Thermodynamics

856 | ThermodynamicsFIGURE 17–43Still frames from schlierenvideography illustrating thedetachment of an oblique shock from acone with increasing cone half-angle din air at Mach 3. At (a) d 20 and(b) d 40, the oblique shock remainsattached, but by (c) d 60, theoblique shock has detached, forminga bow wave.Photos by G. S. Settles, Penn State University.Used by permission.Ma 1d(a) (b) (c)corresponds to a normal shock, and the weak case, b b min , representsthe weakest possible oblique shock at that Mach number, which is calleda Mach wave. Mach waves are caused, for example, by very smallnonuniformities on the walls of a supersonic wind tunnel (several can beseen in Figs. 17–36 and 17–43). Mach waves have no effect on the flow,since the shock is vanishingly weak. In fact, in the limit, Mach waves areisentropic. The shock angle for Mach waves is a unique function of theMach number and is given the symbol m, not to be confused with thecoefficient of viscosity. Angle m is called the Mach angle and is found bysetting u equal to zero in Eq. 17–46, solving for b m, and taking thesmaller root. We getMach angle: m sin 1 11>Ma 1 2(17–47)Since the specific heat ratio appears only in the denominator of Eq.17–46, m is independent of k. Thus, we can estimate the Mach number ofany supersonic flow simply by measuring the Mach angle and applyingEq. 17–47.FIGURE 17–44Shadowgram of a one-half-in diametersphere in free flight through air at Ma 1.53. The flow is subsonic behindthe part of the bow wave that is aheadof the sphere and over its surface backto about 45. At about 90 the laminarboundary layer separates through anoblique shock wave and quicklybecomes turbulent. The fluctuatingwake generates a system of weakdisturbances that merge into thesecond “recompression” shock wave.Photo by A. C. Charters, Army Ballistic ResearchLaboratory.Prandtl–Meyer Expansion WavesWe now address situations where supersonic flow is turned in the oppositedirection, such as in the upper portion of a two-dimensional wedge at anangle of attack greater than its half-angle d (Fig. 17–45). We refer to thistype of flow as an expanding flow, whereas a flow that produces an obliqueshock may be called a compressing flow. As previously, the flow changesdirection to conserve mass. However, unlike a compressing flow, an expandingflow does not result in a shock wave. Rather, a continuous expandingregion called an expansion fan appears, composed of an infinite number ofMach waves called Prandtl–Meyer expansion waves. In other words, theflow does not turn suddenly, as through a shock, but gradually—each successiveMach wave turns the flow by an infinitesimal amount. Since eachindividual expansion wave is isentropic, the flow across the entire expansionfan is also isentropic. The Mach number downstream of the expansionincreases (Ma 2 Ma 1 ), while pressure, density, and temperature decrease,just as they do in the supersonic (expanding) portion of a converging–diverging nozzle.Prandtl–Meyer expansion waves are inclined at the local Mach angle m,as sketched in Fig. 17–45. The Mach angle of the first expansion wave iseasily determined as m 1 sin 1 (1/Ma 1 ). Similarly, m 2 sin 1 (1/Ma 2 ),