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Research Papers: Flows in Complex Systems

Galerkin Boundary Elements for a Computation of the Surface Tractions in Exterior Stokes Flows

[+] Author and Article Information
Jorge D'Elía

CIMEC (UNL-CONICET),
Predio Conicet-Santa Fe, Colectora RN 168,
El Pozo, Santa Fe S3000GLN, Argentina
e-mail: jdelia@unl.edu.ar

Laura Battaglia

CIMEC (UNL-CONICET),
GIMNI, UTN-FRSF,
Lavaisse 610, Santa Fe S3000GLN, Argentina
e-mail: lbattaglia@santafe-conicet.gob.ar

Alberto Cardona

Professor
CIMEC (UNL-CONICET),
Predio CONICET-Santa Fe
Colectora Ruta Nac 168,
Pa. El Pozo Santa Fe,
Sante Fe S3000GLN, Argentina
e-mail: acardona@unl.edu.ar

Mario Storti

Professor
CIMEC (UNL-CONICET),
Predio CONICET-Santa Fe
Colectora Ruta Nac 168,
Pa. El Pozo Santa Fe,
Santa Fe S3000GLN, Argentina
e-mail: mario.storti@gmail.com

Gustavo Ríos Rodríguez

CIMEC (UNL-CONICET),
Predio CONICET-Santa Fe
Colectora Ruta Nac 168,
Pa. El Pozo Santa Fe,
Santa Fe S3000GLN, Argentina
e-mail: gusadrr@santafe-conicet.gov.ar

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 2, 2012; final manuscript received May 10, 2014; published online September 4, 2014. Assoc. Editor: Ye Zhou.

J. Fluids Eng 136(11), 111102 (Sep 04, 2014) (16 pages) Paper No: FE-12-1600; doi: 10.1115/1.4027685 History: Received December 02, 2012; Revised May 10, 2014

In the computation of a three–dimensional steady creeping flow around a rigid body, the total body force and torque are well predicted using a boundary integral equation (BIE) with a single concentrated pair Stokeslet- Rotlet located at an interior point of the body. However, the distribution of surface tractions are seldom considered. Then, a completed indirect velocity BIE of Fredholm type and second-kind is employed for the computation of the pointwise tractions, and it is numerically solved by using either collocation or Galerkin weighting procedures over flat triangles. In the Galerkin case, a full numerical quadrature is proposed in order to handle the weak singularity of the tensor kernels, which is an extension for fluid engineering of a general framework (Taylor, 2003, “Accurate and Efficient Numerical Integration of Weakly Singulars Integrals in Galerkin EFIE Solutions,” IEEE Trans. on Antennas and Propag., 51(7), pp. 1630–1637). Several numerical simulations of steady creeping flow around closed bodies are presented, where results compare well with semianalytical and finite-element solutions, showing the ability of the method for obtaining the viscous drag and capturing the singular behavior of the surface tractions close to edges and corners. Also, deliberately intricate geometries are considered.

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References

Figures

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Fig. 1

Sketch of a closed and piecewise smooth surface A with an exterior flow domain Ωe: the field point x, the source point y, the relative position r = x-y, the unit normals n(x),n(y), and the differential areas dAx,dAy

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Fig. 2

Colormaps of the nondimensional relative traction τ1(GBEM)/τ1(analytic) (a) and L2 norm of the relative percent error and (b) on the sphere surface using GBEM with a Q22 quadrature rule and uniform BEM meshes 5 (left) and 11 (right)

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Fig. 3

Colormaps of the nondimensional traction τ1(GBEM)/τ1(analytic) (a) and L2 norm of the relative percent error and (b) on the sphere surface using GBEM with a Q22 quadrature rule and perturbed BEM meshes 8 (left) and 11 (right)

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Fig. 4

Computational cost as a function of the number of elements E in the sphere case: elapsed real time [s] (a) and RAM [GB] and (b) employing a Q22 quadrature rule and a direct solver on 4 of the 6 cores of a Xeon W3690. Collocation (BEM) and Galerkin (GBEM)

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Fig. 5

Sketch for the axisymmetric creeping flow around a torus

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Fig. 6

Sketch of the quasi 2D steady creeping flow pattern around the rigid cube [32]: (a) parallel flow along edges J-J"'; u∞ = (1,0,0) m/s and (b) splitting flow u∞ = (1,0,1) m/s (symmetric on edges K,K' and antisymmetric on edges L,L')

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Fig. 7

Meridian (a) equatorial, (b) diagonal, and (c) coordinates on the unit cube. The Cartesian coordinate system O(x,y,z) = O(x1,x2,x3) is centered.

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Fig. 8

Quasi 2D steady creeping flow across the middle section of the unit cube: nondimensional traction τ as a function of the coordinate: equatorial EF (a) front meridian AB (b) and rear meridian CD (c). GBEM computation (Q21 and Q22 rules, mesh 11, Table 1) and semianalytical laws O(sp) [32], as a function of the distance s to the edge singularity.

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Fig. 9

3D steady creeping flows of the 90 deg vertex of the unit cube: nondimensional traction τ as a function of the coordinate along the diagonal WI on the top plane. GBEM computation (Q21 and Q22 rules, mesh 11, Table 1) and semianalytical laws O(sp) [32], as a function of the distance s to the vertex singularity.

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Fig. 10

Cartesian τ1 component of the nondimensional traction for the parallel flow u = (U,0,0) m/s, as a function of the meridional and equatorial coordinates sm (a) and se (b), respectively, with FEM (solid line) and GBEM (crosses), using the Q22 quadrature rule and mesh 11. See Figs. 7(a) and 7(b), for the position of the points A-I.

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Fig. 11

Absolute value of the relative error |er| of the nondimensional body force F∧1 and body torque C∧3 as a function of the number of degrees of freedom M and the QIJ quadrature rule with smooth meshes on the unit sphere: F∧1 uniform flow (a), C∧3 shear flow, (b), and F∧1 paraboloidal flow (c). Collocation (BEM, top) and Galerkin BEM (GBEM, bottom).

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Fig. 12

Absolute value of the relative error |er| of the nondimensional body force F∧1 and body torque C∧3 as a function of the number of degrees of freedom M and the QIJ quadrature rule with smooth meshes on the unit cube: F∧1 uniform flow (a), C∧3 shear flow (b), and F∧1 paraboloidal flow (c). Collocation (BEM, top) and Galerkin BEM (GBEM, bottom).

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Fig. 13

Other geometries with edges and corners: concave cube (a), hollow cube (b) [41], and sculpted sphere (c) [41]. Colormaps of the τ1 nondimensional traction for the parallel flow u∞ = (1,0,0) m/s using the Q22 quadrature rule. Collocation (top) and GBEM (bottom).

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Fig. 14

Deliberately intricate 3D geometries using wrapped tubes [42]: pentagon knots (a), circle knots (b) and triangle knots (c). Collocation (top) and GBEM (bottom).

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Fig. 15

Master triangles p and q for the simplex coordinates

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