Unsteady boundary layer flows generated in an incompressible, homogeneous, nonrotating viscous fluid bounded by a rigid wavy plate are studied theoretically. The Laplace transform method is employed to obtain exact solutions of the unsteady boundary layer equations in a wavy plate configuration. The structures of the unsteady velocity distribution and the associated boundary layers are determined explicitly and several particular solutions are recovered as special cases of this analysis. The physical interpretation of the mathematical results are examined.

Unsteady boundary layer flows induced by a wavy plate

The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part. These actual flows show a special characteristic, denoted as turbulence. The character of a turbulent flow is most easily understood the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. i% order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity. Actually, however, one ob~erves that, for larger Reynolds numbers, the pressure drop increases almost with the square of the velocity and is very much larger then that given by the Hagen Poiseuille law. One may conclude that the actual flow is very different from that of the Poiseuille flow.

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Boundary layer theory

Fourier integrals for practical applications

The work presented here is an extension of that of Blasius (1908) on boundary-layer growth at a cylinder started from rest. It is shown that if an infinite plane moves in its own plane in a viscous fluid, the velocity distributions are similar at different times if the velocity V{t) of the plane is of the form V(t) = Atα or V(t) = A ect, where t is the time. These cases are then applied to the theory of boundary-layer growth and the second approximation to the velocity in the boundary layer is calculated. From this we find a first approximation for the distance travelled by the cylinder before separation starts. The second approximation to the separation distance was calculated by Blasius for α = 1, by Goldstein & Rosenhead (1936) for α = 0, and is found here for the case V(t) — A ect. Another approximate method for finding the separation distance, when separation starts at the rear stagnation point, is developed and applied to the impulsive start. The method is based on the momentum equation and assumes that the velocity profile near the rear stagnation point will always be similar to an initial profile. The numerical results are presented in the tables, and include the variation with a of the separation distance (according to the first approximation). It is hoped to use the method of Gortler (1944) to evaluate the drag on a circular cylinder and thus to discuss the initial motion of such a cylinder.

Boundary-Layer Growth

Unsteady boundary layer flows generated in a homogeneous, non-rotating viscous fluid are considered. The method of Laplace transform is used to obtain exact solutions of the unsteady boundary layer equations in a more general situation. The structures of the unsteady velocity field and the associated boundary layers are determined. Several particular solutions are recovered as special cases of the present general theory. The physical implications of the mathematical results are investigated.

Unsteady boundary layer flows induced by a wavy plate

References

Boundary layer theory

Fourier integrals for practical applications

Boundary-Layer Growth

Some exact solutions of unsteady boundary layer equations-I

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