Dynamics of single point mooring in deep water

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Archiet

kelweg 2, 2C8 CO Deift

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015 7873

Discussion by Subrata K. Chalerabarti

SUBRATA K. CHAKRABARTI.3The authors have attempted to obtain the analytical formulation for the complex problem of mooring. However, as pointed out by the authors, the validity of the theories developed In a wide range of applications is doubtful because of the lack of experimental data.

Some of the assumptions made In their analysis are questionable to the

writer.

In order to obtain the equations of dynamics for the buoy, the wave pres-sure forces were calculated based on the Airy wave theory. Note that the Airy theory makes the assumption that the surface boundary condition is satisfied

at the still water level. Thus the expression for pressure is not valid for

positive z (Fig. is). In the examples given the buoy was above SWL for part of the cycle, but the expression for pressure is not applicable there. Accord-ing to the linear theory, the pressure at a point z below the SWL [20] is given by

-

[

-

in which 'y = the specific weight of water; k = the wave number; and i = the surface elevation from the SWL. This expression is valid for a 0 and gives

(46) in which the wave is crossing the SWL (z = 0 and , = 0). At other points, a = 0 gives

(47) and thus the decay above SWL may be assumed linear (hydrostatic). The pres-sure distributions at the crest, SWL and trough are shown in Fig. 15. In this case the pressure is forced to satisfy the free surface boundary condition and (45)

allovember, 1970, by John 11. Nath and Michael P. Feltx (Proc. Paper 7700). 3ftend, AnalyUcal Group. Marine Roaearch and Development. Chicago Bridge & Iron Co., Plainfleld, Ill,

SWL

FIG. 15.PRESSURE DISTRIBUTIONLINEAR WAVE THEORY

zI (48)

FIG. 16.PRESSURE DISTRIBUTIONMODIFIED LINEAR WAVE THEORY equation, the denominator may be assumed as constant for the integration

and differentiation with respect to I, x, and y. This formulation seems to be a better representation of the ,pressure distribution since the body is near the water surface.

No basis was given by the authors for their choice of the inertia and drag coefficients. The hydrodynamic coefficients are functions of geometry, veloc-ity and acceleration and other parameters. The exact relationships have not yet been established. Prototype or model test results wiil provide information

at a few discrete points and these values may not be valid at all other points.

WW 3 DISCUSSION 589

the distribution is only approximate. The writer assumes that the authors used this distribution in the analysis.

On the other hand, the expression of pressure due to linear theory may be modified so that the dynamic pressure satisfies the boundary condition com-pletely (even though the expression does not conform with Laplace's equa-tion). The expression will have the form

F

coshk(h+z)

= '

L cosh k (is + '1)

-and the pressure distributions are as shown In Fig. 16. To satisfy Laplace's

588 August, 1971 WW 3

Action.Proceedings of aSymposium. Vol. Ii,Delft, The Nctherlands,July, 1969.

Hogg, R. V., and Craig, A. T.. Introduction to Mathematical Statistics, The Macmillan

Comp.,New York, N.Y., Second Edition 1965,Section1.5.

Venis, W. A., discussion of "Wave Forces on the Eider Evacuation Sluices,"Research ml

It is not clear to the writer how the expressions for the wave drag forces in the X2 and Z2-directlons (refer to Eqs. 40 and 43) were obtained. Accord-ing to the Morison formula, the drag force Is proportional to the projected area in the direction of the velocity. In the X2-dlrection-the expression for drag force should be taken as

FDX2 = .CDRADAPPTVXaIVXaI

in which A is the projected area in the X2-direction,

A = n(40)l _(so)

in which I = the height (average for small wave lengths) of the buoy below water. In the Z2-dlrection, the expression for drag force does not seem cor-rect. The drag force in the Z2-directlon, as stated In their proceeding paper (21) (without proof), was found to be

1 ,r(40) I

FDZS =

CJ1,

_{E PT}

In the preceding expression the unit of CDAXI. cannot be that of force, even though CDAXL is not dimensionless anymore. Also it is not explained in the paper how the authors derived this empirical relationship.

Appendix. References.

20. Ippen, A. 1., "Estuary and Coastline Hydrodynamics," McGraw Hill Book Comp., Inc.. New York, N.Y., 1966.

2i. Nath, J. M., andFelix, M.P.,"Dynamics of Single Point Mooring in Deep Water," Proceedings,

Civil Engineering in the Oceans II. ASCE, 1969, pp. 45-64.

(49)

FLUME EXPERIMENTS OF ALTERNATE BAR FORMATIONS

Discussion by Frank Engelund

FRANK ENGELUND.4The authors should be congratulated with the inter-esting and carefully performed investigation. Experiments with transport of

low density material are few and give an interesting check on the basic

reasoning.

5Febary, 1971, by Hal-Yam Chang, Daryl B. Simons und David A. Woolhisur

(Proc. Paper 7869).

Prof. of Hydraulics,Hydraulic Lab., Tech. Univ. of Denmark, Copenhagen, Denmark.

On one particular point the writer finds that the- authors' conclusion is misleading. Quoting Hansen's result (Ref. 3) the authors apply a definition, Eq. 2, of the friction factor, f, based on some hypothetical diameter related to circular pipe flow, while the factor f in Hansen's formula, Eq. 1, is based on the hydraulic radius or mean depth of the flow, corresponding to

s=fF

in which S = the slope and F = the Froude number,

If the authors results are replotted in double logarithmic scale, see Fig. 10, the points are nicely grouped round a straight line with the inclination 2.

FIG. 10 This leads to the relation

As

= constant F2 ₍₁₀₎

Substitution of Eq. 9 then gives

A = constant

_(ii)

Thus, contrary to the authors, the writer finds a close correlation between the experimental results and a formula of the Hansen type. However, it is

re-markable that the constant in Eq. ills much smaller than 14 (about 5 to 6)' and this calls for an explanation. In this connection it should be noted that _the Instability is In fact very sensitive to changes in the resistance law. Flume

590 August, 1971 WW 3 WW 3 DISCUSSION 591 1.0 0 S 0.I I _p 0.l _{ir I.0}