Modified Lagrangian vortex method with improved boundary conditions for water waves past a thin bottom‐standing barrier

Herein, the modified Lagrangian vortex method (LVM), a hybrid analytical‐numerical algorithm per se, is devised to simulate the process of vortex formation and shedding from the sharp edge of a zero‐thickness vertical plate under linear water‐wave attack. Application of the Helmholtz decomposition facilitates a convenient switch between the inviscid‐ and viscous‐flow models, thereby enabling easy incorporation of vorticity effects into the potential‐flow calculations for the viscous‐dominated region. In evaluating the potential‐flow component, making good use of the quickly convergent technique with singular basis functions, correctly capturing the singular behavior in velocity fields near the tip of the plate, leads to a considerable reduction of computational burdens and to 12‐decimal‐place accuracy. The viscous correction is carried out via the meshless LVM with improved boundary conditions. Comparisons with previously published results show good agreement. Simulations of vortex generation and evolution illuminate the ability of the present method, and provide a supplement to pertinent experimental works. The hybrid scheme proposed herein allows flexibility for the former LVM and convenience in the code development. Such a compromise fits particularly well for the high‐resolution modeling of sharp‐edged vortex shedding without heavy numerical developments. Copyright © 2014 John Wiley & Sons, Ltd.     

Viscous waves and wave-structure interaction in a tank using adapting quadtree grids

Viscous waves and waves over a submerged cylinder in a stationary tank are simulated using a volume-of-fluid numerical scheme on adaptive hierarchical grids. A high resolution interface-capturing method is used to advect the free surface interface and the Navier–Stokes equations are discretised using finite volumes with collocated primitive variables and solved using a Pressure Implicit with Splitting of Operators (PISO) algorithm. The cylinder is modelled by using the technique of Cartesian cut cells. Results of flow of a single fluid past a cylinder at Reynolds number Re=100 are presented and found to agree well with experimental and other numerical data. Viscous free surface waves in a tank are simulated using uniform and quadtree grids for Reynolds numbers in the range from 2 to 2000, and the results compared against analytical solutions where available. The quadtree-based results are of the same accuracy as those on the equivalent uniform grids, and retain a sharp interface at the free surface while leading to considerable savings in both storage and CPU requirements. The nonlinearity in the wave is investigated for a selection of initial wave amplitudes. A submerged cylinder is positioned in the tank and its influence on the waves as well as the hydrodynamic loading on the cylinder is investigated.     

A simple SPH algorithm for multi‐fluid flow with high density ratios

In this paper, we describe an SPH algorithm for multi‐fluid flow, which is efficient, simple and robust. We derive the inviscid equations of motion from a Lagrangian together with the constraint provided by the continuity equation. The viscous flow equations then follow by adding a viscous term. Rigid boundaries are simulated using boundary force particles in a manner similar to the immersed boundary method. Each fluid is approximated as weakly compressible with a speed of sound sufficiently large to guarantee that the relative density variations are typically 1%. When the SPH force interaction is between two particles of different fluids, we increase the pressure terms. This simple procedure stabilizes the interface between the fluids. The equations of motion are integrated using a time stepping rule based on a second‐order symplectic integrator. When linear and angular momentum should be conserved exactly, they are conserved to within round‐off errors. We test the algorithm by simulating a variety of problems involving fluids with a density ratio in the range 1–1000. The first of these is a free surface problem with no rigid boundaries. It involves the flow of an elliptical distribution with one fluid inside the other. We show that the simulations converge as the particle spacing decreases, and the results are in good agreement with the exact inviscid, incompressible theory. The second test is similar to the first but involves the nonlinear oscillation of the fluids. As in the first test, the agreement with theory is very good, and the method converges. The third test is the simulation of waves at the interface between two fluids. The method is shown to converge, and the agreement with theory is satisfactory. The fourth test is the Rayleigh–Taylor instability for a configuration considered by other authors. Key parameters are shown to converge, and the agreement with other authors is good. The fifth and final test is how well the SPH method simulates gravity currents with density ratios in the range 2–30. The results of these simulations are in very good agreement with those of other authors and in satisfactory agreement with experimental results.Copyright © 2012 John Wiley & Sons, Ltd.     

The solution of two‐dimensional free‐surface problems using automatic mesh generation

A new method is described for the iterative solution of two‐dimensional free‐surface problems, with arbitrary initial geometries, in which the interior of the domain is represented by an unstructured, triangular Eulerian mesh and the free surface is represented directly by the piecewise‐quadratic edges of the isoparametric quadratic‐velocity, linear‐pressure Taylor–Hood elements. At each time step, the motion of the free surface is computed explicitly using the current velocity field and, once the new free‐surface location has been found, the interior nodes of the mesh are repositioned using a continuous deformation model that preserves the original connectivity. In the event that the interior of the domain must be completely remeshed, a standard Delaunay triangulation algorithm is used, which leaves the initial boundary discretisation unchanged. The algorithm is validated via the benchmark viscous flow problem of the coalescence of two infinite cylinders of equal radius, in which the motion is due entirely to the action of capillary forces on the free surface. This problem has been selected for a variety of reasons: the initial and final (steady state) geometries differ considerably; in the passage from the former to the latter, large free‐surface curvatures—requiring accurate modelling—are encountered; an analytical solution is known for the location of the free surface; there exists a large body of literature on alternative numerical simulations. A novel feature of the present work is its geometric generality and robustness; it does not require a priori knowledge of either the evolving domain geometry or the solution contained therein. Copyright © 1999 John Wiley & Sons, Ltd.     

A moving mesh algorithm for aero‐thermo‐mechanical modelling in turbomachinery

This paper describes the development of a mesh deformation method used for aero‐thermo‐mechanical coupling of turbo‐engine components. The method is based on the nonlinear solution of an elastic medium analogy, solved using finite element discretisation and modified to let the boundary nodes be free to slide over the deflected surfaces. This sliding technique relies on a B‐spline reconstruction of the moving boundary and increases the robustness of the method in situations where the boundary deflection field presents significant gradients or large relative motion between two distinct boundaries. The performance of the method is illustrated with the application to an interstage cavity of a turbine assembly, subjected to the deformations computed by a coupled thermo‐mechanical analysis of the engine component. Copyright © 2011 John Wiley & Sons, Ltd.     

Study of water waves with submerged obstacles using a vortex method with Helmholtz decomposition

The present study develops a 2‐D numerical scheme that combines the vortex method and the boundary integral method by a Helmholtz decomposition to investigate the interaction of water waves with submerged obstacles. Viscous effects and generation of vorticity on the free surface are neglected. The second kind of Fredholm integral equations that govern the strengths of vortex sheets along boundaries are solved iteratively. Vorticity is convected and diffused in the fluid via a Lagrangian vortex (blob) method with varying cores, using the particle strength exchange method for diffusion, with particle redistribution. A grid‐convergence study of the numerical method is reported. The inviscid part of the method and the simulation of the free‐surface motion are tested using two calculations: solitary wave propagation in a uniform channel and a moving line vortex in the fluid. Finally, the full model is verified by simulating periodic waves travelling over a submerged rectangular obstacle using nonuniform vortex blobs with a mapping of the redistribution lattice. Overall, the numerical model predicts the vortices' evolution and the free‐surface motion reasonably well. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical analysis of moving interfaces using a level set method coupled with adaptive mesh refinement

A novel numerical scheme is developed by coupling the level set method with the adaptive mesh refinement in order to analyse moving interfaces economically and accurately. The finite element method (FEM) is used to discretize the governing equations with the generalized simplified marker and cell (GSMAC) scheme, and the cubic interpolated pseudo‐particle (CIP) method is applied to the reinitialization of the level set function. The present adaptive mesh refinement is implemented in the quadrangular grid systems and easily embedded in the FEM‐based algorithm. For the judgement on renewal of mesh, the level set function is adopted as an indicator, and the threshold is set at the boundary of the smoothing band. With this criterion, the variation of physical properties and the jump quantity on the free surface can be calculated accurately enough, while the computation cost is largely reduced as a whole. In order to prove the validity of the present scheme, two‐dimensional numerical simulation is carried out in collapse of a water column, oscillation and movement of a drop under zero gravity. As a result, its effectiveness and usefulness are clearly shown qualitatively and quantitatively. Among them, the movement of a drop due to the Marangoni effect is first simulated efficiently with the present scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

Control of hydrothermal waves in a thermocapillary flow using a gradient‐based control strategy

The problem of controlling the hydrothermal waves in a thermocapillary flow is addressed using a gradient‐based control strategy. The state equations are the two‐dimensional unsteady incompressible Navier–Stokes and energy equations under the Boussinesq approximation. The modeled problem is the ‘open boat’ process of crystal growth, the flow which is driven by Marangoni and buoyancy effects. The control is a spatially and temporally varying heat flux boundary condition at the free surface. The control that minimizes the hydrothermal waves is found using a conjugate gradient method, where the gradient of the objective function with respect to the control variables is obtained from solving a set of adjoint equations. The effectiveness of choices of the parameters governing the control algorithm is examined. Almost complete suppression of the hydrothermal waves is obtained for certain choices of the parameters governing the control algorithm. The numerical issues involved with finding the control using the optimizer are discussed, and the features of the resulting control are analyzed with the goal of understanding how it affects the flow.Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

Adaptive Lagrangian–Eulerian computation of propagation and rupture of a liquid plug in a tube

Liquid plug propagation and rupture occurring in lung airways can have a detrimental effect on epithelial cells. In this study, a numerical simulation of a liquid plug in an infinite tube is conducted using an Eulerian–Lagrangian approach and the continuous interface method. A reconstruction scheme is developed to allow topological changes during plug rupture by altering the connectivity information about the interface mesh. Results prior to the rupture are in reasonable agreement with the study of Fujioka et al. in which a Lagrangian method is used. For unity non‐dimensional pressure drop and a Laplace number of 1000, rupture time is shown to be delayed as the initial precursor film thickness increases and rupture is not expected for thicknesses larger than 0.10 of tube radius. During the plug rupture process, a sudden increase of mechanical stresses on the tube wall is recorded, which can cause tissue damage. The peak values of those stresses increase as the initial precursor film thickness is reduced. After rupture, the peaks in mechanical stresses decrease in magnitude as the plug vanishes and the flow approaches a fully developed behavior. Increasing initial pressure drop is shown to linearly increase maximum variations in wall pressure and shear stress. Decreasing the pressure drop and increasing the Laplace number appear to delay rupture because it takes longer for a fluid with large inertial forces to respond to the small pressure drop. Copyright © 2010 John Wiley & Sons, Ltd.     

Three‐dimensional wave impact on a rigid structure using smoothed particle hydrodynamics

Navier–Stokes computations of a wave–structure interaction are performed with the aim of assessing the potential of smoothed particle hydrodynamics to accurately estimate impact loading time history. A three‐dimensional dam‐break flow with a rectangular column located downstream is considered. The net force and impulse exerted on the column is monitored throughout the simulation with the results correlating well with existing experimental data. Initial and boundary conditions and numerical parameters are varied and their effect on the column load investigated. The column load is found to be most sensitive to the choice of boundary treatment. Copyright © 2011 John Wiley & Sons, Ltd.     

Vortex‐in‐cell method combined with a boundary element method for incompressible viscous flow analysis

In this study, an immersed boundary vortex‐in‐cell (VIC) method for simulating the incompressible flow external to two‐dimensional and three‐dimensional bodies is presented. The vorticity transport equation, which is the governing equation of the VIC method, is represented in a Lagrangian form and solved by the vortex blob representation of the flow field. In the present scheme, the treatment of convection and diffusion is based on the classical fractional step algorithm. The rotational component of the velocity is obtained by solving Poisson's equation using an FFT method on a regular Cartesian grid, and the solenoidal component is determined from solving an integral equation using the panel method for the convection term, and the diffusion term is implemented by a particle strength exchange scheme. Both the no‐slip and no‐through flow conditions associated with the surface boundary condition are satisfied by diffusing vortex sheet and distributing singularities on the body, respectively. The present method is distinguished from other methods by the use of the panel method for the enforcement of the no‐through flow condition. The panel method completes making use of the immersed boundary nature inherent in the VIC method and can be also adopted for the calculation of the pressure field. The overall process is parallelized using message passing interface to manage the extensive computational load in the three‐dimensional flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled fluid structure analysis for wing 445.6 flutter using a fast dynamic mesh technology

The flow around wing 445.6 was modelled using Navier–Stokes equations and S-A model. The wing vibration and flow mesh deformation were computed using a fast dynamic mesh technology proposed by our own group. Wing 445.6 flutter was analysed through a strong coupling between the wing vibration and flow. The reduced flutter velocity was predicted and results are in good agreement with the experimental data. It is found that the subsonic flutter is mainly induced by the flow separation and the transonic and supersonic flutter are mainly caused by the oscillating shock wave and its induced flow separation. The positive aerodynamic work increases due to the oscillating shock wave when the subsonic flow becomes transonic reducing the flutter velocity. While the positive aerodynamic work induced by the oscillating shock wave decreases when the transonic flow becomes supersonic increasing the flutter velocity. That is why the transonic dip exists.     

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.     

An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Sparse element for shallow water wave equations on triangle meshes

Within the mixed FEM, the mini‐element that uses a bubble shape function for the solution of the shallow water wave equations on triangle meshes is simplified to a sparse element formulation. The new formulation has linear shape functions for water levels and constant shape functions for velocities inside each element. The suppression of decoupled spurious solutions is excellent with the new scheme. The linear dispersion relation of the new element has similar advantages as that of the wave equation scheme (generalised wave continuity scheme) proposed by Lynch and Gray. It is shown that the relation is monotonic over all wave numbers. In this paper, the time stepping scheme is included in the dispersion analysis. In case of a combined space–time staggering, the dispersion relation can be improved for the shortest waves. The sparse element is applied in the flow model Bubble that conserves mass exactly. At the same time, because of the limited number of degrees of freedom, the computational efficiency is high. The scheme is not restricted to orthogonal triangular meshes. Three test cases demonstrate the very good accuracy of the proposed scheme. The examples are the classical quarter annulus test case for the linearised shallow water equations, the hydraulic jump and the tide in the Elbe river mouth. Copyright © 2013 John Wiley & Sons, Ltd.