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Kovasznay flow

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Normalized streamline ( ψ / L U {\displaystyle \psi /LU} ) contours of the Kovasznay flow for R e = 50 {\displaystyle Re=50} . Color contours denote normalized vorticity ω L / U {\displaystyle \omega L/U} .

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.

References

  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.
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