Continuous Adjoint Complement to the Blasius Equation

TL;DR

This paper develops a continuous adjoint framework for the classical Blasius boundary-layer equation, deriving analytical expressions and validating them against numerical Navier-Stokes solutions for flat plate flows.

Contribution

It introduces a novel adjoint complement to the Blasius equation, providing analytical solutions and insights into shape sensitivity and boundary-layer properties.

Findings

Adjoint equations simplify depending on advection treatment.

Analytical expressions for boundary-layer thickness and shear stress are derived.

Numerical validation shows good agreement with Navier-Stokes simulations.

Abstract

The manuscript is concerned with a continuous adjoint complement to two-dimensional, incompressible, first-order boundary-layer equations for a flat plate boundary-layer. The text is structured into three parts. The first part demonstrates, that the adjoint complement can be derived in two ways, either following a first simplify then derive or a first derive and then simplify strategy. The simplification step comprises the classical boundary-layer (b.-l.) approximation and the derivation step transfers the primal flow equation into a companion adjoint equation. The second part of the paper comprises the analyses of the coupled primal/adjoint b.-l. framework. This leads to similarity parameters, which turn the Partial-Differential-Equation (PDE) problem into a boundary value problem described by a set of Ordinary-Differential-Equations (ODE) and support the formulation of an adjoint complement to the classical Blasius equation. Opposite to the primal Blasius equation, its adjoint complement consists of two ODEs which can be simplified depending on the treatment of advection. It is shown, that the advective fluxes, which are frequently debated in the literature, vanish for the investigated self-similar b.l. flows. Differences between the primal and the adjoint Blasius framework are discussed against numerical solutions, and analytical expressions are derived for the adjoint b.-l. thickness, wall shear stress and subordinated skin friction and drag coefficients. The analysis also provides an analytical expression for the shape sensitivity to shear driven drag objectives. The third part assesses the predictive agreement between the different Blasius solutions and numerical results for Navier-Stokes simulations of a flat plate b.-l. at Reynolds numbers between 1E+03 <= ReL <= 1E+05 .