A Taylor–Galerkin‐based algorithm for viscous incompressible flow

Hawken, D. M.; Tamaddon-Jahromi, H.R.; Townsend, Peter; Webster, M.F.

doi:10.1002/fld.1650100307

Cited by 121 publications

(112 citation statements)

References 26 publications

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“…The Taylor-Galerkin scheme is a two-step Lax-Wendroff time stepping procedure (predictor-corrector), extracted via a Taylor series expansion in time [10,11]. The pressure-correction method accommodates the incompressibility constraint to ensure second-order accuracy in time (see [12,13]). A three-stage structure emerges per time-step cycle, which can be expressed in discrete form, see [1].…”

Section: Numerical Algorithmmentioning

confidence: 99%

“…Here, we observe from the background theory [12,16,21], that element-cell contributions may be assembled, via element boolean transformation matrices L e ,…”

Section: B Consistent Mass-matrix Iterationmentioning

confidence: 99%

“…Such a single iteration, with absolute-value row-sum diagonalisation is often referred to in the literature as "mass-lumping" [11,12]. However, and in contrast to M f vl , one can now proceed with further iterations, due to the favourable conditioning of M f el .…”

Section: B Consistent Mass-matrix Iterationmentioning

confidence: 99%

“…However, and in contrast to M f vl , one can now proceed with further iterations, due to the favourable conditioning of M f el . Such time-discretisation is anticipated to improve the time-accuracy of the proposed f v-rhs-discretisation, as observed in the f e-context [10,12]. Within the above framework, adopting M f el , we have options posed as to choice of b sch .…”

Section: B Consistent Mass-matrix Iterationmentioning

confidence: 99%

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Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

Webster

Tamaddon-Jahromi

Aboubacar

2004

Numerical Methods Partial

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Hybrid finite volume/element methods are investigated within the context of transient viscoelastic flows. A finite volume algorithm is proposed for the hyperbolic constitutive equation, of Oldroyd-form, whilst the continuity/momentum balance is accommodated through a TaylorGalerkin finite element method. Various finite volume combinations are considered to derive accurate and stable implementations. Consistency of formulation is key, embracing fluctuation distribution and median-dual-cell constructs, within a cell-vertex discretisation on triangles. In addition, we investigate the effect of treating the time-term in a finite element fashion, using mass-matrix iteration instead of the standard finite volume mass-lumping approach. We devise an accurate transient scheme that captures the analytical solution at short and long time, both in core flow and near shear boundaries. In this respect, some difficulties are highlighted. A new method emerges, with the Low Diffusion B (LDB, with or without mass-matrix iteration) as the optimal choice. We progress to a complex flow application and demonstrate some provocative features due to the influence of true transient boundary conditions on evolutionary flow-structure in a 4:1 start-up rounded-corner contraction problem. c (Year) John Wiley & Sons, Inc.

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Section: Numerical Algorithmmentioning

confidence: 99%

“…Here, we observe from the background theory [12,16,21], that element-cell contributions may be assembled, via element boolean transformation matrices L e ,…”

Section: B Consistent Mass-matrix Iterationmentioning

confidence: 99%

Section: B Consistent Mass-matrix Iterationmentioning

confidence: 99%

Section: B Consistent Mass-matrix Iterationmentioning

confidence: 99%

See 2 more Smart Citations

Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

Webster

Tamaddon-Jahromi

Aboubacar

2004

Numerical Methods Partial

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“…A semi-implicit treatment for diffusion is employed to address linear stability constraints. Full details on this scheme have already been published extensively in the literature [8,9]. The flow is modeled as incompressible, via a pressure-correction scheme.…”

Section: Introductionmentioning

confidence: 99%

Modelling three‐dimensional rotating flows in cylindrical‐shaped vessels

Sujatha

Webster

2003

Numerical Methods in Fluids

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This paper reports on a study concerned with the numerical simulation of dough kneading that arises in the food processing industry. The flows considered are in a complex domain setting. Two dough mixers running at various rotation speeds are studied; one with a single stirrer, the other with two stirrers.Stirrers are fixed on the lid of the vessel and the motion is driven by the rotation of the outer vessel. Two different mixer orientations are considered, generating horizontal or vertical-rotating flow fields. Three-dimensional numerical simulations are performed on the full flow equations in a cylindrical polar co-ordinates system, through a finite-element, semi-implicit time stepping, Taylor-Galerkin pressure-correction scheme. The results reflect excellent agreement against the equivalent experimental findings. The motivation for this work is to develop advanced technology to model the kneading of dough. The ultimate target is to predict and adjust the design of dough mixers, so that optimal dough processing may be achieved notably, with reference to local rate-of-work input.

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A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS

Bell

Surana

1996

Int. J. Numer. Meth. Engng.

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SUMMARYThis paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier-Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier-Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of Co element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the spac*time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space-time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {d} which makes gradient of error functional with respect to {a} a null vector. This is accomplished by using Newton's method with a line search.A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space-time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space-time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space-time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.

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A Taylor–Galerkin‐based algorithm for viscous incompressible flow

Cited by 121 publications

References 26 publications

Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

Modelling three‐dimensional rotating flows in cylindrical‐shaped vessels

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS

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