complex op• 数据种类:无 → cimage(仅由参数决定的算子 —— 不接受图像或数据输入)
• 调用: 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))(要直接调用实现,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));从台账取用则 opsmath.get("potential_flow_joukowski"))
儒可夫斯基翼型周围的无黏流, 以复速度场给出。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
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 的每个算子都先校验输入再计算(不让任何东西无声通过):
• **complex 输入一律 ValueError** —— 强制转成 float64 会无声丢掉虚部(numpy 只发一个 ComplexWarning,然后返回一个「看着合理却是错的」实数)。请显式写出 .real/.imag/abs(),或改用支持复数的 complexops。
• **含被掩元素的 masked array 一律 ValueError** —— 拒绝「剥掉掩码直接使用下面原值」的隐式转换。请显式选择填充还是丢弃。
• **所有输入中的 NaN/Inf 一律 ValueError**(明确给出个数后拒绝 —— 它会污染整个结果)。
• 形状严格:不对 1-D 与 2-D 做隐式提升或广播(向量槽位收到矩阵、矩阵槽位收到向量都是 ValueError;请显式 reshape)。
• 尺寸上限:接受矩阵的算子与 stat_histogram 的 bins,超过 mathops.MAX_ELEMENTS(2^26 ≈ 6700 万个元素)即 ValueError。
• 示例数据目录(下载 URL / 许可证) —— 2-D 用 skimage.data(BSD/公有领域)加合成图,3-D 给出真实数据源(Stanford/PDS 等)的下载 URL。
• 算子来历与参考文献 —— 该算子族所依据的研究/方法出处。
• 算法的正典(作者・年份)与用途见上面的族使用指南。
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
cimage 作为输入)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 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.