Translation of adjoint boundary conditions into modal conditions

Determine how to translate the adjoint boundary conditions Y(x,0) = −K/(12x), Y(L,η) = 0, and Y(x,∞) = 0 for the Blasius boundary layer into explicit boundary conditions for the adjoint eigenfunctions Dk(η) and corresponding constraints on the separation constants σk in the adjoint eigenvalue problem −Dηηη + F0Dηη + σF0,ηDη + 2(σ−1)F0,ηηD = 0 (equation (25)), providing a direct mode-wise formulation consistent with the separated representation Y(x,η) = Σk ak Dk(η) x−σk/2.

Background

In Section 3 the authors derive the adjoint boundary-layer equation and boundary conditions for the Blasius flow and introduce a separated, eigenfunction expansion for the adjoint solution. While the wall condition can be handled by enforcing Dk(0)=0 except for the σ=1 mode, the remaining boundary conditions involve global relations over the infinite sum of modes. The authors note that they do not know how to directly translate these adjoint boundary conditions into conditions on individual modes and separation constants, indicating a gap between the PDE-level boundary conditions and the modal formulation.