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Computational Fluid Dynamics · Chapter 1 of 10 · Beginner

Introduction to CFD

CFD replaces a wind tunnel with a mesh and a solver. Before any of that, two dimensionless numbers decide which physics the simulation must capture, and one habit decides whether to believe the result.

Readiness check

This opening chapter needs fluids sense and unit care. Tick only what you can do closed-notes.

Compute a Reynolds number and classify a flow.
Recall the speed of sound in air.
State conservation of mass and momentum in words.
Read a meshed domain with inlet and outlet.
Convert between m, mm, m/s, and Pa.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit the Reynolds number in Fluid Mechanics, Chapter 1.

3 or more weak itemsReview fluid mechanics fundamentals before continuing.

The core idea

CFD divides a flow domain into many small control volumes, writes the conservation laws on each, and solves the resulting algebraic system. The mesh and the physics models, not the colors, decide the answer.

domain → mesh of control volumesPDEs → algebraic equations per cellRe and Ma set the required physics

A CFD code has three stages: a pre-processor defines the geometry, mesh, models, and boundary conditions; a solver discretizes the governing equations and iterates to a solution; a post-processor visualizes it. The heart is discretization: the continuous Navier-Stokes equations become one algebraic equation per control volume, millions of them, solved together. Before meshing, two numbers frame the problem. The Reynolds number says whether the flow is laminar or turbulent, which decides if a turbulence model is needed; the Mach number says whether density changes matter, which decides between an incompressible and a compressible solver. And no result is trusted until mesh independence and a validation case confirm it.

The skill works when: you let Re and Ma choose the physics, then verify the solution before reporting it.

The skill breaks down when: contours are read off a coarse mesh with default models and no convergence or validation check.

The concept. The flow domain is divided into a mesh of control volumes, finer near the body where gradients are steep. CFD writes and solves a conservation balance on every cell.

The skills, taught in order

Five skills frame what CFD is, how a code runs, the discretization idea, the modeling decisions, and the verification habit.

1.1 What CFD is

Computational fluid dynamics solves the equations of fluid motion numerically over a discretized domain. It complements experiment and theory, giving full-field data that is hard to measure, but at the price of modeling and numerical approximations that must be controlled.

1.2 How a CFD code works

Every code has three parts. The pre-processor builds the geometry, generates the mesh, and sets fluid properties, models, and boundary conditions. The solver discretizes and iterates. The post-processor renders contours, vectors, and integrated quantities like drag and pressure drop.

1.3 The discretization idea

The finite volume method integrates each conservation law over a control volume, turning a partial differential equation into an algebraic relation between a cell and its neighbours. Assembling these for every cell gives a large sparse system solved iteratively.

1.4 Modeling decisions: Reynolds and Mach

Two dimensionless numbers frame the setup. The Reynolds number Re = ρUL/μ decides laminar versus turbulent and whether a turbulence model is required. The Mach number Ma = U/a decides whether compressibility matters: below about 0.3, density is essentially constant and an incompressible solver is appropriate.

Number	Decides	Threshold
Reynolds Re = ρUL/μ	laminar or turbulent	turbulent above a flow-dependent value
Mach Ma = U/a	incompressible or compressible	compressible above about 0.3

1.5 Verify before you trust

A converged, colorful solution can still be wrong. Mesh independence (the answer stops changing on refinement), residual convergence, and comparison with experiment or a known case are what turn a picture into evidence. This habit runs through the whole course and returns in Chapter 10.

Engineering connection: the Reynolds and Mach decisions made here set which equations and models the rest of the course discretizes.

Worked example 1: Reynolds number and the turbulence decision

Air flows through a 0.15 m duct at 12 m/s. With ρ = 1.2 kg/m³ and μ = 1.8×10⁻⁵ Pa·s, find the Reynolds number and decide whether a turbulence model is needed.

Figure 1. The Reynolds number is built from density, speed, the duct diameter, and viscosity. At over 10⁵ the flow is turbulent, so a turbulence model is required.

ProblemFind Re and the modeling decision for the duct flow in Figure 1.
Given / findρ = 1.2 kg/m³, U = 12 m/s, D = 0.15 m, μ = 1.8×10⁻⁵ Pa·s. Find Re and whether to model turbulence.
AssumptionsFully developed internal flow; the diameter is the length scale.
ModelCompute Re = ρUD/μ and compare to the internal-flow transition (about 2300 to 4000).
EquationsRe = ρUD/μ
SolveRe = (1.2 × 12 × 0.15)/(1.8×10⁻⁵) = 2.16/1.8×10⁻⁵ = 120 000. Since Re ≫ 4000, the flow is turbulent.
CheckRe is dimensionless, and 10⁵ is typical of everyday duct flow, which is almost always turbulent. A laminar simulation would be physically wrong here.
ConclusionThe Reynolds number alone mandates a turbulence model (Chapter 8). Getting this decision wrong invalidates the whole simulation.

Result. Re ≈ 120 000, turbulent, so a turbulence model is required.

Worked example 2: Mach number and compressibility

A flow of air moves at 100 m/s, where the speed of sound is 343 m/s. Find the Mach number and decide whether the simulation can treat the flow as incompressible.

Figure 2. The Mach number compares the flow speed to the speed of sound. At 0.29 it sits just below the 0.3 threshold, so density changes are negligible and an incompressible solver is justified.

ProblemFind Ma and the compressibility decision for the flow in Figure 2.
Given / findU = 100 m/s, a = 343 m/s. Find Ma and whether to use an incompressible solver.
AssumptionsAir at standard conditions, so a = 343 m/s; the 0.3 threshold separates the regimes.
ModelCompute Ma = U/a and compare to 0.3.
EquationsMa = U/a
SolveMa = 100/343 = 0.29. Since 0.29 < 0.3, density changes are under about 5 percent, so the flow is treated as incompressible.
CheckThe threshold of 0.3 corresponds to roughly a 5 percent density variation, the usual limit for the incompressible assumption. At 100 m/s the air barely compresses.
ConclusionAn incompressible solver is appropriate, simplifying the equations and avoiding the energy coupling a compressible solver needs. Above 0.3 that choice would flip.

Result. Ma = 0.29 < 0.3, so the incompressible assumption is valid.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Trusting a coarse mesh	Reporting values that change on refinement	"Did the answer change when I refined?"	Run a mesh-independence study before believing a result.
Wrong physics from the start	Laminar run for a turbulent flow	"What does the Reynolds number say?"	Let Re decide laminar versus turbulent.
Compressible by default	Slow runs and convergence trouble at low speed	"Is the Mach number below 0.3?"	Use an incompressible solver when Ma is small.
Colors as proof	Conclusions from a pretty contour plot	"Is this converged and validated?"	Check residuals and a validation case before reporting.

Practice ladder

Level 1 · Direct skill

Water (ρ = 1000, μ = 1.0×10⁻³) flows at 1 m/s in a 0.05 m pipe. Find the Reynolds number.

Show answer

Re = ρUD/μ = (1000 × 1 × 0.05)/1.0×10⁻³ = 50 000. Turbulent, so a turbulence model is needed.

Level 2 · Mixed concept

A fan blade tip moves at 180 m/s in air (a = 343 m/s). Is the flow compressible?

Show answer

Ma = 180/343 = 0.52, above 0.3, so the flow is compressible and density changes must be modeled.

Level 3 · Independent problem

Air at 25 m/s flows over a 2 m car (ρ = 1.2, μ = 1.8×10⁻⁵, a = 343). State whether you would use a turbulence model and an incompressible solver.

Show answer

Re = (1.2 × 25 × 2)/1.8×10⁻⁵ = 3.3×10⁶, turbulent, so yes to a turbulence model. Ma = 25/343 = 0.073 < 0.3, so incompressible.

Level 4 · Transfer to real engineering

Pick a real flow you would simulate (cooling a CPU, airflow over a drone). State its Reynolds and Mach numbers and the solver and turbulence decisions they imply.

What good work looks like

Sensible estimates of Re and Ma with a length scale stated, and explicit laminar/turbulent and incompressible/compressible choices justified by the thresholds.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my Reynolds and Mach numbers justify the solver and turbulence choices."

"Give me four flows; I will pick laminar or turbulent and compressible or not."

"Set up this CFD case for me." Making the modeling decisions yourself is the skill.

"Is this turbulent?" Computing Re and comparing is the point.

Portfolio task

Frame one real CFD case: estimate Re and Ma, choose the solver and turbulence model they imply, and list the verification checks you would run before trusting it.

Must include: a Reynolds number, a Mach number, a solver and model choice, and a verification plan.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What are the three parts of a CFD code?

Pre-processor (geometry, mesh, models), solver (discretize and iterate), post-processor (visualize).

2. What does the finite volume method do?

Integrates each conservation law over a control volume, turning a PDE into an algebraic equation per cell.

3. What does the Reynolds number decide?

Laminar versus turbulent, and whether a turbulence model is needed.

4. What does the Mach number decide?

Incompressible versus compressible; below about 0.3 density is essentially constant.

5. What turns a contour plot into evidence?

Mesh independence, residual convergence, and a validation comparison.

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

+1 dayRe-derive the Re and Ma decisions from a blank page.

+3 daysFrame the physics for three new flows.

+7 daysCarry the equations into Chapter 2, the governing equations.

+30 daysReuse the verification habit when you report any CFD result.

Textbook mapping

Item	Mapping
Primary source	Versteeg and Malalasekera, An Introduction to Computational Fluid Dynamics, Chapter 1 (Introduction)
Cross-reference	Anderson, Ch. 1 · Fluid Mechanics, Ch. 1
Core topics	1.1 What CFD is · 1.2 How a code works · 1.3 Discretization · 1.4 Reynolds and Mach · 1.5 Verify before trusting
Engineering connection	The Re and Ma decisions set the equations and models for the whole course.
Read next	Chapter 2: The Governing Equations.