26 Classification Notes No. 30.
March 2000
6.1.10
Tentative values of the drag coefficient as a function of
Kulegan-Carpenter number for smooth and marine growth
covered circular cylinders for supercritical Reynolds
numbers are given in Figure 6.3, expressed as:
ì 1.45 for K C < 10
ï
ï 2
C D = C D ( k / D) í 1/ 5
for 10 < K C < 37
ï (K C - 5)
Figure 6.7 Interference drag coefficient as function of ï 1 .0 for 37 < K C
î
length solidy ratio ( R e = 6 × 105 ).
The figure is valid for free flow field without any influence
of a fixed boundary. For KC<10 the formula is expected to
be conservative.
6.1.11
The drag coefficient for steady current is equal to the
asymptotic value for KC equal to infinity. For combined
wave and current action, the increase of KC due to the current
Figure 6.8 Area to be considered in evaluating the loads may be taken into account.
due to shock pressure on circular cylinders
6.1.8 6.1.12
Values for the hydrodynamic drag coefficient CD for other To determine the drag coefficients for circular cylinders
smooth cross-sectional shapes in steady flow may be chosen close to a fixed boundary, the drag coefficients given in
equal to the corresponding wind shape coefficients given in 6.1.10 may be multiplied by a correction factor obtained
Tables 5.2-5.6. from Figure 6.4.
6.1.9 6.1.13
Hydrodynamic coefficients for circular cylinder in For several cylinders close together, group effects may be
oscillatory flow with in-service marine roughness, and for taken into account. If no adequate documentation of group
high KC values, may normally be taken as: effects for the specific case is available, the drag coefficients
for the individual cylinder may be used.
Surface condition CD Cm
Multiyear roughness k/D > 1/100 1.05 0.8 6.1.14
Mobile unit (cleaned) k/D < 1/100 1.0 0.8 An increase in the drag coefficient due to cross flow vortex
Smooth member k/D < 1/10000 0.65 1.0 shedding should be accounted, see 7.3.3.
The smooth values will normally apply above MWL+2m 6.1.15
and the rough values below MWL+2m, where MWL is as Tentative values of drag (CD) and drag interference
defined in Figure 4.2. The roughness for a “mobile unit” coefficients (ID) for composite cylindrical shapes are given in
applies when marine growth roughness is removed between Figure 6.6 and 6.7. The coefficients are based on a series of
submersion of members. model tests with shapes as defined in Figure 6.5 and given as
a function of length solidity SL in constant and oscillatory
The dependence on roughness may be interpolated as flow. The Reynold's number and Keulegan-Carpenter
number are referred to pitch diameter, Dp.
ì 0 . 65 ; k/D < 1/10000
ï
ï 0.65 (2.36 + 0.34 log 10 (k/D)) ; 1/10000 < k/D < 1/250 The solidity ratio SL is defined as:
CD = C D (k / D ) = í
ï 1.0 ; 1/250 < k/D < 1/100
ï
î 1.05 ; 1/100 < k/D < 1/25
SL =
åD i
Dp
The above values apply for both stochastic and deterministic
wave analysis when the guidance given in 3.1.3 is followed. The interference drag coefficient is defined as:
CD Dp
ID =
åCi
(i)
D Di
DET NORSKE VERITAS
Classification Notes No. 30.5 27
March 2000
where where:
Di = diameter of individual cylinder, see Figure 6.5 q = angle in degrees, see Figure 6.9
Dp = pitch diameter, see Figure 6.5 C D0 = the drag coefficient for tubular with appropriate
roughness as defined in 6.1.9
C(Di ) = drag coefficients for the individual cylinders
without any interaction C D1 = the drag coefficient for flow normal to the rack
( q =90o), related to the projected diameter, W.
SL = length solidity ratio
C D1 is given by
ID = interference drag coefficient.
6.1.16 ì W
ï1.8 : < 1 .2
For split tube chords (Jack-up leg chords) the hydrodynamic D
ï
coefficients may, in lieu of more detailed information be ï W W
C D1 = í1.4 + : 1.2 < < 1 .8
taken in accordance with Figure 6.9 and corresponding ï 3D D
formulae, as appropriate. ï W
ï2 : 1.8 <
D
î
The added mass coefficient Cm = 1.0 may be applied for all
pD 2
headings, related to the equivalent volume per unit
4
q length.
6.1.17
For a triangular chord (Jack-up leg chords) the
hydrodynamic coefficients may, in lieu of more detailed
information be taken in accordance with Figure 6.10
corresponding formulae, as appropriate.
2.5 q
C W/D=1.4
D 2
1.5
W/D=1.1
1
0.5
0
0 10 20 30 40 50 60 70 80 90
Heading θ (deg)
2.5
C
D2 W/D=1.1
Figure 6.9 Split tube chord and typical values for CD
(SNAME 1997) 1.
CD related to D
For a split tube chord as shown in Figure 6.9, the drag 1
CDpr related to
coefficient CD related to the reference dimension D , the 0.5 projected diameter
diameter of the tubular including marine growth may be
taken as: 0
0 45 90 135 180
Heading angle from rack q
ìCD0 for 0 < q £ 20 o o
ï
CD = í W 2æ 9
( o ö
) o
ïCD0 + (CD1 D - CD0 )sin ç 7 q - 20 ÷ for 20 < q £ 90
o
î è ø Figure 6.10 Triangular chord and typical values of CD
DET NORSKE VERITAS
28 Classification Notes No. 30.5
March 2000
The drag coefficient CD related to the reference dimension D 6.2.3
may be taken as In the vicinity of large bodies the free surface elevations (i.e.
wave height) may be increased. This should be accounted for
D pr
C D = C Dpr (q) in the wave load calculations as well as for estimates of deck
D clearances.
where the drag coefficient related to the projected diameter,
6.2.4
CDpr , is determined from
Hydrodynamic interaction between large volume parts
should be accounted for.
ì1.70 for q= 0 o
ï
ï1.95 for q = 90 o
ï
6.3 Second order wave loads on large volume
C Dpr =í1.40 for q = 105 o structures
ï1.65 for q = 180 o - q 0
ï 6.3.1
ï2.00 for q = 180 o
î Second order hydrodynamic load effects may in many cases
be important for the design of large volume structures in
Linear interpolation is to be applied for intermediate waves. Such load effects should be investigated. The
headings. The projected diameter, Dpr, may be determined different effects are explained in 6.3.2-6.3.6.
from:
D cos (q) : 00 <q <q0 6.3.2
When a linear regular first order wave is interacting with
Dpr Wsin (q) + 0.5D| cos (q)| : q0 <q <1800 -q0 itself and an ocean platform, forces of different characters
are created. In addition to first order linear exciting forces,
D |cos (q)| : 1800 -q0 <q <1800 mean nonlinear second order forces (drift forces) and non-
linear forces varying in time with twice the first order wave
D ù , describes when half the frequency act on the structure. In the present state of the art
The angle, q0 = arctan é
ê ë 2W úû
effects of higher order than two are usually neglected.
rackplate is hidden. The added mass coefficient Cm = 1.0
may be applied for all headings, related to the equivalent 6.3.3
pD 2 Irregular waves are assumed to be composed of an infinite
volume per unit length. number of fundamental frequencies and amplitudes (a wave
4
spectrum). In irregular sea the resulting second order exciting
forces contain three components. These are the mean forces
6.2 Wave loads on large volume structures (drift forces), forces varying in time with the difference
6.2.1 frequencies (often called slow drift forces) and with the sum
frequencies of the wave spectrum (high frequency forces).
The transfer function for linear wave loads on large bodies
should be determined by diffraction theory. The theories may
6.3.4
be based on sink-source methods or finite fluid elements. For
simple geometical shapes analytical solutions may be used. The difference frequency forces may in particular be
The results from sink-source methods should be carefully important for design of mooring and dynamic positioning of
checked for surface piercing bodies such that irregular offshore structures as well as for offshore loading systems.
frequences are avoided. If a new structural concept is For large volume structures with a small waterplane area the
introduced and the loads can not adequately be described by slow drift forces may create large vertical motions.
state of the art methods, model experiments are
recommended. 6.3.5
The sum frequency forces may become an important
6.2.2 excitation source in considering wave load effects on certain
The wave loads on structures composed of large volume offshore platforms as for instance the tension leg concept and
parts and slender members may be computed by a deep water gravity platforms.
combination of wave diffration theory and Morison's
equation. The modifications of velocities and acceleration 6.3.6
due to the large volume parts should however be accounted The second order forces should be determined by a
for when using Morison's equation. consistent second order theory or by model tests.
DET NORSKE VERITAS
Classification Notes No. 30.5 29
March 2000
6.4 Slamming loads from waves For flat bottom slamming taking account cushioning and
three dimensional effects, the slamming pressure coefficient
6.4.1 should not be taken less than Cpa =2p.
Structural members near the water surface (splash zone) are
susceptible to forces caused by wave slamming when the For a wedge shaped body with deadrise angle b above 15
member is being submerged. An important example is a degrees (see Figure 6.11), taking account of three
horizontal member where dynamic response of the member dimensional effects the slamming pressure coefficient should
should be accounted for. not be taken less than:
6.4.2 Cpa =
2.5
For a cylindrical shaped structural member the slamming (tan b)1.1
force per unit length may be calculated as:
where
1
Fs = r Cs D v 2 b =wedge angle at the intersection between body and water
2
surface (see Fig. 6.11 for definition).
where
For a wedged shaped body with 0<b<15 degrees, taking
FS = slamming force per unit length in the direction of account of cushioning and three dimensional effects, a linear
the velocity interpolation between results for flat bottom ( Cpa =2p for
r = mass density of the fluid
b=0 degrees) and b=15 degrees can be applied.
CS = slamming coefficient
D = member diameter
b
v = relative velocity between water and member
normal to the member surface
For a smooth circular cylinder the slamming coefficient
should not be taken less than CS=3.0.
water surface
6.4.3
Space average slamming pressure over a broader area
(several plate fields) can be calculated from: Figure 6.11 Description of symbol
1 6.4.5
ps = r Cp a v 2
2 The fatigue damage due to wave slamming may be
determined according to the following procedure:
where
· Determine minimum wave height, Hmin, which can cause
ps = space average slamming pressure slamming
· Divide the long term distribution of wave heights, in
r = mass density of the fluid excess of Hmin, into a reasonable number of blocks
· For each block the stress range may be taken as:
Cp
a
= space average slamming pressure coefficient
D s j = 2 [a sslam - (s b + s w )]
v = relative normal velocity between water and
surface where
sslam = stress in the element due to the slam load given
6.4.4 in 6.4.2
The space average slamming pressure coefficient should be
determined using recognised theoretical and/or experimental sb = stress due to the net buoyancy force on the
methods. The values given below should not be mixed with element
local pressure coefficients that can be considerably larger. sw = stress due to vertical wave forces on the element
For a smooth circular cylinder the slamming pressure a = factor accounting for dynamic amplification.
coefficient should not be taken less than Cpa =3.0.
DET NORSKE VERITAS
30 Classification Notes No. 30.5
March 2000
· Each slam is associated with 20 approximate linear 6.5.5
decaying stress ranges. The impact velocity (v) should be taken as that
· The contribution to fatigue from each wave block is corresponding to the most probable largest breaking wave
given as: height in n years. The most probable largest breaking wave
height may be taken as 1.4 times the most probable largest
i = 20 k significant wave height in n years.
nj
yj = R
Nj å
i = 20 - n i
æ i ö
ç ÷
è 20 ø
7. Vortex induced oscillations
where
7.1 General
nj = number of waves within block j
7.1.1
Nj = critical number of stress cycles (from relevant S-
N curve) associated with D s j Wind or any fluid flow past a member may cause unsteady
flow patterns due to vortex shedding. This may lead to
ni = number of stress ranges in excess of the limiting vibrations of the member normal to its longitudinal axis.
stress range associated with the cut off level of Such vibrations should be investigated.
the S-N curve
7.1.2
R = reduction factor on number of waves. For a
given element only waves within a sector of 10 At certain critical flow velocities, the vortex shedding
degrees to each side of the normal direction to frequency may coincide with or be a multiple of the natural
the member have to be accounted for. In case of frequency of motion of the member, resulting in harmonic or
an undirectional wave distribution, R equals 0.11 subharmonic resonance vibrations. The following provides
guidance on both methods for determining the motion
k = slope of the S-N curve (in log-log scale). amplitude, the response and the forces on the member.
6.4.6 7.1.3
The calculated contribution to fatigue due to slamming has to In the following the necessary criterion for presence of vortex
be added to the fatigue contribution from other variable shedding is listed. The vortex shedding frequency in steady
loads. current or flow with KC numbers greater than 40 may be
calculated as follows:
6.5 Shock pressure from breaking waves
v
f = St
6.5.1 D
Breaking waves causing shock pressures on vertical surfaces
where
should be considered.
f = vortex shedding frequency (Hz)
6.5.2
In absence of more reliable methods the procedure described St = Strouhal's number
in 6.4.2 may be used to calculate the shock pressure.
v = local flow velocity normal to the member axis
6.5.3 D = diameter of the member
The coefficient Cs depends on the configuration of the area
exposed to shock pressure. A lower limit of Cs for circular Vortex shedding is related to the drag coefficient of the
cylinders is 3.0. member considered. High drag coefficients usually accompany
strong regular vortex shedding or vice versa. This means that
the Strouhal number (St) is a function of Reynolds number (Re)
6.5.4 for a smooth stationary cylinder. The relationship between St
The area exposed to shock pressure may be taken as a sector and Re for a circular cylinder is given in Figure 7.1. For other
of 45 degrees with a height of 0.25 Hb , where Hb is the most cross sectional shapes St may be taken from Table 7.1.
probable largest breaking wave heigh in n years, Figure 3.3.
The region from SWL (see 4.2.4) to the top of the wave crest Rough surfaced cylinder or vibrating cylinders (both smooth
should be investigated for the effects of shock pressure. and rough surfaced) have Strouhal numbers which in practice
can be considered independent of the Reynolds number.
DET NORSKE VERITAS
