Numerical calculations of laminar flows using contravariant velocity fluxes

Xu, Hao; Zhang, C.

doi:10.1016/s0045-7930(99)00006-7

Cited by 5 publications

(8 citation statements)

References 23 publications

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“…In order to do this we compare our results with the results found in the literature. At this point, we would like to note that in the literature among the studies that have solved the skewed cavity flow [3,5,11,13,17,18,20,22,25,27,28,31,32], only Demirdžić et al [5], Oosterlee et al [17], Shklyar and Arbel [22] and Louaked et al [13] have presented tabulated results therefore we will mainly compare our results with those studies. As mentioned earlier, Demirdžić et al [5] have presented solutions for skewed cavity for Reynolds number of 100 and 1000 for skewed angles of α = 45 • and α = 30 • .…”

Section: Resultsmentioning

confidence: 98%

“…The skewed cavity problem is a perfect test case for body fitted non-orthogonal grids and yet it is as simple as the cavity flow in terms of programming point of view. Later Oosterlee et al [17], Louaked et al [13], Roychowdhury et al [20], Xu and Zhang [31], Wang and Komori [28], Xu and Zhang [32], Tucker and Pan [27], Brakkee et al [3], Pacheco and Peck [18], Teigland and Eliassen [25], Lai and Yan [11] and Shklyar and Arbel [22] have solved the same benchmark problem. In all these studies, the solution of the driven skewed cavity flow is presented for Reynolds numbers of 100 and 1000 for only two different skew angles which are α = 30 • and α = 45 • , and also the maximum number of grids used in these studies is 320 × 320.…”

Section: Introductionmentioning

confidence: 99%

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Numerical solutions of 2‐D steady incompressible flow in a driven skewed cavity

Ertürk¹,

Dursun

2007

Z Angew Math Mech

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Section: Resultsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Numerical solutions of 2‐D steady incompressible flow in a driven skewed cavity

Ertürk¹,

Dursun

2007

Z Angew Math Mech

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“…Equations (22), (23) are expressed using non-physical components V j and T j i that are defined with respect to non-normalized bases e j and e i e j . These non-physical components have scale factors associated with them and do not have the same dimensions as the physical quantities.…”

Section: Governing Equations In Curvilinear Coordinates With Mixed Comentioning

confidence: 99%

A new mixed basis Navier–Stokes formulation for incompressible flows over complex geometries

Murali

Rajagopalan

2016

Journal of Computational Physics

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Numerical modeling of complex geometries necessitates the use of curvilinear body fitted coordinates. This article proposes a novel mixed basis formulation of the governing conservation equations for general curvilinear non-orthogonal grids with the physical covariant velocity as the primary solution variable. This results in an algorithm which has many advantages of orthogonal equations. The conservation equations written in this form retains the diagonal dominance of the pressure equation. The newly formed conservation equations are solved on a structured grid using the SIMPLER algorithm and are shown to converge well for non-orthogonal grids. Standard K − model is used for the turbulence closure.In the context of pressure equation, the grid orientation can be leveraged to our advantage by using curvilinear velocity components. Many researchers have used either contravariant or covariant velocity components and have come up with different forms of the governing equation. When curvilinear velocity components are used, the spatially varying nature of the basis vector gives rise to additional flux and source terms in the governing equation. Convective and diffusive terms contain secondary non-directional fluxes that need to be treated explicitly as part of the source term. In high Reynolds number flows the convection terms dominate over the diffusion terms. In the pressure based solution procedures of incompressible flow it is desirable to treat convection terms implicitly while secondary diffusive fluxes may still be treated explicitly. Demirdzic et al. [5], Yang et al. [6], and Graef et al.[7] used physical contravariant velocity for flow calculation in complex geometries. Although such a formulation represents the convective fluxes compactly and enables their implicit treatment with ease, the lack of diagonal dominance of the pressure equation still persist due to the presence of multi-directional pressure gradient in the momentum equations. Sharatchandra and Rhode [8] followed a vector discretization procedure for steady flows with the physical contravariant velocity components as the primary variable. The final discrete equations contain cross-directional pressure gradient terms indirectly. When their solution method is applied to unsteady flows, each discretized momentum equation will have additional time derivative terms of the non-directional velocity components that need to be treated explicitly. They presented test cases that have moderate grid non-orthogonality. Karki and Patankar [9] and Tamamidis and Assanis [10] demonstrated that by using covariant velocity projections as the primary solution variable, governing equations with unidirectional pressure gradient term might be obtained. However, the divergence of convection-diffusion flux tensor expressed in pure covariant components yields far too many terms. The convective flux across a face cannot be represented with a single velocity component alone and contains secondary non-directional convective fluxes. Karki and Patankar [9] avoided the use of te...

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“…Many authors [49,68,69] emphasize that the numerical approximation of these terms can reduce the numerical accuracy.…”

Section: The Integral Expression Of the Shallow-water Equations In Thmentioning

confidence: 99%

“…As a consequence, numerical discretization of the Christoffel symbols can reduce the numerical accuracy. Yang et al [68], Xu and Zhang [69] and Wesseling et al [49] avoid the Christoffel symbols and ensure the strong conservation properties by dotting the strongly conservative motion equations in a vector form by the contravariant base vector after the discretizations.…”

Section: Introductionmentioning

confidence: 99%

Central WENO scheme for the integral form of contravariant shallow‐water equations

Gallerano

Cannata

2010

Numerical Methods in Fluids

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A new Central Weighted Essentially Non-Oscillatory scheme for the solution of the shallow-water equations expressed in contravariant formulation is presented. One of the most efficient methodologies belonging to Central WENO family involves: reconstructions of cell-averaged values of flow variables, reconstruction of point-values of flow variables, advancing from time level t(n) to time level t(n+1) of the cell-averaged values. The extension of the above-mentioned methodology into the contravariant environment implies that the contravariant shallow-water equations must be expressed in an integral form. An element of novelty presented in this paper regards the definition of a formal integral expression of the shallow-water equations in a contravariant formulation, in which the Christoffel symbols are avoided. The WENO reconstructions are performed by a two-dimensional interpolating procedure taking into account the curved coordinate lines. The proposed scheme ensures the satisfaction of the exact C-property. Several two-dimensional test cases are used to verify the good resolution for smooth and discontinuous solutions. Copyright (C) 2010 John Wiley & Sons, Ltd

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Numerical calculations of laminar flows using contravariant velocity fluxes

Cited by 5 publications

References 23 publications

Numerical solutions of 2‐D steady incompressible flow in a driven skewed cavity

Numerical solutions of 2‐D steady incompressible flow in a driven skewed cavity

A new mixed basis Navier–Stokes formulation for incompressible flows over complex geometries

Central WENO scheme for the integral form of contravariant shallow‐water equations

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