Chapter 2 Aerodynamics: Some Fundamental Principles and Equations

1
Chapter 2 Aerodynamics Some Fundamental
Principles and Equations

• SONG, Jianyu
• Feb. 28.2009

2
What will we learn from this chapter?

• How to model the fluid?(3 points)
• How to describe the fundamental principles with
  the model mathematically?(3 points)
• Learn some concepts for studying the fluid.(3
  points)

3
How to model the fluid?
4
Three Approaches

1. Finite Control Volume Approach
2. Infinitesimal Fluid Element Approach
3. Molecular Approach

5
Finite Control Volume Approach

• Finite Control Volume is
• a closed volume drawn with a finite region of the
  flow.
• Denoted by V
• Finite Control Surface is the closed surface
  which bounds the control volume
• Denoted by S
• Figure 2.13 (l and r)
• May be fixed in space
• May be moving with the fluid

6
Infinitesimal Fluid Element Approach

• Infinitesimal Fluid Element is an infinitesimally
  small fluid element in the flow, with a
  differential volume dV
• Remark It has the same meaning as in calculus ,
  however, it should be large enough to contain a
  huge number of molecules so that it can be viewed
  as a continuous medium.
• Figure 2.14 (l and r)
• May be fixed in space
• May be moving with the fluid

7
Molecular Approach

• In actuality, the motion of a fluid is the mean
  motion of its atoms and molecules.
• More elegant method with many advantages in the
  long run.
• However, it is beyond the scope of this book.

8
Fundamental Principles
9
Review some calculus

• Stokes theorem
• Let A be a vector field. The line integral of A
  over C is related to the surface integral of A
  over S
• Divergence theorem
• The surface and volume integrals of the vector
  field A are related
• Gradient theorem
• If p represents a scalar field, a vector
  relationship analogous to the equation

10
Three Fundamental Principles

• Conservation of mass 
• Newtons second law
• Conservation of energy

11
Conservation of mass

• Finite Control Volume
• The fixed model
• V and S is constant with time, but mass in the
  volume may change
• Figure 2.18
• Description
• Edge view of small area A.
• A small enough so that the velocity field V is
  constant

12
Conservation of mass

• Mass can be neither created nor destroyed
• Figure 2.19
• Velocity field V
• vector elemental surface area dS
• the - is for the fact that the time rate of
  decrease of mass inside the control volume
• The last equation is also called
• Continuity equation
• It is one of the most fundamental equations of
  fluid dynamics

13
Conservation of mass

• In the last slide. we get the equation dealing
  with a finite space
• Further, we want to have equations that relate
  flow properties at a given point
• Divergence theorem
• This equation is the continuity equation in the
  form of a partial differential equation.

14
Newtons second law

• Finite Control Volume
• the fixed model
• Force time rate of change of momentum
• Force exerted on the fluid as it flows through
  the control volume come from two sources
• Body force act at a distance on the fluid
  inside V
• Surface forces pressure and shear stress acting
  on the control surface S
• The computation of will be in
  Chapter 7

15
Newtons second law

• time rate of change of momentum
• GNet flow of momentum out of control volume
  across surface S
• HTime rate of change of momentum due to unsteady
  fluctuations of flow properties inside V

16
Newtons second law

• Just for the same reason as the conservation of
  mass, we want to have equations that relate flow
  properties at a given point
• As it is a vector function we only consider the
  x part
• (Fx)viscous denotes the proper form of the x
  component of the viscous shear stresses when
  placed inside the volume integral(Chapter 15)

17
Conservation of energy

• Energy can be neither created nor destroyed it
  can only change in form.
• System a fixed amount of matter contained within
  a closed boundary
• Surroundings the region outside the system.
• Thermodynamics first law
• Apply the first law to the fluid inside control
  volume
• Figure 2.19
• Power equation

An incremental amount of heat be added to the
system The work done on the system by the
surroundings Change the amount of internal
energy in the system
18
Conservation of energy

• Let the volumetric rate of heat addition per unit
  mass be denoted by
• The rate of heat addition to the control volume
  due to viscous effects by
  (Chapter 15)
• -----------------------------------------
• Recall f is the body force per unit mass
• For viscous flow, the shear stress on the control
  surface will also perform work(chapter 15)
• Denote this distribution by
• -----------------------------------------
• Internal energy e (is due to the random motion of
  the atoms and molecules)
• The fluid inside the control volume is not
  stationary, it is moving at the local velocity V

19
Conservation of energy

• In the same way, we can get a partial
  differential equation for total energy from the
  integral form given above.

20
Three Fundamental Principles

• Conservation of mass---continuity equation 
• Newtons second law---momentum equation
• Conservation of energy---energy equation

21
Some Concepts
22
Substantial Derivative

• Figure 2.26
• Show the example of density field
• Local derivative
• Convective derivative
• An interesting analogous P144

23
Pathlines, Streamlines

• Where the flow is going?
• Trace the path of element A as it moves
  downstream from point 1, such a path is defined
  as pathline for element A
• A streamline is a curve whose tangent at any
  point is in the direction of the velocity vector
  at the point.
• A analogue in P148
• Pathline a time-exposure photograph of a given
  fluid element
• Steamline a single frame of a motion picture
• For steady flow (is one where the flow field
  variables at any point are invariant with
  time)they are the same.

24
Pathlines, Streamlines

• Given the velocity field of a flow, how can we
  obtain the mathematical equation for a
  streamline?
• Let ds be a directed element of the streamline
• Knowing u, v, and w as functions of x, y, and z,
  they can be integrated to yield the equation for
  the streamline f(x, y, z)0

25
Pathlines, Streamlines

• Physical meaning of the equation
• Consider a streamline in 2D
• Figure 2.30a
• Streamtube
• Consider the streamlines which pass through all
  points on C
• Figure 2.30b

26
Angular velocity, Vorticity and Strain

• Figure 2.32
• Consider an infinitesimal fluid element moving in
  a flow field.
• it may also rotate and become distorted

27
Angular velocity, Vorticity and Strain

• Figure 2.33
• Consider a 2D flow in x-y plane

28
Angular velocity, Vorticity and Strain

• Define a new quantity vorticity
• write it in a more compact way
• In a velocity field, the curl of the velocity is
  equal to the vorticity

• The above leads to two important definitions
• If ??0 at every point in a flow, the flow is
  called rotational, this implies that the fluid
  elements have a finite angular velocity
• If ?0 at every point in a flow, the flow is
  called irrotational. This implies that the fluid
  elements have no angular velocity rather, their
  motion through space is a pure translation.
• Figure 2.36 for contrast

29
Angular velocity, Vorticity and Strain

• Lethe angle between sides AB and AC be denoted by
  ?
• The strain of the fluid element as seen in the xy
  plane is the change in ?.

30
Thank you!

• QA