2011-7-18 1
School of Jet Propulsion
Beihang University.
2011-7-18 2
Chapter 1 Introduction
1.1 Preliminary Remarks
When you think about it,almost
everything on this planet either is a
fluid or moves within or near a fluid.
-Frank M,White
What is a
fluid?
2011-7-18 3
The concept of a fluid
A solid can resist a shear stress(剪切应力 )
by a static deformation,a fluid can not.
Any shear stress applied to a fluid,no matter
how small,will result in motion of that fluid.
The fluid moves and deforms continuously as
long as the shear is applied.
2011-7-18 4
What is Fluid Mechanics
Fluid Mechanics is the study of fluid either
in motion (Fluid Dynamics 流体动力学 ) or at
rest(Fluid Statics 流体静力学 ) and
subsequent effects of the fluid upon the
boundaries,which may be either solid
surfaces or interfaces with other fluids.
2011-7-18 5
The famous collapse of the
Tacoma Narrow Bridge in 1940
Curved shoot (Banana shoot)
Nospin Spin
why
2011-7-18 6
Boeing 747
70.7× 64.4× 19.41 (m)
395 000kg
An-225
84× 88.4× 18.1 (m)
600,000kg
How can the
airplane fly? Drag & Lift
2011-7-18 7
2011-7-18 8
The engine of a turbofan(涡扇 ) jet
2011-7-18 9；
2011-7-18 10
History and Scope ofFluid Mechanics
Pre-history:
Sailing ships with oars(橹桨 ) and irrigation
system were both known in prehistory
2011-7-18 11
Archimedes(285-212 BC)
Parallelogram law for addition of
vectors
Law of buoyancy
2011-7-18 12
Leonardo da Vinci(1452-1519)
* Equation of conservation of mass
in one-dimensional steady flow
* Experimentalist
* Turbulence
2011-7-18 13
Isaac Newton(1642-1727)
Laws of motion
Laws of viscosity of
Newtonian fluid
2011-7-18 14
18th century
Mathematicians：
Euler（欧拉）,Euler equation
Bernoulli (伯努利 ), Bernoulli equation
Frictionless(无粘 ) flow solutions
D’Alembert（达朗贝尔）：
D’Alembert paradox（佯谬，疑题）
Engineers,Hydraulics （水力学） relaying on
experiment
Channels, Ship resistance,Pipe flows,Wave turbine
Pitot Venturi Torricelli Poiseuille
2011-7-18 15
19th century
Navier (1785-1836)
& Stokes (1819-1905)
N-S equation viscous flow solution
Reynolds (1842-1912)
Turbulence
Famous experiment on
transition
Reynolds Number
2011-7-18 16
20th century
Ludwig Prandtl (1875-1953)
Boundary theory(1904)
To be the single most important tool in
modern flow analysis.
The father of modern fluid mechanics
Vonkarman
(1881-1963)
I.taylor(1886-1975)
Laid foundation for
the present state
of the art in fluid
mechanics
2011-7-18 17
1.2 The Fluid as a Continuum (连续介质 ）
*vv
mlim
v??
??
??
?
Density(密度 )
Elemental volume（流体微团、流体质点）
* Large enough in microscope（微观）
10-9mm3 of air at standard conditions contains
approximately 3× 107 molecules.
So density is essentially a point function and fluid properties can be
thought of as varying continually in space,
* Small enough in macroscope（宏观）,
Most engineering problems are concerned with physical
dimensions much larger than this limiting volume.
2011-7-18 18
The elemental volume must be small enough in macroscope
Such a fluid is called a continuum,which simply means that
its variation in properties is so smooth that the
differential calculus can be used to analyze the substance.
2011-7-18 19
1.3 Some Properties of fluids
1.viscosity(粘性 )
* Definition,When a fluid is sheared（ 剪切 ）,it begins to
move,Subsequently,a pair of forces appear on the
shear surface,which resists the shear motion of the
fluid,This is called viscosity
This resistant force is shear stress.(剪切应力,内摩擦应力 )
In fact,this shear motion of a fluid is a kind of
deformation（ 变形 ）
* The nature of viscosity：
For liquid is cohesion（ 结合 ） (movie)
For gas is the transport of momentum（ 动量输运 ） (movie)
2011-7-18 20
m, Coefficient of viscosity (粘性系数 )[FT/L2]
n ? m / ?,Kinematic viscosity (运动学粘性系数 )[L2/T]
du
dy? ?
Velocity gradient
du
dy?m?
* Newtonian law of viscosity
(牛顿粘性定律，牛顿内摩擦定律 ) U U
u(y)
x
y
Shear
stress
The linear fluid,which follow Newtonian resistance law,is called
Newtonian flow,（ 牛顿流动、牛顿流体 ）
The velocity gradient is in fact a kind of deformation.
Real fluid (Viscous),Ideal fluid (Inviscid & Frictionless)
2011-7-18 21
2,Compressibility(压缩性 )
Incompressible(不可压 ),? = const
Most liquid flows are treated as incompressible.
Only 1 percent increase if pressure increase by 220
Compressible（ 可压缩 ）,? = ? (P.T)
Gases can also be treated as incompressible when
their velocity is less than 0.3 Ma numbers
3,State Relations for Gases
Perfect-gas Law（理想气体状态方程）
P R T??
2011-7-18 22
4.Thermal Conductivity(热传导 )
,heat flux in n direction per unit area
k,coefficient of thermal conductivity
T,temperature
n,direction of heat transfer
Fourier’s law of heat conduction
Tqk
n
???
?
v
qv
2011-7-18 23
1.4 Two different points of view in analyzing problems in mechanics
* The Eulerian view (欧拉观点 )and the Lagrangian
view （拉格朗日观点）
The Eulerian view is concerned with the field of flow,
appropriate to fluid mechanics.
The Lagrangian view follows an individual particle moving
though the flow,appropriate to solid mechanics.
The contrast of two frames
2011-7-18 24
* Flow classification(流动分类 )
According to Eulerian view,any property is function
of coordinates（ space） and time,In Cartesian system
(直角坐标系 ), it can be expressed as
f(x,y,z,t)
x,y,z,t,Eulerian variable component ( 欧拉变数 )
f,Function of only one coordinate component,one-
dimensional ( 一维 1-D）, In the like manner,two-
dimensional （ 二维 2-D）,three-dimensional ( 三维 3-D )
f
,Function of time ~~ unsteady (非定常 )
Otherwise steady (定常 )
f
0tf? ??
0tf? ??
2011-7-18 25
One
Two dimensional
Three
Steady
Unsteady
Compressible
Incompressible
Viscous
Inviscid
? ?
? ?
2011-7-18 26
1.5 Streamline(流线 ),Pathline(迹线 ) & Flowfield (流场 )
* What is a
streamline
A streamline is the line
everywhere tangent to
the velocity vector at a
given instant.
2011-7-18 27
A pathline is the actual path traversed by a given
fluid particles.
For steady flow,Streamline = Pathline
* What is a
pathline
Pathlines in steady flow
Pathlines in unsteady flow
2011-7-18 28
Flow Pattern (流型、流普、流线族）
Stream surface(流面 )& Streamtube (流管）
Flow pattern, a set of streamlines
Streamsurface,a collection of all the streamlines passing
through a line which is not a streamline.
Stream line can not intersect（相交），
except for singularity point(奇点 )
Streamtube, a closed collection of streamlines.
No flow
across
streamtub
e walls
2011-7-18 29
Flow field (流场 ), In a given flow situation,the
properties of the fluid are functions of position and
time,namely space-time distributions of the fluid
properties,
2011-7-18 30
Streamline equation(流线方程 )
c o s (,) uVu V?v v c o s (,) vVv
V?
v v
c o s (,) dxsx ds?v c o s (,)
dysy
ds?
v
ds -> Infinitesimal (无穷小 )
d x d y
uv?
dy
dx
ds? ?
d x d y d z
u v w??
?
2011-7-18 31
Example:
Given the steady two-dimensional velocity distribution
u=kx,v=-ky,w=0,where k is a positive constant.
Compute and plot the streamlines of the flow,including
direction.
Solution,Since time (t) does not appear explicitly,the motion is
steady,so that streamlines,pathlines will coincide.Since w=0,the motion
is two-dimensional.
d x d y
uv?
d x d y
kx ky
??
d x d y
xy
????
l n l n l nx y c? ? ?Integrating,xy c?
Hyperbolas(双曲线 )
2011-7-18 32
Direction,u=kx,v=-ky
Quadrant I （第一象限 ) (x>0,y>0)
u>0,v<0
xy c?
At the point o,u =v =0
Singularity point,(汇 )
x
y
o
2011-7-18 33
1.5 Surface force(表面力 )
and body force（ 质量力，体积力 ）
Surface force acts continuously on the side surfaces
of fluid elements,
Pressure,friction, Contact
surface force per unit area( 单位面积 ) （ 应力 ）
Body force acts on the entire mass of the element.
Gravity,electromagnetic,No cotact
Per unit mass(单位质量 ) g
R X i Y j Z k? ? ? vv vv
2011-7-18 34
Home work
1,Given the velocity distribution:
u = - c y,v = c x,w = 0
Where c is a positive constant,Compute and plot the
streamlines of the flow.
2,Given velocity distribution:
u = x + t,v = - y + t,w = 0 ( t is time)
Find the streamline passing through point(-1,-1) at the
instant t=0,
School of Jet Propulsion
Beihang University.
2011-7-18 2
Chapter 1 Introduction
1.1 Preliminary Remarks
When you think about it,almost
everything on this planet either is a
fluid or moves within or near a fluid.
-Frank M,White
What is a
fluid?
2011-7-18 3
The concept of a fluid
A solid can resist a shear stress(剪切应力 )
by a static deformation,a fluid can not.
Any shear stress applied to a fluid,no matter
how small,will result in motion of that fluid.
The fluid moves and deforms continuously as
long as the shear is applied.
2011-7-18 4
What is Fluid Mechanics
Fluid Mechanics is the study of fluid either
in motion (Fluid Dynamics 流体动力学 ) or at
rest(Fluid Statics 流体静力学 ) and
subsequent effects of the fluid upon the
boundaries,which may be either solid
surfaces or interfaces with other fluids.
2011-7-18 5
The famous collapse of the
Tacoma Narrow Bridge in 1940
Curved shoot (Banana shoot)
Nospin Spin
why
2011-7-18 6
Boeing 747
70.7× 64.4× 19.41 (m)
395 000kg
An-225
84× 88.4× 18.1 (m)
600,000kg
How can the
airplane fly? Drag & Lift
2011-7-18 7
2011-7-18 8
The engine of a turbofan(涡扇 ) jet
2011-7-18 9；
2011-7-18 10
History and Scope ofFluid Mechanics
Pre-history:
Sailing ships with oars(橹桨 ) and irrigation
system were both known in prehistory
2011-7-18 11
Archimedes(285-212 BC)
Parallelogram law for addition of
vectors
Law of buoyancy
2011-7-18 12
Leonardo da Vinci(1452-1519)
* Equation of conservation of mass
in one-dimensional steady flow
* Experimentalist
* Turbulence
2011-7-18 13
Isaac Newton(1642-1727)
Laws of motion
Laws of viscosity of
Newtonian fluid
2011-7-18 14
18th century
Mathematicians：
Euler（欧拉）,Euler equation
Bernoulli (伯努利 ), Bernoulli equation
Frictionless(无粘 ) flow solutions
D’Alembert（达朗贝尔）：
D’Alembert paradox（佯谬，疑题）
Engineers,Hydraulics （水力学） relaying on
experiment
Channels, Ship resistance,Pipe flows,Wave turbine
Pitot Venturi Torricelli Poiseuille
2011-7-18 15
19th century
Navier (1785-1836)
& Stokes (1819-1905)
N-S equation viscous flow solution
Reynolds (1842-1912)
Turbulence
Famous experiment on
transition
Reynolds Number
2011-7-18 16
20th century
Ludwig Prandtl (1875-1953)
Boundary theory(1904)
To be the single most important tool in
modern flow analysis.
The father of modern fluid mechanics
Vonkarman
(1881-1963)
I.taylor(1886-1975)
Laid foundation for
the present state
of the art in fluid
mechanics
2011-7-18 17
1.2 The Fluid as a Continuum (连续介质 ）
*vv
mlim
v??
??
??
?
Density(密度 )
Elemental volume（流体微团、流体质点）
* Large enough in microscope（微观）
10-9mm3 of air at standard conditions contains
approximately 3× 107 molecules.
So density is essentially a point function and fluid properties can be
thought of as varying continually in space,
* Small enough in macroscope（宏观）,
Most engineering problems are concerned with physical
dimensions much larger than this limiting volume.
2011-7-18 18
The elemental volume must be small enough in macroscope
Such a fluid is called a continuum,which simply means that
its variation in properties is so smooth that the
differential calculus can be used to analyze the substance.
2011-7-18 19
1.3 Some Properties of fluids
1.viscosity(粘性 )
* Definition,When a fluid is sheared（ 剪切 ）,it begins to
move,Subsequently,a pair of forces appear on the
shear surface,which resists the shear motion of the
fluid,This is called viscosity
This resistant force is shear stress.(剪切应力,内摩擦应力 )
In fact,this shear motion of a fluid is a kind of
deformation（ 变形 ）
* The nature of viscosity：
For liquid is cohesion（ 结合 ） (movie)
For gas is the transport of momentum（ 动量输运 ） (movie)
2011-7-18 20
m, Coefficient of viscosity (粘性系数 )[FT/L2]
n ? m / ?,Kinematic viscosity (运动学粘性系数 )[L2/T]
du
dy? ?
Velocity gradient
du
dy?m?
* Newtonian law of viscosity
(牛顿粘性定律，牛顿内摩擦定律 ) U U
u(y)
x
y
Shear
stress
The linear fluid,which follow Newtonian resistance law,is called
Newtonian flow,（ 牛顿流动、牛顿流体 ）
The velocity gradient is in fact a kind of deformation.
Real fluid (Viscous),Ideal fluid (Inviscid & Frictionless)
2011-7-18 21
2,Compressibility(压缩性 )
Incompressible(不可压 ),? = const
Most liquid flows are treated as incompressible.
Only 1 percent increase if pressure increase by 220
Compressible（ 可压缩 ）,? = ? (P.T)
Gases can also be treated as incompressible when
their velocity is less than 0.3 Ma numbers
3,State Relations for Gases
Perfect-gas Law（理想气体状态方程）
P R T??
2011-7-18 22
4.Thermal Conductivity(热传导 )
,heat flux in n direction per unit area
k,coefficient of thermal conductivity
T,temperature
n,direction of heat transfer
Fourier’s law of heat conduction
Tqk
n
???
?
v
qv
2011-7-18 23
1.4 Two different points of view in analyzing problems in mechanics
* The Eulerian view (欧拉观点 )and the Lagrangian
view （拉格朗日观点）
The Eulerian view is concerned with the field of flow,
appropriate to fluid mechanics.
The Lagrangian view follows an individual particle moving
though the flow,appropriate to solid mechanics.
The contrast of two frames
2011-7-18 24
* Flow classification(流动分类 )
According to Eulerian view,any property is function
of coordinates（ space） and time,In Cartesian system
(直角坐标系 ), it can be expressed as
f(x,y,z,t)
x,y,z,t,Eulerian variable component ( 欧拉变数 )
f,Function of only one coordinate component,one-
dimensional ( 一维 1-D）, In the like manner,two-
dimensional （ 二维 2-D）,three-dimensional ( 三维 3-D )
f
,Function of time ~~ unsteady (非定常 )
Otherwise steady (定常 )
f
0tf? ??
0tf? ??
2011-7-18 25
One
Two dimensional
Three
Steady
Unsteady
Compressible
Incompressible
Viscous
Inviscid
? ?
? ?
2011-7-18 26
1.5 Streamline(流线 ),Pathline(迹线 ) & Flowfield (流场 )
* What is a
streamline
A streamline is the line
everywhere tangent to
the velocity vector at a
given instant.
2011-7-18 27
A pathline is the actual path traversed by a given
fluid particles.
For steady flow,Streamline = Pathline
* What is a
pathline
Pathlines in steady flow
Pathlines in unsteady flow
2011-7-18 28
Flow Pattern (流型、流普、流线族）
Stream surface(流面 )& Streamtube (流管）
Flow pattern, a set of streamlines
Streamsurface,a collection of all the streamlines passing
through a line which is not a streamline.
Stream line can not intersect（相交），
except for singularity point(奇点 )
Streamtube, a closed collection of streamlines.
No flow
across
streamtub
e walls
2011-7-18 29
Flow field (流场 ), In a given flow situation,the
properties of the fluid are functions of position and
time,namely space-time distributions of the fluid
properties,
2011-7-18 30
Streamline equation(流线方程 )
c o s (,) uVu V?v v c o s (,) vVv
V?
v v
c o s (,) dxsx ds?v c o s (,)
dysy
ds?
v
ds -> Infinitesimal (无穷小 )
d x d y
uv?
dy
dx
ds? ?
d x d y d z
u v w??
?
2011-7-18 31
Example:
Given the steady two-dimensional velocity distribution
u=kx,v=-ky,w=0,where k is a positive constant.
Compute and plot the streamlines of the flow,including
direction.
Solution,Since time (t) does not appear explicitly,the motion is
steady,so that streamlines,pathlines will coincide.Since w=0,the motion
is two-dimensional.
d x d y
uv?
d x d y
kx ky
??
d x d y
xy
????
l n l n l nx y c? ? ?Integrating,xy c?
Hyperbolas(双曲线 )
2011-7-18 32
Direction,u=kx,v=-ky
Quadrant I （第一象限 ) (x>0,y>0)
u>0,v<0
xy c?
At the point o,u =v =0
Singularity point,(汇 )
x
y
o
2011-7-18 33
1.5 Surface force(表面力 )
and body force（ 质量力，体积力 ）
Surface force acts continuously on the side surfaces
of fluid elements,
Pressure,friction, Contact
surface force per unit area( 单位面积 ) （ 应力 ）
Body force acts on the entire mass of the element.
Gravity,electromagnetic,No cotact
Per unit mass(单位质量 ) g
R X i Y j Z k? ? ? vv vv
2011-7-18 34
Home work
1,Given the velocity distribution:
u = - c y,v = c x,w = 0
Where c is a positive constant,Compute and plot the
streamlines of the flow.
2,Given velocity distribution:
u = x + t,v = - y + t,w = 0 ( t is time)
Find the streamline passing through point(-1,-1) at the
instant t=0,
