The Fourth Order Finite Volume For Two Phase (Part 2)

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

The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 1)

Preliminaries

In this chapter, we discussed the detail explanation of the preliminaries of computational fluid dynamics. Here we discussed different special cases to described the motion of fluids. In motion of fluid we discussed about the conservative variables, conservation law of momentum, mass and energy described by the equation of continuity.

Primitive and the Conservative Variables

Plasma, usually called gas mixture, is a kind of energy that makes up the majority of the universe. The solution of a system of equations governing plasma dynamics generally requires technical knowledge. They attract attention and are linked to some key assumptions. The bulk properties of matter are sufficient to provide, and fluid particles remain the same data on a system’s current condition.

The primitive variables are mass, velocity and pressure. The components of velocity are { ${v_{1}}$ , ${v_{2}}$ , ${v_{3}}$ }, the mass of the fluid particles m and the pressure p are called the primitive variables. The other type of variables are the conservative variables that describes the fluid flow. This type of variables included as density of mass ${\rho}$ . The x-momentum, y-momentum, and z-momentum are three different types of momentum are $\rho {{v_1}}$ , $\rho {{v_2}}$ , $\rho {{v_3}}$ . The total energy E, the component of magnetic field in every direction ${B_{1}}$ , ${B_{2}}$ , ${{B_{3}}}$ .

The conservation law of mass, the conservation energy and the Newton second law of motion these all are depends on the conserved quantities.

This comprehensive study delves into the intricacies of two-dimensional two-phase flows, focusing on the fundamental conservation laws that govern these complex systems. We meticulously examine the mass conservation rules, Newton’s second law of motion, and energy conservation rules to understand the underlying dynamics.

The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 1)

To accurately model and analyze these two-phase flows, we employ the fourth-order CESE scheme. A powerful numerical technique known for its accuracy, efficiency, and stability. This scheme allows us to capture the nuances of two-phase flow behavior and gain valuable insights into their complex interactions.

Our research explores various aspects of two-phase flows, including phase interactions, interface dynamics, and flow patterns. We investigate the transfer of mass, momentum, and energy. Between the two phases, analyze the behavior of the interface, and study the different flow regimes that can occur.

Two-phase flows have broad applications in numerous fields, such as chemical engineering, energy systems, and environmental engineering. Understanding these flows is crucial for optimizing processes, improving efficiency, and addressing environmental challenges.

By conducting this comprehensive analysis. We aim to contribute to the advancement of two-phase flow research. And provide valuable insights for researchers and practitioners in various fields.

The fluid hight h, the x-momentum $hv_{1}$ , the y-momentum $hv_{2}$ , and the related magnetic field components in each direction $hB_{1}, hB_{2}$ are among the preserved variables.

Conservation Law

Conservation law relates with physics that deals with the study of measurable property in which the system does not change as the system expand overtime. The mathematical statements explaining the transport of properties of fluids that must be conserved are known as conservation laws. Conservation law also includes conservation law of linear momentum, energy, angular momentum and electric charge etc. Mathematically conservation law is depends on continuity equation, PDE that gives us the relation between the amount and transport of the quantity.

This definition is fulfilled by considered that we have a fixed arbitrary volume V in which fluid is flow in a surface in a specific volume in a given time interval. Due to net change in fluid flow across the surface S, the conserved quantity is zero. In general, a set of conserved variables is referred to as a vector dependent on space and spatial coordinates $(x, t) := (x, y, z, t)$ , is represented as:

(1) $\begin{equation*} u(x, t) = (u_1(x, t), u_2(x, t), ..., u_m(x, t))^{T},\end{equation*}$

where m specifies the vector’s number of components. The flow function is:

(2) $\begin{equation*} f(u) = (f_1(u), f_2(u), ..., f_m(u))^{T}.\end{equation*}$

Within the time range [t{0}, t{0}+dt], the flow across the surface S of an arbitrary volume V produces.

The Fourth Order Finite Volume For Two Phase Flow Model using Space Time CESE Method (Part 1)

(3) $\begin{equation*} \iiint\limits_V \left[u\left( x,t_{0} + dt \right) - u\left( x,t_{0}\right) \right] \partial_v + \int_{t_{0}}^{t_{0} + dt}\iint\limits_V f(u(x,t)).dSdt=0\end{equation*}$

We applied the fundamental theorem of calculus to the first volume integral, which involves the variables $u_{i}, i = 1,……,m$ representing the starting and ending terms. For the second integral, we utilized the divergence theorem over the surface S within the time interval [t0,t0+dt]. This led to the following expression:

(4) $\begin{equation*} \int_{t_{0}}^{t_{0} + dt}\iiint\limits_V [\frac{\partial u}{\partial t}+ \nabla . f(u)].dVdt=0\end{equation*}$

We obtain the conservation laws of the form because the volume is randomly chosen.

(5) $\begin{equation*} \frac{\partial u}{\partial t}+\nabla.f(u)=0.\end{equation*}$

Hyperbolic Conservation Law

A hyperbolic conservation law is depends on time and coupled with the PDE’s. In one dimension it takes the form: $U_{x}$ + $F U_{x}$ = 0

(6) $\begin{equation*} U(x,0) = U_{0}(x)\end{equation*}$

where $U= (u_{1} , u_{2}, …..,u_{n})$ and $f(U)=(f_{1} , f_{2}, …..,f_{n})$ are the vector conserved variables and flux functions. This system is also write in quasi linear form:
where $f'(U)$ = $\frac{\partial f(u)}{\partial x}$ and A(u) is the jaccobian matrix.
This equation is hyperbolic if the eigenvalues of A is m then $\lambda_{1} ,\lambda_{2},………\lambda_{m}$ and the $r_{1},r_{2}……,r_{m}$ are the eigenvectors are the linearly independent.

The Riemann Problem

We also discuss about how to solve Riemann problems in this paper. The IVP of Riemann is discussed in two forms which is $u^{L}$ and $u^{R}$ that is apart by a plane across where jump of discontinuity at x=0 atleast one $i$ , $u_{i}^{L}$ $\not=$ $u_{i}^{R}$ . The difficulty with the starting value is :