Turbulent flow closure problem

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

For turbulent reacting flows, LES introduces an additional closure problem due to filtering of the chemical source term (Cook and Riley 1994 Cook et al. 1997 Jimenez el al. 1997 Cook and Riley 1998 Desjardin and Frankel 1998 Wall et al. 2000). For the one-step... 

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. 

The four equations governing two-dimensional turbulent flow are, therefore, Eqs. (2.95), (2.106), and (2.116). These contain, beside the four mean flow variables u, v, p, and T, additional terms which depend on the turbulence held. In order to solve this set of equations, therefore, information concerning these turbulence terms must be available and the difficulties associated with the solution of turbulent flow problems arise basically from the difficulty of analytically predicting the values of these terms. The set of equations effectively contains more variables than the number of equations. This is termed the turbulence closure problem . In order to bring about closure of this set of equations, extra equations must be generated, these extra equations constituting a turbulence model . 

The unsteady governing equations that apply in laminar flow also apply in turbulent flow but essentially cannot be solved in their full form at present In turbulent flow, therefore, the variables are conventionally split into a time averaged value plus a fluctuating component and an attempt is then made to express the governing equations in terms of the time averaged values alone. However, as a result of the nonlinear terms in the governing equations, the resultant equations contain extra variables which depend on the nature of the turbulence, i.e., there are, effectively, more variables than equations. There is, thus, a closure problem and to bring about closure, extra equations which constitute a turbulence model" must be introduced. 

Discuss the meaning of the following terms (i) The turbulence closure problem, (ii) A turbulence model, (iii) A boundary layer, (iv) Similar flow fields. 

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. 

So far the closure problem for the system of Reynolds equations has not been theoretically solved in a conclusive way. In engineering calculations, various assumptions that the Reynolds stresses depend on the average turbulent flow parameters are often adopted as closure conditions. These conditions are usually formulated on the basis of experimental data, dimensional considerations, analogies with molecular rheological models, etc. 

Equations (16.56) and (16.57) provide a solution to the closure problem inasmuch as the turbulent fluxes have been related directly to the mean velocity and potential temperature. However, we have essentially exchanged our lack of knowledge of u u2 and u 2B for Km and Kt, respectively. In general, both Km and Kj are functions of location in the flow field and are different for transport in different coordinate directions. The variation of these coefficients, as we will see later, is determined by a combination of scaling arguments and experimental data. 

As we have just seen, the closure problem is the fundamental impediment to obtaining solutions for the mean velocities in turbulent flows. In order to progress at all, from a purely mathematical point of view, we must obtain a closed set of equations. The simplest approach to closing the equations is based on an appeal to a physical picture of the actual nature of turbulent momentum transport. 

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