Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T05:28:47.269Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis

Published online by Cambridge University Press:  16 June 2014

M. Luhar ,
A. S. Sharma  and
B. J. McKeon
M. Luhar*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
A. S. Sharma
Affiliation:
Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UK
B. J. McKeon
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We generate predictions for the fluctuating pressure field in turbulent pipe flow by reformulating the resolvent analysis of McKeon and Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) in terms of the so-called primitive variables. Under this analysis, the nonlinear convective terms in the Fourier-transformed Navier–Stokes equations (NSE) are treated as a forcing that is mapped to a velocity and pressure response by the resolvent of the linearized Navier–Stokes operator. At each wavenumber–frequency combination, the turbulent velocity and pressure field are represented by the most-amplified (rank-1) response modes, identified via a singular value decomposition of the resolvent. We show that these rank-1 response modes reconcile many of the key relationships among the velocity field, coherent structure (i.e. hairpin vortices), and the high-amplitude wall-pressure events observed in previous experiments and direct numerical simulations (DNS). A Green’s function representation shows that the pressure fields obtained under this analysis correspond primarily to the fast pressure contribution arising from the linear interaction between the mean shear and the turbulent wall-normal velocity. Recovering the slow pressure requires an explicit treatment of the nonlinear interactions between the Fourier response modes. By considering the velocity and pressure fields associated with the triadically consistent mode combination studied by Sharma and McKeon (J. Fluid Mech., vol. 728, 2013, pp. 196–238), we identify the possibility of an apparent amplitude modulation effect in the pressure field, similar to that observed for the streamwise velocity field. However, unlike the streamwise velocity, for which the large scales of the flow are in phase with the envelope of the small-scale activity close to the wall, we expect there to be a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\pi /2$ phase difference between the large-scale wall-pressure and the envelope of the small-scale activity. Finally, we generate spectral predictions based on a rank-1 model assuming broadband forcing across all wavenumber–frequency combinations. Despite the significant simplifying assumptions, this approach reproduces trends observed in previous DNS for the wavenumber spectra of velocity and pressure, and for the scale-dependence of wall-pressure propagation speed.

JFM classification

Boundary Layers: Boundary layer structure Boundary Layers: Boundary Layers Boundary Layers: Pipe flow boundary layer
Type
Papers
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , pp. 38 - 70
Copyright
© 2014 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

Ahn, B.-K., Graham, W. R. & Rizzi, S. A. 2010 A structure-based model for turbulent-boundary-layer wall pressures. J. Fluid Mech. 650, 443–478.Google Scholar
del Alamo, J. C. & Jimenez, J. 2009 Estimation of turbulent convection velocities and corrections of Taylor’s approximation. J. Fluid Mech. 640, 5–26.CrossRefGoogle Scholar
Bandhyopadhyay, P. R. & Hussain, A. K. M. F. 1984 The coupling between scales in shear flows. Phys. Fluids 27 (9), 2221–2228.CrossRefGoogle Scholar
Bradshaw, P. & Koh, Y. M. 1981 A note on Poisson’s equation for pressure in a turbulent flow. Phys. Fluids 24, 777.CrossRefGoogle Scholar
Bull, M. K. 1967 Wall-pressure fluctuations associated with subsonic turbulent boundary layer flow. J. Fluid Mech. 28, 719–754.Google Scholar
Chakraborty, P., Balachandar, S. & Adrian, R. J. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214.Google Scholar
Choi, H. & Moin, P. 1990 On the space–time characteristics of wall-pressure fluctuations. Phys. Fluids 2 (8), 1450–1460.Google Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75–110.Google Scholar
Chung, D. & McKeon, B. J. 2010 Large-eddy simulation of large-scale structures in long channel flow. J. Fluid Mech. 661, 341–364.Google Scholar
Dinkelacker, A., Hessel, M., Meier, G. E. A. & Schewe, G. 1977 Investigation of pressure fluctuations beneath a turbulent boundary layer by means of an optical method. Phys. Fluids 20 (10), S216–S224.Google Scholar
Foias, C., Manley, O., Rosa, R. & Temam, R. 2001 Navier–Stokes Equations and Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Ghaemi, S. & Scarano, F. 2013 Turbulent structure of high-amplitude pressure peaks within the turbulent boundary layer. J. Fluid Mech. 735, 381–426.CrossRefGoogle Scholar
Guala, M., Metzger, M. & McKeon, B. J. 2011 Interactions within the turbulent boundary layer at high Reynolds number. J. Fluid Mech. 666, 573–604.Google Scholar
Hu, Z. H., Morfey, C. L. & Sandham, N. D. 2002 Aeroacoustics of wall-bounded turbulent flows. AIAA J. 40 (3), 465–473.CrossRefGoogle Scholar
Jacobi, I. & McKeon, B. J. 2013 Phase relationships between large and small scales in the turbulent boundary layer. Exp. Fluids 54, 1481.CrossRefGoogle Scholar
Jimenez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236.CrossRefGoogle Scholar
Johansson, A. V., Her, J. Y. & Haritonidis, J. H. 1987 On the generation of high-amplitude wall-pressure peaks in turbulent boundary-layers and spots. J. Fluid Mech. 175, 119–142.CrossRefGoogle Scholar
Kim, J. 1989 On the structure of pressure-fluctuations in simulated turbulent channel flow. J. Fluid Mech. 205, 421–451.Google Scholar
Klewicki, J., Priyadarshana, P. J. A. & Metzger, M. M. 2008 Statistical structure of the fluctuating wall pressure and its in-plane gradients at high Reynolds number. J. Fluid Mech. 609, 195–220.Google Scholar
Koumoutsakos, P. 1999 Vorticity flux control for a turbulent channel flow. Phys. Fluids 11, 248.CrossRefGoogle Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J.2013 A systems approach to modeling opposition control in turbulent pipe flow. In 43rd Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, June 24–27.Google Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014 Opposition control within the resolvent analysis framework. J. Fluid Mech. 749, 597–626.CrossRefGoogle Scholar
Marusic, I. & Heuer, W. D. C. 2007 Reynolds number invariance of the structure inclination angle in wall turbulence. Phys. Rev. Lett. 99, 114504.CrossRefGoogle ScholarPubMed
Marusic, I., Mathis, R. & Hutchins, N. 2010 Predictive model for wall-bounded turbulent flow. Science 329 (5988), 193–196.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2011 A predictive inner–outer model for streamwise turbulence statistics in wall-bounded flows. J. Fluid Mech. 681, 537–566.Google Scholar
McKeon, B. J., Jacobi, I. & Sharma, A. S. 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25, 031301.Google Scholar
McKeon, B. J., Li, J., Jiang, W., Morrison, J. F. & Smits, A. J. 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.Google Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number $10^7$ . J. Comput. Phys. 186 (1), 178–197.Google Scholar
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling and prediction of the streamwise energy intensity in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.Google Scholar
Monty, J. P., Hutchins, N., Ng, H. C. H., Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442.Google Scholar
O’Farrell, C. & Martin, P. 2009 Chasing eddies and their wall signature in DNS data of turbulent boundary layers. J. Turbul. 10, N15.Google Scholar
Panton, R. L. & Linebarger, J. H. 1974 Wall pressure spectra calculations for equilibrium boundary layers. J. Fluid Mech. 65 (2), 261–287.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199.Google Scholar
Perry, A. E. & Marusic, I. 1995 A wall-wake model for the turbulence structure of boundary-layers. 1. Extension of the attached eddy hypothesis. J. Fluid Mech. 298, 361–388.Google Scholar
Schewe, G. 1983 On the structure and resolution of wall-pressure fluctuations associated with turbulent boundary-layer flow. J. Fluid Mech. 134, 311–328.CrossRefGoogle Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.Google Scholar
Sharma, A. S., Morrison, J. F., McKeon, B. J., Limebeer, D. J. N., Koberg, W. H. & Sherwin, S. J. 2011 Relaminarisation of ${R}e_{\tau }=100$ channel flow with globally stabilizing linear feedback control. Phys. Fluids 23 (12), 125105.CrossRefGoogle Scholar
Snarski, S. R. & Lueptow, R. M. 1995 Wall pressure and coherent structures in a turbulent boundary layer on a cylinder in axial flow. J. Fluid Mech. 286, 137–171.Google Scholar
Thomas, A. S. W. & Bull, M. K. 1983 On the role of wall-pressure fluctuations in deterministic motions in the turbulent boundary layer. J. Fluid Mech. 128, 283–322.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Tsuji, Y., Fransson, J. H. M., Alfredsson, P. H. & Johansson, A. V. 2007 Pressure statistics and their scaling in high-Reynolds-number turbulent boundary layers. J. Fluid Mech. 585, 1–40.CrossRefGoogle Scholar
Tsuji, Y., Imayama, S., Schlatter, P., Alfredsson, P. H., Johansson, A. V., Marusic, I., Hutchins, N. & Monty, J. 2012 Pressure fluctuation in high-Reynolds-number turbulent boundary layer: results from experiments and DNS. J. Turbul. 13 (50), 1–19.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2008 A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81–112.Google Scholar