# Mangler Axisymmetric Transform > Use when you must map the steady laminar boundary layer on a slender axisymmetric body of revolution or a sharp cone into an equivalent 2-D flow with the mangler-transformation: evaluate the Mangler transformed running length xi = integral (r0/L)^2 dx and the transformed normal coordinate from the body radius distribution, the cone-surface radius and the equivalent 2-D length for power-law bodies, and the sharp-cone values at equal running length from flat-plate baseline values passed in: skin friction and wall shear times the sqrt-3 laminar cone factor, the 99-percent, displacement and momentum thicknesses divided by sqrt-3, the thinner higher-shear cone layer at the same station. Produces the cone boundary-layer values and the coordinate mapping in SI units that anchor laminar cone-surface and body-of-revolution boundary-layer estimates. Trigger: mangler-transformation, cone-boundary-layer, axisymmetric-body-boundary-layer, laminar-cone-factor, body-of-revolution-bl. - Skill: `ashfordeou/mangler-axisymmetric-transform` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/mangler-axisymmetric-transform` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/mangler-axisymmetric-transform/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/mangler-axisymmetric-transform --- # Mangler Axisymmetric-Body Transform (aerodynamics/boundary-layer/mangler-axisymmetric-transform) Use when you must map the steady laminar incompressible boundary layer on a slender axisymmetric body of revolution or a sharp cone into an equivalent 2-D flow with the Mangler transformation (Mangler, 1948, in the form Schlichting Boundary-Layer Theory, boundary layers on bodies of revolution, and White Viscous Fluid Flow present it). This leaf implements the geometry mapping: the transformed running length xi = integral (r0/L)^2 dx, the transformed normal coordinate ybar = (r0/L)*y, the power-law body closed forms, and the sharp-cone closed-form ratios at equal running length, in pure Python stdlib, closed form, no iteration. On a sharp cone the transformed flow is the Blasius zero-pressure-gradient layer, so the mapping closes: wall shear and skin friction are sqrt(3) times the flat-plate values at the same running length, and the 99-percent, displacement and momentum thicknesses are 1/sqrt(3) times the flat-plate values, the thinner higher-shear cone layer. It consumes the flat-plate baseline values as arguments from the sibling boundary-layer-theory correlations and outputs only the cone-scaled values and the transform coordinates: it owns no skin-friction or thickness correlation, no stagnation-point layer, no compressible flow and no heat transfer, which stay with the sibling leaves listed below. Laminar incompressible steady flow only, constant nu; the cone edge velocity is constant, and for general slender bodies with varying edge velocity this leaf returns only the transformed coordinates and lengths, leaving the 2-D pressure-gradient layer solution to the layer-evolution siblings. ## Domain quick reference Geometry convention: x is the running length along the surface from the apex (cone) or nose (general body), y is the distance normal to the surface, r0(x) is the body radius at station x, alpha is the cone semi-vertex angle in degrees, L is the Mangler reference length in m. The transform is invariant to L: xi scales as L^-2 and ybar as L^-1, so every physical output is L-free. Module constants: SQRT3 = 1.7320508075688772, INV_SQRT3 = 0.5773502691896258, RHO_AIR = 1.225 kg/m3, NU_AIR = 1.5e-5 m2/s with dynamic viscosity always derived MU_AIR = RHO_AIR*NU_AIR = 1.8375e-05 Pa s, never an input. - Cone geometry: r0(x) = x*tan(alpha), m, exact tan, never the small-angle approximation. - Mangler transformed running length, cone closed form: xi = integral_0^x (r0(t)/L)^2 dt = r0(x)^2*x/(3*L^2), m, so xi scales as x^3 along a cone: the half-length station carries one eighth of the full-station xi. - Transformed normal coordinate: ybar = (r0(x)/L)*y, m, wall to wall (y = 0 maps to ybar = 0); inverse map y = ybar*L/r0(x). - General slender power-law body r0(x) = A*x^n (exponent 0 the cylinder, exponent 1 the cone): xi = A^2*x^(2n+1)/((2n+1)*L^2), closed form. - Equal-running-length cone ratios (laminar, incompressible, constant u_e): tau_w,cone = SQRT3*tau_w,flat and Cf,cone = SQRT3*Cf,flat at the same x (identical ratios, both use the same 0.5*rho*u_e^2 normalization); delta_cone = INV_SQRT3*delta_flat, delta*_cone = INV_SQRT3*delta*_flat, theta_cone = INV_SQRT3*theta_flat at the same x. - Momentum-integral closure: the pair satisfies the axisymmetric zero-pressure-gradient balance d(theta*r0)/dx = r0*Cf/2 exactly; the reversed pair (both factors sqrt(3)) fails by a factor of 3. - Shape factor: H = delta*/theta is preserved by the transform, H_cone = H_flat = 2.591566265 at the Blasius baseline. - Blasius station scaling (consumption helpers, no constants inside): local Cf ~ 1/sqrt(x), cf(x2) = cf(x1)*sqrt(x1/x2); 99-percent thickness ~ sqrt(x), delta(x2) = delta(x1)*sqrt(x2/x1). - Mangler plane-to-physical scaling for the cone: physical shear is the transformed shear times r0(x)/L and physical thickness is the transformed thickness times L/r0(x); composed with the Blasius station scaling these reduce exactly to the SQRT3 and INV_SQRT3 closed forms. ## Workflow 1. Fix the cone state and run the geometry traverse: the semi-vertex angle alpha, the running length x and the reference length L. Read the cone surface radius with cone_radius(x, half_angle_deg) and the equivalent 2-D running length with mangler_xi(x, half_angle_deg, ref_length), where xi = cone_radius^2*x/(3*L^2) carries the x^3 content of the transformation. 2. Map the layer coordinates: the transformed normal coordinate ybar with transformed_normal_coordinate(x, y, half_angle_deg, ref_length), and the equivalent 2-D length of a general slender power-law body with powerlaw_mangler_xi(x, amplitude, exponent, ref_length) (exponent 0 the cylinder, exponent 1 the cone). 3. Consume the flat-plate baseline: take the sibling boundary-layer-theory values at the same running length x and edge velocity (local skin-friction coefficient, wall shear, 99-percent, displacement and momentum thicknesses) as inputs. This leaf derives none of them; they are passed in as arguments. 4. Scale the cone skin friction and wall shear: cone_skin_friction (cf_flat) and cone_wall_shear(tau_w_flat) multiply the flat-plate values by the sqrt-3 laminar cone factor at the same running length. 5. Scale the cone thicknesses: cone_boundary_layer_thickness(delta_flat), cone_displacement_thickness(delta_star_flat) and cone_momentum_thickness(theta_flat) divide the flat-plate values by sqrt-3 (the inverse sqrt-3 factor) at the same running length. 6. Check the shape factor: the cone displacement-to-momentum ratio equals the flat-plate shape factor 2.591566265 on the Blasius baseline; the transformation scales the layer, it does not reshape it. 7. Evaluate the equivalent 2-D layer at the Mangler length xi with the Blasius station-scaling helpers blasius_cf_at_station(cf_at_x1, x1, x2) and blasius_delta_at_station(delta_at_x1, x1, x2), then apply the plane-to-physical scalings (r0/L on shear, L/r0 on thickness); the composed pipeline closes on the cone closed forms within float noise. 8. Confirm the deterministic checks: the momentum-integral closure of the factor pair and the input-rejection traverse of non-physical inputs, with the contract test scripts/test_mangler_axisymmetric_transform.py. ## Worked example Slender cone at half_angle 5.0 deg in standard air nu = 1.5e-5 m2/s, rho = 1.225 kg/m3, constant edge velocity u_e = 30.0 m/s, running length x = 2.0 m, reference length L = 1.0 m. All values below are real outputs of the module. - Reynolds number at the station: Re_x = u_e*x/nu = 4.0000000e6, sqrt(Re_x) = 2000.0 exactly. - Geometry: cone surface radius cone_radius(2.0, 5.0) = 1.749773271e-01 m; Mangler equivalent 2-D running length mangler_xi(2.0, 5.0, 1.0) = 2.041137665e-02 m, the 2 m cone surface maps to a 2 cm flat plate; transformed normal coordinate at the flat-plate layer top transformed_normal_coordinate(2.0, 5.0e-3, 5.0, 1.0) = 8.748866353e-04 m. - Power-law bodies at x = 2.0 m: r0 = 0.05*x^0.5 gives powerlaw_mangler_xi = 5.00000000e-03 m, the cylinder r0 = 0.1 m gives 2.00000000e-02 m, and the cone (amplitude tan(5 deg), exponent 1) reproduces xi = 2.041137665e-02 m. - Flat-plate baseline at x = 2.0 m (inputs consumed from the sibling boundary-layer-theory Blasius correlations): cf_flat = 0.664/sqrt(Re_x) = 3.32000000e-04, tau_w,flat = 0.332*rho*u_e^2/ sqrt(Re_x) = 1.83015000e-01 Pa, delta_flat = 5.0*x/sqrt(Re_x) = 5.00000000e-03 m, delta*_flat = 1.72080000e-03 m, theta_flat = 6.64000000e-04 m. - Cone values at the SAME running length x = 2.0 m: cf_cone = cone_skin_friction(3.32e-4) = 5.750408681e-04, ratio to the plate value 1.732050808 = sqrt(3); tau_w,cone = cone_wall_shear(0.183015) = 3.169912785e-01 Pa, ratio sqrt(3), equal to the Cf ratio; delta_cone = cone_boundary_layer_thickness(5.0e-3) = 2.886751346e-03 m, ratio 0.5773502692 = 1/sqrt(3); delta*_cone = cone_displacement_thickness(1.7208e-3) = 9.935043432e-04 m and theta_cone = cone_momentum_thickness(6.64e-4) = 3.833605787e-04 m, both at ratio 1/sqrt(3). - Shape factor on the cone: H_cone = delta*_cone/theta_cone = 2.591566265, identical to H_flat (the transform scales the layer, it does not reshape it). Momentum-integral closure at the worked station: the analytic d(theta_cone*r0)/dx equals r0*Cf,cone/2 to float noise (ratio 1), the consistency check that the sqrt(3) and 1/sqrt(3) factor pair obeys the axisymmetric boundary-layer momentum balance. Read-off: a 5 deg half-angle cone at 2 m running length in a 30 m/s stream (Re_x = 4e6) carries a laminar skin friction of 5.75e-4, exactly sqrt(3) times the 3.32e-4 of the flat plate at the same station, with its boundary layer squeezed from 5.0 mm to 2.89 mm; the equivalent 2-D flow lives on a 2 cm plate, and it is on that short equivalent plate that the flat-plate correlations of boundary-layer-theory are evaluated before the mapping scales the values back to the cone surface. The sqrt(3) family is the laminar cone signature: thinner layer, higher wall shear. This momentum- only leaf computes no heat transfer; the wall-shear rise is why a laminar sharp-cone surface runs hotter than the flat plate at the same Reynolds number in the heat-transfer analogy, which the high-speed heating leaf owns. ## Verification - Confirm cone_radius(2.0, 5.0) returns 1.749773271e-01 m, equal to 2.0*math.tan(math.radians(5.0)), and mangler_xi(2.0, 5.0, 1.0) returns 2.041137665e-02 m with the x^3 scaling (the half-length station carries one eighth of the xi) and the L^-2 scaling. - Confirm the power-law family: powerlaw_mangler_xi(2.0, math.tan(math.radians(5.0)), 1.0, 1.0) reproduces mangler_xi, the cylinder returns 2.0e-02 m and the r0 = 0.05*x^0.5 body returns 5.0e-03 m. - Confirm the cone closed forms at the worked station: cone_skin_friction (3.32e-4) = 5.750408681e-04 (between 4.0e-4 and 8.0e-4) with ratio SQRT3, cone_wall_shear(0.183015) = 3.169912785e-01 Pa with ratio SQRT3 equal to the Cf ratio, cone_boundary_layer_thickness(5.0e-3) = 2.886751346e-03 m (between 2.0e-3 and 3.5e-3 m, below the 5.0e-3 m plate layer), cone_displacement_thickness(1.7208e-3) = 9.935043432e-04 m and cone_momentum_thickness(6.64e-4) = 3.833605787e-04 m, each with ratio INV_SQRT3. - Confirm the shape factor H_cone = 2.591566265 equals 1.7208/0.664, the Mangler shear and thickness pipelines (the composed plane-to-physical and Blasius station-scaled values) close on the closed forms within 1e-9 relative, and the momentum-integral closure ratio is 1. - Confirm the consumption helpers: blasius_cf_at_station(3.32e-4, 2.0, 4.0) = 2.3475945135e-04 (cf at twice the station is cf/sqrt(2)) and the round trips over (x1, x2) and back hold within float noise. - Confirm every non-physical input raises ValueError: x at 0 or negative on the geometry functions, half_angle_deg at 0 or 90, y negative on the transformed normal coordinate, ref_length at 0 or negative, amplitude 0 or exponent negative on the power-law body, zero baselines or stations on the helpers, and non-positive flat-plate baselines on the cone functions. - Run the contract test offline: python3 scripts/test_mangler_axisymmetric_transform.py (34 tests, deterministic, passes under /usr/bin/python3 and the pyenv 3.13.12 interpreter). ## Pitfalls - Reversing the cone factor pair: wall shear and skin friction carry sqrt(3) but the thicknesses carry 1/sqrt(3). Putting sqrt(3) on the thickness as well breaks the axisymmetric momentum integral d(theta*r0)/dx = r0*Cf/2 by a factor of 3; the cone layer is thinner and more strongly sheared than the plate layer at the same station. - Comparing at equal running length versus equal transformed length: the sqrt(3) family is the equal-x comparison. At equal xi the equivalent 2-D layer and the physical layer are identical by construction (ratios of 1); the factor appears only when two physical stations at the same x are compared. - Applying the factor family to turbulent cone layers: the sqrt(3) and 1/sqrt(3) factors are laminar-only closed forms; the turbulent cone rule has no exact factor and this leaf does not claim one. - Feeding a baseline from the wrong station or edge velocity: the cone functions multiply whatever flat-plate values are passed in, so the baseline must come from the same running length x and the same edge velocity u_e; mixing stations silently corrupts the cone values. - Using the small-angle approximation: the module uses tan(alpha) exactly; replacing it with alpha in radians shifts the radius and hence every transformed length. - Taking the mapping for compressible or heat-transfer work: this leaf is momentum-only laminar incompressible content. The Eckert reference-temperature method, recovery factor and Reynolds-analogy heating belong to flat-plate-skin-friction-heating, and stagnation- point attachment layers belong to stagnation-flow-boundary-layer. ## Related leaves - aerodynamics/boundary-layer/boundary-layer-theory: owns the flat-plate Blasius and 1/7-power correlations whose values this leaf consumes as baseline inputs at the same running length. - aerodynamics/boundary-layer/stagnation-flow-boundary-layer: the Homann axisymmetric attachment-region layer of the nose, a constant-thickness similarity layer with no running length, no cone surface and no transform content. - aerodynamics/boundary-layer/boundary-layer-separation and aerodynamics/boundary-layer/boundary-layer-transition: the Thwaites traverse layer evolution of 2-D flows with varying edge velocity, which this leaf leaves to them for general slender bodies. - aerodynamics/high-speed/flat-plate-skin-friction-heating: the compressible plate-station skin friction and Reynolds-analogy heat transfer of a 2-D high-Mach stream, the regime fence of this incompressible momentum-only geometry mapping. - aerodynamics/high-speed/hypersonic-flow: the Newtonian cone axial force at Mach well above 5, the pressure side of cone flow, not the viscous side. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_mangler_axisymmetric_transform.py The 34 tests cover the worked-example anchors with magnitude bounds (cone radius, Mangler xi with its x^3 and L^-2 scalings, transformed normal coordinate, power-law bodies), the sqrt(3) cone factor on skin friction and wall shear with ratio checks, the inverse sqrt(3) factor on the three cone thicknesses with the thinner-than-plate bound, the shape-factor preservation, the Mangler shear and thickness pipelines closing on the closed forms, the Blasius station-scaling helpers with round trips, the momentum-integral closure (including the failure of the reversed factor pair), deterministic repeat calls, module stdlib hygiene, and the input-rejection traverse of non-physical running lengths, angles, normal distances, reference lengths, power-law amplitudes and exponents, station helpers and flat-plate baselines. ## Compliance - Standards referenced, not reproduced: NACA TR-824 is the cited reference for the viscous boundary-layer family (the sibling precedent for this pack in standards-map.yaml); the classical transform treatment follows Mangler 1948 as presented by Schlichting Boundary-Layer Theory (boundary layers on bodies of revolution) and White Viscous Fluid Flow (Mangler transformation section), whose material is cited through the report. Summary-only methodology per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.