Dear CFD-ers,

One of my colleagues would like to compute incompressible flow past an array of spheres in a box. The array is orientated along the x-direction. Periodic boundary conditions are used in the y and z-directions.

I have available a 3-D incompressible flow code based on the pressure-velocity formulation. Naturally the code requires the velocity to be specified.

My question is how to best compute the flow for a *prescribed pressure drop* over the array of cylinders? So far, I have obtained approximate answers iteratively by prescribing the velocity, checking the pressure drop, correcting the velocity and so on.

Any suggestions, references to known work and other comments welcome.

Happy New Year from down under,

Margot Gerritsen University of Auckland New Zealand

m.gerritsen@auckland.ac.nz

The pressure drop normally is the solution of the problem. For incompressible flow problems, the solution is fixed when the inlet velocity is prescribed. And if you also specify the pressure at a point, the whole pressure field is also known because it is related solely to the velocity field only. But you can run several cases to establish the pressure drop coefficient, that is force difference = (0.5* rho * V*V) * area * Cd. this Cd may depend on the Reynolds number. Once you have established the Cd characteristics, you can plug into the above formula to find the velocity V (say, the inlet velocity) when the pressure drop is prescribed. Since Cd is a function of Reynolds number as Cd( Re=rho * V * L / mu ), you have to solve for V iteratively from this algebraic equation. So, I think what you are doing is the right approach, but, to make the effort useful, you need to establish the Cd( Re) coefficient first, similar to the drag coefficient Cd(Vcar) of a car moving at a constant speed Vcar.

An easier non-iterative solution is to use a pressure-specified inflow condition rather than velocity-specified inflow. How you do this depends on your numerical algorithm. I have experience only with FV methods using the Rhie-Chow approach for pressure-velocity coupling. In this technique, the mass flows across faces depends not just on velocity but also on pressure differences. The mass flows are easily found at pressure boundaries in the same way as at internal faces (but the pressure gradients are one-sided rather than centered). For pressure inflows it is also necessary to supply an additional contraint in the form of flow direction.

Hope this helps,

Phil

A possible approach, valid in the limit of fully developed flow conditions, would be to express the pressure field as the sum of a (periodic) local pressure, and an overall pressure drop:

P = p_per + Beta x

The (prescribed) overall pressure gradient Beta is then introduced as a source term in the streamwise component of the momentum equation, ending up with periodic BC also along x for the velocity and the (reduced) pressure field.

I stress again that this is valid only for fully developed flow conditions (eg. a large number of cylinders at low to moderate Re values).

Regards,

Enrico Nobile

There is a paper of Heywood, Rannacher and Turek on pressure drop and fluid fluxes on inlets- outlets. May be the papers of Heidelberg group on http://iwr.gaia.uni-heidelberg.de can also help.

Good luck, Maxim