☰
These guidelines are based on the working folder located here.
MHD modelling
1) Enabling/disabling MHD
In the
twoTemperatureDictFile "<constant>/thermo2TModel";
mhdDictFile "<constant>/mhdProperties";
temperatureBounds
{
Tlow 200;
Thigh 15000;
}
This
active true;
mhdModel lowReMag;
hallEffect false;
electricalConductivityModel Bush;
constantElectricalConductivityCoeffs
{
value 5100;
}
BushCoeffs
{
sigma0 5100;
T0 12000;
n 2;
}
The first switch,
2) MHD models
The electric current density, j (in A/m2), is given by Ohm's law
`j = σ (E + U × B)`
where
- `σ` is the scalar or tensor electrical conductivity in S/m (see §3)
- `E` is the electric field (in V/m)
- `B` is the total magnetic field, sum of the imposed and induced magnetic fields (in T)
- `U` is the velocity field.
The Lorentz force appearing as a source term in the momemtum equations is then defined as
`F^{L} = j × B`
and the Joule heating source term appearing in the total energy equation can be expressed as
`Q^{J} = j ⋅ E`
2.1 Low magnetic Reynolds number model
The low magnetic Reynolds number model can be selected as follows
mhdModel lowReMag;
Under the inductionless approximation, `Re_{m}` << 1, the influence of the velocity field on the magnetic field can be neglected. The total magnetic field is thus the imposed magnetic field.
Current limitations:
- constant total magnetic field (the magnetic induction equation is not solved)
- Poisson equation not solved for the electric potential
3) Electrical conductivity models
3.1 Constant electrical conductivity
A constant electrical conductivity, in S/m, can be prescribed using
the
electricalConductivityModel constantElectricalConductivity;
constantElectricalConductivityCoeffs
{
value 5100;
}
3.2 Bush
This model can be implemented as
electricalConductivityModel Bush;
BushCoeffs
{
sigma0 5100;
T0 12000;
n 2;
}
In the
`σ = σ_{0}*(T/T_{0})^n`
where
- `σ_{0}` is a reference electrical conductivity in S/m
- `T_{0}` is a reference temperature in K
- `n` is a temperature exponent
- `T` is the local trans-rotational temperature in K.
3.3 Chapman-Cowling
This model can be implemented as
electricalConductivityModel ChapmanCowling;
In the
`σ = 4.0227904*10^{-18}*n_{e^{-}}/sqrt(T)`
where
- `T` is the local trans-rotational temperature in K
- `n_{e^{-}}` is the local electron number density in m-3 defined as
`n_{e^{-}} = p_{e^{-}}/(k_{B}*T)`
where
- `k_{B}` is the Bolzmann constant in J/K
- `p_{e^{-}}` is the local electron pressure in Pa
3.4 Raizer
This model can be implemented as
electricalConductivityModel Raizer;
In the
`σ = 83.0*exp({-3.6*10^{4}}/T)`
where `T` is the local trans-rotational temperature in K.
3.5 Spitzer-Harm
This model can be implemented as
electricalConductivityModel SpitzerHarm;
In the
`σ = 1.56*10^{-4}*T^{1.5}/ln(1.23*10^{4}*T^{1.5}/sqrt(n_{e^{-}}))`
where
- `T` is the local trans-rotational temperature in K
- `n_{e^{-}}` is the local electron number density in m-3 defined as
`n_{e^{-}} = p_{e^{-}}/(k_{B}*T)`
where
- `k_{B}` is the Bolzmann constant in J/K
- `p_{e^{-}}` is the local electron pressure in Pa
4) Hall parameter
Hall effects can be accounted for using the
hallEffect true;
constantHallParameter 1.0; // optional, default: -1.0
When the optional key
`β = σ*B/(e*n_{e^{-}}`
and the scalar electrical conductivity is multiplied by the following tensor to yield the tensor electrical conductivity
where
`D = B^2 (1 + β^2)`
5) Creation of an initial magnetic field and electric potential
You can set-up the initial magnetic field and electric potential field manually or by either using funkySetFields or foamCalc.
For the magnetic field, the file header is
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
location "0";
object B;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [1 0 -2 0 0 -1 0];
while it is as follows for the electric potential
FoamFile
{
version 2.0;
format ascii;
class volScalarField;
location "0";
object elecPot;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [1 2 -3 0 0 -1 0];