Transport equation velocity

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... 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

Gibaldi et al. [45] postulated that convective forces may be present in the GI tract during in vivo dissolution. This study took advantage of the well-defined hydrodynamics of the rotating disk, incorporating the solutions for the velocity profile and transport equations of Cochran [50] and Levich [51] to obtain... 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

This transport equation cannot be solved directly because it involves several unclosed terms. The SGS flux ucj> represents the spatial transport of by the unresolved velocity fluctuations. Models for this term can generally be written in the form of a generalized transport equation ... 

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]). 

As another application of this method of asymptotic integration, we shall consider the problem of the Fourier coefficients pfU(P t) i 1 the limit of long times. As mentioned above, we do not wish to give here a detailed proof of the transport equation for pk] p] t) (see, for instance, Ref. 31). The main result of this analysis is, however, very simple in the limit of long times (t —> oo), the correlations are entirely determined by the velocity distribution function p< p t). One has ... 

As it stands, the time-independent transport equation (105) still applies to the complete iV-particle velocity distribution... 

The transport equation (441) has been studied in another context j30.35.36 it is out of place to give this analysis here and we shall merely quote the conclusion in the case of small gradients, Eq. (440) is entirely equivalent to the well-known Stokes-Navier equation of hydrodynamics.f More precisely, the average velocity w, which from Eqs. (424), (426), and (439) is given by ... 

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 first term on the right-hand side of (2.61) is the spectral transfer function, and involves two-point correlations between three components of the velocity vector (see McComb (1990) for the exact form). The spectral transfer function is thus unclosed, and models must be formulated in order to proceed in finding solutions to (2.61). However, some useful properties of T (k, t) can be deduced from the spectral transport equation. For example, integrating (2.61) over all wavenumbers yields the transport equation for the turbulent kinetic energy ... 

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. 

The transport equation for the Reynolds stresses can be found starting from the governing equation for the velocity fluctuations ... 

Thus, in Chapter 6, the transport equations for /++ x, t) and the one-point joint velocity, composition PDF /u+V, + x. / ) are derived and discussed in detail. Nevertheless, the computational effort required to solve the PDF transport equations is often considered to be too large for practical applications. Therefore, in Chapter 5, we will look at alternative closures that attempt to replace /++ x, t) in (3.24) by a simplified expression that can be evaluated based on one-point scalar statistics that are easier to compute. 

In one-point models for turbulent mixing, extensive use of conditional statistics is made when developing simplified models. For example, in the PDF transport equation for /++ x, r), the expected value of the velocity fluctuations conditioned on the scalars appears and is defined by... 

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