Oblique shock wave

The formation of a shock wave is dependent on the objects that affect the flow field. The conservation of mass, momentum, and energy must be satisfied at any location. This is manifested in the formation of a shock wave at a certain location in the flow field to meet the conservahon equations. In the case of a blunt body in a supersonic flow, the pressure increases in front of the body. The increased pressure generates a detached shock wave to satisfy the conservation equations in the flow field to match the conserved properties between the inflow and outflow in front of the body. The velocity then becomes a subsonic flow behind the detached shock wave. However, the shock wave distant from the blunt body is less affected and the detached shock wave becomes an oblique shock wave. Thus, the shock wave appears to be curved in shape, and is termed a bow shock wave, as illustrated in Fig. C-1. 

When a two-dimensional wedge is placed in a supersonic flow, a shock wave that 

Though the velocity component parallel to the shock wave remains unchanged, Vi = V2, the velocity component normal to the shock wave, Ui — M2, changes through the shock wave. The change in the normal velocity component through the oblique 

The angle between the inflow streamline and the oblique shock wave, p, is expressed by 

Since the Mach number of the inflow to the shock wave is given by Mj = wjui and that of the outflow from the shock wave is given by M2 = iV2/a2. the Mach number of the normal velocity perpendicular to the oblique shock wave, Mj, is represented by 

Since the Mach number of the inflow to the shock wave is given by and 

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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).

Figure C-3. Oblique shock wave formed by a two-dimensional wedge.

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 ... 

The entropy change through the oblique shock wave is given by Sj - Si = Cpln(T2/Ti) - ln(p2/pi)... 

It is shown that two p and two 0 correspond to one Mj. When p is small, the static pressure ratio P2/P1 is small, and the shock wave is weak. On the other hand, when P is large, a strong shock wave is formed, for which P2/P1 is large. The Mach number behind the oblique shock wave becomes supersonic for weak shock waves and subsonic for strong shock waves. 

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). 

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. 

Figure D-4. Compression by the formation of three oblique shock waves with three ramps.

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.

Erkman describes in Ref 8a, experiments conducted at Poulter Labs, Stanford Research Institute, Menlo Park, Calif, in which A1 plates were caused to spall by explosively induced oblique shock waves. The work was principally directed toward developing techniques for performing reproducible experiments with A1 and for testing the scaling laws for spalling. During these experiments was also studied decay of explosively-induced shock waves in solids... 

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

Ibid, 28, 1437-41(1957) (Plane shock waves) and Ibid 29, 167-70(1958) (Oblique shock waves) 52) Dunkle s Syllabus (1957-1958). Properties of Shock Waves, which include Supported Shock Waves (pp 50-1) General Properties of Shock Waves (51-2) Detachment of Shock Waves (52-3) Conditions Behind Shock Front (53-6) Variability of Specific Heats (56-7) Relaxation Processes, Ionization, and Chemical Reaction (57-60) and Shock Waves in Solids (60). 

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