Fractional time stepping

In the first fractional time step, the micromixing term is solved separately for each notional particle 153... 

The overall fractional-time-stepping process can be represented by... 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

The intra-cell processes are common to all PDF codes, and are treated the same in both Eulerian and Lagrangian PDF codes.8 On the other hand, inter-cell processes are treated differently in Eulerian PDF codes due to the discrete representation of space in terms of x . In PDF codes, fractional time stepping is employed to account for each process separately. Methods for treating chemical reactions and mixing are described in Section 6.9. Thus we will focus here on the treatment of inter-cell processes in Eulerian PDF codes. 

In the MC simulation, these equations are treated numerically using fractional time stepping. For unsteady flow, the simulation time step At is determined from the FV code by... 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

The basic idea of the MC approach lies in the discrete representation of the joint PDF by an ensemble of stochastic particles. Each particle carries an array of properties denoting position, velocity and scalar composition. During a fractional time stepping procedure [6] the particles are submitted to certain deterministic and stochastic processes changing each particle s set of properties in accordance with the different terms in the PDF evolution equation. Afterwards the statistical moments may be derived in the simplest case by averaging from the ensemble of particles. 

The hybrid algorithm we use in the actual case is a combination of two sub-models Conventional SIMPLE approach together with a, k — e model and elliptic velocity-composition joint PDF scheme [6]. The sub-models interact as follows The CFD model supplies mean velocity fields, V(p and arrays of turbulent kinetic energy and dissipation rates as input for the PDF part. Having obtained these quantities as input, the fractional time step algorithm provides scalar composition and density as final output. The averaged density-field is finally handed back to the CFD sub-model. 

By introducing a selective imphdt/explicit treatment of various parts of the equations, a certain separating at each time step of the set of eqrrations may be obtained to improve computational efBdency. This suggests the possibility to apply devoted solvers to subproblems of each fractional time step [31],... 

---