Reynolds transport equation

As we shall see in Chapter 4, models for izf have much in common with those used for IZij in the Reynolds stress transport equation. Indeed, as shown using transported PDF methods in Chapter 6, the model for uniquely determines the model for 7zf. 

The Reynolds-stress transport equation then becomes... 

The reduction of the turbulent-reacting-flow problem to a turbulent-scalar-mixing problem represents a significant computational simplification. However, at high Reynolds numbers, the direct numerical simulation (DNS) of (5.100) is still intractable.86 Instead, for most practical applications, the Reynolds-averaged transport equation developed in... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

Given (5.292) and the mixture-fraction PDF, the chemical source term in the Reynolds-averaged transport equation for (7) is closed ... 

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. 

The final form of the Reynolds-stress transport equation will then depend on the choice of Gij. For example, in the simplified Langevin model (SLM)... 

Numerous investigators have elaborated the theory by including bulk transport and surface diffusion of the surfactant molecules [45], or the effect of shear and dilatation surface viscosity [49], The general conclusion is that the rate of thinning predicted by Reynolds equation is too low (the reader may find the details in ref. [237]). Hopefully our case is different and we do not have to enter into these considerations (see Section 3.2). 

Within the homogeneous layer treatment, the thinning observed is decelerated with respect to Reynolds prediction (Fig. 3.35) and thus one may ignore any refinements of the Reynolds equation taking into account bulk and/or surface transport (a lower estimate of FI(/t) is thus obtained). However, there is a quantitative disagreement between the bulk viscosity and its effective value deduced from the slope of the linear part of h 2(t). 

Comparison with (1.385) shows that the equation for the mean velocity is just the Navier-Stokes equation written in terms of the mean variables, but with the addition of the term involving v -v -. Thus, the equations of mean motion involve three independent unknowns Fj, p and v -v -. This is perhaps the best known version of the closure problem. Equation (1.387) is the Reynolds equation and the term v v j is the Reynolds stress. This term represents the transport of momentum due to turbulent fluctuations. 

In this section the heat and mass transport coefficients for turbulent boundary layers are examined. In this case the model derivation is based on the governing Reynolds averaged equations. In these equations statistical covariances appear which involve fluctuating velocities, temperatures and concentrations. The nature of these terms is not known a priori and their effects must by estimated by semi-empirical turbulence modeling. The resulting parameterizations allow us to express the unknown turbulent fluctuations in terms of the mean flow field variables. It is emphasized that the Reynolds equations are not actually solved, merely semi-empirical relations are derived for the wall fluxes through the inner boundary layer. 

Reynolds-stress transport equation Models for conditional acceleration... 

The transport equations describing the instantaneous behavior of turbulent liquid flow are three Navier-Stokes equations (transport of momentum corresponding to the three spatial coordinates r, z, in a cylindrical polar coordinate system) and a continuity equation. The instantaneous velocity components and the pressure can be replaced by the sum of a time-averaged mean component and a root-mean-square fluctuation component according to Reynolds. The resulting Reynolds equations and the continuity equation are summarized below ... 

The Reynolds equations, the continuity equation, which is turned into an equation for pressure correction [10], and the transport equations for the turbulence quantities k and e, are integrated over the respective finite volume elements resulting from the discretization of the stirred tank domain. The convection and diffusion terms in the transport equations are approximated using the hybrid-scheme of Patankar [10]. The resulting algebraic equations are then solved with the aid of the commercial CFD software PHOENICS (Version 2.1). 

Similarly, we have the wj equation. Let w- equation be multiplied by wj and wj equation be multiplied by u p then, the two resulting equations are added and averaged to yield the following Reynolds stress transport equation ... 

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