A numerical study on high pressure gradient flows

Due to the developments in computer sciences, numerical solutions to flow related problems are more interesting and developing area. In the last thirty years CFD has become a feasible tool to be integrated in design process. It allows engineers to do simulations for highly complex designs and investigate complex phenomena for which experiments are too expensive or impossible. One of those investigation areas is underwater explosion. One may divide the underwater explosion phenomenon in to two main parts. If the first one is the response of the structure to the acceleration due to an underwater explosion, the second one is the propagation of the shockwave due to the explosion. In this thesis the second part is investigated. The propagation of the shockwave due to an explosion can be divided into two parts also. One may investigate methods to simulate the time dependent amplitude of the shockwave due to the underwater explosion. This subject may include bubble collapse, secondary shockwaves, cavitation etc. Also there is a problem of capturing a fully developed shockwave's propagation. In literature and in industry the first part of this problem is commonly captured with JWL equation (Jones-Wilkins-Lee equation), which is an EOS (Equation of State) for explosives. There are several coefficients in the JWL equation and the values of those coefficients are given extensively in many references for different kind of explosives. The present thesis is focused on the second part, the propagation of the fully developed shockwave. Shockwave can be defined as an abrupt disturbance in the media. There are two characteristic properties of the shockwave. First one is the shockwave's propagation speed is faster or equal to the speed of sound in that media. The second characteristic property of the shockwave is that the disturbance in the media is nearly un-continuous. Numerical methods that are used for capturing the shockwave, due to its speed and nearly uncontinuous nature, need to be highly complex. The best discretization of the PDE's and the discretization of the field variables are yet to be found. For example the research that is done by Kanamori & Suzuki (2011) shows that high order schemes tend to oscillate near gradient region and low order schemes tends to smooth the discontinuity. Also because of the high propagation speed and pressure gradient, the mesh that needs to be used is highly dense. As a result of the dense mesh computational effort is considerably high for these kind of calculations. Furthermore the CFL number (time step that is used) needs to be small because of the stability issues which results in long computation time. In this thesis different methods and approaches for simulation of high gradient flows are employed in order to overcome the aforementioned difficulties for calculating high pressure gradient flow (shock propagation). Two different numerical methods are used for calculating this phenomenon. The first one is Finite Volume Method (FVM) and the second one is Smoothed Particle Hydrodynamics (SPH). In both methods conservative integral forms of Euler equations are used. According to Hirsch (2007), the integral form is fully capable of defining un-continuities such as shockwaves. This is done in order to eliminate the stability issues related with the nature of the phenomena. Implicit time stepping scheme is used for FVM in order to use higher CFL numbers (time steps) for calculations and reducing the solution time. SPH method, because that it is a Lagrangian solution method, results in fewer computational nodes (particles) which creates a smaller solution matrix and reduces the solution time. Also according to Monaghan (1988), particle based methods are more suitable to define the physical phenomena's because that every particle represents the atomic structure of the fluid. In the present thesis after a brief introduction to the problem in chapter one, the meaning of conservative form, the definition and the physical meaning of flux and the derivations of Euler equations (continuity, momentum and energy equations) in conservative form are explained at length in chapter two. In chapter three, the methodology used for capturing high pressure gradient flow using FVM is explained at length. The discretization methods for both field variables and flux, the preconditioning technique and advantages, flux treatment and its application to a simple PDE (partial differential equation), the meaning of CFL condition and the requirement to fulfil in order to solve the problem at hand and the equation of state used to model compressible fluid (in gas state) are explained and mathematically shown. Fundamentals of SPH method and the methodology for capturing high pressure gradient flow using SPH method is explained in chapter four. SPH is a meshless Lagrangian method which is developed by astrophysicists in the late 70's. This is a relatively new method and shows a great promise on violent flows. In chapter four, after a brief introduction, Lagrangian and Eulerian approaches to engineering problems are discussed and the related formulations, the difference between them and the advantages/disadvantages of both are given. Characteristic features of SPH formulization are introduced. Integral definition of any field variable is given and the definition and usage of kernel function is introduced. The particle approach and its implementation is discussed. After the fundamentals, the particle approach is applied to Euler equations and the optimum forms of the equations for capturing high pressure gradient flow using SPH method are given. In chapter five, both FVM and SPH method are employed to the same problem and the results are presented with comparison to the study that is done by Wardlaw (1998). The aforementioned study is done in 2-D and with a compressible fluid which can be defined with ideal gas equation of state. The calculations as a part of present thesis are done using the same operating conditions. In order to simulate the benchmark case with FVM, a well-documented commercial code, ANSYS FLUENT is used as the solver. Domain is created by ANSYS Design Modeller and the generation of the mesh is performed by the pre-processor, ICEMCFD. The mesh convergence study is done and the solver preferences are given for the simulation. For simulating the same benchmark case by SPH method, a solver code is written as a part of this thesis. In chapter five the manipulation of the solution domain and the distribution of real and virtual particles are shown. Solver algorithm and the results are shared. In the end, the computations performed show that both the numerical methods and the solution strategies that are applied to the benchmark case as a part of this thesis are suitable for capturing shockwaves. The advantages and disadvantages of both methodologies are discussed in chapter six.