11.1.3. Numerical Parameters¶

This section contains keywords for defining numerical parameters. Keywords related to solving the governing equations (e.g., LEQ_IT, DISCRETIZE, UR_FAC, etc.) are dimensioned for the ten types of equations:

Index	Equation Type
1	Gas Pressure
2	Solids Volume Fraction
3	Gas and Solids U Momentum Equation
4	Gas and Solids V Momentum Equation
5	Gas and Solids W Momentum Equation
6	Gas and Solids Energy Equations (Temperature)
7	Gas and Solids Species Mass Fractions
8	Granular Temperature
9	User-Defined Scalar and K-Epsilon Equation
10	DES Diffusion Equation

For example, LEQ_IT(3) = 10, specifies to use 10 iterations within the linear equation solver for the U Momentum Equations (of type 3).

11.1.3.1. MAX_NIT¶

Data Type: INTEGER

Maximum number of iterations [500].

11.1.3.2. NORM_G¶

Data Type: DOUBLE PRECISION

Factor to normalize the gas continuity equation residual. The residual from the first iteration is used if NORM_G is left undefined. NORM_G=0 invokes a normalization method based on the dominant term in the continuity equation. This setting may speed up calculations, especially near a steady state and for incompressible fluids. But, the number of iterations for the gas phase pressure, LEQ_IT(1), should be increased to ensure mass balance.

11.1.3.3. NORM_S¶

Data Type: DOUBLE PRECISION

Factor to normalize the solids continuity equation residual. The residual from the first iteration is used if NORM_S is left undefined. NORM_S = 0 invokes a normalization method based on the dominant term in the continuity equation. This setting may speed up calculations, especially near a steady state and incompressible fluids. But, the number of iterations for the solids volume fraction, LEQ_IT(2), should be increased to ensure mass balance.

11.1.3.4. PPG_DEN¶

Data Type: DOUBLE PRECISION

Denominator used to scale initial fluid pressure correction equation residual when Norm_g=0. Default value is 10. Increase to tighten tolerance, decrease to loosen.

11.1.3.5. EPP_DEN¶

Data Type: DOUBLE PRECISION

Denominator used to scale initial solids volume fraction correction equation residual when Norm_g=0. Default value is 10. Increase to tighten tolerance, decrease to loosen.

11.1.3.6. TOL_RESID¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (continuity + momentum) [1.0d-3].

11.1.3.7. TOL_RESID_T¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (Energy) [1.0d-4].

11.1.3.8. TOL_RESID_X¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (Species Balance) [1.0d-4].

11.1.3.9. TOL_RESID_TH¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (Granular Energy) [1.0d-4].

11.1.3.10. TOL_RESID_SCALAR¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (Scalar Equations) [1.0d-4].

11.1.3.11. TOL_RESID_K_EPSILON¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (K_Epsilon Model) [1.0d-4].

11.1.3.12. TOL_DIVERGE¶

Data Type: DOUBLE PRECISION

Minimum residual for declaring divergence [1.0d+4]. This parameter is useful for incompressible fluid simulations because velocity residuals can take large values for the second iteration (e.g., 1e+8) before dropping down to smaller values for the third iteration.

11.1.3.13. TOL_SUM_RESID_ABS¶

Data Type: DOUBLE PRECISION

Maximum residual at convergence (sum of all residuals) [1.0d-10]. If sum of all residuals is below this tolerance at the first inner iteration, convergence is declared. This helps run faster when steady state is reached.

11.1.3.14. DETECT_STALL¶

Data Type: LOGICAL

Reduce the time step if the residuals stop decreasing. Disabling this feature may help overcome initial non-convergence.

Table 11.16 Valid Values¶
Name	Default?	Description
`.FALSE.`		Continue iterating if residuals stall.
`.TRUE.`	◉	Reduce time step if residuals stall.

11.1.3.15. LEQ_METHOD(EQUATION ID NUMBER)¶

Data Type: INTEGER

\(1 \le Equation ID Number \le 10\)

LEQ Solver selection. BiCGSTAB is the default method for all equation types.

Table 11.17 Valid Values¶
Name	Default?	Description
`1`		SOR - Successive over-relaxation
`2`	◉	BiCGSTAB - Biconjugate gradient stabilized.
`3`		GMRES - Generalized minimal residual method
`5`		CG - Conjugate gradient

11.1.3.16. LEQ_TOL(EQUATION ID NUMBER)¶

Data Type: DOUBLE PRECISION

\(1 \le Equation ID Number \le 10\)

Linear Equation tolerance [1.0d-4].

11.1.3.17. LEQ_IT(EQUATION ID NUMBER)¶

Data Type: INTEGER

\(1 \le Equation ID Number \le 10\)

Number of iterations in the linear equation solver.

Default value: - 20 iterations for equation types 1-2 - 5 iterations for equation types 3-5,10 - 15 iterations for equation types 6-9

11.1.3.18. LEQ_SWEEP(EQUATION ID NUMBER)¶

Data Type: CHARACTER

\(1 \le Equation ID Number \le 10\)

Linear equation sweep direction. This applies when using GMRES or when using the LINE preconditioner with BiCGSTAB or CG methods. ‘RSRS’ is the default for all equation types.

Table 11.18 Valid Values¶
Name	Default?	Description
`RSRS`	◉	(Red/Black Sweep, Send Receive) repeated twice
`ISIS`		(Sweep in I, Send Receive) repeated twice
`JSJS`		(Sweep in J, Send Receive) repeated twice
`KSKS`		(Sweep in K, Send Receive) repeated twice
`ASAS`		(All Sweep, Send Receive) repeated twice

11.1.3.19. LEQ_PC(EQUATION ID NUMBER)¶

Data Type: CHARACTER

\(1 \le Equation ID Number \le 10\)

Linear preconditioner used by the BiCGSTAB and CG LEQ solvers. ‘LINE’ is the default for all equation types.

Table 11.19 Valid Values¶
Name	Default?	Description
`NONE`		No preconditioner
`LINE`	◉	Line relaxation
`DIAG`		Diagonal Scaling

11.1.3.20. UR_FAC(EQUATION ID NUMBER)¶

Data Type: DOUBLE PRECISION

\(1 \le Equation ID Number \le 10\)

Underrelaxation factors. Default value: - 0.8 for equation types 1,9 - 0.5 for equation types 2,3,4,5,8 - 1.0 for equation types 6,7,10

11.1.3.21. UR_F_GS¶

Data Type: DOUBLE PRECISION

The implicitness calculation of the gas-solids drag coefficient may be underrelaxed by changing ur_f_gs, which takes values between 0 to 1.

0 updates F_GS every time step
1 updates F_GS every iteration

11.1.3.22. UR_KTH_SML¶

Data Type: DOUBLE PRECISION

Underrelaxation factor for conductivity coefficient associated with other solids phases for IA Theory [1.0].

11.1.3.23. DISCRETIZE(EQUATION ID NUMBER)¶

Data Type: INTEGER

\(1 \le Equation ID Number \le 10\)

Discretization scheme used to solve equations.

Table 11.20 Valid Values¶
Name	Default?	Description
`0`	◉	First-order upwinding.
`1`		First-order upwinding (using down-wind factors).
`3`		Smart.
`2`		Superbee (recommended method).
`5`		QUICKEST (does not work).
`4`		ULTRA-QUICK.
`7`		van Leer.
`6`		MUSCL.
`8`		minmod.
`9`		Central (often unstable; useful for testing).

11.1.3.24. DEF_COR¶

Data Type: LOGICAL

Use deferred correction method for implementing higher order discretization.

Table 11.21 Valid Values¶
Name	Default?	Description
`.FALSE.`	◉	Use down-wind factor method (default).
`.TRUE.`		Use deferred correction method.

11.1.3.25. CHI_SCHEME¶

Data Type: LOGICAL

This scheme guarantees that the set of differenced species mass balance equations maintain the property that the sum of species mass fractions is one. This property is not guaranteed when a flux limiter is used with higher order spatial discretization schemes. Note: The chi-scheme is implemented for SMART and MUSCL discretization schemes. Darwish, M.S., Moukalled, F. (2003). Computer Methods in Applied Mech. Eng., 192(13):1711-1730.

Table 11.22 Valid Values¶
Name	Default?	Description
`.FALSE.`	◉	Do not use the chi-scheme.
`.TRUE.`		Use the chi-scheme correction.

11.1.3.26. CN_ON¶

Data Type: LOGICAL

Temporal discretization scheme.

Table 11.23 Valid Values¶
Name	Default?	Description
`.FALSE.`	◉	Implicit Euler based temporal discretization scheme employed (first-order accurate in time).
`.TRUE.`		Two-step implicit Runge-Kutta method based temporal discretization scheme employed. This method should be second- order accurate in time excluding pressure terms and restart time step which are first-order accurate. However, as of 2015, testing shows that second-order accuracy is not observed.

11.1.3.27. MAX_INLET_VEL_FAC¶

Data Type: DOUBLE PRECISION

The code declares divergence if the velocity anywhere in the domain exceeds a maximum value. This maximum value is automatically determined from the boundary values. The user may scale the maximum value by adjusting this scale factor [1.0d0].

11.1.3.28. DO_TRANSPOSE¶

Data Type: LOGICAL

Solve transpose of linear system. (BICGSTAB ONLY).

11.1.3.29. ICHECK_BICGS¶

Data Type: INTEGER

Frequency to check for convergence. (BICGSTAB ONLY)

11.1.3.30. OPT_PARALLEL¶

Data Type: LOGICAL

Sets optimal LEQ flags for parallel runs.

11.1.3.31. USE_DOLOOP¶

Data Type: LOGICAL

Use do-loop assignment over direct vector assignment.

11.1.3.32. IS_SERIAL¶

Data Type: LOGICAL

Calculate dot-products more efficiently (Serial runs only.)

11.1.3.33. DIL_EP_S¶

Data Type: DOUBLE PRECISION

Dilute flow threshold. When the volume fraction of a certain phase in a cell is smaller than this value the momentum equation for that phase is not solved in the cell. It is strongly encouraged to keep the default value of 1E-4.

11.1.3.34. ZERO_EP_S¶

Data Type: DOUBLE PRECISION

Minimum value of solids volume fraction tracked. It is strongly encouraged to keep the default value of 1E-8.

11.1.3.35. ZERO_X_GS¶

Data Type: DOUBLE PRECISION

Small value for species mass fraction for disregarding residual calculation. It is strongly encouraged to keep the default value of 1E-7.

11.1.3.36. TMIN¶

Data Type: DOUBLE PRECISION

Minimum temperature allowed in the system (K).: It is strongly encouraged to keep the default value of 50.0.

11.1.3.37. TMAX¶

Data Type: DOUBLE PRECISION

Maximum temperature allowed in the system (K).: It is strongly encouraged to keep the default value of 4000.0.