Dear all, I got a serious poor convergence problem while dealing with highly viscous fluid flow with viscosity depending on temperature. It seldom diverge actually but oscillate after the nomalized residual of velocities(or pressure) reduced to aroud 0.1. Collocated FVM and SIMPLE is adopted. A simple 3d tube geometry with cold wall temp and hot fluid is tested. The isothermal version of this code shows very good convergence. The nonlinearity caused by the vaiable viscosity may reduce the coupling ability of SIMPLE algorithm, I guess. Any comments or suggestions would be highly appreciated. Thanks.

Hi,

You really should'n have a problem with SIMPLE and variable viscosity - after all, our turbulence models introduce a variable effective viscosity AND are highly nonlinear (and everything works fine). I suspect you may have a problem with high viscosity: say, your diffusion Co number is very high and you're doing transient/under-relaxation based on the convection flux. If your viscosity is high (even locally), it will change the nature of the momentum equation from convection- to diffusion-dominated. Take a look at the Re number (is it ~1?) and the amount of work you're doing in the momentum predictor (it should be almost negligible compared to the rpessure solver).

Hrv

Thanks for your response. Numerically, the diffusoin-dominated problems should be more stable than the convection-dominated problems. In convection-dominated problem, the convective fluxe will contribute negtive values to the coefficient and reduce the diagonal-dominated charateristic of the resulting matrix. That's why the upwinding technique is needed. The diffusion-dominated problems will give an alway diagonal-dominated matrix, and hence the high viscosity should stablize the computation. I use the "deferred correction" technique(1st upwind during iteration and 2nd CDS after converged) to retain the diagonal domination, and indeed very good convergence is observed during inner iteration for each velocity components (only 1-2 iterations are needed to achieve tolerence of 10e-3 by BICGSTAB). However, the variable viscosity perturbs the resulting coefficient matrix considerablely during each SIMPLE outer iteration, and hence reduces the convergence. I ever tried to under-relax the viscosity but it only slightly improve the convergence.

Well, in that case very little can go wrong. Under-relaxation with standard relaxation parameters is tuned for incompressible turbulent flows; maybe you need to re-tune the relaxation parameters (any oscillatory behaviour in residuals?). The only other possibility is that your model is not well posed in the numerical sense or that there is more than one solution and you're switching between valid answers.

Hrv

(1). If you plot the viscosity from cell to cell, what is the ratio of viscosity between cells. What is the range of this ratio? (2). What is the Reynods number of the flow? If it is turbulent, what turbulence model are you using?

If you are using the arthmetic interpolation for viscosity to obtain the nodal viscosity, you should try the harmonic average-based interpolation. This make sense since viscosity is a diffusion coefficient ( same as thermal conductivity in heat conduction).. See Patanker for the Harmonic averaging.

hi John, the viscosity ratio is about 10 in the vincity of wall and 2 elsewhere. Acutally, my system is within creeping flow range. Full Navier-Stokes are solved.

hi Saud. Yes, I ever thinked that the harmonic interpolation should be better according to Patankar, and tried it, but no obvious improvement was obtained.

Hi Dr Jasak, can you tell me the range of the viscosity ratio within your system. will it diverge if the vsicosity ration is too large ? How you approximate the viscosity at the cell face center?

Hi Anthony,

you said that the isothermal case converged well. Have you tested your temp or scalar transport solver independantly of the variable visc part of your momentum solver? Say a given conservative velocity field or with constant viscosity?

There are also some special issues with termperature-velocity coupling see the work by Galpin and Raithby....sorry I do not have my reference list handy but I think in Numerical Heat Transfer about 1986...don't quote me on that!

Regards..................Duane

(1). There are possible areas: one is the algorithm itself (the form of the equation, the solution algorithm), the other is the boundary condition. (2). You can vary the temperature field (thus the viscosity variation) to check the sentivity of the solution. You can also refine the mesh in the area with high viscosity gradients.

Based on your answers to Mr. Chien's questions, I think you have similar problems with mine before. I have been doing Non-Newtonian fluid flow simulations and they have the same poor convergence problem. My viscotiy varies enormously across y-axis for a channel(example). If I use a lower power index in Power-Law fluid model, the convergence deteriorates seriously. It is well known phenomena and there are a few recommendations.

(1) decrease a relaxation parameter (2) if you use transient code, use smaller time step. (3) mix two nonlinear solvers such as Picard and Newton-Raphson. The former has bigger radius convergence. So use the Picard intiallly to get your intermediate solutions close engough to final solutions and switch to the Newton-Raphson, which has much faster convergence rate(2nd order).

It is based on my experience(mainly FEM code). I could not find any more fancy or sophisticated ones that make sense to everyone.

Jaret

You may want to modify SIMPLE to SIMPLET. This modification intended to carry out the temperature effect more effectively when correcting the pressure

Hi, Duane. I ever thinked that It maybe the thermal boundary layer that reduces the convergence. So I tried a constant viscosity case, that is, velocity field is independent on temperature, and the convergence was very good just like that in the isothermal case. Both converged velocity and temperature fields can be obtained. Consquently, I think that it should be the variable viscosity that deteriorates the convergence. The variable viscosity problem should be very common in the field of crystal growth, soldification, casting and polymer processing. Is there anyone try solve these problem with FVM+SIMPLE?

hi Jaret. Actually, I am fighting the non-isothermal, non-Newtonian flow by FVM for my PhD program (Permanent Head Damage ). Several months ago, the suceess in the isothermal code really makes me filled with the confidence that I will soon finish my job. After the numerous fights with non-linearity in the coutless dark nights,however, at present, I thinks I am still far way from the land of the Grail.

Hi, Saud

Can you suggest the literature for SIMPLET.

Zeng