Double checking my notation

Homework Help
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1

In the context of chapter 1 of the textbook, is it true that the star symbol (e.g. A*) means at the throat of the nozzle, while the naught subscript (e.g. p_0) means while stagnant?

2

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

PK Bot’s answer:
Yes, in the context of Chapter 1 of the textbook, the star symbol (e.g., A^*) denotes conditions at the throat of the nozzle, where the Mach number M = 1. This is the point where the flow is choked, meaning the flow speed is equal to the speed of sound. The naught subscript (e.g., p_0) represents stagnation conditions, which are the conditions the fluid would have if it were brought to rest isentropically. These notations are commonly used in the analysis of quasi-1D compressible flow, particularly in the design and analysis of nozzles.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 1.5: Isentropic quasi-1D flow relations Part 2 (HIGHLY RELEVANT)
  2. Chapter 1 Engineering Models: Quasi-1D Compressible Flow (HIGHLY RELEVANT)
  3. Chapter 1.5: Isentropic quasi-1D flow relations Part 1 (RELEVANT)
  4. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (RELEVANT)
  5. Chapter 1 Fundamentals: The Area–Mach-number relation is derived by combining conservation of mass … (RELEVANT)

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3

Technically, the star denotes “sonic” conditions, i.e. conditions when M=1. Stagnation conditions describe the properties that a fluid would have if isentropically slowed to M=0, and sonic conditions describe the properties that a fluid would have if isentropically accelerated/decelerated to M=1.

In a converging-diverging nozzle, the only place where M can be equal to 1 is at the throat. And in normal operation M always equals 1 at the throat. And in normal operation the flow in a nozzle has no shocks and is well approximated as isentropic. With that geometry and with normal operation, sonic values are synonymous with values at the throat.

However, if the p_0/p_\infty ratio is too low and the flow is not choked, the star does not describe conditions at the throat because the throat isn’t sonic. And, if there is a normal shock in the nozzle, then downstream of a normal shock, the starred conditions describe what that fluid’s properties would be if they were isentropically accelerated to M=1, but those values will not match the values at the throat. That normal shock increased the entropy of the fluid.