Scalar flux production term

It is imperative that the same closure for the scalar flux be used in (4.70) to find the scalar mean and the scalar-variance-production term Vj,. 

The terms involving y in the SR model equations correspond to the fraction of the scalar-variance production that falls into a particular wavenumber band. In principle, yn could be found from the scalar-flux spectrum (Fox 1999). Instead, it is convenient to use a self-similarity hypothesis that states that for Sc = 1 at spectral equilibrium the fraction of scalar variance that lies in a particular wavenumber band will be independent of V. Applying this hypothesis to (4.103)-(4.106) yields 31... 

Recall that the production term 7 in (6.285) results from the scalar flux, which is not included in (6.287).) As with the FP model discussed in Section 6.6, the doubly conditioned scalar dissipation rate must be supplied by the user. For example, the conditional scalar PDF / (i// t) generated by... 

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

Now, one of the terms that must be considered when one constructs the scalar dot product of v with convective momentum flux in the equation of motion is... 

Table 1.1 Conjugate pairs of variables in work terms for the fundamental equation for the internal energy U. Here/is force of elongation, Z is length in the direction of the force, <7 is surface tension, As is surface area, , is the electric potential of the phase containing species i, qi is the contribution of species i to the electric charge of a phase, E is electric field strength, p is the electric dipole moment of the system, B is magnetic field strength (magnetic flux density), and m is the magnetic moment of the system. The dots indicate scalar products of vectors.

The terms in this equation have physical interpretations analogous to those in the momentum flux equation (1.394), except for the additional term (i.e., the second term on the RHS), which is a production/loss term related to the mean scalar quantity gradient. Physically, this term suggests production of the scalar quantity flux when there is a momentum flux in a mean scalar quantity gradient. The turbulent momentum flux implies a turbulent movement of the fluid. If that movement occurs across a mean scalar quantity gradient, then the scalar quantity fluctuation would be expected. 

If fluid motion is restricted to the x direction in rectangular coordinates as illustrated above, then the only nonvanishing scalar of convective momentum flux is pVxVx, which has units of momentum per time per area or force per unit area. Hence, pvv contains forces per unit area that are transmitted across the surfaces that bound fluid within the control volume, and terms of this nature due to convection motion of a fluid must be included in the force balance. One must construct the product of each of the nine scalars generated from this second-rank tensor with the surface area across which the force (or stress) is transmitted. Information about these smfaces and the coordinate direction in which the force acts is contained in the subscripts of the velocity components. For the most general type of fluid flow in rectangular coordinates, the nine scalars that one can generate from convective momentum flux are... 

The convective mass flux term V pcoi, is expanded using the product rule for the V operator, which is similar to the product rule for the derivative operator, except for the fact that one must exercise caution in placing the dot between two vectors. It makes no sense to calculate the scalar (i.e dot) product of a vector and a scalar. If p< , v is viewed as a product of < , and pv, then ... 

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