### Abstract

This paper is concerned with an overview of upwinding schemes, and further nonlinear applications of a recently introduced high resolution upwind differencing scheme, namely the ADBQUICKEST [V.G. Ferreira, F.A. Kurokawa, R.A.B. Queiroz, M.K. Kaibara, C.M. Oishi, J.A. Cuminato, A.F. Castelo, M.F. Tomé, S. McKee, assessment of a high-order finite difference upwind scheme for the simulation of convection–diffusion problems, International Journal for Numerical Methods in Fluids 60 (2009) 1–26]. The ADBQUICKEST scheme is a new

TVD version of the QUICKEST [B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Computer Methods in Applied Mechanics and Engineering 19 (1979) 59–98] for solving nonlinear balance laws. The scheme is based on the concept of NV and TVD formalisms and satisfies a convective boundedness criterion. The accuracy of the scheme is compared with other popularly used convective upwinding schemes (see, for example, Roe (1985) [19], Van Leer (1974) [18] and Arora & Roe (1997) [17]) for solving nonlinear conservation laws (for example, Buckley–Leverett, shallow water and Euler equations). The ADBQUICKEST scheme is then

used to solve six types of fluid flow problems of increasing complexity: namely, 2D aerosol filtration by fibrous filters; axisymmetric flow in a tubular membrane; 2D two-phase flow in a fluidized bed; 2D compressible Orszag–Tang MHD vortex; axisymmetric jet onto a flat surface at low Reynolds number and full 3D incompressible flows involving moving free surfaces. The numerical simulations indicate that this convective upwinding scheme is a good generic alternative for solving complex fluid dynamics problems.

TVD version of the QUICKEST [B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Computer Methods in Applied Mechanics and Engineering 19 (1979) 59–98] for solving nonlinear balance laws. The scheme is based on the concept of NV and TVD formalisms and satisfies a convective boundedness criterion. The accuracy of the scheme is compared with other popularly used convective upwinding schemes (see, for example, Roe (1985) [19], Van Leer (1974) [18] and Arora & Roe (1997) [17]) for solving nonlinear conservation laws (for example, Buckley–Leverett, shallow water and Euler equations). The ADBQUICKEST scheme is then

used to solve six types of fluid flow problems of increasing complexity: namely, 2D aerosol filtration by fibrous filters; axisymmetric flow in a tubular membrane; 2D two-phase flow in a fluidized bed; 2D compressible Orszag–Tang MHD vortex; axisymmetric jet onto a flat surface at low Reynolds number and full 3D incompressible flows involving moving free surfaces. The numerical simulations indicate that this convective upwinding scheme is a good generic alternative for solving complex fluid dynamics problems.

Original language | English |
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Pages (from-to) | 435-459 |

Number of pages | 35 |

Journal | Mathematical and Computer Modelling |

Volume | 57 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Feb 2013 |

### Keywords

- advective transport
- CBC/TVD stability
- high resolution
- upwinding
- monotonic interpolation
- convection modelling
- boundedness
- flux limiter
- normalized variables
- free surface flows

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## Cite this

Ferreira, V. G., Kaibara, M. K., Lima, G. A. B., Silva, J. M., Sabatini, M. H., Mancera, P. F. A., & McKee, S. (2013). Application of a bounded upwinding scheme to complex fluid dynamics problems.

*Mathematical and Computer Modelling*,*57*(3-4), 435-459. https://doi.org/10.1016/j.mcm.2012.06.021