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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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