Timeline for Infinite expresssion 1/0 encountered when use NDSolve for 3D axisymmetric Navier-Stokes (Euler) equations
Current License: CC BY-SA 4.0
15 events
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| Apr 11, 2023 at 6:32 | comment | added | Alex Trounev | @mike Actually we can't reproduce results from Thomas paper one to one, since he used FDM, while we use FEM. | |
| Apr 10, 2023 at 8:40 | comment | added | mike | Hi Alex: I noticed an interesting difference between Thomas numerical solution in Ref.1 and your solution in the answer. If you let $z1=4*10^{-3}} and do Plot[u[40][x,z1],{x,-0.02,0.02}], you will see that u[t(40)][x=0,z1]>1. But from inspection of Figure (3.1) in Ref.1, you will see that |u[t(40)],x=0,z1)|<1/10. This also true for w. As you now that (Thomas)u1==(your)u, (Thomas)omega1==(your)w. | |
| Apr 10, 2023 at 8:27 | comment | added | mike | In the answer to question "mathematica.stackexchange.com/questions/188101/…", @xzczd mentioned that "We can use r=0 as the left boundary because "FiniteElement" is able to handle the removable singularity there". This probably explained why FEM worked here as well. | |
| Apr 7, 2023 at 4:45 | comment | added | Alex Trounev |
Actually the "MaxCellMeasure" and dt are only parameters to improve solution.
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| Apr 7, 2023 at 2:02 | comment | added | mike | Alex: How can I improve the quality of your NDSolve/FEM solution. (1) set "MaxCellMeasure" to 0.0001 or even smaller; reduce dt to 1/10 of its original value and increase the total time steps tp 400. Can I set "InterpolationOrder" to 4? Best regards- | |
| Apr 6, 2023 at 5:05 | comment | added | Alex Trounev |
@mike DirichletCondition[{w[x, z] == 0, f[x, z] == 0, u[x, z] == 0}, True] are boundary conditions same as yours BC, while initial conditions are U[0][x_, z_] := 12000*(1 - x^2)^18*(Sin[2*Pi*z]/(1 + (25/2)*Sin[Pi*z]^2)); W[0][x_, z_] := 0; F[0][x_, z_] := 0;.
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| Apr 6, 2023 at 2:11 | comment | added | mike | I also have question about DirichletCondition[{w[x, z] == 0, f[x, z] == 0, u[x, z] == 0}, True]. Is this an initial condition? Why do we not need boundary conditions? | |
| Apr 5, 2023 at 23:19 | comment | added | mike | You are right. Since doubling the domain in $x$ will not automatically reduce the error near $x=0$, I go back to original domain in Hou's paper $D=\{(x,z):0\leq x\leq 1,0\leq z\leq 1/2\}$. The solution looks similarly nice. $1/x$ term will not cause trouble. | |
| Apr 5, 2023 at 16:11 | comment | added | Alex Trounev | @mike Actually this model defined at $x\ge 0$ only. As I understand, solution at $x<0$ used for convergence only. | |
| Apr 5, 2023 at 15:08 | comment | added | mike | I see. These are the lower-left and up-right corners. Have you checked plots like "Plot[U[40][x, 0], {x, -0.05, 0.05}, PlotRange -> All]" and "Plot[U[40][x, 0], {x, -0.05, 0.05}, PlotRange -> All]"? Why are they anti-symmetric in x? | |
| Apr 5, 2023 at 14:16 | comment | added | Alex Trounev |
@mike In your notation Rectangle[{-1, -1/2}, {1, 1/2}] is $D_2=(x, z): -1\le x\le 1, -1/2\le z \le 1/2$.
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| Apr 5, 2023 at 12:40 | comment | added | mike | The thing people want to now is whether the singularity will form on the z axis (x=0) in finite time. Thus the thing that is less ideal in Hou's paper, in my opinion, is that x=0 is the boundary of their simulation. This may introduce large error. In my equations above, (x=0,z=0) is in the middle of the domain $D_2={(x,z): -1<=x<=1,-1/2<=z<=1/2}$, I was hoping to see that by putting the potential singular region in the middle of the simulation domain, we would see more accurate behavior of the solutions (finite time blowup singlarity formation) | |
| Apr 5, 2023 at 12:39 | comment | added | mike | Very impressive! I would suggest that you publish it. I do not quite understand that "Rectangle[{-1, -1/2}, {1, 1/2}]". The solutions are symmetric in $x$ and anti-symmetric in $z$. I can follow you if you solve the PDEs for $-1<=x<=0,0<=z<=1/2$ and then symmetrize it wrt $x$ and anti-symmetrized wrt $z$. But why "Rectangle[{-1, -1/2}, {1, 1/2}]"? What do is mean? | |
| Apr 5, 2023 at 12:03 | vote | accept | mike | ||
| Apr 5, 2023 at 11:32 | history | answered | Alex Trounev | CC BY-SA 4.0 |