Upload
jasper-thornton
View
243
Download
0
Embed Size (px)
Citation preview
1
White: Chapter 2
潘锦珊: 第一章
Chapter 2
Pressure Distributionin a fluid
(Fluid Statics Basic)
2
A
Fp
A
lim
0Definition:
Unit:2mNPa (SI)
2cmkg atm
PSI
mmHg
OmmH2(Pound per Squire Inch)
Vertical to the surface and point into it.
At any point, pressure is independent of orientation.
Properties of Pressure
Pressure
3
At any point in a static fluid, pressure is independent of orientation.
Verification:
zgpp
pp
nz
nx
2
1
0zWhen pppp nzx
np
y
x
z
xzp
xp
)2
1( zxbgG
(up)
sin
sin
ybpybp nx
sin
ybpP nn
ybpP xx
Forces on left and up surface:
:x
4
Fluid Mechanics
AerodynamicsFluid at rest
Fluid Statics
Fluid Dynamics
Fluid in motion
5
Pressure is the only surface force.
Pressure distribution relates to body force only.
Dams(水坝) Buoyancy related instrument(利用浮力的装置) Fluid power system(液压驱动系统)
Connected vessel (连通器) ……
Applications:
§ 2.1 Fluid @ rest
6
Consider a cube in a static fluid
Pressure at the center is p;
Body forces are kZjYiXR
dxdy
dzrightpleftp p
2
2dx
x
ppp
dx
x
ppp
left
left
§ 2.2 Equilibrium of a Fluid Element
7dx
dy
dzXdxdydz
Zdxdydz
Ydxdydz
dxdy
dz 2
dx
x
pp
2
dx
x
pp
p
Pressure:
Body force:
8
022
Xdxdydzdydzdx
x
ppdydz
dx
x
pp
In x direction:
Force on left surface
Force on right surface
Body force in x direction
Xx
p
Zz
p
Yy
p
Euler Equilibrium Equations
(Euler 1775)
)( ZdzYdyXdxdp
9
Zz
p
Yy
p
Xx
p
Pressure increase in the direction of body force.
Surfaces in fluid with same pressure, vertical to body force everywhere, in gravity field it is a horizontal plane.
Equipressure surface(等压面)
Rp
)( kZjYiXkz
pj
y
pi
x
p
10
)( ZdzYdyXdxdp
x
z
z
hz0
p0
Zdzdp )( gZ gdz
1 Basic rule:
Cgzp General solution
2 Boundary condition:
00 , ppzz 00 gzpC
)( 00 zzgpp
hpghpp 00
§ 2.3 Pressure Distribution under Gravity
11
ghphpp 00
Pressure at free surface
Pressure due to weight on top
Any point with the same depth h under free surface has the same pressure.
equipressure surface(等压面)
Free surface is an equipressure surface
12
p0
Water manometer(水柱压力计)
p
h
Pressure source
Connected water tube
Application ?
Absolute pressure (绝对压力)
Relative pressure(相对压力)
Gauge pressure(表压力)
Vacuum degree(真空度)
Pressure measurement
h (mmH2O)
h+p0 (mmH2O) pA 绝压
pG 表压
13
P2.7
Homework:
14
AA
AhAppP dd 0
sinyh
dAhppdAdP )( 0
AA
AyApAyApP dsindsin 00
A
ydAy
A
xdAx A
cA
c
, Centroid(形心,重心)
AhApAyApP cc 00 sin
hc
y cC
h
y
x
A
α
y
x
p0
Find total force P
§ 2.4 Hydrostatic Force on Plane Surface
15
AhpP c )( 0 hc: depth of centroid
The force on a submerged plane equals the pressure at the plate center times the plate area, independent of the shape of the plate or the angle .
Center of pressure
Is the center of pressure at centroid ?
hc
y cC
hd
y dD
h
y
x
A
α
y
x
p0
16
Ay
Iyy
C
CCD
A
ydAayp )sin( 0
A
c dAyAyp 20 sin
AyIA 2 AyII CC
2
hc
y cC
hd
y dD
h
y
x
A
α
y
xp0
AA
ydAhpydp )( 0
Moment to x axis
17
Example
The gate is 5m wide,is hinged at point B,and rest against a smooth wall at point A.
Find (a) The force on the gate
exerted by seawater pressure,
(b) The horizontal force Px exerted by the wall at point A
m8
m6
A
B
mN
210050
m15
cxP
18
AhF c
1051210050
N1003.6 6
(a) Centroid: 3m above B
Solution:
3
12
1bLIc
AL
ILL
c
ccd
6 xd PFL(b)
B
PxA
cc p
F
L c
LNPx 106059.4 6
m8
m6
A
B
mN
210050
m15
cxP
19
Hoover Dam Channel
1. Select a dA and find the three forces on it
2. Integration
§ 2.5 Hydrostatic Forces on Curved Surfaces
20
Conclusion:
x
1. Horizontal forces:
xcx Ahpp )( 0
xO
z
AAx
2. Vertical forces:
VApp zz 0
O
z
V
21
Example:
Find the forces acting on the hemi-spherical covers.
R
F
O x
y
H
45o
Solution:
2
2
2gHRAHF xx
sy gVF
FFF yX22
F
F
x
ytan 1
22
2
245sin RRAx
45cos3
2 23 HRRVs
22
§ 2.6.1 Uniform Linear Acceleration (恒加速度直线运动)
aX= -a
g
x
GravityBody force
Inertia Force
§ 2.6 Fluid in rigid body motion
23
a
z
g
xa
x
xagdzdxax
Boundary condition:
0,0 ppzx
)(0 gzxapp x
0pp At free surface(自由液面) xg
az x
Equipressure surface(等压面) Cxg
az x
Euler Equilibrium Equations
)( ZdzXdxdp
24
A cup of coffee is 7cm deep at rest.
1. Whether it will spill out while ax=7m/s2?
2. Gage pressure of point A?
Example:
cmD
z 14.2tan2
g
axtanSolution:
AA ghP
31010 mkg
Pa906
)0214.007.0(8.91010
Az
cm6
cm7
cm3
g
27 smax
xaIt will not spill out !
25
§ 2.6.2 Rigid body rotation (整体旋转)
g
ω2r
z
f
O
ω2y
ω
x
y
R
O
f
ω2xry
xθ
gZyYxX ,, 22
Body force:
)( ZdzYdyXdxdp
)( 22 gdzydyxdx
Equipressure surface: dp=0
022 gdzydyxdx
Cgzyx )(2
1 222
26
Cyxg
z )(2
1 222 Crg
z 22
2
1
Parabola dish (抛物面)
Free surface: 0,0 zzr s
022
2
1zr
gzs g
ω2r
z
f
O
z0
How to find z0?
旋转抛物面体的体积是同底面积和高的圆柱体积的一半。
27
P2.64
P2.97 (selective)
P2.147
P2.152
Homework: