Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T08:24:11.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-dimensional models for compressible temporally developing shear layers

Published online by Cambridge University Press:  15 August 2013

Bashar R. Qawasmeh  and
Mingjun Wei
Bashar R. Qawasmeh
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA
Mingjun Wei*
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A methodology to achieve extremely-low-dimensional models for temporally developing shear layers is extended from incompressible flows to weakly compressible flows. The key idea is to first remove the slow variation (i.e. viscous growth of shear layers) through symmetry reduction, so that the model reduction using proper orthogonal decomposition (POD)-Galerkin projection in the symmetry-reduced space becomes more efficient. However, for the approach to work for compressible flows, thermodynamic variables need to be retained. We choose the isentropic Navier–Stokes equations for the simplicity and the availability of a well-defined inner product for total energy. To capture basic dynamics, the compressible low-dimensional model requires only two POD modes for each frequency. Thus, a two-mode model is capable of representing single-frequency dynamics such as vortex roll-up, and a four-mode model is capable of representing the nonlinear dynamics involving a fundamental frequency and its subharmonic, such as vortex pairing and merging. The compressible model shows similar behaviour and accuracy as the incompressible model. However, because of the consistency of the inner product defined for POD and for projection in the current compressible model, the orthogonality is kept and it results in simple formulation. More importantly, the inclusion of compressibility opens an entirely new avenue for the discussion of compressibility effect and possible description of aeroacoustics and thermodynamics. Finally, the model is extended to different flow parameters without additional numerical simulation. The extension of the compressible four-mode model includes different Mach numbers and Reynolds numbers. We can clearly observe the change in the nonlinear interaction of modes at two frequencies and the associated promotion or delay of vortex pairing by varying compressibility and viscosity. The dynamic response of the low-dimensional model to different flow parameters is consistent with the vortex dynamics observed in experiments and numerical simulation.

JFM classification

Boundary Layers: Free shear layers Compressible Flows: Compressible Flows Nonlinear Dynamical Systems: Low-dimensional models
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 364 - 393
Copyright
©2013 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

Amsallem, D. & Farhat, C. 2008 Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (7), 1803–1813.Google Scholar
Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15 (1), 345–389.Google Scholar
Aref, H. & Siggia, E. D. 1980 Vortex dynamics of the two-dimensional turbulent shear layer. J. Fluid Mech. 100 (04), 705–737.Google Scholar
Batchelor, G. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bogdanoff, D. W. 1983 Compressiblity effects in turbulent shear layers. AIAA J. 21, 926–927.CrossRefGoogle Scholar
Bradshaw, P. 1977 Compressible turbulent shear layers. Annu. Rev. Fluid Mech. 9, 33–54.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Colonius, T., Lele, S. K. & Moin, P. 1993 Boundary conditions for direct computation of aerodynamic sound generation. AIAA J. 31, 1574–1582.Google Scholar
Colonius, T., Lele, S. K. & Moin, P. 1997 Sound generation in a mixing layer. J. Fluid Mech. 330, 375–409.CrossRefGoogle Scholar
Day, M. J., Mansour, N. N. & Reynolds, W. C. 2001 Nonlinear stability and structure of compressible reacting mixing layers. J. Fluid Mech. 446, 375–408.CrossRefGoogle Scholar
Deane, G. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3, 2337–2354.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Elliott, G. S., Samimy, M. & Arnette, S. A. 1995 The characteristics and evolution of large-scale structures in compressible mixing layers. Phys. Fluids 7 (4), 864–876.Google Scholar
Freund, J. B. 1997 Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound. AIAA J. 35 (4), 740–742.Google Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365–424.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Lieu, T. & Farhat, C. 2007 Adaptation of aeroelastic reduced-order models and application to an F-16 configuration. AIAA J. 45 (6), 1244–1257.Google Scholar
Lieu, T. & Lesoinne, M. 2004 Parameter adaptation of reduced order models for three-dimensional flutter analysis. AIAA Paper 2004-0888.CrossRefGoogle Scholar
Lock, R. C. 1951 The velocity distribution in the laminar boundary layer between parallel streams. Q. J. Mech. Appl. Maths 4 (1), 42–63.Google Scholar
Luchtenburg, D. M., Günter, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and actuated flows around a high-lift configuration. J. Fluid Mech. 623, 283–316.Google Scholar
Monkewitz, P. A. & Huerre, P. 1982 Influence of the velocity ratio on the spatial instabilty of mixing layers. Phys. Fluids 25, 1137–1143.CrossRefGoogle Scholar
Moser, R. D. & Rogers, M. M. 1993 The three-dimensional evolution of a plane mixing layer: pairings and transition to turbulence. J. Fluid Mech. 247, 275–320.Google Scholar
Noack, B., 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.Google 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
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer: an experimental study. J. Fluid Mech. 197, 453–477.CrossRefGoogle Scholar
Potter, O. E. 1957 Laminar boundary layers at the interface of co-current parallel streams. Q. J. Mech. Appl. Maths 10 (3), 302–311.CrossRefGoogle Scholar
Rogers, M. M. & Moser, R. D. 1992 The three-dimensional evolution of a plane mixing layer: the Kelvin–Holmholtz rollup. J. Fluid Mech. 243, 183–226.CrossRefGoogle Scholar
Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flow using POD and Galerkin projection. Physica D 189 (1–2), 115–129.Google 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.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.CrossRefGoogle Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Annu. Rev. Fluid Mech. 11, 95–122.Google Scholar
Sandham, N. D. 1994 The effect of compressibility on vortex pairing. Phys. Fluids 6 (2), 1063–1072.Google Scholar
Sandham, N. D. & Reynolds, W. C. 1990 Compressible mixing layer: linear theory and direct simulation. AIAA J. 28, 618–624.Google 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.Google Scholar
Schmidt, R. & Glauser, M. 2004 Improvements in low dimensional tools for flow-structure interaction problems: using global POD. AIAA Paper 2004-0889.Google Scholar
Tadmor, G., Lehmann, O., Noack, B. R. & Morzynski, M. 2010 Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22, 034102.Google Scholar
Thurow, B., Samimy, M. & Lempert, W. 2003 Compressibility effects on turbulence structures of axisymmetric mixing layers. Phys. Fluids 15 (6), 1755–1765.Google Scholar
Urban, W. D. & Mungal, M. G. 2001 Planar velocity measurements in compressible mixing layers. J. Fluid Mech. 431, 189–222.CrossRefGoogle Scholar
Wei, M. & Freund, J. B. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123–152.Google Scholar
Wei, M., Qawasmeh, B. R., Barone, M., van Bloemen Waanders, B. G. & Zhou, L. 2012 Low-dimensional model of spatial shear layers. Phys. Fluids 24 (1), 014108.Google Scholar
Wei, M. & Rowley, C. W. 2009 Low-dimensional models of a temporally evolving free shear layer. J. Fluid Mech. 618, 113–134.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63 (02), 237–255.CrossRefGoogle Scholar
Zank, G. P. & Matthaeus, W. H. 1991 The equations of nearly incompressible fluids. I. Hydrodynamics, turbulence, and waves. Phys. Fluids A 3 (1), 69–82.Google Scholar