Calculation of Flow Properties Across a Normal Shock

• The easiest way to analyze a normal shock is to consider a control surface around the wave as shown in Fig. 41.2. The continuity equation (41.1), the momentum equation (41.2) and the energy equation (41.3) have already been discussed earlier. The energy equation can be simplified for an ideal gas as

(40.9)

• By making use of the equation for the speed of sound eq. (39.5) and the equation of state for ideal gas eq. (38.8), the continuity equation can be rewritten to include the influence of Mach number as:
  

(40.10)


Introducing the Mach number in momentum equation, we have

Therefore ,

(40.11)

Rearranging this equation for the static pressure ratio across the shock wave, we get

(40.12)

• As already seen, the Mach number of a normal shock wave is always greater than unity in the upstream and less than unity in the downstream, the static pressure always increases across the shock wave.
  
  
• The energy equation can be written in terms of the temperature and Mach number using the stagnation temperature relationship (40.9) as

(40.13)

Substituting Eqs (40.12) and (40.13) into Eq. (40.10) yields the following relationship for the Mach numbers upstream and downstream of a normal shock wave:

(40.14)

Then, solving this equation for as a function of we obtain two solutions. One solution is trivial , which signifies no shock across the control volume. The other solution is

(40.15)

in Eq. (40.15) results in

Equations (40.12) and (40.13) also show that there would be no pressure or temperature increase across the shock. In fact, the shock wave corresponding to is the sound wave across which, by definition, pressure and temperature changes are infinitesimal. Therefore, it can be said that the sound wave represents a degenerated normal shock wave. The pressure, temperature and Mach number (Ma₂) behind a normal shock as a function of the Mach number Ma₁, in front of the shock for the perfect gas can be represented in a tabular form (known as Normal Shock Table). The interested readers may refer to Spurk[1] and Muralidhar and Biswas[2].