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Effects of base flow modifications on noise amplifications: flow past a backward-facing step

Published online by Cambridge University Press:  20 April 2015

X. Mao
X. Mao*
Affiliation:
School of Engineering and Computing Sciences, Durham University, Durham DH1 3LE, UK
*
†Email address for correspondence: [email protected]
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Abstract

Amplifications of flow past a backward-facing step with respect to optimal inflow and initial perturbations are investigated at Reynolds number 500. Two mechanisms of receptivity to inflow noise are identified: the bubble-induced inflectional point instability and the misalignment effect downstream of the secondary bubble. Further development of the misalignment results in decay of perturbations from $x=28$ onwards (the step is located at $x=0$), as has been observed in previous non-normality studies (Blackburn et al., J. Fluid Mech., vol. 603, 2008, pp. 271–304), and eventually limits the receptivity. The receptivity is found to be maximized at an inflow perturbation frequency of ${\it\omega}=0.50$ and a spanwise wavenumber of ${\it\beta}=0$, where the inflow noise takes full advantage of both mechanisms and is amplified over two orders of magnitude in terms of the velocity magnitude. In direct numerical simulations (DNS) of the flow perturbed by optimal or random inflow noise, vortex shedding, flapping of bubbles, three-dimensionality and turbulence are observed in succession as the magnitude of the inflow noise increases. Similar features of linear and nonlinear receptivity are observed at higher Reynolds numbers. The Strouhal number of the bubble flapping is 0.08, at which the receptivity to inflow noise reaches a maximum. This Strouhal number is close to reported values extracted from DNS or large eddy simulations (LES) at larger Reynolds numbers (Le et al., J. Fluid Mech., vol. 330, 1997, pp. 349–374; Kaiktsis et al., J. Fluid Mech., vol. 321, 1996, pp. 157–187; Métais, New Trends in Turbulence, 2001, Springer; Wee et al., Phys. Fluids, vol. 16, 2004, pp. 3361–3373). Methods to further clarify the mechanisms of receptivity and to suppress the noise amplifications by modifying the base flow using a linearly optimal body force are proposed. It is observed that the mechanisms of optimal noise amplification are fully revealed by the distribution of the base flow modification, which weakens the bubble instabilities and misalignment effects and subsequently reduces the receptivity significantly. Comparing the base flow modifications with respect to amplifications of inflow and initial perturbations, it is found that the maximum receptivity to initial perturbations is highly correlated with the receptivity to inflow noise at the optimal frequency ${\it\omega}=0.50$, and the correlation reduces as the inflow frequency deviates from this optimal value.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Boundary layer receptivity Flow Control: Instability control
Type
Papers
Information
Journal of Fluid Mechanics , Volume 771 , 25 May 2015 , pp. 229 - 263
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Copyright
© 2015 Cambridge University Press 

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References

Armaly, B., Durst, F., Pereira, J. & Schönung, B. 1983 Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473–496.Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 1435–1458.Google Scholar
Barkley, D., Gomes, M. G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.Google Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.CrossRefGoogle Scholar
Blackburn, H. M. & Sherwin, S. J. 2004 Formulation of a Galerkin spectral element–Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197 (2), 759–778.Google Scholar
Boiko, A., Dovgal, A. & Sorokin, A. 2012 Instability of a backward-facing step flow modified by stationary streaky structures. Phys. Fluids 24, 104104.CrossRefGoogle Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.Google Scholar
Brandt, L., Sipp, D., Pralits, J. O. & Marquet, O. 2011 Effects of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.CrossRefGoogle Scholar
Cathalifaud, P. & Luchini, P. 2000 Algebraic growth in boundary layers: optimal control by blowing and suction at the wall. Eur. J. Mech. (B/Fluids) 19, 469–490.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Gavarini, M. I., Bottaro, A. & Nieuwstadt, F. T. M. 2004 The initial stage of transition in pipe flow: role of optimal base-flow distortion. J. Fluid Mech. 517, 131–165.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Guégan, A., Schmid, P. & Huerre, P. 2006 Optimal energy growth and optimal control in swept Hiemenz flow. J. Fluid Mech. 566, 11–45.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Kaiktsis, L., Karniadakis, G. & Orszag, S. 1996 Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. J. Fluid Mech. 321, 157–187.Google Scholar
Kaiktsis, L. & Monkewitz, P. 2003 Global destabilization of flow over a backward-facing step. Phys. Fluids 15, 3647–3658.Google Scholar
Kaltenbach, H. & Janke, G. 2000 Direct numerical simulation of flow separation behind a swept, rearward-facing step at $Re_{H}=3000$ . Phys. Fluids 12, 2320–2337.Google Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414–443.Google Scholar
Lanzerstorfer, D. & Kuhlmann, H. 2012 Global stability of the two-dimensional flow over a backward-facing step. J. Fluid Mech. 693, 1–27.Google Scholar
Lashgari, I., Tammisola, O., Citro, V., Juniper, M. & Brandt, L. 2014 The planar X-junction flow: stability analysis and control. J. Fluid Mech. 753, 1–28.Google Scholar
Le, H., Moin, P. & Kim, J. 1997 Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech. 330, 349–374.CrossRefGoogle Scholar
Lesshafft, L. & Marquet, O. 2010 Optimal velocity and density profiles for the onset of absolute instability in jets. J. Fluid Mech. 662, 398–408.Google Scholar
Mao, X., Blackburn, H. M. & Sherwin, S. J. 2012 Optimal inflow boundary condition perturbations in steady stenotic flows. J. Fluid Mech. 705, 306–321.Google Scholar
Mao, X., Blackburn, H. M. & Sherwin, S. J. 2013 Calculation of global optimal initial and boundary perturbations for the linearised incompressible Navier–Stokes equations. J. Comput. Phys. 235, 258–273.Google Scholar
Marquet, O., Sipp, D., Chomaz, J. M. & Jacquin, L. 2008a Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008b Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Mcgregor, O. & White, R. 1970 Drag of rectangular cavities in supersonic and transonic flow including the effects of cavity resonance. AIAA J. 8, 1959–1964.Google Scholar
Métais, O. 2001 Large-eddy simulation of turbulence. In New Trends in Turbulence, Springer.Google Scholar
Park, H., Jeon, W., Choi, H. & Yoo, J. 2007 Mixing enhancement behind a backward-facing step using tabs. Phys. Fluids 19, 105103.Google Scholar
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251–288.Google Scholar
Schäfer, F., Breuer, M. & Durst, F. 2009 The dynamics of the transitional flow over a backward-facing step. J. Fluid Mech. 623, 85–119.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schrader, L., Brandt, L. & Henningson, D. S. 2009 Receptivity mechanisms in three-dimensional boundary-layer flows. J. Fluid Mech. 618, 209–241.Google Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex shedding at low Reynolds numbers. J. Fluid Mech. 218, 71–107.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Vanneste, J. 1999 A spatial analogue of transient growth in plane Couette flow. J. Fluid Mech. 397, 317–330.Google Scholar
Wee, D., Yi, T., Annaswamy, A. & Ghoniem, A. 2004 Self-sustained oscillations and vortex shedding in backward-facing step flows: simulation and linear instability analysis. Phys. Fluids 16, 3361–3373.Google Scholar
Yanase, S., Kawahara, G. & Kiyama, H. 2001 Three-dimensional vortical structures of a backward-facing step flow at moderate Reynolds numbers. J. Phys. Soc. Japan 70, 3550–3555.Google Scholar
Yokoyama, H., Tsukamoto, Y., Kato, C. & Iida, A. 2007 Self-sustained oscillations with acoustic feedback in flows over a backward-facing step with a small upstream step. Phys. Fluids 19, 106104.Google Scholar
Zaki, T. A., Wissink, J. G., Rodi, W. & Durbin, P. A. 2010 Direct numerical simulations of transition in a compressor cascade: the influence of free-stream turbulence. J. Fluid Mech. 665, 57–98.Google Scholar
Zhang, H., Fan, B. & Chen, Z. 2010 Optimal control of cylinder wake flow by electro-magnetic force based on adjoint flow field. Eur. J. Mech. (B/Fluids) 29, 53–60.Google Scholar