Chapter 2
Aerodynamics: Some Fundamental Principles and Equations
There is so great a difference between a fluid and a collection of solid particles that the laws of pressure and of equilibrium of fluids are very different from the laws of the pressure and equilibrium of solids . Jean Le Rond d’Alembert, 1768
(a) The control volume fixed in space : ●fixed →no change in shape or volume ●particles, therefore mass passing through boundary ●forces interacting on boundary ●energy may exchange through boundary (work, heat)
A ds A dS
C S
Where area S is bounded by the closed curve C.
The surface integral of A over S is related to the volume integral of ▽· (divergence of A) over V by divergence’ A theorem (also called Gauss Formula):
2.3.2 Infinitesimal fluid element approach
Definition of infinitesimal fluid element: Imagine an infinitesimally small fluid element of a differential volume dV (微元体积, 体积微元). The fluid element is infinitesimal in the same sense as differential calculus; However it is large enough to contain enough fluid particles, i.e., contains a huge number of molecules. ●A fluid element may have shape, volume. (a) May be fixed in space (fluid moving through) (b) May move with fluid (always the same fluid particles)
Aerodynamics is a fundamental science, steeped (浸，泡，浸透）in physical observation. So we should make every effort to gradually develop a “physical feel” for the material.
1. Invoke （诉诸于）three fundamental physical principles which are deeply entrenched (deeply-believed, in-grained根深蒂固相 信地) in our macroscopic observations of nature, namely, a. Mass is conserved, that’s to say, mass can be neither created nor destroyed.
volume V, and the closed surface which bounds the control volume is defined as control surface S.
Fixed control volume and moving control volume: (a) A control volume may be fixed in space with the fluid moving through it; (b) Alternatively, the control volume may be moving with the fluid such that the same fluid particles are always inside it.
Emphasis of this section:
1. What is a suitable model of the fluid? 2. How do we visualize this squishy substance in
order to apply the three fundamental principles? 3. Three different models mostly used to deal with aerodynamics. finite control volume （有限控制体） infinitesimal fluid element （无限小流体微团） molecular （自由分子）
2.2 Review of Vector relations
2.2.1 to 2.2.10 Skipped over 2.2.11 Relations between line, surface, and volume integrals
The line integral of A over C is related to the surface integral of ▽×A (curl of A) over S by Stokes’ theorem (Stokes formula):
How to make reasonable judgments on difficult problems: An important virtue of all successful aerodynamicists is that they have good “physical intuition” based on thought and experience, which allows them to make reasonable judgments on difficult problems.
b. Newton’s second law: force=mass☓acceleration
c. Energy is conserved; it can only change from one form to another 2. Determine a suitable model of the fluid. 3. Apply the fundamental physical principles listed in item 1 to the model of the fluid determined in item 2 in order to obtain mathematical equations which properly describe the physics of the flow.
PART
I
(Chapters 1 and 2)
FUNDAMENTAL PRINCIPLES （基本原理）
In part I, we cover some of the basic principles that apply to aerodynamics in general. These are the pillars on which all of aerodynamics is based
In this chapter, basic equations of aerodynamics will be derived.
(How are the equations developed?)
Phed with the development of these equations
Which one to use depends on the focus of our investigation on fluid flow.
Notes:
In some other text books, !!!!!define or call the control volume fixed in space ☛control volume (控制体), !!!!! whereas call the moving control volume containing the same particles ☛system (系统）(归柯庭,汪军,等：《工程流体力学》，科学出版社) or 质量体 (张兆顺,崔桂香:《流体力学》,清华大学出版社) Besides, in these books, they use Reynolds Transport Theorem (雷诺输运定理) to derive the continuity equation, momentum equation, and energy equation.
A dS ( A)dV
Where volume V is bounded by the closed surface S. If p represents a scalar field, a vector relationship analogous to divergence theorem is given by gradient theorem:
(b) The control volume moving with the fluid ●moving with fluid →may change in shape and volume ●no particles, therefore no mass passing through boundary (no change in mass) ●forces interacting on boundary ●exchange of energy, work, heat may exist through boundary