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cole Polytechnique Laboratoire d'Hydrodynamique (LadHyX) Thse prsente pour obtenir le grade de DOCTEUR DE L'COLE POLYTECHNIQUE spcialit : mcanique parYongyun Hwang

Large-scale streaks in wall-bounded turbulent ows: amplication, instability, self-sustaining process and control

Soutenue le 17 dcembre 2010 devant le jury compos de: M. Carlo Cossu M. Bruno Eckhardt M. Uwe Ehrenstein M. Stphan Fauve M. Patrick Huerre M. Jean-Christophe Robinet M. Pierre Sagaut Directeur de thse Examinateur Rapporteur Examinateur Examinateur Examinateur Rapporteur cole Polytechnique & IMFT, Toulouse Universit de Marburg, Germany Universits de Provence, Marseille cole Normale Suprieure, Paris cole Polytechnique, Palaiseau ENSAM, Paris UPMC, Paris

Yongyun Hwang

Large-scale streaks in wall-bounded turbulent ows: amplication, instability, self-sustaining process and control

AcknowledgementFinancial support for this work was provided by the French Ministry of Foreign Aairs through a Blaise Pascal Scholarship and from cole Polytechnique through a Gaspard Monge Scholarship. Parts of this work were done in collaboration with A. P. Willis and J. Park, and the author deeply appreciates their kind help. The use of channelflow and diablo codes are also gratefully acknowledged.

ContentsContents 1 Introduction1 2 3 4 Streaky motions in laminar and transitional ows Streaky motions in wall-bounded turbulent ows Motivations and objectives of this work . . . . . Organization of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Linear non-normal amplication of coherent streaks1 2 3 4 5 Equations for small coherent motions . . . . . . . . Optimal perturbations . . . . . . . . . . . . . . . . Base ows . . . . . . . . . . . . . . . . . . . . . . . Optimal amplications and associated perturbation Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Instability of large-scale coherent streaks1 2 3

The streaky base ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Stability of secondary perturbations . . . . . . . . . . . . . . . . . . . . . . . . 33 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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4 The existence of self-sustaining process at large scale1 2 3 4 Background . . . . . . . . . . . . . . . . . . . . . . . . The reference simulation . . . . . . . . . . . . . . . . . The numerical experiment with increased Smagorinsky Dynamics in the minimal box . . . . . . . . . . . . . .

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5 Articial forcing of streaks: application to turbulent drag reduction1 2 3 4 5 Motivation . . . . . . . . . . . . . . Direct numerical simulation . . . . . Response to nite amplitude optimal Skin-friction drag reduction . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusion and outlooki

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CONTENTS

Appendix

Numerical tools . . . . . . . . . . . . . . 1 Numerical simulations . . . . . . . 2 The optimal amplication . . . . . 3 Stability of nite amplitude streaks

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Bibliography

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Chapter 1

Introduction1 Streaky motions in laminar and transitional owsTransition to turbulence in wall-bounded ows such as plane Couette, pressure-driven channel, pipe and boundary layer ows has been an elusive problem for a long time. Traditionally, the st step is linear stability analysis, which seeks exponentially growing modes in time or space. However, the critical Reynolds numbers for linear instability do not agree with those at which transition is observed. For example, the pressure-driven channel ow is stable for Reh < 5772 (Orszag, 1971) and plane Couette ow is stable for all Reynolds numbers (Romanov, 1973). However, experiments have shown that the channel ow undergoes transition to turbulence for Reh as low as 1000 (e.g. Patel & Head, 1969) and for the plane Couette ow the transitional Reynolds number is in the range 325 < Reh < 370 (e.g. Lundbladh et al., 1992). This discrepancy between the experimental observations and linear stability analysis has led to numerous eorts to explain transition without a primary linear modal instability.

Figure 1.1. Smoke visualization of streaks in transition under the high-level free-stream turbulence in a boundary layer (from Matsubara & Alfredsson, 2001).

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CHAPTER 1. INTRODUCTION

Flow

z,opt / Gmax Rmax 5.3(a) 3.9(a) V 4.5(b) 3.5(b) 3.9(a) 3.1(a) 6.3(c) 3.3(d)

Couette Channel ow Pipe ow Boundary layer

Table 1.1. Optimal spanwise wavelengths of the maximum responses to initial perturbation Gmax , harmonic forcing Rmax , and stochastic excitation V in laminar wall-bounded ows. Results from (a) Trefethen et al. (1993), (b) Jovanovi & Bamieh (2005), (c) Schmid & Henningson (1994), (d) Butler & Farrell (1992). Here, is half of the channel height for Couette and channel ows, the radius for pipe ow, and the boundary-layer thickness for boundary layer.

Under high-level free-stream noise, transition often occurs without the linear instability waves (bypass transition). In such an environment, the streaks are often observed as a prominent feature (Kendall, 1985; Matsubara & Alfredsson, 2001), and they are shown in Fig. 1.1. The streaks consist of a spanwise alternating pattern of high and low streamwise velocity which is elongated in the streamwise direction. The appearance of streaks is now understood with the `lift-up' eect which transforms streamwise vortices into streaks by taking energy from the base ow, and it is an important process leading the large energy growth in the stable laminar ows (Moatt, 1967; Ellingsen & Palm, 1975; Landahl, 1980, 1990). This mechanism is essentially associated with the nonnormal nature of linearized Navier-Stokes operator, and the growth of the streaks has been extensively investigated in most of the canonical wall-bounded laminar ows by optimizing three types of perturbations: initial conditions (Butler & Farrell, 1992; Reddy & Henningson, 1993; Trefethen et al., 1993; Schmid & Henningson, 1994), harmonic forcing (Reddy & Henningson, 1993; Reddy et al., 1993; Trefethen et al., 1993) and stochastic excitation (Farrell & Ioannou, 1993a ,b , 1996; Bamieh & Dahleh, 2001; Jovanovi & Bamieh, 2005). The optimal perturbations leading the largest growth of the streaks are found to be almost uniform in the streamwise direction with a well-dened band of amplied spanwise wavelengths. Table 1 reports the spanwise wavelengths of the optimal streaks, which have been shown to correspond well to the ones observed in transitional ows (Matsubara & Alfredsson, 2001). When the streaks reach suciently large amplitudes via the lift-up eect, they can sustain the growth of secondary perturbations. This secondary growth appears through inectional modal instability (Walee, 1995; Reddy et al., 1998; Andersson et al., 2001) or secondary transient growth (Schoppa & Hussain, 2002; Hpner et al., 2005; Cossu et al., 2007), and is often dominated by the sinuous mode originated from the spanwise shear of the streaks. The secondary growth leads the breakdown of the amplied streaks and the ow eventually develops into the turbulent state. The development of streaks is a crucial element for the bypass transition, but wellcontrolled streaks are also found to be useful for delaying transition. Cossu & Brandt (2002, 2004) have shown that moderate amplitude streaks stabilize Tollmien-Schlichting waves in laminar boundary layers. This theoretical prediction was recently conrmed by the experiment of Fransson et al. (2004, 2005), and Fransson et al. (2006) have shown that the transition via the Tollmien-Schlichting wave can be delayed by articially driven streaky ows.

2. STREAKY MOTIONS IN WALL-BOUNDED TURBULENT FLOWS

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2 Streaky motions in wall-bounded turbulent owsTurbulent ows are phenomenologically much more complicated than transitional ows due to their multi-scale nature, and wall-bounded turbulent ow are even less understood than the other important canonical ows such as free shear ows or isotropic turbulence (Jimnez, 2007). The main reason essentially stems from the presence of the wall, which connes the size of the energy-containing large eddies to their wall-normal locations. Therefore, the length scale of those large eddies varies with the wall-normal location, and this lays at the core of the complexity of wall-bounded turbulent ows.

2.