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The decay of isotropic turbulence preserving a general type of self similarity. (English) Zbl 1392.76018

Summary: 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 A. M. Obukhov [“Spectral energy distribution in a turbulent flow”, C. R. (Dokl.) Acad. Sci. URSS 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 [J. O. Hinze, An introduction to its mechanism and theory. New York, NY: McGraw-Hill (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.

MSC:

76F05 Isotropic turbulence; homogeneous turbulence
76F20 Dynamical systems approach to turbulence
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