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A ‘reciprocal’ theorem for the prediction of loads on a body moving in an inhomogeneous flow at arbitrary Reynolds number

Published online by Cambridge University Press:  20 October 2011

Jacques Magnaudet
Jacques Magnaudet*
Affiliation:
Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INPT/UPS 5502, Allée Camille Soula, 31400 Toulouse, France
*
†Email address for correspondence: magnau@imft.fr
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Abstract

Several forms of a theorem providing general expressions for the force and torque acting on a rigid body of arbitrary shape moving in an inhomogeneous incompressible flow at arbitrary Reynolds number are derived. Inhomogeneity arises because of the presence of a wall that partially or entirely bounds the fluid domain and/or a non-uniform carrying flow. This theorem, which stems directly from Navier–Stokes equations and parallels the well-known Lorentz reciprocal theorem extensively employed in low-Reynolds-number hydrodynamics, makes use of auxiliary solenoidal irrotational velocity fields and extends results previously derived by Quartapelle & Napolitano (AIAA J., vol. 21, 1983, pp. 911–913) and Howe (Q. J. Mech. Appl. Maths, vol. 48, 1995, pp. 401–426) in the case of an unbounded flow domain and a fluid at rest at infinity. As the orientation of the auxiliary velocity may be chosen arbitrarily, any component of the force and torque can be evaluated, irrespective of its orientation with respect to the relative velocity between the body and fluid. Three main forms of the theorem are successively derived. The first of these, given in (2.19), is suitable for a body moving in a fluid at rest in the presence of a wall. The most general form (3.6) extends it to the general situation of a body moving in an arbitrary non-uniform flow. Specific attention is then paid to the case of an underlying time-dependent linear flow. Specialized forms of the theorem are provided in this situation for simplified body shapes and flow conditions, in (3.14) and (3.15), making explicit the various couplings between the body’s translation and rotation and the strain rate and vorticity of the carrying flow. The physical meaning of the various contributions to the force and torque and the way in which the present predictions reduce to those provided by available approaches, especially in the inviscid limit, are discussed. Some applications to high-Reynolds-number bubble dynamics, which provide several apparently new predictions, are also presented.

JFM classification

Vortex Flows: Vortex dynamics

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Type
Papers
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Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 564 - 604
Copyright
Copyright © Cambridge University Press 2011

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References

1. Auton, T. R. 1987 The lift force on a spherical body in rotational flow. J. Fluid Mech. 183, 199–218.CrossRefGoogle Scholar
2. Auton, T. R., Hunt, J. C. R. & Prud’homme, M. 1988 The force exerted on a body moving in an inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.CrossRefGoogle Scholar
3. Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.CrossRefGoogle Scholar
4. Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
5. Biesheuvel, A. & Hagmeijer, R. 2006 On the force on a body moving in a fluid. Fluid Dyn. Res. 38, 716–742.CrossRefGoogle Scholar
6. Brenner, H. 1963 The Stokes resistance of an arbitrary particle. Chem. Engng Sci. 18, 1–25.CrossRefGoogle Scholar
7. Burgers, J. M. 1920 On the resistance of fluids and vortex motion. Proc. K. Akad. Wet. Amsterdam 23, 774–782.Google Scholar
8. Chan, C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92, 131–170.CrossRefGoogle Scholar
9. Chang, C. C. 1992 Potential flow and forces for incompressible viscous flow. Proc. R. Soc. Lond. A 437, 517–525.Google Scholar
10. Chang, C. C. & Chern, R. L. 1991 Vortex shedding from an impulsively started rotating and translating circular cylinder. J. Fluid Mech. 233, 265–298.CrossRefGoogle Scholar
11. Chang, C. C., Yang, S. H. & Chu, C. C. 2008 A many-body force decomposition with applications to flow about bluff bodies. J. Fluid Mech. 600, 95–104.CrossRefGoogle Scholar
12. Cox, R. G. & Brenner, H. 1968 Lateral migration of solid particles in Poiseuille flow. I. Theory. Chem. Engng Sci. 23, 147–173.CrossRefGoogle Scholar
13. Dériat, E. 2002 Inviscid shear flow around a cylinder close to a wall. C. R. Mec. 330, 35–38.CrossRefGoogle Scholar
14. Eames, I. 2010 Momentum conservation and condensing vapour bubbles. Trans. ASME: J. Heat Transfer 132, 091501.CrossRefGoogle Scholar
15. Eames, I. & Hunt, J. C. R. 1997 Inviscid flow around bodies moving in weak density gradients without buoyancy effects. J. Fluid Mech. 353, 331–355.CrossRefGoogle Scholar
16. Galper, A. & Miloh, T. 1994 Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field. Proc. R. Soc. Lond. A 446, 169–193.Google Scholar
17. Galper, A. & Miloh, T. 1995 Dynamic equations of motion for a rigid or deformable body in an arbitrary non-uniform potential flow field. J. Fluid Mech. 295, 91–120.CrossRefGoogle Scholar
18. Grotta Ragazzo, C. & Tabak, E. 2007 On the force and torque on systems of rigid bodies: a remark on an integral formula due to Howe. Phys. Fluids 19, 057108.CrossRefGoogle Scholar
19. Harper, J. F. & Moore, D. W. 1968 The motion of a spherical liquid drop at high Reynolds number. J. Fluid Mech. 32, 367–391.CrossRefGoogle Scholar
20. Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365–400.CrossRefGoogle Scholar
21. Howe, M. S. 1989 On unsteady surface forces, and sound produced by the normal chopping of a rectilinear vortex. J. Fluid Mech. 206, 131–153.CrossRefGoogle Scholar
22. Howe, M. S. 1991 On the estimation of sound produced by complex fluid–structure interactions, with application to a vortex interacting with a shrouded rotor. Proc. R. Soc. Lond. A 433, 573–598.Google Scholar
23. Howe, M. S. 1995 On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high and low Reynolds numbers. Q. J. Mech. Appl. Maths 48, 401–426.CrossRefGoogle Scholar
24. Howe, M. S. 2007 Hydrodynamics and Sound. Cambridge University Press.Google Scholar
25. Howe, M. S., Lauchle, G. C. & Wang, J. 2001 Aerodynamic lift and drag fluctuations of a sphere. J. Fluid Mech. 206, 131–153.CrossRefGoogle Scholar
26. Kambe, T. 1987 A new expression of force on a body in viscous vortex flow and asymptotic pressure field. Fluid Dyn. Res. 2, 15–23.CrossRefGoogle Scholar
27. Kang, I. S. & Leal, L. G. 1988 The drag coefficient for a spherical bubble in a uniform streaming flow. Phys. Fluids 31, 233–237.CrossRefGoogle Scholar
28. Kochin, N. E., Kibel, I. A. & Roze, N. V. 1964 Theoretical Hydromechanics. Wiley.Google Scholar
29. Kok, J. B. W. 1992 Dynamics of a pair of gas bubbles moving through liquid. Part I. Theory. Eur. J. Mech. (B/Fluids) 12, 515–540.Google Scholar
30. Korotkin, A. I. 2009 Added-Masses of Ship Structures. Springer.CrossRefGoogle Scholar
31. Lamb, H. 1945 Hydrodynamics. Cambridge University Press.Google Scholar
32. Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81–126.CrossRefGoogle Scholar
33. Legendre, D., Magnaudet, J. & Mougin, G. 2003 Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid. J. Fluid Mech. 497, 133–166.CrossRefGoogle Scholar
34. Levich, V. G. 1949 Bubble motion at high Reynolds numbers. Zh. Eksp. Teor. Fiz. 19, 18–24.Google Scholar
35. Levich, V. G. 1962 Physico-Chemical Hydrodynamics. Prentice Hall.Google Scholar
36. Lighthill, M. J. 1986a An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press.Google Scholar
37. Lighthill, M. J. 1986b Fundamentals concerning wave loading on offshore structures. J. Fluid Mech. 173, 667–681.CrossRefGoogle Scholar
38. Lorentz, H. A. 1907 Ein allgemeiner satz, die bewegung einer reibenden flüssigkeit betreffend, nebst einigen anwendungen desselben (A general theorem concerning the motion of a viscous fluid and a few applications from it). Abh. Theor. Phys. 1, 23–42.Google Scholar
39. Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.CrossRefGoogle Scholar
40. Magnaudet, J. & Mougin, G. 2007 Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311–337.CrossRefGoogle Scholar
41. Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a vertical wall. J. Fluid. Mech. 476, 115–157.CrossRefGoogle Scholar
42. Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Macmillan.CrossRefGoogle Scholar
43. Miloh, T. 2003 The motion of solids in inviscid uniform vortical fields. J. Fluid Mech. 479, 279–305.CrossRefGoogle Scholar
44. Miloh, T. 2004 Fluid–body interaction in the presence of uniform vorticity and density gradient. Phys. Fluids 16, 22–28.CrossRefGoogle Scholar
45. Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.CrossRefGoogle Scholar
46. Moore, D. W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.CrossRefGoogle Scholar
47. Mougin, G. & Magnaudet, J. 2002 The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Intl J. Multiphase Flow 28, 1837–1851.CrossRefGoogle Scholar
48. Palierne, J. F. 1999 On the motion of rigid bodies in incompressible inviscid fluids of inhomogeneous density. J. Fluid Mech. 393, 89–98.CrossRefGoogle Scholar
49. Noca, F., Shiels, D. & Jeon, D. 1996 Measuring instantaneous fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 11, 345–350.CrossRefGoogle Scholar
50. Noca, F., Shiels, D. & Jeon, D. 1999 A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 13, 551–578.CrossRefGoogle Scholar
51. Pan, L. S. & Chew, Y. T. 2002 A general formula for calculating forces on a 2-D arbitrary body in incompressible flow. J. Fluids Struct. 16, 71–82.CrossRefGoogle Scholar
52. Pozrikidis, C. 1997 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
53. Protas, B., Styczek, A. & Nowakowski, A. 2000 An effective approach to computation of forces in viscous incompressible flows. J. Comput. Phys. 159, 231–245.CrossRefGoogle Scholar
54. Quartapelle, L. & Napolitano, M. 1983 Force and moment in incompressible flows. AIAA J. 21, 911–913.CrossRefGoogle Scholar
55. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
56. Sherwood, J. D. 2001 Steady rise of a small spherical gas bubble along the axis of a cylindrical pipe at high Reynolds number. Eur. J. Mech. (B/Fluids) 20, 399–414.CrossRefGoogle Scholar
57. Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distortions. Phys. Rev. Lett. 77, 4102–4104.CrossRefGoogle ScholarPubMed
58. Taylor, G. I. 1928 The forces on a body placed in a curved or converging stream of fluid. Proc. R. Soc. Lond. A 120, 260–283.Google Scholar
59. Tollmien, W. 1938 Über kräfte und momente in schwach gekrümmten oder konvergenten strömungen. Zng.-Arch. 9, 308–326.Google Scholar
60. Vasseur, P. & Cox, R. G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78, 385–413.CrossRefGoogle Scholar
61. van Wijngaarden, L. 2005 Bubble velocities induced by trailing vortices behind neighbours. J. Fluid Mech. 541, 203–229.CrossRefGoogle Scholar
62. Wu, J. C. 1981 Theory for aerodynamic force and moment in viscous flows. AIAA J. 19, 432–441.CrossRefGoogle Scholar