Homogeneous Turbulence Dynamics
Illustrating the main trends using computational and experimental results, this book summarizes the most recent theoretical, computational and experimental results dealing with homogeneous turbulence dynamics.
TABLE OF CONTENTS Homogeneous Turbulence Dynamics
Chapter 13: Linear Theories. From Rapid Distortion Theory to WKB Variants
13.1 Rapid Distortion Theory for Homogeneous Turbulence
The essentials for RDT were introduced in Chapter 5. More details are given in this chapter for deriving basic equations and calculating statistics. One considers the pure incompressible case in the presence of a mean flow with admissible mean-velocity-gradient matrix A ij( t). In the linear limit, the fluctuating field ( 
, p ?) satisfies modified equation (2.29) with the advection distortion parts written in terms of A ij( t):
13.1.1 Solutions for ODEs in Orthonormal Fixed Frames of Reference
The solution of Eq. (13.1) is most easily obtained by Fourier analysis, with elementary components of the form
where ? 2 = ?1. Evolution equations for the amplitudes are easily obtained from Eq. (13.1). They can be written as follows:
Time dependency of the wave vector k allows one to simplify the advection term by setting
According to the mathematical treatment of partial derivative equations, this equation also gives the characteristic lines of the operator 
in Fourier space. Because the mean trajectories are the characteristic lines of the advection operator 
, it is clear that eikonal equation (13.5) is the counterpart of 
? A ij x j in Fourier space, changing A ij into ? A ji and x into k. A more physical interpretation will be given in the following discussion.
More classically, the pressure contribution b is removed from consideration by the incompressibility constraint, which is equivalent...
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