importance for stability investigations, not because it is treated in paper 1-5. We have limited knowledge of this area and therefore content ourselves with some remarks to give perspective to the next two sections which are directly connected with paper 1-5. For conservation laws in one space dimension, to our knowledge, direct numerical simulation has not been used much for the specific purpose of testing shock wave stability. However, see [18] for simulation of the linearized time dependent problem. General purpose" computational methods for hyperbolic conservation laws is of course a very big research area [17]. To attack the shock wave stability problem which is posed in the unbounded domain R it is necessary to truncate to a bounded 15

24 16 Chapter 4. Computational problems related to stability interval. The choice of artificial boundary conditions at the introduced boundary the is an important issue. When this choice has been made a finite difference method with implicit time stepping, see e.g. [20], would be an easily implemented and reasonably effective choice of numerical method. For the Navier-Stokes equation on the other hand, direct numerical simulation is a major research branch. In the case of periodic boundary conditions or channel flows spectral methods is a viable choice of numerical method, [8]. In the channel case trigonometric polynomials are used in the x 1 and x 3 directions (using the coordinate system introduced in section 3.2). In the normal (x 2 ) direction the boundary conditions forces a different choice of basis functions. Usually Chebyshev polynomials are used, not because they satisfy the boundary conditions but for more subtle approximation properties which make it possible to still obtain spectral accuracy, see [8] or paper 3. In the investigation of the threshold amplitude for e.g. plane Couette flow the most straightforward approach is to simply try different initial perturbations, perform a direct numerical simulation and see if turbulence occurs. This approach is used in many papers of which we only mention [25] because of its close relation to paper 4 and 5. The drawbacks are that it is computationally expensive, even with state of the art numerical methods and supercomputers, to try many different initial data. Also, if one calculates the eigenvalues and eigenvectors for the linear problem, then the slowest decaying mode is found. To find this with trial and error from the initial value problem is more difficult. The last point however also indicate the advantage that with direct simulation that one gets all information including the nonlinear coupling. 4.2 Calculation of eigenvalues and eigenvectors We write the linearized problem in the abstract form du dt = Au; (4.1) where u 2 X (a Banach space) and A is a linear operator. For a definition of A in the conservation laws case we refer to paper 1 and in the plane Couette flow case paper 5. The eigenvalues and eigenvectors of A which we intend to compute is of obvious importance for the problem (4.1). If we have found an eigenvalue and an eigenvector ' then, of course, u(t) ='e t ; solves the initial value problem. In the general framework of (4.1) the situation is however much more complicated than eigenvalues and eigenvectors. For instance the spectrum consists of the point spectrum, the continuous spectrum and the residual spectrum, see e.g. [51]. To make the discussion more specific we treat the viscous shock wave case and the plane Couette case in different subsections below. The difficulties of these two eigenvalue problems lie in quite different areas.

25 4.2. Calculation of eigenvalues and eigenvectors Viscous shock waves Here the eigenvalue problem has the following form B d2 ' dx 2 d (A') = '; x 2 R (4.2) dx ' 2 L 2 : The matrices A and B are the same as in (2.8). The condition, ' 2 L 2, replaces boundary conditions (there is no boundary). As discussed above in the nonlinear context the unbounded domain must be truncated to a bounded interval ( L; L) before discretization and artificial boundary conditions must be chosen. This is a difficult problem and our solution depend on lemma 2 of paper 1. The lemma implies that if Re >0 then a corresponding eigenfunction (if it exists) must decay exponentially as x!±1. This motivates the following simple choice of boundary conditions. '(±L) =0 We refer to [5] for a detailed investigation of this type of reduction to a bounded interval. We remark that, for the investigations motivated by paper 1 and contained in paper 2, only eigenvalues in the right half plane (Re >0) are of interest. If there are none then the result of paper 1 implies nonlinear stability. Paper 2 is dedicated to an investigation of one of very few known examples of unstable viscous shock waves. In that paper we compute the (only) unstable eigenvalue. The truncated problem is of the simplest kind, there are countably many eigenvalues accumulating at infinity (for Re! 1). In paper 2 we discretize the problem with a second order finite difference approximation. The problem is then reduced to finding the eigenvalues of a tridiagonal matrix. The best algorithm for this problem (tridiagonal, non-symmetric matrix) when only a few eigenvalues are of interest is inverse iterations with the restarted Arnoldi method [46]. This allows us to compute the most unstable eigenvalues (Re large) efficiently. The problem for a fixed L is thus easily solved. The limit L!1must also be studied. In the limit (L = 1) the spectrum is no longer a countable point spectrum, however in the right/unstable half plane there is only a point spectrum, see paper 1 and 2 for more information on this. Another approach which is used in [6] is to rewrite equation (4.2) to a first order system. Then a certain canonical choice of solutions of the rewritten problem are taken as columns of a matrix. The determinant of this matrix evaluated at x =0is an analytical function (called the Evans function) which can be computed to yield stability information Plane Couette flow The eigenvalue problem in this case is more computationally expensive because the linear operator in this case is a partial differential operator in three space

26 18 Chapter 4. Computational problems related to stability dimensions. The divergence constraint (div ' = 0) also makes the situation more complicated than the standard elliptic case. There is a well-known reformulation of the problem which leads to less computations and also removes the difficulty with the divergence constraint. In investigations of the spectrum it is standard to study this reformulation. We sketch it briefly here. Details are given in paper 4, see also the books [35], [11] and [43]. Fourier transformation is applied in the x 1 and x 3 directions, we denote the corresponding Fourier variables" by ο 1 and ο 3. In the next step two equations are derived, a fourth order uncoupled equation for u 2 (the normal velocity), this equation is called the Orr-Sommerfeld equation. The second equation is a second order equation for the normal vorticity. This equation is forced by u 2 but does not couple back to the equation for u 2. This analytical reformulation leads to the following generalized eigenvalue problem depending on the two parameters ο 1 and ο 3 (for the moment we consider a fixed Reynolds number). A11 0 u2 B11 0 u2 = 0; 1 <x iο 1 A < 1 2 u 2 (±1) = u 0 2(±1) = 0 (4.3) 2 (±1) = 0 Here the A 11 is a fourth order differential operator and A 22 and B 11 are second order differential operators. For each fixed pair (ο 1 ;ο 3 )wehave a point spectrum f k (ο 1 ;ο 3 )g 1 k=1. To solve (4.3) we could apply a finite difference approximation as for the conservation law case, instead we choose a spectral Chebyshev method. This is motivated by the following three considerations. First, it is a well established and tested method for this problem, it was introduced in [39] to compute the eigenvalues of the Orr-Sommerfeld problem. Second, the high (spectral) accuracy leads to very few unknowns in the discrete problem, so the matrix eigenvalue problem is easily solved. The third reason requires more explanation. We program the discretization ourselves but then rely on available linear algebra routines in Matlab which in turn uses functions from LAPACK [2]. Difference methods would maybe be competitive if one uses linear algebra routines designed for sparse matrices, (Chebyshev discretization leads to full matrices). Sparse eigenvalue solvers are indeed available for the standard and generalized eigenvalue problems. In the next section however we will calculate the pseudospectra. Then the operation of calculating the square root of a (positive definite) matrix must be performed. We must also calculate the singular values of a matrix which is a product of four factors, one of which isthe inverse of a sparse matrix. For these more complicated linear algebra operations it is difficult (if possible) to use the sparsity ofthe matrices. For the Chebyshev method on the other hand there is no problem, we just perform the operations on our (full) matrices and then call the singular value routine. This is the third motivation for our choice of method.

27 4.3. Computation of pseudospectra Computation of pseudospectra In this section we discuss pseudospectra in the simplest possible setting of a linear ordinary differential equation, and the corresponding resolvent equation du dt = Au; (A 2 R n n ) (4.4) u(0) = u (0) ; (A si)u = f: How to compute the pseudospectra in the case Av = v 00 + ffxv on the interval ( 1; 1) with boundary conditions v(±1) = 0 is described in detail in paper 3. Paper 4 deals with the plane Couette flow case. Other works which are concerned with pseudospectra related to hydrodynamical stability are [40] and [50]. We emphasize the fact from linear algebra that even if all eigenvalues of a linear system are distinct and lie well inside the stable half-plane, inputs to that system may be amplified by arbitrarily large factors if the eigenfunctions are not orthogonal to one another. A connection between the transient growth of solutions of (4.4) and the norm of the resolvent is the following inequality from [43]. sup t>0 fl fl fl e Atfl sup Re s>0 k(a si) 1 k =: fl Usually fl is what one is trying to compute. The Euclidean matrix norm equals the largest singular value of the matrix. Since we are interested in the norm of the inverse we instead calculate the smallest singular value. To obtain fl we then maximize over s. Here it is sufficient to consider imaginary s. The reason for this is that the resolvent cannot have a local maximum in the resolvent set, [12] p.230. Also, the norm decays as jsj!1. A combination of these two facts implies that the maximization can be restricted to imaginary s. The number fl can also be used to bound the solution of (4.4). The formulas expressing this involve a certain time integral of the solution. For completeness we now define the ffl-pseudospectrum of A as the following subset of the complex plane Λ ffl (A) = ρ ff s 2 C : k(a si) 1 k 1 : ffl As mentioned above we use the matrix norm which is subordinate to the Euclidean vector norm which, in canonical coordinates is given by kxk = ψ nx k=1 jx k j 2! 1=2 :

28 20 Chapter 4. Computational problems related to stability Finally we explain how to calculate matrix norms based on other vector norms defined by inner products. In the canonical basis we have the following expression for such a norm. kxk 2 m = x T Mx = nx nx k=1 l=1 m kl x k x l Here m kl denotes the matrix elements of the positive definite matrix M. We denote the square root of M by F. We thus have M = FF (and F = F T ). Using this we obtain kxk 2 m = x T FFx = kf xk 2 : The following derivation then shows that the m-norm of A is given by the Euclidean norm of FAF 1, i.e. the largest singular value of this matrix. kak = sup x kaxk m kxk m = sup x ftake : y = F xg = sup y kfaxk kf xk = kfaf 1 yk kyk = kfaf 1 k: The techniques described in this section for matrices are used in paper 4 and 5 in the more complicated context of differential operators.

29 Chapter 5 Summary of the papers 5.1 Paper 1: Stability of viscous shock waves for problems with non-symmetric viscosity matrices The stability of viscous shock waves has been much studied. It is both of an intrinsic mathematical interest and of obvious physical importance. In this first paper we prove a theorem giving sufficient conditions for the nonlinear stability of viscous shock wave solutions to systems of conservation laws. The analysis applies to strong shock waves of Lax type but is restricted to perturbations with zero mass. A well-known difficulty in this problem is that the eigenvalue = 0 (due to translational invariance of the conservation law) is not separated from the rest of the spectrum. This excludes the use of standard semi-group methods. We use the Laplace transform and reduce the problem to a spectral condition on the resolvent equation of the linearized problem. Precise estimates of the solution of the resolvent equation are then transformed back to estimates for the linearized equation. Finally these are used to prove full nonlinear stability. The paper is accepted for publication in SIAM Journal on Mathematical Analysis. In the bibliography this is entry [34]. 5.2 Paper 2: Numerical investigation of examples of unstable viscous shock waves In this paper we numerically investigate the stability of a one-parameter family of viscous shockwaves. The PDE is a perturbed version of cubic model, a 2 2 system of conservation laws. The situation is related to the theorem of paper 1 in that 21

30 22 Chapter 5. Summary of the papers the shock wave violates the most interesting sufficient condition for stability. The work is motivated by the fact that there are very few known examples of unstable shockwaves and also by the recent growing interest in stability problems for viscous strong shock waves. The investigation proceeds by first computing the shock profile and then solving the truncated eigenvalue problem. We not only verify the instability but also give the rate of exponential growth of the perturbation. The convergence and performance of the numerical methods is also studied. This paper is accepted for publication in the proceedings to the 8th International Conference on Hyperbolic Problems. Theory, Numerics, Applications.", Magdeburg, In the bibliography this is entry [24]. 5.3 Paper 3: A Chebyshev tau spectral method for the calculation of eigenvalues and pseudospectra This paper is motivated by the investigations of the resolvent of plane Couette flow in paper 4. Paper 3 is concerned with the numerics of the Chebyshev tau spectral method for a rather narrow class of boundary value problems for ODEs (including the Orr-Sommerfeld and Squire equations). This method, or very similar methods, are used in several works starting with the classical paper of Orszag (1971). All details of the method, in particular concerning the implementation of boundary conditions and the computation of pseudospectra, are however not given in the papers reporting results obtained with the method. Because of this we believe that the detailed account of this paper will be of use for researchers interested in the method or its applications. This paper is published with the report number: TRITA-NA-0125, KTH, In the bibliography this is entry [31]. 5.4 Paper 4: Analytical and numerical investigation of the resolvent for plane Couette flow In this paper we have collected both analytical estimates of the norm of the resolvent and computational results. The region of interest concerning the resolvent is the unstable complex half-plane and we determine the Reynolds number dependence of the bounds. The analytical estimate is derived in a sector of the complex half-plane and capture both the dependence on the complex variable and the Reynolds number. The sector covers all but a bounded part of the unstable half-plane.

31 5.5. Paper 5: Bounds of the threshold amplitude for plane Couette flow 23 The numerical investigation concerns the resolvent inthe unstable half-plane. We test several weighted norms with coefficients depending on the Reynolds number. The norm of the Resolvent grows like R fl in all cases, the different weighted norms give different exponents fl. For the energy norm we have fl = 2, in this paper we find a norm which lead to the exponent fl =1. This is used is paper 5 to improve earlier bounds in the literature on the threshold amplitude for stability of plane Couette flow. In this paper we apply the methods which where investigated for a model problem in paper 3. This paper is submitted to SIAM Journal on Applied Mathematics. In the bibliography this is entry [32]. 5.5 Paper 5: Bounds of the threshold amplitude for plane Couette flow In this paper we prove nonlinear stability for finite amplitude perturbations of plane Couette flow. We use a bound from paper 4 of the solution of the resolvent equation in the unstable complex half-plane to estimate the solution of the full nonlinear problem. The result is a lower bound, including Reynolds number dependence, of the threshold amplitude below which all perturbations are stable. This is an improvement of the corresponding result in [25]. We use the norm found in paper 4 which lead to a linear growth of the (maximum in the unstable half-plane) resolvent with the Reynolds number. The resolvent estimate is transformed back to an estimate for the linearized equation, which in turn is used to prove nonlinear stability. The general approach has much in common with that of paper 1. The actual form of the resolvent estimate and the nonlinearity however leads to substantial differences in the proofs. This paper is submitted to Journal of Nonlinear Mathematical Physics. In the bibliography this is entry [33].