The structure of isotropic turbulence at very high Reynolds numbers

Abstract

A recapitulation is first given of a recent theory of homogeneous turbulence based on the condition that the Fourier amplitudes of the velocity field be as randomly distributed as the dynamical equations permit. This theory involves the average infinitesimal-impulse-response functions of the Fourier amplitudes and employs a new kind of perturbation method which yields what are belived to be exact expansions of third- and higher-order statistical moments of the Fourier amplitudes in terms of second-order moments and these response functions.

In the present paper the theory is applied in lowest approximation (called the direct-interaction approximation) to stationary isotropic turbulence of very high Reynolds number. The characteristic wave-number $k_0 = \epsilon |v^3_0$ and Reynolds number $R_0 = v_0 k^{-1}_0 |v$ where v₀ is the r.m.s. velocity in any given direction, ε is the power dissipated per unit mass, and ν is the kinematic viscosity, are introduced. For $R^{\frac {1}{3}}_0 \gg 1$ it is found that the inertial and dissipation ranges extend over wave-numbers k satisfying k₀ [Lt ] k [Lt ] R₀k₀. The time-correlation and average infinitesimal-impulse-response functions of the Fourier amplitudes in these ranges are evaluated. They are found to be asymptotically identical and given by J₁(2v₀kτ)/v₀kτ, where τ is the time interval.

The energy spectrum in these ranges is determined by a non-linear integral equation, involving the time-correlation and response functions, which is suitable for solution by iteration. The solution is of the form E(k)/v₀v = (k/k_d) $^{- \frac {3} {2}}$f(k/k_d), where E(k) is the three-dimensional spectrum function $k_d = R^{\frac {2}{3}}_0 k_0$ is a wave-number characterizing the dissipation range, and f(k/k_d) is a universal function. In the inertial range, E(k) = f(0) (ε V₀)^½$ k^{- \frac {3} {2}}$, asymptotically. Theparameter f(0) can be obtained by quadratures, without solving the integral equation for E(k). Spectral energy transport throughout the inertial and dissipation ranges is found to proceed by a cascade process essentially local in wave-number space; the direct power delivered by all modes below k to all modes above k′ [Gt ] k is of order ε(k/k′)$ ^{ \frac {3} {2}}$ if k and k′ both lie within the inertial range. The mean-square velocity derivatives of all orders are found to be finite. For $R^{\frac {1}{3}}_0 \gg 1$ the skewness factor of the distribution of the nth-order longitudinal velocity derivative is found to have the asymptotic form $A_n R^{-\frac {1}{6}}_0$ where A_n is a universal constant.

The theory is compared with experiment and is found to be slightly better supported than the Kolmogorov theory. However, it is stressed that extreme caution must be exercised in interpreting the experimental evidence as support for either theory.

An analysis is given of the relations between the Kolmogorov theory, Heisen-berg's heuristic theory, the analytical theories of Heisenberg and Chandrasekhar, the theories of Proudman & Reid and Tatsumi, and the present theory.