December 23, 2016

In 1904 Prandtl showed that the effects of viscosity at a high Reynolds number could be represented by approximating the Navier–Stokes equations with the celebrated boundary-layer equations, which for two-dimensional steady flow reduce to: ... ... ... ... Here, ... and ... are the coordinates parallel and perpendicular to the body surface, respectively. For a semi-infinite wedge with an angle of taper ... , one can prove that far from the wedge the potential flow is given by ... , where ... or ... , and ... is a scale factor. The above boundary-layer equations admit a self-similar solution such that the velocity profiles at different distances ... can be made congruent with suitable scale factors for ... and ... . This reduces the boundary-layer equations to one ordinary differential equation. Let us introduce a function ... such that: ... where ... . Then, we have from continuity equation: ... . The boundary-layer equations can be written as follows: ... with ... and ... The above equation can be solved for a user-set value of parameter ... when ... and ... using the shooting technique. The limiting case ... is flow over a flat plate (Blasius problem). Using the following definitions of ... and ... , one can obtain the particle trajectories (streamlines) for different starting points. This Demonstration plots such trajectories and also shows the growth of the boundary-layer ... (red curve) in a separate plot for any value of the wedge angle. The evolution of the ... -velocity component and its congruent properties with the growth of the boundary layer is also shown.