Hello It is about the same problem I posted early about the difficulty to get a converged solution(CONVERGENCE). There is a temperature difference vertically across the cell. Due to the presence of inlet and outlets on the 2-vertical wallls opposte to each other, there is a pressure difference as well. I tried to solve the equations in a dimensionalised form and the solution did not converge. Then I thought to simplify them by non-dimensionalising in terms of Rayleigh number(because I assume it was buoyancy dominated flow). All the properties were expressed in terms of Ra. As I stated in the early message I did find problems with convergence. It converged for a small Ra but as I go beyond Ra=1E4 solution just starts to diverge Then I tried to non-dimensionalise w.r.t. Reynolds number and still the problem is the same.

I discussed with a mathematician friend that would 3D simulation be easy to converge as someone responded to my earlier message that I should try 3D. He told me that convergence is a numerical problem if 2D system of equations is not converging there is no chance of getting a converged solution by extending that set of equations into 3D. Any suggestions to handle flows with mixed convection?

From this and your previous posts, it seems that, given the value of the Rayleigh number of your problem, you are likely in a turbulent regime (horizontal walls differentially heated) or at least transitional (horizontal walls differentially heated).

From my past experiences, high-Ra buoyant flows pose considerable challenge to any attempt to tackle the problem using a steady-state approach, with or without a turbulence model. In brief, my suggestions to *aid* the convergence are: 1) Perform several load/restart steps, i.e. try to get a converged solution for a lower Ra, and use this as the starting point for a higher Ra value; 2) Use first a higly dissipative scheme, 1st order upwind, before switching to a higher-order scheme; 3) Under-relax heavily during the intermediate phases; 3) If (1-3) don't make it work, then switch to a transient approach, starting from a pseudo-transient (no convergence at each time-step), to a full transient (force convergence at each time-step). If both fails, try to reduce the time-step value, at least during the transients where the Ra value is increased.

I will not discuss issues like turbulence modelling in buoyancy-dominated flows, resolution requirements, time-scales etc. but let me add that, for these type of problems, the *quality* of the results depends on what are your objectives: for example, flow pattern and thermal stratification can be reasonably obtained, while the correct prediction of heat fluxes at the walls is considerably more difficult.

Some people suggested to move from 2D to 3D, and maybe consider all the details of the geometry. This is likely to only mitigate some of the difficulties, but would significantly increase the computing effort. Therefore, I would suggest to use a 2D approach (quicker, more intuitive overview of the flow physics, easier visualization and post-processing etc.) before attempting a 3D solution. Obviously, a full 3D, high-resolution, time-dependent model will not face *convergence difficulties* (give a look at http://www-dinma.univ.trieste.it/~nirftc/research/), but, even assuming that it can be done at all, it would be absolutely meaningless for your *practical* problem, where maybe a parametric study has to be performed in a short time and with minimal/reasonable resources.

Hope it helps,

I shall try as per your suggestions. However I need to understand bit more as how to handle the physics of the problem. What is the criteria to decide that free or forced convection is dominant? I tried to scale my problem first w.r.t. Ra and then Re. What if both effects are important? I understand even very small difference in temperature Ra becomes very high (due to L cubed)and therefore buoyancy effect can not be ignored. If I don't non-dimensionalised the problem it is bit messy. But may be that is the only way. Is it? By the way I need to find the characteristics for building up of startification in enclosures with inlets/outlets and heat sources/sink etc. I am not that bothered about heat fluxes at this stage.

As your problem is 3d and transient you can not expect to get a converged solution for a steady state calculation.

If you run it in 3d and steady state you dont get a converged solution but maybe some usefull insight in the flow at all.

Then you have to decide if a transient calculation is more usefull to solve your technical problem.

First, the physics of the problem:

1) In general, for a heat transfer problem neglecting viscous heating with perscribed geometry and boundary conditions:

U*=U*(Re,Gr,Pr,x*,y*,z*,t*) P*=P*(Re,Gr,Pr,x*,y*,z*,t*) T*=T*(Re,Gr,Pr,x*,y*,z*,t*)

Where the * quantities are non-dimensionalized and x*,y*,z*,t* are the non-dimensional space and time variables.

Note 1. Other forms can be made by multiplying the dimensionless numbers by each other and including the product in place of one of the multipliers. eg: Re*Pr=Pe, replace Pr or Re by Pe: U*=U*(Re,Gr,Pe,x*,y*,z*,t*)

Note 2. We seldom consider both Re and Gr. A coupling between the momentum and energy equation occurs of the order of Gr/(RE)^2 (also known as the Archemedes No Ar. So for moderatly forced flows the coupling is one way and not dependant upon Gr. But in mixed convection, we must consider both!

Note 3. We often deal with long time averaged (for steady state) and want to have integral quantities ie Cd, h-bar, Nu-bar, etc. and in these cases the integration is performed over time and space and we have forms like:

Nu bar=f(Re,Pr)...for forced convection Nu bar=f(Re,Gr,Pr)...for mixed convection Nu bar=f(Gr,Pr)...for free convection

So, in your non-dimensionalizing, for mixed convection, you should have a dependance on Re,Gr,Pr!!

In your problem: Is there a prescribed velocity(or pressure difference) at the boundaries which represents the velocity scale for forced convection? If so it should show up somewhere in your equations.

To quote Eckert from p525 "Analysis of Heat and Mass Transfer": "A flow situation is called free- or natural convection flow when it is created solely by body forces. In this case, then, no prescribed velocity is available which might be used to define a Reynolds number. As a consequence, the Reynolds parameter drops out of the dimensionless equations, and heat transfer, for instance, is described by an equation of the form:

Nu=f(Ra,Pr)

note: here he replaced Gr by Gr*Pr ie same as Nu=f(Gr,Pr)

Is the flow through your inlets a function of the body forces (and therefore a function of Gr) or is it prescribed??? If the pressure dfference between these openings is a function of the body forces (Gr dependance) then this is really not a forced component but rather a natural convection scaled component that flows into the region!

A thourough read through Eckert's discussion on dimensional analysis and free vs mixed convection is advised.

Second, the numerical problem: In a very similar problem "Finite Element Analusis of Incompressible Fluid Flow Incorporating Equal Order Pressure and Velocity Interpolation" by Schneider, Raithby, and Yovanovich, p 89--102 in Numerical Methods in Laminar and Turbulent Flow, ed, Taylor, Morganm, and Brebbia.

They found convergence of non-linear iterations increadingly difficult to Ra=1000 and divergence at Ra = 10 000. The discussion is quite interesting, and the issue is summed by:

"...summarized by observing that the desired solution to the problem may satisfy a given non-linear set of equations, but it does not automatically follow that this set of equations can be used in an iterative scheme (to remove the non-linearities) to obtain that solution.

What are the details of the solution algorithm that you are using?

good luck..............................Duane

thanks Duane I had problems with the network so could not respond to you earlier. Where I can get hold of Eckert's discussion. Could you give more details of the book on Numerical method you mentioned.

Hi,

did you have any success in getting the references that I gve you?

Eckert's text is in all of the libraries.

Numerical Methods in Laminar and Turbulent Flow, ed, Taylor, Morganm, and Brebbia will probably need to be an inter-library loan!

regards........................................... .Duane

-- .................................................. .. Duane Baker P.Eng., B.Sc.(Mec E), M.A.Sc.(Chem E), M.A.Sc.(Mech E) CFD Research Engineer Alberta Research Council Email: baker@arc.ab.ca .................................................. ...