feat(cfd): vorticity standard deviation (variance identity)#419
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Add FlowSolution::vorticity_std_dev(), the standard deviation of the interior vorticity: sigma_omega = sqrt( <(omega - <omega>)^2> ) (1/s) the spatial fluctuation of the central-difference vorticity about its area-average, completing the mean / rms / std vorticity-statistics triad (mean_vorticity #400, rms_vorticity #406). By the variance identity it satisfies sigma^2 = omega_rms^2 - <omega>^2. Mirrors pressure_std_dev's deviation-sum form (mean via mean_vorticity, then sum (omega-mean)^2); returns 0 for a grid too small (nx < 3 or ny < 3). Analytic test vorticity_std_dev_completes_the_variance_identity: (a) uniform shear u(y)=gamma*y has constant omega = -gamma -> sigma = 0; (b) quadratic shear u(y)=a*y^2 has omega = -2a*y varying in space -> sigma > 0, and the variance identity sigma = sqrt(omega_rms^2 - <omega>^2) threads mean_vorticity + rms_vorticity (three separate loops, non-tautological); (c) grid too small for an interior difference -> 0. Pure method on FlowSolution, no lib.rs change. cfd-native 124 lib tests (was 123), cargo clippy --all-targets -D warnings clean.
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Add beam::circular_polar_second_moment_of_area(diameter), the polar second moment of a solid round shaft about its longitudinal axis: J = pi * d^4 / 32 (m^4) the torsion constant of a round bar -- the J that feeds polar_section_modulus (Z_p = J/(d/2)) and torsional_rigidity (GJ), neither of which previously had a way to compute J for a circle. By the perpendicular-axis theorem it is exactly twice the bending circular_second_moment_of_area (J = Ix + Iy = 2*I, since Ix = Iy for a circle). Guard mirrors the bending companion. Analytic test circular_polar_second_moment_of_area_is_pi_d4_over_32: (a) worked d=0.1 -> J = pi*0.1^4/32 ~= 9.8175e-6 m^4; (b) threads circular_second_moment_of_area (#419) (non-tautological, perpendicular-axis theorem): J = 2*I; (c) threads polar_section_modulus: Z_p = J/(d/2) = pi*d^3/16; (d) threads torsional_rigidity (#410): GJ of a steel shaft; (e) quartic in diameter; (f) 0-sentinel guards for non-positive d and non-finite input. Re-exported from lib.rs alphabetically in the beam block. valenx-fem 261 lib tests (was 260), cargo clippy --all-targets -D warnings clean.
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Add beam::circular_plastic_section_modulus(diameter), the plastic section modulus of a solid round section: Z = d^3 / 6 (m^3) the section property for fully-plastic limit bending of a shaft (M_plastic = sigma_y * Z) -- the circular companion to rectangular_plastic_section_modulus (#426). For a solid circle the shape factor is Z/S = 16/(3*pi) ~= 1.698 (Z = d^3/6 vs the elastic S = pi*d^3/32). Guard mirrors circular_second_moment_of_area. Analytic test circular_plastic_section_modulus_is_d3_over_6: (a) worked d=0.1 -> Z = 1.6667e-4 m^3; (b) threads elastic_section_modulus + circular_second_moment_of_area (#419) (non-tautological): Z = (16/3pi)*S and the shape factor Z/S = 16/(3pi); (c) cubic in diameter; (d) 0-sentinel guards for non-positive d and non-finite input. Re-exported from lib.rs alphabetically in the beam block. valenx-fem 263 lib tests (was 262), cargo clippy --all-targets -D warnings clean.
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…442) Add beam::hollow_circular_second_moment_of_area(outer_diameter, inner_diameter), the bending second moment of a tube/pipe cross-section: I = pi * (D^4 - d^4) / 64 (m^4) the annulus = outer disc minus inner disc -- the I of structural tubing, far stiffer per unit weight than a solid bar since the removed core carries little bending stress. Guard: D > 0, 0 <= d < D, finite. Analytic test hollow_circular_second_moment_of_area_is_pi_d4_minus_d4_over_64: (a) worked D=0.1, d=0.05 -> I ~= 4.6019e-6 m^4; (b) threads circular_second_moment_of_area (#419) (non-tautological): the annulus is I(D) - I(d); (c) solid limit: a zero bore is a solid circle; (d) monotonicity: a larger bore removes more material so I drops; (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 265 lib tests (was 264), cargo clippy --all-targets -D warnings clean.
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What
Adds
FlowSolution::vorticity_std_dev()— the standard deviation of the interior vorticity:// sigma_omega = sqrt( <(omega - <omega>)^2> ) (1/s)Why
The vorticity statistics had
mean_vorticity(#400) andrms_vorticity(#406) but no standard deviation.sigma_omegacompletes the mean / rms / std triad — mirroring the pressure family'smean/std_dev(#388) /rms(#394) — and satisfies the variance identitysigma^2 = omega_rms^2 - <omega>^2. It is0exactly for a spatially uniform vorticity and grows as the swirl becomes more non-uniform. Mirrorspressure_std_dev's deviation-sum form (mean viamean_vorticity, then sum(omega - mean)^2); returns0for a grid too small (nx < 3orny < 3).Test
vorticity_std_dev_completes_the_variance_identity:u(y)=gamma*yhas constantomega = -gamma->sigma = 0;u(y)=a*y^2hasomega = -2a*yvarying in space ->sigma > 0, and the variance identitysigma = sqrt(omega_rms^2 - <omega>^2)threadsmean_vorticity+rms_vorticity(three separate loops, non-tautological);0.Pure method on
FlowSolution, nolib.rschange.valenx-cfd-native124 lib tests (was 123);cargo clippy -p valenx-cfd-native --all-targets -- -D warningsclean. Research-grade post-processing reducer.