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3 - Hydrodynamic Flow Stability I: Linear Instability

Published online by Cambridge University Press:  27 October 2021

Tim C. Lieuwen
Tim C. Lieuwen
Affiliation:
Georgia Institute of Technology
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Summary

In Chapter 2 we showed that flow disturbances can be decomposed into vorticity, entropy, and dilatational/acoustic fluctuations. The next two chapters focus on the evolution of vorticity in flows, and how vorticity in one region of the flow interacts with other regions of vorticity to influence hydrodynamic flow stability, leading to self-organization into concentrated regions of vorticity and flow rotation. Such large-scale structures, embedded on a background of acoustic waves and broadband, smaller-scale turbulence, dominate the unsteady flow fields in combustors. These large-scale structures play important roles in processes such as combustion instabilities, mixing and entrainment, flashback, and blowoff. For example, we will discuss vortex–flame interactions repeatedly in discussions of combustion instabilities in later chapters.

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Unsteady Combustor Physics , pp. 81 - 112
Publisher: Cambridge University Press
Print publication year: 2021

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References

Alfons, M., Survey on jet instability theory. Progress in Aerospace Sciences, 1984, 21(0): pp. 159–199.Google Scholar
Kabiraj, L., Sujith, R.I., and Wahi, P., Bifurcations of self-excited ducted laminar premixed flames. Journal of Engineering for Gas Turbines and Power, 2012, 134: 031502.Google Scholar
Rhode, M., Rollins, R., Markworth, A., Edwards, K., Nguyen, K., Daw, C., and Thomas, J., Controlling chaos in a model of thermal pulse combustion. Journal of Applied Physics, 1995, 78(4): pp. 2224–2232.CrossRefGoogle Scholar
Huerre, P. and Monkewitz, P.A., Local and global instabilities in spatially developing flows. Annual Review of Fluid Mechanics, 1990, 22(1): pp. 473–537.CrossRefGoogle Scholar
Godréche, C. and Manneville, P., Hydrodynamics and Nonlinear Instabilities. 1998, Cambridge University Press.CrossRefGoogle Scholar
Blackburn, H., Barkley, D., and Sherwin, S., Convective instability and transient growth in flow over a backward-facing step. Journal of Fluid Mechanics, 2008, 603: pp. 271–304.Google Scholar
Monkewitz, P., Huerre, P., and Chomaz, J., Global linear stability analysis of weakly non-parallel shear flows. Journal of Fluid Mechanics, 1993, 251: pp. 1–20.Google Scholar
Chomaz, J., Huerre, P., and Redekopp, L., A frequency selection criterion in spatially developing flows. Studies in Applied Mathematics, 1991, 84: pp. 119–144.Google Scholar
Yildirim, B. and Agrawal, A., Full-field measurements of self-excited oscillations in momentum-dominated helium jets. Experiments in Fluids, 2005, 38(2): pp. 161–173.Google Scholar
Emerson, B., Lundrigan, J., O’Connor, J., Noble, D., and Lieuwen, T., Dependence of the bluff body wake structure on flame temperature ratio, in Proceedings of 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 2011, Orlando, FL: AIAA.Google Scholar
Schmid, P.J., Nonmodal stability theory. Annual Review of Fluid Mechanics, 2007, 39: pp. 129–162.CrossRefGoogle Scholar
Busse, F., Bounds on the transport of mass and momentum by turbulent flow between parallel plates. Journal of Applied Mathematics and Physics (ZAMP), 1969, 20(1): pp. 1–14.Google Scholar
Schmid, P.J. and Henningson, D.S., Stability and Transition in Shear Flows, Marsden, J.E. and Sirovich, L., eds. Vol. 142. 2001, New York: Springer.Google Scholar
Balasubramanian, K. and Sujith, R., Non-normality and nonlinearity in combustion–acoustic interaction in diffusion flames. Journal of Fluid Mechanics, 2008, 594: pp. 29–57.Google Scholar
Kulkarni, R., Balasubramanian, K., and Sujith, R., Non-normality and its consequences in active control of thermoacoustic instabilities. Journal of Fluid Mechanics, 2011, 670: pp. 130–149.CrossRefGoogle Scholar
Criminale, W.O., Jackson, T.L., and Joslin, R.D., Theory and Computation of Hydrodynamic Stability. 2003, Cambridge University Press.Google Scholar
Bagheri, S., Schlatter, P., Schmid, P.J., and Henningson, D.S., Global stability of a jet in crossflow. Journal of Fluid Mechanics, 2009, 624: pp. 33–44.CrossRefGoogle Scholar
Betchov, R. and Criminale, W.O.J., Stability of Parallel Flows. Applied Mathematics and Mechanics, Frenkiel, F.N. and Temple, G., eds. Vol. 10. 1967, Academic Press.Google Scholar
Ho, C. and Huerre, P., Perturbed free shear layers. Annual Review of Fluid Mechanics, 1984, 16(1): pp. 365–422.Google Scholar
Wang, S. and Rusak, Z., The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown. Journal of Fluid Mechanics, 1997, 340: pp. 177–223.CrossRefGoogle Scholar
Gaster, M., A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. Journal of Fluid Mechanics, 1962, 14(02): pp. 222–224.Google Scholar
Fjortoft, R., Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geofysiske Publikasjoner, 1950, 17(6): p. 1.Google Scholar
Menkes, J., On the stability of a shear layer. Journal of Fluid Mechanics, 1959, 6(4): pp. 518–522.CrossRefGoogle Scholar
Widnall, S.E., Bliss, D.B., and Tsai, C.-Y., The instability of short waves on a vortex ring. Journal of Fluid Mechanics, 1974, 66(1): pp. 35–47.Google Scholar
Michalke, A., Survey on jet instability theory. Progress in Aerospace Sciences, 1984, 21: pp. 159–199.Google Scholar
Howard, L., Note on a paper of John W. Miles. Journal of Fluid Mechanics, 1961, 10(04): pp. 509–512.Google Scholar
Høiland, E., On Two-Dimensional Perturbation of Linear Flow. Vol. 18. 1953, Grøndahl & søns boktr., I kommisjon hos J. Dybwad.Google Scholar
Balsa, T., On the spatial instability of piecewise linear free shear layers. Journal of Fluid Mechanics, 1987, 174: pp. 553–563.Google Scholar
Michalke, A., On the inviscid instability of the hyperbolictangent velocity profile. Journal of Fluid Mechanics, 1964, 19(04): pp. 543–556.CrossRefGoogle Scholar
Monkewitz, P. and Huerre, P., Influence of the velocity ratio on the spatial instability of mixing layers. Physics of Fluids, 1982, 25: pp. 1137–1143.Google Scholar
Tammisola, O.J. and Juniper, M.P., Coherent structures in a swirl injector at Re = 4800 by nonlinear simulations and linear global modes. Journal of Fluid Mechanics, 2016, 792: pp. 620–657.Google Scholar
Meliga, P., Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. Journal of Fluid Mechanics, 2017, 826: pp. 503–521.CrossRefGoogle Scholar
Barkley, D., Linear analysis of the cylinder wake mean flow. Europhysics Letters, 2006, 75(5): pp. 750–756.Google Scholar
Mantic-Lugo, V., Arratia, C., and Gallaire, F., A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Physics of Fluids, 2015, 27(7): p. 074103.Google Scholar
Mezic, I., Analysis of fluid flows via spectral properties of the Koopman operator. Annual Review of Fluid Mechanics, 2013, 45: pp. 357–378.CrossRefGoogle Scholar
Williamson, C.H.K., Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Physics of Fluids, 1988, 31: p. 2742.Google Scholar
Sipp, D. and Lebedev, A., Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. Journal of Fluid Mechanics, 2007, 593: pp. 333–358.Google Scholar
Turton, S.E., Tuckerman, L.S., and Barkley, D., Prediction of frequencies in thermosolutal convection from mean flows. Physics Review E, 2015, 91(4): p. 043009.Google Scholar
Manoharan, K., Frederick, M., Clees, S., O’Connor, J., and Hemchandra, S., A weakly nonlinear analysis of the precessing vortex core oscillation in a variable swirl turbulent round jet. Journal of Fluid Mechanics, 2019, 884, A29.Google Scholar
Oberleithner, K., Sieber, M., Nayeri, C.N., Paschereit, C.O., Petz, C., Hege, H.C., Noack, B.R., and Wygnanski, I., Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction. Journal of Fluid Mechanics, 2011., 679: pp. 383–414.CrossRefGoogle Scholar
Rosenhead, L., The formation of vortices from a surface of discontinuity. Proceedings of the Royal Society of London, 1931, 134(823): pp. 170–192.Google Scholar
Abernathy, F.H. and Kronauer, R.E., The formation of vortex sheets. Journal of Fluid Mechanics, 1962, 13(1): pp. 1–20.Google Scholar
Crighton, D., The Kutta condition in unsteady flow. Annual Review of Fluid Mechanics, 1985, 17(1): pp. 411–445.CrossRefGoogle Scholar
Mack, L.M., Review of linear compressible stability theory, in Stability of Time Dependent and Spatially Varying Flows, Dwoyer, D.L. and Hussaini, M.Y., eds. 1987, Springer, pp. 164–187.CrossRefGoogle Scholar
Drazin, P.G. and Reid, W.H., Hydrodynamic Stability. 2nd ed. 2004, Cambridge University Press.Google Scholar
Brouillette, M., The Richtmyer–Meshkov instability. Annual Review of Fluid Mechanics, 2002, 34(1): pp. 445–468.Google Scholar
Mohaghar, M., Carter, J., Musci, B., Reilly, D., McFarland, J., and Ranjan, D., Evaluation of turbulent mixing transition in a shock-driven variable-density flow. Journal of Fluid Mechanics, 2017, 831: pp. 779–825.Google Scholar
Drazin, P. and Reid, W., Hydrodynamic Stability. 1st ed. 1981, Cambridge University Press.Google Scholar
Day, M.J., Mansour, N.N., and Reynolds, W.C., Nonlinear stability and structure of compressible reacting mixing layers. Journal of Fluid Mechanics, 2001, 446: pp. 375–408.CrossRefGoogle Scholar

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