武科大流体力学Chapter3.ppt

Chapter 3. FLUID FLOW
CONCEPTS & BASIC EQUATIONS
第三章流体流动概念和基本方程组
双语课程

2
2023/6/1
The relations between the fluid motion and the
forces on it will be treated in this chapter.
Objective:
1) to build the basic equations with ideal fluid;
2) to resort to the data of experimentation to revise
the solutions;
3) to obtain the results to go with the actual situation.
Abstract

3
2023/6/1
3.1 Flow Characteristics
3.2 The Concepts
3.3 Reynolds Transport Equation #
3.4 Continuity Equation ++
3.5 The Bernoulli Equation ++
3.6 Flow Losses & Steady Flow Energy Equation
3.7 Application of the Energy Equation ++
3.8 Applications of the Momentum Equation ++
Home Work
Outline

4
2023/6/1
3.1 FLOW CHARACTERISTICS
3.1 流动特性

5
2023/6/1
The space pervaded (弥漫,充满) the flowing
fluid is called flow field (流场).
 velocity u,
 acceleration a,
 density ,
 pressure p,
 temperature T,
 viscosity force Fv , and so on.
Motion parameters:

6
2023/6/1
3.1.1 The method to describe fluid motion
Lagrangian viewpoint and Eulerian description are
the two main approaches to describe fluid motion.
 Lagrange method focus on the fluid particle
(流体质点) or fluid parcel (流体微团).
 Eu1er method focuses on the spatial location
of flow field (流场).
Eu1er method is used in our textbook.

7
2023/6/1
Velocity u=u(x, y, z, t) (3.1)
Pressure p=p(x, y, z, t) (3.2)
Acceleration
t
z
u
y
u
x
u
dt
d
z
y
x













u
u
u
u
u
a (3.3)
For a fluid particle
substantial derivative (质点导数) or
material acceleration (质点加速度)

8
2023/6/1
t
u
z
u
u
y
u
u
x
u
u
dt
du
a i
i
z
i
y
i
x
i
i












 (3.3a)
convective acceleration (迁移加速度): the
difference in velocities of different fluid
particles at different spatial position.
local acceleration (当地加速度) or time-varying
acceleration (时变加速度): the rate of the velocity
of a fluid particle u in a given point with respect to
time t in the direction i.
Component form:

9
2023/6/1
3.1.2 Laminar flow and turbulent flow
Reynolds (雷诺), a British scientist, showed that there are two
basic types of fluid flow: laminar and turbulent flow in 1883.
laminar flow (层流):
The fluid flows orderly.
One layer of fluid particle moves smoothly over another layer.
The particles are not mixed with each other.
The viscous force meets the Newton’s law of viscosity,v=du/dy.
Turbulent flow (紊流, 湍流)：
Very complicated and irregular motion trajectories;
There is transverse motion as well;
The viscous frictions of turbulent flow consist of not only the
viscous friction but also the turbulent shear resistance, t=uxuy

10
2023/6/1
3.1.3 Steady flow and unsteady flow
For steady flow (定常流), motion parameters independent of
time.
u=u(x, y, z)
p=p(x, y, z)
Steady flow may be expressed as
The motion parameters are dependent on time, the flow is
unsteady flow (非定常流).
0



t

0



t
u
0



t
p
0



t
T
u=u(x, y, z, t)
p=p(x, y, z, t)

11
2023/6/1
a) steady flow b) unsteady flow
Figure 3.1 Steady flow and unsteady flow
0
0
H
water supply
overfall(溢流道)
orifice(孔口)
0
0
H
3
H
2
H
1
t1
t2
t3
orifice
Example
The velocity and
pressure in the
orifice (孔口) do
not vary with time
and the shape of
effluent remains a
constant jet of flow.
The flow in orifice
is variable with
the water level, in
which the velocity
and the pressure
both vary with
time.

12
2023/6/1
When the fluctuations (波动) in a turbulent flow are
small, the instantaneous velocity(瞬时速度) can be
replaced with the mean value in a span of time. The
mean value is called temporal mean velocity (时均速度),
3.1.4 Temporal mean（时间平均）
0
1 t
u udt
t
  (3.4)
So, the flow can be treated as a steady flow.

13
2023/6/1
A uniform flow (均匀流) is the one in which all velocity
vectors (direction and magnitude) are identically the same
everywhere in a flow field for any given instant.
Flow such that the velocity varies from place to place at any
instant is nonuniform flow (非均匀流) .
3.1.5 Uniform flow and nonuniform flow

14
2023/6/1
3.1.6 Rotational and irrotational
a) irrotational b) rotational
Figure 3.2 Rotational and irrotational
In the fluid parcel two
marked infinitesimal
[,infini’tesəməl] （无
限小）line elements
that are at right angles
（直角）to each other
are as shown in Fig. 3.2.
y
x
y
x
0 0
If the fluid particles rotate with any axis, the flow was said
to be rotational flow, or vortex flow (涡旋流); otherwise it is
irrotational flow (无旋流).

15
2023/6/1
3.2 THE CONCEPTS
3.2 概念

16
2023/6/1
One dimension——all main variables in the flow field can be
completely specified by a single coordinate if the variation of flow
parameters transverse to the mainstream direction can be neglected.
Flow in parallel planes is two-dimensional because the
motion of fluid particle normal to these planes is restricted for the
existence of the planes.
Three-dimensional flow is the most general flow.
3.2.1 Flow dimensionality （流的维度）

17
2023/6/1
A path line (迹线, 轨迹线) is the trajectory of an individual
fluid particle in flow field during a period of time.
Streamline (流线) is a continuous line (many different fluid
particles) drawn within fluid flied at a certain instant, the
direction of the velocity vector at each point is coincided with
（与…一致） the direction of tangent at that point in the line.
3.2.2 Path line and streamline
path line Streamline

18
2023/6/1
1) In unsteady flow, the direction and shape of streamline is
variable with time.
2) In steady flow, the streamline keeps steady all along; hence
the streamline coincides with the path of particle.
3) streamline can not be intersected (相交, 交叉) or replicated
(折转, 反折).
1. Characteristics of streamline
a
a
b
b
1
1
a
a
b
b
1
1
a
a
b
b
1
1
a) Gate valve b) Sudden enlargement tube c) cylinder
Figure 3.3 Flow patterns (流线谱) in various boundary conditions
Fluid moves along the boundary.

19
2023/6/1
z
y
x u
dz
u
dy
u
dx


For example, a known plane flow field:
2
2
y
x
ky
ux


 2
2
y
x
kx
uy

 0

z
u
x
dy
y
dx


making a integral,
x2+y2=C
2. Differential equation of streamline
Various streamlines can be obtained by assigning different constant
C, and the streamlines are a cluster of circles in various radii.

20
2023/6/1
1. Stream tube (流管)
The stream tube is composed of the streamlines passing through
every point in a closed curve (not the streamline) C0 which is
drawn in the flow field.
2. Mini-stream tube (微小流管)
The stream tube with an infinitesimal section is said to be mini-
stream tube. Streamline is the extreme case of mini-stream tube.
3. Total flow (总流)
Total of countless mini-stream
tubes is called total flow.
3.2.3 Stream tube, mini- stream tube, total flow
Figure 3.4 Stream tube
C0
y
z
0
x

21
2023/6/1
3.2.4 Cross section, flow rate and average velocity
1. Flow section (通流面)
The flow section is a section that every area element in the
section is normal to mini-stream tube or streamline. The flow
section is a curved surface（曲面）. If the flow section is a
plane area（平面）, it is called a cross section (横截面).
1
1
2
2
I
II
u1
u2
dA1
dA2
Figure 3.5 Flow section

22
2023/6/1
The amount of fluid passing through a cross section in unit
interval is called flow rate or discharge.
)
s
/
m
(
)
kg/s
(
)
N/s
(
3




G
M
Q
Q
M
gQ
G




weight flow rate
volumetric flow rate
mass flow rate

 A
udA
Q (3.7)
For a total flow
2. Flow rate (流量)

23
2023/6/1
3. Average velocity (平均速度)
umax
V
Figure 3.6 Distribution of velocity over cross section
A
udA Q
V
A A
 

The velocity u takes the maximum umax on the pipe axle and
the zero on the boundary as shown in Fig. 3.6.
The average velocity V according to the equivalency of flow
rate is called the section average velocity (截面平均速度).
According to the equivalency of flow rate, VA=∫A udA=Q,
therewith,

24
2023/6/1
A water pump yields Q=0.65m3/min. Find the average velocity if
inner diameter of water pipe is d=100mm and convert the
volumetric flow rate to the rate of mass flow and the rate of weight
flow.
(1) average velocity V
)
m/s
(
38
.
1
1
.
0
)
4
/
(
60
/
65
.
0
2





A
Q
V
(2) rate of mass flow M
M=ρQ =1000×0.65=650 (kg /min) =10.83 kg/s
(3) rate of weight flow G
The special weight of water  is 9810 N/m3, so
G=Q=9810×0.65=6376.5 (N/min) =106.3 N/s
EXAMPLE 3.1

25
2023/6/1
3.3 REYNOLDS TRANSPORT EQUATION
3.3 雷诺输运方程 #

26
2023/6/1
3.3.1 System
A system (系统) is a set of definite fluid particles selected in
the interest of researcher.
System boundary is the surroundings that distinguish the
set of particles (system) from the other material (environment)
and named it the surface of system.
( )
d m
dt
 
u
F (3.3.1)
The dynamics of a system can be expressed by
ΣF - the resultant of all external forces acting on the system;
m - the mass of the definite particles, a fixed value;
u - the velocity at the center of mass of the system.

27
2023/6/1
3.3.2 Control volume
Control volume (cv, 控制体) is defined as an invariably
hollow volume or frame fixed in space or moving with constant
velocity through which the fluid flows.
The boundary of control volume is called control surface
(cs, 控制面).
For a cv:
1) its shape, volume and its cs can not change with time.
2) it is stationary in the coordinate system. (in this book)
3) there may be the exchange of mass and energy on the cs.

28
2023/6/1
Name Definition Characters Method
System
a collection of fluid
matter of fixed identity
(i.e. always the same fluid
particle) that will move,
flow, and interact with its
surroundings. (closed)
 it’s shape, volume
and boundary can
change with time;
 it can move;
 has the exchanges
of energy, but no
mass.
Lagrangian
Control
Volume
(cv)
a geometrically defined
volume in space through
which fluid particles may
flow in or out of the
control volume. Hence it
will contain different fluid
particles at different
points in time. (opened)
 it’s shape, volume
and boundary can
not change;
 it is fixed;
 has the exchanges
of mass and energy.
Eulerian
System vs Control Volume

30
2023/6/1
3.3.3 Reynolds transport equation
Figure 3.7 A system flow through control volume
—— control surface ------- system boundary
(cv)
a) time t
y
x
z
o
b) time t1>t
u
n
cs
dA
d) Inflow area Ai
(adb)
u
dA
n
cs
c) Outflow area Ao
(acb)
cs
a
b
c
d

31
2023/6/1
N(t) is a term of a general property of the system, such as
mass, momentum and so on. The velocity of fluid particle at a
spatial point in cv at time t is represented by u(r, t), r=r(x, y, z)
is radius vector of spatial point relative to coordinate origin.
The general property per unit mass is η(r, t)=N(t)/m.
The increment of system in time (t1t) is

 )
(
)
( 1 t
N
t
N 
 2
1
)
,
( 1



 d
t
r 


3
1
)
,
(



 d
t
r

32
2023/6/1

dt
t
dN )
(  
1
1
1
1
1
lim ( ( , ) ( , ))
t t
t t d
t t 
   

 
 
 
 
  
 
 r r








 

2 3
1
)
,
(
)
,
(
1
lim 1
1  



 d
t
d
t
t
t
t
t
r
r
From Lagrange mean value theorem
)
(
)
,
(
)
,
(
)
,
( 1
)
(
1 1
t
t
t
t
t t
t
t 



 

 





r
r
r ( 0≤θ≤1)






 d
t
t)
,
(r
 
1
1
1
1
1
lim ( ( , ) ( , ))
t t
t t d
t t 
   

 
 
 
 
  
 
 r r




 t
t
t
N
t
N
dt
t
dN
t
t
1
1 )
(
)
(
lim
)
(
1








 
 

2
1 3
1
1
)
,
(
)
,
(
1
lim 1
1 
 




 d
t
d
t
t
t
t
t
r
r
When (t1−t) comes near zero, volume τ1 approaches to volumeτ.

33
2023/6/1




2
1
)
,
(
1
lim 1
1 

 d
t
t
t
t
t
r 1
1 1 1
1
1
lim ( , ) ( , ) ( )
o
o
t t
A
t t d t t
t t



  r u r A
( , ) ( , ) ( , ) ( , )( )
o o
o o
A A
t t d t t dA
 
 
 
r u r A r u r n
1
3
1
1
1
lim ( , )
t t
t d
t t 
 
   r ( , ) ( , )
i
i
A
t t d

  r u r A
Let A=Ai +Ao, so the net rate of flux (净流通率) is:

 o
)
,
(
)
,
( A
r
u
r d
t
t
o
A
 )
)
,
(
)
,
(
( i
A
r
u
r d
t
t
i
A

  A
r
u
r d
t
t
A
)
,
(
)
,
(

 

dt
t
dN )
(







d
t
t)]
,
(
[ r A
r
u
r d
t
t
A
)
,
(
)
,
(
 (3.13)
outer normal
unit vector
Outflow rate of flux (流出流通率):
Inflow rate of flux (流入流通率):
Reynolds Transport Equation (RTE):

34
2023/6/1
( ) ( , )
( , )[ ( , ) ]
cv cs
dN t t
dV t t dA
dt t

 

 

 
r
r u r n
The total rate of
change of an arbitary
generalized property
N(t) of the system
The time rate
of change of the
property N(t)
within the cv
The net rate
of flux of the
property N(t)
through the cs
在某一时刻 t ，系统中某一物理量随时间的变化率，
等于该瞬时与系统重合的控制体内所含同一物理量随时
间的变化率与相应物理量通过控制面的净流通率之和。
Meaning of RTE

35
2023/6/1
3.3.4 Application of the Reynolds transport equation
1. Continuity equation
Let N = m, than  = 1
According to the principle of conservation of mass (质量守
恒定律）, 0

dt
dm
From Eq. (3.13) 0
A
d d
t


 

  

  u A
For a steady flow 0
t




0



A
dA
u

So，
0
d
t




 


Namely the net rate of mass inflow and
outflow to the cs is zero.

36
2023/6/1
2. Equation of momentum
Let N = mu, than  =u.
From Eq. (3.9), (3.13)

 






A
d
d
t
dt
m
d
A
u
u
u
u
F )
(
)
(




In other words, the resultant external force acting on
a system is equal to the sum of variation of the
momentum within the cv and the net rate of flux of
momentum from the cs in unit time.

37
2023/6/1
3. Energy equation
QH —— input heat of a system;
W —— the work done by the system;
N1, N2 —— energy in the initial and final states of system
QH W = N2N1 (3.16)
Taking N =Ei + mu2/2, and then N per unit mass is =ei+u2/2,

 
























A
i
i
H
d
u
e
d
u
e
t
dt
dN
t
W
t
Q
A
u
2
2
2
2








The law of conservation of energy (能量守恒定律) is:
internal energy kinetic energy internal energy per unit mass

38
2023/6/1
For steady flow without energy input and output
 




 0
)
( d
t
Ws/t=0

t
QH


 











A
i d
u
e
gz
p
A
u

 2
2
Therewith
If the flow is adiabatic (绝热的), QH/t=0; then
0
2
2














A
i d
u
e
gz
p
A
u


So Const.
2
2



 i
e
p
u
gz

(3.22a)
δW= δWp+δWs +δWg
work by
pressure
work by
shear force
work by
mass force
(3.20)

39
2023/6/1
3.4 CONTINUITY EQUATION
3.4 连续性方程 (重点)

40
2023/6/1
As fluid is continuous medium, it is thought
that the whole space is occupied fully and
continuously with the fluid and without a point
source (点源) or a point sink (点汇) when the
fluid motion is dealt.
The continuous condition of fluid motion

41
2023/6/1
3.4.1 Steady flow continuity equation of 1D mini stream tube
dA1
u1
dA2
u2
A1
A2
Figure 3.8 One-dimensional stream tube
The net mass inflow
dM=(lu1dA12u2dA2) dt
For compressible steady flow dM=0
lu1dA1=2u2dA2
If incompressible, ρl =ρ2=ρ
u1dA1= u2dA2
The formula is the continuity equation for incompressible
fluid, steady flow along with mini-stream tube.
(3.23)

42
2023/6/1
3.4.2 Total flow continuity equation for 1D steady flow

  2
1 2
2
2
1
1
1 A
A
dA
u
dA
u 

lmV1A1=2mV2A2 (3.24)
For incompressible fluid flow, ρ is a constant.
Q1=Q2 or V1A1=V2A2
dA1
u1
dA2
u2
A1
A2
Figure 3.8 One-dimensional stream tube
Making integrals at both sides of Eq.(3.23)
Integrating it
average density
average velocity
The total flow continuity equation for
the incompressible fluid in steady flow.

43
2023/6/1
V1
V2
d
1
A
B
L
d
2
Figure 3.9 Two hydraulic cylinders
in series
Two hydraulic cylinders (液压缸) in a hydraulic system are shown in
Fig.3.9. The inner diameters of the two hydraulic cylinders are
d1=75mm and d2=125mm respectively. The velocity of piston A is
V1=0.5m/s. Find the velocity of piston B V2 and flow rate Q in the
pipe L.
/s)
(m
10
21
.
2
5
.
0
075
.
0
4
4
3
3
2
1
2
1









V
d
Q
Solution
(m/s)
0.18
125
.
0
10
21
.
2
4
4
/ 2
3
2
2
2 







d
Q
V
EXAMPLE 3.2

44
2023/6/1
3.4.3 Three-dimensional continuity equation *
* 不讲内容，只讲结论
y
z
o
y
u
 dx
x
u
u
y
y



)
(

y
x
x
z
m
dx
dy
dz
y
u
x
u
z
u
Figure 3.10 A hexahedron as control volume

45
2023/6/1
0
)
(
)
(
)
(












z
ρu
y
ρu
x
ρu
t
ρ z
y
x
(3.25)
If /t＝0
0
)
(
)
(
)
(









z
ρu
y
ρu
x
ρu z
y
x
0









z
u
y
u
x
u z
y
x
If ＝const.
(3.27) be used for both steady and unsteady incompressible flow.
ux/x represents the rate of linear deformation (线变率) of
fluid particle in the x direction.
(3.26)
(3.27)
(3.26) be used for compressible fluid in steady flow.
(3.25) be used for compressible fluid flow.

46
2023/6/1
3.5 THE BERNOULLI EQUATION
3.5 伯努利方程 (重点)

47
2023/6/1
3.5.1 Euler’s equation in one-dimensional flow for ideal fluid
x
z
p
s
s
p
p 



δA
A
s

g

Figure 3.11 A cylinder as control
volume
δ z














cos
cos
A
s
g
A
s
s
p
A
s
g
A
s
s
p
A
p
A
p
Fs

















dt
du
A
s
Fs 


du/dt=uu/s+u/t，
cos =z/s = z/ s
s
u
u
t
u





s
z
g
s
p








1
For steady flow, u/t =0
0


 gdz
dp
udu

(3.28)
This is Euler’s equation in differential.

48
2023/6/1
3.5.2 Bernoulli’s equation
Eq.(3.8) is used for ideal fluid flows along a streamline in steady.
Bernoulli’s equation can be obtained with an integral along a
streamline :
C
gz
p
u




2
2
(m2/s2) (3.29)
This is an energy equation per unit mass.
It has the dimensions (L/T)2 because mN/kg=(mkgm/s2)/kg=m2/s2.
The meanings
u2/2 ——the kinetic energy per unit mass (mu2/2)/m.
p/ ——the pressure energy per unit mass.
gz ——the potential energy per unit mass.
Eq. (3.29) shows that the total mechanical energy per unit mass
of fluid remains constant at any position along the flow path.

49
2023/6/1
The Bernoulli’s equation per unit volume is
(N/m2) (3.30)
1
2
2
C
gz
p
u


 

Because the dimension of u2/2 is the same as that of pressure, it
is called dynamic pressure (动压强).
The Bernoulli’s equation per unit weight is
2
2
2
C
z
g
p
g
u




(mN/N, or, m) (3.31)
For arbitrary two points 1 and 2 along a streamline,
1
1
2
1
2
z
g
p
g
u



2
2
2
2
2
z
g
p
g
u




(3.32)

50
2023/6/1
u
h
u
Figure 3.12 Velocity head hu
3.5.3 Geometry expression of Bernoulli’s equation
the Bernoulli’s equation per unit weight is
2
2
2
C
z
g
p
g
u




(mN/N, or, m) (3.31)
Dimension:
z —— L，called the elevation head
(位置水头).
p/(g)——MLT2L2/(ML3LT2)=L,
called the pressure head (压力水头).
u2/(2g) ——L2T2/(LT2)=L, called the
velocity head (速度水头), denoted as hu
The sum of them is called the total head (总水头) , denoted as H.

51
2023/6/1
Figure 3.13 Total head line and piezometer head line
z1
z
0
0
z2
g
p

1
g
p
 g
p

2
g
u
2
2
g
u
2
2
1
g
u
2
2
2
z
a b
g h
e
f
m
u1
u
u2
2
2
1
1
n
k total head line
总水头线
gkh
piezometer head line
测压水头线
enf
elevation head line
位置水头线
amb
datum line
基准线
0-0
H

52
2023/6/1
A venturi meter (文丘里流量计), consisting of a converging portion,
a throat portion in constant diameter and a gradually diverging
portion, is connected in series in a pipeline and used to determine
flow rate in the pipe as shown in Fig. 3.14. Venturi tube is very short;
the flow losses can be neglected. The pipeline is horizontally placed,
namely z1=z2. Determine the flow rate in the pipe.
Solution 1
1
2
2
u1 u2
0 0
Figure 3.14 Venturi tube
z1=z2=0,
g
p
g
u

1
2
1
2

g
p
g
u

2
2
2
2


u1A1=u2A2
  
)
(
2
/
1
1 2
1
2
1
2
2
p
p
A
A
u



  

)
(
2
)
(
2
/
1
2
1
2
2
2
1
2
1
2
1
2
1
2
2
2
2
p
p
A
A
A
A
p
p
A
A
A
u
A
Q







EXAMPLE 3.3

53
2023/6/1
A hollow tubule A with the opening end directed upstream is
installed in the pipeline in Fig. 3.15. The tubule A is called
piezometer for total pressure, or pitot tube (皮托管). Determine the
velocity u1 of liquid in streamline 0-0 at section 1-1 .
Solution
0
2
1
2
1


g
p
g
u

0
0 


g
pm

2
/
2
1
1 u
p
pm 






gh
p
p
u m )
(
2
)
(
2 m
2
1




gh
p
pm )
( m
2 
 


0
0
1
1
h
2 B
2
m
u1
A

m
Figure 3.15 Pitot tube in a straight pipe
Apitot tube; Bpiezometer
m stagnation point(滞止点)
EXAMPLE 3.4
,
2
1 p
p 

54
2023/6/1
3.6 FLOW LOSSES AND STEADY
FLOW ENERGY EQUATION IN THE
DIFFERENTIAL FORM
3.6 流动损失和定常流
能量方程微分形式 #

55
2023/6/1
3.6.1 Reversible and irreversible（可逆和不可逆）
If the system could be reverted to the original
state with the elimination of effect on the
surrounding caused in the former process, this
process is reversible, otherwise it is irreversible.
The irreversible process is universal in the real
system due to the friction.

56
2023/6/1
3.6.2 Energy equation in differential form for a steady flow
i
s
H de
udu
gdz
p
d
w
q 















From thermodynamics(热力学) Tds=dei+pd
s --- the entropy (熵) per unit mass of fluid.
noticing d(p/)=d(p)=  dp+pd =dp/+pd,
udu
gdz
dp
Tds
w
q s
H 






 (3.35)
(3.20)
For a steady flow,
(3.33)
2 2
2 2
H s i i
A
p u p u
Q W gz e d t gz e m
    
 
   
         
   
   
 u A
2
2
H s
H s i
Q W p u
q w gz e
m
 
 

     
Eq.(3.20) in page 50 can be written as

57
2023/6/1
3.6.3 Energy equation for a reversible process
For a reversible process, qH=Tds,
0


 udu
gdz
dp

(3.36)
Equation (3.36) is same with the Euler’s equation in
differential form - Eq.(3.28) in page 55, and it also hold true
for the reversible process in an adiabatic [,ædiə’bætik] flow
（绝热流）, in which
qH=0, ds=0.
if  ws=0
0
s
dp
w gdz udu


   
Energy equation in
differential form for
reversible process

58
2023/6/1
3.6.4 Energy equation for an irreversible process
For a irreversible flow, the Clausius inequality (克劳修斯不等式)
Tds qH >0
Let the flow losses per unit weight of fluid be hw,
ghw= Tds qH
0




 w
s gh
udu
gdz
dp
w


so
If ws=0
0



 w
gh
udu
gdz
dp

(3.38)
For incompressible fluid in the adiabatic flow, =const., dei=cvdT
and qH=0. If ws=0,
0



 dT
c
udu
gdz
dp
v

(3.39)
Energy losses make
increase of temperature
(3.37)
Energy equation in
differential form for
irreversible process

59
2023/6/1
3.6.5 The Bernoulli’s equation for real system
hs——the shaft work(轴功) per unit weight of fluid, hs=ws/g,
hw——the energy losses per unit weight of fluid from section 1-
1 to section 2-2 and always positive.
w
2
2
2
2
2
h
z
g
p
g
u





s
1
1
2
1
2
h
z
g
p
g
u



 (3.40)
“+” denotes the energy added to the system,
“” denotes the energy gotten away the system to the
surrounding.
From Eq.(3.37) s w 0
dp
w gdz udu gh


    
It shows that the total energy decreases gradually in
the downstream (顺流) direction.

60
2023/6/1
u1dA1= u2dA2
Q1=Q2 or V1A1=V2A2
Review
A
udA Q
V
A A
 

C
gz
p
u




2
2
1
2
2
C
gz
p
u


 

2
2
2
C
z
g
p
g
u



 1
1
2
1
2
z
g
p
g
u



2
2
2
2
2
z
g
p
g
u




BERNOULLI EQUATION:
CONTINUITY EQUATION:

61
2023/6/1
3.7 APPLICATION OF THE ENERGY
EQUATION TO STEADY FLUID-FLOW
3.7 定常流体流动能量方程应用 (重点)

62
2023/6/1
Equation (3.33) in section 3.6 can only be applied to
a mini-stream tube or a streamline.
In the engineering practice it is not enough to know
what the energy equation is for a streamline, and the
most important is what the energy equation for a total
flow would be.
i
s
H de
udu
gdz
p
d
w
q 














 (3.33)

63
2023/6/1
3.7.1 Gradually varied flow（缓变流）
Cross section

R
Figure 3.17 Cross section of gradually
varied flow
On the flow section the
angle β between two
streamlines is very small
(infinitesimal), or the
radius of curvature (曲率)
R of each streamline is
very large (infinity).
1) the streamlines tend to be the parallel beelines (平行直线);
2) the acceleration is very small;
3) the flow section can be considered a plane surface.
Definition:
Essence[‘esns](Core):

64
2023/6/1
b
d
x
z
p
dA
dAdz
g

dz
R
u
z
a
c
n
n
0
Figure3.18 Control volume in a flow section
of gradually varied flow
p+dp
(p+dp)dApdA+gdAdz=0
0

 dz
g
dp

For  =const.
0









 z
g
p
d

const









 z
g
p

For steady flow, u/t=0, and the sum of all forces in the n
direction is zero too.
Same with Eq.(2.7) in page 23

65
2023/6/1
3.7.2 The Bernoulli’s equation for the real-fluid total flow
Energy for the flow with weight dG=gudA in a mini- stream tube
dG
dE
z
g
p
g
u
e 




2
2
the total energy per unit time through the cross section

 











A
A
gudA
z
g
p
g
u
dE
E 

2
2
In gradually varied flow on the section
 











A A
dA
u
udA
g
z
g
p
E 3
2



Because of A udA=Q
gQ
z
g
p
udA
g
z
g
p
A



 

















 

66
2023/6/1
A
udA
A
Q
V A



An average velocity V is employed
Q
V
A
V
dA
V
dA
u
A
A
2
3
3
3


 

 

2
2
2
3 Q
V
dA
u
A



 gQ
g
V


2
2

gQ
z
g
p
E 
 








 gQ
g
V


2
2

The energy over the section in the gradually varied flow is
in which  is the kinetic-energy correction factor. For turbulent
flow in a pipe, =1.05～1.10. For laminar flow in a pipe, =2.
Let ,
3
3
V
u 

weight flow rate

67
2023/6/1
The mechanical energy per unit weight over the section in
gradually varied flow is
Let hw be the energy losses per unit weight of fluid from 1-1 to
2-2, the Bernoulli’s equation for a total flow is
Let hs be the shaft work per unit weight of fluid, the Bernoulli’s
equation for a real system is
w
s h
z
g
p
V
h
z
g
p
V






 2
2
2
2
1
1
2
1
g
2
g
2 



(3.45)









 z
g
p
e
 g
αV
2
2

w
h
z
g
p
V
z
g
p
V





 2
2
2
2
1
1
2
1
g
2
g
2 



(3.44)
2
u

68
2023/6/1
3.7.3 Applications of Bernoulli’s equation
Conditions for application:
1)The fluid is incompressible and the flow must be a
steady flow.
2)The selected cross section should be in the
gradually varied flow or in uniform flow.
3)If there is input or output of shaft work between
two sections, the calculation can be carried out with
Eq. (3.45)
4)The flow rate remains a constant along the flow
path.

70
2023/6/1
0
1
1
d
he
0
Figure 3.20 Pumping water
A centrifugal water pump (离心式
水泵) with a suction pipe (吸水管)
is shown in Fig. 3.20. Pump output
is Q=0.03m3/s, the diameter of
suction pipe d=150mm, vacuum
pressure that the pump can reach is
pv/(g)=6.8 mH2O, and all head
losses in the suction pipe
hw=1mH2O. Determine the utmost
elevation (最大提升) he from the
pump shaft to the water surface on
the pond.
EXAMPLE 3.6

71
2023/6/1
Solution
1) two cross sections and the datum plane are selected.
The sections should be on the gradually varied flow, the
two cross sections here are: 1) the water surface 0-0 on the
pond, 2) the section 1-1 on the inlet of pump. Meanwhile, the
section 0-0 is taken as the datum plane, z0=0.
1
0
w
1
1
2
1
1
0
0
2
0
0
g
2
g
2






 h
z
g
p
V
z
g
p
V





72
2023/6/1
2) the parameters in the equation are determined.
The pressure p0 and p1 are expressed in relative pressure
(gauge pressure).
0
g
0


p
O)
mH
(
8
.
6
g
g
2
v
1






p
p
，and
So,V0=0.
1 2
0.03
1.7(m/s)
0.15 / 4
Q
V
A 
  

Let =1
hw0-1=1 mH2O
0
g
0


p
，and
，and

73
2023/6/1
3) calculation for the unknown parameter is carried out.
By substituting V0=0，p0=0，z0=0，p1=pv，z1=he，α1=1 and
V1=1.7 into the Bernoulli’s equation, i.e.,
1
0
w
v
2
1
2
0
0
0 






 h
h
g
p
g
V
e

1
0
w
2
1
v
g
2



 h
V
g
p
he

The values of pv/(g) and V1 are substituted into above formula
and it gives
6.8 0.15 1.0 5.65(m)
e
h    
namely,

74
2023/6/1
3.8 APPLICATIONS OF THE LINEAR-
MOMENTUM EQUATION
3.8 动量方程应用 (重
点)

75
2023/6/1
The linear-momentum equation was obtained by Reynolds transport
equation in Sec. 3.3 in page 49,
For a steady flow of incompressible fluid, if the control surfaces,
normal to the velocity wherever it cuts across the flow, have only
the two surfaces
(3.48)

 






A
d
d
t
dt
m
d
A
u
u
u
u
F )
(
)
(




(3.15)

 





A
x
x
x d
u
d
u
t
A
u
F )
(



The component form, say the x direction
1
1
1
2
2
2 A
u
A
u u
u
F 
 

 )
( 1
2 u
u 
 Q

(3.47)

76
2023/6/1
In the actual flow the velocity over a plane cross section(横截平面)
is not uniform
dA
V
u
A A
2
)
(
1



A
V
dA
u
A
2
2

 
 or
If the un-uniform in velocity over the section is taken into account,
Eq. (3.48) may be rewritten as
)
( 1
1
2
2 V
V
F 

 

 Q
 is momentum correction factor (动量修正系数).
 = 4/3 in laminar flow for a straight round tube.
 = 1.02~1.05 in turbulent flow and it could be taken as 1.
A
V
dA
u
A
2
2

 
 or dA
V
u
A A
2
)
(
1



A
V
dA
u
A
2
2

 
 or (3.49)
(3.50)

77
2023/6/1
A spool valve consists of a spool and sleeve as shown in Fig. 3.21. A
flow of oil with density  in a flowrate Q is flowing from channel A
into the circular annuli and then the flow of oil leaves the annular
orifice B, the outflow of oil will give a force on the spool. Now
determine the force.

Figure 3.21 Steady-state flow force along the axis direction
1 sleeve；2 spool
Fx
A B
1 2
a) Spool valave b) Control
volume
y
x
V1 V2
EXAMPLE 3.7

78
2023/6/1
Solution
Fx acting on the control volume in the x direction



 cos
)
90
cos
cos
( 2
1
2 QV
V
V
Q
Fx 



The oil exerts a force on the spool (the surrounding of CV).

 cos
2
QV
F
P x
x 



The force Px is called the steady-state flow force (稳态液动力).
also called Bernoulli’s force (伯努利力) in oil-hydraulics.
The resultant force is zero in the y direction.

79
2023/6/1
A steel bend (弯头) used to supply
water is shown in Fig.3.21. The inner
diameter of bend is d=500mm. The
angle between bend and horizontal
plane is 45. The flow rate in pipe is
Q=0.5m3/s; the pressure of center
point at the section 1-1 is
p1=108kN/m2. The elevation of center
point at the section 2-2 is z2=0.7m.
The flowing losses of two section is
0.15V2
2/(2g) m. Neglecting the mass
of bent and water, determine the force
applied to the bend by the flowing
water.
Figure 3.22 Forces on a bent
p2
z
x
p1
1
1
2
2
45
v
o
z2
EXAMPLE 3.8

80
2023/6/1
Solution
1) Select the CV:
The inside surface of bend
and both sections 1-1 and 2-2. p2
z
x
p1
1
1
2
2
45
v
o
z2
A1=A2=A. )
m
(
196
.
0
5
.
0
4
2
2




A
V1=V2=V , (m/s)
55
.
2
196
.
0
5
.
0



A
Q
V
Rx
Ry R
P

2) Analyse the forces on CV
3) calculate the parameters in equation
p2=?
2
1
2
2
2
2
1
2
1
1
2
2 





 w
h
z
g
V
g
p
z
g
V
g
p



81
2023/6/1
p2
z
x
p1
1
1
2
2
45
v
o
z2
Rx
Ry R
P

(Pa)
10
6
.
100
)
15
.
0
(
2
)
(
3
2
2
1
1
2





 V
z
z
g
p
p


)
45
cos
(
45
cos
2
2
1
1 V
V
Q
A
p
R
A
p
F x
x 




 


(N)
7599
)
45
cos
(
45
cos
2
2
1
1




 V
V
Q
A
p
A
p
Rx



4) Solution for forces

82
2023/6/1
)
0
45
sin
(
45
sin
2
2 


 

V
Q
R
A
p z 
(N)
14844
)
0
45
sin
(
45
sin
2



 

V
Q
P
Rz 
(N)
16676
2
2


 z
x R
R
R
14844
7599
tan 

z
x
R
R


1
.
27


P =  R = 16676N
p2
z
x
p1
1
1
2
2
45
v
o
z2
Rx
Ry R
P


83
2023/6/1
In Fig.3.23, a nozzle is connected to a pipe D. The exit diameter of
nozzle is d=25mm, the inner diameter of pipe is D=75mm. The
pressure at the entrance of nozzle is p1=0.7MPa. The losses in the
nozzle is hw=0.3V2
2/(2g). The fluid is water. Determine the tension
force T exerted by the nozzle on the pipe.
p2A2
a) nozzle b) CV
d
D
1
1
2
2
1
1
2
2
V2
V1 p1A1 R
cv
y
x
Figure 3.23 Nozzle and its control volume
EXAMPLE 3.9

84
2023/6/1
g
V
g
V
z
g
p
g
V
z
2
3
.
0
0
2
2
2
2
2
2
2
1
2
1
1 






V2=(D/d)2V1=9V1 and z1 = z2
0
806
.
9
1000
10
7
.
0
)]
3
.
0
1
(
9
1
[
806
.
9
2
6
2
2
1







V
p2A2
1
1
2
2
V2
V1 p1A1 R
cv
y
x
V1=3.66 m/s and V2=32.94 m/s
)
( 1
2
1
1 V
V
Q
R
A
p
Fx 



 
)
N
(
2619
)
66
.
3
94
.
32
(
10
16.17
10
075
.
0
4
10
7
.
0 3
3
2
6








 

R
Solution

85
2023/6/1
2
2
2V
A
P 

)
(N/m
10
085
.
1
)
94
.
32
(
1000 2
6
2
2
2
2
2
2
2
2






 V
A
V
A
A
P
p 

The impact force of jet P
The impact pressure of jet p
Solution Procedure for a problem with momentum equation:
1) selection for a proper control volume;
2) thorough analysis to the external forces;
3) application of the momentum equation in component form
for the coordinate axis;
4) caution with the sign of force (plus sign is the same direction
with the selected coordinate axis, otherwise a minus sign is used).

86
2023/6/1
Homework
3.1, 3.4, 3.6,
3.8, 3.10, 3.11

武科大流体力学Chapter3.ppt

武科大流体力学Chapter3.ppt

Recommended

Recommended

More Related Content

Similar to 武科大流体力学Chapter3.ppt

Similar to 武科大流体力学Chapter3.ppt (20)

Recently uploaded

Recently uploaded (20)

武科大流体力学Chapter3.ppt