hi, I want to know whether power law differencing scheme provides better accuracy than upwind differencing scheme for 3-D convection-diffusion problems for laminar flows Actually, I am looking for better differencing scheme to get rid of convergence problems related to energy balance. This may not be also a remedy to get a converged solution for viscous flows. I am just seaching all possibilities causing a convergence problem.

Thank you for all considerations

nuray

I think the power law scheme is somewhat a measure to raise the accuracy of interpolation whereas the upwind scheme considers physics such the flow diection or wave propagation direction. I believe that a scheme considering physics will make the convergence better. To get higher accuracy, you can use higher order upwind scheme. Maybe you treat subsonic flow, then the scheme QUICK is quite good.

X. Ye

I am dealing with very viscous flow i.e., glass currents at high temperatures

Then you can try firstly the low order upwind scheme. Such scheme has a problem in terms of numerical diffusion. But since you treat high-viscous flow, the physical diffusions are much higher than the numerical diffusion. Such low order upwind scheme is described well in the book of Patankar "Numerical Heat Transfer and Fluid Flow"

X. Ye

First of all thank you for your interest.

I am still using the upwind schme well desribed in the book of Patankar. Therefore I am searching for alternative scheme

nuray

One alternative can be the QUICK scheme. See B.P. Leonard (1979): A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Computer Methods in Applied Mechanics and Engeneering Vol. 19, p 59-98.

Good luck

X. Ye

I AM AWARE OF QUICK SCHEME. DO YOU HAVE OR KNOW ANY EXPERIENCE ON THE APPLICATION OF THIS SCHEME FOR VISCOUS FLOWS. I WANT TO SAVE TIME BECAUSE MODIFICATION OF THE CODE FROM ONE SCHEME TO ANOTHER WILL TAKE TIME.

NURAY

I think you must use the Hybrid Linear Paroabolic Aproximation (HLPA). I have implemented it in free surface flows, and it works very well. But it is boring to implement.

Valdemir.

I THINK YOU MEAN HYBRID DIFFERENCING SCHEME OF SPALDING WHICH IS BASED ON A COMBINATION OF CENTRAL AND UPWIND DIFFERENCING SCHEMES . IS THAT TRUE ? IF NOT, COULD YOU GIVE ME A RELATED REFERENCE? NURAY

Power Law and Hybrid Method, as described in Patankar are not much more accurate than Upstream. What someone wants is to use a high accuracy discretisation in order to save Kontrollvolumes and receive a fast solution for his problem. for a defined number of grid points you receive almost the same results with all methods. Despite of it, if you are unlucky you get oscillations or a divergence. There is an other Method ( flux blending Method ) which uses a linear Kombination of Upstream and central deferences. This methos is easy to use but won't impove your solution much. You may get oscillations if your central deference part is getting big but this might be a first step. As it is said viscous flows are well treated with upwind

THABK YOU FOR YOUR COMMENTS. SUPPOSE THAT THE CONVERGENCE OF THE PROBLEM UNDER CONSIDERATION IS VERY DEPENDENT ON THE GIVEN INPUT DATA ESPECIALLY DENSITY AND THERMAL EXPANSION COEFFICIENT. ANY CHANGE IN THESE DATA CAUSE CONVERGENCE PROBLEM RELATED TO ENERGY BALANCE. THIS MEANS THAT IT IS POSSIBLE TO GET CONVERGED SOLUTION FOR 10 TIMES MORE VISCOUS FLOWS USING UPWIND SCHEME. BUT THE SITUATION IS DIFFERENT WITH LESS VISCOUS FLOW. ( VISCOSITY IS 11.93 KG/MS AT 1700 K) I AM TRYING TO GET RID OF THIS INSTABILITY BY A NUMERICAL APPROACH. CHANGING THE DIFFERENCING SCHEME WAS THE FIRST TRIAL.

NURAY

I send you a subroutine of the QUICK scheme written in FORTRAN in a separate email. But it is a 2D version. You have to extend it to 3D.

X. Ye

I used QUICK and later extended it to a third order scheme using spline methods in the early 80's for my Masters - unpublished work. The extension from hybrid to QUICK is a matter of one week at most - there are published reports that spell out how to put each term where.

I have to warn you that if you look at the phase plots of these "standard" higher order schemes you'll see that the window (the range) within which they are stable is smaller than that of upwind and/or hybrid schemes. If I remember correctly Patankar shows an example of the Von-Neumann diagrams for the hybrid/upwind/central difference schemes. So, while QUICK is more accurate, the stability constraint usually limits your choice of grid size, etc. more than for hybrid.

Another problem that you need to be aware of with QUICK is the application of the boundary conditions. The stability of the solution is absolutely dependent on the type of scheme you use for the B.C.'s. You need higher order (compatible) extrapolation schemes to "blend in" the interior points with the boundary points. Basically, don't expect the upwind/hybrid B.C. schemes to work with QUICK - you'll just get garbage even for flows as "simple" as backward-facing step, let alone heat transfer problems.

So, if you want to improve stability I seriously doubt that using QUICK is the answer.

Adrin Gharakhani

(1) What is a typical Reynolds number of your problem?

(2) Are you using pressue smoothing (collocated variables). This can be a source of all sorts of problems.

(3) If your Reynolds number is low. Why not use a finite element approach? This have been well developed for creeping non-Newtonian flows for a great many years.

Hi,

1)The accuracy of the upwind differencing scheme is only first order in terms of Taylor series truncation error. Although this scheme is stable, it may lead to serious numerical diffusion when the grid lines and the flow path are not aligned.

2)Such errors can be minmised by using higher order discertization schemes such as Quick.

3)The mathematical formulation of Quick scheme looks more complex than the Upwind, but it can be easily implemented in the for of a deferred correction to the Upwind scheme( basically all you need is an extra source term in your discretized eqution).

4)However, the Quick scheme may give overshoots or undershoots in regions where there are steep gradients. To remedy this problem the Quick scheme needs to be bounded.

(1). I like it. I really like it. (2). High accuracy means that if your solution is near the right answer and if the condition is good, it will give you better results. Otherwise, diverged answers.

(1). I know that you are anxious to find a solution to your problem. (2). The answer to your first question is : after you have used the upwind method, I think, the power law method ( or the exponential function method) will give you a better solution,( if you can bring the converged solution using the upwind method.) (3). It is a little bit confusing at this point, because your problem is related to the energy equation. Normally, the energy equation is the easiest one to solve, unless you have heat release like combustion. (4). Now we can narrow down the problem areas. It looks like that the source of the problem is from the thermodynamic properties of the material. The large variation of the properties can also create stability or convergence problem. (it definitely will create oscillation and convergence problem!) (5). I think, you will have to go back to the equations and study the stability issue to include the variation of the properties in addition to the convection terms. You are now in the research end of algorithm development. (6). Basically, the stability of the algebraic equations can be related to the physical convection terms, the source terms and the coefficient terms from the geometry or the properties. Each one has to be handled separately. (7). You are doing just fine, except that the problem is the real-world problem. Hope that you can see the problem more clearly. good luck.

No. But it is similar to HYBRID.

If you is happy with HYBRID, I sugest you go on and take into accont the diffusion. Also, take care with oscilations in the solution by using the HYBRID.

Valdemir.

Sorry. At the end the last message, you must read QUICK and not HYBRID.

Good luck. Valdemir.

All comments are very useful for me. Let me clarify the problem a little bit. Laminar viscous flow at high temperature range (1100-1500 C). The main problem is due to large variation of viscosity within the temperature range under operating condition. It shows exponential behaviour.

viscosity = (10.**(term1+(term2/(temp-term3)))) Therefore steepness at the begining of this function cause convergence problem.

However, it is possible to get converged solution with a small variation in density and thermal expansion coefficient.

Then I want to apply a suitable differencing scheme in regions where there are steep gradients.

Nuray