title: Kovasznay flow
text: # 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
  
      ${\frac {u}{U}}=1-e^{\lambda x/L}\cos \left({\frac {2\pi y}{L}}\right),\quad {\frac {v}{U}}=-{\frac {\lambda }{2\pi }}\sin \left({\frac {2\pi y}{L}}\right)$
  
  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
  
      ${\frac {\omega }{U/L}}=Re\lambda e^{\lambda x/L}\sin \left({\frac {2\pi y}{L}}\right),\quad {\frac {\psi }{LU}}={\frac {y}{L}}-{\frac {1}{2\pi }}e^{\lambda x/L}\cos \left({\frac {2\pi y}{L}}\right).$
  
  Similar exact solutions, extending Kovasznay's, has been noted by Lin and Tobak and C. Y. Wang.
  
  