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

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.

Two-Parametric Analysis of Anti-Plane Shear Deformation of a Coated Elastic Half-Space

The anti-plane shear deformation problem of a half-space coated by a soft or a stiff thin layer is considered. The two-term asymptotic analysis is developed motivated by the scaling for the displacement and stress components obtained from the exact solution of a model problem for a shear harmonic load. It is shown that for a rather high contrast in stiffness of the layer and the half-space Winkler-type behaviour appears for a relatively soft coating, while for a relatively stiff one, the equations of plate shear are valid. For low contrast, an alternative approximation is suggested based on the reduced continuity conditions and the fact that the applied load may be transmitted to the interface. In case of a stiff layer, a simpler problem for a homogeneous half-space with effective boundary condition is also formulated, modelling the effect of the coating, while for a relatively soft layer a uniformly valid approximate formula is introduced.

Thermal Turbulent Flow in Leading Edge Grooved and Conventional Tilting Pad Journal Bearing Segments—A Comparative Study

A comparative study between a conventional- and leading edge grooved (LEG) tilting pad journal bearing (TPJB) segment is performed. The developed model uses the Shear Stress Transport (SST) turbulence model, coupled with the energy equation and a partial differential equation for the fluid domain mesh displacement to predict the thermal flow characteristics. Instead of using an effective boundary condition to determine the inlet temperature of the LEG pad and excluding the additional LEG portion, as is common practice, the whole geometry of the LEG is modeled. Several sizes of the LEG portion is investigated and it is shown to have quite a small influence on pressure, temperature, film thickness and turbulence intensity. Moreover, the results also show that the conventional pad gives rise to higher levels of turbulence in the mid plane compared to its LEG counterpart, while the latter has a marginally higher value of turbulence when the volume average value is computed. The maximum value of turbulence is however present in the conventional model.

An approximate secular equation of Rayleigh waves in an elastic half-space coated by a thin weakly inhomogeneous elastic layer

In this paper, the propagation of Rayleigh waves in a homogeneous isotropic elastic half-space coated with a thin weakly inhomogeneous isotropic elastic layer is investigated. The material parameters of the layer is assumed to depend arbitrarily continuously on the thickness variable. The contact between the layer and the half space is perfectly bonded. The main purpose of the paper is to establish an approximate secular equation of the wave. By applying the effective boundary condition method an approximate secular equation of second order in terms of the dimensionless thickness of the layer is derived. It is shown that the obtained approximate secular equation has good accuracy.

Zonally Symmetric Adjustment in the Presence of Artificial Relaxation

Numerical experiments, presented in a companion paper, have been performed in which the zonal-mean state of the stratosphere in a comprehensive, stratosphere-resolving, general circulation model is strongly relaxed (or “nudged”) toward the evolution of a reference sudden warming event in order to investigate its influence on the freely evolving troposphere below. Similar approaches have been used in a number of other studies. This raises the question of whether such an artificial relaxation induces the adiabatic and diabatic adjustments expected below the region of nudging, even in the absence of the stratospheric wave driving responsible for the reference event. Motivated by this question, the zonally symmetric quasigeostrophic diabatic response to zonal forces (representing wave driving) in a system nudged to a time-dependent reference state is studied. In the presence of wave driving in the nudging region that differs from the reference state, the meridional mass circulation of the reference state is reproduced only in the region below the nudging up to a correction that is inversely proportional to the strength of the nudging. The anomalous circulation is confined because of an effective boundary condition at the interface of the nudging layer. The nudging also produces an artificial “sponge-layer feedback” immediately below the region of the nudging in response to differences in the tropospheric wave driving. The strength of this artificial feedback is closely related to the strength of the effective boundary condition; however, the time scale required for the sponge-layer feedback to be established is typically much longer than that required for the confinement.

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

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.

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

An effective boundary condition (EBC) is introduced as a novel technique to predict tsunami wave run-up along the coast and offshore wave reflections. Numerical modeling of tsunami propagation at the coastal zone has been a daunting task since high accuracy is needed to capture aspects of wave propagation in the more shallow 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 sloping bathymetry. 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.