Integrated Computation of Finite Time Lyapunov Exponent During ...

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Computing the backward time flow map, Φ t 0 t0 +T The Eulerian Treatment [Leung, 2011]: A natural way to evolve the backward time flow map in forward time. Treat each component <strong>of</strong> backward time flow map as an Eulerian scalar field on the fixed grid. dΦ t 0 t0 +T ,i + (u · ∇)Φ t 0 t0 +T ,i = 0 (5) dt Solve three level-set equations using a simple, semi-Lagrangian scheme [Staniforth and Côté, 1991] At time t = t 0 , set Φ t0 t0+T = x cv everywhere Evolve each scalar up to t = t 0 + T by solving level set equation. At time t = t 0 + T , Φ t0 t0+T is the backward time T flow map. Finn & Apte FTLE & DNS Integration 7
Efficient composition <strong>of</strong> evolving FTLE fields When computing evolving FTLE fields, redundant tracer integrations eliminated by splitting time T flow map into N time h substeps. Φ t 0+T t 0 Φ t 0+T t 0 = Φ Nh (N−1)h ◦ · · · ◦ Φt 0+2h t 0 +h ◦ Φt 0+h t 0 (6) = IΦ t 0+Nh t 0 +(N−1)h ◦ · · · ◦ IΦt 0+2h t 0 +h ◦ Φt 0+h t 0 (7) [Brunton and Rowley, 2010] Finn & Apte FTLE & DNS Integration 8
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 and 10: Definition and typical computation
Page 11 and 12: The flow solver Originally develope
Page 13: Computing the forward time flow map
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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