the structure of turbulent ow in a rectangular channel containing a cylindrical rod - part 2 phase-averaged measurements

1、Experimental Thermal and Fluid Science 23 (2000) 75?91www.elsevier.nl/locate/etfsThe structure of turbulent ?ow in a rectangular channel containing a cylindrical rod ? Part 2:phase-averaged measurementsM.S. Guellouz 1, S. Tavoularis *Department of Mechanical Engineering, University of Ottawa, Ottawa

2、, ON, Canada K1N 6N5 Received 4 April 2000; received in revised form 28 July 2000; accepted 14 August 2000AbstractIn Part 1, we have presented the Reynolds-averaged turbulence characteristics of iso-thermal ?ow in a rectangular channel containing a cylindrical rod such that it forms a narrow gap wit

3、h a plane channel wall. Part 2 focuses on conditionally sampled measurements of the phase-averaged characteristics of the coherent vortical structures that produce ?ow pulsations across the gap and formulates a tentative physical model of their typical shapes. ? 2000 Elsevier Science Inc. All rights

4、 reserved.Keywords: Rod bundle; Gap ?ow; Coherent structures; Conditional sampling; Phase-averaging1. Introductioncores towards and across the gap region. The question then arises whether these structures can be classi?ed ascoherent, of the type encountered in other free and bounded turbulent ?ows 3

5、. The detection and docu- mentation of coherent structures have been the subject of intense eorts by the turbulence research community, culminating in the development of several sophisticated pattern recognition and phase-averaging methods. To our knowledge, the present study is the ?rst attempt to

6、apply such methods to the characterization of coherent structures in channels containing narrow axial ?ow re- gions. Based on the available literature and our previous experience in rod bundle studies, wedecided to consider?rst a relatively simple con?guration and, thus, to avoid complications which

7、 might obscure the main phenom- enon of interest. The selected geometry consists of a rectangular channel containing a suspended circular rod with a diameter small enough, compared to the channel height and width, for the ?ow away from the gap region to be relatively free of gap eects. In Part 1, we

8、 have presented Reynolds-averaged measurements of the mean velocity, the wall shear stress and the turbulent stresses in the channel, as well as two-point correlations and spectra documenting the formation of quasi- periodic?owpulsationsacrossthegapandthedependenceof their frequency and amplitude on

9、 the gap-to-rod- diameter ratio. The present paper focuses on the phase- averaged features of the coherent structures and the presentation of a plausible physical model. It is hoped that the present work will contribute to the under- standing of this important ?ow phenomenon and willIn a companion p

10、aper (1; hereafter referred to as Part 1), we have explained that the objectives of the present experimental programme are to characterize experimentally the coherent structures that form in the narrow region of a compound channel and to develop a physical model that explains their apparent eects. M

11、otivation for this study was provided by the need to account for the eects of such structures in the predic- tion and safety analyses of the thermal-hydraulic per- formance of the cores of current and planned nuclear power reactors. These, almost universally, consist offuel elements (rods) arranged

12、in bundles and cooled by a?uid that ?ows axially in the subchannels formed among neighbouring fuel elements or between outer elements and the surrounding pressure tube. In Part 1, we have provided a brief historical outline of the discovery of?ow pulsations across narrow gaps in rod bundle ?ows and

13、have summarized the available experimental results as well as previous eorts to model the phenomena that cause these pulsations. All available evidence points to the formation of an array of counter-rotating vortical structures, which transport ?uid from the subchannel* Corresponding author. Tel.: +

14、1-613-562-5800 ext.6271; fax:+1-613-562-5177.E-mail address: taveng.uottawa.ca (S. Tavoularis).1 Current address: Atomic Energy of Canada Ltd., 2251 Speakman Drive, Mississauga, Ont., Canada L5K 1B2.0894-1777/00/$ - see front matter ? 2000 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 -

15、1 7 7 7 ( 0 0 ) 0 0 0 3 9 - X76M.S. Guellouz, S. Tavoularis / Experimental Thermal and Fluid Science 23 (2000) 75?91ratio of 2/3 containing an aluminium pipe (rod) with a diameter of D ? 101 mm and suspended in a way that it would form an adjustable gap with the channel base. The hydraulic diameter

16、of the test section was 1:59D and its length was 54:0D. The channel was supplied with air produced by a blower. The detection of coherent structures was made by a triggering hot-wire probe, which was of the cross-wire type and was placed at the centre of the gap, approximately 3D upstream of the mai

17、n hot-wire probe, which had three sensors and was placed near the exit of the channel. The triggering probefacilitate the development of methods incorporating its eects in the prediction and design of rod bundles and other complex engineering systems.2. Experimental facility and instrumentationThe e

18、xperimental facility and instrumentation used have been described in Part 1 and, in more detail, in 2. A sketch of the ?ow channel is shown in Fig. 1. The test section consists of a rectangular channel with an aspectFig. 1. Sketch of the ?ow facility.Nomenclaturex; y; zaxial, transverse and spanwise

19、 coordi- Drod diameter, mmnates, respectively, mmDhhydraulic diameter, mmxjspatial coordinates,mm kturbulent kinetic energy per unit mass,Greeksm=s2ascaling factor, dimensionlessNnumber of coherent events, dimensionlessQrandom variable, units depend on theaaaverage scaling factor, dimensionless prop

20、ertyaoptoptimal scaling factor, dimensionlessqmean-free ?uctuation of a random vari-jthreshold level, dimensionlessable, units depend on the propertyrastandard deviation of the scaling factor,Ttime interval, sdimensionlessttime, sr2localized variance of the random variableQtjdetection time, ssQ, uni

21、ts depend on the propertytime during a coherent event, relative toU; V ; Waxial, transverse and spanwise velocitythe detection time, s components, respectively, m/sVtresultant velocity on the transverse plane,Other notationm/s? ?time-averaged quantityUbbulk velocity, m/s? ?variable-interval time ave

22、rage Ucmean convection velocity of coherent? ?0root mean square ?uctuationstructures, m/shicoherent component according to theu; v; waxial, transverse and spanwise velocitydouble decomposition?uctuations, respectively, m/s? ?coherent component according to the tri- Wsum of rod diameter and gap width

23、 (seeple decompositionFig. 1), mm?. .?rnon-coherent componentM.S. Guellouz, S. Tavoularis / Experimental Thermal and Fluid Science 23 (2000) 75?9177tj; j ? 1; 2;. ; N, corresponding to the same relative instance in the duration of a coherent event and de?ned as the midpoint of each time interval dur

24、ing which Eq. (3) was satis?ed. Then, the conditional (ensemble) average of property Q at a particular instance s during the average coherent event, also known as phase- average, was de?ned asXNwas mainly used to measure the spanwise velocity vari- ation, while the main probe was used to measure si-

25、 multaneously all three velocity components. The resolution and measuring uncertainty of the hot-wire probes have been discussed in Part 1. Considering that, in the present context, the interest focuses upon rela- tively large-scale features of the ?ow, the experimental accuracy is deemed to be su?c

26、ient. 1hQ?xi ; s?i ? NQ?xi ;jt ? s?:?4?j?13. Principle of conditional samplingThe above method, which is the classical VITA algo- rithm, detected individual structures producing signals varying not only in magnitude, but also in duration and waveform. As a result, the computed ensemble averages had

27、relatively weak magnitudes and erroneously large periods, the values of which seemed to depend on the threshold level. This prompted the development of an iterative enhancement method, to be referred to asenhanced VITA method, which utilized the above conditional sampling technique only to obtain a

28、?rst estimate of the ensemble average to be termed as theraw average. In a second pass through the same data, the signal of each detected structure was correlated with the raw average. Then, the time-axis of each signal was scaled by a factor a. This process was repeated for dif- ferent values of a

29、and an optimum a o,pct orresponding to the correlation with the highest maximum, was de- termined. An optimum time shift, corresponding to the time at which the maximum of the correlation function, was also determined. Finally, the signal of each detected event was time-shifted and scaled opti- mall

30、y with respect to the average event to account for variations in phase and duration of individual events. At the end of this process, a new, enhanced, en- semble average was obtained by ensemble averaging the optimally scaled and time-shifted signals. Since this procedure forces this new ensemble av

31、erage to have the same duration as the raw average, its time scale has to be appropriately scaled to re?ect the duration of an average structure. This is performed by using the aver- age aa of the aopt of all retained structures. This new average was used instead of the raw average during another pa

32、ss through the same signals and the proce- dure was repeated until convergence of successive en- semble averages was achieved. In most cases, three passes were su?cient for convergence. The enhance- ment method reduced substantially the dependence of the results on the threshold value, thus improvin

33、g the objectivity of the technique as far as individual struc- tures are concerned. On the other hand, the enhance- ment method also phase-shifted and time-scaled the structures that preceded or followed the detected,central one, but not optimally, as it used the time shift and scale that were optim

34、al for the central struc- ture. For this reason, presentation of the features of a typical, single, structure were based on the enhanced VITA method, while presentation of reconstructed views of a sequence of structures were based on the classical VITA method.Conditional sampling and phase-averaging

35、 methods are under continuous development and new ideas are being tested by several laboratories. The present ap- proach was to adopt a relatively simple method that has been successful in extracting the main features of co- herent structures and to improve on it, when necessary. The conditional sam

36、pling technique employed in this work is based on the classical variable interval time averaging (VITA) technique of Blackwelder and Kaplan 4. In this technique, the variable-interval time average of a random function Q?xi; t? of position xi and time t is de?ned asZt?T =21Q?xi ; t; T? ? TQ?xi; s? ds

37、;?1?t T=2where T is the averaging time interval, which is chosen to be of the order of magnitude of the time scale of the phenomenon under study. For a stationary random variable, the variable-interval time average approaches the conventional time average, Q,as T !1. Eq. (1) may also be applied to t

38、he square, Q2, of the randomvariable to obtain its variable-interval time average, Q2.The localized variance, de?ned asr2 ?xi; t; T ? ? Q2?xi ;t; T ?Q2?xi; t; T?2?Qis a measure of the local turbulent energy.As shown in Part 1, the presence of ?ow pulsationswas most clearly manifested by the nearly p

39、eriodic os-cillations in the spanwise velocity, W, measured in the gap. This signal was, therefore, used for the detection and phase determination of coherent structures. The presence of a coherent event was detected when the lo- calized variance of W exceeded a preset level, propor- tional to the c

40、onventional variance, w2. The same coherent event was considered to be terminated when the localized variance dropped below that level. More precisely, a coherent event was considered to exist dur- ing each time interval for whichr2 jw2 oW =ot 0;?3?Wwhere j is an adjustable threshold level; the cond

41、ition on a positive velocity derivative was added in order to discriminate between the beginning and the end of a coherent event, and, thus, to ensure proper phase matching among dierent events. In order to phase- match N detected coherent events before averaging, each event was time-shifted by the

42、detection time,78M.S. Guellouz, S. Tavoularis / Experimental Thermal and Fluid Science 23 (2000) 75?914. Eduction of the coherent and non-coherent componentsSimilarly, it can be shown thatUj?i ? h?Ui ? uri?Uj ? urj?i? UiUj ? huriurji:h?UiUi?UjThe conventional Reynolds decomposition of the instantane

43、ous value, Q, of a stationary random process is into a time-average, Q, and a mean-free ?uctuation, q, as?10?Eqs. (9) and (10) allow the calculation of the time- average and the phase-average of the non-coherent stresses from quantities that are easier to evaluate.The terms UiUj are evaluated from t

44、he time series of the corresponding velocity components by integrating the products UiUj over a time interval corresponding to the passage of a pair of vortices, namely the time dif- ference between two consecutive occurrences of a mini- mum of the spanwise coherent velocity component.Q?xi; t? ? Q?x

45、i; t? q?xi; t?:?5?Previous analyses of turbulent ?ow ?elds containing coherent structures have utilized two dierent kinds of decompositions into coherent and non-coherent com- ponents 3. These are the double decomposition and the triple decomposition. For the double decomposi- tion, the instantaneou

46、s value, Q, of a random variable isdecomposed into a coherent component and coherent component, respectively, as follows:Q?xi; t? ? hQ?xi; t?i ? qr?xi; t?:a non-5. Detection and characterization of the coherent struc- tures?6?5.1. Experimental conditionsFor the triple decomposition, the instantaneou

47、s value is decomposed into a time-average component (equal to the conventional Reynolds average), a coherent com- ponent, andanon-coherent component, respectively, as follows:All measurements presented here were performed at a Reynolds number of 108,000 (based on the channel bulk velocity Ub? 10:1 m

48、/s and hydraulic diameter D ?h 160:6 mm; see Part 1). The rod?wall distance was ?xed at 0:100D ?W =D ? 1:100; see Fig. 1), which resulted in strong ?ow pulsations across the gap 1. The average convective speed of the structures was Uc 0:78;Ub 7:9 m/s; it was used to convert the time signals obtained

49、 with ?xed probes into streamwise variations. This approach is justi?ed in view of the evidence pre- sented in Part 1 that coherent structures were convected downstream relatively unchanged. 110,000 structures were detected and phase-averaged byboth the classical and the enchanced VITA techniques. T

50、he optimal av- eraging time interval, T, for the detection of structures, was determined as the time interval that resulted in the most pronounced localized variance of a signal; this was found to be 0.025 s, which is approximately equal to half the period of the average pulsation. A number of tests

51、 were conducted to determine the most appropriate threshold level j that would permit the detection of nearly all coherent structures, but not of any non- coherent ?uctuations; this was found to be 0.7. TheQ?xi; t? ? Q?xi? ? Q?xi; t? ? qr?xi; t?:?7?The non-coherent components in the double and tripl

52、e decompositions are identical, but the coherent compo- nents are dierent; the latter are related asQ ? hQiQ?8?the doubleCompared to the triple decomposition,decomposition has the limitation that it cannot describethe temporal evolution of coherent structures or their energy exchange with the mean ?

53、ow and the non- coherent turbulence. On the other hand, the distinction between time-mean ?ow and coherent structures is not clear and it has been pointed out that coherent struc- tures are not mere perturbations of the time-mean ?ow but constitute the entire non-random motion 3.Following a procedur

54、e analogous to the derivation of continuity and momentum equations for Reynolds- decomposed turbulent ?ow, one may derive corres- ponding equations for the time average, coherent and non-coherent components according to the double and triple decompositions 3. Compared to the Reynolds equations, thes

55、e equations contain the additional termsaoptdistribution of the optimum scaling factorforindividual coherent events was nearly Gaussian, with therelatively small standard deviation ofra ? 0:28aa,which shows that the structures were fairly repeatable. For practical purposes, only events with scale fa

56、ctors in the interval ?aa3ra; aa ? 2ra? were retained; this excluded only a small percent of all detected events. The incre- ment in a for the iterative process to determine its optimum was 0:1ra.U U ; hu u i, and u u ; i; j ? 1; 2; 3. The equationi jri rjri rjforms are simpli?ed by the following as

57、sumptions: the phase-average and the time-average of non-coherent quantities are zero; the phase-average of the time-aver- age is equal to the time-average; and the coherent and non-coherent components are uncorrelated, so that the time-average and the phase-average of their products5.2. Waveforms o

58、f the coherent velocitiesThe classical and the enhanced VITA techniques were applied to the time series of the three velocity components and the corresponding average waveforms were extracted and compared. For positions close to the gap centre, the former technique failed to detect a peakvanish. Then, one can express nolds stresses asthe conventional Rey-u u ? U U ? u u :?9?i