Physics of Fluids

2.1 Aerodynamic Forces

Because the objective of aerodynamics is the determination of forces acting on a flying object, it is necessary that we clearly identify their source. Lift and drag forces, for example, are the result of interactions between the airflow and vehicle surfaces. Part of the force must be a result of pressure variations from point to point along the surface; another part must be related to friction of gas particles as they scrub the surface. Clearly, the key to understanding these forces is found in details of the fluid motions. The application of simple molecular concepts provides considerable insight into these motions.

Modeling of Gas Motion

As a branch of fluid mechanics, aerodynamics is concerned with the motion of a continuously deformable medium. That is, when acted on by a constant shear force, a body of liquid or gas changes shape continuously until the force is removed. This is unlike a solid body, which only deforms until internal stresses come into equilibrium with the applied force; that is, a solid does not deform continuously.

To understand the motion of a fluid, it is necessary to apply a set of basic physical laws, which consist of some or all of the following:

• conservation of mass (the continuity equation)

• Newton’s Second Law of Motion (the momentum equation)

• First Law of Thermodynamics (the energy equation)

• Second Law of Thermodynamics (the entropy equation)

One skill that the student must develop is the effective application of approxi­mation and simplification methods. A proper set of approximations may make unnecessary the use of some laws to produce a practical yet accurate solution for a given problem. This approach is possible only if a clear understanding of the physics of a fluid motion is attained. It is, of course, possible to construct a mathematically correct solution to an incorrectly formulated problem or a solution that is based

on an inappropriate set of assumptions. In such cases, the results can be confusing or misleading or can even lead to costly mistakes. There is no substitute in aero­dynamics for a sound understanding of the fundamental physical laws on which fluid mechanics is based.

It also may be necessary to introduce additional mathematical models or relationships to supplement those in the preceding list. For example, it often is necessary to use equations that characterize a working fluid. These equations of state, or constitutive equations, describe the physical attributes of a fluid. For example, it often is the case in practical problems that the fluid is an ideal or perfect gas, for which the equation of state is:

p = pRT, (2.1)

which is a special relationship among the thermodynamic-state variables, pressure p, density p, and temperature T, needed to describe the behavior of a gas. R is the gas constant—a constant of proportionality—that is determined by the molecular con­figuration of the gas. This and other equations are discussed in considerable detail as needed throughout this chapter.

With regard to details of the molecular structure of a working fluid, we can choose to approach problems from the standpoint of the molecular motion, or a continuum model can be applied that does not attempt to address directly the actual small-scale particle motion. The former is called statistical mechanics, or the kinetic theory of gases. These are fascinating disciplines; however, we do not need a full description of molecular motion to predict forces on an aerodynamic vehicle unless it is flying at an extremely high altitude. In this book, we concentrate almost exclu­sively on a continuum model, which takes advantage of simplifications that result from the relatively large geometrical scale in realistic engineering situations when compared with an atomic or molecular-length scale.

In contrast, it is important to be aware of the molecular origin of physical quantities. In an ideal gas, the size of individual molecules is small when com­pared with the average distance between them. On this basis, the change in kinetic energy due to mutual attraction is negligible. Also, collisions between molecules can be considered as perfectly elastic so that there is no loss of energy due to per­manent deformation of the molecular configuration. The average distance traveled between collisions is the mean free path, X, which is important in the kinetic theory of gases. The mean random velocity of molecules is represented herein by the symbol c.