## Unstable PDE's

spacedout
Posts: 103
Joined: 30 Mar 2020, 23:27
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### Unstable PDE's

Hello

I created a new thread since the name is more suitable than the previous thread I started on ModelPDE.

I am trying to find out where the problem arises with my plasma physics equations.

So for all solvers I replaced

Linear System Solver = Iterative

by

Linear System Solver = Direct

and for all nonlinear equations, I replaced

Nonlinear System Relaxation Factor = 1

by

Nonlinear System Relaxation Factor = .3

But to no avail: The norm in the nonlinear iterations quickly become infinite within a microsecond.

Is that almost a mathematical proof that this is not a numerical problem but that the PDE's themselves have unstable coupling terms (for example like a term with a wrong sign which grows forever) ?

Wish you all a nice weekend

P.S.

the deltaT is 1.0e-8 sec and the Mach number is .8 (272 m/sec around 1 atm and T = 300 Kelvin)
Rich_B
Posts: 52
Joined: 24 Aug 2009, 20:18

### Re: Unstable PDE's

Hello,

Can you post a minimum working example? Or at least a minimum almost working example?

Rich.
spacedout
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### Re: Unstable PDE's

Actually I cannot ! I tried Elmer's original HeatSolver and FlowSolver both with Linear System Solver = Direct on a standard NACA0012 airfoil and the norm in the nonlinear iterations of the FlowSolver still becomes infinite within microseconds.

There are no other solvers called in the .sif (They are a lot of them but with Exec Solver = Never)

(the deltaT is 1.0e-8 sec and the Mach number is .8 (272 m/sec around 1 atm and T = 300 Kelvin)
I use velocity = 0 Heat Flux = 0.0 boundary conditions on the airfoil itself. And I use
velocity = 272 m/sec, pressure = 1 atm and temperature = 300 Kelvin as boundary conditions everywhere in the farfield.
(These are also the initial conditions everywhere in the domain)

Either these conditions are unphysical or Elmer cannot handle highly compressible flows.
kevinarden
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Joined: 25 Jan 2019, 01:28
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### Re: Unstable PDE's

FlowSolver by itself generally does not converge with high velocity flows. It can be coupled with k-epsilon for imprvement

viewtopic.php?f=3&t=6995&hilit=turbulent
spacedout
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### Re: Unstable PDE's

Turbulence models involve the energy equation but FlowSolver only handles continuity and momentum equations. Shouldn't HeatSolver also look at the turbulence keywords in the .sif ? Not according to its description under ElmerModelsManual. Moreover, I found keywords under ElmerTutorialFilesGUI that were different from those of ElmerModelsManual.

Best,
Marc
kevinarden
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### Re: Unstable PDE's

In Elmer, I believe the heat only follows the flow (convection), except for conduction and radiation, and P=VRT. I believe since FlowSolver only handles continuity and momentum is the reason it can't converge on turbulent flow conditions. Therefore if the flow is turbulent other equations must be introduce. The software is developed and update daily, however the manuals, and tutorials are not, so the documentation lags significantly behind the development.
spacedout
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Joined: 30 Mar 2020, 23:27
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### Re: Unstable PDE's

The theory of characteristics for compressional flows in 3D shows that to avoid instabilities and other problems for subsonic flow ( M <=.8 in my case) then,

we must prescribe at most 4 characteristic variables from the freestream values at the inflow boundary (the 5th variable being extrapolated from the interior domain) whereas at the outflow boundary it is the 5th variable which is prescribed from its freestream value and and all other 4 variables are extrapolated from the interior domain.

We can of course get the primitive variables on the boundaries from the characteristic variables.

The above is standard procedure in finite-volume CFD books but Elmer utilizes a finite element approach. (although finite volume is a special case of finite element methods)

Is this theory included in Elmer ?