Incompressible sph method for simulating newtonian and non-newtonian flows with a free surface
TL;DR: In this article, an incompressible smoothed particle hydrodynamics (SPH) method is presented to simulate Newtonian and non-Newtonian flows with free surfaces.
Abstract: An incompressible smoothed particle hydrodynamics (SPH) method is presented to simulate Newtonian and non-Newtonian flows with free surfaces. The basic equations solved are the incompressible mass conservation and Navier–Stokes equations. The method uses prediction–correction fractional steps with the temporal velocity field integrated forward in time without enforcing incompressibility in the prediction step. The resulting deviation of particle density is then implicitly projected onto a divergence-free space to satisfy incompressibility through a pressure Poisson equation derived from an approximate pressure projection. Various SPH formulations are employed in the discretization of the relevant gradient, divergence and Laplacian terms. Free surfaces are identified by the particles whose density is below a set point. Wall boundaries are represented by particles whose positions are fixed. The SPH formulation is also extended to non-Newtonian flows and demonstrated using the Cross rheological model. The incompressible SPH method is tested by typical 2-D dam-break problems in which both water and fluid mud are considered. The computations are in good agreement with available experimental data. The different flow features between Newtonian and non-Newtonian flows after the dam-break are discussed.
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TL;DR: A review of the applications of smoothed particle hydrodynamics (SPH) to incompressible flow can be found in this article, where the authors focus on the applicability of SPH to complex physical problems.
Abstract: This review focuses on the applications of smoothed particle hydrodynamics (SPH) to incompressible or nearly incompressible flow. In the past 17 years, the range of applications has increased as researchers have realized the ability of SPH algorithms to handle complex physical problems. These include the disruption of free surfaces when a wave hits a rocky beach, multifluid problems that may involve the motion of rigid and elastic bodies, non-Newtonian fluids, virtual surgery, and chemical precipitation from fluids moving through fractured media. SPH provides a fascinating tool that has some of the properties of molecular dynamics while retaining the attributes of the macroscopic equations of continuum mechanics.
643 citations
TL;DR: A new divergence-free ISPH approach is proposed here which maintains accuracy and stability while remaining mesh free without increasing computational cost by slightly shifting particles away from streamlines, although the necessary interpolation of hydrodynamic characteristics means the formulation ceases to be strictly conservative.
Abstract: The stability and accuracy of three methods which enforce either a divergence-free velocity field, density invariance, or their combination are tested here through the standard Taylor-Green and spin-down vortex problems. While various approaches to incompressible SPH (ISPH) have been proposed in the past decade, the present paper is restricted to the projection method for the pressure and velocity coupling. It is shown that the divergence-free ISPH method cannot maintain stability in certain situations although it is accurate before instability sets in. The density-invariant ISPH method is stable but inaccurate with random-noise like disturbances. The combined ISPH, combining advantages in divergence-free ISPH and density-invariant ISPH, can maintain accuracy and stability although at a higher computational cost. Redistribution of particles on a fixed uniform mesh is also shown to be effective but the attraction of a mesh-free method is lost. A new divergence-free ISPH approach is proposed here which maintains accuracy and stability while remaining mesh free without increasing computational cost by slightly shifting particles away from streamlines, although the necessary interpolation of hydrodynamic characteristics means the formulation ceases to be strictly conservative. This avoids the highly anisotropic particle spacing which eventually triggers instability. Importantly pressure fields are free from spurious oscillations, up to the highest Reynolds numbers tested.
572 citations
TL;DR: Comparisons of a semi-implicit and truly incompressible SPH (ISPH) algorithm with the classical WCSPH method are presented, showing how some of the problems encountered inWCSPH have been resolved by using ISPH to simulate incompressable flows.
Abstract: In the smoothed particle hydrodynamics (SPH) discretisation method for the Navier-Stokes equations the most widespread method to solve for pressure and mass conservation is the weakly compressible assumption (WCSPH). This includes hydraulics applications and leads to some drawbacks such as severe artificial pressure fluctuations and a limitation to very small time steps related to the WCSPH Mach number and explicit method. This paper presents comparisons of a semi-implicit and truly incompressible SPH (ISPH) algorithm with the classical WCSPH method, showing how some of the problems encountered in WCSPH have been resolved by using ISPH to simulate incompressible flows. Mathematical models are presented before describing SPH formalism. Several standard boundary conditions are introduced and special attention is given to tracking the surface particles. The lid-driven cavity flow (Re=400 and 1000) is performed as a benchmarking test. A bluff body test case (a square cylinder in a closed channel, Re"d=20 and 100 based on the cylinder diameter) shows that pressure fields extracted from WCSPH are very unreliable whereas ISPH predict pressures and forces in closer agreement with classical finite volume CFD methods. Dam-breaking cases, with dry or wet beds downstream, are then presented to highlight free-surface flow and rapid dynamics effects. The WCSPH and ISPH results are generally verified with reference data from experiment and/or another numerical method. All the comparisons show improvement with ISPH and good agreement in general.
538 citations
Cites background from "Incompressible sph method for simul..."
...They are regularly distributed at the initial state and have zero velocity through the whole simulation, while several layers of dummy particles [19] are built as an extension of the edge particles surrounding the solid boundaries to ensure the same order of discretisation (in terms of kernel compact support) for particles located close to those boundaries, as for particles located in the core of the domain....
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...There are several ways to prevent this phenomenon, for example, by using mirror particles [4] or some repulsive forces [14], or dummy particles [19]....
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TL;DR: In this article, the Drucker-Prager model with associated and non-associated plastic flow rules is implemented into the smoothed particle hydrodynamics (SPH) code to describe elastic-plastic soil behavior.
Abstract: Simulation of large deformation and post-failure of geomaterial in the framework of smoothed particle hydrodynamics (SPH) are presented in this study. The Drucker–Prager model with associated and non-associated plastic flow rules is implemented into the SPH code to describe elastic–plastic soil behavior. In contrast to previous work on SPH for solids, where the hydrostatic pressure is often estimated from density by an equation of state, this study proposes to calculate the hydrostatic pressure of soil directly from constitutive models. Results obtained in this paper show that the original SPH method, which has been successfully applied to a vast range of problems, is unable to directly solve elastic–plastic flows of soil because of the so-called SPH tensile instability. This numerical instability may result in unrealistic fracture and particles clustering in SPH simulation. For non-cohesive soil, the instability is not serious and can be completely removed by using a tension cracking treatment from soil constitutive model and thereby give realistic soil behavior. However, the serious tensile instability that is found in SPH application for cohesive soil requires a special treatment to overcome this problem. In this paper, an artificial stress method is applied to remove the SPH numerical instability in cohesive soil. A number of numerical tests are carried out to check the capability of SPH in the current application. Numerical results are then compared with experimental and finite element method solutions. The good agreement obtained from these comparisons suggests that SPH can be extended to general geotechnical problems. Copyright © 2008 John Wiley & Sons, Ltd.
528 citations
TL;DR: The algorithm is based upon Fick's law of diffusion and shifts particles in a manner that prevents highly anisotropic distributions and the onset of numerical instability, and is validated against analytical solutions for an internal flow at higher Reynolds numbers than previously.
Abstract: The incompressible smoothed particle hydrodynamics (ISPH) method with projection-based pressure correction has been shown to be highly accurate and stable for internal flows and, importantly for many problems, the pressure field is virtually noise-free in contrast to the weakly compressible SPH approach (Xu et al., 2009 [31]). However for almost inviscid fluids instabilities at the free surface occur due to errors associated with the truncated kernels. A new algorithm is presented which remedies this issue, giving stable and accurate solutions to both internal and free-surface flows. Generalising the particle shifting approach of Xu et al. (2009) [31], the algorithm is based upon Fick's law of diffusion and shifts particles in a manner that prevents highly anisotropic distributions and the onset of numerical instability. The algorithm is validated against analytical solutions for an internal flow at higher Reynolds numbers than previously, the flow due to an impulsively started plate and highly accurate solutions for wet bed dam break problems at zero and small times. The method is then validated for progressive regular waves with paddle motion defined by linear theory. The accurate predictions demonstrate the effectiveness of the algorithm in stabilising solutions and minimising the surface instabilities generated by the inevitable errors associated with truncated kernels. The test cases are thought to provide a more thorough quantitative validation than previously undertaken.
513 citations
Cites methods from "Incompressible sph method for simul..."
...The method of Shao and Lo [27], although stable, is quite noisy and inaccurate....
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...This instability has been avoided in various ways: by imposing invariant density rather than zero velocity divergence [27], by imposing zero velocity divergence and density invariance together [12], and by imposing the kinematic constraint of constant volume for each particle through non-thermodynamic pressure [9]....
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References
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TL;DR: In this paper, the concept of a fractional volume of fluid (VOF) has been used to approximate free boundaries in finite-difference numerical simulations, which is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations.
Abstract: Several methods have been previously used to approximate free boundaries in finite-difference numerical simulations. A simple, but powerful, method is described that is based on the concept of a fractional volume of fluid (VOF). This method is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations. To illustrate the method, a description is given for an incompressible hydrodynamics code, SOLA-VOF, that uses the VOF technique to track free fluid surfaces.
11,567 citations
TL;DR: In this paper, a new technique is described for the numerical investigation of the time-dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier-Stokes equations are written in finite-difference form, and the solution is accomplished by finite-time step advancement.
Abstract: A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement The primary dependent variables are the pressure and the velocity components Also used is a set of marker particles which move with the fluid The technique is called the marker and cell method Some examples of the application of this method are presented All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired
5,841 citations
TL;DR: A finite-size particle scheme for the numerical solution of two-and three-dimensional gas dynamical problems of astronomical interest is described and tested in this article, which is then applied to the fission problem for optically thick protostars.
Abstract: A finite-size particle scheme for the numerical solution of two- and three-dimensional gas dynamical problems of astronomical interest is described and tested. The scheme is then applied to the fission problem for optically thick protostars. Results are given, showing the evolution of one such protostar from an initial state as a single, rotating star to a final state as a triple system whose components contain 60% of the original mass. The decisiveness of this numerical test of the fission hypothesis and its relevance to observed binaries are briefly discussed.
5,508 citations
TL;DR: In this paper, a finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced, which is equally applicable to problems in two and three space dimensions.
Abstract: A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.
4,991 citations
TL;DR: In this paper, the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed, focusing on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.
Abstract: In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.
3,623 citations