Bernoulli function for planar constant-vorticity flow ($|\mathbf{u}{|}^2/2+\omega \psi +p/\rho =C$)

With the streamfunction convention $(u,v)=({\psi }_y,-{\psi }_x)$, constant scalar vorticity gives $\mathbf{u}\times \mathbf{\omega }=-\omega \nabla \psi$. For steady inviscid constant-density flow without body force, this integrates tothroughout a connected fluid region. Reversing the streamfunction convention changes the sign of its term; the declared convention matters.