Hostname: page-component-7b9c58cd5d-9klzr Total loading time: 0 Render date: 2025-03-20T04:53:47.182Z Has data issue: false hasContentIssue false
  • English
  • Français

Controlling the onset of turbulence by streamwise travelling waves. Part 1. Receptivity analysis

Published online by Cambridge University Press:  08 September 2010

RASHAD MOARREF  and
MIHAILO R. JOVANOVIĆ
RASHAD MOARREF
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
MIHAILO R. JOVANOVIĆ*
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: mihailo@umn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the efficacy of streamwise travelling waves generated by a zero-net-mass-flux surface blowing and suction for controlling the onset of turbulence in a channel flow. For small-amplitude actuation, we utilize a weakly nonlinear analysis to determine base-flow modifications and assess the resulting net power balance. Receptivity analysis of the velocity fluctuations around this base flow is then employed to design the travelling waves. Our simulation-free approach reveals that, relative to the flow with no control, the downstream travelling waves with properly designed speed and frequency can significantly reduce receptivity, which makes them well suited for controlling the onset of turbulence. In contrast, the velocity fluctuations around the upstream travelling waves exhibit larger receptivity to disturbances. Our theoretical predictions, obtained by perturbation analysis (in the wave amplitude) of the linearized Navier–Stokes equations with spatially periodic coefficients, are verified using full-scale simulations of the nonlinear flow dynamics in the companion paper (Lieu et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000340X).

JFM classification

Flow Control: Flow Control Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 663 , 25 November 2010 , pp. 70 - 99
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13 (11), 3258–3269.CrossRefGoogle Scholar
Bensoussan, A., Lions, J. L. & Papanicolaou, G. 1978 Asymptotic Analysis for Periodic Structures. North-Holland.Google Scholar
Bewley, T. R. 2009 A fundamental limit on the balance of power in a transpiration-controlled channel flow. J. Fluid Mech. 632, 443–446.CrossRefGoogle Scholar
Bewley, T. & Aamo, O. M. 2004 A ‘win–win’ mechanism for low-drag transients in controlled two-dimensional channel flow and its implications for sustained drag reduction. J. Fluid Mech. 499, 183–196.CrossRefGoogle Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Bewley, T. & Henningson, D. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552, 167–187.CrossRefGoogle Scholar
Cochran, J. & Krstic, M. 2009 Motion planning and trajectory tracking for three-dimensional Poiseuille flow. J. Fluid Mech. 626, 307–332.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Currie, I. G. 2003 Fundamental Mechanics of Fluids. CRC Press.Google Scholar
Del Alamo, J. C. & Jimenez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Fardad, M. & Bamieh, B. 2008 Perturbation methods in stability and norm analysis of spatially periodic systems. SIAM J. Control Optim. 47 (2), 997–1021.CrossRefGoogle Scholar
Fardad, M., Jovanović, M. R. & Bamieh, B. 2008 Frequency analysis and norms of distributed spatially periodic systems. IEEE Trans. Autom. Control 53 (10), 2266–2279.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5 (11), 2600–2609.CrossRefGoogle Scholar
Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14, L73.Google Scholar
Fukagata, K., Sugiyama, K. & Kasagi, N. 2009 On the lower bound of net driving power in controlled duct flows. Physica D: Nonlinear Phenom. 238 (13), 1082–1086.CrossRefGoogle Scholar
Hœpffner, J., Chevalier, M., Bewley, T. & Henningson, D. 2005 State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows. J. Fluid Mech. 534, 263–294.Google Scholar
Hœpffner, J. & Fukagata, K. 2009 Pumping or drag reduction? J. Fluid Mech. 635, 171–187.Google Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 a Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175.CrossRefGoogle Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 b Relaminarization of R eτ = 100 turbulence using linear state-feedback control. Phys. Fluids 15 (11), 3572–3575.Google Scholar
Jovanović, M. R. 2008 Turbulence suppression in channel flows by small amplitude transverse wall oscillations. Phys. Fluids 20 (1), 014101.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.Google Scholar
Jovanović, M. R. & Fardad, M. 2008 H 2 norm of linear time-periodic systems: a perturbation analysis. Automatica 44 (8), 2090–2098.Google Scholar
Jovanović, M. R., Moarref, R. & You, D. 2006 Turbulence suppression in channel flows by means of a streamwise traveling wave. In Proceedings of the 2006 Summer Program, pp. 481–494. Center for Turbulence Research, Stanford University/NASA.Google Scholar
Kim, J. 2003 Control of turbulent boundary layers. Phys. Fluids 15 (5), 1093–1105.Google Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Math. 28, 735–756.CrossRefGoogle Scholar
Lee, C., Min, T. & Kim, J. 2008 Stability of a channel flow subject to wall blowing and suction in the form of a traveling wave. Phys. Fluids 20 (10), 101513.CrossRefGoogle Scholar
Lee, K. H., Cortelezzi, L., Kim, J. & Speyer, J. 2001 Application of reduced-order controller to turbulent flows for drag reduction. Phys. Fluids 13 (5), 1321–1330.CrossRefGoogle Scholar
Lieu, B. K., Moarref, R. & Jovanović, M. R. 2010 Controlling the onset of turbulence by streamwise travelling waves. Part 2. Direct numerical simulation. J. Fluid Mech. 663, 100–119.CrossRefGoogle Scholar
Marusic, I., Joseph, D. D. & Mahesh, K. 2007 Laminar and turbulent comparisons for channel flow and flow control. J. Fluid Mech. 570, 467–477.CrossRefGoogle Scholar
McComb, W. D. 1991 The Physics of Fluid Turbulence. Oxford University Press.Google Scholar
Min, T., Kang, S. M., Speyer, J.L. & Kim, J. 2006 Sustained sub-laminar drag in a fully developed channel flow. J. Fluid Mech. 558, 309–318.Google Scholar
Odeh, F. & Keller, J. B. 1964 Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math. Phys. 5, 1499–1504.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21 (1), 015109.Google Scholar
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillations. J. Fluid Mech. 521, 251–271.CrossRefGoogle Scholar
Reynolds, W. C. & Hussain, A. K. M.F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3: Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (2), 263–288.Google Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow with application to Malkus's theory. J. Fluid Mech. 27 (2), 253–272.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shanks, D. 1955 Nonlinear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, 1–42.Google Scholar
Sidi, A. 2003 Practical Extrapolation Methods: Theory and Applications. Cambridge University Press.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S.C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Vazquez, R. & Krstic, M. 2007 a A closed-form feedback controller for stabilization of the linearized 2D Navier–Stokes Poiseuille flow. IEEE Trans. Autom. Control 52 (12), 2298–2312.Google Scholar
Vazquez, R. & Krstic, M. 2007 b Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation. Birkhäuser.Google Scholar