Let us consider two initial close particles advected in a two-dimensional turbulence flow. We can introduce Eulerian coordinates as
$$\begin{aligned} {\varvec{x}}={\varvec{\varPhi }}\left( t;t_0,{\varvec{\xi }}\right) \end{aligned}$$
(1)
where
\({\varvec{\varPhi }}\) is the flow map and
\({\varvec{\xi }}\) the Lagrangian coordinates. The trajectories of the particles can be obtained by solving the following set of two ordinary differential equations with appropriate initial conditions
$$\begin{aligned} \frac{d{\varvec{x}} }{d t} = {\varvec{u}} \left( {\varvec{x}} ,t \right) \end{aligned}$$
(2)
where
\({\varvec{u}} =(u,v,w)\) is the velocity field. We can evaluate Eq. (
2) on a finite time interval
\(\left[ 0,T\right]\) in order to compute the final distance that particles can experience. Therefore, if we consider as initial conditions
\({\varvec{\xi }}_0\) and
\(\varvec{\xi }_0+{\varvec{\epsilon }}\) we can evaluate the final distance between the two particles applying a linearisation
[
1]:
$$\begin{aligned} \varDelta {\varvec{x}} \left( T\right) ={\varvec{\varPhi }}\left( T;0,\varvec{\xi }_0\right) -{\varvec{\varPhi }}\left( T;0,{\varvec{\xi }}_0+{\varvec{\epsilon }}\right) \approx \nabla {\varvec{\varPhi }}\left( T;0,{\varvec{\xi }}_0\right) \varvec{\epsilon } \end{aligned}$$
(3)
The magnitude of the final distance can be evaluated as
[
20]:
$$\begin{aligned} \begin{aligned} |\varDelta {\varvec{x}} \left( T\right) |&=\sqrt{\varDelta {\varvec{x}} \left( T\right) \cdot \varDelta {\varvec{x}} \left( T\right) }=\sqrt{\left[ \nabla {\varvec{\varPhi }}\varDelta {\varvec{x}} \left( 0\right) \right] \cdot \left[ \nabla {\varvec{\varPhi }}\varDelta {\varvec{x}} \left( 0\right) \right] }= \\&=\sqrt{\varDelta {\varvec{x}} \left( 0\right) \cdot \left[ {\varvec{C}} \varDelta {\varvec{x}} \left( 0\right) \right] }=\sqrt{ {\varvec{\epsilon }}\cdot \left( {\varvec{C}} {\varvec{\epsilon }}\right) } \end{aligned} \end{aligned}$$
(4)
where
\({\varvec{C}}\) is the Cauchy-Green tensor defined as
\({\varvec{C}} =\left( \nabla {\varvec{\varPhi }}\right) ^T\nabla {\varvec{\varPhi }}\) where
\(\left( \cdot \right) ^T\) denotes the transpose. It is possible to prove that matrix
\({\varvec{C}}\) is positive definite and symmetric. Since we analyse 2D velocity fields,
\({\varvec{C}}\) has two eigenvectors
\({\varvec{e}} _1\) and
\({\varvec{e}} _2\) associated with two eigenvalues
\(0<\lambda _{1}\le \lambda _{2}\), respectively.
Among the infinite directions, in order to experience the maximum separation the two particles must be aligned along the direction pointed out by the eigenvector
\({\varvec{e}} _{2}\) associated with the maximum eigenvalue of the Cauchy-Green tensor. Therefore,
$$\begin{aligned} \text {max}\left| \varDelta {\varvec{x}} (T)\right| \approx e^{T\sigma ^{t_0+T}_{t_0}}\left| \varDelta \widetilde{{\varvec{x}}}(t_0)\right| \end{aligned}$$
(5)
where the superscript
\(\widetilde{\cdot }\) indicates alignment with the eigenvector
\({\varvec{e}} _{2}\) and
$$\begin{aligned} \sigma ^{t_0+T}_{t_0} \left( {\varvec{x}} \right) = \frac{1}{\left| T \right| } \log \sqrt{ \left( \lambda _{2} \right) } \end{aligned}$$
(6)
is the Finite-Time Lyapunov Exponent (FTLE) calculated over the finite-time interval
T. FTLE can be considered a finite-time average of the maximum expansion rate that a pair of close initial particles advected by the flow can experience in a finite-time interval
T.
An analysis based on velocity separations
\({\varvec{s}}\) between particles
[
12] in a two-dimensional turbulent flow with stationary statistics can be enlightening for joining Eulerian and Lagrangian perspectives. The separation velocity
\({\varvec{s}}\) (i.e., the Lagrangian velocity difference) can be written in terms of Eulerian velocity as:
$$\begin{aligned} \overline{|{\varvec{s}} (\varvec{\varDelta })|^2}=\overline{|{\varvec{u}} ({\varvec{x}} +\varvec{\varDelta },t)-{\varvec{u}} ({\varvec{x}} ,t)|^2} \end{aligned}$$
(7)
where
\(\varvec{\varDelta }=(\varDelta _x,\varDelta _y,\varDelta _z)\) is the separation vector with magnitude
\(\left| \varvec{\varDelta }\right| =\varDelta\) and the overline represents the expected value,
\(\overline{\left( \cdot \right) }=E[\left( \cdot \right) ]\). If we note that
\(\varDelta (t)=\left| \varDelta {\varvec{x}} (t)\right| \approx e^{t\sigma ^{t_0+t}_{t_0}}|\varDelta {\varvec{x}} (t_0)|\) and assuming that the integration is forward in time, we get (the dependence from
t is dropped for simplicity):
$$\begin{aligned} \overline{|{\varvec{s}} (\varvec{\varDelta })|^2}=\overline{\left| \frac{d\varvec{\varDelta }}{dt}\right| ^2}\approx \overline{\left| \frac{d}{dt}e^{t\sigma ^{t_0+t}_{t_0}}\right| ^2}|\varDelta {\varvec{x}} (t_0)|^2 \end{aligned}$$
(8)
The behavior of separation velocities, i.e. of the structure function, at the varying of the separation has already been analysed by
[
23]. In the present contribution we would like to analyse the links with Lyapunov exponents. The fundamental relation between the energy spectrum
E(
k) and the auto-correlation of velocity is written adopting Taylor’s hypothesis of frozen turbulence. Therefore, the energy spectrum is obtained in terms of wavenumbers
k, dividing frequencies by the time and space-averaged surface velocities
\(U_s\),
\(k=\frac{f}{U_s}\). As a result, we carry out the analysis along the streamwise direction
x
[
11]:
$$\begin{aligned} E(k)= & {} \frac{1}{2\pi }\int _{-\infty }^{+\infty }R(\varDelta )e^{-ik\varDelta }d\varDelta \end{aligned}$$
(9)
$$\begin{aligned} R(\varDelta )= & {} \int _{-\infty }^{+\infty }E(k)e^{ik\varDelta }dk \end{aligned}$$
(10)
where
$$\begin{aligned} R(\varDelta )=\overline{u(x+\varDelta ,t)u(x,t)} \end{aligned}$$
(11)
Therefore, we can write (all quantities refer to the streamwise component now):
$$\begin{aligned} \overline{|s(\varDelta )|^2}=\overline{|u^2(x+\varDelta ,t)+u^2(x,t)-2u(x+\varDelta ,t)u(x,t)|} \end{aligned}$$
(12)
If we recall that:
$$\begin{aligned} \overline{u^2(x,t)}=\int _{-\infty }^{+\infty }E(k)dk=E_t \end{aligned}$$
(13)
and we assume independence from a translation of the coordinate system, we also get that:
$$\begin{aligned} \overline{u^2(x+\varDelta ,t)}=\int _{-\infty }^{+\infty }E(k)dk=E_t \end{aligned}$$
(14)
As a result, we could compute the Fourier transform of
\(\overline{|s(\varDelta )|^2}\) as:
$$\begin{aligned} \begin{aligned} F(k)=&\frac{1}{2\pi }\int _{-\infty }^{+\infty }\overline{|s(\varDelta )|^2}e^{-ik\varDelta }d\varDelta = \\&=-\frac{1}{2\pi }\int _{-\infty }^{+\infty }2R(\varDelta )e^{-ik\varDelta }d\varDelta \\&\quad +\frac{1}{2\pi }\int _{-\infty }^{+\infty }\overline{u^2(x+\varDelta ,t)}e^{-ik\varDelta }d\varDelta + \\&\quad +\frac{1}{2\pi }\int _{-\infty }^{+\infty }\overline{u^2(x,t)}e^{-ik\varDelta }d\varDelta = \\&= - 2E(k)+2E_t\delta (k) = -2 E(k) \quad for \quad k >0 \end{aligned} \end{aligned}$$
(15)
where
\(\delta (k)\) is the Dirac delta since:
$$\begin{aligned} \int _{-\infty }^{+\infty }e^{-ikx}dx=2\pi \delta (k). \end{aligned}$$
(16)
As a result, substituting Eq.
8 in
15, we obtain:
$$\begin{aligned} F(k)= & {} \frac{1}{2\pi }\int _{-\infty }^{+\infty }\overline{|s(\varDelta )|^2}e^{-ik\varDelta }d\varDelta \nonumber \\\approx & {} \frac{1}{2\pi }\int _{-\infty }^{+\infty }\overline{\left| \frac{d}{dt}e^{t\sigma ^{t_0+t}_{t_0}}\right| ^2}e^{-ik\varDelta }d\varDelta \propto E(k). \end{aligned}$$
(17)
We define
F(
k) as the eulerian spectrum of finite-time Lyapunov exponents since we compute such a spectra from a fixed frame, i.e from an eulerian one. Using Parseval’s theorem, it is possible to compute
E(
k) directly from the streamwise velocity. Thus, we can write:
$$\begin{aligned} E(k)=\frac{1}{2\pi }\int _{-\infty }^{+\infty }R(\varDelta )e^{-ik\varDelta }d\varDelta =\lim _{X\rightarrow \infty }\left| \frac{1}{2\pi }\int _{-X}^{+X}u(x,t)e^{-ikx}dx\right| ^2=S_u \end{aligned}$$
(18)
where
X is the lenght of the domain we take into consideration. Since we cannot extend the above limit to infinity, we have to admit that the above equality holds also for a finite space. The final objective consists in comparing
\(S_u\) and
F(
k) in order to underline the connection between the Lagrangian and the Eulerian framework as shown in Eq.
17. Energy spectra will be evaluated in Sect.
4.