scholarly journals Effective boundary condition for the reflection of shear waves at the periodic rough boundary of an elastic body

2018 ◽  
Vol 40 (4) ◽  
pp. 303-323 ◽  
Author(s):  
Agnès Maurel ◽  
Jean-Jacques Marigo ◽  
Kim Pham
Keyword(s):  
Boundary Condition ◽  
Elastic Body ◽  
Shear Waves ◽  
Homogenization Method ◽  
Rough Boundary ◽  
The Real ◽  
Effective Boundary ◽  
Effective Boundary Condition ◽  
Flat Edge ◽  
The Time Domain

We present a homogenization method to treat the problem of the reflection of waves at the free boundary of an elastic body, the edge being structured periodically at the subwavelength scale. The problem is considered for shear waves and the wave equation in the time domain is considered. In the homogenized problem, a boundary condition at an equivalent flat edge is obtained, which links the normal stress to its derivatives, instead of the usual traction free condition. The problem of the position of the equivalent flat boundary with respect to the real roughnesses is addressed and this is done considering the equation of energy conservation in the homogenized problem and considering the accuracy of the homogenized solution when compared to the real one.

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.

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.

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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