Lift polar with adjoint slopes

PassedExample: lift polar with adjoint slopes5 of 5 checks passed · 686.1 s · quick suite

A warm-started sweep of the angle of attack for a NACA 2412 at Mach 0.4 in one solver process. At every point one adjoint solve gives the lift-curve slope, drawn as a tangent; the checks compare the slopes with a small-step central difference and with differences of neighbouring points.

all points convergedyesexpected truepass
positive lift at zero incidence (camber)0.2821expected > 0.15pass
adjoint slope vs small-step central difference at 4 deg (rel.)9.86e-07expected < 1e-5pass
adjoint slope vs two-degree secants of the polar (worst, rel.)0.0127expected < 0.03pass
warm starts: Newton steps after the first point (max)6expected < 14pass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:58 · cpu · quick suite · record (JSON)Same code as this build

Lift polar and adjoint tangentsExample: lift polar with adjoint slopes

Data table

series	α [deg]	cl
cl	-2	0.0185712
cl	-1	0.148671
cl	0	0.282053
cl	1	0.41544
cl	2	0.544452
cl	3	0.666028
cl	4	0.773825
adjoint tangents	-2.35	-0.0268269
adjoint tangents	-1.65	0.0639692
adjoint tangents	—	—
adjoint tangents	-1.35	0.102697
adjoint tangents	-0.65	0.194645
adjoint tangents	—	—
adjoint tangents	-0.35	0.234873
adjoint tangents	0.35	0.329232
adjoint tangents	—	—
adjoint tangents	0.65	0.369435
adjoint tangents	1.35	0.461446
adjoint tangents	—	—
adjoint tangents	1.65	0.500347
adjoint tangents	2.35	0.588558
adjoint tangents	—	—
adjoint tangents	2.65	0.625372
adjoint tangents	3.35	0.706685
adjoint tangents	—	—
adjoint tangents	3.65	0.734292
adjoint tangents	4.35	0.813358
adjoint tangents	—	—

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:58 · cpu · quick suite · record (JSON)Same code as this build

Each point is a converged flow over a NACA 2412 at Mach 0.4; the short segments are tangents whose slope is the adjoint derivative $\mathrm dc_l/\mathrm d\alpha$ at that point. They line up with the curve: the gradient is the gradient of the function being evaluated.

PolarExample: lift polar with adjoint slopes

α [deg]	cl	cd	cm	dcl/dα [1/rad]	dcl/dα [1/deg]	Newton steps	converged
-2	0.0185712	0.00100944	-0.0572344	7.43176	0.129709	14	yes
-1	0.148671	8.9199e-04	-0.0583837	7.52605	0.131354	6	yes
0	0.282053	9.2904e-04	-0.0601193	7.72338	0.134798	6	yes
1	0.41544	0.00102287	-0.0618284	7.53123	0.131445	6	yes
2	0.544452	0.00117546	-0.0626012	7.22024	0.126017	6	yes
3	0.666028	0.00142276	-0.061664	6.65559	0.116162	6	yes
4	0.773825	0.00180472	-0.0575159	6.47169	0.112952	6	yes

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:58 · cpu · quick suite · record (JSON)Same code as this build

How it was run#

yaml

- id: polar
  uses: sweep
  with:
    case: naca2412_euler
    parameter: alpha_deg
    values: [-2, -1, 0, 1, 2, 3, 4]
    outputs: [cl, cd, cm]
    gradients_wrt: [alpha]

python

with GyreClient(URL).session("naca2412_euler") as flow:
    for a in (-2, -1, 0, 1, 2, 3, 4):
        out = flow.evaluate(alpha_deg=a)["outputs"]
        slope = flow.gradients(of=["cl"], wrt=["alpha_deg"])["cl"]["alpha_deg"]

python

flow = Case(preset("naca2412_euler")).discipline
for a in (-2, -1, 0, 1, 2, 3, 4):
    flow.set_inputs(alpha=math.radians(a))
    flow.solve()
    slope = flow.totals(of=("cl",), wrt=("alpha",))["derivatives"]["cl"]["alpha"]

Nonlinear iterations per pointExample: lift polar with adjoint slopes

The first point starts from the free stream; the others are warm starts.

Data table

α [deg]	steps
-2.0	14
-1.0	6
0.0	6
1.0	6
2.0	6
3.0	6
4.0	6

gyre 0.4.0 · 39979fcf90 · 2026-11-12 11:58 · cpu · quick suite · record (JSON)Same code as this build

The first point starts from a uniform free stream and ramps the pseudo-time step. Every later point starts from the previous converged flow directly in the Newton regime and converges to the same absolute residual level in a few iterations.