Boundary Layer Transition

Boundary Layer Transition (aerodynamics/boundary-layer/boundary-layer-transition)

Use when you must predict where the laminar boundary layer turns turbulent on a two-dimensional body from its edge-velocity distribution. This leaf implements the standard integral estimate: grow the laminar layer with the Thwaites relation to get the momentum thickness theta at each station, then apply the Michel empirical criterion on the local Reynolds numbers to locate the natural transition point. The logic is pure Python, stdlib only, and covers a clean two-dimensional surface (no roughness, sweep or suction inputs and no Tollmien-Schlichting wave-growth integration; the Michel criterion replaces an eN envelope). It pairs with aerodynamics/boundary-layer/boundary-layer-theory, which owns the flat-plate thickness and skin-friction correlations on either side of the transition.

Domain quick reference

• Edge-velocity Reynolds number: Re_x = Ue(x) * x / nu with Ue the inviscid edge velocity and nu the kinematic viscosity (SI units throughout, m, m/s, m2/s).

• Thwaites integral relation for the laminar momentum thickness:

  theta(x)^2 = THWAITES_C * nu / Ue(x)^6 * integral_0^x Ue(xi)^5 d(xi)
  

  with THWAITES_C = 0.45. The integral is evaluated cumulatively with the 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, which makes the constant-velocity flat plate exact.

• Momentum-thickness Reynolds number: Re_theta = Ue * theta / nu.

• Michel transition criterion (empirical):

  Re_theta,tr = MICHEL_A * (1 + MICHEL_B / Re_x) * Re_x**MICHEL_P
  

  with MICHEL_A = 1.174, MICHEL_B = 22400.0, MICHEL_P = 0.46. Transition onset occurs at the first station where Re_theta reaches the threshold; the margin m = Re_theta - threshold crosses zero there.

• Flat-plate exact solution (verification helper): with constant Ue the Thwaites relation closes to Re_theta = sqrt(0.45) * sqrt(Re_x) = 0.6708 * sqrt(Re_x), so theta = 0.6708 * sqrt(Re_x) * nu / Ue.

• Natural-transition distance on a smooth flat plate follows from the Michel crossing, typically Re_x ~ 1.7e6 at low free-stream turbulence, an order of magnitude beyond the textbook 5e5 landmark that boundary-layer-theory quotes for the onset of instability.

• NACA TR-824 frames the boundary-layer context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Assemble the station grid: xs (m) strictly increasing and the edge velocities ues (m/s) at each station, equal length, all ue > 0.
2. Grow the laminar layer with momentum_thickness_profile(xs, ues, nu): returns the Thwaites theta at every station.
3. Build the Reynolds history with re_theta_profile(xs, ues, theta_list, nu) and the Michel threshold with michel_threshold(re_x) at each station.
4. Run the one-call sweep transition_location(xs, ues, nu): returns theta_list, re_theta_list, criterion_margin_list, x_transition (first station where the margin is non-negative, or None), the transition_index and interp_x_transition (linear interpolation of the margin to zero between the bracketing stations).
5. For a quick flat-plate check use flat_plate_transition(nu, ue, x_max), the analytic natural-transition distance on a constant-velocity plate, or michel_criterion(re_x, re_theta) for a single point test.
6. Confirm the deterministic checks with the contract test scripts/test_boundary_layer_transition.py.

Worked example

Smooth flat plate in low-turbulence flow: nu = 1.46e-5 m2/s, Ue = 30 m/s.

• Momentum thickness from momentum_thickness_profile over x = 0.25, 1.0, 2.0 m: theta = 2.3399e-4, 4.6797e-4, 6.6182e-4 m. At x = 1 m the value matches the exact flat-plate value 0.6708 * sqrt(2.0548e6) * nu / Ue = 4.680e-4 m to six digits, and Re_theta(1 m) = 961.6 equals 0.6708
  • sqrt(Re_x) (the trapezoid Thwaites integral is exact for constant Ue).
• Michel threshold at x = 1 m: michel_threshold(2.0548e6) = 951.15, so the margin there is 961.6 - 951.15 = +10.44. On the coarse grid the margins are -38.16 (x = 0.25 m), +10.44 (x = 1.0 m) and +58.60 (x = 2.0 m): transition_location reports x_transition = 1.0 m (station value) with interp_x_transition = 0.8389 m between the bracketing stations.
• Exact-solution helper: flat_plate_transition(1.46e-5, 30.0, 2.0) returns x_tr = 0.8126 m (inside the 0.6-1.1 m band), i.e. Re_x,tr = 1.670e6 with Re_theta at the crossing = 866.8 (inside 800-1100). The margin is negative just below x_tr and non-negative at it, as expected for a first-crossing scan.

Verification

• Confirm momentum_thickness_profile on the three-station flat plate is monotonic and matches 0.6708 * sqrt(Re_x) * nu / Ue within 1e-6 relative at every station.
• Confirm the flat-plate Re_theta identity re_theta_profile(...) equals 0.6708 * sqrt(Re_x) within 1e-9 relative.
• Confirm flat_plate_transition(1.46e-5, 30.0, 2.0) lies in 0.6-1.1 m and the Re_theta at the crossing lies in 800-1100.
• Confirm ValueError rejection of fewer than two stations, unequal-length xs/ues, x below zero or not strictly increasing, ue <= 0, nu <= 0, re_x <= 0 and re_theta < 0.
• Confirm determinism: repeated calls on identical inputs return identical results.
• Run the contract test offline: python3 scripts/test_boundary_layer_transition.py (32 tests, deterministic, under 20 s).

Related leaves

• aerodynamics/boundary-layer/boundary-layer-theory: the laminar and turbulent flat-plate thickness and skin-friction correlations that bookend the transition point.
• aerodynamics/airfoil/xfoil-analysis: viscous airfoil analysis that can supply the edge-velocity distribution and an independent transition estimate for comparison.
• aerodynamics/drag-polars/parasite-drag: uses the laminar versus turbulent split downstream of transition for wetted-area drag.

Pitfalls

• Feeding a grid that is not strictly increasing, starts below zero, or pairs xs and ues of unequal length: the Thwaites integral is only exact for the constant-velocity plate; the module rejects such grids with ValueError, so validate the station list before calling.
• Quoting the station value as the transition location: on the coarse grid transition_location reports x_transition = 1.0 m while the linear interpolation of the margin places the crossing at 0.8389 m; use interp_x_transition for the location estimate.
• Confusing the Michel natural-transition distance with the 5e5 instability-onset landmark: on a smooth flat plate the criterion crosses near Re_x ~ 1.7e6, an order of magnitude beyond the boundary-layer-theory textbook value for the onset of instability.
• Applying the model to forced or roughness-triggered transition: this leaf covers clean two-dimensional natural transition only; the Michel criterion replaces an eN envelope and takes no roughness, sweep or suction input.
• Treating a never-crossed margin as a zero-length run: when the criterion is never met, x_transition is None and the body stays laminar over the supplied stations - report that outcome rather than forcing a crossing.
• Ignoring the margin sign convention at the first station: the sweep returns the first station with non-negative margin, so the bracketing stations and the sign flip define the crossing, not the station where the margin is largest.

Behavior contract (gate 3)

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

python3 scripts/test_boundary_layer_transition.py

The test covers the Thwaites momentum thickness against the flat-plate exact solution, the worked-example theta at x = 1 m (4.680e-4 m), the Michel threshold and criterion logic including the equality boundary, the transition sweep bounds (x_tr in 0.6-1.1 m, Re_theta in 800-1100) with the station versus interpolated crossing distinction, the margin sign flip across the crossing, the never-crossed and first-station- crossed cases, the exact-solution flat_plate_transition helper and its bounds, ValueError rejection of every non-physical input in the validation list, and determinism.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is named for context only; the Thwaites relation and Michel criterion above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.