ANL-93/15ARGONNE NATIONAL LABORATORY9700 South Cass Avenue, Argonne, Illinois 60439

Distribution Category:Engineering, Equipment,

and Instruments(UC-406)

Motion-Dependent Fluid Forces Acting on

Tube Arrays in Crossflow

by

S. S. Chen, S. Zhu, and J. A. Jendrzejczyk

Energy Technology Division

June 1993

Work si p ported by

U.S. DEPARTMENT OF ENERGYOffice of Basic Energy SciencesandTaiwan Power CompanyTaiwan

~)i' ~~eJ(k%.~ M~ *

MAP wl

+.. 1'nIA

Contents

A b stra c t ................................................................................................................... 1

1 Introduction ........................................................................................................ 1

2 Unsteady Flow Theory of Motion-Dependent Fluid Forces .............................. 2

2.1 Quasistatic Flow Theory .............................................................................. 22.2 Quasisteady Flow Theory............................................................................ 32.3 Unsteady Flow Theory ................................................................................. 4

3 Brief Review of M otion-Dependent Fluid Forces ............................................... 4

3.1 Analytical M ethods....................................................................................... 43.2 N um erical M ethods ..................................................................................... 53.3 Experim ental Techniques............................................................................ 5

4 Experim ental Setup............................................................................................ 7

4.1 W ater Channel............................................................................................. 74.2 Force Transducers ........................................................................................ 74.3 Test Section................................................................................................. 10

5 Test Procedures and Data Analysis.................................................................. 13

6 Test Results ......................................................................................................... 14

6.1 Added M ass Coefficients .............................................................................. 146.2 Vortex Shedding........................................................................................... 176.3 M otion-Dependent Fluid Forces .................................................................. 186.4 Fluid-Force Coefficients............................................................................... 206.5 Effects of Oscillation Amplitudes and Excitation Frequencies................. 366.6 Sym m etry and Antisym m etry of Fluid Forces ........................................... 49

7 Unsteady Flow Theory for Fluidelastic Instability of Tube Arrays.......... 51

7.1 Equations of M otion.................................................................................... 517.2 Dynam ic Instability ................................................................................... 537.3 Sim plified Case ............................................................................................. 55

8 Closing Rem arks................................................................................................. 56

11

A ck n ow ledgm en ts.................................................................................................... 58

R eferen ces ................................................................................................................ 5 8

Figures

1 Tube array in crossflow ............................................................................... 3

2 T est ch annel................................................................................................. 8

3 W ater ch annel............................................................................................. 9

4 Schematic representation of force transducer .......................................... 10

5 Force-to-displacement ratio as a function of excitation frequency for

typical cases in which a single tube is oscillating in water...................... 11

6 Tube row, square array, and triangular array.......................................... 12

7 Flow diagram of instrumentation for data analysis.................13

8 S-Thematic representation of first tube array tested, i.e., row oftubes in crossflow ........................................................................................ 14

9 Setup for testing a row of tubes................................................................. 15

10 Force-to-displacement ratio as a function of RMS displacement whenfrequency is 0.5 and 1 Hz.......................................................................... 16

11 Strouhal number for tube row as a function of pitch-to-diameter ratio.. 18

12 Fluid-force components acting on tube row .............................................. 19

13 Fluid force g1 as a function of tube displacement vi for frequenciesof0 .15 an d 2 H z ............................................................................................ 20

14 Magnitude of fluid force and phase angle between fluid force and tubedisplacement ui due to oscillation of Tube 1 in the x and y directions.... 21

15 Fluid-damping coefficients axh, a, , ' t 11 t 12, and '13 forcetransducer Type A ..................................................................................... 24

iv

16 Fluid-damping coefficients p 2 1 , 1, and a3 forcetransducer Type A .................................................................................... 25

17 Fluid-stiffness coefficients alh, a 21a31, tll 21, and t31 forcetransducer T ype A .................................................................................... 26

18 Fluid-stiffness coefficients P l P 3, a1, a"1, and al ) forcetransducer T ype A ..................................................................................... 27

19 Fluid-damping coefficients ah, a a x, 1' r1 , t21, and r' forcetransducer Type B, reduced flow velocity from 2 to 160 .......................... 28

20 Fluid-damping coefficients 1 , 31 1 ' C21 , and a3 1 forcetransducer Type B, reduced flow velocity from 2 to 40 .......................... 29

21 Fluid-stiffness coefficients acr.'2, a a1, a , 'l, t2 1, andri forcetransducer Type B, reduced flow velocity from 2 to 160 ........................ 30

22 Fluid-stiffness coefficients 1 1, P 1,P31, al a 1, and a i forcetransducer Type B, reduced flow velocity from 2 to 40 ........................... 31

23 Fluid-damping coefficients a', ca. 1, Tl'r1 2 1 , and T' forcetransducer Type B, reduced flow velocity from 2 to 200 ........................... 32

24 Fluid-damping coefficients P , p2>31' a1, a21, and a31 forcetransducer Type B, reduced flow velocity from 2 to 200 ......................... 33

25 Fluid-stiffness coefficients all, a a , 1or, o1, and oi forcetransducer Type B, reduced flow velocity from 2 to 200 ........................ 34

26 Fluid-stiffness coefficients Pil1 , p 1, a1 1 , a 1, and al forcetransducer Type B, reduced flow velocity from 2 to 200 ........................... 35

27 Fluid-damping coefficients a'1 , aT, a1' 1 , . t21, andr3 as afunction of RMS tube displacement oscillating at 0.12 Hz, forcetransducer Type A ..................................................................................... 37

28 Fluid-stiffness coefficients alh, a21 a31,1,)21, and T3 1 as afunction of RMS tube displacement oscillating at 0.12 Hz, forcetransducer T ype A ..................................................................................... 38

V

29 Fluid-damping coefficients aci, al, ca3,' 1 ' land 3 as afunction of RMS tube displacement oscillating at 1.2 Hz, forcetransducer Type A ..................................................................................... 39

30 Fluid-stiffness coefficients all, a2 1, a31, r 1 , T 1, and t l as afunction of RMS tube displacement oscillating at 1.2 Hz, forcetransducer Type A ..................................................................................... 40

31 Fluid-damping coefficients a'11, c 3, a317T' 1)' 1,andr3 as afunction of RMS tube displacement oscillating at 0.06 Hz, forcetransducer T ype B ..................................................................................... 41

32 Fluid-damping coefficients Cy'a, 1 0 11, 01, and c31 as afunction of RMS tube displacement oscillating at 0.06 Hz, forcetransducer T ype B ..................................................................................... 42

33 Fluid-stiffness coefficients a l, a 1, ci, 11, r21, and -t l as afunction of RMS tube displacement oscillating at 0.06 Hz, forcetransducer Type B ..................................................................................... 43

34 Fluid-stiffness coefficients l, pP 1P31 ai 1 , a1, and ali as afunction of RMS tube displacement oscillating at 0.06 Hz, forcetransducer Type B ..................................................................................... 44

35 Fluid-damping coefficients a a T, '1, T 1, t'1, and 3 as afunction of RMS tube displacement oscillating at 0.12 Hz, forcetransducer T ype B ..................................................................................... 45

36 Fluid-damping coefficients h, P21, 231011, 1, and a 31 as afunction of RMS tube displacement oscillating at 0.12 Hz, forcetransducer Type B ..................................................................................... 46

37 Fluid-stiffness coefficients all, a21 , a 3 1, t 1 , 1 , and T3 1 as afunction of RMS tube displacement oscillating at 0.12 Hz, forcetransducer Type B ..................................................................................... 47

38 Fluid-stiffness coefficients PIl, P2131,IalyI, a~l, and ayi as afunction of RMS tube displacement oscillating at 0.12 Hz, forcetransducer T ype B ..................................................................................... 48

vi

Tables

1 Calibration constants for three circular-cross-section and threehexagonal-cross-section tubes.................................................................... 10

2 Experimental and theoretical values of added mass coefficientsw ith excitation at 0.5 H z ............................................................................ 16

vii

Motion-Dependent Fluid Forces Acting onTube Arrays in Crossflow

by

S. S. Chen, S. Zhu, and J. A. Jendrzejczyk

Abstract

Motion-dependent fluid forces acting on a tube array were measured as afunction of excitation frequency, excitation amplitude, and flow velocity. Fluid-damping and fluid-stiffness coefficients were obtained from measured motion-dependent fluid forces as a function of reduced flow velocity and excitationamplitude. The water channel and test setup provide a sound facility for obtainingkey coefficients for fluidelastic instability of tube arrays in crossflow. Once themotion-dependent fluid-force coefficients have been measured, a reliable designguideline, based on the unsteady flow theory, can be developed for fluidelasticinstability of tube arrays in crossflow.

1 Introduction

The components of many heat exchangers and steam generators comprise agroup of tubes submerged in crossflow. Fluid flow is a source of energy that caninduce vibration and instability. In general, the excitation forces can be divided intotwo groups. When a tube array is rigid, it disturbs the flow field and the fluid forcesacting on the tubes (called fluid excitation forces) are the result of the fluid flow.For example, steady and fluctuating drag and lift forces are typical fluid excitationforces. When the tubes in an array oscillate in the flow, the oscillating motion willdisturb the flow field and the fluid forces acting on the tubes will depend on themotion of the tubes. All the fluid-force components that are a function of tubemotion are called motion-dependent fluid forces. Typical examples are fluid addedmass, fluid damping, and fluid stiffness.

Fluid excitation forces will excite tube vibration and cause forced vibration andresonance. In general, they do not change tube characteristics. On the other hand,motion-dependent fluid forces can change the characteristics of coupled tube/fluidsystems and may induce instability. Mathematically, it can be stated that fluidexcitation forces appear on the right-hand side of the differential equations thatdescribe the coupled fluid/tube system, whereas motion-dependent fluid forcesappear on the left-hand side of the equation. The main objective of this study is topresent the motion-dependent fluid forces that act on a tube array.

2

2 Unsteady Flow Theory of Motion-Dependent Fluid Forces

Consider a group of n tubes vibrating in a flow as shown in Fig. 1. The axes ofthe tubes are parallel to one another and perpendicular to the x-y plane. The radiusR of each tube is the same, and the fluid is flowing with a gap flow velocity U. Thedisplacement components of tube j in the x and y directions are uj and vj,respectively. The motion-dependent fluid-force components acting on tube j in the xand y directions are, respectively, fj and gj, and are given by Chen (1987b) as

fj= -p7R2Xjajk Uk + k + pU2 jkau + (j1k avk = 2at2 j=1 Ita( )

n

+pU 2 (cc"kuk + ajkvk)j=1

and

Ej = -p n k 1 uk + ja2vk + k2 kkn

n+pU 2 2(j'kuk +fP kvk), (2)

j=1

where p is fluid density; t is time; u is circular frequency of tube oscillations; ajk,

Djk, cjk, and tjk are added mass coefficients; %k' Pjk,Jk) and tik are fluid-damping coefficients; and ajk, > k, ak, and rjk are fluid-stiffness coefficients.

Motion-dependent fluid forces depend on deviation from a reference state ofsteady flow, which can be grouped according to three different theories: quasistaticflow, quasisteady flow, and unsteady flow.

2.1 Quasistatic Flow Theory

At any instant in time, tAe fluid-dynamic characteristics of tubes oscillating ina flow are equal to the characteristics of the same stationary tubes whoseconfiguration is identical to the actual instantaneous configuration. The fluid forcesdepend on the deviation from a reference state of steady flow, i.e., the fluid forcesdepend only on uj and vj, not on 6ij, ij, 0j, and Vj (the dot denotes differentiation withrespect to t), so that

3

U(z)

N\XXX\NN\X\X\X\N\XXNX \N\X\XK\Q>

X 0000000000

000000000000000

g; v

Fig. 1. Tube array in crossflow

fj= pU2 t(akuk +ajkvk)j=1

and

gj = pU 2 (t'kuk + rkvk)-j=1

In this case, the fluid forces are determined uniquely by the tube configuration.

2.2 Quasisteady Flow Theory

At any instant in time, the fluid-dynamic characteristics of tubes moving inflow are equal to the characteristics of the same tubes moving with constantvelocities equal to the actual instantaneous values. The fluid forces depend on tubeconfiguration and are proportional to tube motion. This is reflected by the changesof amplitude and phase of the fluid force with respect to tube motion. In this case,

(3)

(4)

4

the fluid-force components are given by Eqs. 1 and 2. The fluid-stiffness coefficientmatrices are constant,i.e.,a" it'ko k and t'k and are independent of thematice ar costat, je. %k jk 'Jk) jk'reduced flow velocity. The fluid-damping coefficient matrices are functions of thereduced flow velocity.

2.3 Unsteady Flow Theory

In general, the fluid-force components are nonlinear functions of uj, vj, iij, Vj, uj,and ij. The general expressions for the fluid-force components are given in Eqs. 1and 2; however, the fluid-force matrices and fluid-coefficient matrices are functionsof U, uj, vj, jvjdj , and j.

3 Brief Review of Motion-Dependent Fluid Forces

The crux of stability analysis of tube arrays in crossflow is the information onmotion-dependent fluid-force coefficients, in particular, fluid-damping coefficientsand fluid-stiffness coefficients. Several approaches can be used to obtain thesecoefficients, including analytical and numerical methods and experimentaltechniques.

3.1 Analytical Methods

A series of attempts has been made to analyze the fluid forces acting on tubearrays oscillating in crossflow. Theoretically, this analysis should be based on theNavier-Stokes or Reynolds equation. Various approximate methods are used toanalyze this problem because of the difficulty in solving the problem exactly.

Potential-flow theory. An investigation of fluid forces based on the potential-flow theory was performed by Chen (1976). The main conclusion of theinvestigation was that the potential-flow theory is applicable to added masscoefficients but, in general, it is not applicable to fluid-damping and fluid-stiffnesscoefficients. Additional studies were made by Paidoussis, Price, and Mavriplis(1984) and Van der Hoogt and Van Campen (1984).

Quasistarac flow theory. Typical examples are those by Connors (1970) andBlevins (1977); however, the fluid-stiffness forces are measured experimentally. Atthis time, no analytical solution is available for the quasistatic flow theory.

Quasisteady flow theory. Several forms of fluid forces based on thequasisteady-flow theory were presented by Price and Paidoussis (1983 and 1984)

5

and Paidoussis, Price, and Mavriplis (1984). Some heuristic parameters are alsoincluded to improve the quasisteady flow theory.

Unsteady flow theory. Lever and Weaver (1982, 1984) formulated a one-

dimensional model based on mass and momentum conservation equations, andconducted a series of experiments to verify the basic assumption. Additionalstudies, by Yetisir and Weaver (1988, 1992), were conducted to improve the theory.Another approach, which uses a vorticity formulation for two-dimensional fluidmotion, was developed by Marn and Catton (1991).

3.2 Numerical Methods

Effective methods for working with moving boundary problems with small-to-medium deformations are based on various algorithms, including Eulerian,Lagrangian, and Eulerian-Lagrangian. For example, the wake interference behindtwo flat plates normal to the flow was numerically calculated by a finite-elementmethod for two Reynolds numbers, 80 and 160 (Behr, Tezduyar, and Higuchi 1991).The flow patterns were found to vary with Reynolds number and gap size. At thistime, although several algorithms are available to work with the motion-dependentfluid forces, no systematic study has been conducted for tube arrays.

3.3 Experimental Techniques

The assumption of quasisteady flow was considered by Price, Paidoussis, andSychtera (1988) for two tubes, one behind the other. It was found that thediscrepancy between the quasisteady and unsteady fluid-force coefficients dependsnot only on reduced flow velocity but also on the position of the leeward tuberelative to the windward tube. When reduced flow velocity is large, the quasisteadyflow theory is applicable; however, greater caution is required when the quasisteadyflow assumption is applied to small reduced flow velocity.

One method to obtain fluid forces is to base the calculation of force coefficientson measured structural responses, such as accelerations and displacements. Insome cases, because direct measurement of fluid forces is difficult, e.g.,measurements of high buildings or heat exchanger tubes, some attempts have beenmade to use the inverse method (Xie 1988, and Holzdeppe and Ory 1988). Thetechnique is useful when detailed dynamics of the system are well known, such asin a simple bridge section (Xie 1988). Granger (1990) also used an inverse methodto obtain fluid forces. He studied a flexible tube in an otherwise rigid tube bundlethat was subjected to water crossflow. Some fluid forces compare reasonably wellwith the data of Tanaka and Takahara (1981) and Tanaka, Takahara, and Ohta

6

(1982). For a group of flexible tubes, it will be difficult to obtain all fluid-forcecoefficients.

Several experiments focused on measuring motion-dependent fluid forcesdirectly. Teh and Goyder (1988) measured fluid forces acting on a tube that wasexcited to oscillation. These fluid forces were related to the oscillating tube only;therefore, they can be used for fluid-damping-controlled instability only. Hara(1987) measured unsteady fluid forces acting on a tube row and studied the detailedflow field. Funakawa et al. (1990) performed an experimental study of unsteadyfluid forces acting on tube arrays with a pitch ratio of 1.41 in two-phase flow.However, these authors only measured a single component of the fluid forces for aspecific motion. The results provide some insights into the instability of tube arraysin two-phase flow but cannot be used for practical prediction of instability. Themost extensive measurements of motion-dependent fluid forces were by Tanaka$1980); Tanaka and Takahara (1981); and Tanaka, Takahara, and Ohta (1982), whomeasured motion-dependent fluid forces for tube rows and square arrays with pitchratios of 1.33 and 1.42. This technique was also used by Jendrzejczyk and Chen(1987).

In this study, we used the unsteady flow theory. Fluid-force coefficients can bedetermined by measuring the fluid forces acting on the tubes that are due tooscillations of a particular tube. For example, if tube k is excited in the y direction,its displacement in the y direction is given by

vk = v cos cot. (5)

The fluid force acting on tube j in the x direction can be written

fj = 1PU2cjk cOS((t + $jk)v, (6)2

where cjk is the fluid-force amplitude and $jk is the phase angle by which the fluidforce acting on tube j leads the displacement of tube k.

With Eqs. 1 and 5, we can also write the fluid-force component as

fj =(pnR2W2ajk + pU2 j k )vcosWt - pU2akvsinot. (7)

Combining Eqs. 6 and 7 yields

. 1 (8)=.jk -cjk cos $jk -- U.jk

2 Ur

7

and

jk jk sin4jk, (9)

where Ur is the reduced flow velocity (Ur = nU/oR).

The added mass coefficient ajk in Eq. 8 can be calculated by applying thepotential-flow theory (Chen 1975, 1987a). Then aIk and a:'k can be calculated fromEqs. 8 and 9, when the force amplitude cjk and phase angle $jk are measured.Other fluid-force coefficients can be obtained in the same manner.

Fluid-force coefficients depend on tube arrangement, tube pitch, oscillationamplitude, oscillation frequency, and flow velocity. For a given tube array, fluid-force coefficients are a function of oscillation amplitude (d/D) and reduced flowvelocity (Ur), where d is vibration amplitude and D is tube diameter. For small-amplitude oscillations, fluid-force coefficients can be considered a function ofreduced flow velocity only.

4 Experimental Setup

4.1 Water Channel

The test channel is shown in Figs. 2 and 3. Water is pumped into an inputtank. The flow passes through a series of screens and honeycombs and then into arectangular flow channel. The water level is controlled by standpipes in the outputtank and the flow is controlled by the running speed of the pump motor.

Flow velocity is measured with a current flow meter. The rate of propellerrotation is directly proportional to stream velocity and therefore the sensor outputsignal is not effected by other factors, such as water conductivity, temperature, andsuspended particulates.

4.2 Force Transducers

A schematic representation of a force transducer is shown in Fig. 4a. Therelatively rigid main bodies of the tubes are constructed from stainless steel tubingwith a 2.54-cm (1-in.) OD, a 0.071-cm (0.028-in.) wall thickness, and a 38.1-cm(15-in.) length. Thin brass end caps are soldered to both ends of each tube and asmaller, relatively flexible tube, with a 0.635-cm (0.25-in.) OD, a 0.089-cm (0.035-in.) wall thickness, and a 12.07-cm (4.75-in.) length, is fastened to the upper end capof each tube (Fig. 4b). Two sets of strain gauges are placed on the outer surface of

8

DOWNSTREAM TANK TEST AREA UPSTREAM TANK

FLOW METER CONTROL VALVE

Fig. 2. Test channel

the smaller tube where the outer surface of the tube has been machined to a smallerdiameter or a hexagonal section iFig. 4c). When the cross section is circular thetransducer is denoted Type A; when it is hexagonal, the transducer is denotedType B. The two sets of strain gauges measure the force components in the twoperpendicular directions with a sensitivity of =1 V for 0.05 N (0.01 lb) of force actingon the middle of the active tubes.

The force transducers are calibrated by two methods, static and dynamic. Inthe static method in air, the active tube is held fixed at the supported end and agiven force is applied to the middle of the active length. In the dynamic method inair and water, the tube is excited at a given frequency and amplitude in air orwater. Then, the inertia forces due to the sinusoidal oscillations are used todetermine the calibration constant.

All calibration constants used in the measurement are determined in water.For a single tube vibrating uniformly in water, the inertia force is given by

fj = (mt + pirD 2 I/4) o2 d sin(cot), (10)

where mt is tube mass, I is tube length, p is water density, o is oscillation frequency(rad/sec), and d is oscillation peak amplitude. In testing, the inertia force fj ismeasured for a given excitation (o and vibration amplitude d. In the calibration, 0ovaries from =0.25 Hz to 2.0 Hz and d varies from 0.5 to 5.0 mm. From Eq. 10, theratio of fj/d (or g/d) is proportional to 0o2. Figure 5 shows g/d as a function ofexcitation frequency for typical cases in which a single tube oscillates in water inthe x and y directions. From the data, a power-law curve is used to determine theconstant. Theoretically, the ratio fj/d (oi g/d) should be proportional to (o 2; theactual power is very close to 2.0. All force transducers are calibrated in twodirections. The results of the calibration are given in Table 1.

Ik r.f I'A

a

KS

N-o

Ie -0

~A-~ *~

Al

-

'C.

V

I'

m7JZ-zii'4

A j :~**ii

-l VItsi4JAS

4

Fig. 3. Water channel

10

SUPPORT PLATE

2.54-cm-OD TUBE STRAIN GAUGES

38.1 cm 1.c

(a)

STR AIN GAUGES

~ I 0.635 cm

(b)

0 0

(c)

Fig. 4. Schematic representation of force transducer:(a) force transducer, (b) flexible tube, (c) crosssection of flexible tube for strain gauges,

Table 1. Calibration constants for three circular-cross-sectionand three hexagonal-cross-section tubes

x Direction y DirectionTube Cross Section Tube Number (g/v) (g/v)

Circular 1 4.325 4.246Circular 2 7.643 7.631Circular 3 6.244 5.646

Hexagonal 1 2.770 2.597Hexagonal 2 2.266 2.258Hexagonal 3 2.059 2.173

4.3 Test Section

An array of tubes is assembled in the test area. Figure 6 shows several tubearrangements: tube row, square array, and triangular array. An array of tubes,

11

6

E 4E

0

10

8

E

4

2

0

6

E 4

E

...2

00 0.5 1.5 2

- Tube 1, X DIRECTION

FREQUENCY, Hz

Fig. 5. Force-to-displacement ratio as a function of excitation frequency for typicalcases in which a single tube is oscillating in water

I -r

0 0.5 1 1.5 2FREQUENCY, Hz

-+- Tube2, X DIRECTION

-=-0

0 0.5 1 1.5 2 2.1

FREQUENCY, Hz

Tube 3, X DIRETI

- -

-- -. - .- - I . "r- - - -7 -o0

6EEm4

of2

0

6

6

EE

20

8

6

EE

4

2

0

-- Tube 1, Y DIRECTION

0 0.5 1 1.5 2 2.5

FREQUENCY, Hz

- Tube 2. Y DIRECTION

0 0.5 1 1.5 2FREQUENCY, Hz

S Tube 3. Y DIRECTION

0 0.5 1 1.5 2 2.5

FREQUENCY, Hz

I

12

Tube Row

p x

000 "X

O- 10 y

Rectangular Array

00000- 00000 x

00000000

oee o- 4 00000 Y

00000-0000~oooo

Triangular Array

0 0 0 0

0 00 0 0 0 1 1'-4. 0 000

-4. 0 0

0 Rigid Tube* Force Transducer0 Excitation Tube

Fig. 6. Tube row, square array, and triangular array

denoted by solid and shaded circles, is active, while the others, denoted by opencircles, are dummy tubes. All tubes except the middle one, denoted as Tube 1 by asolid circle, are clamped to a support plate with a nut attached to a smallersupporting tube. Tube 1 is not attached to the support plate, but passes through acircular hole in the support plate and is connected to an electromagnetic shaker.The shaker provides the support for Tube 1. In addition, specific oscillations can beassigned to Tube 1 in the x or y direction, and both the oscillation amplitude andfrequency of the shaker can be controlled in the appropriate range.

During tests, the water surface is kept at such a leve. tuat the active length ofthe tubes is submerged in the flow. Normally, only a small portion of thesupporting tube (less than 1.3 cm) is submerged in water. Therefore, the straingauges do not require waterproofing.

13

5 Test Procedures and Data Analysis

A flow diagram of the instrumentation and exciter is shown in Fig. 7. Theexciter provides sinusoidal displacement at a frequency varying from =0.02 Hz to2.0 Hz. Displacement and force signals are first filtered by band-pass filters toeliminate low- and high-frequency noises and then digitized and stored in theanalyzer. These signals are analyzed to obtain the oscillation displacement of thetube, the magnitudes of the forces acting on the active tubes, and the phase betweenthe motion-dependent fluid force and tube displacement.

In Fig. 7, the test facility, instrumentation, and data analysis systems areshown ready for a series of tests. The first tube array tested was a row of tubeswith a pitch-to-diameter ratio (T/D, where T is the gap) of 1.35 (Fig. 8). Motion-dependent fluid forces were measured for Active Tubes 1, 2, and 3, with Tube 1oscillating in the lift (x) or drag (y) direction. A photograph of the setup is shown inFig. 9.

Displacement Transducer

Electromagnetic Exciter Dsl c m n a d

IOIC*Transducer ->pass -:ss.;:... .::..::...: . Electronics Filter

Strain

Gauges

Strain Band-LGauge - pass --

Amplifier Filter

Strain Band--- a Gauge -O- pass -

Ampifier Filter

Outputs:" Displacement .4-- Analyzer" Force

" Phase Angle

Fig. 7. Flow diagram of instrumentation for data analysis

14

3

FLOWW

SHAKER

Fi .L J K _ _ _

Fig. 8. Schematic representation of first tubearray tested, i.e., row of tubes incrossflow

6 Test Results

6.1 Added Mass Coefficients

The added mass coefficients for a row of tubes were measured as a function ofexcitation amplitude and frequency. Typical results are shown in Figs. 10a and b,which show the ratio of measured force to tube displacement as a function of tubedisplacement for two frequencies, 0.5 Hz and 1.0 Hz. Tube 1 was excited in the xdirection, and the forces acting on Tubes 1-3 in the x direction were measured. Theamplitude ratios for the three tubes were independent of the excitation amplitudeand the forces acting on Tubes 2 and 3 were equal.

The calculation of the added mass coefficients is based on the measured forces.The results are given in Table 2 for excitation at 0.5 Hz.

The added mass coefficients were raeasured for a series of excitationamplitudes, ranging from =0.8 to 5 mm, which correspond to =10-60% of the gapbetween the tubes. The effects of vibration amplitude on the added masscoefficients in this range of motion are not significant. The added mass coefficientswere also measured for various excitation frequencies. For example, the values forthe added mass coefficient a,, obtained for three excitation frequencies, 0.5, 1.0,

15

0

16

1.0

EE

w

0.5

0.0

3

EE

-

2

0

--- Tube I f-0.5 Hz

--- Tube 2

Tube 3

2 3 4

RMS DISPLACEMENT, mm

-+--Tube I f-lHz (b)

--- Tube 2Tube 3

2 3

RMS DISPLACEMENT, mm

Fig. 10. Force-to-displacement ratioas a function of RMSdisplacement when frequencyis (a) 0.5 and (b) 1 Hz

Table 2. Experimental and theoretical values of addedmass coefficients with excitation at 0.5 Hz

Added Mass Theoretical Results ExperimentalCoefficients of Chen 1975 Data

all 1.113 1.098

a12 -0.282 -0.282

a13 -0.282 -0.298

all 1.114 1.089

1l2 0.340 0.336

113 0.340 0.336

I

17

and 1.5 Hz, were 1.098, 1.122, and 1.144, respectively. The experimental dataconfirm two characteristics: (1) at low excitation amplitude, the linear flow theory,based on the potential flow theory, will provide added mass coefficients withsufficient accuracy. (2) at high kinetic Reynolds number (which is equal to thecircular frequency of oscillation multiplied by tube diameter squared and fluidkinetic viscosity, Chen 1987a), the oscillation amplitude is insignificant.

6.2 Vortex Shedding

One of the main excitation sources for tube vibration in crossflow is vortexshedding. For a tube row in crossflow, vortex shedding is important. Tests areconducted for the tube row by measuring the resultant lift and drag forces acting onthe three active tubes at different flow velocities. From the frequency spectra of thefluid excitation forces in the lift direction, the vortex shedding frequencies can beidentified and the Strouhal number St can be calculated:

St = fD/U, (11)

where f represents the frequencies corresponding to the peaks in the power spectraof lift forces. In the drag direction, the frequencies are twice those of the liftdirection; therefore, from the frequencies corresponding to the peaks in the powerspectra of drag forces,

St = fD/2U. (12)

Based on the power spectra of the lift forces and drag forces at different flowvelocities, three Strouhal numbers were calculated: 0.105, 0.22 and 0.43. Thedetailed flow field across the tube row was not studied in this test. Differing vortexshedding frequencies are associated with differing flow patterns for steady flowacross a tube row.

Many studies have focused on the vortex shedding process for a single tube incrossflow (Chen 1987a). The Strouhal number is a function of the Reynolds numberand in the subcritical region, it is equal to =0.2. In this test, Reynolds numbervaries from =2500 to 3800. It is in the subcritical region. Various investigatorshave found different vortex shedding frequencies for tube rows in crossflow (Chen1968; Borges 1969; Clasen and Gregorig 1971; Ishigai, Nishikawa, and Yagi 1973;Auger and Coutanceau 1978). Figure 11 shows results obtained in the presentstudy and published data, as a function of the pitch-to-diameter ratio (T/D) of tuberows. The three values obtained during this study compare reasonably well withthe published data.

18

0.5 - Chen (1968)0 Clasen and Gregorig (1971)A Borges (1969)

0.4 - Ishigai at al. (1973)a Auger and Coutanceau (1978)v Present Study

Auger and Coutanceau (1978)0.3- a -. - Ishigai at al. (1973)

A 0

02

0.1-

0.0' I

1.5 2 2.5 3T/D

Fig. 11. Strouhal number for tuberow as a function of pitch-to-diameter ratio

Vortex shedding for a flow across a tube array is fairly complicated. In thisstudy, the vortex shedding frequencies were determined from the frequency spectraof the resultant lift and drag forces and not from the direct measurements of flowvelocity or fluid pressure. Figure 11 shows that there are five vortex sheddingfrequencies for a tube row with T/D = 1.35. Two vortex shedding frequencies thatcan be measured in the wake do not appear in the resultant forces acting on thetubes. To understand the detailed vortex shedding process, direct measurements oflocal flow velocity and fluid pressure behind the tube row, as well as visualization offlow patterns, provides a better understanding. Measurements of resultant fluidforces are useful for predicting tube response in crossflow.

6.3 Motion-Dependent Fluid Forces

Fluid forces on the tubes are relatively small except for the force on the tubebeing excited by the shaker. The magnitude of the forces depends on the excitationamplitude, excitation frequency, and flow velocity. Figure 12 shows fluid forcecomponents fl, f2, g1 , g2, acting on Tubes 1 and 2 as a result of the motion of Tube 1in the y direction, vi. The flow velocity was 0.15 m/s and the excitation frequency toTube 1 was u. i5 Hz. The signals presented in Fig. 12 have been filtered. Thedominant frequency of fluid forces is the same as the excitation frequency; however,the fluid force components are not in phase with the tube displacement vi.Depending on the phase between the tube displacement and fluid forces, the fluidforces may be an excitation mechanism or energy dissipation mechanism.

19

6

E 3

-3

-60.5

0.0

-0.50.5

0.0

0.5

cv 0.0

-0.5

V//3

0.5

0.0 1 -\I

-0.550 60 70 so 90 100

Timp, s

Fig. 12. Fluid-force components actingon tube row

Figure 13 shows the loci of the unsteady fluid forces g, at excitationfrequencies of 0.15 and 2 Hz. At 0.15 Hz, the component gi is negative when thedisplacement is positive and vice versa; therefore, it acts as an excitation force. At2.0 Hz, the component gi is positive when the displacement is positive; therefore, itacts as a damping force. A damping force will contribute to energy dissipation,which will reduce tube response, whereas an excitation force will contribute tonegative damping, which may result in fluidelastic instability.

The phase angle is defined as the angle between the fluid-force component and thetube displacement. This angle can be obtained from the correlation between the twotime histories of fluid force and tube displacement, or from the time history plotsgiven in Fig. 12. Figure 14 shows typical magnitude fluid forces (RMS values) andphase angCs as a function of reduced flow velocity. When the reduced flow is small,=<10, the fluid forces decrease rapidly with increasing reduced flow velocity. This isattributed to the effect of the inertia force of added mass of fluid. At higher reducedflow velocity, the magnitude of the fluid force is almost independent of the reducedflow velocity. Similar trends are noted for the phase angle. At low reduced flowvelocity, the phase angle varies more dramatically with reduced flow velocity,

20

0.3tAS0.15Hz

0.2 -

j o

-0.2

- -3 0 3 6

DISPLACEMENT v , m m

15.0

10.0 7 -

-5.0

-100

-3 -2 1 o 1 2 2

DISPLACEMENT . mm

Fig. 13. Fluid force g, as a functionof tube displacement vi forfrequencies of 0.15 and 2'Hz

while at large reduced flow velocity, the variation of the phase angle with reduced

flow velocity is much smaller.

6.4 Fluid-Force Coefficients

The critical elements for predicting fluidelastic instability of tube arrays in

crossflow are motion-dependent fluid-force coefficients. A reliable method to obtain

these coefficients will contribute significantly to the development of accurateprediction techniques. From the motion-dependent fluid forces acting on the

exciting tube itself and the surrounding tubes and their phase angles with respectto the displacement of the excitation tube, as well as the added mass coefficients

based on the potential flow theory, fluid-damping and fluid-stiffness coefficients canbe calculated by the method described in Section 3. Several series of tests were

performed for the tube row with a pitch-to-diameter ratio of 1.35 with Type A and B

force transducers. The excitation amplitude was set at different values bycontrolling the voltage at the shaker. The displacement of the excited tube depends

912~1

6

6

Tube 2

EQ-D 20 40 60

REDUCED FLOW VELOCOFY

Tube 3

20 40

REDUCED FLOW VELOCTYr

225

150

C7

W 75

C.

Tube 1

20 40 60

REDUCED FLOW VELOCITY

on

30

0 -20Z

-.70

-120

100

CD -25zW

60

fTube 1

3 20 40 60

REDUCED FLOW VELOCIFY

20 40 so

REDUCED FLOW VELOCITY

Tube 3

0

cm

4

1.5

1.0

C

-N }-

D

0.5

0.0

1.5

1.0

0.5

00

0

0 6020 40

REDUCED FLOW VS.OCfTY

(a)

Fig. 1 1. Magnitude of fluid force and phase angle between fluid force and tube

displacement ul due to oscillation of Tube 1 in (a) the x and (b) the y

directions

Tube 2

22

8

0

1.0

0)

0.5

0.0

1.0

" 0.5

0.020 40

REDUCED FLOW VELOCITY

0

0

.-80

z

-160

-240

a

Tube 1

0 20 40 60REDUCED FLOW VELOCITY

Tube 2

-12060 0

(b)

Tube 1

20 40

REDUCED FLOW VELOCITY

Tube 2

0

0 20 40 60

REDUCED FLOW VELOCITY

Tube 3

60

0

W -60

".

-120

60

-60

CL,

Fig. 14. (Cont'd)

20 40 60

REDUCED FLOW VELOCfTY

cm

cm4

0 20 40 6

REDUCED FLOW VELOCITY

Tube 3

0

CC

23

on the excitation frequency. For example, at 0.1 m/s, the excited RMS tubedisplacements with an excitation force denoted by 1.5 v vary from 2.65 mm at0.1 Hz to 1.68 mmat 2 Hz.

Type-A force transducers: Fluid forces were measured for three flow velocities,with the reduced flow velocity varying from =2 to 60 for excitation in the x direction.Only one flow velocity was used in the y direction. The results are given inFigs. 15-18.

Type-B force transducers: The flow velocity is set at 0.1 m/s. Two series oftests were performed with Type-B force transducers:

* The excitation in the x direction was given at three excitation levelsand the reduced flow velocity varied from =2 to 160. In the y direction,the excitation was given at five excitation levels and the reduced flowvelocity varied from =2 to 40. The results are given in Figs. 19-22.

* The reduced flow velocity varied from =2 to 200. The excitation was setat two different levels. The results are given in Figs. 23-26.

Several interesting characteristics of the fluid-force coefficients were noted.

* At high reduced flow velocity, all fluid-force coefficients areapproximately constant. In this range, fluid-force coefficients obtainedat a particular reduced flow velocity are applicable for all values ofreduced flow velocity. This characteristic was also noted for structuralcomponents with other shapes (Chen 1987b).

* Fluid-force coefficients obtained at different flow velocities areapproximately the same. Therefore, for relatively small-amplitudeoscillations, fluid-force coefficients are a function of reduced flowvelocity only. In the measurement of motion-dependent fluid forces, anappropriate combination of flow velocity and oscillation frequency canbe utilized to obtain the best results.

* At small reduced flow velocity, all fluid-force coefficients are a functionof reduced flow velocity and the fluid-force coefficients must bemeasured for all values of reduced flow velocity. In this range ofreduced flow velocity, the functional dependence of fluid forcecoefficients with reduced flow velocity is difficult to characterize.

24

5.0

2.5

0.0 -

"2.5

1.0

0.0

-' "1.0

-2.0

-3.0

1.0

0.5

0.0

-0.5

"1.0

".s.

0 10 20 30 40 50 64EDLCED FLOW VfELwCTY

-+- u " 0.16 WAt*--- "0.15 mmt

- - "0.10 met

0 10 20 30 40 50 SIDJCED FLOW vELmCT

- - U " 0.15 mri.

-+- u .0.16 mm

-+- u " 0.10 m/s

0 10 20 30 40

REDUCED FLOW VELOCfTY

1.0

0.5

0.0

-0.5

"1.0

OH

V.3

0.0

-0.5

-1.0

"1.5

"2.0

1.5

1.0

0.5

0.0

"0.550 60

- -I-. - -I - - - - -I . - I - - - - - - --- - -I . - I - -

-"- U " 0.1 5 mes-+- U " 0.15 m/s

-- U " 0.10 mis

0 10 20 30 40 50 6FMDUED ROW VELOCITY

0 10 20 30 40 50 do

REDUCED FLOW VELOCITY

0 10 20 30 40

REDUCED FLOW VELOCfTY

50 60

Fig. 15. Fluid-damping coefficients (I, a2I, aI 31, T1 > T 1, and r'l, forcetransducer Type A

0

I-r-- - .1 r

-+- U .0.15 m/

-.- U -0.10 mVS

00

A4

0 0

-- u -0.15 m-+- u 0.16 mt

-- U.=0.10 m 4

-- +- U " 0.1 5 M&

-+- U " 0.15 moo- U " 0.10 mis

25

2

b

0

a1o 10 20 20 40 50 60

MUCED FOW VELOCITY

0

-2

.3D 10 20 30 40 50 60

EDUCED FLOW V9 flY

1.0

0.5

0.0

"0.50 10 20 30 40 50 60

FEuCED FLOW VELOCTY

.2

.3

0 10 20 30 40 50 60

EoUCED FLOW VELOCITY

Fig. 16. Fluid-damping coefficients P, $21j,1' 11, , and a, force transducerType A

-+-- U 0.1 S

-T

0

0 10 20 30 40 50 60FEDUCED FOW VEOCTY

-- u -0.is m s

0 10 20 30 40 50 60FEDUED FLOW VELOCITY

- - - r -nv

0.3

0.0

"1.0

"1.5

0.5

0.0

.0.5

"1.0

-- w- U ".1 m/s

IL

- -bBbpp: .0

f a..".I

I-+- U . 0.1 S 4

I

26

i.0

2.5 r

- 0.0

-2.5

"5.0

3.0

2.0

" 1.0

0.0

"1.0

1.5

1.0

0.5

" 0.0

.0.5

-1.5

1.0

0.5

0.0

-0.5

"1.00 10 20 30 40 50 s

E D UCE- ---vaocn -

-- }- U " 0.15 nis

-U"0.15mmI

0 10 20 30 40 50 609EDUCED ROW VELOITY

" -+- Uu "0.15mnu- -U -0.15m/s

-+- U . 0.10 m/s

0 10 20 30 40REDUCED ROW VELOCITY

0.5

0.0 f

-0.

-1.0

1.0

0.5

0 10 20 30 40 50 5FuCED FLOW VoITY

r.......-.......-...-

0

0 10 20 30 40 50 SI

FEDUCED ROW VECCfTY

0.0

-0.5

"1.050 60 0 10 20 30 40

REDUCED ROW VELOCITY50 60

Fig. 17. Fluid-stiffness coefficients c f, to1,t , and tto, force transducer

Type A

.- +U - 0.15 is--- U 0.15 n~s -

" -- s-- U " 0.10 mm'"

owl

-+- U . 01-+i- U " 0.15 Wes

' -+- U -0.10 m/s

-"- U " 0.15 Wes-- "- u . 0.10 We

D0

--- 0.15 MM

-- -U . o.10 f,,,

a 0

27

1

0

.1

.2

.30 10 20 30 40 50 60

FoUCw FOw VaOCt

3

2

0

.10 10 20 30 40 60 60

DUCE FLOW VWOC

0 10 2Q 30 40

EDUCED FLOW VELOCIY

0.5

0.0

n

"0.5

"1.050 60

0 10 20 30 40 50 SG

REIDJED FLOW VELCITmY

0 10 20 30 40 50 SO

REUCED FLOW VELOCITY

U -0.1Is we

0 10 20 30 40

REDUCED FLOW VEOCrn

50 60

Fig. 18. Fluid-stiffness coefficients P11, P 1 aP1 al , and a 1 , force transducerType A

. .

-.. 0.3 e

0

.1

.2

.3

0.5

0.0

m "0.5

-1.0

"1.6

0

.1

-.-. 1.0., .

. -.. , .U ..I ..I .V .I -I- V. I I. .---

.-

0

b~ I

iI

0

. 1

- .

-- - U " 0.16 A

28

6.0

4.0

2.0

0.0

"2.0

1.0

0.0

.1.0

"2.0

2.0

0.0

.2.0

"4.0

0 20 40 60 60 100 120 140

REDUCED FLOW VELOCY

0.5

-- +- a -1.s v-- e-- a -1. v

-1.0

160

3.0

2.0

1.0

-+- a . 1.2 v--- a. 1.5 v

0 20 40 60 60 100 120 140 160

FEJCED FLOW VELOCY

0 20 40 60 50 100 120 140 160REDUCED FLOW VELOCRIY

0 20 40 60 60 100 120 140 160

REDUCED FLOW VELOCITY

Fig. 19. Fluid-damping coefficients CX, I21, X 1 TI 11 t21, and t'1 , forcetransducer Type B, reduced flow velocity from 2 to 160

.. --

-+- "a . 1.2 v

-- + - s.1.8 v

0 20 40 60 30 100 120 140 ISO

EDUCED FlOW VELOCmTY

-+- a 1.2 v-- +- a"-1.3 v-+--s-1.8v

-0.5

-1.0

1.0

0.0

- - -- "-1.2 v-- +- a - 1.3v

-+-a-t1v

-+- a"1.2 v--- .a 1.5 v

0 20 40 so 50 100 120 140 160

FEDUcED FLOW VaCrTY

I

00

r

"'.V

v.u

29

3

2

0

.1

-- a 35 v

-FL F- a "W1.9 v-"- a -2.0 v

REDUCED ROW VEL CITY

1.0

0.0

-1.0

-2.0

"2.0

"4.0

0.5

0.0

-0.5

-1.0

-1.5

"2.0

1.0

0.0

-1.0

-l2.0

"3.0

1.0

0.0

-1.0

-2.0

e.0

4.0

2.0

0.0

.00 5 10 15 20 ?5 30 35 40

ECFD FLOW VELocTY

0 5 10 15 20 25 10 35 40

FD1CED FLOW VELOCITY

-- - - a-I -v-+- a " 1.5 v

-- a - 1.9 v-- "- a " 1.9 v~~- --- a 2.9 v

0 5 10 15 20 25 30 35 40

REDUCED ROW VELOCITY

0 5 10 15 20 25 30 35 40

FDUCED ROW VELOCITY

Fig. 20. Fluid-damping coefficients B, 1 31, pof 11a1, and a', force transducerType B, reduced flow velocity from 2 to 40

-- - a " 1.2 v- - a -1.5 '

--e- a "2.0 v

- ~-f- a " 1..2 v

-"- a " 1.9 v

0 5 10 15 20 25 30 35 40

EDucED ROW VEOCfTY

-0-4. -1.2 v

- -+- a - 1.8 V-+- a -1.9 v-+- a -2.0 v

-+- a " 1.2 v

- - -2.0 v

. 4

r

30

1.0

0.5

0.0

. 5

-- +- a - 1. v

0 20 40 60 60 100 120 140 160

MCMXDFLOW VEL1Y

2.0

0.0

-2.0

-4.0

-e.0

2.0

1.0

0.0

4.0

2.0

0.0

0.0

-0.5

0 20 40 60 S0 100 120 140 160

FEDUCED FLOW VaOCITY

2.0

1.0

0.0

"1.0

0 20 40 60 50 100 120 140 160

REDUCED FLOW VELOCIY

0 20 40 60 0 100 120 140 160

EDUCED FLOW VELOCrY

0 20 40 60 "0 100 120 140 160

REDUCED FLOW VELOCITY

Fig. 21. Fluid-stiffness coefficients all, x1, 3 t 1 , and r 1 ., force transducerType B, reduced flow velocity from 2 to 160

- - --- - 6 ... ,

--i- OR-.2*--+ -a - .

-+- a -1.8v

0 20 40 so "o 100 120 140 160

FELJCED FLOW VaOCfTY

2.0

--- a -"1.5v-- +- a -1. v

-+ - a . 1.2 v"-+- a 1.3 v

"-- a - 1.$ v

-- "-" -1.2 v

-- +- a " 1.5 v

- - - - - - - - - - - - -14

V.Q

-- a"-1.2 v-- +- a.- 1.5 v

-- +- a -1.8 v

I

31

2

-- o-- a " 1.2 v-+- a" 1.3v

-+- a - .8v

0 5 10 15 20 25 30 35 40

FEDUCED FOW VacxnY

1.0

0.0

-1.0

-2.0

ca.3.0

-4.0

-5.0

"d.0

0 5 10 15 20 25 30 35 40

FEDUCED FLOW VELOCIfY

2.0

t.5

1.0

0.5

0.0

"0.5

-

-+- a " 1.2 v

--- a " I.J v

0 5 10 15 20 25 30 35 40

REDUCED FLOW VELOCITY

1.0

0.0

1.0

0.5

0.0

-0.5

-1.0

"1.5

1.0

0.0

-.10

-2.0

"3.00 5 10 15 20 25 30 3i 40

REDUCED FLOW VELOCITY

-o-a - 1.2v-- a - 1.5 v

-+- a - 1.8 v+-a - 1.9 v---a - 2.0 v -

--a"1-

0 5 10 15 20 25 30 35 40

FEDCED FLOW VELOCITY

0 5 10 15 20 25 30 35 40

REDUCED FLOW VELOCITY

Fig. 22. Fluid-stiffness coefficients 1 p 1,)pf1, aj" a) and , force transducerType B, reduced flow velocity from 2 to 40

.1

.2

-- a - 1.6 v

-- a - 1.9v

-- a - 2.0 v

-2.0

"3.0

-+- a - 1.2 v

-+- a " 1.8v

- - : -- - -a-- 1.-2-v

-.- a- 1.5w-- 4- a " 1.1 V- a " 1./v

-- a 2.0 v

- 46

32

1 .5

0.0

"1.s

4.0

2.0

0.0

-2.00 50 100 150 20(

M-D-CE FLOW VELOCITY

2.0

-2.0

-- a 1.5 v-- -a 1. v

0 so 100 150 200

FEDCED FLOW VELOCITY

0 50 100 150

EDUCED FLOW VELOC TY

1.0

0.0

-1.0

.0e 1L0 50 100 150

REDUCED FLOW VELOCf1Y

0.V

-3.o I

0 50 100 150

F¬DUCED FLOW VELOCfY

3.0

2.0

0.0

"1.0

200

Fig. 23. Fluid-damping coefficients aXI,(X1,transducer Type B, reduced flow velocity

.---- a. 1.8

LA'

0 50 100 IS0 200

REDUCED FLOW VELOCrY

31) 1 1)'213, and t31, forcefrom 2 to 200

200

200

.-- r .1.t

L 1,St

-+- a"-t.5v

I-+- a - 1.$ v

-- 3

33

0.5

0.0

-0.5

-1.0

.1.5

-2.0

-2.5

-J.0

.3.s

0.5

0.0

-0.5

-1.0

"1.5

1.0

3.0

2.0

1.0

0.0

-, - - - :

---: - ----- -

-1r

-- +- a " 1.b v

200

1.0

0.0

-1.0

-2.0

-3.0

-+- a .1.5

-+- a . v

0 50 100 150 201

REDUCED FLOW VELOCfiY

0 50 100 150

FEDED FLOW VELOCITY

4.0

2.0

0.0

"2.0200

-- a . 1. 0-- +-EaOFOV.O v

0 50 100 150 20C

REDUCED FLOW VELOCITY--- a " 1.S v

- -- . v

0 20050 100 150

REDUCED FLOW VELOCITY

Fig. 24. Fluid-damping coefficients [',h p21', f31, a1 2, and a, force transducerType B, reduced flow velocity from 2 to 200

"1.00 50 100 150

F DUCED FLOW VELOCITY

-1.0

-2.0

-- -a-15v

0.0

34

2.0

0.0

50 100 150

REiE E FLOW vaCnY0

0 50 100 150

REDUCED FLOW VELOCTY

1.0

0.0

-+- a. 1.v

0 s0 100 150 20(

MX FLOW VBC"Y

-+- a 1.5 -

0.0

-0.5

-1.0

200

-+- a.1.5 v

0 50 100 150 200

REDUCED FLOW VELCITY

-- a"-1.5 v-a"-1./ v

0 50 100 150 20(

M2.0CED FOW VLY

2.r*

1.0

O.0 I

200 0 50 100 150

REDUCED FLOW VELOCITY

Fig. 25. Fluid-stiffness coefficients t, It of t ., and 1, force transducerType B, reduced flow velocity from 2 to 200

..

.4.0

.. 0o

3.0

2.0

: 1.0

0.0

"1.l

4.,

-+- a -1. v

2.0

0.0

S-+- a "1.Sv-+- a " t1.A v

200

0l

35

1.0

0.0

-1.0

-2.0

-3.0

-4.0

"5.0

0 50 100 150

REDUCED FLOW VELOCITY200

-- +- a " 1. V

0 50 100 150 20(

0 50 100 150 200

ECED FLOW VEOCITY-+-a" s -fvy

-+- a" 1." v

b~ 0.5

0.0

"0.5

1.0

0.0

"1.0

1.0

0.5

0.0

-0.5

"1.0

1.0

0.0

"1.0

-2.0

1.0

0.0

"1.0

-2.0

"3.0

1.5

1.0

2000 50 100 150

REDUCED FLOW VELoY

Fig. 26. Fluid-stiffness coefficients $11, 2, 131, a1, a1, and a l, force transducerType B, reduced flow velocity from 2 to 200

+- 1.0 v

0o 0 100 150 200FEV FLOW VELOCflY

- - -1.5 v

0 50 100 150 200

REDUCED FLOW VELOCffY-- -a -15v

-2.0

a

i

36

Fluid-force coefficients obtained in this study agree reasonably well with thosebased on Tanaka's data (Tanaka 1980). However, some of the details are not incomplete agreement. This is probably due to different experimental setup. Theexperimental parameters in the two tests were not identical. The flow across a tuberow is fairly complicated, as demonstrated in the measurement of vortex shedding.Different flow patterns will affect the motion-dependent fluid forces. For example,the direction of f2 due to the motion of ul is different from and opposite to the onemeasured by Tanaka and his colleagues (1980, 1981, 1982). However, the directionof f3 in this test is the same as f2 in the test by Tanaka and his colleagues (1980,1981, 1982). It is apparent that the flow patterns in the two tests might be exactlyopposite to each other.

6.5 Effects of Oscillation Amplitudes and Excitation Frequencies

To obtain reliable fluid forces, the excitation amplitudes of Tube 1 must varywith flow velocity and excitation frequency. Several series of measurements of fluidforces for various tube motions were performed to understand the effect of tubeoscillation amplitude. Fluid force coefficients were obtained for various values ofoscillation amplitude (d), which is the RMS displacement of Tube 1.

Figures 27-30 show the fluid force coefficients obtained with a Type-A forcetransducer for d varying from 1 mm to 4.5 mm at 0.12 Hz and from 1.2 mm to 3.7mm at 1.2 Hz. The flow velocity is 0.15 m/s; therefore, the corresponding reducedflow velocities are 50 and 5 for 0.12 and 1.2 Hz respectively.

Two flow velocities, 0.13 m/s and 0.1 m/s, and two frequencies, 0.06 and 0.12Hz were tested with a Type-B force transducer. The corresponding reduced flowvelocities were 42.5 and 85 for 0.13 m/s at 0.12 and 0.06 Hz and 32.8 and 65.5 for0.1m/s at 0.12 and 0.06 Hz. The results are given in Figs. 31-38.

From Figs. 27-38, it is evident that there is no noticeable difference. Themaximum peak magnitude of 6.36 mm (RMS 4.5 mm) is =70% of the gap betweentubes. The amplitude of tube displacement has been tested at higher values thanthose of Tanaka and his colleagues (1980, 1981, 1982); the results shown agree wellwith their data.

Although fluid-force coefficients are not affected by ail oscillation amplitude70% of the gap, larger oscillation amplitudes are expected to have a significant

influence. As demonstrated in a stability study (Chen 1987a), two instability limits,intrinsic and excited, are encountered. The difference between these two instabilitylimits is partially due to the dependence of fluid-force coefficients on oscillationamplitude. This can be noted in Figs. 19-22. At the largest displacement, all fluid-

37

1.0

0.0

-1.0

1.0

0.0

l .1lJ

1 2 3 4 5DSPLACEMENT, mm

1 2 3 4 5DISPLACEMENT, mm

2 3

DISPLACEMENT, mm4

- -. -.-. -I ---v- -------

0.0

-1.05 2 3

DISPLACEMENT. mm

4 S

Fig. 27. Fluid-damping coefficients x11, c2, cC 31 and c' as a functionof RMS tube displacement oscillating at 0.12 Hz, force transducer Type A

-1.0

1.0

0.0

"1.0

1.0

0.0

"1.0

1.0

r 0.0

-1.0

1.0

. . I -- . - .I . - - . I .,.

i 1_ 1 i

1 2 3 4 5DISPLACEMENT, mm

1 2 3 4 5

DISPLACEMENT. nmm

I I

38

1.0

0.0

l aJ

1 2 3 4 5DLACEMENT, mm

"1.0

1.0

' 0.0

"1.01 2 3 4 6

PLACEMENT, mm

2 3

DSPLACEMENT, mm4

.

0.0

-1.05

1 2 3 4

DISPLACEMENT, mm

1 2 3 4 5DISPLACEMENT. mm

- - - - -- - -

3

DISPLACEMENT. mm4 5

Ftg. 28. Fluid-stiffness coefficients aC1 1, cU1, a 1, T 1, t21 , and1 as a function ofRMS tube displacement oscillating at 0.12 Hz, force transducer Type A

0.0

-1.0

"2.0

1 .0

0

.1.0

1.0

0.0

.1.0

--. - - - -. - -I - -. - - - - - - - - . . . . . . . . . . . ..

-- +1r

A .......... 16 1- . I .- .- A I .L l

I5

F-

-

1

5

1 0

39

1.0

0.0

2.0

1.0

0.0

2.0

1.0

1 2 3 4

' DISPACEMENT. mm

1 2 3 4

DISPLACEMENT, mm

1.1--- --

2 3

DISPLACEMENT. mm

-0.5

--1.5

-2.5

2.0

1.0

4

t 2 3 4

DMSLACEMENT. mm

1 2 3 4

DISPLACEMENT. mm

1 - - - - - - ---r ----: ---' -----

0.2 3

DISPLACEMENT, mm

Fig. 29. Fluid-damping coefficients cx 1, cxi, xJ)3x1) >t2 1, and -c' as a functionof RMS tube displacement oscillating at 1.2 Hz, force transducer Type A

"1.0

0.0

1.0

0.0

-1.04

- : - - - - - - - - - : - - - - - - - - -- -- - - . - - - - - - . - - . - . . . . . . ....

L-- . 11.

1

1

.... ... ......1

4

I4

40

1.0

0.0

-4.0

" -5.0

2.0

M 1.0

0.0

2.0

S 1.0

0.0

1 2 3 4

DISPACEMENT, nvn

1 2 3 4

D0JLACEMENT mm

2 3

DISPLACEMENT, mm

1.0

0.0Na

-1.0

2.0

1.0

0.04

1 2 3 4

DISPLACEMENT, nn

r - --

1 2 34

DSPLACEMENT, mn

.I -----

2 3DISPLACEMENT, TWn

Fig. 30. Fluid-stiffness coefficients all, a 1,x31,1t 1 , and as a function ofRMS tube displacement oscillating at 1.2 Hz, force transducer Type A

"1.0

I. . . . . . . . . .

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

i 4 4

1I 4I

41

1.0

-- -Ug " 0.10 MA- Ug " 0.13 MA

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, mm

1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mm

1.0 1.5 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT, mm

1.0

0.0

"1.0

1.0

- 0.0

1.0

' 0.0

"1.04.0 4.5

1.00.0

0.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, mmi

-+f- Ug . 0.10 m a-+- Ug - 0.13 mh

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, mm

"1.0L

1.04.5 1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mm

4.0 4.5

Fig. 31. Fluid-damping coefficients aI IX, u2, i I 1 21,and r' as a functionof RMS tube displacement oscillating at 0.06 Hz, force transducer Type B

"1.0

Ug- 0.10 Imm

- -- Ug " 0.13 m e

"1.0'1.0

1.0

0.0

"1.0

- t

-+ - Ug " 0.10 m/s-*- Ug- 0.13 /

- - U9 " 0.10 mhs-E- U9 . 0.13 m/s

-- --. --. .. .- . . .. . . -.......

- --.-.-.-.-.-.- I-I-.-.-.-.-,-.-.-.-.-,-.-.-.-.-,-.-.-.-.-,-.-I-.

--- "-b

1 l l l

-- +-- Ug " 0.10 m/s- Ug " 0.13 mft

I

k0.0

I

42

1.0

- 0.0

1.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, mm

1.0 1.5 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT. mm

1.0

"1.05

1.0

0.0

"1.0

1.0

0.0

4.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT. mm

-- ug - 0.10 m-- U9 - 0.13 m/s

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

D SPACEMENT. mm

- -Ug a 0.10 m/s-- Ug - 0.13 mis

a-.

1.0 1.5 2.0 2.5 3.0

DISPLACEMENT, mm

Fig. 32. Fluid-damping coefficients , f2>3,' c1> a2Y, and Cy' as a function ofRMS tube displacement oscillating at 0.06 Hz, force transducer Type B

I-+- Ug0.10m-+-Ug -0.13 ms

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DISPLACEMENT, mm

r-- Ug . 0.10 mm I- -Ug " 0.13 s

0.0

-1.0

1.0

^ 0.0

-1.0

-- wU9 " 0.10 m/1---U9 " 0.19 Wes

3.5 4.0 4.5

-- -U9 - 0.10 nh-+- Uq " 0.13 m/s

. $ .-. -4--- .- _$.___$

5

I

I

n

43

0.0

.1.0

"2.0

1.0

a 0.0

1.0

0.0

-i- Ug 0.10 MM-+- U, " 0.13 mde

OW-N

5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DISPLACEMENT.,mm

1.0

g 0.0

-+- Ug . 0.10 mme-- Ug " 0.13 m h

"1.0 I.

1.. 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, rmm

1.0

0.0

-+ - Ug " 0.10 mme-+ - Ug " 0.13 m e

I "1.0

1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.C

DISPLACEMENT. mm

5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENTmm

1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mm

4.0 4.5

Fig. 33. Fltvid-stiffness coefficients c 11, x21 W3 1, T1 1, T21 , andT3 as a function ofRMS tube displacement oscillating at 0.06 Hz, force transducer Type B

I

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

D PLACEMENT, rrmn

1.0

0.0

-1.01.0

- -

-+- Ug -0.10 m/-+- Ug . 0.13 Wes

-- Up - 0.10 Mfg-- +-- Ug 0.13 m/s

rI- r

-i.0

1-"- Ug . 0.10 mis-- +-- Ug - 0.13 we

I

i

44

1.0

-+- Ug - 0.10 WA-+- Up - 0.13 M&

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, nvn

1.0

0.0

-1.0

1.0

2 0.0

"1.0

-- }- "0.10 Wos

- - U9 -0.13 mes

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, rm

1.0

0.0

"1.0

-- - Ug - 0.10 w/e-+- Ug - 0.13 mft

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, n n

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, nvn

-+- ug -0.10 is-j u -1.3 mJ

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT,mm

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT. mm

Fig. 34. Fluid-stiffness coefficients P fIP31, , ao1 , a1, and a"1 as a function ofRMS tube displacement oscillating at 0.06 Hz, force transducer Type B

-1.0

1.0

0.0

"1.0

1.0

0.0

"1.0

- - Ug - 0.10 mis-- Ug - 0.13 mft

- - Ug - 0.10 mis-- "- Ug - 0.13 mis

0.0

45

1.0

0.0

1.0

2 0.0

1.0

0.0

"1.o1.0 1.3 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT, mm

1.0

0.0

I

s

1.0

a 0.0

-1.05

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DSPLACMENT mm

-Ug. 0.10 mis-+ Ug a 0. 13 mAt

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DPLACEMENT, rrmm

---ug a 0.10 M/s+-Ug a 0. 13 mft

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DSPLACEMENT, mm

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DISPLACEMENT. mm

"1.01 .a4.5 1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mm

.5

4.0 4.5

Fig. 35. Fluid-damping coefficients xh', x , cx 1 C 1 t21 , and -c' as a functionof RMS tube displacement oscillating at 0.12 Hz, force transducer Type B

I- - I---- .-* : :---

-+- Ug - 0.10 n/s

-- +- U - 0.13 m

1.0

+, % 0.0

.- +- Ug -0.10 m/s

-+- Ug .0.13 m/

-- +- Ug - 0.10 RUG--- Ug .0.13 mIs

-- - - -- ---. .. -. , . - . -, - .

-- Ug .0.10 ms-- e- Ug " 0.13 m a

. . . . I . -.-. . - I . . . . I . . A A- I - I - . - . I

-1.0

I A

0

46

1.0

0.0

1.01.0 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT, nn

1.0

0.0

- - U9 - 0.10 nt

-*- U9 - 0.13 ihs

4.5

--- U--Ug - 0.10 M- -- Ug " 0.13 m a

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, mm

1.0

0.0

1.0

1.0

0.0

.1.01.5 2.0 2.5 3.0 3.5 4.0 4.5

DISPLACEMENT, nvn

1.5 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT, mm

.0 1.5 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT, mm

1.0 1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mm

Fig. 36. Fluid-damping coefficients Phf1, fP21 ,P3 ahI a2, and a' as a function ofRMS tube displacement oscillating at 0.12 Hz, force transducer Type B

1.5

-+- Up - 0.10 -m

- -Ug - 0.13 m a

4.5

4.5

1.0

0.0

-1.0

1.0

"1.01.1

-- U9 " 0.10 m/U9 " 0.139 We

-+- Ua . 0.10 m/s

-:- Ug - 0.13 m/s

U

4.0 4.5

1.0s-._._1_ 1 1 1 1

- - ~ ~ ~ ~ ---.----, --. ----

i-+- Ug - 0.13 m e

1 l 1 1 1 1

0.0

0

47

-1.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

D SPACEMENT, mm

-+- U9 . 0.10 m/s

-+- Ug " 0.13 mti

-2.0

1.0

W 0.0

-1.0

1.0

C 0.0

"1.01.0 S.5 2.0 2.5 3.0 3.5 4.0

DISPLACEMENT, mm

1.0

0.0

-1.0.s

1.0.

"0.0

10.5

1.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4

D 'LACEMENT, mm

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DISPLACEMENT. mm

A 0.0

ug-0.10 We" -*- Ug . 0.19 Ris

"1.04.5 1.0 1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mm

.5

4.0 4.5

Fig. 37. Fluid-stiffness coefficients axh, x1, c3 1, 11, or 1, and 1 1 as a function ofRMS tube displacement oscillating at 0.12 Hz, force transducer Type B

0.0

.0 1.5 2.0 2.5 3.0 3.5 4.0 4

DISPLACEMENT, mm

- 3- Ug " 0.10 MIS-* - Ug " 0.13 m/s

5

-- Ug " 0.10 nth

-+- U9 " 0.13 mh

-- Ug - 0.10 MIS-- Ug . 0.13 mis

1 1 l 1

--- Uo " 0.10 rmis

- - Ua - 0.13 We

48

2.0 2.5 3.0 3.50 .c nm

1.0

0.0

"1.04.0 4.5

1-+- us . 0.10 aWB

1-"- Ug - 0.13 aws

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.

DISPLAcEMENT. nm

1.0

0.0

"1.0

1.0

0.0

-1.0

1.0

o" 0.0

"1.05

1.0

0.0

"1.04.0 4.5

1.0

,d 0.0

1.0 1.5 2.0 2.5 3.0 3.5

DISPLACEMENT, mnm4.0 4.5

Fig. 38. Fluid-stiffness coefficients ,P21, I31, ah, a21, and agl as a function ofRMS tube displacement oscillating at 0.12 Hz, force transducer Type B

tU. a. aft UO -0.13 a t

-U-

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5DISPACEMENT. nvn

-- U9 a 0.10 s-- Ug a 0.13 u

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5DISPLACEMENT. nvn

-- Ug a 0.10 4

-- U9 a 0.13 m/s

"1.0 -1.0 1.5

2.0 2.5 3.0 3.5

OISPtACEMENT. wimm1.0 1.5

U"6... , . . . - .

I-- Ug " 0.10 nth-- Ug -0.13 mus

49

force coefficients begin to change significantly. To protect the integrity of the forcetransducers, larger amplitude oscillations were not tested.

The effect of tube oscillation amplitude can be summarized as follows. Becausefluid-force coefficients are independent of tube oscillation amplitude if the tubedisplacement is less than =70% of the gap, small-amplitude oscillations of the tubesdue to other excitation sources are not expected to affect the threshold of fluidelasticinstability. However, if the tubes are excited to perform large-amplitudeoscillations, the nonlinear characteristics of fluid forces become important. Inpredicting tube motion in the post critical region, nonlinear fluid-force coefficientswill be needed to predict nonlinear behavior, including chaotic motion, moreaccurately.

6.6 Symmetry and Antisymmetry of Fluid Forces

The symmetric or antisymmetric characteristics of the fluid-force coefficientsprovide some insights into their effects on fluidelastic instability of tube arrays incrossflow. The details are given by Chen (1987a) in the form of a generalized theoryfor fluidelastic instability of a fluid-structure system. The symmetric orantisymmetric properties of added mass, fluid damping, and fluid stiffness forfluidelastic instability of tube arrays considered here are based on the data obtainedin this test.

The added mass coefficients are symmetric and have been proved by thepotential flow theory for small-amplitude oscillations (Chen 1987a). In a viscousfluid, it is believed that the added mass coefficients are also symmetric for small-amplitude oscillations. Although this has not been proved rigorously, numericalresults and experimental data are in agreement with this observation. Based on thepotential flow theory, the added mass matrix for a row of three active tubes withT/D = 1.35 can be written as

111 -0.28 -0.28 0 0 0

-0.28 11 -0.04 0 0 0

Gjk (Tjk -0.28 -0.04 111 0 0 0------------- ---------------

Ljk : jkJ 0 0 0 '113 0.34 0.340 U 0 '10.34 1.13 0.13

j,k=1, 2,30 0 0 :0.34 0.13 1.13 (13)

It is apparent that the added mass matrix is symmetric.

50

For fluid-damping and fluid-stiffness coefficients, no such symmetric propertyexists. In the past, the use of symmetric and antisymmetric properties has beenbased on physical and geometrical considerations. The coefficient matrices for atube row are assumed to be of the following form:

Matrix for fluid-damping coefficient:

' k jk1

Ltk >jkJ

a1 1 a 12 a12 0 a 12

12 a1 1 0 -a 12 0

0 12 ~-12 2P1 012

-t'2 0 0 s' 01

.12 11 @2 0

Matrix for fluid-stiffness coefficient:

II I ,j

ajk 'ajtI'jkf 1=

all a12

a12 a1 1

~1 0

0

0Xt 12 0

12 : a 12 -a 12

0 -a" 0a' ' 12 0

a12 11 012 012

0 s112 1i 0

o 12 0 11

Test results from this experiment can be used to verify the validity of theseassumptions.

In general, Figs. 15-26, which show fluid force coefficients adi, a;,, a',, a' 1 , T 1 ,1, p, O, al, a 1, ao, a 31, t1 o31, , and of, indicate that

(16)

=~j01 3 cj ~,C2,c~y213101Y P1

and

-a 1 r

0

0

R120

011 .

(14)

(15)

51

cz~i = a

(17)

1 1=-T31,

2 1 =P31.

These equations agree with Eqs. 13 and 14. As pointed out earlier, there is a minordifference between our test and that of Tanaka and his colleagues (1980, 1981,1982), in which the signs of 'r are opposite at high reduced flow velocities. Thisdifference is believed to be associated with the different patterns used by us andthem.

7 Unsteady Flow Theory for Fluidelastic Instability ofTube Arrays

7.1 Equations of Motion

Consider a group of n identical tubes with radius R (= D/2) subjected tocrossflow as shown in Fig. 1. The variables associated with the tube motion in the xand y directions are flexural rigidity El, tube mass per unit length m, structuraldamping coefficient C,, and displacements uj and vj. The equations of motion fortube j in the x and y directions are (Chen 1987b)

E I +4 u C +C 2 +k 2ui n R 2 ( D u k + jk D2v k ' (18)-WE+ m X+ p UR ak -U kjkk k k

k=a1 a 2 kk=I1tn ___ pauk 2 S (

I -Otk -+ a'k _ I pU2cJkuk + C~kvk)=0

a~ -Ljm 2 v i +n 2 2uk a2___EI C4 a+M 2 +k 1pnR2 tjk +Pjk ta2v(19)

nk ( k + pUr jk -k pU t kuk+Ik)=Ok CO at C() at k=

52

Note that fluid-damping coefficients and fluid-stiffness coefficients are functions ofreduced flow velocity Ur (= U/ffD; ff is the oscillation frequency of the tubes in flow).

The in-vacuum variables are mass per unit length m, modal damping ratio (v,and natural frequency fy (= cov/2c). The values for fy and (y can be calculated fromthe equation of motion and appropriate boundary conditions or from tests invacuum (practically in air). The modal function of a tube vibrating in vacuum andin fluid is N(z);

J'2(z dz = 1, (20)

where i is the length of the tubes. Let

uj(z,t) = aj(t)w(z)

and (21)

vj(z,t)= bj(t)v(z),

where aj(t) and bj(t) are functions of time only. Calculation of Eqs. 18 and 19 yields

d2aj dat pR2 n d2ak d2bk21+2( v d+oaj+ 1 jk k +ak d 2

dt2 Vdt v In k dt2 Kdt2

- I (jk dt +k _j U-I((kak +a kbk)=0 (22)k=1 Jd d nk=1 j

and

d2b v db t 2 pnR2 n d2 ak d2 bk2 +2(cov=+(O bj+ 1 k 2 +jk 2dt2 dt v Inbk=.1+ jkdt2 dt

pU 2 k(,Ik dak bk pU2 k ' kak+ bk . (mo_ Ik=1 dt+ Ok ) Im (dktak + P kkJ=-. (23)

mw k~l i -dt i dt Mk=1 k j

When the dimensionless parameters are

53

U = (24)vfvD'

Y _= pR2 .(25)m

Equations 22 and 23 become

'2 v

(j+YI jksk +ajkbk)+ 2Uovaj-3U(Cjkak+ + kbk)k=1 k=1

n

+(0vaj-LUc 2 kak +akbk)=0 (26)V TO V Vk = 1 ( c k

and

n Y ' 2) n

j+ Y Tjkdk + jkbk)+2ovbj -L U2 ) k k + jkbk)k=1 7C3 V k=+

n

+ , 2bj - Uf22(2Tkak + pkbk)=0, (27)V n3 V Vk=1( k j

where the dot denotes differentiation with respect to t.

7.2 Dynamic Instability

Once the fluid-force coefficients are known, it is straightforward to analyze thestability of a tube array in crossflow. Equations 26 and 27 can be written

[M] (Q} + [C] (Q,) +[K] (Q} = (0), (28)

or

[Ms + Mf] (Q} + [Cs + Cf] (Q,) +[Ks + Kfi (Q} = (0), (29)

where IQ) represents the displacement vectors consisting of aj and bj; [M] is themass matrix, including structural mass [Ms] and added mass [Mf]; [C] is the

54

damping matrix, including structural damping [Cs] and fluid damping [Cf]; and [K]is the stiffness matrix, including structural stiffness [Ks] and fluid stiffness [Kf].

By premultiplying by {Q}T and forming the symmetric and antisymmetriccomponents of the matrices,

[Nil]=[M]+[M]T), [M21]= [M][M]T)

[C1]= ([C]+[C]T), [C2 ]1= ([C]- [C]T), (30)

[K1=([K] +[K]T), [K21]= [K] -[K]T),

the terms may be separated, giving

}T [M 1]{Q}+ (Q}T[C2]{Q) +{}T[Kj]{Q}

= - (O)T[M 2 ]{Q) + Q)T[Cl]Q})+{(Q}T[K2](Q)). (31)

Equation 31 equates rates of work. Those terms on the right-hand side of theequation produce a net work resultant when integrated over a closed path throughspace IQ}, with the magnitude depending on the path taken. The forcescorresponding to the matrices [M 2 ], [C2], and [K2], appearing on the right-hand side,are thus, by definition, the nonconservative parts of the forces represented by [M],[C], and [K]. The terms on the left-hand side similarly can be shown to give rise to azero work resultant over any closed path; therefore, together, they are the sum ofthe rates LFd work from the potential forces and the rate of change of kinetic energy.

Fluidelastic instability of tube arrays is caused by high-velocity flow; its effectsare contained in the matrices [Ci], [C2], [K1 ], and [K2 ]. These matrices arefunctions of fluid-damping and fluid-stiffness coefficients. Various types of dynamicinstability can be classified according to the dominant terms in Eq. 31 as follows:

Fluid-damping-controlled instability (single-mode flutter). The dominantterms are associated with the symmetric damping matrix [C1]. The instabilityarises because the fluid-dynamic forces create negative damping. Parameters ccand P. play the most important role in determining the stability-instabilityboundaries.

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Fluid-stiffness-controlled instability (coupled-mode flutter). The dominantterms are associated with the antisymmetric stiffness matrix [K2]. It is calledcoupled-mode flutter because a minimum of two modes are required to produce it.In this case, ak and -r for j = k play the major role in determining the stabilitycharacteristics.

From Eqs. 26 and 27 or 28, the natural frequencies and modal damping ratiosof the coupled tube/fluid system can be calculated as follows:

= ff(y,yv,Uv),(32)

Note that fluid-damping and fluid-stiffness coefficients are functions of Ur (= U/ffD);therefore, a numerical method is needed to calculate ff and .

The nondimensional parameters in Eqs. 26 and 27 are y, v, Uv, ajk, ajk, tjk, Ijk,

ajk 7 k' $kI jk k' k> k, and tik. Therefore, the critical flow velocity can bewritten in a functional form as

UV/),(,vCOv ,jk jk ,tjk ,Pcj'k'Ijk>k k> kjk> k

In the unsteady flow theory, all fluid forces are included and characterizedproperly and the theory is applicable in all cases regardless of different flow regimesor system parameters. On the other hand, quasistatic and quasisteady flow theoriesare applicable in some specific parameter ranges, depending on the case. Therefore,the unsteady flow theory can be used to assess the validity of other approximateflow theories of the fluidelastic instability of tube arrays in crossflow.

7.3 Simplified Case

For a given array of tubes, if fluid-force coefficients are independent of Ur, then

Uv = F(y,4v)

or (34)

Ur = -L-F(y, y ),fr

i.e., the critical flow velocity is a function of mass ratio and damping ratio only.

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In a light fluid, fluid inertia and damping associated with the quiescent fluidcan be neglected. Equations 26 and 27 can be written

n2

j +12 3c ov j_+[(,ajc vk0 k +ajkbk)+(occ"kak +a"kbk1=0_13Vk=1 J kk j kk

(35)

2 2 v 2bj+ 2 vvov j+3bj - U3v kik + 3kk)+COv(tjkak +P1kbk) =0.

k=4.

In a light fluid, all fluid-force coefficients are approximately independent of Ur andthe oscillation frequency is _=ov (i.e., co, = Wx); yU, plays the same role as (v, bothof them contributing to system damping. The modal damping for a particular modecan be written

f =Cy -CYU2, (36)

where C depends on fluid-damping and fluid-stiffness coefficients. Instabilityoccurs if f = 0; i.e.,

UVy = 0.5(37)

or

/ 0.5U I2ni ~(38)

fyD 0c pD2

Thus, the critical flow velocity is a function of the mass-damping parameter andproportional to its half-power.

8 Closing Remarks

In the unsteady flow theory, all fluid force components are included andquantified properly and the theory is applicable in all cases for all systemparameters. On the other hand, quasistatic and quasisteady flow theories areapplicable in some specific parameter ranges because of the approximations made in

57

the characterization of flow effect. Therefore, the unsteady flow theory is the onlytheory that can be used to assess the validity of other approximate flow theories onfluidelastic instability of tube arrays in crossflow.

Tanaka and his colleagues (1980, 1981, 1982) in Japan were the first group tomeasure the motion-dependent fluid forces acting on tube arrays, which are thecritical elements of fluidelastic instability. In their studies, no effort was made todevelop the fluid-force coefficients. It appears that the group at Argonne NationalLaboratory was the first team to develop the nondimensional motion-dependentfluid-force coefficients for tube arrays in crossflow. It also has been verified that athigh reduced flow velocity, these coefficients are independent of reduced flowvelocity. Knowing this characteristic significantly reduces the effort necessary todevelop prediction models.

In earlier studies, only a portion of the motion-dependent fluid forces wasmeasured. In the applications of fluid forces to predict fluidelastic instability,several assumptions must be made about the symmetry or asymmetry of fluidforces. In this study, all fluid-force components were measured. The symmetry orasymmetry of motion-dependent fluid forces is not necessarily correct, and in thefuture, measurements should be performed to verify the symmetry or asymmetry offluid forces. Because of the symmetry or asymmetry of fluid forces, the instability oftube arrays subject to reversed flow may be different from that for uniform flow.

Motion-dependent fluid forces are a function of the amplitude of tubeoscillations. The amplitudes of oscillations range up to 17% of tube diameter (=50%of tube gap) in tests performed in Japan. The fluid forces measured were practicallyindependent of oscillation amplitudes in this range. In the study reported here, theoscillation amplitudes were up to =70% of the tube gap. The effect of oscillationamplitudes on fluid force coefficients was still insignificant.

To develop a reliable design guideline for fluidelastic instability of tube arraysfor application to heat exchangers and steam generators, the unsteady flow theoryis the only reliable theory to use. The key elements are the motion-dependent fluidforce coefficients. In this study, a reliable method was developed and shown capableof providing all necessary fluid force coefficients. A systematic study can now beperformed to measure all fluid force coefficients for various tube arrays, includingsquare and triangular arrays with differing pitch-to-diameter ratios. Once thenecessary fluid force coefficients are determined, the unrLeady flow theory can beapplied to practical components for the evaluation of fluidelastic instability.

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Acknowledgments

This work was jointly funded by Argonne National Laboratory as an FY-1992Argonne Laboratory Directed Research and Development (LDRD) Program titled"Unsteady Flow Theory for Fluidelastic Instability of Tube Arrays in Crossflow,"LDRD Proposal No. 92-046; by Taiwan Power Company under an agreement withthe U.S. Department of Energy, contract agreement 31-109-Eng-38-85847; and bythe U.S. Department of Energy Office of Basic Energy Sciences, Office ofEngineering and Geosciences, under Contract W-31-109-Eng-38.

The authors are grateful to Mr. Roger Smith for his help in setting up the testfacility and instrumentation and conducting the experimental work.

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62

Distribution for ANL-93/15

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Y. Cai D. M. RoteS. S. Chen (20) R. A. ValentinH. Drucker M. W. Wambsganss (3)J. A. Jendrzejczyk (5) R. W. WeeksC. A. Malefyt S. Zhu (5)R. B. Poeppel TIS Files

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Manager, Chicago Field Office, DOEDirector, Technology Management Div., DOE-CHD. L. Bray, DOE-CHA. L. Taboas, DOE-CHErargy Technology Division Review Committee:

H. K. Birnbaum, University of Illinois at Urbana-Champaign, UrbanaR. C. Buchanan, University of Cincinnati, Cincinnati, OHM. S. Dresselhaus, Massachusetts Institute of Technology, Cambridge, MAB. G. Jones, University of Illinois at Urbana-Champaign, UrbanaC.-Y. Li, Cornell University, Ithaca, NYS.-N. Liu, Fremont, CAR. E. Smith, Engineering Applied Sciences, Inc., Trafford, PA

W. H. Chang, Taiwan Power Company, TaiwanP. C. Chen, Power Research Institute, TaiwanJ. S. Coleman, DOE, Washington, DCS. K. Datta, DOE, Washington, DCChing-Nan Fan, Institute of Nuclear Energy Research, TaiwanY. C. Liao,'Taiwan Power Company, TaiwanMing-Hui Lin, Taiwan Power Company, Taiwan0. P. Manley, DOE, Washington, DC1. C. Moon, Cornell University, Ithaca, NY