A Mass-Conserving Vorticity-Velocity Formulation with Application to Axisymmetric Laminar Methane Flames

摘要

In a commonly implemented version of the vorticity-velocity formulation, the governing equations for the fluid dynamics are expressed as two Poisson-like velocity equations together with the vorticity transport equation. For some flows with large vorticity gradients, especially those involving combustion and containing burner walls that protrude into the domain of interest, spurious mass loss or gain can be observed. This lack of adherence to continuity is often negligible but in some cases it can be so significant as to overwhelm the predicted solution, rendering it unphysical. In order to conserve mass, a modification to the vorticity-velocity formulation is proposed, involving the substitution of the kinematic definition of vorticity in certain terms of the fluid-dynamic equations. This modified formulation results in a broader computational stencil when the equations are in a second-order-accurate discretized form, and a stronger coupling between the predicted vorticity and the curl of the predicted velocity field. The resulting system of elliptic equations - which includes the energy and species transport equations - is discretized with finite differences on a nonstaggered grid and is then solved using Newton's method. In order to validate the modified formulation, both the unmodified and modified versions are applied to incompressible, axisymmetric fluid flow through a suddenly expanding pipe. Mass-conserving results generated with the modified formulation show excellent agreement with experimental data. Both formulations are then applied to a confined, axisymmetric laminar flame with detailed chemistry and multi-component transport, generated on a burner whose inner tube extends above the burner surface. The modified formulation effectively eliminates the spurious mass loss or gain to within an acceptable tolerance. The modified formulation will also be applied to an axisymmetric oscillating laminar diffusion flame.