Fluctuating velocity field closure

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. 

G.Gouesbet, A.Berlemont. Prediction of turbulent fields. Including fluctuating velocity correlations and approximate spectra, by means of a simplified second order closure scheme the round free Jet and developped pipe flow. Second Int.Conf. on Numerical Methods In Laminar and Turbulent flow, Venice, Italy, pp 205-216, Plnerldge, Swansea, 1981. 

Closure of the mean scalar field equation requires a model for the scalar flux term. This term represents the scalar transport due to velocity fluctuations in the inertial subrange of the energy spectrum and is normally independent of the molecular diffusivity. The gradient diffusion model is often successfully employed (e.g., [15, 78, 2]) ... 

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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