Discrete also momentum equation

It is noted that the virtual body force Fp depends not only on the unsteady fluid velocity, but also on the velocity and location of the particle surface, which is also a function of time. There are several ways to specify this boundary force, such as the feedback forcing scheme (Goldstein et al., 1993) and direct forcing scheme (Fadlun et al., 2000). In 3-D simulation, the direct forcing scheme can give higher stability and efficiency of calculation. In this scheme, the discretized momentum equation for the computational volume on the boundary is given as... 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

If the flows are unsteady, the terms containing apo can be added on both sides of Eq. (7.10) (refer to Section 6.4). It must be noted that for multiphase flows, the inflow and outflow terms require considerations of interpolations of phase volume fractions in addition to the usual interpolations of velocity and the coefficient of diffusive transport. The source term linearization practices discussed in the previous chapter are also applicable to multiphase flows. It is useful to recognize that special sources for multiphase flows, for example, an interphase mass transfer, is often constituted of terms having similar significance to the a and b terms. Such discretized equations can be formulated for each variable at each computational cell. The issues related to the phase continuity equation, momentum equations and the pressure correction equation are discussed below. 

Usual interpolation rules and definitions of velocity and pressure corrections, similar to single-phase flows (Eq. (6.29)), can be used to derive a pressure correction equation from the discretized form of the overall continuity (normalized) equation. The momentum equation for multiphase flows (Eq. (7.16)) can also be written in the form of Eq. (6.28) for single-phase flows. Again, following the approximation of SIMPLE, one can write an equation for velocity correction in terms of pressure correction,/ ... 

In the coordinate discretization process one selects the node points in the domain at which the values of the unknown dependent variables are to be computed. In the finite volume method one also selects the location of the grid cell surfaces at which the property fluxes are determined. In this way the computational domain is sub-divided into a number of smaller, nonoverlapping sub-domains. There are many variants of the distribution of the computational node points and grid cell surfaces within the solution domain. The grid arrangements associated with the finite volume discretization of the momentum equation are generally more complicated than the one employed for a scalar transport equation. 

Amberg [17] and Shahani et al. [18] used second-order accurate discretization of spatial derivatives. The velocity field was solved from the momentum equation using the well-known pressure-correction method. The nonlinear convective terms were discretized in a special manner to allow a reasonably large time step and to suppress non-physical oscillations in the solution. The diffusive term was treated implicitly also to avoid a severe limitation of the time-step size. 

In this section, an implicit pressure correction method for incompressible flow is outlined [56, 163, 164, 249]. Implicit methods are preferred for slow-transient flows because they have less stringent time step restrictions than explicit schemes. However, the time step must still be chosen small enough so that an accurate history is obtained. It is further noted that implicit methods can also be used to solve steady problems. In this particular approach, an unsteady form of the problem has to solved until a steady state is reached. Eor the artificial time integration, large time steps are often used intending to reach the steady state quickly. If an implicit method is used to advance the momentum equation in time, the discretized equations for the velocities at the new time step are non-linear. Implicit methods thus require an iterative solution process. Several restrictions must be placed on the coefficient matrix to ensure a stable and efficient solution procedure, most important all the coefficients must be positive [164]. 

Even though the velocity component fields are computed from the discrete momentum equation components, the predicted velocity field may still not be correct. This is because the velocity field is also subject to the constraint that it should satisfy the continuity equation. 

In addition to the quantum approaches mentioned above, classical optimal control theories based on classical mechanics have also been developed [3-6], These methods control certain classical parameters of the system like the average nuclear coordinates and the momentum. The optimal laser held is given as an average of particular classical values with respect to the set of trajectories. The system of equations is solved iteratively using the gradient method. The classical OCT deals only with classical trajectories and thus incurs much lower computational costs compared to the quantum OCT. However, the effects of phase are not treated properly and the quantum mechanical states cannot be controlled appropriately. For instance, the selective excitation of coupled states cannot be controlled via the classical OCT and the spectrum of the controlling held does not contain the peaks that arise from one- and multiphoton transitions between quantum discrete states. 

The theory for a particle having a wavelength is represented by the Schrodinger equation, which, for the particle confined to a small region of space (such as an electron in an atom or molecule) can be solved only for certain energies, ie the energy of such particles is quantized or confined to discrete values. Moreover, some other properties, eg spin or orbital angular momentum, are also quantized. 

So far, we have discussed the solution of general transport equations, by assuming the velocity field to be known. In principle, the equations governing the velocity field (viz., the continuity and the momentum conservation equations) are also of a general conservative nature (see Table 1) and ideally should have been solvable by the standard convection-diffusion discretization methodologies discussed earlier. In reality, however, the strategy does not work in that way, and additional considerations need to be invoked. 

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