Water wave trapping in a long array of bottomless circular cylinders

This study tries to identify wave trapping situations by engaging and properly combining two well established phenomena: (i) the trapped modes induced by arrays of cylinders and (ii) the pumping trapped modes which are known to occur in moonpools. To this end, the fundamental hydrodynamic boundary value problem for arrays of bottomless cylinders was solved using standard domain decomposition. The method employed expansions of the solutions for the velocity potentials in polar harmonics combined with the eigenfunction expansions technique. The solution sought for the velocity potentials is achieved using the “direct” method of approach which accordingly requires the employment of a sophisticated matrix manipulation process.The elaboration of the concerned concept was motivated by three basic tasks: (i) to identify whether arrays of truncated and bottomless cylinders indeed preserve the occurrence of Neumann, Dirichlet and near trapped modes, extensively investigated for bottom-seated cylinders; (ii) to examine whether the expected pumping modes in moonpools modify the characteristics of the hydrodynamic resonance regimes (trapped modes) in the open liquid space between the cylinders and vice versa and (iii) to explore the possibility to suggest relevant configurations as parts of integrated mechanisms for practical applications, focusing a fortiori to clusters of hydrodynamically interacting Oscillating Water Columns (OWCs).The method developed is generic and can be employed for arbitrary configurations of multi-body arrays accommodating bottomless cylinders with uneven geometrical characteristics. Trapped modes are identified numerically as peaks in loading and this fact has been explicitly demonstrated in rows of cylinders. Therefore, the numerical results shown and discussed in the present are based on a specific in-line array that has been investigated in the past for bottom-seated cylinders. The investigated subject, i.e. whether the combined wave trapping induced by the examined configuration could be conceived as an efficient water wave power extraction mechanism is approached and discussed through dedicated computations of the free-surface displacements in the moonpools.     

Nonlinear free surface action with an array of vertical cylinders

Nonlinear diffraction of regular waves by an array of bottom-seated circular cylinders is investigated in frequency domain, based on a Stokes expansion approach. A complete semi-analytical solution is developed which allows an efficient evaluation of the second-order potentials in the entire fluid domain, and the wave forces on the structure. Expressions are derived for the second-order potential in the vicinity of individual cylinders. These expressions have a simple form, thus providing an effective means for investigating the wave enhancement due to nonlinear interactions with multiple cylinders. Based on the present method, the wave run-up and free-surface elevations around an array of two, three and four cylinders are investigated numerically.     

Vortex-induced vibrations of a diamond cross-section: Sensitivity to corner sharpness

This paper studies the fluid–structure interaction of an elastically mounted square cross-section cylinder immersed in a free stream. The cross-section is mounted such that its sides are at 45° to the free stream direction, in a “diamond” configuration, and its motion is constrained to the transverse direction relative to the flow direction. Apart from the cross-section, this setup is the same as the majority of single-degree-of-freedom vortex-induced vibration studies of cylinders. Two-dimensional direct numerical simulations of this system have been performed. The Reynolds number based on the point-to-point distance of the cross-section has been fixed at Re=200). Simulations at this Reynolds number allow a direct comparison with previous results from circular cylinders, and therefore focus directly on the impact of the geometry.The sensitivity of the flow, and therefore the motion of the cylinder, to geometrical effects is considered. This is done by rounding the two side corners (those pointing across the flow) at a given radius. For well-rounded corners, the flow behaviour resembles that of a circular cylinder undergoing vortex-induced vibration. However, below a critical radius, the dynamics are considerably altered. Highly disordered and irregular wakes and body motions are observed, as well as a synchronized, periodic P+S wake mode (Williamson and Roshko, 1988), which consists of a pair of vortices on one side, and a single vortex on the other side, shed per oscillation cycle, which results in a non-zero mean lift force. A period-doubled version of this P+S wake is also presented. The spatial structure, and the spatio-temporal symmetries of each of these modes is reported. The results show that even though the spatio-temporal symmetry of the flow is unaffected by the geometry when the body is rigidly mounted (the flow always saturating to a Kármán vortex street) geometric features such as sharp corners can induce a number of spontaneous symmetry breaking bifurcations when the body is elastically mounted. Which of these various modes is observed is shown to be a function of both the corner radius and the spring stiffness, expressed through the reduced velocity.     

TWO-DIMENSIONAL VORTEX-INDUCED VIBRATION OF CABLE SUSPENSIONS

Resonant responses of suspended elastic cables driven by a steady current are investigated. Phenomenological fluid force models for alternate vortex-shedding are coupled with the nonlinear partial differential equations of cable motion. Decoupled cross-flow and in-line vortex-induced vibrations (VIV) are examined first using linearized and nonlinear cable models. The linearized cable model predicts well the basic characteristics of VIV and the nonlinear cable model captures the hysteresis often observed in experiments. Next, coupled cross-flow and in-line vibrations are evaluated by considering two principal coupling mechanisms: (i) cable structural nonlinearities, and (ii) coupled fluid lift and drag. Attention is focused on a “worst-case” resonant response where the natural frequencies for cable modes in the cross-flow and in-line directions are in the same 1:2 ratio as the excitation frequencies associated with lift and drag. The inclusion of cable structural nonlinearities alone leads to coupled responses that differ qualitatively (i.e., in number and stability of periodic motions) when compared to those of the decoupled model. The inclusion of coupled fluid lift and drag produces non-planar “figure eight” motions of the cable cross-section that exhibit similar characteristics to those previously measured on spring supported cylinders.     

On Rayleigh scattering by a grating

The acoustic diffraction of a plane wave by a periodic row of identical cylinders of arbitrary cross section and characteristic dimension a is calculated for ka ? kd < π, where k is the wave number and d is the wavelength of the array. The reflection and transmission coefficients depend only on d, k, the angle of incidence, and the area and virtual mass of the cross section. The general results are applied to a grating of inclined flat plates.     

Oscillations of elastically mounted cylinders in regular waves

Under the assumption of potential flow and linear wave theory, a semi-analytic method based on eigenfunciton expansion is proposed to predict the hydrody-namic forces on an array of three bottom-mounted, surface-piercing circular cylinders. The responses of the cylinders induced by wave excitation are determined by the equa-tions of motion coupled with the solutions of the wave radiation and diffraction problems. Experiments for three-cylinder cases are then designed and performed in a wave flume to determine the accuracy of this method for regular waves.     

Specific gasdynamical features of spontaneous condensation in an unsteady rarefaction wave

On the basis of numerical modeling, the formation of an unsteady shock wave induced by a condensation shock in a rarefaction wave moving in the high-pressure channel of a shock tube filled with moist air is demonstrated. It is shown that in a fairly long channel a periodic structure consisting of an alternating sequence of condensation shocks and the shock waves they generate may be formed. This structure is a linear unsteady analog of the self-oscillation regime of type IV in the classification [1] for condensing medium flows in the subsonic section of a Laval nozzle. The specific features detected are important for planning and interpreting experiments aimed at investigating spontaneous condensation using a “condensation shock tube”.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.     

Propagation and scattering of waves by dense arrays of impenetrable cylinders in a waveguide

A coupled numerical scheme, based on modal expansions and boundary integral representations, is developed for treating propagation and scattering by dense arrays of impenetrable cylinders inside a waveguide. Numerical results are presented and discussed concerning reflection and transmission, as well as the wave details both inside and outside the array. The method is applied to water waves propagating over an array of vertical cylinders in constant depth extended all over the water column, operating as a porous breakwater unit in a periodic arrangement (segmented breakwater). Focusing on the reflection and transmission properties, a simplified model is also derived, based on Foldy–Lax theory. The latter provides an equivalent index of refraction of the medium representing the porous structure, modeled as an inclusion in the waveguide. Results obtained by the present fully coupled and approximate models are compared against experimental measurements, collected in wave tank, showing good agreement. The present analysis permits an efficient calculation of the properties of the examined structure, reducing the computational cost and supporting design and optimization studies.     

Parametric resonance and pattern selection in an array of microcantilevers interacting through fringing electrostatic fields

Nonlinear Dynamics - We investigate collective resonant dynamics of an array of microcantilevers coupled elastically, through a flexible overhang, and electrostatically, through fringing fields....     

Relation between the Hydrodynamic and Elastic Parameters in Surface Wave Diffraction on a Floating Plate

The problem of surface wave diffraction on a floating elastic plate is considered. The relation between the parameters of the elastic vibrations of the plate and the transmitted and reflected wave amplitudes is investigated. It is shown that the maximum amplitudes of the plate stresses and deflections depend nonmonotonically on the incident wave frequency and are reached simultaneously with the maxima of both the transmission coefficient and the length of the wave penetrating into the plate. This makes it possible to use the transmission coefficient as a parameter for investigating the maximum and minimum amplitudes of the hydroelastic vibrations of the plate. As an example, this criterion is used to minimize the vibrations of a plate whose leading edge is elastically connected to the bottom.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Steady Flow from an Array of Subsurface Emitters: Kornev’s Irrigation Technology and Kidder’s Free Boundary Problems Revisited

Kornev’s (Subsurface irrigation, Selhozgiz, Moscow-Leningrad, 1935) subsurface irrigation with a periodic array of emitting porous pipes is analytically modeled as a steady potential Darcian flow from a line source generating a phreatic surface. The hodograph method is used. The complex potential strip is mapped onto the triangle of the inverted hodograph. An analogy with the Deemter (Theoretische en numerieke behandeling van ontwaterings-en infiltratie stromings problemen (in Dutch). Theoretical and numerical treatment of flow problems connected to drainage and irrigation. Ph.D. dissertation, Delft University of Technology, 1950) drainage problem and Kidder (J Appl Phys 27(8):867–869, 1956) free-surface flow toward an array of oil wells underlain by a “wavy” oil–water interface is drawn. For a half-period of Kornev’s flow, the “wavy” phreatic surface has an inflection point. The “waviness” of the phreatic surface is controlled by the spacing between emitters, the strength of line sources, and the pipe pressure and radius. Numerical modeling with HYDRUS involved two factors which constrained the saturated–unsaturated flow: the positive pressure head at the outlet of the modeled domain and lateral no-flow boundaries, with a qualitative corroboration of analytical solutions for potential (fully saturated) and purely unsaturated flows. HYDRUS is also applied to a generalized Philip’s regime of an unsaturated flow past a subterranean hole, which is impermeable at its top and leaks at the bottom.     

Numerical predictions for unsteady viscous flow past an array of cylinders

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analysed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The centre-to-centre distance between adjacent cylinders is fixed at three diameters, and the rows are staggered. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder. Two cases, for Reynolds numbers of 1000 and 10,000, are presented.     

Stability of the laminar boundary layer flow encountering a row of roughness elements: Biglobal stability approach and DNS

The stability of the laminar boundary layer developing on a flat plate in the presence of a periodic row of roughness elements is investigated. A Direct Numerical Simulation is performed to compute the steady flow downstream of the roughness elements, which contains a pair of two counter-rotating streamwise vortices per element, which can be considered as a “pre-streaky” structure. The linear stability of this base flow is analyzed by means of the so-called “biglobal” stability approach. Three-dimensional eigenmodes are found, which are shown to be the continuation of the Tollmien–Schlichting waves present in the case of an unperturbed boundary layer. Moreover, a stabilizing effect due to the roughness-induced vortices is found. A Direct Numerical Simulation of the interaction between a two-dimensional Tollmien–Schlichting wave and the roughness array is also performed. The computed perturbation traveling downstream of the roughness elements is shown to be a linear combination of the biglobal eigenmodes.     

Modeling Phononic Crystals via the Weighted Relaxed Micromorphic Model with Free and Gradient Micro-Inertia

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure.     

Isolation of the resonant component in acoustic scattering from fluid-loaded cylindrical shells

For the problem of scattering of a plane acoustic wave by a cylindrical elastic shell immersed in a fluid, we demonstrate that a recently proposed “intermediate background” formalism serves to separate the resonant component from the non-resonant background in the scattering amplitude. Numerical calculations are performed for the case of an air-filled aluminium shell in water.     

Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane （with respect to the lattice plane） and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave prop- agation are highlighted in high frequency domains. One im- portant result presented herein is the comparison between the first Bloch wave modes to the membrane and bend- ing/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homog- enized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retro- propagating Bloch wave modes with a negative group veloc- ity, and of the corresponding ＂retro-propagating＂ frequency bands.