complex op• データ種: なし → cimage(引数だけで決まる op —— 画像やデータの入力を取らない)
• 呼び出し: 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)) (実装を直接呼ぶなら 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))、台帳から引くなら 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.
mathops の全 op は入力を検証してから計算する(黙って通さない):
• **complex 入力は ValueError** — float64 への強制変換は虚部を黙って捨てる(numpy は ComplexWarning だけ出して「もっともらしく間違った」実数を返す)。.real/.imag/abs() を明示するか、複素対応の complexops を使う。
• **masked array(masked 要素あり)は ValueError** — マスクを剥がして下の生値を使う暗黙変換を拒否。埋める/落とすを明示する。
• **NaN/Inf は全入力で ValueError**(件数を明示して拒否 — 結果全体に伝播するため)。
• 形状は厳格: 1-D と 2-D を暗黙昇格・ブロードキャストしない(vector 枠に matrix、matrix 枠に vector は ValueError。reshape を明示する)。
• サイズ上限: 行列を取る op と stat_histogram の bins は mathops.MAX_ELEMENTS(2^26 ≈ 6700 万要素)超で ValueError。
• サンプルデータ カタログ(DL URL / ライセンス) — 2-D は skimage.data(BSD/public)+ 合成、3-D は実データ源(Stanford/PDS 等)の DL URL。
• 演算子の来歴・参考文献 — この op 族の元になった研究/手法の出典。
• アルゴリズムの正典(著者・年)と用途は上記ファミリ使い方ガイドに記載。
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
cimage を入力に取れる)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. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.