Discrete Quadrature S-Gamma Model Reference

The Discrete Quadrature S-Gamma model is a phase model for particle size distribution. It accounts for the effects of breakup and coalescence on the predicted particle size distribution in a multiphase phase interaction.

Table 1. Discrete Quadrature S-Gamma Model Reference
Model Name	Discrete Quadrature S-Gamma
Theory	S-Gamma.
Provided By	[phase] > Models > Optional Models
Example Node Path	[phase] > Models > Models > Discrete Quadrature S-Gamma
Requires	Material: one of Gas, Liquid, Multi-Component Gas > Non-reacting, Multi-Component Liquid > Non-reacting, Multiphase > Volume of Fluid (VOF), or Multiphase > Multiphase Mixture (MMP) > , or MultiphaseTwo-Phase Thermodynamic Equilibrium,Dispersed Multiphase (DMP),Multiphase Interaction,Gradients (selected automatically). Flow: Coupled Flow or Segregated Flow
Properties	Minimum Diameter Maximum Diameter Convection Number of S-Gamma Equations Secondary Gradients See Discrete Quadrature S-Gamma Model Properties.
Activates	Physics Models	S-Gamma Breakup and S-Gamma Coalescence are activated as optional models in a phase interaction when one phase has the Discrete Quadrature S-Gamma model activated. See Discrete Quadrature S-Gamma Phase Interaction Models Reference.
	Model Controls (child nodes)	S-Gamma Turbulent Prandtl Number See Model Controls.
	Initial Conditions	Size Distribution Specification See Initial Conditions.
	Boundary Inputs	Size Distribution Specification See Boundary Settings.
	Region Inputs	S-Gamma Source Option See Region Settings.
	Solvers	S-gamma See S-gamma Solver Properties.
	Field Functions	See Field Functions.

Discrete Quadrature S-Gamma Model Properties

Minimum Diameter

Sets the minimum allowed Sauter mean diameter.

Maximum Diameter

Sets the maximum allowed Sauter mean diameter.

Convection

Specifies the order of approximation of the convection term used in the S-Gamma transport equations. See Eqn. (2190) and Eqn. (2194).

1st-order
Selects the first-order convection scheme.
2nd-order
Selects the second-order convection scheme.

Number of Quadrature Points

Quadrature points are the sample values on the curve on which the numerical integration is performed. A larger number of points can produce more accurate results, but requires more calculations and so takes longer to perform.

You can specify any number of points. However, the default value of 8 is suitable for most purposes.

Model Controls

The following child node is available with turbulent viscous regimes only.

S-Gamma Turbulent Prandtl Number: Sets the ratio of kinematic turbulent viscosity and coefficient of turbulent diffusion of the S-Gamma moments. This value is $\Pr$ in Eqn. (2189).

Initial Conditions

Size Distribution Specification

Sets the initial particle size distribution in the dispersed phase and controls whether to include particle size variance in the calculations.

Method	Corresponding Initial Conditions Nodes
Sauter Mean Diameter Set the mean diameter only.	Sauter Mean Diameter Specifies the Sauter mean diameter profile. (See Eqn. (2177)).
Sauter Mean and Variance of Diameter Set the mean diameter and particle size variance.	Particle Size Variance Specifies the particle size variance profile. (See Eqn. (2182)). Sauter Mean Diameter Specifies the Sauter mean diameter profile. (See Eqn. (2177)).

Boundary Settings

Inlets and Pressure Boundaries

Size Distribution Specification

Sets the initial particle size distribution in the dispersed phase and controls whether to include particle size variance in the calculations.

Method	Corresponding Physics Value Nodes
Sauter Mean Diameter Set the mean diameter only.	Sauter Mean Diameter Specifies the Sauter mean diameter profile. (See Eqn. (2177)).
Sauter Mean and Variance of Diameter Set the mean diameter and particle size variance.	Particle Size Variance Specifies the particle size variance profile. (See Eqn. (2182)). Sauter Mean Diameter Specifies the Sauter mean diameter profile. (See Eqn. (2177)).

Region Settings

Applies to the dispersed phase in a fluid region.

S-Gamma Source Option

Provides access to the

S_{0}

and

S_{2}

source terms.

Method	Corresponding Physics Value Nodes
S-Gamma Source Term When activated, you can specify source terms as scalar profiles. When deactivated, the source term calculations use their internal defaults.	S-Gamma S0 Source $S_{0}$ in Eqn. (2172). S-Gamma S0 Source Derivative The $S_{0}$ derivative in Eqn. (2185). S-Gamma S2 Source $S_{2}$ in Eqn. (2179) and Eqn. (2181). S-Gamma S2 Source Derivative The $S_{2}$ derivative in Eqn. (2194).

S-gamma Solver Properties

The S-gamma solver controls the solution update for the Discrete Quadrature S-Gamma model.

The properties in the Expert category are for a temporary debug situation, at the expense of simulation accuracy and higher memory usage. Do not change these properties unless you are thoroughly familiar with the Simcenter STAR-CCM+ discretization techniques.

Implicit Under-Relaxation Factor

Improves stability and convergence of the linear system by using the relaxation factor to increase diagonal dominance of the matrix.

The default value is 0.5.

Explicit Under-Relaxation Factor

Specifies the multiplier that is applied to the provisional increment of the solution. Sharing the relaxation between implicit and explicit factors can be used to improve performance.

The default value is 1.0.

Overall Relaxation Factor

The product of the two under-relaxation factors (that is, Implicit Under-Relaxation Factor times Explicit Under-Relaxation Factor). This (read-only) value is a guide to the combined effect of the relaxation.

Interaction Source Storage Retained

When activated, interaction source storage is retained at the end of the iteration. The breakup and coalescence source terms are made available as field functions.

To understand how breakup and coalescence are working in a particular application, you can examine their contributions to the $S_{0}$ and $S_{2}$ transport equations (Eqn. (2185) and Eqn. (2194) respectively).

Reconstruction Frozen

When On, Simcenter STAR-CCM+ does not update reconstruction gradients with each iteration, but rather uses gradients from the last iteration in which they were updated. Activate Temporary Storage Retained in conjunction with this property. This property is Off by default.

Reconstruction Zeroed

When On, the solver sets reconstruction gradients to zero at the next iteration. This action means that face values used for upwinding (Eqn. (905)) and for computing cell gradients (Eqn. (917) and Eqn. (918)) become first-order estimates. This property is Off by default. If you turn this property Off after having it On, the solver recomputes the gradients on the next iteration.

Solver Frozen

When On, the solver does not update any quantity during an iteration. It is Off by default. This is a debugging option that can result in non-recoverable errors and wrong solutions due to missing storage. See Finite Volume Solvers Reference for details.

Temporary Storage Retained

When On, Simcenter STAR-CCM+ retains additional field data that the solver generates during an iteration. The particular data retained depends on the solver, and becomes available as field functions during subsequent iterations. Off by default.

Field Functions

The following field functions are made available to the simulation when the Discrete Quadrature S-Gamma model is used.

ParticleSizeVariance of [phase]: $σ$ in Eqn. (2182).
Sauter Mean Diameter of [phase]: $d_{32}$ in Eqn. (2181), Eqn. (2182), and Eqn. (2183).
Second Size Distribution Moment of [phase]: $S_{2}$ in Eqn. (2179), Eqn. (2183), and Eqn. (2194).
ThreeZeroDiameter of [phase]: $d_{30}$ in Eqn. (2181), Eqn. (2182), and Eqn. (2183).
Zeroth Size Distribution Moment of [phase]: $S_{0}$ in Eqn. (2172), Eqn. (2171), Eqn. (2183), and Eqn. (2185).