Non-dimensional Wave Number Research Articles

Eigenfunction expansion solution for the radiation problem of a uniform cylinder in front of a partially reflective wall

In this paper, the wave radiation problem of a vertical cylinder in front of a partially reflective vertical wall is solved by combining the eigenfunction expansion method and the mirror image method. To satisfy the partial reflection condition at the vertical wall, a scattering incident wave is introduced, which corresponds to the reduction scattering potential from the mirror image cylinder symmetrically distributed about the partially reflective wall and under symmetrical motion about the vertical wall with respect to the original cylinder. Subsequently, the scattering potentials generated by each of the two cylinders are derived using the eigenfunction expansion method on cylindrical coordinate systems with origins at the centers of the two cylinders. By applying Graf's addition theorem, the wave radiation problem is finally solved by transforming the incident scattering potential in the imaginary cylindrical coordinate system back to the original one of the cylinder itself and satisfying the body surface boundary condition. Using the numerical model developed in this paper, the effects of the parameters of the modulus and phase of the partial reflection coefficients as well as the distance between the partially reflective wall and the center of the cylinder on the added mass and radiation damping are investigated. The numerical results show that there are oscillations present in both the curves of added mass and radiation damping versus nondimensional wavenumber ka, the amplitude of the oscillations increases with the increase in the modulus of the reflection coefficient, and the peaks and troughs of the oscillations move to lower wavenumber ranges with increasing phase shift of the reflection coefficient. In addition, the distance between the cylinder and the vertical wall significantly affects the wavenumber range in which the peaks and troughs of the hydrodynamic coefficient curves appear, and the period of the oscillations decreases as the distance increases.

Comparison between linear and quadratic power take off for a single chamber land-fixed oscillating water column (OWC)

This study examines the influence of turbine response type on the performance of a land-fixed Oscillating Water Column (OWC) under second-order Stokes waves of varying heights. Numerical modelling techniques, including the Finite Element Method, incompressible Reynolds-Averaged Navier–Stokes (RANS) equations with the Shear-Stress Transport (SST) k-ω turbulence model, and an Arbitrary Lagrangian–Eulerian (ALE) scheme, were employed to gain insights into OWC behaviour. The linear turbine consistently displayed reduced efficiency with increasing wave height, maintaining a stable peak non-dimensional wave number (kh). Conversely, the quadratic turbine exhibited peak efficiency within a narrower wave height range, strongly influenced by wave height. Non-dimensional volume flow rate, non-dimensional chamber pressure, chamber volume, and maximum force on the front wall remained unaffected by changes in wave height for the linear turbine. In contrast, the quadratic turbine was notably impacted by variations in wave height. Moreover, the quadratic turbine revealed stronger odd-numbered higher-order harmonic components in both force and pressure due to its non-linear response. Notably, both linear and quadratic response types exhibited sloshing at similar wave numbers, suggesting minimal influence of turbine response on sloshing behaviour. Finally, the derived non-dimensional turbine coefficients proved both turbines could scale OWC models with negligible scaling effects from the turbine.

Exploring the Limits of a Deflection-Based Method for the Estimation of Dynamic Stress in Beams

Estimation of the dynamic stress in structures, such as beams and plates, has previously been made using the relationship between stress and velocity spatial maxima based on farfield assumptions. This paper presents a method for the estimation of dynamic stress in a beam using Euler-Bernoulli beam theory, where deflection data from a grid of measurement points on the surface of the beam is used to estimate the dynamic bending stress in the structure. The limitations of the method are investigated via response data provided by a numerical model of a free-free beam. A non-dimensional wavenumber analysis is used to determine the number of points required for an accurate estimate of stress. Beams with a range of material and geometric parameters are modelled in order to explore the limits of the estimation method, and parameters representative of several real-world materials are used to assess the suitability of the method for practical applications.

Frictional slip wave solutions for dynamic sliding between a layer and a half-space

Interfacial waves propagate along the interface between two elastic solids, and are the subject of basic interest to diverse scientific fields, like, seismology, geophysics, material science and composite structures. The focus of the present work is to analyse the interfacial waves arising due to frictional slipping between an elastic layer and an elastic half-space having dissimilar elastic properties, wherein the deformations are confined to the plane of the solids. The interfacial friction is modelled by employing a slip-dependent friction law. The elastodynamic relations between the displacement discontinuities across the fractured interface and the associated traction components of stress for pure slip events (no crack opening along the interface) are obtained. This allows us to derive a secular dispersion relation governing the interfacial wave solutions. The dispersion relation relating the phase velocity and wavenumber is analysed numerically. A family of solutions are shown to exist depending on the critical slip distance, δc, a parameter governing the interfacial friction law, and on the contrast in material properties. We demonstrate the existence of a number of wave modes, all being dispersive in nature. The fundamental wave mode with the lowest phase velocity exists over the largest range of wavenumbers, while other wave modes exist only at sufficiently high non-dimensional wavenumber. The dispersion curves for the frictional slip waves are found to lie between the dispersion curves for the cases of the smooth and the bonded interface. As a case study, the present model is used to study the dispersion of slip waves propagating in Earth's crust-mantle geophysical model. The field observations of surface wave dispersion appear to be in good agreement with that of the frictional slip waves studied here. Based on our results, we suggest an alternative explanation for the observed surface wave dispersion at the regional and global scales. It is suggested that frictional slip waves at the crust-mantle interface could propagate with phase velocities similar to the Rayleigh and Love surface waves.

On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$ , where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

Computing Theoretical Seismograms from a Point Source in a Spherical Multilayered Medium

Abstract Computing theoretical seismograms from a point source in a given Earth model is essential for modeling and inversion of observed seismic waveforms for Earth’s structure and earthquake source parameters. Here, we derived the propagator matrices and source terms for a spherical multilayered Earth model using the exact earth flattening transformation. We found that their differences from their counterparts in horizontal layered media are inversely proportional to the nondimensional horizontal wavenumber and its higher order. In addition, all the source terms in a spherical layered model have a source-depth dependent scaling factor that differs from in a horizontal layered model by up to 6% for deep earthquakes. The surface displacement produced by a point source can be obtained in a similar form as in horizontal layered media. Computation of theoretical seismograms was implemented using the generalized reflection and transmission coefficients method. Numerical tests show that our formulae and implementation are correct and efficient for computing full-wave seismograms, including the permanent displacements, at teleseismic distances up to 100°. Individual seismic phases can be isolated and analyzed semianalytically because the generalized reflection and transmission method is used. Furthermore, our analytic expression of displacement in terms of the propagator matrices and source terms can be used to derive analytic derivatives of seismograms for full-wave waveform inversion.

Large-eddy simulations of interaction between surface waves and a tidal turbine wake in a turbulent channel

Tidal stream turbines are now being developed for array deployments, largely at sites with relatively shallow water depths on either bed-supported, or floating support structures. Proximity to the free-surface presents design challenges with increased exposure to wave-induced kinematics leading to potential for increased peak- and fatigue-loads. Free surface proximity can also alter wake recovery rates which can influence the siting, and operation, of further turbines. To-date the impact of waves on turbine loading and on wake recovery has receivedlimited attention, generally for specific combinations of conditions reproducible in experimen tal facilities [1-2] or numerical models [3]. Improved understanding of how waves affect both turbine loading, and wake dynamics is necessary to inform the development of appropriate load prediction and mitigation methods and to further parameterise wake recovery to inform array siting.
 This work presents CFD analysis of the loading and wake of a three-bladed horizontal axis tidal turbine within a wide, shallow channel with a turbulent inflow developed using a synthetic eddy method and free surface waves modelled using a Level-Set Method (LSM). The turbine and wave conditions considered are based on prior experimental studies [1] within a channel depth h = 0.45 m and mean flow U = 0.47 m/s. The simulated channel is width 3.67h and length 88h and long-crested regular waves are imposed, with the focus of this study on waves with non-dimensional wavenumber kh = 2.8, comparable with experimental conditions. Simulations are accomplished using the in-house large-eddy simulation (LES) code DOFAS(Digital Offshore Farms Simulator) [4], which adopts an Actuator Line Method (ALM) to resolve the turbine blades. LSM is used to model the air-water interface in an accurate manner [5] with waves can be generated in DOFAS using linear or second-order Stokes theories with an absorption layer placed at the outlet of the domain to avoid wave reflections. At the inlet a mean logarithmic velocity profile is imposed over which artificial turbulence is added. Simulations were run for 400 s of physical time on 8,000 cores using ARCHER2.
 The waves studied are shown to alter the rate of wake recovery, increasing the rate in the near-wake of the turbine and slightly reducing the rate beyond eight diameters downstream. Analysis of turbulence characteristics indicates a significant variation across the water column due to wave action. Propagation over the turbine wake also introduces directionality to the wave field, which is associated with the change of wave-speed over the wake region. Modal decomposition analysis of the velocity fluctuations with Proper Orthogonal Decomposition(POD) reveals that the wake dynamics behind the turbine change due to the waves.

Temporal instability of the liquid film formed by a liquid jet impinging on a curved cylindrical wall

The linear temporal instability of the confined curved liquid film formed by a liquid jet impinging on a curved wall has been investigated. The corresponding dispersion relation between the non-dimensional wave growth rate and wave number was derived. The dispersion relation further involves the influences of the local radius of curvature and the azimuthal angle of the liquid film. The dispersion relation could be verified with previous researches, and the optimal wavelength has been verified with experiments. Temporal stability analysis shows that the effects of gas-to-liquid velocity ratio, gas-to-liquid density ratio, the relative thickness of the liquid phase to the gas phase, and azimuthal wave number are similar to previous researches, but the unstable ranges are smaller than that in previous researches. This paper further studied the effects of the local radius of curvature and azimuthal angle of the liquid film. As the local radius of curvature decreases, the instability of confined curved films increases for the given conditions. And both the growth rate and instability range increase with the increase of the azimuthal angle. The solutions would be helpful to understand the atomization process of the liquid film formed by a liquid jet impinging on a curved wall.

Nonlocal wave propagation analysis of a rotating nanobeam on a Pasternak foundation

In this paper, the wave propagation analysis of a cantilever rotating nanobeam modeled as a thin beam based on the Euler–Bernoulli theory on a Pasternak foundation has been investigated using the nonlocal theory of elasticity. The governing partial differential equation of motion for a uniform rotating nanobeam is derived using the Hamilton principle considering the nonlocal parameter of small scale effect. The Spectrum and dispersion relations of non-dimensional wave number are obtained analytically. The effect of changes of different parameters including nonlocal scale parameter, non-dimensional rotational speed, non-dimensional rotational wave frequency ratio, and shear stiffness of the Pasternak foundation on the wave dispersion behavior of the non-dimensional wave number and phase and group speeds dispersions for the rotating nanotube have been studied. It is observed that the wave dispersion characteristics of the rotating nanobeam are extremely influenced by rotational speed, wave number, nonlocal length scale parameter, and shear stiffness of the Pasternak foundation. Moreover, the propagated flexural wave has been shown to exhibit non-dispersive behavior at very high rotational speeds.

Gravity wave scattering by retrofitted circular breakwaters using dual boundary integral formulation

In the present study, the gravity wave scattering by circular edged breakwaters in the presence of retrofits is analysed within the framework of linearized water wave theory. Various types of retrofits (say, porous shield) such as quarter-circular retrofit, semi-rectangular retrofit, semi-circular retrofit, and rectangular retrofit are investigated to protect the newly developed or existing circular breakwaters from the incident wave stroke. A dual boundary integral formulation is used to solve the boundary value problem considering the quadratic pressure drop boundary condition for the waves past the thin porous barriers. A generalized computer code is developed to accommodate the domain changes and is validated with the known results available in the literature. Various results such as the scattering coefficients (wave reflection, transmission and wave energy loss) and force coefficients (vertical and horizontal force on the breakwater and retrofit) are computed as a function of non-dimensional wave number and discussed. The sharp-edged retrofits (rectangular and semi-rectangular) display a considerable reduction of wave transmission as compared with smooth-edged retrofits, and the semi-circular breakwater with rectangular retrofit is suggested for further development of the porous structures. The effect of porous shield alone, breakwater alone, both breakwater and shield on incident wave energy distribution is also reported. The minimal wave transmission and higher wave damping coefficients attended occurred by the proposed breakwaters are identified when the shield porosity varies within 0.1≤μ≤0.2.

Global spectral analysis of the Lax–Wendroff-central difference scheme applied to Convection–Diffusion equation

In this work, the Global Spectral Analysis (GSA) is applied to the convective Lax–Wendroff based discretization of linear convection–diffusion problem in both 1D and 2D. Contrary to standard numerical analysis approaches, two important physical processes (convection and diffusion) are treated together, thus making GSA a function of multiple non-dimensional numerical parameters, namely, the non-dimensional wavenumber (kh), the Courant–Friedrich–Lewy number (Nc) and the Peclet number (Pe). All the three quantities impact the stability of the numerical scheme by affecting both numerical amplification factor as well as numerical diffusion. Likewise, numerical phase speed and numerical group velocity all become expressions of all three non-dimensional parameters. Leveraging the GSA, an accurate map (property charts) of acceptable range of parameter space is evidenced to obtain the spatio-temporal numerical solution that is stable as well as Dispersion Relation Preserving (DRP). The numerical property charts are shown to be useful to calibrate numerical solutions of both 1D and the 2D convection–diffusion equations on uniform meshes. Finally, as demonstrated while solving the Navier–Stokes equation for a Taylor–Green vortex problem, property charts allow to explain numerical behaviors often observed with real complex flow problems.

Dynamic Analysis of Nanosized Arbitrary Shape Inclusion under SH Waves

The scattering of shear waves (SH waves) by nano-scale arbitrary shape inclusion in infinite plane is studied by complex variable function theory. Firstly, the governing equation and the relationships between stress and displacement are given by classical elastic theory. Secondly, the arbitrary shape inclusion in the two-dimensional plane is transformed into a unit circle domain by conformal mapping, the incident wave field and the scattered wave field are presented. Next, the stress and displacement boundary conditions are established by considering surface elasticity theory, The infinite algebraic equations for solving the unknown coefficients of the scattered and standing waves are obtained. Finally, the influence of surface effect, non-dimensional wave number, Shear modulus and hole curvature on the dynamic stress concentration factor are analyzed by some examples, the numerical results show that the surface effect weakens the dynamic stress concentration. With the increase of wave number, the dynamic stress concentration factor (DSCF) decreases. Shear modulus and hole curvature have significant effects on DSCF.

Acoustic scattering by a finite plate with a poroelastic extension using the unified transform method

This paper investigates sound scattering by a finite rigid plate with various extensions using the unified transform method. Different boundary conditions, including rigid, elastic, perforated, and poroelastic conditions are addressed thanks to this efficient method. Parametric studies for the elasticity, the porosity, and the extension length are also conducted over a wide frequency range to explore the combined effects of these variables. Results show that the observed noise reduction can be attributed to the attenuation of pressure difference across the plate. For the case of a finite plate with a poroelastic extension, at a low frequency (k0⩽0.1, where k0 is the non-dimensional wavenumber), porosity can achieve more noise reduction than elasticity. Elasticity generally enhances noise reductions, especially when combined with low porosities. At a high frequency (k0⩾10), the maximum noise reduction can be achieved through either high elasticity or large porosity, which is, however, still smaller than that at low frequencies. The extension length is more effective in noise reduction when applied with large porosity at a low frequency or high elasticity at a medium or high frequency. This study reveals the potential noise reduction by a poroelastic extension attached to a finite rigid plate, and helps shed more light on the noise reduction mechanisms as well as an optimal design of realistic poroelastic extensions.

A generating‐absorbing boundary condition for dispersive waves

AbstractThe detailed modeling of free‐surface waves and their interaction with bottom‐mounted or floating structures requires large computational resources, which is why efficient boundary conditions with low spurious reflection are desirable. The present work presents a review of existing generating‐absorbing boundary conditions (GABCs) for dispersive waves and their reflection characteristics. Hereafter, an adaptation of the classical Sommerfeld condition is proposed by using a depth‐varying coefficient to improve absorption efficiency over a range of wave numbers. An analytical model is proposed to analyse the reflection characteristics for both propagating and evanescent modes, and a considerable improvement in comparison to the Sommerfeld condition is documented over a broad frequency range (reflection coefficients below 5% for nondimensional wave numbers in the range [0, 10]). The new boundary condition is implemented in OpenFoam (waves2Foam) and the functioning for regular, irregular, solitary, and phase‐focused waves is presented.

Nonlinear wave generation using a heaving wedge

This paper investigates the optimization of second-order control signals required to produce stable non-linear, deep-water waves using a wedge-shaped, plunger-type wave generator. Both numerical and experimental methods are utilized. A fully non-linear and dispersive potential flow (FNPF) solver developed at DTU is used for the numerical work, following improvements that reduce reflection to 1%. The numerical solver is validated against theoretical and experimental data. A defect correction optimization scheme is employed, resulting in optimized control signals for non-dimensional wave numbers of 2.04-8.17 and steepness of 3-7%. These control signals produce waves that are within 2% of stream function theory solutions for the fundamental harmonic component, and within 10% for the second-order harmonic component. The results demonstrate both the applicability of this optimization procedure and the suitability of a heaving wedge for generating stable non-linear deep-water waves.

Linear to turbulent Görtler instability transition

We present results from a highly resolved large-eddy simulation of a freely developing Blasius profile over a concave boundary in a large spanwise domain. Due to the large initial Reynolds and Görtler numbers (Reθ,0 = 1175, Gθ,0 = 75), we observe the onset of two dominant wavelengths: the first dominating in the linear/transition region, λ1, and the second dominating in the turbulent region, λ2. Extending previous linear stability analysis (LSA) to higher Görtler numbers and non-dimensional wavenumbers, both dominant wavelengths of the Görtler instability correspond to predictions of LSA, the latter comparable to laminar theory by replacing the kinematic viscosity with the turbulent viscosity in the definition of the Görtler number. The predicted spatial modes compare well with the computed profiles for both λ1 and λ2. The skin friction coefficient Cf is found heterogeneous in the spanwise direction according to the emerging wavelengths λ1 and λ2 of the Görtler instability. We report a smooth increase in Cf from the theoretical predictions of a laminar boundary layer to those for a turbulent boundary layer over a flat plate. The values only slightly overshoot these predictions in the domain of existence of the second dominant wavelength λ2, very different from that reported at lower Reynolds numbers.

Reflection of Longitudinal Wave in the Micropolar Elasticity with Voids

By considering no more interaction between wryness tensor and change in voids volume fraction in the materials, the reflection problem of plane longitudinal waves at a free boundary of micropolar elastic materials with voids has been investigated. We have obtained the amplitude and energy ratios of reflected waves for the incident longitudinal wave by using appropriate boundary conditions. The effect of void parameters in the nondimensional wavenumber, amplitude and energy ratios are computed numerically for the particular material’s model.

Vibration Power Flow of an Infinite Cylindrical Shell Submerged in Viscous Fluids

In the previous investigations of the vibroacoustic characteristics of a submerged cylindrical shell in a flow field, the fluid viscosity was usually ignored. In this paper, the effect of fluid viscosity on the characteristics of vibration power flow in an infinite circular cylindrical shell immersed in a viscous acoustic medium is studied. Flügge’s thin shell theory for an isotropic, elastic, and thin cylindrical shell is employed to obtain the motion equations of the structure under circumferential-distributed line force. Together with the wave equations for the viscous flow field as well as continuity conditions at the interface, the vibroacoustic equation of motion in the coupled system is derived. Numerical analysis based on the additional-damping numerical integral method and ten-point Gaussian integral method is conducted to solve the vibroacoustic coupling equation with varying levels of viscosity. Then, the variation of the input power flow against the nondimensional axial wave number in the coupled system with different circumferential mode numbers is discussed in detail. It is found that the influence of fluid viscosity on the vibroacoustic coupled system is mainly concentrated in the low-frequency band, which is shown as the increase of the crest number and amplitude of the input power flow curves.

Thermo magnetic response of nonlocal propagation of waves in rotating graphene tubules

Thermo magnetic response of propagation of waves in rotating graphene tubules is studied with the aid of nonlocal Euler–Bernoulli beam theory within the framework of spectral analysis. The governing dynamic equation of nonlocal rotating graphene tubules under thermo magnetic response is formulated with the help of equation of thermal force, centrifugal force and electromagnetic force. The dispersion equation of nonlocal rotating graphene tubules under thermo magnetic field is derived. The numerical value of non-dimensional wave number is computed and is represented in terms of scattered curves. The scattered curves of graphene tubules at different rotating speed, nonlocal parameter, temperature and magnetic field strength are also drawn. The results give useful information in the study and design of rotary nano-devices such as nano motors, nanoturbines, nano robots etc. The dispersion curves of non-rotating graphene tubules in the absence of thermal and magnetic field are drawn and are compared with the existing literature.

Dynamics of acoustic impedance matching layers in contactless ultrasonic power transfer systems

The acoustic impedance mismatch between transducer materials and medium in ultrasonic power transfer systems narrows the transduction bandwidth and causes losses through the back reflection of progressive pressure waves at the boundary between the transducers and medium. Capturing both resonances and losses due to impedance mismatch of interwoven elements is essential for advancing the development of these systems. We present a unified approach, based on the multiplication of a sequence of transfer matrices, to determine an equivalent acoustic impedance. The analytical model couples the properties of the transmitter and receiver with multiple matching layers and a single classical quarter-wave layer in controlled setups with the objective of minimizing reflections through acoustic impedance mismatch alleviation. Losses due to ultrasonic attenuation in the material layers and medium are also considered. The acoustic field at the receiver location constitutes the input to the coupled electro-elastic equations of the fluid-loaded and electrically-loaded piezoelectric receiver. Experiments are performed to identify the input acoustic pressure from a cylindrical transmitter to a receiver disk operating in the 33-mode of piezoelectricity. The results show significant enhancements in terms of the receiver’s electrical power output when implementing a two-layer matching structure. We present the results showing non-dimensional wave number variations versus characteristic impedance, which can be used to calculate the materials’ thicknesses for acoustically matching ultrasonic power transfer systems to an acoustic medium of interest at any desired resonant frequency while considering any type of glue or epoxy as the bonding layer. The derived physical models facilitate the development of high-fidelity matched systems with enhanced contactless power transmission.