By David Oliver
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The curve C (dashed lines) is taken to be a circle in each case. In Fig. 3a the flowfield consists of concentric circular streamlines in the counterclockwise direction. It is clear that along the circular integration path C (Fig. 3a) q and dl in Eq. 3) are positive for all dl and therefore C has a positive circulation. In Fig. 3b the flowfield is the symmetric flow of a uniform stream past a circular cylinder. It is clear from the symmetry that the circulation is zero for this case. To illustrate the motion of a fluid with rotation consider the control volume shown in Fig.
The vortex lines passing through an open curve in space form a vortex surface and the vortex lines passing through a closed curve in space form a vortex tube. A vortex filament is defined as a vortex tube of infinitesimal cross-sectional area. 48) Consider, at any instant, a region of space R enclosed by a surface S. 49) R At some instant in time draw a vortex tube in the flow as shown in Fig. 10. Apply Eq. 49) to the region enclosed by the wall of the tube Sw and the surfaces S1 and S2 that cap the tube.
65)) the pressure does not vary across the boundary layer and is said to be impressed on the boundary layer. Therefore, the surface pressure distribution is taken from the inviscid solution in (1) and inserted into Eq. 64). Also, Ue is taken from the inviscid solution as the tangential component of the velocity at the edge of the boundary layer and is used as a boundary condition in the solution of the boundary layer equations. Solving for a high Reynolds number flowfield with the assumption of an inviscid fluid is therefore the first step toward solving the complete physical problem.