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 {\displaystyle U} be the free stream velocity and let L {\displaystyle L} be the spacing between a two-dimensional grid. The velocity field ( u , v , 0 ) {\displaystyle (u,v,0)} of the Kovaszany flow, expressed in the Cartesian coordinate system is given by
u U = 1 − e λ x / L cos ( 2 π y L ) , v U = λ 2 π e λ x / L sin ( 2 π y L ) {\displaystyle {\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 λ {\displaystyle \lambda } is the root of the equation λ 2 − R e λ − 4 π 2 = 0 {\displaystyle \lambda ^{2}-Re\,\lambda -4\pi ^{2}=0} in which R e = U L / ν {\displaystyle 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
λ = 1 2 ( R e − R e 2 + 16 π 2 ) . {\displaystyle \lambda ={\frac {1}{2}}(Re-{\sqrt {Re^{2}+16\pi ^{2}}}).}
The corresponding vorticity field ( 0 , 0 , ω ) {\displaystyle (0,0,\omega )} and the stream function ψ {\displaystyle \psi } are given by
ω U / L = R e λ e λ x / L sin ( 2 π y L ) , ψ L U = y L − 1 2 π e λ x / L sin ( 2 π y L ) . {\displaystyle {\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.
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