The Decay of Isotropic Turbulence Preserving a General Type of Self Similarity

Abstract

In this paper, we have investigated the decay process of the kinetic energy density spectrum in a homogeneous isotropic turbulence. It has been shown that the spectral equation for the energy density spectrum E(k) with modified form of energy transfer spectrum due to Obukhov (Compt. Rend. Acad. Sci. U. R. S. S 32:22–24, 1941) and as restricted to early-period decay process, admits of a class of self-preserving solution. Such solutions are identified by their asymptotic behaviour e.g, $E(k) \sim k^{\frac{2-3c}{c}} ( k \rightarrow 0)$ where $c( < \frac{2}{3})$ is a parameter and $E(k) \sim k^{-\frac{5}{3}} ( k \rightarrow \infty )$. Numerical computations of some selective spectra, corresponding to values of c e.g., $c=\frac{1}{2}, \frac{2}{5}, \frac{1}{3}$ and $\frac{2}{7}$ are then accomplished over the entire range of wave numbers concerned (excluding the viscous dissipation range). We attempt to find a class of non viscous self-preserving solution using the modified Obukhov form (cf Hinze An introduction to its mechanism and theory, 1975) for the spectrum function F(k, t) and compute them numerically. Finally, stability analysis is carried out for the above mentioned self-preserving spectra and it is shown that they represent different degrees of unstable situations.