Expansion Waves

In the expansion wave, the flow velocity is increased and the pressure, density, and temperature are decreased along the stream line through the expansion fan. Since Oj 02, it follows that Mi M2. The flow through an expansion wave is continuous and is accompanied by an isentropic change known as a Prandtl-Meyer expansion wave. The relationship between the deflection angle and the Mach number is represented by the Prandtl-Meyer expansion equation.l - l 

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Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflec ted, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through sohds in a complex array of types of elastic waves. Sound is charac terized by its amplitude, frequency, phase, and direction of propagation. 

Figure 2.6. (a) Compression wave steepens to a shock wave in a medium for which stability criteria are satisfied, where the trailing part of the wave overtakes the leading part, (b) Expansion wave broadens as the leading part of the wave outruns the trailing part. 

Expansion waves are the mechanism by which a material returns to ambient pressure. In the same spirit as Fig. 2.2, a rarefaction is depicted for intuitive appeal in Fig. 2.7. In this case, the bull has a finite mass, and is free to be accelerated by the collision, leading to a free surface. Any finite body containing material at high pressure also has free surfaces, or zero-stress boundaries, which through wave motion must eventually come into equilibrium with the interior. Expansion waves are also known as rarefaction waves, unloading waves, decompression waves, relief waves, and release waves. Material flow is in the same direction as the pressure gradient, which is opposite to the direction of wave propagation. 

The fact that shock waves continue to steepen until dissipative mechanisms take over means that entropy is generated by the conversion of mechanical energy to heat, so the process is irreversible. By contrast, in a fluid, rarefactions do not usually involve significant energy dissipation, so they can be regarded as reversible, or isentropic, processes. There are circumstances, however, such as in materials with elastic-plastic response, in which plastic deformation during the release process dissipates energy in an irreversible fashion, and the expansion wave is therefore not isentropic. 

Throughout this book, a shock pulse (a steady compression wave followed by an expansion wave) will be represented as a profile, such as in Fig. 2.6. In Fig. 2.8 we show a series of P-x snapshots of pressure versus propagation distance x for an initially square pulse, at a series of times t. For a fluid with... 

For simplicity, we have shown an expansion wave in which the pressure is linearly decreasing with time. This, in general, is not the case. The release behavior depends on the equation of state of the material, and its structure can be quite complicated. There are even conditions under which a rarefaction shock can form (see Problems, Section 2.20 Barker and Hollenbach, 1970). In practice, there are many circumstances where the expansion wave does not propagate far enough to fan out significantly, and can be drawn as a single line in the x t diagram. 

In Seetion 2.8, we noted that most expansion waves are isentropie. It was shown in Seetion 2.4 that the differenee between the Hugoniot and isentrope is small for hydrodynamie materials at small strains. Thus we ean also represent relief waves in the P-u plane with the same eurve used to represent shoek waves, if the strains are not too large. 

This result, called the Riemann Integral, can be applied to unsteady isentropic compression waves as well as to expansion waves. By defining a Riemann function ... 

Application of this procedure to inadvertently ignited safety valve discharges can involve a special problem. Certain combinations of pressure ratio and length of safety valve riser can result in choked flow, with a pressure discontinuity at the exit. The pressure of the jet then adjusts to atmospheric pressure in a system of shock waves or expansion waves over a distance of a few pipe diameters. These waves can affect the local mixing of the jet with the crosswind. Since the calculation procedure incorporates correlations for subsonic jets, it cannot be expected to be entirely accurate in this case. Nevertheless, since the wave system... 

By direct action of the shock or expansion wave from the explosive. 

Regions III and IV give expansion waves, which are the low-velocity waves already classified as deflagrations. That these waves are subsonic can be established from the relative order of magnitude of the numerator and denominator of Eq. (5.6a), as has already been done in Chapter 4. 

One can verify that regions I and II give compression waves and regions III and IV give expansion waves by examining the ratio of Au to // obtained by dividing Eq. (5.12) by the square root of Eq. (5.5) ... 

The extent to which a detonation will propagate from one experimental configuration into another determines the dynamic parameter called critical tube diameter. It has been found that if a planar detonation wave propagating in a circular tube emerges suddenly into an unconfined volume containing the same mixture, the planar wave will transform into a spherical wave if the tube diameter d exceeds a certain critical value dc (i.e., d > dc). II d < d.. the expansion waves will decouple the reaction zone from the shock, and a spherical deflagration wave results [6],... 

When a shock wave propagates from one end (A) to the other end (B) of a solid body, a compression force is exerted in the wake of the shock wave that acts within the solid body. The shock wave is reflected at B and becomes an expansion wavethat propagates towards A. An expansion force is then exerted in the wake of the expansion wave that travels back from B. This process is shown schematically in Fig. 9.6. 

When a compression wave travels into materials such as rock or concrete, no damage is inflicted on the materials because of their high compressive strength. However, when an expansion wave travels within the same materials, mechanical damage results near B. This is because rock and concrete are materials of low tensile strength. Fig. 9.7 shows a pair of photographs of the surface (A) and the reverse... 

Fig. 9.6 Schematic representation of shock wave propagation and the generation of a reflected expansion wave within a solid wall.

When an explosive is detonated in water, a shock wave propagates through the water accompanied by a bubble. The chemical energy of the explosive is converted into shock wave energy and bubble energy. The volume of the bubble is increased by the expansion wave and decreased by the compression wave in an oscillatory fashion. The maximum size of the bubble is determined according toili i i... 

Fig. 12.11 shows the structure of a rocket plume generated downstream of a rocket nozzle. The plume consists of a primary flame and a secondary flame.Fil The primary flame is generated by the exhaust combustion gas from the rocket motor without any effect of the ambient atmosphere. The primary flame is composed of oblique shock waves and expansion waves as a result of interaction with the ambient pressure. The structure is dependent on the expansion ratio of the nozzle, as described in Appendix C. Therefore, no diffusional mixing with ambient air occurs in the primary flame. The secondary flame is generated by mixing of the exhaust gas from the nozzle with the ambient air. The dimensions of the secondary flame are dependent not only on the combustion gas expelled from the exhaust nozzle, but also on the expansion ratio of the nozzle. A nitropolymer propellant composed of nc(0-466), ng(0-369), dep(0104), ec(0 029), and pbst(0.032) is used as a reference propellant to determine the effect of plume suppression. The burning rate characteristics of the propellants are shown in Fig. 6-31. Since the nitropolymer propellant is fuel-rich, the exhaust gas forms a combustible gaseous mixture with the ambient air. This gaseous mixture is ignited and afterburning occurs somewhat downstream of the nozzle exit. The major combustion products in the combustion chamber are CO, Hj, CO2, N2, and HjO. The fuel components are CO and H2, the mole fractions of which at the nozzle throat are co(0.47) and iH2(0.24).

The basic characteristics of a one-dimensional shock wave are described in Chapter 1 of this text. However, the shock waves in supersonic flow propagate not only one-dimensionally but also two- or three-dimensionally in space. For example, the shock waves formed at the air-intake of a ducted rocket are two- or three-dimensional in shape. Expansion waves are also formed in supersonic flow. The pressure downstream of an expansion wave is reduced and the flow velocity is increased. With reference to Chapter 1, brief descriptions of the characteristics of a two-dimensional shock wave and of an expansion wave are given here.Ii-5]... 

When a supersonic flow emerges from a rocket nozzle, several oblique shock waves and expansion waves are formed along the nozzle flow. These waves are formed repeatedly and form a brilliant diamond-Uke array, as shown in Fig. C-5. When an under-expanded flow, i. e., having pressure higher than the ambient pressure is formed at the nozzle exit, an expansion wave is formed to decrease the pressure. This expansion wave is reflected at the interface between the flow stream and the ambient air and a shock wave is formed. This process is repeated several times to form a diamond array, as shown in Fig. C-6 (a). 

Figure C-4. Expansion wave formed in supersonic flow along a wall surface with a corner of negative angle.

Expansion waves in gaseous detonations) 63) J-A. Fay, "The Structure of Gaseous Detonation Waves", 8thSympCombstn... 

Expansion Wave. In aerodynamics expansion wave is a pressure wave that has the effect of decreasing the density of air as the air passes thru it, while the compression wave has die effect of increasing density of air. 

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