Free Surface Problems Research Articles

A numerical study of the Whitham equation as a model for steady surface water waves

The object of this article is the comparison of numerical solutions of the so-called Whitham equation to numerical approximations of solutions of the full Euler free-surface water-wave problem. The Whitham equation ηt+32c0h0ηηx+Kh0∗ηx=0 was proposed by Whitham (1967) as an alternative to the KdV equation for the description of wave motion at the surface of a perfect fluid by simplified evolution equations, but the accuracy of this equation as a water wave model has not been investigated to date.In order to understand whether the Whitham equation is a viable water wave model, numerical approximations of periodic solutions of the KdV and Whitham equation are compared to numerical solutions of the surface water wave problem given by the full Euler equations with free surface boundary conditions, computed by a novel Spectral Element Method technique. The bifurcation curves for these three models are compared in the phase velocity–waveheight parameter space, and wave profiles are compared for different wavelengths and waveheights. It is found that for small wavelengths, the steady Whitham waves compare more favorably to the Euler waves than the KdV waves. For larger wavelengths, the KdV waves appear to be a better approximation of the Euler waves.

Sensitivity analysis of finite volume simulations of a breaking dam problem

Purpose– The present work is a numerical study of a breaking dam problem. The purpose of this paper is to assess the effect of turbulence and surface tension models in the prediction of the interface position in a long-term analysis. Additionally, dimensional effects are analyzed by carrying out both 2D and 3D simulations.Design/methodology/approach– Finite volume simulations performed with the different models are compared between them and contrasted with numerical results computed using other numerical techniques and experimental data.Findings– The reported numerical results are in general in good agreement with experimental results available in the literature. They are also consistent with numerical solutions of other authors obtained using different numerical techniques. The results show that the laminar simulations exhibit strong mesh size dependency, while the turbulence models seem to help in producing mesh-independent solutions. Surface tension modeling does not seem to play a relevant role in the interface evolution.Practical implications– Model validation.Originality/value– The value of the present work encompass the comparison of different flow conditions used to simulate a free surface problem and their validation by contrasting numerical results with experiments. Also, the results shown in the present work are a contribution to the understanding of the role of some specific aspects of the models in the simulation of the proposed problem.

Preface to the special issue on “Thin films and fluid interfaces”

Free-surface problems, where the boundary of a moving body of fluid evolves as part of the dynamics, are one of the most interesting and challenging areas in the field of fluid dynamics. Such problems are ubiquitous in natural settings over a wide range of physical scales, from the propagation of surface waves on oceans to the spreading of raindrops on plant leaves. These systems exhibit intricate behaviors and have inspired many different types of applied studies in physics and engineering such as using fluid wetting to test substrate material properties, improving transport in designs of microfluidic devices, and investigating self-assembly of patterned materials. Likewise, free-surface problems have been a long-standing source of fundamental mathematical questions on how to formulate models, analyze systems of nonlinear partial differential equations, and compute efficient numerical simulations. Within the broad spectrum of free-surface problems, one of the most developed branches concerns the behaviors produced by slender bodies of fluids, such as jets and thin sheets [1]. These problems are fundamental to many physical situations including coating processing that involve the spreading of thin films of fluids on solid substrates [2]. The governing equations for these flows are derived through the use of longwave asymptotics and lubrication theory. The resulting mathematical models are higher-order nonlinear partial differential equations that include the influences of surface tension and the geometry of the problem. Some notable papers have given overviews of physical [3–10], mathematical [11–14], and computational [15,16] aspects of this field. Contributing to this rich and active field of research, the motivation for this special issue of the Journal of Engineering Mathematics came from the workshop on “Thin Liquid Films and Fluid Interfaces: Models, Experiments and Applications” held at the Banff International Research Station (BIRS) on December 9–14, 2012 [17]. The workshop added to the growing list of focused meetings organized by various members of this research community. Some of the other conferences in the field were held at BIRS in 2003–2010 [18–21], UCLA’s IPAM center in 2006 [22], the EUROMECH 490 workshop in London in 2007 [23], two meetings at ICMS in Edinburgh in 1999 and 2009 (EUROMECH 497) [24,25], at the Fields Institute in 2012 [26], and the 2013 programme at the Newton Institute at Cambridge University [27]. Several of these meetings have inspired special issues of journals containing new research results on thin film problems [28–32]. An international and interdisciplinary array of researchers in physics, engineering, and applied mathematics attended the 2012 BIRS workshop. Physical problems discussed at the meeting spanned applications from nanoscale to geophysical settings and physiological to industrial contexts. Mathematical questions on modeling of complex fluids and fluid–structure interactions also led to spirited interactions among the conference participants.

A GPU based compressible multiphase hydrocode for modelling violent hydrodynamic impact problems

This paper presents a GPU based compressible multiphase hydrocode for modelling violent hydrodynamic impacts under harsh conditions such as slamming and underwater explosion. An effort is made to extend a one-dimensional five-equation reduced model (Kapila et al., 2001) to compute three-dimensional hydrodynamic impact problems on modern graphics hardware. In order to deal with free-surface problems such as water waves, gravitational terms, which are initially absent from the original model, are now considered and included in the governing equations. A third-order finite volume based MUSCL scheme is applied to discretise the integral form of the governing equations. The numerical flux across a mesh cell face is estimated by means of the HLLC approximate Riemann solver. The serial CPU program is firstly parallelised on multi-core CPUs with the OpenMP programming model and then further accelerated on many-core graphics processing units (GPUs) using the CUDA C programming language. To balance memory usage, computing efficiency and accuracy on multi- and many-core processors, a mixture of single and double precision floating-point operations is implemented. The most important data like conservative flow variables are handled with double-precision dynamic arrays, whilst all the other variables/arrays like fluxes, residual and source terms are treated in single precision. Several benchmark test cases including water-air shock tubes, one-dimensional liquid cavitation tube, dam break, 2D cylindrical underwater explosion near a planar rigid wall, 3D spherical explosion in a rigid cylindrical container and water entry of a 3D rigid flat plate have been calculated using the present approach. The obtained results agree well with experiments, exact solutions and other independent numerical computations. This demonstrates the capability of the present approach to deal with not only violent free-surface impact problems but also hull cavitation associated with underwater explosions. Performance analysis reveals that the running time cost of numerical simulations is dramatically reduced by use of GPUs with much less consumption of electrical energy than on the CPU.

An upwind least square formulation for free surfaces calculation of viscoplastic steady-state metal forming problems

Despite using very large parallel computers, numerical simulation of some forming processes such as multi-pass rolling, extrusion or wire drawing, need long computation time due to the very large number of time steps required to model the steady regime of the process. The direct calculation of the steady-state, whenever possible, allows reducing by 10–20 the computational effort. However, removing time from the equations introduces another unknown, the steady final shape of the domain. Among possible ways to solve this coupled multi-fields problem, this paper selects a staggered fixed-point algorithm that alternates computation of mechanical fields on a prescribed domain shape with corrections of the domain shape derived from the velocity field and the stationary condition v.n = 0. It focuses on the resolution of the second step in the frame of unstructured 3D meshes, parallel computing with domain partitioning, and complex shapes with strong contact restraints. To insure these constraints a global finite elements formulation is used. The weak formulation based on a Galerkin method of the v.n = 0 equation is found to diverge in severe tests cases. The least squares formulation experiences problems in the presence of contact restraints, upwinding being shown necessary. A new upwind least squares formulation is proposed and evaluated first on analytical solutions. Contact being a key issue in forming processes, and even more with steady formulations, a special emphasis is given to the coupling of contact equations between the two problems of the staggered algorithm, the thermo-mechanical and free surface problems. The new formulation and algorithm is finally applied to two complex actual metal forming problems of rolling. Its accuracy and robustness with respect to the shape initialization of the staggered algorithm is discussed, and its efficiency is compared to non-steady simulations.

Laboratory Study on 3D Flow Structures Induced by Zero-Height Side Weir and Implications for 1D Modeling

AbstractSide weir flows induce complex three-dimensional (3D) structures on the flow field in the main channel. Extensive experiments at the laboratory scale were conducted in order to investigate the flow field for the particular case of a side weir with a zero height crest in both fixed bed and steady flow conditions. Detailed measurements of the flow surface, discharge, and velocity field were performed in the main channel upstream, alongside, and downstream of the side weir location. They allowed detection and analysis of the three-dimensional flow structures. Elaborations of the measurements show the implications of such three-dimensional nature of the flow on integral quantities such as Coriolis and Boussinesq coefficients as well as the specific energy and discharge coefficient that are important in the widely used one-dimensional (1D) models. Moreover, the results may help to better understand other free-surface problems similar to the side weir flow, such as lateral diversions, river bifurcations...

Small-time solvability of primitive equations for the ocean with spatially-varying vertical mixing

The small-time existence of a strong solution to the free surface problem of primitive equations for the ocean with variable turbulent viscosity terms is shown in this paper. In this model, the turbulent viscosity coefficients, which include the Richardson number depending on unknown variables, are explicitly formulated. In addition, following the formulation of practical models, the kinematic condition is assumed on the free ocean surface. As in preceding works, we consider the problem in the three-dimensional strip-like region, and assume the f -approximation. Under some conditions on the initial and boundary data and the topography of the bottom of the ocean, we construct a strong local-in-time solution in Sobolev-Slobodetski˘ i spaces. The boundedness of the temperature and salinity is also shown in the present paper.

Irregular Nonlinear Wave Simulation and Associated Loads on Offshore Wind Turbines

The accuracy of predicted loads on offshore wind turbines depends on the mathematical models employed to describe the combined action of the wind and waves. Using a global simulation framework that employs a domain-decomposition strategy for computational efficiency, this study investigates the effects of nonlinear waves on computed loads on the support structure (monopile) and the rotor–nacelle assembly of a bottom-supported offshore wind turbine. The fully nonlinear (FNL) numerical wave solver is invoked only on subdomains where nonlinearities are detected; thus, only locally in space and time, a linear solution (and associated Morison hydrodynamics) is replaced by the FNL one. An efficient carefully tuned linear–nonlinear transition scheme makes it possible to run long simulations such that effects from weakly nonlinear up to FNL events, such as imminent breaking waves, can be accounted for. The unsteady nonlinear free-surface problem governing the propagation of gravity waves is formulated using potential theory and a higher-order boundary element method (HOBEM) is used to discretize Laplace’s equation. The FNL solver is employed and associated hydrodynamic loads are simulated in conjunction with aerodynamic loads on the rotor of a 5-MW wind turbine using the NREL open-source software, fast. We assess load statistics associated with a single severe sea state. Such load statistics are needed in evaluating relevant load cases specified in offshore wind turbine design guidelines; in this context, the influence of nonlinear wave modeling and its selection over alternative linear or linearized wave modeling is compared. Ultimately, a study such as this one will seek to evaluate long-term loads using the FNL solver in computations directed toward reliability-based design of offshore wind turbines where a range of sea states will need to be evaluated.

Magnetohydrodynamics effect on rotating free surface flow of micropolar fluid in a cylindrical container with porous lining

The problem of free surface on electrically conducting micropolar fluid contained in a rotating vertical cylinder of radius a with inner porous lining under radial magnetic field is presented numerically. The problem is formulated with Beavers and Joseph slip condition at the porous lining boundary. The dimensionless governing equations are solved by employing a finite difference method. The expression of free surface, velocity profiles and the coefficient of skin friction on cylinder are calculated. The effects of various emerging parameters of the present study on velocity, micro-rotation and skin friction are discussed and depicted through plots.

CFD study of capillary jets under superimposed destabilizing conditions for microdroplet formation

Initial numerical studies on the capillary laminar instability of continuous polymeric jets subjected to controlled destabilization and their comparison with experimental results obtained from a microparticle formation technique are addressed. Continuous jet instabilities and drop formation are challenging free-surface flow problems that require sophisticated interface tracking methods. The assessment of a Coupled Volume of Fluid and Level Set (CLSVOF) commercial code and its potential application to more complex systems are considered. The data obtained from the numerical simulations under different operating conditions showed good qualitative results, in agreement with the trends experimentally obtained. Nevertheless, the use of generalized Newtonian models instead of more complex constitutive equations accounting for the polymer viscoelasticity preclude the obtaining of better comparisons. However, it can be concluded that the use of CLSVOF under the described conditions deals reasonably well with the tracking of the interface, making its use adequate in capillary laminar instabilities.

Simulation of Droplet Impacting on Elastic Solid with the SPH Method

The phenomenon of droplet impacting on solid surfaces widely exists in both nature and engineering systems. However, one concern is that the microdeformation of solid surface is difficult to be observed and measured during the process of impacting. Since the microdeformation can directly affect the stability of the whole system, especially for the high-rate rotating components, it is necessary to study this phenomenon. Aiming at this problem, a new numerical simulation algorithm based on the Smoothed Particle Hydrodynamics (SPH) method is brought forward to solve fluid-solid coupling and complex free surface problems in the paper. In order to test and analyze the feasibility and effectiveness of the improved SPH method, the process of a droplet impacting on an elastic plate was simulated. The numerical results show that the improved SPH method is able to present more detailed information about the microdeformation of solid surface. The influence of the elastic modulus of solid on the impacting process was also discussed.

Second-Order Pure Lagrange--Galerkin Methods for Fluid-Structure Interaction Problems

In this paper, we propose a second-order (both in time and in space) pure Lagrange--Galerkin method for the numerical solution of fluid-structure interaction problems. The proposed scheme is written in material coordinates and in terms of displacements in the structure and of displacements and pressures in the fluid. Pure-Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of time, and fluid-structure coupling at the interphase is straightforward. Unfortunately, for moderate- to high-Reynolds number flows, pure-Lagrangian methods can lead to high distortion of the mesh elements, and as a consequence inaccurate approximations can be obtained. Before this happens, it is necessary to remesh and reinitialize the motion. In the present paper, we also deal with this problem by proposing a method to be combined with the pure Lagrange--Galerkin method we introduce that preserves the order. In order to...

직교격자상에서 효율적인 비압축성 자유표면유동 해법

An efficient solution algorithm for simulating free surface problem is presented. Navier-Stokes equations for variable density incompressible flow are employed as the governing equation on Cartesian meshes. In order to describe the free surface motion efficiently, VOF(Volume Of Fluid) method utilizing THINC(Tangent of Hyperbola for Interface Capturing) scheme is employed. The most time-consuming part of the current free surface flow simulations is the solution step of the linear system, derived by the pressure Poisson equation. To solve a pressure Poisson equation efficiently, the PCG(Preconditioned Conjugate Gradient) method is utilized. This study showed that the proper application of the preconditioner is the key for the efficient solution of the free surface flow when its pressure Poisson equation is solved by the CG method. To demonstrate the efficiency of the current approach, we compared the convergence histories of different algorithms for solving the pressure Poisson equation.

An extended validation of the last generation of particle finite element method for free surface flows

In this paper, a new generation of the particle method known as Particle Finite Element Method (PFEM), which combines convective particle movement and a fixed mesh resolution, is applied to free surface flows. This interesting variant, previously described in the literature as PFEM-2, is able to use larger time steps when compared to other similar numerical tools which implies shorter computational times while maintaining the accuracy of the computation. PFEM-2 has already been extended to free surface problems, being the main topic of this paper a deep validation of this methodology for a wider range of flows. To accomplish this task, different improved versions of discontinuous and continuous enriched basis functions for the pressure field have been developed to capture the free surface dynamics without artificial diffusion or undesired numerical effects when different density ratios are involved. A collection of problems has been carefully selected such that a wide variety of Froude numbers, density ratios and dominant dissipative cases are reported with the intention of presenting a general methodology, not restricted to a particular range of parameters, and capable of using large time-steps. The results of the different free-surface problems solved, which include: Rayleigh–Taylor instability, sloshing problems, viscous standing waves and the dam break problem, are compared to well validated numerical alternatives or experimental measurements obtaining accurate approximations for such complex flows.

Simulating free surface problem using isogeometric analysis

This paper presents the application of isogeometric analysis to simulate the incompressible free surface flows as well as dam break problem. The governing Navier–Stokes equations using a pressure projection method are solved in a Lagrangian form. The least squares method is used in both function approximation and discretization of the governing differential equations in space. Non-uniform rational B-splines are also used as basic functions. One of the challenging free surface problems, namely dam break, is solved to check the accuracy of the proposed method. The isogeometric analysis method enjoys the advantage of producing symmetric, positive definite matrixes for the considered cases. The results of the free surface at different times are compared with the characteristic-based split finite element method. The results show the ability of the isogeometric analysis method to solve complex fluid dynamic problems with moving free surface boundaries. Moreover, the velocity fields and the pressure distribution due to the dam break problem at different times are plotted.

DualSPHysics: Open-source parallel CFD solver based on Smoothed Particle Hydrodynamics (SPH)

DualSPHysics is a hardware accelerated Smoothed Particle Hydrodynamics code developed to solve free-surface flow problems. DualSPHysics is an open-source code developed and released under the terms of GNU General Public License (GPLv3). Along with the source code, a complete documentation that makes easy the compilation and execution of the source files is also distributed. The code has been shown to be efficient and reliable. The parallel power computing of Graphics Computing Units (GPUs) is used to accelerate DualSPHysics by up to two orders of magnitude compared to the performance of the serial version. Program summaryProgram title: DualSPHysicsCatalogue identifier: AEUS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUS_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public LicenseNo. of lines in distributed program, including test data, etc.: 121,399No. of bytes in distributed program, including test data, etc.: 12,324,308Distribution format: tar.gzProgramming language: C++ and CUDA.Computer: Tested on CPU Intel X5500 and GPUs: GTX 480, GTX 680, Tesla K20 and GTX Titan.Operating system: Any system with a C++ and NVCC compiler, tested on Linux distribution Centos 6.5CUDA: Tested on versions 4.0, 4.1, 4.2, 5.0 and 5.5 with driver version 331.38.Has the code been vectorised or parallelised?: Different threads of CPU or number of cores of GPU.RAM: Tens of MB to several GB, depending on problemClassification: 4.12.Nature of problem:The DualSPHysics code has been developed to study free-surface flows requiring high computational cost.Solution method:DualSPHysics is an implementation of Smoothed Particle Hydrodynamics, which is a Lagrangian meshless particle method.Running time:6 h on 8 processors of Intel X5500 (15 min on GTX Titan) for the dam-break case with 1 million particles simulating 1.5 s of physical time (more than 26,000 steps).

Transient free-surface seepage in three-dimensional general anisotropic media by BEM

Kinematic boundary condition is usually used when dealing with transient free-surface flow problems in isotropic media. When dealing with anisotropic problems, a transformation can transform the anisotropic media to an equivalent isotropic media for seepage analysis, but the kinematic boundary condition cannot be used directly in the transformed media. A generalization of the kinematic boundary condition along any arbitrary direction is derived for use in the transformed domain for general three-dimensional anisotropic problems. A boundary element method for solving transient free-surface seepage problems is developed and the treatment of the proposed kinematic boundary condition in the boundary element method is given. Three examples have been solved to show the reliability and flexibility of the model. Examples are verified with some available experimental and numerical cases to show the accuracy of the model for predicting the phreatic surface and it is shown that anisotropy has a very important and non-neglecting effect in the behavior and the shape of phreatic surface.

Benchmarking of Navier–Stokes codes for free surface simulations by means of a solitary wave

The paper presents a benchmark of four freely available solvers for Navier–Stokes equations: Gerris, OpenFOAM, Thétis and Truchas. These models are selected because they have been reported to deal successfully with oceanographic and coastal engineering applications. The benchmark includes two free surface problems: propagation of a solitary wave and runup of a solitary wave on a plane beach. The fluids are inviscid, which allows a detailed study of energy conservation and comparison of the results with a reference solution given by a boundary integral solver. The Navier–Stokes solvers use the finite volume discretization and free surface capturing techniques based on the volume-of-fluid (VOF) method. In the first benchmark test, we investigate the influence of numerical dissipation and other spurious effects on the energy balance of the wave. In the second problem, we focus on the runup heights and compare them with the reference solution. The beach in the runup problem is represented with a solid body immersed in the Cartesian mesh. The solid boundaries are described with a VOF type approach or a staircase representation, depending on the features of the solver. In addition to the immersed boundary description a couple of body fitted meshes are tested for the runup case.

Solving seven-equation model for compressible two-phase flow using multiple GPUs

In this paper, the application of an HLLC-type approximate Riemann solver in conjunction with the third-order TVD Runge–Kutta method to the seven-equation compressible two-phase model on multiple Graphics Processing Units (GPUs) is presented. Based on the idea proposed by Abgrall et al. that “a multiphase flow, uniform in pressure and velocity at t=0, will remain uniform on the same variables during time evolution”, discretization schemes for the non-conservative terms and for the volume fraction evolution equation are derived in accordance with the HLLC solver used for the conservative terms. To attain high temporal accuracy, the third-order TVD Runge–Kutta method is implemented in conjunction with operator splitting technique, in which the sequence of operators is recorded in order to compute free surface problems robustly. For large scale simulations, the numerical method is implemented using MPI/Pthread-CUDA parallelization paradigm for multiple GPUs. Domain decomposition method is used to distribute data to different GPUs, parallel computation inside a GPU is accomplished using CUDA, and communication between GPUs is performed via MPI or Pthread. Efficient data structure and GPU memory usage are employed to maintain high memory bandwidth of the device, while a special procedure is designed to synchronize thread blocks so as to reduce frequencies of kernel launching. Numerical tests against several one- and two-dimensional compressible two-phase flow problems with high density and high pressure ratios demonstrate that the present method is accurate and robust. The timing tests show that the overall speedup of one NVIDIA Tesla C2075 GPU is 31× compared with one Intel Xeon Westmere 5675 CPU core, and nearly 70% parallel efficiency can be obtained when using 8 GPUs.

Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier–Stokes equations

In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.