Effective boundary conditions at a rough wall: a high-order homogenization approach
Alessandro Bottaro, Sahrish Batool Naqvi
Abstract
Abstract Effective boundary conditions, correct to third order in a small parameter $$\epsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> , are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is l , and $$\epsilon = l/L \ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>l</mml:mi> <mml:mo>/</mml:mo> <mml:mi>L</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , with L a characteristic length of the macroscopic problem. A multiple scale expansion of the variables allows to recover, at leading order, the usual Navier slip condition. At next order the slip velocity includes a term arising from the streamwise pressure gradient; furthermore, a transpiration velocity $${\mathcal {O}}(\epsilon ^{2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> appears at the fictitious wall where the effective boundary conditions are enforced. Additional terms appear at third order in both wall-tangent and wall-normal components of the velocity. The application of the effective conditions to a macroscopic problem is carried out for the Hiemenz stagnation point flow over a rough wall, highlighting the differences among the exact results and those obtained using conditions of different asymptotic orders.