High frequency analysis of the unsteady Interactive Boundary Layer model

Anne-Laure Dalibard (1) , Helge Dietert (2) , David Gérard-Varet (2) , Frédéric Marbach (1)

1 LJLL - Laboratoire Jacques-Louis Lions
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

The present paper is about a famous extension of the Prandtl equation, the so-called Interactive Boundary Layer model (IBL). This model has been used intensively in the numerics of steady boundary layer flows, and compares favorably to the Prandtl one, especially past separation. We consider here the unsteady version of the IBL, and study its linear well-posedness, namely the linear stability of shear flow solutions to high frequency perturbations. We show that the IBL model exhibits strong unrealistic instabilities, that are in particular distinct from the Tollmien-Schlichting waves. We also exhibit similar instabilities for a Prescribed Displacement Thickness model (PDT), which is one of the building blocks of numerical implementations of the IBL model.

Mots clés: Boundary layer equations; Interactive boundary layer; Prandtl boundary layer equation; Instabilities in fluid dynamics

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]

Anne-Laure Dalibard, Helge Dietert, David Gérard-Varet, Frédéric Marbach. High frequency analysis of the unsteady Interactive Boundary Layer model. SIAM Journal on Mathematical Analysis, 2018, 50 (4), pp.4203-4245. ⟨10.1137/17M1157477⟩. ⟨hal-01615251v2⟩