Realizability of the rapid distortion theory spectrum: The mechanism behind the Kelvin–Townsend equations

Abstract

In the context of homogeneous turbulence, we prove that the Rapid Distortion Theory (RDT) model for the spectral tensor preserves the symmetry, positive semidefiniteness, and integrability properties required in Cramér’s characterization of the spectral tensor of a continuous homogeneous random process. From this, a statistically valid correlation tensor is obtained that returns a Reynolds stress tensor model that satisfies realizability conditions. The number of hypotheses used is kept to a minimum, which allows a flexible use of the model in the applications. The Kelvin–Townsend equations allow us to construct the solution and prove its properties by means of a factorization approach. Since the RDT spectral tensor model is a system of transport equations plus an algebraic restriction due to incompressibility, we deal with the existence, uniqueness, and persistence of solutions in a specific set of functions by using DiPerna–Lions renormalization techniques.