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)

### Single Phase and Two Phase Shallow Flow Model

In this section we discussed the model of 2D shallow water of both single-phase and 2-phase flow. The space and time CESE technique is used to solve shallow flow models in single and two phases. The concept is not a constant enhancement of CFD techniques currently in use and is not even close to being the same as other well-founded approaches. This technique offers a number of unique features, such as a uniform treatment of reality, the conservation elements (CEs), and solution elements (SEs) and a technique for capturing shocks.

First we discuss about the single phase in 2 dimensional and then we describe the model of 2 phase in 2 dimensional.

### Single phase shallow water in two dimensions

The 2 single-phase shallow water in 2-D framework lacking bottom topographical is recovered by omitting the electromagnetic influences. The 2-D single-phase framework has the following form:

(1)

(2)

(3)

The height is , the momentums are and in the direction of and and the component of the fluid speed will be and .

(4)

(5)

The conservative factors are as follows:

(6)

and the fluxes:

(7)

(8)

### 2-dimensional 2 phase shallow flow model

Here we applied CESE scheme for 2 phase flow model. Here we describe the detailed model of 2 phase flow in two dimensional shallow flow.

### The Model of Two Phase Flow

We considering a fluid that is flowing in a horizontal surface,a fluid is the mixture of solid granular material. The densities of solid and fluid materials are and where is less than and their components are assumed that it is incompressible. The flow height is , and the solid particle size is assumed to be . Now we defines the variables:

Now we let that the speeds of the solid and liquid materials that is denoted by and respectively. The fluid is moving in one dimensional so the velocities are moving in x-direction. The momentum of solid and fluid material are and . The model of granular combination of mass and force equations are as follows:

(9)

(10)

(11)

(12)

The height of the bottom from a given position is , where belongs to and momentum is denoted by and is the speed or velocity of the flow towards the and

(13)

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

and are the densities. Now can be written as:

(14)

Fluid velocity is always decreased due to drag forces. It is always opposite to the motion of the fluid. , ; is the capacity of drag power. In hyperbolic terms, we neglected the drag force. In compact form the system are:

(15)

where

(16)

fluxes are:

(17)

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

Moreover,

(18)

The and are the conservative and the non conservative fluxes where l=1,2,3,4 respectively. Similarly and are the solid and the fluid material of the granular material are conservative terms while the non conservative terms are the momentum equations and respectively. The mixture of the momentum can be composed as:

(19)

Where and (q) is defined as:

(20)

In above components 2nd and 4th components of are and that is defined in Eq:3.17. Now we compose the system in semi direct structure or quasi-linear form:

(21)

where A(q) which is known as coefficient matrix is given by

(22)

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

The set of permitted states is:

(23)

or similar or equals to

(24)