Cats2D Multiphysics > Documentation > Driven cavity papers

## Lid driven cavity papers

The square lid driven cavity is one of the simplest, most famous, and most solved problems in computational fluid dynamics. A great many papers have been written to obtain numerical benchmarks of various features of this problem. Yet I think there are a number of interesting features related to the physics of driven cavity flows that aren't widely appreciated. At the end of 1993 I submitted two manuscripts to Journal of Fluid Mechanics focusing on different aspects of this problem. Both were rejected. It was an interesting experience, but I felt that my work did not get a fair hearing there.

Neither of these papers was ever published, which I think is a shame, but ultimately my own fault. In any case, I've decided to resurrect them here. One of them focuses on how flow turns around in blocked channels (6.8 Mb PDF). It's worth checking out just for the high resolution graphics. The other paper has two parts, one establishing a new and accurate benchmark solution to this classic problem, and the other reporting solution multiplicity on slightly non-square cavities. If I was to rewrite this paper I would probably separate it into two papers, but I wouldn't change the content much.

I resubmitted this paper to Journal of Computational Physics in 1996. It was accepted subject to minor revisions in the form posted here.

The lid driven cavity paper(14 Mb PDF)

Since I never revised the paper and it remains unpublished, I am including one of the reviews because it raises some minor objections, some of which I think are valid, but none of which overturns anything of consequence in the paper.

I am also including a review from its submission to JFM. Here is my reply to the editor. There is some hilarity in watching the reviewer hang himself by the comparison to Ralph Goodwin's work, but ultimately this behavior is toxic, and far too common in science. It's obvious that the reviewer barely skimmed the paper before jumping into the deep end.

In 1993 the biggest mesh tested for the benchmark studies was 110x110 elements, which was a huge mesh at the time. Using the new Cats2D I can push the mesh refinement to 400x400 elements (1.75 million unknowns), and I can do it on my mid-range laptop computer in 20 seconds per Newton iteration. You cannot get a cup of coffee in 20 seconds.

I have reproduced the numbers in tables 4 and 5 of the paper by solving the problem on uniform meshes of 200x200 and 400x400 elements. The difference in these two ultrafine solutions is so small that Richardson extrapolation is unnecessary to declare the 400x400 mesh to be the exact solution, to the precision reported in the tables. I've modified the tables to highlight numbers in which the final digit varies by more than +1 or -1 from the ultrafine solutions (the number in red is the error in the final digit of the highlighted entries). The tables are remarkably accurate. But so is Cats2D.

I've recently found some benchmarking studies of high quality that postdate my work. In 1998 Botella and Peyret published accurate solutions for Reynolds number up to 1000 using a special method to subtract the velocity singularity at the corners of the moving lid. In 2005 Erturk and coauthors published solutions for Reynolds number up to 21000 using a stream function-vorticity formulation, and these are very accurate to at least 5000, the highest value that I have benchmarked. Both these studies agree with Cats2D to the significant digits in the tables wherever comparisons can be made. Notably each was based on a completely different formulation of the discrete problem than used in Cats2D.

Cats2D has over 20 years of validation backing up its correctness, robustness, and accuracy in solving complex multiphysics problems in transport phenomena.