Effective boundary conditions at a rough wall: a high-order homogenization approach

• Published in: Meccanica (Milan) Vol. 55; no. 9; pp. 1781 - 1800
• Main Authors: Bottaro, Alessandro, Naqvi, Sahrish B.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.09.2020; Springer Nature B.V
• Subjects: Automotive Engineering; Boundary conditions; Civil Engineering; Classical Mechanics; Engineering; Fluid dynamics; Fluid flow; Homogenization; Incompressible flow; Incompressible fluids; Mechanical Engineering; Microhardness; Order parameters; Slip velocity; Stagnation point; Transpiration; Homogenization; Stagnation point flow; Wall transpiration; Slip boundary condition
• ISSN: 0025-6455; 1572-9648
• DOI: 10.1007/s11012-020-01205-2

Effective boundary conditions, correct to third order in a small parameter ϵ , 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 ϵ = l / L ≪ 1 , 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 O ( ϵ 2 ) 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.