Mathematical Description of Fluid Flows
Several concepts of great value in later derivations of the defining equations are reviewed in this section. It is important that students are familiar with these ideas, especially the mental images that they require. Students probably are already familiar with some or all of these ideas from their fundamental courses in mechanics or thermodynamics.
Lagrangian versus Eulerian Mathematical Description
In studying the motion of particles and rigid bodies in mechanics courses, students learned the Lagrangian approach in which each element of the system is represented by a detailed model of its absolute motion and its motion relative to other parts of the system. In aerodynamics (or fluid mechanics), two different mathematical viewpoints may be adopted. Following the Lagrangian methods used in mechanics, physical laws are applied to individual particles in a flow. Equations then are derived that predict the location and status of individual fluid particles as a function of time as they move through space. The equations of dynamics typically use the Lagrangian viewpoint, time being the only independent variable. In these situations, it is necessary to keep track of the positions and velocities of all elements that comprise the system. However, for fluid-flow problems, so many particles are involved (virtually, an infinite number) in the field that this modeling stratagem usually is not practical.
Aerodynamics is concerned with a continually streaming fluid and the task is to determine distributions of flow velocities and fluid properties within a flow-field region rather than tracking the motion of specific particles passing through the region. For example, pressures at the surface of an airfoil or at points of interest in the flow are the main concern, not the behavior of the individual particles that exert these pressures. For this reason, a field, or Eulerian, representation is preferred in most aerodynamics problems and this viewpoint is taken herein. In this representation, the fluid properties and velocities throughout the flow are expressed in terms of position and time. Thus, flow variables such as pressure and velocity are described in terms of the flow-field coordinates and time, so that for an unsteady flow in 3-dimensional Cartesian coordinates (x, y,z), the pressure p = p (x, y,z, t) and the velocity vector V = V(x, y,z, t).