On rotational gas flows

Introduction. The main body of the science of aerodynamics is based on the classical theory of frictionless, incompressible, irrotational fluids. Recently airplanes have attained such high velocities that this fluid model has proved to be too restricted and interest has centered on the irrotational motion of frictionless, compressible fluids. By the term "compressible fluid" one generally means a fluid for which the density p and pressure p are connected by the isentropic relation pp~^ = const. However, the student of aerodynamics is frequently interested in supersonic phenomena and because of the possible occurrence of shock waves, such flows cannot be described, in general, by isentropic, irrotational flows. Accordingly, it becomes necessary to study the motion of gases under less restricted conditions. Let us call a fluid barotropic when there is a unique functional relationship between the pressure and the density of the fluid. The most important examples are the incompressible fluid where the same constant density belongs to each pressure and the isentropic fluid where the relation pp~y = const, holds. The dynamics of frictionless barotropic fluids is based on a theorem due to Lagrange. If a fluid particle is irrotational at one moment, it will remain so for all subsequent time. One can generally assume in aerodynamics that the air starts from rest. The dynamics of the flow can then be summed up in the single statement that the motion is irrotational. It follows that the velocity distribution admits a potential, and the comparative mathematical simplicity of the dynamics of frictionless barotropic fluids follows from this fact. Classical fluid dynamics deals almost exclusively with the theory of frictionless barotropic fluids. To find an example of frictionless non-barotropic fluids, we turn to the theory of the propagation of waves. When Newton developed his theory of soundwaves, he assumed that the motion of air was isothermal. Later his theory was superseded by a better one which assumes isentropic motion. Thus, both theories assumed that the transmitting medium was barotropic. The mathematical theory of onedimensional large disturbances, a much more difficult problem, was developed first by Riemann, who again assumed that the flow is isentropic. However, the isentropic theory of shock-waves turns out to be fallacious because it can be shown to violate the law of conservation of energy. When shock-waves are considered the fluid model must be extended to include non-barotropic fluids. In this connection, let us draw attention to the thermodynamical aspect of the general theory of compressible fluids. In the case of a three-dimensional flow there are six unknowns: three velocity components, pressure, density and temperature. The laws of conservation of matter and momentum together with the equation of state yield only five equations. To get the missing sixth equation the law of conservation of energy, i.e., the first law of thermodynamics, must be used. Flows will be isentropic only when as a consequence of these laws the entropy turns out to be a constant.