A Transport Equations

To refresh the reader s memory, we briefly consider transport, or balance, equations in this appendix. (If the variable is being conserved in the flow field we can call them conservation equations). The Navier-Stokes equations we mentioned in Chap. 2 are a particular type of transport equation. Transport equations can be derived by balancing a certain quantity, let us call it y , over a differential volume element. The balance is  

We perform the balance in the simple case of one-dimensional flow, on a differential element that has unit width and height (the y and 2 directions), see Fig. 7.A.1 

This figure shows the transport of yp in and out of the element. The rate of accumulation of y in the element is d p/dt)dx and, if S denotes the rate of generation of y (for instance by chemical reaction, S may be positive or negative) per unit volume, the rate of generation of ip in the element is S da . 

Applying the balance from Eq. 7.A.1 and simplifying gives the onedimensional differential balance equation, the steady version of which is used in the main text  

Here we have made two assumptions. The first is that the fluid is incompressible, which means that the derivative of the velocity is zero (same volume flow in and out of the element)  

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One-equation models relax the assumption that production and dissipation of turbulence are equal at all points of the flow field. Some effects of the upstream turbulence are incorporated by introducing a transport equation for the turbulence kinetic energy k (20) given by... 

Equation (12.43) is called an Eulerian approach because the behavior of the species is described relative to a fixed coordinate system. The equation can also be considered to be a transport equation for particles when they are... 

Assuming local thermal equilibrium, i.e. the equality of the averaged fluid and solid temperature, a transport equation for the average temperature results which still contains and integral over the fluctuating component. In order to close the equation, a relationship between the fluctuating component and the spatial derivatives of the average temperature of the form... 

Fick s first law applied to homogeneous membranes at steady state is a transport equation... 

Their G//OST -codc essentially consisted of a mass balance for the gas, a transport equation for the bubble number density nb, and a force balance for a single bubble, respectively, which run as... 

The acid-base reaction is a simple example of using the mixture fraction to express the reactant concentrations in the limit where the chemistry is much faster than the mixing time scales. This idea can be easily generalized to the case of multiple fast reactions, which is known as the equilibrium-chemistry limit. If we denote the vector of reactant concentrations by and assume that it obeys a transport equation of the form... 

It is then no longer necessary to solve a transport equation for Y and the numerical difficulties associated with treating the first reaction with a finite-rate chemistry solver are thereby avoided. 

The energy equation is solved in the form of a transport equation for static temperature. The temperature equation is obtained from the enthalpy equation, by taking the temperature as a dependent variable. The enthalpy equation is defined as,... 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

Rendering a transport equation in finite difference form is a straightforward and well known process (e.g., Peaceman, 1977 Smith, 1986). The derivatives of C in time and space are replaced with finite difference equivalents. The resulting difference equation written at a nodal block (7, J) now includes concentration... 

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... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

Like the turbulent energy spectrum discussed in Section 2.1, a transport equation can be derived for the scalar energy spectrum lipjn. t) starting from (1.27) and (1.28) for an inert scalar (see McComb (1990) or Lesieur (1997) for details). The resulting equation is21... 

Like the Reynolds stresses, the scalar flux obeys a transport equation that can be derived from the Navier-Stokes and scalar transport equations. We will first derive the transport equation for the scalar flux of an inert scalar from (2.99), p. 48, and the governing equation for inert-scalar fluctuations. The latter is found by subtracting (3.89) from (1.28) (p. 16), and is given by... 

This is the approach taken in transported PDF methods, as discussed in detail in Chapter 6. Most of the other closure methods discussed in Chapter 5 require knowledge of the scalar variance, which can be found from a transport equation as shown next. 

For a homogeneous scalar field with an isotropic filter, the conditional expected value of the scalar will have the property (+U,

transport equation can be derived for the residual scalar variance defined by11... 

The next level of turbulence models introduces a transport equation to describe the variation of the turbulent viscosity throughout the flow domain. The simplest models in this category are the so-called one-equation models wherein the turbulent viscosity is modeled by... 

At the next level of complexity, a second transport equation is introduced, which effectively removes the need to fix the mixing length. The most widely used two-equation model is the k-e model wherein a transport equation for the turbulent dissipation rate is... 

Like the Reynolds stresses, the scalar flux obeys a transport equation that was derived in Section 3.3 ... 

Chapter 3 will be employed. Thus, in lieu of (x, t), only the mixture-fraction means ( ) and covariances ( , F) (/, j e 1,..., Nm() will be available. Given this information, we would then like to compute the reacting-scalar means and covariances (require additional information about the mixture-fraction PDF. A similar problem arises when a large-eddy simulation (LES) of the mixture-fraction vector is employed. In this case, the resolved-scale mixture-fraction vector (x, t) is known, but the sub-grid-scale (SGS) fluctuations are not resolved. Instead, a transport equation for the SGS mixture-fraction covariance can be solved, but information about the SGS mixture-fraction PDF is still required to compute the resolved-scale reacting-scalar fields. 

In (5.297), the interpolation parameter is defined separately for each component. Note, however, that unlike the earlier examples, there is no guarantee that the interpolation parameters will be bounded between zero and one. For example, the equilibrium concentration of intermediate species may be negligible despite the fact that these species can be abundant in flows dominated by finite-rate chemistry. Thus, although (5.297) provides a convenient closure for the chemical source term, it is by no means guaranteed to produce accurate predictions A more reliable method for determining the conditional moments is the formulation of a transport equation that depends explicitly on turbulent transport and chemical reactions. We will look at this method for both homogeneous and inhomogeneous flows below. 

A transport equation for could be derived starting from its definition in (5.321). However, the resulting expression would be unduly complicated and not necessarily agree with our assumption of independence between Y and f.133 Instead, Y can be treated as any other scalar so that the transport equation for (Y ) has the form of (3.88) on p. 81 with (j a = T and a chemical source term given by... 

We have seen that the joint velocity, composition PDF treats both the velocity and the compositions as random variables. However, as noted in Section 6.1, it is possible to carry out transported PDF simulations using only the composition PDF. By definition, x, t) can be found from /u,< >(V, 0 x, t) using (6.3). The same definition can be used with the transported PDF equation derived in Section 6.2 to find a transport equation for / (0 x, r). 

The closed PDF transport equation given above can be employed to derive a transport equation for the Reynolds stresses. The velocity-pressure gradient and the dissipation terms in the corresponding Reynolds-stress model result from... 

A variety of specific mathematical formulations of the CTRW approach have been considered to date, and network models have also been applied (Bijeljic and Blunt 2006). A key result in development of the CTRW approach is a transport equation that represents a strong generalization of the advection-dispersion equation. As shown by Berkowitz et al. (2006), an extremely broad range of transport patterns can be described with the (ensemble-averaged) equation... 

Fig. 13 Experimental (symbols) and theoretical (lines) data for the current-density as a function of applied voltage for a polymer film of a derivative of PPV under the condition of space-charge-limited current flow. Full curves are the solution of a transport equation that includes DOS filling (see text), dashed lines show the prediction of Child s law for space-charge-limited current flow assuming a constant charge carrier mobility. From [96] with permission. Copyright (2005) by the American Institute of Physics...

One way to determine the characteristics of these trajectories is by solving a transport equation with different probabilities of hopping of the charge carrier and the corresponding diffusion parameters of the host droplet. Another way, which we shall use here, is based on a visualization of the equivalent static cluster structure. This approach allows us to interpret the dynamic percolation process in terms of the static percolation. 

Mechanistic prediction of foam flow in porous media seems to be impossible without a transport equation governing foam texture, i.e., foam bubble size. 

The first equation in system (26) is, for v fixed, a transport equation in t, so that some upwinding is needed for the practical computation of the solution. This fact was first recognized in [98] where streamline upwinding methods were used, and in [99] where discontinuous Galerkin methods were implemented. We describe, in the following, a finite element approximation of system (26) using discontinuous approximations of t. 

Problem (26) can be viewed as a transport equation in r for given v, and a Stokes system in (v,p) for given t. The fixed point iteration scheme described in Theorem 6.1 does not use this fact. A more natural iterative scheme, which uncouples the r and the v equations, reads as follows ... 

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