High-order visualization of three-dimensional lagrangian coherent structures with DG-FTLE

The two-dimensional DG-FTLE algorithm for computing Finite-Time Lyapunov Exponent (FTLE) fields developed in [Nelson and Jacobs, J. Comp. Phys. 295 (2015)] 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. 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.