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When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets velocity without spurious wiggles and assuming that thermal diffusivity is large enough then answer is no. How are these two conditions quantified in the language of dimensionless quantites?

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Look at the cell Péclet number

$$\mathrm{Pe}_h = h v / K$$

where $h$ is mesh size, $v$ is the magnitude of velocity, and $K$ is diffusivity. It is analogous to cell Reynolds number for the momentum equation and is small when "thermal diffusivity is large compared to advection". It is common common in macro-scale fluid dynamics that thermal diffusivity $K$ is very small. If the cell Péclet is much larger than 1, you need stabilization.

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