Kinematic Laplacian Equation Method: A Velocity-Vorticity Formulation for the Navier-Stokes Equations

[+] Author and Article Information
Fernando L. Ponta1

College of Engineering,  University of Buenos Aires, Paseo Colón 850, Buenos Aires C1063ACV, Argentinafponta@fi.uba.ar


Also at the Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 104 S. Wright Street, Urbana, IL 61801.

J. Appl. Mech 73(6), 1031-1038 (Feb 04, 2006) (8 pages) doi:10.1115/1.2198245 History: Received September 24, 2005; Revised February 04, 2006

In this work, a novel procedure to solve the Navier-Stokes equations in the vorticity-velocity formulation is presented. The vorticity transport equation is solved as an ordinary differential equation (ODE) problem on each node of the spatial discretization. Evaluation of the right-hand side of the ODE system is computed from the spatial solution for the velocity field provided by a new partial differential equation expression called the kinematic Laplacian equation (KLE). This complete decoupling of the two variables in a vorticity-in-time/velocity-in-space split algorithm reduces the number of unknowns to solve in the time-integration process and also favors the use of advanced ODE algorithms, enhancing the efficiency and robustness of time integration. The issue of the imposition of vorticity boundary conditions is addressed, and details of the implementation of the KLE by isoparametric finite element discretization are given. Validation results of the KLE method applied to the study of the classical case of a circular cylinder in impulsive-started pure-translational steady motion are presented. The problem is solved at several Reynolds numbers in the range 5<Re<180 comparing numerical results with experimental measurements and flow visualization plates. Finally, a recent result from a study on periodic vortex-array structures produced in the wake of forced-oscillating cylinders is included.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Interpolation functions of the nine-node isoparametric element showing its natural system of coordinates, node numeration, and three examples of functions: for a corner node (node 3), for a central-lateral node (node 8), and for the central node (node 9)

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Figure 2

An example of a mesh of tri-quadrilateral finite elements obtained from a standard triangular discretization

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Figure 3

Schematic view of the internal topology of the tri-quadrilateral element. Subelements (I)–(III) are modeled by standard nine-node isoparametric interpolation. Numbers 1–19 indicate the in-triangle nodal numeration.

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Figure 4

An example of a mesh of 2828 tri-quadrilateral finite elements and 34,216 nodes used for the present analysis, which gives a total of 14,420 nodes after static condensation (geometrical coordinates are given in diameters)

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Figure 5

Comparison of flow visualizations by Taneda and arrow plots from numerical results for the twin-vortex wake behind a cylinder at (a)Re=13.05 and (b)Re=26

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Figure 6

Comparison of the wake length calculated by the kinematic Laplacian equation method and the experimental measurements by Taneda (23)

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Figure 7

Comparison of flow visualization of a Kármán vortex street behind a cylinder at Re=100 by M. M. Zdravkovich with a gray scale plot of the vorticity field produced by the kinematic Laplacian equation method at the same value of Reynolds number

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Figure 8

Comparison of the Strouhal number calculated by the kinematic Laplacian equation method and the experimental measurements by Williamson (24) for Re<180

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Figure 9

Comparison of flow visualization of a P+S wake of an oscillating cylinder for Re=140 by C. H. K. Williamson (private communication to H. Aref) with a gray scale plot of the vorticity field produced by the KLE method at the same Reynolds number



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