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Incompressible Single and Multi-phase SPH with Improved Boundary Treatment

Final Report Summary - SPH-MESHLESS CFD (Incompressible Single and Multi-phase SPH with Improved Boundary Treatment)


The current research program emphasizes two main interlinked aspects of Smoothed Particles Hydrodynamics (SPH) method; namely addressing current shortcomings of the SPH technique which are elaborated in the body of the proposal, and focusing on the development of SPH code to analyze a wide range of engineering flow problems. The specific objectives of the proposal are; (i) to implement a new improved boundary treatment method to the simulation of complex geometries such as flow over an airfoil, and (ii) to develop an incompressible multiphase SPH (IMSPH) algorithm.

Smoothed Particle Hydrodynamics (SPH) is one of the members of meshless Lagrangian particle methods used to solve partial differential equations widely encountered in scientific and engineering problems. Unlike Eulerian (mesh-dependent) computational techniques such as finite difference, finite volume and finite element methods, the SPH method does not require a grid as field derivatives are approximated analytically using a kernel function. In this technique, the continuum or the global computational domain is represented by a set of discrete particles. Here, it should be noted that the term particle refers to a macroscopic part (geometrical position) in the continuum. Each particle carries mass, momentum, energy and other relevant hydrodynamic properties. These sets of particles are able to describe the physical behavior of the continuum, and also have the ability to move under the influence of the internal/external forces applied due to the Lagrangian nature of SPH. Although originally proposed to handle cosmological simulations, SPH has become increasingly generalized to handle many types of fluid and solid mechanics problems. The SPH method has recently received a great deal of attention for modeling multiphase flow problems owing to its obvious advantages such that it notably facilitates the tracking of multiphase interfaces and the incorporation of interfacial forces into governing equations, allows for modeling large topological deformations in flow, and does not require connected grid points for calculating partial differential terms in governing equations. Due to being a relatively new computational method for engineering applications (roughly for two decades old), like any other CFD methods, the SPH method is not completely free from numerical difficulties either, and there are still a few significant issues with the SPH technique, which need to be scrutinized. For instance, it is still a challenge to model physical boundaries effectively and easily. In addition, there are various ways to construct SPH equations (discretization), and a consistent approach has not gained consensus. Highly irregular particle distributions which occur as the solution progresses may cause numerical algorithms to break down, thereby making robustness a significant issue to be addressed. Namely, it is well-known by SPH developers that when passing from one test case to another, new problems are faced. For example, instabilities due to the clamping of SPH particles which are not in general present in modeling a dam-breaking problem show itself in the simulation of flow over bluff bodies, especially at the leading and trailing edges. Additionally, it is not a trivial issue to model multiphase problems having a large variation in transport parameters of constituents. These and other related challenges are not insurmountable. The underlying factors causing these difficulties can be understood thorough extensive research on the verification of SPH against a wide variety of single phase, multiphase, Newtonian and Non-Newtonian flow problems as being the objective of the current work.

In this study, we have developed an improved 2D ISPH algorithm to address above elaborated challenges through simulating a wide variety of single phase, multiphase Newtonian, and Non-Newtonian flow problems. In this work, the test cases for multiphase problems includes single vortex flow, square bubble deformation under the effect of surface tension force, bubble deformation in shear flow, Newtonian bubble rising in viscous and viscoelastic liquids subjected to the combined effects of surface tension and buoyancy forces, Kelvin Helmholtz instability, Rayleigh Taylor Instability, droplet relaxation on a solid substrate, and finally the deformation of a droplet located at the interface of two immiscible fluids while those for the single phase are the flow over backward facing step, a square obstacle and NACA airfoil geometry.

As for the single phase flow in this work, we have endeavored to numerically model the flow over backward facing step, a square obstacle and NACA airfoil geometry using the incompressible SPH (ISPH), and weakly compressible SPH (WCSPH) methods with an improved solid boundary treatment approach, referred to as the Multiple Boundary Tangents (MBT) method. It was shown that the MBT boundary treatment technique is very effective for tackling boundaries of complex shapes. Also, we have proposed the usage of the repulsive component of the Lennard-Jones Potential (LJP) in the advection equation to repair particle fractures occurring in the SPH method due to the tendency of SPH particles to follow the stream line trajectory. This approach is referred to as the artificial particle displacement method, and is shown to eliminate particle clustering induced instabilities. Numerical results suggest that the improved ISPH method which is consisting of the MBT method, artificial particle displacement and the corrective SPH discretization scheme enables one to obtain very stable and robust SPH simulations. Flow over a backward facing step, the square obstacle and NACA airfoil geometry with the angle of attacks between 0o and 15 o in a laminar flow field with relatively high Reynolds numbers (as high as up to 2000) have been simulated. We illustrated that the improved ISPH and WCSPH methods are able to capture the complex physics of bluff-body flows naturally such as the flow separation, wake formation at the trailing edge, and the vortex shedding. The single phase ISPH and WCSPH results are validated with a mesh-dependent Finite Element Method (FEM) and excellent agreements among the results were observed.

Furthermore, we have developed a multiphase Incompressible Smoothed Particle Hydrodynamics (ISPH) method with an improved interface treatment procedure. To be able to demonstrate the effectiveness of the proposed interface treatment which can handle complex multiphase flow problems with high density and viscosity ratios, we have modeled several challenging two and three phase flow problems; namely, single vortex flow, square bubble deformation under the effect of surface tension force, bubble deformation in non-Newtonian shear flow, Newtonian bubble rising in viscous and viscoelastic liquids subjected to the combined effects of surface tension and buoyancy forces, Kelvin Helmholtz Instability, Rayleigh Taylor Instability, droplet relaxation on a solid substrate, and finally the deformation of a droplet located at the interface of two immiscible fluids. The proposed interface treatment includes the usage of (i) cubic spline kernel function for discretizing equations associated with the calculation of the surface tension force while the quntic spline for the discretization of governing equations and the relevant boundary conditions, (ii) smoothing of transport parameters through weighted arithmetic and harmonic interpolations, and (iii) finally, a new discretization scheme for calculating the pressure gradient. The surface tension force is calculated using the so-called Continuum Surface Force (CSF) model. To our best knowledge, within the context of multiphase SPH, no previous research has considered the utilization of two different kernels for interface and governing equations. It is demonstrated that the new interface treatment is quite effective to model multiphase flow problems with the density and viscosity ratios up to 1000 and 100, respectively and the usage of cubic spline for the CSF model significantly improves the quality of the calculated interface in terms of its thickness and sharpness, thereby eliminating the interphase particle penetrations, and in turn leading to the calculation of more accurate velocity and pressure fields. The new interface treatment method is extensively tested on the above given benchmark problems and the results of these simulations are validated against available numerical and experimental data in literature, and excellent agreement is observed between ISPH and literature results.

A two-dimensional ISPH method for modeling incompressible, immiscible three-phase fluid flows has been developed. Surface tension coefficients are decomposed into phase specific coefficients and surface tension force is exerted by implementing the Continuum Surface Force (CSF) model. To complement this, a unique color function is associated with each phase and then smoothed out to improve the robustness of the method while a threshold has been implemented for choosing reliable normals to increase the accuracy of computed surface tension force. Furthermore, artificial particle displacement has been employed to ensure uniform spread of particles throughout the computational domain. Several test cases have been simulated to ensure the capability of the method in handling various three-phase flow combinations.

We have also developed a 2D Lagrangian two-phase numerical model to study the deformation of a droplet suspended in a quiescent fluid subjected to the combined effects of viscous, surface tension and electric field forces. The electrodynamics phenomena are coupled to hydrodynamics through the solution of a set of Maxwell equations. The relevant Maxwell equations and associated interface conditions are simplified relying on the assumptions of the so-called leaky dielectric model. All governing equations and the pertinent jump and boundary conditions are discretized in space using the incompressible Smoothed Particle Hydrodynamics method with improved interface and boundary treatments.

This research work has lead to the preparation of 9 journal papers, (eight already published, one currently under preparation), and eighteen conference papers, thereby contributing significantly to the state of the art in the SPH field.