On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles

This paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles. A point model is used in the vertical, with a no-slip condition at the sea bed. In the finite difference approach the high-shear bottom layer is resolved using either a regular grid on a logarithmic or log-linear transformed co-ordinate or an irregular grid, varying in such a manner as to retain second-order accuracy. The accuracy of these various grid schemes is considered in detail. The relative merits of using either the Crank-Nicolson or Dufort-Frankel time integration methods are considered; in the case of a fine grid in a high-viscosity region, some numerical problems are found with the Dufort-Frankel method. An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high-shear bottom boundary layer. Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers. The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near-bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.     

An experimental investigation into non-linear wave loading on horizontal axis tidal turbines

Tidal turbines are subject to large hydrodynamic loads from combinations of currents and waves, which contribute significantly to fatigue, extreme loading and power flow requirements. Physical model testing enables these loads and power fluctuations to be assessed and understood in a controlled and repeatable environment. In this work, a 1:15 scale tidal turbine model is utilised to further the fundamental understanding of the influence of waves on tidal turbines. A wide range of regular waves are generated in both following-current and opposing-current conditions. Wave frequencies range from 0.31 Hz to 0.55 Hz & wave heights from 0.025 m to 0.37 m in a fixed 0.81 m/s current velocity. Waves are selected and programmed specifically to facilitate frequency domain analysis, and techniques are employed to isolate the effect of non-linear waves on turbine power and thrust.Results demonstrate that wave action induces large variations in turbine power and thrust compared to current only conditions. For the range of conditions tested, peak values of thrust and power exceed current-only values by between 7%–65% and 13%–160% respectively. These wave-induced fluctuations are shown to increase with wave amplitude and decrease with wave frequency. Following wave conditions exhibit greater variations than opposing for waves with the same wave height and frequency due to the lower associated wavenumbers.A model is developed and presented to aid the understanding of the high-order harmonic response of the turbine to waves, which is further demonstrated using steady state coefficients under assumptions of pseudo-stationarity. This approach is proven to be effective at estimating wave-induced power and thrust fluctuations for the combinations of waves, currents and turbine state tested. The outcome of which shows promise as a rapid design tool that can evaluate the effect of site-specific wave–current conditions on turbine performance.     

On the numerical solution of the turbulence energy equations for wave and tidal flows

This paper deals with the numerical solution, using finite difference methods, of the hydrodynamic and turbulence energy equations which describe wind wave and tidally induced flow. Calculations are performed using staggered and non-staggered finite difference grids in the vertical, with various time discretizations of the production and dissipation terms in the turbulence energy equations. It is shown that the time discretization of these terms can significantly influence the stability of the solution. The effect of time filtering on the numerical stability of the solution is also considered. The form of the mixing length is shown to significantly influence the bed stress in wind wave problems. A no-slip condition is applied at the sea bed, and the associated high-shear bottom boundary layer is resolved by transforming the equations onto a logarithmic or log-linear co-ordinate system before applying the finite difference scheme. A computationally economic method is developed which remains stable even when a very fine vertical grid (over 200 points) is used with a time step of up to 30 min.     

Linear and non-linear vibration and frequency response analyses of microcantilevers subjected to tip-sample interaction

Despite their simple structure and design, microcantilevers are receiving increased attention due to their unique sensing and actuation features in many MEMS and NEMS. Along this line, a non-linear distributed-parameters modeling of a microcantilever beam under the influence of a nanoparticle sample is studied in this paper. A long-range Van der Waals force model is utilized to describe the microcantilever-particle interaction along with an inextensibility condition for the microcantilever in order to derive the equations of motion in terms of only one generalized coordinate. Both of these considerations impose strong nonlinearities on the resultant integro-partial equations of motion. In order to provide an understanding of non-linear characteristics of combined microcantilever-particle system, a geometrical function is wisely chosen in such a way that natural frequency of the linear model exactly equates with that of non-linear model. It is shown that both approaches are reasonably comparable for the system considered here. Linear and non-linear equations of motion are then investigated extensively in both frequency and time domains. The simulation results demonstrate that the particle attraction region can be obtained through studying natural frequency of the system consisting of microcantilever and particle. The frequency analysis also proves that the influence of nonlinearities is amplified inside the particle attraction region through bending or shifting the frequency response curves. This is accompanied by sudden changes in the vibration amplitude estimated very closely by the non-linear model, while it cannot be predicted by the best linear model at all.     

Modelling the initial motion of large cylindrical and spherical bubbles

A two-dimensional, transient, finite difference technique based on a volume fraction specification of the free surface position and accounting for the effects of surface tension is shown to accurately predict the initial motion of large cylindrical and spherical bubbles. The predictions compare very favourably with the experimental data of Walters and Davidson. The initial acceleration of cylindrical and spherical bubbles is properly predicted as g and 2g respectively. The penetration of a tongue of liquid from below is the dominant process by which large deformations from the original shape take place and is well predicted by the model in both cases. For the spherical case the eventual transition into a toroidal bubble is demonstrated and the circulation associated with a rising toroidal bubble as a function of its volume upon release is shown to agree very well with experiments. Iterative linear equation-solving techniques applicable to the special nature of the linear system resulting from such a free surface specification are surveyed and a simple Jacobi iteration based on red-black ordering is found to perform well. The impact of the free surface on the relaxation of the linear system and the convergence criteria is also explored.     

Phase velocity effects of the wave interaction with exponentially sheared current

The nonlinear interaction between the unidirectional bichromatic wave-train and exponentially sheared current in water of an infinite depth is investigated. The model is based on the vorticity transport equation and the exact free surface conditions, without any assumptions for the existence of small physical parameters. Earlier works of the wave–current interaction were mainly restricted to either current acted on the monochromatic wave or irregular waves limited to irrotational current. Different from these previous works, no constraint is made in our model for amplitudes of the primary wave, and the current owns an exponential type profile along the vertical line. To ensure that the effect of vorticity on the phase velocity is consistent with earlier derivation, the case of a small amplitude wave traveling on the exponentially sheared current is examined firstly. Then the effect of nonlinearity on the phase velocity of primary waves in a bichromatic wave-train is considered. Accurate high-order approximations of the phase velocity are obtained under consideration of both the nonlinear wave self–self and mutual interactions. Finally, the combined effect of vorticity and nonlinearity on the phase velocity is investigated through the case of a bichromatic wave-train propagating on an exponentially sheared current. It is found that the characteristic current slope determines the effect of vorticity on the phase velocity caused by nonlinear wave self–self and mutual interactions, and the surface current strength may amplify/reduce this effect.     

On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities

This paper examines the validity of non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. As an example, we treat a hinged-hinged Euler-Bernoulli beam resting on a non-linear elastic foundation with distributed quadratic and cubic non-linearities, and investigate the primary (Ω≈ω_n) and subharmonic (Ω≈2ω_n) resonances, in which Ω and ω_n are the driving and natural frequencies, respectively. The steady-state responses are found by using two different approaches. In the first approach, the method of multiple scales is applied directly to the governing equation that is a non-linear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure, and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained numerically by the finite difference method.     

The effect of non-linear interaction between gas and particle velocities on the hydrodynamic stability in the Blasius boundary layer

A weakly non-linear stability analysis of two phase flow in the Blasius boundary layer has been carried out. Two mathematical models have been established based on the perturbation shape preserved assumption and linear stability model of two phase flow proposed by Stuart [On the non-linear mechanics of hydrodynamic stability, J. Fluid Mech. 4 (1958) 1-21] and Saffman [On the stability of laminar flow of dusty gas, J. Fluid Mech. 13 (1962) 120-128], respectively. The perturbation model and the perturbation energy balance equation are solved numerically with Chebyshev spectral method and artificial boundary condition. The numerical program adopted in the present study is verified by comparison with former works. The results show that the non-linear interaction between mean flow and perturbation reduces the growth rate of perturbation, while the non-linear interaction between particle phase and gas phase increases the growth rate of perturbation amplitude. The distortion of the mean flow caused by the Reynolds stress modifies the rate of transfer of energy from the mean flow to disturbance. The existence of particle alleviates the distortedness. The result also indicates that the weakly non-linear stability theory is consistent to linear stability theory, and the addition of fine and coarse particles reduces and increases the critical Reynolds number.     

Finite element modeling of tire/terrain interaction: Application to predicting soil compaction and tire mobility

Tire/terrain interaction has been an important research topic in terramechanics. For off-road vehicle design, good tire mobility and little compaction on terrain are always strongly desired. These two issues were always investigated based on empirical approaches or testing methods. Finite element modeling of tire/terrain interaction seems a good approach, but the capability of the finite element has not well demonstrated. In this paper, the fundamental formulations on modeling soil compaction and tire mobility issues are further introduced. The Drucker-Prager/Cap model implemented in ABAQUS is used to model the soil compaction. A user subroutine for finite strain hyperelasticity model is developed to model nearly incompressible rubber material for tire. In order to predict transient spatial density, large deformation finite element formulation is used to capture the configuration change, which combines with soil elastoplastic model to calculate the transient spatial density due to tire compaction on terrain. Representative simulations are provided to demonstrate how the tire/terrain interaction model can be used to predict soil compaction and tire mobility in the field of terramechanics.     

激光陀螺相邻模零偏差及不平衡电流零偏差性能分析

通过控制参量提高陀螺精度和成品率,是激光陀螺工程技术研究的主要内容之一。引入相邻模陀螺零偏差及不平衡电流零偏差两个参量,对不同等级表陀螺对应参量的研究,发现陀螺性能与该参量具有确定的关系,通过控制这两个参量,可以有效提高陀螺使用精度和成品率。理论研究了两个参数变化的根源,分别为谐振腔参数及锁区的变化,为提高激光陀螺精度提供了一种有效的技术途径,对于进一步提高抖动陀螺精度具有实用价值。     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

A one-dimensional moving grid solution for the coupled non-linear equations governing multiphase flow in porous media. 2: Example simulations and sensitivity analysis

This paper presents numerical examples for the moving grid finite element algorithm derived in Part Ito solve the non-linear coupled set of PDEs governing immiscible multiphase flow in porous media in one dimension. Examples include single- and double-front simulations for two- and three-phase flow regimes and incorporating a mass sink. The modelling approach is shown to achieve significant savings in computation time and memory allocation when compared with fixed grid solutions of equivalent accuracy. This work includes sensitivity analyses for the parameters which are incorporated in the grid adaptation method, including the curvature weights, artificial viscosity and artificial repulsive force. It is found that the curvature weights are exponential functions of the negative ratio of the square root of the domain length to the number of discrete nodes. These weighting parameters are also shown to depend upon the shape of the front. On the basis of the examined simulations, it is recommended that artificial viscosity be neglected in the solution of the coupled non-linear set of PDEs governing multiphase flow in porous media. Similarly, use of a repulsive force is found to be unnecessary in simulations involving the migration of two liquid phases. For multiphase flows incorporating a gas phase it is recommended to use a non-zero value for the repulslive force to avoid development of an ill-conditioned nodal distribution matrix. An equation to evaluate the repulsive force under these circumstances is suggested.     

The influence of model parameters and of the thermomechanical coupling on the behavior of shape memory devices

Shape memory materials (SMM) are receiving increasing attention for their use in applications that exploit their dynamic behavior. A thermomechanical model for devices with pseudoelastic behavior has been proposed in previous works [11] (Bernardini and Pence, 2005) [15] (Bernardini and Rega, 2005). The model takes into account several aspects of SMM behavior by means of seven model parameters.In this paper the effect of each parameter on the non-isothermal rate-dependent behavior of the device is studied, by paying particular attention to the effect of the thermomechanical coupling. Some overall synthetic indicators of the behavior of the shape memory device are defined in terms of the model parameters. By evaluating such indicators, a lot of information about the mechanical, thermal and thermomechanical effects on the device behavior can be gained before computing explicitly the response of the shape memory oscillator.The present work may provide a guide for the proper utilization of the model for the investigation of the dynamic response.     

Impact of density stratification on the global mode in a swirling jet: Stochastic modelling and Lagrangian coherent structures

In an experimental investigation, the stochastic dynamics of the global mode in a turbulent swirling jet are considered. From the application of the swirling jet in gas turbine combustors, it was observed that a specific density gradient in the flow leads to a suppression of the global mode. This phenomenon was replicated in a generic swirling jet using an electrical heating coil placed inside the breakdown bubble. In the present investigation, the dynamics of the global mode obtained from PIV and pressure measurements are analysed using a stochastic reduced-order model to describe the instability. The stochastic model is necessary to explain the interaction between the deterministic dynamics of the global mode and the perturbations by the background turbulence. The calibration of the stochastic model provides the amplification rate of the global mode that defines the transition of the flow, dependent on the swirling strength and the density difference. The spatial structure of the global mode is further investigated from Lagrangian coherent structures of the flow field which are computed from the 3D time-resolved velocity field reconstruction based on planar PIV measurements. The Lagrangian visualisations and schlieren visualisations are used to explain the absence of the density effects on the global mode at larger Reynolds numbers. The analysis gives a detailed view of the stochastic dynamics of a hydrodynamic instability in a turbulent flow.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses

The available accurate shell theories satisfy the interlaminar transverse stress continuity conditions based on linear strain-displacement relations. Furthermore, in majority of these theories, either influence of the transverse normal stress and strain or the transverse flexibility of the shell has been ignored. These effects remarkably influence the non-linear behavior of the shells especially in the postbuckling region. Furthermore, majority of the buckling analyses performed so far for the laminated composite and sandwich shells have been restricted to linear, static analysis of the perfect shells. Moreover, almost all the available shell theories have employed the Love-Timoshenko assumption, which may lead to remarkable errors for thick and relatively thick shells. In the present paper, a novel three-dimensional high-order global-local theory that satisfies all the kinematic and the interlaminar stress continuity conditions at the layer interfaces is developed for imperfect cylindrical shells subjected to thermo-mechanical loads.In comparison with the layerwise, mixed, and available global-local theories, the present theory has the advantages of: (1) suitability for non-linear analyses, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions, considering the transverse flexibility, and releasing the Love-Timoshenko assumption, (3) less required computational time due to using the global-local technique and matrix formulations, and (4) capability of investigating the local phenomena. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.     

Performance of hydrodynamic journal bearing under the combined influence of textured surface and journal misalignment: A numerical survey

A wisely chosen geometry of micro textures with the favorable relative motion of lubricated surfaces in contacts can enhance tribological characteristics. In this paper, a computational investigation related to the combined influence of bearing surface texturing and journal misalignment on the performances of hydrodynamic journal bearings is reported. To this end, a numerical analysis is performed to test three texture shapes: square “SQ”, cylindrical “CY”, and triangular “TR”, and shaft misalignment variation in angle and degree. The Reynolds equation of a thin viscous film is solved using a finite differences scheme and a mass conservation algorithm (JFO boundary conditions), taking into account the presence of textures on both full film and cavitation regions. Preliminary results are compared with benchmark data and are consistent with a positive enhancement in misaligned bearing performances (load carrying capacity and friction). The results suggest that the micro-step bearing mechanism is a key parameter, where the micro-pressure recovery action present in dimples located at the second angular part of the bearing (from 180° to 360°) can compensate for the loss on performances caused by shaft misalignment, while the micro-pressure drop effect at the full film region causes poor performances. Considering the right arrangement of textures on the contact surface, their contours geometries can have a significant impact on the performance of misaligned journal bearings, particularly at high eccentricity ratios, high misalignment degrees and when the misalignment angle α approaches to $0\mathrm{°}$ or $180\mathrm{°}$.     

Effect of Schmidt number on the structure and propagation of density currents

The results of a numerical study of two- and three-dimensional Boussinesq density currents are described. They are aimed at exploring the role of the Schmidt number on the structure and dynamics of density driven currents. Two complementary approaches are used, namely a spectral method and a finite-volume interface capturing method. They allow for the first time to describe density currents in the whole range of Schmidt number 1 ≤ Sc ≤ ∞ and Reynolds number 10² ≤ Re ≤ 10⁴. The present results confirm that the Schmidt number only weakly influences the structure and dynamics of density currents provided the Reynolds number of the flow is large, say of O(10⁴) or more. On the contrary low- to moderate-Re density currents are dependant on Sc as the structure of the mixing region and the front velocities are modified by diffusion effects. The scaling of the characteristic density thickness of the interface has been confirmed to behave as (ScRe)^−1/2. Three-dimensional simulations suggest that the patterns of lobes and clefts are independent of Sc. In contrast the Schmidt number is found to affect dramatically (1) the shape of the current head as a depression is observed at high-Sc, (2) the formation of vortex structures generated by Kelvin–Helmholtz instabilities. A criterion is proposed for the stability of the interface along the body of the current based on the estimate of a bulk Richardson number. This criterion, derived for currents of arbitrary density ratio, is in agreement with present computed results as well as available experimental and numerical data.      

The effect of the couple stress parameter and Prandtl number on the transient natural convection flow over a vertical cylinder

An analysis is performed to study transient free convective boundary layer flow of a couple stress fluid over a vertical cylinder, in the absence of body couples. The solution of the time-dependent non-linear and coupled governing equations is carried out with the aid of an unconditionally stable Crank-Nicolson type of numerical scheme. Numerical results for the steady-state velocity, temperature as well as the time histories of the skin-friction coefficient and Nus- selt number are presented graphically and discussed. It is seen that for all flow variables as the couple stress control parameter, Co, is amplified, the time required for reaching the temporal maximum increases but the steady-state decreases.     

Transmural pressure effects on the stability of clamped cylindrical shells subjected to internal fluid flow: Theory and experiments

An experimental and theoretical parametric study is undertaken to investigate the effect of transmural pressure on the non-linear dynamics and stability of circular cylindrical shells with clamped ends subjected to internal fluid flow. The theoretical structural model is based on the Donnell non-linear shallow shell theory, and potential flow theory is employed to describe the fluid-structure interaction. It is found that, for low transmural pressures in the range investigated, the shell loses stability by static subcritical divergence, while for higher transmural pressures the loss of stability is supercritical. In addition, there are ranges of flow velocity in which the shell exhibits quasi-periodic or even chaotic behaviour.