Guidelines for the instability flow velocity of tube arrays in crossflow

Journal of Sound and Vibration (1984) 93(3), 439-455

GUIDELINES

FOR

VELOCITY

OF

Components Technology Division, Argonne National Laboratory, Argonne, Illinois 60439,

U.S.,4.

TUBE

THE

INSTABILITY

ARRAYS

FLOW

IN CROSSFLOWt

S. S. CHEN

(Received

14 October 1982, and in revised form 23 June 1983)

Fluid flowing across a tube array can cause dynamic instability. Once large-amplitude oscillations occur, severe damage may result in a short time. Such instability must be avoided in design. This paper presents a brief review of different instability models and stability maps developed based on a semi-analytical model and published experimental data. 1. INTRODUCTION

Fluid flowing across a tube array can induce tube vibration and instability. Small tube vibration due to flow excitations always exists but may not cause detrimental effects. In contrast, dynamic instability may result in large-amplitude oscillations, which can cause extensive damage in a short time. The characteristics of tube vibration and instability vary widely in different tube arrays. In general, at low flow velocity, tube vibration is induced by turbulence and, in some cases, by vortex shedding. Once the flow velocity is increased to a certain value, tubeoscillation amplitude increases rapidly with flow; this flow velocity is called the instability flow velocity or critical flow velocity. The critical flow velocity of a tube array depends on many system parameters. Various formulas have been proposed for calculating the critical flow velocity, and stability maps have been employed to evaluate various equipment design. There is a significant scatter in the experimental data obtained by different investigators; it is recognized, now, that this is attributed to the fact that there are different instability phenomena. These extensive experimental and analytical studies are motivated by the excellent original work of Connors [l] and the need to develop design guidelines. The objective of this paper is to summarize the available data in a systematic manner. First, the various stability theories are discussed. Then, stability maps are developed based on the published experimental data and a semi-analytical model. Finally, future research needs are pointed out. 2. STABILITY 2.1.

FLOW

THEORIES

VELOCITY

detailed flow-velocity distribution in a tube array is difficult to measure or calculate. In this paper, the average flow velocity will be applied for all cases. Flow velocity depends on tube arrangement and tube pitch. The most frequently encountered tube arrays are shown in Figure 1. Average flow velocities defined by different investigators are not The

t The substance Fluid Engineering,

of this paper was presented at the International Reading, England, September 1982. 439

Conference

on Flow Induced

Vibrations

in

440

S. S. CHEN Tube

row

Square

orray

Rotated

190”)

square

array

(45”)

0

A

Trlongulor

array

(30”)

Rotated

trmgular

array

(60”)

0 Figure 1. Tube arrangement.

consistent. In this study, regardless of tube arrangement, U is defined as

the average gap flow velocity

(1)

~=[(PI~)I{(PI~)-1}1~,,

where U, is the approach flow velocity, P is tube pitch, and D is tube diameter (a list of nomenclature is given in the Appendix). 2.2. SYSTEM PARAMETERS The three important parameters used in the stability criteria are mass per unit length m, natural frequency f, and damping ratio J. For a tube array vibrating in flow, the definitions of these three parameters vary widely. These parameters can be defined under the following four different conditions. (1) In vacuum: system parameters are measured in vacuum (practically, in air); the effect of the surrounding fluid is ignored. (2) In quiescent fluid-uncoupled vibration: system parameters are measured for an elastic tube vibrating in a fluid with the surrounding tubes being rigid; the coupling effect of fluid is not taken into account. (3) In quiescent fluid-coupled vibration: system parameters are measured for an array of tubes vibrating in a fluid; the coupling among different tubes due to fluid is included. (4) In flow: system parameters are measured in flow for uncoupled and/or coupled modes; in general, they are dependent on the flow velocity. These parameters under different conditions are summarized in Table 1.

TABLE

1

Effective mass, natural frequency, and modal damping ratio in different conditions In quiescent

Parameters Effective Natural Modal

In vacuum

mass (m) frequency

damping

4

Uncoupled vibration m,

fluid

Coupled vibration 4

In flow (uncoupled and/or coupled modes) mr

(f)

fU

f”

fC

fr

ratio (5)

5,

L

5,

6

INSTABILITY

2.3.

MATHEMATICAL

OF

TUBE

ARRAYS

IN CROSSFLOW

441

MODELS

development of this subject has been very exciting. A summary of the published models is presented in Table 2 [l-9]. There are other models which are not listed in Table 2; for example, the quasi-static model of Whiston and Thomas [lo], which is basically the extension of Blevins’ model, and the empirical correlation of Pettigrew et al. [ 111, which is the same as Connors’ model. Table 2 shows that these models are not in agreement in the following different respects. Instability Mechanisms. Before 1980, the displacement mechanism was used exclusively; therefore, the instability is caused by fluidelastic-stiffness forces. If the instability is attributed to the displacement mechanism, the coupling of fluidelastic-stiffness forces with the neighboring tubes is one of the requirements for instability to occur. During that period, it was thought that a single elastic tube among an array of rigid tubes would not become unstable; although experimental results showed the contrary, no plausible explanation was given. One of the cornerstones is the publication of the fluid-force data by Tanaka et al. [4,5] in resolving this issue. Using Tanaka’s fluid-force data, Chen [6] has shown that, in addition to the displacement mechanism, the velocity mechanism is also very important. Based on the two mechanisms, the discrepancy among different models can now be resolved reasonably well. Stability Criteria and System Parameters. Table 2 shows that different investigators have developed different stability criteria and different parameters are used in different correlations. In some cases, the system parameters are not defined in sufficient detail. Equations ofMotion. Different approaches are used in solving the equations of motion; these include a single equation of motion, multiple equations of motion with assumed modes, and general solution of multiple equations. It is recognized that each model has its merits and deficiency. These can be summarized as follows. Quasi-static Models (Connors, Blevins, and Price and Pai’doussis). These models are applicable for the fluidelastic-stiffness-controlled instability only. Although flow-velocitydependent damping forces have been considered partially in some models, these models appear to be not applicable for the fluid-damping-controlled instability. Anulyticul Model (Lever and Weaver). This is based on the velocity mechanism for a single elastic tube, surrounded by rigid tubes, moving in a specific direction. The model requires only three empirical constants and has demonstrated the existence of the “jump” phenomenon in the critical flow velocity at a certain value of mass-damping parameter. For heavy fluid, the results qualitatively agree well with test results; for light fluid, it predicts that the critical flow velocity U/f0 is proportional to the first power of the mass-damping parameter. The model looks very promising; extending the theory to incorporate the coupling fluidelastic-stiffness forces for multiple elastic tubes will probably improve the model significantly. General Semi-analytical Model (Chen, and Tanaka er al.). This requires measurements or computations to determine the fluid-force coefficients. However, this model predicts very well all the observed characteristics of instability for both light and heavy fluids. At the present time, it appears that Chen’s model provides the most detailed insights of the instability phenomena [6,7,12]: (1) identifying two different instability mechanisms.; (2) resolving the controversy among different investigators; (3) identifying proper parameters to be used in stability criteria; (4) comparing well with experimental data; (5) demonstrating the existence of jump in the critical flow velocity at a certain value of mass-damping parameter and multiple stable and unstable regions; (6) developing different stability criteria for light and heavy fluids; and (7) predicting the effect of different system parameters. Therefore, it will be used as the basis to develop the design guidelines., The

Fluidelastic

Tanaka

[4> 51

et al.

Fluidelastic

Blevins [2,3]

Connors

Fluidelastic

vibration

whirling

vibration

Name of instability

[l]

Author

Displacement velocity mechanisms

Displacement mechanism

Displacement mechanism

Instability mechanism

and Fluid dynamic force including Buidelasticstiffness force and flow-velocity-dependent damping force

Fluidelastic-stiffness force

Fluidelastic-stiffness force

Dominant fluid force

2

(P/D

1.33)

=2.0)

1.33)

fluid:

(P/D=

Heavy

(P/D=

Light fluid:

criterion

in c~ossflow

Instability

of tube arrays

TABLE

Variousmodels for stability

fw%r 5,

li

f,. mu,L or

f”, mu,5,

Parameters used in stability criterion

to obtain criterion

Equations of motion obtained by using measured fluid-force data

Equations of motion for tube rows with assumed mode shapes

Energy consideration of a single tube and experimentally measured mode shape

Method stability

!?5

Q

V

vl

INSTABILITY

OF TUBE

ARRAYS

IN CROSSFLOW

443

444 2.4.

S. S. CHEN A GENERAL

SEMI-ANALYTICAL

MODEL

The detailed description and experimental verifications of the models are presented in references [6] and [7]. The model includes three types of fluid forces, as follows. (1) Fluid inertial forces, proportional to tube acceleration and fluid added mass coefficients, affect system natural frequencies and introduce coupled oscillations among different tubes. (2) Fluid-damping forces, proportional to tube velocity and fluid-damping coefficients, contribute to system damping or may cause instability; in general, the effect on natural frequencies is not very significant. It should be pointed out that fluid-damping coefficients given in references [6] and [7] are defined in different manners. (3) Fluidelastic-stiffness forces, proportional to tube displacement and fluidelasticstiffness coefficients, may affect both system damping and natural frequencies and may cause instability. Based on this model, the instability may be caused by the velocity mechanism or the displacement mechanism. fluid force is proportional to the velocity of (1) Velocity Mechanism. The dominant the tubes. Depending on the reduced flow velocity, fluid-damping force may act as an energy-dissipation mechanism or an excitation mechanism for tube oscillations. When it acts as an excitation mechanism, the system damping is reduced. Once the modal damping of a mode becomes negative, the tubes lose stability. This type of instability is called fluid-damping-controlled instability. The instability criterion is [7] given by U/f,0

= aU(2rrJ,m,/pD2)0~5,

(2)

where (Y, is a function of fluid damping coefficients. Note that equation (2) is different from that given in reference [6], in which the mass-damping parameter is raised to the first power. This is not a contradiction or correction, but results from the difference in defining fluid-damping coefficients (see reference [7] for details). The definition given in reference [7] is more convenient. (2) Displacement Mechanisms. The dominant fluid force is proportional to the displacements of the tubes. The fluidelastic-stiffness force may affect natural frequencies as well as modal damping. As the flow velocity increases, the fluidelastic-stiffness force may reduce the modal damping. When the modal damping of a mode becomes negative, the tubes become unstable; this type of instability is called fluidelastic-stiffness-controlled instability. The instability criterion is given by [6,7] U/fLD = P,(2rr
(3)

where pU is a function of fluidelastic-stiffness coefficients. In general cases, the two mechanisms are superimposed on each other. Then the stability criterion can be written as U/f”0

= F(<,, m,lpD2, P/D, turbulence

characteristics).

(4)

The main differences of the two basic instability mechanisms are summarized in Table 3. It should be emphasized that, in general, both fluid-damping coefficients and fluidelasticstiffness coefficients are functions of the reduced flow velocity U, ( = U/f,D). Therefore, the parameters CY,in equation (2) and /3, in equation (3) are functions of the reduced flow velocity Up For light fluids, the instability occurs at large values of U, It has been shown [7] that for large U, both fluid damping coefficients and fluidelastic-stiffness coefficients are approximately independent of Ur; therefore, for a given tube array, (Y, and & are constants.

INSTABILITY

OF TUBE ARRAYS TABLE

445

IN CROSSFLOW

3

Comparison of two difierent instability mechanisms Fluid-dampingcontrolled instability (velocity mechanism)

Mechanisms

-__

Fluidelastic-stiffnesscontrolled instability (displacement mechanism

j -_.

“-5

Instability

criteria

Dominant

fluid force

Flow-velocity-dependent damping force

Fluidelastic-stiffness force

Fluid coupling

Not necessary

Necessary

Phase relationship of tube oscillations

O”, 180”

O”, *90”, 180”

Effect of detuning

Less significant

More significant

3. EMPIRICAL

3.1.

VARIOUS

PUBLISHED

STABILITY

STABILITY

CRITERIA

CRITERIA

In addition to the analytical and semi-empirical stability criteria given in Table 2, various stability criteria have been proposed based on experimental data. Most of these criteria can be grouped into two classes: (1) the critical flow velocity U/f0 is a function of the mass damping parameter,

U/f0 = (Y,(2rml/pD2)rrz; (2) The critical

flow velocity

is a function

(5)

of mass ratio (m/pD*) and damping

r//fD=P,(m/pD)P2(2~~)Pi. These models have been adopted by various investigators; in Tables 4 and 5 [l, 2, 4, 5, 11, 13-211. Most recently, Price and Pai’doussis [8] proposed another

(2$), (6)

the studies stability

are summarized

criterion:

(7) This criterion appears to be not applicable for tube arrays losing stability at relatively low u,. Experimental data for the critical flow velocity obtained by various experimentalists are not in agreement, and various stability criteria do not correlate well. This is attributed to the following reasons. (1) Different parameters are used by different investigators; some use in-vacuum parameters, and some use in-fluid parameters or in-flow parameters. Even with the same stability criterion, the results will be different when using two different sets of parameters, as illustrated in Table 1. (2) Instability may be caused by different instability mechanisms. In the past, the fluidelastic-stiffness-controlled instability mechanism has been used exclusively. It is not expected that the stability criterion for fluidelastic-stiffness-controlled instability can be used to correlate data for fluid-damping-controlled instability.

446

S. S. CHEN TABLE

4

Values of a1 and a2 Investigators

0.5

Tube row with PI D = 1.42

2(27r)“”

0.5

C, and C, are fluid-elasticstiffness force coefficients

0.5

Re = Reynolds number, p = constant

1.0

For square array, and k determined from fluid force

( cxcy)o’25 Y. N. Chen [13]

P Re-“~25

Gross [ 141

4/k

Gorman [ 151 Savkar [ 161 Connors [ 171

3.3 4.95(P/D)*

0.5

Suggested design guideline

0.5

For triangular arrays

0.37+1.76P/D

0.5

For square array 1.41sP/D~2.12

Pettigrew et al. [ 1 l]

3.3

0.5

Suggested design guideline

Weaver and Grover [18]

7.1

0.21

Rotated triangular array P/D= 1.375

2.49 to 6.03

0.2 to 1.08

3.0

0.75

Chen and Jendrzejczyk

[I91 Tanaka et al. [S]

TABLE

P1

Pai’doussis [ 201 (a) 2.3{(PID)

Tanaka and Takahara [4]

For square array, PID=2.0

and P3

P2

P3

0.5

0.25

0.4

0.4

- l>

Pai’doussis [21] (b) 5.8{(PID)-

For various rectangular arrays and mixed array in water flow

5

Values of PI, A,

Investigators

--

9.9

Connors [l] Blevins [2]

Remarks

Q2

-

11

(a) 0.5

(a) 0.5

(b) 0.333

(b) 0.2

Remarks Using published data (a) Including all data (b) Excluding some data For square array, PjD = 1.33 (a) Low-density fluid (b) High-density fluid

(3) The gap flow velocities defined by different investigators are not consistent with one another, and the critical flow velocities are not determined with the same method. (4) Critical flow velocities of tube arrays depend on tube arrangement, spacing, and other parameters. It is apparent that to develop a universal stability criterion applicable to all cases will be difficult, if not impossible.

INSTABILITY

OF TUBE

ARRAYS

IN CROSSFLOW

447

Based on the semi-analytical model [7], it has been demonstrated that equation (5) is applicable for light fluids with a2 = 0.5 and equation (6) is applicable for heavy fluids. In the published experimental data, very few investigators specify the individual values of mass ratio and damping. Consequently, at the present time, it is difficult to use equation (6) for heavy fluids. Therefore, equation (5) will be used in the development of an empirical stability diagrams for both light and heavy fluids. 3.2.

DIMENSIONLESS

PARAMETERS

In the empirical correlations for critical flow velocities, different system parameters measured in vacuum or in quiescent fluid are used. Most investigators use the modal damping value 5, mass per unit length m, and natural frequency f measured in quiescent fluid, but some use those measured in vacuum (actually in air), or in flowing fluid. From the analysis of the model, it is clear that one can use those parameters determined either in vacuum or in quiescent fluid provided that appropriate fluid-damping coefficients are used [6]. However, the stability criteria based on the in-vacuum or in-fluid values are not identical. For practical applications, it is more convenient to use the in-vacuum parameters, since they are well defined. In-fluid parameters are more difficult to determine. In particular, for heavy fluid, inertia and viscous coupling become important; coupledmode frequencies f. effective mass m,, and modal damping 5, are more difficult to measure. The stability criteria, equations (2), (3) and (4), are expressed in terms of in-vacuum parameters. The effects of various parameters can be briefly summarized as follows: (1) 2r[,m,/pD2 is the most important parameter; the critical flow velocity increases with this parameter; (2) m,/pD2 determines the role of added mass; for heavy fluids, it cannot generally be combined with the damping 27rJ as a single parameter; (3) P/D: fluid-force coefficients depend on tube arrangement; therefore, the critical flow velocity will depend on P/D; (4) turbulence characteristics: fluid-force coefficients depend on incoming turbulence characteristics (intensity and scale); again, the critical flow velocity will depend on the turbulence characteristics.

3.3. EMPIRICAL STABILITY DIAGRAMS Available experimental data for the reduced flow velocity U, ( = U/fD) will be plotted as a function of the mass-damping parameter 6, ( = 2r4’m/pD2) for different tube arrangements. It was shown in reference [6] that either the in-vacuum parameters m,, fu, and lo’,,or the in-fluid parameters, m,, fc, and &‘,,can be used in the stability criteria. In most experiments, the in-vacuum parameters are actually measured in air. The effect of air on those parameters is small. Therefore, m,, fo, and 5” will be based on those measured in air. In liquid-flow tests, the values of in-fluid parameters m,, fu, and J,, are generally measured and, in most cases, in-vacuum parameters m,, f,,, and f;, and in-fluid parameters m,, fc, and &, are not measured. Under such circumstances, the in-fluid parameters of uncoupled modes, m,, f,,, and 5; are used. The stability diagrams summarize the published data for different tube arrangements obtained by different investigators.

3.3.1. Tube row [l, 12, 14, 17, 22-281 The critical flow velocity for a tube row depends on the pitch-to-diameter ratio P/D. Several studies have been made to investigate the effect of P/D on the critical flow velocity. Blevins [29] showed that (or = 2(2n-)“‘5/[(D/P)2{2(D/P)3-(D/P)2}]o’25.

(8)

448

S. S.

CHEN

Equation (8) is applicable for P/ D < 2 only. In a symmetric investigation, Ishigai et al. [22] showed that, for a tube row, a1 = 8{(P/D)-0.375).

(9)

Therefore, the critical flow velocity is proportional to {(P/D) -0.375). A new reduced critical flow velocity incorporating the effect of tube spacing can be defined as follows: r7, = (U/fD)/{(P/D) Then u, is independent

-0.375).

(10)

of tube spacing P/D.

Moss damping parameter.

2rr(mlpD2

Figure 2. Stability diagram for tube rows: 0, lshigai er al. [22]; 0, Tanaka Hartlen [23]; W, Halle and Lawrence [24]; 0, Chen and Jendrzejczyk [12]; [25]; +, Gross [14]; X, Blevins et al. [26]; *, Heilker and Vincent [27].

1283; ??0, Connors [l, 171; A, and Zdravkovich

?? , Southworth

Figure 2 shows the critical flow velocity ur as a function of the mass-damping parameter The in-vacuum parameters are used whenever these data are available. Experimental data obtained in air correlate reasonably well for tube rows with different pitch-to-diameter ratios ranging from 1.19 to 2.68. In liquid flow, with the exception of Chen and Jendrzejczyk [12] who used in-vacuum parameters, other investigators have used different parameters: Connors [17] and Halle and Lawrence [24] used fU, m,, and 5;, while Heilker and Vincent [27] used fr, m,, and &. 2T[m/pD2.

3.3.2. Square array (90”) [4, 11, 14, 15, 17, 19, 23, 26, 27, 30, 311 Square tube arrays with different spacing were tested in air by Soper [31], and in water by Chen and Jendrzejczyk [19]. The results of these tests show that the critical flow velocity is not very sensitive to the variation of tube spacing. Therefore, for square arrays, the critical flow velocity U, will be plotted as a function of 6, regardless of the spacing. The results are given in Figure 3. Although different tests were performed for different flow conditions and tube spacing, the data correlate fairly well. In particular, for in-water tests, different investigatorsused different parameters: Tanaka and Takahara [4] used in-vacuum parameters, Heilker and Vincent [27] used in-flow parameters, and the others used in-fluid parameters.

INSTABILITY

OF TUBE ARRAYS

IN CROSSFLOW

449

Figure 3. Stability diagram for square arrays: 0 0, Tanaka and Takahara [41]; ?? , Chen and Jendrzejczyk [19]; /I, Connors [17]; 0, Gross [14]; V, Hartlen [23]; A, Soper [31]; v, Blevins [26]; *, Gorman [15]; +, Heilker and Vincent [27]; 0, Zukauskas and Katinas [30]; x +, Pettigrew et al. [ll].

3.3.3.

Rotated square array (45”) [ll,

15, 23, 27, 311 Based on Soper’s data [31], the critical flow velocity is approximately proportional to (P/D-0*5). Note that in equation (lo), U, = u, for P/D = 1.375. In order to make ur = U, at P/D = l-375, for this case, u, is defined as follows: CT,= (U/fD)/1*143{(P/D)

-0.5).

(11)

The results are given in Figure 4.

Figure 4. Stability diagram for rotated square arrays: 0, Soper [31]; Cl, Hartlen [23]; ?? , Heilker and Vincent [27]; A, German [15]; V ‘I, Pettigrew er al. [ll].

3.3.4. Triangular array (30”) [ll, 14, 15, 19, 23, 27, 30-331 Following the same procedure as that for rotated square arrays, one defines 0, as 0, = (U/fD)/2.105{(P/D)-0.9).

(12)

At P/D = 1.375, u, = U,. Figure 5 summarizes the results. There is more scattering of the data at low values of 6,. This is attributed to different parameters used by different investigators and may be caused by different spacings. Note that equation (12) is based

Figure 5. Stability diagram for triangular arrays: 0, Gross [14]; ?? , Yeung and Weaver [33]; ‘I, Gorman [15]; V, Hartlen [23]; A, Chen and Jendrzejczyk [19]; 0, Soper [31]; 0, Connors [32]; 4, Heilker and Vincent [27]; 4, Zukauskas and Katinas [30]; X +, Pettigrew et al. [Ill.

”

on Soper’s data obtained in a wind tunnel. The variation of the critical flow velocity with tube spacing in water may be different from that in air. For example, the variation of u, with tube spacing in the data by Zukauskas and Katinas [30] is different from that found by Soper [31]. 3.3.5. Rotated triangular arrays (60”) [ll, 15, 18, 23, 27, 31-331 Soper’s data show that the critical flow velocity varies insignificantly with tube spacing. All available experimental data are plotted in Figure 6 regardless of the tube spacing. They agree reasonably well.

Figure 6. Stability diagram for rotated triangular arrays: 0, Weaver and Grover [18]; ?? , Yeung and Weaver [33]; +, Gorman [15]; A A, Heilker and Vincent [27]; V, Hartlen [23]; 0, Soper [31]; Cl, Connors [32]; X +, Pettigrew [ 111.

4. DESIGN

EVALUATION

METHODS

TO

AVOID

INSTABILITY

Critical flow velocities can be predicted by using the analytical method [6,7] or by using empirical stability diagrams.

INSTABILITY

4.1.

ANALYTICAL

OF TUBE

ARRAYS

IN CROSSFLOW

METHOD

The analysis is described in references [6] and [7]. The stability determined by using the following steps. 4.1.1.

451

of a tube array can be

Fluid-force coejjicients

To carry out the analysis, various fluid-force coefficients must be known. Fluid-added mass coefficients can be calculated based on the potential flow solution; a computer program to calculate those coefficients is available [34]. At present, no analytical method is available to calculate the fluid-damping coefficients and fluidelastic stiffness coefficients. There are only a few sets of experimental data. 4.1.2.

Tube properties

Tube mass and tube natural frequencies in-vacuum can be measured or calculated relatively easily [35,36]. In most cases, tube damping is measured or estimated. If the tubes in an array are not identical, it is generally conservative to assume all tubes have the same properties as the one with lower natural frequency and damping. 4.1.3.

Number of tubes

For an array of II tubes, there are 2n equations. In practical situations, it is not necessary to include all tubes; a finite number of tubes will be sufficient in computing the critical flow velocity. It is suggested the tube array used in computation should include the following: tube row, five tubes; square and rotated square arrays, nine tubes; triangular and rotated triangular arrays, seven tubes. In some cases, a small number of tubes can be used to estimate roughly the critical flow velocity. For example, using one tube only will enable one to calculate the critical flow velocity for fluid-damping-controlled type instability. 4.1.4.

Parametric study

Once the analysis has been carried out, it is straightforward study. This information may be useful to alleviate instability.

4.2.

EMPIRICAL

STABILITY

to conduct

a parametric

DIAGRAMS

The stability diagrams given in Figures 2-6 can be used in design evaluation. (1) Calculation of the Mass-Damping Parameter 6, ( = 2vmJJpD’). Fluid density p, tube diameter 0, and tube mass per unit length m,, are relatively easy to determine. The modal damping ratio &‘Ocan be estimated or measured. (2) Determination of the Lower Bound of the Critical Flow Velocity. The lower bounds for different tube arrays are given in Figures 2-6 by solid lines. These are summarized in Table 6. The critical flow velocity calculated from Table 6 can then be compared with the actual flow velocity. The lower bounds have been established on the basis of the following results: according to the analytical and experimental results, the exponent of the mass-damping parameter is always positive: i.e., the slope of all solid lines given in Figures 2-6 are positive; the slope for large values of the mass-damping parameter is O-5; no experimental data will be larger than the values given by the lower bound. It is recognized that the lower bounds are not uniquely determined by this procedure; nevertheless, these bounds have been established on the basis of the available information at this time. It would be nice to establish an error margin for each case. Unfortunately, the data obtained by different

452

S. S. CHEN TABLE 6 Lower bound on critical flow velocity Parameter range for 6,

Array

!

0.05 < s,,, < 0.3 0.3 < 6,, < 4.0 4.0 < 6, < 300

Tube row

Square (90’)

i

UlfD 1.35(P/D-0.375)6:;,“” 2.30( P/ D -0.375)6!;’ 6.00( P/ D - 0.375)S:;’

0.03 < a,, < 0.7 0.7 < 6, < 300

Rotated square (45”) Triangular

Rotated triangular

3.%(P/D-0.9)6!,’ 6.53(P/D-O.S)So,;’

0.1<6,,<2 2<6,,<300

(30” j

(60”)

investigators are not analyzed have used different parameters margin at this time.

0.01<6,<1 1 < 6,,, < 300

in the same manner. For example, different as given in Table 1. It is difficult to establish

5. CONCLUDING

investigators such an error

REMARKS

The instability of a tube array subjected to crossflow may be attributed to fluid-damping force, fluidelastic stiffness force, or a combination of both. Since a different fluid force will be dominant in different parameter ranges, a single stability criterion cannot be applied for all cases. In general, at low values of mass-damping parameter, instability is often of the fluid-damping-controlled type; and at high values of the mass-damping parameter, instability is frequently of the fluidelastic-stiffness-controlled type. In most cases, both fluid-damping and fluidelastic forces contribute to the instability of a tube array. In the past, the instability was considered to be attributed to the fluidelastic stiffness force only and the instability criterion was that developed by Connors [l]. Different investigators have used Connors’ stability criterion to correlate all experimental data and found the correlation did not collapse the data well. Now it is recognized that some of the instability is attributed to the fluid-damping force. Based on different instability mechanisms and published experimental data, the stability criteria for five different tube arrangements are summarized in this paper. These criteria have accounted for the available information at present and will be improved as soon as more theoretical and experimental results become available. Ideally, the critical flow velocity should be calculated based on the procedures discussed in references [6] and [7]. Because of the difficulty to calculate the fluid-damping coefficients and fluidelastic-stiffness coefficients, it is not possible to perform stch computations in general. Only a few cases, in which these fluid coefficients have been measured, permit one to predict the critical flow velocity in a rigorous manner. In design assessment, it is almost imperative to rely on empirical correlations. The approach to use these correlations is straightforward. For example, consider a square array with a pitch-to-diameter ratio of 1.5. Assume that the tube natural frequency in air is 50 Hz, tube diameter is 2.54 cm, and the mass-damping parameter is 0.2. The critical flow velocity calculated from Table 6 is as follows: (13)

INSTABILITY OF TUBE

ARRAYS

IN CROSSFLOW

453

To improve the stability criteria given in Table 6, the key step is to predict the fluid-force coefficients. The fluid-inertia coefficients can be calculated based on the potential flow theory [34]. In most practical applications, the results from the potential flow theory will be acceptable. However, the potential flow solutions for fluid-damping coefficients and fluidelastic-stiffness coefficients are generally unacceptable. Therefore, the main task is to develop an analytical method to compute these coefficients. This is one of the problems that is certain to be pursued in the field of computational fluid dynamics. These fluid-force coefficients can also be measured by using the technique demonstrated by Tanaka and Takahara [4]. This is a very tedious process. Furthermore, fluid-force coefficients are a function of geometry. It will require a large number of experiments before one can quantify the fluid force for all practical tube arrangements. In addition to the prediction techniques, understanding of the basic fluid dynamics for flow across a vibrating tube array remains a difficult task. Detailed flow measurements and theoretical study of the flow field have to be carried out before one can identify the basic flow effect and the effect of tube motion on the flow field. The interaction process of the tube array and the crossflow is certain to receive more attention in the future. In the past decade, the original work by Connors [l] has given great impetus to the numerous, innovative studies of tube arrays subjected to crossflow. The mechanism described by Connors has been used to interpret different phenomena, and Connors’ stability criterion has been used extensively and misused occasionally. Furthermore, erroneous interpretations of Connors’ criterion by some investigators have been published in different journals. These illustrate the lack of understanding in this subject. Based on the model of references [6,7], the inconsistency among experimental data obtained by different investigators as well as different phenomena reported in literature can now be resolved reasonably well. Although it is still not possible to predict the critical Bow velocity analytically, because of the difficulty of calculating the fluid-force coefficients, there is a sound basis for further development to quantify the instability flow velocity..

ACKNOWLEDGMENTS This work was performed under the sponsorship of the Office of Reactor Research and Technology, U.S. Department of Energy. 1 am indebted to Dr M. W. Wambsganss for his review and suggestions of the original manuscript and Dr B. M. H. Soper of AERE Harwell, England, for providing his data for the critical flow velocity as a function of the mass-damping parameter.

REFERENCES 1. H. J. CONNORS 1970 Flow-induced Vibration of Heat Exchangers, (editor D. D. Reiff). pp. 42-56. New York: ASME. Fluidelastic vibration of tube arrays excited by cross flow. 2. R. D. BLEVINS 1974 Journal of Pressure Vessel Technology 96, 263-267. Fluid elastic whirling of a tube row. 3. R. D. BLEVINS 1979 Flow Induced Vibrations (editors S. S. Chen and M. D. Bernstein), pp. 35-39. New York: ASME. Fluid damping and the whirling instability of tube arrays. 4. H. TANAKA and S.TAKAHARA 1981 Journal of Sound and Vibration 77, 19-37. Fluid elastic vibration of tube array in cross flow. 5. H. TANAKA, S.TAKAHARA and K. OHTA 1982 Flow-induced Vibration of Circular Cylindrical Structures (editors S. S. Chen, M. P. Pdidoussis and M. K. Au-Yang), pp. 45-56. New York: ASME. Flow-induced vibration of tube arrays with various pitch-to-diameter ratios. 6. S. S. CHEN 1983 Journal of Vibration, Acoustics, Stress and Reliability in Design 105,51-58 and 253-260. Instability mechanisms and stability criteria of a group of circular cylinders subjected to cross flow; part I: theory, part II: numerical results and discussions.

454

S. S. CHEN

7.S.S.CHEN and J. A. JENDRZEJCZYK 1983 Nuclear Engineering and Design 75, 351-374. Stability of tube arrays in crossflow. 8. S. J. PRICE and M. P. PA~DOUSSIS 1983 Journal of Vibration, Acoustics, Stress and Reliability in Design 105,59-66. Fluidelastic instability of an infinite double row of circular cylinders subjected to a uniform crossflow. 9. J. H. LEVER and D. S. WEAVER 1982 Journal of Pressure Vessel Technology 104, 147-158. A theoretical model for fluidelastic instability in heat exchanger tube bundles. 10. G. S. WHISTON and G. D. THOMAS 1982 Journal of Sound and Vibration 81, 1-31. Whirling instabilities in heat exchanger tube arrays. 11. M. J. PET~IGREW, Y. SYLVESTRE and A. 0. CAMPAGNA 1978 Nuclear Engineering and Design 48, 97-115. Vibration analysis of heat exchanger and steam generator designs. 12. S. S. CHEN and J. A. JENDRZEJCZYK 1982 Journal of Applied Mechanics 48, 704-709. Experiment and analysis of instability of tube rows subject to liquid crossflow. 13. Y. N. CHEN 1974 Transactions of the American Society of Mechanical Engineers 96, Series B, 1065-1071. The orbital movement and the damping of the fluidelastic vibration of tube banks due to vortex formation, part 2, criterion for the fluidelastic orbital vibration of tube arrays. 14. H. GROSS 1975 Ph.D. Dissertation, Technical University of Hanover. Investigations in aeroelastic vibration mechanisms and their application in design of tubular heat exchangers. 15. D. J. GORMAN 1976 Nuclear Scienceand Engineering 61,324-336. Experimental development of design criteria to limit liquid cross-flow-induced vibration in nuclear reactor heat exchange equipment. 16. S. D. SAVKAR 1977 Journal of Fluids Engineering 99,517-519. A brief review of flow induced vibration of tube arrays in cross-flow. 17. H. J. CONNORS 1978 Journal of Mechanical Design 100,347-353. Fluidelastic vibration of heat exchanger tube arrays. 18. D. S. WEAVER and L. K. GROVER 1978 Journal of Sound and Vibration 59, 277-294. Cross-flow induced vibrations in a tube bank-turbulent buffeting and fluid elastic instability. 19. S. S. CHEN and J. A. JENDRZEJCZYK 1981 Journal of Sound and Vibration 78, 3.55-381. Experiments on fluid elastic instability in tube banks subjected to liquid cross flow. 20. M. P. PAIDOUSSIS 1980 Proceedings of IUTAM-IAHR Symposium on Practical Experiences with Flow Induced Vibrations, Karlsruhe, (editors E. Naudascher and D. Rockwell), pp. l-80. Berlin: Springer-Verlag. Flow induced vibrations in nuclear reactors and heat exchangers. 21. M. P. PAIDOUSSIS 1980 Flow-Znduced Vibration Design Guidelines, (editor P. Y. Chen), pp. 11-46. New York: ASME. Fluidelastic vibration of cylinder arrays in axial and cross flow. state of the art. 22. S. ISHIGAI, E. NISHIKAWA and E. YAGJ 1983 International Symposium on Marine Engineering, Tokyo, 1-5-23-1-5-33. Structure of gas flow and vibration in tube bank with tube axes normal to flow. 23. R. T. HARTLEN 1974 Ontario Hydra, Toronoto, Canada, Report No. 74-309-K. Wind tunnel determination of fluidelastic vibration thresholds for typical heat exchanger tube patterns. 24. H. HALLE and W. P. LAWRENCE 1977 Presented at the ASME-IEEE Joint Power Generation Conference, Long Beach, California. ASME Paper No. 77-JPGC-NE-4. Crossflow-induced vibration of a row of circular cylinders in water. 25. P. J. SOUTHWORTH and M. M. ZDRAVKOVICH 1975 Journal of Mechanical Engineering Science 17,190-198. Cross-flow-induced vibrations of finite tube banks in in-line arrangements. 26. R. D. BLEVINS, R. J. GIBERT and B. VILIARD 1981 6th Conference on Structural Mechanic,s in Reactor Technology, Paper No. B6/9. Experiments on vibration of heat exchanger tube arrays in cross flow. 27. W. J. HEILKER and R. Q. VINCENT 1980 Presented at the Century 2 Nuclear Engineering Conference, San Francisco, California. ASME Paper No. 80-C2/NE-4. Vibration in nuclear heat exchangers due to liquid and two-phase flow. 28. H. TANAKA 1980 Transactions of the Japan Society of Mechanical Engineers 46, 408 (Section B), 1389-1407. A study on fluid elastic vibration of a circular cylinder array (one-ray cylinder array). 29. R. D. BLEVINS 1977 Journal of Fluids Engineering 99, 457-461. Fluid elastic whirling of tube rows and tube arrays. 30. A. ZUKAUSKAS and V. KATINAS 1980 Proceedings of IUTAM-ZAHR Symposium on Practical Experiences with Flow Induced Vibrations, (editors E. Naudascher and D. Rockwell). Berlin: Springer-Verlag. Flow-induced vibration in heat-exchanger tube banks.

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31. B. M. SOPER 1980 Flow-Znduced Heat Exchanger Tube Vibration-1980, (editors J. M. Chenoweth and J. R. Stenner), pp. 1-9. New York: ASME. The effect of tube layout on the fluidelastic instability of tube bundles in cross flow. 32. H. J. CONNORS 1980 Flow-Znduced Vibration of Power Plant Components, (editor M. K. Au-Yang), pp. 93-107. New York: ASME. Fluidelastic vibration of tube arrays excited by nonuniform cross flow. 33. H. YEUNG and D. S. WEAVER 1983 Journal of Mechanical Design 105,76-82. The effect of approach flow direction on the flow induced vibrations of a triangular tube array. 34. S. S. CHEN and H. CHUNG 1976 Argonne National Laboratory, Report No. ANL-CT-76-45. Design guide for calculating hydrodynamic mass, part I: circular cylindrical structures. 35. S. S. CHEN and M. W. WAMBSGANSS 1974 Argonne National Laboratory, Report No. ANL-CT-74-06. Design guide for calculating natural frequencies of straight and curved beams on multiple supports. 36. R. D. BLEVINS 1979 Formulas for Natural Frequency and Mode Shape. New York: Van Nostrand Reinhold.

APPENDIX:

p” T u

u* ur

U, a”, P” I P

tube diameter natural frequency, fv, fc,fu, or ff mass per unit length, m,, m,, m,, or m, tube pitch (see Figure 1) tube pitch in the direction 90” from that associated with the tube pitch P gap flow velocity approach flow velocity reduced flow velocity ( = U/f,D) reduced flow velocity ( = U/fD) instability functions based on in-vacuum parameters mass-damping parameter ( = 2vlm/pD*) modal damping ratio, l,, &, l,, or Jr fluid density

Subscripts

; l4 V

NOMENCLATURE

coupled mode in quiescent fluid in flow uncoupled mode in quiescent fluid in vacuum