Sjr dp Sw r dp Sn r df ss r df SppVp

The second level of approximation concerns estimating the values of variables and gradients of variables (normal to cell faces) at locations other than computational nodes (cell centers). Such an approximation will then result in a set of linear algebraic equations. It must be noted that the source terms appearing in Eq. (6.6), that is will generally be non-linear. It is necessary to linearize such source terms in order to formulate a set of linear algebraic equations. A generic source term linearization can be expressed as

The assumption of a linear profile between the neighboring nodes offers the simplest approximation of the gradient at the face lying between those nodes. For example, the gradient of t at face e can be written:

For a uniform Cartesian grid, this approximation is of second-order accuracy. Even for a non-uniform grid, the error reduction with respect to grid refinement is similar to that of a second-order approximation. Higher order polynomials can be used to estimate the required gradients. For example, a fourth-order approximation for the gradient at face e on the uniform Cartesian grid can be written:

0 0

Post a comment