I'm trying to use immersed boundary method for the 3D flow problem (Navier-Stokes equations), but I'm maybe misunderstood something in this method. Main principles I took from this book. I use the continuous forcing approach and the quasi-rigid immersed boundary, i.e. the boundary force computed from this equation , where k is the stiffness, x ref is the initial node position.
My trouble is that in the first time layer I have zero boundary force (because I don't know this force) and nonzero fluid velocity. So the fluid shifts the all boundary nodes and I have something like this (this is the 3D flow around a cylinder that was projected on the Oz axis)
Because the all nodes were shifted they all have the same force including the right half circle. So the streamlines twisted through the immersed boundary (a cylinder).
How can I avoid this?
I have steady flow problem and maybe it's the source of troubles. I thought that the series of calculation of steady problem can approximates the unsteady and allow to use immersed boundary method, but now I'm not sure about this.
UPD I have reached the more correct flow, when stiffness has been increased. I didn't understand this, because my calculation had not enough time steps.