23.2.8 Numerics of the Discrete Phase Model

The underlying physics of the Discrete Phase Model is described by ordinary differential equations (ODE) as opposed to the continuous flow which is expressed in the form of partial differential equations (PDE). Therefore, the Discrete Phase Model uses its own numerical mechanisms and discretization schemes, which are completely different from other numerics used in ANSYS FLUENT.

Figure 23.2.4: The Discrete Phase Model Dialog Box and the Numerics

The Numerics tab gives you control over the numerical schemes for particle tracking as well as solutions of heat and mass equations (Figure  23.2.4).

Numerics for Tracking of the Particles

To solve equations of motion for the particles, the following numerical schemes are available:

implicit   uses an implicit Euler integration of this equation in the separate Theory Guide which is unconditionally stable for all particle relaxation times.

trapezoidal   uses a semi-implicit trapezoidal integration.

analytic   uses an analytical integration of this equation in the separate Theory Guide where the forces are held constant during the integration.

runge-kutta   facilitates a 5th order Runge Kutta scheme derived by Cash and Karp [ 13].

For additional details, see Section  23.6.

You can either choose a single tracking scheme, or switch between higher order and lower order tracking schemes using an automated selection based on the accuracy to be achieved and the stability range of each scheme. In addition, you can control how accurately the equations need to be solved.

Accuracy Control   enables the solution of equations of motion within a specified tolerance. This is done by computing the error of the integration step and reducing the integration step if the error is too large. If the error is within the given tolerance, the integration step will also be increased in the next steps.

Tolerance   is the maximum relative error which has to be achieved by the tracking procedure. Based on the numerical scheme, different methods are used to estimate the relative error. The implemented Runge-Kutta scheme uses an embedded error control mechanism. The error of the other schemes is computed by comparing the result of the integration step with the outcome of a two step procedure with half the step size.

Max. Refinements   is the maximum number of step size refinements in one single integration step. If this number is exceeded the integration will be conducted with the last refined integration step size.

Automated Tracking Scheme Selection   provides a mechanism to switch in an automated fashion between numerically stable lower order schemes and higher order schemes, which are stable only in a limited range. In situations where the particle is far from hydrodynamic equilibrium, an accurate solution can be achieved very quickly with a higher order scheme, since these schemes need less step refinements for a certain tolerance. When the particle reaches hydrodynamic equilibrium, the higher order schemes become inefficient since their step length is limited to a stable range. In this case, the mechanism switches to a stable lower order scheme and facilitates larger integration steps.

This mechanism is only available when Accuracy Control is enabled.

Higher Order Scheme   can be chosen from the group consisting of trapezoidal and runge-kutta scheme.

Lower Order Scheme   consists of implicit and the exponential analytic integration scheme.

Tracking Scheme   is selectable only if Automated is switched off. You can choose any of the tracking schemes. You also can combine each of the tracking schemes with Accuracy Control.

Including Coupled Heat-Mass Solution Effects on the Particles

By default, the solution of the particle heat and mass equations are solved in a segregated manner. If you enable the Coupled Heat-Mass Solution option, ANSYS FLUENT will solve this pair of equations using a stiff, coupled ODE solver with error tolerance control. The increased accuracy, however, comes at the expense of increased computational time.

Tracking in a Reference Frame

Particle tracking is related to a coordinate system. With Track in Absolute Frame enabled, you can choose to track the particles in the absolute reference frame. All particle coordinates and velocities are then computed in this frame. The forces due to friction with the continuous phase are transformed to this frame automatically.

In rotating flows it might be appropriate for numerical reasons to track the particles in the relative reference frame. If several reference frames exist in one simulation, then the particle velocities are transformed to each reference frame when they enter the fluid zone associated with this reference frame.

Staggering of Particles in Space and Time

In order to obtain a better representation of an injector, the particles can be staggered either spatially or temporally. When particles are staggered spatially, ANSYS FLUENT randomly samples from the region in which the spray is specified (e.g., the sheet thickness in the pressure-swirl atomizer) so that as the calculation progresses, trajectories will originate from the entire region. This allows the entire geometry specified in the atomizer to be sampled while specifying fewer streams in the input dialog box, thus decreasing computational expense.

When injecting particles in a transient calculation using relatively large time steps in relation to the spray event, the particles can clump together in discrete bunches. The clumps do not look physically realistic, though ANSYS FLUENT calculates the trajectory for each particle as it passes through a cell and the coupling to the gas phase is properly accounted for. To obtain a statistically smoother representation of the spray, the particles can be staggered in time. During the first time step, the particle is tracked for a random percentage of its initial step. This results in a sample of the initial volume swept out by the particle during the first time step and a smoother, more uniform spatial distribution at longer time intervals.

The menu for staggering is available in the text user interface(TUI), under

define/models/dpm/options/particle-staggering.

The "staggering factor'' in the TUI is a constant which multiplies the random sample. The staggering factor controls the percentage of the initial time step that will be sampled. For example, if the staggering factor is 0.5, then the parcels in the injection will be tracked between half and all of their full initial time step. If the staggering factor is 0.1, then the parcels will be tracked between ninety percent and all of their initial time step. If the staggering factor is set to 0.9, the parcels will be tracked between ten percent and all of their initial time step. This allows you to control the amount of smoothing between injections.

The default values for the options in the TUI are no temporal staggering and a temporal staggering factor of 1.0. The temporal staggering factor is inactive until the flag for temporal staggering is enabled.