Finite-volume scheme

The DOM has been extensively and successfully applied to photocatalytic reactors by the Santa Fe (Argentina) group (see Cassano and Alfano 2000, and references therein) and verified against experimental results (Brandi et al., 1999 Romero et al., 2003). Also Trujillo et al. (2007) have recently used a variant of the DOM, called finite volume scheme to model the effect of air bubbles injected in a fixed catalyst reactor. 

In this section the numerical conservation properties of finite volume schemes for inviscid incompressible flow are examined. Emphasis is placed on the theory of kinetic energy conservation. Numerical issues associated with the use of kinetic energy non-conservative schemes are discussed [158, 49, 47]. 

All of the finite-volume schemes discussed above can be written in the standard flux form ... 

A realizable finite-volume scheme for bivariate velocity moments... 

Here we apply the finite-volume scheme to simulate two different examples of inhomogeneous kinetic equations. The first example is a non-equilibrium Riemann shock problem with different values of the collision time t. The second example is two ID crossing jets with different collision times. In reality, the collision time is controlled by the number density Moo, which we normalize with respect to unity in these examples. Thus, the reader can interpret the different values of t as different values of the unnormalized number density. As noted above, for the multi-Gaussian quadrature we compute the spatial fiuxes using Ml = 14 and Mo = 4 with a CFL number of unity. 

The finite-volume representation of the mixed moments in each cell is found by computing the volume average of Eq. (8.110). On the right-hand side die volume averages of products such as mo o.o appear. Thus, if Mo is piecewise constant in the cell, the volume-average product depends on the volume-average moments, which are the conserved quantities in the finite-volume scheme. 

Qamar, S. Warnecke, G. 2007 Solving population balance equations for two-component aggregation by a finite volume scheme. Chemical Engineering Science 62, 679-693. 

Vikas, V, Wang, Z. J. Fox, R. O. 2012 Realizable high-order finite-volume schemes for quadrature-based moment methods applied to diffusion population balance equations. Journal of Computational Physics (submitted). 

Newer works report successful application of spatial discretization using finite volumes and the weighted essentially nonoscillatory (WENO) method (von Lieres and Andersson, 2010), high-resolution finite volume schemes (Javeed et al, 2011a), and a discontinuous Galerldn method (Javeed et al, 2011b). 

Saad, B., Saad, M. Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media. SIAM Journal of Numerical Analysis, 1-34 (2013)... 

For non-Newtonian fluids the viscosity p is fitted to flow curves of experimental data. The models for this fit are discussed in the next chapter. The energy equation is also implemented in the code and can be used for temperature-dependent problems, but it is not needed for the simulation of fluid dynamic problems like jet breakup due to the uncoupling of the density in the incompressible formulation. The finite volume scheme uses the Marker and Cell (MAC) method to discretize the computational domain in space. The convective and diffusive terms are discretized with second-order accuracy and the fluxes are calculated with a Godunov-type scheme. 

Here, the scalar field 4> x) describes the porosity of the rock, the vector fields ayj t,x) and ao t,x) are the phase velocities, and Uw t,x) and Uo t,x) are the saturations of water and oil, respectively. Note, that Uy and Ug are the fractions of the pore space, that are filled with water or oil, i.e., 0 < Uw,o < 1-Equations (24) and (25) indicate, that a change of mass for each phase in a given region of a reservoir is equal to the net flux of the phase across the boundary of that region. Therefore, the class of finite volume schemes, such as the proposed ADER schemes, are obviously a natural choice from available numerical methods to solve such problems. 

C.D. Munz and R. Schneider. An arbitrary high order accurate finite volume scheme for the Maxwell equations in two dimensions on unstructured meshes. Forschungszentrum Karlsruhe, Germany, unpublished report. 

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