Finite difference scheme for the solution of fluid flow problems on non‐staggered grids

Barton, I. E.; Kirby, Ray

doi:10.1002/1097-0363(20000815)33:7<939::aid-fld38>3.0.co;2-#

Cited by 20 publications

(12 citation statements)

References 46 publications

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“…If the term is computed in that form the pressure contributes to the cell mass balance with a strict (dissipative) fourth order derivative, as opposed to the third order derivative of the other form due to the (not always slight) variation of A u P |i . The amount of dissipation is sometimes controlled by a coefficient (other than the geometric factor) that is adjusted ad hoc [19] [20] [24]. In the computational exercices to be presented with the unsteady version of this scheme we prefer to be consistent and stick to the correct expression given in Eq.…”

Section: Cell Face Velocity Expression Steady Casementioning

confidence: 99%

“…The actual u * e equation they employed contains some unnecessary approximations that can produce a less accurate solution. Barton et al [19] did not derive a special interpolation procedure for u * e , however they eventually assemble a continuity equation for an unsteady problem that contains factors like ∆V i /A u P |i and not ∆V i / A u P |i 2…”

Section: The Interface Velocity In An Unsteady Flowmentioning

confidence: 99%

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Cell face velocity alternatives in a structured colocated grid for the unsteady Navier–Stokes equations

Pascau

2011

Numerical Methods in Fluids

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The use of a colocated variable arrangement for the numerical solution of fluid flow is becoming more and more popular due to its coding simplicity. The inherent decoupling of the pressure and velocity fields in this arrangement can be handled via a special interpolation procedure for the calculation of the cell face velocity named PWIM (Pressure Weighted Interpolation Method). In this paper a discussion on the alternatives to extend PWIM to unsteady flows is presented along with a very simple criterium to ascertain if a given interpolation practice will produce steady results that are relaxation dependent or time step dependent. Following this criterium it will be shown that some prior schemes presented as time step independent are actually not, although by using special interpolations can be readily adapted to be. A systematic way of deriving different cell face velocity expressions will be presented and new formulae free of ∆t dependence will be derived. Several computational exercices will accompany the theoretical discussion to support our claims.

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Section: Cell Face Velocity Expression Steady Casementioning

confidence: 99%

Section: The Interface Velocity In An Unsteady Flowmentioning

confidence: 99%

Cell face velocity alternatives in a structured colocated grid for the unsteady Navier–Stokes equations

Pascau

2011

Numerical Methods in Fluids

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“…This is zero for the present problem because all boundaries are solid walls. Therefore the sum of the 16 right hand sides of (6.8) should also be zero, but this is not guaranteed by the present method as described so far. The sum Pτ m h,ΡΔΩP equals zero because the truncation error of the approximation of the mass flux through a face f contributes with opposite sign to the continuity truncation errors of the CVs on either side of f. Also, the sum Pτ2h h,m ,Ρ ΔΩP over all CVs P of the coarse grid 2h is zero because the mass fluxes F2h which are calculated from the restricted velocity and pressure fields contribute with opposite sign to the relative truncation errors of the CVs on either side of the face.…”

Section: Flow Between Concentric Cylindersmentioning

confidence: 92%

Estimate of the truncation error of finite volume discretization of the Navier-Stokes equations on colocated grids

Syrakos

Goulas

2005

Int. J. Numer. Meth. Fluids

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A methodology is proposed for the calculation of the truncation error of finite volume discretisations of the incompressible Navier -Stokes equations on colocated grids. The truncation error is estimated by restricting the solution obtained on a given grid to a coarser grid and calculating the image of the discrete Navier -Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed methodology is not a new concept but its application to colocated finite volume discretisations of the incompressible Navier -Stokes equations is made possible by the introduction of a variant of the momentum interpolation technique for mass fluxes where the pressure-part of the mass fluxes is not dependent on the coefficients of the linearised momentum equations. The theory presented is supported by a number of numerical experiments. The methodology is developed for two-dimensional flows, but extension to three-dimensional cases should not pose problems.

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“…e.g. [1][2][3][4][5][6][7]). While finite difference, finite element and finite volume methodologies have been extensively developed, the majority of the early fundamental research work in this field was based on finite difference formulations, whereas the finite volume method dominates more recent activity and the commercial codes used in industry.…”

Section: Introductionmentioning

confidence: 99%

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

Zogheib

Barron

2011

Numerical Methods in Fluids

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SUMMARYA new numerical procedure for solving the two-dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well-established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocitypressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first-and second-order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well-known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward-facing step, and lid-driven cavity flow.

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Finite difference scheme for the solution of fluid flow problems on non‐staggered grids

Cited by 20 publications

References 46 publications

Cell face velocity alternatives in a structured colocated grid for the unsteady Navier–Stokes equations

Cell face velocity alternatives in a structured colocated grid for the unsteady Navier–Stokes equations

Estimate of the truncation error of finite volume discretization of the Navier-Stokes equations on colocated grids

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

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