Compressible Fluid Flow free study notes for Diploma / BTech.

1. Introduction to compressible fluid flow

Compressible fluid flow is the flow of a fluid in which the density of the fluid changes significantly due to variations in pressure and temperature during motion.This type of flow is mainly associated with gases, as gases are highly compressible. In contrast, liquids generally experience negligible density changes under normal operating conditions and are treated as incompressible.

Definition:

Compressible fluid flow is the flow in which the density of the fluid varies appreciably because of changes 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 of Compressible fluid

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

3. Mach Number (Most Important Parameter)

The Mach number is a dimensionless quantity that represents the ratio of the velocity of a moving object or fluid to the local speed of sound in the same medium.

It is one of the most important parameters in compressible fluid flow because it determines the flow regime and indicates whether compressibility effects are significant.

Definition:

“Mach Number is the ratio of the fluid velocity to the local speed of sound”. We can expressed mathematically as.

$M = \frac{V}{a}$

Where:

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

Flow Classification

Flow Type	Mach Number (M)	Characteristics
Incompressible Flow	(M < 0.3)	Density changes are negligible.
Subsonic Flow	(0.3 < M < 0.8)	Flow is slower than the speed of sound; compressibility begins to matter.
Transonic Flow	(0.8 < M < 1.2)	Mixed subsonic and supersonic regions; shock waves may form.
Sonic Flow	(M = 1)	Flow velocity equals the speed of sound.
Supersonic Flow	(1.2 < M < 5)	Flow is faster than sound; shock waves are common.
Hypersonic Flow	(M > 5)	Extremely high-speed flow with significant aerodynamic heating.

4. Speed of Sound

$$a = \sqrt{\gamma R T}$$

Where:

• $\gamma$ = Ratio of specific heats
• $R$ = Gas constant
• $T$ = 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:$$\rho A V = \text{constant}$$

7. Energy Equation

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

Where:

• $h$ = Enthalpy

8. Isentropic Relations

For ideal gases:$$\frac{T_0}{T} = 1 + \frac{\gamma – 1}{2} M^2$$$$\frac{P_0}{P} = \left(1 + \frac{\gamma – 1}{2} M^2\right)^{\frac{\gamma}{\gamma – 1}}$$$$\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma – 1}{2} M^2\right)^{\frac{1}{\gamma – 1}}$$

9. Area-Velocity Relation (Important Concept)

$$\frac{dA}{A} = (M^2 – 1)\frac{dV}{V}$$

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: $M = 1$ (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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