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# Exercises

1. Fluid flows between two non-parallel plane walls, towards the intersection of the planes, in such a manner that if is measured along a wall from the intersection of the planes then , where is a positive constant. Verify that a solution of the boundary layer equation (8.35) can be found such that is a function of only. Demonstrate that this solution yields

where , and

subject to the boundary conditions and . Verify that

is a suitable solution of the previous differential equation, where .

2. A jet of water issues from a straight narrow slit in a wall, and mixes with the surrounding water, which is at rest. On the assumption that the motion is non-turbulent and two-dimensional, and that the approximations of boundary layer theory apply, the stream function satisfies the boundary layer equation

Here, the symmetry axis of the jet is assumed to run along the -direction, whereas the -direction is perpendicular to this axis. The velocity of the jet parallel to the symmetry axis is

where , and as . We expect the momentum flux of the jet parallel to its symmetry axis,

to be independent of .

Consider a self-similar stream function of the form

Demonstrate that the boundary layer equation requires that , and that is only independent of when . Hence, deduce that and .

Suppose that

Demonstrate that satisfies

subject to the constraints that , and as . Show that

is a suitable solution, and that

3. The growth of a boundary layer can be inhibited by sucking some of the fluid through a porous wall. Consider conventional boundary layer theory. As a consequence of suction, the boundary condition on the normal velocity at the wall is modified to , where is the (constant) suction velocity. Demonstrate that, in the presence of suction, the von Kármán velocity integral becomes

Suppose that

where . Demonstrate that the displacement and momentum widths of the boundary layer are

respectively. Hence, deduce that

Consider a boundary layer on a flat plate, for which . Show that, in the absence of suction,

but that in the presence of suction

Hence, deduce that, for a plate of length , suction is capable of significantly reducing the thickness of the boundary layer when

where .

Next: Incompressible Aerodynamics Up: Incompressible Boundary Layers Previous: Approximate Solutions of Boundary
Richard Fitzpatrick 2016-03-31