Fleft = 2 [«left • St - (1 - ^bottom ) • Sx]2 tan 3

FIGURE 7.10 Young's VOF method to determine cell face fluxes (from Delnoij, 1999).

FIGURE 7.12 Results of VOF simulations at 0.12 s (influence of differencing scheme). (a) Power law. (b) QUICK (SUPERBEE).

7.2.2. Eulerian-Lagrangian Approach

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be: (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase.

To calculate dispersed phase particle trajectories it will be necessary to solve a set of coupled ordinary differential equations (Eqs. 4.1 and 4.9). Any standard initial value ODE solvers can be used for this purpose. These methods are not discussed here. Necessary details may be found in texts such as Numerical Recipes (Press et al, 1992). When calculating the trajectories of dispersed phase particles, any other auxiliary equations to account for heat transfer or chemical reactions can also be solved following similar procedures. Care must be taken to ensure that the time steps used for integration are sufficiently small and the trajectory integration is adequately time accurate. It is often necessary to use different time steps to simulate transients in the continuous flow field and trajectories of dispersed phase particles.

Although the methods discussed in the previous chapter and standard ODE solvers can be used to carry out simulations of dispersed multiphase flows using a Eulerian-Lagrangian approach, some important issues deserve additional comments. Use of different time steps for different processes is one important issue. If direct interaction among dispersed phase particles is considered (collisions and bouncing of particles), then it may be necessary to use three different time steps: (1) AtF to resolve transients in the macroscopic flow of the continuous phase; (2) AtP to estimate forces acting on dispersed phase particles (during this time interval, the macroscopic flow field of continuous phase may be assumed to be constant); and (3) AtT to update positions of dispersed phase particles (during this time interval, forces acting on particles may be assumed to be constant. Particle velocities and positions may, however, change due to particle collisions). These three time scales are shown schematically in Fig. 7.13 for a case of a gas-liquid flow. The sequence of collisions needs to be processed one collision at a time. Obviously for each collision, the collision time will be different and needs to be computed after identifying the two colliding (for binary collisions) particles. Efficient numerical techniques such as neighbor list techniques have been developed to minimize computations for the identification of two colliding particles causing the next collision (see Hoomans et al., 1996; Delnoij, 1999 for more details). To account for collisions between dispersed phase particles, both soft particle (for example, Tsuji et al., 1993) and hard particle (for example, Hoomans et al., 1996) models have been used. Collisions act as an important mechanism to transport momentum and energy, in the case of gas-solid flows. For dispersed gas-liquid flows, however, the contribution of collisions to momentum transport may be neglected (Delnoij et al., 1999). In such a case, it may be sufficient to use two time steps, one to update the flow field of the continuous phase and the other for trajectory calculations.


FIGURE 7.13 Time steps used for Eulerian-Lagrangian simulations. Atf|ow: Time step used to resolve macroscopic liquid flowfield, Atbubble: Time step used to estimate forces acting on bubbles (macroscopic liquid flowfield is assumed to be constant), tab: Time step used to displace bubbles and to account for collisions. Forces acting on bubbles are assumed to be constant.

Trajectory calculations require the calculation of net force acting on the dispersed phase particles. To calculate such a net force, local values of pressure, continuous phase velocities, partial and substantial derivative of pressure and partial and substantial derivatives of continuous phase velocities need to be available at the center of mass position of dispersed particles. However, these Eulerian variables and their derivatives are known only at discrete nodes in the computational domain. Therefore, suitable interpolation should be used to obtain the required values at the particle location, using the previously obtained solution of the continuous phase flow equations. As a first-level approximation, a continuous phase velocity may be taken as a computational cell based velocity for all locations within the cell. The accuracy of the results, however, will be significantly influenced by this assumption. A better assumption would be to use appropriate area or volume averaging. The concept may be illustrated by considering a two-dimensional example as shown in Fig. 7.14. The local value of a quantity f at the center of mass of the dispersed phase particle fP) can be calculated using with fn being some Eulerian quantity at node n, and An representing an area:

Extension of these formulae for volume averaging in three-dimensional Eulerian-Lagrangian simulations is straightforward.

by fl dx

Bubble bx

FIGURE 7.14 Area weighting of node values (fn) (Refer Eqs. (7.5) and (7.6)).

So far, discussion has focused on simulating one-way coupling between continuous and dispersed phases. To simulate two-way coupling between multiple phases, iterative procedures need to be employed. The overall procedure is as follows:

• Solution of the continuous phase flow field (in the absence of dispersed phase).

• Calculation of trajectories of dispersed phase particles.

• Recalculation of continuous phase flow using the interphase exchange of momentum, heat and mass determined from trajectory calculations.

• Recalculation of dispersed phase trajectories in the modified continuous flow field.

• Repetition of the previous two steps until convergence is achieved.

For the calculation of continuous phase flow in the presence of dispersed phase particles, again several averaging procedures need to be employed. The volume fraction of the continuous phase in a computational cell is calculated by subtracting the ratio of volume occupied by dispersed phase particles and the volume of the computational cell from unity:


When calculating continuous phase volume fraction, it is very important to account for the possibility of dispersed phase particles overlapping with more than one computational cell. The volume occupied by such particles needs to be distributed over the respective cells. Calculation of the exact distribution of the volume of the dispersed phase particles to the respective cells may become computationally intensive when several particles are considered. Equations to distribute the volume of dispersed phase particles to different computational cells are illustrated in Fig. 7.15 for the two-dimensional case. Delnoij (1999) proposed some approximations based on the lengths

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