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Computational Fluid Dynamics · Chapter 8 of 10 · Advanced

Turbulence Modeling

Turbulence has eddies too small and too fast to resolve, so we average them away and model their effect. The price is a closure problem and a mesh that must be sized to the wall.

Readiness check

This chapter models what cannot be resolved. Tick only what you can do closed-notes.

Recall what makes a flow turbulent (high Reynolds number).
Recognise a time-averaged versus fluctuating quantity.
Recall viscosity as momentum diffusion.
Compute a wall shear stress from a skin-friction coefficient.
Take a square root and form a ratio.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit turbulence in Fluid Mechanics, Chapter 8.

3 or more weak itemsReview the Reynolds number in Fluid Mechanics, Chapter 1.

The core idea

Averaging the equations over the turbulent fluctuations leaves extra Reynolds stresses that need a model. The eddy-viscosity idea models them as an extra turbulent viscosity, computed by the k-epsilon model.

φ = φ̄ + φ′ (mean + fluctuation)μ_t = ρC_μk²/εy⁺ = ρu_τy/μ sizes the first cell

Resolving every turbulent eddy (direct simulation) is far too expensive for engineering, so the velocity is split into a mean and a fluctuation and the equations are time-averaged. This Reynolds averaging produces new unknown terms, the Reynolds stresses, more unknowns than equations: the closure problem. The common fix is the eddy-viscosity hypothesis, modeling the stresses as if turbulence added an extra viscosity μ_t, far larger than the molecular value. The k-epsilon model computes it from two transported quantities, the turbulent kinetic energy k and its dissipation ε, as μ_t = ρC_μk²/ε. Near walls the steep gradients demand care: the dimensionless wall distance y⁺ sets how fine the first cell must be, either resolving the viscous sublayer (y⁺ near 1) or bridging it with a wall function (y⁺ of 30 to 300).

The skill works when: you pick a model for the flow and size the first cell to the chosen y⁺.

The skill breaks down when: a model is used outside its regime, or the near-wall mesh does not match its wall treatment.

The concept. The law of the wall: close to the wall the velocity rises linearly through the viscous sublayer, then follows a logarithmic profile. A mesh resolves the sublayer at y⁺ near 1 or starts a wall function near y⁺ of 30.

The skills, taught in order

Five skills set the closure problem, RANS averaging, the eddy-viscosity model, near-wall treatment, and model selection.

8.1 The closure problem

Splitting each quantity into a mean and a fluctuation and averaging the Navier-Stokes equations leaves correlation terms of the fluctuations, the Reynolds stresses −ρu′_iu′_j. These are new unknowns with no equations of their own, so the system is not closed without a model.

8.2 RANS averaging

The Reynolds-averaged Navier-Stokes equations solve for the mean flow only, with the turbulence represented statistically. This is the workhorse of industrial CFD: affordable and good for mean quantities like drag and pressure drop, though it loses the unsteady eddy detail that LES keeps.

8.3 The eddy-viscosity k-epsilon model

The eddy-viscosity hypothesis writes the Reynolds stresses as a turbulent viscosity times the mean strain rate. The k-epsilon model transports the turbulent kinetic energy k and its dissipation ε and forms μ_t = ρC_μk²/ε, with C_μ = 0.09. It is robust for free shear flows but weaker near walls and in adverse pressure gradients.

Model	Transport equations	Best for
Mixing length	0 (algebraic)	simple shear layers
k-epsilon	2	general, free shear
k-omega SST	2	near-wall, adverse gradients
Reynolds stress (RSM)	7	swirl and anisotropy

8.4 Near-wall treatment and y+

The dimensionless wall distance y⁺ = ρu_τy/μ, built from the friction velocity u_τ = √(τ_w/ρ), measures the first cell's position in wall units. Wall-resolved meshes need y⁺ near 1; wall functions bridge the sublayer and need y⁺ of 30 to 300. The mesh must match the model's wall treatment.

8.5 Model selection

No model is universal. k-epsilon suits free shear and internal flows; k-omega SST is better for boundary layers and separation; Reynolds stress models capture swirl; LES resolves the large eddies when unsteady detail is needed and cost allows. Choosing well, and meshing to match, is the heart of turbulent CFD.

Engineering connection: the y⁺ requirement here directly drives the inflation-layer mesh designed in Chapter 9.

Worked example 1: first cell height for a target y+

Air (ρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s) flows at 20 m/s over a surface with skin-friction coefficient C_f = 0.003. Find the wall shear stress, the friction velocity, and the first cell height needed for a wall-resolved mesh at y⁺ = 1.

Figure 1. Resolving the viscous sublayer to y⁺ = 1 demands a first cell only about 19 micrometres thick, which is why wall-resolved meshes are so fine near surfaces.

ProblemFind τ_w, u_τ, and the first cell height for y⁺ = 1 in Figure 1.
Given / findρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s, U = 20 m/s, C_f = 0.003, y⁺ = 1. Find τ_w, u_τ, y.
AssumptionsTurbulent boundary layer; the skin-friction coefficient gives the wall shear.
Modelτ_w = C_f·½ρU²; u_τ = √(τ_w/ρ); then y = y⁺μ/(ρu_τ).
Equationsτ_w = C_f·½ρU², u_τ = √(τ_w/ρ)y = y⁺μ/(ρu_τ)
Solveτ_w = 0.003 × 0.5 × 1.2 × 20² = 0.72 Pa. u_τ = √(0.72/1.2) = √0.6 = 0.77 m/s. y = 1 × 1.8×10⁻⁵/(1.2 × 0.77) = 1.9×10⁻⁵ m ≈ 19 µm.
CheckA 19 micrometre first cell is extremely thin, hundreds of times finer than the geometry, which is why wall-resolved turbulent meshes are expensive. A wall-function mesh at y⁺ = 30 would allow a first cell about 30 times larger.
ConclusionThe target y⁺ sets the first cell height directly. Matching mesh to model is a non-negotiable step in turbulent CFD.

Result. τ_w = 0.72 Pa, u_τ = 0.77 m/s, first cell ≈ 19 µm for y⁺ = 1.

Worked example 2: turbulent viscosity from k-epsilon

A flow at 20 m/s has a turbulence intensity of 5 percent, giving a turbulent kinetic energy k, and a dissipation rate ε = 10 m²/s³. With ρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s, and C_μ = 0.09, find k and the turbulent viscosity, and compare it to the molecular viscosity.

Figure 2. The turbulent viscosity dwarfs the molecular value: turbulence mixes momentum over a thousand times more effectively than viscosity alone, which is why it dominates the transport.

ProblemFind k and μ_t for the flow in Figure 2 and compare μ_t to μ.
Given / findU = 20 m/s, I = 5 percent, ε = 10 m²/s³, ρ = 1.2, μ = 1.8×10⁻⁵, C_μ = 0.09. Find k, μ_t, and the ratio.
AssumptionsIsotropic turbulence for k from intensity; standard k-epsilon constant C_μ = 0.09.
Modelk = (3/2)(IU)² from the intensity; μ_t = ρC_μk²/ε.
Equationsk = (3/2)(IU)²μ_t = ρC_μk²/ε
Solvek = 1.5 × (0.05 × 20)² = 1.5 × 1 = 1.5 m²/s². μ_t = 1.2 × 0.09 × 1.5²/10 = 0.243/10 = 0.024 kg/m·s. The ratio μ_t/μ = 0.024/1.8×10⁻⁵ = 1350.
CheckA turbulent viscosity more than a thousand times the molecular value is typical of a strongly turbulent flow; it is what makes turbulent mixing so much faster than laminar diffusion.
ConclusionThe k-epsilon model turns two transported scalars into the eddy viscosity that closes the equations. That viscosity, not the molecular one, governs turbulent momentum transport.

Result. k = 1.5 m²/s², μ_t = 0.024 kg/m·s, about 1350 times μ.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Mesh not matched to y+	Wall function used at y⁺ = 1	"Does the first cell match the wall treatment?"	Use y⁺ near 1 for wall-resolved, 30 to 300 for wall functions.
One model for everything	k-epsilon used in strong separation	"Does this model fit the flow?"	Match the model (k-omega SST near walls, RSM for swirl) to the physics.
Treating μ_t as fixed	Turbulent viscosity hard-coded	"Is μ_t computed from k and ε?"	The model transports k and ε to compute μ_t everywhere.
Expecting RANS to give eddies	Looking for unsteady structures in a RANS run	"Did I average the turbulence away?"	RANS gives mean fields; use LES for resolved eddies.

Practice ladder

Level 1 · Direct skill

Find the wall shear stress for C_f = 0.004, ρ = 1.2 kg/m³, U = 15 m/s.

Show answer

τ_w = C_f·½ρU² = 0.004 × 0.5 × 1.2 × 225 = 0.54 Pa.

Level 2 · Mixed concept

For the Worked Example 1 flow, what first cell height gives y⁺ = 30 (a wall-function mesh)?

Show answer

y = 30 × 1.8×10⁻⁵/(1.2 × 0.77) = 30 × 1.9×10⁻⁵ = 5.8×10⁻⁴ m ≈ 0.58 mm, thirty times the wall-resolved value.

Level 3 · Independent problem

Find the turbulent kinetic energy for U = 30 m/s and a turbulence intensity of 4 percent.

Show answer

k = (3/2)(IU)² = 1.5 × (0.04 × 30)² = 1.5 × 1.44 = 2.16 m²/s².

Level 4 · Transfer to real engineering

For a real turbulent flow you would simulate, choose a turbulence model and a target y⁺, and justify both from the flow features.

What good work looks like

A model matched to the physics (separation, swirl, free shear) and a y⁺ target consistent with the wall treatment, with the first cell height estimated.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my first cell height matches the y⁺ my model needs."

"Give me four flows; I will pick a turbulence model for each."

"Which model should I use?" Matching model to physics yourself is the skill.

"What first cell height?" Computing it from y⁺ is the point.

Portfolio task

For a turbulent case, choose a model, compute the first cell height for its y⁺ target, and estimate the turbulent viscosity from typical k and ε values.

Must include: a model choice, a first cell height from y⁺, and a turbulent viscosity estimate.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What is the closure problem?

Reynolds averaging produces extra Reynolds stresses, more unknowns than equations, needing a model.

2. What does the eddy-viscosity hypothesis assume?

That turbulence acts like an extra viscosity μ_t on the mean strain rate.

3. Write the k-epsilon turbulent viscosity.

μ_t = ρC_μk²/ε, with C_μ = 0.09.

4. What does y⁺ control?

The first cell height in wall units; near 1 for wall-resolved, 30 to 300 for wall functions.

5. RANS versus LES?

RANS gives mean fields cheaply; LES resolves the large eddies at higher cost.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the first cell height and μ_t from a blank page.

+3 daysChoose models and y⁺ targets for two new flows.

+7 daysCarry the y⁺ requirement into meshing, Chapter 9.

+30 daysReuse model selection whenever you set up a turbulent case.

Textbook mapping

Item	Mapping
Primary source	Versteeg and Malalasekera, An Introduction to Computational Fluid Dynamics, Chapter 3 (Turbulence and its Modelling)
Cross-reference	Wilcox, Turbulence Modeling for CFD · Fluid Mechanics, Ch. 8
Core topics	8.1 Closure problem · 8.2 RANS · 8.3 k-epsilon · 8.4 Near-wall and y⁺ · 8.5 Model selection
Engineering connection	The y⁺ requirement drives the near-wall mesh of the next chapter.
Read next	Chapter 9: Meshing and Boundary Conditions.