Translation of Adjoint Boundary Conditions into Modal Conditions

Blasius Boundary Layer -- Adjoint Eigenvalue Problem

Category: Fluid Dynamics (physics.flu-dyn) Reference: Lozano et al., arXiv:2601.16718 (2026) Date: Jan 23, 2026

Key Results

2.0000

Leading eigenvalue σ₀

0.6474

Second eigenvalue σ₁

0.3321

Blasius f₀''(0)

Modes computed

4001

Grid points

15.0

η_max

Problem Statement

Determine how to translate the adjoint boundary conditions for the Blasius boundary layer into explicit boundary conditions for the adjoint eigenfunctions D_k(η) and separation constants σ_k.

Y(x, 0) = -K / (12x)
Y(L, η) = 0
Y(x, ∞) = 0

Separated representation: Y(x,η) = Σ_k a_k D_k(η) x^-σ_k/2

Main Result

Theorem: Modal Boundary Conditions

Leading mode (k=0): σ₀ = 2, D₀(0) ≠ 0, a₀ D₀(0) = -K/12
Higher modes (k ≥ 1): D_k(0) = 0, D_k(η → ∞) = 0
Outflow: Automatic for L → ∞ when Re(σ_k) > 0
Spectral correspondence: σ_k = 2λ_k
Biorthogonality: ⟨φ_j, D_k⟩_F₀'' = δ_jk N_k

Eigenvalue Spectrum

Primal--Adjoint Correspondence

Boundary Condition Verification

Wall condition: D_0(0) = 1.0 (nonzero, absorbs wall source), D_k(0) = 0 for higher modes.

Outflow Decay Rates vs. Plate Length

Plate length L: 100

Eigenvalue Table

k	σ_k	λ_k = σ_k/2	Type
0	2.0000	1.0000	Inhomogeneous
1	0.6474	0.3237	Homogeneous

Wall Values D_k(0)

k	D_k(0)	Expected	Status
0	1.0000	nonzero	PASS
1	0.0000	= 0	PASS

Biorthogonality Matrix

B_jk = inner product of primal and adjoint eigenfunctions weighted by F₀'' -- should be approximately diagonal.

Numerical Parameters

Domain	[0, 15.0]
Grid points	4001
ODE solver	RK45
Relative tolerance	10^-12
Absolute tolerance	10^-14
Eigenvalue scan points	2000
Scan range σ	[0.5, 30.0]
Root tolerance	10^-10

Outflow Decay at L = 100

k	σ_k	L^-σ_k/2	Assessment
0	2.0000	0.0100	Strong decay
1	0.6474	0.2252	Moderate decay