91 lines
2.7 KiB
ReStructuredText
91 lines
2.7 KiB
ReStructuredText
=============================================
|
|
Greyscale model
|
|
=============================================
|
|
|
|
The LBPM greyscale lattice Boltzmann model is constructed to approximate the
|
|
solution of the Darcy-Brinkman equations in grey regions, coupled to a Navier-Stokes
|
|
solution in open regions. To use the greyscale model, the input image should be segmented
|
|
into "grey" and open regions. Each particular "grey" label should be assigned both
|
|
a porosity and permeability value.
|
|
|
|
A typical command to launch the LBPM color simulator is as follows
|
|
|
|
```
|
|
mpirun -np $NUMPROCS lbpm_greyscale_simulator input.db
|
|
```
|
|
|
|
where ``$NUMPROCS`` is the number of MPI processors to be used and ``input.db`` is
|
|
the name of the input database that provides the simulation parameters.
|
|
Note that the specific syntax to launch MPI tasks may vary depending on your system.
|
|
For additional details please refer to your local system documentation.
|
|
|
|
***************************
|
|
Model parameters
|
|
***************************
|
|
|
|
The essential model parameters for the single phase greyscale model are
|
|
|
|
- ``tau`` -- control the fluid viscosity -- :math:`0.7 < \tau < 1.5`
|
|
|
|
The kinematic viscosity is given by
|
|
|
|
***************************
|
|
Model formulation
|
|
***************************
|
|
|
|
A D3Q19 LBE is constructed to describe the momentum transport
|
|
|
|
.. math::
|
|
:nowrap:
|
|
|
|
$$
|
|
f_q(\bm{x}_i + \bm{\xi}_q \delta t,t + \delta t) - f_q(\bm{x}_i,t) =
|
|
\sum^{Q-1}_{k=0} M^{-1}_{qk} S_{kk} (m_k^{eq}-m_k) + \sum^{Q-1}_{k=0} M^{-1}_{qk} (1-\frac{S_{kk}}{2}) \hat{F}_q\;,
|
|
$$
|
|
|
|
|
|
The force is imposed based on the construction developed by Guo et al
|
|
|
|
.. math::
|
|
:nowrap:
|
|
|
|
$$
|
|
F_i = \rho_0 \omega_i \left[\frac{\bm{e}_i \cdot \bm{a}}{c_s^2} +
|
|
\frac{\bm{u} \bm{a}:(\bm{e}_i \bm{e}_i -c_s^2 \mathcal{I})}{\epsilon c_s^4} \right] ,
|
|
$$
|
|
|
|
|
|
where :math:`c_s^2 = 1/3` is the speed of sound for the D3Q19 lattice.
|
|
The acceleration includes contributions due to the external driving force :math:`\bm{g}`
|
|
as well as a drag force due to the permeability :math:`K` and flow rate :math:`\bm{u}` with the
|
|
porosity :math:`\epsilon` and viscosity :math:`\nu` determining the net forces acting within
|
|
a grey voxel
|
|
|
|
.. math::
|
|
:nowrap:
|
|
|
|
$$
|
|
\bm{a} = - \frac{\epsilon \nu}{K} \bm{u} + \bm{F}_{cp}/\rho_0 + \epsilon \bm{g},
|
|
$$
|
|
|
|
The flow velocity is defined as
|
|
|
|
.. math::
|
|
:nowrap:
|
|
|
|
$$
|
|
\rho_0 \bm{u} = \sum_i \bm{e}_i f_i + \frac{\delta t}{2} \rho_0 \bm{a}.
|
|
$$
|
|
|
|
Combining the previous expressions,
|
|
|
|
.. math::
|
|
:nowrap:
|
|
|
|
$$
|
|
\bm{u} = \frac{\frac{1}{\rho_0}\sum_i \bm{e}_i f_i + \frac{\delta t}{2}\epsilon \bm{g} +
|
|
\frac{\delta t}{2} \frac{\bm{F}_{cp}}{\rho_0}}{1+ \frac{\delta t}{2} \frac{\epsilon \nu}{K}}
|
|
$$
|
|
|
|
|