2D thermal/isothermal incompressible viscous flows

Nicolás, Alfredo; Bermúdez, Blanca

doi:10.1002/fld.895

Cited by 17 publications

(32 citation statements)

References 12 publications

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“…For the driven cavity problem, results agree very well with those reported in the literature [3][4][5][6], [9,10] and with the second method, introduced here, we were able to reduce processing time for about 30% to 35%, for moderate Reynolds numbers and almost 50% for high Reynolds numbers. It can be seen in Figures 1 and 2, oscillations occur because the Reynolds number is very large, so it is necessary to use smaller values of h [13], numerically for stability and physically to capture the fast dynamics of the flow.…”

Section: Discussionsupporting

confidence: 81%

The Taylor Vortex and the Driven Cavity Problems in the Stream Function-Vorticity Formulation

Juárez

Hernández²,

Sánchez³

2014

ACM

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Abstract:In this work, two problems will be presented: The Taylor Vortex problem and the Driven Cavity problem. Both problems are solved using the Stream function-Vorticity formulation of the Navier-Stokes equations in 2D. Results are obtained using two methods: A fixed point iterative method and another one working with matrixes A and B resulting from the discretization of the Laplacian and the advective term, respectively. This second method resulted faster than the fixed point iterative one.

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Section: Discussionsupporting

confidence: 81%

The Taylor Vortex and the Driven Cavity Problems in the Stream Function-Vorticity Formulation

Juárez

Hernández²,

Sánchez³

2014

ACM

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“…The first one [1] is a simple numerical scheme for the unsteady Navier-Stokes equations in stream function and vorticity variables, based on a fixed point iterative process already used in the bibliography. This scheme worked very well, as shown in [7], [8], [9], [10], but the processing time was, in general, very large, especially for high Reynolds numbers. The second scheme discussed in [2], [3], works not only with the symmetric and positive matrix A resulting from the discretization of the Laplacian, but also with the matrix B resulting from the discretization of the advective term.…”

Section: Introductionmentioning

confidence: 95%

Numerical Solution of the Navier-Stokes Equations at High Reynolds Numbers

Bermúdez¹

2015

IJIRSET

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ABSTRACT:In this work, we are working with the unsteady Navier-Stokes equations in the stream function-vorticity formulation. In order to show that thenumerical schemes used are able to handle highReynolds numbers, we are reporting results for the well known problem of the un-regularized driven cavity. We are dealing with Reynolds numbers in the range of 7500 ≤Re ≤ 50000. The results shown here are obtained using two numerical schemes, the first one, is based on a fixed point iterative process (see [1]) applied to the elliptic nonlinear system that results after time discretization. The second scheme (see [2], [3]) which, as we are going to show in the results, is faster than the first one, solves the transport type equation appearing in the Stream function-vorticity formulation of the Navier-Stokes equations using matrixes A and B; the first one resulting from the discretization of the Laplacian term appearing in the equation, and the second one resulting from the discretization of the advective term. Both schemes, for this problem, have been robust enough to handle such high Reynolds numbers, but the second one has proved to be much faster, especially for high Reynolds numbers. In [4] it has already been said that even though turbulence is a tri-dimensional phenomenon, two-dimensional flows at high Reynolds numbers give some clues of transition to real turbulence. In the case of moderate Reynolds numbers, for instance Re ≤ 7500, the flow approaches to an asymptotic steady state as t tends to ∞, but in the case of high Reynolds numbers, such as the ones presented here, the solution seems to be time-dependent. KEYWORDS:As the Reynolds numberincreases the mesh has to be refined and a smaller time step has to be used: numerically, by stability matters andphysically, to capture the fast dynamics of the flow.It hasalready been pointed out in [4] that to get the right vorticitycontours is more difficult than to get the rightstreamlines of the stream function, dueto oscillations appearing on the top right corner of the cavity because of insufficient mesh refining. So, in this work, we are going to present results for the vorticity contours.

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“…The scheme worked very well, as shown in [1][2][3][4], but the processing time was, in general, very large especially for high Reynolds numbers. Working with matrixes A and B, we are dealing with a non-symetric matrix, but it can be proved that it is strictly diagonally dominant for Δt sufficiently small.…”

Section: Introductionmentioning

confidence: 98%

“…The fixed point iterative method has already been used for solving the Navier-Stokes equations and the Boussinesq system under different formulations, see [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Two Different Formulations for Solving the Navier-Stokes Equations with Moderate and High Reynolds Numbers

Bermúdez¹,

Rangel-Huerta²,

FermínGuerrero-Sánchez³

et al. 2018

Computational Fluid Dynamics - Basic Instruments and Applications in Science

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In this work, we discuss the numerical solution of the Taylor vortex and the lid-driven cavity problems. Both problems are solved using the Stream function-vorticity formulation of the Navier-Stokes equations in 2D. Results are obtained using a fixed point iterative method and working with matrixes A and B resulting from the discretization of the Laplacian and the advective term, respectively. We solved both problems with Reynolds numbers in the range of 3200 ≤ Re ≤ 7500. Results are also obtained using the velocity-vorticity formulation of the Navier-Stokes equations. In this case, we are using only the fixed point iterative method. We present results for the lid-driven cavity problem and for the Stream function-vorticity formulation with Reynolds numbers in the range of 3200 ≤ Re ≤ 7500. As the Reynolds number increases, the time and the space step size have to be refined. We show results for 3200 ≤ Re ≤ 20,000. The numerical scheme with the velocity-vorticity formulation uses a smaller step size for both time and space. Results are not as good as with the Stream function-vorticity formulation, although the way the scheme behaves gives us another point of view on the behavior of fluids under different numerical schemes and different formulation.

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2D thermal/isothermal incompressible viscous flows

Cited by 17 publications

References 12 publications

The Taylor Vortex and the Driven Cavity Problems in the Stream Function-Vorticity Formulation

The Taylor Vortex and the Driven Cavity Problems in the Stream Function-Vorticity Formulation

Numerical Solution of the Navier-Stokes Equations at High Reynolds Numbers

Two Different Formulations for Solving the Navier-Stokes Equations with Moderate and High Reynolds Numbers

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