A mass-conserving vorticity–velocity formulation with application to nonreacting and reacting flows

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. However, for some flows with large vorticity gradients, spurious mass loss or gain can be observed. 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 for the reacting flow case – is discretized with finite differences on a nonstaggered grid and is then solved using Newton’s method. Both the unmodified and modified vorticity–velocity formulations are applied to two problems with high vorticity gradients: (1) incompressible, axisymmetric fluid flow through a suddenly expanding pipe and (2) a confined, axisymmetric laminar flame with detailed chemistry and multicomponent transport, generated on a burner whose inner tube extends above the burner surface. The modified formulation effectively eliminates the spurious mass loss in the two test cases to within an acceptable tolerance. The two cases demonstrate the broader range of applicability of the modified formulation, as compared with the unmodified formulation.