Regular Article
Effective Vorticity-Velocity Formulations for Three-Dimensional Incompressible Viscous Flows

https://doi.org/10.1006/jcph.1995.1197Get rights and content

Abstract

The vorticity-velocity formulation for incompressible viscous flows is studied. The main concern is effective numerical solution of the kinematic Cauchy-Riemann equations for velocity, especially when the divergence-free condition of vorticity is violated due to numerical error. The mathematical formulation of a differential and an integral approaches are revisited. A novel projection of vorticity onto the divergence-free space and its application to the two approaches are studied. As an illustration of the differential approach, the 3D lid-driven cavity problem is solved. The projection scheme is justified through numerical tests by comparing it to a new fully divergence-free scheme. Numerical tests indicate that keeping vorticity divergence-free is important for obtaining correct and accurate solutions. The theory and methods apply to other divergence-free fields of physical interest as well, such as that in electro-magnetic dynamics.

References (0)

Cited by (48)

  • Parallel implementation of a VIScous Vorticity Equation (VISVE) method in 3-D laminar flow

    2021, Journal of Computational Physics
    Citation Excerpt :

    Depending on whether the velocity or the stream–function is involved in the equations, the methods are further classified into the vorticity–velocity formulations [30–33] and the vorticity–stream–function formulations [34,35,1]. The Finite Difference Methods (FDM) were commonly used to discretize the governing equations, yet the FDM shows considerable difficulties in handling non–orthogonal grids, and so the above applications were limited to flow in simple and regular computational domains, such as a rectangular domain [30,32–34,1] or a cylindrical domain [31]. Even though Elshabka and Chung [35] proposed a Finite Element Method (FEM) scheme to solve the 3–D vorticity transport equation, the results presented were still about the 3–D cavity flow in a rectangular domain.

  • A conservative viscous vorticity method for unsteady unidirectional and oscillatory flow past a circular cylinder

    2019, Ocean Engineering
    Citation Excerpt :

    Whether the velocity or the stream function is used to represent the kinematics of the flow further categorizes these methods into the vorticity–velocity formulation and the vorticity-stream function formulation. The velocity field and the vorticity field are usually associated by a Poisson equation (Wu et al., 1995). Remarkably, Wu and Thompson (1973) proposed an integro-differential scheme that enables calculating the velocity by a direct and explicit integral of the vorticity.

  • Vorticity vector-potential method for 3D viscous incompressible flows in time-dependent curvilinear coordinates

    2016, Journal of Computational Physics
    Citation Excerpt :

    A variety of numerical methods based on vorticity–velocity formulation have been developed. To enforce the solenoidal conditions properly, the vorticity was projected into the divergence-free space in [9]. Also, staggered grids have been widely used in the vorticity–velocity formulation to overcome numerical difficulties ever since the pioneering work of Harlow and Welch on MAC scheme [10].

  • Natural vorticity boundary conditions on solid walls

    2015, Computer Methods in Applied Mechanics and Engineering
View all citing articles on Scopus
View full text