Kovasznay flow explained

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.

Flow description

Let

be the free stream velocity and let

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]

	u
	U

=1-e^λ\cos\left(

	2\piy
	L

\right),

	v
	U

	λ
	2\pi

e^λ\sin\left(

	2\piy
	L

\right)

where

is the root of the equation

λ^2-Reλ-4\pi²⁼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

λ=

	1
	2

(Re-\sqrt{Re^2+16\pi^2}).

The corresponding vorticity field

(0,0,\omega)

and the stream function

\psi

are given by

	\omega
	U/L

=Reλe^λ\sin\left(

	2\piy
	L

\right),

	\psi
	LU

	y
	L

	1
	2\pi

e^λ\sin\left(

	2\piy
	L

\right).

Similar exact solutions, extending Kovasznay's, has been noted by Lin and Tobak^[3] and C. Y. Wang.^[4] ^[5]

Notes and 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.