Publisher DOI: 10.1063/5.0037779
arXiv ID: 2011.07583v1
Title: Continuous adjoint complement to the Blasius equation
Language: English
Authors: Kühl, Niklas  
Müller, Peter M.  
Rung, Thomas  
Keywords: Physics - Fluid Dynamics;Mathematics - Optimization and Control
Issue Date: 1-Mar-2021
Source: Physics of Fluids 33 (3): 033608 (2021)
Abstract (english): 
This manuscript is concerned with a continuous adjoint complement to two-dimensional, incompressible, first-order boundary-layerequations for a flat plate boundary layer. The text is structured into three parts. The first part demonstrates that the adjoint complement canbe derived in two ways, following either afirst simplify then deriveor afirst derive and then simplifystrategy. The simplification stepcomprises the classical boundary-layer (BL) approximation, and the derivation step transfers the primal flow equation into a companionadjoint equation. The second part of the paper comprises the analyses of the coupled primal/adjoint BL framework. This leads to similarityparameters, which turn the partial-differential-equation (PDE) problem into a boundary value problem described by a set of ordinary-differential-equations (ODEs) and support the formulation of an adjoint complement to the classical Blasius equation. Opposite to the primalBlasius equation, its adjoint complement consists of two ODEs, which can be simplified depending on the treatment of advection. It is shownthat the advective fluxes, which are frequently debated in the literature, vanish for the investigated self-similar BL flows. Differences betweenthe primal and the adjoint Blasius framework are discussed against numerical solutions, and analytical expressions are derived for the adjointBL thickness, wall shear stress, and subordinated skin friction and drag coefficients. The analysis also provides an analytical expression forthe shape sensitivity to shear driven drag objectives. The third part assesses the predictive agreement between the different Blasius solutionsand numerical results for Navier–Stokes simulations of a flat plate BL at Reynolds numbers between 103 ReL 105. It is seen that thereversal of the inlet and outlet locations and the direction of the flow, inherent to the adjoint formulation of convective kinematics, poses achallenge when investigating real finite length (finiteRe-number) flat plate boundary layer problems. Efforts to bypass related issues arediscussed.
Institute: Fluiddynamik und Schiffstheorie M-8 
Document Type: Article
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