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Space-time Taylor-Hood elements for incompressible flows

Pacheco Douglas R. Q, Steinbach Olaf

Computer Methods in Materials Science

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2019

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Vol. 19, No. 2

64--69

EN

Space-time variational methods differ from time-stepping schemes by discretising the whole space-time domain with finite elements. This offers a natural framework for flow problems in moving domains and allows simultaneous parallelisation and adaptivity in space and time. For incompressible flows, the usual approach is to employ the same polynomial order for velocity and pressure, which requires the use of stabilisation techniques to compensate for the inf-sup deficiency of such pairs. In the present work, we extend to the space-time formulation the idea of the popular Taylor-Hood element for the (Navier-)Stokes equations. By using quadratic interpolation for velocity and linear for the pressure, in both space and time, we attain a stable finite element method which provides optimal convergence for pressure, velocity and stresses.

PL

Przestrzenno-czasowe metody wariacyjne wymagają dyskretyzacji metodą elementów skończonych całej domeny przestrzenno czasowej i tym różnią się od metod wykorzystującej schematy kroków czasowych. To podejście dostarcza naturalnych struktur dla problemów przepływu w poruszających się obszarach i pozwala na równoczesne zrównoleglanie i adaptację zarówno w przestrzeni jak i w czasie. Typowym rozwiązaniem dla przepływów nieściśliwych jest zastosowanie tego samego stopnia wielomianu dla prędkości i ciśnienia, co wymaga wprowadzenia metod stabilizacji w celu skompensowania niedoboru infimum-supremum takich par. W niniejszej pracy rozszerzono sformułowanie przestrzenno czasowe o ideę elementu Taylora-Hooda dla równań (Naviera-)Stokesa. Poprzez zastosowanie kwadratowej interpolacji dla prędkości i liniowej interpolacji dla ciśnienia, zarówno w przstrzeni jak i w czasie, uzyskano stabilną metodę elementów skończonych dającą optymalną zbieżność dla ciśnienia, prędkości i naprężeń.

2

Solutions of incompressible Navier-Stokes equations with the artificial compressibility method for two- and three-dimensional shear-driven cavity flows

Kosma Z.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 1

113-125

EN

An efficient method for simulating laminar flows in complex geometries is presented. The artificial compressibility method was applied to solve two- and three-dimensional Navier-Stokes equations in primitive variables on Cartesian grids. Two numerical approaches were proposed in this work, which are based on the method of lines process in conjunction with transfer of all the variables from the boundaries to the nearest uniform grid knots. Initial value problems for the systems of ordinary differential equations for pressure and velocity components were computed using the one-step backward-differentiation predictor-corrector method or the Galerkin-Runge-Kutta method of third order. Some test calculations for laminar flows in square, half-square, triangular, semicircular, cubic, half-cubic, half-cylinder and hemisphere cavities with one uniform moving wall were reported. The present results were compared with the available data in the literature and the Fluent solver numerical simulations.

3

Finite difference schemes and the method of lines for solving incompressible viscous flow problems

Kosma Z., Motyl P.

International Journal of Applied Mechanics and Engineering

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2011

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Vol. 16, no 1

15-35

EN

For the determination of viscous incompressible flows a pure stream-function formulation for the fourth-order equation, the artificial compressibility method, and velocity correction method are employed. Test calculations are performed for various flows inside square, triangular, semicircular and cubic cavities with one uniform wall, the backward-facing step, double bent channels, the flow around an aerofoil at large angle of attack and for flows over models of buildings. Some complex geometrical configurations can be decomposed into a set of simpler subdomains. A practical methodology for the computation of the Navier-Stokes equations in arbitrarily complex geometries is also considered. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots.