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Computational Fluid Dynamics · Chapter 7 of 10 · Intermediate

Unsteady Flows and Time Stepping

Add time and the solution marches forward step by step. An explicit step is cheap but unstable if you push it too far; the Courant and Fourier numbers tell you exactly how far you can go.

Readiness check

This chapter marches in time. Tick only what you can do closed-notes.

Recall the transient term ∂φ/∂t.
Approximate a time derivative by a finite difference.
Recall thermal diffusivity α = k/(ρc).
Form a dimensionless ratio with Δt and Δx.
Distinguish solving directly from marching forward.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit transient conduction in Heat Transfer, Chapter 4.

3 or more weak itemsReview finite differences in Math: Numerical Methods.

The core idea

The unsteady term advances the solution one time step at a time. An explicit step uses only old values and is cheap but conditionally stable; an implicit step solves a system each step but is stable for any Δt.

Co = uΔt/Δx ≤ 1 (explicit, convection)Fo = αΔt/Δx² ≤ 0.5 (explicit, diffusion)implicit: stable for any Δt

The transient term ∂φ/∂t is approximated by the change over a time step Δt. In an explicit scheme the spatial terms are evaluated at the old time level, so each new value is an immediate formula, but information can only travel so far per step: the Courant number Co = uΔt/Δx must stay at or below one, and for diffusion the Fourier number Fo = αΔt/Δx² must stay at or below one half. Exceed them and the solution blows up. An implicit scheme evaluates the spatial terms at the new time level, requiring a linear solve each step but remaining stable for any time step. Crank-Nicolson averages the two for second-order accuracy. The choice is cost per step against the size of step allowed.

The skill works when: you size an explicit time step from the Courant and Fourier limits, or go implicit for large steps.

The skill breaks down when: an explicit step exceeds the stability limit and the solution diverges.

The concept. An explicit step (left) builds each new value from old neighbours, an immediate formula. An implicit step (right) couples the new values together, needing a solve but stable for any time step.

The skills, taught in order

Five skills discretize the unsteady term, contrast explicit and implicit, add Crank-Nicolson, and state the stability limits.

7.1 The unsteady term

Integrating ∂(ρφ)/∂t over a control volume and a time step gives ρ(φ^new − φ^old)V/Δt. How the spatial terms are evaluated, at the old or new time, defines the scheme and its stability.

7.2 The explicit scheme

Evaluating the spatial terms at the old time level makes each new value an explicit formula, with no system to solve. It is cheap per step but only conditionally stable: the time step is capped by the Courant and Fourier limits.

7.3 The implicit scheme

Evaluating the spatial terms at the new time level couples the unknowns, so a linear system is solved each step. The reward is unconditional stability: any time step is stable, though accuracy still limits how large it should be.

Scheme	Stability	Accuracy	Cost per step
Explicit	conditional (Co ≤ 1, Fo ≤ 0.5)	first order in time	cheap, no solve
Implicit	unconditional	first order in time	a solve each step
Crank-Nicolson	unconditional	second order in time	a solve each step

7.4 Crank-Nicolson

Averaging the old and new time levels gives the Crank-Nicolson scheme, which is second-order accurate in time and unconditionally stable. It can oscillate for very large steps, so it is used with moderate time steps for accurate transients.

7.5 The stability numbers

For explicit schemes the Courant number Co = uΔt/Δx limits convection (a parcel must not cross a whole cell in one step) and the Fourier number Fo = αΔt/Δx² limits diffusion. Refining the mesh tightens both, which is why fine explicit simulations need tiny time steps.

Engineering connection: transient SIMPLE and PISO march the pressure-velocity coupling of Chapter 5 in time, subject to these same limits.

Worked example 1: the Courant stability limit

An explicit convection simulation has flow speed u = 2 m/s on a grid with Δx = 0.01 m. Find the largest stable time step, and the Courant number at Δt = 0.004 s.

Figure 1. The Courant number is the distance a parcel travels in one step divided by the cell width. Keeping it at or below one stops the parcel from skipping a cell, which would destabilise an explicit scheme.

ProblemFind the maximum stable Δt and the Courant number at 0.004 s for the case in Figure 1.
Given / findu = 2 m/s, Δx = 0.01 m, explicit scheme. Find Δt_max and Co at Δt = 0.004 s.
AssumptionsExplicit convection; the stability limit is Co ≤ 1.
ModelCo = uΔt/Δx; set Co = 1 for the limiting step.
EquationsCo = uΔt/ΔxΔt_max = Δx/u
SolveΔt_max = Δx/u = 0.01/2 = 0.005 s. At Δt = 0.004 s, Co = (2 × 0.004)/0.01 = 0.8 ≤ 1, so it is stable.
CheckA time step of 0.008 s would give Co = 1.6, above one, so the parcel would cross more than a cell and the explicit scheme would diverge. The chosen 0.004 s sits safely below the limit.
ConclusionThe Courant number caps the explicit step for convection. Finer meshes force smaller steps, the cost of explicit time marching.

Result. Δt_max = 0.005 s; at 0.004 s, Co = 0.8 (stable).

Worked example 2: the diffusion stability limit

An explicit transient conduction simulation has thermal diffusivity α = 1×10⁻⁵ m²/s on a grid with Δx = 0.002 m. Find the largest stable time step from the Fourier number limit.

Figure 2. For explicit diffusion the Fourier number must stay at or below one half. The grid spacing squared in the denominator means halving Δx quarters the allowed time step.

ProblemFind the maximum stable Δt for the explicit conduction case in Figure 2.
Given / findα = 1×10⁻⁵ m²/s, Δx = 0.002 m, explicit scheme. Find Δt_max.
AssumptionsOne-dimensional explicit diffusion; the stability limit is Fo ≤ 0.5.
ModelFo = αΔt/Δx²; set Fo = 0.5 for the limiting step.
EquationsFo = αΔt/Δx²Δt_max = 0.5 Δx²/α
SolveΔt_max = 0.5 × (0.002)²/(1×10⁻⁵) = 0.5 × 4×10⁻⁶/1×10⁻⁵ = 0.5 × 0.4 = 0.2 s.
CheckThe grid spacing enters squared, so refining to Δx = 0.001 m would cut the allowed step to a quarter, 0.05 s. This quadratic penalty is why fine explicit diffusion runs are expensive and implicit schemes are often preferred.
ConclusionThe Fourier number caps the explicit step for diffusion. When small cells force a tiny step, an implicit scheme removes the limit at the cost of a solve per step.

Result. Δt_max = 0.2 s for explicit stability.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Explicit step too large	Solution blows up	"Is Co ≤ 1 and Fo ≤ 0.5?"	Cap the explicit step by the stricter of the two limits.
Assuming implicit is always accurate	Smeared or oscillatory transient	"Is the step too big for accuracy?"	Unconditional stability is not unlimited accuracy; keep Δt reasonable.
Forgetting the Δx² in Fourier	Diffusion limit miscomputed	"Is the grid spacing squared?"	Fo uses Δx², so refining hits the step hard.
Ignoring the stricter limit	Instability despite checking one number	"Did I check both Co and Fo?"	For convection-diffusion, the smaller allowed step governs.

Practice ladder

Level 1 · Direct skill

For u = 5 m/s and Δx = 0.02 m, find the largest stable explicit time step (convection).

Show answer

Δt_max = Δx/u = 0.02/5 = 0.004 s, the step that gives Co = 1.

Level 2 · Mixed concept

For the Worked Example 2 case, what is the maximum step if the mesh is refined to Δx = 0.001 m?

Show answer

Δt_max = 0.5 × (0.001)²/1×10⁻⁵ = 0.5 × 0.1 = 0.05 s, a quarter of the original because Δx is halved.

Level 3 · Independent problem

A convection-diffusion case has u = 1 m/s, α = 2×10⁻⁵ m²/s, Δx = 0.005 m. Find the explicit step from each limit and say which governs.

Show answer

Convection: Δt = Δx/u = 0.005 s. Diffusion: Δt = 0.5(0.005)²/2×10⁻⁵ = 0.625 s. The convection limit (0.005 s) is smaller, so it governs.

Level 4 · Transfer to real engineering

For a real transient simulation, decide whether you would use an explicit or implicit scheme and justify it from the time step the stability limits allow.

What good work looks like

An estimate of the explicit step from Co and Fo, compared to the physical time scale of interest, and an implicit choice if the stable explicit step would be impractically small.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my time step satisfies both the Courant and Fourier limits."

"Give me four cases; I will say whether the explicit step is stable."

"What time step should I use?" Computing the limits yourself is the skill.

"Why did my run blow up?" Spotting the violated limit is the point.

Portfolio task

For a transient case, compute the Courant and Fourier limits, choose explicit or implicit, and justify the time step against both stability and the physical time scale.

Must include: a Courant limit, a Fourier limit, a scheme choice, and a justified time step.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What distinguishes explicit from implicit?

Explicit evaluates spatial terms at the old time (no solve); implicit at the new time (a solve each step).

2. Write the Courant number and its limit.

Co = uΔt/Δx ≤ 1 for explicit convection.

3. Write the Fourier number and its limit.

Fo = αΔt/Δx² ≤ 0.5 for explicit diffusion.

4. What is Crank-Nicolson?

A time-average of old and new levels: second-order accurate, unconditionally stable.

5. Why are fine explicit runs expensive?

Both limits tighten as Δx shrinks, forcing very small time steps.

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

+1 dayRe-derive the Courant and Fourier limits from a blank page.

+3 daysSize the time step for two new cases.

+7 daysCarry time stepping into turbulence, Chapter 8.

+30 daysReuse the stability limits whenever a transient run diverges.

Textbook mapping

Item	Mapping
Primary source	Versteeg and Malalasekera, An Introduction to Computational Fluid Dynamics, Chapter 8 (Finite Volume Method for Unsteady Flows)
Cross-reference	Anderson, Ch. 4 · Heat Transfer, Ch. 4
Core topics	7.1 Unsteady term · 7.2 Explicit · 7.3 Implicit · 7.4 Crank-Nicolson · 7.5 Stability numbers
Engineering connection	Transient SIMPLE and PISO march the flow in time within these limits.
Read next	Chapter 8: Turbulence Modeling.