Normal shock waves

Consider the propagation of a one-dimensional normal shock wave in a gas medium heavily laden with particles. Select Cartesian coordinates attached to the shock front so that the shock front becomes stationary. The changes of velocities, temperatures, and pressures of gas and particle phases across the normal shock wave are schematically illustrated in Fig. 6.12, where the subscripts 1, 2, and oo represent the conditions in front of, immediately behind, and far away behind the shock wave front, respectively. As shown in Fig. 6.12, a nonequilibrium condition between particles and the gas exists immediately behind the shock front. Apparently, because of the finite rate of momentum transfer and heat transfer between the gas and the particles, a relaxation distance is required for the particles to gain a new equilibrium with the gas. 

The elapsed time for particles to pass through the shock front may be approximated by dv/U. Since U is of the same order of magnitude as the speed of sound in the gas, the ratio of the flying time to the Stokes relaxation time of a particle can be expressed by 

To analyze the shock wave behavior in a gas-solid flow, several assumptions are made  

The following analysis is based on the method suggested by Carrier (1958). The law of conservation of mass requires 

consider the velocities and temperatures in two equilibrium regions, one in front of the shock and the other far behind the shock. Applying Eq. (6.83) and Eq. (6.85) to these two regions yields 

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For a sudden change or normal shock wave to occur, the entropy change per unit mass of fluid must be positive. 

S2 — S is positive when Ma 1 > 1. Thus a normal shock wave can occur only when the flow is supersonic. From equation 4.96, if Mai > 1, then Ma2 < 1, and therefore the flow necessarily changes from supersonic to subsonic. If Mai = 1> Mai — 1 also, from equation 4.96, and no change therefore takes place. It should be noted that there is no change in the energy of the fluid as it passes through a shock wave, though the entropy increases and therefore the change is irreversible. 

The end of regime 2 is when the shock wave occurs at the exit plane, this is shown as condition (e). In regime 2 the shock waves are perpendicular to the flow and are therefore called normal shock waves. In regime 3, for example condition (f), the adjustment from the exit-plane pressure to the back pressure occurs outside the nozzle as an oblique compression shock wave. 

A strong normal shock wave is generated in a shock tube tilled with dry air at standard temperature and pressure (STP). The oxygen and nitrogen behind the shock wave tend to react to form nitric oxide. 

The solution expressed by Eq. (1.36) indicates that there is no discontinuous flow between the upstream 1 and the downstream 2. However, the solution given by Eq. (1.37) indicates the existence of a discontinuity of pressure, density, and temperature between 1 and 2. This discontinuity is called a normal shock wave , which is set-up in a flow field perpendicular to the flow direction. Discussions on the structures of normal shock waves and supersonic flow fields can be found in the relevant monographs. 

It is obvious that the entropy change will be positive in the region Mi > 1 and negative in the region Mi < 1 for gases with 1 < y < 1-67. Thus, Eq. (1.46) is valid only when Ml is greater than unity. In other words, a discontinuous flow is formed only when Ml > 1. This discontinuous surface perpendicular to the flow direction is the normal shock wave. The downstream Mach number, Mj, is always < 1, i. e. subsonic flow, and the stagnation pressure ratio is obtained as a function of Mi by Eqs. (1.37) and (1.41). The ratios of temperature, pressure, and density across the shock wave are obtained as a function of Mi by the use of Eqs. (1.38)-(1.40) and Eqs. (1.25)-(1.27). The characteristics of a normal shock wave are summarized as follows ... 

Thus, the oblique shock-wave equations are obtained by replacing Mi with Mi in the normal shock-wave equations, Eqs. (3.19)-(3.23), as follows ... 

Fig. D-2 shows the shock-wave formation at a supersonic diffuser composed of a divergent nozzle. Three types of shock wave are formed at three different back-pressures downstream of the diffuser. When the back-pressure is higher than the design pressure, a normal shock wave is set up in front of the divergent nozzle and the flow velocity becomes a subsonic flow, as shown in Fig. D-2 (a). Since the streamline bends outwards downstream of the shock wave, some air is spilled over from the air-intake. The cross-sectional area upstream of the duct becomes smaller than the cross-sectional area of the air-intake, and so the efficiency of the diffuser is reduced. The subsonic flow velocity is further reduced and the pressure is increased in the divergent part of the diffuser.

Figure D-2. Normal shock-wave formation of a supersonic diffuser at different back-pressures (a) high back-pressure, (b) optimum backpressure, and (c) low back-pressure.

When the back-pressure in the difTuser is optimized, a normal shock wave is set up at the hp of the diffuser and the pressure behind the shock wave is increased. No air spill-over occurs at the lip of the diffuser and the airflow velocity is as shown in Fig. D-2 (b). The pressure in the diffuser increases and the airflow velocity decreases along the flow direction. When the back-pressure is lower than the design pressure, a normal shock wave is swallowed inside of the diffuser, as shown in Fig. D-2 (c). Since the flow velocity in front of the normal shock wave in the diffuser is increased along the flow direction, the strength of the normal shock wave inside of the diffuser becomes higher than that in the case of the diffuser at the optimized back-pressure shown in Fig. D-2 (b). Thus, the pressure behind the shock wave is lowered due to the increased entropy. 

The results of the analysis indicate that the pressure recovery factor is increased by the combination of several oblique shock waves and one weak normal shock wave in order to minimize the entropy increase at the air-intake. 

If the compression stems from one normal shock wave, M4 = 0.45, P04/P01 = 0.213, Pa/Pi = 14.13, and T4/T1 = 3.32. It is evident that the pressure recovery factor obtained by the combination of oblique shock waves is significantly higher than that obtained by one normal shock wave. 

An internal compression system forms several oblique shock waves and one normal shock wave inside the duct of the air-intake. The first oblique shock wave is formed at the lip of the air-intake and the following oblique shock waves are formed further downstream the normal shock wave renders the flow velocity subsonic, as shown in the case of the supersonic diffuser in Fig. D-1. The pressure recovery factor and the changes in Mach number, pressure ratio, and temperature ratio are the same as in the case of the external compression system. Either external or internal air-intake systems are chosen for use in ramjets and... 

Fig. D-5 shows an external compression air-intake designed for optimized use at Mach number 2.0. Fig. D-6 shows a set of computed airflows of an external compression air-intake designed for use at Mach number 2.0 (a) critical flow, (b) sub-critical flow, and (c) supercritical flow. The pressures at the bottom wall and the upper wall along the duct flow are also shown. Two oblique shock waves formed at two ramps are seen at the tip of the upper surface of the duct at the critical flow shown in Fig. D-6 (a). The reflected oblique shock wave forms a normal shock wave at the bottom wall of the throat of the internal duct. The pressure becomes 0.65 MPa, which is the designed pressure. In the case of the subcritical flow shown in Fig. D-6 (b), the shock-wave angle is increased and the pressure downstream of the duct becomes 0.54 MPa. However, some of the airflow behind the obhque shock wave is spilled over towards the external airflow. Thus, the total airflow rate becomes 68% of the designed airflow rate. In the case of the supercritical flow shown in Fig. D-6 (c), the shock-wave angle is decreased and the pressure downstream of the duct becomes 0.15 MPa, at which the flow velocity is stiU supersonic.

Normal shock waves (Refs 25, 27 52) Oblique shock waves (Refs 51, 52 54) Plane shock waves (Refs 3, 16, 41,... 

On the Thickness of Normal Shock Waves in a Perfect Gas , OOR-Case Inst of Tech, Contract DA-33-019-ORD-1116(1954)... 

Other examples of idealized solutions are one-dimensional flow of an ideal gas through a normal shock wave flow of an ideal gas without viscosity through a pipe of slowly changing cross section (wind tunnel) and one-dimensional finite waves in an ideal gas. Numerous other solutions involve making whatever approximations and assumptions necessary to obtain descriptions of observed flows. 

Weak detonations are believed to represent the condensation shocks observed in supersonic wind tunnels [12], [51]. Supercooled water vapor in a supersonic stream has been observed to condense rapidly through a narrow wave. The amount of liquid formed is so small that the equations for purely gaseous waves are expected to apply approximately. Since a normal shock wave would raise the temperature above the saturation point (thus ruling out the ZND structure, for example), and the flow is observed to be supersonic downstream from the condensation wave, it appears reasonable to assume that condensation shocks are weak detonations. This hypothesis may be supported by the fact that unlike chemical reaction rates, the rate of condensation increases as the temperature decreases. Proposals that weak detonations also represent various processes occurring in geological transformations have been presented [52]. 

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