This chapter introduces compressible flow phenomena relevant to high-speed gas dynamics and CFD. It explains when compressibility effects become important, how density variations couple with momentum and energy transport, and how characteristic flow features such as shock waves, expansion regions, and choking arise. The chapter also introduces numerical strategies used to solve compressible flows, with emphasis on density-based finite volume methods and boundary condition interpretation.

Why Compressibility Matters

Compressible flow effects become important when density variations are no longer negligible. In gases, density changes occur readily because molecular spacing responds strongly to pressure and temperature variations.

In engineering practice, compressibility effects are most often triggered by:

High flow velocities
Large pressure gradients
Strong thermal effects

These situations appear in:

Aerodynamics
Gas turbines
Nozzles and diffusers
High-speed internal flows

Mach Number as a Physical Indicator

The Mach number compares the flow velocity to the local speed of sound and provides a clear physical indicator of compressibility relevance.

Key interpretations:

Low Mach number flows behave similarly to incompressible flows when thermal effects are small
Moderate Mach numbers introduce noticeable density variations
High Mach numbers produce wave-dominated behavior and flow discontinuities

A commonly used engineering guideline places the onset of compressibility effects around Mach numbers of order 0.3 and above, though thermal effects can introduce density changes even at lower speeds.

Thermodynamic Modeling of Compressible Gases

For most engineering gas flows, a calorically perfect gas model is sufficient.

This assumption implies:

A simple relationship between pressure, density, and temperature
Constant specific heats over the temperature range of interest
A constant ratio of specific heats

Air at standard operating conditions satisfies this approximation well, making it suitable for most aerodynamic and turbomachinery applications.

Compressible Governing Principles (Conceptual)

Compressible flow modeling requires tracking:

Mass conservation with variable density
Momentum transport influenced by pressure waves
Energy transport linking temperature, velocity, and pressure

Energy conservation plays a central role because:

Kinetic energy changes feed back into temperature
Temperature variations alter density
Density changes affect momentum and pressure fields

This tight coupling distinguishes compressible flow from incompressible formulations.

Euler Flow as a High-Reynolds-Number Approximation

At sufficiently high Reynolds numbers, viscous and thermal diffusion effects are confined to thin boundary layers. Outside these regions, inviscid (Euler) flow provides useful physical insight.

In this limit:

Flow behavior is dominated by unsteady and convective transport
Information propagates through waves
Boundary conditions require special care

Euler flow concepts underpin the understanding of shocks, expansions, and characteristic-based boundary conditions.

Propagation of Information and Flow Directionality

Compressible flows possess a directional nature in how information travels:

Pressure disturbances propagate at finite speeds
Flow variables influence each other along characteristic paths

This property explains why:

Boundary conditions must be compatible with local flow direction
Over-specification can lead to nonphysical solutions
Inlet and outlet treatment differs from incompressible CFD

Understanding information propagation is essential for stable compressible simulations.

Stagnation Quantities and Energy Interpretation

Stagnation (or total) quantities represent the thermodynamic state achieved if the flow were decelerated without losses.

These quantities:

Remain constant along streamlines under ideal conditions
Provide a convenient reference for comparing flow states
Are widely used in nozzle flows, turbomachinery, and aerodynamics

They offer an energy-based interpretation of compressible flow behavior without requiring local velocity details.

Flow Discontinuities and Their Physical Origin

Compressible flows can support extremely thin regions where flow properties change abruptly. These appear as discontinuities in inviscid flow descriptions.

Two important categories arise:

Tangential discontinuities
Shock waves

Their existence reflects the inability of inviscid flow models to smoothly accommodate certain boundary and flow constraints.

Tangential Discontinuities

Tangential discontinuities occur when:

There is no mass transfer across the interface
Pressure remains continuous
Velocity and temperature may differ tangentially

A common example is a jet discharging into a quiescent medium. In real viscous flows, these discontinuities are replaced by thin shear layers.

Shock Waves

Shock waves appear when the flow normal to a surface becomes supersonic relative to the discontinuity.

Across a shock:

Pressure, density, and temperature increase
Velocity magnitude decreases
Entropy rises

Shock waves represent irreversible processes and play a central role in supersonic and transonic flows.

Normal and Oblique Shocks

Normal shocks

Flow approaches the shock perpendicularly
Strong deceleration occurs
Large thermodynamic changes are observed

Oblique shocks

Flow meets the shock at an angle
Flow direction changes across the shock
Thermodynamic jumps are milder than in normal shocks

Engineering configurations often favor weaker oblique shocks to reduce losses.

Expansion Waves

When supersonic flow turns away from itself, pressure decreases smoothly through expansion waves.

Key characteristics:

Flow accelerates
Temperature and pressure decrease
Entropy remains approximately constant

Expansion fans provide a smooth alternative to shock compression and appear frequently in nozzle and external aerodynamic flows.

Quasi-One-Dimensional Compressible Flow

Many compressible systems can be approximated as nearly one-dimensional:

Nozzles
Diffusers
Duct flows with varying area

In these cases:

Area changes strongly influence velocity and pressure
Flow may choke when sonic conditions are reached
Downstream conditions may lose influence upstream

These ideas explain why nozzles exhibit critical mass flow rates.

Numerical Treatment of Compressible Flow

Compressible CFD solvers must handle:

Strong coupling between density, momentum, and energy
Wave propagation and discontinuities
Stability constraints linked to acoustic speeds

This motivates the use of density-based finite volume solvers, which solve the coupled system simultaneously.

Density-Based Solvers: Conceptual View

Density-based solvers:

Treat density as a primary variable
Solve mass, momentum, and energy in a tightly coupled manner
Are well suited for high-speed and shock-dominated flows

Implicit formulations allow larger time steps and are preferred for steady or quasi-steady problems, while explicit formulations are useful for resolving fast transient wave phenomena

Courant Number and Pseudo-Time Stepping

The Courant number links:

Local wave speed
Mesh size
Numerical time scale

In compressible solvers:

Pseudo-time stepping is used even for steady flows
Courant number controls convergence speed and stability
High values accelerate convergence when stability allows

Managing the Courant number is a central numerical control strategy.

Boundary Conditions in Compressible CFD

Boundary conditions must respect information flow direction:

Inlets may require total conditions
Outlets often require static conditions
Supersonic boundaries behave differently from subsonic ones

Correct boundary specification ensures that waves enter and leave the domain physically, preventing numerical reflection and instability.

Engineering Intuition

Compressibility couples velocity, pressure, temperature, and density tightly
Shock waves represent irreversible energy redistribution
Expansion regions accelerate flow smoothly
Choking limits mass flow independently of downstream pressure
Numerical stability depends strongly on wave propagation control

Compressible CFD requires both physical insight and numerical discipline.

Study Priorities

If short on time, focus on:

Physical meaning of Mach number
When compressibility becomes important
Shock waves versus expansions
Stagnation quantities and energy interpretation
Choking and nozzle behavior
Density-based solver motivation

Key Takeaways

Compressible flow arises from significant density variations in gases.
Mach number provides a practical compressibility indicator.
Shock waves and expansions dominate high-speed flow behavior.
Energy transport is central to compressible dynamics.
Density-based solvers efficiently handle strong variable coupling.
Boundary conditions must respect wave propagation.

Introduction to CFD Chapter 6: Compressible Flow