[{"data":1,"prerenderedAt":131},["ShallowReactive",2],{"$f1fyd8g5m9sr1h":3},{"slug":4,"publishedAt":5,"category":6,"title":7,"summary":8,"image":9,"imageAlt":14,"readingMinutes":15,"serviceId":16,"keywords":17,"imageCaption":22,"sections":23,"sources":121},"mesh-independence-study","2026-10-12","methods","Mesh Independence Study: A Worked GCI Example","A step-by-step mesh independence study: three meshes, the refinement ratio, the observed order and the grid convergence index, ending in one sentence for your report.",{"src":10,"thumbnail":11,"width":12,"height":13},"\u002Fassets\u002Fcfd-cases\u002Fshell\u002Fimages\u002F06.webp","\u002Fbrand\u002Fstudies\u002Fshell.svg",1448,1086,"DEEB-FLOW illustration of pressure drop plotted against mass flow rate",5,"review",[18,19,20,21],"mesh independence study","grid convergence index","GCI CFD","mesh sensitivity study","Concept illustration from the DEEB-FLOW case library. It is not a result from this article.",[24,30,36,48,98,105,115],{"id":25,"title":26,"paragraphs":27},"two-meshes","Why two meshes prove nothing",[28,29],"\"We ran it on two meshes and the contours looked the same.\" That sentence appears in a great many reports, and it does not mean what its author hoped. Two contour plots that look alike can hide a 10% difference in the one number the client needed.","A mesh independence study asks how much your result would change if the mesh were infinitely fine, and answers with a number. This article walks through one, from three meshes to a sentence for the report. The method is the one published by Celik and co-authors in 2008, built on Roache’s grid convergence index.",{"id":31,"title":32,"paragraphs":33},"what-you-find-out","What are you trying to find out?",[34,35],"Every finite-volume solution carries a discretisation error that shrinks as the cells get smaller. If the mesh is fine enough, the error shrinks in an orderly way: halve the cell size with a second-order scheme and the error falls by about four.","With solutions on three meshes you can measure how fast the result is changing, extrapolate to zero cell size, and say how far the fine-mesh result is from that limit. The grid convergence index, GCI, is that distance with a safety factor. It bounds the numerical error only and says nothing about whether the turbulence model is right.",{"id":37,"title":38,"paragraphs":39},"four-rules","Four rules before you start",[40,47],{"list":41,"ordered":46},[42,43,44,45],"**Pick the quantity that matters.** Pressure drop, heat duty, a peak temperature. Do the study on what you will report.","**Refine everywhere, by the same factor.** Change the global cell size. If you only refine where you guessed the gradients were, you are testing your guess.","**Keep the near-wall treatment consistent.** With wall functions, keep the first cell in the valid range on all three meshes. If you resolve the boundary layer, refine it with the rest.","**Converge each run properly.** Iteration error must be well below the differences you are trying to measure.",true,"Use a refinement ratio of at least 1.3 in cell size between meshes. In three dimensions that means about 2.2 times the cell count at each step, and a ratio of 1.5 means about 3.4 times.",{"id":49,"title":50,"paragraphs":51},"worked-example","The worked example",[52,53,72,73,76,77,81,82,85,86,89,90,94,95],"A pipe fitting, steady flow, quantity of interest: pressure drop. Three meshes with a constant refinement ratio of 1.5. The numbers are an example chosen to show the method.",{"table":54},{"head":55,"rows":59},[56,57,58],"Mesh","Cells","Pressure drop (Pa)",[60,64,68],[61,62,63],"3, coarse","0.40 × 10⁶","1297",[65,66,67],"2, medium","1.35 × 10⁶","1262",[69,70,71],"1, fine","4.56 × 10⁶","1245","**Step 1, the refinement ratio.** From the cell counts in three dimensions it is 1.5 between each pair of meshes.",{"formula":74},[75],"r = (N₁ \u002F N₂)^(1\u002F3) = (4.56 \u002F 1.35)^(1\u002F3) = 1.5","**Step 2, the differences.** Both are positive and the second is smaller, so the result is converging monotonically.",{"formula":78},[79,80],"ε₃₂ = 1297 − 1262 = 35 Pa","ε₂₁ = 1262 − 1245 = 17 Pa","**Step 3, the observed order.** The scheme is formally second order and we observe 1.78. That is healthy: real meshes with boundary layers and limiters rarely give exactly 2.",{"formula":83},[84],"p = ln(ε₃₂ \u002F ε₂₁) \u002F ln(r) = ln(35 \u002F 17) \u002F ln(1.5) = 1.78","**Step 4, extrapolate to zero cell size.** This is the estimate of what an infinitely fine mesh would give.",{"formula":87},[88],"φ_ext = φ₁ − ε₂₁ \u002F (r^p − 1) = 1245 − 17 \u002F 1.06 = 1229 Pa","**Step 5, the grid convergence index.** The relative change between fine and medium is 1.37%. With the safety factor of 1.25 recommended for a three-mesh study, the fine mesh has a GCI of 1.6% and the medium mesh 3.3%.",{"formula":91},[92,93],"GCI_fine = 1.25 × 1.37% \u002F (r^p − 1) = 1.6%","GCI_medium = 1.25 × 2.77% \u002F (r^p − 1) = 3.3%","**Step 6, check the asymptotic range.** A value close to 1 says the three meshes sit where the error falls in an orderly way, so the extrapolation can be trusted.",{"formula":96},[97],"GCI_medium \u002F (r^p × GCI_fine) = 3.27 \u002F (2.06 × 1.61) = 0.99",{"id":99,"title":100,"paragraphs":101},"report","What goes in the report",[102,104],{"quote":103},"Pressure drop on the fine mesh is 1,245 Pa with a numerical uncertainty of 1.6% (grid convergence index, three meshes, refinement ratio 1.5, observed order 1.78). The extrapolated value is 1,229 Pa.","And one decision. If 3.3% is tolerable for the job, the medium mesh is enough for the remaining cases, at less than a third of the cost. That saving is the practical reason to do the study at the start of a project.",{"id":106,"title":107,"paragraphs":108},"when-numbers-misbehave","When the numbers misbehave",[109],{"list":110},[111,112,113,114],"**The differences change sign.** That is oscillatory convergence, and the extrapolation is not valid. Check iteration convergence first, then add a finer mesh.","**The observed order is far from the formal one.** Below about 0.5, or well above the order of the scheme. Treat the GCI with suspicion and say so. Common causes are a coarse mesh that misses a flow feature and wall functions leaving their valid range on one mesh.","**The differences are tiny.** If all three results agree to four digits, check that the meshes really differ where it matters.","**The result is still moving and a fourth mesh is not affordable.** Report the fine result with its GCI, state that the asymptotic check failed, and treat the number as provisional.",{"id":116,"title":117,"paragraphs":118},"proportionate-effort","How much effort is proportionate?",[119,120],"A full three-mesh study with a reported GCI is right when an absolute value matters: a guarantee, a validation against test data, a publication. For a quick comparison of two design options, one refinement on the quantity of interest is often enough to show that the ranking does not flip. Call that a sensitivity check; it does not give an uncertainty.","What I would not accept is nothing at all. A result from a single mesh has an unknown error, and unknown is worse than large.",[122,125,128],{"href":123,"label":124},"https:\u002F\u002Fdoi.org\u002F10.1115\u002F1.2960953","Celik et al. (2008), Journal of Fluids Engineering: procedure for estimation and reporting of uncertainty due to discretization in CFD applications",{"href":126,"label":127},"https:\u002F\u002Fdoi.org\u002F10.1115\u002F1.2910291","Roache (1994), Journal of Fluids Engineering: a method for uniform reporting of grid refinement studies",{"href":129,"label":130},"https:\u002F\u002Fwww.grc.nasa.gov\u002Fwww\u002Fwind\u002Fvalid\u002Ftutorial\u002Fspatconv.html","NASA \u002F NPARC: examining spatial (grid) convergence",1791759706699]