Mixing-Length Models for Turbulent Transport

Let us assume a steady turbulent shear flow in which u = u i (xj) and 2 = 3 = 0. We first consider turbulent momentum transport, that is, the Reynolds stresses. The mean flux of xi momentum in the X2 direction due to turbulence is pu. Let us see if we can derive an estimate for this flux. 

FIGURE 16.7 Eddy transfer in a turbulent shear flow. 

we note that, since the flow has been assumed to be steady, u does not vary with time and the corresponding derivatives are zero. Thus (16.49) becomes 

The second- and higher-order terms in (16.50) can be truncated if the distance la over which the eddy maintains its integrity is small compared to the characteristic lengthscale of the u field, leaving 

Multiplying (16.51) by u 2a, the turbulent fluctuation in thex2 direction associated with the same ath eddy is 

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The result of the mixing-length idea used to derive the expressions (16.56) and (16.57) is that the turbulent momentum and energy fluxes are related to the gradients of the mean quantities. Substitution of these relations into (16.46) leads to closed equations for the mean quantities. Thus, except for the fact that KM and Kr vary with position and direction, these models for turbulent transport are analogous to those for molecular transport of momentum and energy. 

At the next level of complexity, a second transport equation is introduced, which effectively removes the need to fix the mixing length. The most widely used two-equation model is the k-e model wherein a transport equation for the turbulent dissipation rate is... 

Equation 19.33 plays a key role in the development of two-equation turbulence models. This relationship suggests that besides the -equation, an additional transport equation for the dissipation, s, is required to make predictions for the mixing length... 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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