Three-dimensional flows

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(M²) from the Janzen–Rayleigh expansion of the compressible potential, whose first correction is derived in closed form in the study module (surface speed 3/2 + 0.3131 M² at the equator) and checked numerically here; the neglected terms are O(M⁴) = 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.

• Janzen–Rayleigh correction solves its equation (largest defect)2.99e-06expected < 1e-5 (finite-difference check)pass
• all solves convergedyesexpected truepass
• block interfaces show all 8 relative face orientations8expected = 8pass
• entropy error falls with p (smallest ratio between consecutive orders)2.494expected > 2pass
• spurious drag |cd| falls with p (smallest ratio)2.103expected > 1.5pass
• spurious drag |cd| at the highest resolution0.003294expected < 0.03pass
• surface cp error against Janzen–Rayleigh falls with p (smallest ratio)3.084expected > 1.5pass
• surface cp, area-weighted L2 difference to Janzen–Rayleigh, highest resolution0.01099expected < 0.03pass
• lift, side force and moments vanish (largest |coefficient|)1.11e-13expected < 1e-8pass

Run 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)³/2 = 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.

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:11 · gpu · NVIDIA GeForce RTX 3060 Ti · quick suite · record (JSON)Same code as this build

Steady three-dimensional Navier–Stokes solve: sphere of diameter 1, Re = 100 on the diameter, M = 0.2, adiabatic no-slip wall, far field at 10 diameters; cubed-sphere mesh of 576 curved hexahedra (geometry degree 3) with mixed block orientations, BR2 viscous fluxes, Newton–Krylov with p-multigrid. At this Reynolds number the flow is steady and axisymmetric with a closed recirculation bubble. References: drag coefficient 1.09 (standard drag curve of Clift, Grace & Weber 1978; Johnson & Patel, JFM 378, 1999, compute 1.08–1.10) and a bubble length of about 0.87 diameters (Johnson & Patel 1999; Taneda 1956). Both references are for incompressible flow in an unbounded domain; at M = 0.2 with the boundary at 10 diameters a difference of the order of one per cent is expected, which the tolerance on the drag covers. Drag is referred to the frontal area π/4.

• all solves converged to a steady stateyesexpected truepass
• drag coefficient, highest order: |cd − 1.09| / 1.090.01409expected < 0.03pass
• wake is axisymmetric: lift, side force and moments (largest |coefficient|)1.77e-15expected < 1e-7pass
• a closed recirculation bubble exists; length in diameters, highest order0.9501expected within 15 % of 0.87pass
• adiabatic wall: integrated heat flux0expected < 1e-10pass

Run notes The mesh has the symmetry of the cube, so the vanishing lift and side force show that the solver preserves a symmetric solution; they do not test the stability of the axisymmetric wake (which is stable below Re of about 210). The bubble length is read from the solution nodes on the wake axis by linear interpolation between the two nodes where the velocity changes sign.

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:14 · gpu · NVIDIA GeForce RTX 3060 Ti · quick suite · record (JSON)Same code as this build

An ellipsoid with three different semi-axes (0.5, 0.35, 0.6) at 4 degrees angle of attack and 3 degrees sideslip, M = 0.3, on a mixed-orientation cubed-sphere mesh: nothing is symmetric, so all six wind-axis force and moment coefficients (drag, lift, side force, roll, yaw, pitch about an offset reference point) and all their derivatives are non-zero. Total derivatives from one adjoint solve per output are compared with central finite differences of re-converged solves with respect to the angle of attack, the sideslip angle, the Mach number (and the Reynolds number for the viscous case), and, as a directional derivative, with respect to a smooth three-dimensional displacement of the surface nodes (the shape input x_surface, propagated into the volume by the mesh deformation). The finite differences are accurate to about 1e-7 in absolute terms (step 1e-5, solves converged 12 orders), so errors are relative to the reference value or, for a derivative that is small through cancellation, to 1 % of the largest derivative with respect to the same input (and at least 1e-3).

• adjoint vs central differences, d/dalpha: largest relative error3.28e-06expected < 1e-5pass
• adjoint vs central differences, d/dbeta: largest relative error1.46e-06expected < 1e-5pass
• adjoint vs central differences, d/dmach: largest relative error2.33e-06expected < 1e-5pass
• adjoint vs central differences, d/dx_surface: largest relative error1.97e-06expected < 1e-4pass
• adjoint linear solves reached their tolerance (largest reduction)9.99e-11expected < 1e-9pass
• the flow responds to the sideslip angle (free stream turned by beta)yesexpected truepass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:38 · cpu · quick suite · record (JSON)Same code as this build

Inviscid flow (M = 0.3) past a sphere on a cubed-sphere mesh of 7680 curved hexahedra (geometry degree 2, mixed block orientations) at p = 2: 1,036,800 degrees of freedom, solved on the GPU (NVIDIA GeForce RTX 3060 Ti) with the standard pseudo-transient Newton–Krylov solver and p-multigrid [2, 1, 0]: residual, Jacobian products and Krylov vectors in double precision, the blocks of the smoother in single precision (solver.precond_precision). Solved to a residual reduction of 1e-8, followed by one discrete adjoint solve for the drag coefficient (derivatives with respect to angle of attack, sideslip, Mach number and the position of each of the 6,146 surface nodes). Wall-clock times include just-in-time compilation unless stated; they depend on what else the machine was doing, which is recorded below. The configuration depends on level and device: quick/CPU 4x4x4 blocks, quick/GPU and full/GPU 16x16x5 (1,036,800 unknowns, single-precision blocks), full/CPU 8x8x5. On the GPU the same solve is first tried with double-precision blocks in a process of its own; what happened is in the table of sizes and is not checked, because it depends on the memory of the device.

• steady solve convergedyesexpected truepass
• residual reduction (orders of magnitude)9.375expected >= 8pass
• adjoint linear solve converged (residual reduction)9.97e-09expected < 1e-7pass
• spurious drag of the inviscid sphere0.01716expected < 0.1pass
• lift and side force vanish1.82e-14expected < 1e-7pass
• shape derivative of the drag for every surface node: finite, and its net transverse component vanishes by symmetry (largest |sum|, relative to the largest entry)1.28e-10expected < 1e-6pass
• degrees of freedom1036800expected >= 0pass

Run notes The timings were taken on a busy machine (other processes used 87 % of the capacity of the 8 CPUs while the study ran; the GPU was 15 % utilized when the study started). What limits the size on this device: the element-block inverses of the block-Jacobi smoother are dense (block size squared per element and level, in double precision) and their construction holds several copies at once; the Krylov basis is small in comparison. The mesh-deformation operator behind the shape derivative is a dense (volume nodes x surface nodes) matrix; above 2e7 entries it is applied matrix-free, with its transpose written out chunk by chunk, and is not stored.

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:22 · gpu · NVIDIA GeForce RTX 3060 Ti · quick suite · record (JSON)Same code as this build

The inviscid NACA 0012 case is solved in 2-D and on a spanwise-periodic extrusion of the same section into curved hexahedra. A correct 3-D implementation — metrics in conservative-curl form, neighbour coupling across hexahedral faces, the Newton–Krylov solver with p-multigrid, force integration and the discrete adjoint — must reproduce the 2-D outputs and derivatives per unit span, with no spanwise flow. A structured extrusion only contains faces whose two sides use the same local axes, so this study cannot detect an error in the face-orientation tables; those are exercised by the imported-mesh study and by unit tests on randomly re-oriented meshes.

• both solves convergedyesexpected truepass
• cl: |3-D − 2-D| / |2-D|3.45e-13expected < 1e-08pass
• cd: |3-D − 2-D| / |2-D|1.64e-13expected < 1e-08pass
• cm: |3-D − 2-D| / |2-D|4.18e-12expected < 1e-08pass
• dcl_dalpha: |3-D − 2-D| / |2-D|1.02e-11expected < 1e-06pass
• dcl_dmach: |3-D − 2-D| / |2-D|4.42e-09expected < 1e-06pass
• spanwise momentum (max |ρw|)7.35e-14expected < 1e-10pass
• spanwise variation of the state6.05e-14expected < 1e-10pass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:03 · gpu · NVIDIA GeForce RTX 3060 Ti · quick suite · record (JSON)Same code as this build