Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

36.2 Fluid Forces

Back

The following is a list of several types of forces that the fluid can exert on a body:

1. Drag force. This is due to the pressure difference between the downstream and upstream flow

region. It can be thought of as the force required to hold a body stationary in a fluid of constant

velocity. The drag force is proportional to the square of the velocity of the fluid relative to the

structure.

2. Inertia force. This is the force exerted by the fluid while it accelerates and decelerates as it passes the

structure. It is also the force required to hold a rigid structure in a uniformly accelerating flow, and

it is proportional to the fluid acceleration. The concept of the inertia force in an inviscid flow was

first formulated by Lamb (1945).

3. Added mass. As the body accelerates or decelerates in a stationary fluid, the body carries a certain

amount of the surrounding fluid along with it. This entrained fluid is called the added, apparent,

or virtual mass. In order to accelerate the body, additional force is required to accelerate or

decelerate the added mass.

4. Diffraction force. This is due to the scattering of an incident wave on the surface of the structure. It

is important when the body is large compared with the wavelength of the incident wave.

5. Froude – Kryloff force. This is the pressure force on the structure due to the incident wave, assuming

that the structure does not exist and does not interfere with the incident wave.

6. Lift force. This is due to nonsymmetrical separation of the fluid or due to vortices that are shed in a

nonsymmetrical way. The component of the force perpendicular to the flow direction is the lift

force.

36-16 Vibration and Shock Handbook

7. Wave slamming force. This is due to a single occasional wave with a particularly high amplitude

and energy, and it may be important at the free surface. Sarpkaya and Isaacson (1981) reviewed the

research on slamming of water against circular cylinders. Miller (1977, 1980) found that the peak

wave slamming force on a rigidly held horizontal circular cylinder is proportional to the square of

the horizontal water particle velocity.

36.2.1 Wave Force Regime

Previously, we discussed various types of forces caused by waves and currents. In some cases, one type of

force may be dominant. Hogben (1976) gave a literature review of the fluid force in various regimes. The

load regime of importance can be demonstrated for the case of a vertical cylinder in Figure 36.11 in terms

of H=D and pD=l; where H is the wave height, D is the cylinder diameter, and l is the wavelength. When

linear wave theory is used, H=D is related to the Keulegan – Carpenter number by

K ¼ pH=D

The Keulegan – Carpenter number gives a measure of the importance of drag force relative to the

inertia force. The term pD=l is called the diffraction parameter, and it determines the importance of the

diffraction effect. As H=D increases, the drag force becomes more important and the inertia force becomes

less important. As pD=l increases, the diffraction force becomes important.

Using linear wave theory, the maximum drag force to the maximum inertia force can be written as

fdrag

finertia ¼

D ¼

2p2

From the last relation, we find that the drag force is 5% of the inertial force when H=D ¼ 0:314: The

Morison equation may be used for D=l , 0:2 and fdrag=finertia . 0:1 or thereabouts. It should be noted

that Figure 36.11 is valid only near the surface. The drag force is predominant for a cylinder that extends

from the bottom to the near surface, so that the Morison equation may be used.

For example, consider a fixed jacket platform with legs with a diameter of 10 m and bracings with a

diameter of 0.8 m. For a 10-year storm with l ¼ 100 m and H ¼ 8 m, the ratios H=D and D=l for the leg

are 0.8 and 0.1, respectively. Similarly, the ratios H=D and D=l for the bracings are 10 and 0.08,

respectively. Figure 36.11 shows that the inertia force is dominant for the legs, and both inertia and the

drag forces are important for the bracings.

0.01

0.1

H/D 1

100

large drag

large inertia

inertia diffraction

D/l = 0.2

pD/l

CD = 0

25% drag

5% drag

no waves

inertia

and drag

0.1 1.0 10

FIGURE 36.11 Load regimes near surface.

Fluid-Induced Vibration 36-17

36.2.2 Wave Forces on Small Structures — Morison Equation

The added mass, MA; can be written as

MA ¼ CAMdisp

where CA is called the added mass coefficient and Mdisp is the mass of the fluid displaced by the structure.

For a cylinder with a diameter, D, and height, h, the displaced fluid mass is pD2h=4: It should be noted

that the added mass is a tensor quantity. That is, we can speak of the added mass force in the xi direction

due to the acceleration of the body in the xj direction, denoted as MA

ij : MA

ij is symmetric so that the added

mass force in the xi direction due to the acceleration in the xj direction is equal to the added mass force in

the xj direction due to the acceleration in the xi direction. The off-diagonal terms are not zero if the crosssection

is not symmetric.

Similarly, the inertia force can be written as

FM ¼ CMMdispw_ ð36:25Þ

where the proportionality constant, CM; is called the inertia coefficient.

It should be noted that the added mass and the inertia effects are often neglected for a body vibrating in

air since the displaced air mass is negligible.

The drag force is proportional to the square of the fluid velocity, w, the density of the fluid, r, and the

area of the body projected onto the plane perpendicular to the flow direction, Af ;

FD ¼

CDrAf wlwl

where CD is the drag coefficient. The absolute value sign is used to ensure that the drag force always acts in

the direction of the flow. For a cylinder with a diameter D and height h, the projected area Af is Dh.

For a body with nonzero velocity, the drag force is given by

FD ¼

CDrAf ðw 2 vÞlw 2 vl ð36:26Þ

where w 2 v is the velocity of the fluid relative to the body.

Morison et al. (1950) combined the inertia and drag terms (Equation 36.25 and Equation 36.26)

so that the fluid force on a body is given by

f ¼

CDrAfwlwl þ CMMdispw_

For a cylinder, the fluid force per unit length can be written as

f ¼

CDrDwlwl þ CMrp

For a moving cylinder with velocity v, the Morison force is given by

f ¼

CDrDðw 2 vÞlw 2 vl þ CMrp

36.2.2.1 Inclined Cylinder

Let us now consider the inclined cylinder shown in Figure 36.13. The direction of the flow makes an angle

of u with the cylinder. Often, only the fluid force in the normal direction is considered. The normal

component is given by

f n ¼

CDrDðwn 2 vnÞlwn 2 vnl þ CMrp

w_ n ð36:27Þ

where the superscript is used for the normal component. The term, wn 2 vn; is the normal component of

the relative velocity of the fluid with respect to the structure. Suppose that fluid is flowing to the right,

36-18 Vibration and Shock Handbook

and the cylinder is also moving to the right, as

shown in Figure 36.12. The normal components of

the fluid and cylinder velocities are

wn ¼ lwl cos u; vn ¼ lvl cos u

In three dimensions, it may be difficult to picture

what the normal component should be. Here, we

can find the normal component using the formula

ðwn 2 vnÞn~ ¼ ~t £ ðw~ 2 v~Þ £ ~t ð36:28Þ

where ~t is the unit vector tangent to the cylinder

and n~ is the unit vector normal to the cylinder.

Note that the normal direction depends on the direction of the flow as well as the inclination of

the cylinder.

In some cases, the tangential drag force may be included, and it can be written as

f t ¼

CTrDðwt 2 vtÞlwt 2 vtl ð36:29Þ

where CT is the tangential drag coefficient. Note that CT is usually a very small number.

The normal component of the fluid force is more dominant than the tangential component. It may

seem strange that the fluid force does not act in the direction of the fluid motion. Instead, the force is

predominantly in the normal direction defined by Equation 36.28. In Section 36.3.1, we will demonstrate

what this means by considering a towing cable.

36.2.2.1.1 Determination of Fluid Coefficients

The drag, inertia, and added mass coefficients must be obtained by experiment. However, for a long

cylinder, CM approaches its theoretical limiting value (uniformly accelerated inviscid flow) of 2, and CA

approaches unity (Lamb, 1945; Wilson, 1984). In reality, the inertia and drag coefficients are functions of

at least three parameters (Wilson, 1984):

CM ¼ CMðRe; K; cylinder roughnessÞ

CD ¼ CDðRe; K; cylinder roughnessÞ

where Re is the Reynolds number and K is the Keulegan – Carpenter number given by

Re ; rf UD

; K ; UT

D ð36:30Þ

where rf is the density of the fluid, U is the free stream velocity, D is the diameter of the structure, m is the

dynamic or absolute viscosity, and T is the wave period.

Sarpkaya looked at the variation of these hydrodynamic coefficients extensively and obtained the

plots shown in Figure 36.13 to Figure 36.15 (Sarpkaya, 1976; Sarpkaya et al., 1977). Figure 36.13 shows

the inertia and drag coefficients for a smooth cylinder as a function of K for various values of Re and

the reduced frequency b, defined by b ¼ Re=K: From this figure, we find that for low Re and b, the inertial

coefficient decreases and the drag coefficient increases at about 10 , K , 15: It is found that the drop and

the increase in these coefficients are due to shedding vortices, which also exert forces perpendicular to the

structure and the flow.

Figure 36.14 and Figure 36.15 show the inertia and drag coefficients for a rough cylinder, whose

roughness is measured by k=D: Figure 36.14a shows a drop in the drag coefficient for Re between

104 and 105, and this is called the “drag crisis.” For a larger Re, the drag coefficient stays constant.

As the surface becomes rougher, the drop occurs at lower Re and the drag coefficients for the

larger Re increases.

Figure 36.14 to Figure 36.16 can be used to obtain proper values of the drag and inertia coefficients for

fluid with known Re, Keulegan – Carpenter number, and cylinder roughness.

Direction of flow

Fluid velocity, w

Cylinder

velocity, v

FIGURE 36.12 Inclined cylinder.

Fluid-Induced Vibration 36-19

36.2.3 Vortex-Induced Vibration

When the flow passes around a fixed cylinder, for a very low Re ð0 , Re , 4Þ; the flow separates and

reunites smoothly. When the Re is between 4 and 40, eddies are formed and are attached to the

downstream side of cylinder. They are stable and there is no oscillation in the flow. For a flow with a

Reynolds number greater than about 40, the fluid near the cylinder starts to oscillate due to shedding

vortices. These shedding vortices exert an oscillatory force on the cylinder in the direction perpendicular

to both the flow and the structure. The frequency of oscillation is related to the nondimensionalized

parameter, the Strouhal number, defined by

St ¼

fv D

U ð36:31Þ

where fv is the frequency of oscillation, U is the steady velocity of the flow, and D is the diameter of

the cylinder. For circular cylinders, the Strouhal number stays roughly at 0.22 for laminar flow

ð103 , Re , 2 £ 105Þ and 0.3 for turbulent flow (Patel, 1989).

The lift force due to these shedding vortices can be written as

fL ¼

CLrAf U2 cos 2pfv t ð36:32Þ

where CL is the lift coefficient, which is also a function of Re, K, and the surface roughness.

The experimental data of the lift coefficients show considerable scatter with typical values ranging

from 0.25 to 1. For smooth cylinders, the lift coefficient approaches about 0.25 as Re and K increase.

2.5

0.3

(a)

0.4

0.5

8370

5260

4480

3123

1985

1107 784

b = 497

Re × 10−3 = 10

Re × 10−3 = 60

15 20 30 40 50

150

CD 1.0

1.5

2.0

3.0

3 5 10 50

150 200 5260 4480 31231985

1107784

30 40 60 80 100 20

100 150 200

0.4

0.5

1.0

1.5

2.0

3.0

3 5 10

50 100 150 200

(b)

FIGURE 36.13 Drag and inertia coefficients as functions of K for various values of Re and b. (Source: Sarpkaya,

1976, Proceedings of the Eighth Offshore Technology Conference. With permission.)

36-20 Vibration and Shock Handbook

It should be noted that the vortex forces are not generally correlated on the entire cylinder length. That is,

the phase of the vortex shedding forces varies over the length. The correlation length — the length over

which vortex shedding is synchronized — for a stationary cylinder is about three to seven diameters for

laminar flow. If sectional forces are randomly phased, the net effect will be small. The total force on a

cylinder of length L will be only a fraction of LfL: This fraction is called the joint acceptance and depends

on the ratio of the correlation length to the total length.

When the flow passes by a cylinder that is free to vibrate, the shedding frequency is also controlled by

the movement of the cylinder. When the shedding frequency is close to the first natural frequency of the

cylinder (^ 25 to 30% of the natural frequency [Sarpkaya and Isaacson, 1981]), the cylinder takes control

of the vortex shedding. The vortices will shed at the natural frequency instead of at the frequency

determined by the Strouhal number. This is called lock-in or synchronization, which is a result of

nonlinear interaction between the oscillation of the body and the action of the fluid. Figure 36.16 shows

the shedding frequency, as a function of flow velocity in the presence of a structure. f1 and f2 are the

natural frequencies of the structure.

The amplitude of the structural response and the range of the fluid velocity over which the lock-in

phenomenon persists are functions of a reduced damping parameter — the ratio of the damping force to

the exciting force (Vandiver, 1985, 1993). If the reduced damping parameter is small, the lock-in can

persist over a greater range of flow velocity.

0.1

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1.9

2.0

1.8

1.6

1.4

1.2

1.0

0.1 0.5 1 5 7

0.5 1 5 7

Re × 10 −5

k /D = 1/50

1/100

1/200

1/ 200

1/400

1/ 800

smooth

(a)

(b)

FIGURE 36.14 Drag and inertia coefficients for a rough cylinder as functions of Re for various values of cylinder

roughness (as measured by k=D) for K ¼ 20: (Source: Sarpkaya et al., 1977, Proceedings of the Ninth Offshore

Technology Conference. With permission.)

Fluid-Induced Vibration 36-21

The existing models for vortex-induced

oscillation for a rigid cylinder include singledegree-

of-freedom models and coupled models.

The single-DoF models assume that the effect of

vortex shedding is an external forcing function,

which is not affected by the motion of the body.

The coupled models assume that the equations

that govern the motion of the structure and the lift

coefficients are coupled so that the fluid and the

structure affect each other (Billah, 1989).

36.2.4 Summary

Some of the fluid forces are discussed briefly, and

the regimes where inertia, drag, and diffraction

forces are important are shown as functions of

the ratio of the structural diameter to the wave

0.2 0.5 1 5 10 15

2.0

1.8

1.6

1.4

1.2

1.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

Re × 10−5

k/D = 1/50

k /D = 1/50

1/100

1/200

1/ 200

1/400

1/800

smooth

(a)

(b)

FIGURE 36.15 Drag and inertia coefficients for a rough cylinder as functions of Re for various values of cylinder

roughness (as measured by k=D) for K ¼ 60: (Source: Sarpkaya et al., 1977, Proceedings of the Ninth Offshore

Technology Conference. With permission.)

fv = StU

FIGURE 36.16 An example of fluid elastic resonance.

36-22 Vibration and Shock Handbook

length, D=l; and the ratio of the wave height to the structural diameter, H=D: The wave forces on small

structures are modeled by the Morison equation, and it is valid for D=l , 0:2 and H=D . 0:63 or

thereabouts. The Morison equation includes the effects of added mass, inertia, and drag. The added

mass is simply

MA ¼ CAMdisp

For a cylinder with transverse velocity, v, the normal and the tangential components of the drag and

the inertia forces are given by

f n ¼

CDrDðwn 2 vnÞlwn 2 vnl þ CMrp

w_ n

f t ¼

CTrDðwt 2 vtÞlwt 2 vtl

The fluid coefficients are at least functions of three parameters: the Reynolds number, the Keulegan –

Carpenter number, and the cylinder roughness. The plots of these coefficients are reproduced in

Figure 36.13 to Figure 36.15.

The frequency of the lift force that is exerted by shedding vortices is closely related to the Strouhal

number given by

St ¼

fv D

The lift force due to these shedding vortices can be written as

fL ¼

CLrAf U 2 cos 2pfv t

If the structure is free to vibrate, then lock-in or synchronization may occur when the shedding frequency

is close to the structure’s natural frequency. The structure takes control of the vortex shedding. Many

nonlinear models are available to capture this phenomenon.

Пресс-релиз популярных книг

36.2 Fluid Forces

Популярные книги

Популярные статьи

Навигация