On the accuracy of the viscous form in simulations of incompressible flow problems

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Abstract

We present a numerical study of drag/lift and flux estimates using two forms of Navier-Stokes equations (NSE) that are equivalent in the continuum formulation but not in the discrete finite element formulation. The two investigated forms of the NSE differ in the viscous term that is represented in one form by νΔu with ν being the viscosity and 2ν∇·∇Su in the other form where ∇S represents the deformation tensor. The study consists of numerical analysis of the two forms and computations of drag/lift, pressure drop on the cylinder problem and computations of flux for the Poiseuille flow. The main objective is to provide a clear comparison of the reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time for the two forms of NSEs. Our computational results of the reference values do not differ significantly between the two forms, but the differences are there. For the Poiseuille flow, the differences in the flux computations were much smaller, and this agreed with the computationally obtained results of the divergence of the velocity field. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 523–541, 2012

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