Liquid sloshing in a horizontally forced vessel with bottom topography

This paper presents a numerical study of the free-surface evolution for inviscid, incompressible, irrotational, horizontally forced sloshing in a two-dimensional rectangular vessel with an inhomogeneous bottom topography. The numerical scheme uses a time-dependent conformal mapping to map the physical fluid domain to a rectangle in the computational domain with a time-dependent aspect ratio Q(t), known as the conformal modulus. The advantage of this approach over conventional potential flow solvers is the solution automatically satisfies Laplace's equation for all time, hence only the integration of the two free-surface boundary conditions is required. This makes the scheme computationally fast, and as grid points are required only along the free-surface, high resolution simulations can be performed which allows for simulations for mean fluid depths close to the shallow water water regime. The scheme is robust and can simulate both resonate and non-resonate cases, where in the former, the large amplitude waves are well predicted.Results of nonlinear simulations are presented in the case of non-breaking waves for both an asymmetrical ‘step’ and a symmetric ‘hump’ bottom topography. The natural free-sloshing mode frequencies are compared with the small topography asymptotic results of Faltinsen and Timokha (2009) (Sloshing, Cambridge University Press (Cambridge)), and are found to be lower than this asymptotic prediction for moderate and large topography magnitudes. For forced periodic oscillations it is shown that the hump profile is the most effective topography for minimizing the nonlinear response of the fluid, and hence this topography would reduce the stresses on the vessel walls generated by the fluid. Results also show that varying the width of the step or hump has a less significant effect than varying its magnitude.     

Improved methods for thin, surface boundary layer investigations

 New techniques are developed to improve the velocity flow-field measurement capability within a free-surface boundary layer region on which progressive capillary-gravity waves are present. Due to the extremely thin but rather vortical characteristics of the aforementioned boundary layer, conventional particle image velocimetry (PIV) methods fail to estimate velocity (and vorticity) vectors at an acceptable detection rate. This failure is a direct consequence of optimal PIV parameters that are difficult to achieve in practice for such flow situations. A new technique, Sub-pattern PIV, is developed. This method has features similar to both the super-resolution PIV (Keane et al. 1995) and the particle image distortion (PID) technique (Huang et al. 1993), but is predicated upon a very differential philosophy. Another difficulty that arises in experiments to investigate surface boundary layer flows is that the oscillating and deforming air–water interface has a mirror-like behavior that affects the images, and generates very noisy data. An alternative experimental setup that utilizes the Brewster angle phenomenon is adopted and the specular effects of the free-surface are removed successfully. This Brewster angle imaging, along with the Sub-pattern PIV technique, is used for the target application – a free-surface boundary layer investigation. It proved to be very effective. The methodology of both techniques is discussed, and the modified PIV procedure is validated by numerical probabilistic simulations. Application to the capillary-gravity wave boundary layer is presented in a subsequent paper. Received: 31 July 1997/Accepted: 4 February 1998     

A PSEUDOSPECTRAL σ -TRANSFORMATION MODEL OF 2-D NONLINEAR WAVES

A pseudospectral matrix-element method is proposed for the analysis of 2-D nonlinear time-domain free-surface flow problems. The Chebyshev expansion technique established by Ku & Hatziavramidis has been used to discretize the σ-transformed governing equations including nonlinear boundary conditions. Simulations of non overturning transient waves in fixed and base-excited tanks are presented. The results are compared with first-and second-order analytical solutions for sloshing and standing waves, respectively. Excellent agreement is achieved at low values of wave steepness, with the high accuracy due to the close coupling between points. As the wave steepness increases, the influence of higher-order nonlinear components becomes significant, and is modelled by the present scheme. The solution is extremely stable, with the σ-transformation exactly fitting the free-surface boundary, unlike other schemes which have to use free-surface smoothing.     

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.     

Multicloud Convective Parametrizations with Crude Vertical Structure

Recent observational analysis reveals the central role of three multi-cloud types, congestus, stratiform, and deep convective cumulus clouds, in the dynamics of large scale convectively coupled Kelvin waves, westward propagating two-day waves, and the Madden–Julian oscillation. The authors have recently developed a systematic model convective parametrization highlighting the dynamic role of the three cloud types through two baroclinic modes of vertical structure: a deep convective heating mode and a second mode with low level heating and cooling corresponding respectively to congestus and stratiform clouds. The model includes a systematic moisture equation where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep convective precipitation and a nonlinear switch which favors either deep or congestus convection depending on whether the troposphere is moist or dry. Here several new facets of these multi-cloud models are discussed including all the relevant time scales in the models and the links with simpler parametrizations involving only a single baroclinic mode in various limiting regimes. One of the new phenomena in the multi-cloud models is the existence of suitable unstable radiative convective equilibria (RCE) involving a larger fraction of congestus clouds and a smaller fraction of deep convective clouds. Novel aspects of the linear and nonlinear stability of such unstable RCE’s are studied here. They include new modes of linear instability including mesoscale second baroclinic moist gravity waves, slow moving mesoscale modes resembling squall lines, and large scale standing modes. The nonlinear instability of unstable RCE’s to homogeneous perturbations is studied with three different types of nonlinear dynamics occurring which involve adjustment to a steady deep convective RCE, periodic oscillation, and even heteroclinic chaos in suitable parameter regimes.     

Interaction of viscous wakes with a free surface

The interaction of laminar wakes with.free-surface waves generated by a moving body beneath the surface of an incompressible viscous fluid of infinite depth was investigated analytically. The analysis was based on the steady Oseen equations for disturbed flows.The kinematic and dynamic boundary conditions were linearized for the small-amplitude free-surface waves. The effect of the moving body was mathematically modeled as an Oseenlet.The disturbed flow was regarded as the sum of an unbounded singular Oseen flow which represents the effect of the viscous wake and a bounded regular Oseen flow which represents the influence of the free surface. The exact solution for the free-surface waves was obtained by the method of integral transforms. The asymptotic representation with additive corrections for the free-surface waves was derived by means of Lighthill‘s two-stage scheme. The symmetric solution obtained shows that the amplitudes of the free-surface waves are exponentially damped by the presences of viscosity and submergence depth.     

On Nonlinear Baroclinic Waves and Adjustment of Pancake Dynamics

Three-dimensional nonhydrostatic Euler–Boussinesq equations are studied for Bu=O(1) flows as well as in the asymptotic regime of strong stratification and weak rotation. Reduced prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure/thermal wind imbalance) are analyzed. We describe classes of nonlinear anisotropic ageostrophic baroclinic waves which are generated by the strong nonlinear interactions between the quasi-geostrophic modes and inertio-gravity waves. In the asymptotic regime of strong stratification and weak rotation we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in the horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak rotation. Vertical slanting of these fronts is proportional to μ^−1/2 where μ is the ratio of the Coriolis and Brunt–V?is?l? parameters. These fronts select slow baroclinic waves through nonlinear adjustment of the horizontal scale to the vertical scale by weak rotation, and are the envelope of inertio-gravity waves. Mathematically, this is generated by asymptotic hyperbolic systems describing the strong nonlinear interactions between waves and potential vorticity dynamics. This frontogenesis yields vertical “gluing” of pancake dynamics, in contrast to the independent dynamics of horizontal layers in strongly stratified turbulence without rotation. Received 8 April 1997 and accepted 29 March 1998     

Averaging over Fast Gravity Waves for Geophysical Flows with Unbalanced Initial Data

Various facets of recent mathematical theories for averaging over fast gravity waves on advective time scales for geophysical flows with unbalanced initial data are presented here including nonlinear Rossby adjustment and simplified reduced dynamics. This work is presented within the context of simplified geophysical models involving the rotating shallow-water equations and the rotating stably stratified Boussinesq equations. Novel mechanisms for enhanced gravity wave dissipation through the catalytic interaction with potential vortical modes are also developed here within the context of the rotating shallow-water equations. Received 2 May 1997 and accepted 20 August 1997     

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

Selective enhancement of oblique waves caused by finite amplitude second mode in supersonic boundary layer

Nonlinear interactions of the two-dimensional(2D) second mode with oblique modes are studied numerically in a Mach 6.0 flat-plate boundary layer, focusing on its selective enhancement effect on amplification of different oblique waves. Evolution of oblique modes with various frequencies and spanwise wavenumbers in the presence of 2D second mode is simulated successively, using a modified parabolized stability equation(PSE) method, which is able to simulate interaction of two modes with different frequencies efficiently. Numerical results show that oblique modes in a broad band of frequencies and spanwise wavenumbers can be enhanced by the finite amplitude 2D second mode instability wave. The enhancement effect is accomplished by interaction of the 2D second mode, the oblique mode, and a forced mode with difference frequency. Two types of oblique modes are found to be more amplified, i.e., oblique modes with frequency close to that of the 2D second mode and low-frequency first mode oblique waves. Each of them may correspond to one type of transition routes found in transition experiments. The spanwise wavenumber of the oblique wave preferred by the nonlinear interaction is also determined by numerical simulations.     

Initial Conditions and Near-Field Dynamics in Turbulent Liquid Sheets

High-speed liquid “curtains” have been proposed to protect solid structures in fusion energy applications. Minimizing free-surface waves and spray formation in such flows is important for effective protection in this application. In this work, free-surface waves and turbulent breakup were studied experimentally in jets of water issuing from a rectangular nozzle into ambient air at a Reynolds number of 1.2 × 10⁵. Laser-Doppler anemometry was used to characterize the streamwise and transverse velocity components in the nozzle for two different flow calming section designs. Planar laser-induced fluorescence was used to measure the free-stream position in the near-field of the sheet. The results suggest that transverse velocity fluctuations in the nozzle are the primary factor in determining the amplitudes of free-surface waves. Removing a small amount of low-speed fluid immediately downstream of the nozzle exit (“boundary-layer cutting”) is shown to both significantly reduce free-surface waves and the amount of spray due to turbulent breakup. Overall, boundary-layer cutting appears to have the greatest benefit when used on a “well-conditioned” turbulent liquid sheet.     

Component Mode Synthesis Using Nonlinear Normal Modes

This paper describes a methodology for developing reduced-order dynamic models of structural systems that are composed of an assembly of nonlinear component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes (NNMs). These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface displacements). A class of systems is used to demonstrate the concept and show the effectiveness of the proposed procedure. Simulation results show that the reduced-order model (ROM) obtained from the proposed procedure outperforms the ROM obtained from the classical fixed-interface linear CMS approach as applied to a nonlinear structure. The proposed method is readily applicable to large-scale nonlinear structural systems that are based on finite-element models.     

Quantification of phase shift in the simulation of shallow water waves

Numerical simulation of nonlinear waves to reproduce the laboratory measurements has been a topic of great interest in the recent past. The results reported in the literature are mainly focused on qualitative comparison or on the relative errors between the numerical simulation and measurements in laboratory and hence lack in revealing the existence of phase shift in nonlinear wave simulation. In this paper, the simulation of nonlinear waves in mixed Eulerian and Lagrangian framework using finite element method (FEM) is investigated by applying two different velocity calculation methods viz, cubic spline and least squares (LS). The simulated wave surface elevation has been compared with the experimental measurements. The coherence analysis has been carried out using the wavelet transformation, which gives a better understanding between the numerical and the experimental results with respect to the time–frequency space, compared with the conventional Fourier transformation. It is observed that the application of cubic spline approach leads to a higher phase difference for steeper waves. The present study has shown that the phase difference exists at the higher modes rather than at the primary period. For waves with steepness (wave height/wave length) higher than 0.04, LS approach is found to be effective in capturing the higher‐order frequency components in the event of nonlinearity. In addition, the comparison of numerical simulations with that from PIV measurements for the tests with solitary waves is also reported. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear interaction of a vortex pair with clean and surfactant-covered free surfaces

Two-dimensional unsteady viscous-flow problem associated with the normal incidence of a counter-rotating vortex pair on a free surface is analyzed. Effects of surface tension and insoluble surfactants on the generation of free-surface vorticity and surface waves are investigated. A recently developed finite-difference method based on boundary-fitted coordinates is used to solve the fully-nonlinear problem. Results show that in the absence of surfactants and at low Froude number (based on circulation strength and initial separation distance of the vortex pair), waves of short lengths are generated. However, secondary vorticity generated in this case is not strong enough to affect the outward translation of the primary vortices. At intermediate Froude number, a transient wave developing outboard of the primary vortex becomes steep, and eventually breaks because of local instability. Consequently, free-surface vorticity inhibits the outward translation of the primary vortices. Surface tension in a clean free surface dampens the steep short waves, hence also the generation of free-surface vorticity. However, variation in surface tension induced by surfactants intensifies the generation of surface vorticity, thereby causing the primary vortices to rebound. The increase in the rotational part of wave motion results in the dampening of overall free-surface deformations. However, it is found that the shear stress associated with a large gradient of surfactant concentration could cause local steepening of the short wave generated outboard of the primary vortex.     

微重环境中三维液体非线性晃动的数值模拟

将ALE（任意的拉格朗日-欧拉）运动学描述关系引入到Navier-Stokes方程中,在时间域上采用分步离散方法中的速度修正格式,利用Galerkin加权余量方法推导了系统的有限元数值离散方程;推导了考虑表面张力效应时有限元边界件的弱积分形式。模拟了考虑表面张力情况下圆筒形贮腔中液体的非线性晃动,揭示了考虑表面张力效应时液体非线性晃动的重要特征。     

Vortex structure simulation for supersonic mixing layers using nonlinear PSE method

The method of nonlinear parabolized stability equations (PSE) is applied in the simulation of vortex structures in compressible mixing layer. The spatially-evolving unstable waves, which dominate the vortex structure, are investigated through spatial marching method. The instantaneous flow field is obtained by adding the harmonic waves to basic flow. The results show that T-S waves do not keep growing exponentially as the linear evolution, the energy transfer to high order harmonic modes, and that finally all harmonic modes get saturated due to nonlinear interaction. The mean flow distortion induced by the nonlinear interaction between the harmonic modes and their conjugate harmonic ones, makes great change of the average flow and increases the thickness of mixing layer. PSE methods can well capture the two- and three-dimensional large scale nonlinear vortex structures in mixing layers such as vortex roll-up, vortex pairing, and Λ vortex.     

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

Suspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudo-spectral method in space. The coupled system is integrated in time with a fourth-order Runge–Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner et al. (2015, Journal of Fluid & Structures 52, 166–180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterized by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.     

On the stability and dissipation of wall boundary conditions for compressible flows

Characteristic formulations for boundary conditions have demonstrated their effectiveness to handle inlets and outlets, especially to avoid acoustic wave reflections. At walls, however, most authors use simple Dirichlet or Neumann boundary conditions, where the normal velocity (or pressure gradient) is set to zero. This paper demonstrates that there are significant differences between characteristic and Dirichlet methods at a wall and that simulations are more stable when using walls modelled with a characteristic wave decomposition. The derivation of characteristic methods yields an additional boundary term in the continuity equation, which explains their increased stability. This term also allows to handle the two acoustic waves going towards and away from the wall in a consistent manner. Those observations are confirmed by stability matrix analysis and one‐ and two‐dimensional simulations of acoustic modes in cavities. Copyright © 2009 John Wiley & Sons, Ltd.