Integrated Computation of Finite Time Lyapunov Exponent During ...
Integrated Computation of Finite Time Lyapunov Exponent During ...
Integrated Computation of Finite Time Lyapunov Exponent During ...
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Definition and typical computation <strong>of</strong> the FTLE<br />
1 Release a grid <strong>of</strong> tracers and see where they go: This gives us the The flow map,<br />
Φ(x 0 , t 0 , )<br />
∫ t1<br />
Φ t 1<br />
t0<br />
(x 0 , t 0 ) = x 0 + u(x(τ), τ)dτ (1)<br />
t 0<br />
2 Differentiate the flow map W.R.T. x 0 : Then, compose the right Cauchy-Green deformation<br />
tensor, ∆<br />
⎡<br />
⎤∗ ⎡ ⎤<br />
∆ t 1<br />
t0<br />
(x 0 , t 0 ) = ⎣ dΦt 1<br />
t0<br />
(x 0 , t 0 )<br />
⎦ ⎣ dΦt 1<br />
t0<br />
(x 0 , t 0 )<br />
⎦ , (2)<br />
dx 0<br />
dx 0<br />
3 Determine the largest eigenvalue <strong>of</strong> this tensor. This is then used to compute the<br />
(maximum) FTLE field.<br />
√<br />
σ t 1<br />
1<br />
t0<br />
(x 0 , t 0 ) =<br />
|t 1 − t 0 | log λ max (∆ t 1<br />
t0<br />
(x 0 , t 0 )) (3)<br />
Finn & Apte FTLE & DNS Integration 4