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

• Published in: Journal of computational physics Vol. 199; no. 1; pp. 355 - 375
• Main Authors: Luo, Haoxiang, Bewley, Thomas R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.09.2004; Elsevier
• Subjects: Computational techniques; Contravariant; Exact sciences and technology; Mathematical methods in physics; Navier–Stokes equations; Physics; Time-dependent curvilinear coordinates; Contravariant; Navier–Stokes equations; Time-dependent curvilinear coordinates; Incompressible flow; Calculation; Coordinate system; Two dimensional flow; Navier-Stokes equations; Calculation methods
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2004.02.012

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.