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S' I . NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1579 A METHOD OF QUADRATURE FOR CALCULATION OF THE LAMINAR AND TURBULENT BOUNDARY LAYER IN CASE OF PLANE AND ROTATIONALLY SYMMETRICAL FLOW* By E. Truckenorodt 1. SUMMARY For calculation of the characteristic parameters of the boundary layer (momentumloss thickness and form parameter for the velocity pro file), two quadrature formulas are given which are valid for the laminar as well as for the turbulent state of flow. These formulas cover both the twodimensional and the rotationally symmetrical case. The calculation of the momentumloss thickness is carried out by a simple integration of the energy theorem. The equation for the form parameter is obtained by coupling of the momentum theorem with the energy theorem. Knowledge of the derivatives of the velocity distribution and of the radius of the body along the length x is not necessary. 2. INTRODUCTION The calculation of the laminar and turbulent boundary layer in case of pressure drop and pressure rise is the decisive problem in the deter mination of the flow loss in channels and pipes and of the flow drag of bodies. We shall treat below the plane problem as well as the rotationally symmetrical one. In these considerations, we shall limit ourselves to incompressible flows. Whereas theoretical treatment of the laminar boundary layer has been fundamentally clarified, one is still dependent on semiempirical connections for the treatment of the turbulent boundary layer. Since the exact methods in the laminar case are rather troublesome, various approxi mation methods have been developed just like for the turbulent case. All these methods are based on the momentum equation given for the first time by Th. von KarmAn (ref. i). *"Ein Quadraturverfahren zur Berechnung der laminaren und turbulenten Reibungsschicht bei ebener und rotationssymmetrischer Stromung." Ingenieur Archiv, Band XX, Viertes Heft, 1952, pp. 16228. NACA TM 1579 The approximation method for the plane laminar case has been per fected by K. Pohlhausen (ref. 2). Later on, it was essentially improved by H. Holstein and T. Bohlen (ref. 5): they introduced as the desired parameter not the boundarylayer thickness as Pohlhausen had done but the momentumloss thickness. More recently, K. Wieghardt (ref. 4) der ived an energy equation in addition to the momentum equation which he uses for developing a twoparameter calculation method. A. Walz (ref. 5) simplified this latter method by reverting to the oneparameter condition as in the methods of Pohlhausen and HolsteinBohlen. The rotationally symmetrical method which is analogous to the Pohlhausen method was indi cated by S. Tomotika (ref. 6). Its simplification in the sense of the HolsteinBohlen statement was carried out by F. W. Scholkemeyer (ref. 7). A method for calculation of the plane turbulent boundary layer was indicated for the first time by E. Gruschwitz (ref. 8). Aside from the momentum theorem, Gruschwitz also uses a semiempirical equation obtained from certain energy considerations for determination of a form parameter marking the velocity profile in the boundary layer; this form parameter characterizes the sensitivity to separation of the boundary layer. The empirical parameters appearing iii Gruschwitz' were investigated once more by A. Kehl (ref. 9). Another method which was similar to Gruschwitz' method was developed by E. Buri (ref. 10). In the United States, a method by A. E. von Doenhoff and N. Tetervin (ref. 11) proved to be use ful. In this method, too, an empirical equation, in addition to the momentum theorem, is used for determination of a form parameter char acterizing the velocity profile. Starting from this report, H. C. Garner (ref. 12), England, developed a method which is superior to that of v. DoenhoffTetervin in its numerical evaluation. The transfer of the momentum theorem to the rotationally symmetrical case was performed by C. B. Millikan (ref. 15). Owing to recent investigations by H. Ludwieg and W. Tillmann (ref. 14) and J. Rotta (ref. 15) oh the theoretical properties of turbulent flows, particularly of the wall shear stress and the energy loss in the boundary layer, it is possible to find a better basis for and to improve the existing semiempirical methods. It is the aim of the present report to develop a calculation method which is equally valid for the four cases of laminar and turbulent as well as plane and rotationally symmetrical flow. The most recent results will be taken into consideration. 5. THE MOMENTUM THEOREM AND ENERGY THEOREM As is well known, Prandtl's oPundarylayer equations and Bernoulli's equation represent the fundamental equations for calculation of boundary NACA TM 1379 layers. As stated in the Introduction, we are going to consider incom pressible flows. Since, however, the following derivations can be given for variable density p without particular difficulties, we shall make the transfer to incompressible flow only in the final result. We assume that, in the rotationally symmetrical case, the radius of the body or pipe R is large compared to the thickness of the boundary layer 5. The equations mentioned then read ax 6y x 6y +(puR) + 6(pvR) 0 (2) 6x oy 6p dU = Pau (3) ox dx Therein, u and v signify the velocity components within the boundary layer 0o y & in x and y direction, U the velocity outside of the boundary layer (y 2 ), figure 1. p is the pressure, assumed constant across the boundarylayer thickness, T the shear stress, and p the density of the flowing medium. For the plane (twodimensional) case, the radius R is to be omitted in equation (2) as well as in all following formulas. The pertaining boundary conditions are y = O : u = 0, v = v0, T = TO, P = (4) y = 6 : u = U, T = O, p = Pa If V0 J 0, one is dealing with the case of suction (v0 < 0) or of blowing (v0 > 0). NACA TM 1579 We now combine the three equations (1) to (5) by adding to equa tion (1) which has been multiplied by ur the equation (2) been multiplied by ur+l/(r + 1)R; one may choose arbitrarily r = 0, 1, 2. One then obtains 1 6A(pur+2R) + i(pur+lv) = (r + 1)urgU dU R 8x y + dx + C) by The continuity equation (2) is integrated over the distance y wall, and one obtains pvp pv = puR dy + p Sdx J0 which has (5) from the (6) If we now assume the velocity distribution distribution T(y) in y direction to be tion (5) over y from y = 0 to y = 6. over, substitute equation (6) u(y) and the shearstress known, we may integrate equa We then obtain, if we, more 1 ._(p Ur+2Rfr) + gr dU er PaUr+2R dx U dx Pa U Therein, the newly introduced abbreviations signify fr = 0 Pa U gr = (r + 1) er = (r + 1) r+1 1 ( dy W 0 U( u u U p U T 'dy y PaUj2 NACA TM 1579 Equation (7) is valid quite generally for laminar and turbulent, incom pressiole and compressiole, plane and rotationally symmetrical flow with and without suction or blowing. As said before, the radius R is to be omitted in the plane case. If r is put equal to zero, there results the wellknown momentum equation of von KarmAn. K. Wieghardt (ref. 4) and J. Rotta (ref. 15) have shown that, for r = 1, equation (7) may ue given a physical interpreta tion, namely that of the energy equation. We shall now assume that the flowing medium is incompressible, p = Constant, and also that neither suction nor blowing occur v0 = 0. Then r becomes r = 0: momentum theorem (v. KarmAn, Pohlhausen) 1 d(U2R u2R dx lRM S5* dU T U dx U2 = fo= J/ 5* = go = u U j 1 dy as momentumloss thickness as displacement thickness TO eO as wall shear stress pU2 r = 1: energy theorem (Wieghardt, Rotta) 1 d (U3R5 ) = 2 +t UR dx pU3 with (10) (11) (12) NACA TM 1579 with 8 = fl d+ t pU3 JO el 2 as energyloss thickness pu2 by \ as shearstress work1 In the laminar case, the shearstress work equals the energy converted into heat (dissipation d). In case of turbulent flows, not all energy is converted into heat; one part still remains asturbulence energy (t) which is, however, usually negligibly small. The fact that, in the case r = 1, the second term on the left side of equation (7) vanishes is important since, with the density assumed to be constant, gl = 0. If we now introduce the boundarylayer thickness ratios H = * . and 8 H= & fl (15) we may write for equations (9) and (12) I d (u2Rc ) + H B dU = , U2R dx U dx pU2 1 d(u55R) = 2 d + t UR dx pu3 (16) (17) We subtract equations (16) from (17) and find2 iThis equation is obtained by partial integration from (8). 2For the laminar case, a corresponding formula has already been given by A. Walz (ref. 5). In the turbulent case, too, one can show that the empirical equations indicated by E. Gruschwitz, A. E. von Doenhoff and N. Tetervin, and also H. C. Garner for calculation of the form param eter of the velocity profile (in Gruschwitz' paper 1 = 1 (uO/U)2, in those of the others H) have a structure very similar to that of equation (18). (13) (14) ul u.' dy U \u NACA TM 1379 7 d ( dU d + t +0 S= (H 1)B + 2 (18) dx Ud pU pu2 As will be shown in the following section, the shear stress TO/pU2 and the shearstress work (d + t)/pU3 may be expressed as functions of the dimensionless quantities U/v (Reynolds number formed with the momen tum thickness) and H. Furthermore, a oneparameter condition is valid in good approximation for the velocity distributions in the boundary layer; that is, there exists a fixed relation between the boundarylayer thickness ratios R and H. With consideration of these facts, the equations (17) and (18) represent two equations for the two unknowns 3 and H. In contrast to the existing methods, we shall calculate the momentum loss thickness not from the momentum theorem (16) but from the energy theorem (17). This offers decisive advantages for the performance of the integration indicated in section 4. We resort to the momentum theorem in connection with the energy theorem for calculating the parameter H, equation (18), which is characteristic for the velocity profile. 4. SHEAR STRESS, SHEARSTRESS WORK, AND BOUNDARYLAYER THICKNESS RATIO In order to be able to operate with the equations (17) and (18), one must know the dependence of the shear stress and the shearstress work on the quantities Uo/v and H, and likewise the connection H(H). (a) Laminar Flow Assuming singleparameter velocity profiles we may write S= fY.H\ whence then follows To = (H) with a1 = U (19) pU2 U8 b O d 0(H) pU3 u6 with H = f 1]d 0 U J S= p/i' H= 0 a 0 \ 2 U y 6 d  6 NACA TM 1379 (20) ul dJ uI (d For evaluation of these formulas, we select the socalled Hartree pro files; these are the exact profiles for the velocity variation U ~ xm. In figure 2, the quantities a, 0, H are plotted against H 3. It is of interest that 0 is almost independent of H. For the case without pressure gradient, one has a = 0.220, 3 = 0.173, and H = 2.60. (b) Turbulent Flow Recently, H. Ludwieg and W. Tillmann (ref. 14), as well as J. Rotta (ref. 15), dealt in detail with the determination of the wallshear stress in case of turbulent boundary layers with pressure gradient. LudwiegTillmann indicate for the range of the Reynolds numbers 1 x 10l < U9/v < 4 x 104 the following interpolation formula T0 pU2 S0.123 100.678H U o0.268 VI1 (22) Rotta finds for the wallshear stress the relation U 1 idH U" B . n 1 + B u* K \ v ' (23) The fol (21) %We took over the numerical values of A. Walz (ref. 5). lowing identities are valid: a2 E*, B1 = D* and R= H32. NACA TM 1579 Therein u* = To/p signifies the shear velocity. Thus u 2 ~(24) OU2 \ ) is valid. In equation (25) 1/< = 2.5 and B = B(II) is a function of the quantity Ii H The function B was evaluated and H u* graphically represented by Rotta. We calculated the shearstress values for various values of H and plotted them against the Reynolds number U3/v in figure 5. The agree ment between the values according to LudwiegTillmann and to Rotta is quite satisfactory. J. Rotta also dealt with the calculation of the shearstress work (dissipation and turbulence energy). For the dissipation, there applies d In H U) + (25) pU U/Lh The function appearing therein, G = G(11), has been evaluated by Rotta. With consideration of equation (25), one may write for equation (25) also 5 = ( 2 1 + u(G B1 (25a) Since u f= fiiUH) and I = H 1 I, the dissipation may also be U \v H u* represented as a function of the Reynolds number U9/v and of the form parameter H; this has been done in figure 4. It is found that the differences for different values H are only slight; whereas, for the wallshear stress, they are very considerable. We see from figure 4 that the dissipation may be approximated by the statement d (26) pU5 U3) n v NACA TM 1379 A suitable value for n is n = 1/6. The dependence of the value Ot on H is only slight so that we may assume for it the constant value S= 0.56 x 102. For calculation of the turbulence energy, J. Rotta indicates an approximation formula. With the aid of this formula, one can show that the turbulence energy is negligibly small compared to the dissipation; this fact has already been pointed out by Rotta. We put, therefore, for our further calculations t =0 (27) pU3 The boundarylayer thickness ratio H (energyloss thickness/momentumloss thickness) may also be determined from J. Rotta's results. The equation 12 (H 1)2 H= H + (28) Il2 H is valid. Therein, 12 is a form parameter which is in a fixed relation (indicated oy Rotta) with 1I. We calculated according to equation (28) the values R in dependence on the quantity H and on the Reynolds num ber Us/v. The influence of the Reynolds number was shown to be vanish ingly small. The result of our calculation is represented in figure 5. K. Wieghardt (ref. 16) also has dealt with the connection R(H). He finds H =AH with A = 1.269 and B = 0.579 (29) H B The resulting curve also has been plotted in figure 5. Except for the larger values of H, the agreement with Rotta's curves is satisfactory. 5. THE APPROXIMATION METHOD We shall summarize the result of the previous section: The repre sentations T= f (L, H) d + t U H) and i = h(H) p p 3 pu V NACA TM 1379 given in figures 2 to 5, are valid. In the laminar case, the values TO/pU2 and d/pU3 are inversely proportional to the Reynolds number. (a) The Calculation of the MomentumLoss Thickness As mentioned above, we shall determine the momentumloss thickness from the energy theorem (17). We write for the shearstress work according to equations (20), (26), and (27) d + t ((H) S(30o) pU5 U )n with n = 1 being valid for the laminar, n = 1/6 for the turbulent case. We now substitute this expression into equation (17) and obtain with n x U3+2nRl+ngl+nUn)3 (31) d = 2(1 + n)B(H)fnU5+2nRl+n (52) dx If we now assume that H is known from x and that H, too, is known by the unique relation H(H), equation (52) may be integrated with respect to x, and there results, if we introduce in addition the quantity e = (U8 (n) ) U (33) and take equation (51) into consideration Kl + E(H)US+2nRl+ndx' e(x) = xl (54) H( +n 35+2n l+n H(x) U(x) R(x) NACA TM 1379 As a new abbreviation we introduced E(H) = 2(1 + n)B(H)Hn (35) We plotted this function in figure 6 for the two cases of laminar and turbulent flow. By way of approximation, we shall now assume constant mean values for E(H) and H. This assumption proves correct in a particularly satisfactory manner for the turbulent case. We then obtain from equa tion (34) Cl + A x US+2nRl+ndx' 9(x) = U 2= 1nl+ (36) Herein A E 2(1 + n) (37) fl+n H signifies a mean value suitable for the range xl < x' < x. The integration constant is determined to be Cl = U3+2nR1+n Ul 1n (38) with n = 1/6 if, starting from the point xl, a turbulent boundary layer is present. As the laminar or, respectively, turbulent mean value for A, we shall choose the value which results when we assume U = UD to be constant and consider the flat plate for fully laminar or fully turbulent flow. Equation (36) then becomes, with x1 = 0 and C1 = 0 Op = Unp = Ax (39) Between the momentumloss thickness Sp(l) and the drag coefficient cf of a plate of the length I wetted on one side and approached by a flow of the velocity Um there exists the connection NACA TM 1579 _p()_ Cf (40) 1 2 If we put in equation (59) x = 1, there follows by comparison of equa tions (59) and (40) A = (U n 1+n (37a) According to the existing flow state, cf is to be taken for laminar or turbulent flow (fig. 7). The notation (57a) offers the additional advantage that the surface roughness also can easily be taken into con sideration, merely by substitution of the corresponding cf values of a rough plate. For smooth surfaces, there result the values of table 1 for the constant A. TABLE 1. THE QUANTITIES n AND A FOR LAMINAR AND TURBULENT FLOW Laminar Turbulent (Blasius) (Falkner)4 n 1 1 6 A 0.441 0.760 x 102 We now solve equation (56) for the momentum thickness 9 and obtain if we introduce, in addition, dimensionless quantities and take equation (57a) into consideration the following expression S + ()+nf U 5+2n l+nd 1'+n L _1 2t (41)(5) d =C* x+0 (.0L R V. M. Falkner, Aircraft Engineering, 19, 1945, p. 65. cf = 0.o6 is valid. NACA TM 1579 The integration constant is determined as C i Ul R1). l+n Ft 1 W 2 (42) Summarizing, we repeat once more: If the flow, starting from the initial point xl = O, is fully laminar or fully turbulent, one has Cl* = 0. One has'to insert accordingly the laminar or turbulent drag coefficients. If a laminar starting length precedes the turbulent boundary layer, the value for in equation (42) is to be taken from the laminar calcula tion (second formula in equation (42)). For the laminar case n = 1, for the turbulent case n = 1/6 is valid. The twodimensional case results if one omits in equations (41) and (42) the radius R. As can be seen from equation (41), the quantity n does not play a significant role in the turbulent case so that one may assume for rough calculations also n = 0. (b) The Calculation of the Form Parameter We shall determine the form parameter H which is characteristic for the velocity profile in the boundary layer from equation (18); we write this equation as follows: e(x) d F(H)r(x) + G(H) dx (43) Here e is given according to equation (55). For the other abbreviations r(x) = dU U dx (44) F(H) = (H 1)H (45) G(H) = (2 d+ t pU3 pU2 ' U 2 R 2d l+n NACA TM 1579 are valid. Furthermore, we transform equation (45) by introducing the substitution L(B) = / L(H) (46) F(l) and the abbreviation K(H) G( ) K(L) (47) F(H) We then obtain a differential equation for the new form parameter L e(x)d = r(x) K(L) (48) dx We determined the quantity L by graphical integration of the func tion 1/F(H) over H. The value L = 0 we have placed in the domain of vanishing pressure gradients (constant pressure, flat plate). In fig ures 8 and 9, we represent the relation H(L), also for the laminar case a(L), and, in figures 10 and 11, the relation K(L) for the cases of laminar and turbulent flow. Whereas no influence of the Reynolds num ber exists for the function K(L) in the laminar case, that influence is rather considerable for the turbulent case. One can show that it is possible to give to the equations for calculation of the form parameter appearing in the reports of E. Gruschwitz, A. Kehl, and H. C. Garner the form of our equation (48)5. The resulting connections for H(L) and K(L) have also been plotted in the figures named above. The differences between the individual methods are therefore based on the deviations of the curves H(L) and K(L). For solution of equation (48), we set up a linear expression for K(L), (compare the figures 10 and 11): K(L) = a(L b) (49) The quantities a and b are obtained, for instance, from table 2. In the turbulent case, b is, in addition, dependent on the Reynolds num ber and therewith on the length x. 5A detailed comparison of the individual methods has been carried out in an unpublished report of the author. NACA TM 1579 If we substitute equation (49) into equation (48), we obtain a linear differential equation of the first order for L which we can solve in closed form. For this purpose, we make the substitution p dx S (x) C + A X US3+2nRl+ndx A X1# (50) The last relation follows from the fact that in formula (56) for E the denominator is exactly equal to the derivative of the numerator multiplied by 1/A. The numerical values a/A for smooth surfaces are also plotted in table 2. Without dealing which we perform, in that the derivative L = L E in detail with the intermediate calculation for addition, a partial integration, in such a manner dU/dk does no longer appear we find finally + n U(_)+ 1 Ul t . (51) At the initial point xl = 0, there is also 1i = 0. The first term in equation (51) then disappears. Especially for the laminar case there applies with b = 0 the simple expression L = n U(' d o u() (52) We shall report later on regarding the initial values for L at the stagnation point xl = 0 and at the transition point. As can be seen from equation (51), one may provide the new vari able t with an arbitrary factor without causing thereby a change in L. We may therefore write a xA I = C + (50a) 1+n x/1 +2n l+n 2 X11 I m jP ) (I b(t') In U ) dE' b' ) U1  NACA TM 1579 TABLE 2. THE QUANTITIES a, b, AND a/A FOR LAMINAR AND TURBULENT FLOW IPressure drop. 2Pressure rise. The calculation of t thus will be very simple, since the expression in brackets already occurs in the calculation of the momentumloss thickness (eq. (41)). A complete calculation is carried out as follows. Prescribed: General quantities: U(x), R(x), U. State of flow l1, L.l Desired: n, Cf, a laminarr or turbulent); initial values A Momentumloss thickness a(x); form parameters L, H(x). First, one calculates the momentumloss thickness according to equa tion (41) by performing a simple quadrature. Hence one forms, if one has to calculate turbulent flows, the Reynolds number U/v with which one then calculates the quantity b according to table 2. Next, one deter mines according to equation (50a) the new variable t over which one integrates the function b In U/U1. According to equation (51), one then obtains the form parameter L and, by means of figure 8 or figure 9, the boundarylayer thickness ratio H. Due to the large exponent a/A in equation (50a), the quantity t increases very rapidly with growing U1, R1, NACA TM 1379 length x. This signifies, however, that the term (ft/ )LI in equa tion (51) loses its significance more and more with increasing distance; therewith it is shown that the initial value L1 has, in case of larger distances x (for instance in the neighborhood of the separation point), only slight influence. The advantage of our method lies in the fact that only simple quadratures have to be performed in the individual case. Beyond that, no derivatives of the initial values U and R with respect to x occur. (c) The Initial Values The initial values for the momentumloss thickness as well as for the form parameter L may be different according to the case to be dealt with. 1. Flow toward a body with stagnation point. The boundarylayer calculation is started at the stagnation point xl = 0. There U = 0 and R = 0. From equation (355) then follows immediately o9 = 0 (53) In the twodimensional as well as in the rotationally symmetrical case, the potential velocity varies linearly with the length x, that is U= ex with c = (54) The constant c is different for the twodimensional and for the rota tionally symmetrical case. For the variation of the radius, there results also a linearity in x, namely R =x (55) The integration constant in equations (54), (36), and (41) disappears X1 = C1 = Cl* = O (56) NACA TM 1379 If we substitute equations (54) and (55) into equation (41), there follows, if we put all quantities which additionally enter the formulas in the rotationally symmetrical case into braces  intermediate calculation x>0: = , after a brief 1n 1 cf (,,/x+n 1 n I, 2F2(2 + n) + 1 + +n *l+n L +. Jj Therein c* = signifies the dimensionless expression for U, \Wx/0 (57) c. In the laminar case (n = 1), there results a value different from zero for the momentumloss thickness in the neighborhood of the stagnation point tO0 1 Cf 2 2(5 + (3) TC* (58) If one takes into consideration that c* = c U6 write for equation (58) also = 1.528=  Cf v c, (O) 3 and cf = 1.328, one may S .470 +r 14 (59) In the turbulent case (n = 1/6), the momentumloss thickness at the stagnation point itself (x = O) has the value zero. For the momentumloss thickness ratio at the stagnation point for equal velocity increase (dU/dx)o in the rotationally symmetrical and in the twodimensional case (c = Constant), one then has NACA TM 1579 1 '/ orot\ 2(2 + n +n 60eb + c=const + n (60) If the numerical values for n are substituted, there results in the laminar case the value 0.867 and in the turbulent case 0.816. The exact value for the laminar flow which one can calculate from the Hartree pro file is 0.84). The initial value of the form parameter L is determined according to equation (48) by putting therein according to equation (55) S = 0, r0 = K(LO) (61) P0 is obtained from the following boundarylayer determina 9 dU Q A FO = lim = lim = xO U dx x>0 x 2(2 + n) + (62) S+ n} In the laminar case, there results with equation (59) 0 c .2 0.220 3+ (1} (62a) and from equation (49) K(LO) =0_ 0.077 a a 3+ 1{ (63) (See table 3.) Although, according to equation (62), in the turbulent case a finite value different from zero does result for pO, it is of no 6It can be shown that this value follows also from equation (52). The value tion NACA TM 1379 help in calculating the value LO according to equation (61) since the values K(L) for the Reynolds numbers U_ >0 are not known. Since the flow at the stagnation point always will be initially laminar, we refer to section 5, c5, where we report on the initial values at the transition point. For the variation of the form parameter L with the length x at the stagnation point, there applies also (as can be derived from equa tion (48)) () = 0 (64) dx 0 2. Flow into a channel. The boundary layer begins at a point x1 = Xk = 0 where the velocity, and likewise the radius in the rotationally symmetrical case, are of finite magnitude. Since again X1 = C1 = CI* = 0, there follows immediately from equations (54), (36), and (41) Gk = O and "k = 0 (65) and from equation (44), when dU/dx #/ rk = 0 (66) The form parameter then follows from equation (48) as K(Lk) = 0 (67) In the laminar case, this form parameter becomes Lk = O (68) that is, the value of the flat plate in longitudinal approach flow. There is no difference between the twodimensional and the rotationally symmetrical case. (See table 3.) For the turbulent flow, there applies what was said above in the discussion of the stagnationpoint flow. The variation of the form parameter L with x is obtained from equation (48) NACA TM 1579 (p) Wk dL k 1 dU A dL k (69) 1 1 dU 1+a k A The numerical value a/A is to be taken from table 2. TABLE 5. LAMINAR INITIAL VALUES. THE EXACT VALUES INSOFAR THEY DO NOT AGREE WITH THE APPROXIMATION VALUES ARE PUT IN PARENTHESES Stagnationpoint flow Channel flow Plane, Plane Rotationally rotationally symmetrical symmetrical S 0 .2 0.271 0.23 k 0 S(.292) (.247) LO .0260 .0195 Lk 0 (.0292) (.0208) H0 2.25 2.52 Hk 2.60 (2.22) (2.50) ao .45 .520 .220 (.560) (.524) k 5. Transition laminarturbulent. Behind a certain transition region, the laminar boundary layer is transformed into the turbulent layer at the point x, = x. From the theory of the origin of turbulence, com pare H. Schlichting (ref. 17), one can find, in agreement with measurements, that the transition occurs at places which lie somewhat downstream with respect to the velocity maximum. Since the phenomena in the transition region have not yet been explored, determination of the initial values for 9 and H or L which are required for the turbulent calculation is only approximately possible. WAat is fundamental in our considerations NACA TM 1379 has been shown in figure 12 on the example of the flat plate. Up to the point xu the boundary layer is laminar and obeys the regularities according to Blasius, that is, ~ ~xT. The corresponding Hvalue is constant and amounts to H = 2.60. Starting from the point xu the boundary layer is fully turbulent. Here applies approximately 4 ~ Su + c(x xu). The corresponding Hvalue depends, in addition, to some extent on the Reynolds number Um~o/ prevailing at the transition point. According to measurements, for instance, by F. SchultzGrunow (ref. 18), also from the similar solutions in case of constant pressure by J. Rotta (ref. 15), one finds about 1.2 < H < 1.4, with the Hvalue being the smaller, the larger the Reynolds number. Thus the Hvalue decreases compared to the laminar value. In the transition region, it must therefore vary continuously from the laminar to the turbulent value. In figure 13, we plotted the difference AC against U/v. For the sake of simplicity, we let xu coincide with xu; then the initial value for the momentumloss thickness of the turbulent calculation is equal to the momentumloss thickness which would result at the point xu if the flow were fully laminar up to this point. We shall therefore put St(xu) = xu = O(Xu) (70) For the Hvalue, we write Ht(xu) = Hu = Hi(xu) CH (71) We shall now assume that AH may be taken from figure 15 also for the cases with pressure gradient. This assumption we deem justified since the transition point lies in the proximity of the point of vanishing pressure gradient and, thus, the values of the flat plate may be used in suitable approximation. Having thus determined Hu, we ascertain according to figure 9 the value Lu = L(xu) which then enters equation (51) as L1. We want here to point out once more that in places lying further downstream from transition point the initial value for L is only of slight signif icance; therefore, a somewhat rougher estimate seems justified when itis a matter of determining the separation point. (d) The Separation Point Knowledge of the position of the separation point also is important. Separation results when the wallshear stress assumes the value zero. Whereas this point in the laminar case corresponding to the prescribed velocity profile is fixed by ca = 0 according to equation (19), it is 24 NACA TM 1379 not yet possible to make a perfectly clear statement regarding the separa tion point in the turbulent case. According to the wallshearstress statements of LudwiegTillmann and Rotta (compare fig. 3), the shear stress decreases with increasing value H, however, without attaining the value zero. As approximation rule one may, for instance, assume according to v. DoenhoffTetervin (ref. 11) that the separation starts at the earliest when H 1.8 7 and has certainly taken place when H has attained the value 2.4. Table 4 presents a compilation of the occur ring values. TABLE 4. VALUES OF THE FORM PARAMETER AT THE SEPARATION POINT (e) Examples The usefulness of the method described above is shown on one example each of laminar and of turbulent flow. 1. Howarth flow (laminar). As example for the laminar flow, we choose the wellknown Howarth flow (ref. 19). In this case, the velocity distribution is U x u =1 U0, (72) If equation (72) is substituted into equation (41) where the radius distri bution R is omitted, the quadrature may be carried out in closed form. There results for the momentumloss thickness c f 1A _( x 2\6 3I (73) Laminar Turbulent H 4.058 1.8 to 2.4 L 0.018 0.15 to 0.18 'This statement corresponds .approximately to Gruschwitz' assumption n = 1 (u./U)2 = 0.8. 0 0.2711 vE) U.11 NACA TM 1579 In figure 14, we show the comparison of this approximation with the exact values of Howarth. Within drawing accuracy, the agreement is perfect. For determination of the form parameter, we first calculate the new variable ( according to equation (50a) a 6 6 =ZA with Z = (1 l 1 ) (74) The form parameter itself also may be calculated analytically according to equation (52). Since we find ourselves, corresponding to the pre scribed velocity distributions, in the region of pressure increase, we choose according to table 2 the value a/A = 8. We obtain after a brief intermediate calculation L= 1 i d' 1 dU 'r 1 Z Z8 dZ 1 Zm S0 ud t' 6z8 0 1 Z' m= m + 8 (75) Taking equation (74) into consideration, we represented L(x) also in figure 14. From these values we determined, with the aid of figure 8, the values a for the wallshear stress. They, too, are represented in figure 14 and are there compared with the exact values of Howarth. The agreement is quite satisfactory. We also showed the curve which results according to the Pohlhausen method.8 One achieves, as already stated by A. Walz, a considerably better determination of the separation point, a = O, if one uses in addition to the momentum theorem also the energy theorem, as in our method. 2. Profile NACA 65(216)222 (turbulent). As an example for the turbulent calculation, we choose the profile NACA 65(216)222 (approxi mately) for which measurements (ref. 11) as well as theoretical calcula tions have been carried out. The conditions refer to the upper side of the profile placed at an angle of attack a = 10.10. The Reynolds num ber is Umt/v = 2.64 x 106. The graphical representation of the velocity distribution, the momentumloss thickness, and the form parameter according to measurements, likewise the result of our calculation, are given in 8This curve we took from A. Walz (ref. 5). NACA TM 1379 figure 15. We started our calculation at the first point measured, that is, x/l = 0.075. The agreement between our approximation and the meas urement may be called satisfactory.' For comparison, we also plotted the results according to the calculation of v. DoenhoffTetervin and according to the methods of Garner, Gruschwitz, and GruschwitzKehl. Regarding the form parameter H, the last two methods deviate greatly from the other methods and from the measurement. 9As A. E. v. Doenhoff and N. Tetervin pointed out, the larger differ ences between measurement and calculation for the momentumloss thickness in the neighborhood of the separation point x/2 0.55 might be caused by systematic measuring inaccuracies. NACA TM 1579 APPENDIX DISCUSSION OF THE FORMULA FOR CALCULATION OF THE MOMENTUMLOSS THICKNESS As mentioned before, the momentumloss thickness was calculated, so far, from the momentum theorem (16). With the use of certain simplifica tions, it is possible to find for the calculation of the momentumloss thickness from the momentum theorem a formula similar to our equation (36). Two possibilities exist: 1. The wallshear stress is determined according to the laws of the flat plate in longitudinal approach flow where the length x is expressed by the momentumloss thickness 8, and the approachflow velocity U, by the local velocity. For the laminar as well as for the turbulent case, one may write TO a If one substitutes, moreover, in the momentum theorem a constant value for the boundarylayer thickness ratio H, one can integrate the momentum theorem in closed form, as was shown by E. Truckenbrodt in an unpublished report (compare H. Schlichting, Boundarylayer theory, page 450; one obtains the result .x C1 + (1 + n)a UaRl+n dx' n x V UaRl+n where a = (1 + n)(H + 2) n. The corresponding numerical values may be taken from table 5. NACA TM 1379 TABLE 5 Laminar Turbulent n 1 !(Blasius) I(Falkner) 4 6 A 0.441  0.0076 Factor ahead of the + n .441 O. .007 Ineral (1 + n)a .441 0.0160 .0076 integral inc .470 .0160  3 + 2n 5.0  .53 Exponent of the velocity + 2 0  3.6 distriutiona 8.2 4.0 5.67 b 5.0 4.0  2. According to K. Pohlhausen for the laminar case and to E. Buri for the turbulent case, the wallshear stress and the boundarylayer ratio are functions of the quantity r dU U dx that is, T0 pU2 fl(r) ()n (n H = f2(r) If one puts fx X* = JX 0 A = 8Rl+n Rl+n dx NACA TM 1579 one can find after some intermediate calculations the following equation: d = ( dx * with SA dU U dx* Therein E(r) = (i + n)fl(r) [ + n + (1 + n)f2F(rr represents a universal function for the laminar or, respectively, turbu lent state which, in the first case, may be calculated analytically (for instance, for the Pohlhausen polynomials) and, in tie second case, may be determined from measurements. In both cases, one may find by way of approximation a linear connection between and r (compare A. Walz, LilienthalBericht 141 (1941) and E. Buri). (r) = br + c If one substitutes this expression in the above equation, the integration may be carried out in closed form and yields after a simple intermediate calculation ( )n x Cl + cJ UbRl1+n dx' U R1+n The numerical values c and b are also indicated in table 5. (Compare H. Schlichting, Boundarylayer theory, pp. 199 and 424.) The constants according to equation (56) also nave been indicated for comparison. Translation by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM4 1579 REFERENCES 1. Von Karman, Ti. : Z. angew. Math. Mech. 1, 1921, p. 255. 2. Pohlhausen, K.: Z. angew. Matii. Mech. 1, 1921, p. 252. 5. Holstein, H., and Bohien, T.: LilienthalBericht S 10, 1940, p. 5. 4. Wieghardt, K.: Ing.Arch. 16, 1948, p. 251. 5. Walz, A.: Ing.Arch. 16, 1948, p. 245. 6. Tomotika, S.: Laminar Boundary Layer on the Surface of a Sphere in a Uniform Stream. ARCRep. 1678, 1955. 7. Scholkemeyer, F. W.: Die Laminare Reibungsschicht an Rotationssym netrischen K6rpern. Diss. Braunschweig, 1945. 8. Gruschwitz, E.: Ing.Arch. 2, 1951, p. 521. 9. Kehl, A.: Ing.Arch. 12, 1945, p. 295. 10. Bari, E.: Eine Berechnungsgrundlage fir die Turbulente Grenzschicht bei Beschleunigter und Verzogerter Stromung. Diss. Zurich, 1951. 11. Von Doenhoff, A. E., and Tetervin, N.: Determination of General Relations for the Behavior of Turbulent Boundary Layers. NACA Rep. 772, 1945. 12. Garner, H. C.: The Development of Turbulent Boundary Layers. ARC Rcp. 2155, 1944. 15. Millikan, C. B.: Trans. Am. Soc. Mech. Eng., Appl. Mech. 54, 1952, [Io. 2, p. 29. 14. Ludwieg, H., and Tillmann, W.: Ing.Arch. 17, 1949, p. 288. 15. Rotta, J.: Ing.Arch. 19, 1951, p. 31 and 20, 1952, p. 195. 16. Wieghardt, K.: Zur Turbulenten Reibungsschicht bel Druckanstieg. Untersuchungen und Mitteilungen der Deutschen Luftfahrtforschung, UM 6617, 1944. 17. Schlichting, H.: GrenzschichtTheorie, Karlsruhe, 1951. 18. SchultzGrunow, F.: Luftfahrtforschung, 17, 1940, p. 259. 19. Howarth, L.: Proc. Roy. Soc. London, Ser. A 919, Bd. 164, 1938, p. 547. NACA TM 1379 Y U(x) S^ u(x, y) 8' x Plane U SU(x) Sx RotationallyU UmO Figure 1. Survey sketch. a.3 0.5 1.7 04 0.3 Q2  2D 3D 4.0 Figure 2. Laminar frictionlayer parameters (Hartree profiles). H S Constant pressure N" %\\ a Separationt SI NACA TM 1579 20 0H=1.2 1.8 06 ^ 04o 2 .0 0.2 :*1 _* Rotta " Ludwieg,Tillimann . I I I t I 103 2 5 104 2 U' 1/ 105 Figure 3. Turbulent wallshear stress (according to H. Ludwieg, W. Tillmann, and J. Rotta). NACA TM 1579 d x Pu o.103 2 5 104 2 5 1o5 Figure 4. Turbulent dissipation (according to J. Rotta). NACA TM 1579 1.0 1.2 1.4 1.6 1.8 2.0 2.2 24 Figure 5. Turbulent boundarylayer thickness ratio (according to J. Rotta and K. Wieghardt). NACA TM 1379 E,Exio2 Figure 6. The function E(H) for laminar and turbulent flow. 0.1 1 10 5 2 S~llKner, Cf Ul U5 2 5 106 2 5 107 2 5 108 2 5 109 Figure 7. Drag law of the smooth flat plate in longitudinal approach flow. NACA TM 1579 Figure 8. Relation between the boundarylayer thickness ratio H and the wall shearstress coefficient a, on one hand, and the form parameter L on the other for laminar flow. 0.2 0.1 0 0.1 Relation between the boundarylayer thickness the form parameter L for turbulent flow. ratio H and Figure 9. NACA TM 1357 Figure 10. The function K(L) for laminar flow.  Truckenbrodt  Approximation (49)  Garner  Gruschwitz  GruschwitzKehl Figure 11. The function KiL) for turbulent flow. NACA TM 1379 Lominar Turbulent Ou x XU.xu Figure 1,. Survey sketch from the transition point of the fiat plate. AH SchultzGrunow (flat plate) 05Rotta (similar solution in the case of constant pressure) 10 2 5 104 2 5 105 v Figure 1. Variation of the form parameter H in the range of transition from laminar to turbulent flow. NACA TM 1579 0.02 Figure 14. The laminar boundarylayer parameters of the Howarth flow. NACA TM 1579 . Measurement. {v. DoenhoffTetervin, Garner fGruschwitz, LGruschwitz Kehl Truckenbrodt 0 0,I 0.2 0.3' 0.4 0.5 0.6 j Figure 15. Turbulent boundary layer on the profile NACA 65(216)222 a. = 10.10, T = 2.64 x 106. Measurement according to NACA Rep. 772. NACALangley 52455 1000 Uco  4 3 4 cO a LS" t S S^ O r g 0 e L C a 0S . E L. a E o3 .i0 WO..,i0 S Lao. u d 00 wr3. ,I 0 s a L2 55 a =i oi o I CO~ X!0 :z' R * * i o i\ 4 4 0 0 I i B3d :.< . 3 jaaf.i I I litli 6 C) a 0 c '1. (5 E SOr ll5  1 z zl.4 m I K a, a 00 Vo S 2 5 h a, iBj s a 0. 0 2 a, * 0 b. j 0 aa Cf o *m S S i *A S 0o = a o a0 .5 &sa o. k g8 S  rib a , 0 1 oI ais g eii i s r* : Ills an .aj~ m  .. c . 0 S. 0. *5 a < <  II c; oi iia f? 0 0 4  UNIVERSITY OF FLORIDA 3 1262 0810 548 3 
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