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Computational Fluid Dynamics · Chapter 4 of 10 · Advanced

Convection-Diffusion and Schemes

Add flow to diffusion and a new danger appears: the obvious central scheme produces wild oscillations once convection dominates. One dimensionless number tells you when, and which scheme to use instead.

Readiness check

This chapter adds the convection term. Tick only what you can do closed-notes.

Recall the diffusion coefficients a_W, a_E, a_P.
Distinguish convection from diffusion.
Form a ratio of two quantities.
Recognise when a coefficient turning negative is a problem.
Read a profile plot for oscillations.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit the diffusion coefficients in Chapter 3.

3 or more weak itemsReview the transport equation in Chapter 2.

The core idea

The cell Peclet number, convection over diffusion across a cell, decides whether the central scheme stays stable. Above about 2 it gives negative coefficients and oscillations, so upwind is used instead.

Pe_cell = F/D = ρu δx/Γcentral a_E = D − F/2 (negative if Pe > 2)upwind a_E = D, a_W = D + F (F > 0)

Convection carries φ with the flow, adding a face flux F = ρu to the diffusive conductance D = Γ/δx. The central differencing scheme averages the two neighbouring nodes, which is second-order accurate but makes the downstream coefficient a_E = D − F/2 go negative once the cell Peclet number F/D exceeds 2. A negative coefficient violates the boundedness a physical solution needs, producing nonphysical wiggles. The upwind scheme takes the value from the upstream node, keeping all coefficients positive and the solution bounded, at the cost of first-order accuracy and a numerical smearing called false diffusion. Choosing a scheme is balancing stability against accuracy.

The skill works when: you compute the cell Peclet number and pick a scheme that keeps the coefficients positive.

The skill breaks down when: central differencing is used at high Peclet number and the wiggles are mistaken for physics.

The concept. For a convection-dominated profile, central differencing oscillates around the exact solution while upwind stays smooth but smears the sharp gradient. The cell Peclet number predicts which behaviour you get.

The skills, taught in order

Five skills add the convection term, expose the central-scheme limit, define the Peclet number, introduce upwind, and weigh accuracy.

4.1 The convection term

Convection adds a face mass flux F = ρu to the balance, carrying φ across each face. The discretized cell equation now has both convective (F) and diffusive (D) contributions, and continuity F_e = F_w keeps a_P consistent.

4.2 Central differencing and its limit

Estimating the face value as the average of the two neighbours gives a_E = D − F/2 and a_W = D + F/2. This is second-order accurate but a_E turns negative when F/2 > D, the point where the scheme loses boundedness and oscillates.

4.3 The cell Peclet number

The cell Peclet number Pe = F/D = ρu δx/Γ measures convection relative to diffusion across one cell. Central differencing is stable only for Pe ≤ 2; the practical fix is either to refine the mesh (shrinking δx and Pe) or to change scheme.

4.4 Upwind differencing

The upwind scheme takes the face value from the upstream node, so for positive flow a_W = D + F and a_E = D, both always positive. It is unconditionally bounded, which is why it is robust, but it is only first-order accurate.

Scheme	a_E (F > 0)	Bounded?	Accuracy
Central	D − F/2	only Pe ≤ 2	second order
Upwind	D	always	first order
Hybrid	switches at Pe = 2	always	first/second
QUICK	quadratic upwind	conditionally	third order

4.5 False diffusion and higher-order schemes

Upwind's robustness comes from added numerical diffusion, which smears sharp gradients, especially when the flow crosses the grid at an angle. Higher-order schemes such as hybrid, power-law, and QUICK reduce this error while trying to keep boundedness, the practical compromise in commercial codes.

Engineering connection: the convection-diffusion discretization is the φ-equation that the SIMPLE pressure-velocity coupling of Chapter 5 solves repeatedly.

Worked example 1: the cell Peclet number and scheme choice

A one-dimensional convection-diffusion problem has ρ = 1 kg/m³, Γ = 0.1 kg/m·s, and a grid spacing δx = 0.2 m. Find the cell Peclet number for flow speeds of 0.1 m/s and 2.5 m/s, and decide whether central differencing is safe in each case.

Figure 1. The cell Peclet number scales with flow speed. The slow case sits safely below 2; the fast case is well above it, so central differencing would oscillate.

ProblemFind the cell Peclet number for both speeds in Figure 1 and judge central differencing.
Given / findρ = 1 kg/m³, Γ = 0.1 kg/m·s, δx = 0.2 m, u = 0.1 and 2.5 m/s. Find Pe and the scheme decision.
AssumptionsUniform grid; constant properties; one-dimensional flow.
ModelCompute F = ρu and D = Γ/δx, then Pe = F/D, and compare to 2.
EquationsD = Γ/δx = 0.1/0.2 = 0.5Pe = F/D = ρu δx/Γ
SolveFor u = 0.1: F = 0.1, Pe = 0.1/0.5 = 0.2 (≤ 2, central safe). For u = 2.5: F = 2.5, Pe = 2.5/0.5 = 5 (> 2, central oscillates).
CheckThe Peclet number scales linearly with u, so the fast case is 25 times the slow one, pushing it well past the limit of 2. Refining the mesh to δx = 0.04 m would drop the fast-case Pe to 1.
ConclusionThe cell Peclet number is the single check before trusting central differencing. Above 2, switch scheme or refine the mesh.

Result. Pe = 0.2 (central safe) and Pe = 5 (central oscillates, use upwind).

Worked example 2: central versus upwind coefficients

For the convection-dominated case (F = 2.5, D = 0.5, flow toward the east), compute the east and west coefficients for central differencing and for upwind, and explain which scheme is usable.

Figure 2. At this Peclet number central differencing gives a negative east coefficient, which guarantees oscillations, while upwind keeps both coefficients positive and the solution bounded.

ProblemFind the coefficients for both schemes in Figure 2 and decide which is usable.
Given / findF = 2.5, D = 0.5, flow toward the east (F > 0). Find a_W, a_E for central and upwind.
AssumptionsOne-dimensional; constant F and D; boundedness requires non-negative coefficients.
ModelCentral: a_W = D + F/2, a_E = D − F/2. Upwind: a_W = D + F, a_E = D.
Equationscentral a_E = D − F/2 = 0.5 − 1.25upwind a_W = D + F = 3, a_E = D = 0.5
SolveCentral: a_W = 0.5 + 1.25 = 1.75, a_E = −0.75 (negative). Upwind: a_W = 3, a_E = 0.5, both positive, with a_P = 3.5.
CheckThe negative central a_E means a higher downstream value would lower the cell value, which is unphysical and seeds oscillations. Upwind's positive coefficients guarantee the cell value stays between its neighbours.
ConclusionOnly the upwind scheme is usable here. The price is first-order accuracy and false diffusion, which a finer mesh or a higher-order bounded scheme reduces.

Result. Central a_E = −0.75 (unusable); upwind a_W = 3, a_E = 0.5 (bounded).

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Central at high Peclet	Oscillating, out-of-range values	"Is the cell Peclet number above 2?"	Use upwind or a bounded higher-order scheme, or refine the mesh.
Wiggles read as physics	Reporting nonphysical overshoots	"Do values exceed the boundary range?"	Bounded schemes keep φ between neighbour values.
Ignoring false diffusion	Smeared gradients with upwind	"Is the flow oblique to the grid?"	Align the mesh with the flow or use a higher-order scheme.
Forgetting the upstream direction	Upwind coefficients on the wrong side	"Which way is the flow going?"	Take the value from the upstream node for the sign of F.

Practice ladder

Level 1 · Direct skill

For F = 1.2 and D = 0.8, find the cell Peclet number and say whether central differencing is safe.

Show answer

Pe = F/D = 1.2/0.8 = 1.5 ≤ 2, so central differencing is safe.

Level 2 · Mixed concept

For the Worked Example 1 fast case, what grid spacing δx would bring the cell Peclet number down to 2?

Show answer

Pe = ρu δx/Γ = 2 gives δx = 2Γ/(ρu) = 2(0.1)/(1·2.5) = 0.08 m. Refining from 0.2 to 0.08 m restores central stability.

Level 3 · Independent problem

For F = 3, D = 1, flow toward the east, find the upwind coefficients and a_P, and confirm they are all positive.

Show answer

a_W = D + F = 4, a_E = D = 1, a_P = a_W + a_E = 5 (with F_e = F_w). All positive, so the scheme is bounded.

Level 4 · Transfer to real engineering

For a real convective flow you would simulate, estimate the cell Peclet number on a reasonable mesh and choose a scheme, justifying the accuracy-stability trade.

What good work looks like

A cell Peclet estimate, a scheme choice (upwind or a higher-order bounded scheme for high Pe), and an awareness of false diffusion versus oscillation.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my cell Peclet number justifies the scheme I picked."

"Give me four cases; I will say whether central differencing stays bounded."

"Which scheme should I use?" Computing Pe and the coefficients yourself is the skill.

"Why is my solution oscillating?" Spotting the negative coefficient is the point.

Portfolio task

For a convection-diffusion problem, compute the cell Peclet number, derive the central and upwind coefficients, and explain your scheme choice and its error trade-off.

Must include: a cell Peclet number, both schemes' coefficients, and a justified scheme choice.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the cell Peclet number.

Pe = F/D = ρu δx/Γ, convection over diffusion across a cell.

2. When does central differencing fail?

When Pe > 2, where a_E = D − F/2 goes negative and the solution oscillates.

3. Give the upwind coefficients for eastward flow.

a_W = D + F, a_E = D, both positive.

4. What is false diffusion?

Numerical smearing added by upwind, worst when the flow crosses the grid at an angle.

5. How do you keep accuracy and boundedness?

Refine the mesh or use a higher-order bounded scheme such as hybrid, power-law, or QUICK.

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

+1 dayRe-derive the Peclet check and coefficients from a blank page.

+3 daysCompare schemes for two new cases.

+7 daysCarry the φ-equation into pressure coupling, Chapter 5.

+30 daysReuse the boundedness idea when judging any CFD solution.

Textbook mapping

Item	Mapping
Primary source	Versteeg and Malalasekera, An Introduction to Computational Fluid Dynamics, Chapter 5 (Convection-Diffusion)
Cross-reference	Patankar, Ch. 5 · Anderson, Ch. 6
Core topics	4.1 Convection term · 4.2 Central differencing · 4.3 Cell Peclet number · 4.4 Upwind · 4.5 False diffusion and higher-order schemes
Engineering connection	The convection-diffusion equation is solved repeatedly inside SIMPLE.
Read next	Chapter 5: Pressure-Velocity Coupling: SIMPLE.