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I'm doing a 2D combustion hydrodynamic simulation and there's a hydrodynamic instability that should be triggered because of the particular physical properties of my system. The key to the instability are temperature gradients perpendicular to the flow direction of my problem. What is a good and "consistent" way to artificially add fluctuations in this perpendicular direction to trigger this instability?

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2 Answers 2

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If the problem is truly unstable, round off error is probably enough to kick off the phenomena, though it might take awhile. If there's a small stability region such that you're stuck in a steady state, you might try restarting the flow from the steady solution with a small random or sinusoidal perturbation in the temperature. I prefer this to messing with the velocity and pressure since there's generally no consistency with the continuity equation to worry about. There may be similar arguments to not perturbing the species concentrations since you don't want to have to worry about consistency in the mass.

Edited to add: If perturbing your temperature field makes your worry about consistency with your equation of state, you can always add a right-hand-side function to your energy equation and put the perturbation function there for a few time steps instead of relying on a perturbed initial condition.

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  • $\begingroup$ I think my problem is stuck in a steady state because it's a 2D slab simulation where both y boundaries are set to symmetric, so without fluctuations you get a flow in the x direction with no gradient in the y direction (i.e. every "slice" across the y direction is the same). If the problem had more irregular geometry I think the roundoff errors would be fine but I'm not sure it will kick off with the setup i have. $\endgroup$
    – mathdummy
    Commented Aug 12, 2015 at 23:59
  • $\begingroup$ @mathdummy, I just realized you might have a problem. If you don't have a $T$ or $\nabla T$ forcing term somewhere, you may not be able to kick the instability off with only perturbations in the temperature field. Do you have gravity? $\endgroup$
    – Bill Barth
    Commented Aug 13, 2015 at 0:02
  • $\begingroup$ Well, the instability is kicked off by a cooling mechanism that is very sensitive to temperature and in turn the equation of state is sensitive to temperature as well. Hence the need to have some sort of temperature gradient. The scales I'm dealing with make gravity irrelevant. $\endgroup$
    – mathdummy
    Commented Aug 13, 2015 at 0:05
  • $\begingroup$ Then your best bet may be adding a gaussian bubble to your temperature, or hitting your energy equation with a gaussian forcing function for a few time steps (aka, hit it with a laser). $\endgroup$
    – Bill Barth
    Commented Aug 13, 2015 at 0:11
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The Gaussian is annoying to add since it isn't compactly supported. The bump function solves that but has nasty higher derivatives that require resolution. I spent a little time once figuring how to get a nicer bump function for just the purpose you describe. Write up athttp://www.scribd.com/mobile/doc/61560918/Behavior-of-the-bump-function-s-derivatives

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