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,[1] to the problem derived by Schlichting in 1933[2] 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 description

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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 (such that the fluid loses memory of the shape and size of the slit far away from the origin, it remembers only the net momentum flux). 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 (the jet is so thin that   varies much more rapidly in the transverse   direction than the streamwise   direction) 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

  as  ,

and because the flow is symmetric about   axis

  at  ,

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

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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.[3][4]

Self-similar solution

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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[5][6][7]  

See also

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References

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  1. ^ Bickley, W. G. "LXXIII. The plane jet." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 23.156 (1937): 727-731.(Original paper:http://www.tandfonline.com/doi/abs/10.1080/14786443708561847?journalCode=tphm18)
  2. ^ Schlichting, Hermann. "Laminare strahlausbreitung." ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 13.4 (1933): 260-263.
  3. ^ Kundu, P. K., and L. M. Cohen. "Fluid mechanics, 638 pp." Academic, Calif (1990).
  4. ^ Pozrikidis, Costas, and Joel H. Ferziger. "Introduction to theoretical and computational fluid dynamics." (1997): 72–74.
  5. ^ Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.
  6. ^ Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.
  7. ^ Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.