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
Continuity
Momentum
Momentum
Stress Tensor
Darcy's law
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.
Darcy's law
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:
Darcy's law
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:
Darcy's law
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:
Darcy's law
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:
Darcy's law
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.
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