D. W. Dunn, W. H. Reid
The problem of heat turbulence in isotropic turbulence with a constant mean temperature gradient is considered during the final period of decay. The Reynolds and Peclet numbers are then very small, and all triple correlation terms can be neglected in the equations for the double correlations. On this basis, it is found that the temperature field ultimately becomes independent of the initial conditions on the temperature and has characteristics determined only by the mean temperature gradient, the physical properties of the fluid, and the characteristics of the turbulence. Detailed analytical and numerical results are obtained for the asymptotic state. The mean turbulent heat transfer is in the direction of the mean temperature gradient, with a magnitude proportional to the magnitude of the latter. Although it approaches zero when the Prandtl number approaches zero, its dependence on the Prandtl number is not large for Prandtl numbers of order unity and larger. This type of Prandtl number dependence is typical for many of the other results depending on both velocity and temperature fluctuations. In fact, the rate of decrease with separation distance of the two-point temperature-velocity correlation varies little over the full range of Prandtl numbers and is always about the same as it is for the double velocity correlation. In contrast, all results involving only temperature fluctuations display a strong dependence on the Prandtl number. For example, for small Prandtl numbers the double temperature correlation falls off much more slowly with separation distance than the velocity correlation does, while for large Prandtl numbers the opposite is true.
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