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Low-dimensional models of a temporally evolving free shear layer

Published online by Cambridge University Press:  10 January 2009

MINGJUN WEI  and
CLARENCE W. ROWLEY
MINGJUN WEI*
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA
CLARENCE W. ROWLEY
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

We develop low-dimensional models for the evolution of a free shear layer in a periodic domain. The goal is to obtain models simple enough to be analysed using standard tools from dynamical systems theory, yet including enough of the physics to model nonlinear saturation and energy transfer between modes (e.g. pairing). In the present paper, two-dimensional direct numerical simulations of a spatially periodic, temporally developing shear layer are performed. Low-dimensional models for the dynamics are obtained using a modified version of proper orthogonal decomposition (POD)/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. Equations are obtained for the rate of change of the shear-layer thickness. A model with two complex modes can describe certain single-wavenumber features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. A model with four complex modes can describe interactions between two wavenumbers (vortex pairing) as well. At least two POD modes are required for each wavenumber in space to sufficiently describe the dynamics, though, for each wavenumber, more than 90% energy is captured by the first POD mode in the scaled space. The comparison of POD modes to stability eigenfunction modes seems to give a plausible explanation. We have also observed a relation between the phase difference of the first and second POD modes of the same wavenumber and the sudden turning point for shear-layer dynamics in both direct numerical simulations and model computations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 618 , 10 January 2009 , pp. 113 - 134
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–339.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn.Cambridge University Press.CrossRefGoogle Scholar
Freund, J. B. 1997 Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound. AIAA J. 35, 740–742.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42. Springer.CrossRefGoogle Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365–424.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223–252.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B. R., Pelivan, I., Tadmor, G., Morzyński, M. & Comte, P. 2004 Robust low-dimensional Galerkin models of natural and actuated flows. In Fourth Aeroacoustics Workshop. RWTH Aachen 26–27 February 2004.Google Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flow. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 57–70.Google Scholar
Rowley, C. W. & Marsden, J. E. 2000 Reconstruction equations and the Karhunen–Loève expansion for systems with symmetry. Physica D 142, 1–19.Google Scholar
Rowley, C. W. & Williams, D. R. 2006 Dynamics and control of high-Reynolds-number flow over open cavities. Annu. Rev. Fluid Mech. 38, 251–276.CrossRefGoogle Scholar
Rowley, C. W., Kevrekidis, I. G., Marsden, J. E. & Lust, K. 2003 Reduction and reconstruction for self-similar dynamical systems. Nonlinearity 16, 1257–1275.CrossRefGoogle Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Annu. Rev. Fluid Mech. 11, 95–122.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory, 8th edn.Springer.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences, vol. 142. Springer.CrossRefGoogle Scholar
Stanek, M. J., Raman, G., Kibens, V., Ross, J. A., Odedra, J. & Peto, J. W. 2001 Suppression of cavity resonance using high frequency forcing – the characteristic signature of effective devices. AIAA Paper 2001-2128.CrossRefGoogle Scholar
Tam, C. K. W. & Block, P. J. W. 1978 On the tones and pressure oscillations induced by flow over rectangular cavities. J. Fluid Mech. 89, 373–399.CrossRefGoogle Scholar
Tam, C. K. W. & Webb, J. C. 1993 Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107 (2), 262–281.CrossRefGoogle Scholar
Thomsen, J. J. 2002 Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening. J. Sound Vib. 253, 807–831.CrossRefGoogle Scholar
Wei, M. & Freund, J. B. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123–152.CrossRefGoogle Scholar