Kovasznay flow

Kovasznay flow corresponds to an exact solution of the Navier–Stokes equations and are interpreted to describe the flow behind a two-dimensional grid. The flow is named after Leslie Stephen George Kovasznay, who discovered this solution in 1948. The solution is often used to validate numerical codes solving two-dimensional Navier-Stokes equations.

Flow description
Let $$U$$ be the free stream velocity and let $$L$$ be the spacing between a two-dimensional grid. The velocity field $$(u,v,0)$$ of the Kovaszany flow, expressed in the Cartesian coordinate system is given by


 * $$\frac{u}{U} = 1- e^{\lambda x/L}\cos\left(\frac{2\pi y}{L}\right), \quad \frac{v}{U} = \frac{\lambda}{2\pi} e^{\lambda x/L}\sin\left(\frac{2\pi y}{L}\right)$$

where $$\lambda$$ is the root of the equation $$\lambda^2-Re\, \lambda -4\pi^2=0$$ in which $$Re=UL/\nu$$ represents the Reynolds number of the flow. The root that describes the flow behind the two-dimensional grid is found to be


 * $$\lambda = \frac{1}{2}(Re-\sqrt{Re^2+16\pi^2}).$$

The corresponding vorticity field $$(0,0,\omega)$$ and the stream function $$\psi$$ are given by


 * $$\frac{\omega}{U/L}=Re\lambda e^{\lambda x/L}\sin\left(\frac{2\pi y}{L}\right), \quad \frac{\psi}{LU} = \frac{y}{L}- \frac{1}{2\pi}e^{\lambda x/L}\sin\left(\frac{2\pi y}{L}\right).$$

Similar exact solutions, extending Kovasznay's, has been noted by Lin and Tobak and C. Y. Wang.