Integrated Computation of Finite Time Lyapunov Exponent During ...

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Definition and typical computation of the FTLE 1 Release a grid of tracers and see where they go: This gives us the The flow map, Φ(x 0 , t 0 , ) ∫ t1 Φ t 1 t0 (x 0 , t 0 ) = x 0 + u(x(τ), τ)dτ (1) t 0 2 Differentiate the flow map W.R.T. x 0 : Then, compose the right Cauchy-Green deformation tensor, ∆ ⎡ ⎤∗ ⎡ ⎤ ∆ t 1 t0 (x 0 , t 0 ) = ⎣ dΦt 1 t0 (x 0 , t 0 ) ⎦ ⎣ dΦt 1 t0 (x 0 , t 0 ) ⎦ , (2) dx 0 dx 0 3 Determine the largest eigenvalue of this tensor. This is then used to compute the (maximum) FTLE field. √ σ t 1 1 t0 (x 0 , t 0 ) = |t 1 − t 0 | log λ max (∆ t 1 t0 (x 0 , t 0 )) (3) 4 Detect FTLE ridges visually or analytically: Forward time ridges ⇒ repelling LCS candidates. Backward time ridges ⇒ attracting LCS candidates. Recent, more exact theory [Haller, 2011] Note: Separate computations for forward (t 1 > t 0 ) and backward (t 1 < t 0 ) time Note: Let T be the absolute value of the integration time, |t 1 − t 0 | We can then write forward and backward time T flow maps as Φ t 0+T t and Φ t 0−T 0 t 0 Finn & Apte FTLE & DNS Integration 4
The flow solver Originally developed for DNS/LES on very large grids in complex configurations [Moin and Apte, 2006], Distributed Memory (MPI) parallelization and good scalability to more than 1000 processors. Co-Located (u, p known at cell centers), finite volume discretization Arbitrary cell shapes possible: Hex, tet, pyramid, etc... Fractional time-stepping, and Algebraic Multigrid (AMG) solver for pressure Poisson equation [Falgout and Yang, 2002] Extensions for Multiphase flow problems: Subgrid scale Euler-Lagrange models [Shams et al., 2010], Resolved rigid body motion with fictitious domain approach [Apte et al., 2009] Finn & Apte FTLE & DNS Integration 5
Page 1 and 2: Integrated Computation of Finite Ti
Page 3 and 4: Main Goal: Enhance DNS capability T
Page 5 and 6: Main Goal: Enhance DNS capability T
Page 7 and 8: Definition and typical computation
Page 9: Definition and typical computation
Page 13 and 14: Computing the forward time flow map
Page 15 and 16: Efficient composition of evolving F
Page 17 and 18: Computational flow of the integrate
Page 19 and 20: Moving Boundaries: Inline Oscillati
Page 21 and 22: Complex configurations: Flow throug
Page 23 and 24: Conclusions 1 We can compute evolvi
Page 25 and 26: Physica D: Nonlinear Phenomena, 149
Page 27 and 28: Double Gyre flow: [Solomon and Goll
Page 29 and 30: Double Gyre flow: [Solomon and Goll

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