TY - JOUR
T1 - Dynamics of hydrodynamically unstable premixed flames in a gravitational field–local and global bifurcation structures
AU - Matsue, Kaname
AU - Matalon, Moshe
N1 - Funding Information:
K.M. gratefully acknowledges partial support of World Premier International Research Center Initiative (WPI), the Applied Math for Energy program of the International Institute for Carbon Neutral Energy Research (WPI-ICNER), sponsored by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Japan. K.M. is also partially supported by Progress100 (Global Leadership Training for Young Researchers) program grant in Kyushu University, and Japan Science and Technology Agency (JSPS) Grant-in-Aid for Young Scientists (B) [grant number JP17K14235] for Scientists (B) [grant number JP21H01001], and for Scientists (C) [grant number JP21K03360]. M.M. acknowledges the partial support of the Division of Chemical, Bioengineering, Environmental, and Transport Systems (CBET) division of the US National Science Foundation [grant number CBET 19-11530].
Publisher Copyright:
© 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
PY - 2023
Y1 - 2023
N2 - The dynamics of hydrodynamically unstable premixed flames are studied using the nonlinear Michelson–Sivashinsky (MS) equation, modified appropriately to incorporate effects due to gravity. The problem depends on two parameters: the Markstein number that characterises the combustible mixture and its diffusion properties, and the gravitational parameter that represents the ratio of buoyancy to inertial forces. A comprehensive portrait of all possible equilibrium solutions are obtained for a wide range of parameters, using a continuation methodology adopted from bifurcation theory. The results heighten the distinction between upward and downward propagation. In the absence of gravity, the nonlinear development always leads to stationary solutions, namely, cellular flames propagating at a constant speed without change in shape. When decreasing the Markstein number, a modest growth in amplitude is observed with the propagation speed reaching an upper bound. For upward propagation, the equilibrium states are also stationary solutions, but their spatial structure depends on the initial conditions leading to their development. The combined Darrieus–Landau and Rayleigh–Taylor instabilities create profiles of invariably larger amplitudes and sharper crests that propagate at an increasingly faster speed when reducing the Markstein number. For downward propagation, the equilibrium states consist in addition to stationary structures time-periodic solutions, namely, pulsating flames propagating at a constant average speed. The stabilising influence of gravity dampens the nonlinear growth and leads to spatiotemporal changes in flame morphology, such as the formation of multi-crest stationary profiles or pulsating cell splitting and merging patterns, and an overall reduction in propagation speed. The transition between these states occurs at bifurcation and exchange of stability points, which becomes more prominent when reducing the Markstein number and/or increasing the influence of gravity. In addition to the local bifurcation characterisation the global bifurcation structure of the equation, obtained by tracing the continuation of the bifurcation points themselves unravels qualitative features such as the manifestation of bi-stability and hysteresis, and/or the onset and sustenance of time-periodic solutions. Overall, the results exhibit the rich and complex dynamics that occur when gravity, however small, becomes physically meaningful.
AB - The dynamics of hydrodynamically unstable premixed flames are studied using the nonlinear Michelson–Sivashinsky (MS) equation, modified appropriately to incorporate effects due to gravity. The problem depends on two parameters: the Markstein number that characterises the combustible mixture and its diffusion properties, and the gravitational parameter that represents the ratio of buoyancy to inertial forces. A comprehensive portrait of all possible equilibrium solutions are obtained for a wide range of parameters, using a continuation methodology adopted from bifurcation theory. The results heighten the distinction between upward and downward propagation. In the absence of gravity, the nonlinear development always leads to stationary solutions, namely, cellular flames propagating at a constant speed without change in shape. When decreasing the Markstein number, a modest growth in amplitude is observed with the propagation speed reaching an upper bound. For upward propagation, the equilibrium states are also stationary solutions, but their spatial structure depends on the initial conditions leading to their development. The combined Darrieus–Landau and Rayleigh–Taylor instabilities create profiles of invariably larger amplitudes and sharper crests that propagate at an increasingly faster speed when reducing the Markstein number. For downward propagation, the equilibrium states consist in addition to stationary structures time-periodic solutions, namely, pulsating flames propagating at a constant average speed. The stabilising influence of gravity dampens the nonlinear growth and leads to spatiotemporal changes in flame morphology, such as the formation of multi-crest stationary profiles or pulsating cell splitting and merging patterns, and an overall reduction in propagation speed. The transition between these states occurs at bifurcation and exchange of stability points, which becomes more prominent when reducing the Markstein number and/or increasing the influence of gravity. In addition to the local bifurcation characterisation the global bifurcation structure of the equation, obtained by tracing the continuation of the bifurcation points themselves unravels qualitative features such as the manifestation of bi-stability and hysteresis, and/or the onset and sustenance of time-periodic solutions. Overall, the results exhibit the rich and complex dynamics that occur when gravity, however small, becomes physically meaningful.
KW - Darrieus–Landau instability
KW - Michelson–Sivashinsky equation
KW - continuation
KW - downward propagation
KW - flame dynamics
KW - global bifurcation
KW - upward propagation
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U2 - 10.1080/13647830.2023.2165968
DO - 10.1080/13647830.2023.2165968
M3 - Article
AN - SCOPUS:85147411167
SN - 1364-7830
VL - 27
SP - 346
EP - 374
JO - Combustion Theory and Modelling
JF - Combustion Theory and Modelling
IS - 3
ER -