Wave Forces on Vertical Cylinder

Marine Structure Designs(Wave Forces on Vertical Cylinder)

Definition Sketch z x

Morison Equationinertia force per unit length of piledrag force per unit length of piledensity of fluid (1025 kg/m3 for sea water)diameter of pilehorizontal water particle velocity at the axis of the pilehorizontal water particle acceleration at the axis of the pileinertia (mass) and drag coefficient, respectivelyhorizontal force per unit length of a vertical cylindrical pile

Usage of Morison Equation Morison's equation is valid for all ratios of pile diameter to wave length: Given d, H andT, which wave theory should be used? For a particular wave condition, what are appropriate values of cd and cm?Two problems:

Drag and Inertia CoefficientsDrag coefficients to be used in Morison's equation can only be obtained experimentally.In theory, the value of the inertia coefficient can be calculated (2.0 for a smooth cylinder in an ideal fluid). However, measured values are used in practice, particularly when drag is the dominant force.One problem facing the user of Morison's equation is the larger scatter in values of the inertia and drag coefficients.There is a useful degree of correlation between the coefficients and two flow parameters: Keulegan-Carpenter number and Reynolds number.

K-C & Reynolds Numbervelocity amplitude of the flowperiod of the flowdiameter of pile kinematic viscosity (approximately 10-5 ft2/sec for sea water).Reynolds numberKeulegan-Carpenter number

Inertia & Drag Coefficients (API,1980)Engineering practice is simply to assume them constant, with the values of the drag coefficient chosen within the range 0.6 to 1.0 and the values of the inertia coefficient within the range 1.5 to 2.0 (API,1980)

Linear Wave Theory wave elevation horizontal water particle velocity

Horizontal Acceleration

Horizontal Force & Momenthorizontal force (F)moment about the mud line (M)

Horizontal Force & Moment (contd.)If the upper limit of integration is zero instead of h andlinear wave theory is used, analytical expression ofKi , Kd , Si , Sd can be obtained (SPM Eq. 7-33 ~ 7-36)

Maximum Forces & Moments maximum inertia force maximum drag force

SPM: Fig. (7-71)

Maximum Total Forces/Moments maximum total force maximum total moment relative depth wave steepness inertia-drag ratio index

SPM: Fig. (7-76)

SPM: Fig. (7-80)

Example Problem: SPM, p. 7-127A design wave with height H=3 m and period T=10 s acts on a vertical circular pile with a parameter D=0.3 m in depth d=4.5 m. Assume that cm=2, cd= 0.7, and the density of seawater r=1025.2 kg/m3.

Find: The maximum total horizontal force and the maximum total moment around the mud line of the pile.

Transverse Forces (Lift Forces)Transverse forces result from vortex or eddy shedding on the downstream side of a pile.Transverse forces were found to depend on the dynamic response of the structure.For rigid structures, transverse forces equal to the drag force is a reasonable upper limit.Eddies are shed at a frequency that is twice the wave frequency

Design Estimates of Lift ForceSPMs recommendation for design lift force:empirical lift coefficient (analogous to the drag coefficient)

Example Problem: SPM, p. 7-133A design wave with height H=3 m and period T=10 s acts on a vertical circular pile with a parameter D=0.3 m in depth d=4.5 m. Assume that cm=2, cd= 0.7, and the density of seawater r=1025.2 kg/m3.

Find: The maximum transverse (lift) force acting on the pile and the approximate time variation of the transverse force assuming that Airy theory adequately predicts the velocity field. Also estimate the maximum total force.

*typos*Figure 7-81 to 7-83 (typo: 7-84)*Need to check K-C number to see whether it is reasonable to use cL=cD (should include Fig. 7-84) of SPM for the future lecture.