EOS and NS equation
 

Could anyone tell me ( or refer to some references) how to couple the equation of state ( for example, Van del Waals gas ) to the NS equation for unsteady problem ?

Thanks!

Re: EOS and NS equation
 

In this case, I mean incompressible gas with large density change due to temperature.

Re: EOS and NS equation
 

For compressible NS, EOS links pressure with density and temperature: P=p(rho,T), e=e(rho,T); For incompressible flow with varying density, you can use Bousinessq approximation, rho= rho_0 + delta rho, where delta rho depends on temperature variation. Van der Waals equation is valid for high density gas. You can also use Bousinessq approximation there.

Re: EOS and NS equation
 

You wrote:

"I mean incompressible gas with large density change due to temperature."

Obviously if you have "large density change", you do NOT have an "incompressible gas". I will assume you mean a low speed flow (M << 0.3), but large temperature variation resulting in large density variation.

In this case, you will need some kind of compressible flow solver (whether a pressure-based SIMPLE type solver, or a low-Mach number preconditioned compressible flow solver) to solver the mass, momentum and energy eqnations. In general, when using a non-ideal equation of state, you should consider making all your thermodynamic relations consistent with your rho=rho(P,T) EOS. A standard way to do this is to use "departure functions", which are described for example in "The Properties of Gases and Liquids" by Reid, Prausnitz and Poling.

Of course, depending on your flow solver, you may also need to rederive various Jacobians to reflect more general thermodynamic relationships {which eventually narrows down to expressions for (del rho)/(del T) at fixed P and (del rho) / (del P) at fixed T}.

The biggest numerical headache is probably the fact that rho=rho(P,T) is not an explicit relationship, but a cubic equation in specific volume or density. In fact, you may want to consider more accurate "cubic equations of state" since the amount of work will be the same, but the accuracy improved with better cubic equations of state.

Re: EOS and NS equation
 

Hi, Versi,

Thank you so much on your post which is really helpful!

But I am still not sure about one thing: How to couple the EOS into the NS equation ?

Could you give me more instruction ?

Thanks in advance!