A/fVfArA\^,2'!> 1 CO o < NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1231 REPORT ON INVESTIGATION OF DEVELOPED TURBULENCE By L. Prandtl Translation of "Bericht iiber Untersuchungen zur ausgebildeten Turbulenz." Zeitschrift fur angewandte Mathematik und Mechanik, vol. 5, no. 2, April 1925. Washington September 19, 1949 XORif;^/ Y ^U ill" ^3> > I . WATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECENICAL MEMQRAOTXJM NO. I23I REPORT ON INVESTIGATION OF DEVELOPED TURBULENCE* By L. Prandtl The recent experiments by Jakob and Erk (reference 1) on the resistance of flowing water in smooth pipes, which are in good agreement with earlier measurements by Stanton and Pannell (reference 2), have caused me to change my opinion that the empirical Blasius law (resist- ance proportional to the 7 A power of the mean velocity) was applicable up to arbitrarily high Reynolds numbers. According to the new testa the exponent approaches 2 with increasing Reynolds number, where it remains an open question whether or not a specific finite limiting value of the resistance factor X Is obtained at R = <». With the collapse of Blasius' law the requirements which produced the relation that the velocity in the proximity of the wall varied in proportion to the 7th root of the wall distance must also become void (reference 3)- However, it is found that the fundamental assumption that led to this relationship can be generalized so as to furnish a velocity distribution for any empirical resistance law. These funda- mental assumptions can be so expressed that for the law of velocity distribution in proximity of the wall as well as for that of friction at the waJ.1, a form can be found in which the pipe diameter no longer occurs, or in other words, that the processes in proximity of a wall are not dependent upon the distance of the opposite wall. For the velocity u (time average value) at y distance from the wall only one nondimensional can then be formed, namely, y (v = kine- matic viscosity), giving for u a formula of the form C9(lf) (1) where C is a velocity and cp an arbitrary fiinction. The shearing stress at the wall then must be ^PC2 (2) where C is a constant. * 3ericht liber Untersuchungen zur ausgebildeten Turbulenz.' Zeitschrift fur angewandte Mathematik und Mechanik, vol. 5^ no. 2j April 1925, pp. 136-139. 2 NACA TM No. 12 3I d in y To o'btain the function cp we proceed from the value , , , which "by theory would "be constant = 7 and put ^ ^n y ,. d Zn u where a = In^^. With to u = t) and Zn y = a - t) + Znv^ formula (3) "becomes and, after integration and removal of logarithms, u - Ce>/^-^4^_ (^) "" - ^^ f(a) + 1 which, after including ve'' u gives the velocity profile in parameter representation. p The" empirical law for wall friction reads t= Xpu , with X, a function of — (u = mean velocity, a = pipe diameter) . Dlscovinting for simplicity the difference "between the mean velocity and the velocity in the center ui and assuming for It the value from formula (U) , although it is no longer exactly true in the pipe center, u-|_a we put with ajL = Zn-^y— - ZnX = g(a-L; (6) By (2) and {k) we get i = -^ = ^— --^ = \e 2 21 -*- da ^ = >J f(a) ^ 1 NACA TM No. I23I or do ZnS = - g(ai) + 2/ f(a) + 1 = const. (7) hence after differentiation with respect to a-. With this the protlem is solved and it is readily seen that f(a), which for g'(o) = r assumes the value 7 as before, increases with decreasing g'(a). A more accurate experimental check is awaited, hut even so it is plainly seen that at Reynolds numbers of about 200,000 the 8th root of the wall distance is definitely better than the 7th root- For g'(o) = 1, f(a) = 1, which corresponds to the laminar boundary zone. Furthermore I would like to speak of a formula intended for a hydrodynamic calculation of the distribution of the bn.se flow of a turbulent motion under the most varied conditions. After various fruitless attempts gratifying success was attained. In addition it was found that the formula for the apparent shearing stress T produced by the interchange of momentum, lends- itself to a very clear explanation. In the Boussinesq formula t = per—, e is a measure for the turbulent "exchange" and in its dimension, which is the same as that of V, it is the product of a length and a velocity. The velocity is the transverse velocity w at which on the average the fluid bodies advancing from both sides pass through the layer with the time average value of the velocity = u. The liquid bodies coming from the side of the greater velocity entertain higher values of velocity u, those from the side of smaller velocities, smaller values, with the result that more and more momentum Is transported in one direction than In the other (excepting the point of Umax) • The desired length Z is cheiracterized by the fact that it Indicates the distance of the particular layer, in which the average u velocities, which the liquid bodies have at their passage, are found as the time average value of the flow velocity. Approximated these velocities k NACA TM Wo. 1231 are u + I— and u - I~. (incidentally I is in agreement in order of meignitude with the diameter of the fluid todies (more acciorately it is the decelerating path of the fluid hodies in the remaining fluid, which is, however, proportional to the diameter).) As to the length 2 it can, for the present, only he stated that it must approach zero at the wall, where only todies of smaller diameter than the wall distance can move as discussed. Elsewhere I is to have a very regular distri- bution. If 3 Is the average proportional share of the surface occupied by the fluid bodies entering from one side, a momentum PpwZ— per second passes at this side through the unit surface, and approxi- mately the same amount from the other side. This confirms the Boussinesq. theorem, so we can put e = 23wZ. The next problem is to find a practical formula for the mixing speed w. This mixing speed is rapidly reduced and must be continuously renewed. Hence the assumption that it is produced at the concurrence of two bodies of different velocity u and therefore proportional to the velocity difference^ that Is, the magnitude of I—. With this, however, if all unknown numerical factors are thrown on the not more accurately known length I, the apparent shearing stress becomes T = pl2 du dy du dy (9) dy that Is This formula necessitates a correction for the case that — = 0. dy For the creation of velocity w the neighborhood in a certain width cooperates; it does not become zero when — = 0; it rather can be put proportion al to a. statistical average value of proportional to \j('7~) • If the velocity profile varies in flow direc- tion, as in convergent divergent channels, the points over which the averages are made must be shifted upstream for a certain amount, since the process of development of the velocity w takes time- Formula (7) has already proved Itself in many respects. In a pipe, for instance, the shearing stress is according to the equilibrium con- ditions proportional to the distance r from the center, hence T = ,pZ y = cr NACA TM No. I23I 5 Assuming I as constant gives u = A - Br3/2 (-j^Q) for r > 0, the reflection for r < (fig- 1) • At r = the radius of curvature is then equal to 0; taking instead of — the previously discussed statistical value that can be approximately written as + I' f^^^^l the curvature at r = is finite. The actual velocity distrihutions (reference 1, p. 21) exhibit the singular behavior of the center very plainly (fig. 1). The refinement Just mentioned needs to be applied only where increased accuracy requirements are involved. The total velocity distribution in the pipe is fairly accu- rately obtained when (in the range of the I/7 power law) I is put proportional to [(a - r) (a + r)] '' , a = pipe radius. The foregoing formulas have also been applied to the case of the free turbulence, that is, flows without confining walls, as for example, of a fluid Jet diffusing in a chamber and to the intermingling of a homogeneous air stream with the adjacent air at rest, which is entrained by it. For stationary flows of this type the formula Z = ex is appropriate, x being the distance from the point where the mixing starts. The case reproduced in figure 2 leads to the differential equation for the stream function F('^ j 2cF"f"' ± FF" = which is solved by f" =0 or by 2cF"' ± F = 0. The two solutions of uniform velocity and variable velocity abut in F'" with discontinuity. This and other calculations were numerically carried out by Tollmien, who is to publish an article on it. The agreement of his calculations with experimental data is excellent. Further experimental studies included the velocity distributions in channels of other than circular sections with very unusual results and for which the explanation has not yet been found- Slightly It is approximately so that u' plotted against the cross section of the channel gives a sloping surface. NACA TM No. I23I divergent and convergent smooth -walled channels have also been investi- gated, rough-walled channels are In preparation. It is hoped that our formula will prove itself here also. Translated hy J. Vanier National Advisory Committee for Aeronautics REFERENCES 1. Jakoh and Erk: Der Druckabfall in glatten Rohren und die Durchflussziffer von Normaldiisen. (The Pressure Drop in Smooth Pipes and the Discharge Factor in Standard Nozzles.) Forschungsarbelten des VDI, Heft 26?, 192i<-. 2. Stanton and Pannell: Phil. Trans. Roy. Soc. London (A), vol. 214, I9U, p. 199. 3. Von Karman: Ueher laminare und turbulente Relbung. (On Laminar and Turbulent Friction.) Z.A.M.M. 1, 1921, p. 223. NACA m 1231 u o -► r Figure 1 Figure 2 NACA-Langley - 9-19-49 - 900 Digitized by tlie Internet Arcliive in 2011 witli funding from University of Florida, George A. 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