Compressible Flow and Mach Number

Linearize mass and momentum conservation around a uniform fluid at rest. Small pressure and density perturbations are related through a derivative taken at fixed specific entropy when acoustic disturbances are effectively adiabatic and reversible. Their propagation speed squared is that derivative. The derivative is positive for the stable equilibrium states considered here. An isothermal derivative describes a different thermodynamic response, so substituting it without explaining the timescale and heat transfer changes the physical model.

For a calorically perfect ideal gas, the derivative reduces to gamma times R times absolute temperature, where R is the specific gas constant and gamma is the ratio of specific heats. This specialization assumes the ideal-gas equation of state and constant heat capacities over the relevant range. Real gases, reacting mixtures, and strongly varying molecular excitation need a more careful thermodynamic model. The useful intuition survives: sound speed is a property of the local state, not a universal fixed number assigned to air everywhere.

Continuity states that steady mass flux rho U A remains constant along a quasi-one-dimensional streamtube. That does not determine the separate density and velocity changes. Momentum conservation supplies another relationship, and energy conservation plus an equation of state closes the thermodynamic problem. In an adiabatic steady flow without shaft work or significant elevation change, stagnation enthalpy h plus U squared divided by two is constant. For constant specific heat at constant pressure, enthalpy becomes cp times temperature.

Small velocity-driven density changes in a smooth isentropic flow scale with Mach number squared, explaining why a low-Mach approximation can work when thermal and compositional density variations are also small. It is not a universal rule that every slow flow has constant density: heating or mixing may cause large changes. In the waves lab, predict cancellation at equal amplitude and 180 degrees phase difference. The two traveling sine components illustrate linear pressure-perturbation superposition; their controls do not change mean velocity, sound speed, or Mach number.

For smooth steady quasi-one-dimensional inviscid isentropic flow, combine continuity with streamwise momentum and the acoustic derivative to obtain dA/A = (M squared minus one) dU/U. Thus a converging passage accelerates a subsonic stream, whereas a diverging passage accelerates a supersonic stream. A smooth sonic crossing in a suitable nozzle occurs at a throat. Geometry alone does not guarantee that crossing: reservoir conditions and downstream pressure determine the realized solution and whether choking occurs.

The area–Mach relation below assumes a calorically perfect ideal gas with fixed gamma and fixed stagnation conditions. A star denotes the hypothetical sonic area for the same mass flow and stagnation state, even when no actual sonic section is present. An area ratio larger than one generally has both subsonic and supersonic branches. Shocks produce entropy and invalidate a single isentropic connection across them; friction, heat transfer, and boundary-layer displacement also require extensions. Never select the faster branch simply because an algebraic solver returns it.