How to speed up flow

Theory and Concepts
1

Why does decreasing area of a pipe increase the speed of subsonic flow, but increasing the area of a pipe increase the speed of supersonic flow?

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This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
To understand why decreasing the area of a pipe increases the speed of subsonic flow while increasing the area increases the speed of supersonic flow, we can refer to the area-velocity relation derived from conservation laws.

For subsonic flow (where the Mach number M < 1), when the cross-sectional area of the pipe decreases, the fluid accelerates. This is because the flow must conserve mass, and as the area decreases, the velocity must increase to maintain the mass flow rate. Mathematically, this is expressed as:

\frac{1}{A} \frac{dA}{ds} = \left(M^2 - 1\right) \frac{1}{U} \frac{dU}{ds}

In this case, since M^2 - 1 is negative, a decrease in area (dA < 0) leads to an increase in velocity (dU > 0).

On the other hand, for supersonic flow (where the Mach number M > 1), the behavior is reversed. When the area of the pipe increases, the fluid continues to accelerate. This is because, in supersonic flow, the flow behaves differently: as the area increases, the velocity must also increase to maintain the mass flow rate. Here, M^2 - 1 is positive, so an increase in area (dA > 0) results in an increase in velocity (dU > 0).

In summary:

  • Subsonic Flow (M < 1): Decreasing area increases speed.
  • Supersonic Flow (M > 1): Increasing area increases speed.

This behavior is crucial for the design of nozzles and other fluid machinery, as it dictates that a converging-diverging nozzle geometry is what is needed to accelerate flow in a rocket nozzle.

In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:

Textbook sections:

  1. Chapter 11.3: Poiseuille Flow (RELEVANT)
  2. Chapter 1 Fundamentals: The area–velocity relation is derived by combining conservation of mass, co… (RELEVANT)
  3. Chapter 1.3: Acoustics and Compressible Flow (RELEVANT)
  4. Chapter 1.5: Isentropic quasi-1D flow relations (RELEVANT)
  5. Chapter 6 Engineering Models: Fully Developed Flow (RELEVANT)
  6. Chapter 6.5: Losses in Pipes (RELEVANT)
  7. Chapter 1.2: 1D Conservation Equations (RELEVANT)
  8. Chapter 11 Engineering Models: Fully Developed Flow (RELEVANT)
  9. Chapter 1.4: Properties of sound waves (RELEVANT)
  10. Chapter 6.2: Modified Bernoulli equation (RELEVANT)

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