Stability of Plane Couette Flow and Pipe Poiseuille Flow PER-OLOV ÅSÉN

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1 Stability of Plane Couette Flow and Pipe Poiseuille Flow PER-OLOV ÅSÉN Doctoral Thesis Stockholm, Sweden, 2007

2 TRITA-CSC-A 2007:7 ISSN ISRN KTH/CSC/A--07/07--SE ISBN CSC Skolan för datavetenskap och kommunikation SE Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 25 maj 2007 kl i D3, Huvudbyggnaden, Kungl Tekniska högskolan, Lindstedtsvägen 3, Stockholm. c Per-Olov Åsén, May 2007 Tryck: Universitetsservice US AB

3 iii Abstract This thesis concerns the stability of plane Couette flow and pipe Poiseuille flow in three space dimensions. The mathematical model for both flows is the incompressible Navier Stokes equations. Both analytical and numerical techniques are used. We present new results for the resolvent corresponding to both flows. For plane Couette flow, analytical bounds on the resolvent have previously been derived in parts of the unstable half-plane. In the remaining part, only bounds based on numerical computations in an infinite parameter domain are available. Due to the need for truncation of this infinite parameter domain, these results are mathematically insufficient. We obtain a new analytical bound on the resolvent at s = 0 in all but a compact subset of the parameter domain. This is done by deriving approximate solutions of the Orr Sommerfeld equation and bounding the errors made by the approximations. In the remaining compact set, we use standard numerical techniques to obtain a bound. Hence, this part of the proof is not rigorous in the mathematical sense. In the thesis, we present a way of making also the numerical part of the proof rigorous. By using analytical techniques, we reduce the remaining compact subset of the parameter domain to a finite set of parameter values. In this set, we need to compute bounds on the solution of a boundary value problem. By using a validated numerical method, such bounds can be obtained. In the thesis, we investigate a validated numerical method for enclosing the solutions of boundary value problems. For pipe Poiseuille flow, only numerical bounds on the resolvent have previously been derived. We present analytical bounds in parts of the unstable half-plane. Also, we derive a bound on the resolvent for certain perturbations. Especially, the bound is valid for the perturbation which numerical computations indicate to be the perturbation which exhibits largest transient growth. The bound is valid in the entire unstable half-plane. We also investigate the stability of pipe Poiseuille flow by direct numerical simulations. Especially, we consider a disturbance which experiments have shown is efficient in triggering turbulence. The disturbance is in the form of blowing and suction in two small holes. Our results show the formation of hairpin vortices shortly after the disturbance. Initially, the hairpins form a localized packet of hairpins as they are advected downstream. After approximately 10 pipe diameters from the disturbance origin, the flow becomes severely disordered. Our results show good agreement with the experimental results. In order to perform direct numerical simulations of disturbances which are highly localized in space, parallel computers must be used. Also, direct numerical simulations require the use of numerical methods of high order of accuracy. Many such methods have a global data dependency, making parallelization difficult. In this thesis, we also present the process of parallelizing a code for direct numerical simulations of pipe Poiseuille flow for a distributed memory computer.

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5 Preface This thesis contains five papers and an introduction. Paper I: Per-Olov Åsén and Gunilla Kreiss, A Rigorous Resolvent Estimate for Plane Couette Flow, J. Math. Fluid Mech., accepted Published online. DOI: /s The author of this thesis contributed to the ideas, performed the numerical computations and wrote the manuscript. This paper is also part of the licentiate thesis [1]. Paper II: Malin Siklosi and Per-Olov Åsén, On a Computer-Assisted Method for Proving Existence of Solutions of Boundary Value Problems, Technical Report, TRITA-NA 0426, NADA, KTH, The theoretical derivations were done in close cooperation between the authors, both of them contributing in an equal amount. The author of this thesis had the main responsibility for the computer implementations and wrote section 5 in the report. Malin Siklosi had the main responsibility for the literature studies and wrote sections 1-4 in the report. This paper is also part of the licentiate thesis [1]. Paper III: Per-Olov Åsén and Gunilla Kreiss, Resolvent Bounds for Pipe Poiseuille Flow, J. Fluid Mech., 568: ,2006. The author of this thesis contributed to the ideas, performed the mathematical derivations and wrote the manuscript. Paper IV: Per-Olov Åsén, A Parallel Code for Direct Numerical Simulations of Pipe Poiseuille Flow, Technical Report, TRITA-CSC-NA 2007:2, CSC, KTH, Paper V: Per-Olov Åsén, Gunilla Kreiss and Dietmar Rempfer, Direct Numerical Simulations of Localized Disturbances in Pipe Poiseuille Flow, submitted to Theoret. Comput. Fluid Dynamics The author of this thesis contributed to the ideas, performed the simulations and wrote the manuscript. v

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7 Acknowledgments First and foremost, I would like to thank my advisor Professor Gunilla Kreiss, for her guidance, support and encouragement throughout my work at KTH, making the completion of this thesis possible. It has been a privilege and pleasure being her student. The second paper of this thesis was done in collaboration with Dr. Malin Siklosi, and I thank her for the rewarding experience of working with her. The third paper of this thesis was partly done while visiting Professor Peter Schmid at the University of Washington, Seattle. I thank him for the stimulating experience of working with him. I would like to thank the people at the department of mechanics and the Linné flow centre. Especially, Professor Dan Henningson, Dr. Luca Brandt and Dr. Philipp Schlatter provided invaluable help on the fifth paper of this thesis. Paper 5 would not have been possible without the excellent serial code developed by Professor Dietmar Rempfer and Dr. Jörg Reuter and I am grateful for having been able to work with and develop this code. Also, the computations in the paper were inspired from experiments by Professor Tom Mullin and Dr. Jorge Peixinho and I thank them for fruitful conversations. The Swedish National Infrastructure for Computing provided computer time and the Center for Parallel Computers at KTH provided support during computations and development, all of which I am grateful for. Especially, I would like to thank Dr. Ulf Andersson for helping in the parallelization of the code. I would like to thank all present and former colleagues at CSC for providing a stimulating environment to work in. Finally, I would like to thank my family for help and support throughout the years. Financial support has been provided by Vetenskapsrådet and is gratefully acknowledged. vii

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9 Contents Contents ix 1 Introduction 1 2 The Navier Stokes Equations Wall bounded shear flows Stability of shear flows Stability by resolvent analysis Direct numerical simulations Computer-Assisted Proofs Basic Ideas Relation to Paper Summary of Papers Paper I: A Rigorous Resolvent Estimate for Plane Couette Flow Paper II: On a Computer-Assisted Method for Proving Existence of Solutions of Boundary Value Problems Paper III: Resolvent Bounds for Pipe Poiseuille Flow Paper IV: A Parallel Code for Direct Numerical Simulations of Pipe Poiseuille Flow Paper V: Direct Numerical Simulations of Localized Disturbances in Pipe Poiseuille Flow Bibliography 27 ix

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11 Chapter 1 Introduction The main topic of this thesis is the stability of incompressible plane Couette flow and pipe Poiseuille flow. Plane Couette flow is the stationary flow between two infinite parallel plates, moving in opposite directions at a constant speed, and pipe Poiseuille flow is the stationary flow in an infinite circular pipe, driven by a constant pressure gradient in the axial direction. The mathematical model describing both flows is the Navier Stokes equations. The reason for studying these flows is that they are simple examples of shear flows and the steady analytical solution is known in both cases. A better understanding of the stability of plane Couette flow and pipe Poiseuille flow could provide information useful for more complicated flows. Inducing or avoiding turbulence by active or passive control would have significant impact in various areas. For example, when mixing fuel and air in an engine, a high level of turbulence intensity is desired for an efficient mixing. The air flow around a moving car or airplane is turbulent in large regions which results in high skin friction. By reducing the areas where the flow is turbulent, lower fuel consumption could be achieved. Which mechanisms are important in the transition to turbulence is not understood. A further insight in this area is crucial for controlling turbulence. Paper 1 concerns the stability of plane Couette flow by bounding the norm of the resolvent at the point s = 0 in the unstable half-plane. Previously derived bounds have been based on numerical computations in parts of an infinite parameter domain. We present new analytical results for the resolvent. These results imply a sharp bound on the resolvent in all but a compact subset of the infinite parameter domain. By reducing the domain where computations are needed to a compact set, it is possible to derive a mathematically rigorous bound by using a validated numerical method. This is the topic of paper 2, where we evaluate a method for proving existence and enclosures of solutions of boundary value problems by using numerical computations. Hence, paper 2 provides a way of making the bound on the resolvent in paper 1 rigorous. Papers 3 5 concern the stability of pipe Poiseuille flow. In paper 3, resolvent bounds are derived in parts of the unstable half-plane. The remaining domain 1

12 2 CHAPTER 1. INTRODUCTION grows with the Reynolds number. We also derive resolvent bounds for perturbations satisfying certain conditions. These bounds are valid in the entire unstable halfplane. No numerical computations are used in paper 3. In order to analyze the stability of pipe Poiseuille flow in detail, direct numerical simulations (DNS) can be used. Even in simple geometries like a pipe, such computations require the use of massive computing resources. The use of high order methods in codes for DNS can make parallelization difficult. This is the topic of paper 4, in which we describe the parallelization of a DNS code for pipe Poiseuille flow and show results of good parallel performance. In paper 5, results from the parallel DNS code are presented. Especially, we consider a spatially highly localized disturbance which in experiments have shown to be efficient in triggering turbulence. The initial chapters in the thesis give a brief background to the topics of the five papers. In chapter 2, the Navier Stokes equations are introduced, and some previous results are presented. The literature available on the Navier Stokes equations is vast, and some references to books are given for further reading. Also, some classical cases of wall bounded shear flows are introduced in this chapter. In chapter 3, the stability of flows are discussed, with emphasis on the two shear flows considered in this thesis. Some previous results are presented, both of experimental and mathematical type. Special attention is given to the methods for analyzing stability considered in this thesis. In chapter 4, the use of numerical, approximate solutions for mathematical proofs is discussed. The idea of using computers for mathematically rigorous proofs is almost a contradiction. The inherent rounding errors in floating-point calculations and the necessity of finite dimensional models in a computer seem impossible to overcome. We give the basic ideas of how these obstacles can be conquered by using well known results from functional analysis and by using a different representation of real numbers when stored in a computer. The ideas are focused on the method implemented in paper 2. We also describe in detail how the method in paper 2 can be used to make the numerical part of the proof in paper 1 rigorous. Chapter 5 contains short summaries of the five papers in the thesis. The summaries are slightly more extensive than the corresponding abstracts and are included for the readers convenience.

13 Chapter 2 The Navier Stokes Equations A mathematical description of the flow of a viscous incompressible fluid was first derived in the early 19th century by Navier. Shortly after, others gave the equations a more firm mathematical foundation. The result was the widely known Navier Stokes equations. Given a domain Ω R n, let u(t, x) = (u 1 (t, x),..., u n (t, x)) be the velocity and p(t, x) the pressure at (t, x) = (t, x 1,..., x n ). The non-dimensionalized Navier Stokes equations give the evolution of the flow as u t + (u )u + p = 1 R u, (2.1) u = 0. Here, R is the Reynolds number given by R = V L/ν, where V and L are typical velocity and length scales, respectively, and ν is the kinematic viscosity of the fluid. The equations must also be supplemented with initial and boundary conditions. It is well known that in two space dimensions, (2.1) has a unique solution for all times under some restrictions on the initial condition. In three space dimensions, there are local (in time) existence results which can be extended to global existence results if the initial condition is small enough in some suitable norm, see e. g. [44] p Since the analytical solution of (2.1) is only known in a few special cases, obtaining the solution usually involves the use of some numerical method. Existence and uniqueness results are then valuable. For further reading about the mathematical properties of the Navier Stokes equations, we refer to [14, 44, 54]. 2.1 Wall bounded shear flows In shear flows, the fluid motion is dominated by sheets of fluid moving in different velocities parallel to each other. Although seemingly a substantial simplification, shear flows are present in more complicated flows when considering the flow sufficiently close to an object. In this section, we describe three classical wall bounded 3

14 4 CHAPTER 2. THE NAVIER STOKES EQUATIONS shear flows which have been studied extensively; the main reason for their achieved popularity being that they all are analytical solutions of the Navier Stokes equations. All three flows concern the flow in simple geometries. Pipe Poiseuille flow, also known as Hagen Poiseuille flow, is the (incompressible) flow in an infinite pipe of constant radius. The flow is driven by a constant non-zero pressure gradient in the axial direction. The other two classical wall bounded shear flows are plane Couette flow and plane Poiseuille flow. Both flows concern the flow between two infinite parallel plates. In plane Couette flow, the plates are moving in opposite directions at a constant speed and in plane Poiseuille flow, the plates are stationary and the flow is driven as in pipe Poiseuille flow, i. e. by a constant non-zero pressure gradient in the streamwise direction. The stationary solutions of these flows are parallel flows, where the velocity only depends on the distance from the wall; for plane Couette flow, the solution is a velocity profile which varies linearly between the velocities of the two plates and in plane and pipe Poiseuille flow, the solution is a parabolic velocity profile in the direction of the negative pressure gradient, see Figure 2.1. Figure 2.1: Stationary solution of plane Couette flow (left) and plane and pipe Poiseuille flow (right) The length scale used in the definition of the Reynolds number is in pipe flow the diameter of the pipe and in the channel flows half the distance between the plates. The velocity scale used is half the velocity difference between the plates in plane Couette flow and the maximum and mean velocity in plane and pipe Poiseuille flow, respectively. For the channel, the coordinate system is chosen such that x 1 is the streamwise direction, x 2 the direction normal to the plates and x 3 the spanwise direction. The plates are located at x 2 = ±1, i. e. the domain is Ω c = {x R 3 1 x 2 1}. In the case of pipe Poiseuille flow, cylindrical coordinates, (r, φ, z), are used and the pipe radius is one, yielding the domain Ω p = {(r, φ, z) 0 r 1, 0 φ 2π, z R}. The stationary solutions are now

15 2.1. WALL BOUNDED SHEAR FLOWS 5 given by x 2 e x1, for plane Couette flow, U = (1 x 2 2)e x1, for plane Poiseuille flow, (1 r 2 )e z, for pipe Poiseuille flow. Plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow have been extensively studied by applied mathematicians throughout the years, mainly because they are some of the simplest examples of flows available. However, despite their simplicity much is still unknown about the effects of perturbations on the stationary flows. A better understanding of the important mechanisms in these flows could have implications for other, more complicated, flows.

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17 Chapter 3 Stability of shear flows The field of hydrodynamic stability concerns the stability of various flows when subjected to disturbances. This is an important concept since a stationary unstable flow can not exist in reality. Also, a flow can be stable to some disturbances while unstable to others. A disturbance generating a perturbation which grows with time might lead to turbulence. Quantifying for which disturbances a flow is stable is of great importance in various applications. For an introduction to the field, we refer to the books [8, 9, 41]. Given a flow U, P which solves (2.1), the effect of a disturbance, u 0, can be investigated by considering the equations for the perturbed state. Let the deviation from the given flow be denoted by u, p. Since both the given flow U, P and the perturbed state U + u, P + p satisfy (2.1), subtracting the equations yields u t + (u )u + (U )u + (u )U + p = 1 R u, (3.1) u = 0, with initial condition u(x, 0) = u 0. In order to define stability, we need a norm to measure the size of the perturbation. The most commonly used norm is the L 2 -norm, since it corresponds to the kinetic energy of the perturbation. However, any norm can be used and in some cases other choices of norms might be more suitable. The flow U is called stable to the disturbance u 0 if the norm of the resulting perturbation becomes arbitrarily small as time increases, i. e. if lim u(t) = 0. (3.2) t If the flow is stable to all disturbances, it is called globally stable. Usually, the flow is only stable to all disturbances which are small enough, i. e. to all disturbances satisfying u 0 < γ for some γ > 0. This is known as conditional stability. The type of stability a flow exhibits typically depends on the Reynolds number, R. At low R, the flow might be globally stable, while being conditionally stable at 7

18 8 CHAPTER 3. STABILITY OF SHEAR FLOWS higher R. Especially, some flows have a critical Reynolds number, R C, such that for R > R C, the flow is not conditionally stable. This means that there exists at least one infinitesimal disturbance such that the flow is not stable. Determining how the stability depends on the Reynolds number for different flows is of central interest in hydrodynamic stability. Substantial insight in the stability properties of a flow can be obtained by experimental investigations. Numerous experiments have been performed over the years, both for plane Couette flow and pipe Poiseuille flow. Osborne Reynolds made extensive experimental investigations of pipe flow in the late 19th century; the main achievement of the experiments was the discovery that one non-dimensional number, subsequently named after him, characterized the stability of the flow. He also noted that for Reynolds numbers below R 2000, pipe flow is globally stable. This value has in modern experiments been estimated to R 1800 [30]. Although transition to turbulence may occur at higher Reynolds numbers, laminar flow can be maintained by avoiding disturbances. In highly controlled experiments, laminar pipe flow has been observed at R 10 5 [31]. The highest Reynolds number for which plane Couette flow is globally stable has been determined in experiments to R 350 [45]. From a mathematical point of view, the stability properties of a flow can be investigated in several different ways. The most straightforward way is to consider the eigenvalues of the linearized equations, i. e. of the equations (3.1) without the nonlinear term. If there exists an eigenvalue with positive real part, perturbations with a non-zero component in the direction of the corresponding eigenfunction will exhibit exponential growth. Determining the smallest Reynolds number which allows exponentially growing perturbations gives the critical Reynolds number, R C. However, this does not imply that for subcritical Reynolds numbers, i. e. for R < R C, the flow is stable to all perturbations, since the effect of the nonlinear term is ignored. Hence, the eigenvalues give no information about the possible conditional stability at lower Reynolds numbers. An example of a flow with a critical Reynolds number is plane Poiseuille flow, which becomes linearly unstable at R C 5772 when the so called Tollmien Schlichting wave becomes linearly unstable [28]. However, turbulence typically appears at much lower Reynolds numbers in reality. Also, the perturbation which requires the smallest amplitude for transition to turbulence at subcritical Reynolds numbers is not the Tollmien Schlichting wave [35]. Hence, the spectrum gives poor information about the influence of different perturbations. Even more misleading are the eigenvalues in the cases of plane Couette flow and pipe Poiseuille flow. Romanov proved in 1973 that plane Couette flow is linearly stable at all Reynolds numbers [38]. In experiments however, turbulence has been observed at Reynolds numbers as low as R 350. Pipe Poiseuille flow is believed to be linearly stable at all Reynolds numbers, although formal proofs have only been derived for axisymmetric perturbations [11] as well as for certain non-axisymmetric perturbations [3]. In addition to these proofs, many numerical computations have been made indicating linear stability of pipe Poiseuille flow, see

19 3.1. STABILITY BY RESOLVENT ANALYSIS 9 e. g. [16, 22, 40]. Despite this linear stability, turbulence may still be observed in pipe flow for Reynolds numbers above R 1800 [30] and is the typical state at high Reynolds numbers. In the last two decades, the failure of eigenvalues to predict the stability of these flows has been explained by a phenomenon commonly referred to as transient growth, see e. g. [48] and the review article [39]. If all eigenvalues of the linearized equations have negative real part, linear theory predicts that all perturbations will eventually decay exponentially. However, linear effects may still cause considerable initial growth of a perturbation. This is due to non-orthogonality (in the considered scalar product) of the eigenfunctions of the linearized Navier Stokes operator. Information about this transient growth is not captured by the eigenvalues but can be obtained by considering the resolvent or the ε-pseudospectrum. This is the topic of section 3.1. Since both plane Couette flow and pipe Poiseuille flow are linearly stable, nonlinear effects are necessary for transition to turbulence. Computers are now powerful enough to simulate flows in simple geometries using direct numerical simulations (DNS). The possibility of high control of disturbances and detailed analysis of results makes DNS a powerful tool. Simulations can reveal which mechanisms are the most important during transition to turbulence. Such information is useful in control of turbulence. An improved ability to avoid or induce turbulence would have numerous applications; an efficient mixing of air and fuel in an engine is achieved with a high intensity of turbulence while airplanes would reduce fuel consumption if turbulence could be avoided. In section 3.2, direct numerical simulation is discussed further. 3.1 Stability by resolvent analysis In order to analytically derive conditions for stability, the resolvent can be used. The resolvent is the solution operator of the Laplace transformed linearized problem. Assume that we have a bound on the norm of the resolvent in the entire unstable half-plane. Then it is possible to derive a bound on the solution of the forced linear problem. This bound is given in terms of the bound on the resolvent and the norm of the forcing. The linear bound is then extended to the nonlinear problem by treating the nonlinear term in the equation as part of the forcing. This is only possible if the forcing is sufficiently small. This condition gives a sufficient condition on the size of the perturbation under which nonlinear stability is guaranteed. We first illustrate this method on a simple model problem, similar to the model problem treated in [13], before discussing results for plane Couette flow and pipe Poiseuille flow. For readers who are not interested in the details, the main steps in the proof of conditional stability of the model problem are the following: For the linear problem corresponding to the model problem (3.3), the resolvent bound (3.4) holds in the entire unstable half-plane Re(s) 0. Using Parseval s identity and scalar multiplication, this resolvent bound implies the bound (3.6) for the linear

20 10 CHAPTER 3. STABILITY OF SHEAR FLOWS problem. If the forcing is sufficiently small, a bound for the nonlinear problem is obtained. By doing this also for the differentiated model problem, the bound (3.11) is obtained under the condition (3.12). The bound (3.11) implies stability for the nonlinear problem, i. e. we have proved conditional stability. Model problem Consider the following ordinary differential equation for v = (v 1, v 2 ) T, v t = Lv + g(v) + f(t), v(0) = v 0, (3.3) where L = ( ) ( R 1 0 v1 v 1 2R 1, g(v) = 2 v1 2 ). We are interested in how the stability of this system changes when the positive constant R grows. Consider first the linear, unforced case g = f = 0 with initial condition v 0 = (v1 0, v0 2 )T. Since, with R > 0, the eigenvalues of L are negative, we know that the solution decays exponentially for sufficiently large times. However, the short time behavior can be significantly different. The general solution of this problem is given by ( v1 v 2 ) = v 0 1 ( 1 R ) ( 0 e t/r + (v2 0 v1r) 0 1 ) e 2t/R. We see that v 1 decays exponentially at all times. However, Taylor expanding v 2 at t = 0 shows that v 2 grows linearly for t O(R). This is known as transient growth, and is due to the fact that the operator L is non-normal, i. e. the eigenvectors of L are non-orthogonal. In fact, the eigenvectors of L are (1, R) T and (0, 1) T, i. e. they are increasingly non-orthogonal with increasing R. We now derive conditions for stability of the nonlinear problem. Since the resolvent method uses the Laplace transform, we consider (3.3) with homogeneous initial conditions. Note that (3.1) could be transformed to an equivalent homogeneous problem by e. g. introducing u = v+e δt u 0 for some δ > 0. This would result in a forcing involving the initial perturbation, u 0, in the equations for v. Let and (, ) denote the l 2 norm and inner product of vectors and let denote the corresponding matrix norm. The linear problem corresponding to (3.3) is, after applying the Laplace transform, sṽ = Lṽ + f(s). The solution operator R(s) = (si L) 1 is known as the resolvent. With R > 0, the eigenvalues of L are negative. Hence, the resolvent is well defined in the entire

21 3.1. STABILITY BY RESOLVENT ANALYSIS 11 unstable half-plane, Re(s) 0. For a normal operator, N, the norm of the resolvent, R(s) = (si N) 1, is given by R(s) = sup λ σ(n) s λ 1, where σ(n) is the spectrum of N, see e. g. [12]. However, since L is non-normal, the norm of the resolvent is larger. Straightforward calculations give the sharp bound (si L) 1 CR 2 (3.4) in the entire unstable half-plane. We use this to bound the solution of the linear problem. By using Parseval s identity, it follows that T 0 v(t) 2 dt 0 v(t) 2 dt CR 4 f(t) 2 dt. For t T, the solution v(t) does not depend on f(t) for t > T. Hence, we can set f(t) = 0 for t > T in the above expression, yielding T 0 0 T v(t) 2 dt CR 4 f(t) 2 dt. (3.5) We also need a bound on v(t). Scalar multiplication of the linear equation corresponding to (3.3) with v gives 1 d 2 dt v(t) 2 = (v, v t ) = (v, Lv) + (v, f) C 1 v ( v 2 + f 2 ), where C 1 is a bound on the range of L. Integrating this from t = 0 to t = T and using (3.5) gives T v(t) 2 CR 4 f(t) 2 dt. Hence, we have the following bound for the linear problem, T T v(t) 2 + v(t) 2 dt C L R 4 f(t) 2 dt. (3.6) 0 0 Now, we will treat the nonlinear term as part of the forcing. For the nonlinear term, we have g(v) 2 v 4. (3.7) Assume that the solution of the nonlinear problem (3.3) satisfies v(t) 2 4R 4 K, K 0 = C L f(t) 2 dt, 0 0 (3.8) for all times T [0, ). We prove this assumption by assuming that it is not true, thus deriving a contradiction. Since v(0) = 0, (3.8) must hold with strict

22 12 CHAPTER 3. STABILITY OF SHEAR FLOWS inequality for some initial time interval. Let T 0 > 0 be the smallest time such that there is equality in (3.8) and consider T T 0. From the linear estimate (3.6) and the bounds (3.7) and (3.8), we have T T v(t) 2 + v(t) 2 dt C L R 4 f(t) + g(v) 2 dt 0 0 T 2C L R 4 f(t) 2 + v(t) 4 dt 0 2R 4 K + 8C L R 8 K T 0 v(t) 2 dt. We must now assume that the forcing is sufficiently small. Assume that Then for T T 0, we have the following bound v(t) C L R 8 K 1 2. (3.9) T 0 v(t) 2 dt 2R 4 K. (3.10) Clearly, the assumption of equality in (3.8) at time T 0 can not be true and (3.10) must hold for all times T [0, ). We also need a similar bound for v t. This is obtained by differentiating equation (3.3) with respect to t. It is easily found that g t 2 3( v 2 + v t 2 ). By doing the same derivations as above for v 2 + v t 2, it is found that where v(t) 2 + v t (T) T 0 v(t) 2 + v t (t) 2 dt CR 4 K, (3.11) K = C L f(t) 2 + f t (t) 2 dt. 0 If we assume f(t) H 1 ([0, )), the right hand side of (3.11) is bounded. It follows that v(t) H 1 ([0, )), which implies lim t v(t) = 0. Thus, we have proved nonlinear stability under an assumption similar to (3.9) for K instead of K, i. e. when f(t) 2 + f t (t) 2 dt ĈR 8. (3.12) 0 Plane Couette flow and pipe Poiseuille flow It has been found that the eigenfunctions of the linearized operators of plane Couette flow and pipe Poiseuille flow are highly non-normal in the L 2 -inner product, see e. g. [36, 47]. This can cause significant transient growth, as explained in the

23 3.1. STABILITY BY RESOLVENT ANALYSIS 13 previous section. Therefore, the resolvent or the closely related ε-pseudospectrum has been in focus the last twenty years. The ε-pseudospectrum of a linear operator, L, generalizes the concept of eigenvalues by defining s to belong to the ε-pseudospectrum if (si L) 1 ε 1. Hence, the ε-pseudospectrum gives information of where the resolvent is large, as opposed to the spectrum which only give information of where the resolvent is infinite or non-existing. Computations of the ε-pseudospectrum for plane Couette flow and pipe Poiseuille flow can be found in e. g. [22, 47, 48]. For plane Couette flow, the resolvent, R, has been investigated by numerical and analytical techniques. In [13], the lower bound R CR 2 was proved for the L 2 -norm. Here and below, we use C to denote any constant independent of the Reynolds number. Numerical computations in [13, 17] indicated this asymptotic dependence to hold in the entire unstable half-plane, i. e. R CR 2. An analytical bound on the L 2 -norm of the resolvent was derived in large parts of the unstable half-plane in [17], where also a new norm was introduced. Computations in [17] indicated R CR in the new norm. This is an optimal R-dependence since there is an eigenvalue with real part Re(λ) R 1 [48]. For pipe Poiseuille flow, fewer analytical results about the resolvent have been derived. Numerical computations in [22] indicate the same dependence as for plane Couette flow, i. e. R CR 2 for the L 2 -norm. As in the example in the previous section, a bound on the resolvent can be used to prove nonlinear stability. Using this technique, the upper bound β 5.25 in the threshold amplitude dependence R β was proved for wall bounded shear flows in [13], under the assumption of the resolvent bound R CR 2. Although this upper bound on β is not sharp, it serves as the only analytical proof of conditional stability of wall bounded shear flows. However, since the resolvent bounds available both for plane Couette flow and pipe Poiseuille flow are based on numerical computations in an infinite parameter domain, the proof is not fully rigorous. This is the motivation of the first paper of this thesis [4]. We present a new sharp bound on the resolvent at the believed maximum s = 0. The bound is based on analytical estimates in all but a compact subset of the parameter domain. In this compact set, we use numerical computations to obtain a bound. Since the set is compact, the numerical bound can be made rigorous by using validated numerical methods. We explain in detail how this can be done in the next chapter. Using the same technique, we hope to bound the resolvent in the remaining part of the unstable half-plane in the future. Moreover, analytical bounds can provide more precise information about the resolvent than just the maximum in the unstable half-plane. Such information could be used to improve the upper bound on β, i. e. to sharpen the threshold amplitude for nonlinear stability. The third paper of this thesis [3] concerns the resolvent of pipe Poiseuille flow. We derive analytical bounds on the resolvent in large parts of the unstable halfplane. Also, a bound valid in the entire unstable half-plane is derived for perturbations which satisfy certain relations involving the Reynolds number and the

24 14 CHAPTER 3. STABILITY OF SHEAR FLOWS wave numbers in the axial and azimuthal directions. Especially, this bound on the resolvent is valid for the perturbation which computations indicate to be the perturbation which exhibits largest transient growth [47]. 3.2 Direct numerical simulations In order to investigate the nonlinear behavior of a flow, direct numerical simulations (DNS) can be used. DNS means that the full nonlinear Navier Stokes equations are solved such that all length scales are resolved. This requires large amounts of computer resources as well as numerical methods with high order of accuracy. DNS are therefore so far only possible at moderate Reynolds numbers and in simple geometries, and should therefore not be confused with an engineering tool for realworld problems. DNS has, however, proven to be an excellent tool in research. For example, the threshold amplitude below which perturbations eventually decay has been examined by DNS; using different disturbances, the threshold was found to behave as R β, with 1 β 1.25 for plane Couette flow and 1.6 β 1.75 for plane Poiseuille flow [13, 19, 35]. The Reynolds number below which pipe Poiseuille flow is globally stable has been verified to R 1800 by using DNS [52]. Also, DNS have yielded important understanding of the mechanisms behind transition to turbulence in boundary layers, see e. g. the review article [23]. Most DNS have so far been performed in planar geometries, since Cartesian coordinates can then be used. In pipe flow, high order methods can be used by considering the equations in cylindrical coordinates. However, this introduces smaller grid-cells near the center of the pipe, requiring a smaller time step. Also, additional difficulties arise from the polar singularity in the discretization. Some DNS codes have been developed for pipe Poiseuille flow, see e. g. [10, 18, 20, 21, 27, 29, 42, 49, 56]. However, these codes are typically of rather low order of accuracy and almost exclusively written for serial computers. In the fourth paper of this thesis [2], we present the process of parallelizing a code for DNS of pipe Poiseuille flow for a distributed memory computer. The code is based on compact finite differences of high order of accuracy in the axial direction and Fourier and Chebyshev expansions in the azimuthal and radial directions, respectively [37]. These numerical techniques are computationally efficient but introduce a global data dependency. This makes parallelization difficult, since there is no way to divide the problem into smaller problems which are almost independent. We present our strategy of parallelization and show results on good efficiency of the parallel code. The fifth paper of this thesis [5] concerns DNS of pipe Poiseuille flow. We use the parallel code developed in paper 4 in order to simulate a disturbance which is highly localized in space. The disturbance is a combination of suction and blowing through two small holes located close to each other and aligned such that they form a 45-degree angle with the pipe axis. The motivation for the simulations is

25 3.2. DIRECT NUMERICAL SIMULATIONS 15 that experiments have shown that this disturbance is efficient in triggering turbulence. Our results show an initial formation of so called hairpin vortices, which are known to play a central role in transition to turbulence in boundary layers [6]. The hairpins are initially advected downstream in an ordered and localized way. After approximately 10 pipe diameters, the perturbation changes from being localized to a globally disordered state. Our results show good agreement with the experiments.

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27 Chapter 4 Computer-Assisted Proofs The invention of the computer has had a tremendous impact on the field of applied mathematics. Problems that were practically impossible to solve 50 years ago are solved in fractions of a second today. However, these solutions are almost never true solutions. A numerical solution of a problem usually suffers from errors. One source of error is that the mathematical model might have infinite degrees of freedom, making finite dimensional approximations necessary. Deriving explicit bounds on the errors made by the approximations is usually difficult. Another source of error is the rounding error. Numbers like π, 2 can not be stored exactly in a computer. Even for numbers that are stored exactly, floating-point arithmetic is not closed. This means that even if x and y can be stored exactly, there is no guarantee that e. g. x + y can be stored exactly, making rounding necessary. In this chapter, we give the basic ideas of how to prove existence and enclosures of solutions of elliptic boundary value problems. This is the topic of the second paper of this thesis [43]. We also explain why this topic is relevant for the first paper of this thesis. 4.1 Basic Ideas In this section, we describe two methods for proving existence of solutions of elliptic boundary value problems. The first method was proposed by Nakao, and has been successfully used in various applications [25, 46, 50, 51, 53]. This is the method used in paper 2 of this thesis. The second method was proposed by Plum, and has also proved successful [7, 15, 32, 33, 34]. The methods are quite similar in some parts, and a combination of them has been used by Nagatou, Yamamoto and Nakao [26]. Both methods rely on an approximate, numerical solution, u h, which can be derived by any numerical method. From the approximate solution, a suitable fixedpoint equation, w = T(w), for the error, w = u u h, is derived. The idea is to prove that w = T(w) has a solution in a subset of a Banach space. The subset consists of 17

28 18 CHAPTER 4. COMPUTER-ASSISTED PROOFS functions with norm smaller than an explicitly derived upper bound. This upper bound gives bounds on the magnitude of the error in the approximate solution, u h. In order to prove the existence of a solution of the fixed-point equation, Nakao and Plum use the well known Schauder fixed-point theorem or Banach fixed-point theorem. The theorems state, see e. g. [55], Theorem (Schauder fixed-point theorem). Let W be a non-empty, closed, bounded, convex subset of a Banach space X. If T : W W is a compact operator, then there exists a w W such that w = T(w). Theorem (Banach fixed-point theorem). Let W be a non-empty, closed subset of a complete metric space X. If T : W W is a contraction on W, then there exists a unique w W such that w = T(w). Note that Theorem ensures a unique solution in W, which is not the case for Theorem Verifying that Theorem or Theorem can be applied to the derived fixed-point equation and finding a suitable subset are done in different ways in the approaches by Nakao and Plum. In Nakao s method, the fixed-point equation is divided into a finite dimensional part and an infinite dimensional part. The finite dimensional part is rewritten using the linearization, L h, of the finite dimensional projection of the given equation at the approximate solution, u h. This yields an equivalent fixed-point equation which is more likely to map the finite dimensional part of W into itself. Verifying the conditions of Theorem or Theorem for the finite dimensional part is done by explicitly inverting L h. The infinite dimensional part is treated by analytical methods, using e. g. a priori error bounds on the projection into the finite dimensional subspace. Plum s method uses the linearization, L, of the infinite dimensional problem at the approximate solution, u h. Using a lower bound on the norm of L and a bound on the norm of the residual of u h, the conditions of Theorem or Theorem are verified by analytical and numerical techniques. The main difficulty is to derive the lower bound on the norm of L. This is obtained from the eigenvalue of L or L L with smallest absolute value. Deriving an enclosure of this eigenvalue can be done by solving related finite-dimensional matrix eigenvalue problems which is suitable for computer implementation. In both methods, the effect of the rounding errors in computations must be accounted for. This can be done by using interval arithmetic [24]. Interval arithmetic represents real numbers as closed intervals, where the upper and lower bounds of the intervals are floating-point numbers. Thus, all real numbers can be represented. By defining an arithmetic for the intervals, the effect of the rounding error can be rigorously accounted for in each arithmetic operation. This can be extended to all elementary functions used in computations, such that the functions take intervals as arguments and return intervals which encloses the range of the functions over the argument intervals.

29 4.2. RELATION TO PAPER Relation to Paper 1 In the first paper of this thesis, a bound on the resolvent for plane Couette flow is derived at the point s = 0. This is done by obtaining analytical bounds in all but a compact subset of an infinite parameter domain consisting of wave numbers in two space directions and the Reynolds number. In the remaining compact set, we use standard numerical computations for a finite set of these parameter values. However, although the subset of the parameter domain is bounded, it consists of infinitely many parameter values. Thus, this part of the proof is not rigorous. In this section, we describe how the method in paper 2 could be used to make also the numerical part of the proof rigorous. We first summarize the procedure of making the proof in paper 1 rigorous. There are two separate problems in obtaining a rigorous bound. First, for a given choice of parameter values, how is a rigorous bound on the resolvent obtained by numerical computations? The solution to this is rather straightforward. In paper 1, the problem is reduced to solving a one-dimensional boundary value problem, computing quantities which depend on the solution, and showing that these quantities fulfill certain conditions. This can be done using the method described in paper 2. The second problem is that a rigorous resolvent bound is not only needed for one choice of parameter values but for infinitely many parameter values. This problem is solved by analytical means, resulting in Lemma In short, the lemma states that if a rigorous resolvent bound is derived for one choice of parameter values, then this bound is valid in some neighborhood of the chosen parameter values. The size of this neighborhood is explicitly computable. Hence, rigorous resolvent bounds need to be derived only for a finite set of parameter values. In the rest of this section, we describe this strategy in further detail. In paper 1, the numerical part of the proof concerns the boundary value problem w (x) (iax + b 2 )w(x) = 0, w( 1) = 1, (4.1) w(1) = 0, in the compact parameter domain Σ = {a, b R a [1/16, 40 3 ], b 2 [0, a 2/3 ]}. For every combination of a and b in Σ, we need to prove two things about the solution of (4.1). First, we need to prove that the L 2 -norm of the solution is bounded. Later in this section, we show that this holds for all parameter values in Σ, see the remark after the proof of Lemma Secondly, we need to prove that the matrix ( ) u J = ( 1) (u ) (1) u (1) (u ) (4.2) ( 1) is non-singular. Here, (u )(x) denotes the complex conjugate of u(x). The matrix

30 20 CHAPTER 4. COMPUTER-ASSISTED PROOFS elements are given by where u ( 1) = u (1) = f b (σ) = g b (σ) = f b (σ)w(σ)dσ, (4.3) g b (σ)w(σ)dσ, (4.4) { sinh(b(σ 1)) sinh(2b), b 0, σ 1 2, b = 0, (4.5) { sinh(b(σ+1)) sinh(2b), b 0, σ+1 2, b = 0. (4.6) Note that the matrix (4.2) is non-singular if and only if u ( 1) u (1). Hence, for a given pair of parameters, a and b, we need to enclose the solution w(x) of (4.1) and then derive rigorous enclosures of the absolute values of the integrals (4.3) and (4.4). This can be done with the method used in the second paper of this thesis. However, since we implemented the method in MATLAB, we were not able to obtain rigorous bounds when a is large. Using e. g. Fortran would hopefully be sufficient for covering the parameter domain we are interested in. We are still left with the problem of having an infinite number of parameter values in Σ. This can be handled with analytical techniques. By using information about how far from singular J is at a given point in Σ, we are able to prove that J is non-singular in a neighborhood around this point. The result is summarized in the following lemma. Lemma If for a = A and b = B, the solution W(x) of (4.1) is such that the matrix elements (4.3) and (4.4) satisfy U ( 1) U (1) α (4.7) for some α > 0, then the matrix J given by (4.2) is non-singular for all parameter values a and b satisfying 8β( f B + g B ) + (8β + 1)( f b f B + g b g B ) < where f and g are given by (4.5) and (4.6) and where β = a A + b 2 B 2. Here, is the L 2 -norm on Ω = { 1 x 1}. α W, (4.8) Before proving the lemma, note that it is not obvious that the quantity on the left hand side of (4.7) should be positive. However, numerical experiments indicate

31 4.2. RELATION TO PAPER 1 21 this to always be the case. Of course, if the left hand side of (4.7) would be negative for some parameter combination, a similar lemma could be derived handling this case. Proof. Consider some parameter values a and b satisfying (4.8) and denote the corresponding solution of (4.1) by w(x). From (4.1), the difference w = w W satisfies w (iax + b 2 ) w = (i(a A)x + (b 2 B 2 ))W, w(±1) = 0. Taking the L 2 -inner product of this equation with w, using integration by parts and taking the real part yields w 2 + b 2 w 2 ( a A + b 2 B 2 ) W w = β W w. Using a Poincaré inequality for w and the relation cd c 2 /(2µ) + d 2 µ/2, valid for all c, d R, µ > 0, we thus have the bound ( ) w b2 w 2 4β 2 W 2. (4.9) Now, evaluating u ( 1) from (4.3), using w = w + W and (4.9) gives 1 u ( 1) = (f B (σ) + f b (σ) f B (σ)) ( w(σ) + W(σ))dσ 1 U ( 1) f B w f b f B ( w + W ) (4.10) Similarly, using (4.4) yields U ( 1) 8β f B W (8β + 1) f b f B W. u (1) U (1) + 8β g B W + (8β + 1) g b g B W. (4.11) By (4.7), (4.8), (4.10), and (4.11), we have u ( 1) u (1) > 0 and thus J is non-singular. Remark. We stated earlier in this section that the L 2 -norm of the solution of (4.1) is bounded for all a and b in Σ. Since (4.9) also holds when W is the solution with A and B outside Σ, we can especially chose A = B = 0. Clearly, W is then bounded, and it follows from (4.9) that w = w+w is bounded in any bounded parameter domain. Hence, Lemma and the method used in paper 2 provides a possibility of deriving a rigorous bound on the resolvent in Σ, where the bound in paper 1 is not rigorous. One needs to find a finite set of points in Σ such that J is non-singular for these points and such that the neighborhoods, given by Lemma 4.2.1, cover Σ. In order for Σ to be covered, we must ensure that the measures of the neighborhoods do not become arbitrarily small even if J is non-singular. This can only

32 22 CHAPTER 4. COMPUTER-ASSISTED PROOFS happen if α in (4.7) becomes arbitrarily small somewhere in Σ. However, from (4.10) and (4.11), we know that the function γ(a, b) u ( 1) u (1) is continuous with respect to a and b. Since Σ is a compact set, γ(a, b) attains a minimum, α min, in Σ. Hence, if J is non-singular in Σ, we can cover Σ with a finite number of neighborhoods attained from using Lemma Computations made in paper 1 indicate that J is non-singular, and we believe this could be proved with the approach described in this section. Finally, note that when computing the quantities in (4.8), all computations should be rigorous, using e. g. interval arithmetic. Since f B, g B, f b f B and g b g B can be derived explicitly, implementation using interval arithmetic is straightforward.

33 Chapter 5 Summary of Papers 5.1 Paper I: A Rigorous Resolvent Estimate for Plane Couette Flow In this paper, we derive a rigorous bound on the resolvent for plane Couette flow at the point s = 0. We do this analytically by finding approximate solutions of the Orr Sommerfeld equation while keeping track of the errors made by the approximations. This is not possible in the entire parameter domain. However, the remaining domain is bounded, and we use numerical computations to obtain a bound. Previously derived bounds at s = 0 have been based on computations in an infinite parameter domain, making rigorous results impossible. In a bounded domain, rigorous results can be derived by the use of numerical verification methods using interval arithmetic. This paper is published online in Journal of Mathematical Fluid Mechanics and is entry [4] in the bibliography. 5.2 Paper II: On a Computer-Assisted Method for Proving Existence of Solutions of Boundary Value Problems In paper 2, we investigate a method for proving existence of solutions of elliptic boundary value problems. The method was proposed by Nakao. We solve two problems using this method; a linear test problem and the one-dimensional viscous Burgers equation. For the first problem, the method works well. For Burgers equation however, the computational complexity becomes too large when the viscosity decreases. This is not surprising, since Burgers equation linearized at the correct solution rapidly becomes close to singular when the viscosity is decreased. We therefore reformulate the problem by replacing one of the boundary conditions with a global integral condition. This approach drastically reduces the computational complexity. This paper is a technical report and is entry [43] in the bibliography. 23

34 24 CHAPTER 5. SUMMARY OF PAPERS 5.3 Paper III: Resolvent Bounds for Pipe Poiseuille Flow In paper 3, we derive an analytical bound on the resolvent of pipe Poiseuille flow in large parts of the unstable half-plane. This is done by scalar multiplying the linearized Navier Stokes equations (in Cartesian coordinates) with the solution and using integration by parts. We also consider the linearized equations in cylindrical coordinates, Fourier transformed in axial and azimuthal directions. For certain combinations of the wave numbers and the Reynolds number, we derive an analytical bound on the resolvent of the Fourier transformed problem. In particular, this bound is valid for the perturbation which numerical computations indicate to be the perturbation that gives largest transient growth. Our bound has the same dependence on the Reynolds number as the computations give. This paper is published in Journal of Fluid Mechanics and is entry [3] in the bibliography. 5.4 Paper IV: A Parallel Code for Direct Numerical Simulations of Pipe Poiseuille Flow In this paper, we describe the process of parallelizing a serial code for direct numerical simulations of pipe Poiseuille flow for a distributed memory computer. The serial code, developed by Reuter and Rempfer, uses compact finite differences of at least eighth order of accuracy in the axial direction and Fourier and Chebyshev expansions in the azimuthal and radial directions, respectively. While these methods are attractive from a numerical point of view, they give a global data dependency which makes the parallelization procedure complex. In the resulting parallel code, the partitioning of the domain changes between partitioning in the axial direction and partitioning in the azimuthal direction as needed. We present results showing good speedup of the parallel code. This paper is a technical report and is entry [2] in the bibliography. 5.5 Paper V: Direct Numerical Simulations of Localized Disturbances in Pipe Poiseuille Flow. In this paper, we perform direct numerical simulations of pipe Poiseuille flow subjected to a disturbance which is highly localized in space. The disturbance is a combination of suction and blowing in two small holes, located such that they form a 45-degree angle with the pipe axis. We perform direct numerical simulations for the Reynolds number R = The results show a packet of hairpin vortices traveling downstream, each having a length of approximately one pipe radius. The perturbation remains highly localized in space while being advected downstream for approximately 10 pipe diameters. Beyond that distance from the disturbance origin the flow becomes severely disordered.

35 5.5. PAPER V: DIRECT NUMERICAL SIMULATIONS OF LOCALIZED DISTURBANCES IN PIPE POISEUILLE FLOW. 25 The stability of pipe Poiseuille flow is highly dependent on the specific disturbance used. The reason for studying this particular disturbance is that experiments by Mullin and Peixinho have shown that it is efficient in triggering turbulence, yielding a threshold dependence on the required amplitude as R 1.5 on the Reynolds number. The experiments also indicate an initial formation of hairpin vortices, with each hairpin having a length of approximately one pipe radius, independent of the Reynolds number in the range of R = 2000 to Thus, our computations are in good agreement with the experiments. This paper is submitted to Theoretical and Computational Fluid Dynamics and is entry [5] in the bibliography.

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37 Bibliography [1] P.-O. Åsén. A proof of a resolvent estimate for plane Couette flow by new analytical and numerical techniques. Licentiate thesis TRITA-NA 0427, NADA, KTH, [2] P.-O. Åsén. A parallel code for direct numerical simulations of pipe Poiseuille flow. Technical Report TRITA-CSC-NA 2007:2, CSC, KTH, [3] P.-O. Åsén and G. Kreiss. Resolvent bounds for pipe Poiseuille flow. J. Fluid Mech., 568: , [4] P.-O. Åsén and G. Kreiss. A rigorous resolvent estimate for plane Couette flow. J. Math. Fluid Mech., Accepted DOI: /s [5] P.-O. Åsén, G. Kreiss, and D. Rempfer. Direct numerical simulations of localized disturbances in pipe Poiseuille flow. Theoret. Comput. Fluid Dynamics, Submitted. [6] R. I. Bowles. Transition to turbulent flow in aerodynamics. Phil. Trans. R. Soc. Lond. A, 358: , [7] B. Breuer, P. J. McKenna, and M. Plum. Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof. J. Diff. Eq., 195: , [8] P. G. Drazin. Introduction to hydrodynamic stability. Cambridge university press, Cambridge, [9] P. G. Drazin and W. H. Reid. Hydrodynamic Stability. Cambridge University Press, London, [10] J. G. M. Eggels, F. Unger, M. H. Weiss, J. Westerweel, R. J. Adrian, R. Friedrich, and F. T. M. Nieuwstadt. Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment. J. Fluid Mech., 268: , [11] I. H. Herron. Observations on the role of vorticity in the stability theory of wall bounded flows. Stud. Appl. Math., 85: ,

38 28 BIBLIOGRAPHY [12] T. Kato. Perturbation theory for linear operators. Springer, Berlin, [13] G. Kreiss, A. Lundbladh, and D. S. Henningson. Bounds for threshold amplitudes in subcritical shear flows. J. Fluid Mech., 270: , [14] H.-O. Kreiss and J. Lorenz. Initial-Boundary Value Problems and the Navier- Stokes Equations, volume 136 of Pure Appl. Math. Academic Press, Boston, [15] J.-R. Lahmann and M. Plum. A computer-assisted instability proof for the Orr- Sommerfeldt equation with Blasius profile. Z. Angew. Math. Mech., 84(3): , [16] M. Lessen, S. G. Sadler, and T.-Y. Liu. Stability of pipe Poiseuille flow. Phys. Fluids, 11: , [17] M. Liefvendahl and G. Kreiss. Analytical and numerical investigation of the resolvent for plane Couette flow. SIAM J. Appl. Math., 63: , [18] P. Loulou, R. D. Moser, N. N. Mansour, and B. J. Cantwell. Direct numerical simulation of incompressible pipe flow using a b-spline spectral method. TM , NASA, [19] A. Lundbladh, D. S. Henningson, and S. C. Reddy. Threshold amplitudes for transition in channel flows. In M. Y. Hussaini, T. B. Gatski, and T. L. Jackson, editors, Transition, Turbulence and Combustion, volume 1, pages Kluwer, Dordrecht, Holland, [20] B. Ma, C. W. H. Van Doorne, Z. Zhang, and F. T. M. Nieuwstadt. On the spatial evolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at Re = Part 1. Subcritical disturbance. J. Fluid Mech., 398: , [21] A. Meseguer and F. Mellibovsky. On a solenoidal Fourier-Chebyshev spectral method for stability analysis of the Hagen-Poiseuille flow. Appl. Numer. Math., DOI: /j.apnum [22] A. Meseguer and L. N. Trefethen. Linearized pipe flow to Reynolds number J. Comput. Phys., 186: , [23] P. Moin and K. Mahesh. Direct numerical simulation: A tool in turbulence research. Ann. Rev. Fluid Mech., 30: , [24] R. E. Moore. Interval Arithmetic and Automatic Error Analysis in Digital Computing. PhD thesis, Stanford University, [25] K. Nagatou and M. T. Nakao. An enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems. Lin. Alg. Appl., 324:81 106, 2001.

39 29 [26] K. Nagatou, N. Yamamoto, and M. T. Nakao. An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. and Optimiz., 20(5 & 6): , [27] N. V. Nikitin. Statistical characteristics of wall turbulence. Fluid Dyn., 31: , [28] S. A. Orszag. Accurate solutions of the Orr-Sommerfeld stability equation. J. Fluid Mech., 50: , [29] P. L. O Sullivan and K. S. Breuer. Transient growth in circular pipe flow. I. Linear disturbances. Phys. Fluids, 6: , [30] J. Peixinho and T. Mullin. Decay of turbulence in pipe flow. Phys. Rev. Lett., 96:094501, [31] W. Pfenninger. Boundary layer suction experiments with laminar flow at high Reynolds numbers in the inlet length of a tube by various suction methods. In G. V. Lachmann, editor, Boundary layer and flow control, volume 2, pages Pergamon, [32] M. Plum. Computer-assisted existence proofs for two-point boundary value problems. Computing, 46:19 34, [33] M. Plum. Computer-assisted enclosure methods for elliptic differential equations. Lin. Alg. Appl., 324: , [34] M. Plum and C. Wieners. New solutions of the Gendalf problem. J. Math. Anal. Appl., 269: , [35] S. C. Reddy, P. J. Schmid, J. S. Baggett, and D. S. Henningson. On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech., 365: , [36] S. C. Reddy, P. J. Schmid, and D. S. Henningson. Pseudospectra of the Orr- Sommerfeld operator. SIAM J. Appl. Math., 53:15 47, [37] J. Reuter and D. Rempfer. Analysis of pipe flow transition. Part I. Direct numerical simulation. Theoret. Comput. Fluid Dynamics, 17: , [38] V. A. Romanov. Stability of plane-parallel Couette flow. Funct. Anal. Appl., 7: , [39] P. J. Schmid. Nonmodal stability theory. Annu. Rev. Fluid Mech., 39: , [40] P. J. Schmid and D. S. Henningson. Optimal energy density growth in Hagen- Poiseuille flow. J. Fluid Mech., 277: , 1994.

40 30 BIBLIOGRAPHY [41] P. J. Schmid and D. S. Henningson. Stability and Transition in Shear Flows, volume 142 of Appl. Math. Sci. Springer, New York, [42] H. Shan, B. Ma, Z. Zhang, and F. T. M. Nieuwstadt. Direct numerical simulation of a puff and a slug in transitional cylindrical pipe flow. J. Fluid Mech., 387:39 60, [43] M. Siklosi and P.-O. Åsén. On a computer-assisted method for proving existence of solutions of boundary value problems. Technical Report TRITA-NA 0426, NADA, KTH, [44] H. Sohr. The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Berlin, [45] N. Tillmark and P. H. Alfredsson. Experiments on transition in plane Couette flow. J. Fluid Mech., 235:89 102, [46] K. Toyonaga, M. T. Nakao, and Y. Watanabe. Verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators. J. Comp. Appl. Math., 147: , [47] A. E. Trefethen, L. N. Trefethen, and P. J. Schmid. Spectra and pseudospectra for pipe Poiseuille flow. Comput. Meth. Appl. Mech. Engng., 1926: , [48] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll. Hydrodynamic stability without eigenvalues. Science, 261: , [49] R. Verzicco and P. Orlandi. A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys., 123: , [50] Y. Watanabe, N. Yamamoto, and M. T. Nakao. A numerical verification method of solutions for the Navier-Stokes equations. Reliable Computing, 5: , [51] Y. Watanabe, N. Yamamoto, M. T. Nakao, and T. Nishida. A numerical verification of nontrivial solutions for the heat convection problem. J. Math. Fluid Mech., 6:1 20, [52] A. P. Willis and R. R. Kerswell. Critical behavior in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett., 98:014501, [53] N. Yamamoto. A numerical verification method for solutions of boundary value problems with local uniqueness by Banach s fixed-point theorem. SIAM J. Numer. Anal., 35(5): , 1998.

41 31 [54] V. I. Yudovich. The Linearization Method in Hydrodynamical Stability Theory, volume 74 of Trans. Math. Monogr. American Mathematical Society, Providence, [55] E. Zeidler. Nonlinear functional analysis and its applications: Fixed-Point Theorems. Springer, New York, [56] O. Y. Zikanov. On the instability of pipe Poiseuille flow. Phys. Fluids, 8: , 1996.

42 J. math. fluid mech. c 2005 Birkhäuser Verlag, Basel DOI /s Journal of Mathematical Fluid Mechanics On a Rigorous Resolvent Estimate for Plane Couette Flow Per-Olov Åsén and Gunilla Kreiss Communicated by W. Nagata Abstract. We derive a rigorous bound of the solution of the resolvent equation for plane Couette flow in three space dimensions. We combine analytical techniques with numerical computations. Compared to earlier results, our analytical techniques cover a larger part of the parameter domain consisting of wave numbers in two space directions and the Reynolds number. Numerical computations are needed only in a compact subset of the parameter domain. Mathematics Subject Classification (2000). 76E05, 35Q35, 47N20, 35P05. Keywords. Hydrodynamical stability, Couette flow, resolvent estimate. 1. Introduction The field of hydrodynamic stability has been thoroughly studied by numerous scientists since the late 19th century. It concerns the stability of laminar flows when subjected to perturbations. We refer to [7] and [19] for an introduction to and overview of the field. In this paper we consider stability of plane Couette flow, which is the flow of a viscous incompressible fluid, modeled by the Navier Stokes equations, between two infinite planes moving in opposite directions at constant velocity. The stationary solution is a linear velocity profile in the stream wise direction. Romanov showed in 1973, [18], that plane Couette flow is linearly stable, i.e. all eigenvalues of the linearized problem are in the stable half plane. He also showed nonlinear stability for sufficiently small perturbations. However, finite perturbations may lead to turbulence. Hence, there is a threshold for the size of the perturbations, below which the flow will eventually relaminarize. The motivation of this paper is to provide further insight in how this threshold depends on the Reynolds number, R. In 1993, a dependence of the threshold as R ρ, (ρ > 0), was suggested, [20]. Computations made in e.g. [15] suggest ρ 1.25, while the asymptotic analysis in [5] indicates ρ 1. In [10], the upper bound ρ 5.25 is proved, under the condition This research was supported by the Swedish Research Council grant

43 2 P.-O. Åsén and G. Kreiss JMFM that the L 2 -norm of the resolvent of the linearized problem is bounded in the entire unstable half plane. Using the same technique but a norm with different weights for different velocity components, the upper bound ρ 4 is proved in [14]. The norm used was introduced in [13], where it was found that weighting the normal velocity with R results in an optimal bound of the resolvent. The same norm was later considered in [4], yielding the same resolvent bound. Analytical bounds of the resolvent are only known in parts of the unstable half plane, [12]. See also [4] and [3], where similar results for perturbations with and without span wise variations are presented. In [10], [13], [4] and [3], resolvent bounds based on numerical computations in the parameter domain, consisting of wave numbers and the Reynolds number, are presented. Although there are analytical results for certain wave numbers, [3], [4], [13], the computations do not cover the remaining infinite parameter domain. The papers do not discuss the behavior of the resolvent for parameters outside the domain where computations are done. Further, there is no rigorous analysis of the accuracy of the computations or any discussion of the conditioning of the equations solved. In this paper we present new analytical estimates which give a sharp bound of the L 2 -norm of the resolvent in new parts of the parameter domain. We also show that these estimates reduce the remaining parameter domain to a compact set, which makes it possible to obtain a rigorous bound of the resolvent in the entire parameter domain. Further, the new estimates provide detailed information of the resolvent which could be useful in deriving a sharper upper bound of the threshold for nonlinear stability. In our approach, we consider the Orr Sommerfeld equation, derived by Fourier transformation in the two infinite directions. The Orr Sommerfeld equation is a fourth order ordinary differential equation with the Reynolds number and wave numbers in two directions as parameters. By using asymptotic techniques, we are able to analytically bound the resolvent in a large part of the parameter domain. Together with the above mentioned results in [13], we have analytical bounds in all but a compact subset of the infinite parameter domain. For parameter values in this set, the solution is well behaved and we use standard numerical techniques to obtain a bound. This compact subset can also be reduced to a finite set of parameter values by using analytical estimates of how the solution of a one dimensional, second order boundary value problem depends on the coefficients, see [2]. Combining this analysis with validated numerical methods would result in a rigorous bound of the resolvent in the entire parameter domain. This would of course not be possible in an unbounded parameter domain.

44 On a Rigorous Resolvent Estimate for Plane Couette Flow 3 2. The problem The coordinate system is chosen such that x 1 is the stream wise direction, x 2 the direction normal to the planes and x 3 the span wise direction. With appropriate scaling, a stationary solution of the Navier Stokes equations is given by in the domain x 2 U = 0 0 Ω = {x R 3 : 1 x 2 1}. (1) Linearizing the Navier Stokes equations at the stationary solution (1) and applying the Laplace transform gives u su + x 2 + u p = 1 u + f, x 1 R 0 (2) u = 0, u = 0, x Ω, where R is the Reynolds number. Equation (2) is known as the resolvent equation and the solution operator R(s) : f u (3) is known as the resolvent. We are interested in bounding the L 2 -norm of the resolvent. It is sufficient to consider forcing f C 0 (Ω) such that div f = 0. Estimates for less regular forcing can be obtained by closure arguments for densely defined continuous operators. A non-solenoidal forcing can be divided into a solenoidal part and a non-solenoidal part, where the non-solenoidal part only affects the pressure, [21] p. 48. Romanov proved in [18] that all eigenvalues, λ, for plane Couette flow satisfy Re λ < δ/r, for some δ > 0. Hence, the resolvent is well-defined in the entire unstable half-plane Res 0 at all Reynolds numbers. In [13], the L 2 -norm of the resolvent was proved to satisfy R(s) CR in the domain Σ = {s C : Re s + 1/(2R) Im s 3}. Combining this bound with the maximum principle, [8], implies that the L 2 -norm of the resolvent is maximized for some s with Re s = 0. Computations in e.g. [10] and [13] suggest that the L 2 - norm of the resolvent is proportional to R 2 and maximized when s = 0. In this paper, we show how this bound of the resolvent can be proved for s = 0, i.e. we show how to prove that the solution of (2) with s = 0 satisfies u CR 2 f,

45 4 P.-O. Åsén and G. Kreiss JMFM which is known to be a sharp result, [10]. We will use and (, ) to denote the L 2 -norm and the L 2 -inner product, respectively, and to denote the L -norm. Although the domain will vary, the notation will remain the same. The domain considered will be clear from the context and in most cases also specified. We will also need the max-norm of a matrix, which will be denoted by. 3. Transformation of the problem There is a well-known reformulation of (2) to one fourth order equation for the normal velocity, u 2, and one second order equation for the normal vorticity, η 2. For a detailed derivation, see e.g. [19]. The coefficients in the reformulated problem only depend on x 2, thus making it suitable to apply the Fourier transform in the x 1 - and x 3 -directions. The Fourier transformed equations for the normal velocity and the normal vorticity are known as the Orr Sommerfeld equation and the Squire equation, respectively. Let η = u and g = f. With {ξ 1,ξ 3 } as the dual variables of {x 1,x 3 }, the transformed problem is ( LOS 0 irξ 3 L SQ )( û2 ˆη 2 ) Rs )( ) ( ) û2 ˆ ˆf2 = R, ˆη 2 ĝ 2 û 2 = û 2 = ˆη 2 = 0, x 2 = ±1, ( ˆ where ˆ = 2 x 2 2 k 2, L OS = ˆ 2 iξ 1 Rx 2 ˆ, LSQ = ˆ iξ 1 Rx 2, and k 2 = ξ 2 1 +ξ 2 3. Considering ξ 1 and ξ 3 as parameters and introducing, in analogy with (3), the mapping ˆR(s,ξ 1,ξ 3 ) : ˆf û, it can be shown, see [13], that the resolvent is obtained from R(s) = max ξ 1,ξ 3 ˆR(s,ξ 1,ξ 3 ). Observe that the L 2 -norm is over Ω on the left-hand side and over x 2 [ 1,1] on the right-hand side. Using div u = 0, the Fourier transforms of the velocities in the x 1 - and x 3 - direction can be related to û 2 and ˆη 2 by û 1 = iξ 1û 2 iξ 3ˆη 2 k 2, û 3 = iξ 3û 2 + iξ 1ˆη 2 k 2, where prime denotes derivative with respect to x 2. The relations (5) allows us to evaluate the norm of the Fourier transformed velocity field as 1 û 2 = ( û 2 (x 2 ) 2 + 1k 2 û 2(x 2 ) 2 + 1k ) 2 ˆη 2(x 2 ) 2 dx 2. (6) 1 (4) (5)

46 On a Rigorous Resolvent Estimate for Plane Couette Flow 5 Since f is divergence free, (6) also holds for ˆf 2 with ˆf 2 and ĝ 2 on the right-hand side. Hence, in order to bound the resolvent, we only need to bound û 2, û 2 and ˆη 2 in terms of R, ˆf 2, ˆf 2 and ĝ The resolvent estimate In this section, we derive our main result which is summarized in the following theorem. Theorem 4.1. For s = 0, the resolvent is bounded by R(0) = max ξ 1,ξ 3 ˆR(0,ξ 1,ξ 3 ) CR 2, (7) where C is a constant independent of ξ 1,ξ 3 and R. The proof is divided into two parts. Since the parameter domain for ξ 1, ξ 3 and R is infinite, mere numerical computations are insufficient. We derive analytical bounds in all but a compact subset of the parameter domain. In the remaining part, we use numerical computations Previous results Here we state some results that are identical or similar to results previously derived. The proofs can be found in the corresponding references as well as in [1], which is a more extensive version of this paper. As previously mentioned, we need to find bounds for û 2, û 2 and ˆη 2. Note in (4) that the first equation does not depend on ˆη 2. For ˆη 2 we have the following bound, found in [13]. Lemma 4.2. There is a constant C, independent of ξ 1, ξ 3, R and Re s 0, such that ˆη 2 2 CR2 1 + k 2 ( û ĝ 2 2 ). It remains to find bounds for û 2 and û 2. This will generally not be as easy as for the normal vorticity. However, for some parameter values it can be done in a straightforward way, yielding the following lemma which is similar to Lemma 4.1 in [13]. Lemma 4.3. There is a constant C, independent of ξ 1, ξ 3, R and Re s 0, such that if either ( ξ 1 R) 2/3 ξ1 2 + ξ3 2 or ξ 1 R 1/16 or both inequalities hold, then ( (1 + k 2 ) û (1 + k 4 ) û 2 2 CR 2 k 2 ˆf ) k 2 ˆf 2 2. (8)

47 6 P.-O. Åsén and G. Kreiss JMFM Note that (8), Lemma 4.2 and (6) give the desired bound of the resolvent, (7). Thus, Theorem 4.1 is already proved for the parameter values covered by Lemma 4.3. Remark. In the 2-dimensional case, ξ 3 = 0, we have ˆη 2 2 CR 2 (1+k 2 ) 1 ĝ 2 2 in Lemma 4.2. Then it follows from Lemma 4.3 and (6) that R(0) CR instead of (7). This is the resolvent bound presented in [3]. In order to obtain bounds of û 2 and û 2 in part of the remaining parameter domain, we introduce a new function, v. Consider (4) for û 2. Modifying the boundary conditions makes it possible to use integration by parts. Taking the inner product of (9) with ˆ v, using integration by parts and a Poincaré inequality yields the following lemma. Lemma 4.4. The solution of the auxiliary problem ˆ 2 v (iξ 1 Rx 2 + Rs)ˆ v = R ˆ ˆf 2, satisfies v(±1) = ˆ v(±1) = 0 v 2 + (1 + k 2 ) v 2 + (1 + k 2 ) 2 v 2 Ck2 R k 2 The constant C is independent of ξ 1, ξ 3, R and Re s 0. ( ˆf k 2 ˆf 2 2 ). (9) The properties of the auxiliary problem are discussed in [6] p. 17, referencing Orr, [17]. In [3], the same auxiliary problem is introduced, and bounds similar to the bounds in Lemma 4.4 are derived Reformulation of the problem The reason for introducing the function, v, is to move the inhomogeneity of the problem from a forcing in the equation to inhomogeneous boundary conditions. This is done by considering û 2 = v + σ 1 u 1 + σ 2 u 2, where σ j are scalars, v satisfies (9), and u j satisfy ˆ 2 u j (iξ 1 Rx 2 + Rs)ˆ u j = 0, u j (±1) = 0, ˆ u 1 ( 1) = 1, ˆ u1 (1) = 0, ˆ u 2 ( 1) = 0, ˆ u2 (1) = 1. The boundary conditions û 2(±1) = 0 impose the conditions ( ) ( ) σ1 v J = ( 1) v, (11) (1) σ 2 (10)

48 On a Rigorous Resolvent Estimate for Plane Couette Flow 7 where J = ( ) u 1 ( 1) u 2( 1) u 1(1) u. (12) 2(1) For the right-hand side of (11), we use the Sobolev type inequality, see e.g. [11] p. 380, v 2 µ v 2 + C 1 µ 3 v 2, valid for 0 < µ 1. By choosing µ = (1 + k 2 ) 1/2 and using Lemma 4.4, we obtain v 2 Ck2 R 2 (1 + k 2 ) 3/2 ( ˆf k 2 ˆf 2 2 ). (13) The introduced functions, u j, are similar to those used in [3]. However, in [3] the inhomogeneous boundary conditions are on the first derivative. This might be practical when using only numerical computations to bound u j. The boundary conditions in (10) have the advantage that the fourth order equation can be solved as two coupled second order equations. We will use this when we derive analytical bounds of u j. In order to obtain a bound of the resolvent, we need to derive bounds for u j, u j and σ j in the parameter domain not covered by Lemma 4.3. Note that this parameter domain is infinite in all three parameters ξ 1,ξ 3 and R. By introducing the new variables y = Lx 2, L = ( ξ 1 R) 1/3, δ = ( k L )2, (14) we obtain a problem with only one parameter with infinite domain. To avoid confusion, we will use prime,, to denote derivative with respect to x 2, while derivative with respect to y is denoted by subscript, y, or y. With the new variables, the equations for u j become ( y 2 δ 2 ) 2 u j i(y + γ)( y 2 δ 2 )u j = 0, u j (±L) = 0, yu 2 1 ( L) = 1/L 2, yu 2 1 (L) = 0, yu 2 2 ( L) = 0, yu 2 2 (L) = 1/L 2, (15) where iγ = Rs/L 2. We consider L 1/ 3 16 and δ [0,1], since this includes the parameter domain not covered by Lemma 4.3. With s = 0 γ = 0, equation (15) for u 1 can be written as ( 2 y δ 2 )u 1 = w, u 1 (±L) = 0, (16) ( 2 y δ 2 )w iyw = 0, w( L) = 1/L 2, w(l) = 0. (17) Since symmetry gives u 2 (y) = u 1( y) when γ = 0, all bounds for u 2 will follow from the corresponding bounds for u 1. Note that the equation in (17) can be transformed to the Airy equation by a simple change of variables, see e.g. [16]. We will derive properties of the solution directly from the equation.

49 8 P.-O. Åsén and G. Kreiss JMFM 4.3. New analytical bounds In this section, we derive new analytical bounds which give a sharp bound of the resolvent in all but a compact subset of the parameter domain. The results are summarized in the following lemma. Lemma 4.5. There is a constant C, independent of ξ 1, ξ 3 and R, such that if both ( ξ 1 R) 2/3 ξ1 2 + ξ3 2 and ξ 1 R 40 3 hold, then for s = 0 û û 2 2 Ck2 R 2 ( (1 + k 2 ) ˆf 3/ k 2 ˆf 2 ) 2. As was the case for Lemma 4.3, Lemma 4.5 gives the sharp bound of the resolvent, (7), by using Lemma 4.2 and (6). Note that with the variables (14), Lemma 4.3 and Lemma 4.5 cover all but the compact set L [1/ 3 16,40], δ [0,1]. We will prove Lemma 4.5 by proving that for the parameter values covered and with J given by (12), the solutions of (10) with s = 0 satisfy J 1 2 ( u j 2 + u j 2 ) C, j = 1,2, (18) where C is independent of ξ 1, ξ 3 and R and J 1 is the max-norm of the matrix J 1. Indeed, if (18) holds, then Lemma 4.5 follows by using Lemma 4.4, (11) and (13). In order to prove (18), we consider (17) and (16). We will derive analytical bounds which, after transforming back to the original variables, give (18). The analytical bounds will be obtained by first finding an approximate solution, w app, of (17), then solving (16) with w app on the right-hand side, and finally bounding the errors made by the approximations. This will only be possible when L 40. An approximate solution of (17) can be obtained by writing the equation as ( wy w ) y = ( 0 δ 2 + iy 1 0 ) ( wy w ). (19) When y is large, the eigenvalues of this matrix are well separated and the system can be diagonalized. Let us for a moment consider y [ L, l 0 ] where L > l 0 > 0. Introduce new variables ( wy w ) = ( λ λ 1 1 )( z1 z 2 ), (20) where λ = δ 2 + iy. From (19) and (20), the system for z = (z 1,z 2 ) T becomes ( ) ( ) λ 0 b b z y = z + ε z, (21) 0 λ b b where ( ) 2 ε = λ 2 0, b = i λ0, λ 0 = δ 2λ 2 il 0. (22) In order to diagonalize (21), we use a special case of Lemma 2.6 in [9].

50 On a Rigorous Resolvent Estimate for Plane Couette Flow 9 Lemma 4.6. Consider the system (21) for y [ L, l 0 ]. If l 0 1, there exists a transformation ( ) 1 ε q z = εq 1 + ε 2 z (23) q q such that z y = ( ) λ + p 0 z, (24) 0 λ + p where p = εb + ε 2 bq and p = εb ε 2 bq. Both q and q depends smoothly on y and are bounded by q 1 + l2 0, (25) (2l 0 ) 5/2 q l 2 0 (2l 0 ) 5/2 2. (26) Proof. A reconstruction of the proof is given in Appendix A. We use Lemma 4.6 to derive an approximate solution of (17). Since the lemma is valid for y [ L, l 0 ], we have the exact solution of (17) in this interval by solving (24). For y [ l 0 + 1,L] we set w app 0. Note that this is a solution of (17) and fulfills the right boundary condition. In the remaining interval, y [ l 0, l 0 + 1], we use a function, w int, connecting w app ( l 0 ) to w app ( l 0 + 1) = 0. The choice of w int must be such that w app is at least twice differentiable. A solution of (24) is z 1 = 0, z 2 = Ce 2 3 i((δ2 +iy) 3/2 (δ 2 il) 3/2) e y L p(ω)dω. Hence, using (20) and (23) and imposing the left boundary condition w( L) = 1/L 2 gives the approximate solution 1+ˆq(y) w app L e 2 (y) = 2 3 i((δ2 +iy) 3/2 (δ 2 il) 3/2) y e L p(ω)dω, y [ L, l 0 ], w int (y), y [ l 0, l 0 + 1], 0, y [ l 0 + 1,L], (27) where ˆq(y) = ε q(y) + ε2 q(y) q(y) ε q( L) ε 2 q( L) q( L) 1 + ε q( L) + ε 2. (28) q( L) q( L) Although this approximate solution is valid for l 0 1, we must choose l 0 large enough such that the contributions from ˆq and p are small enough. Also, to be able to bound the error introduced by w int, L must be sufficiently large compared to l 0. Let us choose l 0 = 30 and consider L 40. From (22), we have ε 1/l 0 and b 1/4. Using this with (28) and Lemma 4.6 with l 0 = 30, it follows that p 1 100, (29)

51 10 P.-O. Åsén and G. Kreiss JMFM ˆq (30) The rapid decay of w app away from y = L makes the choice of w int less important. Therefore, the following derivation will be brief. The simplest choice is to take a fifth degree polynomial, thus ensuring continuity up to second derivative. We use three polynomials p n (y), n {0,1,2}, satisfying n y p n ( l 0 ) = 1 and otherwise homogeneous conditions up to second derivative at both ends. It is easily verified that the polynomials are given by p 1 (y) = 1 10(y + l 0 ) (y + l 0 ) 4 6(y + l 0 ) 5, p 2 (y) = (y + l 0 ) 6(y + l 0 ) 3 + 8(y + l 0 ) 4 3(y + l 0 ) 5, (31) p 3 (y) = 1 ( (y + l0 ) 2 3(y + l 0 ) 3 + 3(y + l 0 ) 4 (y + l 0 ) 5). 2 Using these polynomials, w int can be written as 2 w int (y) = y n w app ( l 0 )p n (y), y [ l 0, l 0 + 1]. (32) From (27), we have n=0 y w app ( l 0 ) = ( ) ˆqy ( l 0 ) 1 + ˆq( l 0 ) + p( l 0) λ 0 w app ( l 0 ). (33) Since l 0 = 30 and δ [0,1], we have δ 2 l 0 /30 which gives λ 0 l 0 (901/900) 1/4. By differentiating (28) with respect to y and using Q = Q = 0 with Q and Q given by (43) and (44), we obtain an expression for ˆq y ( l 0 ) in terms of q, q, ε, λ 0, b and p. Using Lemma 4.6 and (22), the parentheses on the right-hand side of (33) is easily bounded by 2 l 0 (it is actually bounded by l 0, but the constant is of minor importance due to the exponential decay of w app (y)). Proceeding in the same way, differentiating (27), replacing derivatives of ˆq using (28), (43) and (44), differentiating p and λ and finally using the known bounds of all quantities, we obtain a bound of 2 yw app ( l 0 ). Summarizing, we have the following bounds y w app ( l 0 ) 2 l 0 w app ( l 0 ), 2 yw app ( l 0 ) 2l 0 w app ( l 0 ). Now, using (30), (32) and (34) and integrating p n (y), p n (y) 2, 2 yp n (y) 2, with p n (y), n {0,1,2}, given by (31), gives l0+1 (34) l 0 w int (y) dy 3 L 2 e 2 3 i((δ2 il 0) 3/2 (δ 2 il) 3/2) e l 0 L p(ω)dω, (35) l0+1 l 0 w int (y) 2 dy 7 L 4 e 2 3 i((δ2 il 0 ) 3/2 (δ 2 il) 3/2) e l 0 L p(ω)dω 2, (36) l0+1 l 0 2 yw int (y) 2 dy 3000 L 4 e 2 3 i((δ 2 il 0) 3/2 (δ 2 il) 3/2) e l 0 L p(ω)dω 2. (37)

52 On a Rigorous Resolvent Estimate for Plane Couette Flow 11 In (36) we used the inequality p 0 (y) + p 1 (y) + p 2 (y) 2 4 p 0 (y) p 1 (y) p 2 (y) 2. This was also used for (37) with 2 yp n instead of p n. This completes the derivation of an approximate solution of (17). Since (16) is a constant coefficient problem, there is a well known solution formula. Solving (16) using the approximate solution w app gives the following lemma. Lemma 4.7. If L 40, l 0 = 30 and δ [0,1], the solution of ( 2 y δ 2 )u app 1 = w app, y [ L,L], (38) with w app given by (27) and the boundary conditions u app 1 (±L) = 0 satisfies y u app u app 1 2 C L 5, y u app 1 ( L) 0.6 L 5/2, y u app 1 (L) 1.4 L 4, where C is independent of L and δ, and is the L 2 - norm over y [ L,L]. Proof. The proof is given in Appendix B. In order to bound the errors made by using the approximate solution, w app, we introduce a new function. Let u 1 = u app 1 + u corr 1. From (16), (17) and (38), u corr 1 must satisfy ( y 2 δ 2 ) 2 u corr 1 iy( y 2 δ 2 )u corr 1 = (( y 2 δ 2 )w app iyw app ) F(y), (39) u corr 1 (±L) = yu 2 corr 1 (±L) = 0. Integration by parts yields the following lemma. Lemma 4.8. The solution u corr 1 of (39) satisfies u corr L 8 F 2, y u corr L 6 F 2, 2 yu corr L 4 F 2, where is the L 2 -norm over y [ L,L]. Proof. The proof is given in Appendix C. In order to estimate the contribution from u corr 1, we need to bound F. Remember that the construction of w app was done such that w app solves (17) when y / [ l 0, l 0 + 1]. Hence, the only contribution to F is from w int and we obtain the following lemma.

53 12 P.-O. Åsén and G. Kreiss JMFM Lemma 4.9. Consider w app given by (27) and F(y) given by (39). If L 40, l 0 = 30 and δ [0,1], then F(y) satisfies F L 4 where is the L 2 -norm over y [ L,L]. e 0.94(l 3/2 0 L 3/2), Proof. The proof is given in Appendix D. By Lemma 4.9, all bounds in Lemma 4.8 are exponentially small. Evaluating numerically with l 0 = 30 and L 40 gives L 8 2 yu corr L 8 y u corr L 5 u corr , y u corr 1 (±L) 10 5 L 4, where the last result follows from the Sobolev type inequality, see e.g. [11] pp , ( 1 y u corr 1 2 µ yu 2 corr µ + 1 ) y u corr 1 2, µ > 0. 2L Using these results, we are finally able to prove Lemma 4.5. Proof of Lemma 4.5. With the variables (14), the parameter domain covered by Lemma 4.5 is covered by L 40 and δ [0,1]. Since u 1 = u app 1 + u corr 1, using Lemma 4.7, (40) and transforming from y to x 2 gives (40) ( ξ 1 R) 4/3 u ( ξ 1 R) 2 u 1 2 C, u 0.5 1( 1), (41) ( ξ 1 R) 1/2 u 1(1) 1.5 ( ξ 1 R), where is the L 2 -norm over x 2 [ 1,1]. Here we have used, from (14), that ( ξ 1 R) 1/3 u 1 2 x 2 = u 1 2 y and ( ξ 1 R) 1/3 u 1 2 x 2 = y u 1 2 y, where subscripts denotes the domain over which the L 2 -norm is evaluated. The last two bounds of (41) give, using (12) and the symmetry u 2 (x 2 ) = u 1( x 2 ), the bound J 1 2 u 1( 1) + u 2 1(1) u 1 ( 1) 2 u 1 (1) 2 C( ξ 1 R). (42) The bound (18) follows from (41) and (42). As previously mentioned, Lemma 4.5 follows from (18) by using Lemma 4.2 and (6). This implies that Theorem 4.1 is proved in the parameter domain covered by Lemma 4.3 and Lemma 4.5.

54 4.4. Numerical computations On a Rigorous Resolvent Estimate for Plane Couette Flow 13 It remains to prove Theorem 4.1 in the parameter domain ξ 1 R [1/16,40 3 ] and ( ξ 1 R) 2/3 ξ 2 1+ξ 2 3. Hence, we want to show that (18) holds also in this parameter domain. Using the variables (14), it is enough to consider L [1/ 3 16,40] and δ [0,1], which is a bounded parameter domain. Here we rely on numerical computations. We solve (16) and (17) using second order finite differences, and then compute the quantities on the left-hand side of (18). Note that we only wish to bound the left-hand side of (18) by any constant. Since (16) and (17) are linear ordinary differential equations with δ only appearing in front of lower order terms, we would not expect the solution or its derivatives to become infinite on a domain with length bounded from below and from above. A potential problem of fulfilling (18) is that J might become singular. However, since the parameter domain is small, the solution can easily be sufficiently resolved. We computed solutions to (16) and (17) for the parameter values δ [0,1] with step size dδ = 0.005, L [0.3,10] with step size dl = 0.1, L [11,15] with step size dl = 1, and L [20,40] with step size dl = 5. After transforming from y to x 2, we computed the quantities on the left-hand side of (18) and the results are shown in Fig. 1. The figures do not contain all the parameter values considered. We used the spatial discretization h = of y [ L,L] in all computations. Clearly, (18) is fulfilled also for L [1/ 3 16,40] and δ [0,1] and Theorem 4.1 is proved. We note that in order for this part of the proof to be rigorous, validated methods using interval arithmetic should be used in the numerical computations L δ δ L Fig. 1. Numerical computations in the parameter domain L [1/ 3 16,40], δ [0, 1]. The left figure shows u 1 + u 1 and the right figure shows J 1. Here denotes the L 2 -norm over x 2 [ 1, 1].

55 14 P.-O. Åsén and G. Kreiss JMFM Remark. We found that already at L 3, J 1 behaves like L 3/2, as predicted by (42). Also, u 1 shows good agreement with the, from (41), asymptotic L 3 for L 10. However, the predicted behavior of L 2 for u 1 is not sharp. Instead, we found that u 1 is proportional to L 9/4. In order to ensure that the solutions were sufficiently resolved, we used several different discretizations of y [ L,L]. Let u h denote the solution on a mesh with step size h and r h denote the restriction to a mesh with step size h. We estimated the error in the solutions by e h = u h r h u h/2 and similarly for the error in the derivative, y e h. For J 1, the error, e J h, was estimated in the same way, though without use of any restriction. Figure 2 shows the estimated errors as function of the step size, h, for L = 40 and δ = 0. Similar plots were obtained for many different values of L and δ. Figure 2 indicates that the solution is sufficiently resolved when h = and, as expected, that the method is second order accurate Fig. 2. Estimated error as function of step size, h. The solid line is e h h, the dashed line is ye h h and the dash-dotted line is e J h. Here, h denotes the discrete L 2 -norm over y [ L, L]. h A simple analysis shows that accurate solutions to (16) and (17) are difficult to obtain numerically when L is large. Consider the case δ = 0. Then, (16) is just the Poisson equation, which is easily solved if w is sufficiently resolved. For large L, we have by (27) that L 2 y w( L) L 1/2. Since L 2 w( L) = 1, we need a spatial step size h L 1/2 to resolve w near the left boundary. Using a uniform mesh to discretize y [ L,L], the computational complexity, N, will be at least

56 On a Rigorous Resolvent Estimate for Plane Couette Flow 15 N L 3/2. Although this can be reduced by using adaptive methods, the problem is increasingly ill-conditioned with growing L. However, in the parameter domain L [1/ 3 16,40], δ [0,1], the solution is easily resolved using standard numerical techniques. 5. Discussion We have shown that the L 2 -norm of the resolvent for plane Couette flow is proportional to R 2 at s = 0. This is known to be a sharp bound in three space dimensions, see e.g. [10]. The two dimensional result R R, which is also sharp, follows from considering the case of no variation in the span wise direction, see the remark after Lemma 4.3. The proof is rigorous outside a compact subset of the parameter domain. This compact set can be reduced to a finite set of parameter values, [2]. Using validated numerical methods in this finite set would result in a rigorous proof of the resolvent bound in the entire parameter domain. This is something we hope to address in the near future. In order to obtain a bound of the resolvent in the entire unstable half plane, it is enough to consider s with Re s = 0. This is due to the maximum principle and analytical bounds in parts of the unstable half plane, see e.g. [8]. All results prior to Lemma 4.5 are valid for all s with Re s 0. For imaginary s, we can derive the bound (18) for u app j by using the transformation ỹ = y + γ, γ = R Im s/l 2. The only result that does not hold for Im s 0 is (40), since the exponential term in Lemma 4.9 will contain (L γ) 3/2, while we still will have high powers of L in the bounds in Lemma 4.8. Thus, we need to use a different strategy when bounding the errors introduced by using the approximate solution. Appendix A. Reconstruction of the Proof of Lemma 4.6 Introducing (23) into (21) gives ( λ εb + ε z y = 2 bq + ε 2 qq ε(ε 2 q 2 Q Q) εq λ εb ε 2 bq ε 2 qq where ) z, Q = q y + 2λq + ε 2 bq 2 b, (43) Q = q y 2(λ + ε 2 bq) q b. (44) Solving Q = 0 and Q = 0 will give the desired result. Consider the linear equation q l y = a(y)q l + f(y), y [ L, l 0 ], q l ( L) = 0,

57 16 P.-O. Åsén and G. Kreiss JMFM where a and f are smooth functions and Re(a) τ < 0. This problem has a unique smooth solution satisfying q l τ 1 f. Solving Q = 0 with q( L)=0 must give q 1 in some interval y [ L, L + κ]. Using this bound of q and (22) yields Re( 2λ) Re( 2λ 0 ) 2l 0, b ε 2 bq l2 0 (2l 0 ) 2. Treating the nonlinear term in (43) as part of the forcing gives a smooth solution satisfying (25). Clearly, choosing l 0 1 gives q 1, and the arguments above can be extended to the entire interval y [ L, l 0 ]. Using q 1 and (22), we have Re(2(λ + ε 2 bq)) 2 2l 5/2 0 1, which is positive for l 0 1. Solving the linear equation (44) with Q = 0 and the condition q( l 0 ) = 0 gives a smooth solution satisfying (26). 2l 2 0 Appendix B. Proof of Lemma 4.7 First we need the following lemma concerning exponential functions. Lemma B.1. If δ [0,1], l 0 > 3 and L l 0, then for y [ L, l 0 ], e e 2 3 i((δ2 +iy) 3/2 (δ 2 il) 3/2) 2 e 3 (( y)3/2 (L) 3/2), (45) e 2 3 i((δ2 +iy) 3/2 (δ 2 il) 3/2 ) 2 3 (1 i)(( y)3/2 (L) 3/2 ) (y + L). (46) l 0 Proof. Hence, we are looking for e 2 3 i((δ2 +iy) 3/2 (δ 2 il) 3/2) = e 2 3 Im((δ2 +iy) 3/2 (δ 2 il) 3/2). [ min Im (δ 2 + iy) 3/2 (δ 2 il) 3/2] min g(δ,y), y [ L, l 0]. δ [0,1] δ [0,1] With r(δ,y) = δ 4 + y 2, θ(δ,y) = arctan(y/δ 2 ) and ϕ(, ) = 3/2θ(, ), we have g(δ,y) = Im [r(δ,y) 3/2 e 3 2 iθ(δ,y) r(δ, L) 3/2 e 3 iθ(δ, L)] 2 = r(δ,y) 3/2 sin (ϕ(δ,y)) r(δ, L) 3/2 sin (ϕ(δ, L)).

58 Differentiating with respect to δ gives On a Rigorous Resolvent Estimate for Plane Couette Flow 17 g δ h(δ,y) = = 3δ(h(δ,y) h(δ, L)), cos (ϕ(δ,y)) y + sin (ϕ(δ,y)) δ2 (δ 4 + y 2 ) 1/4. When l 0 > tan(π/3) = 3 and δ [0,1], we have θ(δ,y) [ π/2, π/3) y [ L, l 0 ], i.e. ϕ(δ,y) [ 3π/4, π/2). This gives h(δ,y) 0. Also, since h y = δ3 δ 4 cos(ϕ(δ,y)) + 2yδ 2 sin(ϕ(δ,y)) y 2 cos(ϕ(δ,y)), 2 (δ 4 + y 2 ) 5/4 which is positive for y < 3, we have that h(δ, L) h(δ,y), giving g/ δ 0. Hence, g(δ,y) is minimized with respect to δ at δ = 0 y [ L, 3), and (45) follows. Taylor expansion of the exponent around δ = 0 gives e 2 3 i((δ2 +iy) 3/2 (δ 2 il) 3/2) 2 = e 3 (1 i)(( y)3/2 L 3/2) R(δ,y). (47) From (45), it follows that R(δ,y) 1. Also, since R(δ, L) 1 = 0, we have R(δ,y) 1 R/ y (y + L). Differentiating (47) with respect to y yields ( R y = δ 2 + iy 1 ( 1 + i) ) y R(δ,y). 2 Taylor expansion gives δ2 + iy δ=0 = 1 2 (1 i) y + i 2 2 (1 i) 1 y δ 2 where ξ(y) [0,δ]. Using this, we obtain R y 1 2 y δ 4 y y (ξ(y) 2 + iy) 7/2 < 3 5 y, i.e. R/ y < 3/(5 l 0 ) and (46) follows. We also need some general results concerning integrals. 5δ 4 128(ξ(y) 2 + iy) 7/2, Lemma B.2. If l 0 = 30, L 40, y [ L, l 0 ], β 2/23 and arg β = π/4 or arg β = 0, then I(y,β) l0 y f( y,β) = 2eβ(( y)3/2 L 3/2 ) 3β y where ε 2 (z) is bounded by e β(( σ)3/2 L 3/2) dσ = f( y,β) f(l 0,β), ( ) β( y) + ε 2( β( y) 3/2 ), 3/2 ε 2 (z) 1 z 2. (48)

59 18 P.-O. Åsén and G. Kreiss JMFM Proof. The integral can be written as l0 I(y,β) e β(( σ)3/2 L 3/2) dσ = e 2σΓ(2 βl3/2 3, ] σ= l0 β( σ)3/2 ), y 3( β( σ) 3/2 ) 2/3 σ=y where Γ(a,z) is the incomplete gamma function defined by Γ(a,z) z t a 1 e t dt. The result follows from the expansion, see e.g. [16] pp , ( n 1 Γ(a,z) = e z z a 1 s=0 (a 1)(a 2) (a s) z s + ε n (z). In order to obtain (48), we use the results in [16]. With z = β( σ) 3/2 and σ < 0, arg β = π/4 and arg β = 0 corresponds to arg z = 3π/4 and arg z = π respectively. When a = 2/3, n = 2, z > 0 and arg z = 3π/4, we have the bound ε 2 (z) 4 9sin(3π/4) z 2, and (48) follows. When a = 2/3, n = 2 and arg z = π, ε 2 is bounded by ε 2 (z) 1 z 2 4 9[ 2/ ln( 2)/(3 z )] (49) when z > 28ln( 2)/ The last term on the right-hand side of (49) is bounded by one if z 10, and (48) follows. With z = β( y) 3/2, the restriction z 10 is fulfilled for y [ L, l 0 ], l 0 = 30 when β 2/23. Note that if y L, the modulus of the integral decays exponentially with L. Also, since the integrand is positive for real β, it must hold that f( y,β) f(l 0,β). The restriction on β is more than enough to ensure f(l 0,β) > 0 for β R. Similarly, using the same expansion of the incomplete gamma function, we obtain the following lemma. Since the proof is similar to the proof of the previous lemma, we only sketch the proof. Lemma B.3. Under the conditions of Lemma B.2 and for β R, we have l0 (σ + L)e β(( σ)3/2 L 3/2) dσ 1 ( 4 L 9β L 3/2 β + 2 ) 3 3L 3 β 4. L

60 Proof. The integral can be written as l0 L On a Rigorous Resolvent Estimate for Plane Couette Flow 19 (σ + L)e β(( σ)3/2 L 3/2) dσ = e βl3/2 2 σe β( σ)3/2 + 2 σγ( 1 3 β( σ)3/2 ) 2σLΓ(2 3, ] σ= l0 β( σ)3/2 ). 9β( β( σ) 3/2 ) 1/3 3( β( σ) 3/2 ) 2/3 σ= L Now, the same expansion of the incomplete gamma function is used as in the previous lemma. The first term in the brackets above indicates that the integral should be proportional to L/β. However, this term will be canceled when expanding Γ( 2 3, β( σ)3/2 ). Noting that the terms containing l 0 can be omitted due to their sign, the result follows. Finally, we need the following result for products of exponential functions. Lemma B.4. Given γ > 0, β > 0, l 0 > 0 and C L l 0, if β satisfies then for all L C L β β γ(c L l 0 ) C 3/2 L, l3/2 0 e γ(y+l)+β(( y)3/2 L 3/2) e β (( y) 3/2 L 3/2), y [ L, l 0 ]. (50) 3β Note that if C L is sufficiently large, β can be chosen to be positive. Proof. Obviously, it is enough to consider y ( L, l 0 ]. Then (50) is equivalent to β γ(y + L) ( y) 3/2 L 3/2 + β, y ( L, l 0]. Note that the first term on the right-hand side is always negative. Differentiating with respect to y gives ( ) γ(y + L) = γ 2L3/2 + ( y) 3/2 3 yl γ f y ( y) 3/2 L 3/2 2(( y) 3/2 L 3/2 ) 2 g. Since f( L) = 0 and f/ y = 3(y + L)/(2 y) > 0 for y ( L, l 0 ], we have f,g > 0, i.e. the hardest constraint on β is at y = l 0. Clearly, if β is chosen for L = L, the inequality also holds L L and the result follows. Now we have the lemmas needed for bounding the various norms. When we use β and γ, they denote constants in an exponential term on the same form as the left-hand side of (50). Note that β must not always be real, e.g. when using Lemma B.2. If not stated, we assume that l 0 = 30, L 40 and δ [0,1].

61 20 P.-O. Åsén and G. Kreiss JMFM The solution of (38) can be written as u app 1 (y) = u h (y) + u p1 (y) + u p2 (y), where u h (y) = sinh(δ(y L)) sinh (2δL) l0+1 L 1 δ sinh(δ(σ + L))wapp (σ)dσ, u p1 (y) = 1 l0+1 δ sinh(δ(y + L)) cosh(δ(σ + L))w app (σ)dσ, (51) y u p2 (y) = 1 l0+1 δ cosh(δ(y + L)) sinh(δ(σ + L))w app (σ)dσ. y We start with proving the bounds of y u app 1 (±L). From this, most of the results for the norms will follow. The derivative at y = L can be written as l0+1 y u app 1 ( L) = L When σ [ L, l 0 ], we rewrite w app given by (27) as where Using (45) we have sinh(δ(σ L)) w app (σ)dσ. (52) sinh(2δl) w app = w app + w app (w e 1), (53) w app (σ) = 1 + ˆq(σ) L 2 e 2 3 i((δ 2 +iσ) 3/2 (δ 2 il 3/2 )), w e σ (σ) = e L p(ω)dω. w app (σ) 1 + C q 2 L 2 e 3 (( σ)3/2 L 3/2), where, from (30), C q = 1/400. From (52) and (53) we obtain y u app 1 ( L) l0 L w app (σ)dσ l0 w app (σ) w e (σ) 1 dσ L l0 L l0+1 ( sinh(δ(σ L)) sinh(2δl) ) + 1 w app (σ) dσ l 0 w int (σ) dσ I 1 I 2 I 3 I int, where we have used sinh(δ(σ L))/sinh(2δL) 1 in I 3 and I int. Consider first I 2. Let ( ) sinh(δ(σ L)) v(δ,σ) = + 1. sinh(2δl) Then v(δ, L) = 0 and v σ = δ cosh(δ(σ L)) sinh(2δl) δ cosh(2δl) sinh(2δl). (54)

62 On a Rigorous Resolvent Estimate for Plane Couette Flow 21 By considering the cases δl 1 and δl > 1, it is easily shown that σ v 1.05, i.e. v(δ,σ) < 1.05(σ + L). With L 40 and β = 2/3, the parentheses on the right-hand side of the inequality in Lemma B.3 is bounded by 2.04, giving I (1 + C q)2.04 L L 3. (55) In order to bound I 3, we wish to bound w e 1 using, from (29), p 1/100. Note that w e ( L) 1 = 0. Obviously, given two smooth, real functions f,g, such that f( L) = g( L) = 0 and σ f(σ) σ g(σ) > 0 for σ [ L, l 0 ], it follows that f(σ) g(σ) on the same interval. Also, for a complex function we have σ f(σ) σ f(σ), which can be shown using polar coordinates. Thus, we have σ e σ L p(ω)dω 1 σ (e σ L p(ω)dω 1) = p(σ)e σ L p(ω)dω p e p (σ+l) 2 p cosh( p (σ + L)). Using p 1/100 < 1, i.e. cosh( p (σ + L)) cosh(σ + L), it follows that w e (σ) 1 2 p sinh(σ + L). (56) For σ [ L, L + α], we have sinh(σ + L) sinh(α)(σ + L)/α. Using this and Lemma B.3 with β = 2/3 gives L+α sinh(σ + L) w app (σ) dσ L (1 + C q) C 1 L 3, (57) C 1 = sinh(α) ( α + 27 ) 2L 3/2 2L 3. For the rest of the interval, we have sinh(σ+l) e σ+l /2. When using Lemma B.4 with β = 2/3, γ = 1, l 0 = 30 and L 40, it is sufficient to choose β = 2/4. Using this with Lemma B.2 gives l0 sinh(σ + L) w app (σ) dσ L+α 1 + C q 2L 2 f(l α, 2/4) f(l 0, 2/4). (58) Since f(l 0, 2/4) > 0, this term can be ignored. Thus, from (57) and (58) we have l0 sinh(σ + L) w app (σ) dσ L 1 + C ( q L 3 C 1 + L 2 f(l α, ) 2/4), f(l α, 2/4) 4 2 ( 2e 4 ((L α)3/2 L 3/2 ) 3 2 ) (L α) 3(L α) + 8 3/2 (L α) 3, where C 1 is given by (57). It is clear that C 1 is decreasing with growing L and increasing with growing α. Also, for sufficiently large L and α, Lf(L α, 2/4)

63 22 P.-O. Åsén and G. Kreiss JMFM is decreasing with both growing α and growing L, since the decreasing exponential term will eventually dominate. Simple numerical experiments show that for α 0.5, Lf(L α, 2/4) is decreasing with L for L 40. Taking α = 1 and evaluating the constants numerically at L = 40 gives l0 sinh(σ + L) w app (σ) dσ C q L L 3. (59) L Using this and (56) with the definition (54) of I 3 gives I 3 2 p 2.62 L L 3. (60) Note that if β 0.18, an estimate similar to (59) can be derived, although with a different constant on the right-hand side. Indeed, β 0.18 is enough to ensure β 2/23 when using Lemma B.4 with γ = 1. Hence, Lemma B.2 may be used, and we obtain a bound similar to (58). Choosing α > 0, the right-hand side of (58) is exponentially small, although possibly with a large constant. Also, β 0.18 is enough to derive an estimate similar to (57), again possibly with a large constant on the right-hand side. Hence, (59) is valid for all β 0.18 with a different value of the constant. We will use this later when bounding the norms. In order to bound I 1 from below, we rewrite I 1 as l0 l0 I 1 ŵ app (σ)dσ ŵ app (σ) w app (σ) ŵ app (σ) 1 dσ II 1 I1 II, where L L ŵ app (σ) = 1 + ˆq(σ) 2 L 2 e 3 (1 i)(( σ)3/2 L 3/2). From (46) and Lemma B.3 with β = 2/3, L 40 and l 0 = 30, we have I II l 0 For I I 1, we have I I 1 1 L 2 l0 L l0 L (σ + L) ŵ app (σ) dσ 2.04(1 + C q)3 5 l 0 L L 3. (61) 2 e 3 (1 i)(( σ)3/2 L 3/2) dσ C q L 2 l0 L 2 e 3 (( σ)3/2 L 3/2) dσ. Using Lemma B.2 with β = (1 i) 2/3 and β = 2/3, we obtain ( I1 I 1 L 5/ i 2 2L 3/2 3 L 2L 3 2 e 3 (l3/2 0 L 3/2 ) l i l 3/2 2l C 1 q ) 2L 3/2 2L 3.

64 On a Rigorous Resolvent Estimate for Plane Couette Flow 23 Evaluating the parentheses for L 40, l 0 = 30 and C q = 1/400 gives I1 I (62) L5/2 For I int, using (35) with (45) and then Lemma B.4 with β = 2/3 and γ = p = 1/100, making it sufficient to choose β = 0.47, gives I int 3 L 2 e0.47(l3/2 0 L 3/2) 1, (63) L13 where the last inequality follows from numerical evaluation with l 0 = 30 and L 40. From (54), (55), (60), (61), (62) and (63), we thus have y u app 1 ( L) ( ) L5/2 L L21/2 L, 5/2 which proves the third bound of Lemma 4.7. At the right boundary, using δ/ sinh(2δl) 1/(2L), (59), (56) and the expression (53) for w app, we obtain δ sinh(2δl) ( l0 2L L p L y u app 1 (L) = l0+1 1 L δ sinh(δ(σ + L))wapp (σ)dσ ) sinh 2 (σ + L) w app (σ) dσ + el l0+1 I int, 2 where I int is bounded by (63). Using Lemma B.4 with β = 0.47 and γ = 1 gives β = 0.35, which bounds the last term on the right-hand side of (64) by e L l0+1 I int 3e 3/2 2 2L 2 e0.35(l 0 L 3/2) 1, (65) L10 where the last inequality is obtained by numerical evaluation. For the second term on the right-hand side of (64), proceeding as when deriving (59) gives l0 L sinh 2 (σ + L) w app (σ) dσ 1 + C q L 3 (sinh(α)c 1 + f(l α, 2/6) ) 5.79 L 3, where C 1 is given by (57) and f(, ) is defined by Lemma B.2. The last inequality in (66) follows from choosing α = 1.1, which is obtained by numerical experiments. Using (64), (65) and (66) yields y u app 1 (L) 1 2L 4 ( ) L L 4, which proves the fourth bound of Lemma 4.7. Since we are not interested in values of constants when bounding the norms, we will give these proves in less detail. The constant C will be used to denote (64) (66)

65 24 P.-O. Åsén and G. Kreiss JMFM any constant independent of L and δ. The contribution from integrating over y [ l 0, l 0 + 1], i.e. integrating over w int, will always be exponentially small. Therefore, we will only consider the contribution from w app when y [ L, l 0 ]. Also, instead of dividing w app into w app and w e, we will use w app directly. Using (45) and Lemma B.4 with β = 2/3 and γ = p = 1/100, making it sufficient to chose β = 0.47, we have L w app (y) 1 + C q L 2 e 0.47(( y) 3/2 L 3/2). From (51) and assuming w int 0, we obtain L u h 2 sinh(δ(y L)) 2 2 l0 1 sinh (2δL) δ sinh(δ(σ + L)) wapp (σ) dσ dy. L As previously mentioned, (59) is valid also for β = 0.47, although with a slightly larger constant. Also, we have 2 2 sinh(δ(y L)) sinh (2δL) y L 2L, since for a fixed y ( L,L), the denominator grows faster than the numerator with δ. Using this and (59) gives u h 2 C L 2 y L L 6 2L dy C L 5. (67) L L Since u p1 = 0 for y [ l 0,L] if we assume w int 0, we have l0 u p l0 sinh(δ(y + L)) δ cosh(δ(σ + L)) w app (σ) dσ dy. Using Lemma B.4 with β = 0.47 and γ = 1 gives β = 2/4. It follows from Lemma B.2 that 2 l0 cosh(δ(σ + L)) w app (σ) dσ C L 4 e 2 2 (( y)3/2 L 3/2). (68) y Since the exponential term decays faster than w app, we may use the result from (66) which gives For u p2 we have u p2 l0 L y u p1 2 C L 5. (69) 2 l0 cosh(δ(y + L)) 2 1 δ sinh(δ(σ + L)) wapp (σ) dσ dy. y

66 On a Rigorous Resolvent Estimate for Plane Couette Flow 25 Again, we consider two intervals. For the first interval, we have from (59) L+α 2 l0 cosh(δ(y + L)) 2 1 δ sinh(δ(σ + L)) wapp (σ) dσ dy C L 6, L L where C only depends on the choice of α. Using (68), since sinh(δ(σ + L))/δ < cosh(δ(σ + L)), bounding cosh(δ(σ + L)) 2 with an exponential function and using Lemma B.2, the contribution from integrating over y [ L + α, l 0 ] is exponentially small. Hence, we have Using (67), (69) and (70) gives u p2 2 C L 6. (70) u 1 2 C L 5, which proves the first bound of Lemma 4.7. It remains to prove the bound of y u 1. Using (51), we can write the derivative as y u app 1 (y) = y u h (y) + y u p1 (y) + y u p2 (y), where cosh(δ(y L)) l0+1 1 y u h (y) = δ sinh (2δL) δ sinh(δ(σ + L))wapp (σ)dσ, y u p1 (y) = cosh(δ(y + L)) y u p2 (y) = sinh(δ(y + L)) L l0+1 y l0 +1 y cosh(δ(σ + L))w app (σ)dσ, (71) sinh(δ(σ + L))w app (σ)dσ. As when bounding u 1, we only consider the contribution from w app when y [ L, l 0 ], since all contributions from integrating over y [ l 0, l 0 + 1] is exponentially small. From (71) we have L y u h 2 L)) 2 2 l0 δcosh(δ(y 1 sinh (2δL) δ sinh(δ(σ + L)) wapp (σ) dσ dy. L L As was motivated previously, (59) also holds for w app with a different constant. Hence, the integral over σ is bounded by C/L 3. The integral over y can easily be bounded by one, giving L y u h 2 C L 6. (72) For y u p1, we have by (71) l0 2 l0 y u p1 2 cosh(δ(y + L)) 2 cosh(δ(σ + L)) w app (σ) dσ dy. y

67 26 P.-O. Åsén and G. Kreiss JMFM When using Lemma B.4 with β = 0.47 and γ = 1, it is enough to choose β = Using this in the last integral above and using Lemma B.2, we obtain 2 l0 2 cosh(δ(σ + L)) w app (σ) dσ C L 2 L. L We use this when y [ L, L+α]. From (68), Lemma B.4 and Lemma B.2, it follows that the contribution from integrating over y [ L+α, l 0 ] is exponentially small. Hence, we have Finally, (71) gives y u p2 2 l0 L y u p1 2 C L 5. (73) 2 l0 sinh(δ(y + L)) 2 sinh(δ(σ + L)) w app (σ) dσ dy. Since sinh(δ(y + L)) < cosh(δ(y + L)), (73) also hold for u p2. Hence, using (72) and (73), we have y y u 1 2 C L 5, which proves the second bound of Lemma 4.7. Appendix C. Proof of Lemma 4.8 The proof is based on integration by parts, the Poincaré inequality h 2 4L 2 h 2, valid for continuous differentiable functions h(x) defined on x [ L, L], such that h(±l) = 0, and the relation ab 1 2µ a2 + µ 2 b2, µ > 0, (74) where a,b,µ R. We introduce the notation ( y 2 δ 2 ). Taking the inner product of (39) corr with u 1, using integration by parts and taking absolute value of the real part yields corr y u δ 2 corr u 1 2 u corr 1 F. u corr Using (74) and the Poincaré inequality for 1, we obtain ( ) 1 2 corr 1 y u L 2 + δ2 corr u 1 2 4L 2 F 2. (75) Using integration by parts we have y u corr δ 2 u corr 1 2 = (u corr 1, u corr 1 ) u corr 1 u corr 1.

68 On a Rigorous Resolvent Estimate for Plane Couette Flow 27 Again, using (74) and the Poincaré inequality for u corr 1, we obtain the inequality (75) with y u app 1 2 and u app 1 2 corr on the left-hand side and u 1 2 on the righthand side. Combining the two inequalities gives the first two bounds of the lemma. The last bound follows from (75) and the relation corr u 1 2 = yu 2 corr δ 2 y u corr δ 4 u corr 1 2. Appendix D. Proof of Lemma 4.9 Since F(y) = 0 when y / [ l 0, l 0 + 1], we only need to consider the contribution from w int. The triangle inequality and the relation (a + b) 2 2a 2 + 2b 2 yield F 2 4 l0 +1 l 0 2 yw int (y) 2 dy + (4δ 4 + 2l 2 0) l0+1 l 0 w int (y) 2 dy. Using (36), (37), δ 1 and l 0 = 30, we have F e 2 L 4 3 i((δ2 il 0) 3/2 (δ 2 il) 3/2 )+ p (y+l) 2. The lemma follow from using (45) and Lemma B.4 with γ = 2 p = 1/50, β = 2 2/3, l 0 = 30 and L 40, making it sufficient to choose β = References [1] P.-O. Åsén and G. Kreiss, A rigorous resolvent estimate for plane Couette flow, Technical Report TRITA-NA-0330, KTH, Stockholm, [2] P.-O. Åsén, A proof of a resolvent estimate for plane Couette flow by new analytical and numerical techniques, Licentiate thesis, TRITA-NA-0427, KTH, Stockholm, 2004, aasen/lic.pdf. [3] P. Braz e Silva, Resolvent estimates for 2-dimensional perturbations of plane Couette flow, Elec. J. Diff. Eq (2002), [4] P. Braz e Silva, Resolvent estimates for plane Couette flow, SIAM J. Appl. Math. 65 (2005), [5] S. J. Chapman, Subcritical transition in channel flows, J. Fluid Mech. 451 (2002), [6] J. Denzler, A study of the spectral theory of the Orr Sommerfeld equation for plane Couette flow, Habilitationsschrift, TU München, [7] P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press, London, [8] N. Dunford and J. Schwartz, Linear operators, part I, Interscience Publ., New York, [9] G. Kreiss, H.-O. Kreiss and N. A. Petersson, On the convergence to steady state of solutions of nonlinear hyperbolic-parabolic systems, SIAM J. Numer. Anal. 31 (1994), [10] G. Kreiss, A. Lundbladh and D. S. Henningson, Bounds for threshold amplitudes in subcritical shear flows, J. Fluid Mech. 270 (1994), [11] H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier Stokes Equations, Pure Appl. Math. 136, Academic Press, Boston, [12] M. Liefvendahl, Stability results for viscous shock waves and plane Couette flow, PhD thesis, KTH, Stockholm, 2001.

69 28 P.-O. Åsén and G. Kreiss JMFM [13] M. Liefvendahl and G. Kreiss, Analytical and numerical investigation of the resolvent for plane Couette flow, SIAM J. Appl. Math. 63 (2003), [14] M. Liefvendahl and G. Kreiss, Bounds for the threshold amplitude for plane Couette flow, J. Nonlinear Math. Phys. 9 (2002), [15] A. Lundbladh, D. S. Henningson and S. C. Reddy, Threshold amplitudes for transition in channel flows, in: M. Y. Hussaini, T. B. Gatski, and T. L. Jackson (eds.), Transition, Turbulence, and Combustion, Vol. I, , Kluwer, Dordrecht, Holland, [16] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, [17] W. M. Orr, The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Proc. of the Royal Irish Academy A27 ( ), [18] V. A. Romanov, Stability of plane-parallel Couette flow, Funct. Anal. Appl. 7 (1973), [19] P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Appl. Math. Sci. 142, Springer, New York, [20] L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993), [21] V. I. Yudovich, The Linearization Method in Hydrodynamical Stability Theory, Trans. Math. Monogr. 74, American Mathematical Society, Providence, RI, Per-Olov Åsén and Gunilla Kreiss Department of Numerical Analysis and Computer Science KTH Stockholm S Sweden aasen@nada.kth.se gunillak@nada.kth.se (accepted: April 20, 2005)

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102 J. Fluid Mech. (2006), vol. 568, pp c 2006 Cambridge University Press doi: /s Printed in the United Kingdom 451 ResolventboundsforpipePoiseuilleflow ByPER-OLOVÅSÉN AND GUNILLAKREISS Department of Numerical Analysis and Computer Science, Royal Institute of Technology (KTH), S , Stockholm, Sweden (Received 18 October 2005 and in revised form 5 June 2006) We derive an analytical bound on the resolvent of pipe Poiseuille ow in large parts of the unstable half-plane. We also consider the linearized equations, Fourier transformed in axial and azimuthal directions. For certain combinations of the wavenumbers and the Reynolds number, we derive an analytical bound on the resolvent of the Fourier transformed problem. In particular, this bound is valid for the perturbation which numerical computations indicate to be the perturbation that gives the largest transient growth. Our bound has the same dependence on the Reynolds number as given by the computations. 1. Introduction Since the pioneer work on pipe ow by Reynolds in the late nineteenth century, hydrodynamical stability theory has experienced great advances. However, some of the most fundamental questions remain unanswered, such as the mechanisms responsible for transition to turbulence. Even in the few simple cases of shear ows where the analytical solutions of the Navier Stokes equations are available, much is still unknown. It has been shown that plane Couette ow is linearly stable at all Reynolds numbers (Romanov 1973) and that plane Poiseuille ow becomes linearly unstable at R 5772 (Orszag 1971). For pipe Poiseuille ow, laminar ow has been observed at R 10 5 in highly controlled experiments (Pfenninger 1961) indicating that the ow is linearly stable. Also, numerous numerical computations have been done (see e.g. Lessen, Sadler & Liu 1968; Salwen, Cotton & Grosch 1980; Schmid & Henningson 1994; Trefethen, Trefethen & Schmid 1999) without nding any unstable eigenvalues of the Navier Stokes equations linearized at the stationary parabolic velocity pro le of pipe Poiseuille ow. However, a formal proof of linear stability exists only for axisymmetric disturbances (Herron 1991). Hence, despite the long history of the problem, the question of linear stability of pipe Poiseuille ow remains an open problem. Even more complicated and unresolved is the question of conditional nonlinear stability of pipe Poiseuille ow. Despite the believed linear stability at all Reynolds numbers, experiments have shown that nite-amplitude perturbations may lead to turbulence at Reynolds numbers larger than the critical Reynolds number R c 2000 (see Draad, Kuiken & Nieuwstadt 1998, and references therein). There is a threshold for the amplitude of the perturbation, below which the ow is stable to all perturbations. This threshold is assumed to behave as R β, with β 1, as R (Trefethen et al. 1993). Determining the correct value of β has proved to be a challenge. Experiments and computations have indicated values in the range 1 β 3/2 (see e.g. Hof, Juel & Mullin 2003; Meseguer 2003; Shan, Zhang & Nieuwstadt 1998). By careful asymptotic analysis, Chapman (2002) argues that β =1 and β =3/2 for

103 452 P.-O. Åsén and G. Kreiss plane Couette ow and plane Poiseuille ow, respectively. In work not yet published, discussed by e.g. Meseguer & Trefethen (2003), Chapman uses the same technique for pipe Poiseuille ow, with β = 1 as the resulting asymptotic exponent. In the last decade, much attention has been devoted to linear transient growth as a possible mechanism for transition to turbulence in shear ows (see e.g. Reddy & Henningson 1993; Trefethen et al. 1993, and references therein). This transient growth is due to the non-normality of the operator of the linearized Navier Stokes equations. More importantly, the operator is increasingly non-normal with increasing Reynolds number. Hence, a small perturbation can exhibit severe short-time growth, owing to linear mechanisms, thus triggering nonlinear effects which lead to turbulence. The transient growth cannot be captured by considering the eigenvalues, since they predict only the exponential decay which eventually follows. More information can be obtained by considering the ε-pseudospectrum or the resolvent. The ε-pseudospectrum is a generalization of the spectrum. For a linear operator, L, the ε-pseudospectrum is the set of complex numbers, s, such that (si L) 1 ε 1. Clearly, all eigenvalues are in the ε-pseudospectrum for any value of ε. If the operator is highly non-normal, the ε-pseudospectrum will include large areas around each eigenvalue for small values of ε. This is an indication that the eigenvalues probably give poor information about the short-time behaviour. Also, the ε-pseudospectrum can be used to derive a lower bound on the transient growth (Trefethen et al. 1993). Numerical computations of the ε-pseudospectrum for pipe Poiseuille ow have been done by Trefethen et al. (1999) and Meseguer & Trefethen (2003). The term R(s)=(sI L) 1 in the de nition of the ε-pseudospectrum is known as the resolvent of L. Hence, the resolvent is the solution operator of the Laplace transformed initial-value problem u t = Lu. Deriving a bound on the norm of the resolvent in the entire unstable half-plane implies linear stability of the initialvalue problem. Also, this bound includes the effects of transient growth and it can also be used for proving conditional nonlinear stability. This was done by Kreiss, Lundbladh & Henningson (1994) who, assuming the bound R(s) CR ρ in the entire unstable half-plane Re(s) 0, proved nonlinear stability of shear ows for perturbations with amplitudes smaller than CR 2ρ 5/4. This serves as the only proof of an upper bound on β in the threshold for nonlinear stability of shear ows. Here, we consider the resolvent of pipe Poiseuille ow. Computations by Meseguer & Trefethen (2003) indicate that the L 2 -norm of the resolvent is maximized at s =0 and depends on the Reynolds number as R(0) R 2. A proof of this bound in the entire unstable half-plane would, besides proving linear stability, make the nonlinear stability result mentioned above directly applicable. The rst result in this paper is a bound on the L 2 -norm of the resolvent, obtained from the Laplace transformed and linearized Navier Stokes equations, in large parts of the unstable half-plane. However, the size of the remaining part grows with R,and the bound is not valid at the point s =0.Inordertoobtainaboundintheremaining part of the unstable half-plane, we consider the equations in both Cartesian and cylindrical coordinates. When using cylindrical coordinates, there is a well-known reformulation of the equations involving the radial velocity and the radial vorticity. The advantage of this formulation is that the number of unknowns reduces to two. The equations are homogeneous in the axial and azimuthal directions. Hence, Fourier transformation can be used in these directions, with dual variables α and n, respectively. The norm of the resolvent of the original problem can be obtained by maximizing the norm of the resolvent of the Fourier transformed problem with

104 Resolvent bounds for pipe Poiseuille flow 453 respect to the wavenumbers. Numerical computations indicate that this maximum occurs when α =0 and n=1 (Trefethen et al. 1999). The second result of this paper is an analytical bound on the resolvent for certain combinations of the wavenumbers and the Reynolds number. The bound is valid in the entire unstable half-plane. In particular, the bound is valid for α =0andn =1,i.e. when the resolvent is believed to be maximized. Our analytical bound has the same R dependence as computations have indicated, i.e. the L 2 -norm is proportional to R 2. This paper is a step towards proving the linear stability of pipe Poiseuille ow. In the case of plane Couette ow, a similar strategy has proved successful. Liefvendahl & Kreiss (2003) derived results for plane Couette ow which are similar to the results in this paper. A resolvent bound for a different combination of the wavenumbers and the Reynolds number has been proved for s =0 by As en & Kreiss (2005). In the remaining bounded parameter domain, numerical computations, which could be made rigorous by using interval arithmetic, yield a resolvent bound ( As en 2005). Together, the computed and the analytical results prove a resolvent bound for s =0. The paper is organized as follows. In 2, we state the problem and introduce some notation. A resolvent bound in large parts of the unstable half-plane is derived in 3. In 4, we derive bounds for certain combinations of the wavenumbers and the Reynolds number. This is done by considering the equations in both Cartesian and cylindrical coordinates. We also show that the resolvent of the original problem can be obtained by maximizing the resolvent of the Fourier transformed problem with respect to the wavenumbers. We discuss the relation between the norm of the resolvent and transient growth of energy in 5. In 6, we discuss what further results are required in order to obtain a bound on the resolvent in the entire unstable half-plane. Finally, we present our conclusions in Theproblem We chosethe (Cartesian) coordinatesystemsuchthat x is the streamwise direction and the pipe radius is one, i.e. the domain is given by D = (x,y,z) 3 : y 2 + z 2 1. (2.1) The (normalized) stationary solution of pipe Poiseuille ow is then given by U = U 0 = 1 y2 z 2 0. (2.2) 0 0 In Cartesian coordinates, we use the notation u =(u, v, w) T = ue x + ve y + we z for the perturbation, where e x, e y and e z are the unit vectors in the x, y and z directions, respectively. Linearizing the Navier Stokes equations at the stationary solution (2.2) and applying the Laplace transform gives 2yv +2zw su + U u x 0 + p = 1 0 R u + f, (2.3a) u =0, (2.3b) u =0, (x,y,z) Γ. (2.3c)

105 454 P.-O. Åsén and G. Kreiss Here, R = U c a/ν is the Reynolds number, where U c is the centreline velocity, a the pipe radius and ν the kinematic viscosity, and Γ = (x,y,z) 3 : y 2 + z 2 =1 is the boundary of D. The resolvent, R(s), is the solution operator of (2.3a c) for a given forcing f,i.e. R(s) : f u. We are interested in bounding the L 2 -norm of the resolvent in the unstable half-plane, Re(s) 0. In particular, we are interested in how the norm of the resolvent depends on the Reynolds number. We assume that u L 2 is a smooth solution, i.e. u 0as x, so that boundary terms vanish when using integration by parts. Without any restriction, we assume f =0and f C0. A non-solenoidal forcing can be divided into a solenoidal part and a part affecting only the pressure (Yudovich 1989). Results for less regular forcing follow from closure arguments. Also, since we are interested in the linear stability for large Reynolds numbers, we consider only R 1. We will derive bounds on the resolvent in large parts of the unstable half-plane using the formulation (2.3a c) and integration by parts. However, in other parts of the unstable half-plane, this is not possible, at least not in a straightforward way. In those parts, we will derive bounds for some combinations of wavenumbers and the Reynolds number. The geometry of the domain suggests that cylindrical coordinates might be useful, and we will return to this later in the paper. We use u, v and u = u, u 1/2 to denote the L 2 -inner product and L 2 -norm, respectively. In Cartesian coordinates, the L 2 -inner product is de ned as u, v = u v dx. As mentioned above, we also use cylindrical coordinates. To avoid confusion, we introduce the corresponding equations and notation later. D 3. Aresolventboundinpartsoftheunstablehalf-plane In order to obtain a bound on the resolvent in large parts of the unstable half-plane, we consider the linearized Navier Stokes equations in Cartesian coordinates, (2.3a c). First, we de ne the following parts of the complex plane (see gure 1). { Σ = s :Re(s) 7+ 1 } Im(s) 0, (3.1a) 2R Σ = s / Σ :Re(s) 0. (3.1b) We can derive a bound on the resolvent using only integration by parts. The result is summarized in the following theorem. Theorem 3.1. If s Σ, where Σ is defined by (3.1a), then the resolvent is bounded by R(s) CR, where C is a constant independent of R. Also, for s Σ we have R(s) 0 as s. Proof. Scalar multiply (2.3a) with u. For the term involving the pressure, we have by using integration by parts, (2.3b) and(2.3c) sothat u, p = u,p =0.

106 Resolvent bounds for pipe Poiseuille flow 455 Im (s) 14R Σ Σ Re (s) 14R Figure 1. The L 2 -norm of the resolvent of pipe Poiseuille ow is bounded by R(s) CR when s Σ. Also, the resolvent tends to zero as s when s Σ. Using the triangle inequality and a + b 2 a 2 + b 2 yields 2 u, yv + zw 2 u yv + zw 2 u ( v + w ) 2 u 2 u < 3 u 2 and, since U is real and independent of x, wealsohave u,uu x = u x,uu = u,uu x u,uu x Im. Hence, using integration by parts and taking the real part gives (Re(s) 3) u R ( u x 2 + u y 2 + u z 2 ) u f. (3.2) Similarly, using integration by parts, taking the imaginary part and using u,uu x u 2 /4+ u x 2 gives ( Im(s) 1 4 3) u 2 u x 2 u f. (3.3) Dividing (3.3) by R, adding to (3.2) and dividing both sides by u yields ( Re(s) 3+ 1 ) ( 13 Im(s) u 1+ 1 ) f. R 4R R Using R 1, it follows that ( Re(s)+ 1 ) R Im(s) 25 u 2 f. 4 Hence, if s Σ, we have u 4R f. (3.4) Im(s)

107 456 P.-O. Åsén and G. Kreiss We also have from (3.2) that 1 u f. (3.5) Re(s) 3 Using either (3.4) or (3.5), depending on s, the result follows. Remark. As seen from (3.5), it is enough that Re(s) > 3 (actually Re(s) > 2 2) for the resolvent to be bounded. Hence, we could make the part of the unstable halfplane where theorem 3.1 does not hold, i.e. Σ de ned by (3.1b), somewhat smaller if desired. However, since Σ grows as R increases, this is of minor interest. Also, the estimate (3.5) gives an R-independent bound on the resolvent when Re(s) > 3. In order to prove linear stability and also nonlinear stability for sufficiently small perturbations, we wish to bound the resolvent in the entire unstable half-plane. The rest of the paper is concerned with how a bound on the resolvent could also be derived in the part of the unstable half-plane not covered by theorem Resolventboundsforcertainwavenumbers Here, we consider the Fourier transformed linearized Navier Stokes equations. We derive resolvent bounds for certain combinations of wavenumbers in relation to the Reynolds number. First, we bound the resolvent when the wavenumber in the axial direction, α, is sufficiently large compared to the Reynolds number. For this, we use the linearized Navier Stokes equations in Cartesian coordinates and Fourier transformed in the axial direction. Next, we derive a resolvent bound when the product of the axial wavenumber, α, and the Reynolds number, R, is sufficiently small. In this case, we use the linearized Navier Stokes equations in cylindrical coordinates. Finally, we bound the resolvent when the azimuthal wavenumber, n, is sufficiently large compared to the product of the axial wavenumber, α, and the Reynolds number. In this case, we use a well-known formulation involving the radial velocity and the radial vorticity. At the end of this section, we discuss the relation between the norm of the resolvent to the original problem, (2.3a c), and the norm of the resolvent to the Fourier transformed problems Cartesian coordinates Since the coefficients in (2.3a c) are independent of x, we may apply the Fourier transform, yielding 2yˆv +2zŵ iα ˆp sû +iαuû 0 + = 1 0 R ˆ û + ˆf, ˆp y ˆp z (4.1a) iαû + ˆv y + ŵ z =0, (4.1b) û =0, (y,z) Γ. (4.1c) Here, ˆ =( y 2 + z 2 α 2 ) and the domain is D= (y,z) 2 : y 2 + z 2 1 with boundary Γ = (y,z) 2 : y 2 + z 2 =1. We de ne, in analogy with the original problem, ˆR(s,α) to be the solution operator of (4.1a c) where α is to be considered as another parameter, i.e. R(s,α): ˆ ˆf û. Note that now denotes the L 2 -norm over the two-dimensional unit disk.

108 Resolvent bounds for pipe Poiseuille flow 457 Using integration by parts, we obtain the following lemma. Lemma 4.1. For all α and R such that the bound α 2 4R, ˆR(s,α) 1 (4.2) holds in the entire unstable half-plane Re(s) 0. Also, ˆR(s,α) 0 as α. Proof. As in the proof of theorem 3.1, scalar multiplying (4.1a) with û, using integration by parts, (4.1b), (4.1c) and taking the real part gives (Re(s) 3)R û 2 + û y 2 + û z 2 + α 2 û 2 R û ˆf R 2 û 2 + R 2 ˆf 2. Rearranging the terms and using Re(s) 0 yields ( û 2 α 2 7R ) 1 R 2 2 ˆf 2 (4.3) and the lemma easily follows Cylindrical coordinates Here, we derive bounds when the product of the wavenumber in the axial direction, α, and the Reynolds number, R, is sufficiently small and when the wavenumber in the azimuthal direction, n, is sufficiently large. We use the notation u =(u, v, w)=ue x + ve r + we θ for the perturbation, where e x, e r and e θ are the unit vectors in the x, r and θ directions, respectively. The stationary solution is now given by U = U 0 = 1 r In cylindrical coordinates, the coefficients in the linearized and Laplace transformed Navier Stokes equations depend only on r. Hence, we may apply the Fourier transforminthex and θ directions, with dual variables α and n, respectively. The resulting equations are sũ +iαuũ 2rṽ +iα p = R 1 ũ + f x, (4.4a) sṽ +iαuṽ + p = R 1 ( ṽ r 2 ṽ 2inr 2 w)+ f r, (4.4b) s w +iαu w + r 1 in p = R 1 ( w +2inr 2 ṽ r 2 w)+ f θ, (4.4c) iαũ + r 1 (rṽ) +inr 1 w =0, (4.4d) ũ =0, r =1, (4.4e) where the prime denotes differentiation with respect to r. Here, the forcing is f =( f x, f r, f θ ), the domain is D= r : r [0, 1] and the Laplacian is given by = 1 ( r ) α 2 n2 r r r r. 2

109 458 P.-O. Åsén and G. Kreiss We de ne R(s,α,n) to be the solution operator of (4.4a e) with α and n as parameters, i.e. R(s,α,n) : f ũ. Clearly, u is periodic in the azimuthal direction. Hence, n only takes integer values, i.e. n, and, as before, α. The only remaining space dimension is r, and since we use cylindrical coordinates, the scalar product is given 1 by ũ, ṽ = ũ ṽr dr. (4.5) 0 In this section, denotes the norm induced by (4.5), i.e. over the one-dimensional domain r [0, 1]. Using Parseval s formula, this norm can be related to the L 2 -norm over the unit disk and to the L 2 -norm over the entire three-dimensional domain (2.1). This is discussed in 4.3. We rst consider axisymmetric perturbations, i.e. n = 0, in which case we derive the following lemma. Lemma 4.2. When n =0 and αr 1/16, the bound R(s,α,0) CR holds in the entire unstable half-plane, Re(s) 0. Here, C is a constant independent of α and R. Proof. First, with the scalar product (4.5) we have by using (4.4d), (4.4e) andn =0 that ũ, iα p + ṽ, p = iαũ + r 1 (rṽ), p =0, iα( ũ, Uũ + ṽ,uṽ + w,u w ) Im. Note that the boundary term from the integration by parts vanishes by using (4.4e) at r = 1 and by using that ṽ, p are bounded at r =0, i.e. ṽ pr r=0 = 0. In the rest of this proof, we use (4.4e) and the fact that ũ and ũ are bounded at r = 0 in rṽ order to remove boundary terms appearing when using integration by parts. Now, scalar multiplying (4.4a) with ũ, (4.4b) with ṽ and (4.4c) with w, using integration by parts, taking the real part and adding the resulting equations yields (Re(Rs)+α 2 ) ũ 2 + ũ 2 + r 1 ṽ 2 + r 1 w 2 R ũ f x + R ṽ f r + R w f θ + 2R ũ, 1 4 ũ 2 +2R 2 f 2 + 2R ũ, rṽ. (4.6) For the last term on the right-hand side, we required the inequality rũ 2 ũ + ũ, (4.7) where denotes the L norm. In order to prove (4.7), note that since ũ is continuous, there exist r m and r M such that r m ũ(r m ) = min r [0,1] rũ(r) rũ ũ, r M ũ(r M ) = max r [0,1] rũ(r) = rũ. Now, (4.7) follows from rm rm r M ũ(r M ) 2 r m ũ(r ) 2 m = (r 2 ũ 2 ) dr = 2rũ 2 +2r 2 ũũ dr r m r m 2 ũ 2 +2 ũ ũ 3 ũ 2 + ũ 2 and (4 ũ 2 + ũ 2 ) 1/2 2 ũ + ũ.

110 Resolvent bounds for pipe Poiseuille flow 459 With n = 0 we have, from (4.4d), (rṽ) = iαũr or rṽ = iα r ũs ds. Using this 0 and (4.7) yields 1 ( r ) 1 2R ũ, rṽ =2R ũ iα ũs ds r dr 2 αr rũ ũ r 2 dr 2 αr (2ũ+ũ ) ( 1 ) 1/2 ( 1 ) 1/2 0 ũ 2 r dr r 3 dr 0 =2 αr (2ũ+ũ αr ( )ũ1 3ũ ũ ) 2. (4.8) We also need the Poincar e type inequalityũ2 4ũ 1 2. (4.9) Consider one component of ũ,e.g.ũ.usingũ= 1 r ũ ds yields 1 ũ2 = 0 r 1 ũ 2 [ 1 2 ũ ds] r dr 1 0 ln(r) r dr = 1 4ũ 2. 0 [ ( 1 ) 1/2 ( 1 1/2 ] 2 r ũ 2 s ds s ds) 1 r dr r Doing the same for ṽ and w gives (4.9). From (4.6), (4.8) and (4.9) we have, using Re(Rs) 0, ( α αr )ũ2 + ( αr )ũ 2 2R 2 f2. The condition αr 1/16 is more than enough to ensure that the term in parentheses on the left-hand side is positive, and the lemma follows. In order to obtain a resolvent bound when the azimuthal wavenumber, n, is sufficiently large, we consider a well-known reformulation of the problem. This formulation is obtained by eliminating the pressure and formulating equations for the radial velocity, ṽ, and the radial vorticity, η (Burridge & Drazin 1969). The resulting equations are 1 R T2 Φ (iαu + s)tφ + k 2 iαr ( U k 2 r ) Φ 2αn TΩ = Tξ, R (4.10a) 1 2αn inu SΩ (iαu + s)ω + R Rk 4 r4tφ + k 2 r Φ = χ. 3 Here, the prime denotes differentiation with respect to r, k 2 =α 2 + n 2 /r 2 and Φ = irṽ, Ω = αr w nũ k 2 r 2 = η ik 2 r, (4.10b) ξ = ir f r, χ = αr f θ nf x, k 2 r 2 T=k 2 r ( ) 1 k 2, S= 1 ( k 2 r 3 ) k 2. r k 2 r r k 2 r 3 r r (4.11a,b)

111 460 P.-O. Åsén and G. Kreiss The corresponding boundary conditions are given by (see e.g. Schmid & Henningson 2001), r =1: Φ = Φ = Ω =0, r =0,n=0: Φ = Φ =0, r =0,n=1: Φ = Ω =0,Φ (4.12) nite, r =0,n 2: Φ = Φ = Ω =0. The two variables Φ and Ω completely describe the system. The original variables can be recovered from ũ = α k 2 r Φ nω, ṽ = iφ r, w = n + αrω. k 2 r 2Φ Hence, the L 2 -norm can be computed as ũ2 =r 1 Φ2 +k 1 r 1 Φ 2 +krω2, (4.13) and similarly, f2 can be computed as f2 =r 1 ξ2 +k 1 r 1 ξ 2 +krχ2. Using (4.10a, b), we obtain the following lemma. Lemma 4.3. For all α, n and R such that n 2 16αR, the bound R(s,α,n) CR 2 holds in the entire unstable half-plane, Re(s) 0. Here, C is a constant independent of α, n and R. Also, for any fixed α and R, R(s,α,n) 0 asn. Proof. The rather lengthy proof is given in the Appendix The relation between the original problem and the Fourier transformed problems Here, we discuss how the bounds onˆr(s,α)and R(s,α,n)can be used to derive a bound onr(s). The arguments closely follow those used by Liefvendahl & Kreiss (2003). We start by proving the following theorem. Theorem 4.2. For all α, n and R such that at least one of the inequalities αr 1 16, α3 4αR, n 2 16αR (4.14a c) hold, there is a constant C, independent of α, n, R and Re(s) 0, such that R(s,α,n) CR 2. (4.15) Also, R(s,α,n) 0 asα+n. Proof. Lemma 4.3 gives (4.15) when (4.14c) holds. Also, using lemma 4.3 when n 0 and lemma 4.2 when n = 0 yields (4.15) when (4.14a) holds. Proving that lemma 4.1 is also valid for R(s,α,n) for any value of n will give (4.15) when (4.14b) holds, and will thus be the equivalent of proving the theorem. When we derived lemma 4.1, we used the linearized Navier Stokes equations in Cartesian coordinates, Fourier transformed in the axial direction. Clearly, the L 2 - norm over the unit disk of û and ˆf is the same when using cylindrical coordinates as when using Cartesian coordinates. Hence, we may assume that ˆR(s,α) in lemma 4.1 is the solution operator of the once Fourier transformed problem given in cylindrical coordinates.

112 Resolvent bounds for pipe Poiseuille flow 461 In order to prove that lemma 4.1 also holds for R(s,α,n), we require Parseval s formula, given in this case by 2π 1 û2 = 0 0û(r, α, θ)2 r dr dθ =2π n= 1 0ũ(r, α, n)2 r dr =2π n= ũ2. Assume that lemma 4.1 does not hold for R(s,α,n) foralln. This means that there exists an α withα > 2 R,ann and a forcing f (r) with f =1 suchthat R(s,α,n ) f > 1. (4.16) Denote by ũ (r) the corresponding solution. Now, consider the inverse transform of f (r), i.e. the forcing ˆf (r, θ)= f (r) exp(in θ) with corresponding solution û (r, θ). Using Parseval s formula and (4.16), we then have û 2 =2π n= ũ 2 =2π R(s,α,n ) f 2 > 2π. Sinceˆf =2π, this would implyˆr(s,α )> 1, i.e. that lemma 4.1 does not hold for R(s,α) ˆ either. Hence, the bound (4.2) of lemma 4.1 holds also for R(s,α,n) for all values of n. The proof that R(s,α,n) 0asα is almost identical. From (4.3), we have ˆ R(s,α) (2α 2 7R) 1 R. By the same arguments as above, this bound also holds for R(s,α,n)for all n. Hence, it follows that R(s,α,n) 0 asα for all values of n. This proves that lemma 4.1 holds also for R(s,α,n) for all values of n and theorem 4.2 is thus proved. Next, we show that R(s,α,n) is related to R(s) by the following relation R(s)=max α,n R(s,α,n). (4.17) Note thaton the left-hand side denotes the L 2 -norm over the entire threedimensional domain (2.1) and on the right-hand side denotes the norm induced by the scalar product (4.5). The proof of (4.17) is straightforward. Note that in the proof of theorem 4.2, we actually proved ˆ R(s,α) max (4.18) n R(s,α,n). Here, we have used max instead of sup since R(s,α,n) 0asn by lemma 4.3, i.e. R(s,α,n)attains a maximal value with respect to n. The opposite inequality of (4.18) follows ( fromũ2 = R(s,α,n) f2 max n R(s,α,n)) 2 f2. Using this with Parseval s formula givesˆr(s,α) max n R(s,α,n), and we thus have ˆ R(s,α)=max n R(s,α,n).

113 462 P.-O. Åsén and G. Kreiss In order to prove (4.17), we must now prove R(s) =max ˆR(s,α). (4.19) α For this, we require R(s,α) 0as α, ˆ in order to ensure that R(s,α) ˆ attains a maximal value with respect to α; but this follows from lemma 4.1. Now, the rest of the proof is similar to the proof above, although some care must be taken because α, i.e.α does not only take integer values as n does. The proof of (4.19) was done by Liefvendahl & Kreiss (2003) and (4.17) follows. 5. Relation between the resolvent and transient growth For pipe Poiseuille ow, numerical computations concerning the energy of an initial perturbation as a function of time have been done by e.g. Schmid & Henningson (1994) and Meseguer & Trefethen (2003). The results show that a substantial initial growth of energy is possible despite the stable eigenvalues. In this section, we relate the norm of the resolvent to this transient growth of energy. Consider the initial-value problem u t = Lu, u(0) = u 0, (5.1a) (5.1b) where L is a linear operator independent of time. If we denote the solution operator of (5.1a b) bye tl, we have u(t)=e tl u 0. (5.2) Since u(t) e tl u 0, we may use e tl as a measure of the largest possible growth (in the norm used) as a function of time. In hydrodynamic stability, the L 2 -norm is typically used, since the square of the L 2 -norm can be interpreted as an energy. Assume that the spectrum of L is in the left half-plane, i.e. lim t u(t) =0. Clearly, this implies lim t e tl =0. Now, if L is a normal operator, we have e tl 1, t 0. This means that no growth of the norm of the solution is possible for any initial data, u 0. However, if L is non-normal, the norm of the solution can experience an initial growth, i.e. e tl > 1 for some times t>0, before eventually decaying. In order to derive a relation between the resolvent and the solution operator, e tl, we apply the Laplace transform to (5.1a). The solution can then be written as ũ(s)=(si L) 1 u 0 R(s)u 0, (5.3) where I is the identity operator and R(s)=(sI L) 1 is the resolvent of L. Applying the Laplace transform to (5.2) and comparing with (5.3), we nd from the de nition of the Laplace transform that R(s)= 0 e st e tl dt. (5.4) If e tl is large, one also expects the norm of the resolvent to be large. Note that the resolvent integrates the effects of transient growth over time. Thus, the norm of the resolvent can be signi cantly larger than e tl. The effects of transient growth can be illustrated by a simple example taken from Schmid & Henningson (1994), and we refer to this paper for further details. Consider

114 Resolvent bounds for pipe Poiseuille flow 463 the 2 2 model problem (5.1) for u =(u, v) T and L given by ( ) 1/R 0 L =. (5.5) 1 2/R Clearly, the eigenvalues of L are in the left-hand half-plane for all R>0. The solution operator of this model problem is ( ) e tl e = t/r 0 ( e 2t/R e t/r) R e 2t/R. From this, we see that sup t 0e tl R and that the maximum is attained at a time t R. Hence, we expect the norm of the resolvent to be proportional to R 2. The resolvent of (5.5) is given by R 0 R(s)=(sI L) 1 = sr +1 R 2 R. (5.6) (sr +1)(sR +2) sr +2 It follows thatr(0) R 2, which is what we expected frome tl. For pipe Poiseuille ow, numerical computations by Schmid & Henningson (1994) indicate that a perturbation with α =0 and n =1 gives the largest transient growth. For this perturbation, the numerical results of both Schmid & Henningson (1994) and Meseguer & Trefethen (2003) are sup tll 2 R, t>0e with the maximum occurring at a time t R. We may thus expect the L 2 -norm of the resolvent (at least at s = 0) to be proportional to R 2, which is con rmed by the extensive numerical computations of Meseguer & Trefethen (2003). Thus, it is likely that for this perturbation, i.e. for α =0andn=1, our resolvent bound in theorem 4.2 is sharp. For further results relating the transient growth to the ε-pseudospectrum and the resolvent, see Reddy, Schmid & Henningson (1993) and Trefethen et al. (1993). 6. Discussion In4, we derived bounds on the resolvent for certain combinations of the wavenumbers and the Reynolds number. We also showed how these results can be used to give a bound on the resolvent of the original problem by using (4.17). Here, we discuss what further results are required to obtain a rigorous bound on the resolvent in the entire unstable half-plane. By theorem 3.1, we already have a bound on the resolvent when s Σ. Inthe remaining part of the unstable half-plane, Σ de ned by (3.1b), we would like to use (4.17) to obtain a bound. The resolvent of the Fourier transformed problem depends on four parameters, α, n, R and s. For convenience, we choose instead the parameters α, n, αr and sr. We would thus like to bound R(s,α,n)in the parameter domain Υ =α, αr, n,sr. The bound should be valid at least for all s Σ, i.e. we may assume Re(s) 7, s CR etc. if needed.

115 464 P.-O. Åsén and G. Kreiss Deriving an analytical bound on the resolvent in the entire parameter domain, Υ, would probably be extremely complicated. Instead, assume that for s Σ,wecould prove that there is some large constant C such that R(s,α,n)is bounded when sr+αr C 2. (6.1) Using theorem 4.2, it would be sufficient if the proof were valid forα3 4αR. Since, by theorem 4.2, we already have a bound whenαr n 2 /16, this would imply a bound also whenn 4C for all values of α, αr and sr. Also, since we assume R 1, (6.1) holds whenα C 2. Hence, in order to cover the entire parameter domain, Υ, it would be sufficient to obtain bounds in the parameter domain Υ = α [0,C 2 ],αr [1/16,C 2 ],n [0, 4C],sR [0,C 2 ]". This is a bounded parameter domain which opens for the possibility of using rigorous numerical computations to cover it. These computations would have to be combined with analytical results, since Υ still contains an in nite number of parameter values. The analytical results should be such that if a numerical bound on the resolvent is valid at a point (α,αr,n,sr ) Υ, a bound follows in some neighbourhood of this point. That is, given the numerical bound, an analytical bound follows for all combinations of α, αr, n and sr such that g(α,αr,n,sr) ε, where g is a continuous function with g(α,αr,n,sr ) = 0. The value of ε could depend on the point chosen, but should be explicitly computable. Also, note that all computations would have to be done with rigorous numerical methods using interval arithmetic. For plane Couette ow, a resolvent bound has been derived under a condition similar to (6.1) at the point s =0 ( As en & Kreiss 2005). The remaining parameter domain is bounded. Analytical results of the type described above were derived by As en (2005), making it possible to prove a rigorous bound on the resolvent at the point s = 0 in the unstable half-plane. Remark. Numerical computations by Schmid & Henningson (1994) indicate that when n 0, the transient growth decreases with increasing αr. Also, computations by Meseguer & Trefethen (2003) suggest that the resolvent is maximized at s =0. This indicates that a resolvent bound could be derived analytically when (6.1) holds, if C is chosen large enough. Remark 2. The results derived in this paper can easily be improved; Σ can be made smaller and theorem 4.2 can cover a larger parameter domain. In order to keep the technicalities at a minimum, we have not aimed at making the results as sharp as possible. However, if the desired analytical results discussed in this section are derived and rigorous numerical computations are to be used in a bounded parameter domain, making the results as sharp as possible could be important in order to reduce the amount of computation required. 7. Conclusions In this paper, we derive bounds on the resolvent of pipe Poiseuille ow. In a large part of the unstable half-plane, a bound is obtained by using integration by parts, see theorem 3.1. However, the size of the remaining part increases with increasing Reynolds number. Also, the theorem does not cover the point s =0, which is where numerical computations indicate that the resolvent is maximized (Meseguer & Trefethen 2003).

116 Resolvent bounds for pipe Poiseuille flow 465 In order to obtain a bound on the resolvent in the remaining part of the unstable half-plane, we consider the linearized Navier Stokes equations, Fourier transformed in the axial and azimuthal directions. We show, as was done by Liefvendahl & Kreiss (2003), that the norm of the resolvent of the original problem is obtained by maximizing the norm of the resolvent of the Fourier transformed problem with respect to the two wavenumbers, α and n. We derive bounds on the norm of the resolvent for different combinations of the axial wavenumber, α, the azimuthal wavenumber, n, and the Reynolds number, R. The results are presented in theorem 4.2. In particular, the theorem is valid for perturbations with α =0 and n = 1, which from numerical computations is believed to yield the largest transient growth (Schmid & Henningson 1994) and the largest resolvent (Trefethen et al. 1999). Also, our resolvent bound, R(s,α,n) CR 2,has the same dependence on the Reynolds number as the results from the numerical computations. The conditions (4.14a c) of theorem 4.2 include perturbations of different physical properties. For instance, structures with weak streamwise dependency are covered by (4.14a) or (4.14c). The velocity eld of the perturbation that gives the largest transient growth, determined by Schmid & Henningson (1994), is of this type. It consists of two counter-rotating vortices near the centre of the pipe. Further, (4.14b) is valid for perturbations with large axial wavenumber compared to the Reynolds number, i.e. for perturbations which are severely affected by viscosity. The remaining parameter domain for α, n, R and s is still unbounded. We brie y discuss what further results are required in order to obtain a bounded parameter domain. Under the conditions for deriving such a result, we also discuss how rigorous numerical computations could be used to obtain a bound on the resolvent in the remaining bounded parameter domain. This would result in a bound on the resolvent in the entire unstable half-plane, which would also serve as the rst proof of linear stability of pipe Poiseuille ow. Part of this work was done while visiting Professor Peter Schmid at the University of Washington, Seattle, USA, and we are thankful for his help and comments on the subject. The work was supported by Swedish Research Council grant Appendix. Proofoflemma4.3 We will use (4.10a,b) and integration by parts to prove lemma 4.3. More precisely, we will show that there is a constant C independent of α, n and R, such that r 1 Φ 2 + k 1 r 1 Φ 2 + krω 2 CR4 n 2 ( r 1 ξ 2 + k 1 r 1 ξ 2 + krχ 2 ) (A1) holds when n 2 16 αr. Here, is the norm induced by the scalar product (4.5). The bound R(s,α,n) CR 2 then follows from (4.13). Also, this proves that R(s,α,n) 0as n and the lemma is proved. When n = 0, the lemma is valid only for α = 0. In this case, lemma 4.2 gives the desired resolvent bound. Thus, we assume n 1 in the remainder of the proof. We use a prime to denote differentiation with respect to r. Although we use only integration by parts, the r appearing in the scalar product (4.5) makes the proof somewhat technical. Also, since r appears in the denominator at several places in the equations, the boundary terms appearing from using integration by parts must be handled with care.

117 466 P.-O. Åsén and G. Kreiss First, we multiply (4.10a) with k 2 r 2 R, scalar multiply with Φ and take the real part. Note that ( ) Φ, k2 iαrr U Φ Im k 2 r 2 k 2 r and, using integration by parts and the boundary conditions (4.12), ( Re Φ, sr ) k 2 r 2TΦ = Re(sR)(#r 1 Φ#2 +#k 1 r 1 Φ #2 ). Since Re(Rs) 0, we thus have ( ) ( 1 Re Φ, Φ k 2 r 2T2 Re Φ, iαur TΦ ) k 2 r ( 2 + Re Φ, 2αn ) ) k 2 r 2TΩ + ( Φ, Re R k 2 r 2Tξ. (A 2) Using integration by parts, we will derive a lower bound on the term on the left-hand side of (A 2) and upper bounds on the terms on the right-hand side of (A 2). Using the de nition of T (4.11a), we have 1 Φ, Φ k 2 r 2T2 = 1 0 ( ) 1 Φ k 2 r (TΦ) dr Φ, 1r 2TΦ. (A 3) The integral is rewritten, using integration by parts, as 1 ( ) [ 1 Φ 0 k 2 r (TΦ) dr = Φ 1 ] r=1 1 k 2 r (TΦ) Φ 1 r=0 0 k 2 r (TΦ) dr [ = Φ 1 k 2 r (TΦ) Φ 1 ] r=1 1 ( k 2 r TΦ + Φ 1 ( ) ) 1 k 2 r + Φ TΦ dr. (A 4) k 2 r r=0 From the de nition of T, (4.11a), we have 1 kr TΦ = 1 kr Φ + k 0 ( ) 1 Φ k k 2 r r Φ. Using this, the integral on the right-hand side of (A 4) can be written as 1 ( Φ 1 ( ) ) 1 1 ( 1 0 k 2 r + Φ TΦ dr = k 2 r 0 kr TΦ + k ) 1 r Φ TΦr dr kr =#k 1 r 1 TΦ#2 + Φ, 1r 2TΦ. (A 5) It follows from (A 3), (A 4) and (A 5) that [ 1 Φ, Φ =#k 1 r 1 TΦ#2 + Φ 1 k 2 r 2T2 k 2 r (TΦ) Φ 1 ] r=1 k 2 r TΦ. (A 6) r=0 We have kept the boundary terms which appear from using integration by parts, since it is not obvious that they are zero or even that they are bounded. We will now show that the boundary terms are zero.

118 Resolvent bounds for pipe Poiseuille flow 467 Expanding the boundary terms in (A 6) gives Φ 1 k 2 r (TΦ) Φ 1 k 2 r TΦ = 2n2 ΦΦ ΦΦ + 4n4 ΦΦ 6n2 ΦΦ + ΦΦ k 2 r 4 r k 6 r 7 k 4 r 5 k 2 r 3 + 2n2 ΦΦ k 4 r 4 ΦΦ k 2 r + ΦΦ + Φ Φ 2n2 Φ Φ + Φ Φ 2 k 2 r r k 4 r 4 k 2 r Φ Φ 2 k 2 r. At r = 1 we have the boundary condition, (4.12), Φ = Φ = 0 and all terms above are thus zero at r =1.Atr =0, we have for$n$=1thatφ =0andΦ < and for$n$ 2 that Φ = Φ = 0. Also, since n%=0,k a r b$r=0 is bounded if a = b and zero if a>b. Hence, the remaining terms as r 0are Φ 1 k 2 r (TΦ) Φ 1 k 2 r TΦ At r = 0 we have byl'hospital'srule and = 2n2 ΦΦ r 0 k 2 r 4 ΦΦ r + Φ Φ r + 4n4 ΦΦ 6n2 ΦΦ k 6 r 7 k 4 r 5 + Φ Φ k 4 r 4 k 2 r 2 2n2 Φ Φ + ΦΦ k 2 r 3 r 0. (A 7) the boundary conditions that Φ Φ ΦΦ = Φ Φ + Φ Φ Φ Φ ΦΦ =0. (A 8) r r 1 r 0 r 0 Again, usingl'hospital'srule, k = n 2 k 1 r 3 and k a r a$r=0 = n a, we have (below, we assume r 0 in all expressions) 2n 2 ΦΦ =2n 2 Φ Φ + ΦΦ Φ +2Φ Φ + ΦΦ k 2 r 4 4k 2 r 3 2n 2 r =2n2Φ =2Φ Φ r=0, (A 9a) 4(3k 2 r 2 2n 2 ) 2n 2 4n 4 ΦΦ =4n 4 Φ Φ + ΦΦ Φ k 6 r 7 7k 6 r 6 6n 2 k 4 r 4 =4Φ, (A 9b) n 2 r=0 6n2 ΦΦ = 6n 2 Φ Φ + ΦΦ k 4 r 5 5k 4 r 4 4n 2 k 2 r = Φ 2 6Φ n 2, (A 9c) r=0 ΦΦ k 2 r = Φ Φ + ΦΦ = Φ Φ, (A 9d) 3 3k 2 r 2 2n 2 n 2 r=0 2n2 Φ Φ = 2 Φ Φ, (A 9e) k 4 r 4 n 2 r=0 Φ Φ r 2 k = Φ Φ. (A 9f) 2 n 2 r=0 Hence, from (A 7), (A 8) and (A 9a/f), it follows that Φ 1 k 2 r (TΦ) Φ 1 k 2 r TΦ =2 r 0 (1 1n 2 ) Φ Φ $r=0 =0, where in the last step we use 1 n 2 =0 if$n$=1 and Φ $r =0=0 if$n$ 2. We have thus shown that the boundary terms in (A 6) are zero, and it follows that Φ, 1 k 2 r 2T2 Φ =1k 1 r 1 TΦ12. (A 10)

119 468 P.-O. Åsén and G. Kreiss We will now use integration by parts to derive an expression involving the desired terms r 1 Φ and k 1 r 1 Φ which appear on the left-hand side of (A 1). First, note that since max r [0,1] n 2 /k 2 1, we have From the de3nition of 0 n 2 k 2 r 1 TΦ 2 k 1 r 1 TΦ 2. (A 11) T (4.11a), we have 1 (( ) 1 k 2 r 1 TΦ 2 = k 2 r Φ 1 r Φ 0 = r 1 Φ 2 + (k 2 r 1 Φ ) 2 1 ( ) 1 1 Φ k 2 r Φ dr 0 )(( ) 1 k 2 r Φ 1 ) r Φ r dr 0 ( ) 1 Φ k 2 r Φ dr. (A 12) Using integration by parts and the boundary conditions gives 1 ( ) [ 1 Φ k 2 r Φ dr = Φ 1 ] r=1 1 k 2 r Φ + Φ 1 r dr = k 1 r 1 Φ 2. (A 13) k 2 r 2Φ r=0 Thus, using (A 10), (A 11), (A 12) and (A 13), we have the following lower bound on the term on the left-hand side of (A 2). 1 Re Φ, Φ n ( 2 r 1 Φ 2 +2 k 1 r 1 Φ 2 + (k 2 r 1 Φ ) 2). (A 14) k 2 r 2T2 For the3rstterm on the right-hand side of (A 2), integration by parts yields Φ, iαu 1 ( ( ) 1 1 k 2 r 2TΦ = Φiα(1 r 2 ) 0 r k 2 r Φ 1 ) r 2Φ r dr [ =iα( Φ 2 r 1 Φ 2 )+ Φiα(1 r 2 ) 1 ] r=1 k 2 r Φ r=0 1 ( ) iα Φ 1 (1 r 2 ) Φ2r k 2 r Φ dr 0 0 =iα( Φ 2 r 1 Φ 2 + k 1 Φ 2 k 1 r 1 Φ 2 ) +2iα 1 0 Φ r Φ r kr k r dr. Since max r [0,1] r/k 1/ n, this gives ( Re Φ, iαur ) k 2 r TΦ 2 αr r 1 Φ k 1 r 1 Φ. (A 15) 2 n Similarly, using integration by parts, the following results for the other terms in (A 2) are easily obtained Re Φ, 2αn k 2 r 2TΩ 2 α n ( r 1 Φ k 2 rω + k 1 r 1 Φ krω ), (A 16a) Φ, Re R k 2 r 2Tξ R( r 1 Φ r 1 ξ + k 1 r 1 Φ k 1 r 1 ξ ). (A 16b)

120 Resolvent bounds for pipe Poiseuille flow 469 Hence, using (A 14), (A 15) and (A 16a,b) in (A 2) yields n 2 ( r 1 Φ 2 +2 k 1 r 1 Φ 2 ) 2 αr r 1 Φ k 1 r 1 Φ n + 2 α n ( r 1 Φ k 2 rω + k 1 r 1 Φ krω ) + R( r 1 Φ r 1 ξ + k 1 r 1 Φ k 1 r 1 ξ ). Using ab a 2 /(2ε)+εb 2 /2, valid for ε>0, on the right-hand side gives n 2 ( r 1 Φ 2 +2 k 1 r 1 Φ 2 ) 2 αr n 3 n2( 1 4 r 1 Φ 2 + k 1 r 1 Φ 2) + n2 4 r 1 Φ 2 + 4α2 n 4 k2 rω 2 + n2 2 k 1 r 1 Φ 2 + 2α2 n 4 krω 2 + n2 8 ( r 1 Φ 2 + k 1 r 1 Φ 2 ) + 2R2 n 2 ( r 1 ξ 2 + k 1 r 1 ξ 2 ). The condition n 2 16 αr (and n 1) is more than enough to ensure that n 2 ( r 1 Φ 2 +2 k 1 r 1 Φ 2 ) 8α2 n 4 k2 rω 2 + 4α2 n 4 krω 2 + 4R2 n 2 ( r 1 ξ 2 + k 1 r 1 ξ 2 ). (A 17) We will now derive a similar result for Ω by using (4.10b). We multiply (4.10b) by ( k 2 r 2 R), scalar multiply by Ω and take the real part. Since Ω,iαURk 2 r 2 Ω Im, Re( Ω,sRk 2 r 2 Ω )=Re(sR) krω 2, and Re(Rs) 0, this yields ( ) ( Re Ω,k 2 r 2 SΩ Re Ω, 2αn 2TΦ ) k 2 r ( + Re Ω, RinU Φ ) + Re( Ω,k 2 r 2 Rχ ). (A 18) r de<nition For the term on the left-hand side of (A 18), using integration by parts and the of S (4.11b) yields 1 ( ) 1 Ω,k 2 r 2 SΩ = Ω r (k2 r 3 Ω ) k 4 r 2 Ω r dr = k 2 rω 2 0 [ Ωk 2 r 3 Ω ] 1 r=1 + Ω k 2 r 2 Ω r dr = k 2 rω 2 + krω 2. (A 19) r=0 0 The terms on the right-hand side of (A 18) are easily bounded from above by ( Re Ω, 2αn ) k 2 r 2TΦ 2 α n ( krω k 1 r 1 Φ + k 2 rω r 1 Φ ), (A 20a) ( Re Ω, RinU Φ ) 2R krω r 1 Φ, (A 20b) r

121 470 P.-O. Åsén and G. Kreiss ( Ω,k Re 2 r Rχ ) 2 R krω krχ. (A 20c) Using (A 19) and (A 20aBc) in (A 18) thus gives k 2 rω 2 + krω 2 2 α n ( krω k 1 r 1 Φ + k 2 rω r 1 Φ ) +2R krω r 1 Φ + R krω krχ. As before, using ab a 2 /(2ε)+εb 2 /2 on the right-hand side gives k 2 rω 2 + krω krω 2 + 4α2 n 2 k 1 r 1 Φ k2 rω 2 + 8α2 n 2 r 1 Φ krω 2 +2R 2 r 1 Φ krω 2 +2R 2 krχ 2. Collecting terms and using krω n 2 krω k 2 rω yields k 2 rω 2 +3 krω 2 16α2 n 2 ( k 1 r 1 Φ 2 +2 r 1 Φ 2) From (A 17), we have 8R 2 r 1 Φ 2 64α2 R 2 n 6 k 2 rω α2 R 2 n 6 krω 2 +8R 2 r 1 Φ 2 +8R 2 krχ 2. (A 21) + 32R4 n 4 ( r 1 ξ 2 + k 1 r 1 ξ 2 ). (A 22) Using (A 22) on the right-hand side of (A 21) and adding the result to (A 17) gives after rearranging the terms ( n α2 + n 4 ) r 1 Φ 2 + n 2 (2 16α2 (1 8α2 64α2 R 2 n 4 n 6 n 4 ) k 2 rω 2 + C(R 4 ( r 1 ξ 2 + k 1 r 1 ξ 2 )+R 2 krχ 2 ). ) k 1 r 1 Φ 2 ) (3 4α2 32α2 R 2 krω 2 n 4 n 6 The condition n 2 16 αr (and R 1, i.e. n 2 16 α ) is enough to ensure that the term in parentheses on the left-hand side is positive. Using also n 2 krω 2 k 2 rω 2, (A 1) follows and the lemma is proved. 12,264J265. REFERENCES 451,35J97. Åsén, P.-O A proof of a resolvent estimate for plane CouetteDow by new analytical and numerical techniques. Licentiate thesis, aasen/lic.pdf. Åsén, P.-O. & Kreiss, G On a rigorous resolvent estimate for plane CouetteDow. J. Math. Fluid Mech. Accepted. 85,269J ,267J312. Burridge, D. M. & Drazin, P. G Comments on Stability of pipe Poiseuille flowf. Phys. Fluids Chapman, S. J Subcritical transition in channeldows. J. Fluid Mech. Draad, A. A., Kuiken, G. D. C. & Nieuwstadt, F. T. M. 1998LaminarJturbulent transition in pipedow for Newtonian and non-newtonianduidṣj. Fluid Mech. Herron,I.H.1991 Observations on the role of vorticity in the stability theory of wall bounded Dows. Stud. Appl. Maths

122 1404a ,175a198. Resolvent bounds for pipe Poiseuille flow 471 Hof,B.,Juel,A.&Mullin,T.2003 Scaling of the 63,801a817. turbulence transition threshold in a pipe. Phys. Rev. Lett. 91 (24), Kreiss, G., Lundbladh, A. & Henningson, D. S Bounds for threshold amplitudes in subcritical shear\ows. J. Fluid Mech. Lessen, M., Sadler, S. G. & Liu, T.-Y Stability of pipe Poiseuille\ow. Phys. Fluids 11, 689a ,1203a1213. Liefvendahl, M. & Kreiss, G Analytical and numerical investigation of the resolvent for plane Couette\ow. SIAM J. Appl. Maths Meseguer, Á Streak breakdown instability in pipe Poiseuille\ow. Phys. Fluids Meseguer, Á. & Trefethen, L. N Linearized pipe\ow to Reynolds number J. Comput. 252,209a238. equation. J. Fluid Mech. 50, 53,15a47. OraSomerfeld Pfenninger, W Boundary layer suction experiments with laminar\ow at 7,137a146. high Reynolds numbers in the inlet length of a tube by various suction methods. In Boundary Layer and Flow Control (ed. G. V. Lachmann), vol. 2, pp.961a980ṗergamon. Reddy, S. C. & Henningson, D. S Energy growth in viscous channel\ows. J. Fluid Mech. 277,197a25. 98,273a284. Reddy, S. C., Schmid, P. J. & Henningson, D. S Pseudospectra of the operator. SIAM J. Appl. Maths Romanov, V. A Stability of plane-parallel Couette\ow. Funct. Anal. Applic. Salwen,H.,Cotton,F.W.&Grosch,C.E.1980 Linear stability of Poiseuille\ow 19,320a325. in a circular pipe. J. Fluid Mech. Schmid,P.J.&Henningson,D.S.1994 Optimal energy density growth in HagenaPoiseuile\ow. J. Fluid Mech. Schmid,P.J.&Henningson,D.S ,578a ,413a420. Stability and Transition in Shear Flows. Appl. Math. Sci., vol Springer. Shan, H., Zhang, Z. & Nieuwstadt, F. T. M Direct numerical simulation of transition in pipe\ow under thein\uenceof wall disturbances. Intl J. Heat Fluid Flow Trefethen,A.E.,Trefethen,L.N.&Schmid,P.J.1999 Spectra and pseudospectra for pipe Poiseuille\ow. Comput. Meth. Appl. Mech. Engng. Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A Hydrodynamic stability without eigenvalues. Science Yudovich, V. I The Linearization Method in Hydrodynamical Stability Theory. Trans.Math. Monogr., vol. 74. American Mathematical Society, Providence. 186,178a197. Phys. Orszag,S.A.1971 Accurate solutions of theorasomerfeldstability

123 A Parallel Code for Direct Numerical Simulations of Pipe Poiseuille Flow Per-Olov Åsén KTH School of Computer Science and Communication SE Stockholm SWEDEN March 30, 2007 Abstract In this report we describe the process of parallelizing a serial code for direct numerical simulations of pipe Poiseuille flow for a distributed memory computer. The serial code, developed by Reuter and Rempfer, uses compact finite differences of at least eighth order of accuracy in the axial direction and Fourier and Chebyshev expansions in the azimuthal and radial directions, respectively. While these methods are attractive from a numerical point of view, they give a global data dependency which makes the parallelization procedure complex. In the resulting parallel code, the partitioning of the domain changes between partitioning in the axial direction and partitioning in the azimuthal direction as needed. We present results showing good speedup of the parallel code. 1 Introduction The stability of pipe Poiseuille flow was first studied experimentally by Osborne Reynolds in the late 19th century. Since then, numerous experimental and some analytical results have given further insight on the stability properties of pipe flow. In the last decades, computers have become powerful enough to simulate flows in simple geometries using direct numerical simulations (DNS). DNS means that the full non-linear Navier Stokes equations are solved numerically such that all length-scales are resolved. This requires numerical methods of high order of accuracy as well as vast amounts of computer resources. DNS has in some cases proven to be an excellent supplement to experiments. For example, both for plane Couette flow and plane Poiseuille flow, DNS has been used to determine bounds on the stability threshold and also to investigate different transition scenarios, see e.g. [6, 9, 19]. So far, only a handful of codes for DNS of pipe Poiseuille flow have been developed. This is mainly due to the additional complexity of solving the equations in cylindrical coordinates, as opposed to channel flow, where Cartesian coordinates are used. Some of the existing codes are based on methods of low order of accuracy in one or more spatial directions, e.g. [2, 15, 25, 26]. Also, most codes with high order of accuracy are, to the authors knowledge, written 1

124 for serial computers, e.g. [8, 12, 17], although parallel, high order codes exist, e.g. [10] (which is based on the serial code presented in [22]). A typical application where serial codes of high order of accuracy can be used is when to consider an initially disturbed state consisting of relatively large structures, e.g. stream-wise independent structures which are optimally chosen to trigger instability, as in [11]. The evolution of the initial disturbance is simulated for sufficiently long time such that it can be decided if turbulence is triggered or not. Such simulations are of great practical and theoretical interest and may even inspire new experimental approaches. However, they are not easily compared with experiments which are usually initialized with disturbances of relatively large amplitude applied in small holes in the pipe wall. Such large and spatially localized disturbances adds additional requirements on the resolution of a simulation mimicking the experiments. This makes the use of a serial code insufficient. In this report, we present the process of parallelizing a serial DNS code for pipe Poiseuille flow for a distributed memory computer. The serial code was developed by Reuter and Rempfer, [20], and is based on compact finite differences of at least eighth order of accuracy in the axial direction and Fourier and Chebyshev expansions in the azimuthal and radial directions, respectively. The reason for parallelizing the code is to be able to perform highly resolved computations in order to investigate the stability properties of pipe flow. Despite the long history of research in pipe flow, not even the question of linear stability has been fully resolved. Only partial analytical proofs have been derived, stating that pipe Poiseuille flow is linearly stable to axisymmetric disturbances, [3], as well as for certain non-axisymmetric disturbances, [27]. Also, numerous numerical computations have been done for the linearized equations without finding any unstable eigenvalues, see e.g. [7, 13, 21]. The question of non-linear stability of pipe Poiseuille flow is even more complicated. It has been suggested that pipe flow is stable to disturbances with amplitudes smaller than some threshold value, which scales with the Reynolds number as R β for some β 1, [23]. Determining the correct value of β has proven to be a difficult task. Experimental results presented in [4, 14] both indicate β = 1. In [4], the flow was disturbed by injection of fluid through six equally spaced small holes around the pipe while in [14], injection through a single hole was used. However, it was also noted in [14] that if suction through a single hole was used, the required amplitude for triggering turbulence was typically two orders of magnitude larger than in the case of injection. Clearly, the stability of pipe flow is highly dependent on the disturbance used. We aim to use the parallel code to determine the value of β for pipe flow. Especially, we wish to investigate how efficient different combinations of injection and suction are at triggering turbulence. The report is organized as follows: In section 2, we present the mathematical formulation of the problem. Section 3 briefly describes the serial code, developed by Reuter and Rempfer, [20]. This section is a brief summary of [20] and we use the same notation as in [20]. In section 4, we discuss parallelization in general and how this code was parallelized. Efficiency and speedup results for two test problems are presented in section 5 and in section 6 we present our conclusions. 2

125 2 Mathematical formulation We consider the flow of an incompressible fluid in an infinite circular pipe where the flow is driven by a constant pressure gradient in the axial direction. The governing equations are the Navier-Stokes equations. Let the vector U denote the velocity field in the pipe and let the scalar valued function P denote the pressure relative to a reference pressure. Then, the equations describing the evolution of U and P are U t + U U = 4 R e z P + 1 U, R (1) U = 0, (2) where e z is the unit vector in the axial direction. On the boundary, we enforce no-slip conditions, i.e. U = 0 on the pipe wall. Equations (1) and (2) have been non-dimensionalized using the Reynolds number, R, which in this case is defined as R = 2Ūza ν, (3) where Ūz is the mean velocity in the axial direction, a is the pipe radius and ν is the kinematic viscosity of the fluid. Note that the quantities on the right hand side of (3) are dimensional quantities while U and P in (1) and (2) are non-dimensional. If equations (1) and (2) are written in cylindrical coordinates, a stationary solution, (V, P s ), is given by V (r) = (V r, V φ, V z ) = V r e r + V φ e φ + V z e z = (1 r 2 )e z, P s (z) = 4 R z + P 0, where P 0 is a constant and e r, e φ and e z are the unit-vectors in the radial, azimuthal and axial directions, respectively. Let v(t, r, φ, z) = (v r, v φ, v z ) and p(t, r, φ, z) denote the perturbation of the stationary solution. By inserting U = V + v and P = P s + p into (1) and (2), corresponding equations for the perturbations are obtained. However, in cylindrical coordinates, the Laplacian is not diagonal, i.e. e r v and e φ v both involve v r and v φ. Using the well-known reformulation, see e.g. [16], u ± = v r ± iv φ, the Laplacian is diagonalized. The resulting two equations for u ± are u ± u ± + v r t r + v ( ) φ u± r φ ± iu ± + (V z + v z ) u ± z = p r i p r φ + 1 [ ( 1 r u ) ± + 1 ( 2 ) ] u ± R r r r r 2 φ 2 ± 2i u ± φ u ± + 2 u ± z 2. The axial velocity, v z, can be obtained from (2), which in cylindrical coordinates is given by 1 rv r r r + 1 v φ r φ + v z = 0. (5) z 3 (4)

126 An equation for the pressure is obtained by applying the divergence operator to (1) and using (2) to simplify the obtained equation. The resulting Poisson equation is 1 r r ( r p ) 1 2 p r r 2 φ p z 2 = 1 rζ r r r 1 ζ φ r φ ζ z z, (6) where v r ζ r = v r r + v ( φ vr r v φ ζ φ = v r r + v φ r ζ z = v r (V z + v z ) r ) φ v φ ( vφ φ + v r + v φ r + (V z + v z ) v r z, ) + (V z + v z ) v φ z, v z φ + (V z + v z ) v z z. The no-slip conditions on the pipe wall implies v(t, 1, φ, z) = 0. 3 The serial code The serial code was developed by Jörg Reuter and Dietmar Rempfer. This section outlines the code and we refer to [20] for further details. In order to avoid confusion, we use the same notation as in [20]. Equations (4 6) are solved numerically in the domain shown in Figure 1. A disturbance is introduced by applying non-homogeneous boundary conditions in one or more disturbance slots as shown in the figure. Inflow Outflow Disturbance slot Damping zone Figure 1: Computational domain. The serial code has been verified by Reuter and Rempfer, [20], by comparing with results from linear theory as well as experimental data. 3.1 Spatial discretization Let w {v r, v φ, v z, p} denote any of the components of the solution. In the azimuthal direction, the solution is 2π-periodic. Hence, it can be approximated 4

127 by a finite Fourier series as w(t, r, φ, z) = N n= N ŵ n (t, r, z)e inφ. (7) The requirement that the sums must be real implies ˆv n = ˆv n, where denotes complex conjugate. Therefore, only the coefficients with n 0 are solved for. Inserting (7) in (4 6) yields û ±,n t + ˆζ ±,n = ˆp n r ± nˆp n + 1 r R 1 r r ( r ˆp ) n n2ˆp n r r 2 n + n = n n, n N 1 r rˆv r,n r [ 1 r r ( r û ) ±,n (1 ± n)2 û ±,n r r û ±,n z 2 (8) + 2ˆp n z 2 = 1 rˆζ r,n inˆζ φ,n ˆζ z,n r r r z, (9) + inˆv φ,n r + ˆv z,n z = 0, (10) where the non-linear terms are given by ˆζ ±,n = ˆζ r,n ± iˆζ φ,n and ( ˆv r,n in ˆζ ˆv r,n ˆv φ,n r,n = ˆv r,n + ˆv φ,n + ˆv + ˆv r,n z,n r r z ˆζ φ,n = ˆζ z,n = Here, n + n = n n, n N n + n = n n, n N ), ( ˆv φ,n ˆv r,n + in ˆv ) φ,n ˆv r,n + ˆv φ,n + ˆv + ˆv φ,n z,n, r r z ( ˆv + z,n ˆv r,n r ˆv + z,n = in ˆv z,n + ˆv φ,n r { Vz + ˆv z,0 if n = 0, ˆv z,n if n 0. + ˆv + z,n ˆv z,n z ). (11) In the radial direction, the solution is expanded in a Chebyshev series as K { 1 ŵ n (t, r, z) = c k w n,k(t, z)t k (r), c k = 2 if k = 0, (12) 1 if k > 0. k=0 The Chebyshev polynomials are defined as T k (x) = cos(k arccos(x)) for 1 x 1. They are especially suitable to resolve boundary layers near walls, since the spacing of collocation points is O(1/K 2 ) near x = ±1. However, the solution is only defined for 0 r 1. This problem can be handled in two ways, see e.g. [16]. The Chebyshev polynomials can be rescaled to the domain 0 x 1. This approach has the disadvantage of introducing O(1/K 2 ) spacing also at r = 0. The second approach is to consider the even or odd continuation of the solution to the entire domain 1 r 1. Since the Chebyshev polynomials, T k, are even/odd functions for k even/odd, the series (12) can be restricted to even or odd indices as L ŵ n (t, r, z) = c 2l+σ w n,l (t, z)t 2l+σ (r), σ {0, 1}. l=0 5 ],

128 Here, σ {0, 1} determines whether even or odd polynomials are used and should be chosen such that û +,n r n+1, û,n r n 1, ˆv z,n r n, ˆp n r n, (13) as r 0, see e.g. [18]. So, for n even, σ = 1 in the expansions of û ±,n and σ = 0 in the expansions of ˆv z,n and ˆp n while the opposite holds for n odd. In the axial direction, differentiation and integration with respect to z are done by compact finite differences of at least eighth order of accuracy. 3.2 Time integration Near the centerline of the pipe, the terms in (8) involving 1/r will become large, requiring small time steps if an explicit time integration scheme is used. A fully implicit scheme would require the solution of a large non-linear system in each time step. Therefore, the serial code uses a third order, four step semi-implicit Runge-Kutta scheme, [1]. The non-linear terms and the axial diffusion are integrated explicitly while the rest of the viscous terms are integrated implicitly. 3.3 Boundary conditions At the inflow boundary laminar flow is assumed, i.e. v z=zi = p z=zi = 0. At the end of the domain, a damping zone is used, see Figure 1. The solution is gradually damped by multiplying v r and v φ with a decaying function. This means that the solution will become independent of z and suitable boundary conditions are v z = p z=zo z = 0. z=zo On the pipe wall, no-slip conditions are imposed on the velocity outside of the disturbance strip, in which the velocity is given by the applied disturbance. For the pressure, the influence matrix method, [5], is used to obtain a boundary condition at the wall. The use of cylindrical coordinates requires boundary conditions at r = 0. From (13), it follows that û ±,n r=0 = 0, if n 1, û ±,n r = 0, if n / {0, 2}, r=0 ˆv z,n r=0 = ˆp n r=0 = 0, if n 0, ˆv z,n r = ˆp n r=0 r = 0, if n 1. r=0 4 Parallelization of the code There exists two different types of parallel computers; shared memory computers, where all processors share the same memory, and distributed memory 6

129 computers, where the processors have their own memory which is not accessible to other processors. Parallelizing a serial code is usually more straight forward for shared memory computers. In this case, the process of parallelizing typically amounts to inserting instructions, using e.g. OpenMP, at the places in the serial code where tasks can be run in parallel. No sending of data between the processors is necessary, since they all have access to the same memory. However, shared memory computers in general have poor scalability; adding additional processors slows down memory access time for all processors. Therefore, shared memory systems are usually not larger than processors, although larger systems exist. Distributed memory computers do not suffer from this problem, since each processor has its own memory bus. However, when data in one processor is needed in another processor, the data must explicitly be sent, using e.g. MPI instructions, through the high-speed network connecting the processors. This adds a level of complexity to the parallelization. The superior scalability of distributed memory computers makes them attractive despite the increase in parallelization complexity. Indeed, most existing parallel systems are of distributed type. At the Royal Institute of Technology, there is a distributed memory computer with approximately 600 processors open for general use. We therefore chose to parallelize the code for distributed memory computers. 4.1 Parallelization for a distributed memory computer When parallelizing a serial code for a distributed memory computer, the work and data must be divided among the processors. The complexity of this vary drastically depending on the problem at hand; it varies from embarrassingly parallel problems, which are such that they can easily be divided into many small independent problems, to inherently serial problems, for which parallelization is virtually futile. As an example of a problem which can easily be divided into smaller problems that are almost independent, consider differentiation of a function. Assume we want to compute the derivative of f(x) on a uniform grid, {x i }, i = 0..N + 1, using central finite differences. We then want to evaluate f (x i ) (f(x i+1 ) f(x i 1 ))/(2 x), i = 1..N, where x is the distance between two grid points. For simplicity, let N be even and let two processors be used for the differentiation. Partition the grid into two parts of equal size, i.e. processor one has the grid {x i }, i = 0..N/2 and processor two has the grid {x i }, i = N/ N + 1. Now, both processors can independently of each other compute the derivative in all their points except at the points closest to where the grid is divided, see Figure 2. In order to compute f (x N/2 ), processor one needs the value of f(x N/2+1 ) which is located in processor two. Similarly, processor two needs the value of f(x N/2 ) from processor one in order to compute f (x N/2+1 ). By explicitly using sending and receiving routines, these values are sent between the processors and the final derivatives are computed. The extension to more than two processors is straight forward. As long as the grid is large enough, such that each processor contains a fair amount of grid points, the time it takes to send data between processors will be negligible compared to the time spent on computations. 7

130 f (x i ) f (x f N ) f (x N (x j ) ) x i x N 2 x N x j Processor 1 Processor 2 Figure 2: Computing f (x) on two processors. The values of f(x) at x N 2 x N 2 +1 are sent between the processors. and 4.2 Parallelization of the pipe Poiseuille code In the serial code described in the previous section, there is no obvious way to divide the work among processors. For example, differentiation with respect to r is done by recursion relations for the Chebyshev coefficients. If f l, l = 0..L are the Chebyshev coefficients of the function f(r), then the coefficients of g(r) = f/ r are given by g l 1 g l+1 = 2l f l, l 1, (14) with g L = 0 and g L 1 = 2L f L. Parallelizing (14) for two processors is easy, since the even and odd coefficients of g(r) are independent of each other. However, using more than two processors severely increases the parallelization complexity. Computing (14) amounts to solving a banded linear system. The straight forward approach of backward and forward substitution is inherently serial. Although there are (direct) methods for solving banded systems in parallel, [24], they generally show poor performance. The same problem is present in the axial direction. Integration and differentiation with respect to z is done by compact finite differences which also requires solving banded linear systems. In the azimuthal direction, the convolution sums (11) are not easily computed in parallel. In order to overcome these problems, we chose to divide the domain in both the axial and the azimuthal directions. Note that if the non-linear terms, ˆζ,, in (8) and (9) were somehow known, the problem of solving (8 10) is embarrassingly parallel. All wave numbers, n, would then be independent of each other and we could distribute an equal amount of wave numbers on each processor. Clearly, the non-linear terms are not known but must be computed from (11). As mentioned above, this is not a trivial task if different wave numbers are located on different processors. However, if the domain is divided only in the axial direction, evaluating (11) is straight forward with the exception of the terms with derivatives with respect to z. Our partitioning strategy thus involves dividing the domain in two directions but not in both directions simultaneously. The partitioning of data changes between being divided in the azimuthal direction and the axial direction as needed. In practice, we do this by storing duplicate solutions. One solution is partitioned in the axial direction so that each processor gets approximately the same number of grid points. The other solution is partitioned in the azimuthal direction such that each processor gets the same number of wave numbers. When 8

131 operations are performed on the solution, the solution which is appropriately partitioned for that operation is used. If the result needs to be partitioned in the other direction, this is done by communication between the processors. Consider for example the problem of computing ˆζ r,n using (11). We want the result partitioned in the azimuthal direction among the processors. Computing ˆζ φ,n and ˆζ z,n in the same way, we would then be in the embarrassingly parallel case of solving (8 10) until the non-linear terms need to be recomputed. Assume for simplicity we have partitioned the work on two processors called P 0 and P 1. The solution should then be partitioned in two parts both in the axial direction, z, and in the azimuthal direction, n. Now, computing ˆζ r,n is done using the following steps, where the numbers refer to the numbers in Figure 3 (since there is no parallelization in the radial direction, we have omitted this direction from the figure). 1. Compute ˆvr,n z using the solution partitioned in the azimuthal direction. 2. Change the partitioning of the domain for ˆvr,n z. This is done by P 0 and P 1 sending the required values to each other. 3. Compute ˆζ r,n partitioned in the axial direction. The other quantities in (11) are already available or easily computed in each processor. Only ˆv r,n z is difficult to compute when data is partitioned in the axial direction, which is why we computed it in step Change the partitioning of the domain for ˆζ r,n. As in step 2, this is done by sending and receiving the required values. n n n P 0 P 1 P 1 P 0 P 1 ˆv r,n z 1 ˆv P r,n 0 z 2 ˆv r,n z ˆv r,n z z z z n P 0 P 1 n ˆζ r,n ˆζ r,n P 1 ˆζ r,n 3 4 ˆζr,n P 0 z z Figure 3: Computing ˆζ r,n on two processors. The numbers at the arrows correspond to the operations listed above the figure. 4.3 Advantages and potential problems As previously mentioned, this choice of partitioning the work means that the total required storage increases by approximately a factor of two. This is due to 9

132 the duplicate storing of the solution. This is a minor problem since typically, a serial code is not parallelized for using two processors only. A potentially more severe problem is the large amounts of communication needed. Advancing the solution one time step requires sending the entire solution (or therefrom derived quantities) between the processors several times. This could be devastating to the performance of the parallel code. As we will see in the next section, our tests show that the time spent on communication is still small compared to the computational time. This is due to the fast communication network on the computer used. Also, performance is not the only issue when deciding on how to divide the work among processors. If we instead had chosen to partition the data only in the axial direction, the required communication would have reduced significantly. However, we would then have been forced to solve the banded linear systems from the compact finite differences in parallel. As mentioned above, direct parallel solvers for banded systems are typically inefficient. Instead, iterative solvers are usually preferred in parallel. Although this could have resulted in an efficient code as well, the amount of changes to the serial code would have been significantly larger. 5 Performance of the parallel code In order to investigate the efficiency of the parallel code, we solved two problems using different number of processors. For both problems, we computed the speedup, S P, and efficiency, E P, defined as S P = T S T P, E P = S P P, where T S is the execution time for the serial code and T P is the execution time for the parallel code using P processors. Ideally, the speedup should be linear, i.e. S P = P, which gives the optimal efficiency E P = 1. Even if linear speedup is achieved for small P, it will eventually deteriorate as P increases. Obviously, this occurs when P is larger than the total amount of work available, leading to the fact that some processors remain idle during the entire execution. In reality, linear speedup typically deteriorates long before this point since the communication time and increased overhead becomes increasingly significant with increasing number of processors. It is even possible to achieve better than linear speedup, so called super linear speedup. This is due to memory effects. A computer has a hierarchical structure of different memory types. The fastest and smallest ( 1 MB) memory is the cache memory (actually, there are two or even three different cache memories on a computer). Obviously, a typical application on a computer is too large to be stored in the cache memory alone. Instead, most parts are stored in the RAM memory, which is larger ( 1 GB) but also several times slower than the cache memory. When an application is too large for the RAM memory, disk storage is used. Disk storage is large ( 100 GB or more) but also orders of magnitude slower than the RAM memory. Super linear speedup can easily be achieved when a problem is so large that disk storage must be used in the serial 10

133 case. By dividing the problem on several processors, each processor can use only the RAM memory for its local part of the problem. The orders of magnitude in reduced memory access time easily makes up for the extra work required for solving the problem in parallel. Even when disk storage is not required, the increase in total cache memory when using several processors can give super linear speedup. We will see an example of this in the second test problem in the next section. For both problems, we used the Dell Xeon cluster Lenngren at the Center for Parallel Computers at the Royal Institute of Technology, Stockholm, Sweden. Lenngren consists of 442 nodes (although approximately 300 nodes are open for general use), each having two 3.4 GHz Nocona Xeon processors and 8 GB of main memory. MPI communication uses an Infiniband network from Mellanox with a bi-directional peak bandwidth of 2 GB/s. Since the computer has two processors per node, they will share the communication and memory bandwidth when both are used. Hence, using only one processor per node gives faster communication and memory access. We solved the problems using one as well as both processors on each node. The difference in speedup between the two cases illustrate the importance of fast communication and memory access. 5.1 The first test problem First, we consider a relatively small problem with 150 points in the axial direction, 40 Fourier modes in the azimuthal direction and 20 Chebyshev polynomials in the radial direction, giving a total of degrees of freedom. The speedup and efficiency for this problem are shown in Figures 4 and SP P Figure 4: Speedup, S P, for the first test problem. The (*) are the results using one processor per node and the ( ) are the results using two processors per node. The dashed line corresponds to linear speedup. When using only one processor per node, an efficiency above 0.75 is achieved up to 10 processors. The large difference in speedup when one or both processors on the nodes are used clearly shows the need of fast communication and memory access. When 20 processors are used, the speedup is rather poor. This is not surprising, since the problem is quite small. 11

134 EP P Figure 5: Efficiency, E P, for the first test problem. The (*) are the results using one processor per node and the ( ) are the results using two processors per node. 5.2 The second test problem In the second test, we use 600, 150 and 150 degrees of freedom in the axial, azimuthal and radial directions, respectively, giving a total of unknowns. The problem size is chosen such that the serial code uses most of the available RAM memory. Therefore, the problem can not be solved in parallel on a single node using two processors, since the parallel code requires approximately twice as much memory. The measured speedup and efficiency are shown in Figures 6 and SP P Figure 6: Speedup, S P, for the second test problem. The (*) are the results using one processor per node and the ( ) are the results using two processors per node. The dashed line corresponds to linear speedup. We observe good speedup up to 15 processors; when using one processor per node, the speedup is even super linear. This is due to memory effects discussed previously in this section. As in the first test problem, the speedup changes significantly when one or both processors per node are used. 12

135 EP P Figure 7: Efficiency, E P, for the second test problem. The (*) are the results using one processor per node and the ( ) are the results using two processors per node. 6 Conclusions We have successfully parallelized an advanced serial code for direct numerical simulations of pipe Poiseuille flow to a distributed memory parallel computer. We have chosen to divide the data and work among the processors by alternating between a decomposition of the domain in the azimuthal direction and the axial direction. The advantage of this is that data can always be ordered in a way which makes each step in advancing the solution embarrassingly parallel. This is not only desirable from the implementational point of view, but also allows for efficient serial algorithms, such as e.g. direct solution of banded linear systems, to be used in any direction. A problem with this choice of domain decomposition could be the vast amounts of communication needed. However, our tests show good speedup and we expect good performance on even larger problems as well. In order to maximize the speedup, special attention should be given to the specific architecture of the computer used. Our tests were run on a cluster where each node has two processors that share communication and memory bandwidth. On this computer, superior speedup was obtained when using only one processor per node. References [1] U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicit-explicit Runge- Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 1997, pp [2] J. G. M. Eggels, F. Unger, M. H. Weiss, J. Westerweel, R. J. Adrian, R. Friedrich and F. T. M. Nieuwstadt, Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment, J. Fluid Mech., 268, 1994, pp [3] I. H. Herron, Observations on the role of vorticity in the stability theory of wall bounded flows, Stud. Appl. Math., 85, 1991, pp

136 [4] B. Hof, A. Juel and T. Mullin, Scaling of the turbulence transition threshold in a pipe, Phys. Rev. Lett., 91(24), 2003, [5] L. Kleiser and U. Schumann, Treatment of incompressibility and boundary conditions in 3-D numerical simulations of plane channel flows, Proc. third GAMM.-conference on numerical methods in fluid mechanics, 1980, pp [6] G. Kreiss, A. Lundbladh and D. S. Henningson, Bounds for threshold amplitudes in subcritical shear flows, J. Fluid Mech., 270, 1994, pp [7] M. Lessen, S. G. Sadler and T.-Y. Liu, Stability of pipe Poiseuille flow, Phys. Fluids, 11, 1968, pp [8] P. Loulou, R. D. Moser, N. N. Mansour and B. J. Cantwell, Direct numerical simulation of incompressible pipe flow using a b-spline spectral method, Technical Memorandum TM , NASA, [9] A. Lundbladh, D. S. Henningson and S. C. Reddy, Threshold amplitudes for transition in channel flows, Transition,Turbulence, and Combustion, M. Y. Hussaini, T. B. Gatski, and T. L. Jackson, eds., Vol. I, Kluwer, Dordrecht, Holland, 1994, pp [10] B. Ma, C. W. H. Van Doorne, Z. Zhang and F. T. M. Nieuwstadt, On the spatial evolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at Re = Part 1. Subcritical disturbance, J. Fluid Mech., 398, 1999, pp [11] F. Mellibovsky and A. Meseguer, The role of streamwise perturbations in pipe flow transition, Phys. Fluids, 18, 2006, [12] A. Meseguer and F. Mellibovsky, On a solenoidal Fourier-Chebyshev spectral method for stability analysis of the Hagen-Poiseuille flow, Appl. Numer. Math. (2006), doi: /j.apnum [13] A. Meseguer and L. N. Trefethen, Linearized pipe flow to Reynolds number 10 7, J. Comput. Phys., 186, 2003, pp [14] T. Mullin and J. Peixinho, Transition to turbulence in pipe flow, J. Low Temp. Phys., 145, 2006, pp [15] N. V. Nikitin, Statistical characteristics of wall turbulence, Fluid Dyn., 31, 1996, pp [16] S. A. Orszag and A. T. Patera, Secondary instability off wall-bounded shear flows, J. Fluid Mech., 128, 1983, pp [17] P. L. O Sullivan and K. S. Breuer, Transient growth in circular pipe flow. I. Linear disturbances, Phys. Fluids, 6, 1994, pp [18] V. G. Priymak and T. Miyazaki, Accurate Navier-Stokes investigation of transitional and turbulent flows in a circular pipe, J. Comput. Phys., 142, 1998, pp

137 [19] S. C. Reddy, P. J. Schmid, J. S. Bagget and D. S. Henningson, On stability of streamwise streaks and transition thresholds in plane channel flows, J. Fluid Mech., 365, 1998, pp [20] J. Reuter and D. Rempfer, Analysis of pipe flow transition. Part I. Direct numerical simulation, Theoret. Comput. Fluid Dynamics, 17, 2004, pp [21] P. J. Schmid and D. S. Henningson, Optimal energy density growth in Hagen-Poiseuille flow, J. Fluid Mech., 277, 1994, pp [22] H. Shan, B. Ma, Z. Zhang and F. T. M. Nieuwstadt, Direct numerical simulation of a puff and a slug in transitional cylindrical pipe flow, J. Fluid Mech., 387, 1999, pp [23] L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261, 1993, pp [24] E. F. Van de Velde, Concurrent scientific computing, Springer, [25] R. Verzicco and P. Orlandi, A finite-difference scheme for threedimensional incompressible flows in cylindrical coordinates, J. Comput. Phys., 123, 1996, pp [26] O. Y. Zikanov, On the instability of pipe Poiseuille flow, Phys. Fluids, 8, 1996, pp [27] P.-O. Åsén and G. Kreiss, Resolvent bounds for pipe Poiseuille flow, J. Fluid Mech., 568, 2006, pp

138 Theoretical and Computational Fluid Dynamics manuscript No. (will be inserted by the editor) Per-Olov Åsén Gunilla Kreiss Dietmar Rempfer Direct numerical simulations of localized disturbances in pipe Poiseuille flow Received: date / Accepted: date Abstract We consider pipe Poiseuille flow subjected to a disturbance which is highly localized in space. Experiments by Mullin and Peixinho have shown this disturbance to be efficient in triggering turbulence, yielding a threshold dependence on the required amplitude as R 1.5 on the Reynolds number, R. The experiments also indicate an initial formation of hairpin vortices, with each hairpin having a length of approximately one pipe radius, independent of the Reynolds number in the range of R = 2000 to We perform direct numerical simulations for R = The results show a packet of hairpin vortices traveling downstream, each having a length of approximately one pipe radius. The perturbation remains highly localized in space while being advected downstream for approximately 10 pipe diameters. Beyond that distance from the disturbance origin the flow becomes severely disordered. Keywords direct numerical simulations, pipe Poiseuille flow, hydrodynamical stability, incompressible Navier Stokes equations PACS j Ft ek 1 Introduction Hydrodynamical stability concerns the stability of various flows when subjected to disturbances. A given stationary flow is called stable if the perturbation generated by a disturbance eventually vanishes, i. e. if the flow returns to the stationary state. If the flow is not stable to a disturbance, it may eventually develop into a turbulent state. Despite a long history of research in the area, the onset of turbulence is not well understood. Controlling the presence of turbulence would be of significant importance in many real-world applications. A better understanding of the onset of turbulence is crucial for achieving this. Pipe Poiseuille flow was first studied experimentally by Osborne Reynolds at the end of the 19th century. He found that the sensitivity to disturbances could be characterized by one non-dimensional number, the Reynolds number, R. For Reynolds numbers smaller than about 2000, the flow was observed to be stable to all disturbances. He also found that laminar flow could be maintained at higher Reynolds numbers by avoiding disturbing the flow; increasing Reynolds number required increasing care in avoiding disturbances. In the experiments, laminar flow was observed up to Reynolds numbers around The findings of Reynolds P.-O. Åsén Linné Flow Centre, CSC, KTH, Lindstedtsv. 3, SE Stockholm, Sweden. Tel.: Fax: aasen@csc.kth.se G. Kreiss Department of Information Technology, Uppsala University, Box 120, S Uppsala, Sweden. D. Rempfer Illinois Institute of Technology, 243c Engineering1, 3110 South State Street, Chicago, IL 60616, USA.

139 2 Per-Olov Åsén et al. have since then been verified in numerous experiments. A critical Reynolds number, below which the flow is stable to all disturbances, around R c 1800 has been found [24] and in highly controlled experiments laminar flow has been maintained up to R = 10 5 [25]. From a mathematical point of view, the most straightforward way to investigate the stability properties of a stationary flow is to consider the linearized Navier Stokes equations. If the linear operator associated with these equations has, at some Reynolds number, an eigenvalue with positive real part, then the flow is linearly unstable. This would imply that there exist infinitesimal disturbances that will make the flow unstable. However, numerous numerical computations have been done without finding any unstable eigenvalues for pipe Poiseuille flow, see e. g. [13, 19, 29]. Also, rigorous results have been proven that show that pipe Poiseuille flow is linearly stable, at any Reynolds number, to axisymmetric disturbances [8] as well as to certain nonaxisymmetric disturbances [3]. Despite the believed linear stability of pipe Poiseuille flow, disturbances may exhibit severe initial growth due to linear effects. This is commonly referred to as transient growth, and is due to the non-normality of the operator of the linearized Navier Stokes equations [31]. To understand transition to turbulence, nonlinear effects must also be taken into account; assuming the flow is linearly stable, it is clear that nonlinear effects are necessary at some point in the transition to turbulence. The question of nonlinear stability of pipe Poiseuille flow has long been a main point of interest. It has been suggested that linearly stable shear flows are stable to all disturbances with amplitudes smaller than some threshold value, which scales with the Reynolds number as R β for some β 1 [31]. Determining the correct value of β for pipe Poiseuille flow has proven to be a difficult task. Which mechanisms are dominant in the evolution of a perturbation is also far from understood. Further knowledge in this would not only assist in determining β, but also provide insight in the process of transition from laminar flow to turbulence. In the last decades, computers have become powerful enough to simulate flows in simple geometries using direct numerical simulations (DNS). DNS means that the full nonlinear Navier Stokes equations are solved numerically such that all length-scales are resolved. This requires numerical methods of high order of accuracy as well as vast amounts of computer resources. The precise control of the disturbance and the possibility to analyze the evolution of the resulting perturbation in detail makes DNS an excellent tool in investigating transition to turbulence. For example, both for plane Couette flow and plane Poiseuille flow, DNS has been used to determine upper bounds on β and also to investigate different transition scenarios, see e. g. [12,15,27]. Also, a critical Reynolds number around R c 1800 in pipe flow has been verified using DNS [34]. For pipe Poiseuille flow, experiments in e. g. [9,20] indicate β 1. In [9], the flow was disturbed by injection of fluid through six equally spaced small holes around the pipe while in [20], injection through a single hole was used. In more recent (not yet published) experiments, Mullin and Peixinho disturbed the flow by a combination of injection and suction through two holes located close to each other. The experiments showed that the required amplitude for triggering turbulence using this disturbance is an order of magnitude smaller than when disturbing with injection in a single hole. Moreover, the experiments showed that the location of the holes relative to each other have significant effect; when the holes are either aligned with or perpendicular to the pipe axis, the resulting threshold exponent is β 1.3, while locating the holes such that they form a 45-degree angle with the pipe axis results in β 1.5. Clearly, the stability of pipe flow is highly dependent on the disturbance used. So far, only a handful of codes for DNS of pipe Poiseuille flow have been developed. This is mainly due to the additional complexity of solving the equations in cylindrical coordinates, as opposed to channel flow, where Cartesian coordinates are used. Some of the existing codes are based on methods of low order of accuracy in one or more spatial directions, e. g. [6,21,33,35]. Also, most codes with high order of accuracy are, to the authors knowledge, written for serial computers, e. g. [14, 18, 23], although parallel, high order codes exist (see [16], which is based on the serial code presented in [30]). Simulations have previously been used in order to determine a value of β for pipe Poiseuille flow; extensive computations in [17] suggest β = 1.5. However, the disturbance considered in [17] corresponds to an initial condition of streamwise independent structures which are chosen to optimally trigger instability. Experimental investigation of such structures is difficult, since disturbances in experiments are typically applied using relatively high velocity injection or suction in small holes in the pipe wall. Such large and spatially localized disturbances add additional requirements on the resolution of a simulation mimicking the experiments. This makes the use of a serial code insufficient. In this paper, we demonstrate the possibility of using direct numerical simulations for disturbances with high spatial locality, making comparisons with experiments possible as well as providing detailed insight into the early development of the perturbations. In particular, we present results from a simulation at R = 5000

140 Direct numerical simulations of localized disturbances in pipe Poiseuille flow 3 where the perturbation is triggered by a disturbance similar to the most efficient disturbance used in the experiments of Mullin and Peixinho, i. e. when the holes make a 45-degree angle with the pipe axis. The simulation was done using a parallelized version of a serial code developed by Reuter and Rempfer [28], based on compact finite differences of at least eighth order of accuracy in the axial direction and Fourier and Chebyshev expansions in the azimuthal and radial directions, respectively. The paper is organized as follows: In section 2, we present the mathematical formulation of the problem. Section 3 summarizes the numerical methods in the serial code developed by Reuter and Rempfer [28], and also the parallelization of the code [2]. In section 4, we describe the experiments by Mullin and Peixinho and how the experiments are modeled in the simulations. The results of the simulations are presented in section 5, followed by our conclusions in section 6. 2 Mathematical background We consider the flow of an incompressible fluid in an infinite circular pipe where the flow is driven by a constant pressure gradient in the axial direction. The governing equations are the Navier-Stokes equations. Let the vector U denote the velocity field in the pipe and let the scalar valued function P denote the pressure relative to a reference pressure. Then, the equations describing the evolution of U and P are U t +U U = 4 R e z P+ 1 U, R (1) U = 0, (2) where e z is the unit vector in the axial direction. On the boundary, we enforce no-slip conditions, i. e. U = 0 on the pipe wall. Equations (1) and (2) have been non-dimensionalized using the Reynolds number, R, which in this case is defined as R = 2Ū z a ν, (3) where Ū z is the mean velocity in the axial direction, a is the pipe radius and ν is the kinematic viscosity of the fluid. Note that the quantities on the right hand side of (3) are dimensional quantities while U and P in (1) and (2) are non-dimensional. If equations (1) and (2) are written in cylindrical coordinates, a stationary solution, (V,P s ), in the domain Ω = {(r,φ,z) 0 r 1,0 φ 2π,z R} is given by V (r) = (V r,v φ,v z ) = V r e r +V φ e φ +V z e z = (1 r 2 )e z, P s (z) = 4 R z+p 0, where P 0 is a constant and e r, e φ and e z are the unit-vectors in the radial, azimuthal and axial directions, respectively. Let v(t,r,φ,z) = (v r,v φ,v z ) and p(t,r,φ,z) denote the perturbation of the stationary solution. By substituting U = V +v and P = P s + p into (1) and (2), corresponding equations for the perturbation are obtained. However, in cylindrical coordinates, the Laplacian is not diagonal, i. e. e r v and e φ v both involve v r and v φ. Using a well-known reformulation, see e. g. [22], u ± = v r ± iv φ, the Laplacian is diagonalized. The resulting two equations for u ± are u ± u ± + v r t r + v ( ) φ u± r φ ± iu ± +(V z + v z ) u ± z = p r i p r φ + 1 [ ( 1 r u ) ± + 1 ( 2 u ± R r r r r 2 φ 2 ± 2i u ± φ u ± ) + 2 u ± z 2 The axial velocity, v z, can be obtained from (2), which in cylindrical coordinates is given by 1 rv r r r + 1 v φ r φ + v z = 0. (5) z ]. (4)

141 4 Per-Olov Åsén et al. An equation for the pressure is obtained by applying the divergence operator to (1) and using (2) to simplify the obtained equation. The resulting Poisson equation is ( 1 r p ) 1 2 p r r r r 2 φ p z 2 = 1 rζ r r r 1 ζ φ r φ ζ z z, (6) where v r ζ r = v r r + v ( φ vr r ζ φ = v r v φ r + v φ r ζ z = v r (V z + v z ) r ) φ v φ +(V z + v z ) v r z, ( ) vφ φ + v r +(V z + v z ) v φ z, + v φ v z r φ +(V z + v z ) v z z. The no-slip conditions on the pipe wall implies v(t,1,φ,z) = 0. 3 Numerical methods The code used in this paper is based on a serial code developed by Reuter and Rempfer which we parallelized for a distributed memory computer. For details of the serial code, we refer to [28], and in [2], details concerning the parallelization can be found. This section is a brief summary of these two papers. Equations (4 6) are solved in the domain D = {(r,φ,z) 0 r 1,0 φ 2π,z i z z o } for given z i and z o, see Figure 1. A disturbance is introduced by applying non-homogeneous boundary conditions in one or more disturbance slots as shown in the figure. The serial code was verified by Reuter and Rempfer by Inflow Outflow Disturbance slot Damping zone Fig. 1 Computational domain. comparing with results from linear theory as well as experimental data. 3.1 Discretization Let w {v r,v φ,v z, p} denote any of the components of the solution. In the azimuthal direction, the solution is approximated by a finite Fourier series as w(t,r,φ,z) = N n= N Since the sums must be real, they can be restricted to n 0. ŵ n (t,r,z)e inφ. (7)

142 Direct numerical simulations of localized disturbances in pipe Poiseuille flow 5 Inserting (7) in (4 6) yields û ±,n t + ˆζ ±,n = ˆp n r ± n ˆp n + 1 r R ( 1 r ˆp n r r r [ 1 r ( r û ) ±,n (1 ± n)2 û ±,n r r r ] û ±,n z 2, (8) ) n2 ˆp n r ˆp n z 2 = 1 r ˆζ r,n in ˆζ φ,n ˆζ z,n r r r z, (9) 1 r ˆv r,n + in ˆv φ,n + ˆv z,n = 0, (10) r r r z for n = 0,1,...N, where the nonlinear terms are given by ˆζ ±,n = ˆζ r,n ± i ˆζ φ,n and ˆζ r,n = ˆζ φ,n = ˆζ z,n = n + n = n n, n N n + n = n n, n N n + n = n n, n N ( ˆv r,n in ) ˆv r,n ˆv φ,n ˆv r,n + ˆv r φ,n + ˆv + ˆv r,n r z,n, z ( ˆv φ,n ˆv r,n + in ) ˆv φ,n ˆv r,n + ˆv r φ,n + ˆv + ˆv φ,n r z,n, z ( ˆv + ) z,n ˆv in ˆv z,n r,n + ˆv r φ,n + ˆv + ˆv z,n r z,n. z (11) Here, { ˆv + Vz + ˆv z,n = z,0 if n = 0, ˆv z,n if n 0. Note that the nonlinear terms (11) result in interactions between different Fourier modes, e. g. modes n = 1 and n = 2 act as a forcing for the n +n = 3 mode. This interaction is disregarded for modes with n +n > N. Therefore, the number of Fourier coefficients used in the computations must be sufficiently large, such that this truncation of nonlinear interaction has negligible effect on the final result. In the radial direction, the solution is expanded in a finite Chebyshev series. The Chebyshev polynomials, T k (x) = cos(k arccos(x)) for 1 x 1, are even/odd functions for k even/odd. The parity relation observed in e. g. [22] is used in order to avoid clustering of points near the center of the pipe. This means that each component of the solution is expanded in even or odd Chebyshev polynomials as ŵ n (t,r,z) = L l=0 { 1 c 2l+σ w n,l (t,z)t 2l+σ (r), c 2l+σ = 2 if 2l + σ = 0, 1 if 2l + σ > 0. Here, σ {0,1} determines whether even or odd polynomials are used and should be chosen such that û +,n r n+1, û,n r n 1, ˆv z,n r n, ˆp n r n, (12) as r 0, see e. g. [26]. So, for n even, odd polynomials, σ = 1, are used in the expansions of û ±,n and even polynomials, σ = 0, are used in the expansions of ˆv z,n and ˆp n. For odd n the situation is reversed. In the axial direction, differentiation and integration with respect to z are done by compact finite differences. An advantage of compact finite differences is that high order of accuracy can be obtained using narrow stencils. This simplifies the treatment of points near the boundary, where non-symmetric stencils must be used. All formulas used in the code are of eighth order of accuracy or higher. 3.2 Time integration Near the centerline of the pipe, the terms in (8) involving 1/r will become large, thus requiring a small time step if an explicit integration scheme is used. A fully implicit scheme would require the solution of a large nonlinear system in each time step. The serial code uses a third order, four step semi-implicit Runge-Kutta

143 6 Per-Olov Åsén et al. scheme [1]. The nonlinear terms and the axial diffusion are integrated explicitly while the rest of the viscous terms are integrated implicitly. Note that since the axial diffusion is integrated explicitly, the CFL condition gives a necessary bound on R 1 t/ z 2 for stability. Similarly, for the advective term, v r t/ r must be bounded, where r is the distance between collocation points in the radial direction. As mentioned above, expanding in even or odd Chebyshev polynomials avoids the undesired clustering of points near the center of the pipe. However, the clustering is still present, and desired in order to resolve the boundary layers, near the wall of the pipe where r = O(1/L 2 ). If no-slip conditions are enforced on the pipe wall, and the flow is driven by e. g. a volume forcing or a given initial state, this is a minor problem since v r is small near the wall. However, driving the flow with locally large inhomogeneous boundary conditions for v r, the admissible time step is severely restricted. This has been observed in our computations. 3.3 Boundary conditions At the inflow boundary, laminar flow is assumed, i. e. v z=zi = p z=zi = 0. At the end of the domain, a damping zone is used, see Figure 1. The solution is gradually damped by multiplying v r and v φ with a decaying function. Hence, the solution becomes independent of z and suitable boundary conditions are v z = p z=zo z = 0. z=zo On the pipe wall, no-slip conditions are imposed on the velocity outside of the disturbance strip, in which inhomogeneous boundary conditions are used to model an applied disturbance. For the pressure, the influence matrix method [11], is used to obtain a boundary condition at the wall. The use of cylindrical coordinates requires boundary conditions at r = 0. From (12), it follows that û ±,n r=0 = 0, if n 1, û ±,n r = 0, if n / {0, 2}, r=0 ˆv z,n r=0 = ˆp n r=0 = 0, if n 0, ˆv z,n r = ˆp n r=0 r = 0, if n 1. r=0 3.4 Parallelization The numerical methods used in the serial code are of high order of accuracy. However, they yield a global data dependency, i. e. the solution can not easily be divided into parts which are almost independent when integrating one time step. For example, differentiation with respect to r is done by recursion relations for the Chebyshev coefficients. If f l, l = 0..L are the Chebyshev coefficients of the function f(r), then the coefficients of g(r) = f/ r are given by g l 1 g l+1 = 2l f l, l 1, (13) with g L = 0 and g L 1 = 2L f L. Parallelizing (13) for two processors is easy, since the even and odd coefficients of g(r) are independent of each other. However, when using more than two processors the parallelization complexity is severely increased. Computing (13) amounts to solving a banded linear system. Although there are (direct) methods for solving banded systems in parallel [32], they generally show poor performance. The same problem is present in the axial direction. Integration and differentiation with respect to z is done by compact finite differences which also requires solving banded linear systems. In the azimuthal direction, the convolution sums (11) are not easily computed in parallel. In order to overcome these problems, we partition the domain in two directions but not in both directions simultaneously. The partitioning of data switches between a division of in the azimuthal direction and a division of the axial direction as needed. In practice, we make this possible by storing duplicate solutions. One

144 Direct numerical simulations of localized disturbances in pipe Poiseuille flow 7 solution is partitioned in the axial direction so that each processor gets approximately the same number of grid points. The other solution is partitioned in the azimuthal direction such that each processor gets the same number of wave numbers. When operations are performed on the solution, the solution which is appropriately partitioned for that operation is used. For example, when differentiating with respect to z, the solution partitioned in the azimuthal direction is used. If the result needs to be partitioned in the other direction, this is done by communication between the processors. This partitioning strategy requires large amounts of communication. However, our tests have shown good speedup [2]. 4 Background and modeling As mentioned in the introduction, we are interested in doing numerical simulations in order to obtain further understanding of experimental results. In this section, we describe the experimental setup and also the mathematical model of the experiments. 4.1 Experimental setup Experiments using either injection and/or suction to disturb the flow have previously yielded a threshold for stability which scales as R 1 with the Reynolds number [5,9,20]. However, as noted in [20], the required amplitude for triggering turbulence can vary significantly depending on the disturbance used; when using suction in a single hole, the required amplitude for triggering turbulence was typically two orders of magnitude larger than in the case of injection in a single hole. Clearly, the stability of pipe flow is highly dependent on the disturbance used. In more recent experiments, Mullin and Peixinho disturbed the flow by applying injection and suction through two small holes located close to each other. The experiments showed that the location of the holes relative to each other have significant importance on how efficient the disturbance is at triggering turbulence. So far, the most efficient way to trigger turbulence has been when the holes are located such that a line through the center of both holes makes a 45-degree angle with the pipe centerline, see Figure 2; both possible disturbances, i. e. injection applied in the hole further upstream or the hole further downstream with suction applied in the other hole, are equally efficient. The disturbance is applied in holes with diameter d = 0.05D, where D is the diameter of the pipe. The amplitude of the disturbance is defined as the injected volume flux, Φ in j, normalized with the pipe flux, Φ pipe, i.e. the amplitude is Φ in j /Φ pipe. With this definition of amplitude, note that although small amplitudes may be required for triggering turbulence, the maximal velocity might still be large since the disturbance holes are small. These experiments yield a threshold which scales as R 1.5 with the Reynolds number. Why this disturbance is so efficient in triggering turbulence is not known. Flow visualizations in the experiments indicate the formation of so-called hairpin vortices which are known to play a central role in the transition to turbulence in boundary layers [4]. Numerical simulations can give further insight in this matter. D Flow direction d 45 Fig. 2 Location of the disturbance holes in the experiments by Mullin and Peixinho.

145 8 Per-Olov Åsén et al. 4.2 Mathematical model We consider a disturbance applied through holes that are lined up to form a 45-degree angle with the pipe axis and with injection applied in the hole further upstream, i. e. injection and suction is applied in the left and right hole in Figure 2, respectively. The disturbance is highly localized in space, both in the axial and azimuthal directions. We model the injection and suction by setting non-homogeneous boundary conditions as v r (φ,z) r=1 = Ae ((φ φ 0)/ε) 2 f(z), (14) for each disturbance slot. The locality of each disturbance slot in the axial direction is modeled by the function f(z). In order to avoid numerical difficulties, f(z) must be sufficiently smooth. The function we used is f [x(z)] = { (1 x 2 ) 6, x 1, 0, x 1, x = z z s. δ Here, z s is the position of the center and δ is the effective width of the slot. The parameters A, φ 0 and ε are chosen in each disturbance slot to model the maximal velocity and azimuthal shape of the applied injection or suction. Since the diameter of the pipe in the computations is D = 2, ε should be chosen such that the effective width in the azimuthal direction of (14) is approximately 0.05D = 0.1. This is achieved with ε However, since Fourier expansion is used in the azimuthal direction, the boundary condition (14) must be applied on the Fourier coefficients, i. e. (14) must be expanded in a finite Fourier series. In order to obtain a good approximation of (14) with ε = 0.03 as a finite Fourier series, about 100 Fourier coefficients are needed. As previously mentioned, the number of Fourier coefficients used in the computations must be sufficiently large, such that the truncation of the nonlinear interactions between high wave numbers has negligible effect. Hence, applying inhomogeneous boundary conditions on 100 coefficients would require numerical computations using even more (say 200) coefficients. Modeling such a small hole would also require high resolution in the other two spatial directions. In the axial direction, z must be small in order to resolve f(z). A disturbance with a given volume flux requires a high maximal velocity in a small hole, which in turn requires high resolution in the radial direction. Making things even worse, the time step must be small due to the high spatial resolution and the high maximal velocity near the wall. We therefore model the experimental disturbance with ε = 0.1, which gives holes of a diameter of approximately 0.3, i. e. three times larger than the holes in the experiments. The locations in the azimuthal direction of the inhomogeneous boundary conditions are φ 0 = π and φ 0 = π for the injection and suction, respectively and the maximal velocity of both is A = 0.5. We represent the inhomogeneous boundary conditions using 40 Fourier coefficients, which results in the functions seen in Figure 3. Note that although we use 40 Fourier coefficients, the resulting functions are still somewhat oscillatory. (a) 0.1 (b) vr r=1 0.2 vr r= φ φ Fig. 3 Resulting boundary conditions for (a) injection at z = 0.3 and (b) suction at z = 0.3 using 40 Fourier coefficients. Corresponding to a diameter of 0.3, we choose δ = 0.3 and the centers of the injection and suction at z s = 0.3 and z s = 0.3, respectively. This results in the functions f(z) shown in Figure 4.

146 Direct numerical simulations of localized disturbances in pipe Poiseuille flow f(z) z Fig. 4 The function f(z) in the boundary condition (14). Solid is for the injection and dashed is for the suction. 5 Numerical results In this section, we present results from a simulation using the inhomogeneous boundary conditions described above. The inhomogeneous boundary conditions were applied at time t = 0 and kept on during the entire simulation until the end time, t In order to avoid large gradients in the initial stage, a gradual increase from no-slip conditions was used in the time span t [0, 0.25]. In this simulation, the Reynolds number was R = 5000 and the length of the pipe was 60 = 30D. Since the inhomogeneous boundary conditions are very localized is space, a high resolution is necessary. We used a grid with N = 100 Fourier modes, L = 200 Chebyshev polynomials and 1500 grid points ( z = 0.04) in the axial direction. The time integration was done using the time step t , requiring a total of time steps to reach the end time. The simulation was run on the Dell Xeon cluster Lenngren at the Center for Parallel Computers at the Royal Institute of Technology, Stockholm, Sweden, consisting of 442 nodes, each having two 3.4 GHz Nocona Xeon processors and 8 GB of main memory. MPI communication uses an Infiniband network from Mellanox with a bi-directional peak bandwidth of 2 GB/s. We used 50 processors for our computation which required a total of approximately CPU-hours or 1.17 CPU-years. We study and present results at two different stages; the initial stage and the late stage. The initial stage describes the evolution for short times, when the perturbation velocity is still somewhat ordered and localized in the radial direction, while the late stage concerns the transition to a more global and disordered perturbation velocity field. The main question addressed in this paper is the one of why the applied injection and suction is so efficient in triggering turbulence. The initial stage is the most interesting from this point of view, since it is directly connected to the way the perturbation is generated. The later stage has little or no relation to the origin of the perturbation and should be analyzed using averaged quantities which is the subject of upcoming work. Still, we present some results here, mainly to demonstrate the capabilities of the parallel code but also to make plausible that the computations are sufficiently resolved to simulate turbulent flow. 5.1 Initial stage Immediately downstream of the disturbance slots, a pair of counter-rotating vortices are excited close to the wall, see Figure 5. The axial velocity of the laminar flow is higher close to the center of the pipe than near the wall. The vortices force high speed fluid towards the wall and low speed fluid towards the center, generating the high- and low-speed streaks seen in Figure 5. The structure of the vortices can be visualized by a method proposed in [10]. In short, the method is based on studying iso-surfaces of the second eigenvalue, λ 2, of S 2 + Ω 2, where S and Ω are the symmetric and antisymmetric parts of the gradient of the velocity, v. In Figure 6, iso-surfaces of λ 2 = 0.5 are shown at two different times. Also shown are the high-speed streaks located close to the pipe wall and the low-speed streaks located closer to the center of the pipe. As

147 10 Per-Olov Åsén et al. (a) (b) Fig. 5: Perturbation velocity field at (a) z = 1 and (b) z = 5 when t Red corresponds to v z = 0.3 and blue represents v z = 0.5. The spacing between the contour lines is 0.1 and the zero contour is dashed. The arrows show the perturbation velocity in the plane. seen in the figure, the perturbation evolves in the form of hairpin vortices. The hairpins appear in a rather structured fashion, with both individual length and distance between two hairpins of approximately one pipe radius. This is in good agreement with the experiments of Mullin and Peixinho, where hairpins of length one pipe radius, independent of the Reynolds number over the range 2000 to 3000, were found. The high- and low-speed streaks created by the hairpin vortices results in localized layers of high shear. The development of these high-shear layers are shown in Figure 7. The hairpins are created close to the wall, and as they are advected downstream, they move closer to the center of the pipe Numerical resolution In order to investigate if the resolution of the simulation was sufficient, the solution was compared to a solution obtained from a simulation in a shorter pipe using the higher resolution of N = 150 Fourier modes, L = 300 Chebyshev polynomials and z = Also, the time step was reduced by a factor of two to t In Figure 8, the streaks and λ 2 structures obtained using the high resolution are shown at time t The figure shows good agreement with Figure 6 (b). Although there are some differences in the details between the two figures, the main structures are very similar. We believe that the main reason for the differences between Figures 6 (b) and 8 is the increased resolution in the axial direction. The volume flux in each disturbance slot corresponds to the integral of the nonhomogeneous boundary condition, (14). Using more Fourier modes does not affect the volume flux since, with the exception of mode zero, Fourier modes have zero mean. However, increasing the resolution in the axial direction gives a slightly different volume flux. Even a small difference in the volume flux of the injection or suction will have a noticeable impact on the disturbance evolution. Therefore, exact agreement is not to be expected and we feel confident that the computation is sufficiently resolved. 5.2 Late stages When the flow has been advected downstream over a sufficient distance, the perturbation s spatial locality in the radial direction is replaced by a more global appearance. This first occurs when t 40 at z 20 and becomes more prominent as time increases. In Figure 9 (a), the rather dramatic change of the locality of the perturbation can be seen at t A low speed streak is starting to appear at z 17, and at z 26, the streak occupies large areas in the center of the pipe.

148 Direct numerical simulations of localized disturbances in pipe Poiseuille flow 11 (a) (b) Fig. 6: Streaks and vortices when (a) t 9.4 and (b) t In both figures, gray represents λ 2 = 0.5, red are high-speed streaks, v z = 0.15, and blue are low-speed streaks, v z = This is also visible in Figure 10. At z = 17, the perturbation is still mainly localized close to where the injection and suction was applied. Traveling further downstream, the perturbation gradually fills the entire cross section of the pipe. In Figure 11, iso-surfaces of high and low speed streaks show this transition from the highly localized streaks to a more globally disordered state. Figures 9 (b) and 12 show the perturbation further downstream. At this point, little or no information about the origin of the perturbation remains. Typical characteristics of turbulent pipe flow, i. e. high speed streaks at the walls and low speed streaks at the center, can clearly be seen, and we believe the resolution in the computation is sufficient also for fully developed turbulent pipe flow.

149 12 Per-Olov Åsén et al. r (a) z r (b) z r (c) z r (d) z r (e) z Fig. 7: Instantaneous contours of (V z +v z )/ r in the plane φ = π, i. e. in the plane where injection is applied at z = 0.3, at times (a) t 6.3, (b) t 9.4, (c) t 12.6, (d) t 15.7, (e) t In all figures, the contours range from 6 (black) to 1 (white) with steps of 2. 6 Conclusions In this paper, we demonstrate the possibilities of using direct numerical simulations for perturbations generated by locally large inhomogeneous boundary conditions which are highly spatially localized. In particular, we consider the simulation of a disturbance which in experiments by Mullin and Peixinho has shown to be efficient in triggering turbulence, yielding a threshold dependence of the required amplitude as R 1.5 on the Reynolds number, R. The disturbance is a combination of injection and suction through two small holes placed such that they form a 45-degree angle with the center line of the pipe, see Figure 2.

150 Direct numerical simulations of localized disturbances in pipe Poiseuille flow 13 (a) Fig. 8: The same plot as in Figure 6 (b) using a grid with N = 150, L = 300 and z = (b) Fig. 9: Axial perturbation velocity, v z, in the plane where injection is applied when t In both figures, red and blue correspond to velocities v z = 0.55 and v z = 0.67, respectively.

151 14 Per-Olov A se n et al. (a) (b) (c) (d) Fig. 10: Perturbation velocity field at (a) z = 17, (b) z = 20, (c) z = 23 and (d) z = 26 when t Colors represent axial velocity ranging from vz = 0.6 (red) to vz = 0.6 (blue). The spacing between contour lines is 0.1 and the zero contour is dashed. We present numerical results for R = The results show an initial formation of hairpin vortices. The vortices are ordered in a structured way, each having an approximate length of one pipe radius. This is in good agreement with the experiments of Mullin and Peixinho. The formation of a local perturbation in the form of hairpins and streaks is subsequently followed by transition to a globally disordered state after approximately 10 pipe diameters. We have performed simulations using different resolutions, thus indicating that the initial formation of hairpins is correct. After breakdown, the large structures of the flow show good agreement with well-known characteristics of turbulent pipe flow. We therefore believe that the resolution used is sufficient also for fully developed turbulence at this Reynolds number. A transition scenario via formation of hairpin vortices is well established in boundary layers, see e. g. the review article [4]. Also in pipe flow, both experimental [7] and numerical [28] results have verified such a scenario. However, previous numerical results have been for perturbations generated by disturbances which are not so localized in the azimuthal direction. Our results confirm the experimental results by Mullin and Peixinho, i. e. that hairpin vortices play a central role in transition to turbulence for the specific disturbance used in these experiments.

152 Direct numerical simulations of localized disturbances in pipe Poiseuille flow 15 Fig. 11: Streaks at t Red are high-speed streaks, v z = 0.15, and blue are low-speed streaks, v z = (a) (b) Fig. 12: Perturbation velocity field at (a) z = 40 and (b) z = 45 when t The color coding and contour lines are the same as is Figure 10. Acknowledgements The authors are grateful for obtaining experimental results from Professor T. Mullin and Dr. J. Peixinho prior to publication. We also thank Professor D. S. Henningson, Dr. L. Brandt and Dr. P. Schlatter for suggestions and fruitful discussions concerning the simulations. References 1. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, (1997) 2. Åsén, P.O.: A parallel code for direct numerical simulations of pipe Poiseuille flow. Tech. Rep. TRITA-CSC-A 2007:2, CSC (2007) 3. Åsén, P.O., Kreiss, G.: Resolvent bounds for pipe Poiseuille flow. J. Fluid Mech. 568, (2006) 4. Bowles, R.I.: Transition to turbulent flow in aerodynamics. Phil. Trans. R. Soc. Lond. A 358, (2000) 5. Draad, A.A., Kuiken, G.D.C., Nieuwstadt, F.T.M.: Laminar-turbulent transition in pipe flow for Newtonian and non- Newtonian fluids. J. Fluid Mech. 377, (1998)