# Immersed boundary method

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?

UPD

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.

• Your question is too vague. Try rephrasing it by including formulas that describe what exactly you are doing. – Wolfgang Bangerth Jan 10 '14 at 1:32
• I've updated my question, but the method contains a main part of formulas. What exactly I miss? – erthalion Jan 10 '14 at 13:38
• It sounds like there is a bug in your code. Are you sure your code is working correctly? What benchmark tests have you ran already? Have you tried simulating an oscillating cylinder? See arxiv.org/abs/1305.3976 for details. – Eldila Jan 13 '14 at 4:54
• Note, the immersed boundary equations are very stiff. Furthermore, the stiffness increases as k increases. Simulating a rigid body will require a very small time step to keep the simulation stable. – Eldila Jan 13 '14 at 6:21
• I've updated my question one more time with the results of my experiments. That was not a bug, but the lack of the time steps and stiffness. – erthalion Jan 13 '14 at 13:44