Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T09:03:29.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Cavity flow characteristics and applications to kidney stone removal

Published online by Cambridge University Press:  07 September 2020

J. G. Williams Open the ORCID record for J. G. Williams [Opens in a new window] ,
A. A. Castrejón-Pita Open the ORCID record for A. A. Castrejón-Pita [Opens in a new window] ,
B. W. Turney Open the ORCID record for B. W. Turney [Opens in a new window] ,
P. E. Farrell Open the ORCID record for P. E. Farrell [Opens in a new window] ,
S. J. Tavener ,
D. E. Moulton Open the ORCID record for D. E. Moulton [Opens in a new window]  and
S. L. Waters Open the ORCID record for S. L. Waters [Opens in a new window]
J. G. Williams*
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG, UK
A. A. Castrejón-Pita
Affiliation:
Department of Engineering Science, University of Oxford, Parks Road, OX1 3PJ, UK
B. W. Turney
Affiliation:
Nufffield Department of Surgical Sciences, University of Oxford, OX3 9DU, UK
P. E. Farrell
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG, UK
S. J. Tavener
Affiliation:
Department of Mathematics, Colorado State University, Oval Drive, CO80523, USA
D. E. Moulton
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG, UK
S. L. Waters
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Ureteroscopy is a minimally invasive surgical procedure for the removal of kidney stones. A ureteroscope, containing a hollow, cylindrical working channel, is inserted into the patient's kidney. The renal space proximal to the scope tip is irrigated, to clear stone particles and debris, with a saline solution that flows in through the working channel. We consider the fluid dynamics of irrigation fluid within the renal pelvis, resulting from the emerging jet through the working channel and return flow through an access sheath. Representing the renal pelvis as a two-dimensional rectangular cavity, we investigate the effects of flow rate and cavity size on flow structure and subsequent clearance time of debris. Fluid flow is modelled with the steady incompressible Navier–Stokes equations, with an imposed Poiseuille profile at the inlet boundary to model the jet of saline, and zero-stress conditions on the outlets. The resulting flow patterns in the cavity contain multiple vortical structures. We demonstrate the existence of multiple solutions dependent on the Reynolds number of the flow and the aspect ratio of the cavity using complementary numerical simulations and particle image velocimetry experiments. The clearance of an initial debris cloud is simulated via solutions to an advection–diffusion equation and we characterise the effects of the initial position of the debris cloud within the vortical flow and the Péclet number on clearance time. With only weak diffusion, debris that initiates within closed streamlines can become trapped. We discuss a flow manipulation strategy to extract debris from vortices and decrease washout time.

JFM classification

Biological Fluid Dynamics: Biomedical flows Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 902 , 10 November 2020 , A16
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alleborn, N., Nandakumar, K., Raszillier, H. & Durst, F. 1997 Further contributions on the two-dimensional flow in a sudden expansion. J. Fluid Mech. 330, 169–188.CrossRefGoogle Scholar
Amestoy, P. R., Duff, I. S., L'Excellent, J. Y. & Koster, J. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Applics. 23 (1), 15–41.CrossRefGoogle Scholar
Amestoy, P. R., Guermouche, A., L'Excellent, J. Y. & Pralet, S. 2006 Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32 (2), 136–156.CrossRefGoogle Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W. D, Kaushik, D., et al. 2018 PETSc users manual. Tech. Rep. ANL-95/11 – Revision 3.9. Argonne National Laboratory.CrossRefGoogle Scholar
Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. 1997 Efficient management of parallelism in object oriented numerical software libraries. In Proceedings of Modern Software Tools in Scientific Computing (ed. Arge, E., Bruaset, A. M. & Langtangen, H. P.), pp. 163–202. Birkhäuser.CrossRefGoogle Scholar
Battaglia, F., Kulkarni, A. K., Feng, J. & Merkle, C. L. 1998 Simulations of planar flapping jets in confined channels. AIAA J. 36 (8), 1425–1431.CrossRefGoogle Scholar
Battaglia, F., Tavener, S. J., Kulkarni, A. K. & Merkle, C. L. 1997 Bifurcation of low Reynolds number flows in symmetric channels. AIAA J. 35 (1), 99–105.CrossRefGoogle Scholar
Dalcin, L. D., Paz, R. R., Kler, P. A. & Cosimo, A. 2011 Parallel distributed computing using Python. Adv. Water Resour. 34 (9), 1124–1139.CrossRefGoogle Scholar
Deuflhard, P. 2011 Newton Methods for Nonlinear Problems. Springer.CrossRefGoogle Scholar
Drikakis, D. 1997 Bifurcation phenomena in incompressible sudden expansion flows. Phys. Fluids 9 (1), 76–87.CrossRefGoogle Scholar
Farrell, P. E., Birkisson, Á. & Funke, S. W. 2015 Deflation techniques for finding distinct solutions of nonlinear partial differential equations. SIAM J. Sci. Comput. 37 (4), A2026–A2045.CrossRefGoogle Scholar
Farrell, P. E., Mitchell, L., Scott, L. R. & Wechsung, F. 2020 A Reynolds-robust preconditioner for the Reynolds-robust Scott–Vogelius discretization of the stationary incompressible Navier–Stokes equations. J. Comput. Maths (submitted) arXiv:2004.09398.Google Scholar
Farrell, P. E., Mitchell, L. & Wechsung, F. 2019 An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier–Stokes equations at high Reynolds number. SIAM J. Sci. Comput. 41 (5), 1--26.CrossRefGoogle Scholar
Fearn, R. M., Mullin, T. & Cliffe, A. K. 1990 Nonlinear flow phenomena in a symmetric sudden expansion. J. Fluid Mech. 211, 595–608.CrossRefGoogle Scholar
Franca, L. P., Frey, S. L. & Hughes, T. J. R. 1992 Stabilized finite element methods: I. Application to the advective–diffusive model. Comput. Meth. Appl. Mech. Engng 95 (2), 253–276.CrossRefGoogle Scholar
Guevel, Y., Allain, T., Girault, G. & Cadou, J. M. 2018 Numerical bifurcation analysis for 3-dimensional sudden expansion fluid dynamic problem. Intl J. Numer. Meth. Fluids 87 (1), 1–26.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Kluwer Academic Publishers.Google Scholar
Hawa, T. & Rusak, Z. 2000 Viscous flow in a slightly asymmetric channel with a sudden expansion. Phys. Fluids 12 (9), 2257–2267.CrossRefGoogle Scholar
Hendrickson, B. & Leland, R. 1995 A multilevel algorithm for partitioning graphs. In Supercomputing ’95: Proceedings of the 1995 ACM/IEEE Conference on Supercomputing (CDROM), p. 28. Association for Computing Machinery.CrossRefGoogle Scholar
Jelić, N., Kolšek, T. & Duhovnik, J. 2007 Numerical investigation of a confined jet flow in a rectangular cavity. Intl J. Multiphys. 1 (2), 245–258.CrossRefGoogle Scholar
John, V., Linke, A., Merdon, C., Neilan, M. & Rebholz, L. G. 2017 On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59 (3), 492–544.CrossRefGoogle Scholar
Kirby, R. C. & Mitchell, L. 2018 Solver composition across the PDE/linear algebra barrier. SIAM J. Sci. Comput. 40 (1), C76–C98.CrossRefGoogle Scholar
Kolšek, T., Jelić, N. & Duhovnik, J. 2007 Numerical study of flow asymmetry and self-sustained jet oscillations in geometrically symmetric cavities. Appl. Maths 31 (10), 2355–2373.Google Scholar
Kum, F., Mahmalji, W., Hale, J., Thomas, K., Bultitude, M. & Glass, J. 2016 Do stones still kill? An analysis of death from stone disease 1999–2013 in England and Wales. BJU Intl 118 (1), 140–144.CrossRefGoogle ScholarPubMed
Lamb, H. 1916 Hydrodynamics. Cambridge University Press.Google Scholar
Mitchell, L. & Müller, E. H. 2016 High level implementation of geometric multigrid solvers for finite element problems: applications in atmospheric modelling. J. Comput. Phys. 327, 1–18.CrossRefGoogle Scholar
Mizushima, J., Okamoto, H. & Yamaguchi, H. 1996 Stability of flow in a channel with a suddenly expanded part. Phys. Fluids 8 (11), 2933–2942.CrossRefGoogle Scholar
Mizushima, J. & Takahashi, H. 1999 Transitions of flow in a distributor cavity with one inlet and two outlets. J. Phys. Soc. Japan 68 (11), 3514–3519.CrossRefGoogle Scholar
Moore, R. G. & Bishoff, J. T. 2005 Minimally Invasive Urological Surgery. Taylor & Francis.CrossRefGoogle Scholar
Mullin, T., Shipton, S. & Tavener, S. J. 2002 Flow in a symmetric channel with an expanded section. Fluid Dyn. Res. 33, 433–452.CrossRefGoogle Scholar
Neofytou, P. & Drikakis, D. 2003 Non-Newtonian flow instability in a channel with a sudden expansion. J. Non-Newtonian Fluid Mech. 111, 127–150.CrossRefGoogle Scholar
Raffel, M., Willert, C., Wereley, S. & Kompenhans, J. 1998 Particle Image Velocimetry: A Practical Guide. Springer.CrossRefGoogle Scholar
Rathgeber, F., Ham, D. A., Mitchell, L., Lange, M., Luporini, F., McRae, A. T. T., Bercea, G.-T., Markall, G. R. & Kelly, P. H. J. 2016 Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw. 43 (3), 24:1–24:27.Google Scholar
Rhines, B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.CrossRefGoogle Scholar
Saeidi, S. M. & Khodadadi, J. M. 2006 Forced convection in a square cavity with inlet and outlet ports. Intl J. Heat Mass Transfer 49 (11–12), 1896–1906.CrossRefGoogle Scholar
Scales, C. D., Smith, A. C., Hanley, J. M. & Saigal, C. S. 2012 Prevalence of kidney stones in the United States. Eur. Urol. 62 (1), 160–165.CrossRefGoogle ScholarPubMed
Scott, L. R. & Vogelius, M. 1985 Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. ESAIM: Math. Model. Numer. Anal. 19 (1), 111–143.CrossRefGoogle Scholar
Silvester, D., Elman, H. & Wathen, A. 2014 Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press.Google Scholar
Smith, A. D., Preminger, G. M., Kavoussi, L. R., Badlani, G. H. & Rastinehad, A. R. 2018 Smith's Textbook of Endourology, 4th edn.John Wiley & Sons.Google Scholar
Sobey, I. J. & Drazin, P. G. 1986 Bifurcations of two-dimensional channel flows. J. Fluid Mech. 171 (6), 263–287.CrossRefGoogle Scholar
Stamatelou, K. K., Francis, M. E., Jones, C. A., Nyberg, L. M. & Curhan, G. C. 2003 Time trends in reported prevalence of kidney stones in the United States: 1976–1994. Kidney Intl 63 (5), 1817–1823.CrossRefGoogle ScholarPubMed
Taylor, G. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186–203.Google Scholar
Thielicke, W. & Stamhuis, E. J. 2014 PIVlab – towards user-friendly, affordable, and accurate digital particle image velocimetry in MATLAB. J. Open Res. Softw. 2, 1–10.CrossRefGoogle Scholar
Walter, J. A. & Chen, C. J. 1992 Visualization and analysis of flow structures in an open cavity. Mod. Phys. Lett. B 114, 819–826.Google Scholar

Williams et al. supplementary movie

A movie of the experimental technique to `switch’ the direction of the jet, to complement Figure 3 in our manuscript. The movie is at 100 fps.

Download Williams et al. supplementary movie(Video)
Video 305.9 MB