I have difficulty in the calculation of velocity and pressure fields for the transient flow. I am using the SIMPLE method to solve the velocity and pressure field for compressible flow. When the density is only function of pressure (isothermal) there is no problem, while when I solve the energy equation and the density is function of both pressure and temperature I get negative pressures. In a general form for the transient term in the countinuity I write:

d/dt (rho)=d/dP (rho) . dP/dt+ d/dT (rho). dT/dt

Now for ideal gas we have: rho=P/(kT)

d/dP (rho)= 1/(kT) (k is molar gas constant)

d/dT (rho)= -P/(kT^2)

The flow is laminar and the mach number is very small, however the density variation is big. Is there any trick to handle the transient part in the pressure equation?

Thanks for the help.

although i am not fully clear about your problem, it appears that the source term needs attention - i mean the linearisation.

How should I do that?

suggest consult the book by Patankar.

Thanks. I guess I have already done that. for example as I said I have written:

d/dt (rho)= d/dP (rho) .dP/dt + d/dT (rho) . dT/dt

thus, for the ideal gas we get:

d/dt (rho)= 1/(kT).dP/dt- P/(kT^2) . dT/dt

Now I have no problem with term 1/(kT) dP/dt, but the problem apears to be with P/(kT^2) .dT/dt. I can either used it in the central coefficient or put it in the source term using the perivious pressure. In either case it gives me negative pressure and wrong velocities!

Can you give a specific solution? Thanks again

Hi Mike,

Have a look in the following paper:

K.C. Karki and S.V. Patankar, Pressure Based Calculation Procedure for Viscous Flows at All Speeds in Arbitrary Configurations, AIAA J V27 N9, 1989, pp 1167-1174.

Although steady-state is treated, it might be helpful.

Rami

At low Mach numbers the term in the momentum equation you are having problems with makes the momentum equation very stiff. There are several ways of handling variable density low Mach number flows and I would suggest you perform (or look up) a low Mach number asymptotic analysis of your set of equations. Subsequent to this you can differentiate the resulting equation of state in order to derive a reasonably well behaved RHS for whichever form of the pressure/continuity equation you adopt.

Andy:

Your suggestion seems very interesting. What do you mean with RHS? Do you have any reference (Paper or book) regarding this issue?

Many thanks.

By RHS I mean Right Hand Side or the source term of the elliptic equation which is solved to enforce continuity in a typical low speed code.

I suspect most texts describing the details of a split/segrated/projection/approximate factorization type numerical method for low speed variable density flow is likely to discuss or refer to the point. I guess combustion is probably the most widely researched low speed variable density flow and I would suggest starting with a few papers in this area.

Andy,

Thanks a lot. As I said it seems the main problem appears from the countinuity transient term. I will try to find some papers in the area you said. Do you have any specific paper in mind?