Simulation of flows around cylinders by a Lagrangian vortex scheme

A method is presented for calculating flows around cylinders of arbitrary shape, which may be in an array. Vorticity is discretised and is moved in a Lagrangian time-stepping procedure by the vortex-in-cell method. The zero-velocity surface condition is maintained at each time step by representing the surface by a vortex sheet through a boundary integral calculation. Vorticity diffusion is simulated through random walks. Results are presented for an isolated circular cylinder in steady and oscillatory incident flows.     

A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling

We introduce a mixed discontinuous/continuous finite element pair for ocean modelling, with continuous quadratic layer thickness and discontinuous velocity. We investigate the finite element pair applied to the linear shallow-water equations on an f-plane. The element pair has the property that all geostrophically balanced states which strongly satisfy the boundary conditions have discrete divergence equal to exactly zero and hence are exactly steady states of the discretised equations. This means that the finite element pair has excellent geostrophic balance properties. We also show that the element pair applied to the non-rotating linear shallow-water equations does not have any spurious small eigenvalues. We illustrate these properties using numerical tests and provide convergence calculations which show that the numerical solutions have errors which decay quadratically with element edge length for both velocity and layer thickness.     

Bifurcation structure of a wind-driven shallow water model with layer-outcropping

The steady state bifurcation structure of the double-gyre wind-driven ocean circulation is examined in a shallow water model where the upper layer is allowed to outcrop at the sea surface. In addition to the classical jet-up and jet-down multiple equilibria, we find a new regime in which one of the equilibrium solutions has a large outcropping region in the subpolar gyre. Time dependent simulations show that the outcropping solution equilibrates to a stable periodic orbit with a period of 8 months. Co-existing with the periodic solution is a stable steady state solution without outcropping.A numerical scheme that has the unique advantage of being differentiable while still allowing layers to outcrop at the sea surface is used for the analysis. In contrast, standard schemes for solving layered models with outcropping are non-differentiable and have an ill-defined Jacobian making them unsuitable for solution using Newton’s method. As such, our new scheme expands the applicability of numerical bifurcation techniques to an important class of ocean models whose bifurcation structure had hitherto remained unexplored.     

Inleakage of a vortex-ring bunch onto a plane screen

A method is suggested for simulating axisymmetric laminar or turbulent flows formed during the motion of a vortex-ring bunch of given geometry and circulation toward a plane screen. Earlier, similar problems were simulated with the numerical solution of the Navier-Stokes equations for laminar flows. Turbulent flows have remained unconsidered until now. When a vortex ring approaches the screen, the secondary nonstationary flow is induced near the screen’s surface and this secondary flow causes the formation of the radial boundary layer (provided that air viscosity is taken into account). First, the medium spreads out from the critical point at the screen’s center with the negative pressure gradient along the radial coordinate and then detaches in the region of the positive pressure gradient. This radial wall flow and the corresponding boundary layer are considered in the quasi-stationary approximation. When the boundary layer detaches at successive instances, the flow is replenished with the radially moving secondary vortex rings whose circulations have the sign opposite to that of the circulation of the primary vortex ring. It is the interaction of the primary and secondary vortices that governs process dynamics, which differs substantially from that in the case when the formation of secondary vortices is disregarded. The suggested method is based on the method of discrete vortices (a perfect liquid) and the boundary-layer (laminar or turbulent) theory. During the development of the flow under investigation, the nonstationary ascending flow in the direction perpendicular to the screen’s plane is formed and then this flow decays and dissipates. Simulations for large Reynolds numbers corresponding to the formation of the turbulent boundary layer show that the velocity of ascending vortices in the plane of the initial vortex bunch is less than one-tenth of the initial velocity of the descending vortex ring. The boundary layer is introduced into calculations with the sole goal of determining the parameters of the secondary vortex rings formed during boundary-layer detachments. The interaction of the primary and secondary vortices is then considered within the framework of a perfect medium. Simulations for large Reynolds numbers corresponding to the formation of the turbulent boundary layer on the screen were correlated with the available data obtained in laboratory experiments for small Reynolds numbers. Qualitative agreement between the simulations and experiments is fairly satisfactory. The simulation for one combination of the circulation and vortex-ring geometry takes at most 10–15 min with the use of an average PC.     

Notes on linear and nonlinear barotropic flows in a polar circular basin

Barotropic flows in a circular ocean are studied. Flows are driven by an inflow and an outflow through openings at the circumference. A linear, steady state solution is interpreted in terms of dissipating planetary waves. A weakly nonlinear, steady state solution is obtained numerically. It differs remarkably from the linear solution; an intense anticyclonic polar gyre extending over the whole basin is formed. The nonlinear term is essential to the gyre and can not be neglected, although the Rossby number is small.     

Vortex shedding suppression for flow over a circular cylinder near a plane boundary

In this study, the Navier-Stokes equations and the pressure Poisson equation for two-dimensional time-dependent viscous flows are solved with a finite difference method in a curvilinear coordinate system. With this numerical procedure, the vortex shedding flow past a circular cylinder near a wall is investigated. The flow is calculated for a broad range of gap ratios for different Reynolds numbers ranging from 80 to 1000. Based on the numerical solutions, the vortex shedding is observed using various methods, and the mechanism for the vortex shedding suppression at small gap ratios is analyzed. The critical gap ratio at which the vortex shedding is suppressed is identified at different Reynolds numbers.     

Vortex dynamics of a fluid near a boundary with a circular cavity

The motion of a point vortex along a rectilinear boundary with a circle cavity, which models the coastline of a bay, and associated fluid particle advection are studied within a model of barotropic inviscid fluid. Using an analytical expression for the complex potential through which the velocity field is determined, we show that fluid particles start moving irregularly when the vortex is passing the cavity due to the nonstationarity of the velocity field generated by the vortex. Some of the fluid particles which were initially inside the vortex atmosphere leave it due to the irregularity and remain within the cavity vicinity. Depending on the initial position of the vortex and a parameter that determines the cavity size, the fraction of these fluid particles can differ significantly from fluid particles initially uniformly distributed within the vortex atmosphere. The escape of fluid particles from the vortex atmosphere is shown to be most efficient in the case of a relatively closed cavity under the condition that the initial vortex atmosphere area should be significantly smaller than the cavity area.     

ATMOSPHERE-OCEAN INTERACTION AND ITS RELATION TO OCEAN CURRENTS AND STORM SURGES

The theory of barotropic, shallow-sea ocean currents and storm surges considered from the viewpoint of air-sea interaction is presented. A 4-layer quasi-balanced dynamical model for a simulation of the structures of both the atmospheric and the marine planetary boundary layers is obtained by an analytical approach. A simplified scheme governing the storm surges induced by a slowly travelling circular atmospheric vortex is deduced.     

基于角动量模型的流场涡旋提取方法

分析海洋流场数据进行流场涡旋提取是流场可视化的重要方法。通常实际存在的涡旋具有不规则性,导致涡旋的提取通常需要复杂的计算。本文分析流场数据中涡旋的特点,提出一种基于角动量最大值模型提取涡旋的方法。建立角动量模型计算获得近似涡旋中心点,然后通过空间聚集分析和流场信息熵的计算来提取区域内涡旋。实验结果表明,与传统的提取方法相比,该方法不需要通过复杂计算即可提取涡旋区域并保证准确度,可以应用于不同空间分辨率的数据,同时解决了流场信息熵计算时鞍点对涡旋影响的问题,对于弱涡旋也有较好的提取效果。     

亚临界区雷诺数下圆柱绕流场电磁力控制数值研究

在亚临界区雷诺数下,采用脱体涡模拟方法对弱电解质中电磁力作用下圆柱绕流场及其升阻力特性进行了数值模拟与分析。结果表明,电磁力可以提高圆柱体边界层内的流体动能,延缓圆柱体近壁面流动分离,减弱绕流场中流向和展向大尺度涡的强度,减小圆柱体阻力及其升力脉动幅值;当电磁力作用参数大于某个临界值后,流动分离角消失,在圆柱体尾部产生射流现象,电磁力产生净推力作用,出现负阻力现象,而且升力脉动幅值显著减小且接近于零。     

Form Drag Caused by Topographically Forced Waves in a Barotropic β Channel: Effect of Higher Mode Resonance

The relationship between form drag and the zonal mean velocity of steady states is investigated in a very simple system; a barotropic quasi-geostrophic β channel with sinusoidal topography. When a steady solution is calculated by the modified Marquardt method, keeping the zonal mean velocity constant as a parameter, the characteristic of the solution changes at a phase speed of a wave with a wavenumber higher than that of the bottom topography. For velocities smaller than this critical value, there exists a stable quasi-linear solution similar to the linear solution. For larger velocities, there exist three solutions whose form drag is very large. In addition, the resonant velocity of the mode, whose wavenumber is the same as the bottom topography, has no effect on these solutions. When the quiescent fluid is accelerated by a constant wind stress, acceleration stops around the critical velocity for a wide range of the wind stress. If the wind stress is too large for the acceleration to stop, the zonal current speed continues to increase infinitely. It is implied that the zonal velocity of equilibrium is mainly determined, not by the wind stress, but by the amplitude of the bottom topography and the viscosity coefficient.     

An approximate solution of wave-modified Ekman current for gradually varying eddy viscosity

An approximate steady solution of the wave-modified Ekman current is presented for gradually varying eddy viscosity by using the WKB method with the variation of parameters technique. The parameters involved in the solution can be determined by the two-dimensional wavenumber spectrum of ocean waves, wind speed, the Coriolis parameter and the densities of air and water. The solution reduces to the exact solution when the eddy viscosity is taken as a constant. As illustrative examples, for a fully developed wind-generated sea with different wind speeds and a few proposed gradually varying eddy viscosities, the current profiles calculated from the approximate solutions are compared with those of the exact solutions or numerical ones by using the Donelan and Pierson wavenumber spectrum, the WAM wave model formulation for wind input energy to waves, and wave energy dissipation converted to currents. It is shown that the approximate solution presented has an elegant form and yet would be valid for any given gradually varying eddy viscosity. The applicability of the solution method to the real ocean is discussed following the comparisons with published observational data and with the results from a large eddy simulation of the Ekman layer.     

Nonlinear dynamics of the circulation variability in the western ocean

The nonlinear dynamics of the low-frequency variability of a mid-latitude ocean are studied. The mechanism of the separation of the western boundary current from the western wall, as well as the meridional displacements of the separation point and the separated eastward jet, is analyzed. A regional barotropic quasigeostrophic eddy-resolving numerical model is used for the analysis. The flow in a rectangular domain is simulated by the constant inflow and outflow of fluid through the boundaries. A regime when the nonlinearity prevails over the dissipation and the advection and β terms are of the same order of magnitude is considered, which is characteristic of the actual ocean. When the nonlinearity exceeds the threshold value, a periodic solution is obtained. The solution is determined by the nonslip boundary condition at the western wall. The solution obtained is studied in detail. The meridional displacements of the western boundary current separation point with respect to the western wall and the separated eastward jet can reach a few hundred kilometers. Their intensities and the intensity of the recirculation gyre in the western boundary current are found to oscillate with a period of about five years.     

Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section

Most off-shore oil platforms are supported by vertical cylinders extending to the ocean floor. An important problem in off-shore engineering is the calculation of the wave loading exerted on these vertical cylinders. Analytical solutions have been found for the case of plane incident waves incident on a circular cylinder by MacCamy and Fuchs [(1954), Wave forces on piles: a diffraction theory. U.S. Army Corps of Engineering, Beach Erosion Board, Technical Memorandum No. 69] and also for short-crested waves incident on a circular cylinder by Zhu [(1993), Diffraction of short-crested waves around a circular cylinder. Ocean Engng 20, 389–407]. However, for a cylinder of arbitrary cross-section, no analytic solutions currently exist. Au and Brebbia [(1983), Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Modelling 7, 106–114] proposed an efficient numerical approach to calculate the wave loads induced by plane waves on vertical cylinders by using the boundary element method. However, wind-generated waves are better modelled by short-crested waves. Whether or not these short-crested waves can induce larger wave forces on a structure is of great concern to ocean engineers. In this paper wave loads, induced by short-crested incident waves, on a vertical cylinder of arbitrary cross-section are discussed. For a cylinder of certain cross-section, the wave loads induced by short-crested waves can be larger than those induced by plane waves with the same total wave number.     

Difference-frequency wave loads on a large body in multi-directional waves

The second-order difference-frequency wave forces on a large three-dimensional body in multi-directional waves are computed by the boundary integral equation method and the so-called FML formulation (assisting radiation potential method). Semi-analytic solutions for a bottom-mounted vertical circular cylinder are also developed to validate the numerical method. Difference-frequency wave loads on a bottom-mounted vertical cylinder and stationary four legs of the ISSC tension-leg platform (TLP) are presented for various combinations of incident wave frequencies and headings. These force quadratic transfer functions (QTF) can directly be used in studying slowly varying wave loads in irregular short-crested seas described by a particular directional spectrum. From our numerical results, it is seen that the slowly varying wave loads are in general very sensitive to the directional spreading function of the sea, and therefore wave directionality needs to be taken into account in relevant ocean engineering applications. It is also pointed out that the uni-directionality of the sea is not necessarily a conservative assumption when the second-order effects are concerned.     

Two-layer quasi-geostrophic model of contour dynamics for a round basin

A quasi-geostrophic contour dynamics model permitting one to study flows induced by a system of vortex patches in a two-layer ocean with round shore boundaries in the presence of specified background flows caused by the bottom relief, the β effect, and sources and sinks at the boundaries is proposed. The principal relations of the model are presented and the algorithm of its numerical realization is described. Some experimental results of the study of the evolution of unstable two-layer vortices are demonstrated.     

Response of a two-layer ocean to typhoon passage in the western boundary region

Effect of the typhoon passage on the western boundary region of a two-layer ocean with bottom topography is studied. The ocean is initially at rest and is set in motion by a typhoon passing parallel to the west coast. Equations that represent barotropic and baroclinic modes of motions are solved numerically by means of the method of finite differences. Motions of the barotropic mode are assumed to be horizontally non-divergent. In this mode, an elongated vortex is produced by the typhoon and propagates toward the south after passage of the typhoon. Behavior of the vortex may be interpreted as continental shelf waves. It is found that the formation and propagation of continental shelf waves are hardly affected by the density stratification. As for the baroclinic response, the typhoon causes considerable interface displacements along its track. The interface displacements are associated with geostrophic motions and remain for long time, though they are formed on the continental slope. Besides the large scale baroclinic response, internal Kelvin waves are induced along the artificial east wall.     

A diagnostic stabilized finite-element ocean circulation model

A stabilized finite-element (FE) algorithm for the solution of oceanic large scale circulation equations and optimization of the solutions is presented. Pseudo-residual-free bubble function (RFBF) stabilization technique is utilized to enforce robustness of the numerics and override limitations imposed by the Babuška–Brezzi condition on the choice of functional spaces. The numerical scheme is formulated on an unstructured tetrahedral 3d grid in velocity–pressure variables defined as piecewise linear continuous functions. The model is equipped with a standard variational data assimilation scheme, capable to perform optimization of the solutions with respect to open lateral boundary conditions and external forcing imposed at the ocean surface. We demonstrate the model performance in applications to idealized and realistic basin-scale flows. Using the adjoint method, the code is tested against a synthetic climatological data set for the South Atlantic ocean which includes hydrology, fluxes at the ocean surface and satellite altimetry. The optimized solution proves to be consistent with all these data sets, fitting them within the error bars.The presented diagnostic tool retains the advantages of existing FE ocean circulation models and in addition (1) improves resolution of the bottom boundary layer due to employment of the 3d tetrahedral elements; (2) enforces numerical robustness through utilization of the RFBF stabilization, and (3) provides an opportunity to optimize the solutions by means of 3d variational data assimilation. Numerical efficiency of the code makes this a desirable tool for dynamically constrained analyses of large datasets.     

Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem

A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1], [2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.     

Effects of bottom topography and density stratification on the formation of western boundary currents

A wind-driven, general circulation for a two-layer ocean with continental shelf-slope along the western boundary is studied numerically. Special attention is focused on the formation process of the western boundary current in the subtropical gyre. The western boundary current develops in the upper layer along the western boundary on the shelf-slope with a bottom trapped poleward flow in the lower layer. The poleward undercurrent is concentrated approximately along the contour lines of the potential vorticity,f/D, wheref is the Coriolis parameter andD the depth of the ocean. The separation of upper- and lower-layer flows on the shelf-slope represents a typical transient response. As the response approaches a steady state, the poleward undercurrent decreases in amplitude, and the motion tends to be confined to the upper layer. The flow pattern becomes similar to that found in a flat bottom ocean. A steady-state response is expected to be isostatic (no motion in the lower layer), even on the shelf-slope, as conservation of potential vorticity would suggest.The remarkable increase in transport of the western boundary current produced by the formation of an anticyclonic vortex on the shelf-slope extending throughout the hemisphere (Holland, 1973) does not occur in the wind-driven general circulation.