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

Syrakos, Alexandros; Goulas, A.

doi:10.1002/fld.1038

Cited by 29 publications

(47 citation statements)

References 21 publications

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“…For instance, Ferziger and Peric [2] and Syrakos and Goulas [3] accounted for skewness by using the value and its derivatives at the centre of the line linking two neighbouring cells to interpolate the value at the face centre. Zang et al [4] employed a third-order upwind quadratic scheme to interpolate velocity at the face in curvilinear coordinates.…”

Section: Introductionmentioning

confidence: 99%

A Taylor series‐based finite volume method for the Navier–Stokes equations

2008

Numerical Methods in Fluids

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SUMMARYA Taylor series-based finite volume formulation has been developed to solve the Navier-Stokes equations. Within each cell, velocity and pressure are obtained from the Taylor expansion at its centre. The derivatives in the expansion are found by applying the Gauss theorem over the cell. The resultant integration over the faces of the cell is calculated from the value at the middle point of the face and its derivatives, which are further obtained from a higher order interpolation based on the values at the centres of two cells sharing this face. The terms up to second order in the velocity and the terms up to first order in pressure in the Taylor expansion are retained throughout the derivation. The test cases for channel flow, flow past a circular cylinder and flow in a collapsible channel have shown that the method is quite accurate and flexible.

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Section: Introductionmentioning

confidence: 99%

A Taylor series‐based finite volume method for the Navier–Stokes equations

2008

Numerical Methods in Fluids

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“…Truncation error, E t , measures the accuracy of the discrete approximation to the differential equations (2.2). For FVD schemes, the traditional truncation error is usually defined from the time-dependent standpoint [19,22]. In the steady-state limit, it is defined (e.g., in [10]) as the residual computed after substituting Q into the normalized discrete equations (2.1),…”

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confidence: 99%

Notes on accuracy of finite-volume discretization schemes on irregular grids

Diskin

Thomas

2010

Applied Numerical Mathematics

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Abstract. Truncation-error analysis is a reliable tool in predicting convergence rates of discretization errors on regular smooth grids. However, it is often misleading in application to finitevolume discretization schemes on irregular (e.g., unstructured) grids. Convergence of truncation errors severely degrades on general irregular grids; a design-order convergence can be achieved only on grids with a certain degree of geometric regularity. Such degradation of truncation-error convergence does not necessarily imply a lower-order convergence of discretization errors. In these notes, irregular-grid computations demonstrate that the design-order discretization-error convergence can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all.1. Introduction. These notes are a response to the recently published article [18]. The article applies a truncation-error analysis to evaluate accuracy of finitevolume discretization (FVD) schemes on general unstructured grids. The analysis is accompanied by computations performed on regular and irregular grids. While we agree with the analysis and its conclusions in application to regular smooth grids, we consider the truncation-error analysis and some of the conclusions derived from it invalid in application to irregular-grid computations.On regular grids, convergence of truncation errors (also referred as local errors in the literature) is an accurate indicator of convergence of discretization errors (also referred as global errors). However, the truncation-error convergence is often misleading for FVD schemes defined on irregular (e.g., unstructured) grids. As shown in [18] and before this in [17], the second-order convergence of truncation errors for some commonly used FVD schemes can be achieved only on grids with a certain degree of geometric regularity. Other studies, e.g., [3,6,7,10,13,14,15,16,20,21], showed that truncation-error convergence degradation on irregular grids does not necessarily imply a degradation of discretization-error convergence. Examples shown in the following sections confirm that on irregular grids, the design-order discretization-error convergence can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all. Note that these results do not contradict the Lax theorem, which states that consistency (convergence of truncation errors) and stability are sufficient (not necessary) for convergence of discretization errors. In fact, for some formally inconsistent discretization schemes, it has been rigorously proved that the discretization errors converge [3,6,7,13,16,21].Article [18] applied a truncation error analysis to FVD schemes for the Poisson equation. A thin-layer approximation was analyzed. It was shown that the truncation error is O(1) (i.e., does not converge) in grid refinement unless the grids are regular. The discretization error of the scheme was inferred to be non-convergent. By coincidence, the particular thin-layer FVD s...

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“…The equations given in the previous Section were solved using a finite volume method, which was developed on the foundation of an existing method for generalised Newtonian and viscoplastic flows [49,50,22]. In the present Section the method will only be summarised, while the extensions pertaining to viscoelastic flow will be described in detail in a separate publication.…”

Section: Methodsmentioning

confidence: 99%

Theoretical study of the flow in a fluid damper containing high viscosity silicone oil: Effects of shear-thinning and viscoelasticity

2018

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The flow inside a fluid damper where a piston reciprocates sinusoidally inside an outer casing containing high-viscosity silicone oil is simulated using a Finite Volume method, at various excitation frequencies. The oil is modelled by the Carreau-Yasuda (CY) and Phan-Thien & Tanner (PTT) constitutive equations. Both models account for shear-thinning but only the PTT model accounts for elasticity. The CY and other generalised Newtonian models have been previously used in theoretical studies of fluid dampers, but the present study is the first to perform full two-dimensional (axisymmetric) simulations employing a viscoelastic constitutive equation. It is found that the CY and PTT predictions are similar when the excitation frequency is low, but at medium and higher frequencies the CY model fails to describe important phenomena that are predicted by the PTT model and observed in experimental studies found in the literature, such as the hysteresis of the force-displacement and force-velocity loops. Elastic effects are quantified by applying a decomposition of the damper force into elastic and viscous components, inspired from LAOS (Large Amplitude Oscillatory Shear) theory. The CY model also overestimates the damper force relative to the PTT, because it underpredicts the flow development length inside the piston-cylinder gap. It is thus concluded that (a) fluid elasticity must be accounted for and (b) theoretical approaches that rely on the assumption of one-dimensional flow in the piston-cylinder gap are of limited accuracy, even if they account for fluid viscoelasticity. The consequences of using lower-viscosity silicone oil are also briefly examined. This is the accepted version of the article published in:

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

Cited by 29 publications

References 21 publications

A Taylor series‐based finite volume method for the Navier–Stokes equations

A Taylor series‐based finite volume method for the Navier–Stokes equations

Notes on accuracy of finite-volume discretization schemes on irregular grids

Theoretical study of the flow in a fluid damper containing high viscosity silicone oil: Effects of shear-thinning and viscoelasticity

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