il\ci\;ri^-i^li 
 
 H «« ?.!•* 
 
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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1218 
 
 LECTURE SERIES «BOU]SIDARY LAYER THEORY" 
 
 PART n - TURBULENT FLOWS 
 
 By H. SchlLchting 
 
 Translation of "Vortragsreihe" W.S. 1941/42, Luft- 
 fahrtforschungsanstalt Hermann Goring, Braunschweig 
 
 Washington 
 April 1949 
 
 ^^SHY OF FLORIDA 
 roajMEMTS DEPARTMENT 
 
 eAINESV(LLE,FL 32611-7011 udA 
 
-EkCk TM Wo. 1218 
 
 TABLE OF CONTENTS 
 
 Chapter XIII. GENERAL REMARKS ON TURBULENT FLOWS 1 
 
 a. TurlDulent Pipe Flow 1 
 
 b. TurlJulent Boundary Layers 3 
 
 Chapter XIV. OLDER THEORIES 1^ 
 
 Chapter XV. MORE RECENT THEORIES; MIXING LENGTHS 11 
 
 Chapter XVI. PIPE FLOW I8 
 
 a. The Smooth Pipe I8 
 
 h. The Rough Pipe 25 
 
 Chapter XVII. THE FRICTION DRAG OF THE FLAT PLATS IN LONGITUDINAL 
 
 - FLOW '. 30 
 
 a. The Smooth Pipe 3I 
 
 b. The Rough Pipe J+O 
 
 c. The AdmisBihle Roughness l+l 
 
 Chapter XVIII. THE TURBULENT FRICTION LAYER IN ACCELERATED, 
 
 RETARDED FLOW U3 
 
 Chapter XIX. FREE TURBULENCE • I+9 
 
 a. General Remarks: Estimations . . . . » h9 
 
 h. The Plane Wake Flow 56 
 
 c. The Free Jet Boundary 6I 
 
 d. The Plane Jet 65 
 
 Chapter XX. DETERMINATION OF THE PROFILE DRAG- FROM THE LOSS OF 
 
 MDMENTUM ■ . . 66 
 
 a. The Method of Betz 69 
 
 h. The Method of Jones 72 
 
 Chapter XXI. ORIGIN OF TURBULENCE 7I+ 
 
 a. General Remarks 7^ 
 
 h. The Method of Small Oscillations 76 
 
 c. Results 91 
 
 Chapter XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION 
 LAYER ACCORDING TO THE MEl'HOD 01'" GRUSCHWITZ 
 
 (R-EFERENCE 78) 93 
 
 a. Integration of the Differential Equation of the Turbulent 
 
 Boundary Layer 9j 
 
 b. Connection Between the Form Parameters rj and H = i3/5* of 
 
 the Boundar-y Layer 96 
 
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 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
 http://www.archive.org/details/lectureseriesbouOOunit 
 
MTIONAL AUVTSOEY COMMITTEE FOE AEEOKAliriCS 
 
 TECinnCAL MEMOEAMUM NO. 12l8 
 
 LECTUEE SERHS "BOIMDAEY LAYER THEOEY" 
 Part II - Turbulent Flows* 
 By H. Schlichting 
 
 CHAJTEE XIII. GEHERAL REMAEES ON TUKBIOnST FLOWS 
 a. Turbulent Pipe Flow 
 
 The flow laws of the actual flows at high Eeynolds numbers differ 
 considerably from those of the laminar flows treated In the preceding 
 part. These actual flows show a special characteristic, denoted as 
 "turbulence." 
 
 The character of a turbulent flow is most easily imderstood in 
 the case of the pipe flow. Consider the flow through a straight pipe of 
 circular cross section and with a smooth wall. For laminar flow each 
 fluid particle moves with uniform velocity along a rectilinear path. 
 Because of viscosity, the velocity of the particles near the wall is 
 smaller than that of the particles at the center. In order to maintain 
 the motion, a pressure decrease is required which, for laminar flow, is 
 proportional to the first power of the mean flow velocity (compare 
 chapter I, Part I) . Actually, however, one observes that, for larger 
 Eeynolds numbers, the pressure drop Increases almost with the square of 
 the velocity and is very much larger than that given by the Hagen— 
 Polseuille law. One may conclude that the actual flow is very different 
 from that of the Polseuille flow. 
 
 The following test. Introduced by Eeynolds, . is very instructive: 
 If one inserts into the flowing fluid a colored filament one can observe, 
 for small Eeynolds numbers, that the colored filament Is maintained down- 
 stream as a sharply defined thread. One may conclude that the fluid 
 actually flows as required by the theory of laminar flow: a gliding 
 along, side by side, of the adjoining layers without mutual mixing 
 (laminar = layer flow). For large Eeynolds numbers, on the other hand, 
 one can observe the colored filament, even at a small distance downstream 
 from the inlet, distributed over the entire cross section, that Is, mixed 
 
 *"Tortragsrelhe 'Grenzschichttheorle. ' Tell B: Turbulente Stromungen." 
 Zentrale fiir wlssenscliaftliches Berichtswesen der Luftfahrtforschimg des 
 Goneralluftzeugmeisters (ZWB) Berlln-rAdlershof , pp. 15^+— 279. The 
 original language version of this report is divided Into two main parts. 
 Tell A and Tell B, which have been translated as separate WACA Technical 
 Memorandums, Nos. 1217 and 12l8, designated Part I and Part II, respectively. 
 This report is a continuation of the lecture series presented in part I, the 
 equations, figures, and tables being numbered in sequence from the first 
 part of the report. For general information on the series, reference 
 should be made to the preface and the Introduction of Part I. 
 
2 NACA TM no. 12l8 
 
 to a great extent vlth tlie rest of the fluid. Thus the flow character 
 lias changed completely for large Eeynolds numbers: A pronounced 
 transverse mixing of adjacent layers takes place. Irregular additional 
 velocities in the longitudinal and transverse directions are superposed 
 on the main velocity. This state of flow is called turhulant. As a 
 consequence of the mixing the velocity is distributed over the cross 
 section more uniformly for turhulent than for laminar flow (compare fig. 17, 
 part -I) . For turhulent flow there exists a very steep velocity increase 
 in the immediate neighhorhood of the wall and almost constant velocity in 
 the central regions. Consequently the wall shearing stress is considerably 
 larger for turhulent than for laminar flow; the same applies to the drag. 
 This follows also from the fact that in turbulent flow a considerable 
 part of the energy is used up in maintaining the turbulent mixing motion. 
 
 The exact analysis of a turbulent flow shows that at a point fixed 
 in space the velocity is subjected to strong irregular fluctuations with 
 time (fig. 72). If one measures the variation with time of a velocity 
 component at a fixed point in space, one obtains, qualitatively, a 
 variation as shown in figure 72. The flow is steady only on the average 
 and may be interpreted as composed of a temporal mean value on which the 
 irregular fluctuation velocities are superimposed. 
 
 The first extensive experimental Investigations were carried out 
 by Darcy (reference 60) in connection with the preliminary work for a 
 large water-distributing system for the city of Paris. The first quanti- 
 tative experiments concerning laminar pipe flow were made by Hagen 
 (reference 95)- The first systematic tests regarding the transition from 
 laminar to turbulent flow were made by Osborn Eeynolds (reference 6l) . 
 He deteimlned by experiment the connection between flow volume and pressure 
 drop for turbulent flow and investigated very thoroughly the transition 
 of the laminar to the turbulent form of flow. He found, in tests of 
 various velocities and in pipes of various diameters, that transition 
 
 always occurred at the same value of the Eeynolds number: — . This 
 
 Eeynolds number is called the critical Eeynolds number. The measurements 
 gave for the pipe flow: 
 
 Ke.^i^ = (^) = 2300 (13.1) 
 
 crit v^ y /crit 
 
 For Ee < Ee^^-j^^ the flow is laminar, for Ee > I^e^-^j^-j., turbulent. Later 
 on it was ascertained tliat the numerical value of Ee^^-j^j^^ is, moreover, 
 very dependent on the particular test conditions. If the entering flow 
 was very free of disturbances, laminar flow could be maintained up to 
 Ee = 24000. However, of main interest for the technical applications 
 is tlio lowest critical Eeynolds number existing for an arbitrary disturbance 
 of tlie entering flow, due either to irregularities in the approaching flow 
 or to vortices forming at the pipe inlet. Concerning the drag law of the 
 pipe Eeynolds foimd tliat the pressure drop is proportional to the 1.73 
 power of tlie mean flow velocity: 
 
 -1-73 
 A"D ---. u 
 
NACA TM Wo. 1218 
 
 Id. Turtulent Boundary Layers 
 
 Becently it was determined that the flow along the surface of a body 
 (■boimdary layer flow) also can be turbulent. We had found, for instance, 
 for the flat plate in longitudinal flow that the drag for laminar flow is 
 
 proportional to \J '^q (compare ecjuation (9.I8), Part I.) However, towing 
 tests on plates for large Reynolds numbers carried out by Froude 
 
 1 B'l 
 (reference 96) resulted in a drag law according to which W ~ Uq ' ' . More- 
 over, the drag coefficients in these measurements remained considerably 
 higher than the drag coefficient of the laminar plate flow according to 
 equation (9.I9), Part I. Presumably this deviation is caused by the 
 turbulence of the boundary layer. 
 
 A clear decision about the turbulent flow in the boundary layer was 
 obtained by the classical experiments of Eiffel and Prandtl concerning 
 the drag of spheres in 19114. (reference 62). These tests gave the following 
 results regarding the drag of spheres (compare fig. 73). The curve of 
 drag against velocity shows a sudden drop at a definite velocity V^pit.* 
 although it rises again with further increasing velocity. If one plots 
 the drag coefficient c^^ = w/f ^ U^ (F = frontal area) against the 
 Reynolds number Uod/v, c^ shows a decrease to 2/5 of its original value 
 at a definite Reynolds number (^®crit^' ^s^^L'tl explained this phenomenon 
 
 in 1911j-. He was able to show that this drag decrease stems from the laminar 
 boiindary layer changing to turbulent ahead of the separation point. The 
 resulting considerable rearward shift of the separation point causes a 
 reduction of the vortex region (dead water) behind the sphere (fig. 7k). 
 This hypothesis could be confirmed by experiment: by putting a wire ring 
 on the sphere (sphere diameter 28 centimeters, wire diameter 1 millimeter) 
 one could attain the smaller drag at smaller V^^^^j^^ and Pe^j,^.^^. The wire 
 
 ring is put on slightly ahead of the laminar separation point; it causes a 
 vortex formation in the boundary layer, which is thus made turbulent ahead 
 of the separation point and separates only farther toward the rear. By 
 means of the wire ring the boundary layer is, bo— to— say, "infected" with 
 turbulence. Due to the mixing motions which continually lead high velocity 
 air masses from the outside to the wall, the turbulent boundary layer is 
 able to overcome, without separation, a larger pressure increase than the 
 laminar boundary layer. 
 
 The turbulence of the friction layer is of great Importance for all 
 flows along solid walls with pressure increase (diffuser, wing suction 
 side). It is, however, also present in the flow along a flat plate where 
 the pressure gradient is zero. There the flow in the boundary layer is 
 laminar toward the front, experiencing transition to the turbulent state 
 further downstream. Whereas the laminar boundary layer thickness increases 
 downstream with xl/^^ the turbulent boundary layer thickness increases 
 
k WACA TM No. 1218 
 
 approximately as x ' ; that is, for the t'orb-olent bo'ondary layer the 
 increase of the houndary layer thickness is considerably larger, (fig. 75). 
 The position of the transition point x ., is given by (fig. 75): 
 
 (13.2) 
 
 In comparing the critical Eeynolds numbers for the pipe and the plate 
 one must select r and S, respectively, as reference lengths. The 
 equation for the flat plate is, according to Blasius (reference 8) 
 (compare equation (9.21a)) 
 
 or 
 
 U^5 _ . U^x 
 
 ■'o 
 
 y (13.3) 
 
 n 
 
 Thus, with (— ) = 25000 : 
 ^ ^ ^crit 
 
 (^) = 5.500 = 2500 (flat plate) (13.^) 
 
 ^ -^crit 
 
 ^max ^ 
 This critical Eeynolds number must be compared with — at the 
 
 transition point for the pipe. Due to the parabolic laminar velocity 
 
 distribution in the pipe ri ^ = 2 u, and because r = i d, then, for 
 
 the pipe, u r/v = ud/v. According to equation (I3.I), (•^) = 23OO 
 
 crit 
 for the pipe. Thus the comparable critical Reynolds numbers for pipe and 
 plate show rather good agreement. 
 
 CHAPTER XrV. OLDER THEORIES 
 
 The first efforts toward theoretical calculation of the turbulent 
 flows go back to Reynolds. One distinguishes in the theory of turbulence 
 two main problems: 
 
 1. The flow laws of the developed turbulent flow: 
 
 The space and time velocity variations affect the time average of 
 the velocity; they act like an additional internal friction. The problem 
 is to calculate the local distribution of the time average of the velocity 
 components, and thus to gain further information concerning, for instance, 
 the friction drag. 
 
NACA TM No. I2l8 
 
 2. Origin' of turbulence: 
 
 One Investigates under what conditions a small disturbance, 
 superposed on a laminar flow, increases with time. According to whether 
 or not the disturbance ir. "reases with time, the laminar flow is called 
 unstable or stable. The :':vestigation in question is therefore a stability 
 investigation, made to clarify theoretically the laminar— turbulent transi- 
 tion. These investigations aim particularly at the theoretical calculation 
 of the critical Eeynolds number. They are, in general, mathematically 
 rather complicated. 
 
 The first problem, since it is the more important one for general 
 flow problems will be our main concern. The second will be discussed 
 briefly at the end of the lecture series. 
 
 As to the first problem, that of calculation of the developed 
 turbulent flow, one may remark quite generally that a comprehensive 
 theoretical treatment, as exists for laminar flow. Is not yet possible. 
 The present theory of the developed turbulent flow must be denoted as 
 semi— empirical. It obtains its foundations to a great extent from 
 experiment, works largely with the laws of mechanical similarity, and 
 always contains several or at least one empirical constant. Nevertheless 
 the theory has contributed much toward correlating the voluminous experi- 
 mental data and also has yielded more than one new concept. 
 
 For the numerical treatment one divides the turbulent flow, unsteady 
 in space and time, into mean values and fluctuation quantities. The mean 
 value may a priori be formed with respect to either space or time. We 
 prefer, however, the time average at a fixed point in space, and form 
 such mean values of the velocity, pressure, shearing stress, etc. In 
 forming the mean values one must not neglect to take them over a suffi- 
 ciently long time interval T so that the mean value will be independent 
 of T. Let the velocity vector with its three mutually perpendicular 
 components be 
 
 •»v = iy + Jv + kw (1^-1) 
 
 For a tijrbulent flow the velocity components are therefore functions of 
 the three space coordinates and the time: 
 
 u = u (x, y, z, t) 
 
 V = V (x, y, z, t) (li+.2) 
 
 w = w (x, y, z, t) 
 
 The time average for the component u, for instance, is formed as follows: 
 
 (li^.3) 
 
 *nTroughout the text, underscored letters are used in place of 
 corresponding German script letters used in the original text. 
 
WACA TM No. 1218 
 
 If u, V, w are independent of t^ and T, the motion is called steady 
 on the average J or quasisteady. A steady turbulent flow^ in the sense 
 that the velocity at a point fixed in space is perfectly constant, does 
 not exist. The velocity fluctuations are then defined by the equations 
 
 u = u + u' 
 
 T = V + V 
 
 w = w + w 
 
 (ih.k) 
 
 and in the same way for the pressure: 
 
 P + P' 
 
 (1^.5) 
 
 The time average of the fluctuation quantities equals zero, according to 
 definition, as the following consideration will show immediately: 
 
 to+T 
 
 ,to+T 
 
 u'dt = - 
 
 Ito+T 
 
 udt-^ 
 
 dt = u - u = {Ik. 6) 
 
 ThuE 
 
 u- 
 
 (ik.V 
 
 The Additional "Apparent" Turbulent Stresses 
 
 As a result of the velocity fluctuations additional stresses 
 ( = apparent friction) originate in the turb'olent flow. This is readily 
 illustrated for instance by the case of the simple shearing flow u = u(y) 
 (fig. 76). Here ~ = 0; however, a fluctuation velocity v' in the 
 transverse direction is present. The latter causes a momentum transfer 
 between the adjoining layers across the main flow. This momentum transfer 
 acts like an additional shearing stress t. Whereas in laminar flow the 
 friction is brought about by the molecular momentum exchange, the turbulent 
 exchange of momentum is a macroscopic motion of, mostly, much stronger effect. 
 
 The equations of motion of ths turbulent flow, with this tiirbulent 
 appa.rent friction taken Into consideration, can be obtained from the 
 Navler— Stokes differential equations by substituting equation (lli.lj-) into 
 the latter and then forming the timc^ averages in the Wavier— Stokes 
 ilfferentlal equations. To that purpose the Navier— Stokes differential 
 equations (3.I6) are written in the form: 
 
NACA TM No. 12l8 
 
 'Sa ^(u^) S(uv) S(uw) 
 
 p|^ " ~ax 
 
 az 
 
 P T— + — ^ + + — ^ - 
 
 [at Bx Sy hz 
 
 St Bx Sy Sz 
 
 ap /S^u B^u sV 
 ^ + ^^ ( -2 + -^ + — 2 
 \Sx Sy Sz / 
 
 Sp /S2t B2v S2v^ 
 — + n' — - + + 
 
 \Sx Sy Sz > 
 
 \Sx Sy Sz 
 
 -— + — + — = 
 ox Sy Bz 
 
 > (1^.8) 
 
 By introducing equations (l^A) and (lli-.5) and forming the time averages 
 one first ottains from the continuity equation: 
 
 5u Sv c)w _ 
 Sx Sy Sz 
 
 (11..9) 
 
 and thus also: 
 
 Su' St' Sw' 
 + + = 
 
 Sx Sy 
 
 (11^.10)- 
 
 By introduction of equations (lii.U) into the left side of equation (14.8) 
 one obtains expressions asj for instance, 
 
 2 2 2 — 2 
 
 u = (u + u' ) = u + 2 u u' + u' etc. 
 
 In the subsequent formation of the time average the squared terras in 
 the "barred quantities remain unchanged since they are already constant 
 with respect to time. The mixed terms^ as for Instance u u', . . . 
 and also the terms that are linear in the fluctuation quantities are 
 eliminated in forming the average "because of equation (1^.7). However, 
 
 2 
 
 the terms that are quadratic in the fluctuation quantities as u' , 
 
 a*v*, . . . remain. Thus one o"btains from the equation system 
 (equation (ll4-.8))j after forming the time average the following system 
 of equations: 
 
8 
 
 KACA TM No. 1218 
 
 V dx 'dj 3z 
 
 /_5t -Bv — Sv 
 Sx 5y Sz 
 
 — + uAu — p < 
 
 Sx hj Sz 
 
 ^ + MAT -p p^^ + ^ll- + ^'^' 
 Sy Sx ^ Sz 
 
 /_^ _^ -5w 1 Bp .- 
 plu^- + V-— + w— 1= - -ii + pAw -p 
 \ ox oj oz / oz 
 
 2 
 
 I Sx Sy Sz 
 
 ^ 
 
 > (14.11) 
 
 The left side now formally agrees with the Navier-Stokes differential 
 equations for steady flow if one writes instead of u, y, w the time 
 averages of these quantities. On the right side additional terms which 
 arise from the fluctuations have been added to the pressure and friction 
 terms . 
 
 Remembering that in deriving the Wavier— Stokes differential equations 
 one could write the resultant surface force per unit volume "by means of 
 the components of a stress tensor according to equation (3.7) in the form 
 
 5x Sy 
 
 St Sa Bt 
 
 ,jl^Z^ y. yz 
 
 Sx 
 
 k \ + — ^— + 
 
 Sy Sz 
 
 -^ Bx 
 
 
 ^ 
 
 Bz> 
 
 > 
 
 ilk.l2) 
 
 one recognizes "by comparison with (equation (lU.ll)) that one may introduce 
 for the quantities added by the fluctuation motion a syiimietrical stress 
 tensor in the following manner: 
 
 ,2 
 
 a = — pa' 
 
 X ^ 
 
 T = — pu'v' 
 xy 
 
 V " " ^'^'^' 
 
 ay = - pv'2 
 
 T = — pu'w' 
 XZ ^ 
 
 yz 
 
 pv^^'ir 
 
 tTJ« 
 
 T = — pu'w' 
 XZ 
 
 Ty^ = - PV'W' 
 
 a^ = - pw»2 
 
 ^ 
 
 J 
 
 (14.13) 
 
 One has therefore^, for the mean values of the quaslsteady flow, the 
 following equations of motion: 
 
NACA TM No. 12l8 
 
 , -Su -Su -Su 
 
 PJU— + T— + W— 
 
 3X aj oz 
 
 /-Civ' _Sv -Sv 
 
 P(U + T + W 
 
 Sx Sy Sz 
 
 Ai^ + v^ + w^V 
 
 Bx Sy 
 
 Sp ^- ^^x 
 ox dx 
 
 St 
 
 Bt 
 
 xy 
 
 Sy 
 
 St 
 
 Sp A— " xy 
 
 By Sx By 
 
 Bp + ^v + -^ + -J- 
 Bz Bx By 
 
 JLZ 
 
 xz 
 
 Bz 
 Bt 
 
 + 
 
 yz 
 
 Ba. 
 
 z 
 
 Bz 
 
 > (1^.1^) 
 
 y 
 
 The continuity equation (equation (li<.,9)) also enters. The boundary 
 conditions are the same as for laminar flow: adhering of the fluid to the 
 wall, that is, on the solid walls all velocity components equal zero. 
 According to equation (l^.lif) the mean valuep of the turbulent flow obey 
 the same equations of motion as the velocity component's of a laminar flow, 
 with the friction forces, however, increased by the apparent stresses of 
 the turbulent fluctuation motion. But since the fluctuation velocities 
 u*, v', . . . and particularly their space distribution are unknown, 
 equations (lij-.lij-) and (l4.13) are, at first, rather useless for tlie 
 calculation of a turbulent flow. 
 
 Only when one will have succeeded in expressing the fluctuation 
 quantities "' 
 
 u' ~, u'v', ... in a suitable manner by the time averages 
 will it be possible to use equations (l4.lU) to calculate. 
 
 in particular, the mean values u, v, w. 
 
 A first expression of this irind which brought however little saccess 
 was originated by Boussinesq (reference 6k). He introduced, aside from 
 the ordinary viscosity coefficient, a new viscosity coefficient of the 
 apparent turbulent friction. In analogy to the stress tensor for 
 laminar flow which is, according to equation (3.I3): 
 
 cr T T 
 
 X 'xy xz 
 
 ''xy ^y ""yz 
 
 ""xz "^jz ^z 
 
 
 p 
 
 
 — — 
 
 p 
 p 
 
 + u- 
 
 Bu Bu Bu 
 
 Bx By Bz 
 
 Bv Bv Bv 
 
 Bx By Bz 
 
 Bw Bw Bw 
 
 Bx ^y Bz 
 
 + 1^ 
 
 Bu Bv 
 
 Bw 
 
 Bx 
 
 Bx 
 
 Bx 
 
 Bu 
 By 
 
 Bv Bw 
 By By 
 
 Bu 
 Bz 
 
 Bv 
 Bz 
 
 Bw 
 Bz 
 
 ^^^ (14.15) 
 
 Boussinesq puts for the apparent turbulent friction: 
 
 Or. Bu ^ _ „ / Bu , Bv 
 X dx xy \$,j 3z 
 
 (1I+.16) 
 
 Then there corresponds to the laminar viscosity coefficient n the mixing 
 factor pg; 
 
10 NACA TM No. 12l8 
 
 H~pe or v~e 
 
 The kinematic viscosity of the turhulent flow (apparent friction) is 
 usually very much larger than that for the ordinary laminar friction. 
 (Hundred— or thounsandfold or more). In general, one may therefore 
 altogether neglect the ordinary viscosity terms mAu, ... in 
 equation (lij-.li<-). Only at the solid walls where due to the no— slip 
 condition 
 
 u=v=w=0 as well as u* = v' = w' =0 
 
 the apparent turbulent friction disappears, does the laminar friction 
 again hecome dominant. Thus there exists in every turbulent friction 
 layer in the immediate neighborhood of the wall a very narrow zone where 
 the flow is laminar. The thickness of this laminar sublayer is only a 
 small fraction of the turbulent boundary layer thickness. 
 
 One can easily understand from the example of the simple shear flow 
 according to figure 76 that in a turbulent flow the mean value u'v» is 
 different from zero. For this case, a correlation exists between the 
 fluctuation velocities u' and v' in the following manner: The 
 particles with negative v' have "mostly" a positive u*, since they 
 come from a region of larger mean velocity u.* The parts with positive 
 v', on the other hand, have "mostly" a negative u', because they come 
 from a region with smaller u, and retain in the transverse motion 
 approximately the x-momentum of the layer from which t hey come. Thus, 
 "mostly" u'v' < and, therefore, the time average u'v* < 0. Therefore, 
 the shearing stress is this flow is: 
 
 = - p u'v' > 
 xy 
 
 In measuring turbulent flows one usually measiH'es only the mean 
 values u, V, . . . since only they are of practical interest. However, 
 in order to obtain deeper insight into the mechanism of the turbulent 
 flow, the fluctuation quantities have recently been measured and also 
 their mean squares and products: 
 
 V^.2 v^ v^ 
 
 According to measurements by Eelchardt (reference 65) in a rectangular 
 tunnel (width 1 millimeter, height 2k centimeters) the maximum value 
 
 ^"Mostly" is to indicate that particles with different signs, 
 thoui^Ji not excluded, are in the minority. 
 
NACA TM No. 12l8 
 
 11 
 
 \[^'- 
 
 of V li* J for instance, equals 0.1 3 vl^, the majcimum -value of 
 
 equals 0.05 u^^^. Both naxima lie in the neighborhood of the wall. 
 
 One may saj, therefore, that in this case the turhulence is strongest 
 near the wall. 
 
 In a flow that is homogeneous (wind t'onnel), turbulent fluctuations 
 are also always present to a varying degree. They determine the so— called 
 degree of turbulence of a wind tunnel. Since the measurement of the 
 fluctuation quantities is rather difficult (hot— wire method), a more ■ 
 convenient measuring method has been chosen, for the present, for determin- 
 ation of the degree of turbulence of a wind tunnel: namely, the determin- 
 ation of the critical Eeynolds number for the sphere from force measure- 
 ments or pressure distribution measurements. One defines as critical 
 Reynolds number the one where the drag coefficient c^ = 0.3. It 
 
 becomes clear that a unique connection exists between the critical Reynolds 
 number and the turbulent fluctuation velocity in the sense that the 
 critical Reynolds number of the sphere is the lower, the higher the 
 turbulent fluctuation velocity. According to American measorements 
 (reference 97) the connection between the longitudinal fluctuation and 
 the measured critical Reynolds number for the sphere is as shown in the 
 following table: 
 
 i.'/s 
 
 O.OOij- 
 
 0.0075 
 
 0.012 
 
 0.017 
 
 0.026 
 
 «%rlt ^°"^ 
 
 2.8 
 
 2.k 
 
 2.0 
 
 1.6 
 
 1.2 
 
 In addition to the apparent Increase of viscosity, the turbulent 
 fluctuation motion has other effects: It tends to even out any tempera- 
 ture differences or variations in concentration existing in a flow. 
 The diffusion of heat, for instance, is much larger than for laminar flow, 
 because of the exchange motions which are much stronger in turbulent 
 flow. A close connnection therefore exists, for Instance, between the 
 laws of flow and of heat transfer from a heated body to the fluid flowing 
 
 ■by. 
 
 Ninth Lecture (February 2, 19i<-2) 
 CHAPTER XV". MORE RECENT THEORIES j MIXING LENGTH 
 
 In order to make possible a quantitative calculation of turbulent 
 flows, it is necessary to transform the expressions for the apparent 
 turbulent stresses (equation (ll4-,13)) in such a manner that they no longer 
 contain the unknown fluctuation velocities but contain the components of 
 the mean velocities. Consider, for that purpose, a particularly simple 
 
12 MCA TM No. 12l8 
 
 flow, namely a plane flow which has the same direction everywhere and a 
 ■velocity varying only on the different stream, lines. The main— flow 
 direction coincides with the z— direction; then: 
 
 u = il(y) V = w = (15.1) 
 
 Of the shearing stresses, only the component t = t" ia present, for 
 
 which from equation (114-.13) as well as from Boussinesq's equation, 
 equation (llf. 15). there resiilts: 
 
 du 
 
 T = - p u'v» = p e ii^ (15.2) 
 
 dy 
 
 This formula shows that | t |/p equals the square of a velocity. One 
 putSj therefore, for use in later calculations. 
 
 u»v' 
 
 (15.3) 
 
 and denotes v.,,. as shearing stress velocity. Thus this shearing stress 
 velocity is a measure of the momentum transfer "by the turhulent 
 fluctuation motion. 
 
 According to Prandtl (reference 66), one may picture the turhulent 
 flow mechanism, particularly turhulent mixing, in the following simplified 
 manner: Fluid particles, each possessing a particular motion, originate 
 in the turbulent flow; they move for a certain distance as coherent masses 
 maintaining their velocity (momentum). One now ass-jmes that such a fluid 
 particle which originates in the layer (y, — l) and has the velocity 
 u(y-| — l) moves a distance I = mixing length normal to the flow (fig. 77)' 
 If this fluid particle maintains its original velocity In the x— direction 
 it will have, in its new location y^, a smaller velocity than Its new 
 surroundings, the velocity difference heing 
 
 -u' = uij^) - u(y-^ - I) with v' > 
 
 Likewise a fluid particle coming from the layer (y-, + l) to y-|^ has 
 at the new location a greater velocity than the surroundings there; the 
 difference Is 
 
NACA TM No. 12l8 13 
 
 Tij and u^ give the turbulent velocity fluctuation in the layer y, . 
 One ohtalns for the mean value of this velocity fluctuation 
 
 |u'| = i (|u^|+ lu^l) = I 
 
 (t)l 
 
 (I5.i^) 
 
 From thla equation one ohtains the following physical interpretation for 
 the mixing length Z: 
 
 The mixing length signifies the distance in the transverse direction 
 which a fluid particle must travel at the mean velocity of its original 
 layer so that the difference "between its velocity and the velocity of the 
 new location equals the mean velocity fluctuation of the turhulent flow. 
 It is left open whether the fluid particles in their transverse motion 
 fully maintain the velocity of their original layer, or whether they have 
 partly assumed the velocity of the traverse layer and then travelled 
 larger distances in the transverse direction. The Prandtl mixing length 
 which is thereby introduced has a certain analogy to the mean free path 
 of the kinetic theory of gases, with, however, the difference that there 
 one deals with microscopic motions of the molecules, here with macroscopic 
 motions of larger fluid particles. 
 
 One may picture the origin of the transverse fluctuation velocity v' 
 in the following way: 
 
 Two fluid particles flowing from the layers (y-, + l) and (y-, — Z) 
 
 meet in the layer y-[_ in such a manner thab one lies behind the other: 
 the faster (y-|_ + Z) behind the slower (y-i — 2). They then collide 
 
 with the velocity 2u' and glveway laterally. Thereby originates the 
 transverse velocity V , directed away from the layer y-, to both sides. 
 
 If, conversely, the slower of the two particles is behind the faster, 
 they withdraw from each other with the velocity 2u' . In this case the 
 space formed between them is filled up out of the surroundings. Thus 
 originates a transverse velocity v* directed toward the layer y-,. 
 One concludes from this consideration that v' and u' are of the same 
 order of magnitude and puts 
 
 |v'| = number |uM = ntunber Z — (I5.5) 
 
 In order to express the shearin g str ess according to equation (I5.2) one 
 has to consider the mean value u*v* more closely. The following 
 conclusions can be drawn from the previous considerations. 
 
Ik 
 
 NACA TM No. 12l8 
 
 The particles arrlTing in the layer y-, with positive t* (from 
 
 "below, fig. 77) have "mostly" a negative u' so that u'v' is negative. 
 For the particles arriving with negative v'j u* is "mostly" positive, 
 so that u'v' is again negative. "Mostly" signifies that particles with 
 different sign are not wholly excluded, but are strongly outnumbered. 
 The mean value u'v» is therefore different from zero and negative. Thus 
 one puts 
 
 u»v« = - k lu'l Iv'l 
 
 (15.6) 
 
 with k^O; 0<k<l. The niimerical factor k, also called the 
 correlation coefficient is not known more closely. According to 
 equation (I5.5) ^^^ (15»^) one now obtains 
 
 u«v' = 
 
 r,\2 
 
 - number Z^/duV 
 
 (15.7) 
 
 the "number" in this equation being different from the one in 
 equation (15.5)* If on.e includes the "number" in the unknown mixing 
 length, one can also write 
 
 ^TTTT = _ 22/duJ 
 
 (15.8) 
 
 and thus finally obtains for the turbulent shearing stress according to 
 equation (I5.2) 
 
 P 2 
 
 2/du\2 
 
 Considering that the sign of t also must change with the sign of 
 it is more correct to write 
 
 du 
 
 T=pi2 
 
 du 
 
 iy 
 
 du Prandtl's 
 ^7 formula 
 
 (15.9) 
 
 This Is the famous Prandtl mixing length formula which has been very 
 successful for the calculation of turbulent flows. 
 
 If one compares this formula (equation (15.9)) with the equations 
 of Boussinesq where one had put t = e -r— ( e = mixing factor = turbulent 
 analogue of the laminar viscosity \x) , one has for the mixing factor 
 
RACA TM Ro« 12l8 15 
 
 (15.10) 
 
 c = p2^ 
 
 du 
 
 The turlaulent mixing factor e Is in moat cases larger than the laminar 
 ■viscosity [x "by several powers of ten. Moreover the mixing factor e 
 is dependent on the velocity and on the location and tends toward zero 
 near a wallj "because there the mixing length goes toward zero. 
 
 If one compares Prandtl's formula equation (I5.9) with Boussinesq«s 
 equation (15.2) one could f)erhap8 think at first that not much has been 
 gainedj since the unknown quantity e ( = apparent viscosity) has been 
 replaced by the new unknown I = mixing length. Nevertheless Prandtl's 
 formula is considerably better than the old formula for the following 
 reason: It is known from tests that the drag for turbulent flow is 
 proportional to the square of the velocity. According to equation (15.9) 
 one obtains this square law for drag by assuming the mixing length to be 
 independent of the velocity, that is, by assuming the mixing length to 
 be purely a function of position. It is considerably easier to make a 
 plausible assumption for the length I = mixing length than for the 
 apparent turbulent viscosity e, and therein lies the considerable 
 superiority of Prandtl's formula equation (I5.9) over Boussinesq's 
 equation (I5.2). 
 
 In many cases the length 2 can be brought into a simple relation 
 to the characteristic lengths of the respective flows. For the flow along 
 a smooth wall I must, at the wall itself, equal zero, since all trans- 
 verse motions are prevented at the wall. For the flow along a rough wall, 
 however, the limiting value of I at the wall equals a length of the 
 order of magnitude of the height of the roughness. 
 
 It would be very useful to have a formula permitting the determina- 
 tion of the dependence of the mixing length on the position for any 
 arbitrary flow. Such an attempt has been made by v. Karman (reference 68). 
 V. Karman makes the assumption that the inner mechanism of the turbulent 
 flow is such that the motion at various points differs only with respect 
 to time— and length-scale, but is otherwise similar (similarity hypothesis). 
 Instead of the units of time and length one may select those of velocity 
 and length. The velocity unit that is Important for the turbulent motion 
 is the shearing stress velocity v^ according to equation (I5.3). The 
 corresponding unit of length is the mixing length Z. 
 
 In order to find the quantity I from the data of the basic flow 
 u(y), V, Karman applies the Taylor development* for u(y) in the neigh- 
 borhood of the point y-, . 
 
 u(y) = u(y^) + (y - y^) ^0) ^ir('^- ^1^^ (^) + ^15.11) 
 
 * In the following, the bar over the mean velocity will be omitted, 
 for simplification. 
 
16 
 
 NACA TM No. 12l8 
 
 The length 2 cannot depend on the velocity ^.(y-,), since according 
 
 to Newton's principle of relativity the addition of a constant velocity 
 has no influence on the course of motion. Thus 
 
 ■du\ 
 
 and the higher derivatives remain as characteristic data of the "basic 
 flow. The simplest length to "be formed from it is 
 
 du 
 dy 
 
 d!u 
 dy^ 
 
 V. Earman puts therefore 
 
 (15.12) 
 
 According to this formula I is not dependent on the amount of velocity 
 "but only on the velocity dlstrl"butlon. Thus 2 is a pure position 
 function as required ahove. In equation (15.I2) k, is an empirical 
 
 constant which must "be determined from the experiment. To arrive further 
 at the tur"bulent shearing stress, v. Kannan also maintains Prandtl's 
 equation (15.9). 
 
 In generalizing equation (I5.9) one o"btains, according to Prandtl, 
 the complete expression for the tur"bulent stress tensor of a plane flow 
 in the form 
 
 /CT^ T, 
 
 \:^ % 
 
 \= pi' 
 
 du 
 
 dy 
 
 Sx 
 
 Su I ^ 
 Si 
 
 ^ 
 
 (15.13) 
 
 du Sv 
 
 ^ dx '" Sy 
 
 2 ^ 
 
 The common factor on the right side signifies the tur"bulent mixing 
 factor according to equation (15.IO). 
 
NACA TM No. 12l8 I7 
 
 Tenth Le6ture (February 9, 19k2) 
 Flow Along a Smooth Wall 
 
 ¥e will immediately make a first application of Prandtl's formula 
 (equation (15.9)) for the flow along a smooth wall. The normal distance 
 from the wall is denoted as y. Let the wall coincide with the x-axis. 
 For the Telocity distribution then, u = u(y). For this case one sets 
 the mixing length in the neighborhood of the wall proportional to the 
 distance from the wall 
 
 I = K.J ' (15.1^) 
 
 the constant k must he determined from the experiment. Moreover one 
 makes the assumption that the shearing stresb t is constant in the 
 entire flow region; then the shearing stress Telocity v^ according to 
 equation (15.3) also is constant. If one further neglects the laminar 
 friction^ one obtains from equations (I5.2), . (I5.9), (15.14) 
 
 or 
 
 ^ = I* 
 dy Kj 
 
 and by integration 
 
 u = — 2n y + constant (I5.I5) 
 
 In determining the constant of integration one must pay attention to the 
 fact that the turbulent law equation (I5.9) does not apply right up to 
 the wall but that Tery near to the wall an extremely thin laminar layer 
 is present. From the laminar Tlscoslty \x and the turbulent shearing 
 stress Telocity t^ one can form the length v/t^. The constant of 
 Integration in equation (I5.I5) is determined from the condition that 
 u = for y = yQ. Thus there results, according to equation (I5.I5) 
 
 u = ^ (2n y - Zn y^^) (15-16) 
 
 The as yet unknown distance from the wall yQ is set proportional to 
 the length v/t^, thus 
 
18 NACA TM No. 12l8 
 
 yo = P =fe - (15.17) 
 
 where 3 signifies a dimensionless constant. Thus one finally ottains 
 for the velocitj disbritution at the smooth wall 
 
 T^ 
 
 y^* 
 
 u = ^ (in ±^ - in ^) (15.18) 
 
 that is, a logarithmic Telocity distribution law. It contains two 
 empirical constants k. and p. According to measurements «; = O.k. 
 From equation (I5.I8) one can see that the dimensionless velocity u/v>. = cp 
 can he represented as a function of the dimensionless distance from the 
 wall T\ = v^(.y/v. The latter is a sort of Eeynolds number, formed with the 
 distance from the wall y and the shearing stress velocity v^.. Thus one 
 obtains for larger Reynolds numbers from equation (I5.I8) the following 
 universal velocity distribution law 
 
 Cp(Ti) = A In T\ + B (15.19) 
 
 with A = l/n = 2.5. For smaller Eeynolds numbers, where the laminar 
 friction also has a certain influence, tests gave the velocity distribution 
 larf 
 
 Cp(Tl) = C Tl'' (15.20) 
 
 or 
 
 t - ^ (^Y (15.20a, 
 
 with the exponent n equalling about l/?. These universal velocity dis- 
 tribution laws according to measurements for pipe flow are given in 
 figure 78. They will be discussed in more detail in the following chapter. 
 
 CHAPTER rVI. PIPE FLOW 
 a. The Smooth Pipe 
 
 Among the various turbulent flows of practical importance, pipe 
 flow was investigated with particular thoroughness because of its great 
 practical importance. We shall therefore consider the pipe flow first. 
 It will be noted at this point that the flow laws of the pipe flow may 
 be applied to other cases, as for instance the plane plate in longituainal 
 flow. Consider a straight pipe of circular cross section and with a smooth 
 wall. Let y be the radial coordinate measured from the pipe axis. The 
 
WACA TM No. 1218 19 
 
 "balance of forces "between the shearing stress t and the pressiire 
 
 drop P-, — Pp on a piece of pipe of length L yields as "before for the 
 
 laminar flow according to equation (2..1a)j the relation: 
 
 -^-^1 (16.1) 
 
 This formula applies equally to laminar and turbulent flow. In it t 
 now signifies the sum of the laminar shearing stresses and of the apparent 
 turhulent shearing stresses. Over a cross section, t is proportional 
 to y. The shearing stress at the wall Tq may "be determined 
 experimentally "by measurement of the pressure drop: 
 
 Pn - P2 r 
 
 r^=-^—^L . (16.2) 
 
 For the tur'bulent flow the connection "between pressure drop and flow 
 volume Q = irr u must "be obtained from tests.* In the literature there 
 exists a very great number of pipe resistance formulas. Only those serve our 
 purpose which satisfy Eeynolds' law of similarity. One of them is the 
 formula of H. Blasius (reference 69), set up partic-ularly carefully, 
 which is valid for a smooth wall and for Eeynolds numbers 
 Ee = ud/v^ 100 000. 
 
 If one introduces, as "before in equation (2.6), the dimensionless 
 pipe resistance coefficient X "by the equation 
 
 !Ll^ = iPu2 (16.3) 
 
 L d 2 . 
 
 X is, according to Blasius: 
 
 ,- ,x-iA 
 
 X = 0.316U(^^J (l6.k) 
 
 Comparing equations (l6.2) and (I6.3) one finds: 
 
 ^o = I P ^^ ■ (16-5) 
 
 and therefore according to equation (iG.k): 
 
 _7/k lA -lA 
 Tq = 0.03955 p a'^ V ' d (16.6) 
 
 *In the following, u is, for the pipe flow, the mean flow velocity 
 at the cross section, as distinguished from the time average in the previous 
 sections. 
 
20 NACA TM No. 12l8 
 
 If one introducsBj moreover, instead of the diameter d the radius r, 
 
 the nximerical factor in this linear equation must be divided by 
 
 1/4 
 2 ' =1.19. Thus T becomes: 
 
 o 
 
 ^o = 0.03325 P u^/^ V^A r-lA = p v^2 (ig^^) 
 
 where the shearing stress velocity is defined by the wall shearing stress: 
 
 v^ = ^ (16.8) 
 
 If one finally factors the quantity v^,. in equation (16.7) into v^^. ' x 
 v^ ' , one obtains: 
 
 This equation is very similar to equation (15.20a); however, the mean 
 velocity now takes the place of the local velocity and the pipe radius 
 takes the place of the distance from the wall. One passes first from 
 the mean velocity to the maximum velocity u-|_; based on measurements of 
 
 Wikuradse (reference 70) u = 0.8 u,, and therewith follows from 
 equation (16.9): 
 
 u, f^*r\^/^ 
 
 If this formula is assumed to be valid for any distance from the wall, 
 one obtains: 
 
 ^^ , B.lk (^-ff (16.10) 
 
 or 
 
 <p = 8.1k y^l'^ (16.11) 
 
 This is the so— called 1/7— power law for the velocity distribution; its 
 form was already given in equation (15.20a). The coefficients n and t]. 
 
NACA TM No. 12l8 21 
 
 still uiLlaaown there^ have now "been determined on the tasis of the 
 resistance law of the pipe flow. Figure 78 shows, according to measure- 
 ments of Nikuradse (reference 70) that this law is well satisfied in the 
 range of Reynolds numhers up to 100,000. Naturally this law of velocity 
 distribution can apply only to the region of Reynolds n^jimbers where the 
 pipe resistance law given by equation {l6.h) is valid, since it was 
 derived from this law. 
 
 For pirrposes of later calculations we shall derive from 
 equation (16.IO) the shearing stress velocity v^. One obtains: 
 
 7/8. V 1/8 
 v^ = 0.150 u (^] (16.12) 
 
 7 /fi 1 
 
 with 8.7^ = 6.65 and = O.I5O. From equation (16.I2) follows: 
 
 6.65 
 
 Tq = p rf = 0.0225Pu'^/^ (tT^^ (I6.I3) 
 
 \y/ 
 
 This relation will be needed later. 
 
 Comparing measured velocity distributions with equation (16.IO) 
 one can state that outside of the range of validity of equation (16.IO), 
 namely for Re > 100,000, a better approximation is obtained by the 
 power 1/8, l/9j or l/lO instead of l/7- The measurements concerning 
 the pipe resistance (fig. 8I) show an upward deviation from the formula 
 of Blasius for larger Reynolds numbers. 
 
 The logarithmic velocity distribution law, equation (15.19)^ derived 
 in the previous chapter has been verified by Nilniradse (reference 70) on 
 the basis of his meas'jrements for the smooth pipe. For this purpose from 
 the measured press'ore drop for each velocity profile one first determines 
 the wall shearing stress according to equation (l6.2) and from tha t, 
 according to equation (16.8), the shearing stress velocity v^ = \/t /p. 
 
 Then the dimensionless velocity cp = u/v^ can be plotted against the 
 dimensionless distance from the wall t] = yv^(./v. The measurements of 
 Nikuradse in a very large range of Reynolds numbers. He = k X 10^ up to 
 32^4-0 X 103, lie very accurately on a straight line if one plots cp against 
 log r] (fig. 78). The straight line has the equation: 
 
 cp = 2.5InTi + 5.5 
 
 (l6.11f) 
 
 This gives, by comparison with equation (I5.I8), the following nimerical 
 values for the coefficients K and P 
 
22 NACA TM No. 12l8 
 
 K = oAoo p = 0.111 (16.15) 
 
 Mixing Length 
 
 From the measured Telocity distribution and the measirred pressure 
 drop the distribution of the mixing length over the pipe cross section 
 can he determined according to equations (l6.2)j (I6.I), and (15.9). 
 
 T = Tq — (y = distance from the pipe axis). This determination of the 
 
 mixing length from the meaS'jrements in the pipe was made hy Wikuradse 
 (reference 70). For large Eeynolds numbers^ where the influence of 
 Tiscosity is negligihle, one ohtains a distribution of the mixing length 
 l/r over y/r which is independent of the Ee-number (fig. 79). The 
 following interpolation formula can be given for this distrihution: 
 
 - = 0,11+ _ 0.08 f 1 - ^ ) - 0.06(1 - ^) (16.16) 
 
 In this equation y signifies the distance from the wall. The develop- 
 ment of equation (16.16) for small y/r (neighborhood of the wall) gives 
 
 2 
 
 I = o.kj - o.k-k — + . . . (16.16a) 
 
 In the neighhorhood of the wall the mixing length is, therefore, propor- 
 tional to the distance from the wall. Equation (16.I6) for the distri— 
 hution of the mixing length applies not only to the smooth pipe, hut, 
 according to the measurements of Nikuradse (reference 71) also to the 
 rough pipe, as can he seen from figure 79- From this fact one can derive 
 in a very simple manner a universal form for the law of velocity distri- 
 hution, valid for the smooth as well as for the rough pipe. One puts 
 
 for the mixing length distrihution: I = Kj f('^')with f ^^^-^ 1 for ^ -^' 0. 
 
 Furthermore follows from the linear distrihution of the shearing stress 
 over the cross section: 
 
 T = T^ (1 — ^j (y= distance from the wall) 
 
 together with equation (15-9) 
 
 <iy I Up 
 
 ^ ^ (16.17) 
 
 " 7f(y/r) 
 
 and hence by integration: 
 
MCA TM Wo. 1218 23 
 
 u = ^ J! 1_1 (16.18) 
 
 the lower limit of Intergration Jq where the velocity equals zero is, 
 according to the considerations of the previous section, proportional 
 
 to v/v^j thus: Jq/t = F-|_f— \ From equation (16.18) follows: 
 
 r \ r 
 
 and therefo'rej from equations (I6.I8) and (l6.l8a) 
 
 %ax - u = ^* F Cy/r) 
 
 (16.19) 
 
 This law, with the same fimctlon F(y/r), applies equally to smooth and 
 rough pipes. It states that the curves of the velocity distrihution over 
 the pipe cross section for all Eeynolds numbers and all roughnesses can 
 "be made congruent "by shifting along the velocity axis, if one plots 
 (u^q y — u/v^ against y/r (fig. 80). The explicit expression for the 
 
 function F(y/r) is ohtainsd immediately from equation (l6.1i<-)j according 
 to which ■ 
 
 u - u = 2.5v. to ^ = 5.75 -^^ log - (16.20) 
 
 max jy 
 
 Universal Resistance Law 
 
 According to their derivation the velocity— distribution— law 
 (equations (16.I9) and (l6.20)) are to he regarded as valid for arbitrary 
 Eeynolds number since the laminar viscosity was neglected as compared 
 with the turbulent viscosity. We shall now derive from the velocity- 
 distribution— law equation (l6.20) a resistance law which, in contrast to 
 Blasius*, applies up to Reynolds numbers of arbitrary magnitude. 
 
2k WACA TM No. 12l8 
 
 From equation (l6.20) one may determine "by integration over the 
 cross section the mean flow velocity u. One finds: 
 
 ^ = "majc - 3.75 v» (l6.21) 
 
 The test results of Nikuradse (reference 70) gave a number slightly 
 different from 3-75^ namely: 
 
 According to equation (16.5): 
 
 X. = 8 (^) ^ (16.23) 
 
 From the uniyersal Telocity distribution law of the smooth pipe 
 equation (l6.l4) follows: 
 
 %ax = ^* {2-5 in ~ -^ 5.5 
 
 and hence the connection with equation (16. 2l): 
 
 u = T^ J2.5 In ^ + 1.75}* (16.24) 
 
 The Reynolds numher enters into the calculation by msans of the identity; 
 
 III ^ 1 ud T* ^ ud \[X 
 V - 2 V ^ V i^^ 
 
 Thus results from equations (16.23) and (l6.2k): 
 
 8 • 
 
 X = 
 
 2.5 Zn (^^ \lxj - 2.5 In 1^ \|2 + 1.75 
 
 (2.035 log(^ \fx)-0.9l| 
 
WACA TM No. 1218 25 
 
 or: 
 
 -^ = 2.035 log (^ ^) - 0.91 (16.25) 
 
 Accordingly a straight line must result for the resistance law of the smooth 
 
 pipe, if one plots l/ \/7, against logf^>/Xj. This is Tery well confirmed 
 
 hy Nikuradse's measurement (fig. 8I). The numerical values according to the 
 measurements differ only slightly from those of this theoretical derivation. 
 From Nikuradse's measurements was found: 
 
 (16.26) 
 
 Universal Resistance La v For Smooth Pipes 
 
 This is the final resistance law for smooth pipes. On the basis of 
 its derivation it may he extrapolated up to Reynolds numbers of arbitrary 
 magnitude. Thus measurements for larger Reynolds numbers than those of 
 Nikuradse's tests are not required. Up to Re = 100,000 this 'juiiversal 
 resistance law Is In good agreement with the Blasius law according to 
 equation (16.U). For higher Reynolds numbers the Blasius law deviates 
 considerably from the measurements (fig. 8I). 
 
 Concerning the determination of X from equation (16.26), where it 
 appears on both sides, it should be added that it can be easily obtained 
 by successive approximation. 
 
 Eleventh Lecture (February 16, 19i4-2) 
 
 b. The Rough Pipe 
 
 The ch.aracteristic parameter for the flow along a rough wall is the 
 ratio of grain size k of the roughness to the boundary layer thickness, 
 particularly to the thickness of the laminar sublayer 5 which is 
 
 always present within the turbulent friction layer in the immediate 
 neighborhood of the wall. The thickness of the laminar sublayer is 
 
 5, = number — . The effectiveness of roughness of a certain grain size 
 depends, therefore, on the dimensionlese roughness coefficient k/6,«, 
 
 In the experimental investigations of the resistance of turbulent flows 
 over rough walls_, the rough pipe has been studied very thoroughly since it 
 is of great practical importaiice. Besides depending on the Reynolds number, 
 the resistance of a rough pipe is a function of the relative roughness r/k. 
 One distinguishes for the resistance law of a rough pipe three regions. 
 
26 NACA TM No. 12l8 
 
 The subsequently given "boundaries of these regions are Talid for sand 
 roughness kg like those investigated "by Nikuradse (reference 71). 
 
 1. Hydraulically smooth: The grain size of the roughness is so 
 small that all roughnesses lie within the laminar aulilayer. In this 
 case the roughness has no drag increasing effect. This case exists for 
 small Eeynolds numbers and for values of the characteristic roughness 
 
 numher: 2. — 5. 
 
 2. Fully developed roughness flow: The grain size of the roughness 
 is so large that all roughnesses project from the laminer auh— layer. The 
 friction drag then consists predominantly of the form drag of the single 
 roughness elements. A purely square drag law applies. For the pipe the 
 resistance coefficient X, is then independent of Re and only dependent 
 on the relative roughness k/r. This law exists for very large Resmolds 
 
 numbers. For sand roughness this law applies for: — — §■ ^ 70. 
 
 3. Intermediate region: Only a fraction of the roughness elements 
 project from the laminar suhlayer. The drag coefficient depends on r/k 
 as well as on Re. This law exists for medium Reynolds numters andj for 
 
 ^ v^ ^ 
 the sand roughness, for: 5 — 70. 
 
 The dependence of the pipe resistance coefficient on the Reynolds 
 number and on the relative roughneBS according to the measurements of 
 Nikuradse (reference 71) can he seen from figure 82 as well as, in 
 particular, the three laws Just given. 
 
 The velocity distrihution on a rough wall is given, hasically, in 
 the same way as for the smooth wall hy equation (I5.I6). One has only 
 to substitute for the constant of integration y^ another value: yQ 
 proportional to the roughness grain size. One puts for sand roughness 
 yQ = 7 kg and hence obtains from equation (I5.I6) 
 
 (16.27) 
 
 
 
 f 
 
 
 
 ■V 
 
 u 
 
 1 1 
 
 = — ^ 
 
 7,n 
 
 J 
 
 - 7,n 
 
 y ? 
 
 V* 
 
 K 
 
 ^ 
 
 ^a 
 
 
 ) 
 
 The constant 7 is, moreover, a function of the roughness form and the 
 roughness distribution. Comparison with experiments of Nikuradse 
 (reference 71) on pipes roughened artificially by sand yields for the 
 velocity distribution the general formula: 
 
 2.5 In ^ + B ] (16.28) 
 
MCA TM No. 1218 27 
 
 the constant B teing different in each domain descrlhed above; it 
 depends on v^k /v . 
 
 For the fully developed roughness flow B = 8.5, thus: 
 
 u = v^ 
 
 ^2.5 to ^ + 8.5") (fully rough) (I6.29) 
 
 whence follows: 
 
 %ax = ^*(2-5 ^n^+ 8.5) (16.30) 
 
 and: 
 
 u - u = v^ 2.5 2n - (16.30a) 
 
 in agreement with equation (l6.20). Thus there applies also to rough 
 pipes, as equation (l6.21) did hefore to the smooth pipe, the relation: 
 
 '^ = %ax - 3-75 ^* (16.31) 
 
 From here one can, by a calculation which is perfectly analogous to the 
 previous one for the smooth pipe, easily arrive at the resistance law of 
 the rough pipe for fully developed roughness flow. By insertion of u^^^ 
 
 according to equation (16.3O) into equation (I6.3I) one obtains: 
 
 or: 
 
 or; 
 
 u = v^ ("2.5 2n ^ + 1^.75) (16.32) 
 
 /v \2 8 
 
 x = Q[^) =, 2 (16.33) 
 
 (2.5 in ^.^.75) 
 
 X = -, Tp (16. 3i^) 
 
 (2.0 log — + 1.68j 
 
28 
 
 NACA TM No. 12l8 
 
 This is the square resistance law of the fully deTeloped roughness flow. 
 Comparison with the test results of Nikuradse (fig. 83) shows that one 
 ohtains "better agreement if one changes the number 1.68 to 1.7i^. Thus 
 the resistance law of the pipe flow for fully developed roughness is: 
 
 (16.35) 
 
 X 
 
 
 
 
 1 
 
 
 
 
 (^ 
 
 
 
 log 
 
 V 
 
 ^s 
 
 + 
 
 1 
 
 -f 
 
 In the plots of -j= against log r/kg, (fig. 83) the test results fall 
 very aGcxirately on a straight line. 
 
 For flow along a rough wall in the intermediate region the constant 
 B in equation (16.28) is, moreover, a function of the roughness coeffi- 
 cient T^ /v . For this case also one can derive the resistancfe law 
 
 immediately from the velocity distribution. According to equation (16.28); 
 
 B = — - 2.5 2n -i = 
 
 u 
 
 ^2£ _ 2.5 to ^ 
 
 * 8 
 
 (16.36) 
 
 On the other hand, according to equations (16.3I) and (16.23) 
 
 u 
 
 max 
 
 ^» ^* 
 
 - + 3.75 = ^+ 3.75 
 
 (16.37) 
 
 so that one ohtains from equation (I6.36) 
 
 B 
 
 V 
 
 u o I. -,_ y_ ^ 2V2 _ o c -,„ r 
 
 = ^^ - 2.5 In 
 V-. k 
 
 2.5 2n i^ + 3.75 (16.38) 
 ^s 
 
 One can, therefore, determine the constant B as a function of v^k^S' 
 
 either from the velocity distribution or from the resistance law. The 
 plot in figure 8k- shows good agreement hetween the values determined 'dj 
 these two methods. At the same time the determination of the resistance 
 law from the velocity distrlhution is confirmed. 
 
 The formula for B includes the case of the smooth pipe. B is, 
 according to equation (l6.1i4-). 
 
NACA TM Wo. 1218 29 
 
 u T "^^^s 
 B = — - 2.5 Zn f- = 2.5 In + 5.5 (16.39) 
 
 Thus a straight line results for B in the plot against log t^ /v. 
 
 Other Eoughnesses 
 
 Because of the great practical importance of the roughness— pr oh lem 
 a few data concerning roughnesses other than the special sand roughness 
 will he given. Fikuradse's sand roughness may also he characterized hy 
 the fact that the roughness density was at its maximum value^ because 
 the wall was covered with sand as densely as possible. For many practical 
 roughnesses the roughness density is considerably smaller. In such cases 
 the drag then depends, for one thing, on form and height of the roughness, 
 and, moreover, on the roughness density. It -is useful to classify any 
 arbitrary roughness in the scale of a standard roughness. Nikuradse's 
 sand roughness suggests itself as roughness reference (roughness scale) 
 because it was investigated for a very large range of Reynolds numbers 
 and relative roughnesses. Classification with respect to the roughness 
 scale is simplest for the region of fully developed roughness. According 
 to what was said previously, for this region the velocity distribution 
 is given by: 
 
 ^ = 5.75 ZOg ^ + Bg Bg = 8.5 (16.40) 
 
 and the resistance coefficient by: 
 
 r \2 
 2.0 log — + 1.1k j 
 
 (16. Ul) 
 
 One now relates to an arbitrary roughness k an equivalent sand roughness 
 kg by the ratio 
 
 kg = a k (16.42) 
 
 Where by equivalent sand roughness kg is meant that grain size of sand 
 roughness which has, according to equation (l6.4l) the same resistance 
 as the given ro-oghness k. 
 
 Basically, of course, the equivalent sand roughness kg can be 
 determined by a reslstajice meas'jrement on the pipe. However, such measurements 
 
30 NACA TM No. 12l8 
 
 for arbitrary roughneases are difficult to perform. Measijrements on 
 arl)itrary roughnesses In a tunnel with plane walls are more convenient. 
 To this purpose an exchangealile wall of a tunnel with rectangular cross 
 section Is provided with the roughness to he Investigated (fig. 85). 
 From the measurement of the velocity distrihution in sach a tunnel with 
 a rough and a smooth longitudinal wall one ohtalns, for the logarithmic 
 plot against the distance from the wall, a triangiolar velocity distrihu— 
 tion (compare fig. 85). From the logarithmic plot of the velocity dlstrl- 
 hutlon over the rough wall 
 
 u = n^ log y + m^ (I6.U3) 
 
 one ohtalns "by comparison with the universal law according to 
 equation (I6.28) for the shearing stress velocity at the wall: 
 
 *"" 5.75 
 
 Further, one determines for the roughness to "be investigated the constant 
 B of the velocity distrlhution law, namely: 
 
 B = — - 5.75 log I (I6.if5) 
 
 ^#r ^ 
 
 By comparison of equation (iS.k^) with (16.40) one ohtalns for the 
 equivalent sand roughness-: 
 
 ]£ 
 
 5.75 Zog -^ = 8.5 -B ' (16. 46) 
 
 k 
 
 In this way one may determine the drag for arhitrary roughnesses from a 
 simple measurement in the roughness tunnel. This jaethod may "be also 
 carried over to the case of the intermediate region. 
 
 CHAPTER XVTI. THE FRICTION DRAG OF THE FLAT PLATE 
 IN LONGrmOINAL FLOW 
 
 The tur"bulent friction drag of the plate in longitudinal flow is 
 of very great practical importance, for Instance as friction drag of wlngs^ 
 airplane fuselages, or ships. The exact measurement of the friction drag 
 for the large Reynolds numhers of practice is extremely difficult. Thus 
 
WACA TM No. 1218 31 
 
 it is particularly important that one can, according to Prandtl 
 (references 73 bxiA. 7^)? calculate the friction drag of surfaces from the 
 results of pipe flow studies. This conversion from the pipe to the plate 
 can he made for the smooth as well as for the rough plate. 
 
 a. The Smooth Plate 
 
 One assumes, for simplification, that the boundary layer on the 
 plate is turbulent from the leading edge. Let the coordinate system he 
 selected according to figure 86. The boundary layer thickness 6(x) 
 increases with the length of run x. Let b be the width of the plate. 
 For the transition from pipe to plate the free stream velocity Uq of 
 
 the plate corresponds to the maximum velocity u,^ in the pipe, and 
 the boundary layer thickness 6 to the pipe radius r. 
 
 One now makes the fundamental assumption that the same velocity 
 distribution exists in the boundary layer on the plate as in the pipe. 
 This is pertalnly not exactly correct since the velocity distribution in 
 the pipe is influenced by a pressure drop, whereas on the plate the 
 pressure gradient equals zero. However, slight differences In velocity 
 distribution are insignificant since it is the momentum Integral which 
 is of fundamental Importance for the drag. For the drag W(x) of one 
 side of the plate of length x, according to equations (lO.l) and (10.2) 
 
 ¥(x) = b T (x) dx = bp 
 
 p5(x) 
 
 u (Uq - u) dy (17.1) 
 
 whence 
 
 1 dW 
 b dx 
 
 ^o(x) (17.1a) 
 
 The equation (I7.I) can also be written in the form 
 
 il 
 
 W(x) = bpu/6(x)j -^ ^1 - -^j d i (17.2) 
 
 For the velocity distribution in the boundary layer one now ass^jmes 
 the 1/7-power law found for the pipe. Replacing u^^^ by U^ and r 
 
 by 5]^ one may write this law, according to equation (16.II): 
 
32 
 
 NACA TM No. 12l8 
 
 
 = V 
 
 1/7 
 
 (17.3) 
 
 Hence the momentum. Integral "becomes 
 
 il 01 
 
 J = 
 
 u 
 
 1 - 
 
 t) 
 
 d i 
 
 s 
 
 .''% 
 
 l/7\ 
 
 Tl« dV 
 
 ^ (17A) 
 
 and thus 
 
 W(x) = ^ hpTj/5(x) 
 Hence follows^ according to equation (17.1a) 
 
 (17.5) 
 
 ° - 72 P^o dl 
 
 (17.6) 
 
 On the other hand, one had found "before for the smooth pipe, equation (16.I3) 
 
 again replacing r "by 5 and u "by U : 
 
 TTWY " o 
 
 Tq = 0.0225 pUo'^^ (^^' 
 
 (17.7) 
 
 By equating equations (17.6) and (17.7) results: 
 
 7 2d5 "^A/vN^A 
 
 12 P^o i = °-°225 PTJ, (5 
 
 This is a differential equation for 5(x). The integration yields; 
 
 k 5A 72 /v\^A 
 
 _5 =-0.022,1-) X 
 
 > 
 
 (17.8) 
 
 / X / v\l/5 i^/5 
 6(x) = 0.37(^1 ' X ^^ 
 
MCA TM No. 1218 
 
 33 
 
 5(x) 
 
 _ 0.37X 
 
 Ee, 
 
 V 
 
 (17.9) 
 
 For the turliulent "boundary layer the boundary layer thickness, therefore, 
 increases with x^/5. The corresponding equation for the laminar flow 
 was, according to equation (9.21a), 6 = 5\/vx/Uq. 
 
 By suhstitution of equation (17.8) into equation (17. 5) one ohtains 
 
 2 -^/5 
 W(x) = 0.036 hpUg X (Re^) 
 
 or for the drag coefficient c„ = W/^ U xb: 
 
 Cj. = 0.072 (Ee ) 
 
 -1/5 
 
 Comparing this result with test results on plates one finds the numerical 
 value 0.072 to he somewhat too low. 
 
 c„ = 0.074 (Ee,^) 
 
 -1/5 
 
 valid for 
 5 X lo5 < Ee < 10''' 
 
 (17.10) 
 
 corresponds "better to the measurements. This law holds true only 
 
 7 
 for Ee < 10 , corresponding to the fact that the Blasius pipe resistance 
 
 law and the l/7— power law of the velocity distri'bution, which form the 
 hasis of this plate drag law, are not valid for large Eeynolds numbers. 
 This law is represented in figure 87 together with the laminar— flow law 
 according to equation (9.19)« The initial laminar flow on the front part 
 of the plate can he taken into consideration by a subtraction, according 
 to Prandtl (reference 73): 
 
 c-f = 0.07ij- Ee;j 
 
 -1/5 
 
 1700 
 Ee 
 
 5 X 105 < R 
 
 < 10 
 
 7 
 
 (17.11) 
 
 The plate drag law for very large Eeynolds numbers can be obtained 
 in essentially the same way by starting from the universal logarithmic 
 law for the velocity distribution equation (l6.l4-l) which, according to 
 
3h 
 
 NACA TM No. 12X8 
 
 its derlTatioiij is valid up to Eeynolds numbers of arbitrary magnitude. 
 Here the calculation TDecoines considerably more complicated. The 
 development of the calculation is clarified if one first introduces the 
 Telocity distribution in a general form. We had introduced for the pipe 
 
 y^* 
 
 The values at 
 
 flow the dimensionless variables cp = -^ and ti 
 
 the edge of the boundary layer are to be denoted by the index 0, thus 
 
 ^o = 
 
 
 5v. 
 
 ^o = 
 
 (17.12) 
 
 Then: 
 
 ^ = ^*^ = ^o ^ 
 
 (17.13) 
 
 y = 
 
 
 dy 
 
 V^o 
 U 
 
 dT, 
 
 (17.14) 
 
 From the equation t = follows, with W according to equation (17.1) 
 
 o b dx 
 
 P ^* P dE <> ' ^(u. - ^) dy ;> 
 
 r 05 
 
 ^^, 
 
 \^ =T^T-< ^(Uo - u) dy 
 
 dx dTio 
 
 ''o 
 
 and according to equation (17.13) 
 
 U_ 
 
 vcp 
 
 o d 
 
 9o 
 
 2 " ^o ^% 
 
 u(Uo - u)dTi 
 
 dx 
 
NACA TM Wo. 1218 
 
 35 
 
 o _ Jo _d_ 
 
 (p^2 " ^° dx dTlo 
 
 ,no 
 
 t^-t'^^ 
 
 ^ 
 
 " dX dTl^ ^O V "Pr 
 
 dn 
 
 U 6.1] 
 _Q_ = V —a -d— 
 
 cp^2 dx dri^ 
 
 cp-^)d, 
 
 (17.15) 
 
 2\ 
 
 In forming the integral -= — P f cp — ^ j drj one must note the following 
 
 facte: The differentiation with respect to the upper limit gives zero, 
 since for t\ = t)q, cp = cpQ. In the differentiation of the integrand 9 
 
 is to he regarded as constant, and cp as a function of t) . Therefore 
 
 •iTlo 
 
 rih 
 
 ^1o 
 
 Tie 
 
 9 
 
 dT) 
 
 9 
 
 Thus follows from equation (17.15) 
 
 1% 
 
 dx dT), 
 
 (17.16) 
 
 One puts, for simplification. 
 
 dcp^ 
 
 1% 
 
 F(T,^) = ^ I 9% 
 
 (17.17) 
 
 and obtains from equation (17.16) 
 
36 
 
 NACA TM No. 12l8 
 
 ^ = xf ^^^o) '^\ 
 
 If one assijines this law to "be valid from ths leading edge of the plate 
 (x = 0), that iSj that the flow is turtulent starting from the front, 
 
 the integration gives: 
 
 X = X , (,^) 
 
 (17.18) 
 
 with 
 
 1^1 
 
 (m) = ^^^o)^ 
 
 (17.19) 
 
 'ao=° 
 
 Equation (17. l8) can also he written so that the Ee-mxmber formed with 
 the length of run x appears: 
 
 U X 
 Ee^ = -^ = $ (r,^) 
 
 (17.20) 
 
 with: 
 
 (Til) 
 
 dcp 
 
 '%=° 
 
 R^. 
 
 9 dTi dTi 
 
 ,"o 
 
 (17.19a) 
 
 Equation (17.20) gives the relation hetween ths dimensionless houndary 
 layer thickness t^ = v^b/v and the Re-number U x/v . 
 
 The drag remains to "be calculated. From 
 
 follows J "because ^ - \> P r ^ dji 
 
NACA TM No. 12l8 
 
 37 
 
 1^1 
 
 ¥ = -b p Uo V 
 
 F(Tlo) 
 
 Jt1q=o 
 
 ^r 
 
 dTiQ = b p Uq vt (ti^) 
 
 (17.21) 
 
 where 
 
 t 
 
 (tI^) 
 
 P^l 
 
 F(no) 
 
 ^, 
 
 dTi 
 2 'o 
 
 n.=° 
 
 (17.22) 
 
 /p 2 
 
 The drag coefficient c = ¥/^ Uq "b x "becomes finally 
 
 c. = l^i, . = 2 
 
 ^ V (Til) «(t1i) 
 
 (17.23) 
 
 Hence C|. also turns out to he a function of t]-,. Equations (17.20) 
 
 and (17.23) together give a parametric representation of c as a function 
 
 ^ v„5 
 of Ee-jj-j where the parameter is the boundary layer thickness r\ = — ^— . 
 
 Numerical Results 
 
 In order to arrive at numerical results, one must introduce a 
 special function for 9(1)). For the l/7— power law according to equation 
 (16.11) that is, with cp = 01]/', one would obtain the drag law according 
 to equation (17.IO). One uses the universal logarithmic velocity distri- 
 bution law, equation (l6.l4). 
 
 cp = 2.5 In T] + 5.5 
 
 In order to make the carrying out of the integrations more convenient, 
 one writes 
 
 cp = 2.5 In (1 + 9ti) 
 
 Then 9 becomes, for t) = 0, cp = 0. The adding of the one changes 
 cp(ti) a little, only for very small r\ and has only litble influence 
 on the integrals. If one writes the law at first in the general form 
 
38 
 
 • NACA TM No. 12l8 
 
 cp = a 2n (1 + "bT)) 
 
 (17.21+) 
 
 the calculation of the Integrals equations (17.17)^ (17. 19)^ (17.22), 
 with z = 1 + Tdt] yields 
 
 F(t]) = a^ (zn^z _2toz + 2-- 
 
 ^(t) = — (2 2n z — kz Zn z — 2 In z + 6z — 6 j 
 
 ^^) = =h S^ + 1 - 
 
 2(z -1) 
 2n z 
 
 > (17.25) 
 
 With the numerical values 
 
 a =2.493 t = 8.93 
 
 one ohtains for the drag lav the following tahle: 
 
 103 
 
 -6 
 Ee 10 
 
 c 10^ 
 f 
 
 0.500 
 
 0.337 
 
 5.65 
 
 1.00 
 
 0.820 
 
 J+.75 
 
 2.00 
 
 1.96 
 
 i+.05 
 
 3.00 
 
 3.25 
 
 3.71 
 
 5.00 
 
 6.10 
 
 3.3i^ 
 
 12.0 
 
 17.7 
 
 2.81 
 
 20.0 
 
 32.5 
 
 2.57 
 
 50.0 
 
 96.5 
 
 2.20 
 
 100.0 
 
 217.5 
 
 1.96 
 
 500.0 
 
 li+Ol.O 
 
 1.55 
 
 This tatle can he replaced by the following interpolation formula: 
 
 Cj = 
 
 0.472 
 
 (Zog Ee^.) 
 
 2.58 
 
NACA TM No. 12l8 
 
 39 
 
 Comparison -with test resiilts shows that the agreement improTes if the 
 number 0.472 is slightly varied, hy putting 
 
 °f = 
 
 O.k^^ 
 
 (log Ee^) 
 
 2.58 
 
 6 9 
 
 10 < Ee^< 10^ 
 
 (17.26) 
 
 Prandtl - Schlichtlng's Plate Drag Law 
 
 The laminar approach length may again "be taken into consideration 
 "by the same sobtraction as before; thus: 
 
 Cf = 
 
 OA55 
 
 _ ITOO 
 (log Ee^) ^ ^ 
 
 5 X lo5 
 
 < Ee^ < 10-- 
 
 (17.27) 
 
 Whereas the system of formulas equation (17.25) Is valid up to Ee-^iumbers 
 of arbitrary magnitude, the interpolation formulas, eq^uations (I7.26) 
 
 and (17.27), have the upper limit Ee = 10 . However, this limit taJces 
 care of all Ee— numbers occurring in practice. The theoretical formula 
 (equation (17.27)) is also plotted in figure 87. Figure 88 gives a 
 comparison with test results on plates, wings, and airship bodies. The 
 agreement is quite good. 
 
 Very recently this plate drag law has been somewhat improved by 
 Schultz-Grunow (reference 89). Until then, the turbulent velocity profile 
 measured in the pipe (l/7— power law, logarithmic law) had been carried 
 over directly to the plate, mainly because accurate velocity distribution 
 measurements of the plate boundary layer did not exist. The exact 
 measurement of the plate boundary layer showed, however, that the plate 
 profile does not completely coincide with the pipe profile. The teat 
 points show, for large distance from the wall, a slight upward deviation 
 from the logarithmic law found for the pipe. Thus the loss of momentum 
 on the plate is somewhat smaller than that calculated with the logarithmic 
 law. Schultz-G-runow repeated the calculation of the drag law according 
 to the formula system given above with the velocity distribution law 
 for the plate measured by him. His result Is represented by the 
 interpolation formula 
 
 °f = 
 
 O.lj-27 
 
 (- 0.1+07 + ^og Ee„) 
 
 2.61+ 
 
 6 
 10 < Ee, 
 
 < 10' 
 
 (17.28) 
 
I^Q NACA TM No. 12l8 
 
 This law Is also plotted in figure 87. The differences from the Prandtl- 
 Schlichting law are only slight.* 
 
 The corresponding rotationally— symmetrical problem, that is, the 
 turbulent boundary layer on a body of revolution at zero incidence, was 
 treated by C. B. Millikan (reference 79). The l/7-power law of the 
 Telocity distribution was taken as basis. Application to the general 
 case has not yet been made. 
 
 Twelfth Lecture (February 23, 1914-2) 
 
 b. The Rough Plate 
 
 The conTersion from pipe resistance to the plate drag may be carried 
 out for the rough plate in the same manner as described previously for the 
 smooth plate. One assumes a plate uniformly covered with the same rough- 
 ness k. Since the boundary layer thickness 5 increases from the front 
 toward the rear, the ratio k/5 which is significant for the drag decreases 
 from the, front toward the rear. Behind the initial laminar run, therefore, 
 follows at first the region of the fully developed roughness flow; the so- 
 called Intermediate region follows and farthest toward the rear there is, 
 finally, if the plate is long enough, the region of the hydraullcally 
 smooth flow. These regions are determined by specification of the n-umerical 
 values for the roughness coefficient v^k /v. In order to obtain the drag 
 
 of the rough plate, one must perform the conversion from pipe flow to plate 
 flow for each of these three regions individually. This calculation was 
 carried out by Prandtl and Schlichting (reference 76), based on the results 
 of Nikuradse (reference 71) for the pipe tests with sand roughness. For 
 this conversion one starts from the universal velocity distribution law of 
 the rough pipe according to equation (16.28), the quantity B being 
 dependent also on the characteristic roughness value v^k /v^ according 
 
 to figure Qk. The calculation takes basically the same course as described 
 in detail for the smooth plate in chapter XVTIa. It is, however, rather 
 complicated and will not be reproduced here. One obtains as final result 
 for the total drag coefficient of the sand— rough plate a diagram (fig. 89) 
 • which represents the drag coefficient as a function of the Reynolds 
 number U^Z/v with the relative roughness Z/kg as parameter. Just as 
 for the pipe, a given relative roughness 2 /kg has a drag Increasing 
 effect not for all Ee-numbers, but only above a certain Re-Hiumber. This 
 diagram is applicable also for roughnesses other than sand roughness, if 
 one uses the equivalent sand roughness. In the diagram (fig. 89) the 
 square drag law is attained. Just as for the pipe, for every relative 
 
 * The tables pertaining to the plate drag formulas are given in 
 table 7 5 chapter XXII. 
 
NACA TM No. 12l8 
 
 kl 
 
 roughness 2 As provided the Ee-number is sufficiently large, 
 interpolation formula 
 
 The 
 
 Cj = [1.89 + 1.62 Zog ^ j 
 
 2 A-2.5 
 
 (17.29) 
 
 applies to this law. 
 
 c. The Admlssitle Eoughness 
 
 The problem of the admissible roiighness of a wall in a flow is 
 very important in practice since it concerns the effort that might 
 reasonably be expended in smoothing a surface for the purpose of drag 
 reduction. Admissible roughness signifies that roughness above which a 
 drag increase would occur in the given turbulent friction layer (which, 
 therefore, still is in effect hydraulically smooth). The admissible 
 relative roughness kg/z decreases with increasing Re— number UqZ/v 
 
 as one can see from figure 89. It is the point where the particular 
 curve Z/kg diverges from the curve of the smooth wall. One finds the 
 values for the admissible relative roughness according to the following 
 table; they can also be combined into the one formula 
 
 ^o ■'^s admlss. 
 
 102 
 
 (17.30) 
 
 U I 
 
 
 V 
 
 lo5 
 
 xo« 
 
 10^ 
 
 108 
 
 105 
 
 admlss. 
 
 10-3 
 
 10-^ 
 
 10-5 
 
 10^ 
 
 10-^ 
 
 From equation (17.3O) one recognizes that the admissible roughness height 
 is by no means a function of the plate length. This fact is significant 
 for instance for the admissible roughness of a wing. Equation (17.3O) 
 states that for equal velocity the admiesible roughness height is the 
 same for a full scale wing as for a model wing. Let us assume a numerical 
 example : 
 
 Wing: chord 2 = 2m 
 
 Velocity U = 3OO km/h = 83 m/sec 
 
k2 KACA TM No. 12l8 
 
 From equation (17. 30) results an admissible magnitude of 
 
 roughness kg = 0.02 mm. This degree of smoothness is not always attained 
 "bj the wing surfaces manufactured in practice, so that the latter have a 
 certain roughness drag. In the considerations just made one deals with 
 an increase of the friction drag in an a priori turbulent friction layer. 
 
 However, the roughness may also change the drag "by disturbing the 
 laminar friction layer to such an extent that the point of laminar/ 
 turbulent transition is shifted toward the front. Thereby the drag can 
 be increased or reduced according to the shape of the body. The drag is 
 increased by this displacement of the transition point" if the body in 
 question has predominant friction drag (for instance wing profile). The 
 drag might be reduced, circumstances permitting, for a body with pre- 
 dominant pressure drag (for instance, the circular cylinder). One calls 
 the roughness height which causes the transition the "critical roughness 
 height". According to Japanese measurements (reference 77) this critical 
 roughness height for the laminar friction layer is given by 
 
 ^*^crlt 
 * -^^^ = 15 (17.31) 
 
 A numerical example follows: 
 Assume, as prescribed before, a 
 
 wing Z = 2m 
 
 U = 300 km/h = 83 m/sec 
 
 then Re = UqI/v = lo'^. Consider the point of the wing x = 0.12, thus 
 Re^j. = U x/V = 10 . Up to this point the boundary layer might remain 
 laminar under the effect of the pressure drop. The wall shearing stress 
 for the laminar boundary layer is according to equation (9.17) 
 
 Tq ^ ^^^ ^^ 2.rir" 6900 m^ 
 
 -^ = 0.332 Uq \^-^ = 0.332 -^— = 2.29 p 
 
 P VUo^ 103 sec^ 
 
 hence: 
 
 T* = V^o/P = 1*52 m/sec 
 and according to equation (17.3I) 
 
 k = 15 T^ = -^^- i 10'~'^ = 0,14 mm 
 crit ^* 1.52 7 
 
NACA TM No. 12l8 k3 
 
 The critical roughness height causing the transition Is^ therefore^ ahout 
 ten times as high as the roughness height admissible In the turhulent 
 friction layer. The laminar friction layer therefore "tolerates" a 
 greater roughness than the turhulent one. 
 
 The following can he said ahout the Influence of the roughness on 
 ^the form drag: Sharp— edged "bodies are Indifferent to surface roughnesses 
 "because for them the transition point Is fixed hy the edges, as for 
 Instance for the plate normal to the flow. Short ciirved bodies, on the 
 other hand, as for instance the circular cylinder, are sensitive. For 
 the circular cylinder the critical Reynolds number, for which the known 
 large pressure drag reduction occurs, is largely dependent on the 
 roughness. With increasing relative roughness k/R (E = radius of the 
 circular cylinder) Ee^j^j^^ decreases. According to British measurements 
 
 (reference 9O) "tiie drag curves for a circular cylinder with different 
 relative roughnesses have a course as indicated in figure 90. The 
 boundary layer is so disturbed by the rot^hness that the laminar /turbulent 
 transition occurs for a considerably smaller Ee-number than for the 
 smooth cylinder. The roughness has here the same effect as Prandtl's 
 trip wire, that is, in a certain region of Ee-^iumbers it decreases the 
 drag. It is true, however, that the supercritical drag coefficient is 
 then always larger for the rough circular cylinder than for the smooth 
 one. 
 
 CHAPTER XVTII. THE TUEBULENT ERICTICJN LAYER IN 
 ACCELEEATED AND RETARDED FLOW 
 
 The cases of turbulent friction layer treated so far are relatively 
 simple insofar as the velocity outside of the friction layer along the 
 wall is constant. Here as for the laminar flow the case of special 
 interest is where the velocity of the potential flow is variable along 
 the wall (pressure drop and pressure rise). As for laminar flow, the 
 form of the boundary layer profile along the wall is variable. In 
 practice this case exists for Instance for the friction layer on the 
 wing, on the turbine blade, and in the dlffuser. Of special interest is 
 the question of whether separation of the boundary layer occurs and, 
 if so, where the separation point is located. The problem consists 
 therefore for a prescribed potential flow in following the turbulent 
 friction layer by calculation. The calculation of the turbulent friction 
 drag is of Importance. The corresponding problem for the laminar friction 
 layer was solved by the Pohlhausen method (chapter X). 
 
 For the turbulent friction layer the method of Gruschwitz 
 (reference 78) proved best. Gruschwitz makes the asumption that the 
 velocity profiles of the turbulent boundary layer for pressure drop and 
 
hk NACA TM No. 12l8 
 
 rise can be represented as a one— parameter family, if one plots u/u 
 against y/-a. -d signifies the momentum thickness which is, according 
 to equation (6.32), defined "by: 
 
 U^^ =1 u(U - u) dy (18.1) 
 
 As form parameter one selects 
 
 (18.2) 
 
 u('3) denoting the velocity in the friction layer at the distance from 
 the wall J = -a. That t[ actually is a serviceable form parameter can 
 he recognized from figure 9I where a family of turbulent boundary layer 
 profiles is plotted according to Gruschwitz. Gruschwitz found from his 
 measurements that the turbulent separation point Is given by 
 
 Ti = 0.8 (Separation) (I8.3) 
 
 The form parameter r] is analogous to the Pohlhausen— parameter X of 
 the laminar friction layer. However, a considerable difference exists 
 between -q and X: for the laminar friction layer an analytical relation 
 exists between X and the press''jre gradient and the boundary layer 
 thicknessj namely according to equation (10.41) 
 
 g2 
 
 X = -^^ (18. i^) 
 
 V dx 
 
 Such a relation is thus far lacking for the t^lrbulent boundary layer, 
 since one does not yet possess an analytical expression for the turbulent 
 velocity profiles*. One needs therefore an empirical equivalent for 
 equation (l8.J|). 
 
 For the special case of the turbulent friction layer without 
 
 1/7 
 pressure gradient where the l/7— power law u/U = (y/5) applies for 
 
 the velocity profile, one finds from equations (I8.I) and (l8.2) 
 
 * Compare, however, chapter XXIIb, where under certain assijmptions 
 such an analytical connection is indicated. 
 
NACA TM Wo. 1218 
 
 i^5 
 
 1 
 6 
 
 J_ 
 72 
 
 ^ = -L T) = 0.1+87 
 
 (18.5) 
 
 Since in the case of the turhulent "boundary layer, an analytical 
 expression for the velocity dlBtrnDutlon does not exist, the calculation 
 is limited to the determination of the four characteristics of the friction 
 layer: form parameter t], wall shearing stress t displacement 
 thickness 5*, momentum thickness t3 . Four equations are required for 
 their calculation. 
 
 As for the laminar 'boundary layer, the momentum theorem yields 
 the first equation; the momentum theorem may, according to equation (10. 36) 
 be written in the form: 
 
 (I) 
 
 (18.6) 
 
 The second equation is yielded "by the function 
 
 T = ^^^) 
 
 (II) 
 
 (18.7) 
 
 obtained by Gruschwitz by evaluation of the measured velocity profiles 
 (fig. 92) J and regarded as generally valid. It can be derived also by 
 calculation from the form of the velocity profile (compare appendix 
 chapter XXIl) and yields: 
 
 Tl = 1 - 
 
 H - 1 
 H(H + 1) 
 
 H-1 
 
 (18.8) 
 
 The third equation is empirically derived by Gruschwitz from his 
 measurements. He considers that the energy variation of a particle moving 
 parallel to the wall at the distance y = i3 is a function of u^, U, t3, V. 
 Dimension considerations suggest the following relation: 
 
 ^^=F(„Re) 
 q dx 
 
 (18.9) 
 
 = |n2 
 
 y = ^ 
 
 and g-, = p + § u^^ signify the total pressure in the layer 
 The evaluation of the test results showed that a dependence on 
 
1^6 
 
 NACA TM Wo. 1218 
 
 the Ee— number is practically non-existent, and that one can represent 
 equation (18.9) in the following manner: 
 
 ■^ i = O.OO89U T] - O.OOlj-61 
 
 <1 djc 
 
 (18.10) 
 
 Furthermore, the identity 
 
 go - 
 
 V^^^-V 
 
 P 2 
 2^^ 
 
 = £u2fl-]£i i.ciT, 
 
 if 
 
 is valid. One puts 
 
 1 T] = ^ 
 
 ■ (18.11) 
 
 and has therefore 
 
 dg-L 
 dx 
 
 dC 
 dx 
 
 (18.12) 
 
 Now equation (I8.IO) can "be written: 
 
 d^ 
 
 ■a ^ = - 0.00894 ^ + 0.00i|6l q 
 
 (III) (18.13) 
 
 The fourth equation is still missing and is replaced by the following 
 estimation of Tq: According to the calculations for the plate in 
 longitudinal flow, equation (IT-T) was: 
 
 ^ - 0.0225(— j = 0.0225 (Eeg)' 
 
 -lA 
 
 (18.14) 
 
 If one takes into consideration that for the l/7— power law of the Telocity 
 distribution: 
 
NACA TM No. 12l8 
 
 ^7 
 
 6.=|5, 
 
 i3 = -1- 5 
 72 
 
 one can write equation (l8.lU) also; 
 
 Tn -lA -lA 
 
 ° - 0.01338 (Reg^) = 0.01256 (Ee^) 
 
 01/ 
 
 (17) (18.15) 
 
 For calculation of -a and t] (t, = Q. r\, respectlTsly) one must now solve 
 the following system of equations: 
 
 — + 0.00894 -^ = 0.01+61 i 
 dx V V 
 
 i^ ^ A ^ H\ .fl dg 
 djc I 2y q djc 
 
 PU^ 
 
 (18.16) 
 
 2 T 
 
 = •^17 is a given function of x; H and — ^ 
 2 pU^ 
 
 are given functions 
 
 of T] = ^/q or i3, respectively. This system of equations is to be 
 solved downstream from the transition point. 
 
 Initial values: As initial value for i3 one takes the value from 
 the laminar friction layer at the transition point: 
 
 i3 
 
 oturh. 
 
 •a 
 
 olam. 
 
 (18.17) 
 
 This is "based on the consideration that the loss of momentum does not 
 vary at the transition point since it gives the drag. The initial value 
 of r\ is somewhat arbitrary. G-ruschwitz takes 
 
 T1_ = 0.1 
 
 and states that a different choice has little Influence on the result. 
 
 With these initial values the system of equations (18.16) may he 
 solved graphically, according to a method of Czuher (compare appendix. 
 Chapter XXII, where an example is given) . A first approximation for i3 
 is ohtained by first solving the second equation with constant values 
 
 for Tq /pU and H; 
 
1^8 WACA TM No. 1218 
 
 L2. 
 2 
 PU 
 
 = 0.002; H = 1.5 (18.18) 
 
 are appropriate. Thsre'by the second equation Is a differential eq^uation 
 of the first order for -d. This first approximation i3t (x) is then 
 
 suhstituted into the first equation, vhich then 1)60011168 a differential 
 equation of the first order for ^(x); let its solution te denoted 
 hy L(x). Thus one has also a first approximation for tj: t]-,(x). 
 
 With T|, (x) one determines the course of H(r|) according to figure 92 
 
 and is now atle to improve t according to equation (I8.I5). These 
 
 values of hoth H and t are now inserted in the second equation, and 
 
 a second approximation -Boi^) is obtained. By sutstitution of -SoCx) 
 
 into the first equation one ohtains the second approximation ^p(x), etc. 
 
 The method converges so well that the answer is essentially attained in 
 the second approximation. 
 
 The separation point is given ty 
 
 Ti = .0.8 
 
 Incidental to the houndary layer calculation one obtains the following 
 characteristic values of the friction layer as functions of the arc 
 length X : 
 
 ■bM, b*(x), Ti(x), To(x). 
 
 The "boundary layer calculation for the profile J OI5, c^^ = is given 
 
 as example in figure 93 • The transition point was assumed at the velocity 
 maximum. The calculation of the laminar "boundary layer for the same case 
 was Indicated in chapter XII. The details of this example are compiled 
 in the appendix, chapter XXII. 
 
 It should "be mentioned that the calculation for the tur'bulent 
 "boundary layer must "be performed anew for every Ee-numier UqI/v, whereas 
 
 only one calculation was necessary for the laminar "boundary layer. The 
 reasons are, first, that the transition point travels with the Ee-number, 
 and second, that the initial val ue of i3/t varies with Ee, since for 
 
 the laminar boundary layer t- \ — ^=^ at the transition point Is fixed. 
 
 "t y V 
 
 It must be noted that the values obtained for Tq become incorrect 
 in the neighborhood of the separation point: At the separation point t 
 
 must equal zero, whereas equation (18.I5) gives everywhere t / 0. 
 
MCA TM No. 1218 kg 
 
 B oundary Layer Without Pressure Gradient 
 
 In this case q(x) = Constant. Equation (18.I3) can te written: 
 
 1 i^ = _ 
 
 - ^ = - o.oo89ij-Ti + 0.001^61 (18.19) 
 
 or, "because q.(x) = constant; 
 
 ^ = iMl = q U (18.20) 
 
 djc dx ^ djc 
 
 Thus equation (18.I9) becomes; 
 
 ^ ^ = - 0.0089UT1 + o.ooi)-6i (18.21) 
 
 A solution of this equation is: 
 
 . 0,00461 ^ g (^8.22) 
 
 ' 0.0089!^ ^ ^ 
 
 Since at the ■beginning of the tiirbulent friction layer t\ is smaller 
 than this value (transition point t) = 0.1) and since according to 
 equation (l8.2l) drj/djc > 0, t] must in this case approach the value 
 T] = 0.516 asymptotically from "below. For the velocity profile of the 
 1/7— power law, t\ = O.U87 (compare equation (18.15)). The profile 
 attained asymptotically for uniform pressure (p = constant) therefore 
 almost agrees with the l/7— power law that was previously applied to the 
 plate in longitudinal flow. 
 
 A great many "boundary layer calculations according to this method 
 are performed in the dissertation "by Pretsch (reference 80). 
 
 Thirteenth Lecture (March 2, 19^12) 
 
 CHAPTER XIX. FREE TURBULENCE 
 a. General iRemarks; Estimations 
 
 After considering so far almost exclusively the turbulent flow 
 along solid walls, we shall now treat a few cases of the so— called free 
 turbulence. By that one understands turbulent flows where no solid walls 
 
50 WACA TM No. 12l8 
 
 are present. Examples are the spreading of a Jet and its mixing with the 
 surrounding fluid at rest; or the wake flow behind a "body towed thi^ough 
 the fluid at rest (fig. 9^)- Qualitatively these turbulent flows take a 
 course similar to that for the laminar case (compare chapter IZ) ; quanti— 
 tatively, howerer, considerable differences exist, since the turbialent 
 friction is very much larger than the laminar friction. In a certain way, 
 the cases of free turbulence are, with respect to calculations, simpler 
 than turbulent flows along a wall, since the laminar sublayer is not 
 present and the laminar friction as compared with the turbulent one can 
 therefore be neglected for the entire flow domain. The free turbulence 
 may be treated satisfactorily with Prandtl's concept of the turbulent 
 shearing stress according to equation (I5.9): 
 
 .1^ 
 
 du 
 
 Su , 
 
 — (19.1) 
 
 oy 
 
 the mixing length I being assumed a pure position function. The 
 turbulent friction has the effect of making the Jet width increase and 
 the velocity at its center decrease with increasing distance along the 
 Jet. 
 
 We now perform rough calculations, according to Prandtl (reference 2, 
 Part I), for a few cases of free turbulence which give Information about 
 the laws governing the increase of width and the decrease of "depth" with 
 the distance x. 
 
 It has proved useful for such turbulent Jet problems to set the 
 mixing length I proportional to the Jet width b: 
 
 1 = p = constant (19.2) 
 
 b 
 
 Furthermore, the following rule has held true: The increase of the 
 width b of the mixing zone with time is proportional to the fluctuation 
 of the transverse velocity v' : 
 
 ^~v. (19.3) 
 
 , D S S 
 
 D/Dt signifies the substantial derivative: thus: — = u ^r— + v :r— . 
 
 Dt dx dy 
 
 According to our previous estimation, equation (I5.5): v' = 2 
 Therefore : 
 
 
MCA TM No. 1218 51 
 
 Dt dy 
 Furthermore, the mean value of ^ equals approximately: 
 
 - number — =^ 
 
 Sy 
 
 and thus: 
 
 =rr = nuuiber t- u = number B u (I9.5) 
 
 Dt D max max v ^ v; 
 
 Jet (Plane and Circular] 
 
 We shall estimate , "by means of these relations, how the width 
 increases with the distance x and the velocity at the center decreaeea. 
 At first, for the circular as well as for the plane Jet: 
 
 =rr = number u -r- (19 -o) 
 
 Dt max dx \ ^ j 
 
 It follows, "by comparison with equation (I9.5): 
 
 -T— = number r- = number 
 dx b 
 
 b = number x + constant 
 
 If the origin of coordinates is suitably selected (it need not coincide 
 with the orifice) one has therefore: 
 
 b = number x 
 
 (plane and circular Jet) (I9.7) 
 
 The relation between u^^y and x Is obtained from the momentum 
 
 theorem. Since the pressijire is constant, in the x— direction, the x— 
 momentum must be independent of x, thus: 
 
 J-p/u^dF = constant 
 
52 
 
 NACA TM Nc 12l8 
 
 whence follows for the circular jet: 
 
 or; 
 
 2 2 
 J = nuniber p u b 
 
 u = niniber t- \| — 
 max ^ \ P 
 
 and "because of equation (I9.7) 
 
 ^^max 
 
 = number 
 
 X Yp 
 
 (circular Jet) (19.8) 
 
 For the plane Jet, if J' signifies the momentum, per unit length of the 
 Jet: 
 
 J' '= numher p u h 
 
 1 y J' 
 
 "^^ = ^™*er — \) -p- 
 
 y^ 
 
 and because of eq.uation (I9.7): 
 
 (plane Jet) (19-9) 
 
 Wake (plane and circular) 
 
 The calculation for the wake is somewhat different, since the 
 momentum which gives directly the drag of the "body must he calculated in 
 a slightly different way. The momentum integral is now (compare 
 equation (9.14.0)) : 
 
 W=J=p/u(Uo-u)dF 
 
 (19.10) 
 
 At large distance from the body u* = Uo — u is small compared with U(- 
 (fig. 95) so that u» « Uq and 
 
NACA TM No. 12l8 
 
 53 
 
 u(Uo - u) = (Uq - u' ) .u» Z Uq u« 
 
 Thus one obtains for the circiilar wake: 
 
 § c,., F U^2 «, p TJ u« jt h2 
 
 2 w ■" "'o 
 
 t c F 
 u* w 
 
 — — /w — ^^— 
 
 o no 
 
 (19.11) 
 
 Instead of equation (19.6) now applies: 
 
 Dt ° dbc 
 
 (19.12) 
 
 and instead of equation (19.5) 
 
 =rr = number t- u' 
 Dt D 
 
 (19.13) 
 
 Equating of equations (19.12) and (19.I3) gives: 
 
 tJ ^ ~ i u' = p u» 
 o djc l3 
 
 (19.1^) 
 
 By comparison with equation (I9.II) one obtains: 
 
 dx jt 
 
 p c^ F X 
 
 (Wake Circular) 
 
 (19.15) 
 
 By insertion in equation (I9.II) results: 
 
 u' .. 1 /"v ^ 
 ^o 4p2,2 
 
 1/3 
 
 (Wake circular ) 
 
 (19.16) 
 
'^h 
 
 liACA TM No. 1218 
 
 For the plane wate "behind a long rod, wing, or the like with the 
 diameter d and the length L,W = c — U^, Ld and 
 
 ¥ = J^pUqU'13L 
 
 and hence; 
 
 c d 
 
 (19.17) 
 
 and from this in combination with equation (I9.IU) 
 
 2 h — ~ p c d 
 dx w 
 
 h ~ \fpTT7 
 
 (wake plane) 
 
 (19.18) 
 
 By substitution in equation (I9.I7) results; 
 
 
 
 
 ^c 
 
 d^ 
 
 ^1/2 
 
 u' 
 
 
 1 
 
 w 
 
 
 
 /V 
 
 — 
 
 
 — 
 
 
 Uo 
 
 
 2 
 
 ^p 
 
 ^/ 
 
 / 
 
 (Wake plane) 
 
 (19.19) 
 
 ThuSj for the circular wake, the width increases with \^y^, and. 
 
 -2/3 
 the velocity decreases with x ; for the plane wake, the width 
 
 increases with \f^, and the velocity decreases with x ' . 
 
 The power laws for the width and the velocity at the center are 
 compiled once more in the following tahle. The corresponding laminar 
 cases which were partially treated in chapter IX, are included. More- 
 over, the case of the free Jet boundary is given, that is the mixing of 
 a homogeneous air flow with the adjoining air at rest. (Compare 
 figure 97. ) 
 
NACA TM Wo. 1218 
 
 55 
 
 
 Lami nar 
 
 TurlDulent 
 
 Case 
 
 Width 
 
 Velocity at Center 
 
 u 
 max 
 
 or u' 
 
 respectively 
 
 Width 
 h 
 
 Velocity at Center 
 
 u 
 max 
 
 or u' 
 
 respectively 
 
 Plane Jet 
 
 .2/3 
 
 X-V3 
 
 X 
 
 x-1/2 
 
 Circular Jet 
 
 X 
 
 x-1 
 
 X 
 
 x-1 
 
 Plane wake 
 
 xl/2 
 
 x-1/2 
 
 xl/2 
 
 x-1/2 
 
 Circular wake 
 
 xl/2 
 
 ^1 
 
 xl/3 
 
 x-2/3 
 
 Free Jet "boundary 
 
 xl/2 
 
 x° 
 
 X 
 
 x° 
 
 POWER LAWS FOE THE INCEEASE WUH WIDTH AND THE DECREASE OF TELOCITY 
 
 WITH THE DISTANCE x 
 
 For a few of the cases treated here the velocity distribution will 
 he calctilated explicitly "below. The calculation on the "basis of the 
 Prandtl mixing length theorem was performed for the free Jet "boundary, 
 the plane Jet, and the circular Jet "by W. Tollmien (reference 81), for 
 the plane wake "by H. Schlichting (reference 82) and for the circular 
 wake "by L. M. Swain (reference 83). 
 
 The equations of motion for the plane stationary case are, according 
 to equation (lU.lU), if the laminar friction terms are completely 
 neglected: 
 
 = _i^ + i^+i ^Ih. 
 
 ^ p ^ p Sx p Sy 
 
 u ^ + V ^^ 
 
 Sv Sv 
 Sx hj 
 
 Su Sv 
 S^ 3y 
 
 1 M+ 1 ^+ 1 ^ 
 p ^ p dx p Sy 
 
 (19.20) 
 
 y 
 
56 MAC A TM No. 12l8 
 
 Td. The Plane Wake Flow 
 
 \Je shall now calculate the velocity diatrihution for the plane wake 
 flow. A cylindrical "body of diameter d and bt.zii h is considered. 
 Further, let 
 
 u' = U^ - u (19.21) 
 
 he the wake velocity. One applies the momentum theorem to a control 
 area according to figure 95 ^ "the rear ho-jndary B C of which lies at 
 such a large distance from the hody that the static pressure there has 
 the undisturhed value. As shown in detail in chapter H., equation (g.kO), 
 one ohtains: 
 
 W = h p / u (Uq - u) dy 
 
 BC 
 
 (19.22) 
 
 ¥ = h p / (Uq - u' ) u' dy 
 
 J 
 BC 
 
 For large distances "behind the "body u' « Uq so that one may 
 
 2 , . 
 
 approximately neglect the term u» in comparison with 17^ u' m 
 
 equation (19.22). Hence: 
 
 (19.22a) 
 
 Since, on the other hand W = c„ d h ^ U "", there "becomes: 
 •' " 2 o 
 
 i+h 
 
 ^' iy = I c^ i u^ (19.23) 
 
 -b 
 
 This prohlem can "be treated "rfith the boundary layer differential equations; 
 they read according to equation (I9.2O) with the Prandtl expression for the 
 turhulent shearing stress according to equation (I5.9), with p and a^ 
 neglected: 
 
SiACA TM Wo. 1218 
 
 57 
 
 ^ 
 
 Su Su 1 St ^,2 Su S a 
 u — + V — = = 22 TT 
 
 Sx By ^ Sy 
 
 5u I 5v _ Q 
 
 Sy By' 
 
 Sx Sy 
 
 / 
 
 For ths mixing length one puts, according to equation (19.2) 
 
 (I9.2J+) 
 
 2 = p b 
 
 (19.25) 
 
 Further : 
 
 U' = U-, + Up + . 
 
 ^ = h 
 
 (19-26) 
 (19.27) 
 
 For the wake "velocity u-i and the width "b the power laws for the 
 decrease and increase, respectively, with x were already foimd in 
 equations (19.I8) and (19,19), One therefore writes: 
 
 u 
 
 -1/2 
 
 u ych^ ^^^) 
 
 o \ w 
 
 1 
 
 (19.28) 
 
 h = B (cwd x) 
 
 1/2 
 
 (19.20) 
 
 According to equations (19.27) and (19.29) 
 
 Sn 
 
 _ 1 Bti 
 
 ^y B(cwd x)^/2 ^^ 
 
 It 
 
 2 X 
 
 The estimation of the terms in equation (I9.2ij-) with respect to their 
 order of magnitude in x gives: 
 
 Bu -3/2 bv -3/2 Su -1 S u _ -3/2 
 ^:— ~x'; — ~x-^'; — ~x ; ~ x -^' ; 
 
 Sx By ' Sy .2 
 
 V ~ X 
 
 *The terms u^ • - • signify additional terms of higher approximation, 
 which disappear- according to a higher power of x than does u-^. 
 
58 
 
 WACA TM No. 1218 
 
 Hence the term v ^ ^ x whereas the largest terms ~ x . 
 equation (l9.2i|-) is simplified to: 
 
 1 2 ^^1 ^ ^1 
 - U — - -^2 1 
 
 ° Sx 
 
 ^ ^' 
 
 Thus 
 
 (19.30) 
 
 The neglected terms are taken into account only in the next approximation. 
 The further calculation gives: 
 
 ^^1 / X \-l/2 
 
 S; 
 
 c d 
 
 _ i a f I - 1. f 
 
 2 X 2x 
 
 SU' 
 
 Sy 
 
 = u. 
 
 ^-1/2 
 
 \\i J 
 
 f 
 
 B(c^dx) 
 
 1/2 
 
 S^u 
 
 ^y 
 
 1 n r-^V'^'f" ^ 
 
 w 
 
 ° ' c d ) ^2 
 
 B c d X 
 w 
 
 2 Z 
 
 2 Bu-, S U-, 
 
 2 p^ U^^/^-^^^ ^ f'f" 
 
 !/c c 
 
 B/'c^dxj 
 
 1/2 
 
 After Insertion in equation (I9.3O) and eliminating the factor 
 
 2 1 / X \"^/^ / N 
 
 U — (-rV) "the following differential equation results for f(Ti) 
 
 i (f + Ti f) = ^ f«f" 
 
 B 
 
 (19.31) 
 
 The houndary conditions are: 
 
 Su 
 y = h: u = 0; — - 
 ■^ dy 
 
 
 
NACA TM No. 12l8 59 
 
 That is: 
 
 Ti = 1: f = f • = (19.32) 
 
 Th3 differential— eq^uatl on (I9.3O) may immediately te integrated once 
 and gives: 
 
 2 
 
 ifTi= — f'^ + Constant 
 2 B 
 
 Because of the hoondary conditions, the integration constant must equal 
 zero; thus: 
 
 1 3 2 
 
 - f Ti = — f ' 
 
 2 ^ B . 
 
 This may he Integrated in closed form: 
 
 ^1/2 1/2 ^ \ |2pf df 
 B dTi 
 
 df J B ,^ , 
 
 if V2P' 
 
 2fl/2 = \ pi 2 3/2 ^ c„^g^3^^ 
 
 -^N5^"^^^' 
 
 Because f = for ti = 1, C = - i J^ and hence: 
 
 3 y2p2 
 
 ,2 
 
 = l^(l_,^3/2) ■ (19.33) 
 
 q _„2 \ / 
 
 f = i 
 
 9 2p2 
 
60 NACA TM Wo. 12l8 
 
 The condition f = for T] = 1 is simultaneously satisfied according 
 
 to equation (19'33)« In. f", that Is — —, a singularity results at the 
 
 Sy2 
 
 center (t) = O) and on the edge. For ti = 0, f" =oo; the velocity 
 profile there has zero radius of curvature. At the edge there exists 
 a discontinuity in curvature. In contrast to the laminar "boundary 
 layer solutions, where the velocity asymptotically approaches the value 
 of the potential flow, one obtains here velocity profiles which adjoin 
 the potential flow at a finite distance from the center. 
 
 The constant B remains to he determined: 
 
 n+h Oh ph ni 
 
 u' dy =*2 u' dy = 2 / u^ dy = 2U^ c^d B / f(Ti) dii 
 
 J-h Jo t/o Jo 
 
 From equation (19-33) one finds: 2 / (^ — 'H ) '^'H = iT^ 
 
 UO 
 
 and hence; 
 
 and, by comparison with equation (19-23) 
 
 or: 
 
 b2 1 
 
 = i; B = \/lO p (19.3^) 
 
 2 2 
 
 Thus the final result for the width of the wake and the velocity 
 distribution from equations (19-28) and (19.29) is: 
 
 '^Eeviewer's note: Integrating from —1 to +1, as was done in the 
 original German version, results in an imaginary term, which was avoided 
 in the translation by integrating from to +1 and doubling the result. 
 
WACA TM No. 1218 61 
 
 "b = \ 10 (3 /"c d zS^'^ 
 
 ^1 .. VlO / X \-l/2 
 
 2 (19.35) 
 
 Uo 18P \c^^ 
 
 1-(J''^' 
 
 The constant 3 = z/b is the only empirical constant of this theory; it 
 must "be determined from the measurements. 
 
 Comparison with the tests of Schlichting (reference 82) shows that 
 the two power la^-s (equations (I9.28) and (I9.29)) are well satisfied, 
 and also bhat the form of the velocity distrihution shows good agreement 
 with equation (19.35)^ figia^e 96. The constant 3 is determined as 
 
 p = 1 =,0.207 
 
 The solution found is a first approximation for large distancesj according 
 to the meas'jrements it is valid for x/c^d > 50' For smaller distances 
 
 -1 -1 /2 
 one may calculate additional -terms which are proportional to x , y: ' , 
 
 . . . for the wake velocity in equation (19.26). 
 
 The rotationally— symmetrical wake problem was treated by 
 Miss L. M. Swain (reference 83). For the first approximation results 
 exactly the same function for the velocity distribution; only the power 
 laws for the width b and the velocity at the center u^' are different, 
 
 namely b «' x ' and u^' ~ x ' , as already Indicated in 
 equations (19.15) aJ^i (I9.16). 
 
 Fourteenth Lecture (March 9, 19^2) 
 
 c. The Free Jet Boundary 
 
 The plane problem of the mixing of a homogeneous air streaiu with 
 the adjoining air at rest shall also be treated somewhat more accurately 
 (fig. 97). It is approximately present for instance at the edge of the 
 free Jet of a wind tuiinel. The problem was solved by Tollmlen 
 (reference 8I). 
 
 The velocity profiles at various distances x are affine. One 
 Gets 
 
 u = U^f(Ti) = UJ'(ti) (10.^,6) 
 
62 NACA TM No. 12l8 
 
 where 
 
 Ti = |; (t~ x) (19.37) 
 
 and 
 
 nr\) =/f(n) ^T 
 
 Furthermore set 
 
 2 = 01 = 01) (19.38) 
 
 The equation of motion reads: 
 
 u^+T^ = i^ = 2l2^^ (19.39) 
 
 Sx dy p Sy Sy ^ 2 
 
 One integrates the continuity equation hy the stream function: 
 
 ilf=j'udy=U^xj' f(Ti) dTi = U^ X F(ti) (I9.i^0) 
 
 Then: 
 
 |li = __£^F"; ^ = ^Y"; ^ = ^F'" 
 ax X ' ' By X ' ^^2 ^2 
 
 T = -^ = -U^ (F - TiF') (19.J^0a) 
 
 Suhatitution into the equation of motion (19.39) gi"^es, after division 
 hy Uo^/x; 
 
 FF" + 2c^ F"F'" = (19.^1) 
 
NACA TM No. 12l8 
 
 63 
 
 The boundary conditions are: 
 at the inner edge: 
 
 Tl = \' 
 
 u = U : 
 
 
 F» = 1 
 
 
 1^- 
 
 F" = 
 
 
 T = 0: 
 
 F = Tl 
 
 at the outer edge: 
 
 
 
 T) = TI2: 
 
 u = 0: 
 
 F« = 
 
 
 ^^ = 6: 
 oy 
 
 F" = 
 
 > (19-^2) 
 
 J 
 
 Since the "boundary points r\^ and T)p are still free, these five hoimdary 
 
 conditions can Tae satisfied by the differential equation of the third order 
 equation (I9.J4-I). By introduction of the new variable: 
 
 Tl* = 
 
 ^y^ 
 
 (19A3) 
 
 the differential equation (19.41) is transformed into (' = differentiation 
 with respect to t)*) 
 
 F F" + F" F'" = 
 
 (19.i^i^) 
 
 The solution F" = 0, which gives u = Constant, is eliminated. The 
 general solution of the linear differential equation 
 
 F + F'" = 
 
 (l9.i+5) 
 
 is 
 
 F = e 
 
 \-x\> 
 
 with \ signifying the roots of the equation X,-^ + 1 = 0, thus: 
 
6U NACA TM Wo. 12l8 
 
 Hence the general solution is: 
 
 F = 
 
 r^ 
 
 ^ 
 
 VJ„J , . 2 ^^. /\fy 
 
 = C^ e ' +0^6 cosl ^ T]*y + C e sin\^^ t]*; (19.^^6) 
 
 If J moreover, one measitreB the ti— coordinate from the inner houndary 
 point, thus puts: 
 
 j^* = j^* - Ti*^ 
 
 the solution (equation (I9A6)) can also "be written: 
 
 Hi ... 21 
 
 =Tpf 2 
 F = d-| e + dp e cos 
 
 f ^) . Ija 2 =1. (f V 
 
 From the to-andary conditions (equation (I9.U2)) res'jlt for the constants 
 the values: 
 
 Tl* = 0.981 ; Tl* = - 2.01+ ; Tl* = - 3.02 
 
 d-L = -0.0062; d2 = O.987J d^ = 0.577 
 For the width of the mixing region one oh tains: 
 
 t = ^[\ - \) = ^^2c2 ^^* - n|^ 
 
 h = 3.02X^/20 2 X 
 
 The constant c must he determined from experiments. From measurements 
 it is found that 
 
 h = 0.255 X (19.^7) 
 
MCA TM No. 1218 65 
 
 Hence 
 
 ^ 
 
 2c^ = 0.08ij-5j c = 0.0174 
 
 and 
 
 ^ = 0.0682 (19-i^8) 
 
 It is striking that here the ratio z/b is essentially smaller than for 
 the wake. 
 
 The distrihution of the velocity components u and y over the 
 width of the mixing zone is represented in figure 98. 
 
 From the second equation of motion one may calculate the pressure 
 difference "between the air at rest p ' and the homogeneous air stream p-, . 
 
 One finds: 
 
 p -p = o.ooJ+8 S u (19.49) 
 
 i o 20 \ ^ ^J 
 
 Thus an excess pressure of one— half percent is present in the Jet. For 
 the inflow velocity of the entrained air one finds according to 
 equation ( I9 . 40a ) : 
 
 ^- CO = - FCrig) Uq = + 0.379 V 2c^ Uq 
 
 and with the measured value of c; 
 
 v_ „ = 0.032 Uq (19.49a) 
 
 d. The Plane Jet 
 
 In a similar manner one may also calculate the plane turbulent jet 
 flowing from a long narrow slot (compare fig. 94). The laws for the 
 increase of the width and the decrease of the center velocity have 
 already been given in equations (I9.7) and (I9.9): "b «v Xj Uj^ ~ rj=- 
 The calculation of the velocity distribution was carried out by Tollmien 
 (reference 8I); it leads to a non— linear differential equation of the 
 second order the integration of which is rather troublesome. Measurements 
 for this case were performed by Forthmann (reference 91)- In figure 99 
 
66 
 
 MCA TM No. 1218 
 
 the measurements are compared with "t^he theoretical curTe. The agreement 
 Is rather good. Only in the neighhorhood of the velocity maximum is 
 there a slight systematic deviation. There the theoretical curve is 
 more pointed than the meas'Jired curve; the theoretical curve^ namely, 
 again has at the maximum a vanishing radius of curvature. 
 
 According to the Prandtl formula, equation (15.9)^ the exchange 
 hecomes zero at the velocity maximum, whereas actually a small exchange 
 is still taking place. 
 
 e. Connection Between Exchange of Momentum, 
 
 Heat and Material 
 
 In concluding the chapter on turlsulent flows I should like to 
 make a few remarks ahout the connection between the turbulent exchange 
 and the heat and material transfer in a turbulent flow. 
 
 In the Prandtl theorem equation (I5.9) for the apparent turbulent 
 stress: 
 
 T = p Z 
 
 one can interpret: 
 
 A 
 
 P I 
 
 
 Su 
 
 Bu _ . Su 
 
 kg sec I 
 
 m 
 
 (19.50) 
 
 as a mixing factor. It has the same dimension as the laminar viscosity [x. 
 Furthermore, the shearing stress t may be interpreted as a momentum flow: 
 
 momentum 
 T = = momentum flow 
 
 2 
 m sec 
 
 (19.51) 
 
 Momentum = mass X velocity = [kg sec] . 
 
 Another effect of the turbulent mixing phenomena, besides the 
 increased apparent viscosity by transport of momentijm,is the transport 
 of all properties inherent in flowing matter, as heat, concentration of 
 impurities, etc. If this concentration is not ■'jnlform, more heat or 
 imparity is carried away by the turbulent exchange from the places of 
 higher concentration than is brought back from the places of lower concen- 
 tration. Thus there results, on the average, transfer from the places of 
 higher to those of lower concentration. 
 
NACA TM No. 12l8 6? 
 
 This results J for temperat-ure differences^ in a turliulent heat 
 transfer; for concentration differences (for instance, of salt), in a 
 turhiolent diffusion. They can, in analogy to eq^uation (I9.5O) be expressed 
 as follows: 
 
 ,, , „-. momentum transport „ d / moment omN 
 
 Momentum flow = = A,- — I — —, 1 
 
 2 ' dv Vunit massy 
 
 m sec '' 
 
 Heat flow = heat transport = a^ ^ ( heat_\ 
 
 2 Q dy \unit mass/ 
 
 m sec 
 
 material N 
 
 „, ri . . -, transport of material . d / 
 
 Flow of material = = A, — ( ■ 
 
 2 ^ dy V 
 
 cLy \unlt mass/ 
 m sec 
 
 The heat content of the unit mass is -c-e ( = temperatiire. 
 
 c = specific heat 
 
 cal m 
 
 For chemical or mechanical 
 
 2 
 kg sec degree _ 
 
 concentrations the concentration of material per unit mass is called the 
 
 concentration c; it is therefore the ratio of two masses and therefore 
 
 dimensionless. Thus the ahove equations may also be written in the 
 
 following forms: 
 
 T = A^ — 
 T^dy 
 
 > (19.52) 
 
 , ^- \ dy 
 
 The question arises as to whether A^, Aq, Aw are numerically the same 
 
 or different. If the momentum is transported exactly like heat or material 
 concentration — Prandtl's theorem is "based on this assumption — it would 
 follow that A^ = Aq = Aj, and, for instance, the velocity and temperature 
 
 distributions in a turbulent mixing region would have to be equal. 
 However, measurements show partially different behavior. 
 
 One has to distinguish between wall turbulence and free turbulence. 
 Concerning free turbulence, calculations of G-. I. Taylor (reference 92) 
 and measurements of Fage and Falkner (reference 93) showed for the velocity 
 and temperature profile of the plane wake flow 
 
 Aq 
 
 -^ = 2 (free turbulence) (I9.53) 
 
 At- 
 
68 NACA TM No. 12l8 
 
 The heat exchange Is, therefore, larger than the momentum exchange. 
 Conseq^uently the temperature profile is wider than the Telocity profile. 
 The theory given for that phenomenon by G. I. Taylor operates with the 
 conception that the particles, in their turbulent exchange moTements, 
 do not maintain their momentum (Prandtl), hut their vortex strength 4^. 
 
 (Prandtl's momentum exchange theory »« Taylor's vorticity transfer 
 theory). However, there are cases not satisfied "by the Taylor theory 
 (for instance the case of the rotationally symmetrical wake). That the 
 heat exchange for free turbulence is considerably larger than the momentum 
 exchange is also shown by experiments of Gran Olsson (reference 88) 
 concerning the smoothing out of the temperature and velocity distributions 
 behind grids of heated rod's. With increasing distance behind the grid 
 the temperature differences even out much more rapidly than the differ- 
 ences in velocity. 
 
 For wall turbulence the difference between the mixing factors for 
 momentum and temperature is smaller. H. Eeichardt (reference 8?) was 
 able to show, from measurements of the temperature distribution in the 
 boundary layer on plates in longitudinal flow by Elias (reference 86) 
 and in pipes by H. Lorenz (reference Ik), that here 
 
 A 
 
 -^ = l.k to 1.5 (wall turbulence) (19. 5U) 
 
 T 
 
 Herewith we shall conclude the considerations of free turbulence. 
 
 CHAPTER XX: DETERMINATION OF THE PROFILE DRAG FROM 
 THE LOSS OF MOMENTUM 
 
 The method, previously discussed in chapter IX, of determining 
 the profile drag from the velocity distribution in the wake is rather 
 important for wind tunnel measurements as well as for flight tests; we 
 shall therefore treat it in somewhat more detail. The determination of 
 the drag by force measurements is too inaccurate for many cases, in the 
 wind tunnel for instance due to the large additional drag of the wire 
 suspension; in some cases (flight test) it is altogether impossible. In 
 these cases the determination of the drag from the wake offers the only 
 serviceable possibility. 
 
 The formula derived before in chapter IX, equation (9.^1) for 
 determination of the drag from the velocity distribution in the wake is 
 valid on.ly for relatively large distances behind the body. It had been 
 assumed that in the rear control plane (test plane) the static pressure 
 equals the pressure of the undistijrbed flow. However, in practically 
 carrying out such tests In the wind tunnel or in flight tests one is 
 
NACA TM No. .1218 
 
 69 
 
 forced to approach the "body more closely. Then the static pressiire gives 
 rise to an additional term in the formula for the drag. For measurements 
 close behind the "body (for instance, for the wing, for x < t) this term 
 is of considerable importance, so that it .must he known rather accurately. 
 A formula was indicated, first hy Betz (reference 84), later hy B. M. Jones 
 (reference 85) which takes this correction into consideration. Although 
 at present most measurements are evaluated according to the simpler Jones 
 formula, we shall also discuss Betz' formula since its derivation in 
 particular is very interesting. 
 
 a. The Method of Betz 
 
 One imagines a control surface surrounding the Tsody as shown in 
 figure 100. In the entrance plane I ahead of the "body there is flow 
 with free— stream total pressure g^, hehind the "body in plane II, the 
 total pressijre gp < g . The lateral boundaries are to lie at so large 
 
 a distance from the "body that the flow there is undisturhed. In order 
 to satisfy the continuity condition for the control surface the velocity 
 U2 in plane II must he partially greater than the undisturhed velocity 
 
 Ur 
 
 Consider the plane prohlemj let the tody have the height h. 
 
 Application of the momentum theorem to the control surface gives: 
 
 W = h 
 
 n+ 00 «+ 00 
 
 (^1 ^ ^\> - (^ 
 
 u — 00 J — 00 
 
 Pg + Pu 2 ' *^ 
 
 (20.1) 
 
 In order to make this formula useful for test evaluation the integrals 
 must he transformed in such a manner that the Integrals need to he extended 
 only over the "wake". For the total pressures 
 
 A 
 
 at infinity: 
 
 in plane I: 
 
 In plane II: 
 
 O O d O 
 
 P 2 
 80 = Pi + 2 ^1 
 
 P 2 
 
 (20.2) 
 
 / 
 
 Outside of the wake the total pressure everywhere equals g^. Hence 
 equation (20.1) becomes 
 
70 
 
 NACA TM Wo. 1218 
 
 W = li 
 
 1+ CO 
 
 (so - S2) 
 
 dy + ^ 
 
 u, — 
 
 u^ ) dy 
 
 (20.3) 
 
 Thus the first Integral already has the desired fomij since the integrand 
 differs from zero only within the wake. In order to give the same form 
 to the second integral, one introduces a hypothetical substitute flow 
 u.2'(y) ^^ plane II which agrees with Ug everywhere outside of the wake, 
 
 Ug within the wake "by the fact that the total pressure 
 
 g . Thus 
 
 hut differs from 
 for u„' equals 
 
 + ^u». 
 
 (20A) 
 
 Since the actual flow 
 
 Up satisfies the continuity equation the 
 
 ^2_' "2 
 flow volume across section II for the hypothetical flow u.^ , u ' is too 
 
 large. It shows a source essentially at the location of the "body which 
 has the strength 
 
 Q = ^J (^* - ^2) ^ 
 
 (20.5) 
 
 A source in a frictionless parallel flow experiences a forward thrust 
 
 E = - P U„ Q 
 
 (20.6) 
 
 One now again applies the momentum theorem according to equation (20. 3) 
 for the hypothetical flow with the velocity u at the cross section I 
 
 and the velocity u'p at the cross section II. Since go = gQ and the 
 
 resultant force, according to equation (20.6), equals R, one ohtains 
 
 - P TT, 
 
 oft = 2\ 
 
 A 
 
 Tin 
 
 U 
 
 •2> 
 
 • (20.7) 
 
 By subtraction of equation (20.7) from equation (20. 3) there results 
 
 W + p Uq Q = h 
 
 /(So - S2)^y ^ 2/(^'2' - -2') ^y 
 
 (20.8) 
 
NACA TM lo. 1218 
 
 71 
 
 or because of equations (20. 5) and (20.6): 
 
 W = h 
 
 /(go - S2)^y ^ |/("'2' - ^2')^ - P ^o /(-'2 - -2X 
 
 One may now perform each of these integrations only across the wake, since 
 outside of the wake u» = u . Due to u'g^ - ug^ = C u» - u Vu» + u ) 
 a transformation gives the following formula: 
 
 (20.9) 
 
 Betz' Formula 
 
 In order to determine W according to this equation, one has to measure 
 in the test cross section "behind the "body the following values: 
 
 1. Total pressure gj (therewith g is the value of gp outside 
 
 of the wake). 
 
 2. Static pressure Pp. 
 
 Furthermore, p = static pressure at infinity. 
 
 Hence one obtains all quantities required for the evaluation of 
 equation (20.9 ). 
 
 It is useful for the evaluation of wind tunnel tests to introduce 
 dimenslonless quantities. With F = ht as area of reference for the 
 drag: 
 
 V = c^ h t I u/ 
 
 and hence from equation (20.9) 
 
 (20.10) 
 
 For the case in which p = p =0, at the test cross section one can 
 
 2 o 
 
 write this equation, "because g^ = q^: 
 
72 
 
 NACA TM No. 12l8 
 
 1 - 
 
 ^^--JH-m 
 
 (20.11) 
 
 This agrees with equation (^.kl).* Thus in this case Betz' formula 
 changes, as was to he expected, into the previous simple formula. 
 
 h. The Method of Jones 
 
 Later B. M. Jones (reference 85) indicated a similar method which 
 in its derivation and final formula is somewhat simpler than Betz* method. 
 
 Let cross section II (fig. 101) (the test cross section) lie close 
 hehind the hody; there the static pressure p^ is still noticeably 
 
 different from the static pressure p^. Let cross section I he located 
 so far hehind the hody that the static pressure there equals the undis— 
 turhed static pressure. Then there applies for cross section I according 
 to equation (9.UI)* 
 
 W = h p 
 
 / ^1 (Uq - ^1) ^1 
 
 (20.12) 
 
 In order to relate the value of u-, hack to measurements at cross 
 section II, continuity for a stream filament is first applied: 
 
 P ^1 iyn = P u„ dy^ 
 
 (20.13) 
 
 * In chapter IX the total drag of the hody (both sides of the 
 plates) was designated by 2 Wj here the entire drag equals W! 
 
NACA TM No. 12l8 
 
 73 
 
 Jones makes the further assumption that the flow from cross section II 
 to cross section I is without loss, that is, that the total pressure is 
 constant along each stream line from II to I: 
 
 So = Sn (20. Ill) 
 
 First, according to equations (20.12) and (20.13); 
 
 W = hp /u2(u^-ujdy. 
 
 (20.15) 
 
 Furthermore : 
 
 P 2 
 p+— U =g=q 
 ■^o 2 o °o ,^o 
 
 Po + - ^■ 
 
 P 2 
 
 2\ =h = ^2 
 
 P p 
 p + — u '^ = go 
 2 2 2 
 
 with p =0 
 o 
 
 > (20.16) 
 
 and hence 
 
 ^2 \ I ^2 ~ P2 
 U 
 
 Un 
 
 U_ 
 
 (20.17) 
 
 n P 
 
 From equation (20.12) follows, with W = c t h ^ U^ : 
 
 w 
 
 -Sf^ 
 
 ^l^ . ^2 
 
 ^r~ 
 
 and hecause of equation (20.1?) 
 
 (20.18) 
 
74 NACA TM Wo. 1218 
 
 Formula of Jones 
 
 Thus all quantities may te measured in cross section II close to 
 the 'bod.j. This formula is simpler for the evaluation than Betz' formula, 
 equation (20.10). 
 
 In the limit, when the static pressure in the test cross section 
 "becomes P2 = P , this formula, of course, must also transform into the 
 simple formula equation (20.11). One obtains for po = p^ = from 
 equat i on ( 20 . I8 ) : 
 
 This is in agreement with equation (9.41). *• 
 
 Fifteenth Lecture (March I6, 19lf2) 
 CHAPTER ZXI: ORIGIN OF TURBULENCE 
 a. General Remarks 
 
 In this section a short summary of the theory of the origin of 
 turhulence will "be given. The experimental facts concerning laminar/ 
 tur"bulent transition for the pipe flow and for the "boundary layer on the 
 flat plate have "been discussed in chapter XIII. The position of the tran- 
 sition point is extremely important for the drag pro"blem, for instance 
 for the friction drag of a wing, since the friction drag depends to a 
 great extent on the position of the transition point. 
 
 The so— called critical Reynolds nurriber determines transition. For 
 the pipe (ud/v) . , = 23OO, and for the "boundary layer on the plate 
 
 (Uqx/v) = 3 to 5 X 105. However, experimental investigations show 
 
 the value of the critical Reynolds numher is very dependent on the 
 
 initial distur'bance. The value of Re . , is the higher the smaller the 
 
 crit ° 
 
 initial disturbance. For the pipe flow the magnitude of the initial 
 
 disturbance is given by the shape of the inlet, for the plate flow by 
 
 the degree of turbulence of the oncoming flow. For the pipe, for instance, 
 
 a critical Reynolds number (^<i/'^)crit ~ ^0,000 can be attained with very 
 
 special precautionary measures. 
 
 According to today's conception regarding the origin of turbulence, 
 transition is a stability phenomenon. The laminar flow in itself is a 
 solution of the Navier-Stokes differential equations up to arbitrarily 
 high Reynolds numbers. However, for large Ee-^iumbers the laailnar flow 
 
NACA TM Wo. 1218 75 
 
 ■becomes unetatle, in the sense that small chance disturhances (fluctu- 
 ations ,ln velocity) present in the flow increase with time and then alter 
 the entire character of the flow. This conception stems from Eeynolds 
 (reference 101). Accordingly, it ought to he posslhle to ohtain the 
 critical Eeynolds number from a stahility investigation of the laminar 
 flow. 
 
 Theoretical efforts to substantiate these assumptions of Eeynolds 
 mathematically reach rather far hack. Besides Eeynolds, Eayleigh 
 (reference 102) in particular worked on the problem. These theoretical 
 attempts did not meet with success for a long time, that is, no instability 
 could be established in the investigated laminar flows. Only very 
 recently has success been attained, for certain cases, in the theoretical 
 calculation of a critical Eeynolds number. 
 
 One assumtes for the theoretical investigations that upon the basic 
 flow which satisfies the ITavier— Stokes differential equations a disturbance 
 motion is superimposed. One then investigates whether the disturbance 
 movement vanishes again under the influence of friction or whether it 
 Increases with time and thus leads to ever growing deviations from the 
 basic flow. The following relations will be intorduced for the plane 
 case: 
 
 basic flows: U(x, y)^. Y(x, y); P(x, y) 
 
 disturbance movement: u'(x, y). v' (x, j) ; p'(x, y) 
 resultant movement: U + u* . Y + v' ; P + p' 
 
 > (21.1) 
 J 
 
 P, p* signify pressure. The investigation of the stability of such a 
 disturbed movement was carried out essentially according to two different 
 methods : 
 
 1. Calculation of the energy of the disturbance movement. 
 
 2. Calculation of the development of the disturbance movement with 
 
 time according to the method of small oscillations. 
 
 I am going to say only very little about the first method since it 
 was rather unsuccessful. The second method was considerably more 
 successful and will therefore be treated in more detail later. 
 
 The first method was elaborated mainly by H. A. Lorenz 
 (reference IO3). Thp following Integral expression may be derived for 
 the energy balance- of the disturbance movement: 
 
 ^ / EdV = p / MdY - 11 / NdY (21.2) 
 
16 
 
 WACA TM Wo. 1218 
 
 In it E = -^ (u' + t' ] signifies the kinetic energy of the distur'baiLce 
 
 movement. The Integration is performed over a space which participates 
 in the movement of the basic flow and at the "boundaries of which the 
 velocity equals zero, jrr signifies the substantial derivative. Thus 
 
 one finds on the left side of equation (21.2) the increase with time of 
 the energy of the disturbance movement. On the right slde^ 
 
 M = - 
 
 u' 
 
 ,2 SU 
 
 ax 
 
 ,2 av 
 
 v' + 
 
 Sy 
 
 u'v« + 
 
 VSy 
 
 > (21.3) 
 
 N = 
 
 vSx 3y / 
 
 y 
 
 The first integral signifies the energy transfer from the main to the 
 secondary movement j the second the dissipation of the energy of the 
 secondary movement. If the right side is greater than zero, the intensity 
 of the secondary movement increases with time, and the basic flow is thus . 
 unstable. An assumed disturbance movement u', v* satisfies merely the 
 continuity equation, but no heed is paid to its compatibility with the 
 equations of motion. If one could prove that the right side is negative 
 for any arbitrao^y disturbance movement u*, v* this would serve as proof 
 of the stability of the basic flow. On the other hand, the instability 
 would be proved as soon as the right side is positive for a possible 
 disturbance. Unfortunately general investigations in this direction 
 are very difficult and have not led to much success. H. A. Lorenz 
 (reference IO3) treated as an example the Couette— flow (fig. 102), assuming 
 an elliptical vortex as a superimposed disturbance movement. He found for 
 /U d\ 
 
 this case V — tt— / = 288, whereas Couette' s measurements for this case 
 ^ V /fcrit ' 
 
 gave the value I9OO. 
 
 b. The Method of Small Oscillations 
 
 For the second method (method of small oscillations) the disturbance 
 movement is actually calculated, that is, its dependance on the spatial 
 coordinates x, y and the time t is developed on the basis of the 
 hydrodynamic equations of motion. We shall explain this method of small 
 oscillations in the case of a plane flow. In view of the applications 
 of this method we shall immediately assume a special basic flow: the 
 component U, namely, is to be dependent only on y and t and Y = 0. 
 Such basic flows had been previously called "layer flows". They exist 
 for instance in tunnel flow and pipe flow, approximately, however, also 
 in the boundary layer since here the dependence of the velocity component 
 XJ on the longitudinal coordinate x is very much smaller than the 
 dependence on the transverse coordinate y. One now assumes a basic flow 
 
NACA TM Wo. 1218 
 
 U(y, t); Y= Oj P(x, y) 
 
 77 
 
 (21A) 
 
 This basic, flow, ty itself, then satisfies the Navier-Stokes equations, 
 thus 
 
 ^ 
 
 Su 1 5p S u 
 — + = V — 
 
 Bt P Sx ^^2 
 
 Sy 
 
 = 
 
 (21.5) 
 
 / 
 
 A disturbance movement which is also two— dimensional is superimposed 
 upon this "basic flow:. 
 
 disturbance motion: u»(x,y,t); T»(x,y,t); p*(x,y,t) (21.6) 
 
 One then has as the 
 
 resultant motion: u=U+u'; t=0+t*; p=P+p' (21.7) 
 
 This resultant motion is required to satisfy the Navier— Stokes differential 
 equations and one investigates whether the disturbance motion dies away 
 or Increases with time. The selection of the initial values of the dis- 
 turbance motion is rather arbitrary, but it must of course satisfy the 
 continuity equation. The superimposed disturbances are assumed as "small", 
 in the sense that all quadratic terms of the disturbance components are 
 neglected relative to the linear terms. According to whether the dis- 
 turbance motion fades away or increases with time, the basic flow is called 
 stable or unstable. 
 
 By insertion in the Navier— Stokes differential equations (3.I8) one 
 obtains, neglecting the quadratic terms in the disturbance velocities 
 
 5U 5u» „ Su» 
 
 — + + U + V 
 
 St at Sx 
 
 . Su 1 Sp 1 Sp' 
 
 Sy P ax P Sx 
 
 
 + A u' 
 
 — + u — -— i ^* 
 
 at ax p ay p ay 
 
 ^ « V A V' 
 
 >(2l.8) 
 
 au» av« 
 ax ay 
 
 / 
 
78 
 
 NACA TM No. 12l8 
 
 If one now notes the fact that the 'baBic flow by itself satisfies the 
 Navier— Stokes differential eguationsj equation (21.5) j equation (21.8) 
 is simplified to: 
 
 Su' TT ^' , SU 1 Sp' ^ , 
 + U + t' — + i— = V A u* 
 
 Bt Sdc Sy P Sx 
 
 St 
 
 ^ 
 
 P ay 
 
 SU« , SV« _ Q 
 
 -^ 
 
 &vL+U^t1^1Se1^VAt» > 
 
 (21.9) 
 
 The pertinent "boundary conditions are: Tanishing of the disturbance 
 components u* and t' on the bounding walls. From the system 
 equation (21. 9) of three equations with three unknown quantities u*, 
 t', p» one may at first eliminate p' by differentiating the first 
 equation with respect to y and the second with respect to x and then 
 subtracting the second from- the first. This gives, with continuity taken 
 into consideration: 
 
 S^u' ^ jj B^u' + ^, ^ _ 5^' _ jj SV 
 
 BtBy 
 
 + V 
 
 SxSy 
 
 a^u 
 
 ,^ ^y 
 
 ay^ 
 
 StSx 
 
 » S^u' S^t' 
 
 Sy SxSy 
 
 ax-^ 
 
 Sx- 
 
 > 
 
 > (21.10) 
 
 / 
 
 In addition to this there is the continuity equation (21.9). There are 
 now two equations with two unknown quantities u*, v*. 
 
 Form of the Disturbance Movement 
 
 For cases where the basic flow predominantly flows in one direction 
 as for instance boundary— layer or pipe flow, the disturbance motion is 
 assumed to be a wave progressing in the x-direction (= main flow direction), 
 the amplitude of which depends solely on y. The continuity equation of 
 the disturbance motion may in general be integrated by a disturbance 
 function for which the following expression may be used: 
 
NACA TM No. 12l8 
 
 19 
 
 * (x^y^t) = cp(y)e 
 
 ia(x-ct) 
 
 (21.11)* 
 
 Where: 
 
 X = 2it/a the waTe length of the disturbance (a = real) 
 
 C = C + 1 c. 
 r 1 
 
 c = Telocity of wave propagation 
 
 c^ = amplification factor; c . < 0: stable; c. > 0: unstable 
 
 9(y) = 9 (y) + i 9.(y) = amplitude of the disturbance movement 
 r 1 
 
 From equation (21.11) one obtains for the components of the distirrbance 
 movement 
 
 li' = T^ = 9'(y)e 
 oy 
 
 St , , , ia(x^t) 
 T» =-—- = - ioo:p(y)e 
 dx 
 
 > (21.13) 
 
 / 
 
 By substitution into equation (21.10) one obtains the following differ- 
 ential equation for the disturbance amplitude cp: 
 
 / i 
 
 iafUcp" — ccp" - cpU" + a ccp - Ua cp 
 
 j = v^cp"" - 2a cp" + a 9 j 
 
 or 
 
 (U - c)(9" - a29) - u"q) = =^-i^ (cp"" - 2aV + o-V) (21. U) 
 
 Ow 
 
 ■One introduces dlmensionless quantities into this equation by referring 
 
 all velocities to the maximum velocity U of the basic flow (that is 
 
 m 
 
 for the friction layer the potential flow outside of the boundary layer 
 
 and all lengths to a suitable reference length §) (for Instance, for 
 
 ^he convenient complex formulation is used here. The real part of 
 the flow function, which alone has physical significance, is therefore 
 
 Re(t) 
 
 C-it 
 
 cp COS 
 
 r 
 
 a 
 
 (- - V) 
 
 cp. sin 
 1 
 
 a ( X — 
 
 c t 
 r , 
 
 (21.12) 
 
80 
 
 NACA TM Wo. 1218 
 
 the "boundary layer flow, the "bo-undary layer thickness). Furthermore, 
 differentiation with respect to the dimensionless quantity y/5 will 
 "be designated ty a prime mark ('). 
 
 One then obtains from equation (21.1i|) 
 
 (U - c)(cp" - a^qp) - U"cp = -^ (q)"" - 2aV + a\) 
 
 (21.15) 
 
 DISTURBADJCE Dlb'Jb'KHEETIAL EQUATION 
 
 where E = ■■ ™ . This is the disturbance differential equation for the 
 amplitude cp of the disturbance movement. The boundary conditions are, 
 for instance, for a boiuidary layer flow 
 
 y = (wall): u' = v' = 0: 9 = cp' = 
 y = 00 : u' = "v* = 0: cp = qp' = 
 
 (21.16) 
 
 The stability investigation is an eigenvalue problem of this differential 
 equation for the disturbance amplitude <p(y) in the following sense: A 
 basic flow U(y) is prescribed which satisfies the Navier-Stokes differ- 
 ential equations. Also prescribed is the Eeynolda number R of the basic 
 flow and the reciprocal wave length a = 2it/x, of the disturbance movement. 
 From the differential equation (21. I5) with the boundary conditions 
 equation (21. I6) the eigenvalue c = Cp + i Cj_ is to be determined. The 
 
 sign of the imaginary part of this characteristic value determines the 
 stability of the basic flow. For Cj^ < the particular flow (U, R) 
 
 is, for the particular disturbance a, stable; for Cj^ > 0_, unstable. 
 The case Cj^ = gives the neutrally stable disturbances. One can 
 represent the result of the stability calculation for an assumed basic 
 flow U(y) in an a, R— plane in such a manner that a pair of values 
 Cj., c± belongs to each point of the a, R— plane. In particular the 
 
 : = in the a, R— plane separates the stable from the unstable 
 
 curve 
 
 disturbances. It is called the neutral stability curve (fig. IO3). In 
 view of the test results one expects only stable disturbances to be present 
 at small Reynolds numbers for all wave lengths a, unstable disturbances, 
 however, for at least a few a at large Reynolds numbers. The tangent 
 to the neutral stability curve parallel to the a-^axis gives the critical 
 Reynolds number of the respective basic flow (fig. IO3). 
 
WACA TM No. 1218 81 
 
 Methods of Solution and Gteneral Properties of the Dlstiirban ce 
 Differential Equation 
 
 Since the stability limit (Cj^ = 0) is expected to occur at large 
 Reynolds numbers^ it suggests itself to suppress the friction terms in 
 the general disturhance differential equation and to ohtain approximate 
 solutions from the so— called frlctionless disturbance differential 
 equation which reads 
 
 (U - c) (cp" - a^q)) -U"q) 
 
 (21.17) 
 
 Only two of the four "boundary conditions^ equation (21.l6),of the complete 
 disturbance differential equation can now be satisfied since the friction- 
 less disturbance differential equation is of the second order. The 
 remaining boundary conditions are: 
 
 y = 0: v« = Oj 9 = Oj y = 00: t» = 0: cp = (21. 18) 
 
 The cancellation of the friction terms in the disturbance differential 
 equation is very serious, because the order of the differential equation 
 is thereby lowered from ij- to 2 and thus Important properties of the 
 general solution of the disturbance differential equation of the fourth 
 order possibly are lost. (Compare the previous considerations in 
 chapter IV concerning the transition from the Navler— Stokes differential 
 equations to potential flow. ) 
 
 An Important special solution of equation (21.17) is the one for 
 a constant basic flow, U = constant^ which is needed for Instance for 
 the stability investigation of a boundary layer flow as a Joining 
 solution for an outer potential flow. One obtains from equation (21.17) 
 for 
 
 U = constant: cp = e 
 
 iay 
 
 However, due to the boundary conditions for cp at y = ", the only 
 permissible solution Is 
 
 cp = e^ y (21.19) 
 
 We shall prove at first two general theorems of Eaylelgh on the 
 neutral and unstable oscillations of the frlctionless disturbance 
 differential equation. 
 
82 WACA TM No. 12l8 
 
 Theorem I: The wave Telocity c for a velocity profile with 
 U"(y) — must, for a neutral oscillation (c^ =0, c = c ), egual 
 the hasic velocity at a point so that there exists within the flow a 
 point U — c = 0. 
 
 Proof: (indirect) One makes the assumption c > U (= maximum, 
 velocity of the hasic flow). One then forms from equation (21.17) the 
 following differential expressions: 
 
 2 TT" 
 
 L(cp) = cp" - a cp qp = (21.20a) 
 
 U — c 
 
 and 
 
 2- U" — 
 
 L(cp) = cp" - a cp ^ cp = (21.201)) 
 
 U - c 
 
 L(q)) signifies the expression o"btained from 1(9) , if one inserts 
 everywhere the conjugate complex q^uantities. Because of the "boundary 
 conditions 
 
 y = 0: cp = cp = 
 y=oo. q3 = q) = o 
 
 One forms further the expression 9L(cp) + cp L(cp) and integrates 
 it "between the limits y = and y = co. The integrals may "be taien up to 
 y = 00, since for large y, 9 »« e~^y. Because of equation (21.20aj b) 
 J2_ must then "be 
 
 ■^1 = 
 
 cp L(q)) + cp L(cp) 
 
 dy = (21.21) 
 
 ^y=o 
 
 After insertion of equation (21.20a and "b) results, "because c = c, 
 
 J = j ( cp" cp" + cp"(p - 2a^cp9 - 2 — - — cp9 j dy = 
 'y=o 
 
 or 
 
WACA TM Wo. 1218 
 
 83 
 
 noo 
 
 J = 
 
 1 
 
 qxp' + cp'9 
 
 - 2 
 
 -"o 
 
 llo 
 
 cp'cp»dy - 2 I [ a^ + — ^^ ] cp9 dy = 
 
 U - c 
 
 The first term vanishes due to the houndary conditions, hence there 
 remains 
 
 Jl =-2 
 
 Jy=o 
 
 cp»cp' +( a^ + — 2 — jqxp 
 U - c / 
 
 dy = 
 
 (21.22) 
 
 U" - 
 
 cp'cp' as well as qxp are positive throughout; if U" — and c > U , 
 U"/U' — c — and hence the integrand in eq^uation (21.22) is positive 
 throughout. Thus the integral cannot "become zero. The assumption made 
 at the "beginning c > U therefore leads to a contradiction. 
 
 For "basic flows with "D" — 0, as for instance "boundary— layer flows 
 in a pressure drop, the wave propagation velocity therefore must "be smaller 
 than U^ for neutral disturbances. Hence a point IT — c = exists 
 within the flow. This point is a singular point of the frictionless 
 disturbance differential equation (21.17) and plays as such a special role 
 for the investigation of this differential equation. The wall distance y 
 at which U — c = is called j = Y-y- = critical layer. 
 
 This first Rayleigh theorem proved a'bove applies — as shall "be noted 
 here without proof — in the same manner to flows with tT" > 0. 
 
 Sixteenth Lecture (March 23, 19^2) 
 
 Theorem II: A necessary condition for the presence of amplified 
 oscillations (c^ > 0) is the presence of an inflection point within the 
 
 "basic flow (U" = 0). 
 
 Proof: (indirect) According to assumption, c. ^ 0; thus 
 U - c / for all y. With L(cp) and L(cp) one forms, according to 
 equation (21.20a) and (21.20h), a simdlar expression as "before. This 
 latter, integrated from y = to y = °°, must again give 0, thus 
 
 noo 
 
 J2 = 
 
 [ 
 
 9 L(q)) - cp L(cp) dy = 
 
 ]■ 
 
 y=0 
 
 (21.23) 
 
8U 
 
 NACA TM No. 12l8 
 
 By substitution according to equation (21.20a and b) results with 
 
 c = c — 1 c. 
 r 1 
 
 '2 = 
 
 '7=0 
 
 cpcp" - cptp" - ^(—^ 21. 
 
 VU - c U - c 
 
 dy = 
 
 or 
 
 '2 = 
 
 W - qxp'' 
 
 - 2i c. 
 
 JQ 
 
 u" 
 
 cpcp dy = 
 
 U - c 
 
 {2±.2k) 
 
 The first term again vanishes because of the boundary conditions. Since cp^ 
 is positive thrcaghout and lu — c| / 0, the integral can only vanish if 
 U" changes its sign, that is, an inflection point of the velocity profile 
 U" = must be present within the flow. It has, therefore, been proved: 
 In order to make the presence of amplified oscillations possible, an ■ 
 inflection point mast exist in the velocity profile of the basic flow, or, 
 expressed briefly, such oscillations are possible only for inflection 
 point profiles. 
 
 Later on Toll mi en (reference 110) proved that the presence of an 
 inflection point is not only a necessary but also a sufficient condition 
 for the existence of amplified oscillations. Hence the following simple 
 statement is valid: Inflection point profiles are unstable. It must be 
 mentioned that all these considerations apply in the limiting case B— ^<» 
 since the proofs were obtained from the frictionless disturbance differ- 
 ential eq_uation. 
 
 ¥e know from our previous considerations about the laminar boundary 
 layer that inflection point profiles always exist in the region of pressure 
 rise, whereas in the pressure drop region the boundary layer profiles are 
 always without an inflection point, (fig. lOij-). Hence we recognize that 
 the pressure rise or pressure drop is of decisive significance for the 
 stability of a boundary layer flow. 
 
 The converse of the theorem just set up is also valid, namely, that 
 for E — >oo velocity profiles without inflection point are always stable. 
 From this, however, "one must not conclude that profiles without inflection 
 point are stable for all Reynolds numbers. A closer investigation for 
 Reynolds numbers of finite magnitude shows that there profiles without an 
 inflection point also become unstable. One is faced with the peculiar 
 fact that the transition from Re = » to Ee-number of finite magnitude, 
 that is, the addition of a small viscosity to a frictionless flow, has a 
 
NACA TM No. 12l8 85 
 
 destabilizing effect, whereas one intuitively expects the opposite. As 
 later considerations will show in more detail, the typical difference 
 between the neutral stability curves of a basic flow with and without 
 inflection point appears as represented in figure IO5. For the velocity 
 profile without an inflection point the lower and the upper branch of the 
 neutral curve have , for E — > 00 , the same asymptote a = . For the 
 velocity profile with inflection point the lower and upper branch of the 
 neutral curve have, for E — >», different asymptotes so that for E = 00 
 a certain wave length region of unstable disturbances exists. Furthermore 
 the critical Eeynolds number is smaller for velocity profiles with an 
 inflection point than for those without an inflection point. 
 
 Hence it is to be expected for very large Ee-iiumber, to a first, 
 very rough approximation, that the transition point in the boundary layer 
 of a body lies at the pressure minimum. Figure IO6 shows schematically 
 the pressure distribution for a rather strongly cambered wing profile at 
 a small lift coefficient. The transition point would be expected in this 
 case Just behind the nose on the pressure side, slightly more toward the 
 rear on the suction side. 
 
 Solution of the Disturbance Differential Equation 
 
 In order to perform the actual calculation for the boundary— value 
 problem Just formulated, one needs at first a fundamental system 
 Cp-,, cp. °^ '^^^ general disturbance differential equation (21.15). 
 
 One Imagines the basic flow U(y) given in the form of a power series 
 development : 
 
 U " 
 U(y) = U' y + -^ y2 + . . . . (21.24) 
 
 ° 2! 
 
 If one introduces this expression into equation (21.15) and. then wants 
 to construct a solution from the complete differential equation which 
 satisfies the boundary conditions (equation (21. 16)), one encounters 
 extreme difficulties of calculation, due to the two conditions to be 
 satisfied for y = 00, In order to obtain any solution at all, one has 
 to make various simplifications. The simplificationB concern: 
 
 1. The basic flow: Instead of the general Taylor— series 
 equation (21.2^4-) one takes only a few terms, thus for instance a linear 
 or a quadratic velocity distribution. 
 
 2. The disturbance differential equation: For calculation of the 
 particular solutions the disturbance differential equation Is considerably 
 simplified. 
 
 Eegarding 1, it should be noted that linear velocity distributions 
 frequently have been investigated with reepect to stability, as for 
 
86 NACA TM No. 12l8 
 
 instance the Couette flow according to figiore 102 or a polygonal approxi- 
 mation for cuTTed Telocity profiles according to figure 107. This facili- 
 tates the calculation due to the fact that then the singular point 
 U — c = is avoided in the frictionless dist'ur'bance differential 
 equation (21.17) for neutral disturhances. However, all InvestigationB 
 with linear velocity diBtritutions (references 104, 105, 106) were unsuc- 
 cessful with the frictionless as well as with the complete differential 
 equation. No critical Reynolds number resulted. When one later took 
 for a "basis paraholic profiles, these negative results "became intelligl"ble. 
 One must, therefore, take at least a parahollc distrl'bution as a "basis 
 for the "basic flow. 
 
 Regarding 2, it should "be noted that one can provide approximate 
 solutions for the solutions of the complete differential equation (21.15) 
 from the frictionless differential equation (21.17) since the solutions 
 are required only for large Ee-^iumber E. The frictionless differential 
 equation however can yield no more than two particular solutionsj two 
 more have to "be calculated, taking the largest friction terms in 
 equation (21.15) into consideration. 
 
 The course of the calculation for the particular solutions will "be 
 "briefly Indicated. One limits oneself to neutral dlstjir'bances, assumes 
 a para'bolic velocity distrl'bution^ and imagines the latter developed in 
 the neighborhood of the critical layer. 
 
 y = yg.: U-c=U-c^ = 
 
 U - c = U'^(y - y^) . !^ (y - j^f (21.25) 
 
 The first pair of solutions 9-, j "Pp ^^ then o"btained from the friction- 
 less dlstur"bance differential equation (21.17) "by suhstltution of 
 equation (21.25). According to known theorems a"bout linear differential 
 equations with a singular point a linearly independent pair of solutions 
 has the form 
 
 ^i=(^-^k)^(^-^e) 
 
 u" 
 
 (21.26) 
 
 P, and P are power series with a constant term different from zero. 
 The parttc-ular solution cp is especially interesting. 
 
NACA TM No. 12l8 
 
 87 
 
 9', 
 
 -> °° for 
 
 J = It, 
 
 That Isj the u'— component of the dleturhance velocity becomes infinitely 
 large in the critical layer. This can also he understood directly from 
 the frictionlesB disturhance differential eq^uation (21.17). According 
 to equation (21.17) 
 
 cp" — a cp = 
 
 U" 
 
 U - c 
 
 9 
 
 or 
 
 >p" 
 
 j-^; ^' ~ ic8 (y - y^). If u"^ ^ 
 
 This singular hehavior of the solution cp2 ^^ "the critical layer stems 
 
 of course from neglecting the friction. The frictionless differential 
 eq^uation here no longer gives a serviceable approximation. In the 
 neighborhood of the critical layer the friction must be taken into 
 consideration. Moreover, there is another inconvenience connected with 
 the cpp' For fulfillment of the boundary conditions one req^uires the 
 
 solution for y — y-g- > as well as for y — y„ < 0. However, for qpo 
 
 it is at first undetermined what branch of the logarithm, should be chosen 
 at transition from y — yg; ^ to y — y;g; "^ 0. This also can be clarified 
 
 only if in the neighborhood of y — y^-, at least, the large friction terms 
 
 of the complete differential equation (21.15) are taken into consideration. 
 The details of the calculation will not be discussed here. The calcu- 
 lation leads, as Tollmien (reference 109) tias shown, to the result that 
 one obtains for the solution cp2 a so— called transition— substitution in 
 
 the critical layer which appears as follows: 
 
 U" 
 
 7-J^>0: cpg = P2(y -J^) + —(7- ^k) ^l(y - ^k) ^°e (7 - Y^ 
 
 u" 
 
 y - Jk < 0= ^2 = P2(y - ^k) ^ ^ (y - ^k) ^i(y - ^k) i°8 " 
 
 y - y- 
 
 K 
 
 - in 
 
 (21.27) ' 
 
 If one writes, according to this, the complete u'— component, then in the 
 neighborhood of 7 ~ J-^'- 
 
KACA TM No. 12l8 
 
 J - J^> 0: u' = 
 
 < 0: u« 
 
 U". 
 
 K 
 
 + — 1'°S (j - J-^) COB (ox - pt) 
 
 .5^ .OS 
 
 u" 
 
 K 
 
 cos (ax - pt) S (21.28) 
 
 + It sin (ax — pt) 
 
 y 
 
 One ottalns therefore in the critical layer a phase discontinuity for the 
 u* -component. This is retained eren in going to the limit, E — > 00 . it 
 is lost, however, if one neglects the curvature of the basic flow U" 
 or if one operates only with the frictionless differential equation. This 
 phase discontinuity is very significant for the development of the motion. 
 The loss of the phase discontinuity is the reason that stahility investi- 
 gations neglecting the curvature U" or operating only with the frictionless 
 differential equation remain unsuccessful. 
 
 With this friction correction in the critical layer the pair of 
 
 solutions cp 
 
 9. 
 
 is sufficiently determined. By taking the friction 
 
 '1' -^2 
 
 terms in equation (21.15) into consideration, one then obtains a second 
 pair of solutions 9 , cp, which can he represented hy Hanlcel and Bessel 
 
 functions. Of these -two solutions cp, tends very strongly towards 
 
 infinity and is therefore not used "because of the "boundary conditions, 
 equation (21. 16). cp tends, for large y, towards zero. 
 
 The Boundary Yalue Pro"blem 
 
 The general solution as a linear combination of the four particular 
 solutions is: 
 
 cp = C^cp^ + 
 
 CgiPg + C^cp^ + C^cpj^ 
 
 (21.29) 
 
 Let us consider in particular the case where a "boundary layer profile is 
 investigated with r-espect to sta"bility. For this case the "boundary value 
 pro"blem can "be somewhat simplified. The previous considerations showed 
 that in the disturbance differential equation the friction essentially 
 needs to "be taken into consideration only in the neigh'borhood of the 
 critical layer; also, of course, at the wall, because of non— slip. The 
 critical layer is always rather close to the wall; hence for y > 6, 
 
NACA TM Wo. 1218 
 
 89 
 
 where U = U^j^ = constant^ one may use the frictionlees solution which is 
 
 according to equation (21. I9) 9= e~°''^. Thus the condition that the 
 solution for y = 5 Joins the solution for U = constant is 
 
 9' + a cp = $ = 
 o 00 
 
 (21.30) 
 
 This mixed "boundary condition is therefore to he set up on the outer edge. 
 Furthermore, the particular solution cpi is a priori eliminated in the 
 
 general solution (equation (21.29)), since it grows, for positive y — jy-, 
 
 "beyond all limits j thus C. = 0. Hence there remains for the houndary 
 
 •value according to equation (21.16) 
 
 Ccp +Cq) +Cq) =0 
 1 lo 2 2o 3 30 
 
 C cp» + C cp' + C cp» =0 
 1 Id 2 2o 3 30 
 
 C$ +C<i) +C$ =0 
 ri5 2 26 3^35 
 
 ~\ 
 
 > 
 
 (21.31) 
 
 y 
 
 A further simplification takes place "because of the fact that "because of 
 the rapid fading away of the solution cp on the outer edge y = 8, the 
 
 solution $_ already practically equals zero. In the third equation of 
 
 15 
 equation (21. 3I) 
 
 35 
 
 may therefore he cancelled. The boundary value 
 
 problem actually to he solved is, therefore. 
 
 qp cp cp 
 ^lo ^2o 30 
 
 cp* cp' cp* 
 ^ lo ^ 2o ^ 2o 
 
 15 
 
 $^0 
 25 
 
 = 
 
 (21.32) 
 
 This determinant gives the eigenvalue problem indicated above, which 
 requires - as has been said before - the solution of the following problem: 
 Given 
 
90 NACA TM No. 12l8 
 
 1. "basic flow U(y) 
 
 2. Eeynolds number Ee = U S/v 
 
 3. wave length of the disturhance a = 23t/a. 
 
 One seeks from equation (21.32) the pertinent complex eigenvalue 
 + 1 c . . Therein c gives the velocity of wave propagation and 
 
 c. the anrplification or damping. 
 
 0=0 
 
 r 
 
 Equation (21.32) may formally "be written in the foirTa: 
 
 F(a, c^, c., E; TJ'^, U"^, ....)= (21.33) 
 
 where equation (21.33) signifies a complex equation^ hence is equivalent 
 to two real equations 
 
 f fa, c , c., E; U' , U" , . . . = 
 IV r 1 o o ' 
 
 fJa, , c, E; U» , U" , . . . ) 
 2\~' r' 3.' ^ o-* o-* / 
 
 > (21.3^) 
 
 
 
 If one imagines for Instance Cp eliminated from these two equations, 
 one obtains one equation "between a, E, c. : 
 
 gJa,, c^, E; U»Q, U"^, . . J) = (21.35) 
 
 From this equation Cj_ can be calculated as a function of a and E. 
 The constants U*o, U"o, • • • are parameters of the basic flow. Thus, 
 if equation (21.35) is assumed solved with respect to c.. 
 
 i = gg^a, E; U'q, U"o, . . .j (21. 36) 
 
 Cj = 
 
 Finally one obtains from it, for the neutral disturbances c. = 0, a 
 curve in the a, E— plane, given by the equation 
 
 gg/a, E; U»^, ir^, . . .) (21.37) 
 
WACA TM No. 1218 91 
 
 This is the sought for neutral stahllity curve (compare figure IO3), which 
 separates the unstable from the statle disturhances and also yields 
 the theoretical stahility limit, that is, the critical Eeynolds number 
 
 ^^crif 
 
 The performance of the calculation, here only indicated, is analyt- 
 ically not possible since the quantities a, c , E enter into the 
 
 determinant, equation (21. 32), in a very complicated manner. One has 
 therefore to resort to numerical and graphical methods. The critical 
 Eeynolds number is Tery largely dependent on the form of the velocity 
 profile of the basic flow, in particular on whether the velocity profile 
 of the basic flow has an inflection point, thus on U"(y). 
 
 The critical Eeynolds number found from such a calculation gives 
 exactly the boundary between stability and instability, hence the first 
 occurrence of a neutrally stable disturbance. In comparison with the 
 transition point of test results it is therefore to be expected that the 
 experimental transition point appears only for larger Eeynolds numbers 
 where an amplification of the unstable disturbance has already occurred. 
 
 c. Eesults 
 
 A few results of such stability calculations will be given. The 
 completely calculated example concerns the boundary layer on the flat 
 plate in longitudinal flow with the laminar velocity profile according 
 to Blasius (compare chapter IXa). In figure IO8 the streamline pattern 
 of this plate boundary layer with the superimposed disturbance movement 
 is given for a special neutral distvirbance. Figure IO9 shows, for the 
 same neutral disturbance, the amplitude distribution and the energy 
 balance. Since the distizrbance in question is neutral, the energy 
 transfer from the main to the secondary movement is of exactly the same 
 magnitude as the dissipation of the energy of the secondary movement. 
 Figure 1^0 shows the neutral— stability curve as result of the stability 
 calculation according to which the critical Eeynolds number is referred 
 to the displacement thickness b* of the boundary layer 
 (Uj^S'**'/v )„pj+ = 575- The connection between displacement thickness S* 
 
 and length of run x is for the laminar boundary layer according to 
 equation (9. 21) 
 
 V* 
 
 = 1.73 
 
 Thus a critical Eeynolds number formed with the length of run 
 (U x/v) .^ = 1.1 X lo5 corresponds to the critical Ee-number 
 
 (U 5*/v) = 575. The critical number for this case observed in tests 
 
 ^ ' crit 
 
92 NACA TM No. 12l8 
 
 was 3 to 5x10-^. It was explained atove tJaat it must be larger than 
 the theoretical number. Furthermore^ figure 110 shows that at the 
 stability limit the unstable wave lengths are of the order of magnitude 
 X = 55- The unstable disturbances thus have rather long wave lengths. 
 
 This calculation, carried out by Tollmlen (reference IO9) for the 
 flow without pressure gradient was later applied by Schlichtlng 
 (reference 11^4-, 115) "to boundary layer flows with pressure drop and 
 pressure rise. The boundary layer profiles with pressure rise and 
 pressure drop can be represented in a manner appropriate for the stability 
 calculation as a one— parameter family with the form parameter X-p i^ 
 
 according to Pohlhaueen's approximate calculation. One then obtains for 
 each profile of this family a neutral— stability curve as Indicated in 
 figure 111. Hence the critical Eeynolds number (^r^* / '^) crit ^^ ^ 
 function of the form parameter X-p h according to figure 112. In 
 retarded flow (X-p r <0) the critical Ee-number is smaller than for the 
 plate flow (^pi^ = 0) , for accelerated flow {'>^-pu ^ 0) it is larger. 
 
 With this result of a universal stability calculation the position of the 
 theoretical transition point may be determined conveniently for an 
 arbitrary body shape (plane problem) in the following manner: At first, 
 one has to calculate for this body the potential flow along the contour, 
 ftirthermore one has to carry out, with this potential flow, a boundary 
 layer calculation according to the Pohlhausen method. This calculation 
 yields the displacement thickness and the form parameter Xp^, as 
 
 functions of the arc length along the contour, in the form 
 
 - f^(s) and Xp^ = f2(B) 
 
 Since in general there exists, accelerated flow at the front of the body 
 and retarded flow at the rear X-nj. decreases from the front toward the 
 
 rear. By means of the universal stability calculation according to 
 figure 112 one may determine a critical Reynolds number i^-n^* / "^^ crit 
 for each point of the contour. The position of the transition point 
 for a prescribed Ee-^umber U^t/v is then given by the condition 
 
 ^ ^crlf V V V /crit 
 
 Figure 113 shows, for the example of an elliptic cylinder*, how to find 
 the transition point. The curve (Uj^^-^/v)^^^^ decreases from the front 
 
 *The boundary layer calculation for this elliptic cylinder was given 
 in figure 52. , 
 
NACA TM No. 12l8 93 
 
 toward the rear; the curve Ujj^S^/v for a fixed Ee-^aumber U t/v increases 
 
 from the front toward the rear. The intersection of the two curves gives 
 the theoretical transition point for the respective Ee— number U-t/v. 
 
 By determining this point of intersection. for various U-t/v one ohtains 
 
 the transition point as a function of U t/v. The result is represented 
 
 in figure 114. The transition point travels with increasing Ee-durober 
 from the rear toward the front; however, the travel is considerahly 
 smaller than for the plate in longitudinal flow which is represented in 
 figure llij- for comparison. Finally figure II5 shows the result of such a 
 stability calculation for four different elliptic cylinders in flow 
 parallel to the major axis. The shifting of the transition point with 
 the Ee— number increases with the slenderness of the cylinder. For the 
 circular cylinder the shifting is very slight, which is caused hy the 
 strongly marked velocity maximum. As a last result, figure II6 shows the 
 travel with Ee— number of the transition point on a wing profile for various 
 lift coefficients. The profile in question is a symmetrical Joukoweky 
 profile with lift coefficients Cg^ = to 1. With increasing angle of 
 
 attack the transition point travels, for fixed Ee-nijmber toward the front 
 on the suction side, toward the rear on the pressure side. (compare the 
 velocity distirbutions for this profile, given in figure 5^.) One recog- 
 nizes that the shift of the transition point with the lift coefficient 
 is essentially determined by the shift of the velocity maximum. 
 
 The last examples have shown that it is possible to calculate 
 beforehand the position of the transition point as a function of the 
 Ee— number and the lift coefficient for the plane problem of an arbitrary 
 body immersed in a flow (particularly a wing). Eegarding the comparison 
 with test results it was determined that the experimental transition 
 point always lies somewhat further downstream than the theoretical tran- 
 sition point. The reason is that between the theoretical and the experi- 
 mental transition points lies the region of amplification of the unstable 
 disturbances. This amplification also can be calculated on principle 
 according to methods similar to those previously described. (Compare 
 Schllchting (reference 112) where this was done for the special case of 
 the plate in longitudinal flow. ) Presumably one can obtain a still closer 
 connection between the theoretical instability point and the experimental 
 transition point by applying such calculations to the accelerated and 
 retarded flow. 
 
 CHAPTER XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION 
 LAYER ACCORDING TO THE METHOD OF GEUSCHWTTZ (REFERENCE 78) 
 a. Integration of the Differential Equation of the 
 Turbulent Boundary Layer 
 
 In order to integrate the system of equations (I8.I6), one 
 first introduces dimensionless variables. One refers the lengths to the 
 
9k 
 
 NACA TM No. 12l8 
 
 wing chord t and the velocity to the free stream velocity U^, thus: 
 
 t "" ' 
 
 t 
 
 S 
 
 1^: 
 
 = c* = 
 
 IP 
 
 (22.1) 
 
 Hence the system of equations (equation (I8.I6)) may "be written: 
 
 — + o.oo89i^ -^ = 0.001^61 f-^'f i- 
 
 dx* t3* \U J -d* 
 
 djc* \ 2/ U. dx* 
 
 o 
 
 (22.2) 
 
 First, the second equation is solved with constant values for ''"o/pu 
 
 and H, namely 
 
 ^ 
 
 = 0.002; H = 1.5 
 
 The first approxinjation '"3 *(x*) obtained from that is then substituted 
 
 in the first differential equation. From the latter one obtains a first 
 approximation ^-,*(x*) and from that, according to equation (22.1), 
 Ti-|(x*). With T], (x*) one determines according to figure 92 the course 
 
 of H(t)) and corrects t according to equation (18.I5). Then one 
 
 obtains from the second differential equation a second approximation 
 
 For the solution of the differential equations one uses the isocline 
 method which can be applied for the present case, according to Czuber, 
 in the following manner: Both differential equations have the form: 
 
 g + f (x)y = g(x) 
 
 (22.3) 
 
 As can be easily shown, this differential equation has the property that 
 all line elements on a straight line x = constant radiate from one point. 
 The coordinates of this point (= pole) are: 
 
NACA TM No. 12l8 95 
 
 i = X + -i-; X = 44 (22A) 
 
 f(x)' f(x) 
 
 Thus one has only to calculate a sufficient numiber of these poles and 
 can then easily draw the integral curve. 
 
 Figure 93 indicates the result of such a calculation for the profile 
 J 015; °a ~ ^' '^^^ calculation of the laminar boundary layer for the 
 
 same profile was performed in chapter XII, figure 49 ^ tahle 6. 
 
 Initial values: The transition point was placed somewhat arhitrarily 
 at the velocity maximum of the potential flow (x/t = 0.l4l). It was 
 assumed that: 
 
 Ee = -2_ = 10 
 V 
 
 For the laminar "boundary layer was found: 
 
 U t 
 ^ = 0.141: ^ 1 -^ = 1.56 (tahle 6) 
 
 Hence there results, with S*/i3 = 2.55; 
 
 - = O.lU: (1) = 0.611 X 10-3 (tahle 8) 
 
 The corresponding r] — value was ass'jmed to be 
 
 T] = 0.1 (table 8) 
 
 Calculation to the second approximation suffices. The result is compiled 
 in table 8 and figure 93. A turbulent separation point does not exist 
 since r\ remains below 0.8. From the variation of the shearing 
 stress T along the wing chord the drag coefficient of the surface 
 
 friction may be determined: 
 
 W=2b| T dx (x= measured along chord) 
 
96 
 
 -^kCk TM No. 1218 
 
 or c,^ = W/2 Id t § Uc 
 
 w 
 
 °w = 
 
 (22.5) 
 
 The e-valuation of the Integral gives 
 
 c^ = 0.0090 
 
 t. Coimection Between the Form Parameters ti and H = tS/S* 
 
 of the Tiirhiilent Boundary Layer 
 
 According to Pretsch (reference 80) one may also represent analyti- 
 cally the relation "between the form parameters t] = 1 — (u^/tr)^ and 
 H = 5*/t3 which was found empirically by G-ruschwitz, compare figure 92. 
 A power law is set up for the velocity distribution, of the form: 
 
 H -f-L^ _ n 
 
 U 1^8 
 
 ; 
 
 = z 
 
 (22.6) 
 
 with n = 1/6, 1/1, 1/8 . . 
 Hence results: 
 
 according to the experiments so far. 
 
 5* 
 6 
 
 ni 
 
 
 
 1 - 
 
 n\ n 
 z dz = 
 
 n + 1 
 
 (22.7) 
 
 dy/5=0 
 
 Furthermore ; 
 
 1 
 5 
 
 0^1= ^"(^-^") 
 
 s^-f)^l=l ^- 
 
 dz 
 
 (22.8) 
 
 n _ 7,2n 
 
 z^^ - z 
 
 i\z = _J^ L_ = S 
 
 y n+1 2n+l (n+ l)(2n + 1 
 
 z=0 
 
NACA TM No. 12l8 
 
 97 
 
 From equations (22.7) and (22.8) follows: 
 
 H = ii- = 2n + 1 
 
 or 
 
 n = 
 
 H - 1 
 
 (22.9) 
 
 From equation (22.8) follows further 
 
 i = H - 1 
 5 H(E + 1) 
 
 (22.10) 
 
 The GruBchwitz form parameter t) Is defined according to equation (l8.2) 
 hy: 
 
 'u 
 
 Tl = 1 - 
 
 ■■3\2 
 
 U, 
 
 With equation (22.6) t] becomes: 
 
 1 - 
 
 iT 
 
 (22.11) 
 
 Substitution of equation (22.10) into (22.11) gives; 
 
 (22.12) 
 
 The connection "between H and t\ calculated according to this equation 
 is given in the following table and is also plotted In figure 92. The 
 curve calculated according to equation (22.12) almost coincides with the 
 curve found empirically by Gruschwltz. 
 
98 
 
 NACA TM No. 12l8 
 
 H 
 
 T\ 
 
 1 
 
 
 
 1.1 
 
 0.270 
 
 1.2 
 
 O.i^Oi^ 
 
 1.4 
 
 0.573 
 
 1.6 
 
 0.688 
 
 1.8 
 
 0.772 
 
 2.0 
 
 0.833 
 
 2.2 
 
 0.881 
 
 2.k 
 
 0.916 
 
 2.6 
 
 0.941 
 
 2.8 
 
 0.959 
 
 3.0 
 
 0.972 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
WACA TM Wo. 1218 99 
 
 BIBLIOGRAPBY 
 
 Tart B. — Tiirtulent Flow 
 
 Chapters XIII to XX 
 
 '""60. Darcy: Memoires des Savants strangers. Vol. VII, I858. 
 
 61. Reynolds, OslDorn: Phil. Trans. Eoy. Soc, I883 or Collected papers 
 
 vol. II, p. 51* 
 
 62. Prandtl, L.: Ueter den Luftwlder stand von Ku^eln, Gottingen 
 
 Nachrichten, p. I77, 19114-. 
 
 63. Lorenz, H. A.: Abhandlungen ub. theoretische Physik. Bd. 1, 
 
 pp. k3 - hi, 1907. 
 
 6k. Boussinesq, T. V.: Mem. Pres. par div. Sav., vol XXIII, Paris, I877. 
 Theorie des I'ecoulement tourtillant, Paris I987. 
 
 65. Eeichardt, H. : Mess^jngen turtulenter Schwankungen . Naturw. I938, 
 
 p. h-Ok. 
 
 66. Prandtl, L. : Ueber die ausgebildete Turtulenz. ZAMM I925, p. I36 and 
 
 Verhdl. II. intemat. Kongress f. angew. Mech. Zurich, I926. 
 
 67. Schmidt, W. : Massenaustausch in freier Atmophare und verwandte 
 
 Erscheinungen. Hamburg, I925. 
 
 68. V. Karman, Th. : Mechanlsche Aehnlichkeit und Turtulenz. Nachr. Ges. 
 
 Wiss. Gottingen, Math. Ehys. Elasse, p. 58, I93O. 
 
 69. Blasius, H. : Forschungsheft I3I des Tereins deutscher Ingen. . I9II. 
 
 70. Nituradse, J.: Stromung in glatten Rohren. "VDI— Forschungsheft 356, 
 
 1932. 
 
 71. Nikuradse, J. : Stromungsgesetze in rauhen Rohren VDI— Forschungsheft 
 
 361, Berlin 1933- 
 
 72. Schlichting, H. : Experimentelle XJntersuchungen zum Eauhigkeitsproblem. 
 
 liig.-Arch. Bd. 7, I936, p. 1. 
 
 73. Prandtl, L.: Ueber den Re ibungswlder stand stromender Luft. Results 
 
 of the Aerodynamic Test Institute, Gottingen, III. Lleferung, 
 1927. 
 
 jh. Prandtl, L. : Zur turhulenten Stromung in Rohren und Langs Platten. 
 Restilts of the Aerodynamic Test Institute, Gottingen, IT. 
 Lieferimg, 1932. 
 
 ■"The reference numbers used herein have "been taken directly from 
 the original German Version. 
 
100 NACA TM No. 12l8 
 
 75. v. Karmaxi, Th. : Verhdlg. III. Internat. Mech. Kongreas, Stockholm, 
 
 1930. 
 
 76. Prandtl, L., and Schlichting, H. : Das Widerstandsgesetz rauher Platten. 
 
 Werft, Eeederei, Hafen. 1934, p. 1. 
 
 77. Tanij Hama, Mituiai: On the Permissible Eoughness in the Laminar 
 
 Boundary Layer. Aeron Bes. Inst. Tokyo Eep. I5, U19, I9U0. 
 
 78. G-ruBchwitz, E. : Die turbulente Eeibungsschicht in etener Stromung 
 
 bei Druckabfall und Druckanstieg. Ing.-Arch. Bd.II, p. 321, I93I. 
 
 79- Millikan, C B.: The Boundary Layer and Skin Friction for figure of 
 EeTolution. Trans. Americ. Soc. Mech. Eng., vol. 54, No. 2, 1932. 
 
 80. Pretsch, J. : Zur theoretischen Berechnung des Profilwlderstandes. 
 
 Jb. 1938 d. deutschen Luftfahrtforshung, p. I 6I. 
 
 81. Toll mi en, W. : Berechnung der turbulenten Ausbreitungsvorgange. 
 
 ZAMM Bd. lY, p. 468, 1926. 
 
 82. Schlichting, H. : Ueber das ebene Windschattenproblem. Ing.-Arch. 
 
 Bd. I., p. 533, 1930. 
 
 83. Swain, L. M. : Proc. Eoy. Soc. London. A. 125, 129, P- 647. 
 
 84. Betz, A. : Ein Verfahren zur direkten Ermittlung dea Profilwlderstandes. 
 
 Z. f. M. Bd. 16., p. 42, 1925. 
 
 85. Jones, B. M. : The Measurement of Profile Drag by the Pitot Transverse 
 
 Method. ARC Eep. I6888, I936. 
 
 86. Elias, F. : Die Warmeiibertragung einer geheizten Platte an stromende 
 
 Luft. ZAMM, 1929, p. 434. 
 
 87. Eeichardt, H. : Die Warmeiibertragung in turbulenten Eeib-'jngsschlchten. 
 
 ZAMM 20, 1940, p. 297. 
 
 88. Olsaon, Gran. : Geachw.— und Temperattir— Vert ei lung hinter elnem Gltter 
 
 bel turbulenter Stromung. ZAMM, I936, p. 257. 
 
 89. Schultz-Grunow, F. : Neues Eeibungawiderstandsgesetz fur glatte Platten. 
 
 Luftf. Forschung 1940, p. 239. 
 
 90 : W^ierstandsmeasungen an rauhen Ereiszylindern. 
 
 ARC Eep. 1283, 1929. 
 
 91. Forthmann, E. : Ueber turbulente Strahlausbreitung. Ing.-Arch. Bd. V, 
 p. 42, 1934. 
 
WACA TM No. 1218 . 101 
 
 92. Taylor, G. I.: The Transport of Vorticity and Heat Troiogh Fluids in 
 
 Turtulent Motion. Proc. Boy. Soc. A, Vol. I35, p. 685, 1932. 
 
 93. Fage and Falkner: Note on Experiments on the Temperatiire and Velocity 
 
 in the Wake of a Heated Cylindrical Obstacle. Proc. Roy. Soc. A. 
 Vol.135, P- 702, 1932. 
 
 914-. Lorenz, H. : Zeitschr. F. techn. Physik 15, p. 376, 193^4-. 
 
 95. Hagen: Pogg. Ann. Bd. k6, p. U23, l839- 
 
 96. Froude: Experiments on the Svirface Friction. Brit. Ass. Eep., I872. 
 
 97- Eryden, H. L. and Kuethe, A. M. : Effect of TurTDiolence in Windtunnel 
 Measurements. KACA Eep. Wo. 3l<-2, 1929 . 
 
102 MCA TM No. 12l8 
 
 BIBLIOGRAPHJ ^ 
 Origin of Turbulence (chapter XXI) 
 
 101. Reynolds, 0.: Sci. papers 2, I883. 
 
 102. Eayleigh, Lord: Papers 3, 1887. 
 
 103. LorenZj H. A.: Athandlungen u'ber thsoretische Physik. I k3, 
 
 Leipzig, 1907. 
 
 104. Sommerfeld, A. : Ein Beitrag zur Hydrodynami schen Erklarung der 
 
 turbulenten FlussigkeitslDewegung. Atti. d. IV" congr. int. dei 
 Mathem. Eom I9O9. 
 
 105. V. Miees, P.: Beitrag zum Oszlllationstheorem. Helnrich Weter 
 
 Festschr. I912. Derselte: Zur Turtulenz— Theorie . Jahresbericht 
 d. deutsch.. Mathem. Ver. I9I2. 
 
 106. Hopf , L. : Der Verlauf klelner Schwingungen in einer Stromung 
 
 relbender Flussigkeit. Ann. Phys. kk, p. 1, 191^; aj^i: Zur 
 Theorie der Turhulenz. Ann. Phys., 59 p. 538, 1919. 
 
 107. Prandtl, L. : Bemerkungen liher die Entstehung der Turhulenz. 
 
 Zeitschr. angew. Math. u. Mech. 1, k31, 1921. 
 
 108. Tietjens, 0.: Beitrage zum Turhulenzprohlem. Diss. Gottingen, 1922, 
 
 and Zeitschr. angew. Math. u. Mech 5, 200, 1925. 
 
 109. Toll mi en, W. : Ueber die Entstehung der Turbulenz. Nachr. Ges. Wiss. 
 
 Gottingen. Math. Phys. Klasse I929, p. 21. 
 
 110. Toll Tn ien, W. : Ein allgemeines Kriterium der Instabilitat laminarer 
 
 G-eschwindigkeitsverteilungen. Nachr. Ges. "Wiss. Gottingen. Math. 
 Phys, Klasse Bd. I, Nr. 5, 1935. 
 
 111. Prandtl, L.: Ueber die Entstehung der Turbulenz. Zeitschr. angew. 
 
 Math. u. Mech. 11, k07 , 1931. 
 
 112. Schlichting, H. : Zur Entstehung der Turbulenz bei der Piatt enstromung. 
 
 Nachr. Ges. Wiss. Gottingen. Math. Phys. Klasse, 1933 j P- I8I. 
 
 113. Schlichting, H. : Amplitudenverteilung und Energiebilanz kleiner 
 
 Storungen bei der Plattenstromung . Nachr. Ges. Wiss, Gottingen. 
 Math. Phys. Klasse 1935, P. ^8. 
 
WACA TM Wo. 1218 103 
 
 Hi*-. Schlichtingj H. : Berechnung der kritischen EeynoldsBchen Zalil elner 
 EelTDungBBchicht in beschleunigter mid verzogerter Stromimg. 
 Jahrlsuch 19^0 der deutschen Luftfahrt— Forschung, p. I 97- 
 
 115. Schlichtlng, H. ajid Ulrich, A.: Zur Berechnung des Umschlagea 
 laminar /turtulent. (Preisausschrei'ben 19I1O der Lilienthal 
 Gtesellscliaft). Lilienthal-Gesellschaft Bericht S. 10, 19i4-l. 
 
ioi+ 
 
 NACA TM No. 12l8 
 
 TABLE Vn . - THE BRAG LAM OF THE 3dOOTH PLATE 
 
 I 
 
 '^f 
 
 0.074 
 Eel/5 
 
 la 
 
 ^f 
 
 0.074 1700 
 Eel/5 «- 
 
 II 
 
 °f 
 
 0.k'i'> 
 
 2.58 
 (log Ee) 
 
 
 
 o.4?5 
 
 TJ 1 
 
 Be = 
 
 f „ „ ,2.58 Ee 
 (log Ee) 
 
 HI o 
 
 0.427 
 
 f 2 64 
 
 (- 0.407 + log Ee) ■ 
 
 
 I 
 
 la 
 
 U 
 
 Ha 
 
 HI 
 
 Ee=^ 
 
 Of X 103 
 
 Cf X 103 
 
 cj. X 103 
 
 Cf X lo3 
 
 Of X lo3 
 
 IC? 
 
 7.40 
 
 
 
 7.13 
 
 
 
 7.63 
 
 2 X IC? 
 
 6.43 
 
 
 
 6.11 
 
 
 
 6.50 
 
 3 X IC? 
 
 5.93 
 
 
 
 5.62 
 
 
 
 5.85 
 
 4 X IC? 
 
 5.60 
 
 
 
 5.33 
 
 
 
 5.50 
 
 5 X 10? 
 
 5.37 
 
 
 
 5.06 
 
 
 
 5.23 
 
 6 X IC? 
 
 5.18 
 
 2.35 
 
 4.92 
 
 2.17 
 
 5.06 
 
 8 X lo5 
 
 4.88 
 
 2.76 
 
 4.62 
 
 2.50 
 
 4.74 
 
 10^ 
 
 4.67 
 
 2.97 
 
 4.46 
 
 2.76 
 
 4.51 
 
 2 X 10 
 
 4.07 
 
 3.22 
 
 3.96 
 
 3.11 
 
 3.95 
 
 3 xlO^ 
 
 3.74 
 
 3.17 
 
 3.67 
 
 3.10 
 
 3.68 
 
 4 xlO^ 
 
 3.54 
 
 ■ 3.11 
 
 3.50 
 
 3.07 
 
 3.50 
 
 5 XIO^ 
 
 3.38 
 
 3.04 
 
 3.40 
 
 3.06 
 
 3.33 
 
 6x10^ 
 
 3.26 
 
 2.98 
 
 3.28 
 
 3.00 
 
 3.21 
 
 8 X 10^ 
 
 3.08 
 
 2.87 
 
 3.09 
 
 2.88 
 
 3.07 
 
 lof 
 
 2.94 
 
 2.77 
 
 2.99 
 
 2.82 
 
 2.94 
 
 2 X 10'^ 
 
 2.56 
 
 2.47 
 
 2.67 
 
 2.58 
 
 2.62 
 
 5 X 10^ 
 
 2.13 
 
 2.16 
 
 2.38 
 
 2.35 
 
 2.26 
 
 108 
 
 1.85 
 
 1.83 
 
 2.14 
 
 2.12 
 
 2.03 
 
 2X 10^ 
 
 
 
 
 
 1-93 
 
 1.92 
 
 1.87 
 
 5 xio^ 
 
 
 
 
 
 1.70 
 
 1.70 
 
 1.61 
 
 109 
 
 
 
 
 
 1.56 
 
 1.56 
 
 1.47 
 
 2 X 10^ 
 
 
 
 
 
 1.43 
 
 1.43 
 
 1.33 
 
 5 X 10^ 
 
 
 
 
 
 1.30 
 
 1.30 
 
 1.19 
 
 10^° 
 
 
 
 
 
 1.20 
 
 1.20 
 
 1.10 
 
NACA TM No. 12l8 
 
 105 
 
 
 o 
 
 EH 
 
 O 
 
 Hi 
 fa 
 
 o 
 
 g 
 
 I 
 
 O 
 
 § 
 
 ON 
 
 J- 
 
 
 o 
 o 
 
 § 
 
 o 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 VD 
 
 OJ 
 
 ^ 
 
 H 
 
 no 
 
 H 
 
 CO 
 
 H 
 
 [^ 
 
 -* 
 
 in 
 
 OJ 
 
 l>- 
 
 o 
 
 NO 
 
 
 H 
 
 VD 
 
 -* 
 
 vo 
 
 OJ 
 
 in 
 
 ON 
 
 OO 
 
 ON 
 
 -i* 
 
 rH 
 
 CO 
 
 NO 
 
 ^ 
 
 -* 
 
 OO 
 
 
 
 
 • 
 
 
 • 
 
 
 
 
 • 
 
 • 
 
 • 
 
 
 
 
 
 
 
 C C 
 
 tr- 
 
 I-- 
 
 \o 
 
 VD 
 
 in 
 
 ^ 
 
 -* 
 
 OO 
 
 OO 
 
 OO 
 
 (M 
 
 OJ 
 
 CM 
 
 OJ 
 
 CM 
 
 
 1- t> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 OJ ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 on 
 
 ^- 
 
 u\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 rH 
 
 O 
 
 0\ 
 
 OJ 
 
 t-- 
 
 OJ 
 
 OJ 
 
 in 
 
 t— 
 
 H 
 
 j- 
 
 OO 
 
 NO 
 
 in 
 
 J- 
 
 
 £t^ 
 
 VD 
 
 03 
 
 o 
 
 in 
 
 ON 
 
 in 
 
 OJ 
 
 On 
 
 t— 
 
 CO 
 
 co 
 
 CO 
 
 NO 
 
 CM 
 
 -* 
 
 
 O 
 
 O 
 
 H 
 
 M 
 
 H 
 
 OJ 
 
 OO 
 
 OO 
 
 -rf- 
 
 in 
 
 NO 
 
 t- 
 
 00 
 
 ON 
 
 CJN 
 
 
 
 J- 
 
 VO 
 
 CO 
 
 H 
 
 CO 
 
 ^ 
 
 NO 
 
 ON 
 
 OJ 
 
 in 
 
 t^ 
 
 CA 
 
 ?? 
 
 in 
 
 o 
 
 OJ 
 
 
 in 
 
 o 
 
 i>- 
 
 OJ 
 
 in 
 
 t^ 
 
 ON 
 
 H 
 
 ■^ 
 
 NO 
 
 00 
 
 ON 
 
 H 
 
 (^ 
 
 H 
 
 OJ 
 
 ^ 
 
 ^ 
 
 in 
 
 in 
 
 in 
 
 in 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 1^ 
 
 f- 
 
 •H 
 -P 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 
 cvi 
 
 t~- 
 
 o 
 
 ^ 
 
 o 
 
 o 
 
 NO 
 
 OJ 
 
 in 
 
 H 
 
 OO 
 
 t~- 
 
 in 
 
 en 
 
 t- 
 
 H 
 
 * H 
 
 M3 
 
 o 
 
 ro 
 
 o 
 
 OJ 
 
 rH 
 
 CO 
 
 NO 
 
 OO 
 
 H 
 
 CT\ 
 
 c^ 
 
 NO 
 
 in 
 
 in 
 
 
 H 
 
 -=^ 
 
 VD 
 
 t~- 
 
 c^ 
 
 t^ 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 in 
 
 in 
 
 in 
 
 in 
 
 in 
 
 
 
 
 
 • 
 
 • 
 
 
 
 
 
 
 
 
 
 
 
 
 
 O 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ro 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 O 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r-l 
 
 H 
 
 o 
 
 t^ 
 
 H 
 
 o 
 
 ^ 
 
 c~- 
 
 in 
 
 00 
 
 in 
 
 OO 
 
 NO 
 
 CO 
 
 o 
 
 OJ 
 
 H 
 
 ^ 
 
 M3 
 
 t^ 
 
 CO 
 
 H 
 
 -4- 
 
 t-- 
 
 H 
 
 NO 
 
 rH 
 
 t- 
 
 OO 
 
 CO 
 
 CM 
 
 NO 
 
 t- 
 
 
 ^1^ 
 
 O 
 
 o 
 
 o 
 
 H 
 
 H 
 
 H 
 
 OJ 
 
 OJ 
 
 no 
 
 oo 
 
 -=f 
 
 -=J- 
 
 in 
 
 in 
 
 in 
 
 
 t~- 
 
 OJ 
 
 -* 
 
 ^ 
 
 O 
 
 H 
 
 OO 
 
 NO 
 
 ON 
 
 in 
 
 o 
 
 NO 
 
 OO 
 
 OJ 
 
 H 
 
 
 
 oo 
 
 ro 
 
 1-1 
 
 
 o 
 
 On 
 
 CO 
 
 f- 
 
 NO 
 
 NO 
 
 NO 
 
 in 
 
 in 
 
 in 
 
 m 
 
 
 ,°'^ 
 
 OJ 
 
 OJ 
 
 OJ 
 
 O 
 
 OJ 
 
 r-\ 
 
 H 
 
 M 
 
 H 
 
 H 
 
 H 
 
 rH 
 
 H 
 
 H 
 
 •H 
 
 
 o 
 
 o 
 
 o 
 
 O 
 
 o 
 
 O 
 
 O 
 
 o 
 
 O 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 
 I- ^ 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 O 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 
 H 
 
 UA 
 
 in 
 
 t- 
 
 H 
 
 in 
 
 CO 
 
 ON 
 
 o 
 
 in 
 
 00 
 
 H 
 
 -* 
 
 in 
 
 in 
 
 
 w 
 
 O 
 
 H 
 
 OJ 
 
 m 
 
 -=J- 
 
 -* 
 
 ^ 
 
 ^ 
 
 in 
 
 in 
 
 in 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 
 
 H 
 
 H 
 
 ■H 
 
 H 
 
 f-l 
 
 r-{ 
 
 H 
 
 rH 
 
 H 
 
 rH 
 
 rH 
 
 H 
 
 H 
 
 H 
 
 ,-\ 
 
 
 
 
 ^ 
 
 o 
 
 in 
 
 in 
 
 in 
 
 in 
 
 NO 
 
 On 
 
 OJ 
 
 J^ 
 
 OO 
 
 in 
 
 -* 
 
 t- 
 
 
 H 
 
 
 m 
 
 H 
 
 CO 
 
 OJ 
 
 in 
 
 h- 
 
 ON 
 
 H 
 
 -=)- 
 
 NO 
 
 CO 
 
 CJN 
 
 <.) 
 
 o 
 
 
 P- 
 
 H 
 
 OJ 
 
 -* 
 
 -=*• 
 
 in 
 
 in 
 
 in 
 
 in 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 t~ 
 
 t— 
 
 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O 
 
 O 
 
 O 
 
 O 
 
 o 
 
 o 
 
 o 
 
 
 OJ 
 
 t— 
 
 ^ 
 
 Oi 
 
 VO 
 
 ro 
 
 ON 
 
 OJ 
 
 in 
 
 H 
 
 rH 
 
 ^ 
 
 CM 
 
 in 
 
 CM 
 
 
 * H 
 
 VO 
 
 o 
 
 en 
 
 H 
 
 OJ 
 
 H 
 
 CO 
 
 NO 
 
 OO 
 
 rH 
 
 0\ 
 
 t-- 
 
 NO 
 
 in 
 
 in 
 
 
 -»_n 
 
 H 
 
 -* 
 
 \D 
 
 I^ 
 
 t— 
 
 f- 
 
 NO 
 
 NO 
 
 NO 
 
 NO 
 
 in 
 
 in 
 
 in 
 
 UN 
 
 in 
 
 8 
 
 -P 
 
 i 
 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 O 
 
 O 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 Cv-l 
 
 O 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 iH 
 
 H 
 
 t- 
 
 ^ 
 
 in 
 
 H 
 
 OO 
 
 H 
 
 in 
 
 NO 
 
 O 
 
 f- 
 
 On 
 
 OO 
 
 o 
 
 rH 
 
 M 
 
 /--< 
 
 VD 
 
 ^ 
 
 o 
 
 ro 
 
 NO 
 
 O 
 
 -=J- 
 
 ON 
 
 in 
 
 o 
 
 in 
 
 o 
 
 OO 
 
 -=j- 
 
 
 <Pl-P 
 
 O 
 
 o 
 
 O 
 
 H 
 
 H 
 
 H 
 
 OJ 
 
 OJ 
 
 OJ 
 
 ro 
 
 ^ 
 
 -=h 
 
 in 
 
 in 
 
 in 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CM 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 d 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 II 
 
 in 
 
 • 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 CO 
 
 OJ 
 
 CO 
 
 O 
 
 o 
 
 00 
 
 H 
 
 in 
 
 rH 
 
 t— 
 
 o 
 
 r-[ 
 
 H 
 
 
 Rl^ 
 
 
 H 
 
 a\ 
 
 00 
 
 rrx 
 
 o 
 
 o 
 
 CO 
 
 t- 
 
 in 
 
 OO 
 
 OJ 
 
 OO 
 
 
 
 
 
 OJ 
 
 on 
 
 -=J- 
 
 -=t 
 
 in 
 
 in 
 
 ^ 
 
 ^ 
 
 -* 
 
 -=J- 
 
 -* 
 
 ^ 
 
 -* 
 
 -* 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 
 
 
 
 -b° 
 
 O 
 
 ? 
 
 ? 
 
 ? 
 
 ^ 
 
 ^ 
 
 ^ 
 
 V 
 
 V 
 
 ^ 
 
 f 
 
 Y 
 
 V 
 
 V 
 
 Y 
 
 
 
 r-\ 
 
 t~- 
 
 ITN 
 
 OJ 
 
 OO 
 
 NO 
 
 in 
 
 ro 
 
 -=f 
 
 i>- 
 
 in 
 
 t~- 
 
 ON 
 
 ® 
 
 ^ 
 
 
 M0° 
 
 t-- 
 
 ■ VD 
 
 -* 
 
 H 
 
 t^ 
 
 OO 
 
 CT\ 
 
 in 
 
 H 
 
 t-- 
 
 j- 
 
 H 
 
 o\ 
 
 
 OJ 
 
 OJ 
 
 OJ 
 
 OJ 
 
 H 
 
 M 
 
 O 
 
 o 
 
 O 
 
 0\ 
 
 ON 
 
 ON 
 
 00 
 
 CO 
 
 CO 
 
 
 
 H 
 
 r-{ 
 
 H 
 
 H 
 
 H 
 
 rH 
 
 r^ 
 
 rH 
 
 rH 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 
 
 H 
 
 
 in 
 
 in 
 
 O 
 
 rH 
 
 NO 
 
 H 
 
 OO 
 
 H 
 
 H 
 
 ON 
 
 J- 
 
 o 
 
 o 
 
 
 
 J- 
 
 |>- 
 
 rn 
 
 o 
 
 CO 
 
 NO 
 
 ^ 
 
 OO 
 
 rH 
 
 ON 
 
 NO 
 
 H 
 
 NO 
 
 <^ 
 
 o 
 
 
 Ml-P 
 
 H 
 
 i-l 
 
 OJ 
 
 m 
 
 ro 
 
 J- 
 
 in 
 
 NO 
 
 f- 
 
 t- 
 
 CO 
 
 ON 
 
 ON 
 
 ON 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 O 
 
 o 
 
 o 
 
 o 
 
 O 
 
 O 
 
 o 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 o 
 
 H 
 
 o 
 
 ON 
 
 o 
 
 o 
 
106 
 
 KACA TM Wo. 1218 
 
 Laminar 
 
 /////////^yy//y////////yy 
 
 Figure 71.- 
 
 Turbulent 
 
 Laminar and turbulent velocity distribution 
 in pipe. 
 
 Velocity t/Z'^y 
 
 r 
 
 iJL 
 
 i 
 
 Time t 
 
 Figure 72.- Fluctuation with time of the velocity of 
 turbulent flow at a fixed position. 
 
WACA TM No. 1218 
 
 107 
 
 Velocity y 
 
 1 Cy 
 
 ^ 
 
 Re 
 
 crit 
 
 ***-^ 
 
 Figure 73.- Drag and drag coefficient of a sphere. 
 
 Subcritical 
 Supercritical 
 
 Figure 74.- Flow around a sphere; subcritical 
 and supercritical (schematic). 
 
108 
 
 NACA TM No. 12l8 
 
 0, 
 
 '77yy777yvZ7y77:^pV7y77777777777' 
 
 laminar i turbulent 
 
 ^crit. 
 
 Figure 75.- 
 
 Laminar and turbulent boundary layer on a flat plate in 
 longitudinal flow. 
 
 y 
 
 uM 
 
 V \ \v' 
 
 7 '■'■' 
 
 y 
 
 Figure 76.- Transfer of momentum by the 
 turbulent fluctuation velocity. 
 
 y 
 
 
 u(y,-U_ 
 
 ti(y) 
 
 nn 
 
 i-^u 
 
 Figure 77,- Explanation of the mixing length. 
 
NACA TM No. 12l8 
 
 109 
 
 to 
 
 l<^ 
 
 
 V 
 
 0) 
 
 
 a, 
 
 
 •t-i 
 
 
 u, 
 
 
 5 
 
 
 o 
 
 
 o 
 
 o 
 
 a 
 
 ^i^ 
 
 
 
 5 
 
 
 a 
 
 l^ 
 
 g 
 
 OS 
 
 r— 1 
 
 
 o 
 
 
 • i-H 
 
 
 +-> 
 
 
 ;=) 
 
 
 ^ 
 
 
 • rH 
 
 
 !~f 
 
 c^ 
 
 
 
 •i-H 
 
 ^^ 
 
 t:! 
 
 
 ^ 
 
 
 •■-1 
 
 
 CJ 
 
 
 o 
 
 
 . — 1 
 
 ir^ 
 
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 Figure 85.- Measurement of the drag of an arbitrary roughness. 
 
 
 
 
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 Figure 91.- Velocity profiles in the turbulent friction layer with pressure 
 decrease and pressure increase (according to Gruschwitz [78 j). 
 
120 
 
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 121 
 
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 Figure 93.- Result of the calculation of the turbulent friction layer 
 according to Gruschwitz, Example profile J 015; Ca_ = 0. 
 
122 
 
 NACA TM No. 12l8 
 
 Figure 94.- Free turbulence: free jet and wake. 
 
 Figure 95.- Plane wake flow; explanatory sketch. 
 
NACA TM No. 12l8 
 
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 Figure 97,- The free jet boundary; explanatory sketch. 
 
 
 
 
 
 
 
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 Figure 98.- Free jet boundary; distribution of the longitudinal and 
 
 transverse velocity. 
 
NACA TM No. 12l8 
 
 125 
 
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126 
 
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 Figure 100,- Determination of the profile drag according to Betz, 
 
 Po^^^o 
 
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 Figure 101.- Determination of the profile drag according to Jones. 
 
RACA TM Re. 12l8 
 
 127 
 
 ■^^o 
 
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 Figure 102,- The stability investigation of the Couette flow. 
 
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 Figure 103.- The neutral stability curve as result of a stability- 
 
 investigation (schematic). 
 
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128 
 
 NACA TM Wo. 1218 
 
 Staole 
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 ^ /Profile without inflection point 
 
 ^®crit. ^^crit. 
 
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 Figure 105.- Neutral stability curves for velocity profiles without and 
 
 with inflection point (schematic). 
 
 P/q 
 
 D = Pressure side 
 
 5 = Suction side 
 
 (J = Transition point 
 
 Figure 106.- Wing profile with pressure distribution. 
 
WACA TM No. 1218 
 
 129 
 
 Figure 107.- Approximation of a velocity- 
 profile by a polygon. 
 
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 Figure 108.- Streamline pattern and velocity distribution of the plate 
 boundary layer for neutrally stable disturbance U(y) - basic flow; 
 u(y) = U(y) + u'(x,y,t) = disturbed velocity distribution; 
 A = 27r/a = wave length of the disturbance. 
 
130. 
 
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 Parameters of the disturbance: 
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 277 
 
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 Figure 109.- Neutral disturbance for the friction layer on the flat plate. 
 
 Amplitude of the disturbance velocity: 
 u'(x,y,t) = u-^(y) cos (x - P^t) - U2(y) sin (x - p^t). 
 
WACA TM No. 1218 
 
 131 
 
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 Figure 110.- Neutral stability curve for the friction layer on the flat 
 
 plate in longitudinal flow. 
 
132 
 
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 133 
 
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 -8 -5-4-2 Z it 6 
 
 The critical Reynolds number (Uj^^s 
 
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 of A 
 
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13^ 
 
 NACA TM Wo. 1218 
 
 Figure 113.- Stability calculation for the elliptic cylinder of axis 
 
 ratio a^A^i = 4, 
 
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 of axis ratio a-i /b-]^ = 4. 
 
WACA TM No. 1218 
 
 135 
 
 
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 Flat plate 
 
 Uo-t. 
 
 A = laminar separation point (independent of Rg - -^) 
 
 M = maximum velocity of the potential flow. 
 
 Figure 115.- The position of the instability point as a function of the 
 Reynolds number for the elliptic cylinders of axis ratios 
 a;^/b^ = 1, 2,4,8. 
 
136 
 
 NACA TM No. 12l8 
 
 Symmetrical Joukowsky profile: - = 0.15 
 
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 A = laminar separation point (independent of Re=-^ ) 
 M = maximum velocity of the potential flow 
 S = stagnation point 
 
 Figure 116.- The position of the instability point as a function of the 
 Reynolds number for a Joukowsky profile for lift coefficients of 
 
 Ca = to c^ = 1. 
 
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