R. A. Antonia

Statistical Averaging in Variable Density Fluid Turbulent Motion

This chapter is devoted to deriving statistical transport equations governing averaged single point properties of the flow field. It is only concerned with first (mean) and second order moments. Two different formulations will be mainly considered, referring to mean mass-conservative and non conservative evolutions. It is not intended for extensive and detailed presentations of the different formalisms — using either the classical formulation of the equations (“standard”, “mass-weighted” averaging,...) or the specific volume formulation —, but rather focuses on physical interpretation of density fluctuation correlations, according to mass conservative and non conservative mean flow analysis.

The High-Speed Turbulent Shear Layer

In high speed flows, significant density fluctuations can be generated by the flow itself. In this Chapter, compressibility effects in free shear flows with uniform and non uniform density are discussed in detail The free shear layer is choosen as a useful benchmark for evaluating such effects, since, unlike the jet, neither the Mach number nor the density ratio are decreasing with distance. Recent DNS results are used to illustrate important aspects of the compressible free shear layer, including Mach number effects and variable density effects.

Variable Density Fluid Turbulence: Preamble

Turbulent flows of variable density fluids are widely present in various domains of human activity and natural environment. In actual fact, density changes arise in low-speed flows and high-speed motions as well. Aeronautics is probably one of the most commonly quoted situations, since compressible subsonic, transonic, and supersonic motions are present in many applications, such as high-speed aircraft flight or supersonic combustion ramjet engines, for instance. However, as far as industrial applications involving fluids motions are concerned, variable density fluid flows are part of chemical engineering, thermal engineering, energetic, etc.

Examples of Variable Density Effects in Turbulent Flows

To give a flavor of the subject, the two-fold goal of this chapter is: (i) to provide some illustrative examples of the rather wide range of situations concerned with the matter of the monograph, (ii) to bring out some quantitative information about some salient features of variable density effects in turbulence.

Relative Behaviour of Velocity and Scalar Structure Functions in Turbulent Flows

This chapter reviews in a critical manner the existing analytical framework for describing the behaviour of velocity and scalar structure functions in turbulent flows. The assumptions which underpin this framework are only likely to be validated at very large Reynolds numbers and for relatively homogeneous and isotropic flows. These conditions are unlikely to apply in the laboratory. The major emphasis is on the likely dependence of second-order structure functions (or equivalently spectra) on both the Taylor micro-scale Reynolds number Rλ and other parameters, such as the large scale anisotropy or the dissipation timescale ratio or, more generally, the initial conditions of the flow. Measurements strongly indicate that the influence of Rλ and of the other parameters cannot be ignored. The retention of the non-homogeneity of the flow in the Navier-Stokes and heat transport equations provides a better idea of how large the magnitude of Rλ should be before the “asymptotic” results of Kolmogorov and Yaglom may be attained. Special attention is given to a suitable framework which allows velocity and scalar fluctuations to be compared meaningfully. The analogy between scalar and energy structure functions (or spectra) appears to work well for flows with a continuous injection of turbulent energy and scalar variance.

First-Order Modeling

This chapter opens the last part of the monograph, which is more specifically dedicated to an engineering audience. It begins with a general presentation and discussion of prediction methods for turbulent flows, based on statistical — or Reynolds — averaged Navier-Stokes equations (RANS). Then, the incidence of density changes and the incorporation of variable-density and compressibility effects in first-order closure models are analyzed with respect to (i) “modifications” to incompressible schemes and (ii) introduction of additional “specific contributions” to non-constant density flows. At last, some zero-, one-, two- and three-equation models are reviewed.

The Structure of Some Variable-Density Low-Speed Shear Flows

This chapter focuses on the influence of density contrasts on the development of some basic low-speed shear flows. The specific features of these variable-density flows are best accounted for as seen from their vorticity dynamics. The baroclinic torque, connecting misaligned pressure and density gradients, reorganizes the vorticity field according to the fluid inertia. It is introduced after a short literature survey. Then the particular cases of the mixing layer and the jet are examined. The two-dimensional and some three-dimensional aspects are documented, based on temporally and spatially developing numerical simulations.

Approximate Models for Variable Density Fluid Motions

This chapter aims at giving a comparative overview of some of the various models which, derived from the general Navier-Stokes equations, account for density variations according to several types of approximations. The role of the pressure is first examined. The Helmholtz decomposition is introduced and the linear analysis of Kovasznay compressible modes is presented. Then several models are discussed, referring to Boussinesq’s approximation, and other approximations which are concerned with (i) filtering acoustic effects, (ii) incorporating density variations in pseudo-incompressible formulations and (iii) deriving weakly compressible limits to the general compressible equations.

General Equations and Classification of Variable Density Fluid Motions

The first goal of this chapter is to recall the “general, instantaneous, local” equations governing variable density fluid motions, according to the classical approach of continuum thermo-mechanics and using the local equilibrium hypothesis. On this basis, the classical numbers associated with density changes in a fluid motion are introduced and the departure from the solenoidal condition is discussed. The final part of the chapter is devoted to the presentation of a general time-scale analysis which is relevant to distinguishing and classifying several turbulent flows of variable density fluid.

Some Basic Variable Density Mechanisms in Turbulent Flows

The aim of this chapter is to point out the existence and emphasize the understanding of some of those properties which make variable density turbulent flows different from the incompressible ones.