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On the Development of a Nonprimitive Navier–Stokes Formulation Subject to Rigorous Implementation of a New Vorticity Integral Condition

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Abstract

In this paper, a new integral vorticity boundary condition has been developed and implemented to compute solution of nonprimitive Navier–Stokes equation. Global integral vorticity condition which is of primitive character can be considered to be of entirely different kind compared to other vorticity conditions that are used for computation in literature. The procedure realized as explicit boundary vorticity conditions imitates the original integral equation. The main purpose of this paper is to design an algorithm which is easy to implement and versatile. This algorithm based on the new vorticity integral condition captures accurate vorticity distribution on the boundary of computational flow field and can be used for both wall bounded flows as well as flows in open domain. The approach has been arrived at without utilizing any ghost grid point outside of the computational domain. Convergence analysis of this alternative vorticity integral condition in combination with semi-discrete centered difference approximation of linear Stokes equation has been carried out. We have also computed correct pressure field near the wall, for both attached and separated boundary layer flows, by using streamfunction and vorticity field variables. The competency of the proposed boundary methodology vis-a-vis other popular vorticity boundary conditions has been amply appraised by its use in a model problem that embodies the essential features of the incompressibility and viscosity. Subsequently the proposed methodology has been further validated by computing analytical solution of steady Stokes equation. Finally, it has been applied to three benchmark problems governed by the incompressible Navier–Stokes equations, viz. lid driven cavity, backward facing step and flow past a circular cylinder. The results obtained are in excellent agreement with computational and experimental results available in literature, thereby establishing efficiency and accuracy of the proposed algorithm. We were able to accurately predict both vorticity and pressure fields.

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Acknowledgements

The first author is thankful to Dr. Deepjyoti Goswami, Department of Mathematical Sciences, Tezpur University, India for some intense discussions. Both the authors are grateful to the anonymous reviewers for their comments and suggestions.

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Correspondence to Tony W. H. Sheu.

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Sen, S., Sheu, T.W.H. On the Development of a Nonprimitive Navier–Stokes Formulation Subject to Rigorous Implementation of a New Vorticity Integral Condition. J Sci Comput 72, 252–290 (2017). https://doi.org/10.1007/s10915-016-0355-x

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Keywords

  • Vorticity integral condition
  • Vorticity field
  • Pressure field
  • Navier–Stokes equations

Mathematics Subject Classification

  • 65M06
  • 65N06
  • 76D05
  • 76M20