Rankine Source Solution Research Articles

An improved impermeable solid boundary scheme for Meshless Local Petrov–Galerkin method

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson’s Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition.

Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

In the Lagrangian meshless (particle) methods, such as the smoothed particle hydrodynamics (SPH), moving particle semi-implicit (MPS) method and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R), the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities (such as the viscous stresses). In some meshless applications, the Laplacians are also needed as stabilisation operators to enhance the pressure calculation. The particles in the Lagrangian methods move following the material velocity, yielding a disordered (random) particle distribution even though they may be distributed uniformly in the initial state. Different schemes have been developed for a direct estimation of second derivatives using finite difference, kernel integrations and weighted/moving least square method. Some of the schemes suffer from a poor convergent rate. Some have a better convergent rate but require inversions of high order matrices, yielding high computational costs. This paper presents a quadric semi-analytical finite-difference interpolation (QSFDI) scheme, which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations. The convergence, accuracy and robustness of the present schemes are compared with the existing schemes. It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures, particularly for estimating the Laplacian of given functions.

A sharp-interface treatment technique for two-phase flows in meshless methods

In this paper, a new interface treatment technique for multi-phase meshless methods is proposed for two-dimensional simulations. It enables the interface conditions to be applied effectively on the interface points involving interface tension and high viscosity. This technique is incorporated in the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) and the model predictions are validated through a number of standard test cases. Convergent results that agreed well with both available analytical and numerical solutions from other methods are obtained in simulations of square-droplet deformation, capillary wave, bubble rising and Rayleigh–Taylor instability. In these cases, sharp pressure discontinuity at the interface is well predicted with depressed parasitic current and the capability of the technique in dealing with high viscosity is comprehensively demonstrated.

Comparison of different iterative schemes for ISPH based on Rankine source solution

Smoothed Particle Hydrodynamics (SPH) method has a good adaptability for the simulation of free surface flow problems. There are two forms of SPH. One is weak compressible SPH and the other one is incompressible SPH (ISPH). Compared with the former one, ISPH method performs better in many cases. ISPH based on Rankine source solution can perform better than traditional ISPH, as it can use larger stepping length by avoiding the second order derivative in pressure Poisson equation. However, ISPH_R method needs to solve the sparse linear matrix for pressure Poisson equation, which is one of the most expensive parts during one time stepping calculation. Iterative methods are normally used for solving Poisson equation with large particle numbers. However, there are many iterative methods available and the question for using which one is still open. In this paper, three iterative methods, CGS, Bi-CGstab and GMRES are compared, which are suitable and typical for large unsymmetrical sparse matrix solutions. According to the numerical tests on different cases, still water test, dam breaking, violent tank sloshing, solitary wave slamming, the GMRES method is more efficient than CGS and Bi-CGstab for ISPH method.

MLPG_R method for modelling 2D flows of two immiscible fluids

SummaryThis is a first attempt to develop the Meshless Local Petrov–Galerkin method with Rankine source solution (MLPG_R method) to simulate multiphase flows. In this paper, we do not only further develop the MLPG_R method to model two‐phase flows but also propose two new techniques to tackle the associated challenges. The first technique is to form an equation for pressure on the explicitly identified interface between different phases by considering the continuity of the pressure and the discontinuity of the pressure gradient (i.e. the ratio of pressure gradient to fluid density), the latter reflecting the fact that the normal velocity is continuous across the interface. The second technique is about solving the algebraic equation for pressure, which gives reasonable solution not only for the cases with low density ratio but also for the cases with very high density ratio, such as more than 1000. The numerical tests show that the results of the newly developed two‐phase MLPG_R method agree well with analytical solutions and experimental data in the cases studied. The numerical results also demonstrate that the newly developed method has a second‐order convergent rate in the cases for sloshing motion with small amplitudes. Copyright © 2016 John Wiley & Sons, Ltd.

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

Three meshless methods, including incompressible smooth particle hydrodynamic (ISPH), moving particle semi-implicit (MPS) and meshless local Petrov–Galerkin method based on Rankine source solution (MLPG_R) methods, are often employed to model nonlinear or violent water waves and their interaction with marine structures. They are all based on the projection procedure, in which solving Poisson’s equation about pressure at each time step is a major task. There are three different approaches to solving Poisson’s equation, i.e. (1) discretizing Laplacian directly by approximating the second-order derivatives, (2) transferring Poisson’s equation into a weak form containing only gradient of pressure and (3) transferring Poisson’s equation into a weak form that does not contain any derivatives of functions to be solved. The first approach is often adopted in ISPH and MPS, while the third one is implemented by the MLPG_R method. This paper attempts to review the most popular, though not all, approaches available in literature for solving the equation.

Modified SFDI for fully nonlinear wave simulation

In the Meshless Local Petrove-Galerkin based on Rankine source solution (MLPG-R), a simplified finite difference interpolation (SFDI) scheme was developed for numerical interpolation and gradient calculation (CMES, Vol. 23(2), pp. 75-89). Numerical tests concluded that the SFDI is generally as accurate as the linear moving least square method (MLS) but requires less CPU time. In this paper, a modified SFDI is proposed for numerically modelling of nonlinear water waves, considering the typical feature of the spatial variation of the wave-related parameters. Systematic numerical investigations are carried out and the results indicate that the modification considerably improves the robustness of the SFDI on gradient estimation. Although the scheme is originally derived for meshless method, its feasibility and accuracy in the mesh-based methods are discussed here through the fully nonlinear wave simulation using the Quasi Arbitrary Lagrangian Eulerian Finite Element Method (QALE-FEM), which is based on fully nonlinear potential theory.

Incompressible SPH Based on Rankine Source Solution for Water Wave Impact Simulation

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian meshless particle method. It was originally developed to simulate astrodynamics but has been extended to model dynamics problems with violent motions in many areas. This paper is based on ISPH and its pressure time history is transformed into another equation based on a Rankine source solution. In the new formulation of ISPH, the governing equation for pressure does not include any derivatives of unknown functions and so overcomes the problems associated with direct numerical approximation to second derivatives in existing ISPH formulation, and some important numerical handling techniques will also be included, like free surface particle identification method and solid boundary discrete scheme. The newly improved method will be applied to model various wave and wave impact interaction with different angle slopes, and the results of comparison with experimental data are also given. According to the comparison of pressure time history, this new method can get a good agreement with experimental results.

Incompressible SPH method based on Rankine source solution for violent water wave simulation

With wide applications, the smoothed particle hydrodynamics method (abbreviated as SPH) has become an important numerical tool for solving complex flows, in particular those with a rapidly moving free surface. For such problems, the incompressible Smoothed Particle Hydrodynamics (ISPH) has been shown to yield better and more stable pressure time histories than the traditional SPH by many papers in literature. However, the existing ISPH method directly approximates the second order derivatives of the functions to be solved by using the Poisson equation. The order of accuracy of the method becomes low, especially when particles are distributed in a disorderly manner, which generally happens for modelling violent water waves. This paper introduces a new formulation using the Rankine source solution. In the new approach to the ISPH, the Poisson equation is first transformed into another form that does not include any derivative of the functions to be solved, and as a result, does not need to numerically approximate derivatives. The advantage of the new approach without need of numerical approximation of derivatives is obvious, potentially leading to a more robust numerical method. The newly formulated method is tested by simulating various water waves, and its convergent behaviours are numerically studied in this paper. Its results are compared with experimental data in some cases and reasonably good agreement is achieved. More importantly, numerical results clearly show that the newly developed method does need less number of particles and so less computational costs to achieve the similar level of accuracy, or to produce more accurate results with the same number of particles compared with the traditional SPH and existing ISPH when it is applied to modelling water waves.

A hybrid method for modelling two dimensional non-breaking and breaking waves

This is the first paper to present a hybrid method coupling an Improved Meshless Local Petrov Galerkin method with Rankine source solution (IMLPG_R) based on the Navier–Stokes (NS) equations, with a finite element method (FEM) based on the fully nonlinear potential flow theory (FNPT) in order to efficiently simulate the violent waves and their interaction with marine structures. The two models are strongly coupled in space and time domains using a moving overlapping zone, wherein the information from both the solvers is exchanged. In the time domain, the Runge–Kutta 2nd order method is nested with a predictor–corrector scheme. In the space domain, numerical techniques including ‘Feeding Particles’ and two-layer particle interpolation with relaxation coefficients are introduced to achieve the robust coupling of the two models. The properties and behaviours of the new hybrid model are tested by modelling a regular wave, solitary wave and Cnoidal wave including breaking and overtopping. It is validated by comparing the results of the method with analytical solutions, results from other methods and experimental data. The paper demonstrates that the method can produce satisfactory results but uses much less computational time compared with a method based on the full NS model.

Improved MLPG_R method for simulating 2D interaction between violent waves and elastic structures

Interaction between violent water waves and structures is of a major concern and one of the important issues that has not been well understood in marine engineering. This paper will present first attempt to extend the Meshless Local Petrov Galerkin method with Rankine source solution (MLPG_R) for studying such interaction, which solves the Navier–Stokes equations for water waves and the elastic vibration equations for structures under wave impact. The MLPG_R method has been applied successfully to modeling various violent water waves and their interaction with rigid structures in our previous publications. To make the method robust for modeling wave elastic–structure interaction (hydroelasticity) problems concerned here, a near-strongly coupled and partitioned procedure is proposed to deal with coupling between violent waves and dynamics of structures. In addition, a novel approach is adopted to estimate pressure gradient when updating velocities and positions of fluid particles, leading to a relatively smoother pressure time history that is crucial for success in simulating problems about wave–structure interaction. The developed method is used to model several cases, covering a range from small wave to violent waves. Numerical results for them are compared with those obtained from other methods and from experiments in literature. Reasonable good agreement between them is achieved.

MLPG Method Based on Rankine Source Solution for Modelling 3D Breaking Waves

MLPG Method Based on Rankine Source Solution for Modelling 3D Breaking Waves

MLPG_R Method for Numerical Simulation of 2D Breaking Waves

Following our previous work, the Meshless Local Petrov-Galerin method based on Rankine source solution (MLPG_R) will be extended in this paper to deal with breaking waves. For this purpose, the governing equation for pressure is improved and a new technique called Mixed Particle Number Density and Auxiliary Function Method (MPAM) is suggested for identifying the free surface particles. Due to complexity of the problem, two dimensional (2D) breaking waves are only concerned here. Various cases are investigated and some numerical results are compared with experimental data available in literature to show the newly developed method is robust.

MLPG Method Based on Rankine Source Solution for Simulating Nonlinear Water Waves

MLPG Method Based on Rankine Source Solution for Simulating Nonlinear Water Waves

Calculation of Deep-Water Wash Waves Using a Combined Rankine/Kelvin Source Method

This paper presents a method for computation of far-field wash waves in deep water. The method combines a nonlinear Rankine source method in an inner domain with a Kelvin source method for the far-field waves in an outer domain. Kelvin sources are distributed on a vertical matching wall, positioned at the outer edge of the inner domain. These sources are used to specify a boundary condition for the disturbance velocity potential on the matching wall. The boundary condition is used in the Rankine source solution of the inner domain. The size of the inner domain can be reduced in the transverse direction compared to a method using Rankine sources only, as the wave reflections at the edge of the inner domain are eliminated. Further, the far-field waves can be computed using the solution on the matching wall together with the Kelvin source distribution. The verification of the present method includes a comparison for a single Kelvin point source and a comparison to a Rankine source method at intermediate distances for the Wigley hull and for a catamaran. A grid dependence study for the position, size, and panel density on the matching wall is included for the Wigley hull. Computed and measured longitudinal wave cuts are compared for a catamaran both in the inner and the outer domain. Good agreement is obtained.