THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING GENERALIZATION OF A CLASS OF SOLUTIONS OF THE LAMINAR, INCOMPRESSIBLE BOUNDARY LAYER EQUATIONS Ward 0. Winer Arthur G. Hansen September 1960 IP-464

LIST OF FIGURES Figure Page 1. Orientation of skew coordinates.......................... 3 2. Functions F' (r) and G' (r) plotted against............ 8 3a. Streamlines of flow in skew-linear coord., 0 = 0........ 11 3b. Streamlines of flow in skew-linear coord., coord., 0 = 30~ 12 3c. Streamlines of flow in skew-linear coord., 0 = 60~........ 13 -ii -

NOMENCLATURE a, a', al, b, b' Cp Ci F (r) G (r) g(x,z) m n, n' Pr r s T* To T U, V, w U, W x, y, z X, Y, Z a2, at1' a' constants constants constant pressure specific heat constant (i = 0, 1,...........10) function of similarity parameter j, u = UF'(r) function of similarity parameter q, w = WG'(Q) function of coordinates, x and z thermal conductivity constant constants Prandtl number, Pr = -C constant constant scaled surface temperature, T*(xz) = T - T. temperature of mainstream temperature of surface boundary layer velocity components in x, y, z directions respectively boundary layer velocity components in X, Y, Z directions respectively mainstream velocity components in x, z directions respectively skew-linear coordinate system rectangular coordinate system -iii

NOMENCLATURE (continued) TI similarity space variable, v coefficient of kinematic viscosity Ap coefficient of absolute viscosity p density 0 angle between X and x in surface plane 80* dimensionless temperature, e* = T - Too Tw - Too ~ function of similarity parameter i, 0* = 8 (rq) susperscript prime denotes differentiation with respect to i -iv

INTRODUCTION Exact solutions of the laminar, incompressible boundary-layer equations are generally obtained by the use of similarity transformations. By means of such transformations, the equations of the boundary layer as well as the associated energy equation are reduced to ordinary differential equations which can then be solved by standard numerical methods. The price that is paid for obtaining such solutidns 'is that the method is applicable only for rather restricted classes of mainstream flows (Reference 1). This might well be expected since one of the characteristics of similarity solutions is that the computed boundary layer velocity profiles are allowed to vary only in scale along coordinate directions. This limitation is reflected at once in the permissible types of mainstream flows which give rise to the boundary layers. It can be stated, therefore, that the choice of the coordinate system is one of the key factors in obtaining physically significant similar solutions of the boundary layer equations. Unfortunately, there are also restrictions on the type of coordinate systems which may be employed in similarity analyses. For example, if flow over a plane surface is considered, the permissible orthogonal coordinate systems are rectangular, polar, and spiral (Ref. 1). In the analysis which follows, it will be shown that a considerable degree of generalization can be achieved in formulating similarity solutions of the boundary-layer equations by the simple expedient of relaxing the requirement that the coordinate system be orthogonal.

-2 - The system to be considered will be a linear skew system. It will further be shown that similar solutions of the boundary layer referred to a rectangular coordinate system are carried over directly as solutions in the linear skew systems. Analysis of the Boundary Layer Equations in Linear Skew Coordinates Consider a linear skew coordinate system embedded in a surface as shown in Figure 1. The skew coordinates x, z are related to the rectangular coordinates X, Z by the equations. x = X sec 0 X = x cos 0 (1) z = Z - Xtan 0 Z = z + xsin 0 The y-coordinate remains normal to the surface and hence is unchanged by the above transformation. If mainstream velocity components in the x- and z- directions are designated by U and W, respectively, and the velocity components of the boundary layer in the x-, z- and y- directions by u, w and v respectively, the three-dimensional boundary-layer equations become (c.f. Ref. 2 Chap. 18). uu + vU + w = U + U + v2u (2a) I + vw + + wz + a (2b) ux + Vz + U + Wz 2 b 6y c6z, OK 6z a y

-3 - YY Z,z,u,U X 11-11" LEADING EDGE OF PLATE Figure 1. Orientation of skew coordinates

The continuity equation is au + av + aw= o (2c) Equations (2a) - (2c) are clearly identical to the equations which one would write for a system of rectangular coordinates. It follows that solutions of the boundary layer equations in rectangular coordinates are valid for the equations expressed in linear skew coordinates. The requirements for similarity solutions of the boundary layer equations in rectangular coordinates have been determined in Reference (1). The results can be summarized as follows. TI = g(x,z) (3a) u F1t (r) (3b) U w = G' (r) (3c) W v Sv au- F +U F'-F) +G + W (G'-G)J (3d) g ax x az a6z Substitution of expressions (3b) - (3d) into Equations (2a) - (2c) gives (4a) F' - C1F'2 + (C1-C2)F"F + (C3-C4)F"G - C5G'F' + C1 + C5= 0 (4b) G"' - C3G' + (C3-C4)G"G + (Cl-C2)G"F - C6F'G' + C6 + C = 0

where 1 = u -= g x - U ang C 2 F 9 &m 1 a C = 3 2 x g W aing 4 2 bz g =.w au c5 2 aZ ^"^l C6,. Uaw Tg w2~ Theboundary conditions on F and G are F(O) = G(O) = 0 F' (0) = G'(0) = 0 (l1) = Lim G' (r) = 1 Tq-. co Lim F' T-> 00 (in order equations they are: If it is now required that the Ci (i = 1, 2,...6) be constant, that Equations (4a) and (4b) become ordinary differential in (T) four sets of solutions for U, W and g2 can be found, (Reference 1) Case I U_ aenx m-l U = aenxzmW = benxzTm Case II U = axnzm-1 W = bxn-lzm g2 cU - cb W x a z g2= cU = ca W b z

Case III Case IV U = axn U = aenx W = bxm W = bemx g2_ cU g2= cU x (Four other cases could also be listed by obvious symmetrical interchange of variables) It is interesting at this point to investigate the differences between the requirements on the mainstream flow as referred to rectangular coordinates and those referred to the linear skew coordinates. This can be done readily by expressing the results for the two systems in terms of rectangular coordinates by use of Equations (1). Table I lists expressions for mainstream velocity components and streamlines: Table I - Comparison between mainstream characteristics in rectangular and skew-linear coordinate systems.

Table I (continued) Case III Case IV Rectangular Skew-Linear Rectangular Skew-Linear U aXn'Xn naef aenX | W | bXm| b'Xm bem' bem'X Mainstream m-na+l (mt-n)X reamlinZ=a (m-n+) ZaX +Xtan Z=ale(m-n)X Z=ale +Xtane In order to illustrate more completely the distinctions between solutions expressed in rectangular and skew-linear systems the following example has been chosen from Case III. Let U = UO = const and W = 2Uox. The corresponding mainstream streamlines are then given by z = x2 + const (5) The similarity equations for the boundary layer are 2 + F"' = (6a) G"' + G"F - F'G' + 1 = 0 (6b) F(O) = G(O) = F'(O) = G'() = 0 Lim F' = Lim G' = 1 TI-4 oo 'T|-> o* Solutions for Equations (6a) and (6b) are given in Reference (3). The functions F'(q) and G';(r) are plotted in Figure 2. It is shown in

(P-!x 0 2 3 4 5 6 Figure 2. Functions F' (n) and G' (n) plotted against n

-9 - Reference (3) that the so-called "limiting streamline" (i.e. the envelope of boundary layer streamlines as y -~ 0) is given by z = X2G"(0) + const. = 4.270x2 + const. (7) F"(O) Plots of Equations (5) and (7) appear in Figures 3a - 3c for a rectangular system and linear-skew systems in which the angle inclination of the x-axis and the X-axis is 30~ and 60~ respectively. It can be seen from these plots that when rectangular coordinates are employed the mainstream flow crosses the surface leading edge at right angles while in the skew system the mainflow crosses the leading edge at the angle of inclination of the x-axis. In the latter cases, the surfaces can be considered as inclined to a uniform flow. The relationship between boundary-layer velocity profiles can be obtained readily from Equation (1): u = u cos 0 (8a) v= v (8b) w = w + u sin 0 (8c) where u, v and w have been used to denote the boundary-layer velocity components of a flow in a skew system referred to the XZ - axes. Substituting the relations for u and w from (3b) and (3c) into (8a) and (8c) gives u = U cos 0 F'(r)!. (9a) w = WG'(rq) + U sin 0 F' (T) (9b)

-10 - The velocity component normal to the surface remains identical to that of the orthogonal coordinate system. Thus it is seen that the velocity profiles for a skew system exhibit similarity in the X-direction but not in the Z-direction unless W = U. Requirements for Similar Temperature Profiles Requirements for similar temperature profiles in a skew-linear system can be determined from an analysis of the boundary-layer energy equations under the assumption of constant fluid properties, (c.f. Reference 4). The equation in terms of the skew system variables is: 6,, T CT 6T 62^T 62 ^ o.nu w2 2 ]u pCp (u x + v Wz) = k + w- + 2 sin 0- -y (10) *p ax dy az by b y by by by A non-dimensional temperature ratio is now defined as (c.f. Reference 4) T - T T - T n =,_~m00 = 00 T* Tw - T where T = temperature in the boundary layer TO = temperature of the mainstream Tw = surface temperature T* = T*(x,z) = T - TO If O* is expressed as a function of the similarity variable T], e* = 8O(r) Equations (3a), (3b), (3c), (3d) and (11) can be used to transform Equation (10) into the form 8 "-C7F' -C8G' + Pr(C1-C2) 'F + Pr(C3-C4) 'G + C9F" 2 + C1G 2 + FG =0 (12) o (0) =.1 Lim ( (r) = Ti "4CO

-11 - 9 8 7 6 5 N N I / X//II / 1/ 1/ I,/ / /I.j~/ / _ I __ /</ /F / MAIN-FLOW STREAMLINES /_/_ ------ _ _ L IMAIN-FLOW STREAMLINE ON -- LIMITING STREAMLINE ON SURFACE a/~~~~ I 4 3 2 0 0 I 2 3 X, x Figure 3a. Streamlines of flow in skew-linear coord. _ = 0~

-12 - 9 8 7 6 5 N N 1/ 17 / / / 0 I / / v ------ MAIN FLOW STREAMLINES / - - LIMITING STREAMLINE / /:0 \ ON SURFACE 0/ 8 300 0 I 2 Figure 3b. Streamlines of flow in skew-linear coord, 6 = 30~

-13 - 9 _ _ _ _ _ 8/: I | / ^ --- MAIN-FLOW STREAMLINES ON SURFACE 5 N // =6 0~ Q LI 4 / "9 2 / - -MAIN-FLOW STREAMLINES - LIMITING STREAMLINES ON SURFACE 00 O l Figure 3c. Streamlines of flow in skew-linear coord, 0 = 60~ 2

-14 - where P U 6taT* P U2 C7 r C9 r g2 ax T*C PrW anT* PrW2 C8= o7 -z 0-r g2 7 z T*C p Cll = 2 sin 0 r UW = 2 sin 0 I10Y 4 T*1Cp In order that Equation (12) be an ordinary differential equation C7, C8, C9 and ClO(and hence, Cll) are required to be constant. From the expressions for C9 and Co1 it is readily seen that U must be proportional to W, i.e. the streamlines of the mainstream are straight. However, the coefficients Cg and Clo arise from the viscous dissipation term in the energy equation and if it is assumed that viscous heating is negligible, the requirements imposed by C9 and C10 can be disregarded. If it is also assumed that T* = constant, Equation (12) becomes " + Pr(C1-C2)8 'F + Pr(C3- C4)8 'G = 0 (13) Thus the requirements for similar temperature profiles are precisely those given for the mainstream velocity components and the function g. A similar result is given in Reference (5) for two - dimensional flows over a body of revolution rotating in a fluid at rest and a body of revolution in a rotating fluid,

-15 - If the case of negligible viscous heating but variable wall temperature is considered, the two equations P U aT* C = const. = r g2T* ax g Txa PrW T* C8 = const. =- P g2T* az constitute:.a system df equatinns for T*. The solutions for T* corresponding to the four sets of requirements on U, W and g are as follows: Case I: T* = (const.) erxzs Case II: T* = (const.) xrzs Case III: T* = (const.) xresz Case IV: T* = (const.) erxesz (Again, four additional cases can be obtained by appropriate interchange of independent and dependent variables in all expressions of any of the four cases) At present solutions of Equation (12) corresponding to computed functions F and G have not been carried out by the authors. Obtaining such solutions should be a rather straightfOrwxd".- numerical procedure. In conclusion, it can be said that the requirements for all possible similarity solutions of the boundary layer and energy equations in skew-linear coordinates have been found and presented. The usual Cartesian system is simply a special case.

-16 - REFERENCES 1. Hansen, A. G., "On Possible Similarity Solutions for Three Dimensional, Incompressible, Laminar Boundary-Layer Flows Over Developable Surface and With Proportional Mainstream Velocity Components', NACA TM 1437, 1958o 2o Michal, A. D., "Matrix and Tensor Calculus" John Wiley and Sons Inc., 1947. 35 Hansen, Ao G. and Herzig, H. Z., "Cross Flows in Incompressible Boundary Layers", NACA TN 3651, 1955. 4. Schlichting, H., "Boundary Layer Theory", McGraw-Hill Book Company, Inc., 1955. 5. Stojanovic, D., "Similar Temperature Boundary Layers", Journal of the Aero/Space Sciences, September 1959.