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Application of the fictitious domain method for Navier-Stokes Equations

Computers, Materials & Continua 2022年第10期73卷 2035-2055页

作者： Almas Temirbekov Zhadra Zhaksylykova Yerzhan Malgazhdarov Syrym Kasenov Al-Farabi Kazakh National University AlmatyKazakhstan Abai Kazakh National Pedagogical University AlmatyKazakhstan Sarsen Amanzholov East Kazakhstan University Ust-KamenogorskKazakhstan

To apply the fictitious domain method and conduct numericalexperiments, a boundary value problem for an ordinary differential equation is considered. The results of numerical calculations for different valuesof the iterative parameter τ and the small parameter ε are presented. Astudy of the auxiliary problem of the fictitious domain method for NavierStokes equations with continuation into a fictitious subdomain by highercoefficients with a small parameter is carried out. A generalized solutionof the auxiliary problem of the fictitious domain method with continuationby higher coefficients with a small parameter is determined. After all theabove mathematical studies, a computational algorithm has been developedfor the numerical solution of the problem. Two methods were used to solvethe problem numerically. The first variant is the fictitious domain methodassociated with the modification of nonlinear terms in a fictitious subdomain.The model problem shows the effectiveness of using such a modification. Theproposed version of the method is used to solve two problems at once that arisewhile numerically solving systems of Navier-Stokes equations: the problem ofa curved boundary of an arbitrary domain and the problem of absence of aboundary condition for pressure in physical formulation of the internal flowproblem. The main advantage of this method is its universality in developmentof computer programs. The second method used calculation on a uniform gridinside the area. When numerically implementing the solution on a uniformgrid inside the domain, using this method it’s possible to accurately take intoaccount the boundaries of the curved domain and ensure the accuracy of thevalue of the function at the boundaries of the domain. methodical calculationswere carried out, the results of numerical calculations were obtained. Whenconducting numerical experiments in both cases, quantitative and qualitativeindicators of numerical results coincide.

关键词： fictitious domain method Navier-Stokes equations difference schemes approximation computational algorithm numerical experiment

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The fictitious domain method with Sharp Interface for Elasticity Systems with General Jump Embedded Boundary Conditions

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Advances in Applied Mathematics and Mechanics 2021年第1期13卷 119-139页

作者： Mohamed Kara Salim Mesbahi Philippe Angot Applied Mathematics Laboratory Department of MathematicsFaculty of SciencesFerhat Abbas UniversitySetifAlgeria Fundamental and Numerical Mathematics Laboratory Department of MathematicsFaculty of SciencesFerhat Abbas UniversitySetifAlgeria Aix-Marseille Universit´e Institut de Math´ematiques de MarseilleCNRS UMR 7373MarseilleFrance

In framework of the fictitious domain methods with immersed interfaces for the elasticity problem,the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method.The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain.Here,we present a cell-centered finite volume method to discretize the fictitious domain problem.The proposed method is numerically validated for different test cases.This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.

关键词： fictitious domain method sharp interface elasticity system jump embedded boundary conditions finite volume method

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Numerical simulation of hemodynamic interactions of red blood cells in micro- capillary flow

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Journal of Hydrodynamics 2014年第2期26卷 178-186页

作者：石兴张帅王双连 School of Aeronautics and Aerospace Zhejiang University

The hemodynamic interactions of red blood cells （RBCs） in a microcapillary flow are investigated in this paper. This kind of interaction is considered as the non-contact mutual interaction of cells, which is important in the suspension flow of blood, but not sufficiently understood. The distributed Lagrange multiplier/fictitious domain method in the lattice Boltzmann framework is used to solve the suspension of the RBCs. The modification of the flow due to the cells, the dependence of the cell deformation on the flow and the cell-cell interaction via the fluid are discussed. It is revealed that the long-range hydrodynamic interaction with a long interacting distance, more than about 5 times of the RBC equivalent radius, mainly has effect on the rheology properties of the suspension, such as the mean velocity, and the short-range interaction is sensitive to the shape of the cell in the microcapillary flow. The flow velocity around the cell plays a key role in the cell deformation. In the current configuration of the flow and cells, the cells repel each other along the capillary.

关键词： erythrocyte hydrodynamic interactions fictitious domain method lattice Boltzmann method （LBM） fluid-structure interactions

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Numerical studies of the hysteresis in locomotion of a passively pitching foil

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Journal of Hydrodynamics 2016年第3期28卷 359-368页

作者：邵雪明张晓龙余钊圣 State Key Laboratory of Fluid Power Transmission and Control Department of Mechanics Zhejiang University

The direct-forcing fictitious domain method is extended to simulate the locomotion of a passively pitching foil. Our study focuses on the hysteresis phenomenon that the critical frequency for the reverse of the locomotion direction of the wing in case of decreasing frequency is smaller than that in case of increasing frequency. In our simulations, the hysteresis phenomenon is produced by imposing different initial conditions at a same frequency. Our results indicate that the ratio of the heaving amplitudes of two foil edges is crucial to the direction of the foil＇s horizontal motion, and the amplitude of the leading edge is generally smaller. The critical frequencies for the reverse of the locomotion direction are increased, when the foil-fluid density ratio is decreased or the spring constant is increased. The critical frequencies in the bi-stability regime also depend on the initial velocity imposed, and the hysteresis loop generally becomes larger if the initial velocities are closer to the terminal locomotion velocities of the foil.

关键词： flapping foil passively pitching hysteresis fictitious domain method

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Effects of finite-size heavy particles on the turbulent flows in a square duct

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Journal of Hydrodynamics 2017年第2期29卷 272-282页

作者：林昭武邵雪明余钊圣王连平 State Key Laboratory of Fluid Power and Mechatronic Systems Department of MechanicsZhejiang University Department of Mechanical Engineering University of Delaware

A parallel direct-forcing fictitious domain method is applied in fully-resolved numerical simulations of particle-laden turbulent flows in a square duct. The effects of finite-size heavy particles on the mean secondary flow, the mean streamwise velocity, the root-meansquare velocity fluctuation, and the particle concentration distribution are investigated at the friction Reynolds number of 150, the particle volume fraction of 2.36%, the particle diameter of 0.1 duct width, and the Shields number ranging from 1.0 to 0.2 Our results show that the particle sedimentation breaks the up-down symmetry of the mean secondary vortices, and results in a stronger secondary-flow circulation which transports the fluids downward in the bulk center region and upward along the side walls at a low Shields number. This circulation has a significant impact on the distribution of the mean streamwise velocity, whose maximum value occurs in the lower half duct, unlike in the plane channel case. The flow resistance is increased and the turbulence intensity is reduced, as the Shields number is decreased. The particles accumulate preferentially at the face center of the bottom wall, due to the effect of the mean secondary flow. It is observed that the collision model has an important effect on the results, but does not change the results qualitatively.

关键词： Turbulent duct flow particle-laden flow mean secondary flow fictitious domain method

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Modeling the Sedimentation of Red Blood Cells in Flow under Strong External Magnetic Body Force Using a Lattice Boltzmann fictitious domainmethod

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Numerical Mathematics(Theory,methods and Applications) 2014年第4期7卷 512-523页

作者： Xing Shi Guang Lin School of Aeronautics and Aerospace Zhejiang UniversityHangzhou310027P.R.China Department of Mathematics School of Mechanical EngineeringPurdue UniversityWest LafayetteIN 47907USA Computational Mathematics Group Fundamental and Computational Sciences DirectoratePacific Northwest National Laboratory902 Battelle Blvd.P.O.Box 999WA 99352USA.

Experimental observations show that a strong magnetic field has a dramatic influence on the sedimentation of RBCs,which motivates us to model the sedimentation of red blood cell(RBC)under strong external magnetic body force.To model the sedimentation of a RBC in a square duct and a circular pipe,a recently developed technique derived from the lattice Boltzmann and the distributed Lagrange multiplier/fictitious domain methods(LBM-DLM/FD)is extended to employ the mesoscopic network model for simulations of the sedimentation of a RBC in flow.The flow is simulated by the LBM with a strong magnetic body force,while the network model is used for modeling RBC deformation.The fluid-RBC interactions are enforced by the Lagrange multiplier.The sedimentation of RBC in a square duct and a circular pipe is simulated,which demonstrates the developed method’s capability to model the sedimentation of RBCs in various flows.Numerical results illustrate that the terminal settling velocity increases incrementally with the exerted body force.The deformation of RBC has a significant effect on the terminal settling velocity due to the change in the frontal area.The larger the exerted force,the smaller the frontal area and the larger the RBC deformation become.Additionally,the wall effect on the motion and deformation of RBC is also investigated.

关键词： Sediment erythrocyte fictitious domain method lattice Boltzmann method flowstructure interaction red blood cell

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