S-Gamma Models Reference

The S-Gamma model (see Lo and Rao [505], and Lo and Zhang [507]) is based on predicting the transport of the moments of the particle size distribution:

Zeroth moment: the particle number density, $n$ .
Second moment: related to the interfacial area density.
Third moment: related to the dispersed phase volume fraction.

One transport equation is solved for each moment. As the third moment is based on the dispersed phase volume fraction, it can be derived from the volume fraction equation that is solved by the Segregated EMP Flow solver. You can choose whether to solve for just the second moment, or for the zeroth and second moments, by selecting the required option in the properties of the S-Gamma model. Extensions to the S-Gamma model can be supplied using the source terms for the zeroth and second moments.

Two S-Gamma implementations are provided:

Discrete Quadrature S-Gamma

This model evaluates the integrals associated with breakup, coalescence and bubble entrainments using an adaptive discrete quadrature method.

If the One-Equation option is selected, the integrals are evaluated using a one-point integration at $a = d_{32}$ . If the Two-Equation option is selected, the integration points are distributed log-normally with the appropriate $S_{0}$ , and $S_{2}$ .
Pre-Integrated S-Gamma

This model evaluates the integrals using an analytical method.

If you wish to include the effects of breakup and coalescence on the predicted size distribution, you can select the S-Gamma Breakup and S-Gamma Coalescence models in a multiphase interaction that includes the dispersed phase.

When simulating multiple flow regime flows, if you wish to model the size of newly created bubbles/droplets, you can select the S-Gamma Entrainment model.

Breakup/coalescence or entrainment interaction models for multiple flow regimes are only available with the Discrete Quadrature S-Gamma implementation in a multiphase interaction.

All of the rate models have an overall calibration constant which you can adjust to suit your requirements.

Table 1. S-Gamma Model Reference
Model Names	Discrete Quadrature S-Gamma
	Pre-Integrated S-Gamma
Theory	See S-Gamma Model for Particle Size Distribution.
Provided By	[phase] > Models > S-Gamma Closure
Example Node Path	[phase] > Models > Discrete Quadrature S-Gamma
Requires	Material: Multiphase, Multiphase Interaction (Selected automatically) Multiphase Model: Eulerian Multiphase (EMP), Multiphase Equation of State (Selected automatically), Gradients (Selected automatically) Create two phases. This model is used only in the dispersed phase. Phase model selection (deactivate the Auto-select recommended models checkbox): Optional Models: Particle Size Distribution Particle Size Distribution: S-Gamma
Properties	Key properties are: We Crit and Number of S-Gamma Equations. See S-Gamma Model Properties.
Activates	Physics Models	S-Gamma Breakup and S-Gamma Coalescence are activated as optional models in a Continuous-Dispersed phase interaction when the dispersed phase has the S-Gamma model activated, or in a Multiple Flow Regime phase interaction when one of the phases has the S-Gamma model activated. The S-Gamma Entrainment is activated as an optional model in a Multiple Flow Regime phase interaction when one of the phases has the S-Gamma model activated See Discrete Quadrature S-Gamma Phase Interaction Model Reference and Pre-Integrated S-Gamma Phase Interaction Model 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 S-Gamma Model Field Functions.

S-Gamma Model Properties

Minimum Diameter

Sets the minimum allowed Sauter mean diameter.

Maximum Diameter

Sets the maximum allowed Sauter mean diameter.

Number of S-Gamma Equations

Selects the number of extra equations to solve in addition to the volume fraction.

One Equation
Solves for the transport of interfacial area density (the second moment, $S_{2}$ in Eqn. (2185)).
Two Equation
Solves for both the transport of particle number density and the interfacial area density (zeroth and second moments, $S_{0}$ in Eqn. (2180) and $S_{2}$ in Eqn. (2185)).

Note	When the Fluid Film Wave Stripping or Edge Stripping model is activated, only the Two Equation option is available.

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 5 is suitable for most purposes.

Secondary Gradients

Neglect or include the boundary secondary gradients for diffusion and/or the interior secondary gradients at mesh faces.

On: Include both secondary gradients.
Off: Exclude both secondary gradients.
Interior Only: Include the interior secondary gradients only.
Boundaries Only: Include the boundary secondary gradients only.

Model Controls

The following child node is available for Pre-Integrated S-Gamma and Discrete Quadrature S-Gamma models 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 S-Gamma model.

You are advised to use the same relaxation factor settings for this solver and the Volume Fraction solver.

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.

S-Gamma Model Field Functions

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

ParticleSizeVariance of [phase]: $σ$ in Eqn. (2182).
SauterMeanDiameter of [phase]: $d_{32}$ in Eqn. (2181), Eqn. (2182), and Eqn. (2183).
SecondSizeDistributionMoment 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).
ZerothSizeDistributionMoment of [phase]: $S_{0}$ in Eqn. (2172), Eqn. (2171), Eqn. (2183), and Eqn. (2185).