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On the streamfunction-vorticity formulation in sliding bi-period frames: Application to bulk behavior for polymer blends

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JOURNAL OF COMPUTATIONAL PHYSICS 2006年第1期212卷 268-287页

作者： Anderson, PD Keestra, BJ Hulsen, MA Eindhoven Univ Technol Dutch Polymer Inst NL-5600 MB Eindhoven Netherlands

The Lees-Edwards description of bi-periodic boundary conditions has been extended to the streamfunction and streamfunction-vorticity formulation in sliding bi-periodic frames. The required compatibility conditions are formulated and uniqueness of the solution is shown. The model has been implemented in a spectral element method context to describe bulk shear behavior far away from walls, where no simple periodic boundary conditions can be used. In the numerical model a Lagrangian multiplier is introduced to couple the shearing boundaries. The proposed method has been validated for a mathematical test problem;convergence is shown and the influence of the order of approximation of the Lagrangian multiplier is studied. Finally, results are presented for drop coalescence across the boundaries of the bi-periodic frame. (c) 2005 Elsevier Inc. All rights reserved.

关键词： Lees-Edwards boundary conditions streamfunction-vorticity formulation Lagrangian multiplier polymer blends

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A numerical method for solution of incompressible Navier-Stokes equations in streamfunction-vorticity formulation

COMPUTATIONAL AND MATHEMATICAL METHODS

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COMPUTATIONAL AND MATHEMATICAL METHODS 2021年第6期3卷

作者： Raza, Saad Rauf, Abdul Sabi'u, Jamilu Shah, Abdullah COMSATS Univ Islamabad Dept Math Islamabad 45550 Pakistan Yusuf Maitama Sule Univ Kano Dept Math Kano Nigeria

In this work, we have proposed a numerical approach for solving the incompressible Navier-Stokes equations in the streamfunction-vorticity formulation. The numerical scheme is based on the diagonally implicit fractional-step theta-(DIFST) method used for the time discretization and the conforming finite element method for the spatial discretization. The accuracy and efficiency of the scheme are validated by solving some benchmark problems. The numerical simulations are carried out using the DUNE-PDELab open-source software package. The comparison of DIFST scheme with different time discretization schemes is provided in terms of CPU time. Also, different solvers with preconditioners are investigated for solving the resulting algebraic system of equations numerically.

关键词： conforming finite element method diagonally implicit fractional-step theta-scheme DUNE-PDELab streamfunction-vorticity formulation the incompressible Navier-Stokes equations

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Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2023年 152卷 308-316页

作者： Li, Dinggen Li, Faqiang Xu, Bo Huazhong Univ Sci & Technol Sch Energy & Power Engn Wuhan 430074 Peoples R China

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

关键词： Non-orthogonal Lattice Boltzmann method streamfunction-vorticity formulation

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Direct numerical simulation of settling of a large solid particle during bioconvection

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 2006年第5期51卷 511-530页

作者： Geng, P. Kuznetsov, A. V. N Carolina State Univ Dept Mech & Aerosp Engn Raleigh NC 27695 USA

Settling of a large solid particle in bioconvection flow caused by gyrotactic microorganisms is investigated. The particle is released from the top of the bioconvection chamber;its settling pattern depends on whether it is released in the centre of the bioconvection plume or at its periphery. The Chimera method is utilized;a subgrid is generated around a moving particle, The method suggested by Liu and Wang (Comput. Fluid 2004;33:223-255) is further developed to account for the presence of a moving boundary in the streamfunction-vorticity forrmulation using the finite-difference method. A number of cases for different release positions of the particle are computed. It is demonstrated that bioconvection can either accelerate or decelerate settling of the particle depending on the initial position of the particle relative to the plume centre. It is also shown that the particle impacts bioconvection plume by changing its shape and location in the chamber. Copyright (c) 2005 John Wiley & Sons, Ltd.

关键词： bioconvection Chimera method large particle motile microorganisms moving boundary streamfunction-vorticity formulation subgrid

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Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 2001年第1期37卷 65-86页

作者： Mai-Duy, N Tran-Cong, T Univ So Queensland Fac Engn & Surveying Toowoomba Qld 4350 Australia

A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a,universal approximator' based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of 'finite element-type' discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different;however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of 11 x 11, Re = 400 for the driven cavity flow with a density of 33 x 33 and Rayleigh number Ra = 1000 000 for the natural convection flow with a density of 27 x 27. Copyright (C) 2001 John Wiley & Sons, Ltd.

关键词： mesh-free method Navier-Stokes equations radial basis function networks streamfunction-vorticity formulation

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Dynamics of large solid particles in bioconvective sedimentation

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 2007年第5期53卷 713-733页

作者： Geng, P. Kuznetsov, A. V. N Carolina State Univ Dept Mech & Aerosp Engn Raleigh NC 27695 USA

Settling of one or two large solid particles in a bioconvection flow induced by gyrotactic motile microorganisins is investigated using a 2D numerical model. The results of varying the initial positions of large particles on the bioconvection flow pattern are investigated. The Chimera method is utilized to generate subgrids around the moving particles. It is demonstrated that the introduction of a single large particle displaces bioconvection plume and changes its shape. The introduction of two particles on the same side of the bioconvection plume further displaces the plume while the introduction of two particles oil opposite sides reduces this displacement. The influence of the bioconvection plume Oil the particles' settling paths and particles' settling velocities is investigated. Copyright (c) 2006 John Wiley & Sons, Ltd.

关键词： bioconvection Chimera method large particles motile microorganisms moving boundary streamfunction-vorticity formulation subgrid counter-propagating jets

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Numerical solution of 2D Navier-Stokes equation discretized via boundary elements method and finite difference approximation

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2018年 96卷 64-77页

作者： Sedaghatjoo, Zeynab Dehghan, Mehdi Hosseinzadeh, Hossein Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math 424 Hafez Ave Tehran Iran Persian Gulf Univ Dept Math Bushehr Iran

In this paper boundary elements method (BEM) is equipped with finite difference approximation (FDA) to solve two-dimensional Navier-Stokes (N-S) equation. The N-S equation is converted to a system of ordinary differential equations (ODEs) respect to time in streamfunction-vorticity formulation. The constant direct BEM and 9-point stencil FDA are utilized to handle spatial derivatives, and the final system of ODEs is solved via three numerical schemes, forward Euler, Runge-Kutta and Newton's methods to find a fast ODE solver. Numerical investigations presented in this article show Newton's method is faster than the others and it is able to solve the N-S equation for Reynolds number up to 40,000 when grid points are at most 141 x 141. Thanks to BEM and boundary conditions of lid-driven cavity flow, the final system of ODEs is stable also for higher Reynolds numbers. A new technique is proposed in this article which converts BEM's two-dimensional singular integrals to one-dimensional non-singular ones. The proposed technique reduces computational cost of BEM, significantly, when more accurate results are requested. Numerical experiments show the numerical results are fairly agree with that accurate ones available in the literature.

关键词： Boundary elements method Finite difference approximation Navier-Stokes equation Ordinary differential equations and system of ODEs streamfunction-vorticity formulation High Reynolds number Runge-Kutta formula and Newton's method

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Higher-order unstructured domain decomposition method for Navier-Stokes equations

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COMPUTERS & FLUIDS 1999年第1期28卷 87-120页

作者： Rokicki, J Floryan, JM Warsaw Univ Technol Inst Aeronaut & Appl Mech PL-00665 Warsaw Poland Univ Western Ontario Dept Mech & Mat Engn London ON N6A 5B9 Canada

The unstructured domain decomposition method for the Navier-Stokes equations is described. The solution domain is covered with a union of subdomains of arbitrary shapes with arbitrary (multiple) overlaps. Field equations are discretized using fourth-order compact scheme for curvilinear coordinate systems. Blending functions that define global fourth-order accurate solution are introduced. The global solution is used to transfer information between the subdomains. Various numerical examples confirm the fourth-order accuracy of the results. The algorithm is well suited for parallel implementations. (C) 1998 Elsevier Science Ltd. All rights reserved.

关键词： viscous incompressible fluid flow domain decomposition streamfunction-vorticity formulation compact fourth-order finite-difference schemes orthogonal coordinate system

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