Dear researchers,
I am currently working on the development of a Low Mach Number multi-species solver (a system similar to that used in LMN combustion, but with no reactions at present ...) I am having issues with deriving the exact temperature equation (note that c_p varies for each species...) I emphasize that I am looking for having an equation expressed as a function of the temperature (not enthalpy, not internal nor total energy !)
- Can you please help by providing some references that can help to derive this equation ?
- Shall I start from the conservation equation of the internal or total energy ? should I neglect specific terms ?
- From your experience, is the inter-species diffusion term important and should be taken into account in such a system of GE ?
- Same for the term div(Pu) ? any ideas
- Same for the viscous tensor (u Tau) ... this is so complex to code in fact ....
Thanks for your comments and recommendations !
Regards
Elie
Hi,
I'm currently working on a similar code.
As a suggestion, a good starting point for the temperature equation can be found in Theoretical And Numerical Combustion by Poinsot and Veynante, which derived the temperature equation in the low Mach hypothesis.
The diffusion term is essential, but it depends on which fuels you are targeting. Generally speaking, the diffusion should be taken into account through the full Maxwell equation, but it's prohibitive in terms of computational costs, a more frequent approach is a constant Lewis number diffusion or a Hirshfelder-Curtiss approach.
The divergence of velocity needs a correction depending on how your code solves the full system equation. On that side, there are many approaches. I can suggest you read the Kerkemeier thesis.
Hope to have been helpful.
Regards,
Francesco.
I suggest that you develop your mathematical approach around the fundamental quantities that are conserved rather than secondary ones. For example, temperature is not conserved, while energy is. Force, momentum (angular and linear), mass, and energy are conserved, while velocity isn't. There are also "conservative" forms of partial differential equations (Google "navier-stokes conservative form") and non-conservative ones. For example ∂(ρCT)/∂x vs. ρC∂T/∂x. Use the conservative forms and ∂E/∂x then calculate ρ, C, and T from H. The same thing goes for pressure, which isn't conserved, while dF=d(ma) works even when m and a are both changing. Solving for T as a function of H and composition (instead of H as a function of T) is just another equation to solve and there are many possible approaches.
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