Download Slide 1

2: Introduction and Fundamental Concepts in Mechanics and Fluid Mechanics
1.
Mechanics

2.
Fluid Mechanics

3.
Force-Body-Motion
Fluid as A Continuum / Continuum Assumption

Fluid as A Continuum

Property at A Point

Property Fields
Methods of Description of Motion:

4.
5.
abj
Lagrangian
VS
Eulerian Descriptions
Scalar, Vector, and Tensor Fields

Classification of Fields f ( x, t )

x
1.
Steady VS Unsteady Fields
[Does f at any one point
2.
Uniform VS Non-Uniform Fields
[Does f at any one time t change with the spatial location
the region R?]
change with time t?]

x
in
Classification of Fluid Flows
1
Mechanics
abj
2
Fundamental Concept
Mechanics: Force-Body-Motion
Mechanics
Motion of a Body under the action of Forces.
(Effects of Forces on the Motion of a Body.)

Mechanics:

System-Surroundings-Interactions:
Three main components:
Force - Body - Motion.
Investigate the resulting effects of the mechanical interactions between the system and its
surroundings.
abj

Interactions:
Forces (in FBD) and Work

Effects:
Motion (and other related quantities)
3
Thermodynamics
Thermal Energy and
Thermodynamic Properties of
Substance
Force
Body
Linear Motion
Angular Motion
Translation
(Rigid-Body-Like) Rotation
 Force:
Motion
Deformation
An effort to move a body against its inertia.
(in Newton’s second law aspect.)
abj
4
Fundamental Concept
Three Efforts of Force, Three Types of Body, and Three Types of Motion
Forces (3 Efforts)
Body (3 Types):
Motion (3 Components):
(~ according to the degree of
idealization of permitted motions.)
abj
Force
Particle
Linear Motion (Translation)
Moment of
Force
Rigid Body
Angular Motion (Rigid-Body-Like Rotation)
Intensity of
Force
Deformable Body
Deformation
5
Force-Body-Motion in Equations of Mechanics
Force
1. Force and linear motion


 F  ma
(translation).
2.
Properties of
Body
Motion
(Kinematics)

F 
m

a
M
I

(Moment of) Force and angular motion (rotation).
 M  I
3.
(Intensity of) Force and deformation.

Strain
Stress
(Constitutive Relation)
Hooke’s law
  E


E

 yx 

(du / dy)
Newton’s law of viscosity
 yx   (du / dy)
abj
6
Fluid Mechanics
abj
7
Fundamental Concept: Fluid Mechanics
Fluid Mechanics
=
Mechanics (Force and Motion) and
Thermodynamics (Energy and Energy Transfer)
of Fluid Motion
abj
8
Fluid Mechanics
=
Mechanics (Force and Motion) and
Thermodynamics (Energy and Energy Transfer)
of Fluid Motion
System (Control Volume):
Inlet
Fluid stream only, excluding the solid heater
Heater
Exit
air
Given: At inlet,
Question: At exit,
 (Mass / time , kg / s)
m

 ( x, t ) (Mass / Vol , kg / m3 )
Density profile
 
Velocity profile
V ( x , t ) (Velocity, m / s )

Temperature profile T ( x, t ) (Temperature , K )
Mass flowrate
Mass flowrate
abj
 ?
m

 ( x, t )  ?
Density profile
 
Velocity profile
V ( x, t )  ?

Temperature profile T ( x, t )  ?
9
Fundamental Concept: Definition of Fluids
Simple models for simple solid and fluid.

Solid
finite deformation
under constant shear
 (t)
Fluid
continuous deformation
under constant shear
no matter how small shear is
Definition of Fluid:
A fluid is a substance that deforms continuously under the application of a shear
(tangential) stress no matter how small the shear stress may be. (Fox, et al., 2004)
abj
10
Fundamental Concept
Fluid as A Continuum: Continuum Assumption, Property at A Point
y

 m, V
m
V
x
z
V
• Fluctuation due to random molecular motion
V’
• Continuum assumption breaks down.
Random motion of molecules
significantly affects macroscopic
mean density.
Random motion of molecules
has little effect on macroscopic
mean density.
m
  lim
V V ' V
• Macroscopic spatial variation
Continuum
Assumption
Above this limit:
• Continuum:
A fluid is assumed to be a continuum.
• Property
Its property is assumed at a point, e.g., density at a point.
• Field
at a point:
Function:
A fluid property f is assumed as a continuous field function of

space and time: f  f E ( x, t )
Below
abj
this limit, the continuum assumption breaks down, and the molecular motion of molecules must
11
be taken into account.
Two Methods of Description of Fluid Motion
abj
12
Fundamental Concept
Two Methods of Description of Fluid Motion:
Lagrangian
VS Eulerian Descriptions
Eulerian Description
Lagrangian Description
y
y

X

x
Reference time to
z
Property function
Independent variables

x
abj
x
z
Current time t
Lagrangian Description

f  f L ( X , t)

‘Particle name’
X
Time
Interpretation
f E ( x, t )
x
x
z

y
t
Time evolution of a property f


of a material particle X  X A
is given by


f  f L ( X  X A , t)
Current time t
Eulerian Description

f  f E ( x, t )

x
Spatial position
t
Time
Time evolution of a property f
 
at a fixed point x  x a
in space is given by


f  f E ( x  xa , t)
13
Lagrangian Description

f  f L ( X , t)
reads
the current (at time t) value of the
fproperty

of the material
X

f L ( X , t )is equal to
particle
.
Eulerian Description

f  f E ( x, t )
abj
reads
f at position

is
equal
to
f
(
x
, t) .
t
E
the value of the property

x and time
14
Fundamental Concept: Lagrangian VS Eulerian Views

To put simply
Eulerian view:

f  f E ( x, t )

We watch an identified region in space of interest, and

see what happens in the region.
[Here, we watch a block in the street and see New York cabs passing in and out of our block.]

Lagrangian view: f  f L ( X , t )
abj

We watch/follow the identified mass of interest, and

see what happens to the identified mass.
[Here, we watch and follow, say, one taxicab wherever it goes.]
15
Eulerian/Field Descriptions
Scalar, Vector, and Tensor Fields: Velocity and Property Fields

 

f E ( x , t )  V ( x , t )  V ( x, y , z , t )



 u ( x , t ), v( x , t ), w( x , t ) 

x



 ( x, t ), p( x, t ), T ( x, t )

Space-time point ( x, t )  ( x, y, z, t )
Time t

Common Property Fields:
Scalar Field:
density:
pressure:
temperature:
Vector Field:
velocity:

   ( x, t )   ( x, y, z, t )

p  p( x, t )  p( x, y, z, t )

T  T ( x, t )  T ( x, y, z, t )
  



V  V ( x , t )  u ( x , t ), v( x , t ), w( x , t ) 
Note that it is customary to use (u, v, w) for the (x,y,z)
components of velocity, respectively.
abj
Tensor Field:
stress tensor:

   ( x, t )
16
Steady and Uniform Property Field f

Steadiness: Does f at any one point x change with time t ?
 Steadiness:

Does f at any one point x change with time t ?
Shading represents the value of the property f , say
light = high value,
dark = low value
Steady
t
Physically
Mathematically
abj
t+dt
Unsteady
t
t+dt
does not change
changes
with time t
with time t

f ( x , t )
0
t



f ( x, t )  f ( x )

f ( x , t )
0
t


f ( x, t )
17
Steady and Uniform Property Field f

Uniformity in a region R: Does f at any one time t change with spatial location x ?
 Uniformity in a region R:
Does f at any one time t change with

spatial location x ?
Uniform in R
Non-uniform in R
(at time t)
(at time t)
t
t
Physically
does not change

with spatial location x
Mathematically
abj

f  0


f ( x, t )  f (t )
changes

with spatial location x

f  0


f ( x, t )
18
Example:
Steadiness and Uniformity
Steady ?
?
t
t+dt
t
t+dt
t
?
Uniform at time t + dt?
?
t+dt
t
abj
Uniform at time t?
t+dt
19
Classification of Fluid Flows
Flows
1
Compressible
Incompressible
Indicating Parameters (Otherwise stated, for the latter)
 = constant
 M < 1 - Subsonic
 M = 1 - Sonic;
2
Subsonic, (Tran)Sonic, Supersonic, Hypersonic
 M ~ 1 - Transonic
 M > 1 - Supersonic
 M > ~ 5 - Hypersonic
3
Viscous Flow
Inviscid (Frictionless)
4
One, Two-, Three- Dimensional Flow
5
Laminar
 = 0, or effect of viscous stress can be neglected.
The number of spatial coordinates x, y, z, that is required to specify the velocity field.
Laminar: Smooth and orderly (non-random), can be steady or unsteady.
Turbulent
Turbulent Flow: Random fluctuation of velocity field, inherently unsteady.
6
Internal
External
Internal: Flow that is bounded by solid surfaces.
External: Flow over body immersed in unbounded fluid.
7
abj
Rotational
Irrotational
Vorticity (related to angular velocity)



 :   u  0
20
Laminar-Transition-Turbulence: Subsonic Jet
Laminar: orderly motion
Turbulent: random motion
Transition from laminar flow to turbulent flow via instability in subsonic jet.
From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.
abj
21
Laminar-Transition-Turbulence: Subsonic Jet
Transition from laminar flow to turbulent flow via instability in subsonic jet.
From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.
abj
22
abj
23
Example: Classification and Velocity Field
Question:
1. Classify the following velocity fields by stating whether it is

steady

2.
if unsteady, also find

V
t
unsteady?

1-, 2-, or 3-dimensional?

Then, write down the functional form of the field, e.g.,
Also find

the divergence of the velocity field



V ( x, y , t )

 V
NOTE: The divergence of the velocity field is related to the compressibility of the flow.
the vorticity, which is defined as the curl of the velocity field

abj
or

 :  V

NOTE: The vorticity is a measure of the angular velocity of a fluid element.
24
Example: Classification and Velocity Field
3.
In addition, using MLtT as the set of primary dimensions, also state the dimension of
the constant a, b, c, etc.

V  (ax)iˆ  (by) ˆj  (ct )kˆ

V  [a sin(bect )]iˆ  (dx) ˆj



V  a 1  ( y / b) 2 iˆ
abj
25
Example: Classification and Velocity Field
abj
26
Representation of A Vector Field: A Vector Plot
Let a velocity field be given by

V  (ax) iˆ  (by) ˆj
where the field is defined in the region
m/ s
  x   and y  0 , x and y are given in
meters, and a = 1 s-1 and b = -1 s-1. Sketch a vector plot for this field.
abj
27
Example: A Vector Plot

Sketch a vector plot for the following velocity fields. x and y are given in meters.

V  (ax) iˆ  (by) ˆj
Same as above,
abj
m / s,
a  1 s 1, b  1 s 1
a  2 s 1 , b  2 s 1
28