Solution Methodology

This section describes the methodology that is used to solve the system of equations that the Lagrangian Multiphase and the Lagrangian Phase models generate.

The solution of equations in a Lagrangian framework implies the time-marching of ordinary differential equations (ODEs). The set of equations depends on the models that are invoked for the Lagrangian phase in question; the one common element is an ODE for the position of each parcel, Eqn. (2956).

This section covers:

Unsteady Procedure
Steady Procedure
Time-Step Controls
Two-Grid Procedure

Unsteady Procedure

A time-step in an unsteady simulation advances the solution from time $t$ to time $t + Δ t$ . A single iteration of the unsteady solver for the Lagrangian Multiphase model involves:

Recovering the Lagrangian Multiphase solution at time $t$ .
Advancing the Lagrangian Multiphase solution to time $t + Δ t$ by time-marching. A local time-step is used for each parcel. The time-step is adjusted dynamically according to the time-step controls in force.

These two steps are executed in each iteration of the unsteady solver, before the flow solver. If the Two-Way Coupling model is active, source terms that the Lagrangian phase models compute are stored for subsequent application in their respective transport equations.

In an explicit unsteady case, this procedure means that the Lagrangian Multiphase solution is obtained using the continuous phase solution from the previous time-step.

Steady Procedure

An iteration in a steady simulation updates the steady solution. For the Lagrangian Multiphase model, the steady solution consists of

Tracks, generated by the Track File model.
Cell fields such as volume fraction and sources.
Boundary fields such as incident mass flux.

A steady state is attained when these entities become invariant with further iteration. Despite this steady characteristic of the solution, it results from time-marching, where the time variable is the residence time of the parcels. The procedure in each iteration is:

Deactivate the Lagrangian Multiphase solution.
Generate the Lagrangian Multiphase solution by time-marching each parcel until it has left the computational domain, or has been removed, or until the user-specified maximum residence time is reached. A local time-step is used for each parcel, which is adjusted dynamically according to the time-step controls in force.

These two steps are executed in each iteration of the steady solver, before the flow solver.

Time-Step Controls

The local time-step $δ t_{p}$ controls the accuracy of the Lagrangian Multiphase solution. This time-step is used to time-march each parcel. The time-step is calculated dynamically using user-defined bounds and model time-scales.

You can use the Parcel Time Step field function to obtain the local time-step in field function expressions that are evaluated during substepping, for example in a passive scalar source. Simcenter STAR-CCM+ can use the Parcel Time Step field function for analysis or visualization when temporary storage is turned on or when the Lagrangian Multiphase Solver is running.

User-Defined Bounds

Three parameters are available for you to control the local time-step $δ t_{p}$ . Smaller $δ t_{p}$ can increase accuracy, until temporal error becomes negligible. The three parameters are:

The physical time-step $Δ t$ in unsteady simulations. Clearly $δ t_{p} \leq Δ t$ .

The maximum Courant number $C o_{m a x}$ , which implies:

Figure 1. EQUATION_DISPLAY

δ t_{p} \leq \frac{C o_{m a x} Δ x}{m a x (| v |, | v_{p} |)}

(3326)

where $Δ x$ is a characteristic length-scale of the cell containing the parcel.

The minimum Courant number $C o_{m i n}$ , which implies:

Figure 2. EQUATION_DISPLAY

δ t_{p} \geq \frac{C o_{m i n} Δ x}{m a x (| v |, | v_{p} |)}

(3327)

Model Time-Scales

Models can also propose constraints for $δ t_{p}$ . Among the model time-scales are:

The momentum relaxation time-scale, Eqn. (2964), provided by the Drag Force model.
The thermal relaxation time-scale, Eqn. (3024), provided by the Lagrangian Energy model.
The residual of the eddy interaction time, Eqn. (3001), provided by the Turbulent Dispersion model.
The mass transfer time-scale $m_{p} / {\dot{m}}_{p}$ provided by the Droplet Evaporation model.

Two-Grid Procedure

Two-way coupling assumes that the fluid cell volume is large compared to the particle size. This assumption fails in cases where cells become small due to the demands of flow modeling.

To ensure correct coupling when cell sizes are small, Simcenter STAR-CCM+ provides a two-grid procedure for coupling that clusters groups of contiguous cells together to create a virtual grid of larger cells. The simulation uses these larger cells for calculating parcel interactions with the fluid phase. After calculating interactions, the simulation distributes volume fraction contribution, momentum and energy source terms, and other transferred quantities evenly across the component cells.

The two-grid procedure can also be used for shell regions with interactions between fluid films and Lagrangian parcels.