Hi everyone, I am new in LBM. Now, let’s consider simulating fluid flow in porous media with a periodic boundary driven by body force. How to choose the value of body force so that the flow remains laminar and how to transform the body force from physical units to lattice units. I will really appreciate your response. Thank you very much.
If you consider a system of no-slip boundaries orthogonal to the flow and periodic boundaries at the inlet and outlet, this system would give a parabolic velocity profile if the flow is laminar. In that situation, the maximum velocity of the flow is directly related to the body force applied to the fluid:
u_0 = G*L^2 / (2 * nu)
where u_0 is the maximum velocity, G is the body force, 2*L is the distance between the two no-slip boundaries and nu is the kinematic viscosity of the fluid [Guo et al., Phys Rev E, 65, 046308 (2002)]. The dimensionless Reynolds number (Re) for the system is defined as follows:
Re = u * (2*L) / nu
where u=0.5*u_0 is the characteristic velocity for the flow. For the flow to be laminar, the Reynolds number needs to be no more than 2100. Therefore, if you know the kinematic viscosity of the fluid and the distance between the plates, you can work out the maximum characteristic velocity for laminar flow and then work out the maximum body force.
Since a dimensionless number should work for all unit systems, you can scale the body force in physical or lattice units by ensuring the Reynolds number is the same for both. Note that the kinematic viscosity is usually 1/3 * (tau - 0.5) * (dx)^2 / (dt), where tau is the relaxation time for the fluid and dx and dt are the lattice spacing and time step respectively. (In lattice units, dx and dt are both equal to 1.)
For a porous medium, the existence of obstacles in the flow may change the maximum flow velocity, so some care is needed when choosing an appropriate body force: however, the above may at least give you a starting point.
I hope this is useful.