Decay of turbulence in the final period

Abstract

The final period of decay of a turbulent motion occurs when the effects of inertia forces are negligible. Under these conditions the instantaneous velocity distribution in the turbulence field may be solved as an initial value problem. It is shown that homogeneous turbulence tends to an asymptotic statistical state which is independent of the initial conditions. In this asymptotic state the energy of turbulence is proportional to t$^{-\frac{5}{2}}$ and the longitudinal double-velocity correlation coefficient for two points distance r apart is e$^{-r^{2}/8\nu t}$, where t is the time of decay. The asymptotic time-interval correlation coefficient is found to be different from unity for very large time intervals only, showing the aperiodic character of the motion. The whole field of motion comes gradually to rest, smaller eddies decaying more rapidly than larger eddies, and the above stable eddy distribution is established when only the largest eddies of the original turbulence remain. Relevant measurements have been made in the field of isotropic turbulence downstream from a grid of small mesh. The above energy decay and space-interval correlation relations are found to be valid at distances from the grid greater than 400-mesh lengths and at a mesh Reynolds number of 650. The duration of the transitional period, in which the energy decay law is changing from that appropriate to the initial period of decay to the above asymptotic law, increases very rapidly with R$_{M}$. There is a brief discussion of the criterion for the existence of final period decay, although clarification must wait until the existence and termination of the initial period of decay are better understood.