Compressible Flow Fundamentals

Introduction and Context

In liquid-flow analysis, density is often treated as constant. Gases behave differently: pressure and temperature changes can produce substantial density changes. This is especially important when gas velocity is comparable with the speed at which pressure disturbances travel. Such flows occur in pneumatic control systems, air compressors, turbines, rocket and aircraft engines, gas pipelines, and high-speed nozzles.

This chapter develops the one-dimensional, steady-flow framework needed to analyze compressible gas motion. The central ideas are acoustic velocity, Mach number, stagnation properties, choking, converging and converging–diverging nozzles, diffusers, and normal shocks.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain why gas density variation affects flow analysis.
2. Calculate the acoustic velocity and Mach number.
3. Distinguish static, stagnation, and critical properties.
4. Apply continuity, energy, and isentropic-flow relations.
5. Determine the conditions for choking in a nozzle.
6. Analyze converging and converging–diverging nozzles and diffusers.
7. Calculate the principal property changes across a normal shock.
8. Recognize the assumptions and limitations of ideal compressible-flow models.

1. Physical Basis of Compressible Flow

A fluid is compressible when its density changes appreciably during motion. A useful measure is the fractional density change, ∆ρ/ρ. For many engineering liquid-flow problems this quantity is small; for gases it may become important even at moderate velocities.

A pressure disturbance in a gas travels as a weak acoustic wave. If the flow velocity is much smaller than the wave velocity, pressure information can propagate upstream and the flow adjusts gradually. As the flow velocity approaches the wave velocity, upstream communication becomes increasingly limited. When the flow is faster than the wave velocity, disturbances cannot travel upstream through the moving gas.

For a calorically perfect gas, the equation of state is

where p is absolute pressure (Pa), ρ is density (kg/m³), R is the specific gas constant (J/(kg·K)), and T is absolute temperature (K). For air, R ≈ 287 J/(kg·K).

The specific heats are related by

where c_p and c_v are the specific heats at constant pressure and volume. For air under ordinary conditions, γ ≈ 1.4. The ideal-gas model is generally suitable for air at moderate pressure and temperature, but real-gas effects become important at high pressures, near condensation, or at very high temperatures.

2. Acoustic Velocity and Mach Number

2.1 Speed of sound

The acoustic velocity is the speed of propagation of an infinitesimal pressure disturbance. For a general fluid it is defined by

where a is acoustic velocity (m/s), and the derivative is taken at constant entropy. For an ideal gas undergoing an isentropic disturbance,

Thus acoustic velocity depends on absolute temperature, not directly on pressure, for a perfect gas. At 15 °C, the speed of sound in air is approximately 340 m/s.

2.2 Mach number

The Mach number is the ratio of flow speed to acoustic velocity:

where V is local flow speed (m/s). The usual classifications are:

The value 0.3 is not a physical discontinuity; it is an engineering guideline. Compressibility may matter below this value when high accuracy, large pressure changes, or sensitive equipment is involved.

3. Governing Equations for One-Dimensional Flow

Consider steady flow through a duct or nozzle. The standard one-dimensional model assumes uniform properties over each cross-section, negligible transverse velocity components, and a known area variation.

3.1 Continuity

Conservation of mass gives

where ṁ is mass-flow rate (kg/s), A is cross-sectional area (m²), V is mean velocity (m/s), and ρ is density (kg/m³).

For two stations,

Unlike incompressible flow, area reduction does not alone determine velocity because density may also change.

3.2 Energy equation

For steady adiabatic flow with no shaft work and negligible potential-energy change,

For a calorically perfect gas, h = c_pT, so

or

Here T₀ is stagnation temperature. It is the temperature obtained if the gas were brought to rest adiabatically without shaft work. Stagnation temperature remains constant in an adiabatic, no-work flow, even when pressure and velocity change.

3.3 Momentum equation

For a control volume, the steady momentum equation in the flow direction is

Pressure forces, wall forces, and body forces must be included. This equation is particularly important for determining forces on bends, nozzles, and shock-producing devices.

4. Stagnation and Isentropic Properties

A stagnation state is a hypothetical state reached when the local velocity is reduced to zero reversibly and adiabatically. It is not necessarily a point physically present in the apparatus.

For a perfect gas undergoing one-dimensional, steady, adiabatic, reversible flow, the process is isentropic. The principal relations are

The subscript 0 denotes stagnation conditions, while unsubscripted quantities denote local static conditions. These equations require a perfect gas, constant γ, and isentropic flow. They must not be applied across a shock, because a shock is adiabatic but irreversible.

The inverse pressure relation is useful for finding Mach number from pressure measurements:

For example, with air at M = 2,

and

Thus a Mach-2 stream has a stagnation pressure about 7.82 times its static pressure under isentropic conditions.

5. Area–Velocity Relationship

Combining continuity, the energy equation, and the isentropic relations yields the differential area–velocity relation:

This equation explains nozzle behavior:

• For subsonic flow, M < 1, an area decrease causes velocity to increase.
• For supersonic flow, M > 1, an area increase causes velocity to increase.
• At M = 1, the area is stationary to first order; in a smooth nozzle this location is the throat.

Consequently, a converging passage accelerates subsonic flow but decelerates supersonic flow. A diverging passage decelerates subsonic flow but accelerates supersonic flow. A converging–diverging nozzle can therefore accelerate a gas from subsonic to supersonic speed, provided the pressure ratio and back pressure are appropriate.

6. Choking and Critical Conditions

6.1 Meaning of choking

A nozzle is choked when the mass-flow rate reaches its maximum for the specified upstream stagnation state and throat area. The throat then has M = 1. Lowering the downstream pressure further does not increase the mass flow through the throat, although it can alter the downstream flow pattern and shock position.

For isentropic flow, the critical or sonic pressure and temperature ratios are

The asterisk denotes sonic, or critical, conditions. For air, γ = 1.4, giving approximately

and

Therefore a converging nozzle supplied from a large reservoir chokes when the downstream-to-upstream pressure ratio is approximately 0.528 or less, assuming ideal isentropic behavior.

6.2 Choked mass-flow rate

For a perfect gas, the mass flow at a choked throat is

where A* is throat area (m²), p₀ is upstream stagnation pressure (Pa), T₀ is upstream stagnation temperature (K), R is the gas constant, and γ is the specific-heat ratio. This equation assumes steady, adiabatic, isentropic flow up to the throat and negligible inlet kinetic energy relative to reservoir conditions.

The formula shows that maximum mass flow increases with throat area and upstream pressure, and decreases with the square root of upstream temperature. Real nozzles require discharge or flow coefficients to account for boundary layers, friction, and nonuniformity.

7. Nozzles and Diffusers

A nozzle converts enthalpy into kinetic energy and is designed primarily to increase velocity. A diffuser reduces velocity and converts kinetic energy into pressure. The same geometry can behave differently depending on whether the flow is subsonic or supersonic.

7.1 Converging nozzle

A converging nozzle accelerates subsonic flow. If the pressure ratio reaches the critical value, the exit becomes sonic and the nozzle is choked. Further reduction of back pressure cannot produce a supersonic exit because there is no diverging section.

7.2 Converging–diverging nozzle

A converging–diverging nozzle has a throat at minimum area. Subsonic flow accelerates through the converging section. If the throat reaches sonic conditions, the diverging section can accelerate the flow to supersonic speed. The final state depends strongly on back pressure:

• High back pressure: entirely subsonic flow.
• Intermediate back pressure: a normal shock may stand in the diverging section.
• Lower back pressure: the shock moves downstream and may leave the nozzle.
• Properly matched low back pressure: supersonic exit flow.

The area–Mach relation is

For a specified area ratio, this equation generally has two solutions: a subsonic solution and a supersonic solution. The physically realized branch depends on inlet conditions, back pressure, and the presence of shocks.

7.3 Diffusers

A subsonic diffuser normally expands in area, reducing velocity and increasing static pressure. Excessive diffusion can cause boundary-layer separation, producing pressure loss and unsteady operation. A supersonic diffuser generally requires a converging passage or a sequence of shocks to reduce velocity. The resulting pressure recovery is irreversible and lower than the ideal isentropic value.

8. Normal Shock Waves

A normal shock is a thin, irreversible compression wave perpendicular to the upstream flow direction. The upstream flow is supersonic and the downstream flow is subsonic. Across the shock:

• static pressure increases;
• static temperature increases;
• density increases;
• velocity decreases;
• entropy increases;
• stagnation temperature remains constant for an adiabatic shock;
• stagnation pressure decreases.

The shock is modeled as steady, one-dimensional, adiabatic flow with no shaft work and negligible body-force effects. Conservation of mass, momentum, and energy gives

For a perfect gas, the principal normal-shock relations are

The downstream Mach number is

where M₁ is the upstream Mach number and M₂ is the downstream Mach number. These relations apply only to a normal shock, not to an oblique shock or a gradual compression.

Worked example: normal shock in air

Air with γ = 1.4 enters a normal shock at M₁ = 2. The pressure ratio is

The density ratio is

Therefore,

The downstream Mach number is

so M₂ ≈ 0.577. The flow has therefore changed from supersonic to subsonic, with a substantial pressure and temperature rise.

9. Illustrative Engineering Case: Pneumatic Blow-Off Nozzle

Background

An automated machining cell uses compressed air to remove metal chips from a workpiece. A reservoir supplies air at p₀ = 600 kPa absolute and T₀ = 300 K through a circular nozzle with throat area A* = 1.0 × 10⁻⁵ m². The exhaust discharges to an environment at 100 kPa absolute.

Problem

Determine whether the nozzle chokes and estimate the ideal mass-flow rate. Use air with γ = 1.4 and R = 287 J/(kg·K).

Analysis

The pressure ratio is

For air, the critical ratio is approximately 0.528. Since 0.167 is lower than 0.528, the nozzle is choked.

The ideal choked-flow rate is

Substitution gives

Thus,

or approximately 14 g/s.

Practical interpretation

Reducing the outlet pressure below 100 kPa will not increase the ideal throat mass flow. In practice, the measured value will be lower because of contraction, friction, upstream piping losses, and nonuniform velocity. A manufacturer may represent these effects with a discharge coefficient or a standardized pneumatic flow rating.

Learning points

The case illustrates three design principles: pressure ratio determines choking, throat area controls the maximum ideal flow, and reservoir temperature affects capacity. It also shows why a nominally simple air nozzle requires experimental validation when energy consumption, noise, or chip-removal performance is important.

10. Assumptions, Limitations, and Best Practice

The ideal equations are powerful but conditional. Important limitations include:

1. One-dimensionality: Real velocity and temperature profiles are nonuniform.
2. Steady flow: Valves and pneumatic actuators often experience transients.
3. Adiabatic flow: Heat transfer may matter in long pipes or slow processes.
4. Isentropy: Friction, mixing, and shocks increase entropy and reduce stagnation pressure.
5. Perfect-gas behavior: High-pressure gases may require real-gas equations of state.
6. Constant specific heats: Large temperature changes require temperature-dependent properties.
7. Boundary layers: Separation can invalidate simple area-based predictions.
8. Measurement uncertainty: Pressure taps measure local static or total pressure only if properly designed.

A sound engineering workflow is to define the control volume, identify whether values are static or stagnation properties, select an appropriate gas model, check the Mach regime, test for choking, and then apply loss coefficients or computational/experimental data as needed.

In industrial pneumatic systems, safe design also requires pressure-vessel compliance, relief protection, noise control, guarding against high-velocity particle ejection, and verification of acceptable exhaust temperature and force. Computational fluid dynamics can resolve multidimensional effects, but it does not remove the need for conservation-law checks and validated boundary conditions.

Chapter Summary

Compressible flow analysis is required when gas-density changes influence continuity, energy conversion, or pressure communication. The acoustic velocity of an ideal gas is a = √(γRT), and the Mach number M = V/a identifies the flow regime. Conservation of mass, momentum, and energy provides the foundation for one-dimensional analysis.

Stagnation properties describe the state obtained by ideal adiabatic deceleration to zero velocity. Isentropic relations connect static and stagnation pressure, temperature, density, and Mach number. The area–velocity relation explains why subsonic and supersonic flows respond oppositely to area changes.

A nozzle chokes when its throat reaches M = 1. For air, the critical pressure ratio is approximately 0.528. A converging–diverging nozzle can produce supersonic flow when the throat is choked and the back pressure is sufficiently low. A normal shock converts supersonic flow to subsonic flow, raises static pressure and temperature, and irreversibly reduces stagnation pressure. Real equipment departs from ideal behavior because of friction, heat transfer, boundary layers, separation, nonuniformity, and real-gas effects.

Key Terms

Mach number: Ratio of flow speed to local acoustic velocity.

Acoustic velocity: Speed at which a small pressure disturbance propagates through a fluid.

Compressible flow: Flow in which density changes significantly.

Perfect gas: Ideal gas model satisfying p = ρRT.

Specific-heat ratio: Ratio γ = c_p/c_v.

Static property: Local thermodynamic property moving with the fluid.

Stagnation property: Property associated with ideal adiabatic deceleration to zero velocity.

Isentropic flow: Reversible adiabatic flow with constant entropy.

Nozzle: Device that converts enthalpy into kinetic energy.

Diffuser: Device that reduces velocity and ideally increases static pressure.

Throat: Minimum-area section of a nozzle.

Critical condition: Sonic condition at which M = 1.

Choking: Maximum-flow condition caused by sonic flow at a restriction.

Back pressure: Pressure downstream of a nozzle or flow device.

Converging–diverging nozzle: Nozzle with a converging section, throat, and diverging section.

Normal shock: Shock wave perpendicular to the upstream flow.

Stagnation pressure: Pressure obtained by ideal isentropic deceleration to rest.

Entropy generation: Irreversibility associated with friction, shocks, mixing, or heat transfer.

Calorically perfect gas: Perfect gas with constant specific heats.

Review Questions

Short Answer Questions

1. Why can density variation not generally be neglected in high-speed gas flow?
2. Define acoustic velocity and Mach number.
3. Distinguish static temperature from stagnation temperature.
4. What is meant by choking in a nozzle?
5. Why does a normal shock reduce stagnation pressure?

Long Answer Questions

1. Derive the qualitative area–velocity relationship and explain the behavior of subsonic and supersonic flow in converging and diverging passages.
2. Explain the assumptions behind the isentropic stagnation-property relations.
3. Describe the operation of a converging–diverging nozzle as back pressure is progressively reduced.
4. Derive the principal conservation equations applied across a normal shock and discuss their physical meaning.
5. Compare the functions and geometry of subsonic diffusers and supersonic diffusers.

Analytical / Application Questions

1. Air enters an ideal nozzle from a reservoir at 500 kPa absolute and 290 K. Determine whether a downstream pressure of 200 kPa causes choking. Justify the result using the critical pressure ratio.
2. A flow has M = 0.8 and local temperature T = 280 K in air. Calculate the local acoustic velocity and flow speed using γ = 1.4 and R = 287 J/(kg·K).
3. Air passes through a normal shock at M₁ = 3. Explain, without assuming isentropic flow across the shock, how pressure, temperature, density, velocity, entropy, and stagnation pressure change. Use the normal-shock relations to obtain numerical ratios if data are available.

Multiple Choice Questions with Answers

Question 1

For an ideal gas, the acoustic velocity is given by:

A. a = γRT

B. a = √(γRT)

C. a = RT/γ

D. a = V/M²

Correct Answer: B. a = √(γRT)

Question 2

A flow with M = 0.6 is classified as:

A. Subsonic

B. Sonic

C. Supersonic

D. Hypersonic

Correct Answer: A. Subsonic

Question 3

For steady one-dimensional flow, conservation of mass requires:

A. pV = constant

B. T/A = constant

C. ρAV = constant

D. ρ/A = constant

Correct Answer: C. ρAV = constant

Question 4

At the throat of an ideally choked nozzle, the Mach number is:

A. 0

B. Less than 0.3

C. 1

D. Always greater than 2

Correct Answer: C. 1

Question 5

For air with γ = 1.4, the approximate critical pressure ratio p*/p₀ is:

A. 0.167

B. 0.528

C. 1.400

D. 2.000

Correct Answer: B. 0.528

Question 6

A converging passage accelerates which type of ideal flow?

A. Subsonic flow

B. Supersonic flow only

C. Both regimes in the same manner

D. Neither regime

Correct Answer: A. Subsonic flow

Question 7

In an isentropic flow, stagnation pressure:

A. Always decreases with velocity

B. Always increases with velocity

C. Remains constant

D. Becomes zero at sonic conditions

Correct Answer: C. Remains constant

Question 8

Across a normal shock, the downstream flow is generally:

A. Supersonic if the upstream flow is subsonic

B. Subsonic if the upstream flow is supersonic

C. Always sonic

D. At rest

Correct Answer: B. Subsonic if the upstream flow is supersonic

Question 9

For air at M = 2, the isentropic stagnation-temperature ratio T₀/T is:

A. 0.556

B. 1.400

C. 1.800

D. 3.500

Correct Answer: C. 1.800

Question 10

Which quantity decreases across an adiabatic normal shock?

A. Static pressure

B. Static temperature

C. Entropy

D. Stagnation pressure

Correct Answer: D. Stagnation pressure