[waven]

Hello.

I have been making 2d euler code for airfoil geometry.

I used Roe scheme for flux treatment.

When i used first order scheme for spatial descretization, calculation was well converged.

But i used second order scheme, error showed up.

Specific symptom is below.

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Starting calculation, residual decreased well up to around 0.005 order during several itreations.

But after this iteration, residual increased up to around 0.037 and in the result, error showed up.

I recognized that when residual increase, density on the leading edge become lower than order of density value which is calculated by compressible flow theory.

And then density on the leading edge continue to decrease(=residual continue to increase) and become lower than it of freestream. In the result, error show up.

(Freestream Mach number is 0.7)
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When i used first order scheme, this error phenomenon didnt take place.

If you had experience like this or you could guess cause of this error, please give me an advice about what should i check?


When invicid wall boundary treatment is wrong in second order scheme, this phenomenon can happen?

Thanks.

[FMDenaro]

Are you using second order central or upwind discretization? In the former case have you considered that central discretizations for inviscid flow drive to numerical oscillations and finally can produce the solution to blow-up?
You should provide details..

[waven]

I used the second order upwind as shown in below description.

q(j,k)_left=q(j-1,k)+(q(j,k)-q(j-1,k))/2
q(j,k)_right=q(j+1,k)-(q(j+2,k)-q(j+1,k))/2

[FMDenaro]

Originally Posted by waven

I used the second order upwind as shown in below description.

q(j,k)_left=q(j-1,k)+(q(j,k)-q(j-1,k))/2
q(j,k)_right=q(j+1,k)-(q(j+2,k)-q(j+1,k))/2

And how do you ensure the upwind criterion? I see only fixed stencils

[waven]

My descriptin cant be considered as second order upwind? If that, could you let me know the exact description and texts or paper containing the exact description?

[FMDenaro]

Originally Posted by waven

My descriptin cant be considered as second order upwind? If that, could you let me know the exact description and texts or paper containing the exact description?

upwind means that you adapt the stencil to the velocity direction

[waven]

I used cell centered scheme. so i think that my description means the second order upwind at the east face of cell. if you think i was wrong, could you correct my wrong description? you can correct it easily with copy && paste.

[FMDenaro]

Originally Posted by waven

I used cell centered scheme. so i think that my description means the second order upwind at the east face of cell. if you think i was wrong, could you correct my wrong description? you can correct it easily with copy && paste.

if you consider an FD discretization at the node i, a second order upwind stencil uses i,i-1,i-2 for u>0 and i,i+1,i+2 for u<0.

In a FV discretization, the flux at the face left is discretized over i-2,i-1,i for u>0 and i-1,i,i+1 for u<0, this is at third order of accuracy

[waven]

you suggested third order scheme for FVM, How is second order scheme described?

and what does the 'u' mean? is it normal velocity at the face?

could you let me know the text related to your description? it is hard to totally understand your description without front and back context.

[FMDenaro]

Originally Posted by waven

you suggested third order scheme for FVM, How is second order scheme described?

and what does the 'u' mean? is it normal velocity at the face?

could you let me know the text related to your description? it is hard to totally understand your description without front and back context.

u is the convective velocity, have a look to any CFD textbook for the upwind criterion and to the Leonard papers for the third order upwind scheme (QUICK/QUICKEST)

[apoorv]

Originally Posted by waven

I used the second order upwind as shown in below description.

q(j,k)_left=q(j-1,k)+(q(j,k)-q(j-1,k))/2
q(j,k)_right=q(j+1,k)-(q(j+2,k)-q(j+1,k))/2

You will need limiter to limit your reconstruction. In case of shock q(j+1)-q(j) would be discontinuous and lead to error. Refer to Blazek book for further help.

[naffrancois]

I agree with apoorv, you may read the book of Blazek:

Computational Fluid Dynamics: Principles and Applications

Especially Chapter 4 where you will find many things you need for structured finite volume scheme of Euler equations.

[waven]

Thanks a lot to apoorv and naffrancois.

I refered to Blazek book and studied MUSCL limiter.

But as far as i know, limiter is not needed in subsonic condition like my case such as Mach number 0.4~0.7. Because shock is not generated.

Limiter can solve this problem?

[apoorv]

Yes limiting should help, as it will avoid anyover-shot and undershoot, so as to reduce CFL a bit. Also can you give me detail of solver explicit (how many rk loop) or implicit; what cfl you are running

[waven]

Thanks a lot apoorv.

CFL is 0.15, time integration method is first order explicit.

When i used limiter, flux from south at the cell above first layer cell on the wall had error. wall boundary cell had no problem but right above cell had error due to error at the south-dir flux.

apoorv, could you give me an advice on what should i check?