| Citation: |
Guoqiao You, Changfeng Xue. FAST IDENTIFICATION OF THE HYPERBOLIC LAGRANGIAN COHERENT STRUCTURES IN TWO-DIMENSIONAL FLOWS BASED ON THE EULERIAN-TYPE ALGORITHMS[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 568-590. doi: 10.11948/20210229
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FAST IDENTIFICATION OF THE HYPERBOLIC LAGRANGIAN COHERENT STRUCTURES IN TWO-DIMENSIONAL FLOWS BASED ON THE EULERIAN-TYPE ALGORITHMS
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Abstract
Based on the so-called mixing-based partition, we propose an efficient Eulerian algorithm to identify the hyperbolic Lagrangian coherent structure (LCS) of any given two-dimensional (2D) flow fields, which is framework independent. To extract the required LCS, the proposed algorithm only needs to solve one single partial differential equation. Moreover, data is only required at mesh points in the implementation of the proposed algorithm. In contrast, traditional Lagrangian ray tracing approach needs to solve a system consisting of two ordinary differential equations together with a line integral along the particular particle trajectory. Furthermore, if the velocity data is only available at mesh points, the Lagrangian approach needs to implement interpolation to obtain the velocity and also the velocity gradient at non-mesh points along the particle trajectory taking off from each mesh point, which could be quite time-consuming. Based on the doubling technique, we also propose an efficient iterative Eulerian-type algorithm to identify the longtime LCS for 2D periodic flows. Numerical examples are provided to confirm the accuracy, efficiency and effectiveness of the proposed Eulerian algorithms.
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References
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Guoqiao You, Changfeng Xue. FAST IDENTIFICATION OF THE HYPERBOLIC LAGRANGIAN COHERENT STRUCTURES IN TWO-DIMENSIONAL FLOWS BASED ON THE EULERIAN-TYPE ALGORITHMS[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 568-590. doi: 10.11948/20210229
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Figure 1. Lagrangian and Eulerian interpretations of the function
$ \Psi $ [17]. (a) Lagrangian ray tracing from a given grid location$ {\bf x} $ at$ t = 0 $ . Note that$ {\bf y} $ might be a non-grid point. (b) Eulerian values of$ \Psi $ at a given grid location$ {\bf y} $ at$ t = T $ gives the corresponding take-off location at$ t = 0 $ . Note the take-off location might not be a mesh point. -
Figure 2. (Section 5.1) The mixing-based partition with
$ \epsilon = 0.1 $ at the time level (a)$ t = 1 $ , (b)$ t = 5 $ , (c)$ t = 7 $ and (d)$ t = 10 $ . -
Figure 3. (Section 5.1) (a) The
$ \sigma_0^5( {\bf x}) $ field and (b) the$ \sigma_0^{10}( {\bf x}) $ field with$ \epsilon = 0.1 $ , computed using the proposed$\textsf{Algorithm 1}$ . -
Figure 4. (Section 5.1) (a) The
$ \sigma_0^5( {\bf x}) $ field and (b) the$ \sigma_0^{10}( {\bf x}) $ field with$ \epsilon = 0.1 $ , computed using the Lagrangian approach. -
Figure 5. (Section 5.1) The
$ L_1 $ errors of the solutions$ \sigma_0^{10}( {\bf x}) $ computed using the proposed Eulerian approach (red solid line) and the Lagrangian approach (blue dot dashed line). We also plot a solid black line with slope 2 as a reference. -
Figure 6. (Section 5.1) The mixing-based partition with
$ \epsilon = 0.5 $ at the time level (a)$ t = 1 $ , (b)$ t = 5 $ , (c)$ t = 7 $ and (d)$ t = 10 $ . The white triangles are the locations of the particle taking off from$ (0.5, 0.25) $ at$ t = 0 $ . -
Figure 7. (Section 5.1) (a) The
$ \sigma_0^5( {\bf x}) $ field and (b) the$ \sigma_0^{10}( {\bf x}) $ field with$ \epsilon = 0.5 $ , computed using the proposed$\textsf{Algorithm 1}$ . -
Figure 8. (Section 5.1) The
$ \sigma_0^T( {\bf x}) $ field computed using the$\textsf{Algorithm 2}$ , for (a)$ T = 20 $ , (b)$ T = 40 $ , (c)$ T = 80 $ and (d)$ T = 160 $ , respectively. -
Figure 9. (Section 5.1) The
$ L_1 $ errors of the solution$ \sigma_0^{160}( {\bf x}) $ computed using the proposed Eulerian approach (red solid line). We also plot a solid black line with slope 2 as a reference. -
Figure 10. (Section 5.2) The mixing-based partition at the time level (a)
$ t = 0 $ , (b)$ t = 0.3 $ , (c)$ t = 0.7 $ and (d)$ t = 1.9 $ . -
Figure 11. (Section 5.2) (a) The
$ \sigma_0^1( {\bf x}) $ field and (b) the$ \sigma_0^2( {\bf x}) $ field computed using the proposed Eulerian approach. -
Figure 12. (Section 5.3) (a) The
$ \sigma_0^{10} $ field computed using the proposed Eulerian approach. -
Figure 13. (Section 5.4) (a) The
$ \sigma_0^{20} $ field and (b) the$ \sigma_0^{50} $ field computed using the proposed Eulerian approach. (c) The$ \sigma_0^{50} $ field computed using the Lagrangian approach.