Development and analysis of a well-posed model for the turbulent dispersion tensor

The transport of a passive scalar in high-Reynolds-number turbulent flows is controlled by the flow and is independent from the passive scalar itself. Therefore, turbulent dispersion is a function of the velocity field and Reynolds stresses. On the other hand, the transport equation for a passive scalar is a linear partial differential equation; any modeled form of the equation has to preserve this property and the superposition principle.; It is shown that, in a homogeneous and high-Reynolds-number flow, an evolution equation for the dispersion tensor can be obtained from a model transport equation for the turbulent scalar flux and the definition of the dispersion tensor. This equation shows that the dispersion tensor is a function of the flow field only. It is shown that, in the limit of high Reynolds/Peclet number flows, the symmetric part of the dispersion tensor has to be a positive semidefinite tensor. This is the well-posedness condition for the dispersion tensor. It is observed that the solution of the evolution equation for the dispersion tensor, obtained from the general linear transport equation for the turbulent scalar fluxes, does not satisfy this condition in some cases. A modification to this model is proposed to prevent any solution which is not well posed.; The equilibrium assumption for the normalized dispersion tensor results in an implicit algebraic expression for the dispersion tensor. A new method is proposed to solve the implicit expression in order to obtain an explicit algebraic model for the dispersion tensor. The model constants are found using channel flow DNS data as well as turbulent boundary layer experimental data.; The present model gives good predictions of the heat transfer rates and the temperature profiles in boundary layers with constant wall temperature and with a step change in wall temperature. The model predictions show good agreements with the experimental data on channel flows with constant wall temperature and with a step change in wall temperature. The model is able to predict the variation of the heat transfer rate downstream of a two-dimensional backward-facing step reasonably well. It predicts the location and the value of the peak of the heat transfer rate very well. However, the predicted recovery in the boundary layer, downstream of reattachment, is slower than the experiments show.