Fiveable
Fiveable
Fiveable
Fiveable

Fluid Mechanics

13.2 Isentropic Flow

2 min readLast Updated on July 19, 2024

Isentropic flow is a key concept in fluid mechanics, describing ideal gas behavior without heat transfer or friction. It's crucial for understanding how pressure, density, and temperature change in various flow situations, from nozzles to diffusers.

By applying conservation of energy and ideal gas laws, we can predict flow properties at different points. This helps engineers design efficient systems and analyze complex flow scenarios, making isentropic flow a fundamental tool in fluid dynamics.

Isentropic Flow

Isentropic flow definition and assumptions

Top images from around the web for Isentropic flow definition and assumptions
Top images from around the web for Isentropic flow definition and assumptions
  • Occurs without heat transfer or friction involves a reversible adiabatic process (no heat exchange with surroundings) and no dissipative effects (viscosity or turbulence)
  • Assumes steady, one-dimensional flow of an ideal gas with constant specific heats
  • No shaft work (turbines or compressors) or heat transfer occurs during the flow process
  • Neglects the effects of gravity on the flow

Derivation of isentropic flow relations

  • Derived from conservation of energy and ideal gas law principles
  • Pressure ratio P2P1=(ρ2ρ1)γ=(T2T1)γγ1\frac{P_2}{P_1} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma} = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}} relates pressure PP, density ρ\rho, temperature TT, and specific heat ratio γ\gamma between two points
  • Density ratio ρ2ρ1=(P2P1)1γ=(T2T1)1γ1\frac{\rho_2}{\rho_1} = \left(\frac{P_2}{P_1}\right)^{\frac{1}{\gamma}} = \left(\frac{T_2}{T_1}\right)^{\frac{1}{\gamma-1}} expresses the relationship between density, pressure, and temperature
  • Temperature ratio T2T1=(P2P1)γ1γ=(ρ2ρ1)γ1\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma-1} connects temperature changes to pressure and density variations

Calculation of isentropic flow properties

  • Isentropic relations enable calculation of pressure, density, and temperature at different flow points
  • With two known properties at a point, the third can be found (if pressure and temperature are given, density is ρ=PRT\rho = \frac{P}{RT} where RR is the specific gas constant)
  • Allows determination of flow conditions throughout an isentropic process (nozzles, diffusers)

Nozzle behavior in isentropic flow

  • Converging nozzles:
    1. Flow velocity increases as cross-sectional area decreases
    2. Pressure, density, and temperature decrease along flow direction
    3. Maximum velocity reached at nozzle throat (minimum area)
  • Diverging nozzles:
    1. Flow behavior depends on pressure ratio across nozzle
    2. Subsonic flow (pressure ratio < critical value): velocity decreases, pressure, density, and temperature increase along flow
    3. Supersonic flow (pressure ratio > critical value): velocity increases, pressure, density, and temperature decrease along flow
  • Critical pressure ratio PP0=(2γ+1)γγ1\frac{P^*}{P_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}} determines flow regime, PP^* is throat pressure, P0P_0 is stagnation pressure
© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.