Conditional scalar mean homogeneous flow

If the conditional fluctuations p p are neglected, the homogeneous conditional scalar mean Q(C t) = ( jpIO is governed by (Klimenko 1990 Bilger 1993) (summation is implied with respect to j and k) 

Since (5.299) is solved in mixture-fraction space, the independent variables are bounded by hyperplanes defined by pairs of axes and the hyperplane defined by XS K, = 1. At the vertices (i.e., V = = (0, and e, (/el. AW)), where e, is the Cartesian unit vector for the /th axis), the conditional mean reaction-progress vector is null 121 

On the other hand, on the bounding hypersurfaces the normal diffusive flux must be null. However, this condition will result naturally from the fact that the conditional joint scalar dissipation rate must be zero-flux in the normal direction on the bounding hypersurfaces in order to satisfy the transport equation for the mixture-fraction PDF.122 

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For a non-premixed homogeneous flow, the initial conditions for (5.299) will usually be trivial Q(C 0 = 0. Given the chemical kinetics and the conditional scalar dissipation rate, (5.299) can thus be solved to find ((pip 0- The unconditional means (y>rp) are then found by averaging with respect to the mixture-fraction PDF. All applications reported to date have dealt with the simplest case where the mixture-fraction vector has only one component. For this case, (5.299) reduces to a simple boundary-value problem that can be easily solved using standard numerical routines. However, as discussed next, even for this simple case care must be taken in choosing the conditional scalar dissipation rate. 

This Appendix supplements Section 2.4. The problem is to find the mean concentration field a(x, t) for an arbitrary scalar entity, given a turbulent velocity field u(x, f), a specified source density homogeneous initial and boundary conditions a = 0 on the outer boundary Sq of a region V. Hence, all scalar is introduced into the flow by the sources (p within V. We consider an ensemble of realizations of the turbulent flow, denoted by a superscript co, so that a" , u" and (p " =

Peclet number limit, the relationship between a ", u" and (p is given by Eq. (14), here rewritten as... 

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