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L ... 7 7 ..' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1314 ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE* By K. Wieghardt and W. Tillmann Abstract: As a supplement to the UM report 6603, measurements in tur bulent friction layers along a flat plate with rising pres sure are further evaluated. The investigation was performed on behalf of the Aerodynamischen Versuchsanstalt Gittingen. Outline: 1. SYMBOLS 2. INTRODUCTION 3. TEST SETUP 4. TEST RESULTS 5. ON THE GRUSCHWITZ CALCULATION METHOD 6. ON AN ENERGY THEOREM FOR FRICTION LAYERS 7. SUMMARY 8. REFERENCES 1. SYMBOLS position rearward from leading edge of the plate distance from wall velocity component in xdirection velocity component in ydirection velocity components outside of the friction layer density viscosity kinematic viscosity of the air. static pressure *"Zur turbulenten Reibungsschicht bei Druckanstieg." Zentrale fr wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluft zeugmeisters (ZWB), BerlinAdlershof, Untersuchungen und Mitteilungen Nr. 6617, Kaiser WilhelmInstitut fir Str6mungsforschung, G6ttingen, November 20, 1944. NACA TM 1314 q = u g =p + q Q = U2 Ux/v U62/v 6 1 = 1 )dy displacement thickness 0 62 = (i i dy momentum loss thickness C 83 = u 1 ( dy. energy loss thickness B12 = 61/52 H32 = 63/62 form parameters of the velocity profile [u(b2) 2 T turbulent shearing stress TO wall shearing stress c = TO/ U2 local friction drag coefficient 1 mixing length 2. INTRODUCTION For the calculation of laminar boundary layers in twodimensional incompressible flow, numerous calculation methods have been developed dynamic pressure total pressure dynamic pressure outside of the friction layer Reynolds number Reynolds number of the friction layer friction layer thickness NACA TM 1314 on the basis of Prandtl's boundarylayer equation; in contrast, only the semiempirical method of E. Gruschwitz (reference 1) and its improvements by A. Kehl (reference 2) and A. Walz (references 3, 4, and 5) are avail able for turbulent friction layers. The reason, as is well known, lies in the lack of a mathematical law for the apparent shearing stress T which originates by the turbulent mixing of momentum. Prandtl's expres sion for the mixing length so far has led to success only in cases where a sufficient number of correct (and also sufficiently simple) data on the variation of the mixing length can be obtained directly from the geometry of the flow as for instance in the case of the free jet. 'As a basis for the Gruschwitz method there serve, therefore, besides the momentum equa tion for friction layers, three statements which are partly empirical and lack a theoretical basis. They are as follows: A. The velocity profiles in the turbulent friction layer for vari able outside pressure form, after having been made adequately dimension less, a singleparameter family of curves, if one disregards the laminar sublayer; thus, every profile may be characterized by a single quantity (n). B. A differential equation, likewise derived solely from experi ments, concerning the variation of this parameter in flow direction as a function of the pressure variation and of the momentum thickness. C. An assumption concerning the wall shearing stress. Gruschwitz inserts a constant as first approximation. A. Kehl (reference 2) then improved the Gruschwitz method. According to his measurements which extended over a larger Renumber range than those of Gruschwitz, it xi necessary to insert in statement B, for a higher Renumber of the friction layer, a function of the Renumber U52/v instead of a certain constant b. Furthermore, Kehl obtained better agreement between the calculation and his test results by sub stituting in statement C for the wall shearing stress the value which results for the respective Re(52) number at a flat plate with constant outside pressure. Finally, A. Walz (references 3, 4, and 5) greatly simplified the integration method mathematically so that for prescribed variation of the velocity outside of the friction layer U(x) the mom entum thickness, the form parameter n, and hence the point of separ ation can be calculated very quickly. No statement is obtained regarding the wall shearing stress, since it had, on the contrary, been necessary to make the assumption C concerning To in order to set up the cal culation method at all. Thus it seemed desirable to investigate the friction drag of a smooth plate for variable outside pressure. On one hand, direct interest in this exists in view of the wing drag; on the other hand, one could NACA TM 1314 expect an improvement in the above calculation method if an accurate statement regarding To could be substituted for the assumption C. In order to arrive at the simplest possible laws, friction layers along a flat smooth plate were investigated where a systematically increasing pressure was produced by an opposing plate. By analogy with the behavior of laminar boundary layers the wall shearing stress was expected to decrease in the flow direction up until separation, more strongly than in case of constant outside pressure. Instead, To increased, after a certain starting distance, more or less suddenly to a multiple of the initial value. A brief report on this striking behavior of the friction drag has already been published (reference 6). In the present report, these tests are further evaluated and compared with those of Gruschwitz and Kehl. Considerable deviations result in places; however, it was not possible to develop a better calculation method with this new test mate rial either. Since the application of an energy theorem had proved expedient for the calculation of laminar boundary layers (reference 7), a theoretical attempt in this respect was made for turbulent friction layers, too; however, it did not meet with the same success. Merely an interpretation for the statement B can be obtained in this manner, which is. however, not cogent. 3. TEST SETUP The test setup and program have been described in the preliminary report. A new measuring method was developed where with the aid of a pressure rake and of a multiple manometer (reference 8) turbulent friction layers could be measured quickly and accurately and the computational evaluation greatly simplified. Friction layers were measured at p = 0 for two different veloc ities U = const., four cases, I to IV, with rising, and one, V, with diminishing pressure (figs. 1 to 6). In cases I and II, p increases almost in the entire measuring range linearly with rearward position with respect to the leading edge of the plate x, whereas in cases III and IV the outer velocity U (U ~ xa) decreases with a power of x (figs. 1 to 6). The velocities were about 20 to 60 meters per second; the test section length was 5 meters so that Renumbers up to 107 were attained. The Renumber of the friction layer (formed with the momentum thickness) increased to from 2 to 7 X 105. NACA TM 1314 4. TEST RESULTS Mainly the variation of the wall shearing stress had been described in the brief preliminary report (reference 6); later the measurements were evaluated more thoroughly. First, we plotted figures 1 to 6 for the different pressure variations: the outer velocity U, the local friction drag coefficient cf', the displacement thickness 61, the momentum thickness 62, and the energy loss thickness 83 .treated in section 6; in analogy to the momentum loss thickness 62, 53 is a measure of how much kinetic energy of the flow is lost mechanically due to the friction layer, that is, is converted to heat by the effect of the friction forces. In case of constant outside pressure cf' depends, for a smooth plate, only on the Renumber. The test points from figure 1 lie between the formulas for cf' according to L. Prandtl (reference 9), F. Schultz Grunow (reference 10), and J. Nikuradse (reference 11) which in the test range Ux/v = 3 x 105 to 107 differ only slightly. For rising pressure, a slight decrease of cf' with rearward position results at first, according to figures 2 to 5; however, after a certain distance cf' increases more or less suddenly to a multiple of its original amount and decreases again only at the end of the test section where for test tech nical reasons the pressure increase could no longer be maintained. W. Mangler (reference 12) found the same unexpected behavior in further evaluating the measurements of A. Kehl (reference 2). In contrast, cf' varies only slightly in case of decreasing pressure. Thus, on one hand, S> 0 must be responsible for the strong increase in wall shearing dx stress; furthermore, the Renumber of the friction layer is of importance since first a certain starting distance is required. Therefore, we 52 dp Our measure plotted in figure 7 cf' against the dimensionless Our measure ments resulted in a comparatively narrow bundle of curves; however, the results according to KehlMangler, drawn in in dashed lines, cannot be brought under a common denominator in this manner. The test arrangement of Kehl was more general insofar as he had at first a piece of laminar boundary layer and, moreover, in some cases first a pressure drop, and then an adjoining pressure rise; in our tests, in contrast, the friction layer, starting from the leading edge of the plate, had been made tur bulent by a trip wire and the pressure increased monotonically. NACA TM 1314 In want of a better criterion, we can see from figure 7 that the strong increase in friction drag is not to be expected as long as 2 < 2 x 103 Q dx The displacement and momentum thickness as well as the wall shearing stress represent only a summary of the development of a friction layer. Therefore, we shall consider below the velocity profiles and shearing stress profiles. The variation of R against the rearward position from the leading edge for constant wall distance is particularly illus trative (figs. 8 and 9). In case II (fig. 8), u suddenly drops steeply in the layers near the wall whereas it decreases continuously in case IV. Accordingly, the characteristic lengths 61, 82, and 63 increase at these points more strongly than before. Since the drag coefficient cf' depends essentially on the variation of , one recognizes at once in case II the point of maximum wall shearing stress at x f 3.3 meters. The shearing stresses are obtained from Prandtl's boundarylayer equation which may be transformed with the aid of the continuity equation and of the Bernoulli equation 1 dp = UU valid outside of the friction p dx dU u u U2bt layer. With Ux = ux = b, uy = y, and Q = one obtains = Ux+uUx dy (1) by 2Q U U U U U The integration which is still to be performed yields additionally a control value for the wall shearing stress fo Cf' 6 T ody (2) 2Q 2 6y 2Q However, the value for To obtained in this manner is not as reliable as the one calculated from the momentum theorem because here graphical differentiation is applied more often. According to a suggestion by NACA TM 1314 Professor Betz, the momentum theorem may be transformed for the calcil ation of TO in the following manner (compare reference 6): To 0 1 d +H12 2) (1 1 dU (3) H1 + 12 ~12 Or3) 2Q U2+H12 dx U dx 2 where a suitable mean value of H12 = 51/82 is substituted for H12. Then the second term is small compared to the first and plays only the role of a correction term so that essentially only one graphical differ entiation has to be performed. First, the shearing stress profiles are plotted for constant outside pressure in figure 10. The shearing stress T with the appertaining wall shearing stress To and the wall distance y with the momentum thickness 52 are made dimensionless. These dimensionless profiles are almost completely identical although a systematic variation with the Renumber is recognizable. The profiles T against y for the two cases II and IV with pressure rise follow in figures 11 and 13; figures 12 and 14 show the corresponding dimensionless shearing stress profiles T/To against y/52. In the case II where the pressure p increases approximately linearly, the Tprofiles differ considerably for various rearward positions, especially the wall distance at which T reaches its maximum value is subject to great changes. In contrast, the dimensionless profiles in case IV where U f 53(x/lm)0.27 meters per second is valid are very similar. Only the one profile at X = 1.99 meters stands out sharply without any perceptible reason. According to D. R. Hartree (reference 13), similar velocity and hence also shearing stress profiles result for laminar boundary layer in the case U ~ xa since for laminar flow T is simply T = ~u. In the ay turbulent friction layer, the velocity and hence also the shearing stress profiles thus vary in this case; however, the latter at least can be described although somewhat forcibly by a singleparameter family of curves, according to figure 14. A. Buri (reference 14) attempted to interpret the shearing stress profiles in general as singleparameter family of curves. Buri selected for the parameter Pg the computed wall tangent of the dimensionless shearing stress profile r = 2 because for this quantity a relation may be read off immediately from the boundarylayer equation. NACA TM 1314 For y = 0, u = v = 0 and therewith Ia =  d which yields ro dx ( = d. The tangent to the shearing stress profile thus calcu y = dx lated is known in general to fit very badly the shearing stress variation in the proximity of the wall determined from measurements. In the tur bulent friction layer, the velocity u decreases only in immediate wall proximity in the socalled laminar sublayer to zero so that the tan gent direction deviates from the above calculated value even for very small wall distances. This is shown also in figure 15, which represents the shearing stress profiles in wall proximity and the pertaining wall tangents found by calculation; for constant outside pressure, too, the curves against y apparently begin to drop from the point y = 0 out ward with a definite angle rather than continue with a horizontal tangent (fig. 10). Nevertheless, ra could be used at first as a com puted quantity for the characterization of a definite shearing stress profile. However, in case II, for instance, the same shearing stress profile would have to be present for x = 3.19 meters and for x = 3.79 meters according to figure 15, which is obviously not true according to figure 12. In general, the shearing stress profiles could not be characterized by one other parameter alone either. At least two quantities would be required for this, for instance, the magnitude and the dimensionless wall distance of the maximum Tmax/To Finally, we calculated from the shearing stress the mixing lengths according to Pranatl's expression T = pZ2 ylj and plotted them in figures 16 to 19 against the wall distance y or in dimensionless form 2/82 against y/62. Like the thickness of the friction layer, I increases more and more with further rearward movement; in case IV, Z even increases on approaching the end of the test section to 30 milli meters. The diminishing of the mixing length for large wall distances cannot be specified, as is well known, since I there is computed as the quotient of two small quantities, namely, T and ( Here oy again 2 increases linearly in wall proximity. In the tube, the result had been I = 0.4y and at the plate for constant pressure, according to SchultLGrunow (reference 10) 2 = 0.43y (dashed in figs. 17 and 19). Here, at rising pressure, I increases at first again linearly but far more rapidly. In case II, I attains the maximum value 2 = l.ly and in case IV even I = 2.0y. Although no fixed relation exists between l/y in wall proximity and the wall shearing stress, it is striking in figures 17 and 19 that 1/y is largest just for those rearward positions where the wall shearing stress To also (compare figs. 3 and 5) attains its maximum value. NACA TM 1314 5. ON THE GRUSCHWITZ CALCULATION METHOD Our test material enables us to examine the basis of the Gruschwitz method enumerated under section 1. The assumption A according to which all turbulent velocity profiles concerned form a singleparameter family of curves is again confirmed in figure 20. Here a few velocity profiles of the different cases I to V are plotted for which the evaluation accidentally had resulted in exactly the same ratio H12 = 51/52; the profiles u/U against y/52 with equal H12 and hence equal  (compare fig. 27) are in agreement even for different velocities, rear ward positions, and pressure variations, thus also after different "previous history." This singleparameter quality covers, however, only the "visible" turbulent part of the profile; the velocities in the laminar sublayer may have a different distribution even for equal j; otherwise, a unique connection would necessarily exist between the wall tangent ra and n or H12 which is certainly not the case according to what was said above. The change of the velocity profiles characterized by the param eters n or H12 with change in the rearward position is represented in figures 21 and 22 for all measuring series. The weak but throughout systematic dependence of the profile on the Renumber for constant out side pressure is remarkable; Niku adze (reference 11), on the other hand, obtained here always the same profile with Tr = 0.515 and H12 = 1.302. In order to apply the momentum theorem for the Gruschwitz method, the relation between H12 and q, necessarily unique for a single parameter profile class, must be known. The results for all profiles of our measuring series are plotted in figure 23. All of them lie below the curve indicated by Gruschwitz but the majority of Kehl's points also lie below the original Gruschwitz curve. Since, however, all the results do not greatly deviate from one another, the assumption A may be regarded as correct in good approximation. The long dashed curve was calculated by Pretsch (reference 15) for power profiles; the power pertaining to a certain H12 is given in the figure at the right. Concerning the short dashed curve, compare section 6. Matters are different for assumption B. Gruschwitz had obtained from his test evaluation 52 dg(52) an b, with a = 8.94 x 10' and b = 4.61 X 103 Q dx NACA TM 1314 g(82) is the total pressure at the wall distance 52' 2 g(62) = p + 2 (52 Because of 71 = 1 u(62)2/U2 and the Bernoulli equation outside of the friction layer p + Q = constant, we may also write dg(52) d d(Qq) = [p + Q(I a)] = dx dx + 1 dx Kehl, who investigated a larger Renumber range, found b to be addi tionally dependent on U52/v; therefore, he plots 52 dg(52) 62 d(QT) ( b = an = an + (4)  Q dx Q dx against U62/v. This has oeen done for our measurements in figure 24. In this diagram, Gruschwitz obtained a horizontal straight lihe b = constant = 4.61 x 10' and Kehl the plotted curve which, starting from Ub2/v = 2 x 10 slowly drops. These two lines have been drawn solidly as far as measurements existed. Our measurements show that at least for the cases I and II that is, for linear steep pressure rise  b can be assumed neither as constant nor as a unique function of U62/v since we obtain in this diagram two essentially deviating curves. The same result is obtained also in th cases III and IV for higher Renumbers. Only for UB2/v < 10 the deviations of the measuring points may be interpreted as scatter. It is not the Kehl relation but the simpler Gruschwitz relation which is confirmed here. For U52/v < 2 x 103 Kehl took the drawn variation of b in order to obtain significant calculation results directly behind the transition point from laminar to turbulent flow. The two points with U62/v < 10 in the case V are not an argu ment against this bvariation for the reason that the friction layer had been made artificially turbulent to start with by means of the trip wire. Below, we shall attempt a theoretical interpretation of the relation B which so far has been set up and investigated in a purely empirical manner. NACA TM 131h Since the variation of the wall shearing stress proved to be very complicated, according to figures 2 to 6, we cannot improve upon assump tion C which refers to the wall shearing stress. In order to estimate the effect of this assumption on the calculation, the cases II and IV were calculated with the aid of Walz's simplified integration method under different assumptions regarding cf'. Figure 25 shows the result for the momentum thickness 52. In the momentum theorem, we may put H12 = constant since It appears only in one term 2 + H12 so that even great variations of H12 are of relatively small importance. In case II, one obtains better agreement (at least up to x = 3m) between calculation and tests by using the expression for cf' obtained as a function of U52/V for the plate without pressure rise than by putting Cf' = constant. Conversely, the agreement for case IV is better with the assumption cf' = constant. In view of the actual variation of Cf' which varies from case to case, no generally valid rule can there fore be set up for it, even before the region of the steep rise of cf'. The results of the T calculation are represented in figure 26. The three different calculation methods worked out by Walz are based on the following assumptions: GruschwitzWalz b = constant = 4.61 x 103 and Cf' = constant = 4 x 103 GruschwitzWalz b = constant = 4.61 x 103 and Cf' = 0.0251(U62/v)1/4 0.0164 0.85 GruschwitzKehlWalz b = 064 8 log (U82/v) U12/v 300 and Cf' = 0.0251(U62/v)1/4 NACA TM 1314 As was to be expected according to figure 24, the last, most complicated method is precisely the one that shows the greatest deviations compared to the experiment. In our cases II and IV, the first and simplest cal culation method is the best. The agreement between that calculation and the test is still good even in the region of steep rise cf'. From there on, however, large differences result. 6. ON AN ENERGY THEOREM FOR FRICTION LAYERS Like the approximation methods for laminar boundary layers, the Gruschwitz method is based on Kirmrn's momentum equation which is obtained by integration of Prandtl's boundarylayer equation with respect to the wall distance y. One obtains thereby a statement on the total momentum loss of the friction layer. In analogy, a statement on the total energy loss in the friction layer caused by the friction can be obtained if the boundarylayer equation puux + oVUy = Px + Ty = PUUx + Ty (after addition of (ux + Vy) = 0 because of continuity) is first multiplied by u and then integrated with respect to y  u2) dy x d2 p U x dy = UTy dy 0 0 8 e = U2V = U2 [2v 2 2 u j 8 ux dy 0 0 5 dy d  uTy dy = TUdy = dx 0 0 pu 1 U2 dy 0 This equation signifies that the loss in kinetic energy per unit length transverse to the flow direction in the friction layer equals the rate of doing work of the turbulent shearing stresses. and because of + + [p^J2 0 NACA TM 1314 For the laminar boundary layer T = iuy is valid so that f "uy dy = Po u 2 dy here signifies the dissipation, that is, the energy converted to heat per unit time. In turbulent friction layers, in contrast, no simple relation between the shearing stress and the velocity profile exists; if it did, the essen tially singleparameter family of turbulent velocity profiles would have to include also a singleparameter profile class of the shearing stresses which is not the case according to the test evaluation. If we define, corresponding to the momentum loss thickness, an energy loss thickness 5 j (u2 dy (7) and the following dimensionless work of the shearing stresses T (8) e = n = i dy (8) work against the wall shearing stress To 6y U we can write equation (6) also as 1 d (u3 e I= 1 Iu3s3) (9) c ,U3 dx Since we have thus obtained one new equation with two new unknowns, this energy theorem at first does not help us any further either. How ever, we can calculate the energy loss thickness 53 from the velocity distributions and obtain, due to tne singleparameter quality of the velocity profiles, a fixed relation between the quantities 61, 62, and 63. Plotting thus the ratio H32 = 63/52 against H12 = 61/82 for all profiles measured, we obtain figure 27. All points come to lie, with only very small deviations, on one "street." For power profiles u/U = (y/s)n the long dashed curve AH12 H (10) 32 H12 B NACA TM 1314 would result with A = 4/3 and B = 1/3. A good fairing curve for the actual profiles at pressure rise and d2op results if we put A = 1.269 and B = 0.379. On the basis of this empirical relation, C3 may now be calculated from 81 and 52 with sufficient accuracy. If one more relation could be found for the ratio e as well, it would be possible to calculate from the new equation (9) for instance cf'. The calculation of e from the measuring results is made difficult by the fact that it results as the quotient of two quantities which can both be found only by graphic differentiation. This explains the great variation of the evalues plotted for the different cases in figure 28. On the whole, e varies only comparatively little; even when cf' for instance in case IV increases from 3 to 16 x 103, e remain between 0.8 and 1.2. At first glance, e seems to have, according to the defining equation (8), the significance of an efficiency which could not exceed 1. Actually, however, e may assume any arbitrary value, according to the velocity and shearing stress profile concerned; in the immediate proximity of turbulent separation, above all, where To vanishes, e may assume arbitrary magnitude. Because of the inaccuracy of the calculation from the test data, it was not possible to determine for e a relation to the other boundary layer quantities. However, one can set up an interesting analogy between the energy equation and Gruschwitz's assumption B. If one substitutes the above relation for H32 which was found experimentally in the energy d82 equation and eliminates  with the aid of the momentum theorem, one dx obtains Ux B dB12 H12(A 2e) + 2eB  +  c (11) U H12(H12 B)(H12 1) dx 2AH2(H12 1) On the other hand, one may differentiate out the GruschwitzKehl relation (assumption B), equation (4), and iite Ux 1 d1 dH12 b a (12)  + (12) U 21 dHl2 dx 2T regarding r as a function of H12: 9 = 1(H12). We now assume that the relation B follows from the energy theorem. If this is the case, NACA TM 1314 the left sides of equations (11) and (12) must, first of all, be iden tical. We therefore equate, as an experiment, the left sides of the equations and obtain a differential equation for n(H12)l the solution of which reads (12 l)/(1 B) T = number x H122 (12 )2/(l (13) 2/(1 B) (H12 B) From the H32 H12 relation, we had found for B: B = 0.379. If, furthermore, we put, for adaptation to the test results, the number equal to 0.986, the function H12(T) is, according to figure 23, quite well satisfied by equation (13). Thus, the left sides of the energy equation (11) and of the rela tion B (equation (12)) seem to be identical. Then the right sides also must be equal which solved with respect to b results in HI2(A 2e) + 2eB b = a. + cf'T with A = 1.269 AH12(H1_ 1) and B = 0.379 (14) Therewith, a relation between quantities of the velocity and of the shearing stress profile, based on the energy theorem, has been found for the assumption of the Gruschwitz method. In practice, of course, this equation is of no help, either, since no data concerning the newly intro duced quantity e are available although one deals here, according to the defining equation (8), with a comparatively illustrative quantity. Thus one may conclude from equation (14) only that it is improbable that invariably b = constant (as Gruschwitz assumes) or that b depends solely on U82/v (as Kehl presupposes), for b appears here as a func tion of the respective velocity profile (r and H12) as well as of e and Cf' which likewise is not determined merely by U82/V. However, theoretically this conclusion is not cogent, either. It is, in itself, conceivable that the course of the turbulence mechanism is such that, in spite of different previous history, the complicated relations between n, Cf', and e have precisely the properli..s which cause b to be, for every rearward position, lor instance a function of U52/V only and the test evaluation according to figure 24 actually shows that b, at least within a certain region, hardly varies. NACA TM 1314 7. SUMMARY The report deals with measurements in the turbulent friction layer along a flat plate where the static pressure from the leading edge of the plate onward systematically rises or decreases. In case of pressure rise, there results after a certain starting distance, a large increase in wall shearing stress to a multiple of the initial value; this has already been briefly commented on in the report UM Nr. 6603. The further evaluation of the test material (calculations of the hearing stresses and mixing lengths) also gave qualitative information on this problem. As rule of thumb, it'can merely be said that this strong increase in friction drag does not occur as 2 < 2 x 13 is valid. Q dx Furthermore, the empirical relations on which the Gruschwitz method is based are checked with the aid of the measurements. It is again confirmed that the turbulent velocity profiles form with sufficient accuracy a singleparameter family. Gruschwitz obtains from an empirical relation a differential equation for the variation of that parameter in flow direction; Kehl improved that equation by not fixing a quantity b contained in it as constant, like Gruschwitz, but by considering it as a function of the Renumber of the friction layer. The present measure ments confirm at first the simple relation b = constant. However, for higher Renumbers of the friction layer in the region of strongly increasing wall shearing stresses different values for b result, according to the previous history of the friction layer; here a relation of the form b = b(U62/v) is no longer sufficient, either. It is shown that this GruschwitzKehl relation can be interpreted as statement of the energy theorem applied to friction layers. However, this energy theorem which simply signifies that the work of the turbulent shearing stresses equals the loss of kinetic energy in the friction layer does not provide any practical help (for instance for setting up a calculation method). As long as a sufficient statement for calculation of the shearing stresses themselves is lacking, a link between the newly intro duced total work of the shearing stresses and the known friction layer quantities, d displacement or momentum thickness, etc., also is dx lacking. For the same reason, it is not even possible to calculate the wall shearing stress; an approximation value for it must be inserted in the Gruschwitz method. Thus the result of the present investigation is, NACA TM 1314 17 on the whole, negative with respect to the problem of calculating in advance turbulent friction layers; however, the test material represented in the figures might prove useful for further theoretical considerations. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM 1314 8. REFERENCES 1. Gruschwitz, E.: Die turbulente Reibungsschicht in ebener Str6mung bei Druckabfall und Lruckanstieg. Ing.Archiv, Bd. II, Heft 3, September 1931, pp. 321346. 2. Kehl, A.: Untersuchungen ilber konvergente und divergente, turbulente Reibungsschichten. Ing.Archiv, Bd. XIII, Heft 5, 1943, pp. 293329. (Available as R.T.P. Translation No. 2035, British Ministry of Air craft Production.) 3. Walz, A.: Graphische Hilfsmittel zur Berechnung der laminaren und turbulenten Reibungsschicht. LillenthalGesellschaft fUr Luftfahrtforschung Bericht S 10, 1940, pp. 4574. 4. Walz, A.: Zur theore.tischen Berechnung des H6chstauftriebsbeivertes von TragflUgeiprofilen ohne und mit Auftriebsklappen. ZWB For schungsbericht Nr. 1769, March 1943. 5. Walz, A.: N.herungsverfahren zur Btechr'.ng dei jaminaren und turbulenten Reibungsschicht. Untersuchungen und Mitteilungen Nr. 3060. 6. Wieghardt, K.: Ueber die Wandschubsparinung in turbulenten Reibungs schichten bel ver&nderlichem Aussendruck. Untersuchungen und Mitteilungen Nr. 6603. 7. Wieghardt, K.: Ueber einen Energiesatz zur Berechnung laminarer Grenzschichten. (Available from CADO, WrightPatterson Air Force Base, as ATI 33090.) 8. Wieghardt, K.: Staurechen und Vielfachmanometer fur Messungen in Reibungsschichten. Technische Bericht, Bd. 11, No. 7, 1944, p. 207. 9. Prandtl, L.: Zur turbulenten Str5mung in Rohren und langs Platten. AVAEraebn. IV. Lief., 1932, p. 18. 10. ScnultzGrunow, F.: Neues Reibungswiderstandsgesetz fur glatte Platten. Luftfahrtforschung, Vol. 17, No. 8, Aug. 20, 1940, p. 239. (Available as NACA TM 986.) 11. Nikuradse, J.: Turbulente Reibungsschichten. Publication of the ZWB, 1942. 12. Mangler, W.: Das Verhalten der Wandschubspannung in turbulenten Reibungsschichten mit Druckanstieg. Untersuchungen und Mitteilungen Nr. 3052, 1943. NACA TM 1314 19 13. Hartree, D. R.: On an Equation Occurring in Falkner and Skan's Approximate Treatment of the Equations of the Boundary Layer. Proc. Cambridge Phil. Soc., vol. 33, pt. 2, April 1937, pp. 223239. 14. Buri, A.: Eine Berechnungsgrundlage fir die turbulente Grenzschicht bei beschleunigter und verzogerter Grundstr6mung. EidgenSssische Technische Hochschule, Zurich, 1931. (Available as R.T.P. Translation No. 2073, British Ministry of Aircraft Production.) 15. Pretsch, J.: Zur theoretischen Berechnung des Profilviderstandes. Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Avail able as NACA TM 1009.) NACA TM 1314 0.005 C, f U = const. = 178m/s z1V 0.003J .. 12 0.002 D o 8 0.001 4 0 0 2 3 4 X [m] 5 0.005 0.004 6 Cf' I Cf' U cons. = 33.0m/l 6,4[ 0.003 1 /2 0002 0.001I a 4 0 2 3 4 x [m 5 Figure 1. Constant outer velocity, displacement, momentum, and energy thickness; local drag coefficient. NACA T~ 1314 Figure 2. Pressure rise: Case I. Displacement, momentum, and energy thickness; local drag coefficient. NACA TM 1314 Figure 3. Pressure rise: Case II. Displacement, momentum, and energy thickness; local drag coefficient. NACA T' 1314 Figure 4. Pressure rise: Case EI. Displacement, momentum, and energy thickness; local drag coefficient. NACA TM 1314 x> Figure 5. Pressure rise: Case IV. Displacement, momentum, and energy thickness; local drag coefficient. + NACA TM 1314 F 2 Figure 6. Pressure drop: Case V. Displacement, momentum, and energy thickness; local drag coefficient. NACA TM 1314 x Case I + Case FPressure * Coae LZ rJe * Case IV SPressure v Case V I drop Kehl Mangler oAK A K * K Pressure 7a rise 7b # Figure 7. c' against d 2 for pressure rise and pressure drop; comparison with measurements of Keh comparison with measurements of Kehl. NACA TM.131)L 27 o o ot oV  Cd 41j 0, u 4 >o .4 C o o 0 0 4 0 I, I SIz3 28 NACA TM 1314 o 0000 00 0O 0 1 0 i, C 0 0 aF U') o4 rd * oo o 0 0 0 a, ) ,,I :Iz NACA TM 1314 29 Dimensionless shearing stress profiles for constant outer velocity. Figure 10. 30 NACA TM 1314 o Q, 0) x x W x k / 7 ID x /,, / fi^ +*yf NACA TM 1314 31 N P., N a, 'IL r,4 EE E E E E/ B,. / x x l / x/ I LLL NACA TM 1314 A l) Cd .2 U to 0 4 a 4 NACA IM 1314 CO o04O+Ox OO _________ ________ i Li c/^ $4 o 0 a4 0 m 4b *a I ca o 0 Fl I a ho m l~ Nt NACA TM 1314 0 0.2 0.4 06 06 0 Case LN II / /,, T. / / / // //./ / // / i / ox x.437m UO / // a 1987m 1.0 / / 2.567 m / / + 3.187 m x/ /3.47 m o 4.087 m o 4.367 m I f ) .. .. ... .. . Figure 15. Shearing stress profiles and wall tangents (Buri form parameter). Case i I 7/ 0 0.2 0.6 1.0 2 0 NACA TM 1314 35 E f E S E E I o CO co + C O / tr Ca /00 So a S9. 4 / / a ~')s\ ~ o + 0( 0 0 ~iJIfj 36 NACA TM 1314 x CD X cd 0 +  /' Y/ /+ TI V C r f r f rr to x o.L o co co co jno4 "4 0 x 0 iC co CNt C\j C ^ S ^ c C> Q C NACA M' 1314 37 0 eC O) o < + x y 4 o / / an o x / 0 I I r Ia I I NACA TM 1314 In in U2 a) 0 54 0 C. bD a CI V NACA TM 1314 o CO II I t4  0  l 0 o " N \ '6 4u_ u W j ,) &4 'Il co 4. 44 0~ 'I) C"' 91 1 1 C 4~ 08 7Gr 0.6 04 ,^ 0 / 2 3 4 Figure 21. Variation of T for all measuring se oa 12 1.6 t.4 2 0 / 2 3 4 x [rr Figure 22. Variation of H12 for all measuring series. point markings, compare fig. 23.) ies. ries. (Significance of NACA TM 1314 NACA IM 1314 41 +I 0P i \ e\ \ \ o e x + Ba c,> CO, 42 NACA TM 1314 to  I 1 I I I'14 ' 0 (Q Q0 Os g C\j I p qj II qI q )> NACA TM 1314 Pr) a f 0 O V) oo t C3 00 '4 li *4 O  I I) 0 NACA TM 1314 OO C 'cd "1 a :J Cd u CO p I i 1 Co NACA TM 1314 45 I \1I u Sit /0 c'o /o 1 I I i I g / r / 9 0S / / k6 NACA TM 1314 u oa e o 0 a QJ..a "i S4ca 4o It) t / I')*'^ o0 ___^ 1_____ r NACALangley 10451 1000 CF r i . n So *a StOZN t b ZBB& uL. M y N EX 0 I < Wi ;= c.Jo :;3 0 u w GDN g S . 0 c  I'. g< Z a  mZ s?^S~g a H3f c ) aK U G c0 u n 0 m uD GD V) U ca ra iSSio "3 1' ri o Qoj 'C'L U .' C 5 w Lq" S '" 'L SO C: ~'o PO S.2 o jl o O y E . c ll ul: G c u G0 0 .2os. .0 Ig vl 5^! E i. E a g c ia.. n 0 0 Cg "6 ^ ^ .. g 5iai 2 Z Cc S A Go, I ~aE a. oc G o.0 E cc .0 _ i^J^III^ N " I~ I' ~ 1N cca 'j n CQ a I ^,s L^^  5S f?=B^S 6C2?5^S^l ss^Saisia^ 0 inito aP=.S.. 1. 2 56 r, L 7 cu cu B L . <. s~ ~G? 0 "5^ U ^^' <IZ 7 0  , ZZ e P, l> ra ? e 'a iN~ p ilSli Es~ iah fe CM^Li& '<H5 Ln *~d S * L4,asI I aoo '] .... G4) L u Oo Po e L su Sa 0 0S co r w U iL AD r. U t s 5 h ccg a llil i SUESGD 1 0Lu00 iGD5i Gl i GD t V  la ssaIl l^^.ss^ c;Id^ilis (V!( A s vlS rau agB5 gas 7;En fci IS kt 's u. a u <* e 2 (4 gii .s isi o c ii ga 2 a 4?.;.(V U 5 S (4& h( 4mql (V U, aU Va c; CIS W4 .i:5 i^ BQ t f 0 *3 1 bisl g"B62,Sg a(1 gg ll gllz lrlll 3, o 0e d m. = .5 E4 gggma g 2 < B "4?W ta g 4(t S aV2 0 21 C 4?(V( S;x Ua ~U)5 kSw (V.VL 4?UC (4 V < .e mo Po s~'^ I .......... .. . a UNIVERSITY OF FLORIDA 3 126!J 0L L05 799 3 