question about ideal gas thermal flow

yong

For the hot gas introduced into a cold long tube, if the density of the gas is modeled with the ideal gas, say, P*V = NRT, or density = P/RT, how can we solve the unsteady NS for continuity, momentum ( for 1D for example), and energy (For compressible flow, the pressure appears in the energy equation, but for low speed flow, the pressure contribution to the internal energy can be dropped, so I think we still solve the energy in the following form: d_rho*Cp*T/d_t + d_rho*Cp*T*U/d_x = ... ).

In general, the pressure is calculated from pressure-poission eqn, but in this case, we have to relate the P to density and temperature, and it is very complicated and nonlinear.

Any one has any idea about this ? thanks in advance.