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Computing Viscous Flow in an Elastic Tube

Published online by Cambridge University Press:  09 August 2018

Yi Li ,
Ioannis Sgouralis  and
Anita T. Layton
Yi Li*
Affiliation:
Department of Mathematics, Duke University, Durham, NC, USA.
Ioannis Sgouralis*
Affiliation:
Department of Mathematics, Duke University, Durham, NC, USA.
Anita T. Layton*
Affiliation:
Department of Mathematics, Duke University, Durham, NC, USA.
*
Corresponding author. Email address: yili@math.duke.edu
Email: io-sgou@math.duke.edu
*Email:alayton@math.duke.edu
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Abstract

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.

Keywords

65M0676D0576M20Navier-Stokes flowaxisymmetry immersed boundaryfluid-structure interaction
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 4 , November 2014 , pp. 555 - 574
Copyright
Copyright © Global Science Press Limited 2014

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References

[1] Arthurs, K. M., Moore, L. C., Peskin, C. S., Pitman, E. B., and Layton, H. E.. Modeling arteriolar flow and mass transport using the immersed boundary method. J. Comput. Phys., 147:402–440, 1998.Google Scholar
[2] Beale, J. T. and Layton, A. T.. A velocity decomposition approach for moving interfaces in viscous fluids. J. Comput. Phys., 228:3358–3367, 2009.Google Scholar
[3] Biros, G., Ying, L., and Zorin, D.. An embedded boundary integral solver for the unsteady incompressible navier-stokes equations. SIAM J. Sci. Comput., submitted for publication.Google Scholar
[4] Brown, D. L., Cortez, R., and Minion, M. L.. Accurate projection methods for the incompressible navier-stokes equations. J. Comput. Phys., 168:464–494, 2001.CrossRefGoogle Scholar
[5] Hu, H.H., Patankar, N.A., and Zhu, M.Y.. Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique. Journal of Computational Physics, 169:427–462, 2001.Google Scholar
[6] Huerta, A. and Liu, W.K.. Viscous flow with large free surface motion. Computer Methods in Applied Mechanics and Engineering, 69:277–324, 1988.Google Scholar
[7] Hughes, T.J.R., Liu, W.K., and Zimmermann, T.K.. Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering, 29:329–349, 1981.Google Scholar
[8] Jung, E., Lim, S., Lee, W., and Lee, S.. Computational models of valveless pumping using the immersed boundary method. Comput. Methods Appl. Mech. Engin., 197:2329–2339, 2008.Google Scholar
[9] Kim, J. and Moin, P.. Application of a fractional-step method to incompressible navier-stokes equations. J. Comput. Phys., 59:308, 1985.CrossRefGoogle Scholar
[10] Lai, M.-C.. Fourth-order finite difference scheme for the incompressible Navier-Stokes equations in a disk. Int J Numer Meth Fluids, 42:909–922, 2003.CrossRefGoogle Scholar
[11] Lai, M.-C. and Tseng, H.-C.. A simple implementation of the immersed interface methods for stokes flows with singular forces. Comput Fluids, 37:99–106, 2008.Google Scholar
[12] Lai, MC, Huang, CY, and Huang, YM. Simulating the axisymmetric interfacial flows with insoluble surfactant by immsered boundary method. Int J Numer Anal Model, 8:105–117, 2011.Google Scholar
[13] LeVeque, R. J. and Li, Z.. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31:1019–1044, 1994.Google Scholar
[14] LeVeque, R. J. and Li, Z.. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput., 18(3):709–735, 1997.Google Scholar
[15] Li, Y, Williams, S, and Layton, AT. A hybrid immersed interface method for driven stokes flow in an elastic tube. Numer Math in press, 2012.Google Scholar
[16] Li, Z. and Lai, M.-C.. The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys., 171:822–842, 2001.Google Scholar
[17] Li, Z., Wang, W.-C., Chern, I.-L., and Lai, M.-C.. New formulations for interface problems in polar coordinates. SIAM J. Sci. Comput., 25:224–245, 2003.Google Scholar
[18] Liu, W.K., Chang, H., Chen, J., and Belytschko, T.. Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua. Computer Methods in Applied Mechanics and Engineering, 68:259–310, 1988.Google Scholar
[19] Liu, W.K. and Ma, D.C.. Computer implementation aspects for fluid-structure interaction problems. Comput Meth Appl Mech Eng, 31:129–148, 1982.Google Scholar
[20] Peskin, C. S.. Numerical analysis of blood flow in the heart. J. Comput. Phys., 25:220–252, 1977.Google Scholar
[21] Peskin, C. S.. The immersed boundary method. Acta Numerica, 11:479–517, 2002.Google Scholar
[22] Xiu, D. and Karniadakis, G.. A semi-Lagrangian high-order method for Navier-Stokes equations. J. Comput. Phys., 172:658–684, 2001.Google Scholar
[23] Zhang, L.T., Wagner, G., and Liu, W.K.. Modeling and simulation of fluid structure interaction by meshfree and fem. Communications in Numerical Methods in Engineering, 19:615–621, 2003.Google Scholar
[24] Zhijun, T., Le, D.V., Lim, K.M., and Khoo, B.C.. An immersed interface method for the incompressible Navier-Stokes equations with discontinuous viscosity across the interface. SIAM J. Sci. Comput., 31(3):1798–1819, 2009.Google Scholar