Laminar Far Wake

Laminar Far Wake (aerodynamics/boundary-layer/laminar-far-wake)

Use when you must compute the two-dimensional incompressible laminar far wake downstream of a thin flat plate at zero incidence, the Goldstein (1933) similarity wake that the attached laminar boundary layers shed from the trailing edge, in the form Schlichting Boundary-Layer Theory (wakes and free-shear-layers chapter, wake behind a flat plate) and White Viscous Fluid Flow (laminar free shear layers, plane wake defect solution) present it. This leaf is the eighth member of the boundary-layer pack and owns the downstream free-shear wake region that every attached-layer sibling stops short of: boundary-layer-theory stops at the plate surface quantities and the transition, separation and unsteady siblings never leave the attached layer. Far downstream of the trailing edge the small velocity defect u1 = U - u collapses onto a Gaussian similarity profile whose cross-stream integral conserves the momentum deficit, and the wake-momentum-integral identity D = rhoUintegral u1 dy is the closed-form basis of wake-survey drag measurement. It does not do attached flat-plate boundary-layer thickness or skin-friction estimation on the surface, transition or separation traverses, creeping Stokes sphere flow, unsteady plate Stokes layers, the wind-tunnel solid-blockage and wake-blockage corrections, or turbulent, axisymmetric, compressible and near-wake flows: incompressible constant-property laminar flow only, small defect u1 << U (x/c at least of order ten), two-dimensional thin-plate small-deficit wakes of a symmetric body at zero incidence.

Domain quick reference

The wake coordinate x is measured downstream from the trailing edge, y is the cross-stream coordinate, u1 is the positive velocity defect U - u. Module constants: NU_AIR = 1.46e-5 m2/s, RHO_AIR = 1.225 kg/m3, U_INF = 5.0 m/s, PLATE_CHORD = 1.0 m, BLASIUS_THETA_COEF = 0.664, BLASIUS_DRAG_COEF = 1.328, SIDES = 2.

• Trailing-edge state (attached Blasius layer, one side): theta_c = 0.664sqrt(nuc/U) = 0.664c/sqrt(Re_c) with Re_c = Uc/nu the chord Reynolds number.
• Plate drag per unit span (both sides), the momentum source of the wake: D = 2rhoU^2theta_c = 1.328rhoU^2sqrt(nuc/U), N/m; the drag coefficient C_D = D/(0.5rhoU^2c) = 4*theta_c/c = 2.656/sqrt(Re_c), dimensionless.
• Linearized wake defect equation: Udu1/dx = nud2u1/dy2, the small-defect (u1 << U) form of the boundary-layer equations in the wake, with the momentum invariant D = rhoUintegral_{-inf}^{+inf} u1 dy, the wake-momentum-integral drag identity (exact in the far-wake limit, the closed-form basis of wake-survey drag).
• Gaussian similarity profile (the Goldstein far-wake solution): u1(x, y) = u_c(x)exp(-B(x)y^2) with spread parameter B(x) = U/(4nux), 1/m2, which satisfies the linearized equation identically, hence u_c proportional to x^-1/2.
• Centerline defect from the drag: u_c(x) = (D/(rho*U))sqrt(B/pi), m/s, collapsing for the two-sided Blasius plate to the closed form u_c(x) = (0.664U/sqrt(pi))*sqrt(c/x) (coefficient 0.664/sqrt(pi) = 0.3746218835).
• Defect and wake velocity at (x, y): u1 = u_cexp(-By^2) and u = U - u1, both m/s.
• Widths: half-defect width y_half = sqrt(ln(2)/B) (u1 = u_c/2 there) and one-over-e width y_e = sqrt(1/B) (u1 = u_c*exp(-1) there), both m, growing as x^1/2 since B is proportional to 1/x.
• Momentum integral of the Gaussian: integral u1 dy = u_csqrt(pi/B), m2/s, independent of x and equal to D/(rhoU) = 2Utheta_c at every station; the wake-survey drag is D = rhoUu_c*sqrt(pi/B), N/m.
• Full (nonlinear) momentum deficit, documented for honesty: rhointegral u(U - u) dy = rho*(Uintegral u1 dy - integral u1^2 dy) with integral u1^2 dy = u_c^2sqrt(pi/(2B)) closed form; it lies below the linearized identity by a relative amount of order u_c/U and converges to it downstream (ratios 0.94702046516 at x = 25c, 0.97351023258 at x = 100c, 0.98675511629 at x = 400c).

SI units throughout; Goldstein 1933, Schlichting Boundary-Layer Theory and White Viscous Fluid Flow material cited through NACA TR-824, summary-only.

Workflow

1. Fix the trailing-edge state of the fully laminar plate: chord c, freestream speed U, rho, nu, then the chord Reynolds number (reynolds_number) and the trailing-edge momentum thickness (momentum_thickness_blasius): theta_c = 0.664*c/sqrt(Re_c), the state the far wake carries downstream.
2. Assemble the plate drag, the momentum source of the wake: plate_drag_per_span(U, rho, nu, c) with the default two-sided plate (SIDES = 2, one side carrying rhoU^2theta_c), and reduce it with plate_drag_coefficient, cross-checked against C_D = 2.656/sqrt(Re_c) and C_D = 4*theta_c/c.
3. Choose the downstream traverse station x measured from the trailing edge (x/c of order ten or more for the linearized far-wake regime) and evaluate the Gaussian spread parameter with wake_spread_parameter(U, nu, x): B = U/(4nux).
4. Traverse the centerline defect: centerline_defect_from_drag(D, rho, U, B) is the normalization that conserves the drag exactly, and centerline_defect_blasius(U, c, x) is its closed-form twin for the two-sided Blasius plate, giving the x^-1/2 decay law explicitly.
5. Traverse the Gaussian defect profile across y with velocity_defect_gaussian(u_c, B, y) and recover the wake velocity with wake_velocity(U, u_c, B, y); read the wake widths from half_defect_width(B) and one_over_e_width(B).
6. Check the decay and spreading laws across stations: the invariant u_c*sqrt(x) is constant (x^-1/2 centerline decay) and the half-defect width grows as x^1/2, doubling when x quadruples.
7. Integrate the momentum deficit: defect_integral(u_c, B) = u_csqrt(pi/B) is station-independent, and drag_from_wake(u_c, B, rho, U) = rhoU*integral u1 dy reproduces the plate drag at every station to float noise, the wake-survey drag identity.
8. Run the nonlinear honesty check: full_momentum_deficit(rho, U, u_c, B) lies a few percent below the linearized drag_from_wake and converges to it as the wake spreads; report the residual when the defect is not tiny.
9. Report the drag diagnostic: the wake-survey drag per unit span and the drag coefficient from the traverse, with the profile decay and spreading numbers as the laminar-wake fingerprint.
10. Confirm the deterministic checks: rerun the closed-form identities and reject non-physical inputs (non-positive U, nu, rho, c, x, D, B, u_centerline, sides below one) with the contract test scripts/test_laminar_far_wake.py.

Worked example

Air at standard conditions nu = 1.46e-5 m2/s, rho = 1.225 kg/m3, a flat plate of chord c = 1.0 m at U = 5.0 m/s, wetted on both sides with fully laminar Blasius boundary layers to the trailing edge. All values below are real outputs of the module.

• Trailing-edge state: Re_c = Uc/nu = 342465.7534; momentum thickness at the trailing edge (one side) theta_c = 0.664sqrt(nuc/U) = 1.1346436974e-3 m (1.135 mm); total drag per unit span, both sides, D = 1.328rhoU^2sqrt(nuc/U) = 2rhoU^2theta_c = 6.9496926464e-2 N/m; drag coefficient C_D = D/(0.5rhoU^2c) = 0.00453857479, identical to 2.656/sqrt(Re_c) = 0.00453857479 and to 4theta_c/c = 0.00453857479 (three routes agree).
• Far-wake traverse at x = 100c = 100.0 m downstream of the trailing edge (centerline defect 3.746 percent of U, solidly small-defect): spread parameter B = U/(4nux) = 8.5616438356e2 1/m2; centerline defect u_c = (D/(rhoU))sqrt(B/pi) = 0.18731094174 m/s, identical to the closed form (0.664U/sqrt(pi))*sqrt(c/x) = 0.18731094174 m/s; widths y_half = sqrt(ln(2)/B) = 2.8453398864e-2 m (2.845 cm) and y_e = sqrt(1/B) = 3.4176014981e-2 m, with y_e/y_half = 1.2011224087 = sqrt(1/ln(2)).
• Profile values: u1(y_half) = 9.3655470869e-2 m/s (exactly u_c/2), u1(y_e) = 6.8907844572e-2 m/s (u_cexp(-1)), and the recovered wake-axis velocity u(0) = U - u_c = 4.8126890583 m/s; at y = 6y_half = 0.170720393182 m the wake is indistinguishable from the freestream, u/U = 0.999999999999455.
• Momentum integral: integral u1 dy = u_csqrt(pi/B) = 1.1346436974e-2 m2/s, equal to D/(rhoU) and to 2Utheta_c at every station; the wake-survey drag D = rhoUintegral u1 dy = 6.9496926464e-2 N/m equals the plate drag to float noise (ratio 1.0).
• Decay and spreading across stations: centerline defect 0.37462188348 m/s at x = 25c, 0.18731094174 m/s at x = 100c and 0.093655470869 m/s at x = 400c, so u_c(4x)/u_c(x) = 0.5 exactly and the invariant u_csqrt(x) = 1.8731094174 m/ssqrt(m) is unchanged (x^-1/2 decay); half-defect width 1.4226699432e-2 m at x = 25c, 2.8453398864e-2 m at x = 100c and 5.6906797727e-2 m at x = 400c, so y_half(4x)/y_half(x) = 2 exactly (x^1/2 spreading).
• Nonlinear honesty check: the full deficit rhointegral u(U - u) dy lies 5.297953 percent below D at x = 25c (ratio 0.94702046516), 2.648977 percent below at x = 100c (ratio 0.97351023258, residual 1.8409574184e-3 N/m) and 1.324488 percent below at x = 400*c (ratio 0.98675511629), converging to the linearized identity downstream.

Read-off: a 1 m fully laminar plate at 5 m/s in air drags 6.95e-2 N per metre of span; 100 m downstream the wake axis runs 3.75 percent slow inside a Gaussian defect 2.85 cm wide at half depth, the integrated momentum deficit of that Gaussian recovers the plate drag exactly, and quadrupling the downstream distance halves the centerline defect while the wake doubles in width. A wake-survey traverse at x/c = 100 integrated with the identity D = rhoUintegral u1 dy reports the drag without any force balance.

Verification

• Confirm reynolds_number(5.0, 1.0, 1.46e-5) returns 342465.7534 and momentum_thickness_blasius(5.0, 1.0, 1.46e-5) returns 1.1346436974e-3 m, equal to 0.664/sqrt(Re_c) to float noise.
• Confirm plate_drag_per_span(5.0, 1.225, 1.46e-5, 1.0) returns 6.9496926464e-2 N/m (between 6.5e-2 and 7.5e-2 N/m), the sides = 1 value equals rhoU^2theta_c, and the two-sided value is twice the one-sided value.
• Confirm plate_drag_coefficient(5.0, 1.225, 1.46e-5, 1.0) returns 4.53857479e-3, equal to 2.656/sqrt(Re_c) and to 4*theta_c/c.
• Confirm wake_spread_parameter(5.0, 1.46e-5, 100.0) returns 8.5616438356e2 1/m2 and is 0.25 of the x = 400.0 value.
• Confirm the centerline defect at x = 100.0 m is 0.18731094174 m/s by both routes (drag normalization and closed form, agreeing to float noise), with u_c/U = 0.03746218835 and the invariant u_c*sqrt(x) = 1.8731094174 across x = 25, 100 and 400 m.
• Confirm the decay and spreading laws: u_c(4x)/u_c(x) = 0.5, u_c(2x)/u_c(x) = 1/sqrt(2), y_half(4x)/y_half(x) = 2 and B(4x)/B(x) = 0.25, all exact to float noise.
• Confirm the Gaussian shape at x = 100.0 m: y_half = 2.8453398864e-2 m, y_e = 3.4176014981e-2 m, u1(y_half)/u_c = 0.5, u1(y_e)/u_c = 1/e, u(0) = 4.8126890583 m/s and u/U = 0.999999999999455 at y = 6*y_half.
• Confirm the momentum identity: defect_integral = 1.1346436974e-2 m2/s and drag_from_wake = 6.9496926464e-2 N/m at x = 25, 100 and 400 m, equal to the plate drag to float noise at every station.
• Confirm the nonlinear ratios 0.94702046516, 0.97351023258 and 0.98675511629 (monotone toward one) and the linearized residual 1.8409574184e-3 N/m at x = 100.0 m.
• Confirm every non-positive U, nu, rho, c, x, D, B, every negative u_centerline and every sides below one raises ValueError, and that every function is deterministic on repeat calls.
• Run the contract test offline: python3 scripts/test_laminar_far_wake.py (32 tests, deterministic).

Related leaves

• aerodynamics/boundary-layer/boundary-layer-theory: the attached flat-plate layer owner (Blasius thickness, skin friction, transition on the surface); this leaf consumes the trailing-edge momentum state as the source the wake carries and never estimates surface-layer quantities.
• aerodynamics/boundary-layer/stokes-creeping-flow-drag: the Re << 1 attached-body limit with its no-wake fore-aft symmetric creeping flow, the regime fence that stays away from the finite-Reynolds wake profile of this leaf.
• aerodynamics/boundary-layer/boundary-layer-transition and boundary-layer-separation: the attached-layer traverse leaves that stop at transition onset and separation, never mapping the downstream wake state this leaf owns.
• aerodynamics/wind-tunnel/windtunnel-wall-corrections: uses wake blockage, a scalar tunnel-geometry correction to the dynamic pressure, not the wake velocity-defect profile or the momentum integration of this leaf.
• aerodynamics/wind-tunnel/windtunnel-data-reduction: the raw-run pressure-rake reduction leaf; the wake-survey drag identity of this leaf is the physics behind its rake traverse.

Pitfalls

• Applying the far-wake Gaussian too close to the trailing edge: the similarity profile holds for x/c of order ten or more; immediately behind the trailing edge the near-wake defect follows the Goldstein error-function solution, not this small-defect Gaussian.
• Treating the linearized identity as exact when the defect is not small: D = rhoUintegral u1 dy neglects the u1^2 term, which runs a few percent of the drag at x/c ~ 25 and only converges downstream; use the full_momentum_deficit diagnostic for honesty.
• Mixing the width conventions: y_half = sqrt(ln(2)/B) (defect halved) differs from y_e = sqrt(1/B) (defect at 1/e) by the factor sqrt(1/ln(2)) = 1.2011224087; report which width is meant.
• Forgetting the two-sided plate: the plate drag doubles with SIDES = 2 (each side carries rhoU^2theta_c), and the centerline defect scales with the square root of the drag, so a one-sided mistake shifts every downstream profile value.
• Measuring x from the wrong origin: the wake coordinate x runs downstream from the trailing edge, not from the plate leading edge, and the chord Reynolds number Re_c still uses the full chord.
• Confusing this leaf with the pack's attached-layer owners: it does not estimate displacement thickness, momentum thickness or skin friction on the plate surface, and the wake-blockage scalar of the wind-tunnel leaves is not a velocity-defect profile.
• Extending the Gaussian beyond its regime: turbulent wakes, jets, mixing layers, axisymmetric wakes, compressible wakes and the near-wake error-function layer all lie outside this small-defect laminar solution.

Behavior contract (gate 3)

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

python3 scripts/test_laminar_far_wake.py

The test covers the worked-example contract (Reynolds number, momentum thickness, plate drag with magnitude bound, the three-route drag coefficient, spread parameter, centerline defect by both routes, widths, profile values and wake velocities), the decay and spreading laws across stations (x^-1/2 centerline decay, x^1/2 width growth), the momentum identity at every station (defect integral and wake-survey drag against the plate drag), the nonlinear momentum-deficit honesty check with the linearized residual, deterministic repeat calls, and the input-rejection traverse of non-physical U, nu, rho, c, x, D, B, u_centerline and sides arguments.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is the cited reference for the boundary-layer pack; the laminar wake treatment follows Goldstein 1933, Schlichting Boundary-Layer Theory (wakes and free-shear-layers chapter) and White Viscous Fluid Flow (laminar free shear layers), whose material is cited through the report. Summary- only methodology per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.