Shock and Vibration

Volume 2015, Article ID 467376, 16 pages

http://dx.doi.org/10.1155/2015/467376

## Numerical Simulation of Underwater Shock Wave Propagation in the Vicinity of Rigid Wall Based on Ghost Fluid Method

^{1}School of Civil Engineering, Henan Polytechnic University, Jiaozuo, Henan 454000, China^{2}Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China

Received 25 September 2014; Accepted 17 December 2014

Academic Editor: Chao Tao

Copyright © 2015 Ru-Chao Shi et al. 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

This paper presents numerical simulation of underwater shock wave propagation nearby complex rigid wall. The Ghost Fluid Method (GFM) for the treatment of complex rigid wall is developed. The theoretical analysis on the GFM-based algorithm and relevant numerical tests demonstrate that the GFM-based algorithm is first-order accurate as applied to complex rigid wall. A large number of challenging numerical tests show that the GFM-based algorithm is robust and quite simple in various practical problems. The numerical results on shock wave propagation in the vicinity of rigid wall are verified by comparing to exact solution and the results by body-fitted-grid method.

#### 1. Introduction

Numerical simulation of underwater shock wave propagation nearby rigid wall has been a hot area of research in computational physics [1–3], computational biological fluid mechanics, [4] and computer graphics [5]. On the side of application for practical engineering problems, structure somewhile has to be assumed to be rigid wall to simplify computational models [6–10] or to present comparisons for numerical results on deformable solid and elastic-plastic solid [11–14]. There are three kinds of methods by so far to simulate the shock wave propagation in the vicinity of rigid wall. One of them, which is also the most popular, is body-fitted-grid method that need body-fitted grids to be constructed in curvilinear coordinate system according to the shape of rigid wall [15–19]. After the computational regions are discretized by body-fitted grids, numerical methods such as finite difference method, finite volume method, and finite element method can be employed to solve the flow field. Among these three numerical methods, finite difference method may lead to geometrically induced errors [20] while finite volume method and finite element method have shown stability in numerical simulation; finite element method especially is competent to various complicated problems. Another kind of method is Cartesian-grid methods that require the rigid wall to be treated by special algorithm that may be very complicated and can be divided into first-order method [21–24] and second-order method [25]. The application of this kind of method for underwater shock wave propagation can be found in [1] whereas there are very few applications of the Cartesian-grid methods since this kind of method is quite complicated in numerical implementation. The third kind is meshless methods (e.g., Smooth Particle Hydrodynamic method) that have widely applications for complex computational boundary but also have few applications for underwater shock wave propagation because of the large computational resources taken in numerical modelling and simulation.

In this work, we employ the idea of Ghost Fluid Method (GFM) in Cartesian grids to treat complex rigid wall to simulate underwater shock wave propagation. Actually, GFMs are not accurate for defining boundary condition, such as solid-liquid interface, solid-gas interface, liquid-gas interface, and liquid-liquid interface. However in recent years GFMs have shown wide range applications [26–29] for shock wave propagation while interacting with material interface because of simplification and robustness. GFMs can be divided into original GFM (OGFM) and several improved versions. OGFM is developed by Fedkiw et al. [30] and the essence of nonoscillation has been verified. Nevertheless, OGFM is incapable in numerical simulation of multimedium that is very stiff at interface or where the density ratio is very large though it can be applied for low pressure gas-liquid flow. Then a modified GFM (MGFM) is developed by Liu et al. [31], which is robust for gas-liquid flow where the density ratio of the medium is very large. Next, real GFM (RGFM) [32] is developed to treat moving interface in some extreme situations. Subsequently some attempts in employing GFMs to treat liquid-solid interface and gas-solid interface have been made. A few of simple applications of GFM for fluid/solid coupling are presented by Fedkiw [33] but GFM is still found to be inaccurate in some situations. To overcome this problem, an essentially MGFM for fluid/elastic solid coupling is proposed and developed by Liu et al. [34] by employing Navier equation to solve solid and constructing an approximate Riemann problem at the interface.

As GFM has the advantages of simplification, robustness, and flexibility, we apply the idea and essence of GFM for treatment of complex rigid wall. There are several problems that need to be solved. The dominant problem is how to define Riemann problem for water-rigid wall interface. The second problem is how to evaluate the flow states at ghost grid nodes. The last one is how to analyze the numerical accuracy of the GFM-based algorithm. The text below is arranged as follows. In Section 2, we briefly present the methods, governing equations and numerical schemes, which are employed to solve the flow field. In Section 3, the GFM for the treatment of water-rigid wall interface and analysis on conservation errors are presented in detail. In Section 4, the theoretical solution for underwater shock wave propagation in the vicinity of rigid wall is provided to further test the numerical error of GFM-algorithm developed. In Section 5, various challenging tests are carried out by GFM-based algorithm, body-fitted-grids method, and theoretical analysis for further comparisons and discussions. In last section, brief conclusions are summarized and given.

#### 2. Methods and Governing Equations

In this work, Euler equations are employed to solve flow field. While we apply GFM to rigid wall, level set method and related techniques are used to keep track of water-rigid wall interface and other material interfaces.

##### 2.1. Euler Equations

The flow field is governed by Euler equations:where is density and , , and are velocities at , , and direction, respectively. is total energy which can be expressed asHere, is internal energy per unit mass. 2D Euler equations are obtained by setting . 1D Euler equations are obtained by setting and and can be given aswhere and . The total energy can be written asIn calculations, Euler equations are solved with fifth-order WENO spatial discretization [35] and second-order Runge-Kutta (R-K) time discretization.

##### 2.2. Equations of State

For gas, we employ -law which is expressed aswhere is pressure, is density, is internal energy per unit mass, and is the ratio of specific heats. For high pressure gas, we can also use JWL equation [36]:where is pressure, is density, is internal energy per unit mass, and , , , , and are constants.

For water, Tait equation [36] and isentropic one-fluid model [37] are employed here and given aswhere is the saturated pressure of water, , , , , , , and the initial value of is set to be . For every iteration step, would be updated by if the result of is beyond . Generally, . To succinctly express the relationship between pressure and density , we rewrite the EOS of water that does not include gas phase asOne may find that pressure of water can be determined by density directly. For water, the internal energy per unit mass is also the function of density , which can be expressed as [36]Since the internal energy of water can be determined by the density directly, it is not necessary to solve corresponding energy conservation equation in calculations.

##### 2.3. Level Set Method

We use level set equation [38] to keep track of material interface:where , , and are velocities in , , and directions, respectively. is the sign distance function of arbitrary point in the domain and can be written asFor 2D problems, in (10) is set to be . For 1D problems, both and are set to be . The velocity extension technique [39] is employed to improve the accuracy of results by level set method. In calculations, level set equation is discretized by fifth-order HJ-WENO [40] and second-order R-K method.

##### 2.4. Constant Extrapolation Approach

One of the most important steps of GFM is extrapolation that can extrapolate flow states from real region to ghost region. The partial equations for extrapolations can be expressed as [30, 41]where is artificial time, is the flow states that need to be extrapolated, and is the vector normal material interface and will be given as . The sign “” is taken as “+” if the flow states need to be extended from to . If the flow states are extrapolated from to , we take “.” In calculations, (12) is simply solved by first-order upwind MinMod scheme.

#### 3. The GFM for Rigid Wall

GFMs include assuming that medium also exists outside the boundary of real medium, how to set ghost grid nodes for real medium and then how to update the flow states at these ghost grid nodes. In order to develop the GFM for rigid wall, we may first construct meshes on the entire domain neglecting whether or not the meshes are on the side of fluid (water). Then we extend the values at the grid nodes on the side of fluid (water) across fluid-rigid wall interface to the other side where ghost fluid is assumed to exist and grid nodes are correspondingly set as ghost nodes. While we try to complete the two steps mentioned above, the interface between real fluid and ghost fluid needs to be defined and we choose level set function to describe this interface. The algorithm for updating status of ghost fluid nodes should retrain real fluid from escaping when real fluid interacts with the interface and should behave the essential features of rigid wall on the opposite side of real fluid.

##### 3.1. GFM-Based Algorithm

Supposing fluid is on left side of interface while rigid wall is on the right side of interface, we follow the ideology of [30, 31] and mathematically described the 1D GFM for treating fluid-rigid wall interface aswhere the subscripts and , respectively, represent the status on the left and right side of interface and the sign “” refers to the ghost status at the related grid nodes. The density and velocity on the side of rigid wall can be considered asThe interfacial status at next time step will be predicted. Since GFM needs to behave the features of slip wall boundary for inviscid flow, we havewhere superscript “” indicates that the value of is predicted for next time step. will be extrapolated from interface to ghost region; hence we haveBy combining (14), (15), and (16), we haveSince density and pressure must satisfy continuous condition, we also extrapolate the values at the real grid nodes that are in the neighborhood of interface to ghost grid nodes, which can be expressed asThe illustration for GFM algorithm mentioned above is shown in Figure 1. For multidimensional problem, velocity resolutions are tangential velocity and normal velocity . and are obtained the same as and , and is treated as and :where will be determined byCorrespondingly, the schematic for multidimensional GFM algorithm is shown in Figure 2.