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

The Fourth Order Finite Volume For Two Phase (Part 4)

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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)

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

CESE Scheme of 4th Order

The second order CE/SE scheme is based on a fourth-order CE/SE system with a similar general notation. Consider the following type of situation the time-space. Cartesian coordinates are a type of coordinate system E 3(x, y, t), where x is a variable. The Eq. will be:

(1)   \begin{equation*}\nabla \cdot h_m = 0,\end{equation*}

where \nabla=(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial t}) and h_m is the flux vector in space and time. Integrating equation Eq.1:

(2)   \begin{equation*}\oint_{A(v)} h_m \cdot ds = 0,\end{equation*}

In any space–time region v, the closed boundary surface in E 3(x, y, t), da =(d\delta)n, within the space d\delta and the unit outward normal to the vector n, is denoted by A(v). Furthermore, the entire space–time flow exiting the surface element da is denoted by h_m\cdot ds. The conservation element shows in Figure 4.5, projected as the polygon V {1}V {5}V {7}V {9}. At time level n-\frac{1}{2} on the (x,y) plane, there is a 2D cartesian coordinates grid. The three-dimensional space and time region is separated into conservation elements, which are non-overlapping quadrilaterals. There are four basic CEs in each CE. The associated BCEs are V_{1}V_{2}V_{3}V_{4}, V_{3}V_{4}V_{5}V_{6}, V_{6}V_{7}V_{8}V_{3} and V_{8}V_{3}V_{2}V_{9}.

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

The centroid of each BCE is denoted by the letters C_{i}, while the centroid of CE is denoted by the letters G. The points G and V_{3} coincide due to the uniform grid.
The alternating CESE technique is set up so that the BCE cell centres have a solution. C_{i} is retrieved in the first half of the time step and the vertices’ solution (for our condition G) is modified in the 2nd half of the time step. There is an solution’s element related with each solution point G (SE).

The purpose of the solution element is to obtain flow variables q_{m}, f_{m} and g_{m} with a Taylor expansions. Take for instance, the solution point. C_{i}^{/}. The polygon is the CE that corresponds to it. V_{1}V_{2}V_{3}V_{4}V_{1}^{/}V_{2}^{/}V_{3}^{/}V_{4}^{/} The SE, on the other hand, is the result of the union of three planes. V_{1}^{/}V_{2}^{/}V_{3}^{/}V_{4}^{/}, A_{1}A_{3}A_{1}^{//}A_{3}^{//} and A_{2}A_{4}A_{2}^{//}A_{4}^{//}.
The CEs’ boundries are closer to the cartesian planes, and the axis corresponds to their outward unit ordinary vectors.

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

(\boldmath f, \boldmath g, \boldmath q) travelling over CE and BCE border planes.
Since the conservation law of shallow water MHD conditions applies to any space-time closed area, and the specified CE is similar to the space-time closed region, we can discretize the conservation law. Using the solution point C_{i}^{/} as an example, follow the conservation law Eq.2 to its CE. Unlike the finite difference scheme, we can extend the CESE approach to higher-order schemes using the same grid and related CEs and SEs. This is one reason why the CESE scheme is easy to extend to higher orders. Equation 2 implies that

(3)   \begin{align*}\oint_{S\left({CE}\left({C}{1}\right)\right)} \mathbf{h}{m} \cdot {d} \mathbf{s}=\oint_{S\left({V}{1} {V}{2} {V}{3} {V}{4} {V}{1}^{\prime} {V}{2}^{\prime} {V}{3}^{\prime} {V}{4}^{\prime}\right)} \mathbf{h}{m} \cdot {d} \mathbf{s}. \end{align*}

The point (0,0,- 1) is unit outward normal to the bottom surface. As a result, this surface only allows space-time flux \boldmath q to travel through it.

As a result, these planes only pass through space and time flux \boldmath f. We divided each side of the surface into two sections. For each section, we integrated using the Taylor third-order expansion of fluxes near the solution point.

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

Let us consider that V_{1}^{}V_{1}^{\prime}V_{4}^{}V_{4}^{\prime}, for example V_{1}^{}V_{1}^{\prime}A_{1}^{}A_{1}^{\prime} and A_{1}^{}A_{1}^{\prime}V_{4}^{}V_{4}^{\prime} are the union of the two polygons. The integration V_{1}^{}V_{1}^{\prime}A_{1}^{}A_{1}^{\prime} utilizes the Taylor expansion of f_{m} at V_{1} and the integration A_{1}^{}A_{1}^{\prime}V_{4}^{}V_{4}^{\prime} utilizes the Taylor expansion of f_{m} at V_{4}.

Although (0,0,1) is the top surface’s outward unit vector, using the Taylor expansion we approximated the flux q_{m} at the solution point C_{1}^{\prime}.

(4)   \begin{equation*}q _ { m t } = - f _ { m x } - g_{my}.\end{equation*}

It is necessary to construct down the mixed space and time derivatives by taking derivatives with respects to the space variable x on the two sides and considering them the same.

Spatial Derivatives of Conserved factors

The f_m is the derivative of flux and the time derivative of the conservative variables at this point are all include in the flux derivative. At this point all that requires is to obtain the space derivative factors of q_m, such as q_{m\xi}, q_{m\xi\phi}, and q_{m\xi\phi\chi}\quad\xi=x,y;\quad\phi=x,y;\quad\chi=x,y.

Using q_m acquired from Equation 4, the first derivative of q_{m\xi} is changed at the i_th node. To changed each space of higher order derivative of conserved variables we use the updated value of next derivative of higher order. We must now modify the values of two more derivatives, q {m\xi\phi} and q {m\xi\phi\chi}, for the fourth-order scheme. We can retrieve all derivatives using the corrected value.

Conclusions

Derived and implemented a fourth-order CESE method on a two-phase granular flow model. We compared the results with those obtained using the existing second-order CESE method. We wanted to check how solutions behave by increasing the order of a method. It is always desirable that increasing the order will definitely improve the accuracy of the results. In the case of the CESE method the increase in order does not always produce accurate results. In some cases by changing the limiters improved the results significantly.

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

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