Experimental study of forces on a cylinder in oscillatory flow with a cross current

Experimental study of forces on a cylinder in oscillatory flow with a cross current P. K. S T A N S B Y

Simon Engineering Laboratories, University of Manchester, Oxford Road, Manchester M13 9PL N. EL-KHAIRY and G. N. BULLOCK

Department of Civil Engineering, University of Salford, Salford M5 4 WT Forces on a circular cylinder have been measured with nominally two-dimensional current and oscillatory flow at right angles. Previous results for purely oscillatory flow defined by a KeuleganCarpenter number, Kc, have been extended for reduced velocities, Vr, in the range 3-10. For Kc< 7 modification of the Karman street by oscillation is complex and locking-on has a dominant influence. For Kc>7 simply adding forces due to the current and oscillation as though in isolation generally gives 'conservative results. The 'current' drag shows considerable variation and can even be negative. The Morison fit to the in-line force is generally less satisfactory when there is a current and can be wholly inadequate. Key Words: Flow, cylinder, force, current, oscillation

INTRODUCTION The prediction of wave forces on the cylindrical elements of offshore structures is particularly difficult when wakes are generated. The flow incident on a vertical" cylinder in unidirectional, regular waves has been idealised as planar oscillatory flow by ignoring vertical velocity components. Such flows produced in U-tubes 1-3 have confirmed the suitability of Morison's equation for describing the in-line force and the cross-flow or lift force has been described through maximum and r.m.s, values and through spectra. The drag and inertia coefficients appear to relate well to the wave situation 4-7 as do lift frequenciesfl' 9 The magnitude of lift is particularly sensitive to the three-dimensional nature of a flow but it remains of considerable significance in waves. Forces on a cylinder will obviously be modified by the action of a current. The U-tube is an ideal device for generating oscillatory flow but it is virtually impossible to introduce currents in any direction other than in line with the oscillation. A kinematically identical flow may be achieved in the frame of reference relative to the cylinder by oscillating the cylinder rather than by oscillating the onset flow. The forces in the two situations differ only by the Froude-Krylov force which is zero for the oscillating cylinder. 1° To allow currents in various directions, oscillating the cylinder in the current is thus a convenient way to effect combined oscillatory and current flow relative to the cylinder. Since the aim here is to produce results for flow around a fixed cylinder the term 'oscillatory flow' will be used directly to describe flow relative to the oscillating cylinder; the oscillatory incident velocity will be equal in magnitude and opposite in direction to the cylinder velocity. This paper is only concerned with oscillatory flow normal to the current; oscillation in other directions will Received July 1982. Written discussion closes Dec. 1983. 0309-1708/83/040195-09 $02.00 © 1983 CML Publications

be reported subsequently. Mercier xl has made a similar experimental study over a limited parameter range but casts some doubt on the reliability of his data. The instantaneous force per unit length F(t) may be shown to be defined by

1

2

~pU1D

,

D

,

T'

--

(1)

(see Notation for definitions). UI T/D is the Keulegan-Carpenter number for oscillatory flow and will be denoted by Kc; UoT/D is equivalent to the reduced velocity used to describe the vibration of flexible cylinders in current flow and will be denoted by Vr. 1° Reynolds number Re may be defined by either U1D/v (as equation (1)) or UoD/u; the more appropriate form depends upon whether the current or oscillation is the dominant influence. This also applies to the way in which F(t) is non-dimensionalised. Full-scale Reynolds numbers, which are often greater than 106 , could not be achieved with the rig and values in the range 6 x 103 to 2 × 104 were used. Force coefficients due to a current or an oscillatory flow in isolation show little dependence upon Reynolds numbers ha this range, which is thus suitable for investigating the influence of Kc and Vr.

EXPERIMENTAL ARRANGEMENT The vibrating rig was designed to hold a cylinder vertically in an open-channel flume. The common flume design with a settling chamber leading to a channel with parallel sides had highly irregular flow characteristics with large eddies generated in the settling chamber. An aluminium alloy honeycomb was placed in the channel to 'straighten' the flow and this was followed by a converging section after

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Experimental study of forces on a cylinder in oscillatory flow: P. K. Stansby, N. El-Khairy and G. N. Bullock wind-tunnel practice. The working section downstream of the convergence was 0.5 m wide and the water depth was maintained at 0.28 m. A time-averaged velocity profile across the working section is shown in Fig. 1. Outside the wall boundary layers the turbulence intensity was about 1%. This was the closest approach to steady uniform flow that could be achieved without significant modification to the facility. The velocity at mid-section was used to define Vr. The vibrating rig was designed to perform simple harmonic oscillations with selected magnitudes and frequencies. A ~hp variable-speed d.c. motor was mounted vertically on a massive steel frame which spanned the flume and rested on rubber blocks on the laboratory floor. The rotary motion of a point on the motor fly wheel was converted to the simple harmonic motion of a carriage, which was free to move along straight, parallel, horizontal support rods, through a scotch-yoke mechanism (see photograph in Fig. 2). These rods were supported on the steel frame of the water channel so that they were exactly normal to the flow direction. (Orientations at 15 ° intervals can also be set up). With this arrangement transmission of vibration from the motor to the cylinder was kept to a minimum. A sealed, hollow cylinder of 25.4 mm external diameter was mounted vertically on the oscillating carriage. Strain gauges were attached to the cylinder surface at two vertical levels above the water surface and at 90 ° intervals around the cylinder. In this way orthogonal forces may be

Figure 2.

Photograph of vibrating mechanism

a}

Vertical profile at mid-channel

:)50

200

150

Figure 3. In-line force signal in untreated form (a) and after fihering (b )

I00

measured. The cylinder position was monitored through a displacement transducer. The natural frequency of the cylinder (in still water) was 45 Hz and this is considerably higher than the highest oscillation frequency of 2 Hz. Figure 3 shows a force signal taken directly from the strain gauges. Noise at the natural frequency is clearly present but it is difficult to know how it could be reduced further. Low-pass, matching Butterworth filters were used to condition the data before analysis. (The Filters were thoroughly tested before use and a cut-off of 10 Hz was usually used.) The cylinder was vibrated in air to quantify the inertia force due to its mass so that it may be later subtracted from the force measured in water to give the hydrodynamic component. The conditions at the ends of a cylinder are known to influence force measurements and end plates have been used in wind tunnels. 12 However, this would have created considerable difficulties with this arrangement. A horizontal perspex plate with a hole to allow cylinder oscillation was fixed at the free surface to prevent wave generation and the gap between the lower end of the cylinder and the flume bed was made as small as possible (less than 1 mm) to

E E

50

0.0

0.2

0.5

i

l

I

0.3

0.4

O. 5

tt

m/s

Horizontal profile at mid-depth

0.3 0

~ ]00

I ~00

__

I 300

___

1

400

I 500

Figure 1. An example of the velocity profile at the channel working section

196

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mm

Experimental study of forces on a cylinder in oscillatory flow." P. K. Stansby, N. El-Khairy and G. N. Bullock

Table 1. Variationof drag coefficient, CDO,with Reynolds number, Re = UoD/v, in a current Re CDO

3520 1.04

3990 1.06

4340 0.90

4695 1.00

FORCES

'cURRENT' DRAG~ COEFFICIENT CDo

OSCILLATORY FLOW UlCOS(2~t/T)

CURRENT Uo

10530 1.14

T CD

3rr ~ F1(t). cos(2m/T)

-oU21D dt 0 T _-arc (F~(t). sin(2~t/T) "t

(2a)

1 = ~ .,/

'IN-LINE' FORCE TO WHICH MOHISON EQUATION MAY BE FITTED WITH COEFFICIENTS CD1 AND CA

Figure 4.

6870 0.98

for an oscillatory flow with velocity U1 cos(2m/T), without the contribution of the Froude-Krylov component,

INCIDENT VELOCITIES i

5400 1.02

LIFT, COEFFICIENT CL

CA-- ~-~fi-TJ

p-U~1D

d

(2b)

o

Diagram showing force notation

(Note, with the Froude-Krylov component the inertia coefficient CM = 1 +CA). Mean values for 20 cycles are taken. The mean force per unit length, Fo, in the direction of the current of velocity Uo, will have a 'current' drag coefficient given by

suppress trailing vortices. For current flow, Table 1 gives drag coefficient variation over the range of Reynolds numbers used for which the Strouhal number remained close to 0.2. These results are typical of the nominally two-dimensional flows i° which are the subject of this study. The Strouhal number, S, for the Karman vortex street was obtained by measuring the transverse force frequency,/"1, since S=flD/Uo.fl is often called the vortex shedding frequency as it also describes the frequency with which a pair of vortices is added to the Karman street.

Fo Coo -

1

(3)

2

pUoD

The fluctuating force per unit length, F2(t), superimposed on the mean current drag, Fo, will result mainly from oscillatory flow in these tests and will be called the lift force, non-dimensionahsed as F~(t)

CL(t) -- ½PU21D

RESULTS

(4)

A schematic diagram showing incident velocities and force notation is shown in Fig. 4. For given Kc and Vr, 20 cycles of oscillation were recorded in digital form with a sampling interval of 10 ms. The force variation was thus well defined and the following values were obtained for forces in line with and transverse to the oscillatory flow : mean, standard

With current and oscillatory flows at right angles suitable force terminology must be specified. Forces in line with the oscillatory flow will be represented by the Morison equation with an 'oscillatory' drag coefficient Cot and an added mass coefficient CA obtained through Fourier analysis, la If F~(t) is the 'in-line' force per unit length,

2"5

0

Col

/U-tubedata1 0

2"0

Vr

I.S

S-O ~ _

10----~~

0

o

o

~

I'0

Vr = O-"'l~I

o.s L0

Vr 5 ~/'M 0

2

4 /

~ 6

8

I0

12

14

16

18

20

2:)

Kc

24

-0.5 l Figure5. Variation Of CDl with Kc for Vr=O, 5, 10. fl= 760 for Kc >~12.9 and fl=1170 for Kc <,9.7. Corresponding values from U-tube experiments 1are shown by symbol, o

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Experimental study o f forces on a cylinder in oscillatory flow: P. K. Stansby, N. EI-Khairy and G. N. Bullock deviation, maximum over 20 cycles, average maximum for each cycle and Fourier components (for Morison's equation and lift when periodic). After forces due to the current alone had been measured, forces due to oscillatory flow alone were measured. Values of Col are compared with corresponding U-tube measurements 1 in Fig. 5. Since these devices are of virtually constant period a parameter, 13= Re/Kc, which is independent of U1, is often used instead of Re to define the incident flow. The values of t3 used for these experiments were 760 for Kc>~12.9 and 1170 for Kc<~9.7. Corresponding values of CA are shown in Fig. 6 and an example of the close fit of the Morison formula to the measured force is shown in Fig. 7, for Kc = 15.8. R.m.s. CL and maximum CL are plotted in Figs. 8 and 9. It is apparent that the values of Col are less than, and the values of CA greater than, U-tube values) The values of CL are also less than U-tube values. Since Ca is reduced below the potential-flow value of unity due to the influence of the wake and Col and CL are entirely caused by the wake, all the changes from U-tube values indicate that the influence of the wake has been diminished. This could be due to more exaggerated end effects or unwanted mechanical noise reducing spanwise correlation of the flow and thereby reducing the total forces measured here. The variations of CL with Kc in these experiments show peaks around K c = l O and 18 where the lift frequencies have dominant components at twice and three times the oscillation frequency. (Lift frequencies are given more detailed consideration below). This feature was also observed in the U-tube experiments of refs. 2 and 3 with = 200 and 497 but it is strangely absent from the results of ref. 1 for the ;3 values of these experiments. Figures 5, 6, 8 and 9 include curves for Vr = 5 and 10 to show the influence of a current. However, important details are made more apparent by plotting against Vr. In Fig. 10, the variations of CD1 and CA with Vr for Kc <~6.4 show sharp changes, while those for Kc > 6.4 do not. These jumps around Vr = 5 are associated with the 'locking-on' phenomenon where the 'in-line' force frequency due to the wake is shifted from a value determined approximately by the Strouhal number to the oscillation frequency. Lockingon occurs for a range of Vr dependent upon Kc and outside this range the in-line force due to the wake will not be well represented by the Morison equation which describes a force at the oscillation frequency. These zones are sketched on a Kc/Vr plane in Fig. 11. (The 'M' denotes a region

°V V V /V,VY Y ?

_t~

A,fi

__

I/1

I1

/~ _

-0"5

(~ 0

~

~

~ ~ _'

t

~/

'°

y,, ,y

'

'

-05

O5

-05

(d)

Figure 7. Examples o f in-line force variations with time The dashed lines show the Morison fit. (a) Kc = 15.8, Vr = O. (b) K e = 4 . 7 , V r = 5 . (c) K c = 4 . 7 , Vr=lO. (d) Kc= 15.8, Vr = 10

1-5

I-0 0-5

0

0

6

8

I0

12

4

-0-5I Figure 6.

198

Var&tion o f CA with Kc for Vr = O, 5, 10. Other details as Fig. 5

Applied Ocean P,esearch, 1983, VoL 5, No. 4

16

18

20

22

24

Experimental study o f forces on a cylinder in oscillatory flow: P. K. Stansby, N. El-Khairy and G. N. Bullock I/r=O 1"5

0

o / \

~-

-,,K--.-"-"- U-tube data

o

o

o

r.m,s. C L

o

I'0 ~ r 0'5

0

I

I

1

I

I

I

I

I

I

2

4

6

8

IO

12

14

16

18

I

I

20

I

22

24

Kc

Variation o f r.m.s. CL with Kc for Vr = O, 5, 10. Other details as Fig. 5, except U-tube values have ~ = 1107for

F~ure 8. all Kc

\

0

0

~ i ~

\

0

U.tube data ~

O = L

\ 3-0

o \

\

\\

o

~

o

o ~,,,.

max CL

,,

2"5

CDO U_~

/

7,

\

up

2'0

1'5

I'0

\

\

~

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

~

\

. . . . .

V r = lO

0'5

0

0

I

I

I

I

I

I

I

I

I

I

2

4

6

8

I0

12

14

16

18

20

J

22

|

2~

Kc

Figure 9. as Fig. 5

Variation o f maximum CL (full lines) and Coo (U~/U]) (dashed lines) with Kc, for Vr = O, 5, 10. Other details

where the force in line with the oscillatory flow is predominantly at the oscillation frequency and is thus fitted reasonably by the Morison equation. The 'S' denotes a region where vortex shedding at the Strouhal frequency is' significant.) Even with locking-on, the force time history in Fig. 7 with Vr = 5 and Kc = 4.7 shows that the amplitude of the force can vary markedly while the phase remains unchanged. An upper limit for locking-on at Vr = 9.5 has been measured for Kc = 3.14 The range increases as Kc increased and locking-on can occur up to at least Vr = 10 for Kc = 4.7 and 6.4. The boundaries are, however, not well defined and locked-on and Strouhal frequencies occur intermittently for a range of Vr. This is shown for the force time history with Vr = 1 0 and Kc = 4.7 in Fig. 7. The lower limit is around Vr = 4. For Kc---4.7, the variation of Col and CA with Vr is independent of whether Vr is increasing or decreasing (see Fig. 10). However, for Kc = 6.4 there is a very definite hysteresis; for decreasing Vr the curve attempts to follow

the line for Kc = 7.9 while for increasing Vr it attempts to follow the line for Kc = 4.7, Suggesting that Kc = 6.4 is close to a changeover or critical value. Mercier n investigated Kc up to about 8 but did not show hysteresis. Nor was it observed bv garpkaya is who measured forces with 0<~Kc<<,6.4. In view of this our experiments were repeated and force coefficients were reproduced within -+5%. This Kc range relates to the flow-induced vibration problem for which there is a large amount of data, particularly for small amplitudes of vibration, i.e. very low Kc. These vibrations occur with locking-on and, by a straightforward analysis, maximum response should coincide with maximum negative Col. Maximum response occurs with Vr between 5.5 and 6.5 (see, for example, ref. 10, p. 587) and this clearly does not correspond with the values for ma×imum negative CDI in Sarpkaya's experiments (Fig. 5 in ref. 15). In our experiments, with Kc = 4.7, maximum negative CO1 occurred with Vr = 5.25. This is below the values for maximum dynamic response but higher than the

Applied Ocean Research, 1983, Vol. 5, No. 4

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Experimental study o f forces on a cylinder in oscillatory flow: P. K. Stansby, N. EI-Khairy and G. IV. Bullock

c~ 2"0

Kc= ~

1"5

~'-'Kc

= 4.7

1.0

"t

0-5 :

0

:

Kc

= 22.2

~ ....

-

K c = 15.8

KC

-0"5 -

>'~-

= 7.8

-

-I'0

l

Kc=

i-o

12.9

xZ-:~'-~ --'-

Kc

= 22.2

----~-.---

oI

ratio (and blockage) have a marked influence on drag coefficient: 6 It appears that not only lift forces are sensitive to spanwise correlation of the flow and it could well be impossible to produce a definitive set of data for forces in nominally two-dimensional flow. It was not possible to. attempt an investigation of end effects and aspect ratio here. It should be emphasised that the term 'locking-on' describes the shift of the in-line force frequency from approximately the Strouhal frequency to the cylinder frequency; the wake is a modified form of the Karman street. For large Kc the in-line force frequency will also be predominantly at the cylinder frequency but this is not 'locking-on' since now the wake due to the oscillatory flow is modified by the current. This change appears to have occurred for Kc >~7.9 where the sharp changes in CDI and CA associated with locking-on have disappeared. From these experiments, to the nearest integer, Kc = 7 is considered to be the upper limit for locking-on. For Kc > 7, Figs. 5 and 6 show that current has less influence on CO1 than CA which can be markedly reduced. In general the current causes the Morison formula to give a less close fit to measured data, as shown in Fig. 12 by the goodness-of-fit parameter 1

F20T

2OT

]1/2

O5

ol, -o.s [

,

,

,

CD1 with

Vr for Kc =

/

Region for Morison f i t (M)

M

12

Kc

0

~

Figure 10. Variation o f CA and 4.7, 6.4, 7.9, 12.9, 15.8, 22.2

14

,

/X'O~.'~-t,

I0

8

Phase jump

Intermittent / ~'4~-~ boundary / ~ / ~ v zones /

. -

6

4

~-~ \ \ L o c k i n g - o n / Region for Strouhal frequency (S)

2

0

Tertiary locking-on

Vr

Figure 11. Diagrammatic sketch showing different zones o f wake behaviour on the Kc/Vr plane

value obtained by Sarpkaya at the same Kc. There are, however, various uncertainties about the equivalence of experiments with cylinders which are forced to oscillate and which are responding dynamically, 1° mainly for high amplitudes of vibration. There are separate uncertainties about the influence of end effects which have already been mentioned. While the use of end plates might be thought to eliminate end effects, recent experiments with a fixed cylinder in a wind tunnel have shown that aspect

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Applied Ocean Research, 1983, Vol. 5, No. 4

where FMeas is the measured force (with no Froude-Krylov component) and FMor is the Morison fit. A force time history with Kc = 15.8 and Vr = 10 is given in Fig. 7. Figure 13 shows the ratio of the maximum in-line force given by Morison's equation to the maximum measured force. The ratio usually lies between 0.5 and 0.8; the value of 0.4 for Vr = 10 and Kc = 4.7 corresponds to a situation when locking-on was not occurring and the Morison equation gives a wholly inadequate representation of in-line force. If a number of cycles considerably in excess of 20 were analysed the results would obviously not be identical. However, the results are representative of the different flow regimes. The variation of Coo with Vr is shown in Fig. 14 for different Kc. For Kc = 4.7 there is a maximum near Vr = 5. Since Vr/Kc = Uo/U1, for very large Vr in relation to Kc, Ggo will tend to its value without oscillation, i.e. Coo ~ 1.0. Hysteresis at Kc = 6.4 is again evident and the curve for Kc = 7.9 is of completely different form. The curve for K c = 12.9 is again different and remarkably shows negative values for Vr < 3.6. (Dye visualisation of the wake showed that large repetitive vortex structures were formed downstream of the cylinder as a result of vigorous separation movement). For Kc>~ 12.9 the curves have similar shapes and magnitudes in general increase as Kc increases. Some of these results are also shown in Fig. 9, nondimensionalised by U~ rather than U~, for direct comparison with maximum lift forces; the maximum total force in the current direction may be readily obtained. It is seen that maximum CL becomes greater than CDOroughly where Kc = Vr or /-/1 = Uo. Figures 8 and 9 show how the peaks in CL for V r = O at K c - ~ l O and 18 are removed by the action of the current. The high values for .Vr =~ 0 are due to the way in which previously shed vorticity returns around the cylinder and interacts with the newlyforming wake. With Vr > 3 vorticity is swept downstream and wake structures for high lift are not produced.

Experimental study o f forces on a cylinder in oscillatory flow: P. K. Stansby, N. El-Khairy and G. N. Bullock 60

5O 17,

% 4O

30

20

~-~

Vr=O

"~-~

IO

0

I

I

I

I

I

I

I

I

I

2

4

6

8

I0

12

14

16

IB

I

20

I

I

22

Kc

24

Variation o f goodness-of-fit parameter for Morison's equation [0) with Kc, for Yr = O, 5, 10

Figure 12.

I.O

0"8

0"6

v~-=lo

0.4

0.2

00

I

I

I

I

I

I

I

I

I

2

4

6

8

I0

12

14

16

18

I

20

I

I

22

24

Kc

F~,ure 13. Variation o f the ratio o f the maximum in-line force from Morison's equation to the maximum measured over 20 cycles (¢) with Kc, for Vr = O, 5, 10 Kc = 22.2 6

$

Coo 4

3

Kc

=K~644__ ~

2

I

0

-I

/

Figure 14. Variation o f C ~ with Vr, for Kc = 4. 7, 6.4, 7.9, 12.9, 15.8, 19.0, 22.2

The-lift force is usually approximately periodic and Fourier components at the second and fourth harmonics of the oscillation frequency are shown in Fig. 15. The fourth harmonic for Vr = 5 is not shown in order to avoid confusion. The removal of the peaks in the second harmonic by the action of the current is consistent with the r.m.s, and maximum values. The prominent second harmonics for K c < 7 are consistent with locking-on. Without oscillation fluctuating forces in line with the current are at twice the vortex shedding frequency (as defined by a Strouhal number) and the second harmonics are thus consistent with a vortex shedding frequency locked-on to the oscillation frequency. As Kc is increased through 7 the second harmonics persist without locking-on and they are greatest when there is no current. Around Vr = 5 this continuity through Kc = 7 is i n spite of a marked change in wake behaviour, as shown in Fig. 10. The third harmonic was always negligible for Vr = 10 while it

Applied Ocean Research, 1983, Vol. 5, No. 4

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Experimental study o f forces on a cylinder in oscillatory flow: P. K. Stansby, N. El-Khairy and G. N. Bullock

n = 2

cL., 1"5

"/----2

;.o

O

0

I

I

I

2

4

6

~

t

8

~

....

I0

i

I

I

I

12

14

16

18

I

20

I

I

22

24 Kc

Figure 15. Variation o f Fourier components o f CL with Kc for: Vr : 0 (full lines); Vr = 5 (dashed lines); Vr = 10 (chaindashed lines); CLn is the amplitude o f the nth component showed a prominent peak around Kc = 18 for Vr = 0 and 5. Very wide ranges of Vr and Kc may occur in practice, from zero upwards, and any combination is conceivable. These experiments thus cover a relatively narrow, albeit interesting, range and zones of wake behaviour outside this range are also sketched on the Kc/Vr plane in Fig. 11. For Kc < 3 wake behaviour has been extensively investigated 14'17 and criteria for the boundaries of locking-on and for the jump in the phase of the in-line force in relation to oscillatory velocity (as indicated by the jumps in Col and CA in Fig. 10) are included in Fig. 11. It must be stressed that results are sensitive to minor differences in experimental arrangement and Fig. 11 is intended to be primarily diagrammatic. As well as 'primary' locking-on at the oscillation frequency, 'tertiary' locking-on at one third of the oscillation frequency was also observed, a4 Measurements of mean base pressure 17 indicate that CDOin this range will be much greater than for primary lockingon. Although the range of primary locking-on generally increases as Kc increases, there was no evidence of tertiary locking-on in these experiments with Vr > 3 and Kc >14.7. The Kc limits of this zone are not known and experimental observations only are included in Fig. 11. There is a variety of possible flow mechanisms for Vr < 3. For high values of Kc and Vr there must be a boundary between the S and M regions. This will probably be a zone where the wake-induced, in-line force is intermittently at the Strouhal and oscillation frequency, as it was at the boundaries of locking-on. The zone will probably extend from the upper locking-on zone and a hypothetical area is sketched in Fig. 11. CONCLUSIONS Forces on a circular cylinder in current and oscillatory flow may be described by Vr and Kc and are thought to show little dependence on Reynolds number, over the range used. The Reynolds numbers are lower than full-scale values where the same trends are expected since the basic wake phenomena are similar. Current and oscillatory flow were at right-angles and two broad wake regimes have been identified, roughly divided by Kc = 7 (for 3 < Vr < 10). For Kc < 7 modification of the Karman street by oscillatory flow is complex and force coefficients show a wide range of values. Morison's equation can be inadequate outside the

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Applied Ocean Research, 1983, Vol. 5, No. 4

range of locking-on which is not well-defined. For Kc > 7 high values of C5 are removed by current action (for V r > 3 ) . Coo shows considerable variation but simply adding forces due to the current and oscillatory flow as though in isolation gives total forces which are greater than or close to measured values. The Morison fit is in general less satisfactory when a current is flowing. ACKNOWLEDGEMENTS The authors express their thanks to the University of Jordan for providing a grant for N. E1-Khairy. The experiments were undertaken at the Department of Civil Engineering at the University of Salford and would not have been possible without the expert assistance of the technical staff. Particular mention is given to M. Baker who designed the data handling systems. NOTATION added mass coefficient in Morison's equation drag coefficient in Morison's equation mean drag coefficient in current direction, Coo normalised using U2o lift coefficient, normalised using U~. Lift is the cL fluctuating force normal to the oscillatory flow, superimposed on the mean current drag amplitude of the nth Fourier component of CL CLn inertia coefficient CM cylinder diameter D vortex shedding frequency in a current Ii force per unit length F(t) force per unit length in-line with the oscillatory Fl(t) flow which is fitted by Morison's equation fluctuating lift force per unit length F~(t) mean force per unit length, in current direction F0 Keulegan-Carpenter number Kc Reynolds number Re Strouhal number S time t period of oscillatory flow T ul amplitude of velocity of oscillatory flow current velocity Uo reduced velocity Fr 3=Re/Kc parameter to define U-tube flows kinematic viscosity of water V

G col

Experimental study or forces on a cylinder in oscillatory flow." P. K. Stansby, N. EI-Khairy and G. N. Bullock density of water goodness-of-fit p a r a m e t e r for M o r i s o n ' s e q u a t i o n ratio o f m a x i m u m in-line force f r o m M o r i s o n ' s e q u a t i o n t o m a x i m u m m e a s u r e d force

REFERENCES 1

2 3

4 5 6 7

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