A coupled continuous and discontinuous finite element method for the incompressible flows

Gao, Puyang; Ouyang, Jie; Peng-fei, Dai; Zhou, Wen

doi:10.1002/fld.4358

Cited by 8 publications

(5 citation statements)

References 47 publications

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“…This formulation proposed by [39,40] makes use of the incompressibility constraint ∇ • u = 0, and it is well understood that the rotational formulation significantly improves accuracy as compared to the Laplace formulation [53,52]; nevertheless the latter formulation ∇ 2 u is also used sometimes [61,62,38].…”

Section: High-order Dual Splitting Schemementioning

confidence: 99%

“…Remark One might raise the question why the terms in equation ( 17) coming from the acceleration term evaluate the prescribed boundary data g u instead of simply evaluating the numerical solution u coming from the interior of the domain, since this is also done for the convective and viscous terms in the boundary conditions. From the formulations in [22,24,63,62,38] it is unclear whether the acceleration term in equation (17) should directly evaluate the numerical solution u if the time derivative is not known analytically. The same holds for the BDF time derivative terms in equation (18).…”

Section: High-order Dual Splitting Schemementioning

confidence: 99%

“…with viscosity set to ν = 0.025, and simulated over the time interval 0 ≤ t ≤ T = 1. The computational domain at start time is Ω 0 = [−L/2, L/2] 2 with length L = 1, and is deformed according to the twodimensional mesh movement function (62) with parameters A = 0.08 and T G = 4T (maximum deformation reached at end time t = T ) unless specified otherwise. Again, a mesh as depicted in Figure 2(b) with refinement level l is used.…”

Section: Temporal and Spatial Convergence Behaviormentioning

confidence: 99%

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High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

Fehn,

Heinz,

Wall

et al. 2020

Preprint

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This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both monolithic as well as projection or splitting-type Navier-Stokes solvers. The framework is flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. The Navier-Stokes equations with ALE transport term are solved on the deformed geometry storing one instance of the mesh that is updated from one time step to the next. Discretization in space is applied to the time discrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step. This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as is shown theoretically and demonstrated numerically by the example of the free-stream preservation test. We discuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projectiontype methods and the ingredients needed to preserve high-order accuracy. We show numerically that the formulations proposed in this work maintain the formal order of accuracy of the Navier-Stokes solvers. Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows.

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Section: High-order Dual Splitting Schemementioning

confidence: 99%

Section: High-order Dual Splitting Schemementioning

confidence: 99%

Section: Temporal and Spatial Convergence Behaviormentioning

confidence: 99%

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High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

Fehn,

Heinz,

Wall

et al. 2020

Preprint

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“…According to the splitting scheme [28,29], we can obtain three subequations. We first neglect the terms about pressure and viscous in equation ( 8) and introduce an intermediate velocity u to replace u n+1 .…”

Section: Temporal Discretization and Splitting Schemementioning

confidence: 99%

The Finite Element Numerical Investigation of Free Surface Newtonian and Non-Newtonian Fluid Flows in the Rectangular Tanks

Gao

2020

Mathematical Problems in Engineering

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In this paper, we develop a new computational framework to investigate the sloshing free surface flow of Newtonian and non-Newtonian fluids in the rectangular tanks. We simulate the flow via a two-phase model and employ the fixed unstructured mesh in the computation to avoid the mesh distortion and reconstruction. As for the solution of Navier–Stokes equation, we utilize the SUPG finite element method based on the splitting scheme. The same order interpolation functions are then used for velocity and pressure. Moreover, the moving interface is captured via the concise level set method. We take advantage of the implicit discontinuous Galerkin method to handle the solution of level set and its reinitialization equations. A mass correction technique is also added to ensure the mass conservation property. The dam break-free surface flow is simulated firstly to demonstrate the validity of our mathematical model. In addition, the sloshing Newtonian fluid in the tank with flat and rough bottoms is considered to illustrate the feasibility and robustness of our computational scheme. Finally, the development of free surface for non-Newtonian fluid is also studied in the two tanks, and the influence of power-law index on the sloshing fluid flow is analyzed.

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“…In order to overcome this difficulty, the upwind schemes are widely used for the DG discretization of the convective term such as Vijayasundaram numerical flux, 13,14 the Lesaint-Raviart flux, 15,16 or the local Lax-Friedrichs flux. 17,18 What is more, since the nonlinearity of the equations, numerous works of the temporal discretization formats have been done. [19][20][21] These formats may be classified into three groups of fully explicit scheme, fully implicit scheme, and semiimplicit scheme.…”

Section: Introductionmentioning

confidence: 99%

A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

Ouyang

Wang

et al. 2021

Numerical Methods in Fluids

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In this work, we consider equal-order discontinuous Galerkin (DG) solver for incompressible Navier-Stokes equations based on high-order dual splitting scheme. In order to stay stable, the time step size of this method has been reported that is strictly limited. The upper bound of time step size is restricted by Courant-Friedrichs-Lewy (CFL) condition (Hesthaven and Warburton, 2007) and lower bound is required to be larger than the critical value which depends on Reynolds number and spatial resolution (Ferrer et al., 2014).For high-Reynolds-number flow problems, if the spatial resolution is low, the critical value may be larger than CFL condition, then instability will occur for any time step size. Therefore, sufficiently high spatial resolution is indispensable in order to maintain stability, which increases the computational cost. To overcome these difficulties and develop a robust solver for high-Reynolds-number flow problem, it is necessary to further study the instability problem at small time steps. We numerically investigate the effect of the pressure gradient term in projection step and the velocity divergence term in pressure Poisson equation on the stability for small time step size, respectively, and conclude that the DG formulation of the pressure gradient term has a more significant effect on the stability of the scheme than that of the velocity divergence term. Integration by parts of these terms is essential in order to improve the stability of the scheme.Based on this discretization format, an appropriate penalty parameter for pressure Poisson equation is utilized so as to provide the scheme with an inf-sup stabilization. Moreover, the lid-driven cavity flow is considered to verify that this numerical algorithm enhances the stability without additional stabilization term at small time step size and high-Reynolds number for equal-order polynomial approximations.

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A coupled continuous and discontinuous finite element method for the incompressible flows

Cited by 8 publications

References 47 publications

High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

The Finite Element Numerical Investigation of Free Surface Newtonian and Non-Newtonian Fluid Flows in the Rectangular Tanks

A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

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