Regular Article
A Penalty Method for the Vorticity–Velocity Formulation

https://doi.org/10.1006/jcph.1998.6135Get rights and content

Abstract

We present a new vorticity–velocity formulation and implementation for the unsteady three-dimensional Navier–Stokes equations, based on a penalty method. It relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary. This approach is particularly attractive for high-order methods for which the often-used influence matrix method fails to converge for Δt→0. The accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two- and three-dimensional internal and external flows. In particular, the flow past a finite span cylinder attached to end-plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length.

References (58)

  • S.J Sherwin et al.

    Tetrahedral h-p finite elements: Algorithms and flow simulations

    J. Comput. Phys.

    (1996)
  • C.G Speziale

    On the advantages of the vorticity–velocity formulation of the equations of fluid dynamics

    J. Comput. Phys.

    (1987)
  • B Szabo et al.

    Superconvergent extraction of flux intensity factors and first derivatives from finite element solutions

    Comput. Methods Appl. Mech. Engrg.

    (1996)
  • X.H Wu et al.

    Effective vorticity–velocity formulations for three-dimensional incompressible viscous flows

    J. Comput. Phys.

    (1995)
  • S Abarbanel et al.

    Multi-dimensional Asymptotically Stable 4th-Order Accurate Schemes for the Diffusion Equation

    (February 1996)
  • S Abarbanel et al.

    Multi-dimensional Asymptotically Stable Finite Difference Schemes for the Advection-Diffusion Equation

    (July 1996)
  • B.F Armaly et al.

    Experimental and theoretical investigation of backward-facing step flow

    J. Fluid Mech.

    (1983)
  • R Barrett et al.

    Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed.

    (1994)
  • E.J Chang et al.

    Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion

    J. Fluid Mech.

    (1994)
  • O Daube et al.

    On the velocity-vorticity formulation of Navier-Stokes equations in incompressible flows

    C.R. Acad. Sci. Paris Ser. II

    (1991)
  • H Eisenlohr et al.

    Vortex splitting and its consequences in the vortex street wake of cylinders at low Reynolds number

    Phys. Fluids A

    (1989)
  • P. Fischer, Projection techniques for iterative solution of Ax=b with successive right-hand sides, Tech. Rep, Center...
  • J Fontaine et al.

    Wakes past a finite spanwise plate

    C.R. Acad. Sci. Paris Ser. IIb

    (1995)
  • J.H Gerrard

    The wakes of cylindrical bluff bodies at low Reynolds number

    Philos. Trans. R. Soc. London Ser. A

    (1978)
  • G Guevremont

    Finite Element Vorticity-Based Methods for the Solution of the Incompressible and Compressible Navier–Stokes equations

    (1993)
  • G Guevremont et al.

    Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

    Int. J. Numer. Methods Fluids

    (1990)
  • G Guj et al.

    Numerical solutions of high-Re recirculating flows in vorticity–velocity form

    Int. J. Numer. Methods Fluids

    (1988)
  • M.D Gunzburger et al.

    On finite element approximations of the streamfunction-vorticity and velocity-vorticity equations

    Int. J. Numer. Methods Fluids

    (1988)

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