Taylor-green vortex problem

General
1

Hi all,

I’m a PhD student comparing PyFR to standard CFD tools in a hope to transferring high-order technology to industrial applications.

As a start, I’ve recently tried to run 3D Taylor-Green vortex case (Re=1600, M=0.1 Ref:[1]) using compressible PyFR solver (PyFR 1.7.5). I’m running the case using coarser grids (uniform hexahedra, 16^3 and 32^3 using polynomial order 4) with respect to what would be required by a DNS. I observe a matching kinetic-energy (E) vs. time curve to the reference DNS. However, the time derivative of E (-dissipation rate) is wiggly, the magnitude of wiggles being significant for 16^3 P4 case. Figures are attached for clarity. Has anyone seen similar things before?

Things I’ve tried to ensure the wiggles are not originating from how often I save the data but the simulation itself:

  1. Saving of two very close instances in time (5 time steps apart) and taking time derivative based on these.

  2. Reducing the time step by 1/8th of the one shown in the config file attached.

The above treatment did not change the result.

I also tried over integration in element volume and interfaces using 2x integration points, but in vain. On the other hand the exponential filter helped a bit, although a lot of tweaking in strength and cut-off number was required. Any comments on their usage?

Thank you for your time.

Attachments: config file (also showing over integration approach), mesh file, E vs. time and -dE/dt vs. time curves obtained. The attachments correspond to 16^3 P4 simulation.

[1] B. C. Vermeire, F. D. Witherden, and P. E. Vincent, “On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools,” J. Comput. Phys., vol. 334, 2017.

PyFR_64cube.png
PyFR_64cube.png377×586 21.7 KB

condnFile_TGV.ini (1.51 KB)

tgv_16cube.msh (530 KB)

2

Hi Vishal,

Yes this behaviour is expected and has been reported by other DG and FR research groups . As you coarsen the mesh while keeping the Reynolds number constant you are relying more on numerical dissipation to dissipate energy, rather than physical dissipation. If the mesh is extremely coarse, as it looks like may be the case in your simulations, it can allow aliasing instabilities to grow and ultimately give negative density or pressure. The lower-order schemes tend to be less prone to this in my experience, since they are more dissipative.

As a starting point, you may want to read the Taylor-Green section in paper [1] where we explored different FR schemes in exactly this configuration.

This has also been studied by other groups in the DG and FR community. For example [2] shows plots very similar to yours. You should be able to find other Taylor-Green papers that discuss it in more detail.

[1] B.C. Vermeire, P.E. Vincent, On the properties of energy stable flux reconstruction schemes for implicit large eddy simulation, Journal of Computational Physics, Volume 327, 2016
[2] J.R. Bull, A. Jameson, Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes, AIAA Journal, Vol. 53, No. 9, 2015.

3

Hi again,

Thank you for the explanation.

“”"

Yes this behaviour is expected and has been reported by other DG and FR research groups . As you coarsen the mesh while keeping the Reynolds number constant you are relying more on numerical dissipation to dissipate energy, rather than physical dissipation. If the mesh is extremely coarse, as it looks like may be the case in your simulations, it can allow aliasing instabilities to grow and ultimately give negative density or pressure. The lower-order schemes tend to be less prone to this in my experience, since they are more dissipative.
“”"
What surprised me in my simulations was that non-dealiased results were not much different than the dealiased results. However, I could only test this for poly. order P=4. I’ll try it out for P=3 as well.

“”"

As a starting point, you may want to read the Taylor-Green section in paper [1] where we explored different FR schemes in exactly this configuration.
“”"
I had a chance to go through this paper few weeks ago. Very informative and motivating for using FR schemes! What I really missed in this paper was energy dissipation vs. time results in Taylor-Green vortex section. It’d have been nice to see how the simulations behaved before falling over (or how they continued). For example, in my results attached above, the simulation did not fail but it did not give a stable result either. I understand that there are page limits in the journal papers. It’d be very helpful if you could provide the document where these results are plotted (if it exists).

Thanks again for reference [2]. It certainly is useful.

Best regards,
Vishal

4

Hi Vishal,

In my experience, anti-aliasing does not improve the accuracy of these kinds of simulations very much when compared to other factors, such as the polynomial degree. It is usually good at stabilizing things, but in this case your meshes may just be too coarse for even that.

In the paper I referenced we ran several thousand simulations, so it was not practical to put dissipation plots for them all. However, the meshes and ini files are included in the supplementary material so you should be able to reproduce the dissipation plots quickly using those.