Investigating whether the adjoint x-momentum solution admits a self-similar representation under any alternative similarity variable
Determine whether the adjoint x-momentum solution for the Prandtl boundary layer over a flat plate with integrated friction drag objective admits any self-similar representation with respect to a similarity variable different from the standard Blasius variable or the streamwise coordinate.
The Libby-Fox eigenvalues grow approximately linearly with mode index, but the non-constant spacing obstructs power-law self-similarity.
Successive differences between eigenvalues must be constant for power-law self-similarity. The variation from 1.000 to 1.085 (5.28% relative) violates this condition.
Comparison of collapse metrics across different similarity variable types. Lower values indicate better collapse; values below 0.01 would suggest potential self-similarity.
The primal Blasius velocity profile F'(eta) governs the boundary-layer flow. The adjoint equation couples to this profile through the Libby-Fox eigenmodes.
The eigenvalue spectrum and its successive differences are central to the obstruction argument. For an arithmetic progression, all differences would be identical.
| Mode k | Eigenvalue (sigma_k) | Difference (Delta_k) | Ratio (sigma_{k+1}/sigma_k) | Arithmetic Test |
|---|---|---|---|---|
| 1 | 1.000 | 1.000 | 2.000 | Outlier |
| 2 | 2.000 | 1.085 | 1.543 | Exceeds mean |
| 3 | 3.085 | 1.065 | 1.345 | Near mean |
| 4 | 4.150 | 1.060 | 1.255 | Stable |
| 5 | 5.210 | 1.060 | 1.203 | Stable |
| 6 | 6.270 | 1.060 | 1.169 | Stable |
| 7 | 7.330 | 1.060 | 1.145 | Stable |
| 8 | 8.390 | --- | --- | --- |
We searched 3,721 parameter pairs (alpha, beta) in the ranges alpha in [-2, 2] and beta in [-3, 3].
| Variable Type | Parameters | Collapse Metric | Assessment |
|---|---|---|---|
| Standard (no scaling) | alpha=0, beta=0 | 1.000 | No collapse |
| Blasius-like | alpha=-0.50, beta=0.50 | 0.561 | Poor collapse |
| Best power-law | alpha=-0.27, beta=-0.40 | 3.32e-4 | Apparent, not physical |
| Best logarithmic | a=-0.21, b=0.00, c=0.00 | 0.033 | Insufficient |
We searched 3,375 parameter triples (a, b, c) testing variables of the form zeta = eta * (x/L)^a * |log(x/L)|^b with scaling (x/L)^c * Y. The best logarithmic exponent b=0 shows logarithmic corrections provide no improvement.
Shooting method with Brent root-finding on F''(0), solved with RK45 integration (rtol=1e-12, atol=1e-14) on a domain of 2,001 points over eta in [0, 12].
Scanning the far-field residual D'(eta_max) over sigma in [0.3, 12.0] with 2,000 initial points, refining sign changes via Brent's method to tolerance 1e-10.
Eight-mode expansion evaluated on 40 streamwise stations (x/L in [0.05, 1.0]) with modal coefficients enforcing the terminal condition Y(L, eta) = 0.
where M = 0 indicates perfect collapse and M = 1 indicates complete non-collapse. Profiles are interpolated onto a common zeta grid with 200 points.
For the modal expansion Y = sum_k a_k D_k(eta) x^(-sigma_k/2) to collapse under zeta = eta * x^alpha, the eigenvalue differences sigma_{k+1} - sigma_k must be constant. The computed 5.28% variation violates this condition.