On the growth of Taylor-Görtler vortices along highly concave walls
A. M. O. Smith
Quarterly of Applied Mathematics
Abstract. The primary objective of this study has been to prepare a chart for computing the growth of Taylor-Gortler vortices in laminar flow along walls of both high and low concave curvature. Taylor-Gortler vortices are streamwise vortices having alternate right-and left-hand rotation that may develop in the laminar boundary layer along a concave surface. The equations of motion are derived anew and re-examined with regard to the importance of the various terms. The final equations used in
... paration of the chart are found to be valid for radii of curvature as small as 30 times the boundary layer thickness. Furthermore, it is shown that the equations are not restricted in validity to cases of constant wall curvature, constant free stream velocities, or to boundary layers of constant thickness. Whereas the previous analyses by Taylor and Gortler assumed the vortex to grow exponentially as a function of time, the present study recasts the growth into a more convenient form in which the vortex grows as a function of distance. The solution is an eigenvalue problem, which in the present study has been solved mainly by Galerkin's method-a variational method. Both the eigenvalues and the eigenfunctions are presented, the former in the aforementioned chart. It is possible to compare the solutions for neutral stability with those given by Gortler. The two solutions are in approximate agreement. A second method of solution also is described. This method is believed to offer considerable improvement, provided a high-speed digital computer is available. In the one case checked by both methods agreement was within 2%. Finally, the stability chart was applied to all the known experimental data concerning the effect of concave curvature on the transition point. The well known parameter Rs(d/r)1/2 is shown to be inadequate as an indicator of the transition point. Instead, the experimental data indicate that an apparent amplification factor, exp / /3 d.r, is a much better measure. Available results show that transition of this type will occur when / /3 dx reaches a value of about ten. 2. The flow past a concave plate. A considerable body of indirect evidence indicates that a laminar flow along a concave wall does not remain two-dimensional. Instead, it rolls up into alternate right-and left-hand vortices as indicated in Fig. 1 . To obtain some insight into the forces that cause the formation of these vortices, consider the streaming of an incompressible, viscous fluid past a concave wall, Fig. 2 . If the Reynolds number is not extremely low, a boundary layer will develop. At some arbitrary height, y1 , within the boundary layer, the velocity is . At some other height y2 = yx + Ay the velocity is u2 = ux + (du/dy)Ay. At the height yt , the fluid is under a pressure . By the usual boundary layer equations for two-dimensional '