~~~~~;1I~'N5TiTUTE' O-F-:rCCHNOL(;UY PUBLICATION NO. Reprinted from JOUR="AL OF THE AEROI\AFTICAL SCIEI\CES Copyright, 1055, by the Institute of the Aeronautical Scienec, and reprinted by permission of the ('opyr;g~t owne~, . JA:-IUARY, 10.55 \ or UME ~2, :\0. A Simple Laminar Boundary I---Iayer with Secondary Flow* HENK G. LOOSt California I n..rtitute oj Technolopy SUMMARY and is of considerable importance to the understanding The incompressible laminar boundary layer over a flat plate of complex flow phenomena that occur in turboma­ is studied for the simple case where the stream lines in the free chinery. flow have a parabolic shape. An exact solution of the boundary­ A detailed investigation of the cross flow in the layer equations is derived. No separation occurs, even when laminar boundary layer on an infinite swept wing was there is a strong adverse pressure gradient along the stream lines, carried out by Sears,2 where the pressure gradient in so that in this instance the secondary flow has a favorable in­ fluence. Because of the variation of total pressure from one the free stream was normal to the leading edge and, stream line to another in the free stream, the total pressure within hence, had a component normal to the free-stream the boundary layer at a given point can exceed that of the cor­ flow. Sears calculated the resulting cross flow and responding free stream. showed in particular that the cross flow vanishes for the swept flat plate at zero angle of attack-that is, when SYMBOLS the imposed pressure gradient vanished. The cross x, y, z Cartesian coordinates flow generated by the centrifugal pressure gradient on velocity components in x, y, z directions, respec- a propeller or turbomachine blade was investigated in a tively similar manner by Fogarty,3 where the blade was u, V, W velocity components u, v, w in the free stream treated as an airfoil of infinite span rotating about a T magnitude of the total velocity in the free stream t, n velocity components respectively tangential and point of its span. normal to the free flow direction The study of three-dimensional boundary layers has (J angle between free flow direction and the normal been pursued further by Howarth4 and other workers. to the leading edge in an arbitrary point A problem of particular interest in the study of turbo­ p static pressure machines was discussed by J\lager and Hansen,5 who p kinematic viscosity a, b constants considered the cross flow generated by a curved free stream, carrying its own pressure gradient, which flows INTRODUCTION over a semi-infinite flat plate. When the free stream is a potential vortex, a solution was obtained which is HEN AN AIR STREAM FLOWS over the surface of a valid over distances from the leading edge for which the W flat plate, the curvature of the stream lines turning angle of the free stream is small. within the boundary layer may differ from that of the Now it seems clear that a considerable portion of the free stream because of a gradient of pressure normal to cross flow in boundary layers is directly related to the the direction of free-stream flow. Then, since accord­ fact that the velocity in the boundary layer is less than ing to boundary-layer theory the pressure in the boun­ in the free stream. For example, if a thick boundary ary layer does not vary with distance from the surface, layer is built up by a straight uniform flow over a plate the boundary-layer flow must, because of its lower ve­ and then at some point the free stream is turned, the locity, curve more sharply than the free-stream flow in cross flow that takes place has little to do with local order to balance this pressure gradient. The boundary­ viscous stresses but is simply a matter of transporting layer velocity component normal to the direction of the the vorticity that was generated far upstream. This free stream has become known as the cross flow or the idea was used by Squire and Winter6 to discuss the secondary flow. This secondary flow arising from a secondary flow that occurs in airfoil cascades having a boundary layer operating in an externally produced thick wall boundary layer and by Hawthorne7 in his pressure gradient was initially discussed by PrandtP treatment of the flow in pipe bends. The process becomes simpler when viscous stresses may be neg­ Received February 22,1054. 8 * This work was carried out under the sponsorship of the Office lected, and it has often been suggested (e.g., Hayes ) of Scientific Research, Air Research and Development Command that the general problem of three-dimensional boun­ and represents a portion of an investigation into secondary flow dary-layer flow may be considered in two parts: an in­ in axial compressors. The author wishes to express his appre­ viscid outer part matched with a viscous sublayer. ciation to Drs. H. S. Tsien and F. E. Marble for many stimulating The present work, however, is an exact . calculation discussions. t Research Fellow in Jet PropUlsion, Daniel and Florence Gug­ of the laminar boundary layer developed on a flat plate genheim Jet Propulsion Center. by a flow having stream lines parallel to the plane of the 35 3(j J 0 U R ~ A L 0 F THE A E RON AUT I CAL SCI ENe E S - J A ~ U A R Y. 1 9 5 5 plate and of parabolic shape in this plane. The free g(TJ) = TJIF" (TJI) dTJI (10) stream is rotational, having a constant vorticity that is J:~ directed normal to the plate. This situation is similar Therefore if the velocity component parallel to the to that which occurs in some problems of fluid motion leading edge is assumed to be of the form past turbo machine blades. W = aWo(TJ) + bXWI(TJ) (11) BOUNDARY LAYER WITH PARABOLIC MAIN STREAM Eq. (4) may be transformed to Let U, V, and W denote the components of free­ stream velocity in the x, y, and z directions, respec­ (a/x) [- (1/2) (F'TJ - g)wo' - WOlf] + tively, where x is the distance measured along a semi­ b[ -1 + F'WI - (1/2) X infinite flat plate in a direction normal to its leading (F'TJ - g)WI' - w/] = 0 (12) edge, y normal to the plane of the plate, and z normal to these two coordinates as shown in Fig. 1. If the Furthermore, by partial integration of Eq. (10), the velocity components are chosen to be quantity PTJ - g appearing in Eq. (12) may be written U = constant I F'TJ - g = J:~ F'(TJI) dTJI V = 0 ~ (1) W = a bx ) + so that Eg. (12) becomes with a and b constants, then the stream lines of the (a/x) [- (l/2) Fwo' - WOlf] main flow are of parabolic shape and the flow is of uni­ + form vorticity, n = b, with the vorticity vector in di­ b[ -1 + F'WI - (1/2)FwI' - WI"] = 0 (13) rection of the y axis. The Helmholtz relation for for­ Now if the form assumed in Eq. (ll) is valid, Eq. ticity is obviously satisfied. (13) must be an identity in x and the quantities ~o and The boundary-layer equations for an incompressible WI fulfill the equations fluid are, in this case, where the velocity is independent of z, WOlf + (1/2)Fwo' = 0 (14) uo ou WI" + (1/2)FwI' - F'WI = -1 (15) u + v- (2) ox oy with boundary conditions op/oy = 0 (3) TJ = 0, Wo = { ° (16) ow ow WI = 0 u ,- + v - - bU = (4) TJ = 00, ox oy {:: : ~ (17) (ou/ox) + (ov/oy) = 0 (.5) The Blasius solution for the velocity u = UF'(-q) in the with the boundary conditions direction of x satisfies the differential equation F" F + u = 0 2F'" = 0, with the boundary conditions TJ = 0, F'(O); y = 0, v = 0 (6) TJ = 00, F' = 1. It is obvious that the function Wo = { w=o F'(TJ) satisfies Eq. (14) and the boundary conditions [Eqs. (16) and (17)]. The cross-flow component Wo, y = 00, ( U = U (7) which is related with the constant free-flow component \ W = a + bx W = a, has the same profile as the velocity component Egs. (2) and (5), together with the boundary conditions u. This result was first obtained by Sears in reference for the velocity u, are identical with those for the two­ 2. I t means that in the case of a parallel stream ap­ dimensional laminar boundary layer on a flat plate with proaching a flat plate at an arbitrary angle, the flow zero pressure gradient. Therefore the solution for u direction in the boundary layer is the same as in the and v is the well-known Blasius solution.9 Eq. (4) and free stream. In other words, no cross flow occurs, the boundary conditions for W remain for determination and, consequently, a cross flow in a boundary layer can of the velocity component w. only exist if there is a pressure gradient. Eq. (4) can be reduced to an ordinary differential equation by using TJ = YVU/vx as new independent CALCULATION OF THE CROSS FLOW variable. The Blasius solution may be expressed in The Eq. (15) is linear and nonhomogenous; the the form coefficients F(TJ) and F'(TJ) are tabulated functions.9 (8) Instead of calculating the solution of Eq. (15) with boundary conditions given by Eqs. (16) and (17), u and W will be related with velocity components and n, (9) t which are, respectively, tangential and normal to the A S IMP L E LA M'I N ARB 0 U N DAR Y LAY E R WIT H SEC 0 N DAR Y FLO W 37 8.0 z I w FI~.5 72 I 6,4 - THE FUNCTIONS U J-.