Decay of vorticity in isotropic turbulence

Abstract

The equation describing the rate of change of the mean square vorticity in homogeneous isotropic turbulence is obtained and the terms occurring therein are discussed. A negative contribution to d͞ω 2 / dt arises from the effect of viscosity, while a positive contribution is produced by the tendency for the random diffusive motion to extend the vortex lines. This latter contribution can be related to the skewness of the probability distribution of the rate of extension of line elements of the fluid aligned in any given direction. The results of direct measurements of each of the factors appearing in the vorticity equation are then described. The measurements were made by analyzing electrically the output from a hot-wire anemometer placed downstream from a grid in a uniform stream. Both U 2 / u͞ 2 and ʎ 2 are found to increase approximately linearly with time during decay of the turbulence and their rates of change are consistent with the energy equation. The skewness factor mentioned above is approximately constant during decay, with the same value at all Reynolds numbers. It follows that the rate of increase of ͞ω 2 ¯ due to vortex extension is proportional to ( ͞ω 2 ) 3/2 , and further measurements show that the effect of viscosity has a similar dependence, so that the ratio of the two contributions to d͞ω 2 / dt remains the same throughout the decay. The viscous contribution is always the greater but the contributions tend to equality as the grid Reynolds number increases. The measurements of all terms in the vorticity equation are shown to satisfy the equation with sufficient accuracy. One of the deductions from the measurements is that the double velocity correlation function tends to a cusp at the origin as the Reynolds number increases indefinitely.