For an irrotational, incompressible fluid we can solve Laplace's equation for the velocity potential in a fluid in order to obtain the velocity field. This can be done for flow around a cylinder to obtain $$u_r=U(1-\frac{a^2}{r^2})\cos\theta$$ $$u_{\theta}=-U(1+\frac{a^2}{r^2})\sin\theta$$ as the radial and tangential velocity components, with the far field velocity of magnitude U in the x direction. This seems to be a laminar flow solution. However what is there in the method of solution that causes this to be the case? My notes say that this method holds best in the inviscid limit (high Reynolds number) because in that case an irrotational fluid remains irrotational (Kelvin's circulation theorem holds), however high Reynolds number is actually the turbulent flow regime so this contradicts the above.
Can anybody offer any insight here? Thanks :)