Roller Numerical Methods

From CIRPwiki
Jump to: navigation, search

The surface roller transport equation is solved in CMS-Wave using a finite difference method. The source terms are calculated at the grid cell centers. The advective or transport term is approximated using either the first-order or second-order upwind finite difference scheme. The first order upwind scheme is given by

 


\frac{\partial(S_{sr}c_j)}{\partial x} \bigg|_{i,j} = 
\left\{
\begin{align}
&\frac {(S_{sr}c_j)_{i,j} - (S_{sr}c_j)_{i,j-1}} {\delta x_{i,j-1}} , \text{for }  c_{i,j} > 0 \\
&\frac {(S_{sr}c_j)_{i,j+1} - (S_{sr}c_j)_{i,j}} {\delta x_{i,j}}, \text{for }  c_{i,j} < 0
\end{align}
\right.

(1)

where S_{sr} = 2E_{sr} and i and j indicate the position along either the rows or columns, and \delta x_{i,j} is the cell-center distance between adjacent cells in the jth direction and at position i. The second-order upwind scheme is given by

 


\frac{\partial(S_{sr}c_j)}{\partial x_j}\bigg|_{i,j} = 
\left\{
\begin{align}
&\frac {3(S_{sr}c_j)_{i,j}-4(S_{sr}c_j)_{i,j-1} + (S_{sr}c_j)_{i,j-2}} {\delta x_{i,j} + \delta x_{i,j-1}}, \text{for }c_{i,j} > 0 \\
&\frac{-3(S_{sr}c_j)_{i,j} + 4(S_{sr}c_j)_{i,j+1} - (S_{sr}c_j)_{i,j+2}}  {\delta x_{i,j} + \delta x_{i,j+1}}, \text{for } c_{i,j} < 0
\end{align}
\right.

(2)

The surface roller calculation is achieved by setting the initial roller energy and time-stepping until the steady-state solution is reached. For simplicity, an explicit Euler scheme is used as follows

 


(S_{sr})^{n+1} = (S_{sr})^n + \Delta t_{sr} \left(-D_r + f_e D_{br} - \frac{\partial (S_{sr}c_j)} {\partial x}\right)^n

(3)

where \Delta_{sr} is the surface roller time step and is determined as \Delta t_{sr} = \text{0.5 max}(\Delta x_j / c)\text{ where }\Delta x_j is the cell size in the j^{th} direction. The steady-state solution is typically reached after ~40-80 time steps and takes about 1-2 seconds to run on a desktop personal computer.


Documentation Portal