Flexural gravity wave motion over poroelastic bed

Using Biot’s consolidation theory, effect of poroelastic bed on flexural gravity wave motion is analyzed in both the cases of single-layer and two-layer fluids. The model for the flexural gravity waves is developed using linear water wave theory and small amplitude structural response in finite water depth. The effects of permeability and shear modulus of poroelastic bed and time period on flexural gravity wave motion are studied by analyzing the dispersion relation, phase speed, plate deflection, interface elevation and pressure distribution along water depth. Various results for surface gravity waves are analyzed as special cases. The study reveals that bed permeability retards the hydrodynamic pressure distribution along the water depth significantly compared to shear modulus whilst, floating plate deflection decreases significantly with change in shear modulus compared to permeability of the poroelastic bed. The present study can be generalized to analyze various wave–structure interaction problems over poroelastic bed.     

The effect of bonding layer properties on the dynamic behaviour of surface-bonded piezoelectric sensors

The dynamic behaviour of piezoelectric sensors depends on the bonding condition along the interface between the sensors and the host structure. This paper provides a comprehensive theoretical study of the effect of the bonding layer on the coupled electromechanical characteristics of a piezoelectric sensor bonded to an elastic substrate, which is subjected to a high frequency elastic wave. A sensor model with a viscoelastic bonding layer, which undergoes a shear deformation, is proposed to simulate the two dimensional electromechanical behaviour of the integrated system. Analytical solution of the problem is provided by using Fourier transform and solving the resulting integral equations in terms of the interfacial stress. Numerical simulation is conducted to study the effect of the bonding layer upon the dynamic response of the sensor under different loading frequencies. The results indicate that the modulus and the thickness of the bonding layer have significant effects on sensor response, but the viscosity of the bonding layer is relatively less important.     

螺旋侧板和水深对浮式风机动态响应的影响

为研究漂浮式风力机平台动态响应的优化措施,提出平台附加螺旋侧板的方式。建立基于Spar平台的5MW风力机整机模型,利用有限元软件进行水动力计算,得到不同水深条件下,风力机平台在风、浪、流载荷联合作用下的频域特性,通过与不附加螺旋侧板情况下的动态特性参数对比,探讨螺旋侧板是否对结构的频响特性起到提升作用。结果表明,附加螺旋侧板后,结构的垂荡和纵摇的运动幅值和所受波浪力均得到了显著抑制;与纵荡和纵摇相比,垂荡运动的幅值和所受波浪力所受影响更显著;水深变化对结构响应的幅值有着显著影响。     

Determining the shear modulus of water in experiments with a floating disk

The problem of decaying rotation of a disk floating on the surface of a viscoelastic fluid in a cylindrical container is solved by numerical methods. The motion is found to have the form of decaying oscillations observed previously for water. In addition to the viscosity coefficient, the constructed mathematical model of the viscoelastic fluid has two more independent parameters: shear modulus and time of relaxation of elastic stresses. Elastic parameters of water are determined through comparisons with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 100–103, January–February, 2008.     

Wave propagation through mangrove forests in the presence of a viscoelastic bed

The influence of viscoelastic ocean beds on the characteristics of surface waves passing through mangrove forests is analyzed under the assumption of linearized water wave theory in two dimensions. The trunks of the mangroves are assumed to be in the upper-layer inviscid fluid domain, whilst the roots are inside the viscoelastic bed. The associated equation of motion is obtained by coupling the Voigt’s model for flow within the viscoelastic medium with the equation of motion in the presence of mangroves. The modified dynamic conditions are coupled with the kinematic conditions to obtain the boundary condition at the free surface and the interface of the two fluids consisting of the upper layer inviscid fluid and the viscoelastic fluid bed. To understand the effects of bed viscosity as well as elasticity on energy dissipation, the complex dispersion relation associated with the plane progressive wave is derived and analyzed. Effect of physical parameters associated with mangroves and viscoelastic bed on wave motion in surface and internal modes are computed and analyzed to understand their roles in attenuating wave effects. The present model will be useful in the better understanding of wave propagation through mangroves in the coastal zone having muddy seabed.     

ANALYSIS ON IMPACT RESPONSES OF UNRESTRAINED PLANAR FRAME STRUCTURE(Ⅱ)—NUMERICAL EXAMPLE ANALYSIS

By using the formula derived in Part （ Ⅰ ）, the instant response of an unrestrained planar frame structure subjected to the impact of a moving rigid-body are evaluated and analysed. The impact force-time history between the structure and the moving rigid-body, shear force and bending moment distribution along the beams, axial force distribution along the bars were calculated. The wave propagation phenomena of the longitudinal wave in the bars, the flexural and shear waves in the beams were also analysed. The numerical results show that the time duration of impact force is controlled by the flexural wave and the longitudinal wave ; the shear effect in beams should not be neglected in the impact response analysis of structures.     

ANALYSIS ON IMPACT RESPONSES OF UNRESTRAINED PLANAR FRAME STRUCTURE(Ⅱ)-NUMERICAL EXAMPLE ANALYSIS

By using the formula derived in Part (Ⅰ), the instant response of an unrestrained planar frame structure subjected to the impact of a moving rigid-body are evaluated and analysed.The impact force-time history between the structure and the moving rigid-body, shear force and bending moment distribution along the beams, axial force distribution along the bars were calculated. The wave propagation phenomena of the longitudinal wave in the bars, the flexural and shear waves in the beams were also analysed. The numerical results show that the time duration of impact force is controlled by the flexural wave and the longitudinal wave; the shear effect in beams should not be neglected in the impact response analysis of structures.     

Structural vibration of flexural beams with thick unconstrained layer damping

In this paper, the dynamic behaviour of free layer damping beams with thick viscoelastic layer is analysed. A homogenised model for the flexural stiffness is formulated employing Reddy and Bickford’s quadratic shear in each layer, in contrast to the classical model of Oberst and Frankenfeld for thin beams, which does not take into account shear deformations. The results provided by these two models in free and forced vibration are compared by means of finite element procedures with those of a 2D model, which considers extensional and shear stress, and longitudinal, transverse and rotational inertias.The viscoelastic material is characterised by a fractional derivative model, which takes into consideration the variation of the complex modulus with frequency. To avoid the frequency dependence of the stiffness matrices, the extraction of the eigenvalues and eigenvectors is completed by a new iterative method developed by the authors. The frequency response to a harmonic force is deduced by the superposition of modal contribution functions.From these numerical applications it can be concluded that the model for thick beams provides sufficient accuracy for practical applications, able to reproduce the mechanical behaviour of free layer damping beams with thick viscoelastic layer, reducing the storage needs and computational time with respect to a 2D model.     

Effect of undulating bottom on wave interaction with a floating flexible plate coupled with a flexible porous barrier

Meccanica - In the present study, under the assumption of small amplitude water wave theory and structural response, effect of bed undulation on the wave interaction with a combination of flexible...     

An experimental study on the interactions between surface waves and floating viscoelastic covers

A possible mathematical ice model for the wave interactions in polar seas was developed based on the assumption that an ice cover behaved as a Voigt viscoelastic material. The dispersion relation was found to depend on the rheological properties of the cover. In the present study, an experimental approach was developed that can enable the verification of the theoretical predictions in the laboratory. The approach utilized the blended mixture of white oil and Polydimethylsiloxane (PDMS) material with various mass percentages of a curing agent, to create a floating layer with a range of targeted viscoelastic properties. Due to the large coverage required for wave flume experiments, special curing procedures were also established for the preparation of PDMS material. The rheological results showed that the mechanical behavior of the floating cover was close to a Voigt material. Experiments were conducted to analyze the wave interactions with the floating viscoelastic cover. The measured data showed an obvious change of wavelength when waves propagated along the cover region. It is observed that the change in wavelength can be linked quantitatively to the viscoelastic properties based on the numerical predictions by Wang and Shen (2010). Some differences were however noted for less viscous covers under longer wave periods. A direct comparison of the PDMS covers with a polypropylene (PP) cover was also performed for verification. Only wave lengthening was observed under the PP cover. With a shear modulus more than three orders of magnitude greater than that of PDMS, the theoretical wavelength for the PP cover from Wang and Shen (2010) is very close to that of the thin elastic plate theory from Fox and Squire (1990). Comparison between these two theoretical results and the measured data again deviated with longer wave periods. In both PDMS and PP cases, edge effects and pitching motion of the covers were present at various degrees. In addition, the materials were not strictly a Voigt type. The small deviation from the idealized rheological behavior could also contribute to differences between theoretical and experimental results.     

Rheological properties of disperse systems at low shear stresses

Three-dimensional network structures can be built up in disperse systems due to long-range colloidal interactions between the dispersed particles. The rheological behaviour of such coagulation structures has been studied by means of creep and recovery experiments at low shear stresses, i.e. by measuring the shear strain as a function of time under constant stress and after removal of stress. Measurements of this type give insight into the elastic and viscous deformations and the retardation times necessary to reach equilibrium or steady-state conditions.Results obtained with dispersions of pigments in polymer solutions and with monodisperse polymer latexes indicate the existence of an equilibrium state at low shear stresses with a predominant elastic deformation and a high viscosity suggesting that the disperse systems investigated do not behave exactly as rigid gels but apparently exhibit a dynamic equilibrium of structural break-down and formation under applied stress. This behaviour is approximately described by a 4-parameter-model with an instantaneous and a steady-state compliance, one retardation time, and a viscosity.At higher shear stresses thixotropic structural break-down occurs resulting in a transition from the rheological behaviour described here to a liquid-like state with a comparatively low viscosity. In this stress range the viscoelastic properties become strongly time-dependent.These measurements give evidence of the presence of two types of deformation: an instantaneous, purely elastic deformation attributable to the unperturbed coagulation structure and the creep-recovery behaviour of an elastic liquid apparently related to the breaking and re-forming of bonds.     

Dynamics of a linearly inhomogeneous incompressible isotropic elastic half-space

The study presented in this paper treats the harmonic and transient wave motion of an incompressible isotropic semi-infinite elastic medium with a shear modulus increasing linearly with depth. The medium has a constant mass density and an initial hydrostatic stress distribution due to a constant gravity. In particular, attention is given to the case of a vanishing top rigidity. For this case it is shown that the governing equations resemble the equations governing the deep water motion, and that under normal loading the behaviour of the upper surface resembles that of the upper surface of (deep) water.     

Vibrations of a Floating Elastic Plate Upon Nonlinear Interaction of Flexural-Gravity Waves

The method of many scales was used to construct an asymptotic expansion up to thirdorder terms for the velocity potential of a fluid of finite depth and the flexural deformations of a floating elastic plate arising from the interaction of harmonics of finite-amplitude progressive surface waves. An expression for the second-harmonic amplitude was obtained, and critical values of the wavenumber were determined. Vibrations of plates with different thicknesses and elastic modulus were analyzed. Vertical displacements of the plate under flexural deformation were studied.     

An intriguing empirical rule for computing the first normal stress difference from steady shear viscosity data for concentrated polymer solutions and melts

The Cox–Merz rule and Laun’s rule are two empirical relations that allow the estimation of steady shear viscosity and first normal stress difference, respectively, using small amplitude oscillatory shear measurements. The validity of the Cox–Merz rule and Laun’s rule imply an agreement between the linear viscoelastic response measured in small amplitude oscillatory shear and the nonlinear response measured in steady shear flow measurements. We show that by using a lesser-known relationship also proposed by Cox and Merz, in conjunction with Laun’s rule, a relationship between the rate-dependent steady shear viscosity and the first normal stress difference can be deduced. The new empirical relation enables a priori estimation of the first normal stress difference using only the steady flow curve (i.e., viscosity vs shear rate data). Comparison of the estimated first normal stress difference with the measured values for six different polymer solutions and melts show that the empirical rule provides values that are in reasonable agreement with measurements over a wide range of shear rates, thus deepening the intriguing connection between linear and nonlinear viscoelastic response of entangled polymeric materials.     

Wave trapping by porous barrier in the presence of step type bottom

The present study deals with the trapping of oblique wave by porous barrier located near a rigid wall in the presence of a step type bottom bed. The solution of the physical problem is obtained using the eigenfunction expansion method and multi-mode approximation associated with modified mild-slope equation. Assuming that the porous structure is made of materials having fine pores, the mathematical problem is handled for solution by matching the velocity and pressure at interface boundaries. Various numerical results are computed and analyzed to understand the role of bed profiles, structural porosity, depth ratio, oblique angle of incidence, distance between barrier and step edge and, the distance between the porous barrier and rigid wall in optimizing wave reflection and load on the structure/rigid-wall. A comparison of results on wave trapping by porous barriers over flat and undulated bed reveals that for the same distance between the porous barrier and rigid wall, more number of times optimum reflection occurs in case of undulated bed. The present study is likely to be of immense importance in the design of coastal structures for protecting coastal infrastructures.     

Hydroelastic analysis of surface wave interaction with concentric porous and flexible cylinder systems

The present study deals with the hydroelastic analysis of gravity wave interaction with concentric porous and flexible cylinder systems, in which the inner cylinder is rigid and the outer cylinder is porous and flexible. The problems are analyzed in finite water depth under the assumption of small amplitude water wave theory and structural response. The cylinder configurations in the present study are namely (a) surface-piercing truncated cylinders, (b) bottom-touching truncated cylinders and (c) complete submerged cylinders extended from free surface to bottom. As special cases of the concentric cylinder system, wave diffraction by (i) porous flexible cylinder and (ii) flexible floating cage with rigid bottom are analyzed. The scattering potentials are evaluated using Fourier–Bessel series expansion method and the least square approximation method. The convergence of the double series is tested numerically to determine the number of terms in the Fourier–Bessel series expansion. The effects of porosity and flexibility of the outer cylinder, in attenuating the hydrodynamic forces and dynamic overturning moments, are analyzed for various cylinder configurations and wave characteristics. A parametric study with respect to wave frequency, ratios of inner-to-outer cylinder radii, annular spacing between the two cylinders and porosities is done. In order to understand the flow distribution around the cylinders, contour plots are provided. The findings of the present study are likely to be of immense help in the design of various types of marine structures which can withstand the wave loads of varied nature in the marine environment. The theory can be easily extended to deal with a large class of problems associated with acoustic wave interaction with flexible porous structures.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Uni-modal destabilization of a visco-elastic floating ice layer by wind stress

A destabilization by wind stress of a homogeneous visco-elastic ice layer of infinite horizontal extent and finite thickness floating on a water layer of finite depth is studied. The water is assumed to be weakly compressible; the viscous dissipation in the water layer is shown to be negligible compared to that in the ice layer. In the model, we assume that a homogeneous wind shear stress is applied to the upper surface of the ice layer at near field and the compression within the ice layer is fixed below the maximum admissible value, i.e. below the value above which ice can no longer be treated as an elastic material. The effect of viscosity is shown to stabilize the acoustic mode and all the unstable seismic modes that in a purely elastic model, treated by Brevdo and Il'ichev [Brevdo, L., Il'ichev, A., 2001. Multi-modal destabilization of a floating ice layer by wind stress. Cold Reg. Sci. Technol. 33, 77–89], possess unbounded growth rates for a growing wave number. The buckling mode is unstable in the domain of parameters considered. In all the cases treated, the model is marginally absolutely stable. The localized unstable disturbances propagate against the wind. The spatially amplifying waves in the model amplify in the direction opposite to the wind direction. The stability results are well approximated by those in a Kirchoff–Love thin ice plate model.     

Who conceived the “complex viscosity”?

In small amplitude oscillatory shear flow, we now universally represent the response of a viscoelastic fluid as a complex quantity, which we call “complex viscosity.” This short piece deepens our understanding of the origins of the complex viscosity and of the man who coined this term, now widely used in rheology.     

Effect of a Shock Pulse on a Floating Ice Sheet

The vibrations of a viscoelastic plate lying on an elastic liquid base subjected to pulse loading have been studied theoretically and experimentally. The effect of the variable depth of the reservoir, plate thickness, and strain relaxation time on the value of the plate vibration amplitude and the length and curvature of the flexural gravity wave profile are analyzed. Good agreement of theoretical and experimental results is obtained.