Volume Force Calculation
Given an array of idealized point masses such that the position vector of the ith particle is denoted qi (SI unit: m), the volume force at position r is
(6-13)
where δ is the Dirac delta function, FD,i is the drag force exerted on the ith particle, and N is the total number of particles. The equation for the volume force is unusable in this form, however, for the following reasons:
•
The number of particles may be very large, making it impractical to model all of them.
•
The magnitude of the volume force becomes infinite at the location of each idealized point mass, making the calculation of the volume force infeasible with most numerical methods.
In the following sections we discuss solutions to each of these problems.
Modeling a Representative Sample of Particles
Because the number of real particles may be too large for every particle to be modeled individually, a practical numerical approach is to release a representative sample of model particles, allowing each model particle to make the same contribution to the volume force as an equivalent number of real particles.
For example, instead of allocating degrees of freedom for 107 small particles, it will often suffice to model 104 particles, each of which has a Force multiplication factor of 103, meaning that it exerts a volume force that is 103 times greater in magnitude than the drag force that acts on it.
Simplification for Constant Mass Flow Rate
If particles are released into the fluid at a constant mass flow rate, then a full time-domain calculation of the coupled particle trajectories and field variables may require particles to be released at a large number of time steps until a stationary solution is reached. The calculation of the fluid velocity and pressure at each time step can be needlessly memory intensive and time consuming. An alternative approach is to release particles at time t = 0 and to allow each model particle to represent a continuous stream of real particles per unit time. The number of real particles per unit time represented by each model particle is denoted the effective frequency of release, frel.
The behavior of a continuous stream of particles can be conveniently modeled by defining an expression for the time derivative of the volume force, rather than the volume force itself:
(6-14)
The volume force can then be computed by integrating over time, as long as sufficient time is given so that the particle trajectories can be traced completely through the modeling domain.
The frequency of release can be computed using the current and number of model particles that are specified in release feature settings. For example, for an Inlet node with release current magnitude (SI unit: kg/s) and number of particles per release N (dimensionless), the effective frequency of release is
When the mass flow rate can be assumed to be constant, then the volume force at the last time step includes contributions from particles at every point along their trajectories in the modeling domain. Thus, it can be applied as the volume force term when computing the fluid pressure and velocity.
The treatment of the constant mass flow rate is determined by the Particle release specification list in the settings window for the Particle Tracing for Fluid Flow interface. If Specify release times is selected, the volume force is computed using Equation 6-13 and is determined by the instantaneous positions of all model particles. Thus, it is necessary to solve for the particle trajectories, fluid velocity, and pressure in the time domain. If Specify mass flow rate is selected, the volume force is computed using Equation 6-14 and is determined by the time history of the model particle positions.
The difference between the Specify mass flow rate and Specify release times option in the Particle release specification list is thus analogous to the difference between integration over Elements and time and integration over Elements as described for the Accumulator (Domain) node.
At this point, the effect of a bidirectional coupling between the particle trajectories and fields has not been considered. For the Specify release times option, this does not require special consideration because the trajectories and fields are computed simultaneously. For the Specify mass flow rate option, however, the trajectories and fields are computed using different study types, and an additional feedback mechanism is needed. The Bidirectionally Coupled Particle Tracing study step can be used to generate a solver sequence that does the following:
1
Set the volume force exerted by the particles on the fluid to zero.
2
Compute the fluid velocity and pressure using a Stationary solver, using the value of the volume force computed in the previous step.
3
Compute the particle trajectories and the resulting volume force in the time domain, using the field variables computed in the previous step.
4
Repeat steps 2 and 3 until a specified number of iterations has been reached, or until another user-specified convergence criterion has been satisfied.
If the number of iterations taken by the solver sequence is sufficiently large, the resulting solution will fully account for the bidirectional coupling between the particle trajectories and stationary fields.
Avoiding Infinitely Large Values of the Volume Force
The Fluid–Particle Interaction node defines variables for each component of the volume force exerted by particles on the surrounding fluid. These variables are discretized using constant shape functions that are, in general, discontinuous across boundaries between elements. For a mesh element j with volume Vj, and with the Particle release specification set to Specify release times, the average volume force FV,j is
where ni is the force multiplication factor of the ith model particle. The integral on the right-hand side is a volume integral over element j. The resulting volume force is the average volume force over the mesh element, which may be written more concisely as
where the sum is taken over all particles that are within mesh element j.
If instead the Particle release specification is Specify mass flow rate, each model particle represents a number of particles per unit time which follow along the same path, determined by the effective frequency of release frel. The volume force within the mesh element can then be expressed as the solution to the first-order equation