On the Decay of Homogeneous Turbulence before the Final Period

Abstract

A three‐point correlation equation is used to obtain a relation for the triple correlations applicable at times before the final period. In this case the equation is made determinate by neglecting the quadruple correlations. Higher order approximations, valid at still earlier times, could be obtained by considering four‐ or five‐point correlation equations. In each case the set of equations is made determinate by neglecting the highest order correlation. Only two‐ and three‐point correlation equations are considered here. The correlation equations are converted to spectral form by taking their Fourier transforms. Expressions are obtained for the energy transfer function, which describes the transfer of energy from large to small eddies, and for the energy spectrum function which gives the contributions of the various eddy sizes to the total energy. By integrating the energy spectrum over all wave numbers (or eddy sizes), the following energy decay law is obtained: $$\overline{u^2}{\mathrm{=A(t-t}}_0{)}^{\text{−5/2}}{\mathrm{+B(t-t}}_0{)}^{\text{−7}},$$ where $\overline{u^2}$ is the mean square of the velocity fluctuation, t is the time, and A, B, and t₀ are constants determined by the initial conditions. For large times the last term becomes negligible leaving the well‐known −5/2 power decay law for the final period. Comparison of the decay law with experimental data indicates good agreement for times considerably before, as well as during, the final period.