2nd Order Spatial accuracy: Euler Equation on Unstrutured Grid

AshwaniAssam · January 27, 2015, 04:59

Hi all.

I am trying to get a second order accurate solution for the euler code on an unstructured grid. At present I am using the Green Gauss Gradient reconstruction with max-min and Venkatkrishnan limiter. I am using primitive variable formulation for gradient reconstruction on the faces. Scheme used is explicit AUSM of Liou. 2006.

For a 1-D shocktube flow problem, in the y-direction I am getting a flow with magnitude around 1, whereas for the same scheme and hexahedral grid with structured grid formulation, I was getting in the y-direction a magnitude of around 1e-3.

Because of this I suspect, that when I try to run the code for high Mach number, the code blows.

Can anyone, please help me, in implementing, 2nd order spatial accuracy on an unstructured grid, for compressible flow code.

with regards,
Ashwani

naffrancois · February 7, 2015, 19:06

Hello,

It is not unusual to have non-zero velocity components using non-aligned grids. The magnitude of these components should be however small compared to the main direction of the flow. Actually, as it is related to the truncation error it should decrease with mesh refinement, if it does not well there is probably something wrong. Then, you might have a problem with your gradient computation. There are many ways to check this, the very first test would be to check zero gradients for uniform flow. Then you may also test a shock tube problem in a square (or cubic in 3d) domain and change the direction of the flow.

A deeper test would also involve study of the accuracy of the gradient in an asymptotic sense, taking an exact solution from which you can extract exact gradient and compare it to the numerical gradient on a sequence of increasing grid sizes.

Don't forget that Green gauss gradients are quite inaccurate on stretched tetras if you compute the face values as an arithmetic average. Least-squares do better. But if your grid is fairly good and sufficiently refined I would not bother too much about this at this point.

Keep in mind that comparing 2nd order reconstruction on tetras and hexas will always be quite disapoiting. You will get better resolution of the rarefaction wave mainly and a slightly sharper front at the shock. It will always look better using hex, it is the price to pay

Good luck !

January 27, 2015, 04:59	2nd Order Spatial accuracy: Euler Equation on Unstrutured Grid	#1
AshwaniAssam Senior Member Ashwani Join Date: Sep 2013 Location: Hyderabad Posts: 154 Rep Power: 12	Hi all. I am trying to get a second order accurate solution for the euler code on an unstructured grid. At present I am using the Green Gauss Gradient reconstruction with max-min and Venkatkrishnan limiter. I am using primitive variable formulation for gradient reconstruction on the faces. Scheme used is explicit AUSM of Liou. 2006. For a 1-D shocktube flow problem, in the y-direction I am getting a flow with magnitude around 1, whereas for the same scheme and hexahedral grid with structured grid formulation, I was getting in the y-direction a magnitude of around 1e-3. Because of this I suspect, that when I try to run the code for high Mach number, the code blows. Can anyone, please help me, in implementing, 2nd order spatial accuracy on an unstructured grid, for compressible flow code. with regards, Ashwani

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February 7, 2015, 19:06		#2
naffrancois Senior Member Join Date: Oct 2011 Posts: 239 Rep Power: 16	Hello, It is not unusual to have non-zero velocity components using non-aligned grids. The magnitude of these components should be however small compared to the main direction of the flow. Actually, as it is related to the truncation error it should decrease with mesh refinement, if it does not well there is probably something wrong. Then, you might have a problem with your gradient computation. There are many ways to check this, the very first test would be to check zero gradients for uniform flow. Then you may also test a shock tube problem in a square (or cubic in 3d) domain and change the direction of the flow. A deeper test would also involve study of the accuracy of the gradient in an asymptotic sense, taking an exact solution from which you can extract exact gradient and compare it to the numerical gradient on a sequence of increasing grid sizes. Don't forget that Green gauss gradients are quite inaccurate on stretched tetras if you compute the face values as an arithmetic average. Least-squares do better. But if your grid is fairly good and sufficiently refined I would not bother too much about this at this point. Keep in mind that comparing 2nd order reconstruction on tetras and hexas will always be quite disapoiting. You will get better resolution of the rarefaction wave mainly and a slightly sharper front at the shock. It will always look better using hex, it is the price to pay Good luck !