Orestis mentioned in a post about a year ago (titled ‘lid cavity’) that the Zou & He velocity BC is unstable for 1/tau > 1.8. I have indeed found this out the hard way, but is anyone aware of a reference in which this stability constraint is discussed further?

I have another question about the Zou/He velocity boundary condition that I hope you can help me with. I have found that when using this as an inlet in conjunction with an extrapolated stress free outlet my computational domain monotonically accumulates mass. Have you any experience with this problem, or is it possibly a problem with my implementation?

For a simple 2D Poiseuille flow problem:

I use half-way bounce-back on the top and bottom walls (i.e. Ny=1 and Ny=NYNODS).

I use an extrpolated stress-free outlet at the right hand boundary for f3, f6, and f7
fi(y,NXNODS) = 2*(fi(y,NXNODS-1)) - fi(y,NXNODS-2)

The velocity at the left hand inlet is specified as a parabolic profile in which I assume that the parabolic curve equals zero at y = 0.5 and y = NYNODS-0.5.

All inlet nodes are processed as follows:
rho_in = (1/(1-u_x))(f0+f2+f4+2(f3+f6+f7))
f1=f3+(2/3)rho_inu_x
f5=f7+[(1/6)rho_inu_x]-[(1/2)(f2-f4)]
f8=f6+[(1/6)rho_inu_x]+[(1/2)(f2-f4)]

The top left corner node is then processed again:
rho_in = f0(1,NYNODS-1)+f1(1,NYNODS-1)+ … +f7(1,NYNODS-1)+f8(1,NYNODS-1)
f1=f3
f4=f2
f8=f6
f5=(1/2)(rho_in - f0 - 2(f2+f3+f6)
f7=f5

The bottom left corner node is then processed again:
rho_in = f0(1,2)+f1(1,2)+ … +f7(1,2)+f8(1,2)
f1=f3
f2=f4
f5=f7
f6=(1/2)(rho_in - f0 - 2(f3+f4+f7)
f8=f6

Any advice with this problem would be greatly appreciated!

Thanks again for your reply, it is very much appreciated.

I have also found the instability for the Zou/He velocity BC for moderate Reynolds number, as discussed in your paper. For the back-step flow problem I cannot attain anything much over Re = 100. Unfortunately I don’t have enough time to look into implementing a new velocity BC.