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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1344 
 
 ON THE THEORY OF THE TURBULENT BOUNDARY LAYER 
 
 By J. Rotta 
 
 Translation of "Uber die Theorie der turbulenten Grenzschichten." 
 Mitteilungen aus dem Max -Planck -Institut fiir Stromungsforschung 
 
 (Gottingen) , No. 1, 1950 
 
 Washington 
 February 1953 
 
 / OF FLORIDA 
 
 PS DEPARTMENT 
 
 ON SCIENCE UBRARY 
 
 17011 
 
 7011 USA 
 
2H Q VUW 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 13^ 
 
 ON THE THEORY OF THE TURBULENT BOUNDARY LAYER* 
 By J. Roipta 
 
 INTRODUCTION 
 (By A. Betz) 
 
 In view of the high specialization of scientific research many papers, 
 basically important for further progress, are of interest only for a rela- 
 tively small circle of close colleagues. In normal times, such reports 
 nevertheless could he published in scientific periodicals without diffi- 
 culty. The periodicals published papers from various fields and thus 
 offered to a relatively large circle of readers sufficiently valuable 
 material. Today, this procedure is faced with considerable difficulties 
 which can be traced back to two main reasons: Scientific work has devel- 
 oped enormously so that periodicals had to be greatly increased in number 
 and volume. Thus, on one hand, it takes the reader a great deal of time 
 to follow the literature of his special field; on the other hand, sub- 
 scription to periodicals represents a heavy financial burden. In addi- 
 tion, almost all scientists, especially in Germany, are greatly impover- 
 ished and can no longer carry the increasing financial load; the sale of 
 periodicals is thereby reduced and the costs rise still further. 
 
 Looking for a way out of this difficulty I thought it desirable to 
 relieve the periodicals, first of all, of reports which address only a 
 relatively small circle of interested parties and yet, to be understand- 
 able, have to be somewhat extensive. For such reports the considerable 
 cost expenditure required for issue of a good periodical does not pay; 
 such expenditure is in order only in case of a correspondingly large 
 circulation. In order to acquaint the few specialists with such reports 
 and to make those reports accessible for later need, one can economically 
 recommend only a reproduction method which is relatively cheap also in 
 case of small circulation. On the basis of these considerations, I have 
 decided to print such reports which originate in the Max-Planck-Institute 
 for flow research and, also, a few older reports from this field, which 
 are no longer available by Rota printing method, and to edit them in 
 informal sequence as communications of the institute. Herewith, I give 
 to our colleagues the first issue of these communications. May it fulfill 
 the tasks described. 
 
 Gottingen, March 1950. 
 Albert Betz 
 
 "tiber die Theorie der turbulenten Grenzschichten. " Mitteilungen aus 
 dem Max-Planck-Institut fur Stromungsforschung (Gottingen), No. 1, 1950. 
 
2 NACA TM 13^4 
 
 OUTLINE 
 
 SURVEY 
 SYMBOLS 
 
 1. INTRODUCTION 
 
 2. ENERGY BALANCE OF TURBULENT FLOWS 
 
 3. EQUATIONS OF THE TURBULENT BOUNDARY LAYERS 
 
 3.1. Fundamental Equations 
 
 3.2. Momentum and Energy Theorem 
 
 3.3- Supplementary Considerations Regarding Boundary- Layer Turbulence 
 
 k. THE UNIVERSAL TURBULENT BOUNDARY-LAYER FLOW 
 
 5. EXISTENCE OF SIMILAR SOLUTIONS 
 
 5.1. Differential Equations and Boundary Conditions 
 5-2. Properties of the Similar Solutions 
 
 6. EMPIRICAL BOUNDARY-LAYER PROFILES 
 
 6.1. Velocity Profiles 
 
 6.2. Turbulence Profiles 
 6.3- Dissipation Function 
 
 7. NUMERICAL ESTIMATION OF THE SIMILAR SOLUTIONS 
 
 8. SUMMARY 
 
 9- REFERENCES 
 
 SURVEY 
 
 As a rule, a division of the turbulent boundary layer is admissible: 
 a division into a part near the wall, where the flow is governed only by 
 the wall effects, and into an outer part, where the wall roughness and 
 the viscosity of the flow medium affects only the wall shearing stress 
 occurring as boundary condition but does not exert any other influence 
 on the flow. Both parts may be investigated to a large extent independ- 
 ently. Under certain presuppositions there result for the outer part 
 "similar" solutions. The theoretical considerations give a cue how to 
 set up, by appropriate experiments and their evaluation, generally valid 
 connections which are required for the approximate calculation of the 
 turbulent boundary layer according to the momentum and energy theorem. 
 
NACA TM 13^ 
 
 SYMBOLS 
 
 x,y,z coordinates (x, z parallel to the wall; y perpendicular 
 distance from the wall) 
 
 p density 
 
 > of the flow medium 
 V kinematic viscosity J 
 
 Velocities and pressures, stresses: 
 
 U,V,W velocity components of "basic flow (average values in 
 time) (U in x-direction, V in y-direction, 
 W in z-direction) 
 
 U-i flow velocity in x-direction, outside of the boundary layer 
 
 u,v,w components of the turbulent fluctuation velocities 
 
 p mean value in time of the static pressure 
 
 p fluctuations of the static pressure 
 
 a ,a ,a z , normal stresses 
 
 y (mean values in time) 
 t ,T xz ,T yz shearing stresses j 
 
 (in section 3 following T xy is written t = t) 
 Tq wall shearing stress 
 
 v * = \/ T o/P shearing- stress velocity 
 c^* = 2(v*/U-[) local friction coefficient 
 
 Turbulence quantities: 
 
 E kinetic turbulence energy (per unit mass) 
 
 S energy dissipation (per unit mass) 
 
 pQ ,pQ ,pQ components of the energy diffusion (energy I values 
 
 flow per unit time and unit area) (in sec 
 tion 3 following Q is written Q = Q) 
 
 D = / S dy dissipation function 
 ^0 
 
 in time 
 
NACA TM 13^ 
 
 k,kq, c 
 
 characteristic length of the large turbulence elements 
 (statement Z = ny; K ~ 0.4) 
 
 dimensionless factors according to equations (3.l6), 
 (3.17), (3- 18). (Characterized for the universal 
 boundary- layer flow in the range &w = y « 8 by the 
 index "0") 
 
 Thicknesses of the boundary layer: 
 8 total thickness 
 
 J w 
 
 thickness of the sublayer directly affected by the 
 viscosity and the wall roughness 
 
 1=j (l - U/U^dy displacement thickness 
 
 5i = 
 
 £>2 = / (U/U-j) (l - U/U^dy momentum-loss thickness 
 
 >3 = / (u/uj 
 
 J o 
 
 - (u/uj 
 
 dy energy-loss thickness 
 
 H 12 = S-^bg 
 
 form parameters 
 
 H 
 
 32 = 8 3/ 5 2 
 
 Similar solutions: 
 \|r stream function 
 
 t\ = y/x dimensionless coordinate 
 m exponent of the law prescribed at the outer edge (eq. (5.4) ) 
 
 Re. 
 
 U^x 
 
 Reynolds number formed with the coordinate x 
 
NACA TM 13H 
 
 Empirical boundary- layer profiles: 
 
 -i2 
 
 lyy 
 
 V 5 l U l; 
 
 if^l 
 
 V 5 i u i 
 
 > form parameters 
 
 C 
 K 
 
 Be, ,HA 
 
 constant of the velocity profile near the wall (eq. (U.8)) 
 constant of the outer velocity profile (eq. (6.8)) 
 
 Reynolds number formed with the displacement thickness B-^ 
 
 B 
 
 G 
 
 U 
 
 1 1 
 
 - ± In Re, = C + K 
 
 v* k 1 
 
 A form parameter in equation (6.12) 
 
 d/v* 3 - i In Re-, 
 
 K L 
 
 1. INTRODUCTION 
 
 For evaluation of the flow conditions about a body and in particular 
 for estimation of its flow drag, the behavior of the flow layer bordering 
 on the body, which may be either laminar or turbulent, is of very high 
 importance. Whereas, for the laminar boundary layers, the physical rela- 
 tions have been clarified and the mathematical problems, too, have been 
 worked out sufficiently to have calculation methods at disposal which are 
 serviceable in practice, there are, for the turbulent boundary layers, 
 above all still problems of a physical kind to be solved. 
 
 From the basic hydrodynamic equations, one may derive relations for 
 the time averages of the flow quantities in turbulent boundary layers 
 which are similar to those valid for laminar boundary layers. Further- 
 more, a relation for the energy balance of the turbulent movement is at 
 disposal which was given first by L. Prandtl (ref. l). In spite of 
 these equations, an exact calculation of the turbulent boundary layers 
 
NACA TM 13W 
 
 is not possible since one has not yet succeeded in setting up formulas 
 for essential processes in the mechanism of the turbulent movement. Thus 
 the question arises whether there is, under such circumstances, any sense 
 in a discussion of the boundary- layer equations. Actually, however, a 
 few statements are possible on the basis of the means at disposal, if one 
 considers the following two empirical facts which may be regarded as 
 absolutely certain today: 
 
 (1) The total processes are affected by the kinematic viscosity v 
 
 and the geometrical properties of the wall (wall roughness) only in a 
 
 very thin layer in the neighborhood of the wall; in the remaining domain 
 
 of the boundary layer, the flow appears to be practically independent of 
 
 the viscosity and the wall roughness. If the thickness of the layer in 
 
 which these influences are effective is called 5,. and the thickness of 
 
 w 
 
 the boundary layer 5, one has therefore, as a rule, & w « 5. 
 
 (2) Because of & w « 5 one may expect the conditions for wall 
 
 distances y ^ 5 W to be independent of the flow conditions at the outer 
 
 edge of the boundary layer (y — ^ 5). Inside of the layer S w there 
 
 exists, therefore, a velocity law affected solely by the geometrical 
 properties of the wall; the wall shearing stress Tq represents the 
 
 essential parameter. 
 
 With these assumptions, one may separate the influence of the kine- 
 matic viscosity and that of the wall properties from the other influences. 
 Thereby, it becomes not only possible to discuss various properties of the 
 turbulent boundary layers but also to determine empirically, with the aid 
 of similarity relations, from suitable measurements, the quantities needed 
 for the development of approximation methods for calculation of turbulent 
 boundary layers. Such series of measurements with all required quantities 
 are not available in a desirable form at present; however, it is possible 
 to perform them with today's test techniques. 
 
 In the following sections only flows of an incompressible fluid are 
 considered which are steady on the average. 
 
 2. ENERGY BALANCE OF TURBULENT FLOWS 
 
 Since following the motion to the last detail is not possible in 
 turbulent flows, a statistical treatment must be applied. The flow 
 quantities will be expressed by arithmetical mean values, and the aver- 
 aging in time will be simplest where one deals on the average with 
 stationary flows. The time-averaged velocity with the components U, V, 
 and W in x-, y-, and z-direction is called basic flow. Superimposed 
 
NACA TM 13^ 
 
 on it is the disordered turbulence motion fluctuating with time with the 
 components u, v, and w, which is always three-dimensional even when 
 the basic flow may be regarded as two-dimensional. The velocity fluctu- 
 ations cause fluctuations of the static pressure which will be denoted 
 by p, whereas the time average of the static pressure is expressed 
 by p. 
 
 If one sets up, for the purpose of theoretical treatment, the 
 Navier-Stokes equations of motion for turbulent flows and performs the 
 time averaging, a few mean-value expressions formed from the fluctu- 
 ation velocities, which are to be regarded as new unknowns, remain in 
 the equations. In the search for further relations, in order to estab- 
 lish a mutual connection between these unknowns as well as a connection 
 with the other flow quantities, one can derive from the Navier-Stokes 
 differential equations further equations which partly convey very inter- 
 esting insight into the turbulent flow phenomena. It is not the purpose 
 of this report to discuss this more closely; but an important equation 
 among those mentioned above, which describes the balance of the kinetic 
 energy contained in the velocity fluctuations and for that reason is 
 physically the most graphic one, is utilized subsequently to a very great 
 extent. Since it is not yet to be found in exact form in literature, it 
 will be given herein for general three-dimensional flows. 
 
 With the occurrence of shearing and tensile or compressive stresses, 
 kinetic energy is withdrawn from a basic flow which partly reappears as 
 kinetic energy of the disordered turbulence motion. Let the time average 
 of this turbulence energy, referred to the unit mass, be 
 
 ? ? ? 
 E = U + V + W (2.1) 
 
 with the bars signifying the time averaging. Owing to the viscosity of 
 the fluid, kinetic energy is withdrawn continually from the basic flow 
 as well as from the turbulence motion and is converted into heat (dissi- 
 pation S); moreover, because of the turbulence motion in general, an 
 exchange of turbulence energy takes place between various points of the 
 flow space. If one deals with nonhomogeneous turbulence, which is mostly 
 the case, these exchange processes do not balance and an energy trans- 
 port pQ, occurs which one could compare to a diffusion process. An 
 energy balance for the coordination of these single effects (as first 
 formulated in this manner by L. Prandtl (ref. 1)) expresses that the sum 
 of these contributions equals the total (substantial) variation of the 
 turbulence energy. For a three-dimensional basic flow with the 
 
8 
 
 NACA TM 13^ 
 
 components U, V, and W this energy balance reads in the stationary 
 case quite generally: 
 
 dx by dz 
 
 Total variation of the turbulence energy 
 
 ^ + ... * + a „ m + T ^g t g) + T= (g + g) + Tyz (g t I) 
 
 ax ' "y a y ' " z a z ' ' *y 
 
 Energy withdrawn from the basic flow 
 
 -pS - p 
 
 aq x aQ y sq z 
 
 + — — + 
 
 bx dy Sz 
 Dissipation Energy diffusion 
 
 (2.2) 
 
 Therein 
 
 o x = P 2v 
 
 au 
 ax 
 
 ^ = p2v |- v2 
 
 „ = P 2v^- w 2 
 
 (2.3) 
 
 are the time-averaged normal stresses and 
 
NACA TM 13^ 
 
 T xy = P 
 
 •(£♦#-- 
 
 , g + M) _ ™F 
 
 
 > 
 
 (2.4) 
 
 T yz = p 
 
 \dz dy. 
 
 W 
 
 are the time-averaged shearing stresses. For the dissipation the 
 expression 
 
 S = V 
 
 2 (g) +2 (l) t2 (S) + (t + l)' 
 
 (M) ♦(£♦£) -(I) -(I) -(I 
 
 
 'du + chA /dy_ + du\ 
 
 Sz dx/ \dx dy/ 
 
 (2.5) 
 
 is valid, and the components & , ft , and Q of the energy diffusi 
 have the following form 
 
 on 
 
 Q = _v(^I + ^i_ + ^v + *U») + u fu 2 + v 2 +. v 2 P 
 x \ax dx By Sz / \ 2 P 
 
 
 n [ BE dvu civ dvw | /u + v + \r p\ 
 
 ^ = - V W + §r + ^r + arj + v l — i — + p) 
 
 > (2.6) 
 
 i— ^..2 
 
 Q =_ v Sl + ^li + Swv + Sw_ + w 
 z \Sz dx By dz / 
 
 u 2 + v 2 + w 2 P. 
 2 P 
 
10 NACA TM 13^4- 
 
 Equation (2.2), the derivation of which would be too lengthy here, is 
 attained by addition of the three Navier-Stokes equations of motion 
 after they have been multiplied by u, v, and w, respectively. By 
 application of the continuity equation and several transformations one 
 finally obtains, after having formed the mean values, the form (2.2) 
 with the expressions (2.5) and (2.6). 
 
 After this explanation which applies quite generally for turbulent 
 flows, we shall deal with the special problem of turbulent boundary- 
 layer flow. 
 
 3. EQUATIONS OF THE TURBULENT BOUNDARY LAYERS 
 3.1. Basic equations 
 
 In the following, the x-axis is assumed to lie parallel to the wall 
 and y to be the vertical distance from the wall. The three-dimensional 
 turbulence motion with the components u, v, w is assumed to be super- 
 imposed on the components U and V of the two-dimensional basic flow 
 in x- and y-direction. The boundary- layer equation -for a two-dimensional 
 flow along a plane wall resulting from the Navier-Stokes equations of 
 
 motion then reads with the simplifications introduced by L. Prandtl 
 
 u|u + vMU .a? + |l (3.1) 
 
 ox ay J ox ay 
 
 The theory developed by L. Prandtl in 1904 at first for laminar 
 boundary layers starts from the fact that the processes producing the 
 friction drag take place in a very thin layer on the body. Accordingly, 
 one may assume, for simplification of the problem, that the velocity 
 component V normal to the wall is small compared with the component U 
 acting parallel to the wall; furthermore, the static pressure p may be 
 assumed to be independent of the wall distance. An estimation of the 
 order of magnitude then indicates which terms in the equations may be 
 neglected. In case of turbulent boundary layers the mean value p is 
 
 influenced by the velocity fluctuations (p = p - pv^; p = p for y = 0); 
 in the derivation with respect to x this slight influence is partly com- 
 pensated by the term -ou /ox neglected in equation (3-1) so that op/cix 
 may be regarded as independent of y. 
 
NACA TM 13^ 
 
 11 
 
 Furthermore, one uses the continuity equation for the basic flow 
 
 dx Sy 
 
 (3.2) 
 
 The continuity condition must of course be satisfied also by the fluctu- 
 
 ating motion 
 
 v, w which is already taken into consideration in the 
 
 following formulas. In case of turbulent boundary- layer flows, there 
 applies for the shearing stress t in the xy-plane the expression 
 
 = pfv 
 
 Sy 
 
 UV J 
 
 (3.3) 
 
 Herein uv is the time mean value of the product of the fluctuation 
 components u and v acting vertically to one another. The 
 expression -puv is also denoted as Reynolds stress. 
 
 We now include into our considerations as a further equation the 
 energy balance of the turbulent flows given in section 2. For steady 
 two-dimensional boundary layers expressions (2.2), (2.5), and (2.6) are 
 
 p 
 
 simplified, under the customary assumptions to 
 
 .(„ 
 
 Sx 
 
 + V 
 
 = T 
 
 Sy 
 
 P s 
 
 Sy 
 
 (3A) 
 
 SyJ 
 
 f Su 
 
 + 2 
 
 oyy 
 
 
 c)w . Sv 
 
 dy Sz 
 
 du ( dw 
 
 Sx 
 
 dv t du 
 dx 6~y 
 
 (3.5) 
 
 (3.6) 
 
 See also footnote 1. t is put equal to t and Q, is put equal 
 
 to Q,y. The indices may be omitted here as before in equation (3-1) and 
 later on, since a confusion is quite impossible. 
 
12 NACA TM 13^ 
 
 The boundary- layer equations (3-1), (3-2), and (3-^) have to satisfy 
 the following boundary conditions: 
 
 y = 0: U = 0; V = 0; u 2 =v 2 =w 2 = 
 
 y — > 5: U-^U i; u 2 -^ 0; v 2 — > 0; w 2 — > (3-7) 
 
 U-^ is the velocity outside of the boundary layer which is assumed to be 
 prescribed as a function of x. 
 
 These relations are valid under the assumption of a sufficiently 
 smooth wall. For uneven walls the formulation is considerably more 
 complicated. In section k we shall revert to the treatment of rough 
 walls where unevennesses of a certain mean magnitude are statistically 
 distributed over the surface. 
 
 3.2. Momentum and Energy Theorem 
 If one introduces the quantities 
 
 Displacement thickness 6-j_ = / [1 - — ]dy (3.8) 
 
 poo . 
 
 Momentum-loss thickness &2 = / [^ ~ P^ (3-9) 
 
 ^0 U l\ U l/ 
 
 and wall shearing stress T n = T f° r y = 0> one ma y derive from equa- 
 tions (3-1) and (3-2) the momentum equation 
 
 f L%) + u A 2k = I© ' (3.10) 
 
 dx \ 1 d) 1 1 ax p 
 
 which has proved to be advantageous for the approximated integration of 
 the boundary- layer equation, and which has the same form for turbulent 
 as for laminar boundary layers. 
 
NACA TM 13^ 
 
 13 
 
 By integration over y one may develop from equation (3.^-) a corre- 
 sponding energy equation. The partial integration of the left side of 
 equation (3.^0 yields, with use of equation (3-2) 
 
 /o> 
 
 5x 
 
 ^f) dy 
 
 dy, 
 
 _d_ 
 
 UE dy 
 
 (3.H) 
 
 Furthermore there applies for the basic flow the relation to be derived 
 from equation (3-1) (cf. the reports by K. Wieghardt, ref. 2) 
 
 
 
 t Mdy 
 
 oy . 
 
 " &J IM 
 
 u' 
 
 )<iy 
 
 (3.12) 
 
 The diffusion term --^ in equation (3-^) disappears in the integration 
 
 since the components u, v, w tend toward zero for y = as well as 
 for y — >8. After introduction of the energy-loss thickness 
 
 5 3 = 
 
 u_ 
 
 Ul 
 
 ft 
 
 dy 
 
 (3.13) 
 
 and of the dissipation function 
 
 D = / S dy 
 '0 
 
 (3.1*0 
 
 one then obtains as the energy theorem 
 
 I H^S) ■ 
 
 _>Vw 
 
 dx 
 
 UE dy (3.15) 
 
 
 
 'Energy flow loss of ' '•Dissipation' '■Increase of the ' 
 the basic flow per turbulent energy 
 
 unit length flow per unit 
 
 length 
 
 This equation is significant for the behavior of turbulent boundary 
 layers. The energy losses of the basic flow essentially are first con- 
 verted into kinetic turbulent energy which in turn is transformed into 
 
lk NACA TM 13M+ 
 
 heat by friction; however, the conversion of the basic flow energy into 
 turbulent energy, and the transformation of the turbulent energy into 
 heat need not take place at the same location and at the same time. This 
 state of affairs is expressed in equation (3. 15)- Very frequently the 
 increase of the turbulence energy flow contributes only slightly to 
 equation (3.15) — thus, for instance, in case of ordinary plate flow 
 without pressure gradient — however, there are also cases where this 
 term gains more significant influence besides the dissipation function. 
 
 3.3- Supplementary Considerations Regarding Boundary-Layer Turbulence 
 
 By qualitative considerations one obtains a survey of the connec- ; 
 tions still lacking between the quantities E, uv, S, and Q, occurring 
 in the energy balance (eq. (3.^)). We shall limit ourselves especially 
 to the region y ^ 5 where the viscosity of the flow medium may be 
 
 regarded as arbitrarily small. First of all, the terms with V in equa- 
 tions (3-3) and (3-6) are hereby eliminated; whereas, in equation (3.5) 
 
 the contribution v(dU/dy) of the basic flow to the dissipation 
 becomes negligibly small. 
 
 The amount E of the kinetic energy of the turbulence, which is a 
 quantity suitable for dimensional considerations, is composed of the , 
 contributions of a very large number of turbulence elements of many 
 different orders of magnitude; however, there exist for turbulence two 
 characteristic lengths. One is the characteristic length Z for the 
 dimensions of the large turbulence elements, which for boundary- layer 
 flows is given approximately by the pertinent distance from the wall. 
 The second is the diameter Z^ of the smallest turbulence elements, 
 
 which is determined by the quotient of the kinematic viscosity and the 
 mean fluctuation velocity — thus, approximately by v/\/e. For the fol- 
 lowing considerations, the fact is important that the kinetic energy E 
 is contained chiefly in the large elements and that, therefore, the 
 momentum and energy exchange phenomena are essentially caused by the 
 large elements. If the viscosity is sufficiently small or, more accu- 
 rately, if the Reynolds number i_i. Q f the turbulence is sufficiently 
 
 large, which is the case in the region y ^ 5 W , we need in our consider- 
 ations only to refer to the one characteristic length Z which corre- 
 sponds to the dimensions of the large elements to make the total effects 
 of turbulence independent of the viscosity. 
 
 The apparent shearing stress puv caused by the turbulent fluctu- 
 ation velocities may be traced back to a momentum transport which can be 
 expressed by the form used by L. Prandtl (ref. 1) 
 
 I = _Hv" = k\/EZ ^- (3.16) 
 
 P dy 
 
NACA TM 13^ 15 
 
 wherein k is a dimensionless factor. The product k\/EZ represents 
 
 an apparent kinetic viscosity (cm /secj or, respectively, an exchange 
 quantity — concepts first introduced by J. Boussinesq and W. Schmidt. 
 
 By considerations similar to those on which equation (3-l6) is 
 based, one arrives for the energy transport Q caused by exchange phe- 
 nomena at the expression introduced by L. Prandtl (ref. 1) 
 
 Q = -k-,VEZ ^ 
 
 dy 
 
 Since, however, the turbulence elements of different order of magnitude 
 have different energy density, an energy diffusion takes place, not only 
 when an energy gradient is present but is obviously possible also when 
 the turbulence at adjacent locations differs only by the linear dimensions 
 or by the structure (for instance, the energy spectrum). With this inter- 
 pretation the expression 
 
 Q=— ^ '- (3-17) 
 
 dy 
 
 is justified,-^ which originates by taking the exchange quantity kVEZ 
 for the energy transport under the differential sign. Prandtl' s form 
 is contained in expression (3-17) as a special case. Here again k„ 
 
 is a dimensionless factor which like k in equation (3-l6) is chiefly 
 determined by the structure of the large turbulence elements. 
 
 In contrast, the energy dissipation is caused mainly by the small 
 elements. If one combines again the influences dependent on the struc- 
 ture (that is, the spectrum) in a dimensionless factor c^_, one obtains 
 
 according to equation (3-5)? with omission of the contribution v(dU/dy) , 
 the relation 
 
 VCJ.-2 
 
 E 
 
 V 
 
 •3 
 
 The expressions (3-l6) and (3-17) cannot yet lay claim to full 
 general validity. A further discussion of these questions is not pos- 
 sible within the scope of this report and will, therefore, be the subject 
 of another publication. For the present problem, the expressions (3.l6) 
 and (3.17) are, at any rate, sufficient. 
 
16 NACA TM 13^ 
 
 Herein c-, is determined chiefly by the small turbulence elements and is 
 independent of the Reynolds number only for very small ^^- (cf. ref. 3)- 
 
 For sufficiently large Reynolds numbers, we may express the dissipation 
 process as a wandering of the kinetic energy (taking its course inde- 
 pendently of the viscosity) from larger to smaller elements whereby, 
 however, the energy content of the turbulence motion does not change. 
 The extent of transformation into heat, occurring almost exclusively in 
 the smallest turbulence elements, is guided by the amount of energy 
 supplied to them by the larger elements (cf. the reports by C. F. 
 v. Weizsacker (ref. h) and W. Heisenberg (ref. 5))- Thus for large 
 Reynolds numbers, one may replace in the given expression for S the 
 kinematic viscosity by an apparent kinematic viscosity of the dimension 
 \/E2 one then obtains the relation 
 
 S = c ?— (3.18) 
 
 which is valid for y j? 5 W . The magnitude of the factor c appearing 
 therein is determined mainly by the large elements. 
 
 The dimensionless quantities k, k q , and c, (appearing in equa- 
 tions (3.16), (3.17)> an d (3-l8)) which depend on the structure of the 
 turbulence, will generally assume amounts differing from point to point; 
 however, for sufficiently large Reynolds numbers, they are independent 
 of the kinematic viscosity. Their calculation presupposes complete 
 theoretical mastery of the statistic behavior of the turbulence motion. 
 This goal, for turbulence research, however, is still far remote. For 
 the following investigations, k, k„, and c are therefore introduced 
 
 formally as functions of the location although without selection of 
 special statements. However, it will be possible to assume offhand that 
 they are continuous functions and do not become infinite at any point. 
 
 For the characteristic length I in equations (3>l6), (3-17)> and 
 (3.18) for the large turbulence elements, there is, with consideration 
 of the regions near the wall, the expression 
 
 I = Ky (3.19) 
 
 of advantage where k is a general constant. In this form, the length I 
 is in the region 5 W = y « 5 identical with the mixing length introduced 
 
 by L. Prandtl (ref. 6). For the sake of simplicity, the expression (3.19) 
 is used for the entire boundary layer, although the dimensions of the 
 large turbulence elements for larger distances from the wall no longer 
 increase in proportion to y. The deviations between the actual dimen- 
 sions of the large elements and the expression (3-19) °ne may assume as 
 taken into consideration in the factors k, k , and c. 
 
NACA TM 13^ 17 
 
 By introduction of the relations (3-l6), (3-17), (3-18), and (3-19) 
 into the boundary-layer equations, one may obtain a few statements 
 regarding the behavior of the solutions; no limitation of the general 
 validity seems to be connected with it. 
 
 h. THE UNIVERSAL TURBULENT BOUNDARY- LAYER FLOW 
 
 For wall distances which are small compared with the boundary- layer 
 thickness 5, the shearing stress does not noticeably deviate from the 
 value t~ of the wall shearing stress and the flow conditions are here 
 
 practically independent of the pressure gradient Bp/dx. It has already 
 been mentioned in the introduction that the viscosity and the wall rough- 
 ness exert an immediate influence on the flow phenomena only in a layer 
 of the thickness 6 W adjacent to the wall. If this thickness 6 W is 
 
 sufficiently small, there will certainly exist wall distances y larger 
 than &„, yet so small compared with the boundary-layer thickness 5 that 
 
 in this region a universal boundary- layer flow results for which all flow 
 quantities are determined by only two quantities which have physical 
 dimensions, namely, the shearing stress velocity 
 
 v* 
 
 V^O/P ( k 'V 
 
 and the absolute distance y from a plane of reference which practically 
 coincides with the wall surface. This flow is influenced by the viscosity, 
 the wall properties, and the pressure gradient dp/Bx only insofar as 
 they determine the magnitude of v*. Aside from this influence, the flow 
 in this region is not affected by either the outer boundary conditions or 
 the wall properties and the viscosity. The velocity variation of the 
 basic flow is prescribed for 5 W % y « 5 by the known relation (ref. 6). 
 
 *y ^ = v* (k.2) 
 
 oy 
 
 Therein k ~ 0. k is a universal constant which is identical with the 
 value k in equation (3- 19)- 
 
 Not only the velocity variation is known, however, but also Important 
 statements are possible concerning the turbulence energy and dissipation. 
 Within the validity range of the universal boundary-layer flow no value 
 of y is in any way distinguished. The structure of turbulence (energy 
 
 spectrum, etc. ) is therefore similar in all sections parallel to the wall . 
 _ 
 
 With the exception of the smallest turbulence elements. 
 
18 NACA TM 13H 
 
 Owing to this similarity and the equality of the shearing stress t = -puv, 
 
 the energy E has a value independent of the wall distance y so that the 
 turbulence at different wall distances of this region differs only in its 
 linear dimensions. 
 
 The factors k, k q , c in expressions (3.16), (3-17)j and (3.18) 
 become for S w 5 y « S generally valid constants which we shall emphasize 
 
 by the subscript "0" (^-q, kg , Cq\ . With -uv = v* and Z = ny there 
 follows from the relations (3-l6) and (4.2) 
 
 With this value, the expression (3.17) then yields for the energy trans- 
 port Q, caused by exchange a value also independent of y 
 
 V 
 
 which obviously corresponds to an energy flow in the direction toward the 
 wall. Since, furthermore, under the presuppositions named, the terms on 
 the left side of equation (3-4) become, in first approximation, small com- 
 pared with the expressions on the right side and finally (because Q, = Const, 
 the last term on the right side disappears, the dissipation is, for 
 y « 5, equal to the energy withdrawn from the basic flow: 
 
 S= T _0^ = V * 2 M (4.5) 
 
 Hence results with expressions (3.18), (3-19)? (4.2), and (4.3) the 
 relation 
 
 c 
 
 - ^0 
 
 k 3 (4.6) 
 
 which like equation (4.3) was found by L. Prandtl (ref. 1); on the basis 
 of measurements, the value of k Q was estimated to be k Q = O.56. 
 
NACA TM 13^4 19 
 
 The existence of the universal boundary-layer flow in the region 
 B w ^ y « 6 suggests the division of the boundary-layer flow into a 
 
 part near the wall (0 1» y « 5) and an outer part (y > 5 W ) . The flows 
 of both parts merge asymptotically into the universal boundary-layer 
 flow: The flows of the first part with increasing y, those of the 
 second with decreasing y. The advantage attained by this division is 
 that one is now able to investigate the flow phenomena in each part 
 separately with reduction of the influencing quantities (experimentally 
 or theoretically) and to combine both parts, as occasion demands, since 
 both have the same asymptotic variation at the point of junction. 
 
 For the part near the wall (0 = y « &), there exists a velocity 
 law of the general form (ref. 6) 
 
 U 
 
 r*f(^) (4. 7 ) 
 
 * 
 
 /v*y\ 
 wherein the function fl — — I is dependent not only on -^- but, in 
 
 general, also on the wall roughness. The existing experimental results 
 on smooth and rough walls may be understood and represented in formulas 
 (ref. 7) directly up to the wall, and with aid of L. Prandtl's mixing- 
 length expression. Here we are interested only in the asymptotic form 
 for y > B„ which results from relation (4.2) by integration: 
 
 U = v 
 
 * 
 
 i Zn ^-Z + C 
 K V 
 
 (4.8) 
 
 Therein the constant C is a function of the wall roughness. 
 
 The outer part (y > 5 W ) has to satisfy the boundary- layer equations 
 given in section 3«lj using the relations named in section 3-3, one may, 
 however, neglect herein the kinematic viscosity. In flow problems of 
 the practice, the desideratum usually is the boundary- layer flow, with 
 the velocity at the outer edge U-i (x) and wall properties and viscosity 
 
 prescribed. For theoretical investigations, the problem may be formulated 
 differently: beside U-^(x), the shearing stress velocity v* is pre- 
 scribed as a function of x and the desideratum is the wall condition 
 required in order to produce this variation v*(x). Instead of the 
 boundary conditions indicated in section 3-l> in this problem the fol- 
 lowing conditions for lim y > at the wall (y = 0) must be satisfied 
 
 for the outer part (y > 5 W ) in order to guarantee the connection with 
 expression (4.8): 
 
 lim ^- = A V = 0; *=(£) (4. 9 ) 
 
 y — >0 dy xy \k / 
 
20 NACA TM l^kk 
 
 p 
 
 /v*\ 
 
 Since on one hand the value — — ) which corresponds to the local 
 
 friction coefficient c^' 
 
 (# -■$-.¥ 
 
 may be estimated quite satisfactorily, according to existing approxi- 
 mation formulas (for instance, ref. 10), even without exact knowledge 
 of all boundary-layer details and varies only slowly with x, and since 
 on the other hand the velocity profile of the outer part in case of 
 appropriate normalization seems to be dependent on v*/U^ only to a 
 
 comparatively slight extent, as will be shown later, a treatment of the 
 boundary layer in this manner where the outer part is simply determined 
 with v*(x) and U-^(x) prescribed promises some prospect of success 
 
 also for the first-named problem of practice. 
 
 The presuppositions for the division into two mutually independent 
 regions are, in most cases, satisfactorily fulfilled. This is, however, 
 by no means self-evident and is, therefore, to be checked for the indi- 
 vidual case. For this purpose, we add the following orders of magnitude: 
 The thickness 5 W is for smooth walls 6 W ~ 100 v /v*; the pertinent 
 
 Reynolds number of the turbulence for y = 5 W is \/El/v ~ 100. For 
 
 pronounced wall roughness, c^ is determined by the dimensions of the 
 
 roughnesses. According to the experiments of J. Nikuradse (ref. 8) on 
 sand-rough pipes, 5 W is approximately equal to the grain size of the 
 
 roughness; the y-values are measured in this case from a plane of refer- 
 ence in which U, on the average, disappears. 
 
 5. EXISTENCE OF SIMILAR SOLUTIONS 
 5.1. Differential Equations and Boundary Conditions 
 
 It will now be shown that under certain assumptions so-called similar 
 solutions exist for turbulent boundary layers, too, similar to the case 
 of laminar boundary layers — that is, solutions for which the velocity 
 profile along the wall is distorted only affinely. We investigate the 
 solution of the boundary-layer equations to be expected, with neglect of 
 the viscosity, in the range y >5 . In order to satisfy the continuity 
 
NACA TM 13H 21 
 
 condition, we introduce for the basic flow the stream function \|r from 
 which the components U and V are derived^ by the relations 
 
 After substitution of this function into the equation of motion (3.1), 
 there follows 
 
 i|r i|r - ft =-I p - (U7) (5.1) 
 
 *y Y xy x T yy p *x v 'y . 
 
 and the energy equation (3.^) assumes with the relations (3-17), (3-l8), 
 and (3.19) the form 
 
 w 3/2 / 3/2\ 
 ¥x " *xEy = "UvV -c E - + (k V E jyy (5.2) 
 
 Finally, one obtains for uv according to equation (3-l6) 
 
 -uv = Kk^/Eyt^ (5-3) 
 
 It may now be shown that for velocity distributions prescribed at 
 the outer edge of the boundary layer of the form 
 
 U x = ax 111 (5.4) 
 
 with a and m being constant quantities, and for a prescribed constant 
 
 v*/U^ there exist similar solutions for relations (5-1), (5«2), and (5-3) . 
 
 If the flow is unaffected by the viscosity, the geometric similarity of 
 the flow pattern requires that the boundary- layer thickness for similar 
 
 5 
 Partial derivatives with respect to x and y in this section are 
 
 expressed by subscripts x and y. 
 
 Constant v*/U^ signifies a constant local friction coefficient. 
 
 Under what circumstances and to what extent this assumption can be 
 practically satisfied is shown in section 5-2. 
 
22 
 
 NACA TM 13^ 
 
 7 
 solutions increase linearly with x 1 . For this reason, we introduce 
 
 the dimensionless variable t\ = y/x and make the statements 
 
 ^ = a^ 1 ^ - f(Ti)] 
 
 E = a^ 2 **^) 
 
 -uv = a x^p(tj) 
 
 (5-5) 
 
 Therein f(T])> q? ( -q ) , <!>(t\) are only functions of the variable r\. In 
 order to satisfy the prescribed boundary condition (eq. (5.^0) in case 
 of large y, f(rj) must for large t\ tend asymptotically toward a 
 constant value — that is, lim f'(n) = must be time. Thus, there 
 
 t\ > oo 
 
 follows from equation (5-1) for t\ > oo 
 
 ip = a 2 !^ 2 ^- 1 
 P x 
 
 (5.6) 
 
 which results also directly from relation (5-^) and Bernoulli's equa- 
 tions. After substitution of equations (5-5) and (5-6) into equa- 
 tions (5-1) to (5-3), one obtains after division by 
 
 2 2m-l ? v 2m 
 
 -a x 
 
 ax 
 
 a 3 x 3m-l 
 
 respectively 
 
 2mf» - mf' 2 - (m + 1)(t) - f)f" 
 
 ^P 
 
 (1 - f')2m$ + (1 + m)(f 
 
 >2 
 
 -cpf" 
 
 K ij 
 
 Cp = -KCkT)\/$f" 
 
 $3/2 d . 
 
 4,- 
 
 1!)$' = 
 
 3/2\ 
 
 k aH* 
 
 > 
 
 (5-7) 
 
 The same results also from the momentum theorem (eq. (3-10)) 
 
NACA TM l^hk 23 
 
 In this system of ordinary differential equations, x no longer appears 
 explicitly so that one actually has to expect systems of solution of 
 equations (5-5) where the boundary- layer thickness 5 increases line- 
 
 Q 
 
 arly with x 
 
 For unequivocal determination of a solution, five boundary condi- 
 tions must be prescribed. In order to satisfy the three boundary condi- 
 tions (relations (U.9)) at the inner edge, one has to put 
 
 lim f" = -■&- f(0) = 0; *(o)=^_ (5.8) 
 
 ti — > si _k 2 
 
 with cp = cp(0). Two conditions at the outer edge of the boundary layer 
 are added: 
 
 lim ; f* = 0; $ = (5-9) 
 
 The first insures that the basic flow merges with the prescribed flow; 
 whereas, the second causes the turbulence intensity outside of the 
 boundary layer to die out to zero. 
 
 5.2 Properties of the Similar Solutions 
 
 Owing to the conditions at the inner edge, there appears, in addi- 
 tion to the parameter m occurring in equations (5-7), as a further 
 freely selectable quantity the value cpg which like the ■ local friction 
 
 coefficient c^' is according to relation (U.10): 
 
 (5.10) 
 
 pu x 2 
 
 The solution of the system of equations (5-7) with the boundary condi- 
 tions (relations (5-8) and (5-9)) is, therefore, a two-parameter curve 
 family. The velocity profile of the outer part, most interesting in 
 these solutions, may be represented in the form 
 
 Hi^„,.£W (5 . u) 
 
 v* ^/cp 
 
 Q 
 
 For instance, the displacement thickness S-, according to equa- 
 tion (3-8) is: b 1 = xf(»). 
 
2k NACA TM 13^ 
 
 and is dependent on the two parameters m and v*/U-]_. Likewise, there 
 
 results, of course, for the pertaining "turbulence profile," that is 
 the plotting of the kinetic turbulence energy 
 
 -^(t,) = ' (5.12) 
 
 v* d ^0 
 
 over T), also a two-parameter curve family dependent on m and v*/U^. 
 
 For turbulent flows in a pipe or between parallel walls, the 
 velocity profile corresponding to equation (5-11), plotted over the 
 wall distance y divided by the pipe diameter or, respectively, the 
 mutual distance of the walls (so-called "center law"), is independent 
 of the magnitude of the friction coefficient (compare, for instance, 
 ref. 8). It seems appropriate to point out this difference between 
 turbulent pipe and boundary- layer flow. Furthermore, attention should 
 be called to the difference compared to the laminar boundary layers 
 where the velocity profile is a function of only one parameter, 
 namely m. 
 
 The solution of equations (5-7), valid only for wall distances y s? b^, 
 
 must be supplemented by the wall profile (relation (*)-.7)) in order to 
 obtain from it the complete velocity profiles. The condition for the 
 continuous junction of the outer part to relation (^.7) is obtained by 
 elimination of the quantity U/V*, with the aid of relation (*+.8), from 
 the asymptotic form 
 
 1 ^ =-- In(ri) + kL, —) (5.13) 
 
 V* K \ U]_ / 
 
 resulting for small T)-values by integration of f" according to rela- 
 tions (5-8). In this manner, one obtains 
 
 U l 1 , U l J v*\ 1 , Uix 
 
 — + - Zn — - Km, — J = - In — — • + C (5-1*0 
 
 v* k v* ^ ' U-jJ k V 
 
 The constant K(m, v*/U]) in equations (5.13) and (5.1*0 may be, deter- 
 mined, in case of prescribed parameters m and v*/U-p from the system 
 of equations (5-7). 
 
 The solutions of the outer part herein discussed have real signifi- 
 
 U-jX 
 cance only when the Reynolds number Re x = and the wall roughness, 
 
 the effect of which is expressed in the quantity C, are such that equa- 
 tion (5.1*0 is identically satisfied for all x-values. 
 
NACA TM 13U14- 25 
 
 For extremely large Reynolds numbers, there exists a linear rela- 
 tion between C and the logarithm of the length characterizing the 
 
 roughness (for instance, of the grain size k) , so that the right side 
 of equation (5.1*0 becomes independent of x when the grain size k is 
 proportional to x — that is, when k/x = Const. For hydrodynamically 
 smooth walls and for constant roughness where C is a constant, the 
 condition (eq. (5.1**-)) cannot be rigorously satisfied for all x-values. 
 This would be possible only in the case m = -1 which has, however, no 
 physical significance because then the flow separates from the wall. 
 However, since x in equation (5-1*0 appears in logarithmic form, it 
 will be permissible to regard, for sufficiently large x-values, the 
 expression on the right side of equation (5.1**-) as approximately con- 
 stant from x- interval to x- interval also for arbitrary m. Under this 
 assumption, one may regard the similar solutions with practically suf- 
 ficient accuracy as valid for the individual interval also for smooth 
 walls and for walls with constant roughness. It is, however^ essential 
 that the value 
 
 5 w 5 w v * U l V 
 
 x V v* U-jX 
 
 be so small that the function (u^ - u)/v* at the point y/x = & w /x 
 
 actually deviates only slightly from the asymptotic form (eq. (5-13)) • 
 Otherwise, the method selected, joining the wall law (eq. (*)-.7)) to the 
 solutions obtained with neglect of the viscosity effect, does not lead 
 to useful results. 
 
 Since the required conditions are rarely satisfied in actual cases, 
 the similar solutions will evoke chiefly theoretical interest. One has 
 here a type of solution of the boundary- layer equations which offers a 
 comparatively simple survey and is thus suitable for the study of theo- 
 retical problems. It could be shown that the solutions of the outer 
 boundary- layer part depend on v*/U-^. On the problem regarding the 
 
 extent of this influence, which is one of the next-to-most-important 
 ones, one can, at the time, obtain information only from experiments, 
 as will be shown in section 6. 
 
 It is perhaps necessary to point out that the only assumption for 
 the derivation of these theoretical results was that the Reynolds number 
 of the turbulence should be sufficiently large (except in the thin sub- 
 layer c^-) so that the viscosity may be neglected in the boundary- layer 
 equations; aside from the customary boundary- layer simplifications no 
 
 9 
 For sand-rough walls, there applies, for instance, according to 
 
 the experiments by J. Nikuradse (ref. 8): C = 8.5 - -i In ^^. 
 
 k V 
 
26 NACA TM 13 Wj- 
 
 restricting hypotheses were introduced. The findings thus have general 
 validity. However, if one wants to determine the solutions of equa- 
 tions (5-7) numerically, one cannot forego some hypotheses; that is, 
 one would have to introduce special formulations for c, k, and k q . 
 Thus, one would, for instance, insert constant values for the factors c, 
 k, k_. This we shall not do, however. Instead, we shall attempt to 
 
 obtain a survey of the solutions by empirical method by using the knowl- 
 edge attained from existing test results. Since nowadays measuring 
 series exist where the wall shearing stress was determined by a special 
 measurement (refs. 9 and 10 ), a sorting of the experimental data can be 
 undertaken with greater success than was so far possible. 
 
 6. EMPIRICAL BOUNDARY- LAYER PROFILES 
 6.1 Velocity Profiles 
 
 Theoretically, for the boundary layer on the plate with constant 
 
 external pressure (the constant external pressure appears as special 
 
 case m = in the system of eqs. (5«7)), a family dependent on the 
 
 local friction coefficient, thus a single-parameter family, would result. 
 
 Ui - U , 
 
 However, the plotting of — = against y/S according to F. Schultz- 
 
 v* 
 
 Grunow (ref. ll) shows that the profiles within the considered Reynolds 
 number range may be represented with practically sufficient accuracy by 
 a single curve. Since the boundary- layer thickness 5 used for the 
 plotting is a quantity which can hardly be exactly defined, the expres- 
 sion yv*/(5-[_U-|) instead of y/o is introduced as reference value with 
 
 use of the displacement thickness o^. Thereby the abscissa scale is 
 
 fixed so that the integral value becomes 
 
 \ 00 
 
 u i- u d /yvM =1 
 
 Jo v* \ & l u l/ 
 
 as one can see from a comparison with equation (3-8) • In figure 1, the 
 
 U-, - U 
 
 values — for the flat plate without pressure gradient were plotted 
 
 v* 
 
 yv* 
 against log . The test points of the smooth plate according to 
 
 5 1 U 1 
 reference 11 fall almost into a single curve; nevertheless, close obser- 
 vation shows a small systematic influence of the value v*/U-j. In con- 
 trast, the test points of the rough plate show according to reference 21 
 somewhat large deviations due to the greater variation of v*/U-j_. This 
 
NACA TM 13I44 27 
 
 investigation admits the conjecture that the magnitude of the local 
 friction coefficient is, in case of appropriate normalization of the 
 y-scale, of only moderate influence on the velocity profile. 
 
 The boundary- layer profiles measured for variable course of pres- 
 sure at the wall may be represented in the same manner. Figure 2, in 
 which several profiles of the quoted measurements by H. Ludwieg and 
 W. Tillmann (ref. 10) are represented, shows that the pressure vari- 
 ation exerts a considerably stronger influence on the profile shape 
 than v* /U^ • 
 
 Performance of approximation calculations requires by no means 
 knowledge of the boundary- layer profiles to the last detail; it is, 
 on the contrary, quite sufficient to be oriented regarding the rela- 
 tions between the individual parameters (displacement thickness 5^, 
 momentum- loss thickness 82, energy- loss thickness S3, and others). 
 Various authors (refs. 10, 12, 13 , 1^, and 15) found empirically that, 
 for the profiles of turbulent boundary layers, for arbitrary pressure 
 increase, these relations are almost unequivocal - that is, that the 
 boundary- layer profiles can be described approximately by only one 
 parameter. Further treatment of test material will be based on this 
 presupposition. The relation between the prescribed velocity distri- 
 bution U]_(x) and the profile parameter to be defined more closely is, 
 
 at first, not yet established. This relation is ascertained only by 
 application of the momentum theorem (eq. (3. 10)) and of the energy 
 theorem (eq. (3-15)) - a method which, in principle, has been known for 
 a long time for the calculation of laminar and turbulent boundary layers 
 and has been very successfully applied in approximation methods. How- 
 ever, one should not forget that this type of single-parameter repre- 
 sentation is no more than an approximation as opposed to the fact that, 
 according to the theory, even in the simplest case of similar solutions 
 a two -parameter family is to be expected. The reason for the usefulness 
 of this approximation lies perhaps in the fact that (as was observed in 
 the case of the flat plate without pressure gradient) the influence of 
 one of the two parameters - namely, of the local friction coefficient - 
 is generally probably little noticeable if the y-scale has been suitably 
 normalized. 
 
 The next step is bringing the desired parameters into a form which 
 permits separate consideration of the influence of the viscosity and of 
 the wall roughness. For the momentum- loss thickness 52, there applies 
 according to equation (3-9) 
 
 t *-£%t--&*£k*® r -£( i -$f« <6 - 2) 
 
28 
 
 NACA TM 13 ^ 
 
 which may also be written as 
 
 5 2 = 5]_ 
 
 C 1 " 5? ^) 
 
 (6.3! 
 
 The value 
 
 In = 
 
 1 - I 
 
 -.2 
 
 r yv> 
 
 v 5 i u i, 
 
 (6A) 
 
 is, under the assumptions, made practically independent of the velocity 
 distribution for y < Sy. Likewise there results for the energy- loss 
 thickness according to equation (3-13) 
 
 w * 
 
 (tf'-'&^r'rt-tf*-* 
 
 -f' 
 
 (6.5) 
 
 or 
 
 5 3 = 8 l 
 
 '-^♦(^ 
 
 (6.6) 
 
 where 
 
 u x - U 
 
 >yv> 
 
 ^ 5 1 U 1, 
 
 (6.7) 
 
 also is practically independent of the profile form for y < 5y. 
 
 The profiles of the representation figures 1 and 2 have according 
 to equation (5.13) for y — > &y the form 
 
 U-l - U 
 v* 
 
 1 ,« yv^ 
 
 = -- Zn 
 
 + K 
 
 5 1 U 1 
 
 (6.8) 
 
NACA TM I3I4.I1- 
 
 29 
 
 with the quantity K of a different amount for every profile form. 
 Since, on the other hand, the velocity profiles have for small y- values 
 (Sy. ^ y « o) the form of expression (4.8), there results by substi- 
 tution of expression (4.8) into equation (6.8) 
 
 U l 1 
 
 — = - In Re-, + B 
 
 V* K 1 
 
 (6.9) 
 
 if 
 
 U 1 5 1 
 
 Re-, = 
 
 1 V 
 
 (6.10) 
 
 is the Reynolds number formed with the displacement thickness and B is 
 
 B = C + K 
 
 (6.11) 
 
 The quantities 1-^, I2, and K are pure form parameters which can be 
 immediately derived from the profile form and do not depend on the form 
 of the wall law (eq. (4.7)) if the condition &v « 5 is sufficiently 
 satisfied. For a single-parameter profile family, there exists an 
 unequivocal relation between these quantities which can be determined 
 empirically from existing measurements. For the following consider- 
 ations, we shall regard 1-^ as characteristic form parameter and 
 
 express the others as function of 1-^. 
 
 In order to obtain some sort of numerical basis for this empirical 
 relation and thus to get away from the scatter of the test points, also 
 to facilitate the extrapolation in the region not comprehended in the 
 measurements, the velocity profile (y ^ Oy-) is represented by a simple 
 
 approximation expression which starts out from the wall law (eq. (4.8)) 
 
 U = v* 
 
 K\ V 
 
 I + A| 
 
 + C 
 
 (6.12) 
 
 A is a freely selectable constant. The thickness 5 of the boundary 
 layer is defined for y = & by the condition U = U-j_, so that the quali- 
 fying equation for & reads 
 
 Ui 
 
 v* i(zn 2l& + a) + C 
 
 (6.13) 
 
30 NACA TM 13 kk 
 
 From expressions (6.12) and (6. 13), there follows for the outer part of 
 the velocity profile 
 
 V* K\ 5/ K 5 
 
 For the displacement thickness, there results hence with equation (3.8) 
 
 J l " u /v\ 1 + § 
 
 V d (f) ■ -r <6 - 15) 
 
 s i"i r 1 u i - u ,m x + 5 
 
 5v* J 
 
 Furthermore, the quadratures yield: 
 
 ? 3 1? 
 
 &y* rV u i - u \ d /y\ " + 2 A + 3 
 
 1 s^Jo v v* y u; «(i + ±) 
 
 d (f) - * m < 6 ' 16 ) 
 
 6 + 2i A + n A 2 i a3 
 
 K 2 (l * I) 
 
 Thereby the relation between 1-^ and Ip is given by a parameter 
 
 representation which is plotted in figure 3 for k = O.k and compared 
 with the quoted measurements by H. Ludwieg and W. Tillmann (ref. 10) and 
 F. Schultz-Grunow (ref. ll). 
 
 For the quantity B in equation (6.9), 
 
 1 + A 
 B = C + A . I i n 2 (6-l8) 
 
 The relatively large scatter of the test points for small In -values 
 
 is without practical significance because the term in equation (6.6) 
 dependent on Ig contributes only a very small percentage to 80; the 
 
 scatter is explained by the fact that the experimental It- and Ig-values 
 
 were not obtained directly by quadratures but were calculated backward 
 from the 5-^-, 62-, and So-values determined by quadratures with use of 
 
 the experimentally ascertained v*/Ut_ from equations (6.3) and (6.6). 
 
NACA TM 13^ 31 
 
 would result from expressions (6.12) and (6.13). Although the expres- 
 sion (6.12) renders the velocity profile on the average quite satis- 
 factorily, deviations do appear in details, which take effect chiefly 
 in the quantity K according to equation (6.8). Therefore, the B-value 
 is not satisfactorily represented hy equation (6.18); the modified form 
 
 1 + A 
 B = C + 0.82 - - - Zn £ (6.19) 
 
 K K K 
 
 is more appropriate, as shown by the comparison with measurements repre- 
 sented in figure h for C = 5.2 and k = O.k. 
 
 Figures 3 and h may be regarded as a confirmation for the usefulness 
 of the representation dependent only on the form parameter 1-^ and of 
 
 the expression (6.12). 
 
 6.2. Turbulence Profiles 
 
 If one considers use of the energy theorem (3-15)> one needs data 
 one cannot obtain from the velocity profile alone. In order to ascertain 
 the magnitude of the energy flow, one requires the turbulence profile 
 which in a dimensionless plotting corresponding to figures 1 and 2 is 
 
 represented as E/v* over ( yv*) /(o^U-^) . Herein E/v* tends in the 
 neighborhood of the wall y — > by toward the universally valid value 
 given by expression (^.3) and decreases for y < by very rapidly to 
 zero. Although it is fundamentally possible to determine, with known 
 hot-wire arrangements, the quadratic mean values of all three fluctuation 
 components experimentally and hence, according to relation (2.1), E 
 numerically, evaluable measurements exist only for the component u 
 which, it is true, yields the most essential contribution to E. The 
 
 longitudinal oscillation profiles Vu 2 /v* represented in figures 5 
 and 6 were measured by means of the turbulence-measuring device of 
 W. Tillmann (refs. l6 and 21) developed by H. Schuh. The conjecture 
 following from the universal boundary- layer flow and by « b that the 
 
 V u /v* tends toward a universal value in the same manner as E/v* 2 for 
 y — > 8 W , is only insufficiently confirmed by these measurements. The 
 reason probably lies in inadequacies of measuring technique. 
 
 According to the considerations of section 5, two-parameter curve 
 
 families for E/v* 2 would result for the similar solutions. If, how- 
 ever, the influence of the one parameter v*/U]_ on the velocity profile 
 is small, which is probable according to the preceding section, one may 
 
32 NACA TM 13 kk 
 
 conclude with some certainty from consideration of the third part of 
 equation (5-7) that this parameter exerts only slight influence on the 
 turbulence profile as well. Figure 6 confirms the correctness of this 
 reasoning for the longitudinal-oscillation profiles of the plate flow 
 without pressure gradient for smooth and rough surfaces. Thus, the con- 
 jecture suggests itself that the quantities of interest in turbulence 
 profiles may, like the parameters of the velocity profiles, approxi- 
 matively be described by an unequivocal relation to the form param- 
 eter 1-^. This assumption is taken as the basis of the further 
 
 investigations . 
 
 As could be determined so far, the variation of the turbulence 
 energy flow mostly does not make a very significant contribution for 
 two-dimensional boundary- layer flow, so that a somewhat liberal treat- 
 ment of this influence seems, as a rule, permissible. From a few older 
 measurements by H. Reichardt (ref. 17) in a rectangular channel, by 
 H. C. H. Town end (ref. l8) in a square pipe, and by A. Fage (ref. 19) 
 in a circular pipe, the order of magnitude of the v- and w-variation 
 components can be estimated. Near the wall, the v-component in particu- 
 lar is essentially smaller than the u-component; at larger distance from 
 the wall, the magnitudes of the v- and w-components approach that of the 
 u-component . 
 
 According to definition of the integral expressions practically 
 independent of the wall law (eq. {k.f)) 
 
 the turbulence- energy flow is determined to be 
 
 / UE dy = S 1 U 1 2 v*(l Ti - ^ I T J (6.21) 
 
 For the presupposed single-parameter condition, Lp and I T are only 
 functions of I-j_. According to the existing data, the relation 
 
 poo 
 
 / UE dy = 0.655^^* (6.22) 
 
 seems to be useful for the estimation independently of Iij it is, there- 
 fore, taken as the basis for further evaluations and calculations. 
 
NACA TM 13J+U 33 
 
 6.3- Dissipation Function 
 
 For determination of the dissipation function D occurring in the 
 energy theorem (eq. (3.15)) according to expression (3.1*0, one will 
 again attempt by means of the results represented in section h to 
 express separately the influence of the viscosity and of the wall rough- 
 ness. For this purpose, one may determine S from equation (3-*0 and 
 perform the quadrature for the part near the wall (0 % y « 5) if one 
 puts t/p = v* = Const. and the left side of equation (3-M equal to 
 zero which is admissible for small wall distances. One then obtains 
 for y « 5 
 
 Joy 
 S dy 1 = v* 2 U(y) - Q(y) (6.23) 
 
 
 
 With U according to relation (U.8) and Q according to relation (h.k), 
 there results, hence, if the upper integration limit lies in the region 
 &w % y « & 
 
 6.23a) 
 
 For the outer part of the boundary layer y ^ c^ , one obtains with 
 the relation (3.18) 
 
 r »*..££ r c (e/U2)3/2 tfcn) (*.*) 
 
 J y K J yv*/S 1 U 1 (y , v*)/(B 1 U 1 ) \ S l U l/ 
 
 Since, for 5 W ^ y « 5, E/v* = l/kg is valid according to expres- 
 sion (h.3) and Cq = kg-* according to relation (k.6) , there results 
 from equation (6.2*0 
 
 [" S dy' = v*Yj s - i Zn HLS (6.2ka) 
 
 J Y { K 5 l U lJ 
 
 wherein the value of the integral expression 
 
 j„-if" c ( e/v * 2 ,) 3 afl^\ + l m XHL (6.25) 
 
 K Jyv*/b 1 \J 1 (y'v*)/^!) V 5 1 U 1/ K & 1 U 1 
 
3^ NACA TM l^kk 
 
 ( I 2)3 / 2 
 because of c^VE/v* J = 1 is independent of the lower integration 
 
 limit y if it lies in the range Oy % y « 5. From equations (6.23a) 
 and (6.2Ua), one finally obtains for D the form 
 
 D = / S dy = v* (-i Zn Re ] _ + g) (6.26) 
 
 with 
 
 G = C + J s + k — - (6.27) 
 
 *o 3 
 
 We now again assume that for our single-parameter velocity and 
 turbulence profiles J s , and, therewith for equal wall properties, G 
 as well, is only a function of the parameter 1^ described by expres- 
 sion (6.h). Since nothing is known regarding the behavior of the func- 
 tion c, except for the region 5 W ^ y « 5, J s cannot be calculated 
 
 from expression (6.25), even if the turbulence profile is known. Thus, 
 there remains only the possibility of calculating the function D by 
 differentiation, by means of insertion of the experimentally determinable 
 quantities into the energy equation (3-15); this method suffers, however, 
 from serious uncertainties. The measuring series of F. Schultz-Grunow 
 (ref. ll) on the plate without pressure gradient could be evaluated quite 
 satisfactorily according to this method. The result represented in fig- 
 ure 7 is to be evaluated as a satisfactory confirmation of the correctness 
 of the relation (6.26). 
 
 If boundary- layer measurements with pressure increase are made in a 
 wind tunnel of rectangular cross section, the occurring secondary flows 
 represent a disturbance, as shown by a very careful investigation by 
 W. Tillmann (ref. l6) . These secondary flows have the effect that the 
 flow is not two-dimensional (as had been assumed in the derivation of 
 eq. (3.15)) but at the location of the measurement usually convergent 
 with respect to the planes parallel to the wall. A. Kehl (ref. 13 ) has 
 shown how to consider in the momentum theorem (eq. (3.10)) a convergence 
 or divergence influence. In a similar manner, the energy theorem 
 (eq. (3.15)) for wedge-shaped flow may be ascertained; it is given herein 
 without derivation (compare fig. 8) 
 
 poo 
 
 3 / UE dy 
 
 1 ±L\) + i ^_ = D + A P ue dy /_° (6.28) 
 
 2 dx\ x ->' 2 Xq + x dx J q Xq + x 
 
NACA TM 13H 35 
 
 In the evaluation of the quoted measuring series by H. Ludwieg and 
 W. Tillmann ( ref . 10), the following method was applied: The mean 
 
 measure of convergence — , which is to express summarily the 
 
 Xq + x 
 
 secondary-flow effects in equation (6.28), was estimated with the aid 
 of the momentum theorem given by A. Kehl since all quantities appearing 
 in it, with exception of the measure of convergence, were determined 
 experimentally. This measure of convergence then was introduced into 
 the energy theorem (eq. (6.28)) and, thus, the function D was deter- 
 mined. In this manner, it was possible to eliminate at least approxi- 
 mately the effect of the secondary flows. Aside from these measure- 
 ments, four further measuring series performed by W. Tillmann in the 
 
 same wind tunnel but not published were treated in the same manner 1 . 
 
 The result of this evaluation, which for the first time conveys an 
 
 indication for the magnitude of the dissipation function as a function 
 
 of the profile shape, is shown in figure 9- The scatter is sometimes 
 
 quite considerable; however, on the whole, the test points are grouped 
 
 fairly satisfactorily about a mean curve. Greater accuracy was hardly 
 
 12 
 to be expected in view of the circumstances described . For large 
 
 I-^-values, the results may be approximative ly rendered by 
 
 G = 7.5(1! - 8.2) (6.29) 
 
 In order to make a more reliable determination of the dissipation 
 function (which is very important for the development of approximation 
 methods for the calculation of turbulent boundary layers), measurements 
 would be required for which by avoidance of secondary flows easily sur- 
 veyable flow conditions exist. Measurements in a rotationally symmet- 
 rical wind tunnel probably ensure clear conditions. These measurements 
 would have to include a very exact experimental determination of the 
 turbulence profiles, for instance, by hot-wire measurements. The reason 
 why turbulence measurements of boundary layers have been performed com- 
 paratively rarely can probably be found, amongst other reasons, in that 
 so far no immediate need for quantitative measurements of this kind 
 existed. 
 
 The magnitude of the wall shearing stress which had not been 
 experimentally determined in these measurements could be estimated by 
 means of the relations given by equations (6.16) and (6.19) and figure k, 
 respectively. 
 
 -^Particularly uncertain are the end points of the individual meas- 
 uring series which are denoted by "E" in figures 7 and 9 because the 
 variation of the curve to the differentiated is not exactly fixed at the 
 end of each measuring series. 
 
36 NACA TM 13 hk 
 
 The energy equation in the form (3-15) will probably stimulate 
 carrying out of further turbulence measurements. 
 
 It would mean an essential progress in the determination of the 
 dissipation function if not only the wall shearing stress but also the 
 entire "shearing stress profile" could be determined experimentally. 
 Attempts to measure the mean value of the product uv, which is 
 according to expression (3-3) decisive for the shearing stress by means 
 of hot-wire probes, were made by H. Reichardt (ref. 17) and H. K. 
 
 Skramstad^ , Besides, H. Reichardt (ref. 20) has tried to measure 
 mechanically the mean value uv with an angle probe. Further develop- 
 ment of methods of this type will be of great advantage for the investi- 
 gation of turbulent boundary layers. 
 
 7. FlMERICAL ESTIMATION OF THE SIMILAR SOLUTIONS 
 
 The relations determined from the existing test material may serve 
 for developing approximation methods for calculation of turbulent bound- 
 ary layers with arbitrary pressure gradient. Here we shall use them for 
 quantitatively estimating the conditions for the similar solutions treated 
 in section 5 with the aid of the momentum equation (3. 10) and the energy 
 equation (3 .15) • 
 
 In consequence of the results of section 5> according to which the 
 velocity distribution is prescribed in the form of a power law (rela- 
 tion (5-M) and the boundary- layer thickness 8 increases linearly 
 with x, we make the statements 
 
 \J 1 = ax 111 
 
 \> (7.D 
 
 5 2 = bx 
 
 therein a and b are quantities independent of x. If one takes 
 into consideration that the form parameters of the velocity profiles 
 %2 = ^l/^2 anc ^ ^32 = St/&2 are, according to presupposition, also 
 
 independent of x, there results by substitution of equations (7-l) into 
 
 2 
 the momentum equation (3-10) and after division by the value (ax m ) 
 
 2 
 (2m + l)b + mbH 12 = f—\ (7-2) 
 
 ^National Bureau of Standards, Washington, D. C, USA. 
 
NACA TM 13kk 37 
 
 From the energy theorem (eq. (3-15)), one obtains in the same manner, 
 with use of relation (6.22) 
 
 For the momentum loss thickness cV>, there follows from equation (7-2) 
 
 — = b = V ' 1J (7.^) 
 
 x (2m + 1 + mH 12 ) 
 
 Since the calculation of the boundary layer for a prescribed velocity 
 distribution is troublesome, we choose a more convenient method and 
 determine for prescribed values of the boundary- layer profile the perti- 
 nent velocity variation along x, that is, the exponent m. For this 
 purpose, equation (7-3) is, after elimination of b and with the aid of 
 relation (7 .h) , solved with respect to m: 
 
 H32/2 - (v^U^^d/v* 3 + 0.65H 12 ) 
 m= — (7-5). 
 
 3H 32 /2 - (v*/^)^ 2 + H 12 )D/v*3 + 1.95H 12 J 
 
 This equation is evaluated by calculation of the quantity v*/U-^ for 
 assumed values of the Reynolds number Re-^ = U]_8i/v and of the profile 
 parameter Ii with the aid of the relation (6.9) and figure k. From 
 equation (6.3) then results H-^ = °i/ D 2- With equation {6.6) and fig- 
 ure 3, one may then proceed to calculate o^/o^ and therewith 
 Ho 2 = °3/°2* Equation (6.26) and figure 9 make, furthermore, the deter- 
 mination of J)/v*- > possible. With these quantities, it is finally 
 possible to determine from equation (7-5) the exponent m, from rela- 
 tion (7 .h) the momentum-loss thickness 02/x referred to x, and from 
 
 relation (^.10) the friction coefficient c f ' = 2(v*/U 1 ) 2 . In figure 10, 
 
 the results of such a calculation are compiled for three different 
 Reynolds numbers, with the conditions of smooth walls taken as a basis 
 although they do not exactly satisfy the presuppositions of the similar 
 solutions. 
 
 It is an interesting result that a physically meaningful solution 
 does not exist for all m- values . This state of affairs is not immedi- 
 ately evident from the system of equations (5-7); it follows, however, 
 
38 NACA TM 13I44 
 
 at once from the momentum theorem. If 82 and c f ' are to be positive, 
 one will according to equations (3-8) and (3-9)> because of U/U-^ = 1, 
 always have 5-^ > SgJ thus, H-^ > 1. Negative values of 5g and Cf ' 
 
 can occur only when reverse flow appears near the wall. However, in 
 this case, the boundary- layer theory loses its physical significance 
 since the flow separates from the wall. According to relation (l .h-) , 
 m must therefore be greater than -l/3- Figure 10 shows that the sepa- 
 ration is to be expected approximately in the range of m = -0.2. 
 For comparison, it should be mentioned that, for the corresponding 
 similar solutions of the laminar boundary layers, the separation takes 
 place at m = -0.091. This confirms the well-known empirical fact that 
 turbulent boundary layers can overcome a larger pressure increase than 
 laminar ones. 
 
 Another noteworthy result is the dependence of the profile param- 
 eter H-^2 coordinated to a certain m-value on the Reynolds number. 
 
 The smaller the Reynolds number, the larger is H-^. This dependence 
 
 comes about chiefly due to the fact that the wall law (eq. (k.'j)) corre- 
 sponding to the respective Reynolds number is adapted, according to equa- 
 tion (6.9), to the single-parameter profile of the outer part (y ^ &„.) 
 
 which is independent of Reynolds number. In this manner, the first-order 
 effect of the Reynolds number on the velocity profile is included so that 
 figure 10 actually is based on a two-parameter profile family. The 
 dependence of the outer profile parts on v*/U^_, theoretically proved in 
 
 section 5> may tie regarded as a Reynolds number effect of the second 
 order; this effect was not accurately expressed in the calculation for 
 figure 10. In an investigation by A. E. von Deonhoff and N. Tetervin 
 (ref. 15) who calculated similar solutions with the aid of the approxi- 
 mation method for calculation of turbulent boundary layers indicated by 
 them, a universal relation was found to exist between the exponent of the 
 the velocity law (eq. (7-l)) and the parameter H 12 ; thus no dependence 
 
 on Re existed. However, as figure 10 shows, the influence of the 
 Reynolds number, the expression of which became possible only after one 
 had succeeded in the experimental determination of the wall shearing 
 stress, is rather important for the relation between H-^ and m. 
 
 It need not be explained further that corresponding calculations 
 may be carried out for flows at rough walls as well. For this purpose, 
 one has merely to perform a conversion of the values B and G intro- 
 duced in section 6 corresponding to the modified constant C. In 
 principle, however, these calculations would not offer anything new. 
 
 lit 
 
 The test data at disposal is insufficient for exact determination 
 
 of the m-value corresponding to separation. 
 
NACA TM 13U 39 
 
 8. SUMMARY 
 
 As equations of the turbulent boundary layer, this report indi- 
 cates the customary equation of motion, the continuity equation, and, 
 in addition, a balance for the kinetic turbulence energy from which one 
 may derive for approximation calculations besides the known momentum 
 theorem also an energy theorem for turbulent boundary layers . 
 
 Under the assumption (frequently confirmed by test observations) 
 that the influence of the kinematic viscosity and of the wall roughness 
 takes immediate effect only in a very thin layer &v at the wall, there 
 exists within the turbulent boundary layer a region ( Oy ^ y « 5) in 
 which a universal flow prevails which is determined by the magnitude of 
 the wall shearing stress but, for the rest, is not influenced either by 
 the wall conditions or the velocity distribution U]_(x) prescribed at 
 the outer edge. The presence of this universal boundary- layer flow 
 enables the division of the boundary layer into a part near the wall 
 (0 ^ y ^ &„.) which is affected only by the viscosity and the wall prop- 
 erties and into an outer part (y ^ 5^.) independent of the viscosity in 
 
 which the flow is essentially determined by the velocity distribution 
 prescribed at the outer edge. The flows in these two parts show a 
 mutual influence only insofar as the asymptotic behavior of the inner 
 flow represents a boundary condition for the outer flow. 
 
 With the aid of the indicated boundary- layer equations, it can be 
 proved that, for a prescribed velocity distribution U^ = a x m and a 
 
 local friction coefficient which is almost independent of x, similar 
 solutions exist also for turbulent boundary layers; these solutions 
 depend on two parameters - the exponent m and the local friction 
 coefficient. The boundary- layer thickness increases linearly with x. 
 
 With consideration of the findings obtained, the evaluation of 
 existing test data then yields empirically the relations between the 
 various quantities required for application of the momentum theorem 
 and the energy theorem. Finally, the established relations are used to 
 perform, with the friction laws valid for smooth walls taken as a basis, 
 approximation calculations for the similar solutions. 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
lt-0 NACA TM 13I+U 
 
 9. REFERENCES 
 
 1. Prandtl, L. : Ueber ein neues Formelsystem fur die ausgebildete 
 
 Turbulenz. Nachr.d.Akad.d. Wis sens ch. in Gottingen. Mathematisch- 
 physikalische Klasse aus dem Jahre 19^-5, S. 6. 
 
 2. Wieghardt, K.: Turbulente Reibungsschichten. Monographien uber die 
 
 Fortschritte der deutschen Luftfahrtforschung seit 1939, Gottingen 
 19^6. Teil B5 Oder: Ueber einen Energiesatz zur Berechnung lami- 
 narer Grenzschichten. Ing.-Arch. Bd. 16 ( 19I4-8) , S. 23 1. 
 
 3. Rotta, J.: Das Spektrum isotroper Turbulenz im statistischen 
 
 Gleichgevicht. Ing.-Arch. Bd. l8 (1950), S. 60. 
 
 h. v.Weizsacker, C. F. : Das Spektrum der Turbulenz bei grossen 
 
 Reynolds schen Zahlen. Zeitschr. f .Physik, Bd. 12 It- (19U8), S. 6 lit-. 
 
 5. Heisenberg, W. : Zur statistischen Theorie der Turbulenz. Zeitschr. f. 
 
 Physik, Bd. 12 It- (19^8), S. 628. 
 
 6. Prandtl, L. : Fiihrer durch die Strbmungslehre, 3- Aufl. Vieweg u.Sohn, 
 
 Braunschweig 19^9, S. 105 ff. 
 
 7. Rotta, J.: Das in Wandn'ahe gultige Geschwindigkeitsgesetz turbulenter 
 
 Stromungen. (Erscheint im Ing.-Arch.) 
 
 8. Nikuradse, J.: Strb'mungsgesetze in rauhen Rohren. VDI-Forschungsheft 
 
 36l (1933). (Available as NACA TM 1292.) 
 
 9. Ludwieg, H. : Ein Ger'at zur Messung der Wand sc hub spannung turbulenter 
 
 Reibungsschichten. Ing.-Arch. Bd. 17 (19I+9), S. 207. (Available 
 as NACA TM 128U. ) 
 
 10. Ludwieg, H. and Tillmann, W. : Untersuchungen uber die Wandschub- 
 
 spannung in turbulenten Reibungsschichten. Ing.-Arch. Bd. 17 
 (19^9), s. 288. 
 
 11. Schultz-Grunow, F. : Neues Reibungswiderstandsgesetz fur glatte 
 
 Platten. Luftf . -Forschg. Bd. 17 (19I4-O), S. 239. (Available as 
 NACA TM 986 . ) 
 
 12. Gruschwitz, E. : Die turbulente Reibungsschicht in ebener Strbmung 
 
 bei Druckabfall und Druckanstieg. Ing.-Arch. Bd. 2 (1931), S. 321. 
 
 13. Kehl, A.: Untersuchungen uber konvergente und divergente, turbulente 
 
 Reibungsschichten. Ing.-Arch. Bd. 13 ( 19 1 +3 ) , S. 293. (Available 
 from CADO as ATI 38U29.) 
 
NACA TM I3I4.I1 kl 
 
 1I4. Wieghardt, K. und Tillmann, W. : Zur turbulenten Reibungsschicht 
 bei Druckanstieg. Deutsche Luftfahrtforschung UM 6617 {l^kh) . 
 (Available as NACA TM 131^.) 
 
 15- v.Doenhoff, A. E. and Tetervin, N. : Determination of General 
 
 Relations for the Behaviour of Turbulent Boundary Layers. NACA 
 Rep. No. 772 (19^3). 
 
 l6. Tillmann, W. : Ueber die Wand sc hubs pannung turbulenter Reibungsschichten 
 bei Druckanstieg. Diplomarbeit, Gottingen 19^7 • 
 
 17- Reichardt, H. : Messungen turbulenter Schwankungen . Die 
 Naturwissenschaften, 26. Jg. (1938), S. kok. 
 
 18. Townend, H. C. H. : Statistical Measurements of Turbulence in the 
 
 Flow of Air Through a Pipe. Proc . Roy. Soc . , A. Vol. lV? ( 193M , 
 
 s. 180. 
 
 19. Fage, A.: Turbulent Flow in a Circular Pipe. Phil. Magazine, 
 
 Ser. 7, vol. 21 (1936), p. 80. 
 
 20. Reichardt, H. : Zur-Frage der Schubspannungsmessung in turbulenter 
 
 Stromung. ZAMM Bd. 29 (19I+9), S. l6. 
 
 21. Tillmann, W. : Untersuchungen iiber Besonderheiten bei turbulenten 
 
 Reibungsschichten an Platten. Kaiser-Wilhelm-Institut fur 
 Stromungsforschung UM 6627, Gottingen 19^5- 
 
h2 
 
 NACA TM 13 hk 
 
 Figure 1.- Velocity profiles for constant pressure according to measure- 
 ments of F. Schultz-Grunow (reference 11) on smooth walls and of 
 W. Tillmann (reference 21) on rough walls. 
 
NACA TM 13 kh 
 
 ^3 
 
 Figure 2.- Velocity profiles in case of pressure increase according to 
 measurements of H. Ludwieg and W. Tillmann (reference 10). 
 
uu 
 
 NACA TM 13 H 
 
 * 
 
 > 
 
 >» 
 
 T3 
 fO 
 
 1 
 
 5" 
 
 8T 
 
 CO 
 
 > 
 
 * 
 
 800 
 
 700 
 
 600 
 
 500 
 
 400 
 
 300 
 
 200 
 
 100 
 
 X Schultz-Grunow, dp/dx = 
 o Pressure increase 
 + Pressure increase 
 
 10 
 
 15 
 oo 
 
 l|"| 
 
 20 
 
 30 
 
 5,u, 
 
 H.--I U. 
 
 '12 
 
 v» 
 
 Figure 3.- Relation between the profile parameters I]_ and I„ according 
 to relations (6.16) and (6.17) and according to tests (references 10 and 11). 
 
 H 
 
 12 
 
 = 6 
 
 l/ b 2- 
 
NACA TM 13 hk 
 
 ^5 
 
 30 
 
 25 
 
 ID 
 
 tr 
 
 §• 20 
 
 m 
 iri 
 
 ii 
 
 + 
 
 o 
 
 CD 
 
 A 
 
 10 
 
 ^ 
 
 
 
 ./ 
 
 yf- 
 
 x Schultz - Grunow,d p/d x = 
 + Pressure increase 
 o Pressure increase 
 ■ Pressure decrease 
 
 10 
 
 25 
 
 30 
 
 h = Hi£ 
 
 u, 
 
 '12 
 
 Figure 4.- Connection between the relation v*/Ui, the Reynolds number 
 Re-[_, and the profile parameter 1^ according to relations (6.16) and 
 (6.19) and according to tests (references 10 and ,11) on smooth walls. 
 
1+6 
 
 NACA TM 13^4 
 
 0.05 0.1 0.15 0.2 
 
 0.3 0.35 
 
 yy 
 5 . u , 
 
 Figure 5.- Longitudinal -variation profiles for pressure increase according 
 to measurements by W. Tillmann (reference 16). 
 
NACA TM 13 1*4 
 
 hi 
 
 0.35 
 
 Figure 6.- Longitudinal -variation profiles for constant pressure according 
 to measurements by W. Tillmann (reference 21) on smooth and rough 
 walls. 
 
U8 
 
 NACA TM I3I4-I+ 
 
 24 
 
 23 
 
 ,.*3 
 
 A 22 
 
 21 
 
 20 
 
 19 
 
 
 
 
 
 E 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 * 3 (-0.6 
 
 + 5.75 log Re,) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3.2 
 
 3.4 
 
 3.6 
 
 3.8 
 
 4.0 4.2 
 •log Re, 
 
 4.4 
 
 Figure 7.- Dissipation function for the boundary layer without pressure 
 gradient as a function of the Reynolds number according to measured 
 results by F. Schultz-Grunow (reference 11). E, end point of the 
 measuring series. 
 
 Divergent flow 
 
 Convergent flow 
 
 Figure 8.- Designations regarding the energy theorem (equation (6.28)), 
 
NACA TM l^hk 
 
 ^9 
 
 180 
 
 160 
 
 G 
 
 120 
 
 100 
 
 Figure 9.- Relation between the dissipation function according to 
 
 relation (6.26), the Reynolds number Re^, and the profile parameter 
 
 I-, according to evaluation of measured results on smooth walls. 
 
 Each series of measurements is characterized by a special symbol. 
 E, end points of the individual measuring series. 
 
50 
 
 NACA TM 13^1+ 
 
 
 
 r 6 
 
 
 
 
 
 
 
 1 
 
 1 
 
 3 
 
 4-£- K 
 
 t 
 
 ) 3 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 i \ 
 
 D 
 
 
 
 
 
 
 
 
 \ \ 
 
 
 >o 3 ^ 
 
 
 
 
 
 
 
 
 ^R«i=iC 
 Re,=i0 € 
 
 > 4 —^ 
 
 
 
 
 
 
 
 
 
 
 
 
 -0.4 -0.2 
 
 0.2 0.4 0.6 0.8 1.0 \2 
 
 .m 
 
 1.4 
 
 Figure 10.- Similar solutions of the equations for the turbulent boundary 
 
 layer on smooth walls. 
 
 NACA-Langley - 2-12-53 - 1000 
 
b : 
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