pressure dependent viscosity
 

Dear FOAMers,

I tried implementing pressure dependent viscosity models for incompressible multiphase flow solvers.

First of all, the quick & dirty work around to access the pressure field from the viscosity model (for example in myPouliquenViscoPlastic.C in viscosityModels/myPouliquenViscoPlastic) without changing all related constructors and header files (inspired by http://freefoam.sourceforge.net/doc/...8C_source.html) was:

U_.db().lookupObject<volScalarField>("p_rgh")

The problem is that one gets pressure oscillations even with a very fine grid and small time-step, and at the ECCOMAS 2012 in Vienna it turned out that this is a general problem when solving incompressible Navier-Stokes eq. with pressure dependent viscosities. However I have heard that there are FOAMers that solved this problem so I would like to invite anybody to present his/her solutions here.

At first glance I would be hesitant to use an incompressible formulation for a pressure dependent viscosity. In this case, only the gradient of the pressure is used. The actual magnitudes of the pressure in p_rgh is arbitrary. if you're using a formulation like eta=f(p_rgh), I don't think you can count on accurate results.

It is not that trivial. See http://www.compstat2004.cuni.cz/~mal...ok-prehled.pdf
especially chapter 7.1.
The recently published two-phase debris flow model of Pudasaini uses pressure dependent and shear viscosity for the granular phase by dividing the linearly pressure dependent part through the rate of deformation tensor, so I guess instabillity by pressure is equalized by shear rate and vorticity caused by local viscosity increase...

pressure dependent viscosity model of Pudasaini in OF
 

I implemented the actual Mohr-Coulomb plasticity formulation provided by Pudasaini (2012): 'A general two-phase debris flow model' and tested the formulation with a single phase simulation of sand in a made-up flume. The simulation runs fine. When I apply it to other cases (large rotating drum of Berkley, large scale flume experiments at the USGS flume), the simulation becomes unstable and it is not so obvious why. In 'Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity by Michael Renardy (1986) in 'Communications in partial differential equations' it sais in section 2:
"We conclude that, in contrast to the ordinary Navier-Stokes equations, the initial value problem is not always well-posed. Only as long as the eigenvalues of the symmetric part of the velocity gradient are less than 1/(2 nu') [nu being the pressure dependent viscosity] can we expect to prove a local existence result. If this condition is violated, problems of nonexistence and nonuniqueness occur in the constant coefficient problem and hence can be expected in the full Navier-Stokes system"

Since I am not that familiar with math anymore, I am not shure what this means- could someone see if this could explain successful or failing simulations dependent on initial flow conditions?

He further goes into the "complementing condition" of the boundary with respect to the pressure dependent viscosity. I guess but only guess that the first statement means if viscosity (in this case due to pressure) varies strong spatially, solving incompressible NS might fail dependent on velocity gradient, and I further guess that solving incompressible NS might still be present if boundaries are close (due to "complementing condition" additionnally to ellipticity)
So laboratory experiment-sized simulations might run well, but local pressure inhomogenity for example due to impact to a small obstacle will fail in larger scale.

Anyone who is better informed about partial differential equations could confirm that?

by the way, note that nu' denotes derivative of viscosity with respect to pressure. Got a nice solver by now but still work on getting the stability criteria introduced in a good form.