Lesson 15c: Compressible Flow and Converging-Diverging Ducts

Introduction

• Supersonic flow requires a throat in the duct.
• Key topics include:
  • Area ratio vs. Mach number relationship.
  • Effects of varying back pressure.
  • Example problems.

Mach Number and Area Relationship

• Conservation of Mass Equation: When Mach number = 1, dA = 0 (i.e., area is max or min).
• Sonic Flow: Occurs at Mach number = 1, area must be minimum (throat).
• Subsonic vs. Supersonic Flow:
  • Subsonic: Mach decreases as area increases.
  • Supersonic: Mach increases as area increases.

Converging-Diverging Duct (CD Duct)

• Flow Stages:
  • Upstream of throat: Subsonic flow.
  • At throat: Sonic flow.
  • Downstream of throat: Supersonic flow.
• Conditions: Requires low back pressure (Pb).

Area Ratio vs. Mach Number Relationship

• Flow Equations:
  • Mass flow rate equations for converging duct.
  • Choked and unchoked conditions.
• Deriving Area Ratio: Equating flow equations gives area ratio as a function of Mach number and specific heat ratio.

Example Problem

• Given: Airflows from a pressurized tank through a CD nozzle.
• Calculate:
  • Area at specific Mach numbers (subsonic and supersonic).
  • Area ratio derived from Mach numbers.
• Results:
  • Subsonic and supersonic cases provide the same area ratio.
  • Two roots for Mach number for any area ratio > 1.

Solving for Mach Number

• Equation: Implicit for Mach number when given area ratio.
• Methods:
  • Graphical plot.
  • Trial and error.
  • Excel’s What-if analysis.
  • Newton's method or False Position Method (FPM).
• False Position Method: Recommended for solving implicit equations.

Varying Back Pressure

• Thought Experiment: Lowering back pressure (Pb) affects flow in CD duct.
  • Cases A-G: Ranging from no flow to supersonic jets.
  • Case F: Ideal expansion with supersonic flow throughout.
  • Case D, E: Involves normal shocks.
• Conclusion: Once flow is choked, downstream changes don’t affect upstream flow.

Conclusion

• Understanding flow dynamics in converging-diverging ducts is crucial for applications involving supersonic conditions.
• Upcoming topics include more detailed discussions on shocks.