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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1344 ON THE THEORY OF THE TURBULENT BOUNDARY LAYER SBy J. Rotta : 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 be 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 MaxPlanckInstitute 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 "Uber die Theorie der turbulenten Grenzschichten." Mitteilungen aus .dem MaxPlanckInstitut fUr Stromungsforschung (Gottingen), No. 1, 1950. NACA TM 1344 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 BoundaryLayer Turbulence 4. THE UNIVERSAL TURBULENT BOUNDARYLAYER FLOW 5. EXISTENCE OF SIMILAR SOLUTIONS 5.1. Differential Equations and Boundary Conditions 5.2. Properties of the Similar Solutions 6. EMPIRICAL BOUNDARYLAYER 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 1344 SYMBOLS x,y,z coordinates (x, z parallel to the wall; y perpendicular distance from the wall) p density p of the flow medium V kinematic viscosity of the flow medium Velocities and pressures, stresses: U,V,W velocity components of basic flow (average values in time) (U in xdirection, V in ydirection, W in zdirection) U1 flow velocity in xdirection, outside of the boundary layer u,v,w p Txy, xzyz, TxyT xzyTyz components of the turbulent fluctuation velocities mean value in time of the static pressure fluctuations of the static pressure normal stresses (mean values in time) shearing stresses (in section 3 following Tx is written Tx = T) TO wall shearing stress v* = FOp shearingstress velocity cf' = 2(v*/U) local friction coefficient Turbulence quantities: E kinetic turbulence energy (per unit mass) S energy dissipation (per unit mass) pQ x,PQy pQ components of the energy diffusion (energy flow per unit time and unit area) (in sec tion 3 following Qy is written Q = Q) D =J S dy dissipation function >mean values in time NACA TM 1344 1 characteristic length of the large turbulence elements (statement 1 = KY; K 0.4) k,kq,c dimensionless factors according to equations (3.16), (3.17), (3.18). (Characterized for the universal boundarylayer flow in the range 5w 5 y << by the index "O") Thicknesses of the boundary layer: 5 total thickness 6y thickness of the sublayer directly affected by the viscosity and the wall roughness 81 = (1 U/ l)dy disp 0 82 =f (U/Ul)(1 U/Ul)dy 3 = (u/ul) (U/U) dY lacement thickness momentumloss thickness energyloss thickness H12 1/52 Form parameters H32 53/8 2J Similar solutions: I = y/x m Ulx Re =  x V stream function dimensionless coordinate exponent of the law prescribed at the outer edge (eq. (5.4)) Reynolds number formed with the coordinate x NACA TM 1344 Empirical boundarylayer profiles: > form parameters 12. p d(yvo 12 0 bJ ii Rel  V B U 1 In V* K constant of the velocity profile near the wall (eq. (4.8)) constant of the outer velocity profile (eq. (6.8)) Reynolds number formed with the displacement thickness 61 Re = C + K A form parameter in equation (6.12) G = D/v3 In Re1 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. 1). In spite of these equations, an exact calculation of the turbulent boundary layers 6 NACA TM 1344 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 boundarylayer 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 6, and the thickness of the boundary layer 5, one has therefore, as a rule, 6 << 6. (2) Because of w << 5 one may expect the conditions for wall distances y < 6, to be independent of the flow conditions at the outer edge of the boundary layer (y  5). Inside of the layer by there exists, therefore, a velocity law affected solely by the geometrical properties of the wall; the wall shearing stress TO 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 timeaveraged velocity with the components U, V, and W in x, y, and zdirection is called basic flow. Superimposed NACA TM 1344 on it is the disordered turbulence motion fluctuating with time with the components u, v, and w, which is always threedimensional even when the basic flow may be regarded as twodimensional. 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 NavierStokes equations of motion for turbulent flows and performs the time averaging, a few meanvalue 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 NavierStokes 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 threedimensional 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 u2 ,+2 + 2 E + + (2.1) 2 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 threedimensional basic flow with the NACA TM 1344 components U, V, and W this energy balance reads in the stationary case quite generally: PU LE+ V  + W T o x oy b Total variation of the turbulence energy  A X ay oy SZ + T ( + + dz x y y 3xy +yz Energy withdrawn from the basic flow pS Qy +  + dy (2.2) Dissipation Ox = P 2V a, = (2V az = p (2 Energy diffusion  2) (2.3) are the timeaveraged normal stresses and Therein X u + aw+ TXZ + O ^oz 6x/ aw\ +N 4yj  I6 NACA TM 1344 TLU 6V  pny  + v cy mx/ j y z 6x)z  are the timeaveraged expression shearing stresses. For the dissipation the (2 2 S =v2 ( +2 ( ox 6y z + LxW 6z 6x) + \V 2 12 W _w 2 +  oy/ 2 +2 \Z/ dUz 2 + + 2 avy Oy) I2 oz( 0" ) (b6w + 2 +y oz 'dy 6z +lu +w W2 (6Z ax) is valid, and the components have the following form Qx= E \6x Qz = v ( z \z ax dvu + x 6x + 6 6x Qx' Qy, and Qz u+ OV + u Oy z / sv2 Tby + dz of the energy diffusion J(2 + 2 2 2 P/ + v(2 + 2 + + 2 + Vu2 +i 2 P + 2 P/ (2.4) vdx dx 2 y) (2.5) (2.6) 10 NACA TM 1344 Equation (2.2), the derivation of which would be too lengthy here, is attained by addition of the three NavierStokes 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 xaxis is assumed to lie parallel to the wall and y to be the vertical distance from the wall. The threedimensional turbulence motion with the components u, v, w is assumed to be super imposed on the components U and V of the twodimensional basic flow in x and ydirection. The boundarylayer equationfor a twodimensional flow along a plane wail resulting from the NavierStokes equations of motion then reads with the simplifications introduced by L. Prandtl1 pU + V y   (3.1) \ x oy 6x dy 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 = pO pv2; p = p for y = 0) in the derivation with respect to x this slight influence is partly com pensated by the term u2/6x neglected in equation (3.1) so that 6p/dx may be regarded as independent of y. NACA TM 1344 Furthermore, one uses the continuity equation for the basic flow dU Vy + o bx oy 11 (3.2) The continuity condition must of course be satisfied also by the fluctu ating motion u, v, w which is already taken into consideration in the following formulas. In case of turbulent boundarylayer flows, there applies for the shearing stress T in the xyplane the expression T = p Lu  ( dy U (3.3) Herein uv components expression is the time mean value of the product of the u and v acting vertically to one another. puv is also denoted as Reynolds stress. fluctuation The We now include into our considerations as a further equation the energy balance of the turbulent flows given in section 2. For steady twodimensional boundary layers expressions (2.2), (2.5), and (2.6) are simplified, under the custom simplified, under the customary assumptions to (3.4) pU E + V E = T pS p \ ox oy) y oy +2 L2 +2  S O 2 2 +2 ) \oy / + 2 z 2 wy vdz dy dz) 2 \Oz 6x + (V 2 dax 3y u2 +2 +2 W v + 2 P (3.5) (3.6) 2 See also footnote 1. T is put equal to Txy and Q is put equal to Qy. The indices may be omitted here as before in equation (3.1) and later on, since a confusion is quite impossible. dy Q (E y 12 NACA 4M 1344 :J The boundarylayer equations (3.1), (3.2), and (3.4) have to satisfy the following boundary conditions: y = O: U = 0; V = 0; u2 =v2 w2 = 0 y > 5: U >U; u2  0; v2 0; w2 >0 (3.7) U1 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 4 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 51 = (1 U)dy (3.8) Momentumloss thickness 52 = tu/ UJ)dy (39) and wall shearing stress T0 = T for y = 0, one may derive from equa tions (3.1) and (3.2) the momentum equation (U1 2 + U161 dU1 (3.10) dx. 1 2) 1 1 dx P which has proved to be advantageous for the approximated integration of the boundarylayer equation, and which has the same form for turbulent as for laminar boundary layers. ** . NACA M4 1344 By integration over y one may develop from equation (3.4) a corre sponding energy equation. The partial integration of the left side of equation (3.4) yields, with use of equation (3.2) O'(U E +V 6E dy f S x ay dx0J, UE dy (3.11) Furthermore there applies for the basic flow the relation to be derived from equation (3.1) (cf. the reports by K. Wieghardt, ref. 2) J0O T dy = p ay xJo Ul2 U2)dy (3.12) The diffusion term in equation (3.4) disappears dy since the components u, v, w tend toward zero for for y  After introduction of the energyloss 1 dy "3 =. u1 and of the dissipation function D = S dy "0 in the integration y = 0 as well as thickness (3.13) (3.14) one then obtains as the energy theorem 1 d (U 3 2 dTx_ 53) 0d_ UE dy dx (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 14 NACA TO 1344 4 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 BoundaryLayer 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. (3.4)). 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(6U/dy)2 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 I for the dimensions of the large turbulence elements, which for boundarylayer flows is given approximately by the pertinent distance from the wall. The second is the diameter iR of the smallest turbulence elements, which is determined by the quotient of the kinematic viscosity and the mean fluctuation velocity thus, approximately by v/\/K. 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 of the turbulence is sufficiently V large, which is the case in the region y = 85w, we need in our consider ations only to refer to the one characteristic length I 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) Tv = kEl d (3.16) p oy NACA TM 1344 wherein k is a dimensionless factor. The product k/EZ represents an apparent kinetic viscosity (cm2/sec) or, respectively, an exchange quantity concepts first introduced by J. Boussinesq and W. Schmidt. By considerations similar to those on which equation (3.16) is based, one arrives for the energy transport Q caused by exchange phe ncmena at the expression introduced by L. Prandtl (ref. 1) Q = klEz E 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 SkqiE3/2) 4Q (3.17) ay is justified,3 which originates by taking the exchange quantity kfEZ for the energy transport under the differential sign. Prandtl's form is contained in expression (3.17) as a special case. Here again kq is a dimensionless factor which like k in equation (3.16) 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 cl, one obtains according to equation (3.5), with omission of the contribution v(dU/Py)", the relation E S = vc1  3The expressions (3.16) 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 (3.16) and (3.17) are, at any rate, sufficient. 16 NACA TM 13441 Herein cl is determined chiefly by the small turbulence elements and is independent of the Reynolds number only for very small Cl (cf. ref. 3). V 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. Weizsicker (ref. 4) 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 EZ one then obtains the relation E3/2 S = c (3.18) which is valid for y > 6w. The magnitude of the factor c appearing therein is determined mainly by the large elements. The dimensionless quantities k, kq, and c, (appearing in equa tions (3.16), (3.17), and (3.18)) 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, kq, 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 (3.16), (3.17), and (3.18) for the large turbulence elements, there is, with consideration of the regions near the wall, the expression I = sy (3.19) of advantage where K is a general constant. In this form, the length I is in the region 6w < y << 6 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) one may assume as taken into consideration in the factors k, kq, and c. NACA TM 1344 17 By introduction of the relations (3.16), (3.17), (3.18), and (3.19) into the boundarylayer 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. 4. THE UNIVERSAL TURBULENT BOUNDARYLAYER FLOW For wall distances which are small compared with the boundarylayer thickness 5, the shearing stress does not noticeably deviate from the value TO of the wall shearing stress and the flow conditions are here practically independent of the pressure gradient dp/6x. 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 adjacent to the wall. If this thickness & is sufficiently small, there will certainly exist wall distances y larger than 6w, yet so small compared with the boundarylayer thickness 6 that in this region a universal boundarylayer flow results for which all flow quantities are determined by only two quantities which have physical dimensions, namely, the shearing stress velocity v* =T (4.1) 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 8p/ix 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 5w < y << 5 by the known relation (ref. 6). oy Therein K 0.4 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 boundarylayer 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 wall4 With the exception of the smallest turbulence elements. NACA TM 1344 Owing to this similarity and the equality of the shearing stress TO = 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, kq, c in expressions (3.16), (3.17), and (3.18) become for bw 5 y << 5 generally valid constants which we shall emphasize by the subscript "0" (ko, kqo, Co). With uv = v*2 and I = ny there follows from the relations (3.16) and (4.2) E = (43) (ko/ With this value, the expression (3.17) then yields for the energy trans port Q caused by exchange a value also independent of y k= _v* (4.4) ko 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: 0 2 d (4.5) S = U v (4.5) P Cy oy Hence results with expressions (3.18), (3.19), (4.2), and (4.3) the relation cO = k3 (4.6) which like equation (4.3) was found by L. Prandtl (ref. 1); on the basis of measurements, the value of kO was estimated to be kO = 0.56. NACA TM 1344 The existence of the universal boundarylayer flow in the region w < y << 5 suggests the division of the boundarylayer flow into a part near the wall (0 S y << 5) and an outer part (y > 6w). The flows of both parts merge asymptotically into the universal boundarylayer 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= v*f ) (4.7) wherein the function f 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 > 6B which results from relation (4.2) by integration: U = v* n V + (4.8) Therein the constant C is a function of the wall roughness. The outer part (y > 6,) has to satisfy the boundarylayer equations given in section 3.1; 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 boundarylayer flow, with the velocity at the outer edge Ul(x) and wall properties and viscosity prescribed. For theoretical investigations, the problem may be formulated differently: beside Ul(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 3.1, in this problem the fol lowing conditions for lim y  0 at the wall (y = 0) must be satisfied for the outer part (y > 5w) in order to guarantee the connection with expression (4.8): 2 lim V = ; E = (4.9) y > 0 dy Ky kO 20 NAGA K 1344 Since on one hand the value which corresponds to the local Ul/ friction coefficient cf' v)2 cf' (4.10o) Ul pU2 2 may be estimated quite satisfactorily, according to existing approxi mation formulas (for instance, ref. 10), even without exact knowledge of all boundarylayer 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*/U1 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 Ul(x) prescribed promises some prospect of success also for the firstnamed 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 selfevident and is, therefore, to be checked for the indi vidual case. For this purpose, we add the following orders of magnitude: The thickness w8 is for smooth walls 6, 100 V/v*; the pertinent Reynolds number of the turbulence for y = 6, is EZ/V ~ 100. For pronounced wall roughness, bw is determined by the dimensions of the roughnesses. According to the experiments of J. Nikuradse (ref. 8) on sandrough pipes, Ew is approximately equal to the grain size of the roughness; the yvalues 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 socalled 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 boundarylayer equations to be expected, with neglect of the viscosity, in the range y >w In order to satisfy the continuity NACA TM 1344 condition, we introduce for the basic flow the stream function from which the components U and V are derived5 by the relations U = y; V = Ix After substitution of this function into the equation of motion (3.1), there follows 1  yTi lx r =  p (uv) (5.1) y xy x yy p x y and the energy equation (3.4) assumes with the relations (3.17), (3.18), and (3.19) the form Ex xE = u cE3/2 + (kqyE3/2)yy (5.2) Finally, one obtains for uv according to equation (3.16) uv = Kk Eyijyy (5.3) It may now be shown that for velocity distributions prescribed at the outer edge of the boundary layer of the form U1 = axm (5.4) with a and m being constant quantities, and for a prescribed constant v*/Ul there exist similar solutions for relations (5.1), (5.2), and (5.3)b. If the flow is unaffected by the viscosity, the geometric similarity of the flow pattern requires that the boundarylayer thickness for similar Partial derivatives with respect to x and y in this section are expressed by subscripts x and y. 6Constant v*/U1 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. NACA TM 1344 solutions increase linearly with x7. For this reason, we introduce the dimensionless variable n = y/x and make the statements 4 = axm+l[i f(Ti E =a2x2m() > u, = a22mt(T) (5.5) f(Tq), p(q), 0(D) are only functions of the variable 1. In satisfy the prescribed boundary condition (eq. (5.4)) in case y, f(n) must for large q tend asymptotically toward a value that is, lim f'(r) = 0 must be time. Thus, there Tj 4 m follows from equation (5.1) for q 4 m  p = a2,x2m1 P x (5.6) results also directly from relation (5.4) and Bernoulli's equa After substitution of equations (5.5) and (5.6) into equa (5.1) to (5.3), one obtains after division by a2x2ml a2x2m, a3x3m1 respectively 2mf' mf2 (m + 1)(T f)f" = (1 f')2m0 + (1 + m)(f I)s' = _f" 3/2 2 kq 3/2 cp = KkTnIf" 'p1 (5.7) 'The same results also from the momentum theorem (eq. (3.10)). Therein order to of large constant which tions. tions NACA TM 1344 In this system of ordinary differential equations, x explicitly so that one actually has to expect systems equations (5.5) where the boundarylayer thickness 5 ary ith x arly with x no longer appears of solution of increases line For unequivocal determination of a solution, five boundary condi tions must be prescribed. In order to satisfy the three boundary condi tions (relations (4.9)) at the inner edge, one has to put lim f' = ; S4 0 KT f(o) = o; o(0) .ko2 with 90 = c(0). Two conditions at the outer edge of the boundary layer are added: lim ; n * f' = 0; S= 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 (0 which like thelocal friction coefficient cf' is according to relation (4.10): cf TO 2 pU12 (5.10) The solution of the system of equations (5.7) with the boundary condi tions (relations (5.8) and (5.9)) is, therefore, a twoparameter curve family. The velocity profile of the outer part, most interesting in these solutions, may be represented in the form Uv U v* f'(Tl) JSO (5.11) .5.8) For instance, the displacement thickness 81 according to equa tion (3.8) is: 61 = xf(o). 9 Ul0 24 NACA TM 1344" and is dependent on the two parameters m and v*/U1. Likewise, there results, of course, for the pertaining "turbulence profile," that is the plotting of the kinetic turbulence energy E ( () (5.12) V*2 CPO over 1, also a twoparameter curve family dependent on m and v*/UI. 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 (socalled "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 boundarylayer 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 w, must be supplemented by the wall profile (relation (4.7)) in order to obtain from it the complete velocity profiles. The condition for the continuous junction of the outer part to relation (4.7) is obtained by elimination of the quantity U/v*, with the aid of relation (4.8), from the asymptotic form U1 + K \ (5.13) V* K MUl. resulting for small nvalues by integration of f" according to rela tions (5.8). In this manner, one obtains U1 1 U1 v*\ 1 Ulx + in Km, v = n' + C (5.14) v+K v* K = The constant K(m, v*/U1) in equations (5.13) and (5.14) may bE deter mined, in case of prescribed parameters m and v*/U1, from the system of equations (5.7). The solutions of the outer part herein discussed have real signifi Ulx chance only when the Reynolds number Rex = and the wall roughness, the effect of which is expressed in the quantity C, are such that equa tion (5.14) is identically satisfied for all xvalues. NAC A TM 1344 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)9, so that the right side of equation (5.14) 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. (5.14)) cannot be rigorously satisfied for all xvalues. This wbuld 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 (5.14) appears in logarithmic form, it will be permissible to regard, for sufficiently large xvalues, the expression on the right side of equation (5.14) as approximately con stant from xinterval to xinterval 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 5w 5wv* U1 V x V v* Ulx be so small that the function (U1 U)/v* at the point y/x = 5w/x actually deviates only slightly from the asymptotic form (eq. (5.13)). Otherwise, the method selected, joining the wall law (eq. (4.7)) 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 boundarylayer 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 boundarylayer part depend on v*/U1. On the problem regarding the extent of this influence, which is one of the nexttomostimportant 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 6w) so that the viscosity may be neglected in the boundarylayer equations; aside from the customary boundarylayer simplifications no For sandrough walls, there applies, for instance, according to the experiments by J. Nikuradse (ref. 8): C = 8.5 1 In *k. K V 26 NACA TM 3134 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 kq. Thus, one would, for instance, insert constant values for the factors c, k, kq. 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 refss. 9 and 10), a sorting of the experimental data can be undertaken with greater success than was so far possible. 6. EMPIRICAL BOUNDARYLAYER 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 = 0 in the system of eqs. (5.7)), a family dependent on the local friction coefficient, thus a singleparameter family, would result. Ul U , However, the plotting of 1 against y/5 according to F. Schultz v* Grunow (ref. 11) shows that the profiles within the considered Reynolds number range may be represented with practically sufficient accuracy by a single curve. Since the boundarylayer thickness 6 used for the plotting is a quantity which can hardly be exactly defined, the expres sion yv*/!(1Ul) instead of y/5 is introduced as reference value with use oP the displacement thickness 51. Thereby the abscissa scale is fixed so that the integral value becomes SUl Ud i 1 (6.1) O v4* 5U11 as one can see from a comparison with equation (3.8). In figure 1, the UI U values for the flat plate without pressure gradient were plotted yv* against log The test points of the smooth plate according to 51U1 reference 11 fall almost into a single curve; nevertheless, close.obser vation shows a small systematic influence of the value v*/U1. 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*/U1. This NACA TM 1344 27 investigation admits the conjecture that the magnitude of the local friction coefficient is, in case of appropriate normalization of the yscale, of only moderate influence on the velocity profile. The boundarylayer 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*/U1. Performance of approximation calculations requires by no means knowledge of the boundarylayer 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 51, momentumloss thickness 62, energyloss thickness 53, and others). Various authors refss. 10, 12, 13, 14, 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 boundarylayer 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 Ul(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 I.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 singleparameter repre sentation is no more than an approximation as opposed td the fact that, according to the theory, even in the simplest case of similar solutions a twoparameter 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 yscale 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 momentumloss thickness 52, there applies according to equation (3.9) 2 f ( 1 dy Od= dy fl1 U121 dy (6.2) NACA TM 13 44' which may also be written as (6.3) 62 = 61( I i) The value l1 2 (6.4) is, under the assumptions, made practically independent of the velocity distribution for y < y5. Likewise there results for the energyloss thickness according to equation (3.13) : dy = 2 f 1 dy J0o(1  U2 dy + Ul) 3 1 dy Ul (6.5) 3 = 5 2 3 U I + I2 Uo L d ( also is practically independent of the profile form for The profiles of the representation figures 1 and 2 to equation (5.13) for y ) 6y the form U1 U 1 Iyv + K v* K 61U1 (6.7) y < 8w. have according (6.8) 53 = 0" where (6.6) *ii d 61 b 15 0 3 1 NACA TM 1344' 29 with the quantity K of a different amount for every profile form. .Since, on the other hand, the velocity profiles have for small yvalues (y = y << 6) the form of expression (4.8), there results by substi tution of expression (4.8) into equation (6.8) U1 1 1 In Rel + B (6.9) K if UlI1 Re1 = (6.10) is the Reynolds number formed with the displacement thickness and B is B = C + K (6.11) The quantities I1, 12, 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 5w << 6 is sufficiently satisfied. For a singleparameter 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 I1 as characteristic form parameter and express the others as function of Il. 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 5w) is represented by a simple approximation expression which starts out from the wall law (eq. (4.8)) U = v* (zn y A) ] (6.12) A is a freely selectable constant. The thickness 5 of the boundary layer is defined for y = 6 by the condition U = Ul, so that the quali fying equation for 5 reads U = v* In 6 + A) + (6.13) 30 NACA TM 1344 From expressions (6.12) and (6.13), there follows for the outer part of the velocity profile U1 A y\ 1 yn (6.14) v* K 8/ K & For the displacement thickness, there results hence with equation (3.8) 1I U 1 + Al = l  U d () 2 (6.15) 6v* v* 5 K Furthermore, the quadratures yield: 2 3 1 2 1 5U 1 U U ) U 2 + A + A 3 21 11 2 1 3 12 1U1 J Y d 6 (6.17) 2 1 + A 2 Thereby the relation between II and 12 is given by a parameter representation which is plotted in figure 3 for K = 0.4 and compared10 with the quoted measurements by H. Ludwieg and W. Tillmann (ref. 10) and F. SchultzGrunow (ref. 11). For the quantity B in equation (6.9), 1 + A B = C + A n 2 (6.18) K K K 10The relatively large scatter of the test points for small I1values is without practical significance because the term in equation (6.6) dependent on 12 contributes only a very small percentage to 53; the scatter is explained by the fact that the experimental Il and 12values" were not obtained directly by quadratures but were calculated backward from the 51, 52, and 53values determined by quadratures with use of the experimentally ascertained v*/Ul from equations (6.3) and (6.6). NACA TM 1344 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 Bvalue is not satisfactorily represented by equation (6.18); the modified form A S+A B = C + 0.82 A 1 Zn 2 (6.19) I K It K is more appropriate, as shown by the comparison with measurements repre sented in figure 4 for C = 5.2 and a = 0.4. Figures 3 and 4 may be regarded as a confirmation for the usefulness of the representation dependent only on the form parameter I1 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*2 over (yv*)/(561U). Herein E/v*2 tends in the neighborhood of the wall y > y5 toward the universally valid value given by expression (4.3) and decreases for y < bw very rapidly to zero. Although it is fundamentally possible to determine, with known hotwire arrangements, the quadratic mean values of all three fluctuation components experimentally and hence, according to relation (2.1), E numerically, valuable measurements exist only for the component u which, it is true, yields the most essential contribution to E. The longitudinal oscillation profiles u/v* represented in figures 5 and 6 were measured by means of the turbulencemeasuring device of W. Tillmann refss. 16 and 21) developed by H. Schuh. The conjecture following from the universal boundarylayer flow and 5 << 6 that the u/v* tends toward a universal value in the same manner as E/v*2 for y 4 5w, is only insufficiently confirmed by these measurements. The reason probably lies in inadequacies of measuring technique. According to the considerations of section 5, twoparameter curve families for E/v*2 would result for the similar solutions. If, how ever, the influence of the one parameter v*/U1 on the velocity profile is small, which is probable according to the preceding section, one may 32 NACA TM 134 ': 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 longitudinaloscillation 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 proSiles may, like the parameters of the velocity profiles, approxi matively be described by an unequivocal relation to the form param eter II. 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 twodimensional boundarylayer 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. Townend (ref. 18) in a square pipe, and by A. Fage (ref. 19) in a circular pipe, the order of magnitude of the v and wvariation components can be estimated. Near the wall, the vcomponent in particu lar is essentially smaller than the ucomponent; at larger distance from the wall, the magnitudes of the v and wcomponents approach that of the ucomponent. According to definition of the integral expressions practically independent of the wall law (eq. (4.7)) l, E d and I w U1 U E d(yv* (6.20) 0 v2 1U12 0 v2 the turbulenceenergy flow is determined to be f UE dy = 51U12v* U I T (6.21) For the presupposed singleparameter condition, IT and IT2 are only functions of II. According to the existing data, the relation f UE dy = 0.6551U12v* (6.22) seems to be useful for the estimation independently of 11; it is, there fore, taken as the basis for further evaluations and calculations. NACA TM 1344 33 6.3. Dissipation Function For determination of the dissipation function D occurring in the energy theorem (eq. (3.15)) according to expression (3.14), one will again attempt by means of the results represented in section 4 to express separately the influence of the viscosity and of the wall rough ness. For this purpose, one may determine S from equation (3.4) and perform the quadrature for the part near the wall (0 < y < ) if one puts T/p = v* = Const. and the left side of equation (3.41 equal to zero which is admissible for small wall distances. One then obtains for y << \ S dy' = v2U(y) Q(y) (6.23) 0 With U according to relation (4.8) and Q according to relation (4.4), there results, hence, if the upper integration limit lies in the region ,w < y << SS dy = v n + C + K (6.23a) 0v k 3 For the outer part of the boundary layer y by one obtains with the relation (3.18) S S dy' = / c d 8 (6.24) y yv*/l8U1 (y'v)/( ) / Since, for 6by y << E/v2 = l/k02 is valid according to expres sion (4.3) and cO = ko3 according to relation (4.6), there results from equation (6,.24) SS dy' = v J 1 In Y* (6.24a) wherein the value of the integral expression J1 E/v 2 3/2 y 1 y (.25) J= c d(Y'V* + 1 Zn (y (6.25) is = yv*/1U1 (y,) 1) \10l/ 511u I 1U 1 1 NACA TM 1344 because of cO(E/v*23/2 = 1 is independent of the lower integration limit y if it lies in the range 6wy y << 6. From equations (6.23a) and (6.24a), one finally obtains for D the form D = S dy = v*3(. In Rel + G (6.26) with k G = C + + K (6.27) ko3 We now again assume that for our singleparameter velocity and turbulence profiles Js, and, therewith for equal wall properties, G as well, is only a function of the parameter Il described by expres sion (6.4). Since nothing is known regarding the behavior of the func tion c, except for the region 5w y << 8, J, 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. SchultzGrunow (ref. 11) 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 boundarylayer 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. 16). These secondary flows have the effect that the flow is not twodimensional (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 wedgeshaped flow may be ascertained; it is given herein without derivation (compare fig. 8) 3UE dy 1 d( 38)1 U13 3=D (6.28) + f UE dy + (6.28) 2 d 2 X0 + x dxJ o X + X NACA TM 1344 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 1 which is to express summarily the X0 4 X secondaryflow 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 manner11. 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 2. For large Ilvalues, the results may be approximatively rendered by c = 7.5(I1 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 hotwire 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. 11The 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 4, respectively. 12Particularly 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 1344 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 hotwire probes, were made by H. Reichardt (ref. 17) and H. K. Skramstadl3. 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. NUMERICAL 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 (5.4)) and the boundarylayer thickness 6 increases linearly with x, we make the statements U, = ax" 52 = bx therein a and b are quantities independent of x. If one takes into consideration that the form parameters of the velocity profiles H12 = 51/52 and H32 = 53/52 are, according to presupposition, also independent of x, there results by substitution of equations (7.1) into the momentum equation (3.10) and after division by the value (axm)2 2 (2m + l)b + mbfH12 = ( (7.2) VU1 13National Bureau of Standards, Washington, D. C., USA. NACA TM 1344 37 From the energy theorem (eq. (3.15)), one obtains in the same manner, with use of relation (6.22) 8R32 v* 3 b 0.65 . H,1(3m + 1) = (7.3) 2 U1 Ul1 v*3 For the momentum loss thickness 62, there follows from equation (7.2) 52 (v*/U)2  = b = (7.4) x (2m + 1+ mH12) 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 boundarylayer 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 (7.4), solved with respect to m: H32/2 (v*/Ul)(D/v*3 + 0.65H12) m =(7.5) 3H32/2 (v*/U1)2 + H2)D/v*3 + 1.95H12 This equation is evaluated by calculation of the quantity v*/U1 for assumed values of the Reynolds number Rel = U151/v and of the profile parameter I1 with the aid of the relation (6.9) and figure 4. From equation (6.3) then results H12 = 51/62 With equation (6.6) and fig ure 3, one may then proceed to calculate 63/61 and therewith H32 = 63/52. Equation (6.26) and figure 9 make, furthermore, the deter mination of D/v*3 possible. With these quantities, it is finally possible to determine from equation (7.5) the exponent m, from rela tion (7.4) the momentumloss thickness 62/x referred to x, and from relation (4.10) the friction coefficient cf' = 2(v*/U1)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 mvalues. This state of affairs is not immedi ately evident from the system of equations (5.7); it follows, however, 38 NACA TM 1344 at once from the momentum theorem. If 62 and cf' are to be positive, one will according to equations (3.8) and (3.9), because of U/U1 1, always have 51 > 62; thus, H12 > 1. Negative values of 62 and cf' can occur only when reverse flow appears near the wall. However, in this case, the boundarylayer theory loses its physical significance since the flow separates from the wall. According to relation (7.4), m must therefore be greater than 1/3. Figure 10 shows that the sepa ration is to be expected approximately in the range of m = 0.2.14 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 wellknown 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 H12 coordinated to a certain mvalue on the Reynolds number. The smaller the Reynolds number, the larger is H12. This dependence comes about chiefly due to the fact that the wall law (eq. (4.7)) corre sponding to the respective Reynolds number is adapted, according to equa tion (6.9), to the singleparameter profile of the outer part (y _? 6w) which is independent of Reynolds number. In this manner, the firstorder effect of the Reynolds number on the velocity profile is included so that figure 10 actually is based on a twoparameter profile family. The dependence of the outer profile parts on v*/U1, theoretically proved in section 5, may be 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. (7.1)) and the parameter H12; 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 HI2 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. The test data at disposal is insufficient for exact determination of the mvalue corresponding to separation. NACA TM 1344 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 bw at the wall, there exists within the turbulent boundary layer a region (y, w 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 Ul(x) prescribed at the outer edge. The presence of this universal boundarylayer flow enables the division of the boundary layer into a part near the wall (0 5 y 5 w) which is affected only by the viscosity and the wall prop erties and into an outer part (y 5w) 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 boundarylayer equations, it can be proved that, for a prescribed velocity distribution U1 = a xm 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 boundarylayer 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 ... :... 40 NACA TM 134 .'. 9. REFERENCES 1. Prandtl, L.: Ueber ein neues Formelsystem fur die ausgebildete Turbulenz. Nachr.d.Akad.d.Wissensch. in GCttingen. Mathematisch physikalische Klasse aus dem Jahre 1945, S. 6. 2. Wieghardt, K.: Turbulente Reibungsschichten. Monographien Uber die Fortschritte der deutschen Luftfahrtforschung seit 1939, Gattingen 1946. Teil B5 Oder: Ueber einen Energiesatz zur Berechnung lami narer Grenzschichten. Ing.Arch. Bd. 16 (1948), S. 231. 3. Rotta, J.: Das Spektrum isotroper Turbulenz im statistischen Cleichgewicht. Ing.Arch. Bd. 18 (1950), S. 60. 4. v.Weizsacker, C. F.: Das Spektrum der Turbulenz bei grossen Reynoldsschen Zahlen. Zeitschr.f.Physik, Bd. 124 (1948), S. 614. 5. Heisenberg, W.: Zur statistischen Theorie der Turbulenz. Zeitschr.f. Physik, Bd. 124 (1948), S. 628. 6. Prandtl, L.: Fuhrer durch die Strbmungslehre, 3. Aufl. Vieweg u.Sohn, Braunschweig 1949, S. 105 ff. 7. Rotta, J.: Das in Wandnahe gultige Geschwindigkeitsgesetz turbulenter Stromungen. IErscheint im Ing.Arch.) 8. Nikuradse, J.: Stromungsgesetze in rauhen Rohren. VDIForschungsheft 361 (1933). (Available as NACA TM 1292.) 9. Ludwieg, H.: Ein Gerat zur Messung der Wandschubspannung turbulenter Reibungsschichten. Ing.Arch. Bd. 17 (1949), S. 207. (Available as NACA TM 1284.1 10. Ludwieg, H. and Tillmann, W.: Untersuchungen uber die Wandschub spannung in turbulenten Reibungsschichten. Ing.Arch. Bd. 17 (1949), S. 288. 11. SchultzGrunow, F.: Neues Reibungswiderstandsgesetz fur glatte Platten. Luftf.Forschg. Bd. 17 (1940), S. 239. (Available as NACA TM 986.) 12. Gruschwitz, E.: Die turbulente Reibungsschicht in ebener Stromung 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 (1943), S. 293. (Available from CADO as ATI 38429.) NACA TM 1344 41 14. Wieghardt, K. und Tillmann, W.: Zur turbulenten Reibungsschicht bei Druckanstieg. Deutsche Luftfahrtforschung UM 6617 (1944). (Available as NACA TM 1314.) 15. v.Doenhoff, A. E. and Tetervin, N.: Determination of General Relations for the Behaviour of Turbulent Boundary Layers. NACA Rep. No. 772 (1943). 16. Tillmann, W.: Ueber die Wandschubspannung turbulenter Reibungsschichten bei Druckanstieg. Diplomarbeit, Gottingen 1947. 17. Reichardt, H.: Messungen turbulenter Schwankungen. Die Naturvissenschaften, 26. Jg. (1938), S. 404. 18. Townend, H. C. H.: Statistical Measurements of Turbulence in the Flow of Air Through a Pipe. Proc. Roy. Soc., A. Vol. 145 (1934), S. 180. 19. Fage, A.: Turbulent Flow in a Circular Pipe. Phil. Magazine, Ser. 7, vol. 21 (1936), p. 80. 20. Reichardt, H.: ZurFrage der Schubspannungsmessung in turbulenter Str6mung. ZAMM Bd. 29 (1949), S. 16. 21. Tillmann, W.: Untersuchungen uber Besonderheiten bei turbulenten Reibungsschichten an Platten. KaiserWilhelmInstitut fir Stromungsforschung UM 6627, Gbttingen 1945. NACA TM 1344 < K 5u UU 10 *U rv /< u, t * o Smooth v /U =0.044 a Smooth v/UI= 0.040 4 6 Smooth vU, = 0.037 0 ^ Smooth v'UI = 0.036 4 x Rough vUi = 0.060 O a Rough v U J=0.054 .~ 2 + Rough vU,1 0047 2.4 2.2 2.0 1.8 1.6 1.4 1.0 0.8 0.6 .4 W" log , Figure 1. Velocity profiles for constant pressure according to measure ments of F. SchultzGrunow (reference 11) on smooth walls and of W. Tillmann (reference 21) on rough walls. NACA TM 1344 c log  6,U, uIu t Figure 2. Velocity profiles in case of pressure increase according to measurements of H. Ludwieg and W. Tillmann (reference 10). 44 NACA .TM 1344. 700 600 ,In II N 500 400 300 200 100 0 Sx Schultz Grunow, a p/d x = 0 * 'I o Pressure increase + Pressure increase 5 10 15 20 25 31 0oo F,=j\ v*/ \2,u,/ H,2 v Figure 3. Relation between the profile parameters I1 and 12 according to relations (6.16) and (6.17) and according to tests (references 10 and 11). H12 = 1. 2' K= Q4 *NACA TM 1344 S 0a 0 o.  + t K =0.4 G=5.2 25 PO x Schultz Grunow,8p/d x 0 10  ,++ + Pressure increase So Pressure increase 5 0 Pressure decrease' '"_____ H I Ui H12 vW Figure 4. Connection between the relation v*/U1, the Reynolds number Rel, and the profile parameter 11 according to relations (6.16) and (6.19) and according to tests (references 10 and 11) on smooth walls. ' V NACA TM 1344 0 0.05 0.1 0.15 0.2 SIUI Figure 5. Q3 0.35 Longitudinalvariation profiles for pressure increase according to measurements by W. Tillmann (reference.16). NACA TM 1344 Yr. p r 1 V* f o Smooth v /UI 0.036 3.5 ___ Rough v/U= 0.048  x Rough v'Ul= 0.052 I O 0 0.05 0.1 0.15 Figure 6. Longitudinalvariation profiles for constant pressure according to measurements by W. Tillmann (reference 21) on smooth and rough walls. S y v 6 U, 0.35 ~hLLLLL ~~a~ah~J \~k~~ 48 NACA TM 1344, '.', D 1 +.U 4.2  log Re, Figure 7. Dissipation function for the boundary layer without pressure gradient as a function of the Reynolds number according to measured results by F. SchultzGrunow (reference 11). E, end point of the measuring series. Divergent flow * Flow direction _ T 77~' Convergent flow Figure 8. Designations regarding the energy theorem (equation (6.28)). E 0 7 D =v3 (0.6+5.75 tog Re,) .2 3.4 ..b 3.am " NACA TM 1344 20 Figure 9. Relation between the dissipation function according to relation (6.26), the Reynolds number Rel, and the profile parameter 11 according to evaluation of measured results on smooth walls. 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(CU a) 0 ,P 'Wa) ~i (U Ir .0 r e o c .n a b u < 2 o 0. 0 Us *1 as a) S Cb  I  DWI. r .01 0% 0 a . 05 Cu rt0 i L .C! > La I ii $5m c GO d  9.0 z S crz~QInO  N m 34 CL br, a b . 0 Cu "v .0 E 2, o ^B5 25 E S. mtw w fl 02W '~ .2 0 Hd L 2 uOu ~t 0 c zzolo6u A a m Wr ai 1 i Z F.  B e z . zzs 0 ^1^U,^S U Cu 0 C 6Cu06 i Z ^ S e E02%U~u 0601 E F Go MEh0 o CU g c. 2 .6 '0 > 4c c a, U N * 2 ig A 12 CL 6 .) N' 0 w cc '0 3 *g .s a j r.0 aj. J Q. 0 0 6 g SL E  Cu 4, 3 ; 19 Q0 ^ V' o.'CIC Cu g ., I s WE: l s a^, ilii^5 N C. u V m u r 0. c~ S b' iMS <1*3C 0 D o a CCu 2 I .Ca a .S rI 04 02  b. 0 0rzZ75 02 f2 w 6 gd4 p in a,P~ r ti 0 0 E o CJ IC g .40 Z a. .M  ZZO: O'a P~ rr > 1<* . 5lllSi ^g5  ^Iliiul 0 ' cd axB cd V>S t $4> I^ 'g c uc i s w 0P 35 >>, S III *t> Z z c 0 a lagls^Scdsg CE o ia C ia ru a S m 0$4i 1, 0t a UN v01a Na N.C !,I a= 0 : s i S^ 0 c, ay ~~~c $4 Cu *0 oe cg 0o .0 C d $40 .a Cuu DC 0 t C C, 0 *r0 Cu 0 0' t<3 'W .2 u , .a ' 'adl i  14 D _l l to! G, k, 0 Co C d 0 d QC~ & 54m~ 0. U...  I I 1 (Lu :., ,5 " =41 01 .a ol c c~ I1 I .L II P9E _ o C u Ce o a *  0, d .r 2 x 0 s 2 :5 z CJa iC  04 V 4 H >< 0 5 2!~c < guI m d .a C d:C$ k il 0l 5? .0^' t% S j..4o1 q I r.0 Cu uou>slg boaa E02 .23 E*t o o o a k 0 InI PP zzOiuooE 0 $4 c$ t ' '0 *I 0 d1 M.2 02"' fl'~ S 0  $ .V : C~ S D'a 0 g >. 3: .a a Er ps $i^ ^^s hi C< $4ra 0 '0 > 0< (D 0 V > > a S4 a ahi o h a m B o F>,g 1 0 > U N 2vr F4 .Bi:= i2Z S O C .0 .0~ C . Cu a C 1 3 aX ," 'U m F C 0 al$ N io....s sC*u'aNCs'su m s5 $4C e ~ o au u$ o = ' 4 0 g ka a SC *% '0 0 5. a.= ~ 0 ti u ai U.. Bh r easS^ .a8 I  4u 4) 5 0 Cd ci 4) ga 0 U 00 hfl i* co o 1 0 ci   $4 41 01 0 o 0 k io OC is .00." UNIVERSITY OF FLORIDA 31 I III 11111105 82 8 3 1262 08105 829 8 