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The stability of compressible swirling pipe flows with density stratification

Published online by Cambridge University Press:  23 June 2017

Navneet K. Yadav  and
Arnab Samanta Open the ORCID record for Arnab Samanta [Opens in a new window]
Navneet K. Yadav
Affiliation:
Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
Arnab Samanta*
Affiliation:
Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the spatial stability of compressible, viscous pipe flows with radius-dependent mean density profiles, subjected to solid body rotations. For a fixed Rossby number $\unicode[STIX]{x1D716}$ (inverse of the rotational speed), as the Reynolds number $Re$ is increased, the flow transitions from being stable to convectively unstable, usually leading to absolute instability. If flow compressibility is unimportant and $Re$ is held constant, there appears to be a maximum $Re$ below which the flow remains stable irrespective of any rotational speed, or a minimum azimuthal Reynolds number $Re_{\unicode[STIX]{x1D703}}$ $(=Re/\unicode[STIX]{x1D716})$ is required for any occurrence of absolute instabilities. Once compressible forces are significant, the effect of pressure–density coupling is found to be more severe below a critical $Re$, where as rotational speeds are raised, a stable flow almost directly transitions to an absolutely unstable state. This happens at a critical $Re_{\unicode[STIX]{x1D703}}$ which reduces with increased flow Mach number, pointing to compressibility aiding in the instability at these lower Reynolds numbers. However, at higher $Re$, above the critical value, the traditional stabilizing role of compressibility is recovered if mean density stratification exists, where the gradients of density play an equally important role, more so at the higher azimuthal modes. A total disturbance energy-based formulation is used to obtain mechanistic understanding at these stability states, where we find the entropic energy perturbations to dominate as the primary instability mechanism, in sharp contrast to the energy due to axial shear, known to play a leading role in incompressible swirling flows.

JFM classification

Instability: Absolute/convective instability Compressible Flows: Compressible Flows Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 823 , 25 July 2017 , pp. 689 - 715
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Copyright
© 2017 Cambridge University Press 

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References

Batchelor, G. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.Google Scholar
Bers, A. 1983 Space-time evolution of plasma instabilities—absolute and convective. In Handbook of Plasma Physics: Basic Plasma Physics, vol. 1, chap. 3.2, pp. 451–517. North Holland.Google Scholar
Bridges, T. J. & Morris, P. J. 1984 Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437–460.Google Scholar
Briggs, R. J. 1964 Electron-streaming interaction with plasmas. In Monograph no. 29, MIT Press.Google Scholar
Candel, S., Durox, D., Schuller, T., Bourgouin, J. & Moeck, J. P. 2014 Dynamics of swirling flames. Annu. Rev. Fluid Mech. 46, 147–173.Google Scholar
Chu, B. T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mech. 1 (3), 215–234.Google Scholar
Cotton, F. W. & Salwen, H. 1981 Linear stability of rotating Hagen–Poiseuille flow. J. Fluid Mech. 108, 101–125.Google Scholar
Di Pierro, B., Abid, M. & Amielh, M. 2013 Experimental and numerical investigation of a variable density swirling-jet stability. Phys. Fluids 25 (8), 084104.Google Scholar
Escudier, M. 1988 Vortex breakdown: observations and explanations. Prog. Aerosp. Sci. 25 (2), 189–229.CrossRefGoogle Scholar
Fernandez-Feria, R. & del Pino, C. 2002 The onset of absolute instability of rotating Hagen–Poiseuille flow: a spatial stability analysis. Phys. Fluids 14 (9), 3087–3097.CrossRefGoogle Scholar
Fung, Y. T. & Kurzweg, U. H. 1975 Stability of swirling flows with radius-dependent density. J. Fluid Mech. 72, 243–255.Google Scholar
Gallaire, F. & Chomaz, J. M. 2003 Instability mechanisms in swirling flows. Phys. Fluids 15 (9), 2622–2639.Google Scholar
Gallaire, F., Ruith, M., Meiburg, E., Chomaz, J. M. & Huerre, P. 2006 Spiral vortex breakdown as a global mode. J. Fluid Mech. 549, 71–80.Google Scholar
Garg, V. K. & Rouleau, W. T. 1972 Linear spatial stability of pipe Poiseuille flow. J. Fluid Mech. 54, 113–127.Google Scholar
Hagan, J. & Priede, J. 2013 Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method. J. Comput. Phys. 238, 210–216.Google Scholar
Heifetz, E. & Mak, J. 2015 Stratified shear flow instabilities in the non-Boussinesq regime. Phys. Fluids 27, 086601.CrossRefGoogle Scholar
Herrada, M. A., Pérez-Saborid, M. & Barrero, A. 2003 Vortex breakdown in compressible flows in pipes. Phys. Fluids 15 (8), 2208–2218.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamics and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Huang, Y. & Yang, V. 2009 Dynamics and stability of lean-premixed swirl-stabilized combustion. Proc. Energy Combust. Sci. 35, 293–364.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamics and nonlinear instabilities. In Hydrodynamic Instabilities in Open Flows, pp. 159–229. Cambridge University Press.Google Scholar
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206–229.Google Scholar
Lefebvre, A. H. & Ballal, D. R. 2010 Gas Turbine Combustion: Alternative Fuels and Emissions, 3rd edn. CRC Press.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22 (9), 1192–1206.Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.Google Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769–779.Google Scholar
Liang, H. & Maxworthy, T. 2005 An experimental investigation of swirling jets. J. Fluid Mech. 525, 115–159.Google Scholar
Loiseleux, T. & Chomaz, J. M. 2003 Breaking of rotational symmetry in a swirling jet experiment. Phys. Fluids 15 (2), 511–523.Google Scholar
Lu, G. & Lele, S. K. 1999 Inviscid instability of compressible swirling mixing layers. Phys. Fluids 11 (2), 450–460.Google Scholar
Mackrodt, P. A. 1976 Stability of Hagen–Poiseuille flow with superimposed rigid rotation. J. Fluid Mech. 73, 153–164.Google Scholar
Maslowe, S. A. & Stewartson, K. 1982 On the linear inviscid stability of rotating Poiseuille flow. Phys. Fluids 25 (9), 1517–1523.Google Scholar
Mongia, H. C.2003 TAPS – A 4th generation propulsion combustor technology for low emissions. AIAA Paper 2003-2657.Google Scholar
Oberleithner, K., Paschereit, C. O. & Wygnanski, I. 2014 On the impact of swirl on the growth of coherent structures. J. Fluid Mech. 741, 156–199.Google Scholar
Olendraru, C., Sellier, A., Rossi, M. & Huerre, P. 1999 Inviscid instability of the Batchelor vortex: absolute-convective transition and spatial branches. Phys. Fluids 11 (7), 1805–1820.Google Scholar
Panda, J. & Mclaughlin, D. K. 1994 Experiments on the instabilities of a swirling jet. Phys. Fluids 6, 263–276.Google Scholar
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer: an experimental study. J. Fluid Mech. 197, 453–477.Google Scholar
Pedley, T. J. 1968 On the instability of rapidly rotating shear flows to non-axisymmetric disturbances. J. Fluid Mech. 31, 603–607.Google Scholar
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97–115.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Rusak, Z., Choi, J. J., Bourquard, N. & Wang, S. 2015 Vortex breakdown of compressible subsonic swirling flows in a finite-length straight circular pipe. J. Fluid Mech. 781, 3–27.Google Scholar
Rusak, Z., Choi, J. J. & Lee, J.-H. 2007 Bifurcation and stability of near-critical compressible swirling flows. Phys. Fluids 19 (11), 114107.Google Scholar
Sarpkaya, T. 1995 Turbulent vortex breakdown. Phys. Fluids 7 (10), 2301–2303.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Shrestha, K., Parras, L., Pino, C. D., Sanmiguel-Rojas, E. & Fernandez-Feria, R. 2013 Experimental evidence of convective and absolute instabilities in rotating Hagen–Poiseuille flow. J. Fluid Mech. 716, R12.Google Scholar
Sugimoto, N. 2010 Thermoacoustic-wave equations for gas in a channel and a tube subject to temperature gradient. J. Fluid Mech. 658, 89–116.CrossRefGoogle Scholar
Swaminathan, N. & Bray, K. N. C. 2011 Turbulent Premixed Flames. Cambridge University Press.Google Scholar
Synge, L. 1933 The stability of heterogeneous liquids. Trans. R. Soc. Can. 27, 1–18.Google Scholar
Wang, S. & Rusak, Z. 1996 On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8 (4), 1007–1016.Google Scholar
Wang, S., Rusak, Z., Gong, R. & Liu, F. 2016 On the three-dimensional stability of a solid-body rotation flow in a finite-length rotating pipe. J. Fluid Mech. 797, 284–321.CrossRefGoogle Scholar
Wang, S., Yang, V., Hsiao, G., Hsieh, S.-Y. & Mongia, H. C. 2007 Large-eddy simulations of gas-turbine swirl injector flow dynamics. J. Fluid Mech. 583, 99–122.Google Scholar