[Moreza7]

Hi,


I wrote a MATLAB code to solve the flow over a flat plate using streamfunction-vorticity equations.

The code works fine at low Reynolds numbers (1,10) ,but it diverges at Reynolds=1000 which I want to solve it for.

Even if I use under relaxation factor 0.0005 (!) for streamfuction equation, it still diverges at Reynolds=1000.

I use 600X150 grids. (square grids)

How can I make it work for Reynolds = 1000?

Here are the conditions of the problem:



Boundary Conditions:

[FMDenaro]

I did not read your code but if you are using central second order FD discretization first try to refine the grid up to reach Re_h=1.

Then, from a theoretical point of view, the bottom frontier behind the plate could be not a stream line, the flow could have a vertical velocity.
Have you tried to simulate the full domain with the plat in the center? Of course you should be aware to have double nodes for the vorticity over the two faces.

The outlet BC for the vorticity can be the set as vanishing second normal derivative.

The plate can produce vorticity in the shear that, at high Re number, is unsteady.

[adrin]

I'm testing a similar problem using a much simpler problem (for high-order vorticity transport solver) -> solid wall at bottom, periodic domain in the streamwise direction, and zero velocity at the top (so, basically, there is no vorticity related singularity at the bottom). I'm getting OK solution up to Re = 100, but not beyond (Re = 1000 eventually blows up). I started with a cavity problem and that also blows up beyond Re = 100. The culprit in this case seems to be the singularities at the top corners due to the velocity jump, although I don't understand why a discontinuous solver would have issues with jumps.

With this introduction, I would like to point out that your vorticity boundary condition past the flat plate is definitely physically incorrect (and impossible?). The flat plate has already generated vorticity, which will convect downstream (so, don't expect zero vorticity there). Also, upstream of the flat plate you have a jump in vorticity, which may cause instability if not taken care of correctly in the finite-difference method. I provided my experience above to let others know that I'm seeing similar problems even when using a piecewise discontinuous finite difference solver; so, it would be very valuable for us to share our experiences on this matter

adrin

[FMDenaro]

Originally Posted by adrin

I started with a cavity problem and that also blows up beyond Re = 100. The culprit in this case seems to be the singularities at the top corners due to the velocity jump, although I don't understand why a discontinuous solver would have issues with jumps.


adrin

Generally, the two top corners do not enter into the computatation for second order central FD.

However, I used multidimensional discretizations using the values at the corners and I performed the cases up to Re=10^4 ...

[Moreza7]

Originally Posted by adrin

I'm testing a similar problem using a much simpler problem (for high-order vorticity transport solver) -> solid wall at bottom, periodic domain in the streamwise direction, and zero velocity at the top (so, basically, there is no vorticity related singularity at the bottom). I'm getting OK solution up to Re = 100, but not beyond (Re = 1000 eventually blows up). I started with a cavity problem and that also blows up beyond Re = 100. The culprit in this case seems to be the singularities at the top corners due to the velocity jump, although I don't understand why a discontinuous solver would have issues with jumps.

With this introduction, I would like to point out that your vorticity boundary condition past the flat plate is definitely physically incorrect (and impossible?). The flat plate has already generated vorticity, which will convect downstream (so, don't expect zero vorticity there). Also, upstream of the flat plate you have a jump in vorticity, which may cause instability if not taken care of correctly in the finite-difference method. I provided my experience above to let others know that I'm seeing similar problems even when using a piecewise discontinuous finite difference solver; so, it would be very valuable for us to share our experiences on this matter

adrin

Thanks for your reply.
I used COMSOL to simulate the problem. It gives zero vorticity on bottom boundary past the plate, too.

[adrin]

Did you use the velocity-pressure formulation or vorticity-velocity (or, equivalently, vorticity-stream-function)? I came across a paper that discussed instability for higher Re and using high-order simulation of the steady-state vorticity transport equation (the 2nd-order method is not problematic).

I'm curious what sort of flux limiter(s) you've used successfully; I'm wondering whether this is the potential source of error.

adrin

[FMDenaro]

Originally Posted by adrin

Did you use the velocity-pressure formulation or vorticity-velocity (or, equivalently, vorticity-stream-function)? I came across a paper that discussed instability for higher Re and using high-order simulation of the steady-state vorticity transport equation (the 2nd-order method is not problematic).

I'm curious what sort of flux limiter(s) you've used successfully; I'm wondering whether this is the potential source of error.

adrin

I used the stream function-vorticity formulation without any flux limiter. The instability I found is the Hopf-bifurcation type at Re=10^4. Lower Re numbers provided a steady accurate solution.

The results are shown in my old paper here https://www.researchgate.net/publica...sport_problems

[adrin]

Did you use the velocity-pressure formulation and observed that vorticity is zero post-plate? In this case, what were your boundary conditions for post-plate? I suppose one may impose even non-physical BC, but once vorticity is generated at the plate and convects downstream, it is difficult to imagine a zero vorticity there (the vorticity generated impulsively at the wall is the highest in the field; how can it become zero all of a sudden? For now, I'm having difficulty imagining it)

adrin

[adrin]

Great; I'll take a look at your formulation+implementation

adrin

[Moreza7]

Originally Posted by adrin

Did you use the velocity-pressure formulation and observed that vorticity is zero post-plate? In this case, what were your boundary conditions for post-plate? I suppose one may impose even non-physical BC, but once vorticity is generated at the plate and convects downstream, it is difficult to imagine a zero vorticity there (the vorticity generated impulsively at the wall is the highest in the field; how can it become zero all of a sudden? For now, I'm having difficulty imagining it)

adrin

Maximum vorticity is at the left edge of the plate (Upstream) not at the end of the plate.

COMSOL uses velocity-pressure formulation.
Before and after the plate, I use symmetry boundary condition.

[FMDenaro]

Originally Posted by KevinSpears

Maximum vorticity is at the left edge of the plate (Upstream) not at the end of the plate.

COMSOL uses velocity-pressure formulation.
Before and after the plate, I use symmetry boundary condition.

zero vorticity after the trailing edge is not a physical assumption...what is more is the fact the symmetric condition is a valid assumption only at very small Re number when the flow is so viscosity-prevailing that there is no vortex shedding and you have a steady shear.

[adrin]

I agree the symmetry condition is clearly violated beyond very small Re numbers. However, at low enough Re, due to the symmetry condition (which was not mentioned earlier and was the reason I thought the assigned BC is unphysical) a zero vorticity would make sense. This is because vorticity is negative at the top of the wall and positive at the bottom -> the axis of symmetry would, by default, be zero vorticity.

The exit BC for vorticity is somewhat confusing! In the end, make sure that conservation of circulation is also satisfied and vorticity does not increase/decrease over time

adrin