Air flow is ejected from nozzle (total pressure = 3bar) to ambient air(1bar). If this problem is solved by compressible flow solver using TVD scheme and incompressible flow solver using SIMPLE scheme respectively, both assuming air as an ideal gas, what will be the difference between the results?

Aren't you mixing things up a bit here? TVD, as far as I know, is a discretisation scheme, whereas SIMPLE is an iterative algorithm to couple the pressure and momentum equations.

Moreover, it doesn't make sense to assume an ideal gas in the incompressible case. The ideal gas equation is used to relate the density, temperature, and pressure, but in a presumed incompressible flow, the density is taken to be constant, and the temperature and pressure are uncoupled.

Whether or not the compressibility effects are important in the real case depends on the Mach number. If Ma < 0.3, they are frequently neglected.

Lars Ola

You can assume air as an ideal gas with compressible effects. Incompressible flow simulation in this case is not really suggested if your exit Mach is > 0.4 .If you have Supersonic flow then compressible effects are important and u must have to use compessible flow solver. If u have both schemes available then use TVD comprressible solver, this will give you good results as compared to other one.