Review of Equations of Conservation in Curvilinear Coordinates

Abstract

Momentum equations expressed in the boundary‐fitted curvilinear coordinate systems have been used in the multidimensional modeling of estuaries and lakes. These equations are analyzed and reviewed with focus on the nonlinear advection terms. Imposing the conservation law of momentum on a differential volume in the curvilinear plane, one can derive the momentum equations with components specified in contravariant directions. Geometric information of the curvilinear grid is represented by the transformation coefficients associated with the transformed equations. These mathematical coefficients are defined here with new physical meanings and comparisons made among equations having different forms for the advection terms. Analyses of these comparisons indicate the non‐conservativity of the advection terms in existing equations is caused, not by orthogonality, but curvilinearity of the grid in which the velocities are expressed. We demonstrate that conservative forms of the advection terms cannot exist when momentum is expressed in contravariant directions in a curvilinear, orthogonal or nonorthogonal, grid system.