The self-preserving decay of isotropic turbulence: Analytic solutions for energy and dissipation

Abstract

Analytic solutions for the nonlinear single-point equations, describing the isotropic decay of turbulence, as first formulated by von Kármán and Howarth [Proc. R. Soc. London, Ser. A 164, 192 (1938)] are given. The asymptotic solution indicates the self-preserving k(t)∼t−1 scaling demonstrated in the fixed-point analysis of Speziale and Bernard [J. Fluid Mech. 241, 645 (1992)]. The solutions are exact for a decay with Taylor microscale similarity in which the skewness and palinstrophy coefficients are constant. The approach to the k(t)∼t−1 decay, though not reached in finite time (mathematically speaking), is approached fairly rapidly for typical palinstrophies. An expression for the evolution of the Taylor microscale as a function of the Reynolds number and the palinstrophy is derived. Expansion of the exact solution for small time shows a slow departure from the initial condition, depending on the initial Taylor microscale Reynolds number, as seen in numerical solutions.