[Simbelmynë]

Hey,

I am doing a kinetic energy conservation test for my colocated code and have a question regarding the setup.

Mesh: 33x33

I use periodic boundary conditions with a Taylor vortex initial condition.

The kinetic energy is dissipated if the initial velocities are too high, but if I make sure that they are low then there is no dissipation of kinetic energy.

What may be the cause of this? (some Peclét number violation?)

[FMDenaro]

Originally Posted by Simbelmynë

Hey,

I am doing a kinetic energy conservation test for my colocated code and have a question regarding the setup.

Mesh: 33x33

I use periodic boundary conditions with a Taylor vortex initial condition.

The kinetic energy is dissipated if the initial velocities are too high, but if I make sure that they are low then there is no dissipation of kinetic energy.

What may be the cause of this? (some Peclét number violation?)

The conservation of kinetic energy is a constraint valid in the inviscid limit, thus no other Peclet number than infinity you have ...
Running the Euler equations does not mean you have no numerical viscosity in your method, therefore the magnitude of dissipation can depend on the mesh size.

Please, give more details of what you are doing

[Simbelmynë]

Originally Posted by FMDenaro

The conservation of kinetic energy is a constraint valid in the inviscid limit, thus no other Peclet number than infinity you have ...
Running the Euler equations does not mean you have no numerical viscosity in your method, therefore the magnitude of dissipation can depend on the mesh size.

Please, give more details of what you are doing

Of course you are correct about the Peclét number!

I have a coarse 2d grid and I prescribe periodic boundary conditions and an initial Taylor vortex field. I set the viscosity to zero and use CD for the convective derivatives. Time-stepping is a simple explicit Euler (so very small time-steps are used).

I notice that the magnitude of the initial field have an impact on the kinetic energy conservation of the method. I have not tested finer grids, but the kinetic conservation properties are very good for small values of the initial field (k/k0 less than 1e-7 for a 1 second simulation).

[FMDenaro]

Originally Posted by Simbelmynë

Of course you are correct about the Peclét number!

I have a coarse 2d grid and I prescribe periodic boundary conditions and an initial Taylor vortex field. I set the viscosity to zero and use CD for the convective derivatives. Time-stepping is a simple explicit Euler (so very small time-steps are used).

I notice that the magnitude of the initial field have an impact on the kinetic energy conservation of the method. I have not tested finer grids, but the kinetic conservation properties are very good for small values of the initial field (k/k0 less than 1e-7 for a 1 second simulation).

but the FTCS scheme is unconditionally unstable in the inviscid problem!!

[LuckyTran]

This sounds like numerical diffusion.

Setting viscosity to 0 will eliminate physical/molecular diffusion but you will still have numerical diffusion. Numerical diffusion is hard to limit for these types of problems even with a fine grid, since in Fluent you are limited to a few discretization schemes (2nd order in time, and 2nd upwind or central in space). To control numerical diffusion better, you need either a really fine grid able to resolve all the gradients or use a higher order scheme (4th order, etc).

[Simbelmynë]

Originally Posted by FMDenaro

but the FTCS scheme is unconditionally unstable in the inviscid problem!!

OK, so I just ran a 1000 seconds simulation to test the stability. Works fine. The kinetic energy is growing slowly and end up with 1.3% difference (I was forced to use a time-step that is two orders of magnitude larger than what I would like to use, but for this test I think it is sufficient).

[FMDenaro]

Originally Posted by Simbelmynë

OK, so I just ran a 1000 seconds simulation to test the stability. Works fine. The kinetic energy is growing slowly and end up with 1.3% difference (I was forced to use a time-step that is two orders of magnitude larger than what I would like to use, but for this test I think it is sufficient).

but there is no meaning in testing an unstable method ......

[Simbelmynë]

Originally Posted by LuckyTran

This sounds like numerical diffusion.

Setting viscosity to 0 will eliminate physical/molecular diffusion but you will still have numerical diffusion. Numerical diffusion is hard to limit for these types of problems even with a fine grid, since in Fluent you are limited to a few discretization schemes (2nd order in time, and 2nd upwind or central in space). To control numerical diffusion better, you need either a really fine grid able to resolve all the gradients or use a higher order scheme (4th order, etc).

I have a 4th order CD implemented as well, but I have not compared the results directly to the 2nd order CD. However, it is very clear that the impact is far less than if I modify the magnitude of the initial field.

[Simbelmynë]

Originally Posted by FMDenaro

but there is no meaning in testing an unstable method ......

So you mean that this might be the cause of my observations? I just ran the same simulation with Upwind and the kinetic energy is dissipated a lot (although it is decreasing now instead of increasing).

[FMDenaro]

Originally Posted by Simbelmynë

So you mean that this might be the cause of my observations? I just ran the same simulation with Upwind and the kinetic energy is dissipated a lot (although it is decreasing now instead of increasing).

upwind is stable (c<1) and is ok testing it, but I don't see correct using any unstable method even for few time steps (actually, it is useful but for students that experiment what the numerical instability is)

[LuckyTran]

I am not familiar with how you are characterizing the rate of dissipation of kinetic energy. Are we talking about numerical diffusion or something else?

But, numerical diffusion is proportional to the gradients. With higher initial velocity, the gradients are greater and this would obviously result in "more" numerical diffusion. If you want to see the same result with higher velocity, you would need to increase the size of your domain so that the new combination of the velocity and length scale result in the same velocity gradient.

The 1st order upwind scheme is infamously high in numerical diffusion (which also improves stability) so that is no surprise. The FTCS scheme is unstable w/o numerical diffusion, the presence of numerical diffusion can make it stable. The fact that your solution was not unstable after 1000 s is basically proof that there is numerical diffusion in your problem setup.

[Simbelmynë]

Originally Posted by FMDenaro

upwind is stable (c<1) and is ok testing it, but I don't see correct using any unstable method even for few time steps (actually, it is useful but for students that experiment what the numerical instability is)

Yes, well the simulation did not blow up with the CD even though I took 1e6 time-steps and it produces results that are much better compared to the upwind case.

A blend between UD and CD produces approx 8% decrease in kinetic energy after 1000 seconds @ dt=1e-3.

A CD produces approx 1% increase in kinetic energy for the same case (a 4th order CD slightly less).

[Simbelmynë]

Originally Posted by LuckyTran

I am not familiar with how you are characterizing the rate of dissipation of kinetic energy. Are we talking about numerical diffusion or something else?

But, numerical diffusion is proportional to the gradients. With higher initial velocity, the gradients are greater and this would obviously result in "more" numerical diffusion. If you want to see the same result with higher velocity, you would need to increase the size of your domain so that the new combination of the velocity and length scale result in the same velocity gradient.

The 1st order upwind scheme is infamously high in numerical diffusion (which also improves stability) so that is no surprise. The FTCS scheme is unstable w/o numerical diffusion, the presence of numerical diffusion can make it stable. The fact that your solution was not unstable after 1000 s is basically proof that there is numerical diffusion in your problem setup.

I am taking the total kinetic energy in the domain, k, divided by the initial total kinetic energy in the domain, k0.

Perhaps I go about this problem in the wrong way. How would you suggest to test the kinetic energy conservation properties of a method. Any other suggestions would be most welcome.

[FMDenaro]

the numerical instability is a process that can be slow, especially if you used a very low time step....

Your test setting with Taylor is good, you have simply to use stable schemes and checking the integral of the kinetic energy in time with the same initial condition but for refined grids.

[Simbelmynë]

Originally Posted by FMDenaro

the numerical instability is a process that can be slow, especially if you used a very low time step....

Your test setting with Taylor is good, you have simply to use stable schemes and checking the integral of the kinetic energy in time with the same initial condition but for refined grids.

OK, I just ran a 65x65 (up from the 33x33) and there is a sizable improvement in the Upwind solution. What constitutes as a good solution? I have a paper that claims the Rhie-Chow interpolation method produces poor solutions in terms of kinetic energy conservation. However, their test case is for 1 second and it shows that Rhie-Chow dissipates less than 0.03% in that time-span. In what situations will this be a problem (long simulations using LES?)?

[FMDenaro]

you could check further test-cases as I did in Int. J. Numer. Meth. Fluids 2007; 53:1127–1172.

the LES equation provided a velocity field that is no longer v(x,t) but v_f(x,t). The equation for the energy field v_f.v_f shows that such quantity is no longer conserved