Scalars, conserved

J.B. Bell, C.N. Dawson, and G.R. Shubin, An Unsplit, Higher-Order Godunov Method for Scalar Conservation Laws in Multiple Dimensions, J. Comput. Phys. 74 No. 1 (1988). 

The radiant surface flux vector Q, as computed from Eq. (5-122a), always satisfies the (scalar) conservation condition Im Q = 0 or ... 

Although WRF has several choices for dynamic cores, the mass coordinate version of the model, called Advanced Research WRF (ARW) is described here. The prognostic equations integrated in the ARW model are cast in conservative (flux) form for conserved variables non-conserved variables such as pressure and temperature are diagnosed from the prognostic conserved variables. In the conserved variable approach, the ARW model integrates a mass conservation equation and a scalar conservation equation of the form... 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

TABLE 1 Atmospheric Inverse Methods, Grouped According to the Tcrm(s) in the Scalar Conservation Equation Used as Surrogates for the Source-Sink Term at the Surface. 

May, L.B.H., Shearer, M., and Daniels, K.E. (2010b). Scalar conservation laws with nonconstant coefficients with application to particle size segregation in granular flows. /. Nonlinear Sci., 20, 689-707. 

Shearer, M., Gray, J.M.N.T., and Thornton, A.R. (2008). Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches. Eur. J. Appl. Math., 19, 61-86. 

Consider a two-dimensional, scalar conservation law of the form (1) with solution u(t,x). Within the finite volume framework, each discrete value of the function u is viewed as a cell average U( over a cell. The advantage of the finite volume approach is, that any kind of mesh can be used, i.e., the shape of the control volume can be chosen arbitrarily. Here, we work with a conforming triangulation T with cells Te T, = for which the... 

We presented an extension of the new ADER schemes on adaptive, unstructured triangulations in order to solve linear and nonlinear scalar conservation laws. Originally, the ADER approach based on Arbitrary high order DERiva-tives was introduced by Toro, Millington and Nejad in [38] for linear problems... 

E.F. Toro and V.A. Titarev (2001) Very high order Godunov-type schemes for nonlinear scalar conservation laws. European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Computational Fluid Dynamics Conference, Swansea, Wales, September 2001. 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

Time-reversible energy conserving methods can be obtained by appropriate modifications to the (time-reversible) midpoint method. Two such modifications are (i) scaling of the force field by a scalar such that total energy... 

Conservation of Mass. The general equations for the conservation of mass are the scalar equations (Fig. 21a) ... 

The equation describing the conservation of energy is the scalar equation ... 

This scalar product is conserved in time if and 2 obey the Klein-Gordon equation. It furthermore possesses all the properties usually required of a scalar product, namely... 

Let us consider a dynamically symmetric binary mixture described by the scalar order parameter field < )(r) that gives the local volume fraction of component A at point r. The order parameter < )(r) should satisfy the local conservation law, which can be written as a continuity equation [143] ... 

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

These variables are governed by exactly the same model equations (e.g., (4.103)) as the scalar variances (inter-scale transfer at scales larger than the dissipation scale thus conserves scalar correlation), except for the dissipation range (e.g., (4.106)), where... 

Note that the right-hand side of this expression is an E x I null matrix, and thus element conservation must hold for any choice of e e l,E and i e 1Moreover, since the element matrix is constant, (5.10) can be applied to the scalar transport equation ((1.28), p. 16) in order to eliminate the chemical source term in at least E of the K equations.9 The chemically reacting flow problem can thus be described by only K - E transport equations for the chemically reacting scalars, and E transport equations for non-reacting (conserved) scalars.10... 

In the special case where all chemical species have the same molecular diffusivity, only one transport equation is often required to describe the conserved scalars. The single conserved scalar can then be expressed in terms... 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

The key quantity that determines the number of reacting and conserved scalars is the rank of T. For the non-elementary reaction case, VT = rank(Y) < K.v< The number of reacting scalars will then be equal to Nr, and the number of conserved scalars will be equal to N = K — Nr-... 

Note that Nr = 2. Thus, by applying an appropriate linear transformation, it should be possible to rewrite the scalar transport equation in terms of two reacting and two conserved scalars. 

Note that only Nr < min(7, K) singular values will be non-zero. The columns of Usv corresponding to the non-zero singular values span the reacting-scalar sub-space, and the columns corresponding to the zero singular values span the conserved sub-space. The desired linear transformation matrix is thus... 

Figure 5.2. The composition vector c can be partitioned by a linear transformation into two parts c,., a reacting-scalar vector of length /VT and cc, a conserved-scalar vector of length N. The linear transformation is independent of x and t, and is found from the singular value decomposition of the reaction coefficient matrix Y.

The Nr rows of M that correspond to the non-zero singular values will yield the reacting scalars. The remaining N rows yield the conserved scalars.19 Applying SVD to T in (5.22) yields20... 

In most SVD algorithms, the singular values are arranged in Esv in descending order. Thus, the first Nr rows of M yield the reacting scalars and the remaining N rows yield the conserved scalars. 

In this expression, Vr is the (Nr x I) matrix containing the first Nr rows of Vjv. Note that the reacting scalars cr are coupled to the conserved scalars Cc through (5.33). 

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