Effective boundary conditions for Stokes flow over a rough surface

1996 ◽  
Vol 316 ◽  
pp. 223-240 ◽  
Author(s):  
Kausik Sarkar ◽  
Andrea Prosperetti
Keyword(s):  
Rough Surface ◽  
Stokes Flow ◽  
Smooth Surface ◽  
Traction Force ◽  
Slip Condition ◽  
Partial Slip ◽  
Ensemble Averaging ◽  
Effective Boundary ◽  
Exact Boundary ◽  
Effective Boundary Conditions

Ensemble averaging combined with multiple scattering ideas is applied to the Stokes flow over a stochastic rough surface. The surface roughness is modelled by compact protrusions on an underlying smooth surface. It is established that the effect of the roughness on the flow far from the boundary may be represented by replacing the no-slip condition on the exact boundary by a partial slip condition on the smooth surface. An approximate analysis is presented for a sparse distribution of arbitrarily shaped protrusions and explicit numerical results are given for hemispheres. Analogous conclusions for the two-dimensional case are obtained. It is shown that in certain cases a traction force develops on the surface at an angle with the direction of the flow.

scholarly journals Effective boundary condition for Stokes flow over a very rough surface

2013 ◽  
Vol 254 (8) ◽  
pp. 3395-3430 ◽  
Author(s):  
Youcef Amirat ◽  
Olivier Bodart ◽  
Umberto De Maio ◽  
Antonio Gaudiello
Keyword(s):  
Boundary Condition ◽  
Rough Surface ◽  
Stokes Flow ◽  
Effective Boundary ◽  
Effective Boundary Condition

Effective boundary conditions for the Laplace equation with a rough boundary

1995 ◽  
Vol 451 (1942) ◽  
pp. 425-452 ◽  
Keyword(s):  
Laplace Equation ◽  
Water Waves ◽  
Smooth Surface ◽  
Normal Derivative ◽  
Rough Boundary ◽  
Ensemble Averaging ◽  
Effective Boundary ◽  
Suitable Combination ◽  
Effective Boundary Condition ◽  
The Neumann Problem

The problem of replacing Dirichlet or Neumann conditions on a stochastically embossed surface by approximate effective conditions on a smooth surface is studied for potential fields satisfying the Laplace equation. A combination of ensemble averaging and multiple-scattering techniques is used. It is shown that for the Dirichlet case the effective boundary condition becomes mixed and establishes a relation between the averaged field and its normal derivative. For the Neumann problem the normal derivative on the smooth surface equals a suitable combination of first- and second-order derivatives tangent to the surface. Explicit results are given for small boss concentration and illustrated with the examples of spheroidal and spherical bosses. For the Dirichlet case with hemispherical bosses, direct numerical-simulation results are presented up to area coverages of 75%. An application of the results to the calculation of the added mass of a rough sphere in potential flow, of the capacitance of a rough spherical conductor, and of the transmission and reflection of long water waves at a smooth-rough bottom transition aids in their physical interpretation.

scholarly journals Generalized slip condition over rough surfaces

2018 ◽  
Vol 858 ◽  
pp. 407-436 ◽  
Author(s):  
Giuseppe A. Zampogna ◽  
Jacques Magnaudet ◽  
Alessandro Bottaro
Keyword(s):  
Boundary Condition ◽  
Rough Surface ◽  
Hexagonal Lattice ◽  
Fluid Flows ◽  
Reynolds Numbers ◽  
Slip Condition ◽  
Strain Rate Tensor ◽  
Outer Flow ◽  
Navier Slip ◽  
Roughness Amplitude

A macroscopic boundary condition to be used when a fluid flows over a rough surface is derived. It provides the slip velocity $\boldsymbol{u}_{S}$ on an equivalent (smooth) surface in the form $\boldsymbol{u}_{S}=\unicode[STIX]{x1D716}{\mathcal{L}}\boldsymbol{ : }{\mathcal{E}}$, where the dimensionless parameter $\unicode[STIX]{x1D716}$ is a measure of the roughness amplitude, ${\mathcal{E}}$ denotes the strain-rate tensor associated with the outer flow in the vicinity of the surface and ${\mathcal{L}}$ is a third-order slip tensor arising from the microscopic geometry characterizing the rough surface. This boundary condition represents the tensorial generalization of the classical Navier slip condition. We derive this condition, in the limit of small microscopic Reynolds numbers, using a multi-scale technique that yields a closed system of equations, the solution of which allows the slip tensor to be univocally calculated, once the roughness geometry is specified. We validate this generalized slip condition by considering the flow about a rough sphere, the surface of which is covered with a hexagonal lattice of cylindrical protrusions. Comparisons with direct numerical simulations performed in both laminar and turbulent regimes allow us to assess the validity and limitations of this condition and of the mathematical model underlying the determination of the slip tensor ${\mathcal{L}}$.

Influence of wall roughness on the slip behaviour of viscous fluids

2008 ◽  
Vol 138 (5) ◽  
pp. 957-973 ◽  
Author(s):  
Dorin Bucur ◽  
Eduard Feireisl ◽  
Šárka Nečasová
Keyword(s):  
Boundary Condition ◽  
Viscous Fluid ◽  
Boundary Conditions ◽  
Limit Problem ◽  
Viscous Fluids ◽  
Wall Roughness ◽  
Effective Boundary ◽  
Specific Direction ◽  
Friction Term ◽  
Effective Boundary Conditions

We consider the stationary equations of a general viscous fluid in an infinite (periodic) slab supplemented with Navier's boundary condition with a friction term on the upper part of the boundary. In addition, we assume that the upper part of the boundary is described by a graph of a function φε, where φε oscillates in a specific direction with amplitude proportional to ε. We identify the limit problem when ε → 0, in particular, the effective boundary conditions.

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