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)^2x/(3L^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 = ybarL/r0(x).
• General slender power-law body r0(x) = Ax^n (exponent 0 the cylinder, exponent 1 the cone): xi = A^2x^(2n+1)/((2n+1)*L^2), closed form.
• Equal-running-length cone ratios (laminar, incompressible, constant u_e): tau_w,cone = SQRT3tau_w,flat and Cf,cone = SQRT3Cf,flat at the same x (identical ratios, both use the same 0.5rhou_e^2 normalization); delta_cone = INV_SQRT3delta_flat, delta_cone = INV_SQRT3delta_flat, theta_cone = INV_SQRT3*theta_flat at the same x.
• Momentum-integral closure: the pair satisfies the axisymmetric zero-pressure-gradient balance d(thetar0)/dx = r0Cf/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^2x/(3L^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 = 0.332rhou_e^2/ sqrt(Re_x) = 1.83015000e-01 Pa, delta_flat = 5.0x/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_coner0)/dx equals r0Cf,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(thetar0)/dx = r0Cf/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.