I believe this has to do with the production of vorticity. If you look at the vorticity transport equation (incompressible, barotropic):
$$
\frac{D\vec{\omega}}{Dt} = (\vec{\omega}\cdot\nabla)\vec{v} +\nu\nabla^2\vec{\omega}
$$
you see that if $\vec{\omega} = 0$, then $\frac{D\vec{\omega}}{Dt} = 0$, i.e. there is no production term for vorticity in an incompressible, barotropic flow. Vorticity only enters such a flow at the boundaries, where a vortex sheet is created to satisfy the no-slip condition. If we neglect viscosity and allow for slip at the walls, then vorticity will not be generated and the flow will remain irrotational.
For compressible flows, you can take the curl of the momentum equation, apply Gibbs equation to replace the pressure gradient term, and some algebra to get the following form of the Crocco-Vazsonyi equation [Thompson 1988]:
$$
\frac{D\vec{\omega}}{Dt} = (\vec{\omega}\cdot\nabla)\vec{v} - \vec{\omega}(\nabla \cdot\vec{v}) + \nabla T \times \nabla s + \mu[\text{Lots of other terms}]
$$
The last term is negligible if the viscosity $\mu$ is small. So the only vorticity production term is $\nabla T \times \nabla s$. Since the flow is adiabatic, and viscous dissipation is negligible, we have no entropy production, $\Delta s = 0$. Therefore, if this flow started with $\nabla s = 0$, it will remain so. This means $\nabla T \times \nabla s$ will stay $0$, and an initially irrotational flow will remain irrotational.
References:
- Thompson, Philip A. Compressible-fluid Dynamics. New York, NY: McGraw-Hill
