This chapter introduces the physical nature of turbulent flows and explains why turbulence modeling is necessary in practical CFD. The statistical description of turbulence, the concept of scale separation, and the emergence of the Reynolds-averaged framework are discussed. The chapter concludes with an overview of the RANS–Boussinesq hypothesis and the role of eddy-viscosity models in engineering simulations.

Why Turbulence Dominates Engineering Flows

Most flows encountered in engineering applications operate at sufficiently high Reynolds numbers for turbulence to develop. Examples include internal duct flows, boundary layers on external surfaces, wakes behind bluff bodies, and mixing processes.

Turbulence strongly influences:

Pressure losses and drag
Heat and mass transfer rates
Mixing efficiency
Noise generation
Flow-induced vibrations

As a result, turbulence modeling often determines whether a CFD prediction is useful for design.

Qualitative Characteristics of Turbulent Flow

Turbulent flows exhibit a combination of features that distinguish them from laminar motion:

Three-dimensionality
Velocity fluctuations evolve in all spatial directions, even when the mean flow appears two-dimensional.
Unsteadiness
The instantaneous flow field changes continuously in time, across a wide range of time scales.
Wide range of scales
Large, energetic structures coexist with much smaller, rapidly evolving eddies.
Enhanced transport
Momentum, heat, and species are transported far more efficiently than by molecular diffusion alone.

Because of this complexity, turbulence is best described statistically rather than instantaneously.

Reynolds Number and the Onset of Turbulence

The balance between inertial effects and viscous damping governs whether turbulence can be sustained. At sufficiently high Reynolds numbers, nonlinear inertial mechanisms dominate and enable the formation of unsteady vortical structures.

In practice, the transition to turbulence depends on:

Geometry
Flow disturbances
Surface roughness
Free-stream turbulence levels

This explains why nominal Reynolds-number thresholds vary across different flow configurations.

Statistical Description of Turbulence

Since predicting exact instantaneous flow realizations is impractical, turbulence is approached through statistical averaging.

Reynolds introduced the idea of decomposing flow quantities into:

A mean (time-averaged or ensemble-averaged) component
A fluctuating component capturing turbulent motion

This decomposition allows engineers to focus on predictable mean quantities while accounting for the influence of fluctuations through additional modeled terms.

The Reynolds-Averaged Framework

Applying averaging procedures to the governing equations leads to transport equations for the mean flow. These equations retain the same structure as the laminar equations, with additional stress-like terms arising from velocity fluctuations.

These additional terms represent the momentum transport induced by turbulence and cannot be expressed directly using mean flow variables alone. This difficulty is known as the turbulence closure problem.

Physical Meaning of Reynolds Stresses

The Reynolds stresses describe how turbulent fluctuations transport momentum. Physically, they:

Redistribute momentum across the flow
Modify velocity profiles
Influence separation and reattachment
Control turbulent diffusion rates

Accurate turbulence modeling revolves around providing a reliable approximation for these stresses.

Energy Cascade and Turbulent Scales

Turbulence organizes itself through a hierarchical transfer of energy across scales:

Large scales

Comparable to geometric dimensions
Strongly influenced by boundary conditions and mean flow gradients
Responsible for most of the kinetic energy

Intermediate scales

Less dependent on geometry
Governed primarily by inertial interactions

Small scales

Weakly influenced by geometry
Dominated by viscous effects
Responsible for dissipating kinetic energy into heat

This cascade process explains why resolving all turbulent scales directly is computationally prohibitive for most engineering flows.

Implications for CFD

Fully resolving all turbulent scales requires extremely fine spatial and temporal resolution, which limits direct numerical simulation to research-scale problems.

Engineering CFD therefore, relies on approaches that:

Avoid resolving the smallest scales explicitly
Represent their influence through modeling assumptions

This leads naturally to averaged or filtered formulations.

Approaches to Turbulent Flow Simulation

Three broad strategies exist:

Direct Numerical Simulation (DNS)

Resolves all turbulent scales
Provides the most detailed information
Computationally infeasible for industrial applications

Scale-Resolving Simulations (SRS)

Resolve large-scale unsteadiness
Model only the smallest scales
Include LES, DES, and related methods

Reynolds-Averaged Navier–Stokes (RANS)

Solve equations for mean quantities
Model the full effect of turbulence
Enable steady-state simulations
Remain the dominant industrial approach

This chapter focuses on the RANS framework.

Motivation for Steady RANS Models

Although turbulence is inherently unsteady, many engineering questions involve:

Time-averaged forces
Mean pressure losses
Average heat transfer rates

RANS models enable:

Faster simulations
Reduced computational cost
Simplified post-processing

These advantages explain their widespread use despite known limitations.

Boussinesq Hypothesis and Eddy Viscosity

The Boussinesq hypothesis relates Reynolds stresses to mean velocity gradients through a modeled turbulent viscosity.

This assumption:

Treats turbulence as an enhanced momentum diffusion process
Preserves analogy with laminar stress–strain relationships
Greatly simplifies closure modeling

Its success stems from robustness and efficiency, though it cannot capture all anisotropic turbulence effects.

Families of RANS Turbulence Models

Within the Boussinesq framework, several model classes exist:

Algebraic and zero-equation models

Rely on local flow quantities
Very economical
Limited generality

One-equation models

Introduce a single transported turbulence variable
Well suited for attached wall-bounded flows

Two-equation models

Solve transport equations for turbulence energy and a scale variable
Balance robustness and accuracy
Form the backbone of most industrial simulations

More advanced closures relax the Boussinesq assumption and are addressed in later chapters.

Engineering Intuition

Turbulence modeling trades physical detail for computational feasibility
Mean-flow accuracy depends more on modeling assumptions than solver settings
Eddy-viscosity models perform best when turbulence is close to isotropic
Boundary layers, separation, and curvature challenge simple closures

A clear understanding of turbulence physics helps interpret RANS results realistically.

Study Priorities

If short on time, focus on:

Physical characteristics of turbulent flow
Statistical nature of turbulence
Energy cascade concept
Meaning of Reynolds stresses
Purpose of RANS modeling
Role of the Boussinesq hypothesis

Key Takeaways

Turbulence is a multiscale, three-dimensional, unsteady process.
Statistical descriptions enable practical modeling.
Reynolds averaging introduces unclosed stress terms.
Energy cascades from large to small scales.
RANS models remain essential for industrial CFD.
Eddy-viscosity closures balance robustness and efficiency.

Introduction to CFD Chapter 5: Introduction to Turbulence and RANS Modeling