Dear all,
I have a question in the case of Poiseuille flow with periodic boundary conditions and pressure gradient,
Is there any need to consider pressure drop along the channel in this case?
anyway, when should we consider the pressure drop in poiseuille flow with periodic BCs and how?
Thanks for your help
Hi Casy,
The pressure drop with periodic boundary condition can be calculated as following. Suppose you need to achieve u_final with your simulation. I mean that your final velocity which is involved in Poiseuille velocity distribution for the flow inside plates equals to
u(x)= grad Pressure / visc (x^2/2-H^2/8), where H is Ny in our case - the width of channel. That means that in the center the velocity is gradPressure/viscH^2/8. If it’s equal to u_final we can obtain formula that gradPressure =rho08visc*ufinal/(Ny^2)
Now the force representation is involved. If you’d like to represent the force in the simplest case, like Fc_i/c_s^2, then to get force in terms of pressure you need to multiply this on c_s^2. After that you can obtain that gradPressure=Fc_i.
Therefore force_i term which we need to add to the right hand side of the LBEquation will be gradPressure/6, where 6 is the number of population in which gradient of pressure directs or flow goes (D2Q9 has 6 velocities with x projection).
Finally fpois equals to rho0* 8.0/6.0 * visc * uf / dfloat(ny) / dfloat(ny) and you add it to the ith population as fpois*c_i/c_s^2. You will obtain the u_final as the center velocity in the channel.
Hopefully my math is OK and it can help you a bit,
Alex