Hi, I was able to simulate the compressible Euler equations for a single phase dilute particle cloud subject to body forces as described in:
I used cylindrical coordinates. The form I used is the following:
r = Expression('x', degree=deg) F = (c*(u-u_1) + u*(c-c_1))*r*k_t*v_u*dx + (c*u*u)*v_u*dx + (c*u*u).dx(0)*r*v_u*dx - \ F_drag(u,c)*r*k_mp*v_u*dx - c*F_applied*r*k_mp*v_u*dx + \ (c - c_1)*r*k_t*v_c*dx + (c*u)*v_c*dx + (c*u).dx(0)*r*v_c*dx
The initial conditions are a Gauss function for the initial particle concentration c, an initial velocity u of the form u_0 = k*x and a force field pushing particles towards the center and decreasing to zero at the center.
The simulation gives reasonable results, however it does not need any boundary conditions. I am not sure why this is the case.
My reasoning of why this works is that there is mass conservation for c that keeps the area under the curve constant (therefore no Dirichlet BCs are needed). The u field does not need any Dirichlet BCs because the speed is zero at x=0 because F = 0 there. I added u =0 at x=0 as a BC and get the same result.
Is the above reasoning correct? What about natural BCs? I understand why there might be natural Von Neumann BCs:
And based on the observed results this might apply for c but what about u?
Thank you for your time,