×

High-order visualization of three-dimensional Lagrangian coherent structures with DG-FTLE. (English) Zbl 1390.76342

Summary: The two-dimensional DG-FTLE algorithm for computing finite-time Lyapunov Exponent (FTLE) fields developed in [the authors, J. Comput. Phys. 295, 65–86 (2015; Zbl 1349.76244)] is extended to and tested in three-dimensions and for complex geometries with curved boundaries. In three-dimensions, flow deformation maps, whose eigenvalues determine the FTLE, develop steep gradients over shorter finite times than in two-dimensions. Higher-order DG approximations of the steep gradients yield Gibbs oscillations and poor conditioning of interpolation matrices. In order to reduce Gibbs oscillations, an exponential filter is used that smoothens FTLE fields. \(h\)-adaptivity is shown to improve conditioning. To improve computational efficiency of large scale three-dimensional computations, the DG-FTLE algorithm is adapted to parallel architectures. Three test cases assess the performance of DG-FTLE on curved geometries and in three-dimensions including the two- and three-dimensional viscous flow over a NACA 65-(1)412 airfoil and the ABC flow. The test cases show the efficacy of the exponential filter and \(h\)-adaptivity in removing Gibbs oscillations and improving conditioning. The cases also show the ability of DG-FTLE to visualize FTLE fields in three-dimensional, unsteady flows over complex geometries.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76Nxx Compressible fluids and gas dynamics
76M27 Visualization algorithms applied to problems in fluid mechanics

Citations:

Zbl 1349.76244

Software:

DG-FTLE
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Haller, G.; Yuan, G., Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147, 352-370, (2000) · Zbl 0970.76043
[2] Haller, G., Determining material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149, 248-277, (2001) · Zbl 1015.76077
[3] Haller, G., A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240, 574-598, (2011) · Zbl 1214.37056
[4] Haller, G.; Beron-Vera, F. J., Geodesic theory of transport barriers in two-dimensional flow, Physica D, 241, 1680-1702, (2012) · Zbl 1417.37109
[5] Farazmand, M.; Haller, G., Computing Lagrangian coherent structures from their variational theory, Chaos, 22, 013128, (2012) · Zbl 1331.37128
[6] Shadden, S. C.; Lekien, F.; Marsden, J. E., Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 271-304, (2005) · Zbl 1161.76487
[7] Haller, G., Lagrangian coherent structures, Annu Rev Fluid Mech, 47, 137-162, (2015)
[8] Salman, H.; Hesthaven, J. S.; Warburton, T.; Haller, G., Predicting transport by Lagrangian coherent structures with a high-order method, Theor Comput Fluid Dyn, 21, 39-58, (2007) · Zbl 1170.76346
[9] Cardwell, B. M.; Mohseni, K., Vortex shedding over a two-dimensional airfoil: where the particles come from, AIAA journal, 46, 3, 545-547, (2008)
[10] Lipinski, D.; Cardwell, B.; Mohseni, K., A Lagrangian analysis of a two-dimensional airfoil with vortex shedding, J Phys A, 41, 34, 344011, (2008) · Zbl 1190.76143
[11] Sadlo, F.; Rigazzi, A.; Peikert, R., Time-dependent visualization of Lagrangian coherent structures by grid advection, (Pascucci, V.; Tricoche, X.; Hagen, H.; Tierny, J., Topological Methods in Data Analysis and Visualization, (2011), Springer Berlin), 151-165
[12] Lekien, F.; Ross, S. D., The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds, Chaos, 20, 017505, (2010) · Zbl 1311.76109
[13] Nelson, D. A.; Jacobs, G. B., DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods, J Comput Phys, 295, 65-86, (2015) · Zbl 1349.76244
[14] Nelson, D. A.; Jacobs, G. B., Computation of forward-time finite-time Lyapunov exponents using discontinuous-Galerkin spectral element methods, ASME 2013 International Mechanical Engineering Congress and Exposition, (2013), ASME
[15] Green, M. A.; Rowley, C. W.; Smits, A. J., The unsteady three-dimensional wake produced by a trapezoidal pitching panel, J Fluid Mech, 685, 117-145, (2011) · Zbl 1241.76027
[16] Shadden, S. C.; Astorino, M.; Gerbeau, J.-F., Computational analysis of an aortic valve jet with Lagrangian coherent structures, Chaos, 20, 017512, (2010)
[17] Bettencourt, J. H.; López, C.; Hernández-García, E., Oceanic three-dimensional Lagrangian coherent structures: a study of a mesoscale eddy in the benguela upwelling region, Ocean Modell, 51, 73-83, (2012)
[18] Bettencourt, J. H.; López, C.; Hernández-García, E., Characterization of coherent structures in three-dimensional turbulent flows using the finite-size Lyapunov exponent, J Phys A, 46, 25, 254022, (2013) · Zbl 1351.37117
[19] Knutson, B.; Tang, W.; Chan, P. W., Lagrangian coherent structure analysis of terminal winds: three-dimensionality, intramodel variations, and flight analyses, Adv Meteorol, 2015, (2015)
[20] Garth, C.; Gerhardt, F.; Tricoche, X.; Hagen, H., Efficient computation and visualization of coherent structures in fluid flow applications, IEEE Trans Visual Comput Graphic, 13, 6, 1464-1471, (2007)
[21] Bourgeois, J. A.; Sattari, P.; Martinuzzi, R. J., Coherent vortical and straining structures in the finite wall-mounted square cylinder wake, Int J Heat Fluid Fl, 35, 130-140, (2012)
[22] Sadlo, F.; Peikert, R., Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction, Visual Comput Graphic, IEEE Trans, 13, 6, 1456-1463, (2007)
[23] Lipinski, D.; Mohseni, K., Difficulties in finding Lagrangian coherent structures in 3d flows, 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 1101-1106, (2012)
[24] Kopriva, D. A.; Woodruff, S. L.; Hussiani, M. Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int J Numer Meth Engng, 53, 105-122, (2002) · Zbl 0994.78020
[25] White, F. M.; Corfield, I., Viscous fluid flow, (2006), McGraw-Hill New York
[26] Kopriva, D. A., A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations, J Comp Phys, 143, 125-158, (1998) · Zbl 0921.76121
[27] Kopriva, D. A., Implementing spectral methods for partial differential equations, (2009), Springer New York · Zbl 1172.65001
[28] Hu, F. Q.; Hussaini, M. Y.; Manthey, J., Application of low dissipation and dispersion Runge-Kutta schemes to benchmark problems in computational aeroacoustics, ICASE/LaRC workshop on benchmark problems in Computational Aeroacoustics (CAA), 73-98, (1994), National Aeronautics and Space Administration
[29] Haller, G., Lagrangian coherent structures from approximate velocity data, Phys Fluids, 14, 6, 1851-1861, (2002) · Zbl 1185.76161
[30] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zang, T., Spectral methods in fluid dynamics, (1987), Springer-Verlag New York
[31] Jacobs, G. B.; Kopriva, D. A.; Mashayek, F., Towards efficient tracking of inertial particles with high-order multidomain methods, J Comput Appl Math, 206, 392-408, (2007) · Zbl 1290.76110
[32] Rivlin, T. J., An introduction to the approximation of functions, (1981), Dover New York
[33] Atkinson, K.; Han, W., Theoretical numerical analysis, 39, (2005), Springer
[34] Hlawatsch, M.; Sadlo, F.; Weiskopf, D., Hierarchical line integration, Visual Comput Graphic, IEEE Trans, 17, 8, 1148-1163, (2011)
[35] Camp, D.; Garth, C.; Childs, H.; Pugmire, D.; Joy, K. I., Streamline integration using MPI-hybrid parallelism on a large multicore architecture, Visual Comput Graphic, IEEE Trans, 17, 11, 1702-1713, (2011)
[36] Nouanesengsy, B.; Lee, T.-Y.; Lu, K.; Shen, H.-W.; Peterka, T., Parallel particle advection and FTLE computation for time-varying flow fields, Proceedings of the international conference on high performance computing, networking, storage and analysis, 61, (2012), IEEE Computer Society Press
[37] Deville, M. O.; Fischer, P. F.; Mund, E. H., High-order methods for incompressible fluid flow, 9, (2002), Cambridge University Press · Zbl 1007.76001
[38] Karniadakis, G.; Sherwin, S., Spectral/hp element methods for computational fluid dynamics, (2013), Oxford University Press · Zbl 1256.76003
[39] Hesthaven, J. S.; Warburton, T., Nodal discontinuous-Galerkin methods: algorithms, analysis, and applications, (2008), Springer New York · Zbl 1134.65068
[40] Vandeven, H., Family of spectral filters for discontinuous problems, J Scic Comput, 6, 2, 159-192, (1991) · Zbl 0752.35003
[41] Don, W. S., Numerical study of pseudospectral methods in shock wave applications, J Comput Phys, 110, 1, 103-111, (1994) · Zbl 0797.76068
[42] Brunton, S. L.; Rowley, C. W., Fast computation of finite-time Lyapunov exponent fields for unsteady flows, Chaos, 20, 017503, (2010) · Zbl 1311.76102
[43] Haller, G., An objective definition of a vortex, J Fluid Mech, 525, 1-26, (2005) · Zbl 1065.76031
[44] Sulman, M. H.; Huntley, H. S.; Lipphardt, B.; Kirwan, A., Leaving flatland: diagnostics for Lagrangian coherent structures in three-dimensional flows, Physica D, 258, 77-92, (2013) · Zbl 1356.76108
[45] Nelson, D. A.; Jacobs, G. B.; Kopriva, D. A., Effect of boundary representation on viscous, separated flows in a discontinuous-Galerkin Navier-Stokes solver, Theor Comput Fluid Dynam, 1-23, (2016)
[46] Hunt, J.; Wray, A.; Moin, P., Eddies, streams, and convergence zones in turbulent flows, Stud Turbul Using Numer Simul Databases-I1, 193, (1988)
[47] Okubo, A., Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences, Deep sea research and oceanographic abstracts, 17, 445-454, (1970), Elsevier
[48] Chong, M.; Perry, A. E.; Cantwell, B., A general classification of three-dimensional flow fields, Phys Fluid A, 2, 5, 765-777, (1990)
[49] Jeong, J.; Hussain, F., On the identification of a vortex, J Fluid Mech, 285, 69-94, (1995) · Zbl 0847.76007
[50] Green, M. A.; Rowley, C. W.; Haller, G., Detection of Lagrangian coherent structures in three-dimensional turbulence, J Fluid Mech, 572, 111-120, (2007) · Zbl 1111.76025
[51] Branicki, M.; Wiggins, S., Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponenets, Nonlin Processes Geophys, 17, 1, 1-36, (2010)
[52] Kasten, J.; Petz, C.; Hotz, I.; Noack, B. R.; Hege, H.-C., Localized finite-time Lyapunov exponent for unsteady flow analysis., VMV, 265-276, (2009)
[53] Diamessis, P. J.; Lin, Y.-C.; Domaradzki, J. A., Effective numerical viscosity in spectral multidomain penalty method-based simulations of localized turbulence, J Comput Phys, 227, 17, 8145-8164, (2008) · Zbl 1256.76050
[54] Meiburg, E.; Lasheras, J., Comparison between experiments and numerical simulations of three-dimensional plane wakes, Phys Fluids (1958-1988), 30, 3, 623-625, (1987)
[55] Meiburg, E.; Lasheras, J., Experimental and numerical investigation of the three-dimensional transition in plane wakes, J Fluid Mech, 190, 1-37, (1988)
[56] Lasheras, J.; Meiburg, E., Three-dimensional vorticity modes in the wake of a flat plate, Phys Fluid A, 2, 3, 371-380, (1990)
[57] Williamson, C. H., Vortex dynamics in the cylinder wake, Annu Rev Fluid Mech, 28, 1, 477-539, (1996)
[58] Kerswell, R. R., Elliptical instability, Annu Rev Fluid Mech, 34, 1, 83-113, (2002) · Zbl 1047.76022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.