Dear  Varpe 

Mr Casanova provides you a very good estimator for the uncertainty for the discretization error. THe GCI estimator is good and , to calculate,you will need at least 2 meshes and can provide you very good estimation if it is solving accurately or not, or even not accurate. 

You have to keep in mind that sources of errors or uncertainty are: 

- modeling errors; if you model the physical problem not correctly , you will have errors(and problems) ;

Regarding the numerical method, you have these sources of errors:

- truncation : the term of the Taylor's series that you leave or do not use;/

- iteration: error caused because you have iteration process to solve the linear equation system ;

- rounding : errors due the conversion of the number by the program, language used to coed, hardware and operational system of the computer you use ;

- other errors : errors caused by bad code, not correct data entered , invalid physical parameters, etc.

Depends of the problems you 're solving, truncation is the most cause, but not the only one. There are situations that the truccation errors  can be cancelled and iteration becames bigger than truncation and influencing more than truncation. 

Also, you should do validation, appling some specific "validation tests" or "validation benchmarks" against the code and see if your code runs properly.

I do not know if you are coding or you are using a CFD program to solve. If you're coding, I suggest you read Ferziger and Peric book Computational Methods for FLuid Dynamics and their paper Further discussion of numerical errors in CFD.

ALso, there are very good book by Roache : Verification and Validation in Computational Science and Engineering.

You may read some papers from Carlos Henrique Marchi , Brazilian researcher in Mechanical and AeroSpace Engineering  . For instance 

"Multi-dimensional discretization error estimation for convergent apparent order" (link provided) 

I hope it helps 

http://www.scielo.br/scielo.php?pid=S1678-58782005000400012&script=sci_arttext&tlng=pt

Regarding the uncertainty related only to the CFD model (thus not the uncertainty of your inputs data), first of all you can run a mesh sensitivity analysis and then compare the results of the latest two meshes. The difference will be the error estimate.

Then you can also search for relevant test cases in literature, rebuild it with the same set up (the closest possible to your case) and get the difference between the CFD and the literature.

Finally the best approach could be to run an experimental test, rebuilding with CFD and then compare again.

It is difficult to directly evaluate the numerical uncertainty. However, you can compute error estimates. This would typically include a quantitative study using 3 grid resolutions and an accepted method like Richardson extrapolation, or a modified version such as grid convergence index. See the attached ASME Journal of Fluids Engineering Editorial Policy Statement on the Control of Numerical Accuracy for more details.

Thank you Khaled for providing note and Alessandro for suggestions.

After grid independence study already done,  grid size is chosen. After that the base line model was validated using experimental data for two different performance parameters . Then optimized model with same mesh topology and size was simulated. Now uncertainty has to be estimated for the gain/benefits  with respect to reference case. We donot have an actual experimental data on optimized model or DNS results

Each parameter in model has some range of uncertainty. One can make sensitivity tests with varying values of parameters to study uncertainty of results.  Alternatively one can make tests with different turbulence models (e.g. k-epsilon)

Thank you Ivan and Thierry for sharing views and suggestions. Citations related to the topic  would be highly appreciative.

You can also look for information about the Grid Convergence Index (GCI) to quantify the grid uncertainty.

Dear casanova, here the concern is not of grid sensitivity(already mention) but the result obtained from CFD analysis. What confidence level has to be attached to those relults.

 Grid convergence test/independence study is only the part of whole story. Where numerical errors are ascertained but not the complete flow physics(3d Complex)

In CFD, model validation is very important. you may validate your model using published experimental data for example if you are not able to build the experiment in your lab. Also, you may test the response of your CFD model to some key factors that you have a good background for their impact on your CFD model design. 

Dear Varpe, as far as the confidence levels concerned, I think that the GCI together with the Richardson extrapolation give what you want. Take a look at this page:

http://www.grc.nasa.gov/WWW/wind/valid/tutorial/spatconv.html

Dear  Varpe 

Mr Casanova provides you a very good estimator for the uncertainty for the discretization error. THe GCI estimator is good and , to calculate,you will need at least 2 meshes and can provide you very good estimation if it is solving accurately or not, or even not accurate. 

You have to keep in mind that sources of errors or uncertainty are: 

- modeling errors; if you model the physical problem not correctly , you will have errors(and problems) ;

Regarding the numerical method, you have these sources of errors:

- truncation : the term of the Taylor's series that you leave or do not use;/

- iteration: error caused because you have iteration process to solve the linear equation system ;

- rounding : errors due the conversion of the number by the program, language used to coed, hardware and operational system of the computer you use ;

- other errors : errors caused by bad code, not correct data entered , invalid physical parameters, etc.

Depends of the problems you 're solving, truncation is the most cause, but not the only one. There are situations that the truccation errors  can be cancelled and iteration becames bigger than truncation and influencing more than truncation. 

Also, you should do validation, appling some specific "validation tests" or "validation benchmarks" against the code and see if your code runs properly.

I do not know if you are coding or you are using a CFD program to solve. If you're coding, I suggest you read Ferziger and Peric book Computational Methods for FLuid Dynamics and their paper Further discussion of numerical errors in CFD.

ALso, there are very good book by Roache : Verification and Validation in Computational Science and Engineering.

You may read some papers from Carlos Henrique Marchi , Brazilian researcher in Mechanical and AeroSpace Engineering  . For instance 

"Multi-dimensional discretization error estimation for convergent apparent order" (link provided) 

I hope it helps 

http://www.scielo.br/scielo.php?pid=S1678-58782005000400012&script=sci_arttext&tlng=pt

Thank you Casanova and Abimael for valuable inputs.

I agree with Abimael, there are many sources of uncertainty in the whole modelling process.

Good rules of thumb to check on how accurate your results are could be:

- Test for mesh independence of the results: Se how consistent are your results when you change your mesh from coarser to finer ones.

- Compare your results to similar experiments or your own available data: See if your results are coherent and of the same order of magnitude as from similar cases.

- Fast analytic solutions: Compare your results with analytic solutions you can get from a simplified version of your model.

- Convergence criteria: Play with your convergence criteria in the software you are using, if your results are consistent when meeting with more demanding criteria, this could be an indication of the accuracy of your result.

- Review your results with a verification/validation process: Some organizations have determined some processes to make sure you are doing a good job with your models. For example NASA has some guidelines and there is the ASME V V 20 Standard (Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer) that also will be helpful to review. Some more info at the links below.

http://www.grc.nasa.gov/WWW/wind/valid/tutorial/overview.html

http://files.asme.org/divisions/fed/16302.pdf

Thanks Thierry and Juan. With adequate inputs and literature on the topic, suggested by the responders. Now it seems, home work equivalent to the investigation done exists  for me. Once again thank you All.

Dear Dr. Varpe,

For estimation of uncertainity in computed results, you have been informed about the book by P.J. Roache and websites of NASA  and ASME. You may also consult the following two books.

1. Knupp,P. and Salari, K. "Verification of computer codes in computational science and engineering" Chapman and Hall/ CRC, Boca Raton,2002.

2. Oberkamph, W.L. and Roy C.J. " Verification and validation in scientific computing" Cambridge University Press, Cambridge, U.K. 2010.

With best wishes, E.G.Tulapurkara, I.I.T. Madras, India

You might also want to look at this paper

Article A procedure for the estimation of the numerical uncertainty ...

Our colleagues have already provided excellent references for estimation of the numerical uncertainty, but I would like to add to this. As you are apparently interested in the reliability of your estimated improvement, I will quote the section on V&V in the context of simulation based design from the report of the Resistance committee of the 27th International Towing Tank Conference (You can find the whole report at http://ittc.info/download/proceedings).

"In order to have confidence in a SBD framework the results of the optimisation process should be verified and validated (V&V), i.e. a simulated improvement should correspond to a real-life improvement with a sufficient confidence. It has been proposed that the methodology used for the V&V of single run cases can be extended into a systematic procedure for the V&V of the optimised solution. This V&V process consists of three parts and is based on the difference in performance between a parent and optimised designs. (i) The optimised design is numerically verified, if it can be shown that the magnitude of simulated improvement is larger than the numerical uncertainty. (ii) The optimised design is experimentally verified, if the magnitude of the measured improvement is larger than the experimental uncertainty. (iii) The optimised solution is validated, if the absolute value of the difference between the simulated and measured improvements is less than the combined uncertainty from the simulations and the measurements. (Tahara et al, 2008, 2011)

It should be noted that the methodology is independent of the V&V of the individual solutions for the parent and optimal design and only includes the trend. This is in line with the fundamental goal of the design problem, i.e. to find the optimal design. The absolute values of individual designs can be verified and validated using a single run procedure.

For practical examples of the application of the V&V methodology the interested reader is referred to Tahara et al (2008, 2011) and Kandasamy et al (2013)."

The references cited are

Kandasamy, M., Peri, D., Tahara, Y., Wilson, W., Miozzi, M., Georgiev, S., Milanov, E., Campana, E.F., Stern, F., 2013, "Simulation based design optimisation of waterjet pro-pelled Delft catamaran", International Ship-building Progress, Vol. 60, pp 277-308

Tahara, Y., Peri, D., Campana, E.F., Stern, F., 2008, "Computational fluid dynamics-based multiobjective optimization of a surface combatant using a global optimization method", Journal of Marine Science and Technology, Vol. 13, pp 95–116

Tahara, Y., Peri, D., Campana, E.F., Stern, F., 2011, "Single- and multiobjective design optimization of a fast multihull ship: numerical and experimental results", Journal of Marine Science and Technology, Vol. 16, pp 412–433

Arjan already referred to a paper by Eça and Hoekstra, but I would warmly recommend to have a look at other papers by them as well. They are usually very extensive and detailed.

Dear Varpe,

Computational turbulence models current available are not so much reliable (especially at low turbulence Reynolds number). Before you go for numerical accuracy, it is necessary to refer relevant physical experiments and to compare relative performance of various computational turbulence models. If you are not guided by physical expriment data, you may obtain computational result far from the physical  reality, even if your numerical tool itself is highly accurate - especially at low Reynolds number.      

Let me add one thing. It is supposed that the Launder-Sharma (or, Jones-Launder) low Re k-e model is the least dangerous one when they have no experimental data. It serves usually good for a baseline computation purpose. Good luck! 

Please find this attached thesis regarding the uncertainty that ill help you to resolve your query.

https://www.researchgate.net/file.PostFileLoader.html?id=5454473fd4c118db6b8b492f&key=289a6cd4-bd76-4c23-beda-a9b0abb38169

Thanks Shrish. This is useful is custom codes.How to estimate uncertainty of turbulence models (higher order) in Ansys Fluent. Say perturbing variables in K-omega shear stress model would be cumbersome task.