Transportiveness, discretization method

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

Second method consists of a straightforward discretization method first order (Euler) explicit in time and finite differences in space. Both the time step and the grid size are kept constant and satisfying the Courant Friedrichs Lewy (CFL) condition to ensure the stability of the calculations. To deal with the transport part we have considered the minmod slope limiting method based on the first order upwind flux and the higher order Richtmyer scheme (see, e.g. Quarteroni and Valli, 1994, Chapter 14). We call this method SlopeLimit. 

The numerical solution for the solute-humic cotransport model was obtained by an unconditionally stable, fully implicit finite difference discretization method. The three governing transport Eqs. (38), (48), and (54) in conjunction with the initial and boundary conditions given by Eqs. (39)-(41), (51)—(53), (58) and (59) were solved simultaneously [57]. All flux boundary conditions were estimated using a second-order accurate one sided approximation [53]. 

As development of analytical method for shielding characteristics, a three-dimensional discrete ordinate transport code TORT was broadly tested and verified to possess excellent performance for analysis of complicated configurations. 

The present paper provides the reader with a summary of the effort carried out In Rouen to predict the behaviour of discrete particles transported In turbulent flows.Two approaches (Eulerlan and Lagranglan) are described. Comparisons between Eulerlan predictions, Lagranglan simulations (Including the Influence of particles on turbulence) and analytical, or experimental results. In a large number of situations, show that these methods are very well suited for solving the addressed problem. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

When the transport equation for c is solved with a discretization scheme such as upwind, artificial diffusive fluxes are induced, effecting a smearing of the interface. When these diffusive fluxes are significant on the time-scale of the simulation, the information on the location of different fluid volumes is lost. The use of higher order discretization schemes is usually not sufficient to reduce the artificial smearing of the interface to a tolerable level. Hence special methods are used to guarantee that a physically reasonable distribution of the volume fraction field is maintained. 

This formulation can be compared with the multi-environment (ME) presumed PDF method discussed in Section 5.10. The principal difference between the two approaches is the treatment of turbulent convection. In the transported PDF simulation, turbulent convection is simulated by a random process. In the ME—PDF approach, it is handled using standard FV/FD discretization. 

---