Engineering lesson
Mesh convergence and residuals in an annular CFD calculation
A small nonlinear residual shows that a discrete system was solved. Mesh convergence asks whether a quantity of interest approaches a stable value as spatial resolution improves. These checks address different errors.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Separate algebraic and discretization error
A coarse mesh can have a tiny residual and a biased flow rate. Refining the mesh changes the approximation space, while tightening solver tolerance changes how accurately that discrete problem is solved.
The example uses an explicitly manufactured convergence table to teach the distinction. It is not labelled as measured runtime results from the site's CFD engine.
Compare a defined quantity of interest
Choose flow rate, wall stress or pressure gradient and keep boundary conditions, rheology and geometry fixed. Report both absolute values and changes under refinement.
The linked annular CFD workflow includes its documented mixed formulation and verification diagnostics. Its internal checks are not a user-selectable mesh-refinement study.
Use independent verification where possible
Analytical limiting cases, manufactured solutions and conservation checks can reveal errors that successive-mesh agreement alone misses.
A mesh-independent answer to an inappropriate physical model is still inappropriate. Numerical verification and physical validation remain separate tasks.
Explore the example
Original manufactured convergence exercise with an explicitly known limit. No CFD runtime result or measured convergence order is claimed.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Resolution count n = 16. Quantity of interest Qh: 1.00390625; Discrete residual: 1e-10; Relative error (%): 0.390625.
- Quantity of interest Qh
- Discrete residual
Worked example
Manufactured refinement sequence with a known limit
Teaching quantity Qh = 1 + h², exact limit Q = 1, h = 1/n. Every manufactured discrete residual is 10⁻¹⁰.
Relation and unit basis
Qh = 1 + 1/n²; relative error = (Qh − 1)/1
n is a nominal resolution count. This table is manufactured, not a recorded annular CFD mesh study.
1. Define the baseline
Resolution count n = 16
Teaching quantity Qh = 1 + h², exact limit Q = 1, h = 1/n. Every manufactured discrete residual is 10⁻¹⁰.
2. Select the relation
Qh = 1 + 1/n²; relative error = (Qh − 1)/1
n is a nominal resolution count. This table is manufactured, not a recorded annular CFD mesh study.
3. Substitute the values
At n = 16: Qh = 1 + 1/256 = 1.00390625; error = 0.390625%; residual remains 10⁻¹⁰
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
Quantity of interest Qh = 1.00390625; Discrete residual = 1e-10; Relative error (%) = 0.390625
Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks.
Result and interpretation
Quantity of interest Qh = 1.00390625; Discrete residual = 1e-10; Relative error (%) = 0.390625
Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks.
The transparent manufactured sequence has a known limiting value and deliberately tiny residuals at every resolution.
Exact scenario data
Original manufactured convergence exercise with an explicitly known limit. No CFD runtime result or measured convergence order is claimed.
| Resolution count n | Quantity of interest Qh | Discrete residual | Relative error (%) |
|---|---|---|---|
| 4 | 1.0625 | 1.0000e-10 | 6.25 |
| 8 | 1.015625 | 1.0000e-10 | 1.5625 |
| 16 | 1.003906 | 1.0000e-10 | 0.390625 |
| 32 | 1.000977 | 1.0000e-10 | 0.09765625 |
| 64 | 1.000244 | 1.0000e-10 | 0.02441406 |
Scroll the table to see all columns. The highlighted row is the worked baseline. Downloads retain full numerical precision.
Test your understanding
Work through the question before revealing its answer. Use the stated assumptions and include units in every numerical step.
1. Does a residual of 10⁻¹⁰ prove mesh independence?
Show the worked answer
No. It measures the discrete solve, not the spatial approximation error.
2. Reproduce the baseline Quantity of interest Qh without rounding intermediate values.
Show the worked answer
At n = 16: Qh = 1 + 1/256 = 1.00390625; error = 0.390625%; residual remains 10⁻¹⁰. Quantity of interest Qh = 1.00390625. Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Resolution count n changes from 4 to 64. Other controls remain as stated in the premise. Quantity of interest Qh changes from 1.0625 to 1.00024414.
4. What would be needed to apply this example to a different physical system?
Show the worked answer
Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. n is a nominal resolution count. This table is manufactured, not a recorded annular CFD mesh study. Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks.
Before using the result
- Fix the quantity of interest.
- Keep physical inputs unchanged across meshes.
- Separate residual, refinement and physical validation.
Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks.
Continue learning
Start with the broader workflow guide
Sources and method
The explanation and synthetic examples are original PetroCalcHub teaching material. The references below support the definitions and model context. They do not supply field validation for the synthetic scenarios.