Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-20T11:36:33.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic/transcendental disturbance growth behind a row of roughness elements

Published online by Cambridge University Press:  26 January 2011

M. E. GOLDSTEIN ,
ADRIAN SESCU ,
PETER W. DUCK  and
MEELAN CHOUDHARI
M. E. GOLDSTEIN*
Affiliation:
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH 44135, USA
ADRIAN SESCU
Affiliation:
Department of Mechanical Industrial and Manufacturing Engineering, University of Toledo, Toledo, OH 43606, USA
PETER W. DUCK
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
MEELAN CHOUDHARI
Affiliation:
National Aeronautics and Space Administration, Langley Research Center, Hampton, VA 23681, USA
*
†Email address for correspondence: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is a continuation of the work begun in Goldstein et al. (J. Fluid Mech., vol. 644, 2010, p. 123), who constructed an asymptotic high-Reynolds-number solution for the flow over a spanwise periodic array of relatively small roughness elements with (spanwise) separation and plan form dimensions of the order of the local boundary-layer thickness. While that paper concentrated on the linear problem, here the focus is on the case where the flow is nonlinear in the immediate vicinity of the roughness with emphasis on the intermediate wake region corresponding to streamwise distances that are large in comparison with the roughness dimension, but small in comparison with the distance between the roughness array and the leading edge. An analytical O(h2) asymptotic solution is obtained for the limiting case of a small roughness height parameter h. These weakly nonlinear results show that the spanwise variable component of the wall-pressure perturbation decays as x−5/3 ln x when x → ∞ (where x denotes the streamwise distance scaled on the roughness dimension), but the corresponding component of the streamwise velocity perturbation (i.e. the wake velocity) exhibits an O(x1/3 ln x) algebraic/transcendental growth in the main boundary layer. Numerical solutions for h = O(1) demonstrate that the wake velocity perturbation for the fully nonlinear case grows in the same manner as the weakly nonlinear prediction – which is considerably different from the strictly linear result obtained in Goldstein et al. (2010).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 236 - 266
Copyright
Copyright © Cambridge University Press 2011

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions, National Bureau of Standards.Google Scholar
Acarlar, M. S. & Smith, C. R. 1987 A study of hairpin vortices in a laminar boundary layer. Part 1. J. Fluid Mech. 175, 1–42.CrossRefGoogle Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.Google Scholar
Case, K. M. 1960 Stability of plane Couette flow. Phys. Fluids 3, 143–148.Google Scholar
Choudhari, M. & Duck, P. W. 1996 Nonlinear excitation of inviscid stationary vortex instabilities in boundary layer flow. In Proceedings of the IUTAM Symposium on Nonlinear Instability and Transition in Three-dimensional Boundary Layers (ed. Duck, P. W. and Hall, P.), pp. 409–422 Manchester, UK, 17–20 July 1995. Kluwer.Google Scholar
Choudhari, M. & Fischer, P. 2005 Roughness induced transient growth. In Proceedings of the 35th AIAA Fluid Dynamics Conference and Exhibition. AIAA Paper 2005–4765.CrossRefGoogle Scholar
Ellingson, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.CrossRefGoogle Scholar
Fischer, P. & Choudhari, M. 2004 Numerical simulation of roughness-induced transient growth in a laminar boundary layer. AIAA Paper 2004–2539.CrossRefGoogle Scholar
Fransson, J. M. H., Brant, L., Talamelli, A. & Cossu, C. 2004 Experimental and theoretical investigation of non-modal growth of steady streaks in a flat plate boundary layer. Phys. Fluids 16, 3627.CrossRefGoogle Scholar
Fransson, J. M. H., Talamelli, A., Brant, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 060541.CrossRefGoogle ScholarPubMed
Goldstein, M. E., Sescu, A., Duck, P. W. & Choudhari, M. 2010 The long range persistence of wakes behind a row of roughness elements. J. Fluid Mech. 644, 123–163.CrossRefGoogle Scholar
Kemp, N. H. 1951 The laminar three-dimensional boundary layer and the study of the flow past a side edge. MAeS thesis, Cornell University.Google Scholar
Klebanoff, P., Cleveland, W. G. & Tidstrom, K. D. 1992 On the evolution of a turbulent boundary layer induced by a three-dimensional roughness element. J. Fluid Mech. 237, 101–187.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
Rothmayer, A. P. & Smith, F. T. 1998 Incompressible triple deck theory. In The Handbook of Fluid Dynamics, chap. 23, CRC.Google Scholar
Smith, F. T. 1973 Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803–824.CrossRefGoogle Scholar
Smith, F. T. 1976 a Flow through constricted pipes and channels. Part 1. Q. J. Mech. Appl. Math. 29, 343–364.CrossRefGoogle Scholar
Smith, F. T. 1976 b Flow through constricted pipes and channels. Part 2. Q. J. Mech. Appl. Math. 29, 365–376.CrossRefGoogle Scholar
Smith, F. T., Brighton, P. S., Jackson, P. S. & Hunt, J. C. R. 1981 On boundary layer flow over two dimensional obstacles. J. Fluid Mech. 113, 123–152.CrossRefGoogle Scholar
Tumin, A. & Reshotko, E. 2001 Spatial theory of optimal disturbances in boundary layers. Phys. Fluids 13 (7), 2097–2104.CrossRefGoogle Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar
White, E. B. & Ergin, F. G. 2003 Receptivity and transient growth of roughness-induced disturbances. AIAA Paper 2003–4243.CrossRefGoogle Scholar