## DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods.(English)Zbl 1349.76244

Summary: We present an algorithm for the computation of finite-time Lyapunov exponent (FTLE) fields using discontinuous-Galerkin (dG) methods in two dimensions. The algorithm is designed to compute FTLE fields simultaneously with the time integration of dG-based flow solvers of conservation laws. Fluid tracers are initialized at Gauss-Lobatto quadrature nodes within an element. The deformation gradient tensor, defined by the deformation of the Lagrangian flow map in finite time, is determined per element with high-order dG operators. Multiple flow maps are constructed from a particle trace that is released at a single initial time by mapping and interpolating the flow map formed by the locations of the fluid tracers after finite time integration to a unit square master element and to the quadrature nodes within the element, respectively. The interpolated flow maps are used to compute forward-time and backward-time FTLE fields at several times using dG operators. For a large finite integration time, the interpolation is increasingly poorly conditioned because of the excessive subdomain deformation. The conditioning can be used in addition to the FTLE to quantify the deformation of the flow field and identify subdomains with material lines that define Lagrangian coherent structures. The algorithm is tested on three benchmarks: an analytical spatially periodic gyre flow, a vortex advected by a uniform inviscid flow, and the viscous flow around a square cylinder. In these cases, the algorithm is shown to have spectral convergence.

### 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 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids

DG-FTLE
Full Text:

### 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., An objective definition of a vortex, J. Fluid Mech., 525, 1-26, (2005) · Zbl 1065.76031 [4] 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 [5] Haller, G., A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240, 574-598, (2011) · Zbl 1214.37056 [6] Farazmand, M.; Haller, G., Computing Lagrangian coherent structures from their variational theory, Chaos, 22, 013128, (2012) · Zbl 1331.37128 [7] Haller, G.; Beron-Vera, F. J., Geodesic theory of transport barriers in two-dimensional flow, Physica D, 241, 1680-1702, (2012) · Zbl 1417.37109 [8] 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 [9] Lekien, F.; Shadden, S. C.; Marsden, J. E., Lagrangian coherent structures in n-dimensional systems, J. Math. Phys., 48, 065404, (2007) · Zbl 1144.81374 [10] Jacobs, G. B.; Armstrong, K., Inertial particle dispersion in the Lagrangian wake of a square cylinder, (47th AIAA Aerospace Sciences Meeting, (2009), AIAA) [11] Shadden, S. C.; Astorino, M.; Gerbeau, J.-F., Computational analysis of an aortic valve jet with Lagrangian coherent structures, Chaos, 20, 017512, (2010) [12] Arzani, A.; Shadden, S. C., Characterization of the transport topology in patient-specific abdominal aortic aneurysm models, Phys. Fluids, 24, 081901, (2012) [13] Wang, Y.; Haller, G.; Banaszuk, A.; Tadmor, G., Closed-loop Lagrangian separation control in a bluff body shear flow model, Phys. Fluids, 15, 8, 2251-2266, (2003) · Zbl 1186.76562 [14] Tang, W.; Chan, P. W.; Haller, G., Accurate extraction of Lagrangian coherent structures over finite domains with application to flight data analysis over Hong Kong international airport, Chaos, 20, 017502, (2010) · Zbl 1311.70034 [15] Garth, C.; Gerhardt, F.; Tricoche, X.; Hagen, H., Efficient computation and visualization of coherent structures in fluid flow applications, IEEE Trans. Vis. Comput. Graph., 13, 6, 1464-1471, (2007) [16] 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 [17] Brunton, S. L.; Rowley, C. W., Fast computation of finite-time Lyapunov exponent fields for unsteady flows, Chaos, 20, 017503, (2010) · Zbl 1311.76102 [18] 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 [19] Nelson, D. A.; Jacobs, G. B., Computation of forward-time finite-time Lyapunov using discontinuous-Galerkin spectral element methods, (ASME 2013 International Mechanical Engineering Congress and Exposition, (2013), AIAA) [20] Jacobs, G. B.; Kopriva, D. A.; Mashayek, F., Validation study of a multidomain spectral element code for simulation of turbulent flows, AIAA J., 43, 6, 1256-1264, (2004) [21] Surana, A.; Jacobs, G.; Haller, G., Extraction of separation and reattachment surfaces from 3d steady shear flows, AIAA J., 45, 1290-1302, (2007) [22] Sengupta, K.; Jacobs, G.; Mashayek, F., Large-eddy simulation of compressible flows using a spectral multi-domain method, Int. J. Numer. Methods Fluids, 61, 3, (2009) · Zbl 1170.76020 [23] Sengupta, K.; Shotorban, B.; Jacobs, G.; Mashayek, F., Spectral-based simulations of particle-laden, turbulent flow, Int. J. Multiph. Flow, 35, 9, (2009) [24] Kopriva, D. A., A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations, J. Comput. Phys., 143, 125-158, (1998) · Zbl 0921.76121 [25] Kopriva, D. A.; Woodruff, S. L.; Hussaini, M. Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Methods Eng., 53, 105-122, (2002) · Zbl 0994.78020 [26] Kopriva, D. A., Implementing spectral methods for partial differential equations, (2009), Springer New York · Zbl 1172.65001 [27] 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), (1994), National Aeronautics and Space Administration), 73-98 [28] Mathur, M.; Haller, G.; Peacock, T.; Ruppert-Felsot, J. E.; Swinney, H. L., Uncovering the Lagrangian skeleton of turbulence, Phys. Rev. Lett., 98, 14, 144502, (2007) [29] Jacobs, G. B.; Kopriva, D. A.; Mashayek, F., Towards efficient tracking of inertial particles with high-order multidomain methods, J. Comput. Phys., 206, 392-408, (2007) · Zbl 1290.76110 [30] Haller, G.; Sapsis, T., Lagrangian coherent structures and the smallest finite-time Lyapunov exponent, Chaos, 21, 023115, (2011) · Zbl 1317.37106 [31] Karrasch, D.; Farazmand, M.; Haller, G., Attraction-based computation of hyperbolic Lagrangian coherent structures, J. Comput. Dyn., (2015), in press · Zbl 1353.37161 [32] Rivlin, T. J., An introduction to the approximation of functions, (1981), Dover New York · Zbl 0189.06601 [33] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127, (1999) · Zbl 0926.65090 [34] Hesthaven, J. S.; Warburton, T., Nodal discontinuous-Galerkin methods: algorithms, analysis, and applications, (2008), Springer New York · Zbl 1134.65068 [35] Jacobs, G. B.; Mashayek, F.; Kopriva, D. A., A comparison of outflow boundary conditions for the multidomain staggered-grid spectral method, Numer. Heat Transf., Part B, Fundam., 44, 3, 225-251, (2003)
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.