2.8 Hydrostatic Force on a Plane Surface

Recall that

The normal force, Fn, acting on the submerged surface can be

determined by integrating the pressure as:

dA

dFp n

dA pFn

Ahp

ApdA pF

atm

0

p constp

n

0

From force diagram,

Resultant Force:

That is, the force on a plane

surface caused by a uniform

pressure is equal to the weight

of the volume of liquid above

the surface.

From now on, the contribution

of patm will be excluded in force

camputation caused by the

pressure.

Fn=poA

Fout=patmA

hAFFF outnR

Next consider the more

general case of inclined

surface.

where yc is y-coordinate

of the centroid measured

from x-axis

A y sin

dA y sin

dAsiny

hdAF

c

A

A

AR

A

dA yy A

c

Note: The pressure is always acting along the line perpendicular to the surface.

Rewriting

where

The magnitude of the

resultant force is equal

to the pressure at the

centroid of the area

multiplied by the total

area.

A y sin F cR

A h F cR

sinyh cc

Next let’s determine the

point of action of the

resultant force, called the

center of pressure(CP).

The y-coordinate, yR of

the center of pressure is

determined by of

moments around the x-

axis.

A

2

A

ARR

dA y sin dA siny y

dA p yyF

A sin y F cR

Hence,

where Ix is the second moment of the area (moment of inertia) w.h.t. x-axis

using the parallel axis

theorem,

Ay

I

Ay

dAyy

c

x

c

A

2

R

A

2

x dAyI

2

cxcx y AII

c

c

xc

R yA y

Iy

In summary,

Thus, the resultant force

does not pass through the

centroid but is always

below it.

c

c

xc

R yA y

Iy

A h F cR

The x-coordinate, yR of

the CP is determined by

of moments around the

y-axis.

Hence,

AA

ARR

dA y xsin dA siny x

dA p xxF

Ay

I

Ay

xydAx

c

xy

c

AR

cc

xycR x

Ay

Ix

Ex. 2.6

Ex. 2.7

36

abI

3

xc

2.9 Pressure Prism (Alternative Way of Sect. 2.8)

The magnitude of the resultant force acting on the surface

is equal to the volume of the pressure prism.

The resultant force must pass through the centroid of the

pressure prism.

Ah2

1bhh

2

1 prism pressure of volumeFR

21R FFF

2

hhhh

2

hhhhh

hhhh2

1hhh

2112

12112

1212121

FR = Area of Trapezoid

21

22

12211221R

hh2

hhhh2

hhhh2

1F

Ex. 2.8

2

1FR

1air1 hpp 2air2 hpp

Hoover Dam

- Highest concrete arch-gravity

dam in US

- Depth = 715 ft

- P715 ft = 310 psi

- Thickness at top = 45 ft

- Thickness at bottom = 660 ft

2.10 Hydrostatic Force on a Curved Surface

The x-component is the

same as the hydrostatic

force of the horizontal

projected area, Ax.

The line of action passes

through the center of

pressure of Ax.

x

dA of area projection horizontal :

dAcosdA

x

pdA cospdA

cosdFdF

x

xx,cxx AhpdAF

Ax: Projection area along x-dir.

Hc,x

Fx

The vertical force is equal to

the weight of fluid above the

surface.

The line of action of the

vertical component of the

force is through the centroid

of the volume, V’.

yy pdA sinpdAsindFdF

VydApdAF yyy

where V’ is the volume between the

curved surface and the free surface.

Resultant Force, FR

Fx

Fy

2

y

2

xR FFF

Ex. 2.9

p=H

F1

Pop Bottle

- If Pcoke gas = 40 psi,

Fexerted on the bottom surface = 580 lb

2.11 Buoyancy, Floatation, and Stability

Archimedes’ Principle

- Buoyancy force is caused by

the imbalance of pressures

on the upper and lower

surfaces.

y

y21

ldA

dAppdF

VdVldAF y

(1st)

Buoyant force on a body submerged in a fluid is equal to the

weight of the fluid displaced by the body.

The line of action of the buoyant force passes through the

centroid of the displaced volume. This centroid is called the

center of buoyancy.

Archimedes’ first principle

of buoyancy (287-212 B.C.)

“Eureka”

Archimedes’ Principle (2nd):

For a floating body,

where f is the specific weight

of the fluid and Vs is the

submerged volume.

A floating body displaces a

volume of fluid equivalent to

its own weight.

sf VW

214-class submarine

Flow Analysis of SubmarinePNU ME CFD LAB.

Angle of Attack = 0o

Angle of Attack = 10o

Angle of Attack = 20o

Angle of Attack = 30o

Yawing Angle = 10o 20o

Yawing Angle = 30o

Variation of Angle of Attack(+) : Surfacing

Pressure Contour in Dynamic Motion

Variation of Angle of Attack(+) : Surfacing

Limiting Streamlines in Dynamic Motion

Variation of Angle of Attack(-) : Voyaging

Pressure Contour in Dynamic Motion

Variation of Angle of Attack(-) : Voyaging

Limiting Streamlines in Dynamic Motion

Variation of Yaw Angle(+) : Turning leftward

Pressure Contour in Dynamic Motion

Variation of Yaw Angle(+) : Turning leftward

Limiting Streamlines in Dynamic Motion

Variation of Yaw Angle(-) : Turning rightward

Pressure Contour in Dynamic Motion

Variation of Yaw Angle(-) : Turning rightward

Limiting Streamlines in Dynamic Motion

Neutral Buoyancy

• Research is on going by PNU CFD lab.

Torpedo Propulsor

Limiting Streamlines (Angle of Attack : 10˚)

Limiting Streamlines (Angle of Attack : 30˚)

EX. 2.10

Cartesian Diver

- By pressing the bottle, the

pressure within it is

increased and the air within

the inverted tube is

compressed.

- Then, the additional air

enters into the test tube,

thereby the weight of the

tube to be greater than that

of the surrounding water.

- The tube sinks!!!

Archimedes may have used

mirrors acting as a parabolic

reflector to burn ships

attacking Syracuse(214–

212 BC).

Archimedes’ Screw : can raise the water efficiently

Stability

M: Metacenter

2.12 Pressure Variation in a Fluid with Rigid-Body Motion

Recall that the equation of motion for a fluid in which there are no shear stress.

akp

xax

p

yay

p

zaz

p

Linear Motion (when ax=0)

0x

p

yay

p

zaz

p

Total pressure gradient:

dzagdyadzz

pdy

y

pdx

x

pdp zy

- If ay and az = const, integration yields to

where p0 is the pressure at y=z=0.

- The shape of constant pressure surface is obtained by setting

p=constant.

- So, surface of constant pressure are planes with slope equal to

0zy pzagyap

constantzagya zy

z

y

ag

a

dy

dz

(This slope of constant pressure is also obtained by setting dp=0)

Rewriting,

- Pressure:

- Shape of constant pressure surface:

- Slope equal of constant pressure

surface:

0zy pzagyap

constantzagya zy

z

y

ag

a

dy

dz

- If az is also zero, the free surface has the slope of –ay/g and

the pressure distribution in z-direction is hydrostaic by

0y pgzyap

Ex. 2.11

Rigid-Body Rotation

- Gradient and total derivative in cylindrical coordinate system:

aeez

pe

p

r

1e

r

pp zzr

dzz

prd

p

r

1dr

r

pdp

- Pressure gradient in each direction:

- Total pressure derivative:

22r rra

r

p

0y

p

z

p

dzdrrdzz

pdr

r

pdp 2

0

22 pzr2

1p

- If p is constant,

or, obtained from dp=0,

i.e.,

constantg2

rz

22

g

r

dr

dz 2

constg2

rz

22

Ex. 2.12

constantg2

rz

22

h=h0 at r=0