On the contravariant form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems

The contravariant form of the Navier–Stokes equations in a fixed curvilinear coordinate system is well known. However, when the curvilinear coordinate system is time-varying, such as when a body-fitted grid is used to compute the flow over a compliant surface, considerable care is needed to handle the momentum term correctly. The present paper derives the complete contravariant form of the Navier–Stokes equations in a time-dependent curvilinear coordinate system from the intrinsic derivative of contravariant vectors in a moving frame. The result is verified via direct transformation. These complete equations are then applied to compute incompressible flow in a 2D channel with prescribed boundary motion, and the significant effect of some terms which are sometimes either overlooked or assumed to be negligible in such a derivation is quantified.