Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-07T23:10:58.822Z Has data issue: false hasContentIssue false
  • English
  • Français

Dispersion in a curved tube during oscillatory flow

Published online by Cambridge University Press:  26 April 2006

M. K. Sharp ,
R. D. Kamm ,
A. H. Shapiro ,
E. Kimmel  and
G. E. Karniadakis
M. K. Sharp
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Departments of Civil and Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA.
R. D. Kamm
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
A. H. Shapiro
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
E. Kimmel
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Agriculture Engineering Department, Technion, Haifa 32000, Israel.
G. E. Karniadakis
Affiliation:
Fluid Mechanics Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of curvature on longitudinal dispersion in an axially uniform toroidal tube during oscillatory flow is investigated. The regimes of dispersion and the rate of longitudinal transport are estimated by order-of-magnitude arguments. Experiments are reported for the range, 0.66 < Dn2/α4 < 2.4, 5.4 < α < 26, Sc = 0.68, where Dn is the Dean number, α is the Womersley number and Sc is the Schmidt number. For β2 = α2Sc > 30, curvature causes a sharp increase in the effective diffusivity, relative to that for a straight tube, by a factor of about 6 at Dn2/α4 ≈ 2. The results from two numerical simulation methods are also presented. One, a Monte Carlo simulation (0.01 < Dn < 10, 0.01 < α < 0.32, Sc = 104), predicts the spread of a bolus in quasi-steady flow. The other, a spectral-element method (1 < Dn < 1000, 1 < α < 100, Sc = 0.68), is used to find the dispersion in unsteady oscillatory flow subjected to a constant longitudinal concentration gradient. Two mechanisms are identified by which axial transport is modified by curvature. First, the enhanced lateral transport due to secondary flow decreases axial transport by a factor of up to 5 for low β2 and increases axial transport by an even greater amount for high β2. Second, axial transport is enhanced owing to a form of resonance when the secondary flow circulation time is equal to the cycle period.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 223 , February 1991 , pp. 537 - 563
Copyright
© 1991 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

Berger, S. A., Talbot, L. & Yao, L. S., 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Crank, J.: 1956 The Mathematics of Diffusion, p. 147. Clarendon.
Daskopoulos, P. & Lenhoff, A. J., 1988 Dispersion coefficient for laminar flow in curved tubes. AIChE J. 34, 2052–2058.Google Scholar
Daskopoulos, P. & Lenhoff, A. J., 1989 Flow in curved ducts: bifurcation structure for stationary ducts. J. Fluid Mech. 203, 125–148.Google Scholar
Dean, W. R.: 1928 The streamline motion of fluid in a curved pipe. Phil. Mag. 5, (7), 673–695.Google Scholar
Drazen, J. M., Kamm, R. D. & Slutsky, A. S., 1984 High-frequency ventilation. Physiol. Rev. 64, 505–543.Google Scholar
Eckmann, D. M. & Grotberg, J. B., 1987 Oscillatory flow and mass transport in a curved tube. J. Fluid Mech. 188, 509–527.Google Scholar
Erdogan, M. E. & Chatwin, P. C., 1967 The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube. J. Fluid Mech. 29, 465–484.Google Scholar
Fischer, H. B.: 1972 Mass transport mechanisms in partially stratified estuaries. J. Fluid Mech. 53, 671–687.Google Scholar
Gottlieb, G. O. & Orszag, S. A., 1977 Numerical Analysis of Spectral Methods: Theory and Application. NSF-GBMS Monograph, No. 26. SIAM.
Hino, M., Sawamoto, M. & Takasu, S., 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193–207.Google Scholar
Holley, E. R., Harleman, D. R. F. & Fischer, H. B. 1970 Dispersion in homogeneous estuary flow. J. Hydraul. Engng Dm. ASCE 96, 1691–1709.Google Scholar
Jan, D. L., Shapiro, A. H. & Kamm, R. D., 1989 Some features of oscillatory flow in a model bifurcation. J. Appl. Physiol. 67, 147–159.Google Scholar
Janssen, L. A. M.: 1976 Axial dispersion in laminar flow through coiled tubes. Chem. Engng Sci. 31, 215–218.Google Scholar
Johnson, M. & Kamm, R. D., 1986 Numerical studies of steady flow dispersion at low Dean number in a gently curving tube. J. Fluid Mech. 172, 329–345.Google Scholar
Joshi, C. H., Kamm, R. D., Drazen, J. M. & Slutsky, A. S., 1983 An experimental study of gas exchange in laminar oscillatory flow. J. Fluid Mech. 133, 245–254.Google Scholar
Karniadakis, G. E.: 1989 Spectral element simulations of laminar and turbulent flows in complex geometries. Appl. Numer. Math. 6, 85–105.Google Scholar
Korczak, K. Z. & Patera, A. T., 1986 An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry. J. Comput. Phys. 62, 361–382.Google Scholar
Kurzweg, U. H.: 1985 Enhanced heat conduction in fluids subjected to sinusoidal oscillations. Trans. ASME C: J. Heat Transfer 107, 459–462.Google Scholar
Lyne, W. H.: 1971 Unsteady viscous flow in a curved pipe. J. Fluid Mech. 45, 13–31.Google Scholar
Nunge, R. J., Lin, T.-S. & Gill, N. 1972 Laminar dispersion in curved tubes and channels. J. Fluid Mech. 51, 363–383.Google Scholar
Orszag, S. A. & Kells, L. C., 1980 Transition to turbulence in plane Poiseuille flow and plane Couette flow. J. Fluid Mech. 96, 159–205.Google Scholar
Paloski, W. H., Slosbrrg, R. B. & Kamm, R. D., 1987 Effects of gas properties and waveform asymmetry on gas transport in a branching tube network. J. Appl. Physiol. 63, 892–902.Google Scholar
Patera, A. T.: 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488.Google Scholar
Pedley, T. J.: 1980 The Fluid Mechanics of Large Blood Vessels, pp. 160–234. Cambridge University Press.
Pedley, T. J. & Kamm, R. D., 1988 The effect of secondary motion on axial transport in oscillatory tube flow. J. Fluid Mech. 193, 347–367.Google Scholar
Rhines, P. B. & Young, W. R., 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.Google Scholar
Schroter, R. C. & Sudlow, M. F., 1969 Flow patterns in models of the human branchial airways. Respir. Physiol. 7, 341–355.Google Scholar
Sharp, M. K.: 1987 Dispersion in a curved tube during oscillatory flow. Sc.D. thesis, Department of Mechanical Engineering, MIT.
Smith, R.: 1982 Contaminant dispersion in oscillatory flows. J. Fluid Mech. 114, 379–398.Google Scholar
Sreenivasan, K. R. & Strykowski, P. J., 1983 Stabilization effects in flow through helically coiled pipes. Expts Fluids 1, 31–36.Google Scholar
Strang, G. & Fix, G. J., 1973 An Analysis of the Finite Element Method. Prentice-Hall.
Taylor, G. I.: 1921 Diffusion by continuous movements. Proc. Land. Math. Soc. 20, (2), 196–212.Google Scholar
Taylor, G. I.: 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203.Google Scholar
Topakoglu, H. C.: 1967 Steady laminar flows of an incompressible viscous fluid in curved pipes. J. Math. Mech. 16, 1321–1337.Google Scholar
Watson, E. J.: 1983 Diffusion in oscillatory pipe flow. J. Fluid Mech. 133, 233–244.Google Scholar
Yamane, R., Oshima, S., Sudo, K., Sumida, M., Okamoto, N. & Kizaki, M., 1985 Study of oscillatory flow in curved channel. Bull. JSME 28, 428–435.Google Scholar
Young, W. R., Rhines, P. B. & Garrett, C. J., 1982 Shear-flow dispersion, internal waves and horizontal mixing in the ocean. J. Phys. Oceanogr. 12, 515–527.Google Scholar