Increasing Wave Number Research Articles

Transverse Vibration and Wave Propagation of Functionally Graded Nanobeams with Axial Motion

Based on the nonlocal elasticity theory, the vibration behavior and wave propagation of functionally graded Euler nanobeams with axial motion are investigated. Assuming that the axial velocity of nanobeams is a constant and the graded material parameters vary along the thickness direction in terms of power index, we apply the hypothesis of neutral plane to derive the axial displacement of geometry surface caused by non-uniformity of graded materials. Effects of graded index, axial velocity and nonlocal scale parameter on natural frequencies are analyzed through numerical examples. Also, the relationship between wave propagation frequency, wave velocity and wave number is revealed. The complex mode method is utilized to solve the governing equation, and the natural frequencies and wave velocity are obtained accordingly. To solve the derived transcendental equations, the Newton iteration method is used and a detailed solution flowchart is provided. For vibration, with the increase of non-dimensional axial velocity, non-dimensional nonlocal parameter and gradient index, natural frequencies decrease. For wave propagation, with the increase of wave number, frequency of wave propagation increases. However, with the increase of wave number, velocity of wave propagation decreases. Moreover, increasing the gradient index causes a decrease in frequency and velocity of wave propagation, increasing the axial velocity causes an increase in frequency and velocity of wave propagation, and increasing the nonlocal parameter causes a decrease in frequency and velocity of wave propagation.

Dispersion and Self-Modulation of Waves Propagating in a Solid with Dislocations

In the paper, the basic equations describing ultrasonic wave propagation in a medium with dislocations are derived. Dispersion relations are given under the assumption that dislocations oscillate without damping, i.e., the dislocation component of the general system is conservative. It is shown that ultrasonic wave propagation is characterized by two dispersion branches (acoustic and optical). As the wave number increases, the phase velocity of the wave belonging to the acoustic branch decreases asymptotically from a finite value to zero, while the velocity of the wave belonging to the optical branch decreases asymptotically from infinity to a finite value corresponding to the longitudinal wave velocity. The Nighthill criterion is applied to study modulation instability. The form of the wave packets into which a quasi-harmonic wave is divided due to modulation instability is determined. There can be both periodic stationary wave envelopes and a solitary stationary wave envelope. It is found how the height and width of the wave packet formed due to self-modulation of a quasi-harmonic wave correlate with the basic characteristics of the dislocation structure.

Critical Review of Preconditioners for Helmholtz Equation and their Spectral Analysis

Objectives: The study reviews the iterative solvers for solving Helmholtz equation. It also surveys the preconditioners for Krylov subspace methods for discrete Helmholtz equation. Numerical experiments based conclusions are drawn, after a thorough comparison of various preconditioners. Methods/Statistical Analysis: Five point finite difference scheme for two-dimensional Helmholtz equation is employed to obtain discrete analogue of Helmholtz equation. Krylov subspace method for is considered for iterations. Matrix based preconditioner ILU with variety of fill-in and operator based preconditioner complex shifted Laplace preconditioner (CSLP) are tested. Different shifts are proposed and experimented in CSLP. Optimization of shifts is recommended on basis of results. Spectral analysis is performed to validate results. Findings: For small wave number problem, ILU type preconditioned performs better and is fairly comparable with CSLP. However, increasing wave number makes unaffordable in terms of computational cost. The pure imaginary shift (0,1) and complex shift (1, / 4 π ) costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly. Approximate inversion may affect iterations, but will save solve time for large problems. Extensive spectral analysis, validating results, highlights this work. Application/Improvements: Dependency upon wave number is decreased with this recommended choice of shifts in CSLP. The number of iterations increases at very low rate when wave number is increased at rate of 2. Keywords: Helmholtz Equation, Iterative Methods, Krylov Subspace Methods, Preconditioners, Spectral Analysis

A numerical study on non-homogeneous model for the conjugate-mixed convection of a Cu-water nanofluid in an enclosure with thick wavy wall

Conjugate heat transfer due to the mixed convection and conduction of a Cu-water nanofluid in a thick-wall enclosure is conducted numerically based on the non-homogeneous model for nanofluids. We consider an enclosure with a thick wavy hot left side wall with the right vertical wall being allowed to move vertically downwards to develop a shear driven flow. This in combination with the horizontal temperature gradient leads to a mixed convection within the enclosure. The wavy geometry is transformed to a square domain by the co-ordinate transformation method and the governing equations along with the specified boundary conditions are solved using the finite volume method. The Richardson number and Reynolds number, which governs the mixed convection, are varied up to a moderate range at different choices of the nanoparticle volume fraction and its size, solid-to-fluid conductivity ratio, wave length and amplitude of the wavy interface. The heat transfer characteristics of the nanofluid is analyzed by evaluating the entropy generation and Bejan number. Results show that the impact of the dispersion of nanoparticles on the mixed convection is pronounced for a higher range of the Richardson number and nanoparticle volume fraction. The heat transfer and entropy generation both enhances with the increase of wave amplitude and wave number, however, the heat transfer enhancement rate dominates the entropy generation rate. Entropy generation due to heat transfer is more significant than that of fluid friction irreversibility for all the cases addressed here.

Temperature effects on the propagation and Landau damping of the dust surface waves

The integral of the kinetic electrostatic surface wave dispersion relation is evaluated to determine the wave frequency and the corresponding Landau damping rate for the surface mode of dust ion-acoustic waves including the effect of ion temperature by using the general perturbation and the transverse truncation methods. It is demonstrated that the increase in ion temperature enhances the wave frequency. The effect of ion temperature is more prominent in the range of large wave numbers, and the wave frequency increases in proportion to a quarter power of the ion temperature in the realm of large wave number. The Landau damping of the dust ion-acoustic surface wave is found to be suppressed as the ion temperature increases. However, the effect of ion temperature becomes negligible as the wave number increases. The effects of dust charge and electron density on the Landau damping rate are also presented in this work.

Turbulent enhancement of radar reflectivity factor for polydisperse cloud droplets

Abstract. The radar reflectivity factor is important for estimating cloud microphysical properties; thus, in this study, we determine the quantitative influence of microscale turbulent clustering of polydisperse droplets on the radar reflectivity factor. The theoretical solution for particulate Bragg scattering is obtained without assuming monodisperse droplet sizes. The scattering intensity is given by an integral function including the cross spectrum of number density fluctuations for two different droplet sizes. We calculate the cross spectrum based on turbulent clustering data, which are obtained by the direct numerical simulation (DNS) of particle-laden homogeneous isotropic turbulence. The results show that the coherence of the cross spectrum is close to unity for small wave numbers and decreases almost exponentially with increasing wave number. This decreasing trend is dependent on the combination of Stokes numbers. A critical wave number is introduced to characterize the exponential decrease of the coherence and parameterized using the Stokes number difference. Comparison with DNS results confirms that the proposed model can reproduce the rp3-weighted power spectrum, which is proportional to the clustering influence on the radar reflectivity factor to a sufficiently high accuracy. Furthermore, the proposed model is extended to incorporate the gravitational settling influence by modifying the critical wave number based on the analytical equation derived for the bidisperse radial distribution function. The estimate of the modified model also shows good agreement with the DNS results for the case with gravitational droplet settling. The model is then applied to high-resolution cloud-simulation data obtained from a spectral-bin cloud simulation. The result shows that the influence of turbulent clustering can be significant inside turbulent clouds. The large influence is observed at the near-top of the clouds, where the liquid water content and the energy dissipation rate are sufficiently large.

Short-Crested Wave-Current Forces on Composite Bucket Foundation for an Offshore Wind Turbine

An analytical solution for the diffraction of short-crested incident wave with uniform current on a composite bucket foundation is derived. The influences of the uniform current on wave frequency, wave run-up, wave force, and inertia and drag coefficients on the composite bucket foundation are investigated. The numerical results indicate that the current incident angle and current velocity have significant effects on the short-crested wave run-up, wave force, and inertia and drag coefficients on the composite bucket foundation. For a fixed wave number, the wave frequency, wave run-up, wave forces, and inertia and drag coefficients obviously increase with the increase of current velocity when the relative angle between the current velocity and wave propagation direction is smaller than 90°, whereas they obviously decrease when the relative angle is larger than 90°. It also can be found that the effect of wave-current interaction on the short-crested wave increases with the increase of the total wave number and the decrease of the water depth. The short-crested wave forces will be significantly increased when the current incident angle parallels to the direction of the wave propagating. Therefore, the short-crested wave-current load should be carefully considered in the design of the composite bucket foundation for an offshore wind turbine.

Puffing in planar buoyant plumes: BiGlobal instability analysis and experiments

The present study investigates the puffing behaviour of planar buoyant plumes by employing linear BiGlobal stability analysis and experiments. The BiGlobal instability characteristics of two-dimensional plumes have been explored using stability analysis and compared with the puffing behaviour of both rectangular plumes and square plumes obtained from experiments. In the parameter space investigated, which spans a Richardson number range $0.03<Ri<960$, instability analysis reveals that planar plumes exhibit BiGlobal instability only for varicose perturbations, while they remain stable for sinuous perturbations. The BiGlobal frequency and growth rates of the unstable varicose mode are used to obtain Strouhal number correlation and stability curves. An investigation into the effect of the spanwise wavenumber on BiGlobal instability indicates that planar plumes are more unstable to two-dimensional perturbations than to three-dimensional perturbations. An increase in the spanwise wavenumber tends to stabilize planar plumes without affecting their oscillation frequencies. Experiments suggest that the puffing frequencies in rectangular plumes closely follow the power law obtained from two-dimensional instability analysis while exhibiting a weaker dependence on inlet aspect ratio. To further explore the effect of aspect ratio on puffing behaviour, experiments have been carried out in plumes of aspect ratio 1, i.e. square plumes. Square plumes are found to be more stable and to exhibit higher puffing frequencies than rectangular plumes. The reasons for these differences in puffing dynamics between rectangular and square plumes have been explored from the phase-locked streamwise and spanwise flow visualizations. In addition to puffing, spanwise visualizations in both rectangular and square plumes show the presence of secondary flows at their corners, similar to their constant-density jet counterparts. Finally, from experiments, we deduced a new universal puffing frequency correlation with the hydraulic diameter as the length scale which eliminates the aspect ratio dependence, and is valid for both square and low-aspect-ratio rectangular plumes.

Thermo magnetic response on flexural vibration of nanofibers using Timoshenko beam theory

In this study, the flexural vibration of nanofibers with surface effect in the presence of thermo magnetic field is studied using Modified Timoshenko beam equation. The frequency equations are obtained for nanofiber under surface effect. Numerical calculations are performed and the characteristics of dispersion curves are analyzed. Graphs are drawn for phase velocity with increasing wave number at different radius of nanofibers. The investigation shows that as the wave number increases the corresponding phase velocities increases at different radius. The results are used in designing nanodevices and analysis.

Characteristics of Rayleigh wave propagation in orthotropic magneto-thermoelastic half-space: An eigen function expansion method

• This problem is more general as other problem with different thermoelastic models. • Effect of phase lag due to heat flux vector is more dominating in comparison with other phase lags. • The secular equation magnitude increases with an increasing of wave number. • Rayleigh wave velocity increases with the increase of wave number and magnetic field. • The attenuation coefficient decreases with the increase of wave number and magnetic field. In this article, we theoretically demonstrate the characteristics of Rayleigh surface wave propagation in a homogeneous and orthotropic thermoelastic half-space in the context of three-phase-lag model of generalized thermoelasticity. The influence of magnetic field on Rayleigh wave is analyzed in the framework of two-temperature model. A vector matrix differential equation is formed by employing normal mode analysis, which is then solved by the eigen function expansion method. The frequency equations in closed form are derived and the path of surface particles during Rayleigh wave propagation is found to be elliptical. The results show appreciable differences in phase velocity, attenuation coefficient and specific loss due to the presence of heat-flux phase-lag and is more dominating in comparison with other phase lags.

Noise Calculation Method for Deep Groove Ball Bearing With Considering Raceway Surface Waviness and Roller Size Error

Rolling bearing is a kind of important part of mechanical equipment, and the noise of the rolling bearing is also one of the important criteria for evaluating the quality of a rolling bearing. In the past, researches on rolling bearings are mainly on bearing vibration when people study the noise of bearings. Few studies have established rolling bearing’s noise model. In this paper, a mathematical model for calculating the inner ring’s axis center orbit and ball center’s trajectory of bearing is established based on the mechanics theory with considering raceway waviness and ball size error. Combined with the acoustics theory, a noise calculation model for deep grove ball bearing is established by taking the bearing inner ring and rolling balls as cylindrical sound source and spherical sound source respectively by using the single source compound method. The influences of waviness wave number, waviness amplitude, bearing speed, bearing load and ball size errors on bearing noise at a fixed measuring point are studied by numerical calculation. Results show that with the increase of the waviness wave number, the bearing SPL(Sound Pressure Level) will change with irregular way. With the increase of waviness amplitude, bearing speed, bearing load and ball size error band, the bearing SPL will increase.

Computation of static quantum triplet structure factors of liquid para-hydrogen.

The instantaneous and centroid triplet structure factors, , of liquid (one-center) para-hydrogen are computed on the crystallization line for temperatures T/K ≤ 33. The focus is on salient equilateral and isosceles features, and the methods utilized are path integral Monte Carlo (PIMC) simulations and Ornstein-Zernike (OZ) integral equations, which involve Jackson-Feenberg convolution (JF3) and other distinct closures. Long path integral simulation runs are carried out in the canonical ensemble, so as to obtain sufficiently accurate direct PI triplet results. Conclusions are drawn regarding general triplet structure features and the role and usefulness of the OZ closures. The equilateral features are studied in more detail, and one finds that (a) PIMC results point to the existence of regularity in the centroid main peak amplitudes; (b) some of the studied closures give qualitative descriptions for wave numbers below k ≈ 1 Å-1, but they all fail to describe the main peak amplitude regions (1.75 < k/Å-1 < 2.5); and (c) JF3 plays the role of a limit closure that is valid for increasing wave numbers (k ≥ 2.6 Å-1). In addition, representative isosceles PI features turn out to be reasonably bounded (within Δk = 0.1 Å-1) by those of some closures.

Vibrations of rotating cylindrical shells with functionally graded material using wave propagation approach

Fundamental natural frequencies of rotating functionally graded cylindrical shells have been calculated through the improved wave propagation approach using three different volume fraction laws. The governing shell equations are obtained from Love’s shell approximations using improved rotating terms and the new equations are obtained in standard eigenvalue problem with wave propagation approach and volume fraction laws. The effects of circumferential wave number, rotating speed, length-to-radius, and thickness-to-radius ratios have been computed with various combinations of axial wave numbers and volume fraction law exponent on the fundamental natural frequencies of nonrotating and rotating functionally graded cylindrical shells using wave propagation approach and volume fraction laws with simply supported edge. In this work, variation of material properties of functionally graded materials is controlled by three volume fraction laws. This process creates a variation in the results of shell frequency. MATLAB programming has been used to determine shell frequencies for traveling mode (backward and forward) rotating motions. New estimations show that the rotating forward and backward simply supported fundamental natural frequencies increases with an increase in circumferential wave number, for Type I and Type II of functionally graded cylindrical shells. The presented results of backward and forward simply supported fundamental natural frequencies corresponding to Law I are higher than Laws II and III for Type I and reverse effects are found for Type II, depending on rotating speed. Our investigations show that the decreasing and increasing behaviors are noted for rotating simply supported fundamental natural frequencies with increasing length-to-radius and thickness-to-radius ratios, respectively. It is found that the fundamental frequencies of the forward waves decrease with the increase in the rotating speed, and the fundamental frequencies of the backward waves increase with the increase in the rotating speed. This investigation has been made with three different volume fraction laws of polynomial (Law I), exponential (Law II), and trigonometric (Law III). The presented numerical results of nonrotating isotropic and rotating functionally graded simply supported are in fair agreement with parts of other earlier numerical results.

The onset of nanofluid natural convection inside a porous layer with rough boundaries

Smooth boundaries guarantee the infinitely stable flow when the linear stability analysis is implemented. Hence, the study on rough boundaries is essential when the laminar-turbulent transition is being considered. Because of the importance of rough boundaries, the present paper concentrates on the analysis of the onset of nanofluid natural convection (Rayleigh-Bénard convection) in a porous layer between two rough solid walls. The nanofluid surrounded between these surfaces contains the influence of both Brownian motion and thermodiffusion (thermophoresis). The non-negligible roughness boundary conditions are modeled according to the Saffman interface condition. For this purpose, the roughness is analogized as a shallow saturated fluid porous media. The simulation is performed under the isothermal condition and the zero nanoparticle flux condition for both surfaces. Then the governing equations are transported to an eigenvalue problem, and they are solved by a numerical method. The results prove that boundaries roughness and porosity parameters have a stabilizing influence on the onset of convective heat transfer while nanoparticle-concentration Rayleigh number, thermophoresis parameter, modified nanoparticle-density increment, and Lewis number have the destabilizing effects on the onset of Rayleigh-Bénard convection. For example, by increasing the roughness parameter from zero (i.e., stress-free boundary condition) to 100 (i.e., rigid solid walls with no-sleep condition), the critical convection Rayleigh number and the critical wave number increase around 145% and 41%, respectively.

Oblique Wave Trapping by a Surface-Piercing Flexible Porous Barrier in the Presence of Step-Type Bottoms

The present study deals with the oblique wave trapping by a surface-piercing flexible porous barrier near a rigid wall in the presence of step-type bottoms under the assumptions of small amplitude water waves and the structural response theory in finite water depth. The modified mild-slope equation along with suitable jump conditions and the least squares approximation method are used to handle the mathematical boundary value problem. Four types of edge conditions, i.e., clamped-moored, clamped-free, moored-free, and moored-moored, are considered to keep the barrier at a desired position of interest. The role of the flexible porous barrier is studied by analyzing the reflection coefficient, surface elevation, and wave forces on the barrier and the rigid wall. The effects of step-type bottoms, incidence angle, barrier length, structural rigidity, porosity, and mooring angle are discussed. The study reveals that in the presence of a step bottom, full reflection can be found periodically with an increase in (i) wave number and (ii) distance between the barrier and the rigid wall. Moreover, nearly zero reflection can be found with a suitable combination of wave and structural parameters, which is desirable for creating a calm region near a rigid wall in the presence of a step bottom.

Resonant streaming instability in a non-degenerate quantum plasma including the electron quantum recoil effect

The dispersion relation for the surface waves propagating at the interface of a semi-bounded quantum plasma containing non-degenerate electrons and streaming ions. The electron quantum recoil effect is included in the dielectric permittivity, and the growth rate of the surface waves which are resonant with the streaming ions is derived. It is found that the resonant growth rate decreases with an increase in the plasmon energy; hence, the increase in electron oscillation is found to suppress the instability of the surface wave. However, the resonant growth rate increases with an increase in the electron thermal energy because the energy surplus by electrons is responsible for the unstable mode. As the wave number increases, the rate of decline of the resonant growth rate decreases gradually.

Vibration analysis of circular cylindrical shells made of metal foams under various boundary conditions

This study investigates the free vibration of metal foam circular cylindrical shells under various boundary conditions. The elasticity modulus and mass density of the shells vary gradually and continually in the thickness direction. Two types of porosity distribution are taken into account including symmetrical and unsymmetrical distributions. Love’s shell theory is employed to formulate the governing equations and then the Rayleigh–Ritz method is utilized to solve natural frequencies of the system. The results show that the porosity coefficient has important effect on the natural frequencies of metal foam shells. Its effect also relates to the boundary conditions of the shells. Moreover, different porosity distributions make the metal foam shells possess different vibration characteristics, which is quite obvious at large porosity coefficient. As the circumferential wave number increases, the natural frequencies of the metal foam shells tend to the same under various boundary conditions. Additionally, the present results are verified by the comparison with the published ones in the literature.

Analysis of energy of internal waves in a three­layer semi­infinite hydrodynamic system

Energy characteristics of waves propagation along the contact surfaces in a hydrodynamics system "liquid half-space – layer –layer with a rigid lid" are explored. Based on the solutions of first approximation to a weakly non-linear model, the integral relations for wave motion energy in each layer and for the total energy of the system were obtained. An analysis of energy of wave processes revealed that an increase in wave number causes a decrease in energy of wave motion of the upper layers, and energy of wave motion of the lower half-space at some values of wave number reaches extreme values. In this case, total energy of the system is descending in nature and rather quickly approaches its limit value.The numerical values of energy for three different cases of propagation of progressive waves were obtained: only along the upper contact surface, only along the lower contact surface, along both surfaces simultaneously. Comparison of the obtained values of energy revealed that in the case of waves propagation along both contact surfaces simultaneously, the total energy of the system is close to the sum of energies of the system at waves propagation along one of the surfaces

Effect of catalyst calcination temperature in the visible light photocatalytic oxidation of gaseous formaldehyde by multi-element doped titanium dioxide

The present study investigates the influence of calcination temperature on the properties and photoactivity of multi-element doped TiO2. The photocatalysts were prepared by incorporating silver (Ag), fluorine (F), nitrogen (N), and tungsten (W) into the TiO2 structure via the sol-gel method. Spectroscopic techniques were used to elucidate the correlation between the structural and optical properties of the doped photocatalyst and its photoactivity. XRD results showed that the mean crystallite size increased for undoped photocatalysts and decreased for the doped photocatalysts when calcination was done at higher temperatures. UV-Vis spectra showed that the absorption cut-off wavelength shifted towards the visible light region for the as-synthesized photocatalysts and band gap narrowing was attributed to multi-element doping and calcination. FTIR spectra results showed the shifting of OH-bending absorption bands towards increasing wave numbers. The activity of the photocatalysts was evaluated in terms of gaseous formaldehyde removal under visible light irradiation. The highest photocatalytic removal of gaseous formaldehyde was found at 88%. The study confirms the effectiveness of multi-element doped TiO2 to remove gaseous formaldehyde in air by visible light photocatalysis and the results have a lot of potential to extend the application to other organic air contaminants.

Effect of phase-lags on Rayleigh wave propagation in initially stressed magneto-thermoelastic orthotropic medium

• Thermal boundary condition for generalized thermoelasicity is developed. • Different secular equations are derived which agree with existing literature. • Path of surface particles during Rayleigh wave propagation in orthotropic medium is elliptical. • Rayleigh wave velocity increases with the increase of wave number, initial stress and magnetic field. • Attenuation coefficient decreases with the increase of wave number, initial stress and magnetic field. The present article deals with Rayleigh surface wave propagation in homogeneous magneto-thermoelastic orthotropic medium. Effect of initial stress and magnetic field on Rayleigh waves is studied in the context of three-phase-lag model of generalized thermoelasticity. The normal mode analysis is used to obtain the exact expressions for the displacement components, stresses and temperature distribution. Various frequency equations are derived and compared with the existing literature. The path of surface particles is elliptical during Rayleigh wave propagation. Effect of phase-lags on Rayleigh wave velocity, attenuation coefficient and specific loss are presented graphically. It is observed from graphical presentation that the effect of magnetic field and initial stress on different wave characteristics is pronounced.