complex op• Data kinds: none → cimage (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.potential_flow_joukowski(alpha_deg=5.0, speed=1.0, chord_b=1.0, centre_offset=(-0.09+0.09j), centre=0j, half_width=3.0, shape=(256, 256)) (to call the implementation directly, import mathops; mathops.potential_flow_joukowski(alpha_deg=5.0, speed=1.0, chord_b=1.0, centre_offset=(-0.09+0.09j), centre=0j, half_width=3.0, shape=(256, 256)); from the registry, opsmath.get("potential_flow_joukowski"))
Inviscid flow past a Joukowski aerofoil, as a complex velocity field.
Returns the complex velocity `w(z) = dW/dz = u - i*v` sampled over a
window of the physical plane. Inside the solid body the field is set to
exactly `0 (there is no flow there), so field == 0` is the body mask
and nothing is silently `nan`.
The construction is the classical one: flow past a circle of radius
`a = |chord_b - centre_offset| centred at centre_offset`, plus the
circulation the Kutta condition demands, pushed through the Joukowski
map `z = zeta + chord_b**2 / zeta`. The window is inverted back to the
circle plane by choosing, of the two preimages, the one outside the circle.
★What makes this checkable rather than merely plotted:
• The field is holomorphic outside the body, so this family's own
`cplx_cr_residual` must read ~0 on it — an existing operator is the
oracle, and it is not the formula used to build the field.
• The Kutta condition is the whole point: `dW/dzeta` vanishes at the
trailing edge exactly where `dz/dzeta` does, so the velocity there
stays finite. Perturb the circulation by any amount and the trailing-edge
velocity diverges — the gate has a control group.
• `Re(closed integral of w dz)` around the body is the circulation, and
it is path independent (Cauchy) — a big rectangle and a small one
must agree.
• Far from the body `w -> speed * exp(-i*alpha), decaying like 1/|z|`.
• The zeroed region is the aerofoil, whose area the shoelace formula on
`cplx_joukowski` of the circle gives independently.
Parameters
----------
alpha_deg : float
Angle of attack in degrees.
speed : float > 0
Free-stream speed.
chord_b : float > 0
Half the flat-plate chord; the map is `zeta + chord_b**2/zeta`.
centre_offset : complex
Circle centre. Negative real part thickens, positive imaginary part
cambers. The default is a thin cambered section.
centre, half_width, shape :
The window in the physical plane, as in :func:cplx_plane_grid.
Raises `ValueError: speed <= 0`; non-finite angle; a circle that
does not enclose `-chord_b` (the image would fold over itself) or whose
centre is at or beyond `+chord_b`; degenerate window.
HALCON: no operator.
Every mathops op validates its input before computing (nothing slips through silently):
• **complex input raises ValueError** — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• **masked arrays with masked elements raise ValueError** — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• **NaN/Inf raises ValueError on every input** (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
cimage as input)cplx_cr_residual · cplx_domain_colour · mandelbrot_interior
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski
*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.