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When a small thin two-dimensional body enters a viscous wall layer

Part of: Incompressible viscous fluids Coupling of solid mechanics with other effects

Published online by Cambridge University Press:  16 December 2019

R. A. PALMER Open the ORCID record for R. A. PALMER [Opens in a new window]  and
F. T. SMITH
R. A. PALMER
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK, emails: ryan.palmer.14@ucl.ac.uk; f.smith@ucl.ac.uk
F. T. SMITH
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK, emails: ryan.palmer.14@ucl.ac.uk; f.smith@ucl.ac.uk
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Abstract

If a body enters a viscous-inviscid fluid layer near a wall, then significant effects can be felt from the presence of incident vorticity, viscous forces and nonlinear forces. The focus here is on the response in the outer edge of such a wall layer. Nonlinear two-dimensional unsteady behaviour is examined through modelling, computation and analysis applied for a thin body travelling streamwise upstream or downstream or staying still relative to the wall. The wall layer with its balance between inviscid and viscous effects interacts freely with the body movement, causing relatively high magnitudes of pressure on top of the body and nonlinear responses in the gap between the body and the wall. The study finds explicit solutions for the motion of the body, separation of the flow arising near the wall and possible instabilities occurring over the length scale of any short body.

Keywords

Viscous–inviscid interactionfluid–solid interactionsboundary-layer theory

MSC classification

Primary: 76D09: Viscous-inviscid interaction 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Secondary: 76D10: Boundary-layer theory, separation and reattachment, higher-order effects
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 6 , December 2020 , pp. 1002 - 1028
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Copyright
© The Authors 2019. Published by Cambridge University Press

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References

Anderson, D., Tannehill, J. C. & Pletcher, R. H. (2016) Computational Fluid Mechanics and Heat Transfer, Florida: CRC Press.CrossRefGoogle Scholar
Bhattacharyya, S., Mahapatra, S. & Smith, F. T. (2004) Influence of surface roughness on shear flow. J. Appl. Mech. 71(4), 459–464.CrossRefGoogle Scholar
Dehghan, M. & Basirat Tabrizi, H. (2014) Effects of coupling on turbulent gas-particle boundary layer flows at borderline volume fractions using kinetic theory. J. Heat Mass Transf. Res. 1(1), 1–8.Google Scholar
Einav, S. & Lee, S. (1973) Particles migration in laminar boundary layer flow. Int. J. Multiphase Flow 1(1), 73–88.CrossRefGoogle Scholar
Frank, M., Anderson, D., Weeks, E. R. & Morris, J. F. (2003) Particle migration in pressure-driven flow of a Brownian suspension. J. Fluid Mech. 493, 363–378.CrossRefGoogle Scholar
Gavze, E. & Shapiro, M. (1997) Particles in a shear flow near a solid wall: effect of nonsphericity on forces and velocities. Int. J. Multiphase Flow 23(1), 155–182.CrossRefGoogle Scholar
Gent, R., Dart, N. & Cansdale, J. (2000) Aircraft icing. Philos. Trans. R. Soc. London. Ser. A Math. Phys. Eng. Sci. 358(1776), 2873–2911.CrossRefGoogle Scholar
Hall, G. (1964) On the mechanics of transition produced by particles passing through an initially laminar boundary layer and the estimated effect on the LFC performance of the X-21 aircraft, Technical Report, NASA.Google Scholar
Kishore, N. & Gu, S. (2010) Wall effects on flow and drag phenomena of spheroid particles at moderate Reynolds numbers. Ind. Eng. Chem. Res. 49(19), 9486–9495.CrossRefGoogle Scholar
Lagrée, P.-Y. (1993) Mixed convection at small Richardson number on triple deck scales. Comptes Rendus de l’Academie des Sciences Serie II 318, 1167–1173.Google Scholar
Lagrée, P.-Y. (2007) Interactive boundary layer in a Hele Shaw cell. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik: Appl. Math. Mech. 87(7), 486–498.CrossRefGoogle Scholar
Loisel, V., Abbas, M., Masbernat, O. & Climent, E. (2013) The effect of neutrally buoyant finite-size particles on channel flows in the laminar-turbulent transition regime. Phys. Fluids 25(12), 123304.Google Scholar
Loth, E. & Dorgan, A. J. (2009) An equation of motion for particles of finite Reynolds number and size. Environ. Fluid Mech. 9(2), 187–206.CrossRefGoogle Scholar
Norde, E. (2017) Eulerian Method for Ice Crystal Icing in Turbofan Engines, University of Twente.CrossRefGoogle Scholar
Petrie, H., Morris, P., Bajwa, A. & Vincent, D. (1993) Transition induced by fixed and freely convecting spherical particles in laminar boundary layers, Technical report, Pennsylvania State University Park Applied Research Lab.Google Scholar
Poesio, P., Ooms, G., Ten Cate, A. & Hunt, J. C. (2006) Interaction and collisions between particles in a linear shear flow near a wall at low Reynolds number. J. Fluid Mech. 555, 113–130.CrossRefGoogle Scholar
Portela, L. M., Cota, P. & Oliemans, R. V. (2002) Numerical study of the near-wall behaviour of particles in turbulent pipe flows. Powder Technol. 125(2–3), 149–157.Google Scholar
Pruessner, L. & Smith, F. T. (2015) Enhanced effects from tiny flexible in-wall blips and shear flow. J. Fluid Mech. 772, 16–41.CrossRefGoogle Scholar
Purvis, R. & Smith, F. T. (2016) Improving aircraft safety in icing conditions. In: Aston, P. J., Mulholland, A. J. and Tant, K. (editors), UK Success Stories in Industrial Mathematics, Springer International Publishing, New York, pp. 145–151.CrossRefGoogle Scholar
Rothmayer, A. P. & Smith, F. T. (1998) High Reynolds number asymptotic theories. In: Johnson, R. W. (editor), The Handbook of Fluid Dynamics, Florida: CRC Press, chapter 23–25.Google Scholar
Schmidt, C. & Young, T. (2009) Impact of freely suspended particles on laminar boundary layers. In: 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, p. 1621.CrossRefGoogle Scholar
Schmidt, C., Young, T. M. & Benard, E. P. (2010) The effect of a particle travelling through a laminar boundary layer on transition. In: Seventh IUTAM Symposium on Laminar-Turbulent Transition, Springer, pp. 561–564.CrossRefGoogle Scholar
Smith, F. T. (1976) Flow through constricted or dilated pipes and channels: part 2. Q. J. Mech. Appl. Math. 29(3), 365–376.Google Scholar
Smith, F. T. (2017) Free motion of a body in a boundary layer or channel flow. J. Fluid Mech. 813, 279–300.CrossRefGoogle Scholar
Smith, F. T. & Daniels, P. G. (1981) Removal of Goldstein’s singularity at separation, in flow past obstacles in wall layers. J. Fluid Mech. 110, 1–37.CrossRefGoogle Scholar
Smith, F. T. & Ellis, A. S. (2010) On interaction between falling bodies and the surrounding fluid. Mathematika 56(1), 140–168.CrossRefGoogle Scholar
Smith, F. T. & Johnson, E. R. (2016) Movement of a finite body in channel flow. Proc. R. Soc. A Math. Phys. Eng. Sci. 472(2191), 20160164.Google ScholarPubMed
Smith, F. T. & Jones, M. A. (2003) AVM modelling by multi-branching tube flow: large flow rates and dual solutions. Math. Med. Biol. 20(2), 183–204.CrossRefGoogle ScholarPubMed
Smith, F. T., Ovenden, N. C., Franke, P. T. & Doorly, D. J. (2003) What happens to pressure when a flow enters a side branch? J. Fluid Mech. 479, 231–258.CrossRefGoogle Scholar
Smith, F. T. & Palmer, R. (2019) A freely moving body in a boundary layer: nonlinear separated-flow effects. Appl. Ocean Res. 85, 107–118.CrossRefGoogle Scholar
Smith, F. T. & Servini, P. (2019) Channel flow past a near-wall body. Q. J. Mech. Appl. Math. 72(3), 359–385.CrossRefGoogle Scholar
Smith, F. T. & Timoshin, S. N. (1996) Blade-wake interactions and rotary boundary layers. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 452(1949), 1301–1329.Google Scholar
Smith, F. T. & Wilson, P. L. (2013) Body-rock or lift-off in flow. J. Fluid Mech. 735, 91–119.CrossRefGoogle Scholar
Sobey, I. J. (2000) Introduction to Interactive Boundary Layer Theory, Vol. 3, Oxford Applied Engineering and Mathematics, Oxford University Press, Oxford.Google Scholar
Wang, J. & Levy, E. K. (2006) Particle behavior in the turbulent boundary layer of a dilute gas-particle flow past a flat plate. Exp. Therm. Fluid Sci. 30(5), 473–483.CrossRefGoogle Scholar
Yu, Z., Phan-Thien, N. & Tanner, R. I. (2007) Rotation of a spheroid in a Couette flow at moderate Reynolds numbers. Phys. Rev. E 76(2), 026310.CrossRefGoogle Scholar