Application of the GSMAC-CIP Method to Incompressible Navier-Stokes Equations at High Reynolds Numbers

Makihara, Takafumi; Shibata, Etsutaro; Tanahashi, Takahiko

doi:10.1080/10618569908940834

Cited by 8 publications

(10 citation statements)

References 7 publications

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“…Note that first equations (3) are solved using the CIP solver (Equations (7)- (10)) to obtainv and the spatial derivativesv x andv y for all nodal points. Then for the 4-node element, in the nonadvection phase, the nodal variables v n+1 and p n+1 are calculated from the first equation in (13) and the continuity equation (∇ · v n+1 = 0) using the relaxation method, as given in (17). Then we obtain the spatial derivatives v n+1 x and v n+1 y from the last two equations in (13).…”

Section: Remarks On the Solution Of The Governing Equationsmentioning

confidence: 99%

“…We first calculate the divergence of the velocity, then we obtain the velocity potential from the first equation in (17). Using the second and third equations, the velocity and pressure are updated.…”

Section: The Non-advection Phasementioning

confidence: 99%

“…For the 4-node Galerkin-based element in Figure 1 these shape functions are, using the isoparametric co-ordinates ( , ), [17,20] …”

Section: The Advection Phasementioning

confidence: 99%

“…We use a standard simultaneous relaxation method to update the velocity vector and the pressure [17,20]. Let…”

Section: The Non-advection Phasementioning

confidence: 99%

“…In order for this approximation to be close to the exact value, t must of course be sufficiently small [17]. Using the CIP method, the potential is interpolated using third-order shape functions [18][19][20] (…”

Section: The Advection Phasementioning

confidence: 99%

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The CIP method embedded in finite element discretizations of incompressible fluid flows

Banijamali

Bathe

2006

Numerical Meth Engineering

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SUMMARYQuite effective low-order finite element and finite volume methods for incompressible fluid flows have been established and are widely used. However, higher-order finite element methods that are stable, have high accuracy and are computationally efficient are still sought. Such discretization schemes could be particularly useful to establish error estimates in numerical solutions of fluid flows. The objective of this paper is to report on a study in which the cubic interpolated polynomial (CIP) method is embedded into 4-node and 9-node finite element discretizations of 2D flows in order to stabilize the convective terms. To illustrate the capabilities of the formulations, the results obtained in the solution of the driven flow square cavity problem are given.

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Section: Remarks On the Solution Of The Governing Equationsmentioning

confidence: 99%

Section: The Non-advection Phasementioning

confidence: 99%

“…For the 4-node Galerkin-based element in Figure 1 these shape functions are, using the isoparametric co-ordinates ( , ), [17,20] …”

Section: The Advection Phasementioning

confidence: 99%

“…We use a standard simultaneous relaxation method to update the velocity vector and the pressure [17,20]. Let…”

Section: The Non-advection Phasementioning

confidence: 99%

Section: The Advection Phasementioning

confidence: 99%

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The CIP method embedded in finite element discretizations of incompressible fluid flows

Banijamali

Bathe

2006

Numerical Meth Engineering

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Numerical analysis of moving interfaces using a level set method coupled with adaptive mesh refinement

Kohno

Tanahashi

2004

Numerical Methods in Fluids

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SUMMARYA novel numerical scheme is developed by coupling the level set method with the adaptive mesh reÿnement in order to analyse moving interfaces economically and accurately. The ÿnite element method (FEM) is used to discretize the governing equations with the generalized simpliÿed marker and cell (GSMAC) scheme, and the cubic interpolated pseudo-particle (CIP) method is applied to the reinitialization of the level set function. The present adaptive mesh reÿnement is implemented in the quadrangular grid systems and easily embedded in the FEM-based algorithm. For the judgement on renewal of mesh, the level set function is adopted as an indicator, and the threshold is set at the boundary of the smoothing band. With this criterion, the variation of physical properties and the jump quantity on the free surface can be calculated accurately enough, while the computation cost is largely reduced as a whole. In order to prove the validity of the present scheme, two-dimensional numerical simulation is carried out in collapse of a water column, oscillation and movement of a drop under zero gravity. As a result, its e ectiveness and usefulness are clearly shown qualitatively and quantitatively. Among them, the movement of a drop due to the Marangoni e ect is ÿrst simulated e ciently with the present scheme.

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Single crystal growing problem in the FZ method under microgravity (three-dimensional numerical simulation with the hybrid FEM-BEM applying LSM on the free surface)

Kohno¹,

Tanahashi²

2001

Computational Fluid and Solid Mechanics

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Application of the GSMAC-CIP Method to Incompressible Navier-Stokes Equations at High Reynolds Numbers

Cited by 8 publications

References 7 publications

The CIP method embedded in finite element discretizations of incompressible fluid flows

The CIP method embedded in finite element discretizations of incompressible fluid flows

Numerical analysis of moving interfaces using a level set method coupled with adaptive mesh refinement

Single crystal growing problem in the FZ method under microgravity (three-dimensional numerical simulation with the hybrid FEM-BEM applying LSM on the free surface)

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