scholarly journals Effective boundary condition at a rough surface starting from a slip condition

2011 ◽  
Vol 251 (12) ◽  
pp. 3450-3487 ◽  
Author(s):  
Anne-Laure Dalibard ◽  
David Gérard-Varet
Keyword(s):  
Boundary Condition ◽  
Rough Surface ◽  
Slip Condition ◽  
Effective Boundary ◽  
Effective Boundary Condition

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

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 ${\mathcal{L}}$.

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 Leaky Rigid Lid: New Dissipative Modes in the Troposphere

2013 ◽  
Vol 70 (10) ◽  
pp. 3119-3127 ◽  
Author(s):  
Lyubov G. Chumakova ◽  
Rodolfo R. Rosales ◽  
Esteban G. Tabak
Keyword(s):  
Boundary Condition ◽  
Radiation Condition ◽  
Wave Radiation ◽  
Pseudodifferential Equation ◽  
Piecewise Constant ◽  
Effective Boundary ◽  
Propagating Waves ◽  
Effective Boundary Condition ◽  
Set Up ◽  
Atmospheric Phenomena

Abstract An effective boundary condition is derived for the top of the troposphere, based on a wave radiation condition at the tropopause. This boundary condition, which can be formulated as a pseudodifferential equation, leads to new vertical dissipative modes. These modes can be computed explicitly in the classical setup of a hydrostatic, nonrotating atmosphere with a piecewise constant Brunt–Väisälä frequency. In the limit of an infinitely strongly stratified stratosphere, these modes lose their dissipative nature and become the regular baroclinic tropospheric modes under the rigid-lid approximation. For realistic values of the stratification, the decay time scales of the first few modes for mesoscale disturbances range from an hour to a week, suggesting that the time scale for some atmospheric phenomena may be set up by the rate of energy loss through upward-propagating waves.

Two-phase displacement in Hele Shaw cells: theory

1984 ◽  
Vol 139 ◽  
pp. 291-308 ◽  
Author(s):  
C.-W. Park ◽  
G. M. Homsy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Characteristic Length ◽  
Two Phase ◽  
Phase Displacement ◽  
Effective Boundary ◽  
Small Parameters ◽  
Effective Boundary Condition ◽  
Gap Width

A theory describing two-phase displacement in the gap between closely spaced planes is developed. The main assumptions of the theory are that the displaced fluid wets the walls, and that the capillary number Ca and the ratio of gap width to transverse characteristic length ε are both small. Relatively mild restrictions apply to the ratio M of viscosities of displacing to displaced fluids; in particular the theory holds for M = o(Ca−1/3). We formulate the theory as a double asymptotic expansion in the small parameters ε and Ca1/3. The expansion in ε is uniform while that in Ca1/3 is not, necessitating the use of matched asymptotic expansions. The previous work of Bretherton (1961) is clarified and extended, and both the form and the constants in the effective boundary condition of Chouke, van Meurs & van der Poel (1959) and of Saffman & Taylor (1958) are determined.

scholarly journals Effective coastal boundary conditions for tsunami wave run-up over sloping bathymetry

2014 ◽  
Vol 21 (5) ◽  
pp. 987-1005 ◽  
Author(s):  
W. Kristina ◽  
O. Bokhove ◽  
E. van Groesen
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Shallow Water ◽  
Tsunami Wave ◽  
Nonlinear Phenomena ◽  
Tsunami Propagation ◽  
Effective Boundary ◽  
Daunting Task ◽  
Effective Boundary Condition ◽  
Run Up

Abstract. An effective boundary condition (EBC) is introduced as a novel technique for predicting tsunami wave run-up along the coast, and offshore wave reflections. Numerical modeling of tsunami propagation in the coastal zone has been a daunting task, since high accuracy is needed to capture aspects of wave propagation in the shallower areas. For example, there are complicated interactions between incoming and reflected waves due to the bathymetry and intrinsically nonlinear phenomena of wave propagation. If a fixed wall boundary condition is used at a certain shallow depth contour, the reflection properties can be unrealistic. To alleviate this, we explore a so-called effective boundary condition, developed here in one spatial dimension. From the deep ocean to a seaward boundary, i.e., in the simulation area, we model wave propagation numerically over real bathymetry using either the linear dispersive variational Boussinesq or the shallow water equations. We measure the incoming wave at this seaward boundary, and model the wave dynamics towards the shoreline analytically, based on nonlinear shallow water theory over bathymetry with a constant slope. We calculate the run-up heights at the shore and the reflection caused by the slope. The reflected wave is then influxed back into the simulation area using the EBC. The coupling between the numerical and analytic dynamics in the two areas is handled using variational principles, which leads to (approximate) conservation of the overall energy in both areas. We verify our approach in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia.

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