Technical Notes on CAESAR II Hydrodynamic Loading - CAESAR II - Help

CAESAR II Users Guide

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CAESAR II Version
13

The input parameters necessary to define the fluid loading are described in detail in the next section. The basic parameters describe the wave height and period, and the current velocity. The most difficult to obtain, and also the most important parameters, are the drag, inertia, and lift coefficients: Cd, Cm, and Cl. Based on the recommendations of API RP2A and DNV (Det Norske Veritas), values for Cd range from 0.6 to 1.2, values for Cm range from 1.5 to 2.0. Values for Cl show a wide range of scatter, but the approximate mean value is 0.7.

The inertia coefficient Cm is equal to one plus the added mass coefficient Ca. This added mass value accounts for the mass of the fluid assumed to be entrained with the piping element.

In actuality, these coefficients are a function of the fluid particle velocity, which varies over the water column. In general practice, two dimensionless parameters are computed that are used to obtain the Cd, Cm, and Cl values from published charts. The first dimensionless parameter is the Keulegan-Carpenter Number, K. K is defined as:

K = Um * T / D

Where:

Um = Maximum Fluid Particle Velocity

T = Wave Period

D = Characteristic Diameter of the Element

The second dimensionless parameter is the Reynolds number, Re. Re is defined as

Re = Um * D / n

Where:

Um = Maximum Fluid Particle Velocity

D = Characteristic Diameter of the Element

n = Kinematic Viscosity of the Fluid 1.26e-5 ft2/sec for Sea Water

After you calculate K and Re use the charts to obtain Cd, Cm, and Cl. For more information, see Mechanics of Wave Forces on Offshore Structures by T. Sarpkaya. Figures 3.21, 3.22, and 3.25 are example charts, which display below.

In order to determine these coefficients, the fluid particle velocity (at the location of interest) must be determined. The appropriate wave theory is solved, and these particle velocities are readily obtained.

Of the wave theories discussed, the modified Airy and Stokes 5th theories include a modification of the depth-decay function. The standard theories use a depth-decay function equal to cosh(kz) / sinh(kd),

Where:

k - is the wave number, 2p /L

L - is the wave length

d - is the water depth

z - is the elevation in the water column where the data is to be determined

The modified theories include an additional term in the numerator of this depth-decay function. The modified depth-decay function is equal to cosh(da) / sinh(kd),

Where:

a - is equal to z / (d + h)

The term da represents the effective height of the point at which the particle velocity and acceleration are to be computed. The use of this term keeps the effective height below the still water level. This means that the velocity and acceleration computed are convergent for actual heights above the still water level.

As previously stated, the drag, inertia, and lift coefficients are a function of the fluid velocity and the diameter of the element in question. Note that the fluid particle velocities vary with both depth and position in the wave train (as determined by the applied wave theory). Therefore, these coefficients are in fact not constants. However, from a practical engineering point of view, varying these coefficients as a function of location in the Fluid field is usually not implemented. This practice can be justified when one considers the inaccuracies involved in specifying the instantaneous wave height and period. According to Sarpkaya, these values are insufficient to accurately predict wave forces, a consideration of the previous fluid particle history is necessary. In light of these uncertainties, constant values for Cd, Cm, and Cl are recommended by API and many other references.

The effects of marine growth must also be considered. Marine growth has the following effects on the system loading: the increased pipe diameters increase the hydrodynamic loading; the increased roughness causes an increase in Cd, and therefore the hydrodynamic loading; the increase in mass and added mass cause reduced natural frequencies and increase the dynamic amplification factor; it causes an increase in the structural weight; and possibly causes hydrodynamic instabilities, such as vortex shedding.

Finally, Morrison’s force equation is based the "small body" assumption. The term "small" refers to the "diameter to wave length" ratio. If this ratio exceeds 0.2, the inertial force is no longer in phase with the acceleration of the fluid particles and diffraction effects must be considered. In such cases, the fluid loading as typically implemented by CAESAR II is no longer applicable.

Additional discussions on hydrodynamic loads and wave theories can be found in the references at the end of this article.