Transcript 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:
Lagrangian
VS
Eulerian Descriptions

4.
5.
abj
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
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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)
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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
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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
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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 


Newton’s law of viscosity
 yx   (du / dy)
abj
(du / dy)
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Fluid Mechanics
abj
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Fundamental Concept: Fluid Mechanics
Fluid Mechanics
=
Mechanics (Force and Motion) and
Thermodynamics (Energy and Energy Transfer)
of Fluid Motion
abj
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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 )  ?
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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
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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
Below
abj
at a point:
Function:
A fluid property f is assumed as a continuous field function of

space and time: f  f E ( x, t )
this limit, the continuum assumption breaks down, and the molecular motion of molecules must
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be taken into account.
Two Methods of Description of Fluid Motion
abj
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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  fL ( 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)
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Lagrangian Description

f  f L ( X , t)
reads
the current (at time t) value of the

fproperty of the materialX

f L ( X , t )is equal to
particle
.
Eulerian Description

f  f E ( x, t )
abj
reads
f at position

f
(
x
, t) .
is
equal
to
t
E
the value of the property

x
and time
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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.]
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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 )
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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 )
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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 )
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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
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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.
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Internal
External
Internal: Flow that is bounded by solid surfaces.
External: Flow over body immersed in unbounded fluid.
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abj
Rotational
Irrotational
Vorticity (related to angular velocity)



 :   u  0
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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
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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
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abj
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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.
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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
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Example: Classification and Velocity Field
abj
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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
  x  
m/ s
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
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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
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