URANS simulations for a high-speed transom stern ship with breaking waves

An exploratory study of high-speed surface ship flows is performed to identify modelling and numerical issues, to test the predictive capability of an unsteady RANS method for such flows, to explain flow features observed experimentally, and to document results obtained in conjunction with the 2005 ONR Wave Breaking Workshop. Simulations are performed for a high-speed transom stern ship (R/V Athena I) at three speeds Froude number (Fr) = 0.25, 0.43 and 0.62 with the URANS code CFDSHIP-IOWA, which utilizes a single-phase level set method for free surface modelling. The two largest Fr are considered to be high-speed cases and exhibit strong breaking plunging bow waves. Structured overset grids are used for local refinement of the unsteady transom flow at medium speed and for small scale breaking bow and transom waves at high-speeds. All simulations are performed in a time accurate manner and an examination of time histories of resistance and free surface contours is used to assess the degree to which the solutions reach a steady state. The medium speed simulation shows a classical steady Kelvin wave pattern without breaking and a wetted naturally unsteady transom flow with shedding of vortices from the transom corner. At higher speeds, the solutions reach an essentially steady state and display intense bow wave breaking with repeated reconnection of the plunging breaker with the free surface, resulting in multiple free surface scars. The high-speed simulations also show a dry transom and an inboard breaking wave, followed by outboard breaking waves downstream. In comparison to an earlier dataset, resistance is well predicted over the three speeds. The free surface predictions are compared with recent measurements at the two lowest speeds and show good agreement for both non-breaking and breaking waves.     

Effects of drift angle on model ship flow

The effects of drift angle on model ship flow are investigated through towing tank tests for the Series 60 C_B=0.6 cargo/container model ship. Resistance, side force, drift moment, sinkage, trim, and heel data are procured for a range of drift angles β and Froude numbers (Fr) and the model free condition. Detailed free-surface and mean velocity and pressure flow maps are procured for high and low Fr=0.316 and 0.16 and β=5° and 10° (free surface) and β=10° (mean velocity and pressure) for the model fixed condition (i.e. fixed with zero sinkage, trim, and heel). Comparison of results at high and low Fr and previous data for β=0° enables identification of important free-surface and drift effects. Geometry, conditions, data, and uncertainty analysis are documented in sufficient detail so as to be useful as a benchmark for computational fluid dynamics (CFD) validation. The resistance increases linearly with β with same slope for all Fr, whereas the increases in the side force, drift moment, sinkage, trim, and heel with β are quadratic. The wave profile is only affected near the bow, i.e. the bow wave amplitude increases/decreases on the windward/leeward sides, whereas the wave elevations are affected throughout the entire wave field. However, the wave envelope angle on both sides is nearly the same as β=0°, i.e. the near-field wave pattern rotates with the hull and remains within a similar wave envelope as β=0°. The wave amplitudes are significantly increased/decreased on the windward/leeward sides. The wake region is also asymmetric with larger wedge angle on the leeward side. The boundary layer and wake are dominated by the hull vortex system consisting of fore body keel, bilge, and wave-breaking vortices and after body bilge and counter-rotating vortices. The occurrence of a wave-breaking vortex for breaking bow waves has not been previously documented in the literature. The trends for the maximum vorticity, circulation, minimum axial velocity, and trajectories are discussed for each vortex. Received: 16 September 1999/Accepted: 8 November 2001     

A domain decomposition approach to compute wave breaking (wave‐breaking flows)

A heterogeneous domain decomposition approach is followed to simulate the unsteady wavy flow generated by a body moving beneath a free surface. Attention being focused on complex free surface configurations, including wave‐breaking phenomena, a two‐fluid viscous flow model is used in the free surface region to capture the air–water interface (via a level‐set technique), while a potential flow approximation is adopted to describe the flow far from the interface. Two coupling strategies are investigated, differing in the transmission conditions. Both the adopted approaches make use of the inviscid velocity field as boundary condition in the Navier–Stokes solution. For validation purposes, two different two‐dimensional non‐breaking flows are simulated. Domain decomposition results are compared with both fully viscous and fully inviscid results, obtained by solving the corresponding equations in the whole fluid domain, and with available experimental data. Finally, the unsteady evolution of a steep breaking wave is followed and some of the physical phenomena, experimentally observed, are reproduced. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

旋涡、湍流与自由表面的相互作用

旋涡及自由表面湍流与自由表面的非定常、非线性相互作用是当前流体力学中一个十分活跃的前沿领域的研究课题,它具有深刻的理论意义和重要的实用背景．旋涡、湍流与自由表面的相互作用是一个非常复杂的非线性、非定常过程,涉及到涡－涡、波－涡相互作用,旋涡在自由表面处的断裂、重联、合并及自由表面湍流中的准拟序结构的形成、发展等复杂动力学过程．关于这一领域的研究已经取得了一些重要成果．对旋涡与自由表面的相互作用的基本过程有了较为清楚的认识;但对湍流与自由表面的相互作用中的许多基本物理过程还知之甚少．本文综述此领域的实验、理论和数值模拟研究的最新进展和主要结果,讨论若干有待解决的问题,并指出进一步的研究方向      

Propulsive performance of a flapping plate near a free surface

The propulsive performance, i.e., the time-averaged thrust coefficient or the propulsive efficiency, of a flapping flat plate advancing near an otherwise quiescent free surface (liquid–gas interface) with Re of 1000, Fr of 0.2 and 0.8, and various submergence depths is numerically investigated by employing an adaptive Cartesian cut-cell/level-set method. The flapping kinematics parameters excluding the pitch-leading-heave phase angle were fixed as those commonly seen in literature. Results show that for submergence depth larger than the heave amplitude, the propulsive performance peaks at a smaller pitch-leading-heave phase angle with a shallower submergence for Fr of 0.2 but at the same phase angle for Fr of 0.8. Proximity to the free surface enhances the peak propulsive performance for Fr of 0.2 but the influence is minor for Fr of 0.8. The propulsive performance with Fr of 0.2 increases with decreasing chord-normalized submergence depth for the pitch-leading-heave phase angle smaller than 100°. The trend is reversed for the pitch-leading-heave phase angle larger than 100°. However, the propulsive performance with Fr of 0.8 hardly depends on the chord-normalized submergence depth. For submergence depth equal to the heave amplitude, the temporal variation in the thrust coefficient exhibits characteristics inherently different from those with other submergence depths for Fr of 0.2. Also, the time-averaged thrust coefficient exhibits a unique variation with the pitch-leading-heave phase angle. However, the various characteristics of the propulsive performance are similar to those with other submergence depths for Fr of 0.8. For submergence depth smaller than the heave amplitude and Fr of 0.2, the propulsive performance gains much from exposure of the upper surface of the plate to the gas phase. The efficiency enhancement is linked to the weakening of the leading edge vortices. A second harmonic with significant amplitude is found in the upstream wave for Fr of 0.2 with a typical pitch-leading-heave phase angle.     

A 2D+t SPH model to study the breaking wave pattern generated by fast ships

A 2D+t approach is applied to study the wave pattern generated by high speed slender ships with a sharp stem. This allows approximating the ship motion through a set of equations which are mathematically equivalent to those governing the unsteady 2D free-surface flow generated by a deformable body in the vertical plane transverse to the ship. To describe the body deformation, a proper modeling of the solid boundaries is needed. To this purpose the fixed ghost particle technique recently developed by the authors for 2D SPH schemes is adopted. Then, an in-depth analysis of the bow breaking wave evolution is provided focusing on the different features of the plunging jet as function of the ship forward velocity. The comparison with experimental data proves the proposed SPH scheme to be robust and accurate.     

A level set technique applied to unsteady free surface flows

An unsteady Navier–Stokes solver for incompressible fluid is coupled with a level set approach to describe free surface motions. The two‐phase flow of air and water is approximated by the flow of a single fluid whose properties, such as density and viscosity, change across the interface. The free surface location is captured as the zero level of a distance function convected by the flow field. To validate the numerical procedure, two classical two‐dimensional free surface problems in hydrodynamics, namely the oscillating flow in a tank and the waves generated by the flow over a bottom bump, are studied in non‐breaking conditions, and the results are compared with those obtained with other numerical approaches. To check the capability of the method in dealing with complex free surface configurations, the breaking regime produced by the flow over a high bump is analyzed. The analysis covers the successive stages of the breaking phenomenon: the steep wave evolution, the falling jet, the splash‐up and the air entrainment. In all phases, numerical results qualitatively agree with the experimental observations. Finally, to investigate a flow in which viscous effects are relevant, the numerical scheme is applied to study the wavy flow past a submerged hydrofoil. Copyright © 2001 John Wiley & Sons, Ltd.     

A method to compute ship exhaust plumes with waves and wind

Thermal and concentration transport models are implemented in CFDShip‐Iowa version 4.5, a semi‐coupled solver for air/water free surface flow (Int. J. Numer. Meth. Fluids 2008; 58 (6):591–624), to investigate the exhaust plume around ship superstructures. An incompressible, variable density approximation is implemented where the density can change in all governing equations due to temperature variations only. The thermal and concentration models are tested for the cases of steady and unsteady flow with thermal and solution transport in a 2D square cavity, and for a 3D thermal plume in an open environment, showing good agreement between computational results and experimental data. To test the method in an extreme motions condition, the exhaust plume of the ONR Tumblehome model DTMB 5613 is studied, showing complicated vortical structures in air including a pair of counter‐rotating vortices downstream of the stack for cross‐flow, and bended bird‐plume shape in the symmetry plane and varying arc‐shape in axial sections both for temperature and NO_x concentration fields. Effects of smoke exhaust speed and wind speed on the temperature and concentration distributions are studied. Finally, a smoke downwash computation is performed for a ship free to move in 6 degrees of freedom in a sea state 8 condition. Copyright © 2010 John Wiley & Sons, Ltd.     

Incompressible SPH simulation of wave breaking and overtopping with turbulence modelling

In this paper a truly incompressible version of the smoothed particle hydrodynamics (SPH) method is presented to investigate the surface wave overtopping. SPH is a pure Lagrangian approach which can handle large deformations of the free surface with high accuracy. The governing equations are solved based on the SPH particle interaction models and the incompressible algorithm of pressure projection is implemented by enforcing the constant particle density. The two‐equation k–ε model is an effective way of dealing with the turbulence and vortices during wave breaking and overtopping and it is coupled with the incompressible SPH numerical scheme. The SPH model is employed to reproduce the experiment and computations of wave overtopping of a sloping sea wall. The computations are validated against the experimental and numerical data found in the literatures and good agreement is observed. Besides, the convergence behaviour of the numerical scheme and the effects of particle spacing refinement and turbulence modelling on the simulation results are also investigated in further detail. The sensitivity of the computed wave breaking and overtopping on these issues is discussed and clarified. Copyright © 2005 John Wiley & Sons, Ltd.     

Full-scale two-phase flow measurements on Athena research vessel

Measurements of gas volume fraction, bubble velocity, chord length and bubble size distributions were performed on the research vessel Athena II operating in Saint Andrew Bay in the gulf coast near Panama City, FL. Double tipped sapphire optical local phase-detection probes were used to acquire indicator functions downstream of the breaking bow wave, behind the masker and at the stern. These indicator functions were also taken at different depths, operating speeds and headings respect to the waves. The data processing includes the computation of velocity of individual bubbles and chord lengths, resulting in chord length distributions. These chord length distributions are used to obtain bubble size distributions using a novel procedure described in detail herein. Uncertainty analysis is performed for gas volume fraction, average bubble velocity and chord length. The results indicate that air entrainment increases with ship speed and sailing against the waves at all positions. The bow wave exhibits unsteady breaking that creates bubble clouds, which were characterized and identified by signal processing. At the stern a very strong dependence of bubble size with depth was found, with evidence that small bubbles (smaller than 500 μm) are transported through the bottom of the hull and reach the transom. The roller present at the transom, the associated strong unsteadiness and bubble entrainment are well captured, as indicated by the stern results, showing the frothy nature of the upper layer.     

数值研究平板方舵激波-湍流边界层干扰

数值研究了平板方舵激波-湍流边界层干扰流场。模拟出了分离激波与弓型激波砬撞后形成的“λ”激波结构;消晰地显示了分离区中的旋涡结构,发现流场中会出现二次分离涡,并从理论上分析了流场对称面涡心形态与非定常的关系,得到了涡心为不稳定螺旋点或出现极限环是非定常流动特征的新结论。     

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

An accurate three‐dimensional numerical model, applicable to strongly non‐linear waves, is proposed. The model solves fully non‐linear potential flow equations with a free surface using a higher‐order three‐dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second‐order explicit Taylor series expansions with adaptive time steps. The model is applicable to non‐linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16‐node cubic ‘sliding’ quadrilateral elements, providing local inter‐element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well‐posed in all cases of mixed boundary conditions. Higher‐order tangential derivatives, required for the time updating, are calculated in a local curvilinear co‐ordinate system, using 25‐node ‘sliding’ fourth‐order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio‐temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two‐dimensional solution proposed earlier. Finally, three‐dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

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.     

Numerical simulation of unsteady multidimensional free surface motions by level set method

This paper presents a numerical method that couples the incompressible Navier–Stokes equations with the level set method in a curvilinear co‐ordinate system for study of free surface flows. The finite volume method is used to discretize the governing equations on a non‐staggered grid with a four‐step fractional step method. The free surface flow problem is converted into a two‐phase flow system on a fixed grid in which the free surface is implicitly captured by the zero level set. We compare different numerical schemes for advection of the level set function in a generalized curvilinear format, including the third order quadratic upwind interpolation for convective kinematics (QUICK) scheme, and the second and third order essentially non‐oscillatory (ENO) schemes. The level set equations of evolution and reinitialization are validated with benchmark cases, e.g. a stationary circle, a rotating slotted disk and stretching of a circular fluid element. The coupled system is then applied to a travelling solitary wave, and two‐ and three‐dimensional dam breaking problems. Some interesting free surface phenomena are revealed by the computational results, such as, the large free surface vortices, air entrapment and splashing of the water surge front. The computational results are in excellent agreement with theoretical predictions and experimental data, where they are available. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling the interaction of solitary waves and semi-circular breakwaters by using unsteady reynolds equations

A vertical 2 -D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid( VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater ; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.