Wave Trapping by Dual Porous Barriers Near a Wall in the Presence of Bottom Undulation

Trapping of oblique surface gravity waves by dual porous barriers near a wall is studied in the presence of step type varying bottom bed that is connected on both sides by water of uniform depths. The porous barriers are assumed to be fixed at a certain distance in front of a vertical rigid wall. Using linear water wave theory and Darcy's law for flow past porous structure, the physical problem is converted into a boundary value problem. Using eigenfunction expansion in the uniform bottom bed region and modified mild-slope equation in the varying bottom bed region, the mathematical problem is handled for solution. Moreover, certain jump conditions are used to account for mass conservation at slope discontinuities in the bottom bed profile. To understand the effect of dual porous barriers in creating tranquility zone and minimum load on the sea wall, reflection coefficient, wave forces acting on the barrier and the wall, and surface wave elevation are computed and analyzed for different values of depth ratio, porous-effect parameter, incident wave angle, gap between the barriers and wall and slope length of undulated bottom. The study reveals that with moderate porosity and suitable gap between barriers and sea wall, using dual barriers an effective wave trapping system can be developed which will exert less wave force on the barriers and the rigid wall. The proposed wave trapping system is likely to be of immense help for protecting various facilities/infrastructures in coastal environment.

On the Importance of the Stokes-Brinkman Equations for Computing Effective Permeability in Karst Reservoirs

Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models.Herein,we consider the problem of computing the effective permeability of rock samples based on high-resolution 3DCT scans containingmillions of voxels.We use the Stokes-Brinkman equations in the entire domain,covering regions of free flow governed by the Stokes equations,porous Darcy flow,and transitions between them.The presence of different length scales and large(ten orders of magnitude)contrasts in permeability leads to highly ill-conditioned linear systems of equations,which are difficult to solve.To obtain a problem that is computationally tractable,we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models.We find that,in terms of effective permeability,the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability.All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows,depending on if there do or do not exist percolating free-flow regions.The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.

Finite Element Analysis of Maxwell’s Equations in Dispersive Lossy Bi-Isotropic Media

In this paper,the time-dependent Maxwell’s equations used to modeling wave propagation in dispersive lossy bi-isotropic media are investigated.Existence and uniqueness of the modeling equations are proved.Two fully discrete finite element schemes are proposed,and their practical implementation and stability are discussed.

Numerical Study on Viscous Fingering Using Electric Fields in a Hele-Shaw Cell

We investigate the nonlinear dynamics of amoving interface in aHele-Shaw cell subject to an in-plane applied electric field.We develop a spectrally accurate numerical method for solving a coupled integral equation system.Although the stiffness due to the high order spatial derivatives can be removed using a small scale decomposition technique,the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands.We remove this physically imposed stiffness by employing a rescaling scheme,which accelerates the slow dynamics and reduces the computational cost.Our nonlinear results reveal that positive currents restrain finger ramification and promote the overall stabilization of patterns.On the other hand,negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region.When no fluid is injected,and a negative current is utilized,the interface tends to approach the origin and break up into several drops.We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law(t∗−t)b,where t∗is the estimated pinch-off time.

A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem

In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions.Furthermore,the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution.Therefore,regularization is necessary in order to obtain a stable solution.Numerical results for several benchmark test examples are presented and discussed.

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval.Using an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space.We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear systems.We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

UnsteadyMHD Non-Darcian Flow Over a Vertical Stretching Plate Embedded in a Porous Medium with Thermal Non-EquilibriumModel

An analysis is performed to study the influence of local thermal nonequilibrium(LTNE)on unsteadyMHDlaminar boundary layer flowof viscous,incompressible fluid over a vertical stretching plate embedded in a sparsely packed porous medium in the presence of heat generation/absorption.The flow in the porous medium is governed by Brinkman-Forchheimer extended Darcy model.A uniform heat source or sink is presented in the solid phase.By applying similarity analysis,the governing partial differential equations are transformed into a set of time dependent non-linear coupled ordinary differential equations and they are solved numerically by Runge-Kutta Fehlberg method along with shooting technique.The obtained results are displayed graphically to illustrate the influence of different physical parameters on the velocity,temperature profile and heat transfer rate for both fluid and solid phases.Moreover,the numerical results obtained in this study are compared with the existing literature in the case of LTE and found that they are in good agreement.

The Plane WavesMethod for Numerical Boundary Identification

We study the numerical identification of an unknown portion of the boundary on which either the Dirichlet or the Neumann condition is provided from the knowledge of Cauchy data on the remaining,accessible and known part of the boundary of a two-dimensional domain,for problems governed by Helmholtz-type equations.This inverse geometric problem is solved using the plane wavesmethod(PWM)in conjunction with the Tikhonov regularizationmethod.The value for the regularization parameter is chosen according toHansen’s L-curve criterion.The stability,convergence,accuracy and efficiency of the proposedmethod are investigated by considering several examples.

A Moving Pseudo-Boundary MFS for Three-Dimensional Void Detection

We propose a new moving pseudo-boundary method of fundamental solutions(MFS)for the determination of the boundary of a three-dimensional void(rigid inclusion or cavity)within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary.The algorithm for imaging the interior of the medium also makes use of radial spherical parametrization of the unknown star-shaped void and its centre in three dimensions.We also include the contraction and dilation factors in selecting the fictitious surfaces where the MFS sources are to be positioned in the set of unknowns in the resulting regularized nonlinear leastsquares minimization.The feasibility of this new method is illustrated in several numerical examples.