OCE421 Marine Structure Designs

Lecture #19 (Wave Forces on Vertical Cylinder)

Definition Sketch

d z

z

x

f = f i +fd

1

=cm½¼D2

4dudt

+cd2

½D ujuj

2

f i dz

1

fddz

2

d

z = 0

z = -d

Morison Equation

f i =

fd =

½=

D =

u =dudt =

cm; cd =

1

f = f i +fd

1

=cm½¼D2

4dudt

+cd2

½D ujuj

2

inertia force per unit length of pile

drag force per unit length of pile

density of fluid (1025 kg/m3 for sea water)

diameter of pile

horizontal water particle velocity at the axis of the pile

horizontal water particle acceleration at the axis of the pile

inertia (mass) and drag coefficient, respectively

horizontal 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:

DL

<120

1

Drag and Inertia Coefficients

• Drag 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 Number

K =UmTD

1

velocity amplitude of the flow

period of the flow

diameter of pile

Re=UmD

º

1

Um =

T =

D =

º =

2

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

´ =H2

cos(kx ¡ ¾t)

@2Á@t2

+g@Á@z

=0 at z =0

¾2=gktanhkd

7

u=@Á@x

=¼HT

coshk(d+z)sinhkd

cos(kx ¡ ¾t)

1

horizontal water particle velocity

Horizontal Acceleration

ax =@u@x

dxdt

+@u@z

dzdt| {z }

+@u@t|{z}

3

convective acceleration

local acceleration

ax =u@u@x

+w@u@z

+@u@t

4

u(x;z;t)

2

Note:ax =dudt

1

for small wave steepness

ax ¼@u@t

=2¼2HT2

coshk(d+z)sinhkd

sin(kx¡ ¾t)

1

Horizontal Force & Moment

F =Z ´

¡ df i dz+

Z ´

¡ dfddz

1

=Fi +Fd

2

horizontal force (F)

moment about the mud line (M)

M =Z ´

¡ d(z+d)f i dz+

Z ´

¡ d(z+d)fddz

3

=M i +Md

4

Fi =³ ¼cm

4½gD2H

´K i

1

Fd =³ cd

2½gDH2

´K d

2

M i =(Fi d) Si

Md =(Fd d) Sd

3

M i =(Fi d) Si

Md =(Fd d) Sd

3

dimensionless

Horizontal Force & Moment (contd.)

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

F =Z ´

¡ df i dz+

Z ´

¡ dfddz

1

M =Z ´

¡ d(z+d)f i dz+

Z ´

¡ d(z+d)fddz

3

0 (still water level)

Maximum Forces & Moments

Fi;max =³ ¼cm

4½gD2H

´K i;max

1

Fd;max =³ cd

2½gDH2

´K d;max

2

M i;max =¡Fi;max d

¢Si;max

Md;max =¡Fd;max d

¢Sd;max

3

Using Dean's stream-function theory, graphs [SPM: Fig. (7-71) through Fig. (7-74)] have been prepared and may be used to obtain

K i;max, K d;max, M i;max andMd;max

1

maximum inertia force

maximum drag force

SPM: Fig. (7-71)K i;max vs. d=gT2, for H=Hb=0, 1/4, 1/2, 3/4and1.

1

Maximum Total Forces/Moments

maximum total force

maximum total moment

Fmax =Ám

³½gcdH

2D´

1

Mmax =®m

³½gcdH

2Dd´

2

Ám and®m arefunctionsofd

gT2

HgT2

¯ =cmcd

DH

3

Ám and®m arefunctionsofd

gT2

HgT2

¯ =cmcd

DH

3

Ám and®m arefunctionsofd

gT2

HgT2

¯ =cmcd

DH

3

Ám and®m arefunctionsofd

gT2

HgT2

¯ =cmcd

DH

3

relative depth

wave steepness

inertia-drag ratio index

SPM: Fig. (7-76)

Isolinesof Ám vs. H=gT2andd=gT2, for ¯ =0:05.

1

Figs. (7-77) through (7-79) arefor ¯ =0.1, 0.5and1.

1

SPM: Fig. (7-80)Isolinesof ®m vs. H=gT2andd=gT2, for ¯ =0:05.

1

Figs. (7-81) through (7-84) arefor ¯ =0.1, 0.5and1.

2

Example Problem: SPM, p. 7-127

• A 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 =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 Force

SPM’s recommendation for design lift force:

FL =FL;maxcos(2µ)

1

=³ cL

2½gDH2

´K d;maxcos(2µ)

2

cL =

3

empirical lift coefficient (analogous to the drag coefficient)

K =umaxT

D

4

maximum horizontal velocity (velocity amplitude) averaged over the depth

umax =12

£(umax)bottom+(umax)swl

¤

5

Example Problem: SPM, p. 7-133

• A 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 =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.