MC-Smooth: A mass-conserving, smooth vorticity–velocity formulation for multi-dimensional flows

Abstract

A novel vorticity–velocity formulation of the Navier–Stokes equations – the Mass-Conserving, Smooth (MC-Smooth) vorticity–velocity formulation – is developed in this work. The governing equations of the MC-Smooth formulation include a new second-order Poisson-like elliptic velocity equation, along with the vorticity transport equation, the energy conservation equation, and N_spec species mass balance equations. In this study, the MC-Smooth formulation is compared to two pre-existing vorticity–velocity formulations by applying each formulation to confined and unconfined axisymmetric laminar diffusion flame problems. For both applications, very good to excellent agreement for the simulation results of the three formulations has been obtained. The MC-Smooth formulation requires the least CPU time and can overcome the limitations of the other two pre-existing vorticity–velocity formulations by ensuring mass conservation and solution smoothness over a broader range of flow conditions. In addition to these benefits, other important features of the MC-Smooth formulation include: (1) it does not require the use of a staggered grid, and (2) it does not require excessive grid refinement to ensure mass conservation. The MC-Smooth formulation is a computationally attractive approach that can effectively extend the applicability of the vorticity–velocity formulation.

Keywords:

• vorticity–velocity formulation
• mass conservation
• smoothness
• non-staggered grid
• axisymmetric laminar diffusion flame

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors would like to acknowledge support for this work from the US Department of Energy Office of Basic Energy Sciences [Award Number DE-FG02-88ER13966] (Dr Mark Pederson, contract monitor); the National Science Foundation [CBET-0828802] (Dr Phil Westmoreland, contract monitor); NASA [NNX11AP43A] (Mr Dennis P. Stocker, contract monitor); the US Air Force Office of Scientific Research [AFOSR FA9550-12-1-0157] (Dr Chiping Li and Dr Fariba Fahroo, contract monitors); and the Strategic Environmental Research and Development Program and the Air Force Research Laboratory [agreement number FA8650-10-2-2934].