13.3 Normal Shock Waves

Normal shock waves are abrupt changes in supersonic flow properties. They form when supersonic flow encounters obstacles or sudden area changes, causing a rapid transition from supersonic to subsonic speeds. Understanding these waves is crucial for analyzing supersonic devices and flow systems.

Normal shock relations help calculate flow properties across shocks using conservation laws. These equations relate upstream and downstream conditions, allowing engineers to predict changes in pressure, density, and temperature. This knowledge is essential for designing efficient supersonic flow systems and minimizing energy losses.

Characteristics and Formation of Normal Shock Waves

Characteristics of normal shock waves

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• Thin, non-isentropic regions where flow properties change abruptly
  • Discontinuous increase in pressure, density, and temperature across the shock
  • Decrease in velocity, Mach number, and total pressure downstream of the shock
• Occur when supersonic flow encounters an obstacle or a sudden change in flow area (converging-diverging nozzles, supersonic wind tunnels)
• Form perpendicular to the flow direction, hence the term "normal"
• Require upstream Mach number greater than 1 (supersonic) and downstream Mach number less than 1 (subsonic)
  • Flow must transition from supersonic to subsonic across the shock

Normal Shock Relations and Flow Properties

Normal shock relations calculations

• Derived from conservation of mass, momentum, and energy
• Relate upstream (subscript 1) and downstream (subscript 2) flow properties
  • Pressure ratio: $\frac{p_2}{p_1} = \frac{2\gamma M_1^2 - (\gamma - 1)}{\gamma + 1}$
  • Density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2}{(\gamma - 1)M_1^2 + 2}$
  • Temperature ratio: $\frac{T_2}{T_1} = \frac{[2\gamma M_1^2 - (\gamma - 1)][(\gamma - 1)M_1^2 + 2]}{(\gamma + 1)^2 M_1^2}$
  • Downstream Mach number: $M_2^2 = \frac{(\gamma - 1)M_1^2 + 2}{2\gamma M_1^2 - (\gamma - 1)}$
• Given upstream Mach number and flow properties, downstream properties can be determined
  • Useful in analyzing the performance of supersonic devices (ramjets, scramjets) and flow systems (supersonic wind tunnels)

Entropy and pressure across shocks

• Entropy increases across a normal shock wave due to the abrupt compression and dissipation of kinetic energy
  • Normal shock waves are highly irreversible and non-isentropic processes
  • The increase in entropy is proportional to the strength of the shock wave (higher upstream Mach number)
• Total pressure, also known as stagnation pressure, decreases across a normal shock
  • Total pressure ratio: $\frac{p_{t2}}{p_{t1}} = \left[\frac{(\gamma + 1)M_1^2}{2 + (\gamma - 1)M_1^2}\right]^{\frac{\gamma}{\gamma - 1}} \left[\frac{\gamma + 1}{2\gamma M_1^2 - (\gamma - 1)}\right]^{\frac{1}{\gamma - 1}}$
  • The total pressure loss increases with the strength of the shock wave
• The increase in entropy and loss of total pressure indicate a reduction in the available energy for propulsion (jet engines) or power generation (gas turbines)
  • Minimizing the strength of normal shock waves is crucial for efficient supersonic flow systems

Effects of Back Pressure on Normal Shock Waves in Nozzles

Back pressure effects on shocks

• Back pressure is the pressure downstream of the nozzle exit
• Effect on shock location in converging-diverging nozzles:
  1. If back pressure is lower than the design exit pressure, the flow remains supersonic throughout the nozzle (no shock)
  2. As back pressure increases, a normal shock wave forms and moves upstream in the diverging section
  3. Further increase in back pressure causes the shock to move towards the nozzle throat
• Effect on shock strength:
  • The strength of the normal shock wave depends on the Mach number just upstream of the shock
  • As the shock moves upstream in the nozzle, the upstream Mach number decreases
  • A lower upstream Mach number results in a weaker shock wave (smaller pressure, density, and temperature jumps)
• Implications on nozzle performance:
  • The presence of a normal shock wave in the nozzle reduces the exit velocity and thrust
  • Adjusting the nozzle geometry (area ratio) or controlling the back pressure can help optimize the nozzle performance and minimize shock losses