The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method

### Abstract

In this work, a fourth-order finite volume CESE approach is utilized to solve a two-phase flow model. The model consists of two phases of solid and liquid. Each phase has its mass and momentum equations. The equations are non-linear, non-conservative and coupled together. The exact solution is not available in the literature. Our motivation is to extend the second order CESE method to 4rth order.

We will numerically solve the two-phase flow model to evaluate the performance of the fourth-order CE/SE method. We’ll then compare our results to those previously published for the second-order method. The two-phase flow model presents several numerical challenges, including: As an efficient numerical method can handle these difficulties. The CE/SE strategy is proficient to solve the above-mentioned complexities. Small height changes can induce oscillations in the solution, potentially leading to negative heights.

### Introduction

This chapter gives the motivation about the study of fourth order space time conservation element and solution element(CESE) method for solution of two-phase flow, describe literature, review and resolved the problem numerically, gives the detailed information of this model and provides the summary of contents in this thesis. We briefly define some of the approaches of two-phase flow in section 1.1. In the literature section 1.2, there are many examples of two phase flow.

### Motivation

Two-phase flows occur when two distinct substances or phases of a single substance coexist. Examples include liquid-liquid mixtures (like oil and water), liquid-solid suspensions (like grains and water), and liquid-gas mixtures (like soft drinks). These flows are prevalent in various fields, including ecological research, nuclear energy, and heat transfer systems. The resulting equations from these flows are systems of partial differential equations (PDE’s). These equations are non-linear and in many cases exact solution is difficult to obtain.

These flows consist of mass and momentum conservation equations of solid and fluid phases. These studies have proven that, despite the fact that it is feasible to write accurately the degree of complexity of such forms and the conservation equations, it is not possible to write accurate the degree of complexity of such forms. Their strategy demands a significant amount of competitive advantages and statistical data, which they lack. Therefore in real life situations, it is important to use simpler conservation equations for a variety of functional problems.

### The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 2)

Several numerical difficulties are related to these flows since each phase is studied individually and include the two sets of mass and momentum conservation equations. As an instance, in two-phase if we have two momentum equations that generate many difficulties like loss of hyperbolicity and unreliability that enumerate interphase interactivity between these terms. The most important factor in two phase are volume ratio or mass ratio are represents in terms of concentration. The most basic theory is that the two phases have the same velocity and temperature in the flow channel.

### Literature Review

The two-phase mathematical models present several numerical challenges. Additionally, small changes in physical variables like height and density, especially when dealing with small values, can lead to negative values. Furthermore, physically unrealistic closure relations may hinder the attainment of unique solutions, raising concerns about well-posedness. Lastly, these two-phase models often involve stiff source terms, making their discretization a demanding task. To effectively address these numerical complexities, we require an efficient and robust numerical technique that can accurately handle these challenges and deliver reliable solutions. A well-balanced scheme can serve this purpose.

The present work is related to the extension of second order well-balanced CESE method to fourth order CE/SE method and implementation on shallow granular model with two phases.

We employ a unified approach that treats time and space similarly, drawing inspiration from the finite element discontinuous Galerkin (DG) method. We ensure stability by constraining the fluxes or variables. The conservation element is referred to as the control volume. The CESE scheme has proven effective in various applications.

### The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 2)

Yazhong Jiang initiated the work on extension of 2nd order to higher order shallow water equations. Erduran discussed the fourth order correctness for shallow water equations. Later, Sidrah extended the second order CE/SE technique to fourth order for shallow magnetohydrodynamic equations. In this project we will extend the order of second order CE/SE method to fourth order. And will check the performance by implementing it on two phase shallow granulose flow model.

### Numerical Challenges

To produce the accurate, robust and effective scheme that can be used to solve the fourth order CE/SE of two phase flow model. There are numerous difficulties that arise, some of them are as follows:

1: It is difficult to extend a second order scheme to higher order.

The challenging task is that how accurately

we extend CESE method of order two to fourth order which can produce oscillation free results.

2: There are many numerical methods available to solve the two-phase flows, how CE/SE method is better than the existing methods?

3: How accurately we descritize the source term.

4: How these well balanced laws help us to maintain steady state condition with balancing the behavior of flux terms and source term.

### Proposed Numerical Techniques

This thesis deals with the extension of second order CESE method to fourth order CESE method. The fourth order scheme is then applied to solve two phase granular flow model. The considered model form a complicated set of non linear PDEs.

### CESE Method

The conservation and solution element (CESE) method is a finite volume technique that generates numerical solutions of fluid-dynamic equations and conservation laws in a variety of physical systems. The two-phase flow equations seems to have an easy numerical solution but actually it is not easy. It posses many difficulties due to the involvement of source term. Here we are considering CE/SE scheme which will balance our source terms as well as fluxes and give us an efficient and accurate set of results. The CE/SE scheme, developed by Chang. The difference it has with traditionally using scheme is that it treats time and space in unified manner. Due to its unique methodology, it help to overcome the shortcomings of other schemes.

### The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 2)

For evaluation of fluxes, this scheme use space-time staggered mesh in case of any Riemann problem. In the same way as the traditional, the CESE discretization maintains conservation using the finite volume technique. However, it calculates the spatial derivatives using a different approach with a compact stencil, and it obtains the same results. The CESE method has been proven to be effectively expanded. Applied to all-around of physical issues in the last two decades, particularly nonlinear time dependent hyperbolic systems including the dynamical development of waves and discontinuities. The development of a family of upwind CESE schemes, the improved definitions of CESE for arbitrary meshes, and a new technique to constructing large CESE schemes are all examples of recent CESE advancements.

### The central scheme

For the solution of hyperbolic conservation laws, central schemes (NT) are extensively utilized. These schemes are simple, effective, and well-organized. Central schemes have been used to solve problems in shallow flows, astrophysical, computational fluid dynamics, and many other sectors of science and engineering. These strategies are more accurate and convenient than other upwind methods because they do not utilize Riemann solvers. The foundation of these systems is the Lax-Friedrichs first order scheme.

### The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 2)

It’s a high resolution system that doesn’t need a Riemann solver. Unlike previous upwind techniques, the staggered central scheme does not require the problem’s eigen structure information, making it more efficient and simple. It can also be applied to all systems resulting from conservation rules. One of these schemes’ main drawbacks is that when the time step is very small, there is a lot of numerical dispersion in the solutions. Kurganov and Tadmor propose an enhanced version of the central scheme to address this problem.

As a result, the improved central scheme is a high resolution technique since it provides a trivial amount of dissipation. Because this system is extremely simple and effective, it can be used in a wide range of applications.

### The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 2)

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