Ni\c^ vA-nr] NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1379 A METHOD OF QUADRATURE FOR CALCULATION OF THE LAMINAR AND TURBULENT BOUNDARY LAYER IN CASE OF PLANE AND ROTATIONALLY SYMMETRICAL FLOW By E. Truckenbrodt Translation of *Eiii Quadraturverfahren zur Berechnung der laminaren und turbulenten Reibungsschicht bei ebener und rotationssymmetrischer Stromung. " Ingenieur-Archiv, Band XX, Viertes Heft, 1952. Washington May 1955 ARY n -7011 US/ NATIONAL ADVISORY COMMITTEE FOR AERONAITTICS TECHNICAL MEMORANDUM 1379 A METHOD OF QUADRATURE FOR CALCULATION OF THE LAMINAR AND TURBULENT BOUNDARY LAYER IN CASE OF PLANE AND ROTATIONALLY SYMMETRICAL FLOW* By E . Truckenbrodt 1. SUMMARY For calculation of the characteristic parameters of the boundary layer (momentum- loss 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 two-dimensional and the rotationally symmetrical case . The calculation of the momenti:m-loss 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 oiorselves 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 raomenti^m equation given for the first time by Th. von Karman (ref . l) . "Ein Quadraturverfahren zur Berechnung der laminaren und t\arbulenten Reibungsschicht bei ebener und rotationssymmetrischer Stromung." Ingenieur- Archiv, Band XX, Viertes Heft, 1952, pp. l6-228. 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 boundary- layer thickness as Pohlhausen had done but the momentum- loss thickness. More recently, K. Wieghardt (ref. k) der- ived ein energy equation in addition to the momentum equation which he uses for developing a two-parameter calculation method. A. Walz (ref. 5) simplified this latter method by reverting to the one-parameter condition as in the methods of Pohlhausen and Holstein -Bohlen. 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 Holstein-Bohlen statement was carried out by F . W. Scholkemeyer (ref. 7) • A method for calculation of the plane turbulent bo\indary 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 boundeiry layer . The empirical parameters appearing in 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 momentijm 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. Doenhoff -Tetervin in its numerical evaluation. The transfer of the momentimi theorem to the rotationally symmetrical case was performed by C. B. Millikan (ref. 13) . Owing to recent investigations by H. Ludwieg and W. Tillmann (ref. Ik) and J- Rotta (ref. I5) 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. 3. THE MOMENTUM THEOREM AND ENERGY THEOREM As is well known, Prandtl's boundary- layer 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 incbrapressible flow only in the final result. We ass\:me that, in the rotationally symmetrical case, the radius of the body or pipe R is large compared to the thickness of the boundary layer 6 . The equations mentioned then read P|u|H^v|h dx dy 5p ^ ^T Bx ^y (1) 5(puR) ^ S(pvR) ^ Q hx. Sy (2) dx dx (5) Therein, u and v signify the velocity components within the boundary layer 0< y*^ 6 in x and y direction, U the velocity outside of the boundary layer (y ^ ^) } figure 1. p is the pressure, assumed constant across the boundary- layer thickness, t the shear stress, and p the density of the flowing medium. For the plane (two-dimensional) case, the radius R is to be omitted in equation (2) as well as in all following formulas. The pertaining boundary conditions are y = : u = 0, y = 5 : u = U, ^0^ T = 0, ■\ p = P0_ > p = Pa J (^) If Vq ^ 0, one is dealing with the case of suction (vq < 0) or of blowing (vq > 0) . 1^ NACA TM 1579 We now combine the three equations (l) to (5) by adding to equa- tion (1) - which has been multiplied by \a?" - the equation (2) which has been multiplied by \jf'*'^/{T + l)R; one may choose arbitrarily r = 0, 1, 2, . . . One then obtains i |-(pur+2R) + |-(pur+lv) = (r + l)ur(p^U ^ + |l^ R dx dy \ dx dyj (5) The continuity equation (2) is integrated over the distance y from the wall, and one obtains pv = - I ^J p^ ^y + Po^o (6) If we now assume the velocity distribution u(y) and the shear-stress distribution T(y) in y direction to be known, we may integrate equa- tion (5) over y from y = to y = 5. We then obtain, if we, more- over, substitute equation (6) _1 A.(p u^+2Rf ) ^^dU r dU ^ ^ Pa U (7) Therein, the newly introduced abbreviations signify fr = / _P_ u '0 Pa U - (r + 1) 1- - r+1 dy Pa r-1 dy > ej. = - (r + 1) ul^ a U/ MPaU^ ^Vy (8) NACA TM 1579 Equation (7) is valid quite generally for laminar and turbulent, incom- pressible and compressible, 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 well-known momentum equation of von Kirman . K. V/ieghardt (ref . k) and J. Rotta (ref . I5) have shown that, for r = 1, equation (7) may be 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 vq = . Then r becomes r = 0: momentum theorem (v. KirmAn, Pohlhausen) J^ A(u2r,) + 5^ ^ = I^, u2r d^ U dx pu2 (9) with ^ = f n = 1 - ii U dy as momentum- loss thickness (lO) &* = g, O '0 L 1 - ^ U dy as displacement thickness (H) .U2 = er as wall shear stress r = 1: energy theorem (Wieghardt, Rotta) 1 d u5r dx ^(u5r6) = 2 d + t .u5 (12) with NACA TM 1379 5 = f 1 = 1 - dy as energy-loss thickness (13) d + t pu2 ^yVU/ as sheax-stress work (H^) In the laminax case, the shear-stress work eqiials the energy converted into heat (dissipation d) . In case of turbiilent flows, not all energy is converted into heat; one part still remains as -turbulence 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, g]_ = . If we now introduce the boundary -layer thickness ratios H= 5i and H= 5 (15) we may write for equations (9) and (l2) J_ A(u2r^) + h i ^ = I^, u2r dx U dx u2 (16) _i_ -4-(u5rh^) = 2 ^-^t^ u^R ^ pu5 (17) We subtract equations (16) from (I7) and find^ ■^Dhis equation is obtained by partial integration from (8) . For 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 t] = 1 - (u-d/u)^, in those of the others H) have a structure very similar to that of equation (18) . NACA TM 1579 , ^ = (H - 1)H i |U ^ 2 l±-i - H I2_ (18) As will be shown in the following section, the shear stress Tq/pU and the sheaj--stress work (d + t) /pu5 may be expressed as functions of the dimensionless qiiantities U^/v (Reynolds number formed with the momen- tum thickness) and H. Furthermore, a one-parameter condition is valid in good approximation for the velocity distributions in the boundary layer; that is, there exists a fixed relation between the boundary-layer thickness ratios fl and H. With consideration of these facts, the equations (l7) and (l8) represent two equations for the two unknowns i3 and H. In contrast to the existing methods, we shall calculate the momentum- loss thickness not from the momentum theorem (l6) but from the energy theorem (l7) • This offers decisive advantages for the performance of the integrations indicated in section k . We resort to the momentum theorem in connection with the energy theorem for calculating the parameter H, equation (l8), which is characteristic for the velocity profile. SHEAR STRESS, SHEAR-STRESS WORK, AND BOUNDARY-LAYER THICKNESS RATIO In order to be able to operate with the equations (17) and (I8), one must know the dependence of the shear stress and the shear-stress work on the quantities UiS/v and H, and likewise the connection H(H) . (a) Laminar Flow Assuming single -parameter velocity profiles we may write whence then follows tq ^ a-z(H) pU2 U^ V with a, = — ^ (19) NACA TM 1379 3i(H) pu5 Mi V with h (20) and H ,s/-a 1 - ii U d^ H = '5/i3 (S)' d^ (21) For evaluation of these formulas, we select the so-called Hartree pro- files; these are the exact profiles for the velocity variation U ~ xm. In figure 2, the quantities a, 3> H are plotted against H 5- It is of interest that p is almost independent of H. For the case without pressure gradient, one has a = 0.220, p = 0.175^ ^nd. H = 2.6o. (b) Turbulent Flow Recently, H. Ludwieg and W. Tillmann (ref. 1^), as well as J. Rotta (ref. 15)^ dealt in detail with the determination of the wall-shear stress in case of tirrbulent boundary layers with pressure gradient . Ludwieg -Tillmann indicate for the range of the Reynolds numbers 1 X 105 < Ui3/v < 4 X 10^ the following interpolation formula Q-^^ 10-0 .678H pU2 u^\0.268 (22) [v) Rotta finds for the wall-shear stress the relation JL=iln|HHiV-^ u^ B (23) A-^e took over the nimierical values of A. Walz (ref. 5). The fol- 32- lowing identities are valid: a^ = e*, Pj = D* and H = H NACA TM 1579 Therein u* = \/Tq/p signifies the shear velocity. Thus pU'= U (21^) is valid. In equation (23) 1/k = 2.5 and B = bCI^) is a functi on of the quantity I-|_ H H — — . The function B was evaluated and graphically represented by Rotta. We calculated the shear-stress values for various values of H and plotted them against the Reynolds number U-fl/v in figure 5- The agree- ment between the values according to Ludwieg-Tillmann and to Rotta is quite satisfactory. J. Rotta also dealt with the calculation of the shear -stress work (dissipation and turbulence energy) . For the dissipation, there applies pU- u*l i ink nil + G \ (25) The f\mction appearing therein, G = G(Ii), has been evaluated by Rotta. With consideration of equation (25), one may write for equation (25) also pU- U 1 + f (G - B) (25a) Since ^ = f P^.H and It = ^ " ^ -^, the dissipation may also be U \ V ' / ^ H u* represented as a function of the Reynolds number U-fl/v and of the form parameter H; this has been done in figure h. It is found that the differences for different values H are only slight; whereas, for the wall-shear stress, they are very considerable. We see from figure k that the dissipation may be approximated by the statement d _ ^t(S) .u3 (26) ^) 10 NACA TM 1379 A suitable value for n is n = I/6 . The dependence of the value p^ on H is only slight so that we may assume for it the constant value p = 0.56 X 10-2. 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 = (27) pU^ The boundary-layer thickness ratio H (energy-loss thickness/momentum- loss thickness) may also be determined from J. Rotta 's results . The equation 5 = 3 . H + ^ '" - ^' (28) is valid. Therein, I2 is a form parameter which is in a fixed relation (indicated oy Rotta) with 1^. We calculated according to equation (28) the values H in dependence on the quantity H and on the Reynolds num- ber UtS/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 H(H) . He finds H = — — with A = 1.269 and B = 0-579 (29) H - B The resiilting 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 :^=f(>^.H), l±^.i=g(!«,Hl -d H=h(E) NACA TM 1579 11 given in figures 2 to 5, are valid. In the laminar case, the values Tq/pU^ and d/pU^ are inversely proportional to the Reynolds number . (a) The Calculation of the Momentum -Loss Thickness As mentioned above, we shall detennine the momentiim-loss thickness from the energy theorem (l?) • We write for the shear-stress work according to equations (20), (26), and (27) 5-ti=MHL (30) with n = 1 being valid for the laminar, n = l/6 for the turbulent case . We now substitute this expression into equation (l7) and obtain with X = u5+2nRl+ngl+n/lW^''^ (3^) ^ = 2(1 + n)p(H)Hnu5+2nRl+n (32) 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 = (^f)\ (55) and take equation (5I) into consideration Xl + / E(H)u5+2nRl+n(i^- e(x) = i (5^^) - , , 1+n , . 5+2n , , 1+n H(x) U(x) R(x) 12 NACA TM 1579 As a new abbreviation we introduced E(H) = 2(1 + n)3(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 (3ij-) Ci + A r"" u5+2nRl+n^. e(x) . ( m\ = "^ (56) \ V y y5+2nj^l+n Herein A = _i-= 2(1 + n)g (57) fil+n H signifies a mean value suitable for the range x^ < x' < x. The integration constant is determined to be Ci = Ui5+2nR^l+n^M:)%^ (38) with n = 1/6 if, starting from the point x^, 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 = Uj„ to be constant and consider the flat plate for fully laminar or fully turbulent flow. Equation (36) then becomes, with x-^ = and C3_ = 0p ^Mp)\ = Ax (39) Between the momentum-loss thickness •9p(i) and the drag coefficient Cf of a plate of the length I wetted on one side and approached by a flow of the velocity U„ there exists the connection NACA TM 1379 13 ^p(i) (i^o) If we put in equation (39) x = I, there follows by comparison of equa- tions (39) and (UO) (37a) According to the existing flow state, Cf is to be taken for laminar or turbulent flow (fig. 7)- The notation (37a-) 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 (Blasius) Turbulent, (Falkner)^ n A 1 O.Ul 1 6 0.760 X 10 "2 We now solve equation (36) for the momentum thickness -0 and obtain - if we introduce, in addition, dimensionless quantities and take equation (37a) into consideration - the following expression ^(x) I Ci* + -^ \l+n p^/l ^u\5+2n^R\l+n xjl V"-' U) ^ l_ 1+n UV R W. M. Falkner, Aircraft Engineering, 15, 19^3^ P • 65 . Cj (1^1) .0306 1/7 {¥) is valid . ll^ NACA TM 1579 The integration constant is determined as Cl* = l+n 2 — — l+n (^2) Summarizing, we repeat once more: If the flow, starting from the initial point Xn = 0, is fully laminar or fully turbulent, one has C]_* = 0. One has to insert accordingly the laminar or turbulent drag coefficients. If a laminar starting length precedes the turbulent boimdary layer, the value for d in equation {k2) is to be taken from the laminar calcula- tion (second formula in equation (^2)) . For the laminar case n = 1, for the turbulent case n = I/6 is valid. The two-dimensional case results if one omits in equations (hi) and (U2) the radius R. As can be seen from equation (^1) , 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 : 8(x) ^= F(H)r(x) + G(H) dx (^3) Here 9 is given according to equation (55) • For the other abbreviations r(x) e du U dx (kk) F(H) = (H - 1)H and G(H) = d + t )U5 « m) n > (h^) NACA TM 1379 15 are valid. Furthermore, ve transform equation (Uj) by introducing the substitution ^<« = Jm - """ '"'' and the abbreviation K(H) = - ^ili = K(L) (i^T) F(H) We then obtain a differential eqioation for the new form parameter L e(x)^= r(x) - K(L) (48) We determined the qiiantity L by graphical integration of the func- tion 1/f(H) over H. The value L = 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 ajid 11, the relation K(L) for the cases of laminar and turbiolent 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 tiirbulent 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 (ij-8)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 (k8) , we set up a linear expression for K(L) , (compare the figures 10 ajid 11): K(L) = a(L - b) (1^9) The quantities a and b are obtained, for instaince, from table 2. In the turbulent case, b is, in addition, dependent on the Reynolds num- ber and therewith on the length x . ■^A detailed comparison of the individual methods has been carried out in an unpublished report of the author . 16 MCA TM 1579 If we substitute equation (^9) into equation (h-Q) , 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 i = e ' dx e(x) (Ci + A u3+2nj^l+n^AA (50) The last relation follows from the fact that in formvila (56) for 9 the denominator is exactly equal to the derivative of the numerator multiplied by i/A. The numerical values a/A for smooth surfaces are also plotted in table 2 . Without dealing in detail with the intermediate calculation - for which we perform, in addition, a partial integration, in such a manner that the derivative dU/d^ does no longer appear - we find finally 1 L, -H 1. H(il ^ 1 I u^ i^i- b(r) - in u(r) Un d^' (51) At the initial point x-, = 0, there is also I-, = 0. The first term in equation (51) then disappears. Especially for the laminar case there applies with b = the simple expression L = in^i^idi: (52) We shall report later on regarding the initial values for L at the stagnation point Xj^ = and at the transition point. As can be seen from equation (5l)> one may provide the new vari- able ^ with an arbitrary factor without causing thereby a change in L. We may therefore write — Cl* + /Cj.\ \2/ 1+n ) px/l 5+2n ) 'b 1+n / x' I (50a) NACA TM 1379 17 TABLE 2.- THE QUANTITIES a, b, AND a/A FOR LAMINAR AND TURBULENT FLOW LaminELT Turbiilent L > ^ L < "^ a b a A 2.87 ( 6.5 5-53 8.0 5.01+ X io"2 0.071g — - 0.23 V k.O Ipressixre drop. ^Pressure rise . The calculation of | thus will be very simple, since the expression in brackets already occurs in the calculation of the momentum-loss thickness (eq. {hi)). A complete calculation is carried out as follows . Prescribed; General quantities: U(x) , R(x) , U„i State of flow n, C|., — (laminar or turbulent); initial values U-]_, R^, Desired ; Momentum-loss thickness t3(x); form parameters L, H(x) . First, one calcvilates the momentum-loss thickness according to equa- tion (kl) by performing a simple quadrature. Hence one forms, if one has to calculate turbulent flows, the Reynolds number U-o/v with which one then calculates the quantity b according to table 2. Next, one deter- mines according to equation (50a) the new variable | over which one integrates the fvinction b - In U/U^ . According to equation (51) :> one then obtains the form parameter L and, by means of figure 8 or figure 9^ the boundary-layer thickness ratio H. Due to the large exponent a/A in equation (50a), the quantity | increases very rapidly with growing 18 NACA TM 1579 length X. This signifies^ however, that the terra (^i/|)Lx in equa- tion (51) loses its significance more and more with increasing distance; therewith it is shown that the initial value Lj 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 quadratiores have to be performed in the individual case . Beyond that, no derivatives of the initial values U and R with respect to X occixr. (c) The Initial Values The initial values for the momentum- loss 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 boundary-layer calculation is started at the stagnation point X2_ = . There U = and R = 0. From equation (53) then follows immediately eo = (55) In the two-dimensional as well as in the rotationally symmetrical case, the potential velocity varies linearly with the length x, that is U = ex with c = 2^ (5I1) The constant c is different for the two-dimensional and for the rota- tionally symmetrical case . For the variation of the radius, there resiilts also a linearity in X, namely R = X (55) The integration constant in equations (5^+), (56), and (ill) disappears Xi = Ci = Ci* = (56) NACA TM 1579 19 If we substitute equations (5U) and (55) into equation (^1), there follows, if we put all quantities which additionally enter the formulas in the rotationally symmetrical case into braces ^ k after a brief intermediate calculation -- f = '- — r -^(f r <^^> r , , r ni+n n*i+n ^! 2(2 + n + J 1 + n L I Therein c* = — — signifies the dimensionless expression for c. In the laminar case (n = 1) , there results a value different from zero for the momentum-loss thickness in the neighborhood of the stagnation point f ) = , " ^ (58) *^0 2\/2(3 + {1}) ^ If one takes into consideration that c* = — c and c^ = -^ , one may write for equation (58) also \f^i„ = 1.528 ^E*fl) = i°-'*l° (59) In the turbulent case (n = I/6) , the momentum- loss thickness at the stagnation point itself (x = 0) has the value zero. For the momentum-loss thickness ratio at the stagnation point for equal velocity increase (dU/dx)Q in the rotationally symmetrical and in the two-dimensional case (c = Constant), one then has 20 NACA TM 1379 ^0£^\ ^ l 2{2 + n)' c=const 1 ,1+n %eb I i 5 + 3n > /c=const \ (60) If the numerical values for n are substituted, there results in the laminar case the value O.867 and in the turbulent case O.816. The exact value for the laminar flow which one can calculate from the Hartree pro- file is O.8J+5. The Initial value of the form parameter L is determined according to equation (48) by putting therein according to equation (55) ^n ~ ^ ' To = K(Lo) (61) The value Fq is obtained from the following bo\mdary-layer determina- tion Fq = lim = lim — = — (d2) x_^0 U dx x_^o ^ 2(2 + n) + (l + nj In the laminar case, there results with equation (59) 3+ {1} '0 = -j Oo and from equation (k'^) 6 ^ k(Io) ^^^ 0-077 (65) 5 + {^} (See table 5-) Although, according to equation (62), in the turbulent case a finite value different from zero does result for Fq, it is of no °It can be shown that this value follows also from equation (52) • NACA TM 1379 21 help in calculating the value Lq according to equation (6l) since the values K(L) for the Reynolds numbers — — > are not known. Since V the flow at the stagnation point always will be initially laminar, we refer to section 5> cj, 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 (i+8)) ^^ = {6k) dx/o 2. Flow into a channel . - The bovmdary layer begins at a point X-, = Xi^ = where the velocity, and likewise the radius in the rotationally symmetrical case, are of finite magnitude. Since again X-j^ = C-i = Cj* = 0, there follows immediately from equations (5^), (^6), and {kl) \ = and \ = (65) and from equation (i+^), when dU/dx ^ <» Tk = (66) The form parameter then follows from equation (U8) as K(Lk) = (67) In the laminar case, this form parameter becomes Lk = ' (68) that is, the value of the flat plate in longitudinal approach flow. There is no difference between the two-dimensional and the rotationally symmetrical case. (See table 5-) ^or the turbulent flow, there applies what was said above in the discussion of the stagnation-point flow. The variation of the form parameter L with x is obtained from equation (48) 22 NACA TI4 1379 1 dy\ u dx)^ 1 dU (69) The numerical value a/A is to be taken from table 2 . TABLE 3.- LAMINAR INITIAL VALUES. THE EXACT VALUES - INSOFAR THEY DO NOT AGREE WITH THE APPROXIMATION VALUES - ARE PUT IN PARENTHESES Stagnation-point flow Channel flow Plane Rotationally symmetrical Plane , rotationally symmetrical Ho °0 0.271 ( .292) .0260 ( .0292) 2.25 (2.22) .51+5 ( .360) 0.235 ( .2U7) .0195 ( .0208) 2.32 (2. 30) .320 ( .321+) \ Lk Hk o-k 2.60 .220 3. Transition laminar-turbulent .- Behind a certain transition region _, the laminar boundary layer is transformed into the turbulent layer at the point ^± = x . From the theory of the origin of turbulence^ com- pare H. Schlichting (ref . I7) , 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 t3 and H or L which are required for the turbulent calculation is only approximately possible. What is fundamental in our considerations NACA TM 1579 25 has been shown in figure 12 on the example of the flat plate . Up to the point X.. the boundary layer is laminar and obeys the regularities according to Blasius, that is, ^ ^ \fx. The corresponding H-value is constant and amounts to H = 2.60. Starting from the point x^ the boundary layer is fully tvirbulent . Here applies approximately i3 ~ 0^ + c(x - x^) . The corresponding H-value depends, in addition, to some extent on the Reynolds number Uto-fl/v prevailing at the transition point. According to measurements, for instance, by F. Schiiltz-Grunow (ref. 18), also from the similar solutions in case of constant pressure by J. Rotta (ref. 15), one finds about 1.2 < H < l.k, with the H-value being the smaller, the laj'ger the Reynolds number. Thus the H-value decreases compared to the laminar value. In the treinsition region, it must therefore vary continuously from the laminar to the turbulent value. In figure I3, we plotted the difference M against U^/v. For the sake of simplicity, we let x^^ coincide with x^^; then the initial value for the momentum-loss thickness of the turbxilent calculation is equal to the momentum-loss thickness which would result at the point x^ if the flow were fully laminar up to this point . We shall therefore put ^t(xu) = ^u = ^l(xu) (70) For the H-value, we write Ht(xu) = Hu = Hi(xu) - m (71) We shall now assume that M may be taken from figure 13 also for the cases with pressure gradient. This assimption we deem justified since the transition point lies in the proximity of the point of vanishing pressvire gradient and, thus, the values of the flat plate may be used in suitable approximation. Having thus determined H^, we ascertain according to figure 9 the value L^ = L(xu) which then enters equation (5I) as L^. 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 it- is 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 wall-shear stress assumes the value zero . Whereas this point in the laminar case corresponding to the prescribed velocity profile is fixed by a^ = according to equation {±9), it is 2k NACA TM 1379 not yet possible to msike a perfectly clear statement regarding the separa- tion point in the tiorbulent case . According to the wall-shear-stress statements of Ludwieg-Tillmann and Rotta (compare fig. 3) j 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. Doenhoff-Tetervin (ref . 11) that the separation starts at the earliest when H « 1.8 ''' and has certainly taken place when H has attained the value 2 .^4- . Table k presents a compilation of the occur- ring values . TABLE h.- VALUES OF THE FORM PARAMETER AT THE SEPARATION POINT Laminar Turbulent H L i+.058 -0.018 1.8 to 2.h -0.15 to -0.18 (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 well-known Howarth flow (ref. 19) • In this case, the velocity distribution is Uoo I (72) If equation (72) is substituted into equation (^t-l) where the radius distri- bution R is omitted, the quadrature may be carried out in closed form. There results for the momentum-loss thickness I '.\le 1 -^ ^ 0.2711 1 -^ V --J ^1 (73) 'This statement corresponds approximately to Gruschwitz' assumption T] = 1 - (uyu)2 = 0.8. NACA TM 1579 25 In figure Ik, 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-) I = z^ with z = 1 - (1 - ^ ] = 1 _ (JL) (7I1) 6 , .6 JL 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^i.r idy_,.(^p.^ _j^r z'8 dz' = - ^ y' z^ I Jo ud^' ^^QjQ 1 - z- ^;fe"^ + 8 (75) Taking equation (7^^-) into consideration, we represented L(x) also in figure Ik . From these values we determined, with the aid of figure 8, the values a for the wall-shear stress. They, too, are represented in figure Ik and are there compared with the exact values of Howarth. The agreement is quite satisfactory. We also showed the ciorve which results according to the Pohlhausen method. ° One achieves, as already stated by A. Walz, a considerably better determination of the separation point, a = 0, if one uses in addition to the momentum theorem also the energy theorem, as in our method . 2. Profile NACA 65(2l6)-222 (turbulent) .- As an example for the tijrbulent calculation, we choose the profile NACA 65(216) -222 (approxi- mately) for which measurements (ref . 11) as well as theoretical calcula- tions have been cairried out. The conditions refer to the upper side of the profile placed at an angle of attack a = 10 .1° . The Reynolds num- ber is U^Z/v = 2.64 X 106. The graphical representation of the velocity distribution, the momentum-loss thickness, and the form parameter according to measurements, likewise the resiolt of our calculation, are given in "This curve we took from A. Walz (ref. 5) 26 NACA TM 1579 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 .9 For comparison, we also plotted the results according to the calculation of v. Doenhoff-Tetervin and according to the methods of Garner, Gruschwitz, and Gruschwitz-Kehl . 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 momentum-loss thickness in the neighborhood of the separation point x/l ~ 0-55 might be caused by systematic measuring inacc\jracies . NACA TM 1579 27 APPENDIX DISCUSSION OF THE FORMULA FOR CALCULATION OF THE MOMENTUM-LOSS THICKNESS As mentioned before, the momentum-loss 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 momentum-loss thickness from the momentum theorem a formula similar to our equation (36) . Two possibilities exist: 1. The wall-shear stress is determined according to the laws of the flat plate in longitudinal approach flow where the length x is expressed by the momentum-loss thickness -9, and the approach-flow velocity Uoo by the local velocity. For the laminar as well as for the turbulent case, one may write Tfi a '0 PU2 (^r If one substitutes, moreover, in the momentum theorem a constant value for the boundary-layer 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, Boundary-layer theory, page kyD; one obtains the result Ci + (1 + n)a / U^Rl+n dx' U^\'' „ ^1 e = -) ^ = vJ ya^l+n where a = (l + n) (H + 2) - n . The corresponding numerical values may be taken from table 5. 28 NACA TM 1579 TABLE 5 Laminar Turbulent n 1 — (Blasius) k i(Falkner) 6 Factor ahead of the integral A (1 + n)a c 0.4i+l .kkl .k-jo .0160 .0160 0.0076 .0076 Exponent of the velocity distribution 3 + 2n a b 5.0 8.2 5.0 k.O k.o 3.33 3.67 2 . According to K . Pohlhausen for the laminar case and to E . Burl for the turbulent case, the wall-sheax stress and the boimdary- layer ratio are functions of the quantity P ^ e dU U dx that is, pU^ fi(r) and If one puts H = fp(r) A = BR 1+n and Rl+n dx NACA TM 1379 29 one can find after some intermediate calculations the following equation: dx* with P _ A dU U dx* Therein e(r) = (1 + n)f-L(r) - [2 + n + (1 + n)f2(r2]r represents a universal fimction 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 the 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, Lilienthal-Bericht l4l (19^+1) 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 -(^r- + c U^R-^^'^ dx' u^r1+^ The numerical values c and b are also indicated in table 5- (Compare H. Schlichting, Bo\andaxy-layer theory, pp. 199 axid h2k.) The constants according to equation (56) also have been indicated for comparison. Translation by Mary L. Mahler National Advisory Committee for Aeronautics 30 NACA ™ 1579 REFERENCES 1. Von Karman, Th.: Z. angew. Math. Mech. 1, 1921, p. 253. 2. Pohlhausen, K.: Z. angew. Math. Mech. 1, 1921, p. 252. 5. Holstein, H., and Bohlen, T.: Lilienthal-Bericht S 10, 19^0, p. 5. h. Wieghardt, K.: Ing.-Arch. 16, 19^8, p. 231. 5. Walz, A.: Ing.-Arch. 16, 19^+8, p. 2i+3 . 6. Toraotika, S.: Laminar Boundary Layer on the Surface of a Sphere in a Uniform Stream. ARC -Rep. I678, 1935- 7. Scholkemeyer, F. W.: Die Laminare Reibungsschicht an Rotationssym- metrischen Korpern . Diss. Braiinschweig, 19^3' 8. Gruschwitz, E.: Ing.-Arch. 2, 1931, P- 321. 9. Kehl, A.: Ing.-Arch. 12, 19^3, P- 293- 10. Buri, E.: Eine Berechnungsgrundlage fiir die Turbulente Grenzschlcht bei Beschleunigter und Verzogerter Stroraung . Diss. Zurich, 1931- 11. Von Doenhoff, A. E., and Tetervin, N.: Determination of General Relations for the Behavior of Turbulent Boundary Layers . NACA Rep. 772, 19^3- 12. Garner, H. C: The Development of Tijrbulent Boundary Layers. ARC- Rep. 2133, I9U4. 13. Millikan, C. B.: Trans. Am. Soc . Mech. Eng . , Appl. Mech. ^h, 1932, No. 2, p. 29- Ih. Ludwieg, H., and Tillmann, W.: Ing.-Arch. I7, 19^+9, p. 288. 15. Rotta, J.: Ing.-Arch. I9, 195I, p. 31 and 20, 1952, p. 195- 16. Wieghardt, K.: Zur Turbulenten Reibungsschicht bei Druckanstieg . Untersuchungen und Mitteil\ingen der Deutschen Luftfahrtforschung, UM 6617, 194U. 17- Schlichting, H.: Grenzschicht-Theorie, Karlsruhe, 1951» 16. Schultz-Grunow, F.: Luftfahrtforschung, I7, 19^0, p. 239. 19. Howarth, L.: Proc . Roy. Soc. London, Ser . A 919, Bd. l6k, 1938, p. '^k^ ■ NACA TM 1579 51 \A u(x,y) y^^^T^^ Plane Rotational ly synnmetrical Figure 1.- Survey sketch. Figure 2.- Laminar friction-layer parameters (Hartree profiles). 52 NACA TM 1579 0. Rotta Ludwieg,Tillimann i3 2 )4 2 1/ 10^ Figure 3.- Turbulent wall-shear stress (according to H. Ludwieg, W. Tillmann, and J. Rotta). NACA TM 1379 53 d V ir.3 i 4.0 3 '' *-/ '' n ^^ S!*^-. 1.0 ^==^ ^ 0.8 0.6 H: :|,4/~~ r>=^^^ 1.2 •^,^ O'l Dn ^tn 0.2 r\ 1 H r\c^Vi/N-2 l|3 ^1/ /N lU ? \ 1/6 10' ,3 2 5 10"^ 2 10^ Figure 4.- Turbulent dissipation (according to J. Rotta). 34 NACA TM 1579 1.8 2.0 2,2 2.4 Figure 5.- Turbulent boundary -layer thickness ratio (according to J. Rotta and K. Wieghardt). NACA TM 1579 35 E,Exio^ Figure 6,- The function E(H) for laminar and turbulent flow. 0.074 Prandtl-Schlichting,Cf = -Oil55_ ^*^= — ^ ^ ( U^^\2' rTurbulent \}^^j/-) Figure Y.- Drag law of the smooth flat plate in longitudinal approach flow. 36 NACA TM 1379 2.0 -0.02-0.01 0.01 0.02 0.03 0.04 Figure 8.- Relation between the boundary-layer thickness ratio H and the wall shear-stress coefficient a, on one hand, and the form parameter L on the other for laminar flow. Figure 9.- Relation between the boundary -layer thickness ratio H and the form parameter L for turbulent flow. WACA TM 1579 37 -0.02 -0.01 K 0.01 0.02 0.03 0.04 Approximation (49) Figure 10.- The function K(L) for laminar flow. Truckenbrodt Approximation (49) Garner Gruschwitz Gruschwitz-Kehl -1.0 Figure 11.- The function K(L) for turbulent flow. 38 NACA TM 1379 Xy^Xu Figure 12.- Survey sketch from the transition point of the flat plate. AH 1.5 1.0 05 Schultz-Grunow (flat plate) Rotta (similar solution in the case of constant pressure) I I I Uu^u 10 3 2 10' ,4 2 10= V I'lgure 13.- Variation of the form parameter H in the range of transition from laminar to turbulent flow. NACA TM 1379 39 04 - a i Exact values according to Howarth 0.02 0.04 0.06 0.08 O.IO 0-12 0.I4 0.16 0.02 0.04 0.06 0.08 0.10 ,0.12 014 0.16 J -0.02 Figure 14.- The laminar boundary-layer parameters of the Howarth flow. 1+0 NACA TM 1379 Measurement. |v. Doenhof f-Tetervin , \ Garner Gruschwitz, Gruschwitz-Kehl 0,1 0.2 0.3 0.4 0.5 0.6 Figure 15.- Turbulent boundary layer on the profile NACA 65(216) -222 a = lOjo, — -= 2.64 X 106. Measurement according to NACA Rep. 772. NACA-Langley - 5-24-55 - 1000 -H_ -^ -^ w .2 , 3 <: . 01 5 ° ;, H <; ■ u H Z S -• « -• a S 'SOH siii 2 3 J3 :« "<3 u Ik ri y w m <: K K D D ««; §• c < 0. & o ^ to o o f-< V m w H c = <: o J 5 N to -tr h o >- So; > C — S. 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"> C a* :s °-£ C > " 00 o >" — ' 3 B t, >; c 0) o a « u 3 3 £ O O [H 0] i3 S H c^ S »■ ^" 2 S " o " S 2 s £ S) »*1 TO 0) £ .H .y bo :3 m o ^ to S 3 10 S :« 1> t-5 I' ™ £ CO <_ S ° ° >. I c " S 2 >> 3 - o S i! 2 o c 0) o i, :s o « fc £ . So; I iS H : a gl o ■a ned the If th ^ o •Si " J3 M -M CO 0) 5 .2 B'5 o , 0) > >-•£•- CO a; S t< 3 . ramet m thei the de form pa momentu ledge of l§ « 3r the f the 1 Know s v; o a -c tion ling rem .2g o equa coup theoi ■a -' ^i < u ■< z §■ UNIVERSITY OF FLORIDA 3 1262 08106 548 3 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT ON SCi: IBRAR^ " '1 I ncni L'KIVERSrTY OF MlNfiESOTA ABRO. LAB ftOSEMOUNT RESEAaCH CENTER BOILDIMG 704^» R03EM0UKT. MINNESOTA ATT: PKOFESSOR B.M, LEADON ».? i-^-5-6-7-9-v.u.i«-l5-l7-20.21