The Modified Cumulant Expansion For Isotropic Turbulence At Large Reynolds Numbers

The energy spectrum equation due to the modified zero-fourth-cumulant approximation is solved numerically for large Reynolds numbers R =10 4 ∼10 6 and dealt with analytically. The energy spectrum is found to satisfy different similarity laws in three wavenumber ranges. In the energy-containing range k = O (1), it satisfies an inviscid similarity law and takes the k -5/3 intertial form at higher wavenumbers. In the intermediate range k = O ((ν t ) -1/2 ), it takes the k -2 and k -1 forms at lower and higher wavenumbers respectively. In the energy-dissipation range k = O ( ε 1/4 ν -3/4 ), it satisfies Kolmogorov's similarity law and takes the asymptotic form exp (- b k ) for k →∞. The energy, the skewness, the microscale and the microscale Reynolds number are calculated numerically, and their similarity laws are derived analytically.