Mean velocity field transport 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. 

Finally, using the fact that the mean velocity field is solenoidal, the RANS mean velocity transport equation reduces to... 

However, the mean velocity field will depend on conditional moments of the fluctuating velocity through the mean velocity transport equation,... 

C) and Cgm denote the mean concentration in the occupied zone, concentration at a given point P, the mean concentration in the room, and the concentration at the outlet, respectively. To numerically simulate these parameters, the velocity field is first computed. Then a contaminant source is introduced at a cell (or cells) of a region to be studied, and the transport equation for contaminant C is solved. The transport equation for C is... 

Note that when solving the CFD transport equations, the mean velocity and turbulence state variables can be found independently from the mixture-fraction state variables. Likewise, when validating the CFD model predictions, the velocity and turbulence predictions can be measured in separate experiments (e.g., using particle-image velocimetry [PIV]) from the scalar field (e.g., using planar laser-induced fluorescence [PLIF]). 

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

The mean pressure field (p) appears as a closed term in the conditional acceleration (A, V, ip). Nevertheless, it must be computed from a Poisson equation found by taking the divergence of the mean velocity transport equation ... 

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. 

The Reynolds transport terms (i.e., the terms of the kind v Vy, T v y and c v y) are the mean rates of transport of momentum, heat, and mass across the corresponding CV-surfaces of the balanced element by the turbulent velocity fluctuations. This system of equations can not be solved because these terms are not known. We therefore need additional information regarding the Reynolds transport terms. We may postulate a model for the momentum transport terms, intending to relate the Reynolds transport stresses to the mean flow field variables. 

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]) ... 

Let us temporary assume that the velocity and density fields are known, hence F and Fg can be determined. By that means (12.88) reduces to a transport equation for the property with only one unknown variable. However, in order to solve the convection-diffusion equation we need to approximate the transport property V at the e and w faces. A few classical convec-tion/advection schemes are outlined in the subsequent sections. 

For pressure-based techniques, the lack of an independent equation for the pressure complicates the solution of the momentum equation. Furthermore, the continuity equation does not have a transient term in incompressible flows because the fluid transport properties are constant. The continuity reduces to a kinematic constraint on the velocity held. One possible approach is to construct the pressure field so as to guarantee satisfaction of the continuity equation. In this case, the momentum equation still determines the respective velocity components. A frequently used method to obtain an equation for the pressure is based on combining the two equations. This means that the continuity equation, which does not contain the pressure, is employed to determine the pressure. If we take the divergence of the momentum equation, the continuity equation can be used to simplify the resulting equation. 

One consequence of the continuum approximation is the necessity to hypothesize two independent mechanisms for heat or momentum transfer one associated with the transport of heat or momentum by means of the continuum or macroscopic velocity field u, and the other described as a molecular mechanism for heat or momentum transfer that will appear as a surface contribution to the macroscopic momentum and energy conservation equations. This split into two independent transport mechanisms is a direct consequence of the coarse resolution that is inherent in the continuum description of the fluid system. If we revert to a microscopic or molecular point of view for a moment, it is clear that there is only a single class of mechanisms available for transport of any quantity, namely, those mechanisms associated with the motions and forces of interaction between the molecules (and particles in the case of suspensions). When we adopt the continuum or macroscopic point of view, however, we effectively spht the molecular motion of the material into two parts a molecular average velocity u = (w) and local fluctuations relative to this average. Because we define u as an instantaneous spatial average, it is evident that the local net volume flux of fluid across any surface in the fluid will be u n, where n is the unit normal to the surface. In particular, the local fluctuations in molecular velocity relative to the average value (w) yield no net flux of mass across any macroscopic surface in the fluid. However, these local random motions will generally lead to a net flux of heat or momentum across the same surface. 

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. 

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. 

All symbols have their usual meaning and only more important ones are defined here. Cj is the concentration of component j in the aqueous phase (e.g. polymer, tracer, etc.). The viscosity of the aqueous phase, rj, may depend on polymer or ionic concentrations, temperature, etc. Dj is the dispersion of component j in the aqueous phase Rj and qj are the source/sink terms for component j through chemical reaction and injection/production respectively. Polymer adsorption, as described by the Vj term in Equation 8.34, may feed back onto the mobility term in Equation 8.37 through permeability reduction as discussed above. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (Bondor etai, 1972 Vela etai, 1974 Naiki, 1979 Scott etal, 1987), in order to find the velocity fields for each of the phases present, i.e. aqueous, oleic and micellar (if there is a surfactant present). This pressure equation will be rather more complex than that described earlier in this chapter (Equation 8.12). However, the overall idea is very similar except that when compressibility is included the pressure equation becomes parabolic rather than elliptic (as it is in Equation 8.12). This is discussed in detail elsewhere (Aziz and Settari, 1979 Peaceman, 1977). Various forms of the pressure equation for polymer and more general chemical flood simulators are presented in a number of references (Zeito, 1968 Bondor etal, 1972 Vela etal, 1974 Todd and Chase, 1979 Scott etal, 1987). 

The Eulerian approach relies on solving a transport equation for the particle concentration or number-density. The particles are not considered individually but as a continuous field akin to heat or chemical species, but furthermore characterized by inertia phenomena. The transport equation contains a convection term in which the mean velocities are known from the turbulence model, and also a dispersion term which is at the core of the problem. This dispersion term involves a dispersion tensor or, in a ID-formulation, a dispersion coefficient that we have to determine. 

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