feat(fem): polar second moment of area of a hollow circular shaft#445
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Add beam::hollow_circular_polar_second_moment_of_area(outer_diameter, inner_diameter), the polar second moment of a tube/pipe about its axis: J = pi * (D^4 - d^4) / 32 (m^4) For a CIRCULAR tube this polar second moment IS the genuine St-Venant torsion constant (unlike a non-circular section) -- the torsional stiffness of drive-shaft tubing, far stiffer per weight than a solid shaft. Guard mirrors hollow_circular_second_moment_of_area. Analytic test hollow_circular_polar_second_moment_of_area_is_pi_d4_minus_d4_over_32: (a) worked D=0.1, d=0.05 -> J ~= 9.2038e-6 m^4; (b) threads hollow_circular_second_moment_of_area (#435) (non-tautological, perpendicular-axis theorem): J = 2*I_hollow; (c) threads circular_polar_second_moment_of_area (#423) (non-tautological, annulus): J = J(D) - J(d); (d) solid limit: a zero bore is a solid shaft; (e) 0-sentinel guards for D<=0, d>=D, non-finite, negative bore. Re-exported from lib.rs alphabetically in the beam block. valenx-fem 266 lib tests (was 265), cargo clippy --all-targets -D warnings clean.
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…an) (#464) Completes the scale-separation-ratio pair with the integral-to-Kolmogorov ratio L/η (dimensionless) — the ratio of the energy-containing integral length scale L (integral_length_scale, #445) to the dissipative Kolmogorov length η (kolmogorov_length_scale), the total span of the turbulent cascade from the largest eddies to the smallest. It is the companion to λ/η (taylor_to_kolmogorov_ratio, #455) and the widest scale-separation measure: it grows strongly with Reynolds number (∝ 1/ν^(3/2) for a fixed flow) and in isotropic turbulence satisfies L/η = Re_λ^(3/2)/15^(3/4). Guards η ≤ 0. Analytic test (4 checks) on a pure shear u(y)=γy (ε=νγ²) and a solid-body rotation (ε=0): (a) delegation thread L/η = L ÷ η; (b) non-tautological identity L/η = Re_λ^(3/2)/15^(3/4) threading taylor_reynolds_number (#443); (c) 1/ν^(3/2) scaling — fixed field, quartering ν multiplies the ratio by 8; (d) no- dissipation → 0. valenx-cfd-native 141 lib tests (was 140), clippy clean. Research-grade closed-form diagnostic, not a production turbulence model.
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…e trio) (#466) Completes the turbulence length-scale-ratio trio with the integral-to-Taylor ratio L/λ (dimensionless) — the ratio of the energy-containing integral length scale L (integral_length_scale, #445) to the intermediate Taylor microscale λ (taylor_microscale, #441). It joins λ/η (taylor_to_kolmogorov_ratio, #455) and L/η (integral_to_kolmogorov_ratio, #457). It grows linearly with Reynolds number (∝ 1/ν for a fixed flow) and in isotropic turbulence is exactly L/λ = Re_λ/15. Guards λ ≤ 0 (no dissipation). Analytic test (4 checks) on a pure shear u(y)=γy (ε=νγ²) and a solid-body rotation (ε=0): (a) delegation thread L/λ = L ÷ λ; (b) non-tautological identity L/λ = Re_λ/15 threading taylor_reynolds_number (#443); (c) 1/ν scaling — fixed field, halving ν doubles the ratio; (d) no-dissipation → 0. valenx-cfd-native 142 lib tests (was 141), clippy clean. No UI change. Research-grade closed-form diagnostic, not a production turbulence model.
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What
Adds
beam::hollow_circular_polar_second_moment_of_area(outer_diameter, inner_diameter)— the polar second moment of a tube/pipe about its axis:// J = pi * (D^4 - d^4) / 32 (m^4)Why
For a circular tube this polar second moment is the genuine St-Venant torsion constant (unlike a non-circular section) — the torsional stiffness of drive-shaft tubing, far stiffer per weight than a solid shaft. Guard mirrors
hollow_circular_second_moment_of_area. This completes the hollow circular{bending-I, polar-J}pair.Test
hollow_circular_polar_second_moment_of_area_is_pi_d4_minus_d4_over_32:D=0.1, d=0.05 -> J ~= 9.2038e-6 m^4;hollow_circular_second_moment_of_area(feat(cfd): RMS strain rate (L2 deformation rate) #435) (non-tautological, perpendicular-axis theorem) —J = 2*I_hollow;circular_polar_second_moment_of_area(feat(cad): inertia-tensor trace (rotation invariant) #423) (non-tautological, annulus) —J = J(D) - J(d);D<=0,d>=D, non-finite, negative bore.Re-exported from
lib.rsalphabetically in the beam block.valenx-fem266 lib tests (was 265);cargo clippy -p valenx-fem --all-targets -- -D warningsclean. Research-grade torsion section-property primitive.