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

The Fourth Order Finite Volume For Two Phase (Part 3)

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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)   \begin{eqnarray*}{\partial_t h}+{\partial_x m_1^{x}}+{\partial_y m_2^{x}}=0,\end{eqnarray*}

(2)   \begin{eqnarray*}{\partial_t m_1^{x}}+{\partial_x}\left( \frac{m_1^{2x}}{h}+\frac{g}{2}h^2 \right)+{\partial_y}\left( \frac{m_1^{x} m_2^{x}}{h} \right)=0,\end{eqnarray*}

(3)   \begin{eqnarray*}{\partial_t m_2^{x}}+{\partial_y}\left( \frac{m_1^{x} m_2^{x}}{h} \right)+{\partial_x}\left( \frac{m_2^{2x}}{h}+\frac{g}{2}h^2 \right)=0.\end{eqnarray*}

The height is h, the momentums are m_{1}^{x}=hu and m_{2}^{x}=hv in the direction of x and y and the component of the fluid speed will be u and v.

(4)   \begin{align*}\frac{\partial q_l}{\partial t}&+\frac{\partial f_l}{\partial x}+\frac{\partial g_l}{\partial y}=0, \quad l=1,2,3.\end{align*}

(5)   \begin{align*}{\partial_t \b w}+{\partial_x \b F(\b w)}=0.\end{align*}

The conservative factors are as follows:

(6)   \begin{align*}q_{1} \overset{\mbox{def}}{=}h, \qquad q_{2} \overset{\mbox{def}}{=}m_{1}^{x}=hu, \qquad q_{3} \overset{\mbox{def}}{=}m_{2}^{x}=hv,\end{align*}

and the fluxes:

(7)   \begin{align*}&f_{1}\overset{\mbox{def}} = m_{1}^{x}=q_{2}, \\ & f_{2}\overset{\mbox{def}} = \frac{m^{2x}{1}}{h} +\frac{g}{2}h^2 =\frac{(q_2)^2}{q_1}+\frac{g}{2}(q_1)^2, \nonumber \\ \label{eq10} & f_{3}\overset{\mbox{def}} = \frac{m_{1}^{x}m_{2}^{x}}{h} = \frac{q_{2}q_{3}}{q_1},\end{align*}

(8)   \begin{align*}g_{1}\overset{\mbox{def}}{=}m_{2}^{x}=q_{3}, \quad g_{2}\overset{\mbox{def}} = \frac{m_{1}^{x}m_{2}^{x}}{h}=\frac{q_{2}q_{3}}{q_1}, \nonumber \\g_{3}\overset{\mbox{def}} = \frac{m^{2x}_{2}}{h} +\frac{g}{2}h^2 = \frac{(q_3)^2}{q_1}+\frac{g}{2}(q_1)^2.\end{align*}

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 \rho_{s^{*}} and \rho_{f^{*}} where \rho_{s^{*}} is less than \rho_{f^{*}} and their components are assumed that it is incompressible. The flow height is phi, and the solid particle size phi is assumed to be h. Now we defines the variables:

    \[h_{s^{*}} = \phi(h) \quad and \quad h_{f^{*}} = (1- \phi )h\]

Now we let that the speeds of the solid and liquid materials that is denoted by u_{s^{*}} and u_{f^{*}} respectively. The fluid is moving in one dimensional so the velocities are moving in x-direction. The momentum of solid and fluid material are m {s^{*}}= h{s^{*}}u_{s^{*}} and m {f^{*}}= h{f^{*}}u_{f^{*}}. The model of granular combination of mass and force equations are as follows:

(9)   \begin{eqnarray*} \frac{\partial}{\partial t}h {s^{*}}$ + $\frac{\partial}{\partial x}m {f^{*}}$ = 0 \end{eqnarray*}

(10)   \begin{eqnarray*} \frac{\partial}{\partial t}m_{s^{*}} + \frac{\partial}{\partial x} (\frac{ m_{s^{*}}^{2}}{h_{s^{*}}} + \frac{g}{2} h_{s}^{2} + \frac{g}{2} (1-\gamma) h_{s^{*}}h_{f^{*}}) + \gamma(g)h_{s^{*}} \frac{\partial}{\partial x}h_{f^{*}} = -g h_{s^{*}} \frac{\partial}{\partial x} b + \gamma F^{D}\end{eqnarray*}

(11)   \begin{eqnarray*} \frac{\partial}{\partial t}h_{f^{*}} + \frac{\partial}{\partial x}m_{f^{*}} = 0\end{eqnarray*}

(12)   \begin{eqnarray*} \frac{\partial}{\partial t}m_{f^{*}} + \frac{\partial}{\partial x} (\frac{ m_{f}^{2}}{h_{f^{*}}} + + \frac{g}{2} h_{f^{*}}^{2}) + gh {f^{*}} \frac{\partial}{\partial x}h{s^{*}} = -g h_{f^{*}} \frac{\partial}{\partial x}b - F^{D} \end{eqnarray*}

The height of the bottom from a given position is b=b(x), where x belongs to R and momentum is denoted by m = hu and u is the speed or velocity of the flow towards the x and

(13)   \begin{eqnarray*} \gamma= \frac{\rho_{f^{*}}}{\rho_{s^{*}}}\end{eqnarray*}

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

\rho_{f^{*}} and \rho_{s^{*}} are the densities. Now F^{D} can be written as:

(14)   \begin{eqnarray*} F^{D} = D(h {s^{*}} + h{f^{*}}) ( u_{f^{*}}-u_{s^{*}}).\end{eqnarray*}

Fluid velocity is always decreased due to drag forces. It is always opposite to the motion of the fluid. D = D( \phi,|u_{f^{*}} - u_{s}| ; \sigma) is the capacity of drag power. In hyperbolic terms, we neglected the drag force. In compact form the system are:

(15)   \begin{eqnarray*} \frac{\partial q_{l}}{\partial t} + \frac{\partial f_{l}}{\partial x} = \tau_{l} ,l=1,2,3,4\end{eqnarray*}

where

(16)   \begin{eqnarray*} q_{1} = h_{s^{*}} , q_{2} = m_{s^{*}} = h_{s^{*}}u_{s^{*}} , q_{3} =h_{f^{*}} , q_{4}=m_{f^{*}}=h_{f^{*}}u_{f^{*}}\end{eqnarray*}

fluxes are:

(17)   \begin{eqnarray*} f {1} =m{s}= q_{2}, \\ f_{2}=\frac{m_{s^{*}}^{2}}{h_{s^{*}}} + \frac{g}{2} h_{s^{*}}^{2} + \frac{g}{2}(1-\gamma) h_{s^{*}}h_{f^{*}} = \frac{(q_{2})^{2}}{q_{1}} + \frac{g}{2}(q_{1})^{2}+ \frac{g}{2} (1-\gamma)q_{1}q_{3}, \\f_{3} = m_{f^{*}}=q_{4} , \\ f_{4}= \frac{m_{f^{*}}^{2}}{h_{f^{*}}} + \frac{g}{2}h_{f^{*}}^{2} = \frac{(q_{4})^{2}}{q_{3}} +\frac{g}{2}(q_{3})^{2}.\end{eqnarray*}

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

Moreover,

(18)   \begin{eqnarray*} \tau_{1} =0 = \tau_{3}, \\\tau_{2}= - \gamma g h_{s^{*}}\frac{\partial h_{f^{*}}}{\partial x} - g h_{s^{*}} (\frac{\partial b}{\partial x}) = \gamma g q_{1}(\frac{\partial q_{3}}{\partial x})- g q_{1}(\frac{\partial b}{\partial x}) ,\\\tau_{4} = -g h_{f^{*}}(\frac{\partial h_{s}}{\partial x}) - g h_{f^{*}}(\frac{\partial b}{\partial x}) = -g q_{3}(\frac{\partial q_{1}}{\partial x}) - g q_{3}(\frac{\partial b}{\partial x}).\end{eqnarray*}

Thef_{l} and \tau_{l} are the conservative and the non conservative fluxes where l=1,2,3,4 respectively. Similarly h_{s^{*}} and h_{f^{*}} are the solid and the fluid material of the granular material are conservative terms while the non conservative terms are the momentum equations m_{s^{*} } and m_{f^{*}} respectively. The mixture of the momentum can be composed as:

(19)   \begin{eqnarray*} \frac{\partial }{\partial t}m_{m} + \frac{\partial }{\partial x}f_{m}(q) = -g(h_{s^{*}}+ \gamma h_{f^{*}}) \frac{\partial}{\partial x}b\end{eqnarray*}

Where q=( q_{1}, q_{2}, q_{3}, q_{4} )^{T} and f_{m}(q) is defined as:

(20)   \begin{eqnarray*} f_{m} (q) & = f_{2}(q) + \gamma f_{4}(q)+ \gamma g h_{s^{*}}h_{f^{*}} \\ &= \frac{m_{s^{*}}^{2}}{h_{s^{*}}} + \gamma\frac{m_{f^{*}}^{2}}{h_{f^{*}}}+ \frac{g}{2}(h_{s^{*}}^{2}+ \gamma h_{f^{*}}^{2})+g\frac{(1+ \gamma)}{2}h_{s^{*}}h_{f^{*}}\end{eqnarray*}

In above components 2nd and 4th components of f_{1}q are f_{2}q and f_{4}q that is defined in Eq:3.17. Now we compose the system in semi direct structure or quasi-linear form:

(21)   \begin{eqnarray*} \frac{\partial }{\partial t}(q)+ A(q)+ \frac{\partial}{\partial x}(q) = \psi^{b}(q)\end{eqnarray*}

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

(22)   \begin{eqnarray*} A(q) =\left[ \begin{array}{cccc}0 &1 &0 &0\ -(\frac{q_{2}}{q_{1}})^{2}+gq_{1}+g\frac{1- \gamma}{2}(q_{3}) &2\frac{u_{2}}{q_{1}} &g\frac{1+ \gamma}{2}(q_{1}) &0\ 0 &0 &0 &1\g q_{3} &0 &-(\frac{q_{4}}{q_{3}})^{2}+gq_{3} &2\frac{q_{4}}{q_{3}}\end{array}\right]\end{eqnarray*}

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

The set of permitted states is:

(23)   \begin{eqnarray*} \Omega=\left\lbrace {q \epsilon R^{4} ; h_{s^{*}},h_{f^{*}}\geq 0 ~ and ~ u_{s^{*}},u_{f^{*}} \epsilon R}\right\rbrace\end{eqnarray*}

or similar or equals to

(24)   \begin{eqnarray*} \Omega=\left\lbrace {q \epsilon R^{4} ; h\geq 0, \phi \epsilon\left[ 0,1\right] ~ and ~ u_{s^{*}},u_{f^{*}} \epsilon R}\right\rbrace\end{eqnarray*}

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

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