Solid body motion prediction using a unit quaternion-based solver with actuator disk

A six-Dof motion solver based on unit quaternions and an actuator disk model are implemented for ship hydrodynamics predictions. The six-Dof module is tested using the water entry phenomenon of a free falling sphere. The displacement history and impacting forces are analyzed. A KCS (KRISO container ship) model with the allowances of sinkage and trim is then simulated and validated. The actuator disk model is used to replace a real propeller. The open-water test of a KP458 propeller is first carried out to obtain the thrust and torque coefficients, using both the multi-run and single-run approaches. Oblique Towing Tank (OTT) tests using the actuator disk are conducted at last and the results agree well with the experiments. These models can be used for simulating six-Dof motions and captive model tests of ships.     

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     

Computation of free surface waves around an arbitrary body by a Navier-Stokes solver using the psuedocompressibility technique

A Navier-Stokes equation solver is developed for computing free surface wave and viscous flow around an arbitrary body, in which a free surface model is introduced into the pseudocompressibility solution. The governing equations are classified in a vectorial form, with primitive variables, and a block diagonal system is generated by the discretization of an implicit factorization method. A moving grid system fitted to both the free surface and body surface is generated by an effective cubic spline fitting technique. Two zero-equation turbulence models, namely the Cebeci-Smith model and the Baldwin-Lomax model, are used for turbulent calculations. Numerical simulations are carried out for the free surface viscous flows generated by a submerged hydrofoil and a ship model. Computed results are in reasonable agreement with measurements.     

MPS-FEM数值分析带自由面的流固耦合问题

随着计算科学的发展,研究人员为探索流固耦合问题的物理机理而提出了众多的数值方法。其中,耦合的移动粒子半隐式方法 MPS(Moving Particle Semi-Implicit method)和有限单元法FEM(Finite Element method)为流固耦合问题的数值仿真工作提供了新的途径。本文所有流场的数值模拟工作均采用课题组自主开发的无网格法求解器MLParticle-SJTU来完成。该求解器在原始的MPS法基础上,对核函数、压力梯度模型、压力泊松方程的求解和自由面判断方式等方面进行了改进。此外,在该求解器框架内,基于FEM法拓展了针对结构场进行求解的功能。首先,对MPS和FEM方法的理论模型及其耦合策略进行了介绍。然后,采用该自研MPS-FEM耦合求解器,数值模拟了溃坝流动对弹性结构的冲击及其相互作用的标准问题。通过将结构变形及自由面波型变化等结果与已发表结果进行对比,验证了该求解器在处理带自由面剧烈变化的粘性流体和柔性变形结构的耦合作用问题上的可行性。     

Unsteady RANS method for surface ship boundary layer and wake and wave field

Results are reported of an unsteady Reynolds‐averaged Navier–Stokes (RANS) method for simulation of the boundary layer and wake and wave field for a surface ship advancing in regular head waves, but restrained from body motions. Second‐order finite differences are used for both spatial and temporal discretization and a Poisson equation projection method is used for velocity–pressure coupling. The exact kinematic free‐surface boundary condition is solved for the free‐surface elevation using a body‐fitted/free‐surface conforming grid updated in each time step. The simulations are for the model problem of a Wigley hull advancing in calm water and in regular head waves. Verification and validation procedures are followed, which include careful consideration of both simulation and experimental uncertainties. The steady flow results are comparable to other steady RANS methods in predicting resistance, boundary layer and wake, and free‐surface effects. The unsteady flow results cover a wide range of Froude number, wavelength, and amplitude for which first harmonic amplitude and phase force and moment experimental data are available for validation along with frequency domain, linear potential flow results for comparisons. The present results, which include the effects of turbulent flow and non‐linear interactions, are in good agreement with the data and overall show better capability than the potential flow results. The physics of the unsteady boundary layer and wake and wave field response are explained with regard to frequency of encounter and seakeeping theory. The results of the present study suggest applicability for additional complexities such as practical ship geometry, ship motion, and maneuvering in arbitrary ambient waves. Copyright © 2001 John Wiley & Sons, Ltd.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

A boundary integral equation method for Navier-Stokes equations—application to flow in annulus of eccentric cylinders

A novel Navier-Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non-linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.     

时域Rankine源法求解有限水深船舶在规则波中的水底压力变化

应用势流理论中的Rankine源面元法和时域步进法,求解了有限水深船舶在规则波中运动的水底压力变化。将速度势分解成基本势、局部势和记忆势,以叠模解作为基本势对自由表面条件和物面条件进行了线性化,通过在水底布置面元来满足水底条件。利用研制的水底压力-水面波浪测量系统,测量了不同入射波船模表面波形与水底压力的时历曲线,理论计算与实验结果符合较好,验证了自编程序的正确性。通过对比二者的等高线图发现,水底压力与表面波形的峰谷有较好的一致性,并且压力较波形更为平滑。     

Numerical study on wave loads and motions of two ships advancing in waves by using three-dimensional translating-pulsating source

A frequency domain analysis method based on the three-dimensional translating-pulsating (3DTP) source Green function is developed to investigate wave loads and free motions of two ships advancing on parallel course in waves. Two experiments are carried out respectively to measure the wave loads and the free motions for a pair of side-byside arranged ship models advancing with an identical speed in head regular waves. For comparison, each model is also tested alone. Predictions obtained by the present solution are found in favorable agreement with the model tests and are more accurate than the traditional method based on the three dimensional pulsating (3DP) source Green function. Numerical resonances and peak shift can be found in the 3DP predictions, which result from the wave energy trapped in the gap between two ships and the extremely inhomogeneous wave load distribution on each hull. However, they can be eliminated by 3DTP, in which the speed affects the free surface and most of the wave energy can be escaped from the gap. Both the experiment and the present prediction show that hydrodynamic interaction effects on wave loads and free motions are significant. The present solver may serve as a validated tool to predict wave loads and motions of two vessels under replenishment at sea, and may help to evaluate the hydrodynamic interaction effects on the ships safety in replenishment operation.     

A 3D direct coupling method for steady ship hydroelastic analysis

Asides from the influence of incoming waves, ships can experience steady motions, such as rigid-body sinkage and trim motions, and flexible-body vertical bending motions, due to a constant forward speed even under calm water conditions. In this paper, a novel approach to analyze steady-ship hydroelasticity, particularly for the steady-ship motions and surrounding steady-wave disturbances, is proposed using a three-dimensional (3D) direct coupling method, based on a higher-order boundary element method (HOBEM) and a higher-order shell finite element method (FEM). Within the linearized framework, a solution method is proposed based on a two-step procedure, using two types of Neumann–Kelvin (NK) linear flow models for the fluid part and a virtual work equilibrium equation for the structural part. The first step is to compute a mean position wave-resistance problem using the modified NK equation, the second step is to solve a perturbed position wave-resistance problem, by employing a classical NK model and a virtual work equation based on the first step’s solution. Detailed mathematical formulation and numerical procedures are described, and a few numerical results are illustrated. These include both rigid and flexible steady-ship motions, Von-Mises stress distributions, and wave-resistance coefficients for Froude numbers ranging from 0.15 to 0.5. Furthermore, the numerical results obtained using the present direct coupling method and a modal-based one are compared.     

The Bekker theory of rolling resistance amended to take account of skid and deep sinkage

The Bekker theory of rolling resistance of free rolling, towed, rigid wheels is amended to take account of both skid and deep sinkage without leading to excessive complexity in the predictive equations. Theoretical relationships between skid and sinkage are derived for a free rolling, towed, rigid wheel on a purely cohesive soil ( = 0) and on a purely frictional soil (c = 0) with a sinkage exponent of unity. Generally, good agreement is found between predicted and measured values of rolling resistance and sinkage at a given vertical load, on both sand and clay soils, at shallow and deep sinkage.     

Computation of turbulent free‐surface flows around modern ships

This paper presents the calculated results for three classes of typical modern ships in modelling of ship‐generated waves. Simulations of turbulent free‐surface flows around ships are performed in a numerical water tank, based on the FINFLO‐RANS SHIP solver developed at Helsinki University of Technology. The Reynolds‐averaged Navier–Stokes (RANS) equations with the artificial compressibility and the non‐linear free‐surface boundary conditions are discretized by means of a cell‐centred finite‐volume scheme. The convergence performance is improved with the multigrid method. A free surface is tracked using a moving mesh technology, in which the non‐linear free‐surface boundary conditions are given on the actual location of the free surface. Test cases recommended are a container ship, a US Navy combatant and a tanker. The calculated results are compared with the experimental data available in the literature in terms of the wave profiles, wave pattern, and turbulent flow fields for two turbulence models, Chien's low Reynolds number k–εmodel and Baldwin–Lomax's model. Furthermore, the convergence performance, the grid refinement study and the effect of turbulence models on the waves have been investigated. Additionally, comparison of two types of the dynamic free‐surface boundary conditions is made. Copyright © 2003 John Wiley& Sons, Ltd.     

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

Flow control over an undulating membrane

The flow along a flexible membrane forced to undulate in the form of a streamwise travelling wave pattern is studied experimentally in detail. Flow field and force measurements confirm that the form drag of the wavy wall is significantly reduced when starting the undulatory motion. A mechanical model of an undulating membrane was built, based on previous investigations described in literature, and placed in an open water channel. The motion pattern of the membrane was prescribed in such a way to achieve a downstream travelling wave with increasing amplitude. The exploratory focus laid on the identification of hydrodynamic mechanisms of drag reduction due to undulatory motion. The wave-speed c of the travelling wave was set proportional to the incoming flow velocity U, according to an optimum ratio identified by previous numerical and experimental investigations. Poisson’s equation for the pressure was used to calculate the 2D pressure field from the experimental data of the unsteady flow field. In addition, the integral drag force of the membrane, as a function of c/U, was measured with a force balance to compare with previous published numerical findings. Furthermore, the velocities close to the surface of the membrane were measured, and the boundary layer profiles were determined. The resulting normalised velocity profiles affirm an oscillation between laminar and turbulent flow over one period of the motion. The results are in good agreement with previous experimental and numerical findings. Additionally, the characteristics of the flow along a travelling wave with increasing amplitude are discussed in more detail.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Development of a hybrid finite volume/element solver for incompressible flows

In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier–Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free implicit cell‐centred finite volume method. The Poisson equation derived from the fractional step approach is solved by the node‐based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centres and the auxiliary variable at cell vertices, making the current solver a staggered‐mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady‐state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical solution of velocity distribution for flow through submerged large deflection flexible vegetation

An analytical solution for predicting the vertical distribution of streamwise mean velocity in an open channel flow with submerged flexible vegetation is proposed when large bending occurs. The flow regime is separated into two horizontal layers: a vegetation layer and a free water layer. In the vegetation layer, a mechanical analysis for the flexible vegetation is conducted, and an approximately linear relationship between the drag force of bending vegetation and the streamwise mean flow velocity is observed in the case of large deflection, which differes significantly from the case of rigid upright vegetation. Based on the theoretical analysis, a linear streamwise drag force-mean flow velocity expression in the momentum equation is derived, and an analytical solution is obtained. For the free water layer, a new expression is presented, replacing the traditional logarithmic velocity distribution, to obtain a zero velocity gradient at the water surface. Finally, the analytical predictions are compared with published experimental data, and the good agreement demonstrates that this model is effective for the open channel flow through the large deflection flexible vegetation.     

Droplet break-up through an oblique shock wave

The interaction of a two-phase flow with a wedge where a stationary shock wave is initially settled is studied in a two-dimensional configuration. Before the introduction of the dispersed phase, the flow around the wedge is a supersonic one phase flow such as an attached stationary shock wave is present. Then, the dispersed phase is introduced upstream the initial position of the stationary shock wave. The purpose of this study is to point out two-phase and droplets break-up effects on the oblique shock wave. The two-dimensional equations are solved by a TVD scheme where fluxes are computed by using Riemann solver for the gas phase equations and also for the dispersed phase equations wich is an original approach due to the authors (Saurel et al. 1994). In addition to drag forces and heat and mass transfers, the process of droplets fragmentation based on the particle oscillation is considered. Accepted April 28, 1995     

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.