Bickley jet


In fluid dynamics, Bickley jet is a steady two-dimensional laminar plane jet with large jet Reynolds number emerging into the fluid at rest, named after W. G. Bickley, who gave the analytical solution in 1937, to the problem derived by Schlichting in 1933 and the corresponding problem in axisymmetric coordinates is called as Schlichting jet. The solution is valid only for distances far away from the jet origin.

Flow descriptionKundu, P. K., and L. M. Cohen. "Fluid mechanics, 638 pp." Academic, Calif (1990).Pozrikidis, Costas, and [Joel H. Ferziger]. "Introduction to theoretical and computational fluid dynamics." (1997): 72–74.

Consider a steady plane emerging into the same fluid, a type of submerged jets from a narrow slit, which is supposed to be very small. Let the velocity be in Cartesian coordinate and the axis of the jet be axis with origin at the orifice. The flow is self-similar for large Reynolds number and can be approximated with boundary layer equations.
where is the kinematic viscosity and the pressure is everywhere equal to the outside fluid pressure.
Since the fluid is at rest far away from the center of the jet
and because the flow is symmetric about axis
and also since there is no solid boundary and the pressure is constant, the momentum flux across any plane normal to the axis must be the same
is a constant, where which also constant for incompressible flow.

Proof of constant axial momentum flux

The constant momentum flux condition can be obtained by integrating the momentum equation across the jet.
where is used to simplify the above equation. The mass flux across any cross section normal to the axis is not constant, because there is a slow entrainment of outer fluid into the jet, and it's a part of the boundary layer solution. This can be easily verified by integrating the continuity equation across the boundary layer.
where symmetry condition is used.

Self-similar solutionRosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.Drazin, Philip G.">Philip Drazin">Drazin, Philip G., and [Norman Riley]. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.

The self-similar solution is obtained by introducing the transformation
the equation reduces to
while the boundary conditions become
The exact solution is given by
where is solved from the following equation
Letting
the velocity is given by
The mass flow rate across a plane at a distance from the orifice normal to the jet is