Boundary Layer Separation

Boundary Layer Separation (aerodynamics/boundary-layer/boundary-layer-separation)

Use when you must predict the boundary layer separation location on a two-dimensional body from its edge-velocity or pressure distribution. This leaf grows the laminar layer with the Thwaites integral relation along the edge-velocity traverse and flags the first station where the Thwaites lambda parameter crosses -0.09, the classical laminar separation criterion, and evaluates the Stratford-style pressure recovery criterion to estimate the turbulent separation station and the margin below the 0.35 threshold. Pure Python, stdlib only. It pairs with boundary-layer/boundary-layer-transition, whose Thwaites traverse ends at the natural-transition onset estimated by the Michel rule rather than running to separation, and with boundary-layer/boundary-layer-theory for flat-plate thickness and skin-friction context on the attached side of the flow.

Domain quick reference

• Thwaites integral for the momentum thickness on the laminar side: theta^2 = 0.45 * nu / U^6 * integral_0^x U^5 dx, integrated by the trapezoid rule over the edge-velocity stations (THWAITES_C = 0.45). The traverse starts with theta = 0 at the first station with U > 0; a stagnation head with U = 0 is skipped, which removes the U^6 singularity at the stagnation point.
• Thwaites pressure-gradient parameter: lambda = theta^2 / nu * dU/dx, with dU/dx by central differences (forward at the first station, backward at the last). Accelerating flow keeps lambda positive; a decelerating edge flow drives lambda negative.
• Laminar separation criterion: the first station with lambda <= THWAITES_LAMBDA_SEP = -0.09 is the laminar separation point. None if lambda stays above the criterion for the whole run.
• Stratford-style separation parameter on the pressure recovery: S = C_p * sqrt((x / C_p) * dC_p/dx), evaluated only where C_p > 0 and dC_p/dx > 0 (the recovery side), with the pressure slope taken over the station interval ending at each station (one-sided difference, right-end attribution; this reproduces the verified worked-example anchor S = 0.3505 at station 8).
• Turbulent separation criterion: the first recovery station with S >= STRATFORD_SEP = 0.35 is the turbulent separation station. None if the recovery never reaches the threshold.
• Separation margin: 0.35 - max S on the recovery side before the crossing (headroom up to the separation point, positive while the flow keeps headroom). With no crossing the max is taken over the whole recovery; with no recovery stations at all the margin is the full 0.35.
• Units are SI throughout: x in m, U in m/s, nu in m2/s, C_p dimensionless, theta in m.
• NACA TR-824 frames the laminar boundary-layer correlation context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Build the edge-velocity traverse xs, us (m, m/s) along the body and fix the kinematic viscosity nu. A linearly decelerated run or a digitized airfoil velocity distribution both work as plain station lists.
2. Grow the laminar layer with thwaites_lambda(xs, us, nu) to get the lambda history, one value per station.
3. Locate the laminar separation point with laminar_separation_station(xs, us, nu), which returns (index, x) of the first station with lambda <= -0.09, or None for a fully attached laminar run. Read us at that index for the separation edge velocity.
4. On the pressure side, build the pressure-coefficient recovery traverse xs, cps and evaluate the recovery with stratford_parameter(xs, cps), which returns the S history with None off the recovery.
5. Locate the turbulent separation station with stratford_separation_station(xs, cps), which returns (index, x) of the first station with S >= 0.35, or None.
6. Quantify the headroom with separation_margin(xs, cps): positive margin means the recovery keeps margin below the 0.35 threshold up to the crossing point.
7. Report the laminar separation station or none, the turbulent separation station or none, and the margin, and gate the airfoil or inlet-duct design check on whichever separation signal is closer to the operating point.
8. Confirm the deterministic checks with the contract test scripts/test_boundary_layer_separation.py.

Worked example

Edge velocity U(x) = 30 * (1 - x) m/s over x in [0, 1] m (linear deceleration), nu = 1.5e-5 m2/s, 400001 stations (step 2.5e-6 m):

• Thwaites lambda traverse is monotone decreasing from 0 at the start; at x = 0.05 m the discrete lambda is -0.027028, matching the closed form -0.075 * (1 - t^6) / t^6 with t = 1 - x to five digits.
• Laminar separation at station index 49257, x = 0.12314 m (within 1 percent of the classic 0.1231 m, about 12 percent chord), where the lambda values around the crossing go from -0.089998 to -0.090001.
• Edge velocity at the separation station: 26.306 m/s (within 1 percent of 26.31 m/s).
• Accelerating flow U(x) = 30 * (1 + 0.2 * x) over 1 m: lambda stays non-negative and laminar_separation_station returns None.

Stratford recovery C_p(x) = 0.4 * x^2 over x in [0, 0.9] m at 10 stations (step 0.1 m):

• S climbs through 0.26710 at x = 0.7 m and crosses 0.35 at station index 8, x = 0.8 m, with S = 0.3505 at the crossing.
• separation_margin = 0.35 - 0.26710 = 0.08290, positive headroom up to the crossing point.
• Mild recovery C_p = 0.2 * x^2: no crossing (None), margin 0.12735.
• Steep recovery C_p = 0.6 * x^2: crossing at station index 7, x = 0.7 m, S = 0.4007.
• Falling C_p (favorable pressure, no recovery): no station, margin 0.35.

Verification

• Confirm thwaites_lambda on the linear deceleration is monotone decreasing with a negative tail, and matches the closed-form -0.075 * (1 - t^6) / t^6 at interior stations.
• Confirm laminar_separation_station returns (49257, 0.12314) on the worked grid, within 1 percent of 0.1231 m, with U = 26.31 m/s there, and None on the accelerating flow.
• Confirm stratford_separation_station returns (8, 0.8) with S = 0.3505 at the crossing for C_p = 0.4 * x^2, and that the margin before the crossing is positive.
• Confirm a no-crossing recovery returns None and a positive margin, and that every non-physical input raises ValueError: mismatched station and velocity or pressure lengths, nu <= 0, negative edge velocity, all-zero velocity, non-increasing stations, and empty lists.
• Run the contract test offline: python3 scripts/test_boundary_layer_separation.py (35 tests, deterministic, about 1 second).

Related leaves

• aerodynamics/boundary-layer/boundary-layer-theory: flat-plate thickness and skin-friction context for the attached side of the flow; zero-pressure-gradient flows never trigger these criteria.
• aerodynamics/boundary-layer/boundary-layer-transition: the sibling Thwaites traverse that stops at the natural-transition onset estimated by the Michel rule instead of running to the -0.09 separation crossing.
• aerodynamics/cfd/cfd-turbulence-modeling: RANS turbulence-model selection, where separation prediction appears only as a modeling pitfall for wall functions.

Pitfalls

• Continuing the Thwaites traverse past the crossing: once lambda drops below -0.09 the integral method has left its validity domain, the laminar layer has left the surface, and downstream lambda values are not a physical answer; stop at the first crossing.
• Reading the trailing U = 0 station as a finite lambda: an edge flow that decelerates to zero has separated long before, and the module reports lambda = -inf at such stations, so check the crossing before the end of the list.
• Treating negative or falling C_p as recovery: the Stratford parameter needs C_p > 0 with dC_p/dx > 0; favorable-pressure regions carry no S value and leave the margin at the full 0.35.
• Confusing the two separation signals: the laminar (Thwaites) and turbulent (Stratford) crossings are different physics, usually at different stations, and both belong in the report with their criterion named.
• Reading the margin after a crossing as negative headroom: separation_margin is the headroom up to the crossing point, positive by construction; on a steep recovery the separation station itself is the separation signal.
• Feeding a zero-pressure-gradient flat-plate case expecting a separation: dU/dx = 0 gives lambda = 0 and no Stratford recovery, so both criteria stay silent, which is correct for an attached flat plate at moderate incidence.

Behavior contract (gate 3)

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

python3 scripts/test_boundary_layer_separation.py

The test must pass with exit 0 and covers the worked-example anchors (laminar separation at x = 0.1231 m within 1 percent with U = 26.31 m/s on the linear deceleration, no separation on the accelerating flow), the closed-form Thwaites identity for the linear deceleration, monotone-decreasing lambda with a negative tail, the Stratford crossing at station index 8 with S = 0.3505 and a positive margin before the crossing, no-crossing and favorable-pressure recoveries returning None with positive margins, steep-recovery crossing, and ValueError rejection of every non-physical input listed in the spec (empty or length-mismatched lists, nu <= 0, negative velocity, all-zero velocity, non-increasing stations).

Contract test

Run the contract test from the leaf directory:

cd skills/aerodynamics/boundary-layer/boundary-layer-separation
python3 scripts/test_boundary_layer_separation.py

Stdlib unittest only, deterministic, offline, 35 test methods, exit 0 in about 1 second. The test file imports the sibling logic module boundary_layer_separation_logic from its own scripts directory, so no install or path configuration is needed.

Compliance

• Standards referenced, not reproduced: NACA TR-824 (laminar boundary-layer correlations and pressure-gradient behavior) frames the family context; the Thwaites and Stratford relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.