The influence of Reynolds number on wave forces

PAPER 13

THE INFLUENCE OF REYNOLDS NUMBER ON WAVE FORCES

P.J. RANCE

Hydraulics Research Station. Wallingford. England

SUMMARY

Information, published to date, or. the effect of viscosity on wave forces is very indefinite. A series of experiments carried out in the Pulsating Water Tunnel at the Hydraulics Research Station Wallingford under closely,

controlled conditions clearly defined the variation of primary forces with Reynolds Numbers up to 6 x 105 . Addi-tionally secondary high frequency forces resulting from the shedding of eddies were observed. The findings of this

work cast doubts on the validity of tests on model structures subjected to wave action.

INTRODUCTION

For the past two decades since the proposition of the now familiar Morison Equation

i

there has been a great deal of discussion on the importance of the effects of viscosity on wave forces. Over this period the experimental evidence published has been very inconclusive.

This is understandable to some extent for to measure wave forces over a wide range of the governing parameters under controlled conditions has hitherto not been possible. Experiments on a large scale in the sea are subject to the random conditions found there presenting difficulties in the₂ subsequent analysis. The early work of Harleman and Shapiro assumed the combination of the diffraction theory of MacCormy and Fuchs with drag forces; the latter being based on the steady state values of drag coefficient Cd. In order to support this theory the forces on a pile were determined experimentally with indefinite results: this may have been due, in part, to the assumption of a constant coefficient throughout the cycle. Furthermore, if there is a significant variation of Cd with Reynolds Number the assumption of a

constant coefficient with depth would lead to errors in analy-sis.

Keulegan and carpenter3 after measuring the forces on horizontal cylinders due to standing waves reached the

conclusion that "a correlation between the coefficients and Reynolds number UmDlv does not appear to exist". It is possible that the lack of apparent correlation w~s due to the limitations of the experimental conditions. Again, since the cylinders were of significant si~e compared with the depth of water it is possible that there was a Froude Number effect.

On the other hand, Keim

4

carried out experiments under rectilinear acceleration conditions and found a relationship between the resistance coefficient and Reynolds Number with-in the very limited range of his experiments. The mass of field data collated by Wiegel, Beebe and Moon

5

gives no clear indication of the dependancy of drag coefficient upon Reynolds Number. The conclusion reached by these authors was no more than that the magnitude of the coefficients was of the same order as those for steady state flow.

Many researchers have felt that much of the scatter in results was due to using linear wave theory in the analysis and have resorted to higher order thegries. However it has been suggested by Agerschou and Edens that fifth order theory is not superior to first order theory. Indeed they found indications of the opposite being true.

Other researchers have approached the problem of

analysis from the statistical rather than the deterministic

point of view to bring order from chaos. This approach is not entirely convincing in that the methods used are subject to the validity of the same assumptions made by deterministic methods and are likely therefore to ignore the effects of an unrecognised parameter.

This brief and incomplete review of the subject of the influence of Reynolds Number on wave forces has been given to explain why it was considered necessary to carry out yet another series of experiments under controlled conditions. In 1967 the Hydraulics Research Station, Wallingford carried out a series of experiments in a pulsating water tunnel on behalf of a consortium of firms interested in off-shore structures, organised by the Construction Industry Research and Information Association. The tunnel, designed originally to study forces on a pipe-line resting on the sea bed, has a test section 2.3 metres high by 0.5 metres wide. It has a semi-orbit range of 0 - 2.5 metres and a period range of 4 - 14 seconds. Additionally, a uni-directional current of up to 0.6 mls can also be superposed upon the oscillatory motion. This experimental facility eliminated two of the variables normally encountered i.e. gravity and depth there-by giving a closer control of the experiments. The Reynolds Number range was taken up to 6 x 105 .

In addition to the measurement of drag forces the Hydraulics Research Station, on its own behalf, decided to study the effects of viscosity upon transverse forces. Wiegel, et a1 5 , mention considerable vibrations and the fatigue failure of a pile but on the whole there has been very little information published on this subject.

THE EXPERIMENTS

Eight cylindrical sections were tested with diameters ranging from 0.3 m down to 0.025 m. Each test piece was mounted in turn in the centre of the test section at mid-height. In order to avoid the boundary layer effects due to the walls only the centre 0.3 m of the test pieces were

utilised for force measurements, the 0.1 m matching cheeks on either side being fixed to the tunnel walls. The centre section of the test piece was attached to one side cheek by a cantelever arrangement to which strain gauges were attached. The other side cheek was merely a dummy fixed to the wall and quite free of contact with the specimen. The natural frequency of the system (50-60 cis) was designed to be well above the probable frequency of excitation «20 cis).

The measured forces were functions of five independent variables,

t, the density of water

~ the viscosity of water

a, the semi-orbit length T, the period of motion

and D, the diameter of the test piece

For the plotting of results these variables were grouped **in two dimensionless parameters, aD/Tv and aiD. The former

lS a form of Reynolds Number whilst the latter may be inter-preted in two ways. It may be looked upon as either descri-bing the geometry of the experiment or as being a function of the ratio produced by dividing the drag force by the acceleration force. It is effectively the same as the reci-procal of Iversen's modulus (du/dt) D/u 2 , and the Keulegan parameter UmT/D.

In order to avoid making the assumption that the Morison equation was valid it was decided to use a force parameter rather than deduce drag and mass coefficients. The choice of a suitable force parameter was difficult~ I~ the maximum forces were essentially drag forces then FT Ita D, where F

is the maximum measured force per unit length, would be suit-able. If on the other hand the maximum forces were principally acceleration forces, FT2/~aD2 would be a more suitable

parameter. Since the experiments covered a wide range of conditions a compromise parameter FT2/~D3 was used. This may be considered as scaling the force in terms of test piece diameter and either

2unit V~locity or unit acceleration i.e.

F/~D (T/D)2 or F/~D (TID) . Although this might appear to offer a satisfactory compromise, when plotted against the parameter of aiD it does give undue weight to, the test piece

diameter. However its use was thought to be justified in that it enabled the Reynolds Numb~r effect to be clearly defined. From the Morison Equation a simple relationship between FT2/~D3 and the mass and drag coefficients at the instant of maximum force is obtained i.e.

Cd~2

2 D

+

The values of the force parameter were plotted against

aiD for narrow bands of Reynolds Number. The scatter within each band was surprisingly small being largely attributable to the band width. From the individual plots a composite diagram was constructed which showed clearly the sUbstantial effect that viscosity has on wave forces. In Fig I are shown two typical curves to form the basis of subsequent discussion.

Since transverse forces are primarily a function of

velocit~, the transverse force parameter was taken to be

CL

=

LT 12n2 a 2 D where L is the transverse force. This

parameter plotted against Reynolds Number, Fig. 2 shows considerable scatter reflecting the variable nature of the ~~transverse forces.

These forces are cyclic, oscillating with a frequency much higher than that of the imposed frequency. The para-meter for describing this vibration was taken to be the normal Strouhal Number S

=

ND/V where N is the frequency and V the velocity. It was found that the value of Strouhal Number showed remarkable consistency being 0.2 ~

6%.

There was no variation with aiD or with Reynolds Number.

DISCUSSION

~~ The experimental programme demonstrated clearly the two

effects of viscosity on wave forces viz. the primary forces in-line with the motion of the water and the higher frequency forces. Perhaps the most surprising outcome was the magni-tude of the transverse forces in the lower Reynolds Number range where in some cases they equalled the in-line forces. It was almost as surprising to find that the frequency at which eddies were shed, this being the mechanism giving rise to the transverse forces, was apparently completely indepen-dant of acceleration forces.

In view of the clear definition of the transverse forces it was to be expected that similar high frequency forces

should have appeared in the in-line forces at twice the Strouhal frequency. In fact the force records did show deviations about a mean force line but unfortunately it was found to be extremely difficult to determine the frequency of these deviations. The mean force line was the smoothed force curve obtained by ignoring the higher frequency

oscillations. The deviations appeared to take place at

several discrete frequencies, one superposed upon the others as distinct from a spectrum of frequencies.

The magnitude of the high frequency in-line forces was, generally speaking, low compared with tho main in-line force being less than 10% of the mean measured force. Only at the low end of the Reynolds Number range did they reach the significant proportion of 50% of the mean. On the whole they were approximately one half of the strength of the transverse forces.

Perhaps the significance of these high frequency forces lies not so much in strength as in frequency. In the context of braced structures subject to wave action, if it so

happened that the exciting frequency agreed with the natural frequency of a structural member, albeit a slender member, then fatigue is a possibility.

Returning to the subject of previous investigations it seems probable that the large degree of scatter in the values of published coefficients may be attributed to deviations about mean forces due to eddy shedding. This phenomenon would also explain the result of negative values of Cm, recorded by some authors. In the analysis of the results obtained from the H.R.S. experimental programme the maximum value of the mean force curve was used rather than the abso-lute maximum. Estimates of Cm at times of zero velocity and maximum force, again based on the mean force curve, both

indicated values of 2.0.

The work described in this paper was made necessary by a proposal to build and test small scale models of braced structures and knowledge of the likely scale effects was required. It is fitting, therefore, to conclude by discuss-ing the results in this context and the simplest way of doing this is to consider a hypothetical case.

A braced structure, standing in 60 m of water, is

subjected to

9

m waves with a 10 s period. The maximum force on a

0.3

m diameter member 25 m below may be deduced as

follows. The orbital velocity at this depth would be about 1 mls and thus the Reynolds Number would be about

3.5

x 10

5 .

The semi-orbit length would be 1.7 m giving an aiD value of

5.6.

From Fig. 1. the value of FT2/1D3 appropriate to these values of Reynolds Number and aiD is 400. Hence the maximum force on such a member would be about 12 Kg/m.

If a 1110 scale model were built and subjected to corresponding conditions then the Reynolds Number would be about 0.1 x 10~ and the value of FT2/1D3 1250. In this case the force when scaled up to prototype would be about

37

Kg/m. Thus even with such a large, impracticable model the forecast of forces would be in error by a factor of

3.

Smaller models would give greater er~ors~

The knowledge of this scale effect does not solve the problem of interpreting model results; since orbital

velocities diminish with depth there will also be a varying scale effect with depth. Again, if a braced structure is being tested the variation in sizes of members at a

particu-lar depth will give rise to a corresponding scale effect at that depth. At the moment, therefore, it would seem as

though model tests are not likely to be successful. However it is possible that interference effects between members will negate the effects of viscosity. The Hydraulics Research Station are now planning a programme of research on this aspect of the sUbject.

ACKNOWLEDGEMENTS

This work was part of the research programme of the Hydraulics Research Station, Ministry of Technology and is published with the permission of the Director of Hydraulics Research.

The agreement by the Construction Industry Research and Information Association to publish the limited data on

primary forces is gratefully acknowledged.

REFERENCES

1. Morison, J.R., O'Brien, M.P., Johnson, J.W., and Schaaf, S.A.; The force exerted by surface waves on piles; Petroleum Trans.; 189, TP 2846, 1950, pp. 149-54.

2. Harleman, D.R.F. and Shapiro, W.C.; Experimental and Analytical Studies of Wave Forces on Offshore Structures, Part 1, Results for· Vertical Cylinders; M.I.T. Hydrodynamics Laboratory, Technical Report No. 19, 1955.

3. Keulegan, G.H. and Carpenter, L.H.; Forces on Cylinders and Plates in an Oscillating Fluid; Journal of Research of the National Bureau of Standards; Vol. 60, No.5, 1958.

4. Keim, S.R.; Fluid Resistance to Cylinders in Accelerated Motion; J. Hyd. Div., Proc. ASCE, 82, HY6, Paper No. 1113, 1956.

5. Wiegel, R.L., Beebe, K.E., and Hoon, J., Ocean Wave Forces on Circular Cylindrical Piles; J. Hyd. Div., Proc. ASCE., 83, HY2, Paper No. 1199, 1957. 6. Agerschou, H.A., and Edens, J.J.; Fifth and First

Order Wave - Force Coefficients for Cylindrical Piles; ASCE Coastal Engr. Speciality Conf., 1965, Ch. 10. pp. 239.

F .. Maximum F()I"(;e per foot run

r ..

~ve ~riod

p .. _{Mass Density}

0 ... Diameter of Cylinder

a .. Semi Orbit length of Oscillation

II 10 5 _1-

V

~ J. I

II

i

V

V

f / / I L

I

V

II

RE' No ... 0'1 X 105

-f"-.

V

~

V

/ ~ / It'" _{RE' No ".]·5 x 10}5 / I V

L

/

/

/

FORCES IN THE LINE OF MOTION

2·0 I I I I I

_I

LIT 2

I I I I

I I

% -

D RatIos 2, 5 CL

=

2Tr2 olD

\

1\

- -

-1~ L = Average MaxImum Forces occurmg

)

\~

\

!\

~

1- 8 1- 6

20\

,\

\

_~ - - - ,

\

_~\

_l\

_\

KEY aID n • 'c:: 3 50

'\

~

\\.

0 3-7

'"

Ii. 7-15

'"

,,~

III _15-30 + 30-50

~

x _{::.. 50} + x x

~~

_~

-x~

~

~ - Ab-a 1-2 1-0 0-8 0-6 0-4 ...,; ~ :::,... 10' Re No

DISCUSSION ON PAPER 13

H.N.C. BREUSERS

Delft Hydraulics Laboratory. The Netherlands

The influence of the Reynolds number on wave forces has been demonstrated for a range of aiD values in which the acceleration forces are negligible. As it is known that there is a strong

dependency of the drag coefficient on the Reynolds number in uniform flow, it is not surprising that the same occurs in periodic flow. The values of FT2/pD 3 given for aiD = 10 correspond to CD values of 1.65 and 0.55 for Re = 0.1 • 10

5

and

3.5 .

10

5

respectively. These values are within the range given by WIEGEL. Although it is a good thing to be aware of scale effects, there is no need to mystify them.