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The uniqueness of the solution can be proved as follows: Suppose that 4>i(x) and $z(x) are two solutions such that *i(x) = f(x) + ("" K(x, sfaWds,

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 preparation 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

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The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.

This paper presents a finite-difference method for solving laminar and turbulent-boundary-layer equations for incompressible and compressible flows about two-dimensional and axisymmetric bodies and contains a thorough evaluation of its accuracy and computation-time characteristics. The Reynolds shear-stress term is eliminated by an eddy-viscosity concept, and the time mean of the product of fluctuating velocity and temperature appearing in the energy equation is eliminated by an eddy-conductivity concept. The turbulent boundary layer is regarded as a composite layer consisting of inner and outer regions, for which separate expressions for eddy viscosity are used. The eddy-conductivity term is lumped into a “turbulent” Prandtl number that is currently assumed to be constant. The method has been programed on the IBM 360/65, and its accuracy has been investigated for a large number of flows by comparing the computed solutions with the solutions obtained by analytical methods, as well as with solutions obtained by other numerical methods. On the basis of these comparisons, it can be said that the present method is quite accurate and satisfactory for most laminar and turbulent flows. The computation time is also quite small. In general, a typical flow, either laminar or turbulent, consists of about twenty x-stations. The computation time per station is about one second for an incompressible laminar flow and about two to three seconds for an incompressible turbulent flow on the IBM 360/65. Solution of energy equation in either laminar or turbulent flows increases the computation time about one second per station.

An investigation was made to determine the smallest size of isolated roughness that will affect transition in a laminar-boundary layer. Critical heights for three types of roughness were found in a low-speed wind tunnel. The types were (1) twodimensional spanwise wires, (2) three-dimensional discs, and (3) a sandpaper type. In addition to type of roughness, test variables included the location of roughness, pressure distribution, degree of tunnel turbulence, and length of natural laminar flow.The most satisfactory correlation parameter was found to be the roughness Reynolds Number, based on the height of roughness and flow properties at this height. The value of this critical Reynolds Number was found to be substantially independent of all test variables except the shape of roughness. This parameter also correlates well other published data on critical roughness in low-speed flow. The value of the roughness Reynolds Number necessary to move transition forward to the roughness itself was also determined for the three types of roughness and was found to be approximately constant for a given type of roughness.An investigation of the limited amount of available data on critical roughness in supersonic flow indicates that the effects of roughness may still be correlated by the roughness Reynolds Number. The value of this Reynolds Number depends primarily on the Mach Number at the top of the roughness. When this Mach Number is greater than 1.0, the roughness Reynolds Number based on conditions behind a shock is probably the characteristic parameter. SYMBOLS A = constant a = speed of sound c = chord of airfoil or test surface d = diameter or width of roughness element H = boundary-layer shape parameter 5*/d k = roughness height R = Reynolds Number, UX/J unless otherwise noted R c = chord Reynolds Number U m c/p Ric -roughness Reynolds Number uuk/v R §* = Reynolds Number based on displacement thickness U8*/v Re = Reynolds Number based on momentum thickness Ud/v U = local velocity outside boundary layer u = velocity within boundary layer in x direction v* = friction velocity V Tw /p x = distance along surface measured from forward stagnation point y = distance perpendicular to surface a = angle of attack j3 = shape parameter of Hartree boundary-layer profile, u = cV^2-^ 8 = boundary-layer thickness 5* = boundary-layer displacement thickness 6 = boundary-layer momentum thickness A = length of a wave in the surface fx -dynamic viscosity v = kinematic viscosity p = mass density Subscripts crit = H.W. = k T w = 0

This paper describes a very general method for determining the steady two-dimensional potential flow about one or more bodies of arbitrary shape operating at arbitrary Froude number near a free surface. The boundary condition of zero velocity (solid wall) or prescribed velocity (suction or blowing) normal to the body surface is satisfied exactly, and the boundary condition of constant pressure on the free surface is satisfied using the classic small-wave approximation. Calculations made by the present method are compared with analytic results, other theoretical calculations and experimental data. Examples for which no comparison exists are also presented to illustrate the capability of the method.