  High order reconstruction in the finite volume (FV) approach is achieved by a
more fundamental form of the fifth order WENO reconstruction in the framework
of orthogonally-curvilinear coordinates, for solving the hyperbolic
conservation equations. The derivation employs a piecewise parabolic polynomial
approximation to the zone averaged values to reconstruct the right, middle, and
left interface values. The grid dependent linear weights of the WENO are
recovered by inverting a Vandermode-like linear system of equations with
spatially varying coefficients. A scheme for calculating the linear weights,
optimal weights, and smoothness indicator on a regularly- and
irregularly-spaced grid in orthogonally-curvilinear coordinates is proposed. A
grid independent relation for evaluating the smoothness indicator is derived
from the basic definition. Finally, the procedures for the source term
integration and extension to multi-dimensions are proposed. Analytical values
of the linear and optimal weights, and also the weights required for the source
term integration and flux averaging, are provided for a regularly-spaced grid
in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order
WENO reconstruction for the regularly-spaced grids in the Cartesian coordinates
can be fully recovered in the case of limiting curvature. The fifth order
finite volume WENO-C (orthogonally-curvilinear version of WENO) reconstruction
scheme is tested for several 1D and 2D benchmark test cases involving smooth
and discontinuous flows in cylindrical and spherical coordinates.
