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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.

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 [

Smoothed Particle Hydrodynamics (SPH) method which is a meshless method based on Lagrange’s theory was proposed by Lucy [

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. [

The purpose of this paper is to extend and test the ability of the SPH method for the solid-liquid interaction problem. In Section

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 [

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 [

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 [

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 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 [

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

The flow chart for the numerical simulation (the particle

In the SPH method, density is a critical variable in solution of the governing equations.

In this algorithm, particles of different materials in the influence domain are regarded as neighboring particles, and the numerical results prove that the above algorithm is correct and practical. Moreover, (

The key of momentum equation (

However, nonphysical penetration and particles doping may occur on the fluid-solid interface during the process of impacting. The algorithm of interface proposed by Liu et al. [

Considering the balancing forces, the momentum equation on the fluid-solid interface could be obtained based on (

Consider

In this work, Leap-Frog (LF) method is chosen for solving the time-dependent ordinary differential equations of the SPH method. In order to prevent numerical divergence, the time-step

The two-dimensional Newtonian droplet impacting on a rigid plate as a typical instance for free surface flows has been investigated by other researchers [

In order to compare the results from different methods, the parameters of droplet given in these literatures [^{−1}. The density and dynamic viscosity of droplet are set to ^{−3} and ^{−2}, the droplet begins to fall along the ^{−3},

To check on the reasonable values of

Particle positions of the droplet with different values of

In this section, results from two-dimensional simulations of a droplet impacting on an elastic plate are shown to demonstrate the performance of the proposed SPH method for the fluid-solid coupling problems. Figure

The morphology evolution during the impacting process (the elastic modulus

The stress field evolution during the impacting process (the elastic modulus

The velocity field evolution during the impacting process (the elastic modulus

The shape of the droplet and the deformation of the plate at different dimensionless time are presented in Figures

The stress tensor given here is

With reference to Figure

In order to analyze the influencing factors of the deformation degree, the process of the droplet impacting on the plate is simulated. The plate is regarded as a rigid solid and an elastic solid with different elastic modulus, respectively. The lengths of droplet and plate along

The width of droplet and plate during the impacting process.

The width of droplet

The width of plate

The normal stress tensor

The normal stress

The normal stress

The velocity

The velocity

The velocity

The deformation of droplet in the process of impacting on a rigid wall has been predicted with grid-based method by Tomé et al. [

The computed width of the droplet is compared with those of literatures [

As shown in Figure

Figure

The velocities of droplet and plate along

This paper presents an improved SPH method to deal with the fluid-solid coupling problems. In this work, the SPH method is modified by adding a number of features, such as the correction algorithm of density and artificial viscosity. And the results for a droplet impacting the fixed boundary obtained by this method are in good agreement with those from the FDM of Tomé et al. [

The correction algorithm of density proposed in this paper is used to avoid the distortion of density on the interface. Moreover, an artificial viscosity is also added to the SPH momentum equation to reduce the tensile stress instability. And it is found necessary for the impact problem to retain both terms of artificial viscosity to achieve a better stability and more accurate free surface. The results show that the proposed method is able to accurately capture both the shape of droplet and the microdeformation of elastic solid surface under the force of impacting. The influence of the elastic modulus on the whole process is also discussed. Obviously, the deformation tendency is inversely proportional to the rigidity of the material, which is coincident with the practical law.

Additionally, further more detailed studies are needed for the flow phenomenon of the impacting droplet with different diameters and initial velocities, which will be performed more intensively in future work.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Basic Research Program of China (973 Program, no. 2011CB706602), China National Natural Science Foundation (no. 11072209), and Open Projects of State Key Laboratory for Strength and Vibration of Mechanical Structures (Xi’an Jiaotong University).