Mean Reynolds-stress closure

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. 

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. 

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. 

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. 

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... 

The k,E-model is based on a first order turbulence model closure according to Boussinesq. In analogy to laminar flows, the Reynolds stresses are assumed to be proportional to the gradients of the mean velocities. Transport equations for the turbulent kinetic energy and the turbulent dissipation are developed from the Navier-Stokes equations assuming an isotropic turbulence. The implementation of this model and the parameters used can be found in [10],... 

The averaging process creates a new set of variables, the so-called Reynolds stresses, which are dependent on the averages of products of the velocity fluctuations UjUj (which for i = j simply represent the standard deviations of the velocity components). This creates a closure problem, which is one of the fundamental issues that has to be addressed in the modeling of turbulent flows. Importantly, Equation 3.2.12 also indicates that the Reynolds stress terms, which in line with Taylor s fundamental result should be related to the dispersion parameters, are coupled to the gradients of the mean flow velocity. 

RANS, under which the Reynolds-averaged Navier Stokes equations are solved using some type of closure assumption to account for the Reynolds stress terms. RANS provides the values of the mean wind velocity and estimates of the turbulence statistics within the model domain. 

All numerical models incorporate significant assumptions and approximations, and their predictions must always be regarded as estimates. Solution of the RANS equations, for example, requires some form of closure assumption dealing with the Reynolds stress terms. Since the Reynolds stress terms and the mean flow terms are coupled by the equations, inaccuracies in the closure approximations can affect the predicted mean flow fleld. Furthermore, the boundary conditions imposed on the model require the assumption of velocity profiles and momentum transport rates, which may themselves be approximated. Similar approximations are inherent in any of the various techniques used to compute the wind fleld, with further assumptions being present in each of the dispersion models. 

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. 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

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