"Below Mach 0.3 the flow can be treated as incompressible" is true but usually passed on incompletely. The rule itself is this: at Mach 0.3 the density change is about 5%, and for most engineering decisions that is negligible. What the sentence leaves out is that the threshold applies to the local Mach number.
The real trap: not the inlet velocity, but the highest local one
A flow entering at 40 m/s can easily double through a contraction or over the suction side of a blade. The decision follows not from the mean velocity of the system but from the highest Mach number in the field.
A practical method: solve the problem as incompressible first, take the peak velocity from the result, and compute the Mach number. If it exceeds 0.3, set the problem up as compressible and repeat. This is a cheap check and it is often skipped.
Other cases where the incompressible assumption fails
Large pressure ratio. Even at low Mach, an inlet-to-outlet pressure ratio above 1.2 means density varies from end to end. This is the case in long ventilation ducts and pressurised lines.
Wide temperature range. Velocity is not the only thing that changes density. In a combustion chamber the velocity may be very low while density varies threefold between inlet and outlet.
Possible choking. If Mach 1 is reached at the throat of a relief valve, orifice or nozzle, the mass flow no longer depends on the downstream pressure. An incompressible solution cannot see that limit and overpredicts the flow. In relief sizing that is an error on the unsafe side.
If you decide to solve it compressible
The energy equation is mandatory. In compressible flow temperature, pressure and density are coupled; leaving energy switched off gives an inconsistent solution.
Boundary conditions are given as total quantities. Total pressure and total temperature at the inlet, static pressure at the outlet, is the usual choice. Specifying static pressure at the inlet makes the definition contradict itself as the flow accelerates.
If there is a shock, refine normal to it. The physical shock thickness is on the order of a few mean free paths; the mesh does not resolve that, nor does it need to. But resolution in that region is essential for the shock's position and the pressure jump behind it to come out right. Adaptive refinement earns its keep here more than anywhere else.
Convergence behaves differently. Compressible solutions are more sensitive to the initial condition than incompressible ones. Starting from a sensible initial field, ramping the Mach number up in stages, and using a first-order scheme for the first steps before switching to second order all make convergence noticeably easier.
References
- Ferziger, J. H., Perić, M. & Street, R. L. (2020). Computational Methods for Fluid Dynamics, 4th ed. Springer. DOI: 10.1007/978-3-319-99693-6
- Fay, J. A. & Riddell, F. R. (1958). Theory of stagnation point heat transfer in dissociated air. Journal of the Aerospace Sciences, 25(2), 73–85. DOI: 10.2514/8.7517