On a potential-velocity formulation of incompressible Navier-Stokes equations

Computational fluid dynamics has emerged as an essential investigative tool in nearly every field of technology. Despite a well‐developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging, especially due to the non‐linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably. Recently, by introduction of an auxiliary potential field, a first integral of the two‐dimensional incompressible Navier‐Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original equations [1]. A useful application to free surface simulation was found in [2]. Moreover the new formulation is naturally extendible to three dimensions via tensor calculus, involving a non‐unique symmetric tensor potential. The corresponding degrees of freedom can be used in order to achieve a numerically convenient representation. Finally an efficient staggered‐grid finite difference scheme is applied to a Stokes flow problem in a 3D lid‐driven cavity to demonstrate the capabilities of the new approach. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)