Scalar-conditioned velocity

The composition PDF thus evolves by convective transport in real space due to the mean velocity (macromixing), by convective transport in real space due to the scalar-conditioned velocity fluctuations (mesomixing), and by transport in composition space due to molecular mixing (micromixing) and chemical reactions. Note that any of the molecular mixing models to be discussed in Section 6.6 can be used to close the micromixing term. The chemical source term is closed thus, only the mesomixing term requires a new model. 

The only new unclosed term that appears in the composition PDF transport equation is (Ui ijy>. The exact form of this term will depend on the flow. However, if the velocity and scalar fields are Gaussian, then the scalar-conditioned velocity can be expressed in terms of the scalar flux and the scalar covariance matrix ... 

Spatial transport with scalar-conditioned velocity... 

In this section, we present results found with the scalar-conditioned velocity algorithm introduced in the previous section using the NDF representation in Eq. (8.108). All of these calculations have been done with beta EQMOM using n = 2 nodes for the first quadrature and M = 50 nodes for the second. The initial NDF, where f e [0,1] is the particle volume, has one of two forms. In the first case, referred to as small spread, the NDF is represented by two beta functions, one near = 1/3 and one near f = 2/3, with a small spread parameter cr. The initial moments for this case are shown in Figure 8.28. As can be observed from this figure, the moments are nonzero only within the spatial domain x (-0.8, -0.2). Some statistics computed from the initial NDF are given in Figure 8.29. Note that the initial conditional velocity u(0,x,f) = 1 is independent of the particle size, and the mean and standard deviation of the volume do not depend on x. 

Figure 8.28. Initial moments for the scalar-conditioned-velocity case with small spread.

Figure 8.33. Statistics for the scalar-conditioned-velocity case with small spread, tq = 10 at time t = 1. The conditional velocities are shown for five spatial locations x = -0.8, -0.4, 0, 0.4, and 0.8.

As a final case, we will again consider the joint velocity-scalar NDF governed by Eq. (8.122), but without specifying a functional form for the scalar-conditioned velocity. The moment-transport equation is again given by Eq. (8.122). However, we will now use a scalar-conditioned multivariate EQMOM to reconstruct the joint NDE ... 

---