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7.3 Transport/rheology models
In OpenFOAM, simulations that include flow without energy/heat require modelling of the fluid stress. Many simulations assume a Newtonian fluid in which a viscosity is specified in physicalProperties, e.g. by
viscosityModel constant;
nu 1.5e-05;
- a family of generalisedNewtonian models for a non-uniform viscosity which is a function of strain rate , described in sections 7.3.1 , 7.3.2, 7.3.3 , 7.3.4 , 7.3.5 and 7.3.6 ;
- a set of visco-elastic models, including Maxwell, Giesekus and PTT (Phan-Thien & Tanner), described in sections 7.3.7 , 7.3.8 and 7.3.9 , respectively;
- the lambdaThixotropic model, described in section 7.3.10 .
7.3.1 Bird-Carreau model
The Bird-Carreau generalisedNewtonian model is
|
(7.22) |
viscosityModel BirdCarreau;
nuInf 1e-05;
k 1;
n 0.5;
7.3.2 Cross Power Law model
The Cross Power Law generalisedNewtonian model is:
|
(7.23) |
viscosityModel CrossPowerLaw;
nuInf 1e-05;
m 1;
n 0.5;
7.3.3 Power Law model
The Power Law generalisedNewtonian model provides a function for viscosity, limited by minimum and maximum values, and respectively. The function is:
|
(7.24) |
viscosityModel powerLaw;
nuMax 1e-03;
nuMin 1e-05;
k 1e-05;
n 0.5;
7.3.4 Herschel-Bulkley model
The Herschel-Bulkley generalisedNewtonian model combines the effects of Bingham plastic and power-law behavior in a fluid. For low strain rates, the material is modelled as a very viscous fluid with viscosity . Beyond a threshold in strain-rate corresponding to threshold stress , the viscosity is described by a power law. The model is:
|
(7.25) |
viscosityModel HerschelBulkley;
tau0 0.01;
k 0.001;
n 0.5;
7.3.5 Casson model
The Casson generalisedNewtonian model is a basic model used in blood rheology that specifies minimum and maximum viscosities, and respectively. Beyond a threshold in strain-rate corresponding to threshold stress , the viscosity is described by a “square-root” relationship. The model is:
|
(7.26) |
viscosityModel Casson;
m 3.934986e-6;
tau0 2.9032e-6;
nuMax 13.3333e-6;
nuMin 3.9047e-6;
7.3.6 General strain-rate function
A strainRateFunction generalisedNewtonian model exists that allows a user to specify viscosity as a function of strain rate at run-time. It uses the same Function1 functionality to specify the function of strain-rate, used by time varying properties in boundary conditions described in section 5.2.3.4 . An example specification of the model in momentumTransport is shown below using the polynomial function:
viscosityModel strainRateFunction;
function polynomial ((0 0.1) (1 1.3));
7.3.7 Maxwell model
The Maxwell laminar visco-elastic model solves an equation for the fluid stress tensor :
|
(7.27) |
simulationType laminar;
laminar
{
model Maxwell;
MaxwellCoeffs
{
nuM 0.002;
lambda 0.03;
}
}
7.3.8 Giesekus model
The Giesekus laminar visco-elastic model is similar to the Maxwell model but includes an additional “mobility” term in the equation for :
|
(7.28) |
simulationType laminar;
laminar
{
model Giesekus;
GiesekusCoeffs
{
nuM 0.002;
lambda 0.03;
alphaG 0.1;
}
}
7.3.9 Phan-Thien-Tanner (PTT) model
The Phan-Thien-Tanner (PTT) laminar visco-elastic model is also similar to the Maxwell model but includes an additional “extensibility” term in the equation for , suitable for polymeric liquids:
|
(7.29) |
simulationType laminar;
laminar
{
model PTT;
PTTCoeffs
{
nuM 0.002;
lambda 0.03;
epsilon 0.25;
}
}
7.3.10 Lambda thixotropic model
The Lambda Thixotropic laminar model calculates the evolution of a structural parameter (lambda) according to:
|
(7.30) |
|
(7.31) |
An example specification of the model in momentumTransport is:
simulationType laminar;
laminar
{
model lambdaThixotropic;
lambdaThixotropicCoeffs
{
a 1;
b 2;
c 1e-3;
d 3;
nu0 0.1;
nuInf 1e-4;
}
}