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Predicting transport by Lagrangian coherent structures with a high-order method

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Abstract

Recent developments in identifying Lagrangian coherent structures from finite-time velocity data have provided a theoretical basis for understanding chaotic transport in general flows with aperiodic dependence on time. As these theoretical developments are extended and applied to more complex flows, an accurate and general numerical method for computing these structures is needed to exploit these ideas for engineering applications. We present an unstructured high-order hp/spectral-element method for solving the two-dimensional compressible form of the Navier–Stokes equations. A corresponding high-order particle tracking method is also developed for extracting the Lagrangian coherent structures from the numerically computed velocity fields. Two different techniques are used; the first computes the direct Lyapunov exponent from an unstructured initial particle distribution, providing easier resolution of structures located close to physical boundaries, whereas the second advects a small material line initialized close to a Lagrangian saddle point to delineate these structures. We demonstrate our algorithm on simulations of a bluff-body flow at a Reynolds number of Re = 150 and a Mach number of M = 0.2 with and without flow forcing. We show that, in the unforced flow, periodic vortex shedding is predicted by our numerical simulations that is in stark contrast to the aperiodic flow field in the case with forcing. An analysis of the Lagrangian structures reveals a transport barrier that inhibits cross-wake transport in the unforced flow. The transport barrier is broken with forcing, producing enhanced transport properties by chaotic advection and consequently improved mixing of advected scalars within the wake.

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Author notes

  1. Hayder Salman

    Present address: Department of Mathematics, CB #3250, Phillips Hall, UNC-Chapel Hill, Chapel Hill, NC, 27599, USA

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Hayder Salman & Jan S. Hesthaven

  2. Department of Computational and Applied Mathematics, Rice University, Houston, TX, 77005, USA

    Tim Warburton

  3. Department of Mechanical Engineering, MIT, Cambridge, MA, 02139, USA

    Hayder Salman & George Haller

Authors
  1. Hayder Salman
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  2. Jan S. Hesthaven
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  3. Tim Warburton
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  4. George Haller
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Correspondence to Hayder Salman.

Additional information

Communicated by M.Y. Hussaini

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Salman, H., Hesthaven, J.S., Warburton, T. et al. Predicting transport by Lagrangian coherent structures with a high-order method. Theor. Comput. Fluid Dyn. 21, 39–58 (2007). https://doi.org/10.1007/s00162-006-0031-0

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  • DOI: https://doi.org/10.1007/s00162-006-0031-0

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