Hi every body I am thinking to add heat conduction to Euler equations for solving two dimensional flow. Does this true or the thermal conductivity is related to viscosity which is neglected in Euler equations. if true, any material related to Euler equations with heat conduction.

Hi, two things have to be clear:

1) Euler equation is inviscid Navier-stoker equations, which neglects the viscosity diffussion term. While heat conduction equation is another equation. So not sure "..add heat conduction to Euler equations...".

2) thermal conductivity relates to viscosity by Prandtl number. I don't think there is any material which has no viscosity. Of course you can do that numerically.

thanks jenn for reply

I use conduction for Euler equation to over come the problem of low density region especially when an airfoil at large angle of attack in that region the internal energy comes negative.

Maybe neg energy isn't resulted from absence of heat conduction. Maybe your scheme has not enough dissipation.

Even if heat conduction is added, heat convection still dominates the process of energy transfering. So heat conduction plays little affection.

Qu Kun thanks for reply do you know some material uses Euler equations with heat conduction. best regards

salem:

Let me understand your physical problem first.

1) where is your interested region? In the front area of the airfoil around the stagnation point? I think there is very sharp velocity gradient there or you can say there is boundary layer there. But it DOESN't mean the velocity there is small every where. So you may still put heat convection into the heat equation i.e. u.\grad T as T is the temperature and u is the local velocity. If that region is really thin and there is hugh density drop cross it, there will be shock waves. Then there you have to solve compressible Euler equation combined with heat equation!

2) You mentioned there the internal energy is negative. Internal energy equals the kinetic energy of molecules plus the potential energy among them. Since the molecular energy is alway positive, the negative energy comes from the larger negative potential energy. Or in another word, the distance between two molecules is very big. Is it true in the region you are interested with low viscosity?

Hi jenn 1) May be there is misunderstanding of my question I am not interesting to heat equation, I mean the heat conduction at the energy equation of the Euler equations, q=-k grad (T). 2) the internal energy numerically may comes negative when the kinetic energy dominates the total energy ,e=E-1/2*rho*(U)^2,when one use Conservative variable as COL(Rho,Rho*u,Rho*v,E)and using the Ghost cell boundary condition, reflection boundary condition wall, I mean when there is expansion wave, the vacuum or near vacuum region occurs ,Pressure~=0.0 any inaccuracies in the conserved quantities can easily lead to the kinetic energy exceeding the total energy, causing computational breakdown in from of internal energy being negative .see paper "W.H Hui and S. Kudriakov, on wall overheating and other computational difficulties of shock capturing methods page 22" you can get it online search by google.

Dear salem

I am not familiar with this, although I met the problem of neg energy at the back region of a cylinder, too.

regards

Hi, salem

Maybe this paper is usful for you:

Noh W. F. (1987), Errors for calculations os strong shocks using an artificial viscosity and an artificial heat flux, Journal of Computational Physics, 72:78-120

thanks qu kun