Flow Models

The Flow module solves for conservation of mass and momentum, using the transient Navier-Stokes Equations H.Ding, F.C. Visser, Y.Jiang, and M. Furmanczyk, “Demonstration and Validation of a 3-D CFD Simulation Tool Predicting Pump Performance and Cavitation for Industrial Applications,” FEDSM2009-78256, 2009..

The integral form (conservative) of Reynold’s Averaged Navier-Stokes Equations (RANS) are as follows:

• Continuity

• Momentum

• Stress Tensor

where,

τij | effective shear stress (molecular+turbulent) |

f | body force |

n | surface normal |

ρ | static pressure(Pa) |

t | time |

v | fluid velocity |

vσ | mesh velocity |

Ω(t) | control volume as a function of time |

r | average local fluid density (kg/m3) |

σ | surface of control volume |

µ | dynamic viscosity (Poise or Pa-s) |

µt | turbulent dynamic viscosity |

δij | Kronecker delta(=1 for i=j, =0 for i≠j) |

Viscosity Models

• Constant Dynamic Viscosity—Specifies the fluid viscosity in a selected volume. The unit of dynamic viscosity is Pa-s or N-s/m2.

The value of the dynamic viscosity is specified in the box under the Constant Dynamic Viscosity selection.

• Constant Kinematic Viscosity—Specifies the fluid viscosity in a selected volume. The unit of kinematic viscosity is m2/s. The value of the kinematic viscosity is specified in the box under the Constant Kinematic Viscosity selection.

• Sutherland Law—Specifies the fluid viscosity in a selected volume in terms of the dynamic viscosity (Pa-s). The equation and inputs are as follows:

where,

T | temperature (K) |

µref | viscosity at reference temperature (Pa-s) |

S | Sutherland Temperature (K) |

T is the fluid Temperature (K) required as an input if the energy module is not active. |

Sutherland Law is used to compute the viscosity of an ideal gas as a function of temperature. Sutherland, W. (1893), "The viscosity of gases and molecular force," Philosophical Magazine, S. 5, 36, pp. 507-531 (1893). The following table shows Sutherland's constant and reference temperature for selected gases. Ref: en.wikipedia.org/wiki/viscosity.

Gas | S (K) | Tref (K) | mref (Pa-s) |

air | 120 | 291.15 | 18.27 e-6 |

nitrogen | 111 | 300.55 | 17.81 e-6 |

oxygen | 127 | 292.25 | 20.81 e-6 |

carbon dioxide | 240 | 293.15 | 14.8 e-6 |

carbon monoxide | 118 | 288.15 | 17.2 e-6 |

hydrogen | 72 | 293.85 | 8.76 e-6 |

ammonia | 370 | 293.15 | 9.82 e-6 |

sulphur dioxide | 416 | 293.65 | 12.54 e-6 |

helium | 79.4 | 273 | 19 e-6 |

NonNewtonian Viscosity Models

The nonNewtonian viscosity models are:

• Herschel-Bulkley Model

• Bingham Models

These models provide the appropriate viscosity for various types of fluids that exhibit nonNewtonian flow properties. The Herschel-Bulkley model and Bingham models relate the shear stress to the shear rate as follows:

where,

e0 | critical shear rate |

k | consistency index |

τ0 | yield stress of the fluid |

n | Power Law index. For Bingham model, n=1 |

The shear rate of 0 is the same as the gamma dot in the plot above. |

Resistance Model

Resistance Model is a Flow module option that you can use to set a resistance in a selected volume. The Resistance Model contains the following two models:

• Pressure Loss: based on the following equation:

where,

Cl | linear drag coefficient (Pa-s/m2) |

Cd | quadratic drag coefficient (1/m) |

β | porosity |

ρ | density |

• Darcy's Law: model based on the following equation:

where,

β | porosity |

α | permeability |

µ | dynamic viscosity |

V | velocity |

Cd | quadratic drag coefficient (1/m) |

The velocity used in the resistance equation is the local velocity. F in the equation is measured in the unit N/m3, such as force/volume, or pressure gradient (Dp/Dx), or rg. The pressure drop across the interface is computed by multiplying F by a finite thickness. The porosity is set in the Common module.