Velocity Field Closure

The equations for the mean-velocity field Ui and pressure P in an incompressible fluid with constant density and viscosity are 

Closure is obtained through assumptions that relate the Reynolds stresses Rii to properties of the mean velocity field Ui. The most productive approach has been to use a consitutive equation involving a turbulence length scale, usually called the mixing length. A generalization of the 

if the spatial distribution of I is assumed, Eqs. (1) and (2) form a closed system for the variables 7 and P - - pgV3. Note that the combination of pgV3 with P means that g need not be evaluated. 

Another closure approach used at this level is generalized as 

Mellor and Herring (M2), observing that Eqs. (2) or (5) imply that the Reynolds-stress deviations from Jg 5,y are proportional to the strain rates (and hence that the principal axes of the stress deviation and strain rate are aligned), call these closures Newtonian. Accordingly, we denote them by MVFN. The success of the Newtonian model is remarkable, especially since for even the weakest of turbulent shear flows the principal axes are not aligned (C4). 

---

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

In summary, the mean velocity field (U) could be found by solving (2.93) and (2.98) if a closure were available for the Reynolds stresses. Thus, we next derive the transport equation for lutu ) starting from the momentum equation. 

We shall see that transported PDF closures forthe velocity field are usually linear in V. Thus (/ D) will depend only on the first two moments of U. In general, non-linear velocity models could be formulated, in which case arbitrary moments of U would appear in the Reynolds-stress transport equation. 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-empirical models to express the Reynolds stresses in terms of time-averaged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. 

Another approach that has promise for study of turbulence structure is the fluctuating velocity field (FVF) closure, adopted by Deardorff (D3). Using the analog of a MVF closure for turbulent motions of smaller scale than his computational mesh, Deardorff carried out a three-dimensional unsteady solution of Navier-Stokes equations, thereby calculating the structure of the larger-scale eddy motions. While it is likely that calculations of such complexity will remain beyond the reach of most for some time to come, results like Deardorff s should serve as guides for framing closure models. 

The acronyms for closure type used in this review are as follows FVF, fluctuating velocity field MVF, mean-velocity field MVFN, Newtonian MVF MTE, mean turbulent energy MTEN, Newtonian MTE MTOS, structural MTE MTEN/L, MTEN closure with dynamical length scale equation MRS, mean Reynolds-stress MRS/L, MRS closure with dynamical length scale equation. 

For simple flow geometries, as in the case of stratified two-phase flow between two parallel plates, the velocity field varies only in the perpendicular direction and expressions for the shear stresses, in terms of the phases average velocities, U, U, and the layer depth, H, can be analytically derived by solving the Navier-Stokes equations in the two-phases domains (see, for example, Hanratty and Hershman [61]). Coutris et al. showed that even for the simplest case of fully developed laminar two-layer flow the closure relations which evolve for the wall and inteifacial shear stresses are quite complicated [62]. 

We know from experience with particnlar classes of problems that it is possible to write predictive, deterministic laws for the behavior (predictive over relevant space/time scales that are nseful in engineering practice) observed at the level of concentrations or velocity fields. Knowing the right level of observation at which we can be practically predictive, we attempt to write closed evolution equations for the system at this level. The closures may be based on experiment (e.g., through engineering correlations) or on mathematical modeling and approximation of what happens at more microscopic scales (e.g., the Chapman-Enskog expansion). 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

We shall see in the next section that the conditional-diffusion closure may contribute an additional dissipation term. However, since at high Reynolds numbers die velocity and scalar fields should be locally isotropic, this term will be negligible. 

---