Compressible Fluid Flow free study notes for Diploma / BTech.

1. Introduction

Compressible fluid flow refers to the flow of fluids (mainly gases) in which the density changes significantly due to variations in pressure and temperature.

• Important in high-speed flows
• Typically considered when Mach number > 0.3

Examples

• Airflow in aircraft and rockets
• Gas pipelines
• Jet engines and turbines
• Nozzle and diffuser flows

2. Key Characteristics

• Density is not constant
• Pressure, temperature, and velocity are interdependent
• Flow behavior changes drastically at high speeds

3. Mach Number (Most Important Parameter)

$M = \frac{V}{a}$M=aV​

Where:

• $V$V = Flow velocity
• $a$a = Speed of sound

Flow Classification

• Subsonic Flow: $M < 1$M<1
• Sonic Flow: $M = 1$M=1
• Supersonic Flow: $1 < M < 5$1<M<5
• Hypersonic Flow: $M > 5$M>5

4. Speed of Sound

$$a = \sqrt{\gamma R T}$$a=γRT​

Where:

• $\gamma$γ = Ratio of specific heats
• $R$R = Gas constant
• $T$T = Absolute temperature

5. Types of Compressible Flow

(a) Isentropic Flow

• No heat transfer
• No friction
• Entropy remains constant

(b) Adiabatic Flow

• No heat transfer
• May include friction

(c) Isothermal Flow

• Constant temperature

6. Continuity Equation

For compressible flow:$$\rho A V = \text{constant}$$ρAV=constant

7. Energy Equation

$$h + \frac{V^2}{2} = \text{constant}$$h+2V2​=constant

Where:

• $h$h = Enthalpy

8. Isentropic Relations

For ideal gases:$$\frac{T_0}{T} = 1 + \frac{\gamma – 1}{2} M^2$$TT0​​=1+2γ−1​M2 $$\frac{P_0}{P} = \left(1 + \frac{\gamma – 1}{2} M^2\right)^{\frac{\gamma}{\gamma – 1}}$$PP0​​=(1+2γ−1​M2)γ−1γ​ $$\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma – 1}{2} M^2\right)^{\frac{1}{\gamma – 1}}$$ρρ0​​=(1+2γ−1​M2)γ−11​

9. Area-Velocity Relation (Important Concept)

$$\frac{dA}{A} = (M^2 – 1)\frac{dV}{V}$$AdA​=(M2−1)VdV​

Implications

• Subsonic flow (M < 1): Decrease in area → Increase in velocity
• Supersonic flow (M > 1): Increase in area → Increase in velocity

10. Nozzle and Diffuser Flow

Nozzle

• Converts pressure energy into velocity
• Used in rockets, turbines

Diffuser

• Converts velocity into pressure

Convergent-Divergent Nozzle

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Working

• At throat: $M = 1$M=1 (choked flow)
• Converging section → accelerates subsonic flow
• Diverging section → accelerates supersonic flow

11. Choked Flow

• Occurs when flow velocity reaches Mach 1 at throat
• Maximum mass flow rate achieved

12. Shock Waves

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Types

• Normal Shock: Perpendicular to flow
• Oblique Shock: Inclined

Effects

• Sudden rise in pressure and temperature
• Decrease in velocity
• Increase in entropy

13. Rayleigh and Fanno Flow

Rayleigh Flow

• Heat transfer effects in compressible flow

Fanno Flow

• Friction effects in constant area duct

14. Applications

• Aerospace engineering
• Gas turbines and jet engines
• Supersonic aircraft
• Rocket propulsion
• High-speed wind tunnels

15. Key Points Summary

• Compressible flow involves variable density
• Mach number governs flow behavior
• Nozzle design is crucial for supersonic flow
• Shock waves cause sudden property changes
• Isentropic relations are widely used

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