Reynolds-Stress Closure

In order to compute the structure of the turbulence (i.e., the R,-,), one must employ the dynamical equations for the Rij [ q. (22) ]. This has been the subject of considerable recent interest, though only a few computational experiments have been carried out, and a truly universal general theory has yet to be established. We can expect considerable future activity on this front. 

The problem is again to set up a satisfactory closure structure for the unknown terms in the dynamical equations, here the equations for R.-,-. Some variation in approach is already evident, and interesting debate on the choices is likely over the next several years. 

Examination of Eq. (22) shows that the R, equations contain a pressure-strain-rate correlation term that vanishes in the contraction [Eq. (23)]. The effect of this term must therefore be to transfer energy conservatively between the three components Rn, R22, and R33, and it is generally believed that this transfer tends to produce isotropy in the turbulent motions. Modelings of this term should incorporate this feature. A plausible model of this term, supported somewhat by the data of Champagne et al. (C4) is 

An objection to this model rests on the observation that the fluctuating pressure field is given by a Poisson equation 

This suggests that the P, model should contain terms arising from interactions between the mean and fluctuating velocities, and should somehow reflect the dependence of the pressure fluctuations on distant velocity fluctuations. 

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The next formal level of closure is at the level of the dynamical equations for the turbulent stresses, which we shall call mean Reynolds-stress closures (MRS). There have only been a few experimental calculations at this level, and such closures are not yet tools for practical analysis. 

Gibson M.M. and Younis (1986) Calculation of swirling jets with a Reynolds stress closure, Physics of Fluids, 29, 38-48. 

Speziale, C.G. (1991). Analytical models for the development of Reynolds-stress closures in turbulence. Annual Review of Fluid Mechanics 23 107-157. 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

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. 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

RANS turbulence models are the workhorse of CFD applications for complex flow geometries. Moreover, due to the relatively high cost of LES, this situation is not expected to change in the near future. For turbulent reacting flows, the additional cost of dealing with complex chemistry will extend the life of RANS models even further. For this reason, the chemical-source-term closures discussed in Chapter 5 have all been formulated with RANS turbulence models in mind. The focus of this section will thus be on RANS turbulence models based on the turbulent viscosity hypothesis and on second-order models for the Reynolds stresses. 

Returning to (4.52), it should be noted that many Reynolds-stress models have been proposed in the literature, which differ principally by the closure used for the anisotropic rate-of-strain tensor. Nevertheless, almost all closures can be written as (Pope 2000)... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

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. 

Finally, it is important to reiterate that while the Reynolds-stress model requires a closure for the triple-correlation term (UiUjUk), the PDF transport equation does not ... 

As discussed above, the GLM was developed in the spirit of Reynolds-stress modeling. An obvious extension is to devise large-eddy-based closures for the conditional acceleration. For this case, it is natural to decompose the instantaneous velocity into its resolved and unresolved components 42... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

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. 

An early approach to the closure, typified by the work of Prandtl, represented the Reynolds stress tensor as... 

Turbulent flows with simple closure models (eddy viscosity, mixing length, k-e) or complex closure models (ASM, RSM, RNG) for the Reynolds stresses... 

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