~~~~~;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 over a flat plate of complex flow phenomena that occur in turboma­ is studied for the simple case where the 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 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 , 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 . 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-. F'(?\ h('IJ),F'('IJ)+h('IJ) 5.6 A---- Y FIGURE I COOR~NATE SYSTEM 4,8 \ LEADING FOR FLOW ALONG x EDGE SEMI- INFINITE FLAT PLATE 4.0 \ I \ ~ 3.2 \ / IF\,I+h( z ~ 2.4 ~~'('l \ t ) / } 1.6 J ,/ rh('IJ) ~---t--U / / 1/ 0.8 ./ ~ '------x FIGURE 2 ~ ~ --- VELOCITY COMPONENTS 1.2 IA t AND n FOR STREAM­ LINES NORMAL TO PLATE LEADING EDGE

~~--~-20'--'-~'-~1-~-~

~ FIG.6 1.8/· '\

FIGURE 3 VELOCITY COMPONENTS 1-----+---+/ \ I i t AND n FOR STREAM­ u ~ ! LINES MEETING LEADING EDGE AT ANGLE 90°-e.

L-----x

LEADING EDGE

I P 7 FREE-FLOW STREAMLINE

/ STREAMLINE IN ~ BOUNDARY LAYER / / / ..,. FIGURE 4 STREAMLINES IN VARIATION OF THE COEFFICIENT A WITH 9 TOTAL HEAD FREE FLOW AND IN FOR 9.=-60°, 0°,60° DISTRIBUTION BOUNDARY LAYER 38 J 0 URN A L 0 F THE A E RON AUT I CAL SCI ENe E S - JAN U A R Y. 1 9 5 .5

< local free-flow direction. In Fig. 2 a free-flow stream In the general case where a ~ 0, W = awo(n) + line is shown for the case that the coefficient a is zero- bWI(n), the normal velocity component can be expressed that is, so that the stream lines are perpendicular to as the leading edge of the plate. If {J is the angle between n = [aF' + bx(F' + h)J cos {} - UF' sin {J the free-flow direction and the positive x axis, Now, tan {} = (a + bx)IU, so that t = u cos {J + W sin {J (18) n = bxh(n) cos {} (27) n = W cos {J - u sin (J (19) Further, the quantity bx may be written bx U(tan Now, for a = 0, u = UF'(n) and W = bXWI(n) , and {} - tan (}o), where {}o is the angle between the free-flow 2 2 T = vi U2 + b x is the local free-stream velocity. direction and the x axis at the leading edge (see Then Eqs. (18) and (19) may be written as Fig. 3). Eq. (27) becomes t u2F' + b2x2WI F' + (bxlu)2WI (20) niT = A({J, (}o)h(n) (28) T u 2 + b2 x 2 1 + (bxl U)2 A({}, (Jo) = cos2 {}(tan (J - tan (}o) (29) n bx UWI - bxUF' bx WI - F' In the same wayan expression for tiT may be derived T U2 + b2x 2 U 1 + (bxl U)2 (21) in terms of F'(n), hen), {}, and (Jo, Now it is convenient to define tiT = F'(n) + B({}, (Jo)h(n) (30) (22) B({}, (Jo) = (1/2) sin 2{J(tan {J - tan (Jo) (3l) Then a differential equation for hen) can be derived in the following way: The function WI(n) is a solution of RESULTS WI" + (1/2)FwI' - F'WI = -1, and F(n) satisfies the The functions A({}, (}o) and B({J, (Jo) are plotted in equation F"' + (1/2)FF" = 0. Subtraction of these differential equations gives Figs. 6 and 7 for {Jo = - 60°, 0°, and 60°. For the same values of {Jo, the tangential flow tiT is plotted versus (WI - F')" + F(WI - F')' - F'(WI - F') = n for some values of {Jo. -1 + (F')2 In Fig. 8, {}o = -60° and {J = -60°, -40° ... 60°. For {} = - 40° and {} = - 20°, a "separation" type and taking account of the definition of hen), profile is found, caused by the adverse pressure gradient h"(n) + (1/2)Fh' - F'h = -1 + (F')2 (23) along the direction of the free-stream flow. As will be shown later, the cross flow prevents any actual sepa­ The function hen) is equal to -G'(n) which has been tabulated by Mager and Hansen.5 The normal or ration. cross-flow component [Eq. (21)J, may now be written After passing the point {J = 0, the velocity profile becomes fuller and at the point {} the velocity as = 20° in the upper part of the boundary layer exceeds the n bx hen) free-stream velocity by a small amount. For greater (24) T U 1 + (bxIU)2 values of (J, the maximum velocity in the boundary layer increases more and more, until at {} = 60° the ve­ For small values of x, (bxl U)2 1, and the normal « locity distribution with the highest velocity peak is flow component is the same as that found by Mager reached. For {} > 60°, the maximum velocity in the and Hansen5 for small turning angles. This result is boundary layer drops again, and in the limit {} = 90° obvious, because, for a flow with a = 0, the circular the velocity distribution is just F'(n) + hen), still hav­ stream line coincides in first order with a parabolic ing an increased velocity in the upper part of the boun­ stream line for points close to the leading edge. dary layer. For {Jo = 0° and {Jo = 60° (Figs. 9 and 10) From Eq. (24) it follows that for large turning the velocity profile becomes fuller with increasing angles in "parabolic flow" the distribution of the rela­ {}, and the highest velocity peak in the boundary layer tive normal flow component niT remains the same and is reached at {} = 90°. Of course, for this value of {}o, the magnitude varies as (bxIU)/[1 (bxIU)2]. It is + no "separation" type profile occurs because there is no possible to express niT in terms of the turning angle adverse pressure gradient for the tangential flow. The {} of the free flow. For a = 0, bxl U = tan {} and normal or cross flow niT is distributed over n as A ({J, (bxIU)/[1 + (bxIU)2J = (1/2) sin 2{J (Jo) hen), and so the velocity profile will be always of the shape hen) (see Fig. 5) while the amplitude changes as and therefore A({}, (}o) (see Fig. 6). niT = (1/2) hen) sin 2{J (25) It is of interest to investigate the total head distribu­ tion in the boundary layer. Because, in the boundary­ Further, Eq. (20) can be written as layer approximation, the pressure is constant through tiT = F'(n) + hen) sin 2 (J (26) the boundary layer, the total head loss in the boundary A S IMP L E LAM' I N ARB 0 U N DAR Y LAY E R WIT H SEC 0 N DAR Y FLO W 39 ,., ,-----,-----,~ 1,6 '------,1---,---,--,------, B,,,", I FIG.7 FI1,B 7,2 ! S ,4

5 ,S I 4 B B 4 D ---~ A~ '? ~i F'(,)+h(?J 32 l1 V \\~ 2 .4 .,~~/0 ,btO, V L rr, ) .S 0-2 )) I) V . v ~O.~~V~e.OO 08 / ~ ~ V ~f.-:::::;;; ~r- .(£~~~ r- 0 VARIATION OF 0 02 0.4 O.S t 0.8 1.0 l2 1.4 I.S THE COEFFICIENT B WITH 9 T FOR 9 0 = -600, 0°, 60° VELOCITY DISTRIBUTIONS FOR THE TANGENTIAL FLOW +FOR SOME VALUES OF e AND FOR eo= -60°

8 ,(1 8 ,( 1 I FIG.9 FIG.IO 7. 2 7, 2

6. 6. 4

5 .6 5. 6

I 4:8 4. 8

4.0 J 0 "? 3. 2 /!J ~\ 3. 2 11 ~ V/ 2. A / II \ \ 2.4 / ~V / 10='(,)+ V V ) .6 / ° j, 1.6 L L / / ° rt;'If. ~o ~t&0 tP..o • V. Vao o~ ll) ./ 0~60~ff /"·V Q 8 O. 8 0"~ eZ::: 0~ ~F'('?)+h(?J) ~ $. ~V- ~ ,.....~ ~ ~ 0 ~ 0 0.2 0.4 O.S..1. 0.8 1.0 1.2 1.4 00 0.2 0.4 O,S.1. 0.8 1.0 1.2 1.4 T T VELOCITY DISTRIBUTIONS FOR THE VELOCITY DISTRIBUTIONS FOR THE TANGENTIAL FLOW t FOR OOME VALUES OF a TANGENTIAL FLOW t FOR SOME VALUES OF e AND FOR a.- 00 AND FOR a.=so· 40 J 0 URN A L 0 F THE A E RON AUT I CAL SCI E N C E S - JAN U A R Y, 1 9 5 5 layer can be determined by calculating the kinetic . w . aF' + bx(F' + h) tan t'J' = hm~ = bm---- energy. Neglecting the velocity component v, the ~o u y-o UF' ratio of the kinetic energy in the boundary layer and in the free flow is, for t'J 0, a + bx [1 + lim h(rJ)/F'(rJ)] o = ~-o (UF')2 + (F' + h)2b2x 2 U U2 + b2x2 Now and with lim h(rJ)/ F'(rJ) '1-0 U/VU2 + b2x2 = cos t'J, bX/VU2 + b2x2 = sin t'J is finite so that, for finite values of x, tan t'J' will be finite. E(rJ)/E(oo) = (F')2 cos2 t'J + (F' + h)2 sin 2 t'J Therefore no separation occurs for finite values of x. From this result it follows that the cross flow avoids If the ratio E(-T/)/E( (0) exceeds unity, the total head separation in the present example by transporting fluid at the point of the boundary layer exceeds the free­ with high energy to the "critical" regions. stream value. If rJ = 2.0 is chosen, according to Fig . = .5, F'(2.0) 0.63 and F'(2.0) + h (2.0) = 1.17. Thus if CONCLUSIONS E(2.0)/E( (0) = 0.63 2 cos2 t'J + 1.17 2 sin2 t'J > 1 The boundary-layer equations for the "parabolic" flow along a flat plate can be solved exactly, giving in­ that is, if t'J > 52.10, the energy in the boundary layer formation about the secondary flow (cross flow). The at rJ = 2.0 exceeds the free-stream energy. solution for t'Jo = 0 coincides with that of Mager and This result can be understood by comparing the Hansen' for small turning angles but also remains valid stream lines of the free stream and of the boundary for large angles. There, the formulas of Mager and layer (Fig. 4). Since the stream lines in the boundary Hansen can be used, replacing the turning angle t'J by layer curve more sharply than those in the free stream, (1/2) sin 2t'J. the stream line passing the observation point P at rJ = If -t'Jo is large enough, a "separation" type profile for 2.0 (dashed line in Fig. 4) comes from another upstream the tangential flow will be found, but there will be no location than the free-flow stream line passing through separation because the secondary flow transports fluid the same point P. There is a total head gradient in with high energy to the "critical" regions. According the z direction upstream of the leading edge because of to the total head gradient upstream of the leading edge the vorticity in the parabolic flow, so that upstream (rotation), the total head in the boundary layer can of the leading edge the dashed stream line through P exceed the free-stream value. will have a larger total head. Along the dashed stream line, energy will be dissipated, but it is possible that the REFERENCES energy loss is less than the difference in total head up­ stream of the leading edge. In this case, the total head I Prandtl, L., On Boundary Layers in Three Dimensional Flow. M.A.P. Volkenrode VG89. in some point of P will exceed the free-stream value. 2 Sears, W. R., The Boundary Layer of Yawed Cylinders, Journal Another interesting question is whether there will be of the Aeronautical Sciences, Vol. 1,5, No. I, p. 49, January, 1948. a separation of the boundary layer. Of course, sepa­ 3 Fogarty, L. E., The Laminar Boundary Layer on a Rotating ration cannot be expected for t'Jo > 0 because the pressure Blade, Journal of the Aeronautical Sciences, Vol. 18, No.4, pp. 247-252, April, 19,51. gradient in this case is favorable. But if -t'Jo is suffi­ • Howarth, L., The Boundary Layer in Three Dimensional ciently large, an adverse pressure gradient occurs in the Flow, Part I, Phil. Mag., Ser. 7, Vol. xlii, May, 1951. region t'Jo < t'J < 0, and if there were no cross flow, the • Mager, A., and Hansen, A. G., Laminar Boundary Layer over boundary layer would separate. Actually, a cross Flat Plate in a Flow Having Circular Streamlines, NACA TN. flow occurs, and the separation may be studied by No. 26,58, March, 19,52. employing the solution for the boundary-layer flow • Squire, H. B., and Winter, K. G., The Secondary Flow in Cas­ cade of Airfoils in a Nonuniform Stream, Journal of the Aeronau­ which has been obtained. tical Sciences, Vol. 18, No.4, pp. 271-277, April, 19,51. If three-dimensional separation occurs, then, ac­ 7 Hawthorne, W. R., Secondary Circulation in Fluid Flow, Proc. cording to Hayes,S there is a separation line on the sur­ of The Royal Society of London, Series A, Vol. 206, No. AI086, face which is tangent to all "stream lines" at the sur­ pp. 374-387, May, 1951. face. In the present example, the velocity field is con­ • Hayes, W. D., The Three Dimensional Boundary Layer, NA VORD Report 1313, NOTS 384, May 9, 19,51. stant in the z direction, so that any separation line must 9 Blasius, H., Grenzschichten in Flussigkeiten mit kleiner Rei­ be parallel with the z axis. The flow direction t'J' on bung, Zeitschrift fur Mathematik und Physik, Band .56, p. 1, the surface is given by 1908.