The Computational Accuracy Of Some Finite Element Methods For Incompressible Viscous Flow Problems

Author : Fırat Oğuz Edis
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (85 download)

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Download or read book Efficient Finite Element Computation of Unsteady Incompressible Viscous Flows Using Pseudo-second-order Velocity Interpolation written by Fırat Oğuz Edis and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The objective of this study is to construct and apply a computationally cost-effective Finite Element algorithm for the solution of the unsteady, incompressible Navier-Stokes equations on arbitrarily complex flow domains in two and threespace dimensions. Choice of the interpolation function for the finite element isone of the most important aspects affecting the accuracy and computational costof the resulting scheme. The use of equal order interpolation functions for velocityand pressure can be a source of instability. An example is the Q1Q1 element pair,which employs continuous bilinear interpolation functions for both velocity andpressure. This element pair does not satisfy the div-stability condition and isknown to cause oscillations primarily in the pressure field. However, it is widelyused with the aid of a proper stabilizing term or time integration scheme due to its computational efficiency. Element pairs, coupling piecewise first-order velocity element with piecewise first- order pressure element, and still satisfying the div-stability condition are given in literature for triangular and quadrilateral elements. The quadrilateral element air is also called a pseudo biquadratic velocity and bilinear pressure element (pQ2Q1). The triangular pair is called a P1isoP2/P1 pair in the literature but can also be called analogously to the quadrilateral element as a pP2P1 element. These element pairs satisfy the div-stability condition which is also known asthe 'Ladyzhenskaya-Babuska-Brezzi'(LBB) condition. Fulfilling this conditionensures that no spurious oscillations occur in the pressure field. Furthermore, employment of these elements, when compared to an equivalent first-order pair formulation, is expected to reduce the memory requirement and the CPU timedue to the fewer elements used in the solution of the Poisson equation for pres-sure. However, in spite of these advantages, there is no open literature available presenting computations obtained using the pQ2Q1 element pair and very few forthe pP2P1 elements, discussing the efficiency of the computations in detail andcomparing the results for accuracy. Therefore, the purpose of the present study is: l. to present example computations obtained with pQ2Ql, pP2P1 element pairs, 2. to address special problems arising from the use of these element pairs, 3. to give detailed computation time and storage comparisons. To fulfill these purposes, a finite element formulation based on pQ2Ql, pP2Pl,QlQ1 and PlP1 elements is presented for the computation of two and threedimensional laminar and turbulent unsteady incompressible viscous flows. For each one of these elements, the unsteady Navier-Stokes equations are solvedusing a finite element method, based on a fractional step approach with an ex-plicit time marching scheme. A streamwise upwinding technique is employed tostabilize the convective term for large element Reynolds numbers. For the so-lution of the Poisson equation for pressure, a preconditioned conjugate gradientmethod with an element-by-element technique is employed. The previous value of pressure is used to start the iterations at each time step. This leads to consider- able savings in CPU time compared to auxiliary potential function-based pressure formulations. An algebraic turbulence model, namely the Baldwin-Lomax is im- plemented for the solution of turbulent flows. A modification of the original pseudo-second-order element is realized to accurately represent a curved bound- ary of the domain. Test cases analyzed include the lid-driven cavity flow problem, the flow past an impulsively started circular cylinder at Reynolds numbers 40 and 3000, and turbulent flow over a flat plate at Reynolds number 2 x 10 , in two space di- mensions. In 3D, the lid-driven cavity flow problem at Reynolds number 1000 is analyzed. Results obtained using the pseudo-second-order velocity interpolation elements are compared in terms of accuracy and computational cost with the results obtained using equal-order interpolation pairs. Comparisons with other experimental and numerical data and empirical formula are presented. Results obtained with pP2Pl and pQ2Q1 elements are shown to be as accurate as the results obtained with equal order elements on the same velocity mesh. Comparison of computational efforts for 2D cases indicates CPU time savings up to 68 per cent in favor of the pseudo-second-order velocity interpolation elements. For 3D analyses, savings up to 54 per cent are observed. It is concluded that the use of the pseudo-second-order interpolation for velocity instead of first-order interpolation reduces the computational costs. This is due to the reduction in the size of the stiffness matrix for pressure. The reduction in the computational cost may primarily be in the memory or the CPU time requirements, depending on the programming preferences.