工程流体力学03第三章

For only one inlet and one outlet According to continuity
mout min m
d (mV ) s F m(Vout Vin ) dt
2－out, 1－ in

Fx m(V 2 x V 1x)
y
Fy m(V 2 y V 1 y )
surface(CS)
Basic Laws for system
3.3 The Reynolds Transport Theorem (RTT) 雷诺输运定理
for CV
1122 is CV . 1*1*2*2* is system which occupies the CV at instant t.
Homework: P185 P3.12, P189P3.36
3.4 The Linear Momentum Equation (动量方程) （ Newton’s Second Law ）
ds ( i AiV i )out ( i AiV i )in dt i i
Fz m(V 2 z V 1z )
o
z
x
Example: A fixed control volume of a streamtube in steady flow has a uniform inlet (1,A1,V1 )and a uniform exit (2,A2,V2) . Find the net force on the control volume.
If there are several one-D inlets and outlets :
d s ( i i AV i i ) out ( i i AV i i )in dt i i
Steady , 1-D only in inlets and outlets, no matter how the flow is within the CV .
V 1) mV 1 F sy p1 A1 m(0 F sy p1 A1 mV 1 -4642N
1-D flow : is only the function of s .
( s)
(d )in ( dm)in ( Ads)in ( AVdt )in
In the like manner
(d )out ( AVdt )out
s
ds
t t+dt t t+dt
d s d cv 1 [(d ) out ( d )in] dt dt dt
d cv [( AV )out ( AV )in ] dt
For steady flow :
d cv 0 dt
RTT

ds ( AV ) out ( AV )in dt
Chapter 3 Integral Relations（积分关系式） for a Control Volume in One-dimensional Steady Flows
3.1 Systems (体系) versus Control Volumes (控制体)
System：an arbitrary quantity of mass of fixed identity. Everything external to this system is denoted by the term surroundings, and the system is separated from its surroundings by it‘s boundaries through which no mass across. (Lagrange 拉格朗日) Control Volume (CV): In the neighborhood of our product the fluid forms the environment whose effect on our product we wish to know. This specific region is called control volume, with open boundaries through which mass, momentum and energy are allowed to across. (Euler 欧拉) Fixed CV, moving CV, deforming CV
p1 p 2 4.19 105 N
m 78.5 Kg
998 Kg
s
3
m d 1 10cm, d 2 8cm
2
1
2
m
Neglect the weight of the fluid. Find the force on the water by the elbow pipe. Solution: select coordinate ,control volume
s
t t+dt
t
t+dt
: any property of fluid (m, mV , H , E)
d dm
:The amount of
per unit mass
is :
The total amount of
in the CV
CV cv d cv dm
d ( CV ) dt
p1 2.05 105 N / m2
T1 =865K，V1=288 m/s，A1=0.19㎡；
At the outlet p2 1.143105 N / m2 T2 =766K，A2=0.1538㎡ Please find the mass flux and velocity at the outlet. Given
i
(m )
i
i in
( m i )out
i
Mass flux (质量流量 m )
For incompressible flow:
( A V )
i i
i out
( AiV i )in
i
Qi AiV i Volume flux
体积流量
If only one inlet and one outlet
1-D in & out steady RTT
mV
dmV V dm
(linear momentum)
momentum perunit mass
flux
d (mV ) s ( i AiV iV i )out ( i AiV iV i)in (mi V i )out (mi V i )in i i dt i i
s
t t+dt
1 [ CV (t dt ) CV (t )] dt
t
t+dt
源自文库
1 1 [ s (t dt ) (d )out (d )in] s (t ) dt dt 1 1 [ s (t dt ) s (t )] [(d )out ( d )in] dt dt d s 1 [(d )out (d )in] dt dt d s d cv 1 [(d )out (d )in] dt dt dt
Solution:
F m(V 2 V 1)
m 1 A1V 1 2 A2V 2
2
V1
V2
F x m(V2 x V1x ) m(V 2 V 1 cos )
F y m(V2 y V1y ) mV 1 sin
1
y

o
x
Example:
Given

m const
or
dm 0 dt
dV d ( mV ) F ma m dt dt dH M H (r V ) m dt
dQ dW dE dt dt dt
It is rare that we wish to follow the ultimate path of a specific particle of fluid. Instead it is likely that the fluid forms the environment whose effect on our product we wish to know, such as how an airplane is affected by the surrounding air, how a ship is affected by the surrounding water. This requires that the basic laws be rewritten to apply to a specific region in the neighbored of our product namely a control volume ( CV). The boundary of the CV is called control
3.2 Basic Physical Laws of Fluid Mechanics
All the laws of mechanics are written for a system, which state what happens when there is an interaction between the system and it’s surroundings. If m is the mass of the system Conservation of mass(质量守恒) Newton’s second law Angular momentum First law of thermodynamic
3.3 Conservation of mass (质量守恒)
(Continuity Equation)
f=m
dm/dm=1
dms ( i AiV i )out ( i AiV i)in 0 dt i i
( A V
i i i
i out
)
( i AiV i )in
A1V 1 A2V 2
-------Leonardo da Vinci in 1500
壶口瀑布是我国著名的第二大瀑布。两百多米宽的黄河河面，突然紧缩 为50米左右，跌入30多米的壶形峡谷。入壶之水，奔腾咆哮，势如奔马，浪 声震天，声闻十里。 “黄河之水天上来”之惊心动魄的景观。
Example: A jet engine working at design condition. At the inlet of the nozzle
Newton’s second law
d (mV ) s (m V )out (m V )in i i i i F dt i i
mi V i ：Momentum flux (动量流量)
F
:Net force on the system or CV (体系或控制体受到的合外力)
F m(V 2 V 1)
F x m(V 2 x V 1x ) m V 2


2 1
F sx p 2 A2 mV 2
2 2 4 78.5 d 2 F sx p 2 A2 mV 2 3696 N p 2 2 998 d 2 4

y

o
x
In the like manner
R=287.4 J/kg.K。
gas constant
Solution
According to the conservation of mass
p p1 AV 1 1 AV 45.1 kg / s m AV RT RT1
p1 AV p2 A2V2 1 1 m 1 AV A V RT RT 1 1 2 2 2 1 2 A1 p1 T2 V2 V1 A p T 565.1 m / s 2 2 1