Oceanography - The study of the seas

Delft University of Technology Workgroup Offshore Technology

OCEANOGRAPHY The study of the Seas

TU Deift

Deift UniversitY of Technology

Offshore WJIVERSITEIT Laboratorium voor ScheepsI1ydromechMj Archief Mek&weg 2, 2628_{CD Delft} Te1 OiS- 78873.F O15781836 W.W. Massie

Preface

-1-This is intended only as a help for students attending the classes;

chapter

1 General Oceanografy 1.1 What are the oceans 1.2 History

1.3 Chemicals in the sea 1.4 Relation to density 2 Ocean current dynamics 3 Ocean waves 3.1 Regular waves 3.2 Irregular waves 3.2.1 "storm" spectrum 3.2.2 wave slimatogy 3.2.3 scatter diagram

3.3 Application to deck clearance 3.4 Extensions

4 Hydronamic forces on circular cylinders 4.1 Introduction

4.2 Hydrodynamic force components 4.3 Sloping cylinders

4.4 Parameters and coefficients 4.5 Wave plus currents

4.6 Simplifications 4.7 Additional remarks 4.8 Example

References

Chapter 1: General Oceanography The study of the seas

1.1 What are the oceans?

3

- cover 71% of earth; 1370 x 10 m of water

(if all oceans scaled to total of 1 m3, then N. Sea fits in a beer glass!)

- Mean sea depth: 3800 m, land elev: 840 m 1.2 History:

Count L.F. Marsigli (1725) study of Med. Sea Maury (1855) studied ships logs Challenger (1872-76) first systematic study Biological, Chemical, Geological, Physical branches of study.

1.3 Chemicals in the sea What is in sea water?

Nutrients Dissolved gas

Dissolved organics Suspended solids

The average concentrations of the various elements and their relative values - Ditmar Rule - are quite constant in the oceans. One often speaks of a Salinity value, S.

5 34,6 0/00 for Pacific

34,9 0/00 for Atlantic 38,5 0/00 for Med. Sea 40 0/00 for Red Sea 1.4 Relation to density 4 various metals -Uranium 1,5 g/kg Silver 0,3 Gold 0,006

(There are more than 75 trace elements en the sea!) Seas also contain:

- food for plants - O3 N2

-

sugars, etc. - sand, clay, etc.

Sea water density, p dependent upon S, temperature T, and (slightly) on pressure P.

For sea water 1025 kg/m3

- Chlorine 18,98 g/kg - Sodium 10,56 - Magnesium 1,27 - Sulphur 0,88 - Calcium 0,40 - Potassiam 0,38

T

45

>

As S increases, so does: - density - elect. conductivity - speed of sound - viscosity - surface tension - refractive index - osmotic pressure

Define = -lOOO _{(for convenience)} Fresh water has max. O at T = 4 C

This is not true for salt water:

freeiL

max.p

C-:

5

T

Why we can skate in Friesland _{and not on the North Sea!!}

Other properties: these decrease: - freezing temp. - temp of max. p - heat capacity - thermal conductivity

Density-changes, especially, are very important for ocean circulations.

Because the Mediterranean Sea has higher salinity - evaporation - than the Atlantic Ocean, its density is higher. This causes a desity current at Gibraltar; a continious circulation there eastward _{on the surface,} westward on the bottom of a few million m3/sec.

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Chapter 2: Ocean current dynamics

Forces per unit mass of flowing fluid (constant velocity): Pressure gradient: p/C

Friction: - air-water interface wind shear - internal turbulence

- sea bottom External forces: tide

(example)

Coriolis: ftom motion on a rotating earth Combine these in various ways:

pressure gradient caused by gravity and friction: - "classical" fluid mech.

- velocity // pressure gradient

Examples:

V = C/si Qìezy (channal)

AP = fl/D.½pV2 pipe

II For oceans - even N. Sea - cannot neglet Coriolis: Coriolis force 2 x

Q = angular vel. of earth

V = velocity In scalar form:

Fc fV; f = 2b sin 0 = 1,45 X sin ₀

coriolis force j velocity (to right in N. Hemis). lia Pressure gradient and Coriolis (no friction deep sea):

gi = fv

i in x dir.; vin y dir.

Florida strait: 26 N, 80 km wide y im/sec

A Z = difference in mean sea level across strait _½m.

lib Coriolis and friction (air-sea interface as well as internal turbulence) deep sea:

pfv + A-'- = 0;

-pf u + A-

= O

A is a turbulent "eddy viscosity" u, y are velocity components

Let a wind blow along the + y axis:

- A - T wind:-A =

O

this all yields:

vI.Vs

E

Z7

_{Polar form}

O a ₄₅

-further: q dz

=iQ.

and

isL

wind

Chapter 3: Ocean waves

For another important class of problems, the flow is very time varying, there is no friction and

accelera-tion terms become important.

This leads to wave type problems. We shall consider short waves in which both horizontal and vertical accelerations are important.

3.1 Regular waves BASIC DIFINITIONS

VJce

-ÇJ

Derived:

-9--

-T

H cosh k ( +h) 2 sinh kh H sinh k ( +h) - 2

smb

kh

L(

Peak to through peak to peak peak to peak Relative to SWL cos (kx -L4t) sin (kx -LOt)

Still water level: SWL

Wave heigth : H Wave length : X Wave period : T Wave amplitude : H/Z 2IT/T

f=

L

w

=2Wf

T K = 211/X

In two dimensions, the water surface can be discribed as a sine (or cosine) wave which is a function of both x and t, as well as H.

= f (H,x,t)

H

cos (kx - Wave Profile

The water depth, h, is also a parameter in many kinematic relationships.

u

Horizontal, vertical water velocity components; Z -. O for large h (h > u_

o__-__---

o (-Os b)

e

3(

deep water

subscript0 denotes deep water conditions.

acceleration Components

cci

k (Z

--

h)

i-

_sinh

_kk

(kx

_DE)

;J

z

Water Displacements

Integrate Velocities (left to the student) Results: _{- Movement along closed elliptical paths}

- Major axis horizontal

- Axes shorter with increasing depth - Ellipses become circles in deep water FOR DEEP WATER:

inh

h)

.'rk

kk

CC-) tt r\t

o ERÖ

2.

e kZ

rrv\&.

:c

circte

O5CX

-)

WAVE SPEED (Phase Velocity)

Wave Power

k

Wave Croup Speed

/&

_°

C

c

_--:

deep water:

()/l

25

\1'

2V

shallow water:

(9C

Power dependent upon both energy (H2 ) and wave speed

(period, T)

for deep water:

- _ZTr

b 'T

Wave Length

-

35

j55

T

Ñ s

\

_{t 00 rn}

Wave Energy

f03 Hz

-u--3.2

IRREGULAR WAVES

Define Upward (or downward) zero crossings (easier to determine by computer than peak valves)

Wave length, \

Wave Period, T

Wave Height, H

All H values , O; All H valves different

Average of all wave heights: (seldom used) Average of highest 1/3 of all waves : Hsig

Significant Wave Height

.0

-

Visual Estimate (with experience)

o

Root-Mean-Square wave heiqth: H

=\/i

\-rms

Describes Energy Content Ni 3.2.1 "Storm" spectrum

Re-examine Record For One Storm

Mean water elevation: O

(by definition)

Histogram of water levels shows a -Normal Distribution characterized by:

std. deviation

\O. O.

Variance =

13

-If wave frequencies are more or less constant (they

usually are! ) then the wave heights in this storm

obey a Rayleigh Distribution:

Chance that wave height of given value H in a storm having Hsig is exceeded is:

2

Lk)

For

Pc)

For

P L

I)

O,

00

Alternate Approach - _Spectrum

Consider the irregular wave record to be composec at an infinitive sum of cosine waves, each with

its own amplitude, frequency and phase.

2

B is a constant dependent upon Wind Speed for fully

developed sea.

JONSWAP Spectrum for "young" sea

C

Ç

Area under curve (often denoted M

is a measure of total variance of record.

thus:

¿_

\J-;:::

Mathematical Description Pierson Moskowitz (1964)

óC)

ot

2.Tr)

'5

exp

H

3.2.2. Wave climatology

Atlasses of Hsig, Tsig, Wave direction arid percent exedance for various locations and seasons.

H.O. 700, etc.

This can also be represented by a scatter diagram

T

n; = no observations (storm periods) for which Hsig in the given interval and T in the interval

If data on waves is lacking, it can often be predic-ted from wind data. H, T depend on:

Wind Speed, Fetch Length, Storm Duration and water depth (if not deep).

Wind force 10 (25m/s) gives Hsjg = 25 m, T = 15s, in about 70 h with 3000 km fetch.

This wave can also be general by a higher wind in less time (and distance).

Use of Wave Statistics

Waves occur with various heights, and periods. Often need to choose a worst_case! situation:

- Largest wave? H max.

- certain direction? - certain wave period? - frequent occurence?

The choice depends on the situation: - deck clearance

- foundation loads from structure; ship motions - resonance

- material fatigue

Often a worst case is a combination of wave height., direction and period, for example.

choosing a maximum possible wave heigth - based upon a water depth limitation.

Hsig about 0,6 x water depth

or wave steepness: F-!

is usually overconservative.

Must use a more statistical approach.

CANCE THAT ANDOC DECK HIT

15

-3.2.3 Scatter diagram

eeS oj o

Measure waves for a 6 hr. period Determine Hsig

Determine T (ave.

Plot Scatter diagram with large number of data value combinations (H,T)

--- Such data have only a limited observation period basis (2 years, here) extrapolate for more extreme conditions.

- For our problem, we do not need T data ' Combine via horizontal sums.

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-Such a graph can be extrapolated to lower

P (Hsig) ' Higher Eisig

see I O.c-L TI _cie.

We must combine this with data on how individual waves in a single storm "behave".

Rayleigh Distribution

Chance that an individual wave of heigth H exceeded

in storm of severity Eisig:

-2 /

P (H) = ₎

ri _I3

Combine this with long term distribution.

Convert long term chance of exceedance back to chance of occurence in interval - table col. III

Characterize each interval by a single Eisig value - table col. IV

Nted ave. wave period for each Eisig value to

deter-mine no. of waves in storm: deterdeter-mine from Hjg

using steepness relation - col. IV

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Deck clearance = 174 - 151 = 23 m

H = 2 x 23 46 m too big (asymmetric waves in practice).

choose Hd = 40 m.

Chance that any single given wave in a chosen storm exceeds Hd is

e

col. VI

s'

chance that a wave not greater than Hd:

1 - P(Hd)

chance not eceeded jn.N waves:

Cl - P(H

N

chance at least one wave exceeeds = 1 - chance that none exceed Hd:

E1 =

-

- P(H)

N

Table col. VII

Include storm chance:

E2 E1 . P (Hsig) col. VIII

Note: For small Hsig El small: P(Hd)

For large Hsigt E2 small:

P(Hsig) 0

Overall chance that Ed exceeded at least once in an arbitrarily chosen storm period = E3.

E3 = E2

Chance that no wave in any storm during lifetime of structure exceeds Hd is:

[-i _E3ML

21

M = no. of storms/yr = 1460 L = Life of struct. in years

How to use knowledge of chance Hd exceeded?

Include " insurance prerniumu for losses which occur as a "construction cost investment". Choose design with lowest total investment cost.

3.4 Extensions

- Account for uncertainly of wave height prediction for example.

- Include uncertainties from structural side

- chance that material not up to specifications - chance of fabrication error

- usual variations in material behaviour

This all leads to the concept of

-at the cylinder

4. Hydrodynamic forces on circular cylinders

4.1 Introduction

Since the early 1950's an enormous amount of research has been invested in the determination of the hydrody-namic forces on slender circular cylinders. Progress seems, at times, to be slow and no single method to predict the wave and current forces on structural elements of, say, a jacket structure is universally accepted. In this chapter, an attempt will be made to explain the more popular theories. In the following section, we start by discussing the force components acting on a unit length of cylinder placed perpendi-cular to a two-dimensional flow. Slender, in this discussion implies that the flow characteristics around the cylinder can be characterized by the flow

axis in an undisturbed flow pattern. As such, this implies that the cylinder diameter is much smaller than the wave length,

4.2 Hydrodynamic Force Components

Consider a cylinder of diameter, D, and unit length placed with its axis perpendicular to an infinite, constant uniform velocity field. This unit of cylinder will experience a drag force, _FD of

=

( ½HVV

)(D l)(CD) (4.01)

where: D is the diameter of the cylinder,

V is the undisturbed velocity,

is the mass density of water, and CD is an experimental coefficient.

This drag force is, thus, proportional to the kinetic energy of the undisturbed flow, times the projected area obstructing the flow, times a dimensionless coefficient. Usual values of CD range from about 0.7 to about 1.5. The drag force acts in the same

direction as the velocity, and is caused, primarily by the pressure difference existing between the "front"

and "back" of the cylinder.

A second force component, the lift force, acts along a line perpendicular to the flow direction. It can be described by:

FL ½ V2)(D l)(CL)(sin 2ft) (foa)

where: f is the frequency with which eddies are

shed in the vortex street behind the cylinder,

and CL is an experimental lift coefficient.

F D

25

-The lift force is proportional to the same sorts of quantities as the drag force, but fluctuates in a sinusoidal way with a frequency equal to the fre-quency with which the eddies are shed. The lift force is apparently caused by the alternate eddy formation in the wake of the cylinder. The lift force is only important, thus, when such eddy formation is present. The above two force components are the only ones present in a uniform steady flow.

If, we now allow the undisturbed flow to oscillate as a function of time a third force component, the inertia force, appears.

This force component is described by: F1 = ¼

D2pl)()(CM)

The inertia force is proportional to the accleration of the water times the mass of the water displaced by the cylinder, times an experimental coefficient, CM. The force is directed in the same way as the

instantaneous acceleration.

Morison, et al (1952) seems to be the first to have

suggest(a formula for the wave force acting on a

vertical circular cylinder.

The formula which bears his name is: F = FD + F1

3u

dF = ½ pu u CD Ddl + ¼IT

D2pi CM dL

c)

where dF acts on an element of length dL, and

u is the horizontal component of the velocity in the wave (equation 2.06).

that

Morison assumed, probably uncons ously, that velocity and accelaration components are parallel to the axis of the cylinder do not contribute to the hydrodynamic force in the direction perpendicular to the cylinder

axis.

Why did Morison neglect the lift force? There are probably two reasons:

First, with a vertical cylinder in waves, the line of action of the lift force is perpendicular to the line of action of the other two force components. Secondly: the lift force is directly coupled on the eddy formation in the wake of the cylinder.

Unless a single eddy extends over the entire length of the cylinder - very unlikely in view of the varying

flow conditions under a wave - the resulting lift force - integrated over the cylinder length - will be much less than that predicted by an equation like

4 02

For these reasons lift forces are often neglected

in determination of design loads on an offshore structure as a whole, used, for example, to design the foundation. Lift forces may not be neglected, however, when considering, for example, vibration of an individual structural

EFigure 4.01 shows the inertia and drag force components

on an element of a vertical cylinder orn 1 m length located at a depth op 10 m in infinitely deep water. The cylinder diameter is 0.5 m and the wave height and period are 5 m and 10 seconds, respectively. Values of CM and C are chosen (quite arbitrairily for now), to be

1.9

and 0.7 respectively.

Vibrations play a significant role in the development of important lift forces. A rigid cylinder will expe-rience vortex shedding which is more or less randomly distributed along the cylinder length; the resulting integrated lift force will be small.

If, on the other hand, the cylinder is not rigidly fixed, but is moving perhaps even slightly back and forth perpendicular to the flow direction, then this oscillatory motion will stimulate the development of a wake vortex in the most sheltered location. Since relatively long portions of the cylinder will be oscillating in the same phase, the generation

of long vortices extending over a considerable cylinder length is now stimulated; this increases the magnitude of the lift force integrated over the cylinder length.

If the frequency of vortex shedding, f, is much different form the natural frequency of the transverse vibration, then nothing very spectacular happens. (be sure to

check for a fatique failure, though. ) When, on the

other had, f and the natural frequency of the transverse oscillation are nearly alike, a "locking-in" takes

place - the vortex frequency shifts to agree with

the natural frequency - and a forced resonant vibration

results.

Figure /. 1

EXAMPLE OF NERTIA AND DRAG FORCES

360

P hase

27

-Note that the drag force has a decidedly different character from the velocity. This comes from the fact that it is proportional to the square of the velocity. This non-linearity, a quadratic dependence upon velocity, will lead to many practical problems

when wave forces are to be computed in real random

seas. This will be discussed in more detail later in this chapter.

Of couse, the velocity and acceleration components

are 90 out of phase. This implies that the maximum drag force occurs when the inertia force is zero

and visa versa. Note, also, that the maximum force does not, in general, occur at either of these times.

4.3 Sloping Cylinders

With the advent of the large steel offshore jacket

structures, it has become increasingly important to predict hydrodynamic forces on cylinders having an arbitrary orientation relative to the waves. The

most common procerdure for calculating such lift hem to

and drag forces at present is to attributethe transverse

force components of velocity and acceleration. Recent evidence from studies carried out here in Deift indi-cates that the above approach may not be correct. Unfortunately, testing has not yet progressed far enough to define a better prediction technique.

4.4 Parameteres and Coefficients

The traditional parameter to which drag force coeffi-cients in constant currents have been related for decades in the Reynolds Number, Re. It is defined

as a ratio of viscous forces to inertia forces and

is usually expressed as:

Re = (4.05)

where: O6is the kinematic viscosity of water (usually

about 10 m2/s). Indeed, a reasonably consitent

experimental relationship exists between drag coefficient and Reynolds Number for constant currents.

Such a relationship is less succesful in waves, however. Keulegan and Carpenter (1956) found that for an oscil-lartory flow, both the drag and inertia coefficients could be related to the Keulegan-Carpenter Number or Period Parameter:

KC ûT

D

where: û is the maximum velocity component and

T is the wave period.

If we assume, further, that the velocity component

varies sinusoidally as a function of time, then KC can be expressed as:

KC 2 CM Drag force amplitude CD Inertia force amplitude

Thus, the Keulegan Carpenter Number can be seen as a ratio of drag force to inertia force in waves.

Further, since CM is often a bit larger than C

the two force components contribute about equaly

when KC ' 12.

Another physical interpretation of KC is the ratio

of water displacement to cylinder diameter.

KC -- 2 water displacement amplitude

cylinder diameter

When waves are combined with currents, the Keulegan-Carpenter Number loses significance. Also, as the Keulegan-Carpenter number increases, drag coefficient values approach those for a corresponding Reynolds Number in steady flow.

This seems logical in light of equation 4.07, above.

The current tenclency is to relate the coefficient

values to both Reynolds and Keulegan-Carpenter Numbers. Figures 4.2 and 4.3 summarize the data of design

intrest. It is well to note that many organizations include data such as presented in figures 4.2 and

4.3 in their own guidof recomendend practice.

4.5 Waves Plus Currents

When currents are superimposed on the waves (a tide

superimposed on waves, for example) one must be sure to add the necessary velocity components vectorially

before computing drag forces. The resulting drag

force will be directed perpendicular to the cylinder axis and be in the plane defined by the resulting

velocity vector at that instant and the cylinder

axis.

(4.06)

(4.07)

10 0 25 50 75 100 KC

f

29

-J R

5xlO 25 50 75 100 KC Figure 3

SUGGESTED VALUES OF 0M AND CD AT SUBCRITICAL Re FROM KEULEGAN AND CARPENTER, FOR THE WAVE FORCE NORMAL TO THE AXIS OF A SMOOTH CYLINDER R 5 10

y

30 2.0 1.0 3.0 CD 2.0

Iin

(D Z NJ n o rn -n -n n Fn

z

-1 (n J. U) C z n

-

o

z

o z _(J n n '-t n o (D _u)

I

'D 35 _{25 15} KC

//

12 CD 0.6

//

CMlS CD

/

CMlS /

Choose coefficients ,/Errors

in total and local

from figure 4.3.

"/wave forces greater than _//l00%

are possible.

//

/ 12

CD

06 //

CMlS

CD

Total and local force errors should be less than IíY. except reor IC

15

where error may he large.

CMlS

//

_l//

/ / trrors iii lotal and loca

wave forces greater than

/ l00

are possible. -, r -J q

35i

4 (Don 10 subriitical regime-ø

-Cnt col

regime ø.

//

//

Total forces should be correct to within 2íL; local force errors of more than 50

are possible.

Little data available. Errors

can

be greater than for higher KC

at

sanie Re.

/ / ////////////////

_//

dometer 'wove tength)

O 2 CM 20 drag negligible (C D 0) (diameter/wove length) >02

ditfraction theory should be used

10

106

l0

posteriticot regime

Data above are for smooth iso] ated cylinders

in deep water.

hi

31

-Note that the non-linear character of the drag force makes it incorrect to determine the drga forces

from the wave and constant current seperatly and then to add thes two force components. In the more correct method outlined above, the velocity components are first added as vectors before the resulting drag

force is computed.

4.6 Simplifications

Under certain conditions, the Morison equation (4.04)

can be simplified. Since the non-linear drag term

is the most troublesome, it is helpful to investigate the conditions under which this can be simplified.

If the drag force component is small relative to

the inertia force, then the drag force term in (4.04) can be either neglected or approximated by a linear relationship. Remembering, that the ratio of the

drag force to inertia force is represented by the

Keulegan-Carpenter Number, we can see that KC' must be small if the drag force is to play an unimportant

role in our problem. From equation 4.07 and 4.06

we see that the drag force component is less important when velocity or wave period is small or when the

cylinder diameter is large. In general, the drag

force term can be neglected without significant error

whenever the Keulegan-Carpenter Number is less than about 3. Such low KO values occur often with large floating bodies.

For somewhat larger but still small KO values, the

drag force term can be approximatéd by expressing

V as a Fourier Series and then retaining only the first harmonic. If the velocity can be written as:

V = a sin w t (4.09)

then V

_I'

yields a Fourier Series without a constant term and with exclusively odd harmonics of sin t.

The first termhas amplitude:

8a2

0.8488 a2

3îr

-This means that V V can be approximated by: 8a

3îr

Note that the peak value of the drag force will be somewhat reduced in the linearized approximation. If, on the other hand, the Keulegan-Carpenter Number is very large, the inertia force component becomes relatively unimportant. Such is the case, for example, for a cylinder in a tidal current for which the period is relatively very long. Steady1 data can be used

with succes. '. flow

(4.10)

4.7 Additional Remarks

As we have seen, velocity components under a wave

decreasas we move deeper into the ocean. The most

straightforward practice is to use the computed values of velocity, etc. at each depth to determine the

necessary parameters such as Re and Kc which in turn determine the values of CD and CM to use at that depth . This is popular, but not universal,

practice. An alternative but apperently less correct approach is to evaluate the flow parameters and coeffi-cients at the ocean surface and the use of these

coefficients as constants valid over the entire depth. Figure 4.2 also gives some indication of the uncer-tainty involved in the computation of wave forces.

Note that the uncertainties are greatest for individual structural elements and when the drag force component is relatively more important.

1structures

All of this discussion until now has/been concerned with a smooth cylinder. In reality,/marine growth

soon makes the members of offshore/rough and even larger. Examples of offshore structural elements

whose diameter have been doubled by marine growth

are not hard to find. Often, larger diameters are substituted into the Morison Equation (4.04) when computing forces. Additionally, the roughness tends to increas the drag coefficient, CD somewhat. Minimum CD values of about 0.8 to 1.0 can now be expected. Even slight roughness can often double CD values.

4.8 Example

Since it can be instructive to illustrate a wave force computation, let us compute the hydrodynamic force on a 10 m long element of a structure. The diameter of the element is 2.5 m and it is placed in a vertical position and extends from 95 m below the still water level to 105 m below this level. The design wave has height of 20 meters and a period of 15 seconds. A current of 0.5 rn/sec. flows in the same direction as the waves are propagated. Determine

the maximum force acting on this portion of the structure.

We first determine the relevant flow parameter at

the location of the element.

T

T

) ()

L

-wo)

-

e U.s

33

-where we have assumed conditions at z = -loo m to he typical. Since the constant current acts in the same line as û, we can add it directly. The maximum water velocity will then

be:

0.70 + 0.50 = 1.20 rn/sec (4.13)

The Keulegan-Carpenter Number is, now:

(l.20)(15)

KC-

_D

- 2.5 - 7.2

This implies that both drag and inertia will be important with the latter term dominating slightly.

Checking the Reynolds Number:

uD (1.20)(2.5) 6

Re = = = 3 x 10

10

we see that this is in the postcriterial area.

This allows us to determine CD and CM fromfigure 4.2:

CD = 0.6

(4.16) CM = 1.5

The total velocity at the location of our element now varies

about the constant current velocity. The maximum velocity

- equation 4.13 - is 1.20 m/sec; the minimum velocity is:

0.5 - 0.70 = -0.20 rn/sec (4.17)

or expressing the total velocity, V, as a function of time: V = 0.50 + 0.70 sin

(ht)

(4.18) The acceleration follows from differeritation:

2T dV - (0.70)(-')cos(jt) 2'rr = 0.29 cos (1-g-t) (4.14) (4.15) (4.19)

Now using (4.01) for a 10 m length of cylinder:

FD= (½)025)(0.6)(2.5)0)

[0.5+0.7 sin (t

0.5 + 0.7 sin (t

(4.20) = 7688

0.5 + 0.7 sin (t)

0.5 + 0.7 sin (t)

Also, using (4.03) F1

=¼2.5)2(l0)(l025)(1.5)(0.29)co(t)

(4.22

= 21887 cos (t)

(4.23)

where= 2T/15.

Our suspicion about the dominance of the inertia force seems confirmed.

One can see by inspection that the maximum sum of FD and

FT will occur in the interval during which both terms have

te same sign. Choosing the positive interval (which will

yield the maximum force in this case), then: F = FD + F1

= 7688

.5 + 0.7 sin (t)

2

+ 21887

cos (Lt)

(4.24) dF

This is maximum when = O

d(' t)

Thus, at the maximum:

(2)(7688) .5 + 0.7 sin (t)1 0.7 cos (LOt)

+ 21887 sin

(t) = 0

(4.25)

or:

7688 + 7534 sin (-t) = 0 (4.26)

sin (L-.ìt) =7688 + 7534 sin (Jt) cos ( t)

4.27) 21887

a trial and error solution yields:

o

t = 30

Thus using 4.24:

F = 5558 + 18950 = 24508 N which is our desired answer.

One might like to attack the same problem, but now with the

cylinder placed horizontally parallel to the wave crests

at a depth of 100 m. What will be the maximum horizontal

force acting on this cylinder segment?

The answer is: 24528 N (4.30)

which by chance is not much differnet than the answer to

the first problem.

(4.28)

-R E F E -R EN CES

The following list includes more complete bibliographic data on most (and hopefully all) of the references listed in the text.

Anonymous (1973): Shore Protection Manual: U.S. Army Coastal Engineering Research Center: U.S. Government Printing Office, Washington D.C.

(1976): A Critical Evaluation of the Data on Wave Force Coefficients: The British Ship Research Association Con-tract Report Ib. W 278: Department of Energy Report No. OT/R/7611: August.

Keulegan, G.H.; Carpenter, L.H. (1958): Forces on Cylinders and Plates in an Oscillating Fluid: Journal of Research of the

:otional Bureau of Standards: volume 60, number 5, May. Kinsman, Blair (1965): Wind Waves, Their Generatiòn and

Propaga-tion on the Ocean Surface: Prentice-Hall Inc., Englewood Cliffs, N.J.,U.S.A.

Linnekamp, J. (1977): Hydrodyncnic Forces on a Vertical Cylinder resulting from Irregular Waves: Student Thesis, Coastal Engi-neering Group, Department of Civil EngiEngi-neering, Delft Universi-ty of Technology, Deift, The Netherlands.

In Dutch, original title: Vydrodynrsche krachten tengevolge

van Onregelrnatige Colven op een Verticala Paal.

Massie, W.W. (ed) (1976): Coastal Engineering - volume I, Intro-duction: Coastal Engineering Group, Department of Civil

Engi-neering, Deift University of Technology, Delft, The Netherlands.

Morison, J.R. (1950): Design of Piling: Proceedings of the First

Conference on Coastal Engineering: Long Beach, California,

U.S.A.: Chapter 28, pp 254-258: October.

Osborne, Alfred R.; Brown, J.R. (1977): The Influence of Interna-tional Waves on Deepwater Drilling Operations: Proceedings Ninth Offshore Technology Conference: Volume I, paper 2797:

May.

Saunders, W.R. (1956): h'ydrodynomics in _{Ship Design: The Society} of Naval Architects and Marine Engineers, _{New York, N.Y.,}

U.S.A.

Svedrup, H.U.; Johnson; Fleming, R.H. (1942): The _{Oceczns Their} Physics, T:enistry, and General Biology: Prentice-Hall _Inc.,