Linearization, source terms

In the SR model, Cs = 1, which can be validated using DNS data. Note that at spectral equilibrium TD ea so that the first two terms on the right-hand side of (A.41) function as a linear source term of the form AbU/i )1 2. ... 

Substitution of gradient terms (Eq. (6.8)), interpolated values and the linearized source term in Eq. (6.6) gives the discretized form of Eq. (6.6) for a uniform grid ... 

This formulation ensures that the slope of the linearized source term representation (Eq. (6.7)) is the same as the slope of the non-linear source term at node P, and is always negative. It must be noted that linearization with steeper slopes normally leads to slower convergence. For detailed discussion of the effect of source term linearization on convergence, and on the handling of source terms with non-negative slope with 0, see Patankar (1980). 

Mathematical formulations of various boundary conditions were discussed in Section 2.3. These boundary conditions may be implemented numerically within the finite volume framework by expressing the flux at the boundary as a combination of interior values and boundary data. Usually, boundary conditions enter the discretized equations by suppression of the link to the boundary side and modification of the source terms. The appropriate coefficient of the discretized equation is set to zero and the boundary side flux (exact or approximated) is introduced through the linearized source terms, Sq and Sp. Since there are no nodes outside the solution domain, the approximations of boundary side flux are based on one-sided differences or extrapolations. Implementation of commonly encountered boundary conditions is discussed below. The technique of modifying the source terms of discretized equation can also be used to set the specific value of a variable at the given node. To set a value at... 

When initiating the calculations, turbulent viscosity is set to a small value (usually the same as the molecular viscosity). Linearized source terms for k and s can then be written ... 

The coefficients a consist of all the inflow contributions (convective as well as diffusive) while the coefficients b consist of all the outflow contributions. In the absence of any source or sink, the mass conservation equation dictates that the sum of inflow contributions is equal to the sum of outflow contributions. In the presence of linearized source terms, one can write. 

In spite of the relevance of angular structure in the spectrum (e.g., Ko-men et al. 1984), the radial dependence is integrated out, and the energy balance is solved only as a function of k. After angular integration, the non-linear source term becomes... 

A slow relaxation of micelles was described by Fainerman (1981), Fainerman Makievski (1992a, b), and Fainerman et al. (1993d), using a linear source term and neglecting the transport of micelles. 

The parameters Sq in (10.8) represents the constant part of the linearized source term and Sp in (10.8) is the coefficient of Under-relaxation and iteration within a time step is implied by (10.2). 

Constant part of the linearized source term in FVM discretization... 

Using the Gaussian plume model and the other relations presented, it is possible to compute ground level concentrations C, at any receptor point (Xq, in the region resulting from each of the isolated sources in the emission inventory. Since Equation (2) is linear for zero or linear decay terms, superposition of solutions applies. The concentration distribution is available by computing the values of C, at various receptors and summing over all sources. 

To overcome this difficulty, we can introduce a new variable defined in terms of a linear combination of A and B such that the chemical source term for ( is null. Consider an acid-base reaction of the form... 

Thus, given the weights and abscissas, the micromixing term for the moments is closed. Applying DQMOM, the micromixing source terms (which are added to the right-hand sides of Eqs. (133)—(135)) can be shown to obey for each n — 1,..., N the linear system defined by... 

Note that the right-hand side of this expression contains the closed micromixing term for the moments (Eq. 137). To find the 3 M micromixing source terms (Amn, Bmn, Cmn) from this expression, we must choose a set of 3 M bi-variate moments. Note that because the moment equations are closed when only micromixing is considered, the chosen moments will be reproduced exactly. A convenient choice is to use the uncoupled moments m ) and mak. (Note that this same choice should then be used in Eq. (136).) This yields the linear system... 

The last term on the right-hand side of (1.28) is the chemical source term. As will be seen in Chapter 5, the chemical source term is often a complex, non-linear function of the scalar fields , and thus solutions to (1.28) are very different than those for the z nm-scalar transport equation wherein S is null. 

Another conditional expectation that frequently occurs in closures for the chemical source term is the conditional mean of the composition variables given the mixturefraction. The latter, defined in Chapter 5, is an inert scalar formed by taking a linear combination of the components of 0 ... 

For non-linear chemical reactions, this term leads to new unclosed terms that are difficult to model. For example, even the isothermal second-order reaction, (3.142), where the joint dissipation chemical source term is given by... 

The chemical-source-term closures described in this chapter do not require that the chemical source term be derived from a set of elementary reactions as done here. Indeed, closure difficulties will result whenever S() is a non-linear function of . 

Note that because the columns of T happen to be orthogonal, the linear transformation matrix results in a diagonal form for MT. The reaction rate functions in the transformed chemical source term then act on each of the transformed scalars individually.21... 

Reaction scheme (5.21) is an example of simple chemistry (i.e., Nr = /). In Section 5.5 we show that for simple chemistry it is always possible to choose a linear transformation that puts the chemical source term in diagonal form. Obviously, when Nr < I this will not be possible since the number of reactions will be greater than the number of reacting scalars. 

For simple chemistry, a form for Q( x, t) can sometimes be found based on linear interpolation between two limiting cases. For example, for the one-step reaction discussed in Section 5.5, we have seen that the chemical source term can be rewritten in terms of a reaction-progress variable Y and the mixture fraction f. By taking the conditional expectation of (5.176) and applying (5.287), the chemical source term for the conditional reaction-progress variable can be found to be... 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... 

Computational cost increases linearly with the number of independent variables. Thus a large number of inert scalars can be treated with little additional computational cost. For reacting scalars, the total computational cost will often be dominated by the chemical source term (see Section 6.9). 

The solution for (Eq. 9.9) requires two boundary conditions on c, one on v an initial condition on c and similarly one initial condition on q. Finally we must prescribe the sink/source term for the adsorption. This can be done in the most general case by writing another pde to describe adsorption, which is the transport of the adsorbing species into the crystal structure of the formed adsorbent. This model must be sufficiently broad to allow us to calculate the uptake at any location in the packed bed and at any time during the process. In many cases it wiU be found expedient and quite satisfactory to prescribe the uptake term as some kind of linear driving force model (LDF). 

Within the framework of commercial CFD codes where sequential solution methods are standard, as they need to solve a number of user-specified transport equations, the two potential equations must then be solved through innovative source term linearization. ... 

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