Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-29T08:46:31.278Z Has data issue: false hasContentIssue false

Lagrangian coherent structures in flow past a backward-facing step

Published online by Cambridge University Press:  16 August 2022

Chenyang Huang
Affiliation:
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Marine Numerical Experimental Center, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
Alistair G.L. Borthwick
Affiliation:
School of Engineering, The University of Edinburgh, The King's Buildings, Edinburgh EH9 3FB, UK School of Engineering, Computing and Mathematics, University of Plymouth, Plymouth PL4 8AA, UK
Zhiliang Lin*
Affiliation:
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Marine Numerical Experimental Center, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
*
Email address for correspondence: linzhiliang@sjtu.edu.cn

Abstract

This paper investigates flow past a backward-facing step (BFS) in a duct at Reynolds number $Re = 5080$ based on step height, mean inflow velocity and fluid kinematic viscosity. The flow configuration matches a combustion experiment conducted by Pitz and Daily in 1983. High-resolution velocity fields are obtained in OpenFOAM by direct numerical simulation, and the flow field analysed by Lagrangian approaches. Trajectories of fluid particles in areas of interest are obtained by high-order numerical integration, and used to compute finite-time Lyapunov exponents (FTLEs) and polar rotation angles. Lagrangian coherent structures (LCSs) are extracted using geodesic theory, including hyperbolic LCSs and elliptic LCSs. We use complementary qualitative and quantitative LCS analyses to uncover the underlying flow structures. Notably, we find that a flow pathway in which fluid particles rarely diverge from adjacent particles is opened and closed by FTLE ridges determined by the periodic shedding of vortices from the BFS. Two dominant vortices with significant Lagrangian coherence, generated respectively by the separated boundary layer and shear layer, are self-sustaining and of comparable strength. Hyperbolic repelling LCSs act as transport barriers between the pathway and cores of the coherent vortices, thus playing a major part in the fluid entrainment process. Interactions between these different geometric regions partitioned by LCSs lead to intrinsic complexity in the BFS flow.

Type
JFM Papers
Copyright
© The Author(s), 2022. Published by 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

REFERENCES

Armaly, B.F., Durst, F., Pereira, J.C.F. & Schönung, B. 1983 Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473496.CrossRefGoogle Scholar
Branicki, M. & Wiggins, S. 2010 Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents. Nonlinear Process. Geophys. 17 (1), 136.CrossRefGoogle Scholar
Brown, G.L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64 (4), 775816.CrossRefGoogle Scholar
Cardwell, B. & Mohseni, K. 2007 A Lagrangian view of vortex shedding and reattachment behavior in the wake of a 2D airfoil. In 37th AIAA Fluid Dynamics Conference and Exhibit, AIAA Paper 2007-4231.Google Scholar
Chong, M.S., Perry, A.E. & Cantwell, B.J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2 (5), 765777.CrossRefGoogle Scholar
Elhmaïdi, D., Provenzale, A. & Babiano, A. 1993 Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single-particle dispersion. J. Fluid Mech. 257, 533558.CrossRefGoogle Scholar
Farazmand, M., Blazevski, D. & Haller, G. 2014 Shearless transport barriers in unsteady two-dimensional flows and maps. Physica D 278, 4457.CrossRefGoogle Scholar
Farazmand, M. & Haller, G. 2016 Polar rotation angle identifies elliptic islands in unsteady dynamical systems. Physica D 315, 112.CrossRefGoogle Scholar
Green, M.A., Rowley, C.W. & Haller, G. 2007 Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech. 572, 111120.CrossRefGoogle Scholar
Hadjighasem, A., Farazmand, M., Blazevski, D., Froyland, G. & Haller, G. 2017 A critical comparison of Lagrangian methods for coherent structure detection. Chaos 27 (5), 053104.CrossRefGoogle ScholarPubMed
Hadjighasem, A. & Haller, G. 2016 Geodesic transport barriers in Jupiter's atmosphere: a video-based analysis. SIAM Rev. 58 (1), 6989.CrossRefGoogle Scholar
Haller, G. 2002 Lagrangian coherent structures from approximate velocity data. Phys. Fluids 14 (6), 18511861.Google Scholar
Haller, G. 2011 A variational theory of hyperbolic Lagrangian coherent structures. Physica D 240 (7), 574598.Google Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137162.CrossRefGoogle Scholar
Haller, G. & Beron-Vera, F.J. 2012 Geodesic theory of transport barriers in two-dimensional flows. Physica D 241 (20), 16801702.CrossRefGoogle Scholar
Haller, G. & Beron-Vera, F.J. 2013 Coherent Lagrangian vortices: the black holes of turbulence. J. Fluid Mech. 731, R4.CrossRefGoogle Scholar
Haller, G., Hadjighasem, A., Farazmand, M. & Huhn, F. 2016 Defining coherent vortices objectively from the vorticity. J. Fluid Mech. 795, 136173.CrossRefGoogle Scholar
Haller, G. & Yuan, G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147 (3–4), 352370.CrossRefGoogle Scholar
Holmes, P.J., Lumley, J.L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Hu, R., Wang, L. & Fu, S. 2016 Investigation of the coherent structures in flow behind a backward-facing step. Intl J. Numer. Meth. Heat Fluid Flow 26, 10521068.CrossRefGoogle Scholar
Hunt, J.C.R., Wray, A.A. & Moin, P. 1988 Eddies, streams, and convergence zones in turbulent flows. In Studying Turbulence Using Numerical Simulation Databases-I1, vol. 193. NASA Tech. Rep. 19890015184.Google Scholar
Jeong, J. & Hussein, A.K.M.F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.Google Scholar
Le, H., Moin, P. & Kim, J. 1997 Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech. 330, 349374.CrossRefGoogle Scholar
Ma, X. & Schröder, A. 2017 Analysis of flapping motion of reattaching shear layer behind a two-dimensional backward-facing step. Phys. Fluids 29 (11), 115104.CrossRefGoogle Scholar
Nadge, P.M. & Govardhan, R.N. 2014 High Reynolds number flow over a backward-facing step: structure of the mean separation bubble. Exp. Fluids 55 (1), 122.CrossRefGoogle Scholar
Neamtu-Halic, M., Krug, D., Mollicone, J., Van Reeuwijk, M., Haller, G. & Holzner, M. 2020 Connecting the time evolution of the turbulence interface to coherent structures. J. Fluid Mech. 898, A3.CrossRefGoogle Scholar
Neto, A.S., Grand, D., Métais, O. & Lesieur, M. 1993 A numerical investigation of the coherent vortices in turbulence behind a backward-facing step. J. Fluid Mech. 256, 125.CrossRefGoogle Scholar
Nolan, P.J., Serra, M. & Ross, S.D. 2020 Finite-time Lyapunov exponents in the instantaneous limit and material transport. Nonlinear Dyn. 100 (4), 38253852.CrossRefGoogle Scholar
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17 (3), 445454.Google Scholar
Onu, K, Huhn, F. & Haller, G. 2015 LCS tool: a computational platform for Lagrangian coherent structures. J. Comput. Sci. 7, 2636.CrossRefGoogle Scholar
Peng, J. & Dabiri, J.O. 2009 Transport of inertial particles by Lagrangian coherent structures: application to predator-prey interaction in jellyfish feeding. J. Fluid Mech. 623, 7584.CrossRefGoogle Scholar
Pierrehumbert, R.T. & Yang, H. 1993 Global chaotic mixing on isentropic surfaces. J. Atmos. Sci. 50 (15), 24622480.2.0.CO;2>CrossRefGoogle Scholar
Pitz, R.W. & Daily, J.W. 1983 Combustion in a turbulent mixing layer formed at a rearward-facing step. AIAA J. 21 (11), 15651570.CrossRefGoogle Scholar
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31 (1), 5593.CrossRefGoogle Scholar
Sampath, R., Mathur, M. & Chakravarthy, S.R. 2016 Lagrangian coherent structures during combustion instability in a premixed-flame backward-step combustor. Phys. Rev. E 94 (6), 062209.CrossRefGoogle Scholar
Serra, M. & Haller, G. 2016 Objective Eulerian coherent structures. Chaos 26 (5), 053110.CrossRefGoogle ScholarPubMed
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Maths 45 (3), 561571.CrossRefGoogle Scholar
Suara, K., Khanarmuei, M., Ghosh, A., Yu, Y., Zhang, H., Soomere, T. & Brown, R.J. 2020 Material and debris transport patterns in Moreton Bay, Australia: the influence of Lagrangian coherent structures. Sci. Total Environ. 721, 137715.CrossRefGoogle ScholarPubMed
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93102.CrossRefGoogle Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48 (2–3), 273294.CrossRefGoogle Scholar
Xie, B., Deng, X. & Liao, S. 2019 High-fidelity solver on polyhedral unstructured grids for low-Mach number compressible viscous flow. Comput. Meth. Appl. Mech. Engng 357, 112584.CrossRefGoogle Scholar
Yang, D., He, S., Shen, L. & Luo, X. 2021 Large eddy simulation coupled with immersed boundary method for turbulent flows over a backward facing step. Proc. Inst. Mech. Engrs 235 (15), 27052714.Google Scholar