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.[1] The solution is often used to validate numerical codes solving two-dimensional Navier-Stokes equations.

Normalized streamline (

\psi /LU

) contours of the Kovasznay flow for

Re=50

. Color contours denote normalized vorticity

\omega L/U

.

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[2]

{\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[3] and C. Y. Wang.[4][5]

References

Kovasznay, L. I. G. (1948, January). Laminar flow behind a two-dimensional grid. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 44, No. 1, pp. 58-62). Cambridge University Press.
Drazin, P. G., & Riley, N. (2006). The Navier-Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press. page 17
Lin, S. P., & Tobak, M. (1986). Reversed flow above a plate with suction. AIAA journal, 24(2), 334-335.
Wang, C. Y. (1966). On a class of exact solutions of the Navier-Stokes equations. Journal of Applied Mechanics, 33(3), 696-698.
Wang, C. Y. (1991). Exact solutions of the steady-state Navier-Stokes equations. Annual Review of Fluid Mechanics, 23(1), 159-177.

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[1] Kovasznay, L. I. G. (1948, January). Laminar flow behind a two-dimensional grid. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 44, No. 1, pp. 58-62). Cambridge University Press.

[2] Drazin, P. G., & Riley, N. (2006). The Navier-Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press. page 17

[3] Lin, S. P., & Tobak, M. (1986). Reversed flow above a plate with suction. AIAA journal, 24(2), 334-335.

[4] Wang, C. Y. (1966). On a class of exact solutions of the Navier-Stokes equations. Journal of Applied Mechanics, 33(3), 696-698.

[5] Wang, C. Y. (1991). Exact solutions of the steady-state Navier-Stokes equations. Annual Review of Fluid Mechanics, 23(1), 159-177.