Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T15:02:08.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal energy density growth in Hagen–Poiseuille flow

Published online by Cambridge University Press:  26 April 2006

Peter J. Schmid  and
Dan S. Henningson
Peter J. Schmid
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Department of Applied Mathematics, University of Washington, Seattle, WA 98195. USA.
Dan S. Henningson
Affiliation:
Aeronautical Research Institute of Sweden (FFA), Box 11021, S-16111 Bromma, Swedenand Department of Mechanics, Royal Institute of Technology, S-10044 Stockholm, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and ε-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 277 , 25 October 1994 , pp. 197 - 225
Copyright
© 1994 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

Bergström, L. 1992 Initial algebraic growth of small angular dependent disturbances in pipe Poiseuille flow. Stud. Appl. Maths 87, 61–79.Google Scholar
Bergström, L. 1993 Optimal growth of small disturbances in pipe Poiseuille flow. Phys. Fluids A 5, 2710–2720.Google Scholar
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43a, 697–726.Google Scholar
Burridge, D. M. & Drazin, P. G. 1969 Comments on ‘Stability of pipe Poiseuille flow’. Phys. Fluids 12, 264–265.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Davey, A. 1978 On the stability of flow in an elliptic pipe which is nearly circular. J. Fluid Mech. 87, 233–241.Google Scholar
Davey, A. & Drazin, P. G. 1969 The stability of Poiseuille flow in a pipe. J. Fluid Mech. 36, 209–218.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701–720.Google Scholar
DiPrima, R. C. 1967 Vector eigenfunction expansions for the growth of Taylor vortices in the flow between rotating cylinders. In Nonlinear Partial Differential Equations (ed. W. F. Ames). Academic.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.Google Scholar
Gustavsson, L. H. 1989 Direct resonance of nonaxisymmetric disturbances in pipe flow. Stud. Appl. Maths 80, 95–108.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.Google Scholar
Henningson, D. S. & Reddy, S. C. 1994 On the role of linear mechanisms in transition to turbulence. Phys. Fluids 6, 1396–1398.Google Scholar
Henningson, D. S. & Schmid, P. J. 1992 Vector eigenfunction expansions for plane channel flows. Stud. Appl. Maths 87, 15–43.Google Scholar
Herbert, T. 1977 Die neutrale Fläche der ebenen Poiseuille Strömung. Habilitationsschrift, Universität Stuttgart.
Herron, I. 1991 Observations on the role of vorticity in the stability theory of wall bounded flows. Stud. Appl. Maths 85, 269–286.Google Scholar
Horn, R. & Johnson, J. 1991 Topics in Matrix Analysis. Cambridge University Press.
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Joseph, D. D. & Carmi, S. 1969 Stability of Poiseuille flow in pipes, annuli and channels. Q. Appl. Maths 26, 575–591.Google Scholar
Kato, T. 1976 Perturbation Theory for Linear Operators. Springer.
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206–229.Google Scholar
Lessen, M., Sadler, S. G. & Liu, T.-Y. 1968 Stability of pipe Poiseuille flow. Phys. Fluids 11, 1404–1409.Google Scholar
Lundbladh, A. 1993 Simulation of bypass transition to turbulence in wall bounded shear flows. PhD thesis, Department of Mechanics, KTH, Stockholm.
Mackrodt, P.-A. 1976 Stability of Hagen-Poiseuille flow with superimposed rigid rotation. J. Fluid Mech. 73, 153–164.Google Scholar
Maslowe, S. A. 1974 Instability of rigidly rotating flows to non-axisymmetric disturbances. J. Fluid Mech. 64, 307–317.Google Scholar
Metcalfe, R. & Orszag, S. A. 1973 Numerical calculations of the linear stability of pipe flow. Flow Research Rep. 25. Kent Washington.
O'Sullivan, P. L. & Breuer, K. S. 1992 Transient growth of non-axisymmetric disturbances in laminar pipe flow. Technical Rep., Center for Fluid Dynamics, Brown University.
Pazy, A. 1983 Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer.
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flow. J. Fluid Mech. 252, 209–238.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr-Sommerfeld operator. SIAM J. Appl. Maths 53, 15–45.Google Scholar
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funkcional Anal, i Prolozen. 7. (2), 62–73. (Translated in Functional Anal. & Applics. 7, 137–146.)Google Scholar
Salwen, H., Cotton, F. W. & Grosch, C. E. 1980 Linear stability of Poiseuille flow in a circular pipe. J. Fluid Mech. 98, 273–284.Google Scholar
Tatsumi, T. 1952 Stability of the laminar inlet-flow prior to the formation of Poiseuille regime, II. J. Phys. Soc. Japan 7, 495.Google Scholar
Trefethen, L. N. 1992 Pseudospectra of matrices. In Numerical Analysis 1991 (ed. D. F. Griffiths & G. A. Watson). Longman.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Willke, L. H. 1967 Stability in time-symmetric flows. J. Math. Phys. 46, 151–163.Google Scholar