Curved boundary with slip condition

I am modelling a cylinder in poiseuille flow. There should be momentum exchange between the fluid and the cylinder.
The cylinder is locked in x but allowed to migrate laterally. It also rotates freely.

I have implemented normal bounceback (the easy one) and momentum transfer and it works fine.

Now I want to upgrade the bounceback to take into consideration that the boundary is curved as per this paper:

Lattice Boltzmann Method for 3-D Flows with Curved Boundary
Renwei Mei, Wei Shyy, and Dazhi Yu
University of Florida, Gainesville, Florida

I understand all of it except one thing.
u_w is the velocity at the wall. I have no idea how to get this value or what it’s supposed to mean. Do they mean the velocity of the boundary? In that case it’s fine. Although due to the rotation I would have to split velocities up into the different directions for my D2Q9 model.

So this is my main problem right now.

After I finish this I want to implement slip. I have seen slip models for a straight boundary and I understand it. It’s a quite simple modification for the normal bounce-back scheme. How this would compare to a curved boundary however seems difficult to me.

The way I see it: Particles hit the curved boundary and are deflected (with a probability due to slip) depending on the angle of the boundary, instead of being bounce backed. The problem here is that they would end up somewhere arbitrarily. Not quite sure how to account for this.

Anyone work on something similar or have any ideas?


Regarding u_w: It really is the wall velocity. This means you have to compute it from the center of mass velocity and the rotational velocity which is rather simple for a cylinder in 2D.

Good luck,