The Decay of Isotropic Temperature Fluctuations in an Isotropic Turbulence

Abstract

The approach of von Karman and Howarth (to isotropic turbulence) has been applied to the decay of isotropic temperature fluctuations in a decaying isotropic turbulence, under the restriction that the temperature differences are too small to have any effect on the velocity field. From the correlation equation it is found that, if the scalar functions characteristic of the pertinent double and triple correlations are assumed to descend to zero faster than r~, the product of the second moment of the double temperature correlation coefficient and the mean square temperature fluctuation is a constant during decay. This is analogous to Loitsiansky's theorem for the constancy of the product of turbulent energy and fourth moment of the velocity correlation coefficient in the turbulence decay. For extremely small Peclet Number the correlation equation reduces to the three-dimensional heat conduction equation with spherical symmetry. The corresponding equation for the linear viscous decrement of the velocity field is a quasi-five-dimensional equation. Estimates of decay and scale growth of the temperature fluctuations a t both extremely small and extremely large Peclet Numbers are compared with the corresponding results for the turbulence. Perhaps the most practical result is that the temperature spottiness always seems to die off at a slower rate than the velocity spottiness.