Post on 16-Mar-2020
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