# Are all Irrotational Flows Frictionless and Laminar?

A I understand, Irrotational flows are also inviscid (have zero viscosity). This would lead me to assume that these kind of (purely fictional) flows would be well ordered (Laminar) and be frictionless.

That said. I have a feeling that inviscid flows do not have to be frictionless. Can anyone clarify whether Irrotational Flows have frictionless, laminar or both properties?

Irrotational flows, also called potential flows, are such that their velocity field $\mathbf{u}$ can be written as gradient of a scalar function $\phi$ (called "potential"): $\mathbf{u}=\nabla\phi$. For incompressible flow for which $\nabla\cdot\mathbf{u}=0$ the potential obeys the equation $\nabla^2\phi=0$.
The viscous term in Navier-Stokes equation is $\mu\nabla^2\mathbf{u}$ in which $\mu$ is viscosity. Now if $\mathbf{u}=\nabla\phi$ then $\nabla^2\mathbf{u}=\nabla^2(\nabla\phi)=\nabla(\nabla^2\phi)=0$. Therefore viscous term vanishes even if $\mu$ were non-zero and Navier-Stokes equation reduces to that for inviscid flow.
So to answer your questions: Yes, irrotational flows do not have to be frictionless ($\mu=0$ case). You may find article1 and article2 by D.D. Joseph helpful. However an irrotational flow cannot be called turbulent no matter how complicated the flow because a prime feature of turbulence is that it is intensely vortical (part of the "definition" of turbulence) and that vorticity fluctuations in a turbulent flow occur in all three space-dimensions. A complicated flow but which is two-dimensional and in which therefore vorticity exists only along one direction (normal to flow plane) is not considered a turbulent flow by purists for the latter reason (however the field of 2d-turbulence is picking up; after all it is a matter of definition!). So irrotational flows are "by definition" laminar.