Turbulence Chapter 1: Review of RANS-Boussinesq Models & Statistical Turbulence Description
This chapter introduces turbulence from a statistical and modeling perspective, establishing why turbulence must be modeled rather than resolved in most engineering flows. The chapter explains the decomposition of turbulent motion into mean and fluctuating components, introduces Reynolds averaging, and motivates the Reynolds-Averaged Navier–Stokes (RANS) framework. The physical meaning and limitations of the Boussinesq hypothesis are discussed, followed by an overview of advanced options commonly used to improve two-equation RANS model robustness in industrial CFD.
Why Turbulence Requires Modeling
Turbulent flows are characterized by:
Strong unsteadiness
Three-dimensionality
A wide range of interacting length and time scales
Directly resolving all turbulent scales (DNS) is:
Computationally prohibitive for engineering Reynolds numbers
Infeasible for design and industrial workflows
As a result:
Turbulence is treated statistically
Only averaged effects of turbulence are modeled
This motivates the RANS approach.
Statistical Description of Turbulence
3.1 Instantaneous vs Mean Flow
In turbulent flow:
Velocity, pressure, and scalar fields fluctuate randomly in time
These fluctuations are superimposed on an underlying mean motion
Key idea:
Engineering interest is usually in mean quantities
Fluctuations matter only through their averaged effect on transport
3.2 Reynolds Decomposition
Any instantaneous flow variable is decomposed into:
A time-averaged (mean) component
A fluctuating component with zero mean
Physical interpretation:
The mean represents the large-scale, organized motion
Fluctuations represent turbulent eddies transporting momentum and energy
This decomposition is the foundation of all RANS models.
Reynolds Averaging and Its Consequences
4.1 Reynolds-Averaged Equations
Applying averaging to the Navier–Stokes equations:
Removes explicit time dependence of turbulence
Introduces new terms representing the effect of fluctuations
These new terms are the Reynolds stresses, which:
Act like additional stresses in the momentum equations
Represent turbulent momentum transport
4.2 Closure Problem
The Reynolds stresses are unknowns:
They depend on fluctuating velocities
No new equations are automatically available
This creates the closure problem:
More unknowns than equations
All turbulence models are essentially closure models for these stresses.
Physical Meaning of Reynolds Stresses
Reynolds stresses:
Are not material stresses
Represent momentum flux caused by turbulent motion
Physically:
Large eddies transport high-momentum fluid into low-momentum regions
This enhances mixing and increases effective momentum diffusion
Turbulence therefore behaves like:
An additional, flow-dependent viscosity
But one that is anisotropic and history-dependent
The Boussinesq Hypothesis
6.1 Core Assumption
The Boussinesq hypothesis assumes:
Reynolds stresses are proportional to mean strain rate
Turbulence behaves like an isotropic eddy viscosity
This allows:
Reynolds stresses to be modeled using a scalar turbulent viscosity
Closure of the RANS equations with relatively low cost
6.2 Why It Works (and Why It Fails)
It works well when:
Turbulence is approximately isotropic
Flow curvature and rotation are weak
Separation is mild
It fails when:
Strong anisotropy exists
Swirl, curvature, or rotation dominate
Secondary flows are important
These limitations motivate advanced RANS options and higher-fidelity models.
Two-Equation RANS Models (Context)
Two-equation models:
Solve transport equations for two turbulence quantities
Use them to compute turbulent viscosity
Their popularity comes from:
Robustness
Reasonable accuracy
Low computational cost
This chapter does not focus on individual models, but on how they are modified and extended in practice.
Advanced Options for Two-Equation RANS Models
8.1 Why Advanced Options Are Needed
Standard RANS models:
Are calibrated for simple benchmark flows
Often fail in complex industrial geometries
Advanced options act as:
Corrections
Limiters
Stabilization mechanisms
They do not change the core model, but constrain or adapt it.
8.2 Production Limiters
Purpose:
Prevent excessive turbulence production
Stabilize simulations in stagnation regions or strong strain
Physical meaning:
Turbulence cannot grow arbitrarily fast
Real flows exhibit saturation mechanisms
Production limiters are especially important in:
High-speed flows
Strong acceleration or deceleration zones
8.3 Curvature and Rotation Corrections
Standard RANS models:
Cannot distinguish between strain and rotation
Mis-predict turbulence levels in curved flows
Curvature correction:
Reduces turbulence production in stabilizing curvature
Increases it in destabilizing curvature
Important for:
Cyclones
Turbomachinery
Strongly swirling flows
8.4 Near-Wall Damping and Low-Re Effects
Near walls:
Turbulence is suppressed by viscosity
Length scales collapse
Low-Re and damping functions:
Reduce turbulent viscosity near walls
Allow resolution of viscous sublayer
These options require:
Fine near-wall mesh
Careful y⁺ control
8.5 Model Blending and Adaptivity
Modern RANS approaches increasingly rely on:
Blending between formulations
Context-dependent behavior
This philosophy aims to:
Retain robustness of simple models
Improve accuracy in complex flows
This is a precursor to later hybrid and adaptive models.
Engineering Intuition
RANS models predict average turbulence effects, not turbulence itself
Reynolds stresses represent momentum transport by eddies
The Boussinesq hypothesis is a powerful but restrictive assumption
Advanced options exist to prevent unphysical behavior, not to “fix” turbulence
Most industrial success comes from judicious constraint, not model complexity
Rule of thumb:
If a RANS solution looks unstable or unphysical, the issue is often excessive turbulence production.
Study Priorities
If short on time, focus on:
Reynolds decomposition and averaging
Physical meaning of Reynolds stresses
Closure problem in turbulence modeling
Boussinesq hypothesis and its limits
Purpose of production limiters and corrections
Key Takeaways
Turbulence is treated statistically in engineering CFD.
Reynolds averaging introduces unknown stress terms.
RANS models are closure models for turbulent momentum transport.
The Boussinesq hypothesis enables practical modeling but limits accuracy.
Advanced RANS options constrain turbulence behavior in complex flows.
This chapter sets the foundation for more detailed turbulence models.