Extension of the volume-of-fluid method for analysis of free surface viscous flow in an ideal gas

The volume-of-fluid (VOF) method is a simple and robust technique for simulating free surface flows with large deformations and intersecting free surfaces. Earlier implementations used Laplace's formula for the normal stress boundary condition at the interface between the liquid and vapour phases. We have expanded the interfacial boundary conditions to include the viscous component of the normal stress in the liquid phase and, in a limited manner, to allow the pressure in the vapour phase to vary. Included are sample computations that show the accuracy of added third-order-accurate differencing schemes for the convective terms in the Navier-Stokes equation (NSE), the viscous terms in the normal stress at the interface and the solution of potential flow in the vapour phase coupled with the solution of the NSE in the liquid phase. With these modifications we show that the VOF method can accurately predict the instability of a thin viscous sheet flowing through a stagnant vapour phase.     

Trefftz间接法解自由面渗流问题

渗流问题是岩体水力学研究的重点之一,较常规的有限元法和传统的边界元法而言,Trefftz型边界元法具有程序简单、计算量小及无须奇异积分等特点。本文给出定常和非定常自由面渗流问题的Trefftz解法,计算结果表明,该法收敛性好,结果比较精确。     

An efficient edge‐based level set finite element method for free surface flow problems

We present an efficient technique for the solution of free surface flow problems using level set and a parallel edge‐based finite element method. An unstructured semi‐explicit solution scheme is proposed. A custom data structure, obtained by blending node‐based and edge‐based approaches is presented so to allow a good parallel performance. In addition to standard velocity extrapolation (for the convection of the level set function), an explicit extrapolation of the pressure field is performed in order to impose both the pressure boundary condition and the volume conservation. The latter is also improved with a modification of the divergence free constrain. The method is shown to allow an efficient solution of both simple benchmark cases and complex industrial examples. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

On the accuracy of the desingularized boundary integral method in free surface flow problems

In this paper the numerical properties of the desingularized boundary integral formulation were studied within the framework of free surface potential problems. Several numerical experiments were carried out on simple test cases in order to investigate the effects on the accuracy of the distance between the singularity sheet and the free boundary. The optimum value of this distance was related to the mesh size by simple correlations. Once the desingularized boundary integral formulation had been so calibrated, it was implemented for the solution of two typical free surface flow problems: wave diffraction around a fixed obstacle and wave resistance of submerged bodies. Numerical results are discussed in comparison with experimental data; the computational efficiency and accuracy of desingularized algorithms are confirmed and specified. © 1997 John Wiley & Sons, Ltd.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

Jeffrey fluid flow due to curved stretching surface with Cattaneo-Christov heat flux

The two-dimensional (2D) motion of the Jeffrey fluid by the curved stretching sheet coiled in a circle is investigated. The non-Fourier heat flux model is used for the heat transfer analysis. Feasible similarity variables are used to transform the highly nonlinear ordinary equations to partial differential equations (PDEs). The homotopy technique is used for the convergence of the velocity and temperature equations. The effects of the involved parameters on the physical properties of the fluid are described graphically. The results show that the curvature parameter is an increasing function of velocity and temperature, and the temperature is a decreasing function of the thermal relaxation time. Besides, the Deborah number has a reverse effect on the pressure and surface drag force.     

Wet/dry areas method for interfacial (free surface) flow

In this paper, a new numerical method is developed for two‐dimensional interfacial (free surface) flows, based on the control volume method and conservative integral form of the Navier–Stokes equations with a standard staggered grid. The new method deploys two continuity equations, the continuity equation of the mass conservation for better convergence of the implicit scheme and the continuity equation of the volume conservation for the equation of pressure correction. The convection terms (the total momentum flux) on the surfaces of control volume are accurately calculated from the wet area exposed to the water, and the dry area exposed to the air. The numerical results produced by the new numerical method agree very well with the analytical solution, experimental images and experimentally measured velocity. Copyright © 2012 John Wiley & Sons, Ltd.     

The characteristic‐based split (CBS) meshfree method for free surface flow problems in ALE formulation

A semi‐implicit characteristic‐based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element analysis of transient fluid flow with free surface using VOF (volume-of-fluid) method and adaptive grid

The VOF method is adopted for the finite element analysis of transient fluid flow with a free surface. In particular, an adaptation technique for generating an adaptive grid is incorporated to capture a higher resolution of the free surface configuration. An adaptive grid is created through the refinement and mergence of elements. In this domain the elements in the surface region are made finer than those in the remaining regions for more efficient computation. Also, three techniques based on the VOF method are newly developed to increase the accuracy of the analysis, namely the filling pattern, advection treatment and free surface smoothing techniques. Using the proposed numerical techniques, radial flow with a point source and the collapse of a dam are analysed. The numerical results agree well with the theoretical solutions as well as with the experimental results. Through comparisons with the numerical results of several cases using different grids, the efficiency of the proposed technique is verified. © 1998 by John Wiley & Sons, Ltd.     

Imposing accurate wall boundary conditions in corrective‐matrix‐based moving particle semi‐implicit method for free surface flow

Corrective matrix that is derived to restore consistency of discretization schemes can significantly enhance accuracy for the inside particles in the Moving Particle Semi‐implicit method. In this situation, the error due to free surface and wall boundaries becomes dominant. Based on the recent study on Neumann boundary condition (Matsunaga et al, CMAME, 2020), the corrective matrix schemes in MPS are generalized to straightforwardly and accurately impose Neumann boundary condition. However, the new schemes can still easily trigger instability at free surface because of the biased error caused by the incomplete/biased neighbor support. Therefore, the existing stable schemes based on virtual particles and conservative gradient models are applied to free surface and nearby particles to produce a stable transitional layer at free surface. The new corrective matrix schemes are only applied to the particles under the stable transitional layer for improving the wall boundary conditions. Three numerical examples of free surface flows demonstrate that the proposed method can help to reduce the pressure/velocity fluctuations and hence enhance accuracy further.     

A new fully non‐hydrostatic 3D free surface flow model for water wave motions

A new fully non‐hydrostatic model is presented by simulating three‐dimensional free surface flow on a vertical boundary‐fitted coordinate system. A projection method, known as pressure correction technique, is employed to solve the incompressible Euler equations. A new grid arrangement is proposed under a horizontal Cartesian grid framework and vertical boundary‐fitted coordinate system. The resulting model is relatively simple. Moreover, the discretized Poisson equation for pressure correction is symmetric and positive definite, and thus it can be solved effectively by the preconditioned conjugate gradient method. Several test cases of surface wave motion are used to demonstrate the capabilities and numerical stability of the model. Comparisons between numerical results and analytical or experimental data are presented. It is shown that the proposed model could accurately and effectively resolve the motion of short waves with only two layers, where wave shoaling, nonlinearity, dispersion, refraction, and diffraction phenomena occur. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling jet flows caused by the incidence of a shock wave on a profiled free surface

The problem of the incidence of a shock wave with a front-pressure amplitude of about 30 GPa at the profiled free surface of an aluminum sample is studied. It is shown that in the case of large perturbations (amplitude 1 mm and wavelength 10 mm), jet flows occur on the free surface. The data obtained are described using a kinetic fracture model that takes into account the damage initiation and growth in the material due to tensile stress and shear strain. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 16–23, January–February, 2007.     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical method for the evaluation of hydrodynamic forces of translating bodies under a free surface

This paper presents a numerical method to evaluate the hydrodynamic forces of translating bodies under a free surface. Both steady and unsteady problems are considered. Analytical and numerical studies are carried out based on the Havelock wave‐source function and the integral equation method. Two main problems arising inherently in the proposed solution method are overcome in order to facilitate the numerical implementation. The first lies in evaluating the Havelock function, which involves integrals with highly oscillatory kernels. Particular integration contours leading to non‐oscillatory integrands are derived a priori so that the integrals can be evaluated efficiently. The second problem lies in evaluating singular kernels in the boundary integral equation. The corresponding non‐singular formulation is derived using some theorems of potential theory, including the Gauss flux theorem and the property related to the equipotential body. The subsequent formulation is amenable to the solution by directly using the standard quadrature formulas without taking another special treatment. This paper also attempts to enhance the computational efficiency by presenting an interpolation method used to evaluate matrix elements, which are ascribed to a discretization procedure. In addition to the steady case, numerical examples consist of cases involving a submerged prolate spheroid, which is originally idle and then suddenly moves with a constant speed and a constant acceleration. Also systematically studied is the variation of hydrodynamic forces acting on the spheroid for various Froude numbers and submergence depths. Copyright © 2000 John Wiley & Sons, Ltd.     

Nonlocal-in-time kinetic energy in nonconservative fractional systems,disordered dynamics,jerk and snap and oscillatory motions in the rotating fluid tube

In this article we generalize the basic theoretical properties of nonlocal-in-time kinetic energy approach introduced in the framework of nonlocal classical Newtonian mechanics for the case of fractional dynamical systems explored in the context of the fractional actionlike variational approach. Two independent fractionally Lagrangians weights are considered independently: the Riemann-Liouville fractional weight and the extended exponentially fractional weight. For each weight, the corresponding nonlocal fractional Newton's law of motion is derived. Three main physical applications were discussed in details: free particles, oscillators and dynamics of particles in a rotating tube with earth frame. A number of differential equations depending on fractional and nonlocal-in-time parameters were obtained and their solutions are discussed accordingly. For specific parameters and particular initial conditions, it was observed that the dynamics exhibit a kind of strange phase plot trajectories that indicate the presence of disordered motions. However one of the main results concerns the physics of particles in the rotating tube which display, for specific values of fractional and nonlocal-in-time parameters, oscillatory motions controlled by the nonlocal-in-time parameter.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Some improvements on free surface simulation by the particle finite element method

The Lagrangian method has become increasingly popular in numerical simulation of free surface problems. In this paper, after a brief review of a recent Lagrangian method, namely the particle finite element method, some issues are discussed and some improvements are made. The least‐square finite element method is adopted to simplify the solving of the Navier–Stokes equations. An adaptive time method is derived to obtain suitable time steps. A mass correction procedure is imported to improve the mass conservation in long time calculations and time discretization scheme is adopted to decrease the pressure oscillations during the calculations. Finally, the method is used to simulate a series of examples and the results are compared with the commercial FLOW3D code. Copyright © 2008 John Wiley & Sons, Ltd.     

改进的有自由面非稳定渗流分析复合单元渗透矩阵调整法

应用连续的罚函数,提出了一种改进的自由面边界条件渗流分析复合单元渗透矩阵调整法,优化了此类算法的稳定性,提高了收敛速度和精度,并将此改进算法编制成相应的有限元分析程序,应用于上海地铁4号线董家渡隧道修复基坑降水工程的优化设计,解决了基坑降水过程中复杂三维渗流场的高效精细模拟问题,同时预测了基坑中心水位降至地面以下40 m,位于承压含水层顶面以下,基坑内外自由面落差很大时基坑周边渗流场以及地面沉降的分布特征.