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Chapter 2 Aerodynamics: Some Fundamental Principles and Equations

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Chapter 2 Aerodynamics: Some Fundamental Principles and Equations

SONG, Jianyu

Feb. 28.2009

- 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)

- Finite Control Volume Approach
- Infinitesimal Fluid Element Approach
- Molecular 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

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

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.

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

- Conservation of mass
- Newton’s second law
- Conservation of energy

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

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

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.

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

time rate of change of momentum:

G:Net flow of momentum out of control volume across surface S

H:Time rate of change of momentum due to unsteady fluctuations of flow properties inside V

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)

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

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

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

- Conservation of mass---continuity equation
- Newton’s second law---momentum equation
- Conservation of energy---energy equation

Figure 2.26

Show the example of density field

Local derivative

Convective derivative

An interesting analogous P144

“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.

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

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

Figure 2.32

Consider an infinitesimal fluid element moving in a flow field.

it may also rotate and become distorted

Figure 2.33

Consider a 2D flow in x-y plane

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 ξ≠0 at every point in a flow, the flow is called rotational, this implies that the fluid elements have a finite angular velocity

If ξ=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

Lethe angle between sides AB and AC be denoted by κ

The strain of the fluid element as seen in the xy plane is the change in κ.

Thank you!

Q&A