A theory for flow separation
Authors: Vorus William S. 
Published: April 2006

This paper proposes a high-Reynolds-number theory for the approximate analysis of timewise steady viscous flows. Its distinguishing feature is linearity. But it differs fundamentally from Oseen's (1910) well-known linear theory. Oseen flow is a variation on Stokes flow at the low-Reynolds-number limit.The theory is developed for a %dimensional body moving through an infinite incompressible fluid. The velocity-vorticity formulation is employed. A boundary integral expressing the body contour velocity is written in terms of Green functions of the approximate governing differential equations. The boundary integral contains three unknown boundary distributions. These are a velocity source density, the boundary vorticity, and the normal gradient of the boundary vorticity. The unknown distributions are determined as the solutions to a boundary-integral equation formed from the velocity integral by the prescription of zero relative fluid velocity on the body boundary.The linear integral-equation formulation is applied specifically to the case of thin bodies, such that the boundary condition is satisfied approximately on the streamwise coordinate axis. The integral equation is then reduced to its leading-order contribution in the limit of infinite Reynolds number. The unknown distributions uncouple in the
first-order formulation, and analytic solutions are obtained. A most interesting result
appears at this point: the theory recovers linearized airfoil theory in the first-order infinite-Reynolds-number limit; the airfoil integral equation determines one of the three contour distributions sought.The first-order theory is then demonstrated by application to two classical cases: the zero-thickness flat plate at zero incidence, and the circular cylinder.For the flat plate, the streamwise velocity near the plate predicted by the proposed linear theory is compared with that of Blasius's solution to the laminar boundary-layer equations (Schlichting 1968). The linear theory predicts a fuller profile, tending more toward the character expected of the timewise steady turbulent profile. This character is also exhibited in the predicted velocity distribution across the plate wake, which is compared with Goldstein's asymptotic boundary-layer solution (Schlichting 1968). The wake defect is more severe according to the proposed theory.For the case of the circular cylinder, application of the formulation is not truly valid, since the circular cylinder is not a thin body. The theory does, however, predict that the flow separates. The separation points are predicted to lie at position angles of approximately ± 135°, with angle measured from the forward stagnation point. This compares with the prediction of 109O from the Blasius series solution to the laminar boundary-layer equations (Schlichting 1968).The theory is next applied to the case of a non-zero-thickness flat plate with incidence. From the fully attached flow at zero incidence, the theory predicts that both Ieading-edge separation and reattachment and trailing-edge separation appear on the suction side at small angle. On increasing incidence, the forward reattachment point moves aft, and the aft separation point moves forward. Coalescence occurs near midchord, and the foil is thereafter fully separated.Finally, the first-order contribution to the far-field velocity at high Reynolds number is shown to be identically that corresponding to the ideal flow. This result, coupled with the recovery of linearized thin-foil theory in the near-field limit, is argued to support strongly the physical idea that the ideal flow is, in fact, the limiting state of the complete field flow at infinite Reynolds number. Flow separation can be viewed as present in the ideal flow limit; i t is simply embedded in the infinitesimally thin body-surface vortex sheets into which the entire viscous field collapses as vorticity convection overwhelms vorticity diffusion at the infinite-Reynolds-number limit.