REG NACA-TR-D-1053-1951 Investigation of turbulent flow in a two-dimensional channel.pdf

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4、-,-III!,OfO,01101I_ro_0,0,0_“0+:t=t ffrolrororo+I_9!9_t:_+-4-tO_0,“01,0El00mom00m,m0oI:tm.I:tIIIIIIIIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-c_c.;c_0C.0a_N_,_,.-_-u _ can beestimated and it may be seen that the above relation holdswith reason

5、able accuracy over the large center portion ofthe channel.A new technique for the measurement of X_ has also beenapplied. The method is described in detail in referefice 14and consists in counting the zeros of an oscillograph traceof the u-fluctuations. From these counts X_ may be cal-culated direct

6、ly by assuming a normal and independentdistribution for both u and 5u/Sx:1 9-_-_-XAverage number of zeros of u per secondAz WoIt is known that the distribution of u is closely a Gaussianone even in nonisotropic turbulence (see, for instance, refer-ence 15); however, for the case of bu/Sz a smalldevi

7、ationfrom the normal distribution was found (reference 13). Forthe preliminary measurements of X_ reported presently, nocorrections were applied as yet for this effect. Figure 4shows an oscillograph trace of the u-fluctuation in the mid-dle of the channel at R=30,800. The trace represents aninterval

8、 of approximately 1/20 second.$:x =.:. _.g:;.:f:(_:_.:.V:.:_:. _2.:re:.“_:.:.:.:.:.:x. “:i_ “_._N:“! .:_,$.!iiiiiiiliiiiil , .,: iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii . iilliiiNiiii _.,._: W4 :_:_:_:“ =! =_i_i_i_iiFIC, URE 4.-q?yt)ieal oseillogram of u-lluetuation at R=30,800 and v/d=l.0, lIorizontal line

9、corresponds to u(t) =0; interval is approximately 1/20 second.Measurement of L_.-The following simple procedure _was applied to obtain a rough estimate of scale of turbulencecorresponding to correlations between points along tim x-axis: Denote by Fg-_(n) the fraction of turbulent intensitywhich is c

10、ontributed by frequencies between n and n+dn;that is,u(n) _dn= u_ P7“Z(n) dnand thus_ FT(n) dn = 1Consider now an uncompensated hot-wire. If the timeconstant of this uncompensated wire is M, the responsewill be_ (?_) ncomp.where_u 1+The total intensi W for the uncompensated wire will then begiven by

11、do l+M_n _2 This method was suggested by Dr. H. W. Liepmaml.a Formulas of this general type have been proposed by l(amp_ do Feriet and by Frenkic lfor determining the spectrum of turbulence from uncompensated hot-wire measurements byvarying M (reference 16).981431_52-2Provided by IHSNot for ResaleNo

12、 reproduction or networking permitted without license from IHS-,-,-rOmm,rmroXo=HI!;SoImProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INVESTIGATION OF TURBULENT FLOW IN A TWO-DIMENSIONAL CHANNEL 1145 4% / o._0 I _,08.06.04_020R,-8,000 reference +,o,

13、z_, 300 “-.6/, 600- _.I .2 .3 .4 .5 .6 .7+/dFIGURE 11.- Velocity fluctuations .8 .9 /.0 L .08 “+-“.06“002 I0 .01 .02 .03 .04 .05 .06 .07 .08_/4FI(_URE 12.-Velocity fluctuations _near wall.R_30030, 800.09 ./0 .II.08v .04 _- -e-.02 -/_ o /2,300/ m 80,800, O 61,8000 ./0 .20 .30 .40 .50 .60 .70 .80_/dFI

14、GURE 13,-Distributi0n of velocity fluctuations _.,90 LO /.I.08.o_,_ R :_-_-_:WU.o 2 _ o 12,300E 30,800o 81,COOi0 .10 .20 .30 .,0 .50 .80 .70 .80 .90/.0 I.ItldFK_URE 14.-Distribution of velocity fluctuations .The velocity fluctuations _ relative to local speedsincrease very rapidly near the wall as i

15、s shown in figures 9and 10. Measurements very close to the wall indicate that7_/u reaches a nmxinmm within the laminar boundary layer(yU,/_ _ 17) and it tends toward a constant value at the wallwhich is independent of the Reynolds number. This pointwill be discussed in detail under “Reynolds Number

16、Effect.“The absolute values of the distribution of _ show the samegeneral shape as that obtained by Reichardt (reference 9),having the characteristic maximum near the wall and thusshowing the strong action of viscosity even for values ofy_4a.Using the X-type hot-wire technique for obtaining theveloc

17、ity-fluctuation components v and w, no measurementscould be obtained near the wall. Figures 13 and 14 showthat, while in the center of the channel the magnitudes of_ and _ are the same, _ increases faster toward the wall.This agrees with the ultramicroscope measurements in a pipeby Fage and Townend

18、(reference 18).No length corrections were necessary to the measurementsof u except near the wall; however, no corrections wereapplied in this region since no measurements of kz could bemade. In this region, furthermore, the fluctuations arevery large and the values given in figure 10 must be accepte

19、dwith reserve. The hot-wire response for large velocity fluc-tuations is not well-understood yet and no correction wasattempted. Length corrections were applied to the measure-ments of v and w.CORRELATION COEFFICIENT AND SHEAR DISTRIBUTIONThe correlation coefficient is fairly constant across mostof

20、the channel (fig. 15) as indicated already by WattendorfProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-/9PFM0rao0_2.2_20g_-_Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_Diq$6o_qoNz%I%IProvided by

21、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r.f2CO.NZ$ZooN%CO_-q:)_Ib/-_o:_o-l,/,“,tt_/o:/iIII/,Io_!I!Io._._mI_29Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INVESTIGATION OF TURBULENT FLOW IN A TWO-DIMEN

22、SIONAL CHANNEL 15oz=L z. t2.,I00o 0 Lxo 1“.zo Lyo.I0 .2 .4 . .6 .8 1.0v/a(a) R=12,300.(b) R=30,800.(e) R=61,600.FIGURE 21.-Scale of turbulence distribution.The distribution of Xx obtained by the differentiationmethod shows the same behavior as that of Xy and Xz (fig. 22).It has the characteristic ma

23、ximum at y/d _ 0.7. No measme-ments were made for R=12,300 because of the inaccuracydue to the large noise-to-signal ratios.As a matter of interest the results of some measurements ofX_ obtained with the zero-counting method are comparedwith those described above. (See fig. 22.) Since th e validityo

24、f the assumptions involved in this method is not clarifiedyet, these measurements should be regarded as preliminary.SPECTRUM MEASUREMENTSThe spectra of the velocity fluctuations u _ have been ob-tained at various values of y/d across the channel. Withimproved instrumentation it was possible to reduc

25、e experi-mental scatter by a considerable amount. The accuracy ofthe present measurements is believed to be within 10percent with tbe exception of values comesponding to lowfrequencies (n_100 cps) because of the large-amplitudefluctuations and with the exception of values correspondingto high freque

26、ncies (n_4000 cps) because of noise and ofpossible wire-length effect. A typical spectrum distributiontaken at y/d=1.00 is shown in figure 24. (The measuredspectra at various positions of y/d are given in table I.) Noattempt is made to compare the measured distributions withexisting theories since t

27、he restrictive assumptions of thesetheories (isotropy and flows at high Reynolds numbers) arenot satisfied in the present experiments. The only frequencyrange where comparison is possibly justified is the viscousregion, that is, the high-frequency part of the spectrum whereviscosity plays a dominant

28、 role. Unfortunately here theaccuracy of the measurements is not good enough to affordany definite conclusions. Therefore, the rather close agree-incur of the measured spectrum (for all values of y/d exceptin the laminar sublayer) with the n-Maw predicted by Heisen-berg (reference 2) should be accep

29、ted with reserve (fig. 24). -_ 1- i o _ Zero counf_bg-I I I rb Spectrum . _ D/fferenl/bfion.86 _el . =:_I | n 30,800/5,in. chonnel/0 ./ .2“ .3 .4 .5 .6 .7 . .8 ZO /./v/aFIGURE 22. Mierosea/e measurelilCll_S., _ _k a_4 ! “t,-_.“b-_ , -V/a_L 0 it _6 I v/d) - _-_=, 0 .0 /.5 2.0 2.5 3.0 3.5 4.0 4.5C ITI

30、FIr,IrRE 23.-Ry-correlation. R=30,800. At y/d=l.O, Ly=l.3 centimeters, X_=0.50 centi-meter; at y/d=0.7, L_=l.2 centimeters, kv=0.56 centimeter.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-noNor.?/u)_,=o_-O.18.It is of considerable interest to disc

31、uss the variation of thescale and microscale with Reynolds number. For flows be-hind grids where the turbulence is isotropic the scale is in-dependent of the mean velocity and depends on the meshsize of the grid. Similar behavior was found for the channelflow. Figure 21 shows the distributions of L_

32、 and L= fordifferent velocities. These distributions indicate no con-sistent variation with velocity. Furthermore, measurementsin a 1-inch channel give a value for L_ five times lower and aratio for Ly somewhat larger than the values obtained inthe present investigation.The variation of X depends, o

33、f course, on the velocity andchannel width. The values of X decrease with increasingvelocity; however, the variation of X with channel width isless than that of L.FULLY DEVELOPED CHARACTER OF TURBULENCEThe flow in the channel is called fully developed if timvariations of the mean values of the veloc

34、ity and the meansquares of the velocity fluctuations with x are very small.That the mean velocity profile does not vary downstream isevident from the pressure-gradient measurements (fig. 19).The gradient in x of u 2 was measured on the axis of thechannel. It was found that u _ was indeed decreasing

35、withx. Tim gradient, however, was very small as comparedwith 1_ bp.p bx“bu t2 1 bp- _ 0.01 - -bx p bxHence for all practical purposes bu2/bx can bc neglected.No measureme:nts have been made concerning bu_/bx,since the scatter in the values would cover any effectHowever, there is little doubt that bu

36、v/bx is of the sameorder as bu_/bx and that the use of equations (3a) and (3b)is therefore justified here.ENERGY BALANCE IN FLUCTUATING FIELDThe energy equation for a two-dimensional channel hasthe form given by equation (8) and is valid throughout thecross section of the channel with the exception

37、of a smallduregion near the wall. The term r _ on the left side of equa-tion (8) corresponds to the energy produeed by th.e shearingstresses and it cart be obtained directly from the measure-ments of r and from tile mean velocity profile. The secondterm on the right /_ bxj/ bxj/ expresses the amount

38、 ofenergy that is being dissipated because of tile breakingdown of the larger eddies to smaller ones. The term maybe written explicitly/bu /bv+tw)+tw) +t-Nf/ _z / -SY / by/ bz / _The problem is to express these functions in terms ofeasily measurable quantities. In the case of isotropieturbulence Tay

39、lor solved the problem by introducing themicroseaie of turbulence X and obtained for the dissipationW- 15t_ _ (lO)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_Or_C.mCD“-_.,.,_.m_oIII/-rolrHI-I_-I_o+_0+,-=-F=oI1“_I_1_+_1%_o+_10II-_o0/i/_,omassProv

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