{"id": "sphere_inviscid", "title": "Inviscid flow past a sphere (3-D)", "description": "Subsonic inviscid flow (M = 0.2) past a sphere of diameter 1 in a spherical far field of radius 10, on a cubed-sphere mesh of curved hexahedra (geometry degree 4): six blocks whose local axes are rotated against each other, so that the block interfaces contain all eight relative face orientations and the flow solve itself exercises the orientation tables. The exact solution is isentropic, has no drag (d'Alembert) and, by symmetry, no lift, side force or moment. Its surface pressure is known to O(M\u00b2) from the Janzen\u2013Rayleigh expansion of the compressible potential, whose first correction is derived in closed form in the study module (surface speed 3/2 + 0.3131 M\u00b2 at the equator) and checked numerically here; the neglected terms are O(M\u2074) = 1.6e-03 times a coefficient of order one. Reported per polynomial degree: the L2 entropy error, the drag coefficient (frontal-area reference) and the area-weighted L2 difference of the wall pressure coefficient to that reference.", "passed": true, "checks": [{"name": "Janzen\u2013Rayleigh correction solves its equation (largest defect)", "value": 2.9887802129868746e-06, "expected": "< 1e-5 (finite-difference check)", "passed": true}, {"name": "all solves converged", "value": true, "expected": "true", "passed": true}, {"name": "block interfaces show all 8 relative face orientations", "value": 8, "expected": "= 8", "passed": true}, {"name": "entropy error falls with p (smallest ratio between consecutive orders)", "value": 2.494217526597203, "expected": "> 2", "passed": true}, {"name": "spurious drag |cd| falls with p (smallest ratio)", "value": 2.1034393325218503, "expected": "> 1.5", "passed": true}, {"name": "spurious drag |cd| at the highest resolution", "value": 0.003294028033148043, "expected": "< 0.03", "passed": true}, {"name": "surface cp error against Janzen\u2013Rayleigh falls with p (smallest ratio)", "value": 3.0841963582655727, "expected": "> 1.5", "passed": true}, {"name": "surface cp, area-weighted L2 difference to Janzen\u2013Rayleigh, highest resolution", "value": 0.010990009088485774, "expected": "< 0.03", "passed": true}, {"name": "lift, side force and moments vanish (largest |coefficient|)", "value": 1.1084233064900356e-13, "expected": "< 1e-8", "passed": true}], "charts": [{"id": "cp", "type": "line", "title": "Surface pressure coefficient", "xlabel": "polar angle from the front stagnation point (degrees)", "ylabel": "cp", "xscale": "linear", "yscale": "linear", "series": [{"name": "216 elements, p = 1", "x": [20.75357098368479, 20.75357098368479, 20.75357098368479, 20.75357098368479, 20.75357098368479, 20.75357098368479, 45.99297280436371, 45.99297280436371, 45.99297280436371, 45.99297280436371, 45.99297280436371, 45.99297280436371, 45.99297280436371, 45.99297280436371, 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2.3376577330419162e-06, 5.927829500406654e-07]}, {"name": "|cd|", "x": [8640, 29160, 69120, 135000], "y": [0.12313300448952937, 0.0585388903714675, 0.014441499754388985, 0.003294028033148043]}, {"name": "cp L2 difference", "x": [8640, 29160, 69120, 135000], "y": [0.6057494235442742, 0.1964042989418881, 0.061255887803918246, 0.010990009088485774]}]}], "tables": [{"id": "orders", "title": "Convergence with the polynomial degree", "columns": ["elements", "p", "DOF", "Newton steps", "residual reduction", "entropy error", "cd", "cp L2 diff. (Janzen\u2013Rayleigh)", "cp L2 diff. (incompressible speed)", "cp L2 diff. (incompressible)", "cp max diff.", "largest |cl, cs, cm|", "solve time (s)"], "rows": [[216, 1, 8640, 11, 4.931043809061783e-11, 2.5016539951443297e-05, 0.12313300448952937, 0.6057494235442742, 0.5865552078751347, 0.5936912090307009, 0.9972521460574686, 4.240739575284689e-15, 43.79130268096924], [216, 2, 29160, 12, 6.0815430655632995e-12, 1.002981483558602e-05, 0.0585388903714675, 0.1964042989418881, 0.18221312353933874, 0.1863035511998332, 0.751574224002071, 1.1084233064900356e-13, 54.29572820663452], [216, 3, 69120, 13, 1.4683332482587756e-12, 2.3376577330419162e-06, 0.014441499754388985, 0.061255887803918246, 0.055065099790509796, 0.05508298877818228, 0.3208630018063102, 1.4418514555967942e-14, 39.59684085845947], [216, 4, 135000, 13, 3.516698829745304e-11, 5.927829500406654e-07, 0.003294028033148043, 0.010990009088485774, 0.019267280178455644, 0.011774910563408267, 0.05321356213486972, 6.365526799984621e-14, 111.03891801834106]]}, {"id": "orientations", "title": "Interior faces by relative orientation (transpose, flip first, flip second)", "columns": ["orientation code", "faces"], "rows": [[0, 924], [1, 24], [2, 24], [3, 24], [4, 48], [5, 48], [6, 48], [7, 48]]}], "provenance": {"version": "0.4.0", "commit": "39979fcf902ad6116eb550b881b4278d4599c8e2", "dirty": false, "describe": "39979fcf90", "device": {"platform": "gpu", "device": "cuda:0", "device_kind": "NVIDIA GeForce RTX 3060 Ti", "jax": "0.11.2", "x64": true}, "timestamp": "2026-11-12T11:11:04-0500"}, "wall_time": 424.67, "notes": "The far-field boundary at 20 radii imposes the free stream through a Riemann flux; the potential-flow disturbance there is of relative size (a/r)\u00b3/2 = 6e-5. The drag and entropy thresholds are bounds observed with margin on this mesh, not asymptotic rates: four polynomial degrees on one mesh do not establish an order of convergence.", "level": "quick"}