Sinusoidal flow past a circular cylinder

Sinusoidal flow past a circular cylinder

Coastal Engineerin~ 2 (1978) 149--168 149 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands SINUSOIDAL FLOW PAST A CI...

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Coastal Engineerin~ 2 (1978) 149--168 149 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

SINUSOIDAL FLOW PAST A CIRCULAR CYLINDER

D.J. MAULL and M.G. MILLINER Engineering Department, Cambridge University, Cambridge (Great Britain) (Received January 6, 1978; accepted July 19, 1978)

ABSTRACT Maul, D.J. and Milliner, M.G., 1978. Sinusoidal flow past a circular cylinder. Coastal Eng., 2: 149--168. Unidirectional sinusoidal flow past a circular cylinder has been studied in a U-tube. The forces normal and transverse to the flow direction have been measured as a function of time and the interdependence of the two forces and the production and movement of vortices in the flow has been established.

INTRODUC~ON

A basic c o m p o n e n t o f most off-shore structures is the circular cylinder and a knowledge o f the unidirectional sinusoidal flow r o u n d this shape is required before the mo r e complicated wave-induced flow can be underst ood. This sinusoidal m o t i o n has been generated in three ways: by placing a fixed cylinder beneath a standing wave (Keulegan and Carpenter, 1958), by oscillating a cylinder in still water (see for instance H a m m a n and Dalton, 1971) and by using the resonance o f a large U-tube (Sarpkaya, 1976). The last o f these m e t h o d s is the one used to obtain the results presented in this paper. The forces on the cylinder may be split into t w o com ponent s, a force transverse to the fluid particle direction, the lift force, and one in the particle direction, the drag force. Most a t t e nt i on has been given in the literature to the drag force and the only equation presented has been given, by Morison et al. (1950), for t h a t force. The lift force has been measured by Sarpkaya (1976) and the results presented as a variation o f this force with, effectively, Reynolds n u m b e r and the non-dimensional particle amplitude. The above papers, and m a n y others, are useful in t hat t hey present data for the overall forces but very little m e n t i o n is made of the variation of the forces with the instantaneous particle m o t i o n and any analysis is based u p o n the concept o f the original Morison equation. In this paper we present time variations o f the forces and consider the devel opm e nt of these forces in terms o f the vortices p r o d u ced by the fluid m ot i on.

150

The Reynolds numbers in this experiment are low (of the order of 4.103 ) and are thus much lower than those encountered in full-scale offshore applications. APPARATUS

The U-tube used in the experiments is shown in Fig. 1, the cross-section of the tube is 0.45 m square and the model is placed in the centre of the horizontal limb. The tube is driven by oscillating a small cylinder up and down in the left hand limb at the natural frequency of the tube. The natural frequency can be varied by changing the a m o u n t of water in the tube and the amplitude of the oscillation varied by altering the stroke of the oscillating cylinder. Almost purely unidirectional sinusoidal motion is easily developed in the working section in the horizontal limb and spectral analysis of the fluid motion shows that over the range of flow conditions reported here higher harmonics do not a m o u n t to more than 2% of the fundamental. The circular cylinder models used were made of polished aluminium alloy with diameters of 2.54 and 3.81 cm and horizontally spanned the working section. The forces were measured using strain gauges and the signals recorded on magnetic tape for subsequent analysis on a computer. The water particle oscillation was measured using a capacitance wire in the right hand limb of the tube. Computer programs were available to calculate such quantities as the root-mean-square force, the spectral density distribution and a mean cycle of the force or water particle oscillation. In calculating these mean cycles, simultaneous recordings were made of the water particle oscillations and of the forces. The zero crossing points of the oscillations were identified, thus fix. ing the beginning and end of each cycle of the oscillation and each cycle was

2"65m

WORKING SECTION

.......L.

3 73

Fig. 1. S k e t c h o f U-tube.

m

151

then digitised at equal time intervals. The average of corresponding points during each cycle was then obtained for the force and water particle records and thus mean cycles for these were obtained. For all the results presented here the average cycles obtained were taken from records of about two hundred cycles of the particle oscillation. PRELIMINARY

CONSIDERATIONS

If a cylinder, diameter D, is in a sinusoidal flow which has a particle . amplitude as given by a = a sm (2nt/T), the force per unit length of the cylinder will be given by:

F(t) = f(p, D, ~, t, T, "~) and thus CF = 2F(t)T2/pD 3 = f(2~'~/D, t/T,pD2/pT), where 2~'~/D is the conventional Keulegan-Carpenter number (Kc) usually written as Um T/D, Um being the maximum particle velocity. The term pD2/pT, given the symbol .B by Sarpkaya (1976) is the ratio of the Reynolds number to the KeuleganCarpenter number and occurs naturally in unsteady boundary layer theory. We will in this paper, for convenience, use the above force coefficient CF rather than the more conventional coefficient CF' = 2F(t)/p Um 2D, thus CF = CF'Kc 2 . The reason for this is that during any set of runs in the U-tube, the d i a m e t e r D and the period T were kept constant and the amplitude ~ varied. We will also, when discussing the time variation of a force, use the instantaneous KeuleganCarpenter number 2~a/D instead of the non-dimensional time t/T. Thus an instantaneous force coefficient will be given by:

CF

=

f(2n~/D, 27ra/D, pT/pD 2 )

(1)

and time invariant coefficients such as root-mean-square coefficients will thus be given by:

CF(RM8) = f(2n "a/D, ~ T/pD ~)

(2)

Transverse, or lift, forces per unit length will also be made non-dimensional by dividing by pD 3/2T2. The above eqs. 1 and 2 are sufficient to specify the forces on a cylinder in sinusoidal flow b u t give no insight into the way in which these forces are developed. It is therefore usual to follow the procedure first suggested by Morison et al. and expand the force in the flow direction in the form:

1 F

= Cd--pDu

2

p~D 2 du I u I + CM

4

dt

(3)

that is a drag term proportional to the square of the velocity plus an inertia term proportional to the fluid acceleration. Naturally both of the coefficients

152

Cd and CM may be written as:

Cd, CM = f(2n~/D, 2na/D, pT/pD 2) and for sinusoidal flow it may be shown that, from eq. 3 (assuming Cd and CM constant over T):

C~(RMS)-

Cd2KC 2 + ~4 CM2

(4)

Much experimental effort has been concentrated upon the functional relationships for Cd and CM, usually obtained by measuring the force and then by Fourier analysis calculating the first two terms of the series expansion for F which gives an approximation to Cd and CM (see for example Sarpkaya, 1976). Apart from Morison's eq. 3 another m e t h o d exists for describing the forces on a circular cylinder in unsteady flow. This makes use of the 'discrete vortex' m e t h o d for calculating separated flows which has been used with some success when the main stream is steady and has been reviewed by Clements and Maull (1975). With this m e t h o d the equation for the force in the flow direction (the drag) becomes: d

~D 2 du d~

F =y~Fdt (PY) + 2 p - ~

(5)

where l~ is the circulation round a vortex in the flow field, y is the vertical distance between the vortex and its image in the cylinder and the summation is taken over all the vortices in the field. The coefficient CM of eq. 3 is replaced by its value for potential flow of 2. The equivalent equation for the transverse force (the lift) is: d

L = ~ F - - (px)

(6)

dt

x being the horizontal distance between a vortex and its image. This pair of eqs. 5 and 6 has the advantage that the forces can be explained in terms of the strength and motion of the vortices and further t h a t the lift and drag forces may be correlated since they are produced by the same mechanism. These two equations will be discussed in more detail when the experimental results are presented. RESULTS AND DISCUSSION

Root-mean-square drag force The coefficient

CF(RMS)of

the root-mean-square drag force per unit length,

F(RMS), (CF(RMS) -- 2F(RMS)T 2/pD 3) is shown in Fig. 2 plotted against the Keulegan-Carpenter number, Kc, for one value of the viscous parameter pD 2/U T (- ~) of 200. Also shown on this graph is eq. 4 with constant values of

153

CF|RMS) IOO0

--

/

800

600

400

o/

-

/

oo'''S°°

2oo-

[

0

5

I0

15

20

25

I

30

Kc

Fig. 2. CF(RMS) versus Kc.

CM and C d taken as 2.0 and 1.45. The conclusion from this result, at least for this range of Kc and ~ covered by the experiment, is that constant values of Cd and CM are quite adequate when used with eq. 4 to predict the root~meansquare of the drag force.

Root-mean-square lift force The coefficient of the root-mean-square of the transverse (lift) force as a function of the Keulegan-Carpenter number is shown in Fig. 3 for t w o values of the parameter/~. It is evident from this graph, at least for the data with a value of/3 of 200, that there is considerable scatter of the results especially over some ranges of the Keulegan-Carpenter number. Two force-time traces are shown in Fig. 4 for a Keulegan-Carpenter number of approximately 13, the lower trace giving the higher root-mean-square lift coefficient. The upper trace appears to indicate a modulation of the signal but traces including more cycles show that the force, rather than being modulated, is of an intermittent nature with periods where there is very little force being developed followed by periods of well sustained lift generation. The length of these periods is n o t constant. One m e t h o d of analysing traces such as those shown in Fig. 4 is to produce curves as Fig. 5 for the t w o runs 115 and 116 which were at a KeuleganCarpenter number of 13. This figure shows h o w much time (P(x) expressed as a fraction of the total time) the lift force spends above a fraction x o f the maximum force. Thus for run 115, 0.16 of the total time is spent above a

154

CL (RMS) 400 r

/

A #=443

[] ¢ =

[%671 [O'71 ] ~'~[3 [O'71]

/ 0"791

r

/

0,79, n

[o- 1AEo.7 1

2OO~

A

! I

[O. 76],.,,.......-..~D [ O "82 ]

.":L3Io. 7 9 ]

[]

,, ,~b--,,lo.s~/

[o.Bs]

[o.3~cff [o.461 n . / [O" 3311~,:

| /[3 |o 66 /[0"631 ~;]I~096l O 5

/

>~O. 801 [0"76] ~[0-79]

j

__.

IO

,

15

Fig. 3. CL(RMS) versus

~._

20

,

25 Kc

Kc.

I

Kc =13 Fig. 4. L i f t traces for

Kc =

13, ~ = 200.

value o f 0.2 o f the m a x i m u m force whereas run 116 spends 0.33 o f the total time above a value of 0.2 of the m a xi m um . This t y p e of curve may be used to p r o d u ce a rather crude measure of the i n t e r m i t t e n c y of the lift signal by calculating the fraction of the total time that the signal lies between t w o limits. This has been d o n e for all runs at ~ = 200 shown in Fig. 3 and the intermittency, defined by the fraction of the total time the signal stays in the limit - 0 . 2 ~< x ~< 0.2, calculated (for a sine wave this i n t e r m i t t e n c y is 0.13). These values are shown within brackets at each point on Fig. 3. It is evident, therefore, th at the high value o f the root-mean-square lift in the region between Keulegan-Carpenter numbers of 10 and 15 is mainly due to very stable lift generation with low intermittency. There is an indication in Fig. 3 that different flow situations m a y be occurring at a Keulegan-Carpenter n u m b e r of about 27 but th e intermittencies measured for these t w o runs are approxi m at el y the

155

116

%\ \ I

\r "8

-Xmax I

Fig. 5.

I

I

P(x) v e r s u s

I

x.

same and the large differences in the root-mean-square value of lift is due to large differences in the m a x i m u m value of the lift. The m a x i m u m value is probably of more use to designers and is shown in Fig. 6 for ~ = 200. The Reynolds n u m b e r produced in our apparatus is low (for ~ = 200 and K c = 20, R = 4000) but there is an indication that a similar multiplicity of lift forces occurs at higher values of ~ (or Reynolds number) as shown in Fig. 7 which is taken from the results of Sarpkaya (1976). This multiplicity raises serious d o u b t about any elementary analysis of the lift signal since it must be CLIMAX) I S O0

o

:o../o

IOOO

o/

bJ -

8

j

500

O

Fig.

L

I

l

i

5

IO

15

20

6. CL(MAX) v e r s u s Kc;

i

25 KC

/

156

CL (RMS)

600

/o. /

\\

o o ~o~\

/ / / ~o

400

~%

\\

Oo/O/

o

200

°//

o/

o o," i

0

I

I

5

I0

Fig. 7. CL(RMS) v e r s u s

15

Kc

A 25

20

Kc

(Sarpkaya).

non-stationary in a statistical sense and therefore the root-mean-square value, for instance, will be a function of the length of the record taken. In our case all records covered about 200 cycles of the water particle oscillation which is typically a record of about 10 min.

Frequency analysis of the forces The frequency components of the drag and lift forces have been found by spectral analysis and are shown in Figs. 8, 9 and 10. Fig. 8 shows that the drag force is primarily at the water particle frequency {called order 1) with a C FIAM P ) / , ~ " I000

~i o

o n

L

@xs°

5OO

xo0

~ 0

Fig. 8.

~o + ° o o+ I o ~ I0

CF(AMP)/N/2 v e r s u s Kc

15

qY 8 t~°

°+° 20

o~

o° 25

Kc

f r o m s p e c t r a l a n a l y s i s ; × u o r d e r 1; + c) o r d e r 3.

157

smaller c o m p o n e n t at three times this frequency. It is interesting to note that the magnitude of this component, as a fraction of the first component, is well predicted by Morison's equation if the values of Cd and CM of 1.45 and 2.0 are used. The frequency components of the lift forces are shown in Fig. 9 {orders 1--3) and 10 (orders 4--6) and may be compared with the root-meansquare force shown in Fig. 3. With increasing Keulegan-Carpenter n u m b e r the root-mean-square force reaches a first maximum at approximately K c = 13 and it is at this value that the frequency c o m p o n e n t at twice the particle frequency becomes a maximum (Fig. 9), all other frequency components, with the exception of that at four times the particle frequency being relatively small. The next maximum on the root-mean-square lift coefficient curve is at a KeuleganCarpenter number of a b o u t 18 and this is associated with a large frequency c o m p o n e n t at three times~the particle frequency (Fig. 9) and a smaller comC L {AM P ~/t/'~ 400 = 200 o

Order I

o x

300

2 3

Ix\ Ix\

I

2

20C

f IO0

I

^.,..~o_

\

x

\

//

/o

\\ "

~'xX

8 5

IO

15

20

KC.

Fig. 9. CL(AMP)/~2 versus Kc from spectral analysis for n is 1--3.

C L [AMP)/V/"2 300

200

0

~} = 2 0 0 Order o 4 + 5 o 6

5

I IO

I 15

J 20

o

t 25 Kc

Fig. 10. CL(AMP)/~2 versus Kc from spectral analysis for n is 4--6.

158

ponent at five times the particle frequency (Fig. 10), the even order component being very low. Further increase in the Keulegan-Carpenter number results in a decrease in the root-mean-square lift coefficient which is associated with a decrease in the frequency component at three times the particle frequency and very little increase in the other frequency components. At Keulegan-Carpenter numbers above 25 a further increase in the root-meansquare coefficient is apparent which is now associated with a frequency component at four times the particle frequency. Thus the development of peaks in the root-mean-square lift is closely associated with the progressive occurrence of higher order components in the frequency domain. As we shall see this is a consequence of the development and the motion of vortices in the flow. Force variation during a cycle

In order to try and understand the generation of the forces during a cycle of the water particle oscillation, average force traces over one cycle of the oscillation were produced. This was done by taking a trace of 200 cycles and forming the average cycle as previously described. This procedure is quite justified for the drag forces which are reasonably repetitive but as we have seen the lift forces are far from repetitive in amplitude and an average lift cycle has much less meaning. Two typical sets of average drag, lift and particle oscillation cycles are shown in Fig. 11, corresponding to Keulegan-Carpenter numbers of CF

CF

EJ 0°% 0

CL

o%

0

0

0 0 I

0

0

0

0

x

x

0

o o

L

o2

X)~

'~ %0° Ii

0

X

0 0

1

0

X

×

×

t

% xX×x

Fig. 1 1 . T w o s e t s o f CF, C L a n d w a t e r p a r t i c l e c y c l e s .

~°°oo°

°°

XX

XX

o oo

oOo 0

0

I 0

0 0

CLio

0

O

0

x>~

XX

X

\ X



159

5.85 and 21.1. Whilst this representation is useful it has been found more instructive to plot the force variation over an average cycle as a function of the instantaneous Keulegan-Carpenter number (2~a/D) as shown in Fig. 12 where the drag force is plotted against 2~a/D. According to Morison's equation the inertia c o m p o n e n t of the drag force is given by F = CMp (~D 2/4)/(du/dt), thus CF = -2~ 2 (2~a/D) for Civl = 2. Therefore, a pure inertia force with constant CM appears as a straight line on this type of plot and the above relationship is shown on Fig. 12 as the dashed fine. It is evident from this graph that this line is significant and is an axis of symmetry for all the curves which cover a range CF

C M =2

\

2~o

D -30

30

\

• - I000

Fig. 12. CF versus

2~a/D for various 2~/D.

160

o f m a x i m u m Keulegan-Carpenter numbers from just under 6 to 27. It was stated earlier that on the basis of modelling the flow with discrete vortices:the drag force could be considered to be made up of the inertia force plus a term d e p e n d e n t upon the devel opm e nt and m o v e m e n t of the vortices. The coefficient o f this term (CF - CF + 27r2(27ra/D)) is shown on Fig. 13 p l o t t ed against the instantaneous Keulegan-Carpenter n u m b e r for four values o f the m a x i m u m Keulegan-Carpenter number. It is now possible to explain h o w this drag is f o r m e d in terms of the m o v e m e n t of the vortices which are continuously being developed at the surface o f the cylinder at the separation points and by the m o v e m e n t of the vortices f orm ed in a previous cycle. Three basic vortex mo tio ns are present in these flows, namely: (1) the rolling up of a vortex sheet f r o m the separation poi nt on the cylinder into a 'developed' vortex; (2) the bodily m o v e m e n t of the vortex away from the cylinder; and (3) the translation of a vortex across the cylinder as the flow reverses. These three motions cause changes in the lift and drag and are discussed in the Appendix. Referring now to Fig. 13 and considering the inner curve which is for a m a x i m u m Keulegan-Carpenter n u m b e r of about 7.7. Progressing clockwise r o u n d this curve, at 2na/D of - 7 . 7 the flow which has been from right to left is a b o u t to reverse and vortices which have been form ed in the previous cycle /

-2~'y~

\

~o

-~--~t~-~--

I000

J"

f:::..~

%



- .~-"

/

7

./

Z:-o-o_o_o_ o:,/\

Fig. 13. CF versus

2~ralD.

,ooo_ ./"

<3 L

161

start to sweep over the cylinder. Considering a vortex previously formed at the lower part of the cylinder, this will have a negative vertical velocity and a positive value of circulation giving a negative drag (that is from right to left). However, as the flow develops from left to right and 2~a/D increases from -7.7 new vortices develop from the top and b o t t o m of the cylinder. These cause a positive drag and the effect of these two vortices and that from the previous half-cycle will be a slow increase in the drag force from left to right. This drag will reach a m a x i m u m when the motion of the vortex from the previous halfcycle slows down and the movement of the new vortex from the b o t t o m of the cylinder reaches a maximum at an instantaneous Keulegan-Carpenter number of about 3. The drag now decreases since the motion of the vortices decreases as the free stream velocity reduces to zero at the m a x i m u m KeuleganCarpenter number of 7.7. Since both the lift and drag are caused by the motion and formation of the vortices it is useful to study the lift-drag variation during a cycle rather than the lift-time variation. Such a variation is shown in Fig. 14 for again the maximum Keulegan-Carpenter number of 7.7, where the drag shown is CF, the drag due to the vortices alone. On this curve are also marked the values of the instantaneous Keulegan-Carpenter number (2~a/D) during the cycle. At avalue of 2~a/D of -7.7 the flow is about to reverse and since the previous flow was from right to left and positive lift was developed this indicates t h a t the strongest vortex was formed at the b o t t o m of the cylinder as shown in the sketch of Fig. 14. As shown in the Appendix the m o t i o n of the vortex across the b o t t o m of the cylinder will generate negative lift. However, simultaneously another vortex is being formed at the lower surface which generates positive lift, the result being t h a t the maximum negative lift occurs at an instantaneous Keulegan-Carpenter number of about -4. Subsequent development and movement of this vortex generates positive lift until at a value of 2rra/D of about 3 it moves bodily away from the cylinder and maximum lift occurs. While this vortex has been forming another vortex at the top has been developing which generates negative lift and thus for 5 < 2~a/D ~ 7.7 the total lift decreases. Reversal of the flow at 2~a/D equal to 7.7 sweeps both the immature vortex at the top of the cylinder and the fully formed vortex at the b o t t o m of the cylinder across the cylinder giving a further decrease in the lift. However, another vortex is being formed at the b o t t o m of the cylinder which is causing positive lift and its movement away from the cylinder causes a m a x i m u m lift at 2,raiD = -4. This vortex moves across the cylinder at 2~a/D = - 7 . 7 and the cycle repeats. The vortex movement is shown in Fig. 14, the positive and negative signs associated with each vortex indicating the signs of the lift generated by the vortices, this vortex m o t i o n has been confirmed by flow visualisation. Thus the m a x i m u m lifts occur at approximately the same places in the cycle as the m a x i m u m and minimum vortex drag and the lift force is occurring at twice the frequency of the particle oscillation.

162

CL

f O::o, 6-3/72"~ 150

-4"0

_

-5"6

-I

5S~/~ _1~.6 0

3"2 -15,

Kc

~7 '75

- 4 '4

+

Fig. 14. C L v e r s u s CF f o r

2n~/D=

7.7 a n d v o r t e x m o v e m e n t .

The inter-relationship between the lift and the vortex c o m p o n e n t of the drag may also be seen in Fig. 15 for a m a x i m u m value of the KeuleganCarpenter n u m b e r of 21.1. Again the figures on the graph indicate the instantaneous values of 2rra/D and the sketches under the graph show the vortex m o t i o n and the resulting sign of the lift p r o d u c e d by each vortex. Starting at 2 n a / D = - 2 1 the strong vortex 1 at the top of the cylinder moves across the cylinder as the flow reverses and produces positive lift which is progressively c o u n t e r a c t e d by the negative lift p r o d u c e d by the growth of vortex 2 which moves away f r om the cylinder producing a rapid decrease in the lift at 2 n a / D = O. This m o v e m e n t away from the cylinder will be b o t h in the positive x and y directions and will therefore cause an increase in the drag

163

which is a m a x i m u m at this point in the cycle. Vortex 3 is also forming and producing positive lift and the m a x i m u m lift is produced when it bodily moves away from the cylinder at 2 ~ a / D = 17.5. At the m a x i m u m point of particle amplitude in the cycle ( 2 n a / D = 21) vortex 4 is forming producing negative lift and on reversal of the flow this vortex moves across the cylinder producing positive lift. However, vortex 3, which is stronger than vortex 4, produces negative lift in moving across the cylinder on flow reversal and the lift continues to decrease until 2 ~ a / D = 6 when vortex 5, which generates positive lift, is sufficiently strong to start counteracting this negative lift. The CL 3 0 0 l-

0~_

If

too

1

-'t-300

-,1-

I

,

+

Kc = 21. t

I

C,~ I

Q +

Fig. 15. C L v e r s u s

C for

2~/D

= 21 a n d v o r t e x m o v e m e n t .

164

vortex then moves bodily away from the cylinder at 2~a/D = 0 and maximum lift and drag are generated due to this motion. The subsequent generation and movement of vortex 6 produces negative lift which is a m a x i m u m when 2~a/D = -18. The cycle then repeats at 2:aiD = - 2 1 with the vortex starting to move across the cylinder. This complex m o t i o n produces two vortices moving bodily away from the cylinder in each half-cycle of the particle m o t i o n and thus the lift is at three times the particle frequency as shown in Fig. 9. It has already been stated that over some ranges of the m a x i m u m KeuleganCarpenter number, for instance between 10 and 15, different values of the root-mean-square lift coefficient were obtained at approximately the same value of 2n ~/D. Two lift-vortex drag curves are shown in Fig. 16 for 2~ "~/D approximately equal to 13; in Fig. 16(a) the two lobes are symmetric and a high root-mean-square lift is generated, in Fig. 16(b) the lobes are asymmetric with CL

3oo-

(a) fo--o

o._....~o

K c = 13"29

S -,oS/

'

-2bo~~

/

";\

0/

~oo-~--"~:~o •

:

C 30-

(b

\

Kc

=

13"06

2oi I0-

4oo

40

-20 -30-

Fig. 16.

C L versus

Cl" for

2r~/D

=

13.

165

a low root-mean-square lift. These curves were obtained by analysing runs of 200 cycles each and are thus very crude average cycles. When lift versus vortex drag is presented for either run on an oscilloscope as instantaneous variations it is clear that the run of Fig. 16(a) is very stable in time but that of Fig. 16(b) shows that the lobes slowly change size with time. Fig. 17 shows the same graph for 27r'~/D = 15.7 and this curve is significantly asymmetric and corresponds to a point of Fig. 9 where the lift has almost equal components at twice and three times the particle frequency.

~

%

200t

Fig. 17. C L versus CF f o r

KC=15"7

2~r~/D =

15.7.

CONCLUSIONS

It has been shown that the root-mean-square drag coefficient of a circular cylinder in sinusoidal flow may be divided into two terms as in Morison's equation but with the inertia coefficient taking the potential flow value of two rather than varying with Keulegan-Carpenter number. The variation of the drag coefficient during a cycle may be considered as the addition of two terms, the inertia term again with CM = 2, and a further term which is a function of the movement of the vortices produced. The lift force is also a function of the vortex movement and thus the lift force and t h a t part of the drag force due to vortex movement are strongly related. Since both the lift and vortex drag forces are functions of the vortices and their movement they are very dependent upon the history of the flow as can be seen by the forces produced when the flow reverses and vortices from the previous half-cycle are swept over the cylinder. Thus Morison's equation, which gives the force only as a function of the instantaneous velocity and acceleration, whilst giving reasonable results for sinusoidal flow, may well be in error for flows which are non-periodic.

166

As has been remarked in the Introduction the Reynolds numbers of the flow are much lower than those encountered in full-scale applications. It is thought, however, t h a t the interplay between lift and drag still occurs at higher Reynolds numbers but t h a t the quantitative results presented here may not be of direct application at higher Reynolds numbers.

REFERENCES Clements, R.R. and Maull, D.J., 1975. The representation of sheets of vorticity by discrete vortices. J. Prog. Aerospace Sci., 16: 129--146. Hamman, F.H. and Dalton, C., 1971. The forces on a cylinder oscillating sinusoidally in water. Trans. ASME J. Eng. Ind., 1971, pp. 1197--1202. Keulegan, G.H. and Carpenter, L.H., 1958. Forces on cylinders and plates in an oscillating fluid. J. Res. Nat. Bur. Stand., 60(5). Morison, J.R., O'Brien, M.P., Johnson, J.W. and Schaap, S.A., 1950. The forces exerted by surface waves on piles. J. Petrol. Tech. Am. Inst. Mining Eng., 189: 149. Sarpkaya, T., 1976. Vortex Shedding and Resistance in Harmonic Flow about Smooth and Rough Circular Cylinders at High Reynolds Numbers. Naval Postgraduate School, Monterey, NPS 59 SL 76021. APPENDIX

Consider two-dimensional potential flow about a cylinder of radius a, outside of which is a vortex of strength F located at the point Z i. To satisfy tbe boundary condition on the cylinder an image vortex of strength - F is located at the point Zj. The vortex at Zi moves with the fluid with velocities in the x and y directions of ui and vi and the image vortex moves with velocities uj and vj. It may be easily shown, using the unsteady form of Blasius' equation, that the forces on the cylinder, X and Y, in the x and y directions due to the movements of the vortices are given by:

X=

pF(vi - vj)

Y = - p F ( u i - uj). In addition the force X will contain the usual term proportional to the fluid acceleration of the free stream 2pna2du/dt. Further, the image point Zj is the inverse point of Z i and thus if Z i is the point (x,y) the above forces may be expressed in terms of the point (x,y) and the velocities u, v of the vortex at that point. The forces then become:

X=pFlv

+

(X 2 +

y2)2 [(y2 _ x2)v + 2xyu]

and: l

Y=-pF

a2

U + ( x 2 +y2)2 [(x 2 - y2)u + 2 x y v ] l

167

These two expressions may now be used to describe the forces induced on the cylinder by the moving vortices. The development and subsequent movement of a vortex may be divided into the three types shown in Fig. 18. In the first, developing, stage when there is very little movement of the vortex as a whole, we have along AB a very small v c o m p o n e n t of velocity and a large u component. Thus using the equations for X and Y above, the motion of the vortices along AB will cause Y to be negative and X to be positive. A l o n g B C and CD the u c o m p o n e n t of velocity is small and the v c o m p o n e n t is negative along B C and positive along CD. These vortices will thus produce positive lift due to their motion alongBC and negative lift along CD, the overall effect for the motion from A to D being to produce negative lift. Negative drag is produced by the motion from B to C and positive drag from the motion from C to D giving an overall positive drag for the motion from A to D. The next stage to consider is the bodily movement of the vortex away from the cylinder shown in the second sketch of Fig. 18. Here u is large, v is positive and x greater than y which is of a b o u t the same order as a, the radius of the cylinder. Thus the bracket in the expression for Y is positive and hence the lift is negative, the predominant term in the expression for X is p r v which is positive. The last type of motion which must be considered is the m o v e m e n t of a previously formed vortex across the cylinder when the main flow reverses, which is shown in the third sketch of Fig. 18. The circulation r is negative and initially u and v are positive, y is greater than zero, b u t x is negative. This results in Y being positive and at x = 0, Y = - p F u / ( 1 - a2/y 2) which is positive

A

(I) DEVELOPING STAGE

+++e

f ~ ~/}

+.+O++ "P+C+

+

++++

( . ) aOOlLY

MOVEMENT

(Ill}MOVEMENT AcRoss CYLINDER

~+/~~

Fig. 18. Vortex motions considered.

168

since y > a. As the vortex moves over the top of the cylinder x changes sign and Y remains positive. In the initial part of the m o t i o n the drag will be negative since F is negative and the square bracket in the expression for X is small. .At x = 0, X = p F v / ( 1 + a2/y 2) which is negative and as the vortex passes over the t o p o f the cylinder x becomes positive but the drag still remains negative. Thus the reversal of the flow from left to right will cause positive drag to be developed due to the f o r m a t i o n o f a new vortex but negative drag will be caused due to the m o t i o n o f the vor t ex f or m e d during the previous half-cycle.