0
Research Papers: Flows in Complex Systems

Optimal Perturbations in Boundary-Layer Flows Over Rough Surfaces

[+] Author and Article Information
S. Cherubini

DynFluid, Arts et Metiers ParisTech,
Paris 75013, France;
Dipartimento di Meccanica,
Matematica e Management,
DMMM, CEMeC,
Politecnico di Bari,
Bari 70125, Italy
e-mail: s.cherubini@gmail.com

M. D. de Tullio

e-mail: m.detullio@poliba.it

P. De Palma

e-mail: depalma@poliba.it

G. Pascazio

e-mail: pascazio@poliba.it
Dipartimento di Meccanica,
Matematica e Management,
DMMM, CEMeC,
Politecnico di Bari,
Bari 70125, Italy

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 12, 2012; final manuscript received July 15, 2013; published online September 12, 2013. Assoc. Editor: Zhongquan Charlie Zheng.

J. Fluids Eng 135(12), 121102 (Sep 12, 2013) (11 pages) Paper No: FE-12-1569; doi: 10.1115/1.4025028 History: Received November 12, 2012; Revised July 15, 2013

This work provides a three-dimensional energy optimization analysis, looking for perturbations inducing the largest energy growth at a finite time in a boundary-layer flow in the presence of roughness elements. The immersed boundary technique has been coupled with a Lagrangian optimization in a three-dimensional framework. Four roughness elements with different heights have been studied, inducing amplification mechanisms that bypass the asymptotical growth of Tollmien–Schlichting waves. The results show that even very small roughness elements, inducing only a weak deformation of the base flow, can strongly localize the optimal disturbance. Moreover, the highest value of the energy gain is obtained for a varicose perturbation. This result demonstrates the relevance of varicose instabilities for such a flow and shows a different behavior with respect to the secondary instability theory of boundary layer streaks.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Marusic, I., 2009, “Unravelling Turbulence Near Walls,” J. Fluid Mech., 630, pp. 1–4. [CrossRef]
Cutrone, L., De Palma, P., Pascazio, G., and Napolitano, M., 2007, “An Evaluation of Bypass Transition Models for Turbomachinery Flows,” Int. J. Heat.Fluid Flow, 28, pp. 161–177. [CrossRef]
Cutrone, L., De Palma, P., Pascazio, G., and Napolitano, M., 2008, “Predicting Transition in Two- and Three-Dimensional Separated Flows,” Int. J. Heat Fluid Flow, 29, pp. 504–526. [CrossRef]
Schmid, P., and Henningson, D., 2001, Stability and Transition in Shear Flows, Springer, New York.
Schoppa, W., and Hussain, F., 2002, “Coherent Structure Generation in Near-Wall Turbulence,” J. Fluid Mech., 453, pp. 57–108. [CrossRef]
Brandt, L., Schlatter, P., and Henningson, D. S., 2004, “Transition in a Boundary Layers Subject to Free-Stream Turbulence,” J. Fluid Mech., 517, pp. 167–198. [CrossRef]
Landahl, M. T., 1980, “A Note on an Algebraic Instability of Inviscid Parallel Shear Flows,” J. Fluid Mech., 98, pp. 243–251. [CrossRef]
Landahl, M., 1990, “On Sublayer Streaks,” J. Fluid Mech., 212, pp. 593–614. [CrossRef]
Farrell, B., 1988, “Optimal Excitation of Perturbations in Viscous Shear Flow,” Phys. Fluids, 31, pp. 2093–2102. [CrossRef]
Butler, K. M., and Farrell, B. F., 1992, “Three-Dimensional Optimal Perturbations in Viscous Shear Flow,” Phys. Fluids A, 4, pp. 1637–1650. [CrossRef]
Luchini, P., 2000, “Reynolds Number Independent Instability of the Blasius Boundary Layer Over a Flat Surface: Optimal Perturbations,” J. Fluid Mech., 404, pp. 289–309. [CrossRef]
Schmid, P. J., 2000, “Linear Stability Theory and Bypass Transition in Shear Flows,” Phys. Plasmas, 7, pp. 1788–1794. [CrossRef]
Corbett, P., and Bottaro, A., 2000, “Optimal Perturbations for Boundary Layers Subject to Stream-Wise Pressure Gradient,” Phys. Fluids, 12, pp. 120–130. [CrossRef]
Barkley, D., and Henderson, R. D., 1996, “Floquet Stability Analysis of the Periodic Wake of a Circular Cylinder,” J. Fluid Mech., 322, pp. 215–241. [CrossRef]
Theofilis, V., Hein, S., and Dallmann, U., 2000, “On the Origins of Unsteadiness and Three Dimensionality in a Laminar Separation Bubble,” Philos. Trans. R. Soc. London, 358(1777), pp. 3229–3246. [CrossRef]
Ehrenstein, U., and Gallaire, F., 2005, “On Two Dimensional Temporal Modes in Spatially Evolving Open Flows: The Flat-Plate Boundary Layer,” J. Fluid Mech., 536, pp. 209–218. [CrossRef]
Cherubini, S., Robinet, J.-C., Bottaro, A., and De Palma, P., 2010, “Optimal Wave Packets in a Boundary Layer and Initial Phases of a Turbulent Spot,” J. Fluid Mech., 656, pp. 231–259. [CrossRef]
Acalar, M., and Smith, C., 1987, “A Study of Hairpin Vortices in a Laminar Boundary Layer: Part 1, Hairpin Vortices Generated by a Hemisphere Protuberance,” J. Fluid Mech., 175, pp. 1–41. [CrossRef]
Joslin, R. D., and Grosch, C. E., 1995, “Growth Characteristics Downstream of a Shallow Bump: Computation and Experiment,” Phys. Fluids, 7(12), pp. 3042–3047. [CrossRef]
Lipatov, I., and Vinogradov, I., 2000, “Three-Dimensional Flow Near Surface Distortions for the Compensation Regime,” Philos. Trans. R. Soc. London, Ser. A, 358, pp. 3143–3153. [CrossRef]
Fransson, J., Brandt, L., Talamelli, A., and Cossu, C., 2004, “Experimental and Theoretical Investigation of the Nonmodal Growth of Steady Streaks in a Flat Plate Boundary Layer,” Phys. Fluids, 16(10), pp. 3627–3638. [CrossRef]
Choudhari, M., and Fischer, P., 2005, “Roughness-Induced Transient Growth,” 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Ontario, Paper No. AIAA-2005-4765.
Fransson, J., Talamelli, A., Brandt, L., and Cossu, C., 2006, “Delaying Transition to Turbulence by a Passive Mechanism,” Phys. Rev. Lett., 96(6), p. 064501. [CrossRef] [PubMed]
Piot, E., Casalis, G., and Rist, U., 2008, “Stability of the Laminar Boundary Layer Flow Encountering a Row of Roughness Elements: Biglobal Stability Approach and DNS,” Eur. J. Mech. B/Fluids, 27(6), pp. 684–706. [CrossRef]
Giannetti, F., and Luchini, P., 2007, “Structural Sensitivity of the First Instability of the Cylinder Wake,” J. Fluid Mech., 581, pp. 167–197. [CrossRef]
Bottaro, A., 1990, “Note on Open Boundary Conditions for Elliptic Flows,” Numer. Heat Transfer, Part B, 18, pp. 243–256. [CrossRef]
Verzicco, R., and Orlandi, P., 1996, “A Finite-Difference Scheme for the Three-Dimensional Incompressible Flows in Cylindrical Coordinates,” J. Comp. Phys., 123(2), pp. 402–414. [CrossRef]
O'Rourke, J., 1993, Computational Geometry in C, Cambridge University, Cambridge, UK.
Mohd-Yusof, J., 1997, “Combined Immersed Boundaries/B-Splines Methods for Simulations of Flows in Complex Geometries,” CTR Annual Research Briefs, NASA Ames/Stanford University.
Fadlun, E. A., Verzicco, R., Orlandi, P., and Mohd-Yusof, J., 2000, “Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations,” J. Comput. Phys., 161, pp. 35–60. [CrossRef]
Zuccher, S., Luchini, P., and Bottaro, A., 2004, “Algebraic Growth in a Blasius Boundary Layer: Optimal and Robust Control by Mean Suction in the Nonlinear Regime,” Eur. J. Mech. B/Fluids, 513, pp. 135–160.
Marquet, O., Sipp, D., Chomaz, J.-M., and Jacquin, L., 2008, “Amplifier and Resonator Dynamics of a Low-Reynolds-Number Recirculation Bubble in a Global Framework,” J. Fluid Mech., 605, pp. 429–443. [CrossRef]
Polak, E., and Ribiere, G., 1969, “Note Sur la Convergence de Directions Conjugées,” Rev. Fr. Inform. Rech. Oper., 16, pp. 35–43.
Cherubini, S., De Palma, P., Robinet, J.-C., and Bottaro, A., 2010, “Rapid Path to Transition via Nonlinear Localized Optimal Perturbations,” Phys. Rev. E, 82, p. 066302. [CrossRef]
Von Doenhoff, A. E., and Braslow, A. L., 1961, “The Effect of Distributed Surface Roughness on Laminar Flow,” Boundary Layer Control, Vol. 2, Pergamon, New York.
Monokrousos, A., Akervik, E., Brandt, L., and Henningson, D. S., 2010, “Global Three-Dimensional Optimal Disturbances in the Blasius Boundary-Layer Flow Using Time-Steppers,” J. Fluid Mech., 650, pp. 181–214. [CrossRef]
Orr, W. M., 1907, “The Stability or Instability of the Steady Motions of a Liquid. Part I,” Proc. R. Ir. Acad., Sect. A, 27, pp. 9–68.
Pringle, C. C. T., and Kerswell, R., 2010, “Using Nonlinear Transient Growth to Construct the Minimal Seed for Shear Flow Turbulence,” Phys. Rev. Lett., 105, p. 154502. [CrossRef] [PubMed]
Hoepffner, J., Brandt, L., and Henningson, D. S., 2005, “Transient Growth on Boundary Layer Streaks,” J. Fluid Mech., 537, pp. 91–100. [CrossRef]
Andersson, P., Brandt, L., Bottaro, A., and Henningson, D. S., 2001, “On the Breakdown of Boundary Layer Streaks,” J. Fluid Mech., 428, pp. 29–60. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Sketch of the computational domain: Γ1 and Γ2 indicate the fluid and solid regions, respectively, separated by the body surface, in gray

Grahic Jump Location
Fig. 2

Contours of positive (dark gray) and negative (light gray) streamwise vorticity in the plane y = h0 + 1 for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The black circle represents the bump.

Grahic Jump Location
Fig. 3

Surfaces of the positive (light gray) and negative (dark gray) deviation of the streamwise component of velocity with respect to the spanwise mean for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The black circle represents the bump.

Grahic Jump Location
Fig. 4

Optimal energy gain versus target time for the Blasius base flow (circles) and for the base flows in the presence of the roughness element with height h0 = 0.75 (triangles up), h0 = 1 (triangles down), h0 = 1.5 (squares), and h0 = 2 (diamonds)

Grahic Jump Location
Fig. 5

Optimal perturbations for the Blasius base flow. Top: isosurfaces of the spanwise component of the velocity (dark gray and light gray for negative and positive values, respectively) of the initial (t = 0) optimal perturbation. Bottom: isosurfaces of the streamwise component (light gray and dark gray for negative and positive values, respectively) of the optimal perturbation at t = T = 50.

Grahic Jump Location
Fig. 6

Isosurfaces of the spanwise component of velocity (dark gray and light gray for negative and positive values, respectively) for the initial optimal perturbations (t = 0) obtained at T = 50 for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The inset shows the location and the shape of the roughness element.

Grahic Jump Location
Fig. 7

Contours of the streamwise component of velocity and streamlines in the plane at x = 77.5 for the initial optimal perturbation obtained at T = 50 for the base flow with h0 = 0.75 (top frame) and h0 = 1.5 (bottom frame). The spanwise and wall-normal axis are not on the same scale.

Grahic Jump Location
Fig. 8

Isosurfaces of the streamwise component of velocity (light gray and dark gray for negative and positive values, respectively) for the optimal perturbations at target time T = 50 for four base flows with h0 = 0.75 (first frame), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame)

Grahic Jump Location
Fig. 9

Contours of the streamwise velocity component of the optimal perturbation at target time T = 50 in the plane x = 112.5 for the four base flows with h0 = 0.75 (first frame from top), h0 = 1 (second frame), h0 = 1.5 (third frame), and h0 = 2 (fourth frame). The solid contours represent the streamwise component of the base flow velocity in the range [0.1, 0.9]. Axes are not on the same scale.

Grahic Jump Location
Fig. 10

Isosurfaces for T = 200 of the spanwise component of the velocity (dark gray and light gray for negative and positive values, respectively) for the initial (t = 0) optimal perturbations for the bumps with h0 = 0.75 (top) and h0 = 1.5 (bottom)

Grahic Jump Location
Fig. 11

Isosurfaces of the streamwise component of the velocity (light gray and dark gray for negative and positive values, respectively) for the optimal perturbations at target time t = T = 200 for the bumps with h0 = 0.75 (top) and h0 = 1.5 (bottom)

Grahic Jump Location
Fig. 12

Vectors of the spanwise and wall-normal velocity components and contours of the streamwise velocity component in the plane at x = 77.5 for the initial (t = 0) optimal perturbations obtained for T = 100 and T = 150 (from top to bottom) for the case of the bump with h0 = 1.5

Grahic Jump Location
Fig. 13

Shaded contours of the streamwise velocity component in the planes at x = 133.5 and x = 165 (from top to bottom) for the optimal perturbation at target time t = T = 100 and t = T = 150 (from top to bottom) for the case of the bump with h0 = 1.5. The solid contours represent the streamwise component of the base flow velocity in the range [0.1, 0.9]. Axes are not on the same scale.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In