---
title: "Examples"
summary: "Worked cases with real output: an inviscid aerofoil, a transonic flow with shocks, a separated viscous flow, a turbulent boundary layer, a sphere in three dimensions, a case on a mesh imported from Gmsh, a warm-started polar with adjoint slopes, a coupled aero-structural adjoint, OpenMDAO, and an agent session."
section: examples
order: 0
level: 1
keywords: [examples]
---
<!-- gyre docs: expanded Markdown of examples/index.md (generated tables, verification numbers and chart data are written out). Base level of this page: L1 Essentials; passages at another level are bracketed by 'begin level N' / 'end level N' comments and start with [LN Name]. Built from revision ceb1f4b0be. -->

Every figure, table and field on these pages is produced by the example studies of the verification pipeline and carries the revision that computed it.

- [Inviscid aerofoil](naca0012-euler.md) (L1 Essentials): The complete loop on one case — solve the flow over a NACA 0012, inspect the field and the surface pressure, obtain adjoint derivatives with respect to flow conditions and shape, and estimate the error of the result.
- [Transonic aerofoil](transonic.md) (L2 Practice): A flow with shocks — NACA 0012 at Mach 0.8 — computed with sensor-based shock capturing, converged to machine precision, with adjoint derivatives taken through the shock.
- [Laminar cylinder](cylinder-re40.md) (L1 Essentials): A viscous case with separation — the steady wake of a circular cylinder at Reynolds number 40 — showing no-slip walls, skin friction and the separation point.
- [Turbulent flat plate](turbulent-flat-plate.md) (L1 Essentials): A Reynolds-averaged case with the Spalart–Allmaras model — the zero-pressure-gradient flat plate at a Reynolds number of five million — showing how a turbulent case is specified, what its boundary layer looks like in wall units, and that its drag can be differentiated like any other output.
- [Sphere in three dimensions](sphere.md) (L1 Essentials): Flow past a sphere on an unstructured mesh of curved hexahedra, inviscid and at Reynolds number 100. The same commands, outputs and derivatives as in two dimensions, with three force and three moment coefficients and a sideslip angle.
- [A mesh from Gmsh](imported-mesh.md) (L1 Essentials): Laminar flow past a cylinder in a channel on an unstructured mesh of cubic quadrilaterals written by Gmsh — set up from the mesh's physical groups, solved in Python and through the service, with an adjoint derivative checked against finite differences.
- [Lift polar with adjoint slopes](lift-polar.md) (L2 Practice): A parameter sweep in one warm solver process, with the lift-curve slope from one adjoint solve at every point. Demonstrates warm starts and the consistency of gradients along a path.
- [Coupled aero-structural adjoint](aerostructural-coupling.md) (L2 Practice): A flow coupled to an elastic skin, with the coupled derivative assembled outside Gyre from loads, displacements and reverse total products — the pattern an MDO framework uses, in sixty lines.
- [Inside OpenMDAO](openmdao.md) (L2 Practice): Wrap the flow discipline as a matrix-free OpenMDAO component, in reduced space or at residual level, and let the framework assemble total derivatives in forward or reverse mode. No optimizer involved.
- [An agent at work](agent-session.md) (L2 Practice): The Model Context Protocol exchange behind an agent task — "characterise this aerofoil and tell me how far to trust the drag" — shown as the raw tool calls and results.
- [A remote session called at fixed design points](remote-session.md) (L2 Practice): The calls an optimizer or MDO framework running on another machine would make on a warm session of a deployment — evaluate, gradients, adjoint and tangent products, a finite-difference cross-check, checkpoint and restore — shown at three fixed design points, with no optimizer.