Normal Shocks

• Shock waves are highly localized irreversibilities in the flow .
  
  
• Within the distance of a mean free path, the flow passes from a supersonic to a subsonic state, the velocity decreases suddenly and the pressure rises sharply. A shock is said to have occurred if there is an abrupt reduction of velocity in the downstream in course of a supersonic flow in a passage or around a body.
  
  
• Normal shocks are substantially perpendicular to the flow and oblique shocks are inclined at any angle.
  
  
• Shock formation is possible for confined flows as well as for external flows.
  
  
• Normal shock and oblique shock may mutually interact to make another shock pattern.

Fig 41.1 Different type of Shocks

Figure below shows a control surface that includes a normal shock.

Fig 41.2 One Dimensional Normal Shock

• The fluid is assumed to be in thermodynamic equilibrium upstream and downstream of the shock, the properties of which are designated by the subscripts 1 and 2, respectively. (Fig 41.2).

  Continuity equation can be written as

(41.1)

where G is the mass velocity kg/ m² s, and is mass flow rate

From momentum equation, we can write

(41.2a) (41.2b)

The energy equation is written as

(41.3)

where h₀ is stagnation enthalpy.

From the second law of thermodynamics, we know

To calculate the entropy change, we have

For an ideal gas

For an ideal gas the equation of state can be written as

(41.4)

For constant specific heat, the above equation can be integrated to give

(41.5)

Equations (41.1), (41.2a), (41.3), (41.4) and (41.5) are the governing equations for the flow of an ideal gas through normal shock.

If all the properties at state 1 (upstream of the shock) are known, then we have six unknowns in these five equations.

We know relationship between h and T [Eq. (38.17)] for an ideal gas, . For an ideal gas with constant specific heats,

(41.6)

  Thus, we have the situation of six equations and six unknowns.

• If all the conditions at state "1"(immediately upstream of the shock) are known, how many possible states 2 (immediate downstream of the shock) are there? The mathematical answer indicates that there is a unique state 2 for a given state 1.