Dimensionless Phase Velocity Research Articles

Propagation of torsional surface waves in a double porous layer lying over a Gibson half space

The present study deals with the behavior of torsional surface waves when they propagate through an inhomogeneous fluid saturated porous double layers over a dry sandy inhomogeneous Gibson half space. The inhomogeneities of the porous layers are taken as quadratic and exponential variation with depth in the density, elastic moduli and initial stress. In the half space it varies linearly in the elastic moduli and initial stress. For simplicity of the problem, we have used the separation of variable technique. The dispersion equation has been derived with boundary conditions and solved by an iterative method (Newton Raphson method). We have also converted our dynamical equations into non-dimensional form. It has been observed from the numerical validation of the proposed model that the presence of the initial stress and inhomogeneity of both media affect significantly the phase velocity of torsional surface waves. The effect of initial stress, inhomogeneity parameters, depth ratio, sandy parameter, Biot׳s gravity parameter and porosity of the layer on the dimensionless phase velocity of the torsional surface waves are demonstrated graphically with respect to the non-dimensional wave number kH1 (where k is the wave number and H1 is the thickness of the second porous layer).

Influence of Heterogeneity on the Propagation Behavior of Love-Type Waves in a Layered Isotropic Structure

AbstractThis paper aims to investigate the propagation of Love-type waves in the double layers lying over an isotropic homogeneous half-space. The two layers have been considered of finite width and a different sort of heterogeneities. Debye asymptotic expansion has been used to obtain the closed form of the dispersion equation. It has been found that the heterogeneity and width ratio of layers have a considerable effect on the phase velocity of Love-type waves. In particular, increment in the width ratio decreases the phase velocity of Love-type waves. The established dispersion relation for this study agrees well with the classical results. The comparative study of the present problem with a case in which double layers are homogeneous has been made. Results have been represented graphically, and it has been observed that the dimensionless phase velocity decreases with the increase in dimensionless wave number. It has also been examined that the heterogeneity parameter of uppermost layer and intermediate...

SH-wave velocity in a fiber-reinforced anisotropic layer overlying a gravitational heterogeneous half-space

Purpose – The purpose of this paper is to investigate the existence of SH-waves in fiber-reinforced layer placed over a heterogeneous elastic half-space. Design/methodology/approach – The heterogeneity of the elastic half-space is caused by the exponential variations of density and rigidity. As a special case when both the layers are homogeneous, the derived equation is in agreement with the general equation of Love wave. Findings – Numerically, it is observed that the velocity of SH-waves decreases with the increase of heterogeneity and reinforced parameters. The dimensionless phase velocity of SH-waves increases with the decreases of dimensionless wave number and shown through figures. Originality/value – In this work, SH-wave in a fiber-reinforced anisotropic medium overlying a heterogeneous gravitational half-space has been investigated analytically and numerically. The dispersion equation for the propagation of SH-waves has been observed in terms of Whittaker function and its derivative of second degree order. It has been observed that on the removal of heterogeneity of half-space, and reinforced parameters of the layer, the derived dispersion equation reduces to Love wave dispersion equation thereby validates the solution of the problem. The equation of propagation of Love wave in fiber-reinforced medium over a heterogeneous half-space given by relevant authors is also reduced from the obtained dispersion relation under the considered geometry.

A Dispersive Nonlocal Model for In-Plane Wave Propagation in Laminated Composites With Periodic Structures

In this paper, the problem of in-plane wave propagation with oblique incidence of the wave in an isotropic bilaminated composite under perfect contact between the layers and periodic distribution between them is studied. Based on an asymptotic dispersive method for the description of the dynamic processes, the dispersion equations were derived analytically from the average model. Numerical examples show that the dispersion curves obtained from the present model agree with the exact solutions for a range of wavelengths. Detailed numerical simulations are provided to illustrate graphically the phase and group velocities. Such illustrations allow the identification and comparison of the effects of the unit cell size, wave number and incident angle. It was observed that, as the incident angle increases, the dimensionless quasi-longitudinal phase velocity increases, and the dimensionless quasi-shear phase velocity decreases. In addition, the phase and group velocities decrease as the size of the unit cell increases. The frequency band structure, as a function of the wave-vector components is calculated.

Effect of gravity field, initial stress and rotation on the S-waves propagation in a non-homogeneous anisotropic medium with magnetic field

In this paper, we investigated the propagation of shear wave under the influences of the magnetic field, gravitational field, rotation and initially stressed in anisotropic non-homogeneous incompressible elastic medium. It is well known in the literature that the earth medium is not at all initial stress free and homogeneous throughout, but it is initially stressed and non-homogeneous. The frequency equation that determines the velocity of the shear waves has been obtained. Keeping these things in mind, we have discussed the frequency equation in an initially stressed, non-homogeneous medium with gravity field, magnetic field, and rotation. It has been observed that the inhomogeneity parameter, gravity field, magnetic field, and the initial stress play an important role in the propagation of S-waves propagation. The numerical values on the dimensionless phase velocities are calculated and presented graphically to illustrate the dependences upon gravity field, initial stress, magnetic field, anisotropy and rotation comparatively. The results indicate that the effects of gravity field, initial stress, magnetic field, anisotropy, and rotation are very pronounced.

Propagation of Love waves in a heterogeneous medium over an inhomogeneous half-space under the effect of point source

The present paper deals with the effect of point source on the propagation of Love wave in a heterogeneous layer and inhomogeneous half-space. The upper heterogeneous layer is caused by consideration of exponential variation in rigidity and density. Also in half-space inhomogeneity parameters associated to rigidity, internal friction and density are assumed to be functions of depth. The dispersion equation of Love wave has been obtained by using Green’s function technique. As a special case when the upper layer and lower half-space are homogeneous, our computed equation coincides with the general equation of Love wave. The propagation of Love waves are influenced by inhomogeneity parameters. The dimensionless phase velocity has been plotted against the dimensionless wave number for different values of inhomogeneity parameters. We have observed that the velocity of wave increases with the increase of inhomogeneity parameters.

Shear Wave Propagation in Multilayered Medium including an Irregular Fluid Saturated Porous Stratum with Rigid Boundary

The present investigation is concerned with the study of propagation of shear waves in an anisotropic fluid saturated porous layer over a semi-infinite homogeneous elastic half-space lying under an elastic homogeneous layer with irregularity present at the interface with rigid boundary. The rectangular irregularity has been taken in the half-space. The dispersion equation for shear waves is derived by using the perturbation technique followed by Fourier transformation. Numerically, the effect of irregularity present is analysed. It is seen that the phase velocity is significantly influenced by the wave number and the depth of the irregularity. The variations of dimensionless phase velocity against dimensionless wave number are shown graphically for the different size of rectangular irregularities with the help of MATLAB.

Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space

The present paper is devoted to study the propagation of Love wave in a piezoelectric layer overlying an inhomogeneous half-space. This paper deals with two different piezoelectric layers, one is an electrically open and another is an electrically short circuit. As mathematical tools, the method of separation of variables and Whittaker’s function are applied to obtain the dispersion equation of Love wave. In a particular case the dispersion equation reduces to the classical equation of Love wave when the layer is not piezoelectric and half-space is homogeneous. The numerical values of the dimensionless phase velocities are calculated and presented graphically to illustrate the effects of inhomogeneity, piezoelectricity and dielectric constants. It is observed that the phase velocities decrease with the increase of inhomogeneity parameters and electricity constant. It is also found that the phase velocity increases with the increases of the dielectric constant. Graphical user interface software has been developed by using MATLAB software to generalize the effect of various parameters.

HOMOTOPY SIMULATION OF TWO-PHASE THERMO-HEMODYNAMIC FILTRATION IN A HIGH PERMEABILITY BLOOD PURIFICATION DEVICE

A two-phase thermo-hydrodynamic model is presented for transport in the vertical chamber of a porous media blood filtration device. A non-Darcy drag force formulation was employed. The Marble–Drew fluid–particle suspension model was used to simulate the plasma phase and the suspension (erythrocyte) particle phase. The non-dimensional transport equations were solved using a semi-computational procedure known as the homotopy analysis method (HAM). With the judicious use of the auxiliary parameter ℏ, HAM affords a powerful mechanism to adjust and control the convergence region of solution series. This method provides an efficient approximate analytical solution with high accuracy, minimal calculation and avoidance of physically unrealistic assumptions. Detailed computations are presented for the effects of Grashof number (Gr), momentum inverse Stokes number (Skm), Darcy number (Da), Forchheimer number (Fs), particle loading parameter (PL), buoyancy parameter (B) and temperature inverse Stokes number (SkT) on the dimensionless fluid phase velocity (U), dimensionless particle phase velocity (Up), dimensionless fluid phase temperature (Φ) and the dimensionless temperature of particle phase (Φp). A Prandtl number of 25 was used to simulate blood at room temperature. Excellent correlation was obtained between the HAM and numerical shooting quadrature solutions. The results indicated that there is a strong decrease in fluid phase velocities with increasing Darcian (first order) drag and second-order Forchheimer drag, and a weaker reduction in particle phase velocity field. Applications of the study include porous media bio-filtration devices and dialysis simulations.

Torsional wave propagation in non‐homogeneous layer between non‐homogeneous half‐spaces

SUMMARYThe study of surface wave in a layered medium has their possible application in geophysical prospecting. In the present work, dispersion equation for torsional wave in an inhomogeneous isotropic layer between inhomogeneous isotropic half‐spaces has been derived. Two cases are discussed separately for torsional wave propagation in inhomogeneous layer between homogeneous and non‐homogeneous half‐spaces, respectively. Further, two possible modes for torsional wave propagation are obtained in case of inhomogeneous layer sandwiched between non‐homogeneous half‐spaces. Closed form solutions for displacement in the layer and half‐spaces are obtained in each case. The study reveals that the layer width, layer inhomogeneity, frequency of inhomogeneity, as well as inhomogeneity in the half‐space has significant effect on the propagation of torsional surface waves. Displacement and implicit dispersion equation for torsional wave velocities are expressed in terms of Heun functions and their derivatives. Effects of inhomogeneity on torsional wave velocity are also discussed graphically by plotting the dimensionless phase velocity against dimensionless and scaled wave number for different values of inhomogeneity parameter. Copyright © 2012 John Wiley & Sons, Ltd.

Propagation of shear waves in a poroelastic layer constrained between two elastic layers

In this paper, we investigated the propagation of shear waves in a transversely isotropic poroelastic layer constrained between two elastic layers. Following Biot’s theory, the dispersion equation for shear waves in this structure was derived. The numerical values on the dimensionless phase velocities are calculated and presented graphically to illustrate the dependences upon geometry, anisotropy and porosity comparatively. It is observed that the phase velocities increase with the increase of the porosity and the decrease of the anisotropy. In addition, the geometry in this structure has a significant effect on the phase velocity of the shear waves.

Propagation of torsional waves in an inhomogeneous layer over an inhomogeneous half-space

The present paper is concerned with the study of propagation of torsional waves in an inhomogeneous isotropic layer whose material properties vary harmonically with a space variable, lying over a semi-infinite inhomogeneous isotropic half-space. The closed form solutions for the displacement in the layer and half-space are obtained separately. The dimensionless phase velocity has been plotted against dimensionless wave number and scaled wave number for different values of inhomogeneity parameters. The effects of inhomogeneity have been shown in the dispersion curves using 2D and 3D plot.

Analysis of the numerical dispersion of waves in saturated poroelastic media

This work presents the analysis of the numerical dispersion properties of a finite element method used to approximate the solution of the equations of motion for a fluid saturated porous elastic solid (a Biot medium) in the two-dimensional case and the low frequency regime. The finite element method employed comprises a nonconforming rectangular element for the approximation of each component of the displacement vector in the solid phase, and the Raviart–Thomas–Nedelec mixed finite element space of zero order for the fluid phase. The study is carried out by constructing and analyzing analytic and numerical dimensionless dispersion relations, and by evaluating derived quantities such as dimensionless phase and group velocities and dimensionless attenuation for the three type of waves predicted by Biot’s equations of motion as a measure of the numerical distortion. It is observed that the finite element procedure introduces both numerical dispersion and anisotropy in all three waves in a similar fashion; however, only the slow wave displayed a significant frequency dependent behavior. It was also observed that the loss of accuracy is more important for the dimensionless attenuation than for the dimensionless group or phase velocities. The analysis presented yields lower bounds for the number of points per wavelength needed to reach a desired accuracy in the dimensionless phase and group velocities and attenuation coefficients in Biot media.

Dispersion analysis of a nonconforming finite element method for the three-dimensional scalar and elastic wave equations

We investigate the dispersive properties of a nonconforming finite element method (NC-method) to solve the three-dimensional scalar and elastic wave equations. The study is performed by constructing and analyzing the dispersion relations, and by evaluating derived quantities such as dimensionless phase and group velocities. The behaviour of the present algorithm is compared with that of a conforming finite element method (C-method) with the same order of spatial approximation. It is observed that the NC-method introduces less numerical anisotropy and dispersion than the C-method, allowing to work with fewer than 10 points per wavelength with reasonably small relative errors. On the other hand, the computational cost of the NC-method is higher than that of the C-method. However, the present NC-method is suitable for domain decomposed parallel algorithms; therefore the fact that the NC-method needs more computing time may be partially compensated by its lower transmission load.

Dispersion analysis of a non‐conforming finite element method for the Helmholtz and elastodynamic equations

AbstractWe investigate the dispersive properties of a non‐conforming finite element method to solve the two‐dimensional Helmholtz and elastodynamics equations. The study is performed by deriving and analysing the dispersion relations and by evaluating the derived quantities, such as the dimensionless phase and group velocities. Also the phase difference between exact and numerical solutions is investigated. The studied method, which yields a linear spatial approximation, is shown to be less dispersive than a conforming bilinear finite element method in the two cases shown herein. Moreover, it almost halves the number of points per wavelength necessary to reach a given accuracy when calculating the mentioned velocities in both cases here presented. Copyright © 2003 John Wiley & Sons, Ltd.

Shear waves in a fluid saturated elastic plate

In the present context, we consider the propagation of shear waves in the transverse isotropic fluid saturated porous plate. The frequency spectrum for SH-modes in the plate has been studied. It is observed that the frequency of the propagation is damped due to the two-phase character of the porous medium. The dimensionless phase velocities of the shear waves have also been calculated and presented graphically. It is interesting to note that the frequency and phase velocity of shear waves in porous media differ significantly in comparison to that in isotropic elastic media.

The Gravitational Stability of the Interface between Two Electrorheological Fluids

The propagation of gravitational waves between two semi-infinite superposed electrorheological fluids is investigated. Criteria for stability in the presence of a vertical or a horizontal electric field are examined. A modified Chandrasekhar transcendental dispersion relation is obtained. Stability analysis is discussed throughout the case of a small dimensionless phase velocity. The influence of the nondimensional parameters on the growth rate of the Rayleigh–Taylor instability is discussed. The behavior of the growth rate with respect to the stratified viscosity, as well as the stratified retardation time in the presence or absence of an electric field, is investigated. It is shown that a vertical electric field retards the stabilizing influence of both the retardation time and the viscosity, while the presence of a tangential field suppresses the destabilizing influence of the stratified density. A mechanism that shields against the destabilizing influence of the vertical electric field is observed. This shielding mechanism is due to viscous and elastic effects.

Characteristics of two-dimensional waves on a falling liquid film

Regularly spaced two-dimensional waves are excited with input of controlled disturbances to a water film falling down a vertical surface, and phase velocities u w , wave peak heights h p and wave separations λ are measured for a wide range of flow rates ( Γ = 20 ∼ 86 × 10 −6 m 2/s) and temperatures (5 ∼ 23°C) (corresponding to Reynolds numbers Re = 14 ∼ 90). The variables h p and u w are well described by the relations N uw = 1.13 K F 0.02 N λ 0.31 Re 0.37 and N hp = 0.49 K F 0.044 N λ 0.34 Re 0.46, and also by N hp = 0.425K F 0.019N uw 5 4 and N uw = 1.13K F 0.026N V 1 2 where N uw, N hp. N λ. K F. N V and Re are respectively dimension-less phase velocity, wave peak height, wave separation, physical properties group, volume of liquid carried on a wave and Reynolds number. These correlations show that the two variables λ and Γ can be replaced by one variable, V or u w , and they therefore indicate that the waves are isolated from each other. Comparisons between these correlations and data obtained from the extensive literature on the subject are discussed in relation to the surface elasticity. Shadowgrams of waves are also presented.

Coherence measurements in synthetic turbulent boundary layers

Synthetic turbulent boundary layers were constructed on a flat plate by generating systematic moving patterns of turbulent spots in a laminar flow. The experiments were carried out in a wind tunnel at a Reynolds number based on plate length of 1.7 × 106. Spots were generated periodically in space and time near the leading edge to form a regular hexagonal pattern. The disturbance mechanism was a camshaft that displaced small pins momentarily into the laminar flow at frequencies up to 80 Hz. The main instrumentation was a rake of 24 single hot wires placed across the flow in a line parallel to the surface. The main measured variable was local intermittency; i.e. the probability of observing turbulent flow at a particular point in space and time. The results are reported in numerous (x, z, t)-diagrams showing the evolution of various synthetic flows along the plate. The dimensionless celerity or phase velocity of the large eddies was found to be very nearly 0.88, independent of eddy scale. All patterns with sufficiently small scales eventually showed loss of coherence as they moved downstream. A novel phenomenon called eddy transposition was observed in several flows that contained appreciable laminar regions. The original large eddies were replaced by new eddies at new positions, intermediate to the original ones, while preserving the hexagonal pattern. The present results, together with some empirical properties of a turbulent spot, were used to estimate the best choice of scales for constructing a synthetic boundary layer suitable for detailed study as a model for a natural flow. The values recommended are: spanwise period/thickness ≈ 2.5, streamwise period/thickness ≈ 8.0.

Effect of atmospheric stability on the growth of surface gravity waves

In this paper we study the effect of atmospheric stability on the growth of surface gravity waves. To that end we numerically solved the Taylor-Goldstein equation for wind profiles which deviate from a logarithmic form because stratification affects the turbulent momentum transport. Using Charnock's relation for the roughness height z 0 of the wind profile, it is argued that the growth rate of the wave depends on the dimensionless phase velocity c/u * (where u * is the friction velocity) and a measure of the effect of atmospheric stability, namely the dimensionless Obukhov length gL/u * 2, whereas it only depends weakly on gz t /u * 2 (where z t is the roughness height of the temperature profile). Remarkably for a given value of u * /c, the growth rate is larger for a stable stratification (L > 0) than for an unstable one (L < 0). We explain why this is the case. If, on the other hand, one considers the growth rate as a function of c/U 10 (where U 10 is the windspeed at 10 m), the situation reverses for c/U 10 < 1. For practical application in wave prediction models, we propose a new parameterization of the growth rate of the waves which is an improvement of the Snyder et al. (1981) proposal because the effect of stability is taken into account.