complex op• 資料種類:無 → cimage(僅由參數決定的運算子 —— 不接受影像或資料輸入)
• 呼叫: 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))(要直接呼叫實作,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));從台帳取用則 opsmath.get("potential_flow_joukowski"))
儒可夫斯基翼型周圍的無黏流, 以複速度場給出。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
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 的每個運算子都先檢驗輸入再計算(不讓任何東西無聲通過):
• **complex 輸入一律 ValueError** —— 強制轉成 float64 會無聲丟掉虛部(numpy 只發一個 ComplexWarning,然後回傳一個「看似合理卻是錯的」實數)。請明確寫出 .real/.imag/abs(),或改用支援複數的 complexops。
• **含被遮罩元素的 masked array 一律 ValueError** —— 拒絕「剝掉遮罩直接使用下面原值」的隱式轉換。請明確選擇填補還是丟棄。
• **所有輸入中的 NaN/Inf 一律 ValueError**(明確給出個數後拒絕 —— 它會汙染整個結果)。
• 形狀嚴格:不對 1-D 與 2-D 做隱式提升或廣播(向量槽位收到矩陣、矩陣槽位收到向量都是 ValueError;請明確 reshape)。
• 尺寸上限:接受矩陣的運算子與 stat_histogram 的 bins,超過 mathops.MAX_ELEMENTS(2^26 ≈ 6700 萬個元素)即 ValueError。
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• 演算法的正典(作者・年份)與用途見上面的族使用指南。
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
cimage 作為輸入)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 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.