Effects of Compressibility in Free-Shear Flows. Observations

To understand and model compressibility-induced effects on turbulence is an important topic, as these effects are significant in many engineering applications, particularly in the fields of propulsion and supersonic aerodynamics, which are concerned with jets or wakes subjected to large velocity and density gradients. The compressible plane mixing layer is a generic problem for these applications, and explaining and modeling how compressibility reduces turbulent mixing in a shear layer has motivated the large research effort devoted to this topic during the 1980s and 1990s. Mixing usually refers to interpenetration of two streams. It is characterized by two scales: a length scale δ and a velocity scale ΔU, which evaluate the thickness of the interface and intensity of the fluctuations, respectively. The reduction of mixing by compressibility is illustrated in Fig. 10.1 in which δ is the previously mentioned thickness and Mc = ΔU/a is the convective Mach number, with a the average speed of sound.

There is now a consensus in the literature that the “intrinsic compressibility” (nonzero-velocity divergence in Mach-number-dependent flows) of a turbulent velocity field tends to reduce the amplification rate of turbulent kinetic energy produced by mean-velocity gradients, with respect to the solenoidal case.

Scope of the Book

Turbulence is well known to be one of the most complex and exciting fields of research that raises many theoretical issues and that is a key feature in a large number of application fields, ranging from engineering to geophysics and astrophysics. It is still a dominant research topic in fluid mechanics, and several conceptual tools developed within the framework of turbulence analysis have been applied in other fields dealing with nonlinear, chaotic phenomena (e.g., nonlinear optics, nonlinear acoustics, econophysics, etc.).

Despite more than a century of work and a number of important insights, a complete understanding of turbulence remains elusive, as witnessed by the lack of fully satisfactory theories of such basic aspects as transition and the Kolmogorov k-5/3 spectrum. Nevertheless, quantitative predictions of turbulence have been developed. They are often based on theories and models that combine “true” dynamical equations and closure assumptions and are supported by physical and – more and more – numerical experiments.

Homogeneous turbulence remains a timely subject, even half a century after the publication of Batchelor's book in 1953, and this framework is pivotal in the present book. Homogeneous isotropic turbulence (HIT) is the best known canonical case; it is very well documented – even if not completely understood – from experiments and simple models to recent 4096 full direct numerical simulation (DNS).

Description and knowledge of turbulent flows is advancing well, particularly with the increasing development of numerical resources (Moore's law) and detailed measurements using more and more particle image velocimetry (PIV), stereoscopic particle image velocimetry (SPIV), and particle tracking velocimetry (PTV). Well-documented databases are created that can support techniques of data compression using a dramatically reduced number of modes (POD, wavelet coefficients, master modes, etc.).

Behind this attractive show window, however, the advance of our conceptual understanding of turbulent flows is much less satisfactory. Advances in numerics, experiments, data-compression schemes, are first beneficial to applied studies, for instance those using a smart combination of techniques (often referred to as multiphysics, with hybrid RANS–LES methods, and many others). Turbulent flows are well reproduced in the vicinity of a well-documented “design-point,” but this modeling is questioned far from it (“far” in the parameter's space, or simply in elapsed time for unsteady processes). Efficiency of data-compression schemes, for instance, is ellusive because a low-dimension set of modes, identified and validated near the design-point, can lose its relevance far from it.

We hope that this book will contribute to an honest and up-to-date survey of turbulence theory, with the special purpose of reconciling different angles of attack.

This chapter is devoted to the analysis of the interaction of an initially isotropic turbulence with a normal plane shock wave. Even though this case is very simple from a geometrical viewpoint, it will be seen that it involves most physical mechanisms observed in more complex configurations. It also makes it possible to carry out an extensive theoretical analysis, leading to a deep understanding of the underlying physics.

Brief Survey of Existing Interaction Regimes

Several interaction regimes exist, which can be grouped into two families. The first one, referred to as the destructive interaction family, encompasses all configurations in which the structure of the shock wave is deeply modified during the interaction in the sense that a single well-defined shock wave can no longer be identified, the limiting case being the shock destruction. The second family, i.e., the nondestructive interaction family, is made up of all cases in which the structure of the shock wave is preserved during the interaction. It is important to note that, in the latter case, the shock wave can be strongly corrugated by the incoming turbulence.

Destructive Interactions

The first case of destructive interactions is that of unstable shocks, in which any small disturbances will lead to the destruction of the shock wave because of instability mechanisms.

Physical and Numerical Experiments: Kinetic Energy, RST, Length Scales, Anisotropy

Mean-shear flows are ubiquitous in turbulence. In a real flow, the shear is always created by the no-slip condition on solid walls, except when there is no tangential velocity or when the wall is a belt moving with the same velocity as the flow (shear-free boundary layer). Shear flows are therefore intimately connected with near-wall turbulence dynamics. Nevertheless, many features can be understood in the idealized case of a uniform mean shear in the absence of boundaries, in the context of HAT. The relevance of this idealized model flow was discussed by W. C. Reynolds, among many others. The effect of the wall is to create a mean shear and to block the vertical motion. The arbitrary imposed uniform shear in the HAT framework is also responsible for a reduction of vertical velocity fluctuations (as we shall see with all details in this chapter). Therefore the presence of a wall is not mandatory.

The emphasis in this chapter is put on the departure from isotropy that is due to the application of a constant shear. The main reasons are that it contains all the physical mechanisms present in homogeneous shear flows and that it is the most extensively analyzed flow in this family.

Observations, Propagating and Nonpropagating Motion. Collapse of Vertical Motion and Layering

Turbulent flows can transport passive scalars, such as temperature or concentration. In important applications, such scalar (e.g., temperature, salinity) fluctuations generate a buoyancy force in the presence of gravity, which directly affects the velocity field. In addition, the transport of such “active” scalars by turbulence is altered by a mean-density gradient – intimately related to a mean-scalar gradient – in many applications, especially in atmospheric and oceanic research.

A first sketch of what stable and unstable stratifications are can be understood from a simple displaced-particle argument, as follows. Considering a vertical negative mean-density gradient (the heaviest flow is at the bottom), as in the scheme in Fig. 7.1, if a fluid particle is displaced upward, keeping its density and initially in hydrostatic equilibrium, it must experience a lighter fluid environment: The imbalance between (smaller) buoyancy and (same) weight will result in a downward force. The opposite situation occurs if the particle is moved downward, the imbalance buoyancy–weight will result in a upward force. Accordingly, the buoyancy force acts as a restoring force in this situation of negative mean-density gradient. Vertical oscillations with a typical frequency N are expected (as subsequently rediscussed).

A combination of system rotation and stable stratification is essential for geophysical applications, even if the former effect is significantly smaller than the latter in 3D flows, e.g., for scales much smaller than the synoptic ones in the atmosphere. As for pure rotating turbulence in Chapter 4 and purely stratified turbulence in Chapter 7, linear analysis, i.e., RDT, describes only neutral stability and will lead to the definition of both the wave-vortex eigenmode decomposition and dispersion frequencies of inertia–gravity waves in the present chapter. Nonlinear dynamics is essential, and allows us to revisit a quasi-geostrophic (QG) cascade, which generalizes the toroidal cascade discussed in Chapter 7 with additional Coriolis effects.

Other coupled effects investigated in this chapter can create linear instabilities that can be analyzed within the RDT framework. These instabilities are associated with turbulence-production mechanisms, which are the main striking new physical phenomena when compared with other flows discussed in this book. Therefore only the linear approach will be emphasized in these cases. In the presence of mean shear, barotropic instabilities occur, with a strong analogy between the rotating-shear-flow case and the stratified shear flow. A special case combining the three ingredients, namely the mean shear, system rotation, and stable stratification, is shown to give new insight into the baroclinic instability.

Main Observations

This chapter is devoted to the dynamics of homogeneous turbulent flows submitted to a pure strain. The pure strain case is defined as the case in which the mean-velocity-gradient matrix A is symmetric. As discussed in the rest of this chapter, several experimental setups have been designed during the past few decades that lead to different forms for A. Kinematic aspects, from the design of ducts in experiments to a first insight into RDT (more details are given in Chapter 13), are also introduced in the general case in which A combines a symmetric and an antisymmetric part (mean vorticity) in order to characterize in the simplest way what the specificity is of an irrotational straining process.

Both experiments and numerical simulations lead to the following observations dealing with the dynamics of homogeneous turbulence subjected to pure strain:

• The initially isotropic turbulence becomes anisotropic in the presence of a mean strain, and the principal axes of the RST become identical to those of the A, the axis of contraction for A corresponding to the direction of maximum amplification for the RST. If the strain is applied for a long enough time, anisotropy reaches an asymptotic state. Typical results are displayed in Fig. 5.1.

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