Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T07:03:50.143Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of a uniform through-surface flow on a cylinder and sphere

Published online by Cambridge University Press:  23 March 2016

C. A. Klettner ,
I. Eames ,
S. Semsarzadeh  and
A. Nicolle
C. A. Klettner*
Affiliation:
Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK
I. Eames
Affiliation:
Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK
S. Semsarzadeh
Affiliation:
Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK
A. Nicolle
Affiliation:
Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK
*
†Email address for correspondence: ucemkle@ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of a uniform through-surface flow (velocity $U_{b}$) on a rigid and stationary cylinder and sphere (radius $a$) fixed in a free stream (velocity $U_{\infty }$) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity ${\it\Lambda}\,(=U_{b}/U_{\infty })$ and Reynolds number $Re\,(=2aU_{\infty }/{\it\nu}$, where ${\it\nu}$ is the kinematic viscosity). High resolution numerical calculations are compared against theoretical predictions over the range $-3\leqslant {\it\Lambda}\leqslant 3$ and $Re=1,10,100$ for planar flow past a cylinder and axisymmetric flow past a sphere. For $-{\it\Lambda}\gg 1$, the flow is viscously dominated in a thin boundary layer of thickness ${\it\nu}/|U_{b}|$ adjacent to the rigid surface which develops in a time ${\it\nu}/U_{b}^{2}$; the surface vorticity scales as $Re|{\it\Lambda}|U_{\infty }/a$ for a cylinder and sphere. A boundary layer analysis is developed to analyse the unsteady viscous forces. Numerical results show that the surface pressure and vorticity distribution within the boundary layer agrees with a steady state analysis. The flow downstream of the body is irrotational so the wake volume flux, $Q_{w}$, is zero and the drag force is $F_{D}=-{\it\rho}U_{\infty }Q_{b}$, where ${\it\rho}$ is the density of the fluid and $Q_{b}$ is the normal flux through the body surface. The drag coefficient is therefore $-2{\rm\pi}{\it\Lambda}$ or $-8{\it\Lambda}$ for a cylinder or sphere, respectively. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of stagnation pressure and rate of work by viscous stresses due to sucking. At large $Re|{\it\Lambda}|$, the drag force is largely determined by viscous dissipation for a cylinder, with a weak contribution by the normal viscous stresses, while for a sphere, only $3/4$ of the drag force is determined by viscous dissipation with the remaining $1/4$ due to the flux of stagnation pressure through the sphere surface. When ${\it\Lambda}\gg 1$, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. The vorticity in the boundary layer is weakly dependent on viscous effects and scales as $U_{\infty }/a{\it\Lambda}$ or $U_{\infty }/a{\it\Lambda}^{3/2}$ for a cylinder and sphere, respectively. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity. It is found that the drag coefficient $C_{D}$ for a cylinder (corrected for flow boundedness) is weakly dependent on ${\it\Lambda}$ while for a sphere, $C_{D}$ decreases with ${\it\Lambda}$.

JFM classification

Boundary Layers: Boundary layer structure Boundary Layers: Boundary Layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 793 , 25 April 2016 , pp. 798 - 839
Check for updates
Copyright
© 2016 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

Abramowitz, M. & Stegun, I. A.(Eds) 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Bagchi, P. 2007 Flow past a sphere with surface blowing and suction. Trans. ASME J. Fluids Engng 129, 1547–1558.CrossRefGoogle Scholar
Bar-Lev, M. & Yang, H. T. 1975 Initial flow field over an impulsively started circular cylinder. J. Fluid Mech. 72, 625–647.CrossRefGoogle Scholar
Bashkatov, M. V. & Shabanov, S. I. 1975 Flow around and heat and mass transfer of a sphere with blowing at medium Reynolds numbers. J. Appl. Mech. Tech. Phys. 16, 604–611.CrossRefGoogle Scholar
Basset, A. B. 1888 Treatise on Hydrodynamics, 1st edn. vol. 2. Deighton, Bell and Co.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics, 1st edn. Cambridge University Press.Google Scholar
Berthelsen, P. A. & Faltinsen, O. M. 2008 A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries. J. Comput. Phys. 227, 4354–4397.CrossRefGoogle Scholar
Betz, A. 1925 A method for the direct determination of wing-section drag. Z. Flugtechn. Motorluftschiffahrt. 16, 42–44 (in German).Google Scholar
Bogard, D. G. & Thole, K. A. 2006 Gas turbine film cooling. J. Propul. Power 22, 249–270.CrossRefGoogle Scholar
Castro, I. P. 1971 Wake characteristics of two-dimensional perforated plates normal to an air-stream. J. Fluid Mech. 46, 599–609.CrossRefGoogle Scholar
Cliffe, K. A. & Lever, D. A. 1985 Isothermal flow past a blowing sphere. Intl J. Numer. Meth. Fluids 5, 709–725.CrossRefGoogle Scholar
Crowe, C. T.1976 Conservation equations for vapour-droplet flows including boundary-droplet effects. PhD thesis, Lawrence Livermore Laboratory, University of California.CrossRefGoogle Scholar
Dennis, S. C. R. & Shimshoni, M. 1965 The steady flow of a viscous fluid past a circular cylinder. Ministry of Aviation – Aeronautical Research Council 797, 1–48.Google Scholar
Dukowicz, J. K. 1982 An exact solution for the drag of a sphere in low Reynolds number flow with strong uniform suction or blowing. Phys. Fluids 25, 1117–1118.CrossRefGoogle Scholar
Eames, I. 2010 Momentum conservation and condensing vapor bubbles. Trans. ASME J. Heat Transfer 132, 091501.CrossRefGoogle Scholar
Eames, I. & Duursma, G. 1997 Displacement of horizontal layers by bubbles injected into fluidised beds. Chem. Engng Sci. 52, 2697–2705.CrossRefGoogle Scholar
Fransson, J. H. M., Koniecznyb, P. & Alfredsson, P. H. 2004 Flow around a porous cylinder subject to continuous suction or blowing. J. Fluids Struct. 19, 1031–1048.CrossRefGoogle Scholar
Geuzaine, C. & Remacle, J.-F. 2009 Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Intl J. Numer. Meth. Engng 79, 1309–1331.CrossRefGoogle Scholar
Goward, G. W. 1998 Progress in coatings for gas turbine airfoils. Surf. Coat. Technol. 108–109, 73–79.CrossRefGoogle Scholar
Hunt, J. C. R. & Eames, I. 2002 The disappearance of laminar and turbulent wakes in complex flows. J. Fluid Mech. 457, 111–132.CrossRefGoogle Scholar
Hurley, D. G. & Thwaites, M. A.1951 An experimental investigation of the boundary layer on a porous circular cylinder. Reports and Memoranda No. 2829. British Aeronautical Research Council.Google Scholar
Iacovides, H. & Launder, B. E. 1995 Computational fluid dynamics applied to internal gas-turbine blade cooling: a review. Intl J. Heat Mass Flow. 16, 454–470.CrossRefGoogle Scholar
Klettner, C. A. & Eames, I. 2012 The laminar free surface boundary layer of a solitary wave. J. Fluid Mech. 696, 423–433.CrossRefGoogle Scholar
Klettner, C. A. & Eames, I. 2015 Low-Reynolds-number flow past a cylinder with uniform blowing or sucking. J. Fluid Mech. 780, 1–8.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1966 Fluid Mechanics, 3rd edn. Pergamon.Google Scholar
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Handbook of Experimental Fluid Dynamics (ed. Tropea, C., Yarin, A. & Foss, J. F.), chap. 19. Springer.Google Scholar
Le Clair, S., Hamielec, A. E. & Pruppacher, H. R. 1970 A numerical study of the drag on a sphere at low and intermediate Reynolds numbers. J. Atmos. Sci. 27, 308–315.2.0.CO;2>CrossRefGoogle Scholar
Lee, S. 2000 A numerical study of the unsteady wake behind a sphere in a uniform flow at moderate Reynolds numbers. Comput. Fluids 29, 639–667.CrossRefGoogle Scholar
Levich, V. G. 1949 The motion of bubbles at high Reynolds numbers. Zh. Eksp. Teor. Fiz. 19, 18; see also Physiochemical Hydrodynamics, English translation by Scripta Technica, Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 436.Google Scholar
Ling, G. P. & Fang, J. W. 2002 Numerical study on the flow around a circular cylinder with surface suction or blowing using vorticity–velocity method. Z. Angew. Math. Mech. Appl. Math. Mech. 23, 1089–1096.Google Scholar
Malavard, L. 1984 Un nouveau propulseur éolien de navire, La vie des sciences. C. R. Ser. Générale 1, 57–72.Google Scholar
Mathelin, L., Bataille, F. & Lallemand, A. 2002 The effect of uniform blowing on flow past a circular cylinder. Trans. ASME 124, 452–464.Google Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.CrossRefGoogle Scholar
Nicolle, A. & Eames, I. 2011 Numerical study of flow through and around a circular array of cylinders. J. Fluid Mech. 679, 1–31.CrossRefGoogle Scholar
Nishino, T. & Shariff, K. 2012 Effect of jet nozzle lip momentum loss on circulation control airfoil performance. AIAA J. 50, 551–558.CrossRefGoogle Scholar
Pankhurst, R. C. & Thwaites, B.1953 Experiments on the flow past a porous cylinder fitted with a Thwaites Flap. Aeronautical Research Council Reports and Memoranda. Tech. Rep. R. & M. 2787. 29.Google Scholar
Rayleigh, Lord 1911 On the motion of solid bodies through viscous fluid. Phil. Mag. 21, 697–711.CrossRefGoogle Scholar
Reynolds, O. 1886 On the theory of lubrication and its application to Mr Beauchamp Towers experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. R. Soc. Lond. A 177, 157–234.Google Scholar
Stone, H. A. 1992 An interpretation of the translation of drops and bubbles at high Reynolds numbers in terms of the vorticity field. Phys. Fluids 5, 2567–2569.CrossRefGoogle Scholar
Zienkiewicz, O. C., Taylor, R. L. & Nithiarasu, P. 2005 The Finite Element Method for Fluid Dynamics, 6th edn. Butterworth-Heinemann.Google Scholar
Zhao, F., Harrington, D. C. & Lai, M.-C. 2002 Automotive Gasoline Direct-Injection Engines. Society of Automotive Engineers.CrossRefGoogle Scholar