I am solving for the unsteady turbulent incompressible flow in a duct. The solution plane is the (r,z) plane in a cylindrical coordinate system. The unsteadiness is due to unsteady (electromagnetic) body forces which are sinusoidal with frequency w.

My questions are:

1) Since the N-S equations are nonlinear, is there any reason to expect the solution (say velocity field) to be periodic with frequency w after a "long" time ?

2) and if the answer to (1) is no, then how does one know that we have arrived at the long time solution ?

3) What is the correct b.c for pressure ? Currently, I am using a neumann condition for the pressure on the solid walls. I could use a neumann condition on the axial boundaries, but then I would have all 4 neumann conditions which is inconsistent. On the other hand, I cannot specify the pressure at inflow or outflow since the pressure pulsation will be zero here and that is what I am trying to solve for.

Thanks for any comments.

1) Strictly speaking, no, the flow will never be purely periodic, simply because it's turbulent. However, if I assume you are really solving the RANS equations (not NS), then the flow 'may be' expected to reach a periodic state. That depends on the effect that the forcing has on the flow. To say the least, your frequency w should be orders of magnitude smaller than any frequency associated with turbulence in that pipe. Will the periodicity of the flow also have frequency w? It doesn't have to. There could be nonlinear effects leading to higher harmonics, besides the fundamental frequency.

2) If the flow is aperiodic, there is no unique "long time solution"..

3) I don't see why you shouldn't be able to use Neumann conditions on all boundaries. For incompressible flow only the pressure gradient matters, not pressure itself. But let's back up... how is inompressible pipe flow solved without the body force? That should be a pretty well-known problem.

Mani:

Thanks for your response.

3) The standard way of solving without the body force is to solve it with pressure specified at one end (inflow or outflow) and a neumann condtiion at the other end. That way the problem is unique. With Neumann condition on all 4 boundaries, there are 2 problems:

1) the integral of dp/dn over all boundaries must be equal to the integral of the source term over the domain. Otherwise, there is no solution. This condition is hard to satisfy, in a code.

2) The solution is non-unique up to an additive constant. This is not a serious problem, since the pressure is anyway determined only up to a constant.

Sam

Mani wrote:

But let's back up... how is inompressible pipe flow solved without the body force? That should be a pretty well-known problem.

diaw:

Could you explain this statement in a little more detail, please. What body force would be applied & how?