NACA0012 grid dependence study by Michael Alletto
- contributor: Michael Alletto
- affiliation: WRD MS
- contact: click here for email address
- OpenFOAM versions: v2112
- Published under: CC BY-NC-ND license (creative commons licenses)
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Contents
NACA0012 grid dependence study
Files can be downloaded here.
Results stationary airfoil
As stated in the introduction, the best way to evaluate the performance of a given solver to treat a complex problem, is to have a series of test cases with increasing complexity. Each test case can be evaluated separately allowing to track down the strength and limitations of the solver. Furthermore the interpretation of the results is much easier in a simple case compared to a complex one. The gained knowledge of the simple test cases can then be applied to understand the more complex problem. In this spirit, the first evaluation performed is the one using a fixed grid.
Note that all python scrips which are used to generate the plots shown in this document are included in the provided tutorial.
Grid size variation
This section shows the results of the grid independence study. The size and the labels of the grids evaluate are shown in the table below. The coarsest grid had 170 points in chord wise direction and 40 points in direction normal to the airfoil surface. The finest grid had 400 points in chord wise direction and 100 points in direction normal to the airfoil surface
| grid0 | grid1 | grid2 | grid3 | |
|---|---|---|---|---|
| size | 170x40 | 200x50 | 299x75 | 400x100 |
The next figure shows the coarsest grid used (left) and a zoom of the grid around the airfoil.
The figure below shows the variation of the pressure coefficient (left), the skin friction coefficient (right) and y+ (figure at the bottom left) with the grid size. At the bottom right the evolution of the lift coefficient with time is shown. Since we use local time stepping we have to ensure that the solution is sufficiently converged. For this purpose it is suitable to analyze the evolution of some integral quantity. Since the lift is the quantity we are most interested in, we plotted it over the time. It is evident that the lift coefficient obtained a steady state before the end of the simulation. The experiments of (labeled as exp) and the simulation of (labeled as ref sim) are included as reference. The reference simulation used a grid of approximately 40 000 cells where the wall was fully resolved (all first cell centers lied at y+ < 1). They used the Spalart Allmaras model with the modifications introduced by together with a density based solver. It is evident that already for grid1 there is no noticeable difference to the finer grid regarding the pressure coefficient. The friction coefficient still changes with the grid size. Noticeable is that the present set up with equilibrium wall functions predicts a small shock induces recirculation bubble which reattaches at approximately 0.7 x/c. The extend of the recirculation bubble is similar to the reference simulation which fully resolved the wall. Y+ is in most regions of the flow between 40 and 300 for all grids where the assumption made to derive the wall functions are valid.
![variation of the pressure coefficient (left), the skin friction coefficient (right) and y+ (figure at the bottom) with the grid size. At the bottom right the evolution of the lift coefficient with time is shown. The experiments of [3] (labeled as exp) and the simulation of [2] (labeled as ref sim) are included as reference.](/File:GridSTudy_alettonaca.png)
Since there is almost no difference in the pressure coefficient between grid1 and the finer grids we will continue our evaluation with this grid. This choice is motivated as follows: 1) We are interest in evaluating the performance of the solver to predict the loads on periodically pitching airfoils. Since the loads are predetermined predominantly by the pressure distribution around the airfoil the accurate calculation of the skin friction is not required. 2) We will consider only small angles of attack where the flow remains attached or only a small recirculation bubble appears. For this kind of flows wall functions and a grid which is not too fine is evidently good enough. 3) For the purpose of a tutorial we want to keep the computational time as limited a possible.
At this point we want to spend a few words about what is a grid which is good enough. In this tutorial we do only visual comparison to keep things as simple as possible. In practice one will do quantitative comparison on the quantities of interests, e.g. the lift computed and calculate the error compared to experiments. Once one has obtained an accurate estimate of the error made by the computation, it can be decided if the errors are small enough or too big for a given application.
References
[1] Jack R Edwards and Suresh Chandra. Comparison of eddy viscositytransport turbulence models for three-dimensional, shock-separated flowfields. AIAA journal, 34(4):756–763, 1996.
[2] Michael Iovnovich and Daniella E Raveh. Reynolds-averaged navierstokes study of the shock-buffet instability mechanism. AIAA journal, 50(4):880–890, 2012.
[3] John B McDevitt and Arthur F Okuno. Static and dynamic pressure measurements on a naca 0012 airfoil in the ames high reynolds number facility. 1985.
