Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 350496, 15 pages

http://dx.doi.org/10.1155/2015/350496

## Simulation of Droplet Impacting on Elastic Solid with the SPH Method

^{1}School of Mechanical Engineering, Xinjiang University, Urumqi, Xinjiang 830046, China^{2}School of Electrical Engineering, Xinjiang University, Urumqi, Xinjiang 830046, China^{3}State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an 710049, China

Received 10 September 2014; Revised 11 January 2015; Accepted 11 January 2015

Academic Editor: Stanford Shateyi

Copyright © 2015 Xiao-Jing Ma and Mamtimin Geni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The phenomenon of droplet impacting on solid surfaces widely exists in nature. Simulation of this kind of problems has always been a difficult and important research area in the computational fluid dynamics (CFD). Two basic numerical methods used nowadays are developed from Euler’s theory and Lagrange’s theory, respectively. Over the past few decades, the grid-based methods based on Euler’s theory, such as the finite element method (FEM), the finite volume method (FVM), and the finite difference method (FDM), have been widely applied in the simulation of the flow. However, dealing with large deformations of the moving free surfaces and interaction of the multiphase flows are difficult to achieve for the grid-based methods. Some methods which are good at capturing the free surface and regenerating the grid have been proposed, such as PIC [1], MAC [2], VOF [3], and LS [4] methods. Moreover, mesoscopic simulation methods have been proposed including MPS [5], LBM [6], and DPD [7]. Although these methods can deal with the unsteady flows with complex free surfaces, they also lead to high computational complexity and large amount of data processing.

Smoothed Particle Hydrodynamics (SPH) method which is a meshless method based on Lagrange’s theory was proposed by Lucy [8] and Gingold and Monaghan [9] in the 1970s. The basic idea of the SPH method is to discretize the computed domain into a large number of particles instead of the fixed grids. These particles can freely move according to the governing conservation equations but be restricted by the material properties. The SPH quantities can be obtained as weighted averages from the particle values over the region of interest. Therefore, the SPH method is able to handle the complex free surfaces and the large deformation problems easily. Recently, the SPH method has been adopted for various problems, including Newtonian fluid flows [10], non-Newtonian fluids flows [11–14], incompressible fluids [15–17], multiphase flow [18–20], free surface flows [21, 22], large deformation [23], and the dynamic response of elastic-plastic materials [24–26].

Droplet impacting on a rigid plate is a typical instance of free surface flows and attracts much attention. In recent years, different methods have been used to simulate this problem and some useful conclusions were obtained. For example, Tomé et al. [26] used the finite difference method to simulate unsteady free surface flows. Using MAC, the problem of impacting drops was simulated by Harlow and Shannon [27, 28]. Xie et al. [29] treated the problem of flow phenomenon after droplet with different diameters and velocities impacting the fluid films by using MPS. Prosperetti and Oguz [30] predicted the changing of free surface by BEM when droplet impacted the fluid films. Fang et al. [31], Rafiee et al. [32], and Ma et al. [33] simulated a single droplet impacting a rigid plate by SPH. Various modified algorithms for SPH were employed to improve stability and accuracy in these literatures. However, these methods for simulating the impacting droplet are based on the fixed boundary without considering the deformation of plate. Regarding the plate as a rigid solid surface is a feasible solution for the free surface problem when the hardness of solid is much higher than fluid. In fact, the absolutely rigid solid does not exist. The stability of system will be significantly impaired because of microdeformation of solid or fluid. Taking elastohydrodynamic lubrication, for example, the microdeformation of the wheel gear or bearing can lead to many fatal consequences in practice. Therefore, it is necessary to consider the interaction between solid and fluid to analyze the effect of deformation on the whole system.

The purpose of this paper is to extend and test the ability of the SPH method for the solid-liquid interaction problem. In Section 2, the governing equations are described in detail and the SPH formulation is modified to cope with the special issues. The modifications and improvements include the correction algorithm of density on solid-liquid interface, artificial viscosity, and artificial stress. The process of a droplet impacting an elastic plate with different elastic modulus was simulated and some results are displayed in Section 3. The paper ends with some concluding remarks in Section 4.

#### 2. Introduction of the SPH Method

##### 2.1. Basic Principles of the SPH Method

In the SPH method, the computed domain is discretized into several continuous particles with material properties. These physical quantities are obtained by integral representation of function [34] as follows:where is the approximation of , is the vector position, the smoothing length defines the influence area of the kernel function , and satisfies the following conditions:The integrated form of can be discretized as a summation over all the particles in the influence domain as follows:When the kernel function is differentiable, the approximations derivative of based on (3) is derived as

##### 2.2. Kernel Function

The kernel function is a key element in the SPH method to ensure the accuracy of the algorithm. Many possible forms of the kernel have been analyzed and compared in these literatures [34–37]. In order to balance the computational accuracy and efficiency, the cubic spline kernel is adopted as one of the most widely used kernel functions in SPH method [10]. Therefore, the cubic spline function is chosen as follows:in two dimensions, the normalization factor is .

##### 2.3. Governing Equations

Under the tremendous impact, the solid material deforms obviously and solid particles characterized by SPH method move like fluid. Therefore, the governing equations for high strain hydrodynamics with material strength were proposed in these literatures [24, 38] to simulate the impacting and penetrating problems. Those equations including the mass and momentum conservation equations have exactly the same form as those for transient compressible fluid flow. In a Lagrangian frame, the governing equations are written as [34]where denotes the density, is the mass of particle, is time, and and are the velocity and the acceleration due to external forces of the th component, respectively. is the th component of the stress tensor. Calculation of for solid and fluid is different and more details are shown in Section 2.5.2.

##### 2.4. Artificial Viscosity and Artificial Stress

###### 2.4.1. Artificial Viscosity

In SPH method, the artificial viscosity is employed to allow the algorithm to be capable of modeling shock waves or simply to stabilize a numerical scheme. Considering the artificial viscosity in (7), the momentum equation could be obtained asThe most common form of the artificial viscosity suggested by Monaghan [10, 39] is where , and denotes the sound speed. The parameter for prevents singularities and it should be small enough to prevent severe smoothing of the viscous term in the high density regions. Normally, this is achieved by taking , which means that smoothing of velocity will only take place if the particle spacing is less than [10]. Moreover, the -term produces shear and bulk viscosity, while the -term is similar to the Von Neumann-Richtmyer viscosity to handle high Mach number shocks. The values of and are not critical, but they have significant influence on the computed results. Monaghan [10] suggested their values be around 1.0 and 2.0, respectively. But the -term may result in larger shear viscosity when modeling flows with physical viscosity. It has been proposed that this term should be removed by setting and the -term should be retained to prevent unphysical particle penetration [16]. For the rigid boundary case without the -term of artificial viscosity, there was fracturing phenomenon in the Newtonian impacting droplet, but the computation can continue [31, 32]. However, unlike the rigid-plate case, for considering the deformation of elastic-plate and fluid-solid interaction under impact in this paper, it is found necessary to retain the -term to avoid the divergence and penetration in simulation. The detailed results given in Section 3.2 show that values 1.0 and 2.0 are suitable for and , respectively.

###### 2.4.2. Artificial Stress

In SPH method, the tension force can introduce instability for the solid deformation, while the tensile stress can also become unstable for the free surface. There are many methods to improve the stability of tension and shear. The most commonly used one is the “artificial stress” proposed by Monaghan [40] and Gray et al. [41]. The basic idea of artificial stress is to introduce a small repulsive force between a pair of neighboring particles to prevent them from getting too close when they are in a state of tensile stress. To do this, the momentum equation is modified asThe detailed formulation of the artificial stress is as follows:where , , and is the initial distance of particles. The components of the artificial stress tensor are given by [31, 32]whereIn these equations, is a parameter and it is set to 0.2 for the Newtonian fluid [31, 32].

##### 2.5. Fluid-Solid Coupling

The processing of two-phase coupling is a popular interest of research in computational fluid dynamics. Many techniques have been proposed to accurately and effectively simulate the interface. However, most of them only analyzed the interaction of solid on fluid, such as flow around a cylinder, or the interaction of fluid on solid like feathers in the air movement. Obviously, the interaction between two phases should be analyzed to ensure the accuracy of the simulation results. In this work, an improved algorithm of fluid-solid coupling based on the SPH method is presented. The specific algorithm is as follows and the flow chart for the numerical simulation is shown in Figure 1.