Preface vii
Notations ix
Chapter 1. General Points 1
1.1. Introduction 1
1.2. Tools to analyze and develop control strategies 2
1.2.1. Numerical simulations 2
1.2.2. Sensors 3
1.2.3. Actuators 20
Chapter 2. Summary of the Main Characteristics of Wall Turbulence 23
2.1. Introduction 23
2.2. General equations 23
2.2.1. Eulerian relations 24
2.3. Notations 25
2.4. Reynolds equations 26
2.5. Exact relations and FIK identity 27
2.6. Equations for a turbulent boundary layer 32
2.7. Scales in a turbulent wall flow 34
2.8. Turbulent viscosity closures 35
2.9. Turbulent intensities of the velocity components 47
2.10. Vorticity and near wall coherent structures 51
Chapter 3. Passive Control 65
3.1. Introduction 65
3.2. Large eddy (outer layer) breakup devices, LEBUs (OLDs) 66
3.2.1. General 66
3.2.2. Alteration of the inner structure by outer layer devices 67
3.3. Riblets 72
3.3.1. General 72
3.3.2. Effect of the riblets on the fine structure of wall turbulence 76
3.3.3. Effect of the protrusion height 84
3.4. Superhydrophobic surfaces 93
Chapter 4. Active Control 99
4.1. Introduction 99
4.2. Local blowing 100
4.3. Ad-hoc control 107
4.4. Transverse wall oscillations 115
4.5. Alternated spanwise Lorenz forcing and electromagnetic (EM) control 123
4.6. Extensions of spanwise forcing 131
4.7. Reynolds number dependence 132
4.8. Suboptimal active control 134
4.9. Optimal active control 143
4.10. Optimal linear control 147
4.11. Neural networks 156
4.12. Stochastic synchronization of the wall turbulence and dual control 157
Bibliography 167
Index 185
1
General Points
1.1. Introduction
The active control of wall shear stress (skin friction drag) in turbulent flows is a crucial industrial problem. For pipe and channel flows, 100% of the drag is due to skin friction. At subsonic cruising speeds, approximately half of the total drag of conventional aircraft and 90% of the total drag over an underwater vehicle are due to wall shear stress. A drag reduction of a few percent in civil aerodynamics, for instance, results in several billion dollars of fuel being saved, with a direct environmental impact. According to Kim [KIM 11], reducing fuel consumption by 30% through a drag-reducing control scheme would result in a saving of $38 billion a year for shipping industries. Fifty percent of aviation-related transport energy requirement is related to the turbulent skin-friction on commercial airliners [GAD 00] and the aviation consumes up to 13% of all energy used for transport. A cut in skin friction drag by 20% applied only to all commercial aircrafts operating in the European Community would prevert several million tons of CO2 emissions annually. Successful passive and active turbulent skin-friction control strategies can thus lead to substantial financial and environmental benefits.
The tools necessary to understand near wall turbulence physics and to develop efficient control strategies are the main subjects of this introductory chapter. We first discuss some issues arising from numerical simulations of fully developed turbulent wall flows. This is followed by a short discussion dealing with the progress made during recent decades in the realization of microsensors and microactuators.
1.2. Tools to analyze and develop control strategies
1.2.1. Numerical simulations
The analysis and development of active or passive control strategies requires the detailed resolution of the near wall turbulent flow field. Well-resolved direct numerical simulations (DNSs) constitute an essential tool for these aspects. The publication of the first DNS of a fully developed turbulent channel flow goes back to 1987 [KIM 87]. These data were generated in a small computational domain, and at a low Reynolds number of Here, h is the channel half width, is the shear velocity and ? is the kinematic viscosity. The quantities scaled by the inner variables are denoted as ( )+ hereafter. To our best knowledge, the last DNSs realized in a large domain, of sizes Lx = 8ph in the streamwise and Lz = 3ph in the spanwise directions, have recently been reported by Lee and Moser [LEE 15] and reached Ret = 5200 corresponding to where is the bulk velocity. The computations were realized on an IBM BlueGene/Q system and have an excellent scalability up to 786,432 cores. The entire simulation was conducted over 9 months. Thus, the Reynolds number resolved by DNS increased by a factor of 30 in three decades, due to the progress in scientific computing technology.
However, the DNS of Lee and Moser [LEE 15] generated 140 Terabytes of data [LEE 14]. The management and analysis of this amount of data is delicate, and even the postprocessing requires massively parallelized codes and the use of supercomputers. This explains why the vast majority of the studies dealing with active and passive control and using DNS is limited to low Reynolds numbers of about Ret = 200, as we will see in the following chapters. Trying to cope with this problem by recourse to large eddy simulations (LESs) is obviously hopeless. The main reason is that, not only is the optimal control scheme only acceptable at the subgrid-scale in this case, but in addition the strategies developed by this approach are viable only if the near wall coherent structures are adequately resolved, which is not the case even if recently developed dynamic subgrid models are used. The same Re number limitation is also inherent in experimental control investigations, mainly because of the necessity for performing measurements very close to the wall.
The Reynolds number dependence of the control strategies is undoubtedly an important issue. The effect of the large-scale outer structures on the wall shear stress turbulent intensity is relatively well understood by now (see [TAR 14] and the references within), but the role they play in the regeneration of the mean wall shear stress (drag) is unclear at the moment. Regarding the use of the direct numerical simulations, the DNS around Ret = 1000 obtained in large computational domains seems to be a good compromise. We will, however, see later on that the majority of existing active-passive control investigations are realized in much lower Reynolds number nowadays.
1.2.2. Sensors
The technical implementation of any drag active control method necessitates the use of microsensors and microactuators. Micro system technology has developed tremendously over the last decades. We will shortly review some aspects of microsensors and actuators in this section, which is mainly based on [TAR 10a]. The reader is referred to [GAD 05] for further details.
1.2.2.1. Pressure sensors
Some quantities in near-wall turbulence appear, at first glance, to be simple to determine, quantify and analyze, but are actually phenomenally complex once we get into the details. Although one of these quantities is the wall shear stress, the second is undoubtedly the pressure field. The fluctuating pressure field is directly linked to the hydro/aero-acoustic noise, as clearly shown in the classic analogy of Lightill [LIG 52, LIG 54]. The instantaneous pressure gradient has a direct influence on the structure of the wall flow in a turbulent boundary layer. The information contained in the pressure is global, as it is a volume integral containing instantaneous velocity gradients and a surface integral of the shear. Pressure/velocity correlations, in rapid and slow terms, play a vital role in transport equations [MAN 88, TAR 12].
The local instantaneous pressure gradients are related to the fluxes of vorticity at the wall through
[1.1] where the subindex 0 refers to the wall as usual and ?x and ?z are the local instantaneous streamwise and spanwise vorticity components. The wall normal vorticity flux at the wall is zero, but its flux can be manipulated through
An adverse local pressure gradient at the wall plays the role of a sink of vorticity. Micropressure sensors adequately distributed at the wall can thus clearly be incorporated in any control active scheme to manipulate the vorticity fluxes at the wall.
It is, however, difficult to use the pressure information itself in an active control scheme unless we consider actuators and sensors distributed in the flow, or at the wall, which would specifically act on global characteristics. In isotropic homogeneous turbulence, the intensity of pressure fluctuations is linked to kinetic energy by a function that depends slightly on the turbulent Reynolds number. This is given by
[1.2] where Re? is the Reynolds number based on the Taylor scale, is the kinetic energy and ? is the density [HIN 75]. The constant f(Re?) varies between 0.6 and 0.8. The estimation of temporal and spatial scales of pressure fluctuations is difficult in inhomogeneous and anisotropic flows. Nevertheless, upper limits can be determined in wall flows using the previous relation and the maximum kinetic energy in the internal layer. We therefore have
where we have assumed that at the distance y+ = 12 from the wall, and that C = 0.75. The constant C increases with the Reynolds number, varies between 0.5 and 1, and we consider its median. If to simplify, we opt for a Gaussian distribution of the pressure fluctuations, the sensor has to be able to depict values up to ±10 p', implying a sensitivity of the sensor of about
In isotropic homogenous turbulence, the spatio-temporal pressure correlations implicitly contain the Taylor scale ? related to the longitudinal velocity correlations [HIN 75, p. 309]. The p length scale must be somewhat linked to ?, and the most critical value in the wall layer is the Taylor scale at the wall [ANT 91]. In fact, in the viscous sublayer, dissipation is approximately equal to where we used the asymptotic value of at the wall. The corresponding Taylor scale in the wall normal direction varies, as when y+ 0. The intensity of the wall pressure fluctuations in the small-scale wave number range is inversely proportional to the thickness of the viscous sublayer. The resolution of the inner layer in its totality clearly requires probes that have a dimension d+ that should not exceed five wall units. This statement is in agreement with the conclusion of [GAD 94], involving general Reynolds number dependency of sensor sizes in wall turbulent flows. In order to better describe the effect of probe size on resolution, Figure 1.1 shows the distribution of the intensity of pressure fluctuations reported to the wall shear-stress versus the size of the probe d+, in the range 4. 103 = Re? = 5.103. We notice a large, inevitable decrease in pressure fluctuations for sensors whose size exceeds the thickness of the viscous sublayer. This decrease is clearly due to the spatial averaging effect of active and passive eddies of the buffer layer.
Figure 1.1. The sensor size effect on the measured pressure...
