OFFSHORE TECHNOLOGY CONFERENCE6200 North Central ExpresswayDallas, Texas 75206

PAPERNUMBER 0 TC 2 533

I n - Lin e and Tran s v e r s e For c e s, 0 n Cy lind e r sinOsc i I I atory F low at High Reyno I ds Numbers.

By

Turgut Sarpkaya, Naval Po?tgra~uate_Sc~ool

THIS PAPER IS SUBJECT TO CORRECTION

@Copyright 1976Offshore Technology Conference on behalf of the American Institute of Mining, Metallurgical, and Pe~roleumEngineers, Inc. (Society of Mining Engineers, The Metallurgical Society and Society of Petroleum. Engmee~s),American Association of Petroleum Geologists, American Institute of Chemical Engineers, Amencan Societyof Civil Engineers, American Society of Mechanical Engineers, Institute of Electrical and Electronic~ En­gineers, Marine Technology Society, Society of Exploration Geophysicists, and Society of Naval Architectsand Marine Engineers.This paper was prepared for presentation at the Eighth Annual Offshore Technology Conference, Houston,Tex., May 3-6, 1976. Permission to copy is restricted to an abstract of not more than 300 words. Illustrationsmay not be copied. Such use of an abstract should contain conspicuous acknowledgment of where and bywhom the paper is presented.

ABSTRACT

This paper presents the results of an exten­siv~ experimental investigation of the in-lineand transverse forces acting on smooth and roughcircular cylinders placed in oscillatory flow atReynolds numbers up to 700,000, Keulegan-Carpentetnumbers up to 150, and relative roughnesses from0.002 to 0.02. The drag and inertia coefficientshave been determined through the use of theFourier analysis and the least squares method.The transverse force (lift) has been analysed interms of its maximum, semi peak-to-peak, androot-mean-square values. In addition, the fre­quency of vortex shedding and the Strouhal numberhave been determined.

The results have shown that (a) for smoothcylinders, all of the coefficients cited aboveare functions of the Reynolds and Keulegan andCarpenter numbers, particularly for Reynoldsnumbers larger than about 20,000; (b) for roughcylinders, the force coefficients also depend onthe relative roughness kiD and differ signifi­cantly from those corresponding to the smoothcylinder; and that (c) the use of the 'frequencyparameter' D2/vT and the roughness Reynoldsnumber Umk/v allow a new interpretation of thepresent as well as the previously obtained dataand the establishment of model laws for oscilla-

References and illustrations at end of paper.

tory flow about cylinders at supercriticalReynolds numbers.

INTRODUCTION

The design of structures for the marineenvironment requires the prediction of the forcesgenerated by waves and currents. Much of thepresent knowledge has been obtained by means ofmodel tests at Reynolds numbers generally two tothree orders of magnitude smaller than prototypeReynolds numbers. These model tests have reliedheavily on the so-called Morison formula forexpressing the force as the sum of a drag andinertia force. The values of the drag and iner­tia coefficients to be used in the Morison equa­tion became the subject of many experimentalstudies in the last twenty years. The correla­tion of these coefficients with the relativeamplitude of the waves (or the Keulegan-Carpenternumber) has been generally inconclusive.Furthermore, lift forces which are associatedwith vortex shedding have received relativelylittle attention. It thus became clear that muchis to be gained by considering plane oscillatoryflow about cylinders at high Reynolds numbers inorder to isolate the influence of individualfactors such as relative amplitude, Reynoldsnumber, and the relative roughness on vortexshedding and resistance. It is with this reali­zation that the present investigation was under­taken and the preliminary results obtained with

96 IN-LINE AND TRANSVERSE FORCES ON CYLINDERS IN OSCILLATORY FLOW OTC 2533

smooth cylinders in a small U-shaped water tunneloperating at relatively low Reynolds numbers(2,500 to 25,000) have been previously reported[1 J.

The present paper deals with in-line andtransverse forces acting on smooth and artifi­cially-roughened circular cylinders in harmonicflow at critical and supercritical Reynoldsnumbers.

APPARATUS AND PROCEDURE

Of the two possible methods of generatingrelative harmonic fluid motion about bluff bodiesnamely, oscillating the fluid or the body, theformer has been chosen. The relative merits andshortcomings of the two methods have been amplydiscussed [2J and will not be repeated here.Suffice it to note that large amplitude struc­tural and free surface oscillations commonlyencountered in oscillating the body in a fluidotherwise at rest do not lead to reliable data.The advantages of the apparatus used herein forthe purpose under consideration have already beendemonstrated [lJ and will become further evidentfrom the data to be presented.

The oscillating flow system consisted of alarge U-shaped vertical water tunnel as shown inFig. 1. The cross-section of the two verticallegs is 3 ft by 6 ft and that of the test sectionis 3 ft by 3 ft. The two corners of the tunnelwere carefully streamlined to prevent flowseparation. This design proved to be more thanadequate for no separation was encountered, andalso the desired frequency and amplitude ofoscillation were achieved. The auxiliary compo­nents of the tunnel consisted of plumbing forhot and cold water, butterfly-valve system, andthe air-supply system.

The butterfly-valve system (mounted on topof one of the legs of the tunnel) consisted offour plates, each 18 inches wide and 36 incheslong. All four valves were simultaneously drivenby a simple rack and pinion system actuated by anair-driven piston and a three-way pneumatic valveInitially, the butterfly valves were closed andair was introduced to that side of the tunnel,with an electrically-controlled ball valve, tocreate the desired differential water levelbetween the two legs of the tunnel. Then thevalves were opened with the help of the rack andpinion system and the three-way control valve.This action set the fluid in the tunnel in oscil­latory motion with a natural period of T = 5.272seconds. The elevation, acceleration, and allforce traces were absolutely free from secondaryoscillations so that no filters whatsoever wereused between the outputs of the transducers andthe recording equipment (see Fig. 2).

Throughout the investigation, the monitoringof the characteristics of the oscillations in the

tunnel was of prime importance in view of thefact that most of the difficulties in the pastin the determination of the drag, lift, and iner­tia coefficients resulted from the difficulty ofgenerating a purely harmonic motion free fromvibrations or from applying theoretically derivecrather than measured values for velocities andaccelerations.

Three transducers were used to generatethree independent d.c. signals, each proportionalto the instantaneous value of elevation, velocit)and acceleration. The first one consisted of aplatinum wire stretched vertically in one legof the tunnel. The response of the wire wasperfectly linear within the range of oscillationc

encountered. The second method consisted of themeasurement of the instantaneous acceleration bymeans of a differential-pressure transducerconnected to two pressure taps placed horizon­tally 2 ft apart and 4 ft to one side of the tes1section. The instantaneous acceleration was thercalculated from ~p = ps.dU/dt where ~p is thedifferential pressure, s the distance betweenthe pressure taps and dU/dt is the instantaneousacceleration of the fluid. The third methodagain consisted of the measurement of the differ­ential pressure between two pressure taps placedsYmmetrically on the two vertical legs of thetunnel at an elevation H ft below the mean waterlevel. The linear differential-pressure trans­ducer yielded the instantaneous elevation andhence the amplitude of oscillation since,according to Bernoulli·s equation

A = 2Ao = [~p/yJ 1[1 - (2~/T)2H/gJ (1)maxin which g and T are constant and H is keptconstant.

All three methods gave nearly identicalresults and yielded the amplitude A, the maximumvelocity U~ = 2~A/T, or the maximum accelerationam = (2~/T)2A to an accuracy of about 2% relativEto each other.

The in-line and transverse forces were measured with two identical, cantilever type, forcetransducers, one at each end of the cylinder.The gages had a capacity of 250 lbs and thedeflection of the cantilever end was less than0.008 inches. A special housing was built foreach gage so that it can be mounted on the tunnelwindow and rotated to measure either the in-lineor the transverse force alone. The test cylin­ders were placed in the test section by retrac­ting the gages from their housing and then push­ing them into the bearings mounted on each endof the cylinders. This allowed a gap of 1/32inch between the cylinder and the tunnel wall.The natural frequency of the cylinder and force­transducer combination in water was about 20times larger than the frequency of oscillationof the water column and about 10 times largerthan the largest frequency of vortex shedding.

OTC 2533 TURGUT SARPKAYA 97

It is recognized that the coefficients citedabove are not constant throughout the cycle andare either time-invariant averages or peak valuesat a particular moment in the cycle. A simpledimensional analysis of the flow under consider­ation shows that the time-dependent coefficientsmay be written as

F/(0.5DLpU~)= f(UmT/D, UmD/v, kiD, tiT) (6)

in which F represents the in-line or the trans­verse force. Equation (6), combined with Eq. (2)assuming for now that the latter is indeed valid,yields

and Cmls=Cm• Evidently, the Fourier analysis andthe metnod of least squares yield identical Cmvalues and that the Cd values differ onlyslightly. The details of the modified least­squares method may be found in Ref. [5] and willnot be repeated here.

The transverse force has been expressed interms of various coefficients. Some of these are(a) the maximum lift coefficient defined by CL=(maximum amplitude of the transverse force in acycle)/(0.5LDpU~); (b) the semi peak-to-peakvalue o~ the transverse force normalized by0.5LDpUm; and (c) the normalized root-mean-squarevalue or the transverse force. In addition, thefrequency of the oscillations of the transverseforce and the Strouhal number have been evaluated

Sevencircul'ar cylinders with diametersranging in size from 2 inches to 6.5 inches wereused. The cylinders were turned on a lathe fromaluminum pipes or plexiglass rods and polishedto a mirror-shine surface. Same cylinders werealso used as rough cylinders. For this purpose,sand was sieved to obtain the desired relativeroughness and applied uniformly on the cylindersurface with an air-drying epoxy paint. Aftera series of tests with water at various tempera­tures, the cylinders were polished again andcovered with sand of different size. This pro­cedure was continued until the desired ranges ofall the governing parameters were covered. Theuse of hot water increased the range of theReynolds number by as much as 100%.

Experiments were repeated at least fourtimes for each cylinder and for all suitablyselected amplitudes, roughness hights, and watertemperatures. The in-line and transverse forceswere read every 0.1 second from the traces forthe period of T = 5.272 seconds. Then the drag,inertia, and the lift coefficients, the vortexshedding frequency, Strouhal number, maximumerror between the measured and calculated forces,etc. were evaluated through the use of the appro­priate equations and a computer.

FORCE COEFFICIENTS AND GOVERNING PARAMETERS

Data reduction for the forces in-line withthe direction of oscillation is based on Morisonequation [3] and three different analysis of theforce records, namely, Fourier analysis, leastsquares, and a modified least squares method.

Cd= fl(K, Re, kiD, tiT)

em= f 2(K, Re, kiD, tiT)

(7)

(8)

The in-line force which consists of the drag in which K= UmT/D and Re = UmD/v, and kiD repre­force Fd and the inertia force Fi is assumed to sents the relative roughness.be given by [3]

in which Fm represents the measured force.

The method of least squares consists of theminimization of the error between the measuredand calculated forces. This procedure yields [5]

2wCdl s= -.(8/3w)~ (FmIcose Icose/pDLU~)de (5)

(9)

There is no simple way to deal with Eqs. (7)and (8) even for the most manageable time-depen­dent flows. Another and 'perhaps the only otheralternative is to eliminate time as an independ­ent variable and consider suitable time-invariantaverages as given by Eqs. (3), (4), and (5).Th~s, one has

" Cd\ Cm = f.( K , Re , kiD )" 1\ CL

It appeari, for the purposes of Eq. (9), that theReynolds number is not the most suitable param­eter involving viscosity. The primary reasonsfor this are that the effect of viscosity isrelatively small and that Urn appears in both Kand Re. Thus, replacing Re by Re/K = D2/vT inEq. (9), one has

Ci(a coefficient) = fi(K, a, kiD) (10)

in which a = D2/vT and shall be called the

(2)

(3)

(4)2w

Cm= (2UmT/w3D) .r (Fmsine/pU;LD)deo

F=Fd+Fi=0.5CdLDPIUIU +0.25CmLD2wp.dU/dt

in which Cd and Cm represent respectively thedrag and inertia coefficients and U the instan­taneous velocity of the ambient flow. For anoscillating flow represented by U=-Umcose, withe=2wt/T, the Fourier averages of Cd and Cm aregiven by Keulegan and Carpenter as [4]

2wCd= -0.75 f (FmcoSe/pU~LD)de

oand

‘ OQ IN-LINE AND TRANSVERSE FORCES ON CYLINDERS IN OSCILLATORY FLOW OTC 2533.“

‘frequency parameter’. Evidently, f3‘isconstant The tabulated as well as plotted data for allFor a series of experiments conducted with a force coefficients are given in Ref. [2].:ylinder of diameter D in water of uniform and:onstant temperature since T is constant. Then Figures 7 and 8 show Cd versus Kand ~ verthe variation of a force coefficient with Kmay sus K for five values of 6. Evidently, there is]e plotted for constant values of 6. Subse- very little scatter in the data even though theIuently, one can easily recover the Reynolds figures represent the results of four independ-~umber from Re = KB and connect the points, on ent runs. A summary of the complete data for:ach B=constant curve, representing a given Re. all cylinders is presented in Figs. 9 and 10.

Also shown in Figs. 9 and 10 are the constantLet us now re-examine a set of data previ- Re lines obtained through the use of K= Re/f3.

)uslyobtained by others [4] partly to illus- Evidently,there is a remarkablecorrelationtrate the use and significance of B as one of between the force coefficients, Reynolds number,the governing parameters and partly to take up and the Keulegan-Carpenter number. The smooth-the question of the effect of Reynolds number ness of the constant Re lines is another indica-m the force coefficients. tion of the consistency of the data from one

cylinder to another.The data given by Keulegan and Carpenter

[4] maybe represented by 12 different values Figures 9 and 10 show that Cd and ~ do notof B. The drag and inertia coefficients are vary appreciably with Re for Re smaller thanplotted in Figs. 3 and 4 and connected with about 20,000 and help to explain the conclusionsstraightline segments. Evidently~ the identifi- previously reached by Keulegan and Carpenter [4]cation of the individual data points in terms and Sarpkaya [1].of the cylinder diameter, as was done byKeulegan and Carpenter [4] and also by Sarpkaya The data, similar to those given in Figs. 7[1], irrespective of the B values gives the through 10, are also plotted as a function of Reimpression of a scatter in the data and invites for constant values of K in Figs. 11 and 12one to draw a mean drag curve through all data since it is believed that the Reynolds numberpoints. Such a temptation is further increasedby the fact that the data for each B span over

is the liveliest of all the non-dimensionalparameters. These figures clearly show that Cd

only a small range of K values. Evidently, the decreases with increasing Re to a value ofdrawing of such a mean curve eliminatesthe about 0.5 (dependenton K) and then begins todependence of Cd and/or Cm on f3and hence on Re. increase with further increase in Re. The

Also shown in Figs. 3 and 4 are points rep-inertia coefficient Cm increases with increasingRe, reaches a maximum, and then gradually ap-

resenting four selected Reynolds numbers. The preaches a value of about 1.75. Itwill becorresponding K values for each Re and f?were recalled that the Keulegan-Carpenter data indi-’calculated from K = Re/f3. The points corre- cted an opposite trend. It is believed thatspending to the selected Reynolds numbers are the Keulegan-Carpenter data for Cm are not quitereproduced in Figs. 5 and 6. These figures show , reliable for K > 15. This is also evident fromwithin the range of Re and K values encountered the observation that the data corresponding toin Keulegan-Carpenter data, that (a) Cd depends 8=141 appear to be out of place (see Fig. 4)m both K.and Re and decreases with increasing<e for a given K; and that (b) ~ depends on

relative to those corresponding to 6=97 and 217.Suffice it to say that the results presented in

]oth K and Re for K larger than approximately Figs. 7 through 12 shed new light on the varia-15 and decreases-withincreasingRe. A similar tions of the drag and inertia coefficients andanalysis of Sarpkaya’s data [1] also shows that partly explain the reasons for the large scatter:d and Cm depend on both K and Re and that Cm encountered in the plots of Cd versus Re and Cmincreases with increasing Re. Notwithstandingthis difference in the variation of Cm between

versus Re as compiled by Wiegel [6].

the two sets of data, Figs. 5 and 6 put to rest The data obtained with artificially-rough-the long standing controversy regarding the ened cylinders are presented in Figs. 13 and 14dependence or lack of dependence of Cd and ~ in a manner similar to those given in Figs. 11m Re and show the importance of 6 as one of and 12. It must be noted that there exists twothe governi~g parameters in interpreting thedata, in interpolating the K values for a given

such figures for every value of K. However,the results shown in Figs. 13 and 14 did not

Re, and in providing guide lines for further appreciably vary with K for 30 < K c 60 (seeexperiments as far as the ranges of K and B also Fig. 11).are concerned.

Figures 13 and 14 show that the effect ofRESULTS AND DISCUSSIONS roughness on the resistance to harmonic flow is

quite significant. Not only the presence ofDrag and Inertia Coefficients for Smo@h and vortices on both sides of the cylinder and the

Rough Cylinders - Only the representative data increased level of turbulencebut also thetiillbe presented herein for sake of brevity. roughness bring about an earlier transition to

.

turbulence. Following the transition of theentire boundary layer to turbulence during mostof the cycle, the flow reaches a supercriticalstate and both the drag and inertia coefficientsacquire nearly constant values. Even though theactual events are somewhat more complicated, itseems unlikely that there will be any furthertransitions in the boundary layer.

The reason for the experiments with roughcylinders is of course more than the desire toexamine the effect of relative roughness on Cd>Cm, and CL. It is prompted essentially by anattempt at artificially increasing the Reynoldsnumber to supercritical regime by means of sur-face roughness. Recent experiments [7, 8] withsteady flow over rough cylinders have shown that(a) a change in flow regime takes place at aReynolds-number Vk/v of about 200 independentlyof the diametral Reynolds number; (b) a correctsurface roughness condition provokes supercriti-cal flow for Vk/v > 200, (the condition thatmust be respected is k/D c 0.0022); (c) a smoothcylinder is not a special case but behaves as ifit had a roughness of k/D = 0.000035; and that(d) the apparent diametral Reynolds number isincreased by a factor k/(0.000035.D)for a cylin-der of diameter D and surface roughness k. Theimportance and the consequences of these conclu-sions are self evident for supercritical Reynoldsnumber simulation for flow over circular cylin-ders.

In order to carry over the above ideas toharmonic flow, the data given in Fig. 13werereplotted in Fig. 15 as a function of U k/v forvarious values of k/D. A similar plot ?or Cm hasbeen prepared but will,not be presented here dueto space limitations. Also shown in Fig. 15 arethe mean lines correspondingto steady flow ascompiled by Szechenyi [7]. Figure 15 shows thata change in the flow regime takes place at aroughness Reynolds number of about 130 and thatthe drag coefficient approaches values between0.9 and 1.0 for k/D < 0.002. Evidently, thechange in’the flow regime occurs at higher valuesof U k/v with increasing k/D. Of special inter-

?est or simulation purposes, however, is thesmaller relative roughness.

The magnitude of the apparent increase inthe diametral Reynolds number can be estimatedby fitting the curve obtained for smooth cylin-ders (see Figs. 11 and 13) for K= 50 onto therough cylinder results shown in Fig. 15. Workin!back from the resulting values of Umk/v on theabscissa, this procedure gives an effectiverelative roughness between 0.0004 and 0.0006 forthe smooth cylinder in harmonic flow with K = 50.Further exploration of these ideas will beextremely important in model tests and in thesimulation of supercritical Reynolds numbers.It must, however, be kept in mind that the actualevents in harmonic flow are considerably morecomplicated and that the effect of roughness

cannot be represented by k/D alone. Additionalparameters such as the size distribut’ion,.shapeand packing of grains must be introduced. Alternatively, one can define an equivalent roughnessheight ks [9] or a roughness-length parameterbased on the actual boundary-layer characteris-ticcs.

Transverse Force and Vortex Shedding forSmooth and Rough Cylinders - The data for themaximum lift coefficient for smooth cylindersare summarized in Figs. 16 and 17. The originaldata in plotted and tabulated form are presentedin Ref. [2]. Evidently, the lift coefficientdepends on Re for Re larger than about 20,000and rapidly decreases to about 0.2 for largervalues of Re and K. It is also evident that thelift force is a major portion of the total forceacting on the cylinder and cannot be neglectedin the design of structures.

The frequency of the alternating transverseforce is shown in Fig. 18 in terms of fr=fv/f asa function of K and Re. It is apparentthat fr is not constant and increases withincreasing K and Re. Furthermore, a quick calculation through the use of Fig. 18 shows that theStrouhal number given by fvD/U = fr/K is not

?constant at 0.2, as in steady low, and dependson both Re and K. Experiments with rough cylin-ders yielded similar results for CL, fr, and theStrouhal number [2]. Largest lift as well asthe largest in-line force tended to occur atthetime of maximum velocity. This in turn decreasethe phase angle and hence the inertia coeffi-cient and increased the drag coefficient relativto smooth cylinder. Thus, roughness increasesthe maximum instantaneousforce acting on thecylinder by synchronizing the occurrence of themaximum in-line and transverseforces at or nearthe time of maximum velocity.

CONCLUSIONS

The results presented herein warrant thefollowing conclusions: (a) For smooth cylinders,the drag, lift, and the inertia coefficientsdepend on both the Reynolds and Keulegan andCarpenter numbers; (b) For rough cylinders, thesame force coefficients become independent ofthe Reynolds number above a critical value anddepend only on the Keulegan and Carpenter numberand the relative roughness; (c) Correct artifi-cial roughness may be used to provoke and simu-late supercritical flow in model tests in steadyas well as oscillatory flows; (d) For both smootand rough cylinders, the relationship betweenthe drag and inertia coefficients is not uniqueand depends on the particular value of theKeulegan and Carpenter number; (e) The transvers@force is a significant fraction of the totalresistance at all Reynolds numbers and must beconsidered in the design of structures; (f) Thefrequency of the shedding of the primary vorticesincreases at more or less discrete steps with

1(-in IN-LINE AND TRANSVERSE FORCES ON CYLINDERS IN OSCILLATORY FLOW OTC 2533

Increasing Reynolds and Keulegan-Carpenter num-,bers; (g) The results reported herein and theconclusions arrived at are applicable only tocyJinders in harmonic flow with zero mean veloc-ity. The force coefficients for harmonic flowwith a mean velocity superimposed on it maydiffer significantly from those reported herein;land finally, (h) it is hoped that the datapresented herein will accentuate the need for~actual full scale experiments and enable thoseconcerned to interpret and better understand thefactors effecting the force-transfer coefficientsin wavy flows.

NOMENCLATURE . .. . ..,.,.

AA.CdCLcmDFFdFi

;V9H

;L

~eTt

:mv

BYevP

Amplitude of oscillations at the test sectionAmplitude of oscillations at the free surfaceDrag coefficientMaximum lift coefficientInertia coefficientDiameter of the test cylinderForceDrag forceInertial forceFrequency of oscillations, l/TFrequency-of vortex shedding (first harmonic)Gravitational accelerationElevation (see Fig. 1)Keulegan-Carpenter number, UmT/DRoughness heightLength of the test cylinderPressureReynolds number, UmD/vPeriod of oscillations in the tunneltimeInstantaneous velocityMaximum velocity in the cycleVelocity in steady flow

Frequency parameter, D2/vTSpecific weight of water2rt[TKinematic viscosity of waterDensity of water

AcKNOWLEDGMENT .

The results presented here were obtained inthe course of research supported by a Grant(AG-477) from the National Science Foundation.The skilled experimental assistance of Messrs. N.

I

J. Collins, S. Onur, students at the Naval Post-graduate School, and J. McKay, model maker, isgratefully acknowledged.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

Sarpkaya, T.:“Forces on Cylinders and Spheresin a Sinusoidally Oscillating Fluid”, Journal~f Applied Mechanics, ASME, Vol. 42, No. 1,March 1975, pp. 32=7.

Sarpkaya, T.:“Vortex Shedding and Resistancein Harmonic Flow About Smooth and RouahCircular Cylinders at High Reynolds Numbers’:,Naval Postgraduate School Technical ReportNo. NPS-59SL76021, February 1976, Monterey,California.

Morison. J. R., et al.:’’TheForce Exerted bySurface-Waves on Piles”, Petroleum Trans., -Vol. 189, 1950, pp. 149-157.

Keulegan, G. H. and Carpenter, L. H.:’’Forceson Cylinders and Plates in an OscillatingFluid”, Journal of Research of the NationalBureau of Standards, Research Paper No. 2857,Vol. 60, No. 5, May 1958.

Sarpkaya, T.:“Forces on Cylinders Near aPlane Boundary in a Sinusoidally OscillatingFluid”, Fluid Mechanics in the PetroleumIndustry, ASME, December 1975, pp. 43-47.

Wiegel, R. L.: Oceanographical Engineerinc&Prentice Hall, Inc., Englewood Cliffs, N. J.,1964, pp. 257-260.

Szechenyi, E.:“Supercritlcal Reynolds NumberSimulation for Two-Dimensional Flow OverCircular Cylinders”, Journal of FluidMechanics, Vol. 70, Part 3, 1975, pp. 529-542

Guven, O., Patel, V. C., and Farell, C.:“Surface Roughness Effects on the Mean FlowPast Circular Cylinders”, Iowa Institute ofHydraulic Research Report No. 175, May 1975,Iowa City, Iowa.

Schlichting, H.: Boundary-Layer Theory,McGraw-Hill Book Co., N. Y., 1968, (6-th cd.)pp. 578-586.

4.......................—T........................P 1

1-6’ .—~

.........................

— ~ [-.-.-=--.-.--...--.-”-””’ 16’

P.

“<--”-”””~-””: : : :* :. . , .. It- 30“ ~

FIG, 1- THEU-SHAPED VERTICAL WATER TUNNEL,

FIG, 2- SPNPLEFORCE ANDACCELERATION TRACES,

3.01 ,’ I 1 I 1 I , I I I I I I I I , I , I 1 I . 1 1 II

I -11- 1

I/9= 701::::::;;;;;0::?...

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/ A150000.8 /

0.7 .+935@ 20000

1

0.6 -● 25000 K

0.5 , I 1 I

2 345 678910 20 30 40 50 60 100 150

FIG, 3- RRAGCOEFFICIENT VSTHEKEULEGAN-&PENER NmER FOR THEKEULEGAN-CARPENTERDATA FOR CONSTANTVALUES oF THE FREQUENCyPARAMETER,

3.0 1 I I 1 1 I I I I I 1 1 I 1 [ I I , I 1 I I I I I......... ..97

29~5w“

78 .. . . .. . .. .. . . .. ....... .. -..-.-*::.::::. .. .

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c ...&m

/7=,360+701/4 =3876.”---.---.”z:..-”:=*’z:’-----B.”.

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//. Re

“’f+/ ....@ .. . ..m796

:.; -.,,f,, 987

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0.7 — 2,{;* %3.”;;fo 0 20000

0.6 . K s 25000

I , 1 I t I 1 1 I I I0.5 I I 1 I I 1 I I I i 1 I 1 I I

2150

;IG, ;- iNE~TI;CIE~~;CIENTVSTHl0EULE~L-UflpEN~ER6~~ERFl~0THE I(EULEGAN-CARPENTERi3ATA i=ORCONsTANTVALUES OF THE FREQUENCyPARAMETER

3,0 I I I I I 1’1 I I I I I I I I i I I I I I I I

[

cd2.0

1.0

0.9

0.8

...-/9”.

.........

....,

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10;000 15,000

0.7

0.6 L- —. K 1

0.5 I I I I I Ill I I I I I I I I 1 I I I t I I I I I 12 3 4 5678910 20 30 40 60 60 70 100 150

FIG, 5- DRAGCOEFFICIENT VSTHEWJLEGAN-CANPENTER WIBERFOR‘THE &ULEGAN-CARPENTERDATA FOR CONSTANTREYNOLiIS-NWE.3ERSl

3.0 # i 1 I I I 1 I I I 1 I I I t I 1 I I I I I I I [ II

0.6 — K

0.5 I I I I I I I I I I I 1 I I I I I I I I I I I

2 3 4 5. 678910 20 30 40 50 60 70 100 150

FIG, 6- INERTIACOEFFICIENTvs _hiEKEULEGAN-bRPENTER NIMBERFOR THE kuLEGAkCARpENTER DATA FOR CONSTmT l?mms NmERSj

3.0( I 1 I I [ I I 1 I I I 1 I I I I I I

kto.

0.5t

%4 I I 1 1 I I I 1 I i 1 I 1 I 1 I 1 I

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FIG, 7- IRAG CO;;FICIENTvs W2; kJLE;~-&PENi;R NLMBERFORCONSTWT VALUES OF THE FREWENCY pWETER (pREsENT DATA)s

3.0- 1 I I I I I 1 I , [ I 1 I I I I I I.. cm

I al

1

FIG, 8- INERTIA COEFFICIENT vs THE l&JLEG -CARPENTERNLMBERFORTCONSTANTVALUES OF .THE FREQUENCYPARAMETER PRESENTDATA) I

K

0.3 I I 1 I 1 I I I I I 1 i I I I I 1 I I 1 I2,5345 678910 15 20 30 40 50 100 150’ 200

FIG, 9- DRAG COEFFICIENT vs THE KEULEGAN-CARPENTER NWIBER FORCONSTANT VALUES OF THE FREQUENCY PARAMETER AND THE kYNOU)SN1.14BER,

3.0 I I I I I I 1 I n I I I I 1 I I I

cm

2.0- Rex163=60 $0 100 150 2Q9.. B=5260..... 4480.......,...+,%+%. .........-~~~-. ~.-~l 23“%...... .........-..........

1.5-

~--”””--”- z :

.......%.

““-”-”:”~;mFio;i?:~30 ....... . . .. .1.0 -

... .. . . .

2030 497

......t:~...●.+

0.5-K = UmT/D

i 1 1 i I I I 1

0. 4, 1 I 1 I 1 I 1 1 I 1 1

345 678910 20 30 40 50 100 150 200

FIG, 10- INERTIA COEFFICIENT vs THE KEULEGAN-CARPENTER NLWIBERFOR CONSTANT VALUES OF THE FREQUENCY PARAMETER AND THE kYNOIDSNUMBER ,

1.8 - %.

cd ‘“”%..1.6 -‘%.......+

4 .... ....%.6 ....

1.4 - ‘“”...........

...<....

....‘....

1.2-

......... ~

1.0-8

0.8-

0.6-

0.4-

I I I 1 I I I I I0.2 I I I 1 I 1 1 I I

0.1 0.2 0.5 1 2 345 10

FIG, 11- lkAGCOEFFICIENT vs F&rims NUVIBERFOR CONSTANT VALUES OF K,

2,0 I 1 I I I I I I I 1 , I I I I..........-‘“”-”””----”----- K=6

1.9 . ..............................8 .~~....=....$~-..

.......-’ 0 .- . . ..

I..

0.8 15 ...’’.............................. ........0.7

.......- --...i;.,... . .~0.6 I , I I I I 1 I 1 I 1 , I , 1

0.1 0.2 0.5 1 2 3 4561

[:Gk ~ - INERTIA COEFFICIENT VS kYIWLIXi NIMBER FOR CONSTANTVALUES

1.6 --.,.

1/1 80

_ cd............................................

1.5..

1.4 - ................ .... ... .......... .. ...... l/360

1.3 - :..

1.2 -.t :

. . ...000.....<.. ● * .. . . . . . . . . . . . . . . . . . . ..O. O●*.. ,.:. ● .0..*” ... ..... . . . “.

1.1.........

L “o: .. ... .. ........ ...... ....... ..””””.%.”.”. 1/500: “.;, .

1.0 - ●

t “... ..

1.7 .... i ........... r . .. ... ... .. .... . .. .. .... ..... <k~~= ,)~o r ‘ “1

i I I I I..

i.: . ..”*

Re X lii5 ‘ . . .. OOOO..””O”

0.9 -.,.. ..

.......”

0.8 -

0.7. _

0.6 -

0.5 -

0.4 -

0.3 -

.●

●●✎ STEADY FLOW

..

...●.

0.2 I I I , I I I I I I 1 I 1 I 1 I I 1 1 I 1 J0.1 0.5 1 i 345 10

FIG, 13- DRAGCOEFFICIENT vs PtYNOiDSNLJYBERFORROUGHCYLINDERS,

2.0 , I I I I i I I i I I I II

1.9 -

c1.8 -m

1.7 -

106 -

1.5 -

1.4 -

1.3 -

1.2 -

1.1 -

1.0 -

i

/“ -----.,.””

..”“--..-.-.”.””.--”----”--””-”-=----k/D = 1/500

... .. . . . . . . . .. . . .. . .. . . .. . . .. . . . .. . . ..

#

..*”... ....* .. .“-------------------------------- 1/1 80

......*

0.9 -

Rexlb50.8 I I I I I I I I I 1 I I 1 I I

0.1 0.2 0.3 0.4 0.5 1 2 345

FIG, 14- INERTIA COEFFICIENT vs REYNOLDSNLMBERFOR ROLEiHCYLINDERS,

3.0 I 1 1 I I I I 1 I 1 I I 1 1 I i I I I I 1 I I I

4

2.0 - ‘dki D = 1/500 1/360

..................YV ....... .........

1.0 -

“.-..*.=..=-:..:.Y - - - - -

.●

●

0.5 - “...

0.4 - ..●

. . . . .

0.3 - . . . . .. .OO. t. ●.*..*” k/D=3.5x165 Um k/v

0.2 I I I 1 II I I I I I I [ I I I I I I 1 I I II 145 10 20 30 40 50 100 200 300 1000

FIG, IS - DRAGCOEFFICIENT vs THE ROUGHNESSREYNOLDSNWBERt

4.0I I I I I I I 1 I I I I I I I I I I 1 I

3.0 - fl/9=1107CL

\\

.. --“..+.1985 ,: %.

....; %.....

2.0 -,..

,.. %,

Rex I i53= ~o 40-. .. . . . .. ‘..%.

1.0 -

0.5 -

0.4 -

0.3 -

0.2 –

0.1 I I 1 I I 1 I 1 I 1 I I I I I I I I 1 I3 45 10 20 30 40 50 100 150 200

FIG,16- LIFT COEFFICIENT vs THE KEULEGAN-CARPENTERNWIBERFOR CONSTANTVALUES OF THE FREQUENCYPARPMETERAND THEkYNOLDS NIJ’43ER,

4“ ? , , u

‘CLI I I I 1 1 I 1 I I I I

K=1O..... . .. . . . .. . . . .. ....+..... . . . . .. . . . . . .. . . . ..-~....

3 -15 ““%...*‘. ..*‘. ‘+*

P ;- .. T

-.,

\

“.,“.

‘...,.

........................ ,,-.30 4

40

60

----- ... . .. . . . . . . . . ... . . .. . . . . . . . .. .. . . . . . . . . . . . . . . . . . .. Rex16’~.... .. . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . .. . . . .. . . .. . . .. . .

01 I 1 , 1 I 1 I 1 I 1 I 1 *

10.1 0.5 1 2 345

FIG, 17-kIFT COEFFICIENT VS THE kYNO13S NWIWR FOR CONSTANT

VALUES OF ,

0.055 10 50 100 200

FIG, 18 -NORMALIZEDVORTEXSHEDDING FREQUENCY AS AFUNCTION OF THE kYNOiDSm lkULEGAN-CARPENTERNLM3ERS ,