A theory for the laminar wake of a two-dimensional body in a boundary layer

Abstract

A new theory is developed for the wake far downstream of a cylindrical body of height h, placed with its generators perpendicular to the flow on a surface above which there is a boundary layer of thickness δ. If the streamwise (x) velocity in the wakeis (U + u), then assuming (h/δ) is small enough that the velocity profile in the boundary layer may be regarded as U = αy, and assuming |u| [Lt ] U, linear differential equations governing u are derived. It is found that a constant along the wake is \[ I=\frac{3}{2}\int_0^{\infty} yUu\,dy. \] This result can be used to find an order of magnitude estimate for u, because I is related to the forces on the body producing the wake by the approximate formula \[ I \simeq - C_1/\rho, \] where C₁ is that component of the couple on the body produced by pressure and viscous stresses in the x direction. For the particular case of a small hump on the boundary of height h and length b, such that h [Lt ] b, the above relation is shown to be exact. The perturbation velocity in the wake is found to have a similarity solution \[ u = [I/(xv)]f(y^3/[xv/\alpha]), \] the physical implications of which are discussed in detail. The relevance of the theory to the problem of transition behind a trip wire is also mentioned.