Universal equilibrium range of turbulence

Abstract

The variational approach to the closure problem of turbulence theory, reported in an earlier paper, is applied to the study of the universal equilibrium range of turbulence. Two integral equations are derived for two unknown functions, taking account of viscous dissipation. The equation‐error method of parameter estimation of control theory is used to obtain the approximate solution of the two integral equations. The resulting energy spectrum for the universal equilibrium range is E(k)=ε^2/3k^−5/3F(k/k_d) with F(x)=1.19(1+5.3x^2/3) ×exp(−5.4x^4/3). Here ε is the energy dissipation rate and k_d is the Kolmogorov wavenumber. The corresponding one‐dimensional energy spectrum and dissipation spectrum are calculated and are in agreement with the experimental results. Since F(x) is not a monotonically decreasing function, but has a maximum near x=0.1, the usual experimental values of the Kolmogorov constant will be greater than its theoretical value F(0)≂1.2 and will depend upon the range of k/k_d.