8.5 Finite Difference Methods
227
Since we use the Cartesian vector components, we only need to transform
the derivatives with respect to Cartesian coordinates into the generalized
coordinates:
where pij represents the cofactor of d x i / d t j in the Jacobian J. In 2D this
leads to:
The generic conservation equation, which in Cartesian coordinates reads:
transforms to:
where
is proportional to the velocity component normal to the coordinate surface
= p k j p km - - p l j p l m + p 2 j p m + p3jp3m .
(8.7)
The transformed momentum equations contain several additional terms
that arise because the diffusive terms in the momentum equations contain
a derivative not found in the generic conservation equation, see Eqs. (1.16),
(1.18) and (1.19). These terms have the same form as the ones shown above
and will not be listed here.
Equation (8.5) has the same form as Eq. (8.4), but each term in the
latter is replaced by a sum of three terms in the former. As shown above,
these terms contain the first derivatives of the coordinates as coefficients.
These are not difficult to evaluate numerically (unlike second derivatives).
The unusual feature of non-orthogonal grid is that mixed derivatives appear
227
Since we use the Cartesian vector components, we only need to transform
the derivatives with respect to Cartesian coordinates into the generalized
coordinates:
where pij represents the cofactor of d x i / d t j in the Jacobian J. In 2D this
leads to:
The generic conservation equation, which in Cartesian coordinates reads:
transforms to:
where
is proportional to the velocity component normal to the coordinate surface
= p k j p km - - p l j p l m + p 2 j p m + p3jp3m .
(8.7)
The transformed momentum equations contain several additional terms
that arise because the diffusive terms in the momentum equations contain
a derivative not found in the generic conservation equation, see Eqs. (1.16),
(1.18) and (1.19). These terms have the same form as the ones shown above
and will not be listed here.
Equation (8.5) has the same form as Eq. (8.4), but each term in the
latter is replaced by a sum of three terms in the former. As shown above,
these terms contain the first derivatives of the coordinates as coefficients.
These are not difficult to evaluate numerically (unlike second derivatives).
The unusual feature of non-orthogonal grid is that mixed derivatives appear