Squire Young Profile Drag

Squire-Young Profile Drag (aerodynamics/boundary-layer/squire-young-profile-drag)

Use when you must compute the section profile-drag coefficient of a two-dimensional body or airfoil from the boundary-layer momentum state at its trailing edge with the Squire-Young formula (Squire and Young, ARC R&M 1838, 1938; Schlichting, Boundary-Layer Theory, 7th ed., pp. 158-162; validated against surface-integrated CFD within 2-3 percent in the low-drag range by Coder and Maughmer, Journal of Aircraft 52(3), 2015): c_d,p = 2*(theta_TE/c)*(U_TE/U_inf)**((H_TE+5)/2). This leaf evaluates the trailing-edge factor, maps a known TE momentum state to the profile-drag coefficient, grows the laminar momentum thickness to the trailing edge on the integral growth relation for the fully laminar chain, and reproduces the Blasius flat-plate drag exactly at zero pressure gradient. It is the TE-to-drag member of the boundary-layer pack, complementing the traverses of boundary-layer-separation and boundary-layer-transition, which stop at their separation and transition signals and never map the TE momentum state to drag. It does not do Thwaites lambda traverses, separation or transition criteria, flat-plate skin-friction or thickness correlations, whole-aircraft drag buildup, XFOIL-style polar runs, rotor-blade profile power, or wake-survey instrumentation: incompressible clean 2-D attached flow only, one surface of the section at a time.

Domain quick reference

Module constants (fixed numbers): NU_AIR = 1.46e-5 m2/s (air kinematic viscosity at standard conditions), H_TE = 1.4 (default trailing-edge shape factor H = delta*/theta, documented at about 1.4), LAMINAR_THETA_C = 0.664 and BLASIUS_DRAG_C = 1.328 (= 2*0.664, the laminar flat-plate drag constant), GROWTH_C = 0.664**2 = 0.440896 (the laminar integral-growth constant, pinned so the zero-pressure-gradient plate closes to the Blasius momentum thickness exactly; the sibling Thwaites constant 0.45 would run the flat-plate theta 1.03 percent high because the siblings' theta feeds the Michel criterion while this leaf's theta feeds a drag mapping anchored at Blasius exactness).

• Squire-Young profile drag: c_d,p = 2*(theta_TE/c)* (U_TE/U_inf)**((H_TE+5)/2), dimensionless, ONE surface of the section.
• Edge-velocity factor: f = (U_TE/U_inf)**((H_TE+5)/2). Unity at U_TE = U_inf for any shape factor (the flat plate), below unity for U_TE < U_inf, the attached-flow band of a closed section. The exponent reads (H_TE+5)/2 = 3.2 at the default H_TE = 1.4.
• Zero-pressure-gradient reduction: at U_TE = U_inf the formula is c_d,p = 2theta_TE/c, and with the Blasius TE momentum thickness theta_TE = 0.664c/sqrt(Re_c) it reproduces the Blasius flat-plate drag 2*theta_TE/c = 1.328/sqrt(Re_c) exactly, independent of H_TE because the velocity ratio is unity. This reduction is the leaf's deterministic anchor.
• Laminar integral growth of theta to the trailing edge: theta_TE^2 = GROWTH_C * nu / U_TE**6 * integral_0^c Ue(x)**5 dx, cumulative trapezoid rule over the supplied stations; the segment from the leading edge (x = 0) to the first station keeps Ue at its first-station value, so a constant-velocity plate is exact.
• Linear edge-velocity law Ue(x) = U_inf*(1 - a*x/c): closed form theta_TE = sqrt(GROWTH_C * nu * U_inf**5 * c * (1 - (1-a)6) / (6*a) / U_TE6) with U_TE = U_inf*(1 - a).
• Both surfaces: for a symmetric section at zero lift whose surfaces share the TE state, the total profile drag is twice the one-surface value.

SI units throughout: x in m, U in m/s, nu in m2/s, theta in m, c_d,p dimensionless. The Blasius constants 0.664 and 1.328 appear only as the closed-form zero-pressure-gradient identity targets of the TE mapping, never to compute surface skin-friction distributions or regime classes.

Workflow

1. Fix the section state: chord c, freestream velocity U_inf and kinematic viscosity nu (NU_AIR default at standard conditions); the chord Reynolds number Re_c = U_inf*c/nu places the estimate (2.0548e6 in the worked example).
2. Take the trailing-edge momentum state: the trailing-edge-momentum- thickness theta_TE and the trailing-edge edge velocity U_TE, with the documented trailing-edge shape factor H_TE = 1.4 default (the realistic band runs from about 1.2 to about 2.6).
3. Evaluate the edge-velocity factor with trailing_edge_factor(U_TE, U_inf, H_TE): (U_TE/U_inf)**((H_TE+5)/2), unity on the flat plate, below unity when the TE edge flow still has pressure to recover.
4. Map the momentum state to the section profile-drag coefficient with squire_young_profile_drag(theta_TE, c, U_TE, U_inf, H_TE) = 2*(theta_TE/c)*factor, one surface of the section.
5. Check the zero-pressure-gradient reduction: at U_TE = U_inf the formula collapses to the momentum-integral value 2theta_TE/c and reproduces the laminar flat-plate drag 1.328/sqrt(Re_c) when theta_TE sits at the Blasius value 0.664c/sqrt(Re_c); this closes the deterministic anchor identity of the leaf.
6. For the fully laminar chain, grow the momentum thickness to the trailing edge with momentum_thickness_at_te(xs, ues, nu) over the edge-velocity traverse: theta_TE^2 = GROWTH_C*nu/U_TE6 * integral_0^c Ue5 dx on the cumulative trapezoid rule, U_TE read as the last-station edge velocity.
7. Run the one-call fully laminar chain with fully_laminar_profile_drag (xs, ues, nu, c, U_inf, H_TE): growth then squire-young-formula mapping in one step.
8. Combine surfaces: for a symmetric section at zero lift whose surfaces share the TE state, the total is twice the one-surface value.
9. Run the deterministic checks: rerun the identities above and the input-rejection traverse with the contract test scripts/test_squire_young_profile_drag.py.

Worked example

Air at standard conditions nu = 1.46e-5 m2/s, chord c = 1.0 m, freestream U_inf = 30.0 m/s, so Re_c = 2.0547945205e6. The traverse grids use 4001 stations (step 2.5e-4 m). All values are real outputs of the module.

Case A, direct formula from a TE momentum state: one-surface theta_TE = 3.0e-3 m (theta_TE/c = 0.003), U_TE = 27.0 m/s (edge-velocity ratio 0.9), H_TE = 1.4:

• trailing_edge_factor(27.0, 30.0, 1.4) = 0.71379915616, the (U_TE/U_inf)**3.2 wake-transfer factor, below unity because the TE edge flow still has pressure to recover.
• Raw momentum value 2*theta_TE/c = 6.0e-3; the Squire-Young estimate squire_young_profile_drag(3.0e-3, 1.0, 27.0, 30.0, 1.4) = 4.2827949370e-3 one surface, and 8.5655898739e-3 for both surfaces of a symmetric section at zero lift sharing the TE state.
• H sensitivity: trailing_edge_factor(27.0, 30.0, 2.6) = 0.67007210063, which lowers the one-surface estimate to 4.0204326038e-3, 6.1 percent below the H_TE = 1.4 value at this velocity ratio.

Case B, the anchor identity on the fully laminar flat plate (U_TE = U_inf = 30.0 m/s, zero pressure gradient):

• Blasius TE momentum thickness theta_TE = 0.664*c/sqrt(Re_c) = 4.6321634974e-4 m.
• Reference drag 1.328/sqrt(Re_c) = 9.2643269948e-4.
• squire_young_profile_drag(4.6321634974e-4, 1.0, 30.0, 30.0, 1.4) = 9.2643269948e-4, identity residual 0.0: the formula reproduces the Blasius flat-plate drag exactly when U_TE = U_inf.
• Chain end to end: momentum_thickness_at_te over the 4001-station constant-30 m/s plate = 4.6321634974e-4 m (rel 3.066e-14 against the Blasius value) and fully_laminar_profile_drag(...) = 9.2643269948e-4 (rel 3.066e-14 against 1.328/sqrt(Re_c)): the fully laminar chain reproduces the Blasius drag on the plate.
• Reynolds invariance: c_d,p*sqrt(Re_c) = 1.3280000000 at Re_c and at Re_c/2 (U = 15 m/s), rel residual 3.060e-14.

Case C, the fully laminar 2-D section at low Reynolds number with a decelerated edge flow Ue(x) = 30.0*(1 - 0.1*x/c) m/s over x in [0, 1] m (U_TE/U_inf = 0.9 at the TE), H_TE = 1.4:

• momentum_thickness_at_te over the 4001-station traverse = 5.6151694776e-4 m; the closed form of the linear law, sqrt(GROWTH_CnuU_inf5c(1-0.96)/(6*0.1)/27.0**6) = 5.6151694744e-4 m, quadrature residual 3.220e-13 m (rel 5.734e-10): the integral growth closes to the exact integral.
• Raw momentum value 2theta_TE/c = 1.1230338955e-3, above the flat plate's 9.2643269948e-4 because the deceleration thickens the layer; the Squire-Young mapping fully_laminar_profile_drag(xs, ues, 1.46e-5, 1.0, 30.0, 1.4) = 8.0162064696e-4, identical to the hand recomputation 2(theta_TE/c)*(0.9)**3.2 = 8.0162064696e-4 (residual 0.0).
• The exponent factor carries the whole difference between the raw momentum value and the drag estimate: 0.71379915616 at U_TE/U_inf = 0.9 and H_TE = 1.4, versus unity on the flat plate.

Read-off: a section whose TE layer carries theta_TE/c = 0.003 at an edge-velocity ratio 0.9 and H_TE = 1.4 has a Squire-Young profile-drag coefficient of 4.283e-3 per surface (8.566e-3 total at zero lift), 29 percent below the raw 2theta_TE/c momentum value 6.0e-3; the fully laminar flat plate at Re_c = 2.0548e6 lands exactly on the Blasius value 9.264e-4 through both the direct formula and the integral-growth chain; and a fully laminar section whose edge flow decelerates 10 percent over the chord grows theta_TE to 5.615e-4 m (raw 2theta_TE/c = 1.123e-3) before the edge-velocity-ratio factor 0.7138 brings the estimate to 8.016e-4.

Verification

• Confirm trailing_edge_factor(27.0, 30.0, 1.4) returns 0.71379915616 and trailing_edge_factor(27.0, 30.0, 2.6) returns 0.67007210063, both inside the (U_TE/U_inf)3.2 band, with the factor unity to float noise at U_TE = U_inf for any shape factor and the ratio f(1.4)/f(2.6) equal to (0.9)(-0.6).
• Confirm squire_young_profile_drag(3.0e-3, 1.0, 27.0, 30.0, 1.4) returns 4.2827949370e-3 (magnitude band 4.0e-3 to 4.5e-3) and that it equals 2.0*(3.0e-3/1.0)*trailing_edge_factor(27.0, 30.0, 1.4); the both-surfaces total at symmetric zero lift is 8.5655898739e-3.
• Confirm the anchor identity: with theta_bl = 0.664/sqrt(Re_c) = 4.6321634974e-4 m, squire_young_profile_drag(theta_bl, 1.0, 30.0, 30.0, h_te) equals 1.328/sqrt(Re_c) = 9.2643269948e-4 within 1e-9 relative for h_te in (1.4, 2.6), the reduction being independent of the shape factor at U_TE = U_inf.
• Confirm the chain equals the identity on the flat plate: a 4001-station constant-30 m/s traverse gives momentum_thickness_at_te = 4.6321634974e-4 m and fully_laminar_profile_drag = 9.2643269948e-4 within 1e-6 relative of the Blasius values.
• Confirm Reynolds invariance: on the plate, fully_laminar_profile_drag*sqrt(Re_c) = 1.328 within 1e-9 at U = 30.0 and at U = 15.0 m/s (Re_c/2).
• Confirm the Case C chain: the 4001-station linear-decay traverse ue(x) = 30.0*(1 - 0.1x) gives momentum_thickness_at_te = 5.6151694776e-4 m within 1e-6 relative of the closed-form integral and fully_laminar_profile_drag = 8.0162064696e-4 within 1e-6 relative, equal to 2.0(theta_te/1.0)*(0.9)**3.2.
• Confirm the LE-gap convention: a constant-velocity traverse starting mid-plate (first station above x = 0) still closes to the Blasius theta, because the segment from the leading edge keeps the first-station edge velocity.
• Confirm the input-rejection traverse: non-positive u_te, u_inf, theta_te, chord, nu, a shape factor at or below 1.0, fewer than two stations, unequal or non-increasing station sets, a negative leading station and a non-positive edge velocity all raise ValueError.
• Run the contract test offline: python3 scripts/test_squire_young_profile_drag.py (28 tests, deterministic, passes under both /usr/bin/python3 3.9.6 and the pyenv 3.13.12 interpreter).

Related leaves

• aerodynamics/boundary-layer/boundary-layer-separation: the Thwaites lambda traverse with the -0.09 laminar separation flag and the Stratford turbulent recovery criterion; its traverse stops at the separation signal and never maps the TE momentum state to drag.
• aerodynamics/boundary-layer/boundary-layer-transition: the Michel criterion traverse for the laminar-turbulent transition location; its traverse stops at the natural-transition onset, no TE-to-drag mapping.
• aerodynamics/boundary-layer/boundary-layer-theory: steady flat-plate layer definitions and correlations (Blasius and 1/7-power thicknesses, skin-friction coefficients, regime assignment), flat plate only, no section profile-drag coefficient from a TE momentum state.
• aerodynamics/boundary-layer/stokes-creeping-flow-drag: the exact low-Reynolds-number sphere drag member of the viscous-layer family, the creeping-flow complement to the attached TE-layer mapping here.
• aerodynamics/drag-polars/parasite-drag: the whole-aircraft zero-lift drag buildup over components with form and interference factors, the far-wake level this leaf's TE mapping feeds at section level.
• aerodynamics/airfoil/xfoil-analysis: numerical viscous polar runs, the tool-adjacent route that produces section polars without the closed-form Squire-Young relation.

Pitfalls

• Reading the raw momentum value as the drag: 2*theta_TE/c is the momentum-integral value only when the TE edge velocity equals the freestream; on a decelerated section the edge-velocity-ratio exponent (U_TE/U_inf)**3.2 at H_TE = 1.4 lowers the estimate (0.7138 at ratio 0.9, about 29 percent below the raw value in the worked example).
• Using the Thwaites constant 0.45 in the laminar integral growth: the siblings' theta feeds the Michel criterion, while this leaf's growth constant GROWTH_C = 0.664**2 = 0.440896 is pinned so the flat plate closes to the Blasius theta exactly; 0.45 runs the plate theta about 1.03 percent high and breaks the 1.328/sqrt(Re_c) identity.
• Feeding a shape factor at or below 1.0: H = delta*/theta below unity is not an attached boundary layer (the realistic band runs from about 1.2 to about 2.6), and the module raises ValueError.
• Forgetting the one-surface convention: c_d,p is per surface; a symmetric section at zero lift whose surfaces share the TE state carries twice the value.
• Extending the method beyond its fences: no turbulent or mixed-laminar-turbulent growth model, no compressibility, roughness, sweep, suction or wake-survey instrumentation inputs, and no whole-aircraft buildup; the clean incompressible attached 2-D section state is the whole claim.
• Mixing the momentum-thickness conventions: the trailing-edge- momentum-thickness here feeds a drag mapping, not a flat-plate skin-friction or displacement-thickness correlation, which belongs to boundary-layer-theory.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_squire_young_profile_drag.py

The 28 tests cover the trailing-edge factor anchors and the unity flat-plate reduction, the factor-ratio exponent algebra, the profile-drag coefficient of Case A with its magnitude band and the both-surfaces total, the Blasius anchor identity at both shape factors, the H sensitivity, the flat-plate momentum-thickness and fully laminar chain identities (rel 3e-14 against the Blasius values), the Reynolds invariance at Re_c and Re_c/2, the Case C linear-decay anchors against the closed-form integral, the leading-edge-gap and partial-plate scaling checks, the deterministic repeat calls, the module-constant and no-RNG structure checks, and the full input-rejection traverse of non-physical velocities, shape factors, chords, thicknesses, stations and viscosities. All numeric asserts are tolerance-based (assertAlmostEqual/isclose), so the suite passes identically under /usr/bin/python3 3.9.6 and ~/.pyenv/versions/3.13.12/bin/python3.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is the cited reference for the boundary-layer family context (the report frames the airfoil boundary-layer data context for the relations above). The Squire-Young relation follows Squire and Young, "The Calculation of the Profile Drag of Aerofoils", ARC R&M 1838, 1938, and Schlichting Boundary-Layer Theory, 7th ed., McGraw-Hill, pp. 158-162, the treatment cited for the method by Coder and Maughmer, "Numerical Validation of the Squire-Young Formula for Profile-Drag Prediction", Journal of Aircraft 52(3), 2015, pp. 948-955. The equations are standard engineering methodology, summary-only per standards-map.yaml, and no standards text is reproduced.
• compliance: STANDARDS-REF, gated: false.