Hi. I’ve been working to update my freefem script. Modelling the transient heat equation, combined with incompressible Stokes flow/thermal buoyancy for water in an (MRI) phantom, sphere simplified to 2D circle of radius 10 cm. (low temperature differences)
First two equations in water, third heat equation everywhere:
([Edit] or simplified equations:
With kinematic viscosity /nu (m²/s), velocity u (m/s), pressure p (N/m²), thermal expansion beta (1/K), density ρ (kg/m3), gravity constant g (m²/s), heat capacity Cp (J/kgK), thermal conductivity k (W/mK), air convection heat transfer coefficient h (W/m²K), normal vector n and heat deposition Q (W/m3) modelled as a gaussian distribution in water only.
When comparing to measurements, the results seem to be consistent. (however, quite high h value (of air-phantom convection) and SAR is not directly linked to machine listed wattage)
I tried to perform all factors as good as possible, basing on online information and some meteorology courses from the past (physicist here, no mathematician). I’m a new user, and also used in a critical thinking way, often correcting, LLMs to iterate and correct parts of my script. My questions:
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Would this implementation seem consistent? (or would it be better to restart from scratch?
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Would mesh adaptation using adaptmesh be recommended?
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I’m not totally happy with the viscosity approach I’ve used, which uses very high values in solid materials instead of only solving in water. Would there be a better way? I guess using two fespaces, but I did not get it working… I also did not find a way to use Vh dynamically, thereofore using a function for buoyancy.
If someone could give some critical return, we would be eternally grateful!
(Caveat, only valid for low temperature differences, if temperature differences of the outside are very high (6°C inside, outside 20 °C), at some point I think the viscosity approach of having high viscosity in solid creates issues, and/or combined with too large velocities/inappropriate mesh and also no longer physical true, requiring possibly Navier).
Enclosed an example of the freefem script.
NIST std V5.edp (22.0 KB)
[Edit] Version without SUPG (SUPG = outside of my comprehension, and seems to result in less stable: no SUPG seems more consistent with measurements also):
NIST std V5_noSUPG.edp (22.0 KB)
[Edit2: changed some comments (SAR value) + correct sign when no SUPG]