MS/PhD CFD: Grid Convergence Versus Validation
An MS candidate simulates laminar flow over a backward-facing step at Re = 800 with a finite-volume code. The quantity of interest is the reattachment length x_r/H. Three grids with characteristic cell sizes h, h/2, and h/4 give x_r/H = 6.1, 6.9, and 7.1. The finest grid has 640,000 cells. The experimental value is 7.0 ± 0.2. Residuals fall three orders of magnitude, and the mass imbalance is 10⁻⁵. The supervisor asks whether the finest-grid result is grid-converged and whether the simulation is validated. The code uses second-order upwind convection, SIMPLE pressure–velocity coupling, and a domain extending 10H downstream. The candidate has time for one more run and must choose between a finer grid, a longer domain, or a different outlet boundary condition.
1. Apply Richardson extrapolation to the three reattachment lengths. Estimate the observed order p, the grid-independent value, and the fine-grid GCI. Assume a refinement ratio of 2.
Read the worked answer
With refinement ratio r = 2: p = ln[(f₂ − f₁)/(f₃ − f₂)] / ln r = ln[(6.9 − 6.1)/(7.1 − 6.9)] / ln 2 = ln 4 / ln 2 = 2.
The observed order is 2. Extrapolated value: f_exact = f₃ + (f₃ − f₂)/(rᵖ − 1) = 7.1 + 0.2/(4 − 1) = 7.167.
Fine-grid GCI: GCI = 1.25 |(f₃ − f₂)/f₃| / (rᵖ − 1) = 1.25 (0.2/7.1)/3 ≈ 0.0117, about 1.2%.
2. Does the fine-grid GCI show that the result is validated? Explain the difference between verification and validation, using the experimental uncertainty and at least two other error sources.
Read the worked answer
No. Grid convergence is verification, not validation. The GCI shows the discretization error on the finest grid is about 1.2%, or 0.08 in x_r/H. Validation compares the simulation with experiment, including experimental uncertainty and model error.
The experimental value 7.0 ± 0.2 overlaps the computed 7.1, so the comparison is consistent, but that does not prove the model is correct. Other error sources include the outlet boundary at 10H, the 2D assumption versus spanwise effects in the experiment, inlet turbulence or velocity profile, and the convection scheme. A longer domain or a different outlet condition could shift x_r.
3. Which one additional run should the candidate make, and why? Support with the numbers.
Read the worked answer
Choose the longer domain or outlet-boundary check, not a finer grid. The fine-grid GCI is already 1.2%, while the experimental uncertainty is about 0.2/7.0 ≈ 2.9%. Reducing discretization error further gives diminishing returns.
A domain extending 10H downstream may be too short at Re = 800. If the reattachment length changes by more than 0.1 when the outlet is moved to 20H, that is a model error larger than the grid error. A finer grid would test only the numerical error already estimated. Therefore the one additional run should test a modelling assumption.