10:30
Lagrangian aspects of turbulence 4
Chair: Cedric Beaume
10:30
15 mins
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Effect of gravity on clustering patterns and inertial particles attractors
Franck Nicolleau, Andrzej Nowakowski
Abstract: In this contribution we study the clustering of inertial particles using a periodic kinematic simulation. The systematic Lagrangian tracking of particles makes it possible to identify the particles' clustering patterns for different values of particle's inertia and drift velocity. The different cases are characterised by different pairs of Stokes number $St$ and drift parameter $\gamma$. For the present study $0\leq St \leq 1$ and $0\leq \gamma \leq 2$. The main focus is to identify and then quantify the clustering attractor - when it exists -
that is the set of points in the physical space where the particles settle when time goes to infinity.
Depending on gravity or drift effect and inertia values, the Lagrangian attractor can have different dimensions varying from the initial three-dimensional space to two-dimensional layers and one-dimensional attractors that can be shifted from an horizontal to a vertical position.
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10:45
15 mins
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FINITE TIME LYAPUNOV EXPONENTS AND EXTREME CONCENTRATION FLUCTUATIONS IN 2D TURBULENCE
Hua Xia, Nicolas Francois, Horst Punzmann, Kamil Szewc, Michael Shats
Abstract: The Finite time Lyapunov exponent (FTLE) has been investigated in detail in laboratory 2D experiments. The balance of the forward and backward FTLE suggests the incompressible nature of the turbulence in both the electromagnetic and Faraday wave driven experiments. The tail in the PDF of the FTLE field has been shown to be correlated with the extreme concentration of the passive scale, the ‘unmixing’ event.
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11:00
15 mins
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LAGRANGIAN ANALYSIS OF TURBULENT ROTATING CONVECTION
Hadi Rajaei, Rudie Kunnen, Herman Clercx
Abstract: This study aims to explore how the flow transition from one state to the other in rotating convection will affect the Lagrangian statistics of (fluid) particles. 3D Particle Tracking Velocimetry (3D-PTV) is employed in a water-filled cylindrical tank of equal height and diameter 200 mm. The measurements are performed in the central volume of 50 × 50 × 50 mm3 at a Rayleigh number Ra = 1.28 × 109 and Prandtl number Pr = 6.7. We are reporting the velocity and acceleration pdfs for different Rossby numbers. For different rotation rates, the transverse velocity pdfs show a Gaussian distribution. The vertical velocity pdf has slightly wider tails for stationary and high rotation rate cases, while it approaches the Gaussian distribution for intermediate rotation rates. The acceleration pdfs have significantly wider tails in comparison to those of a Gaussian distribution which is similar to the other turbulent flows. Increasing rotation results in less intermittency in vertical acceleration in the center of RB.
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11:15
15 mins
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Braid Entropy of Faraday Waves driven 2D Turbulence
Nicolas FRANCOIS, Hua Xia, Horst Punzmann, Michael Shats
Abstract: We report new experimental results that use tools from braid theory to characterize two-dimensional turbulent flows driven by Faraday waves. The average topological length of the material fluid lines is found to grow exponentially with time. It allows us to compute the braid’s topological entropy SBraid. We show that SBraid increases as the square root of the turbulence kinetic energy E ~ u^2, where u^2 is the horizontal velocity variance . At long times, the PDFs of Lbraid are positively skewed and present strong exponential tails.
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11:30
15 mins
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LAGRANGIAN AND EULERIAN ROTATING TURBULENCE
Luca Biferale, Irene Mazzitelli, Fabio Bonaccorso, Michel A.T.V Hinsberg, Alessandra S. Lanotte, Stefano Musacchio, Prasad Perlekar, Federico Toschi
Abstract: State-of-the-art direct numerical simulations of rotating turbulence at changing Reynolds and Rossby numbers are presented. Flow is also seeded with millions of particles, with and without inertia, light and heavy. We study two regimes, at high and low rotation. Heavy and light particles are injected along different axis of rotations, allowing to study the combined effects of preferential concentration in presence of Coriolis and Centripetal forces.
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11:45
15 mins
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Clustering of chiral particles in flows with broken parity invariance
Kristian Gustavsson, Luca Biferale
Abstract: \begin{summary}
The dynamics of small particles suspended in turbulent flows is an important problem in Nature and in Science.
Previous work has mainly focused on the motion of spherical particles, while less is known about particles with asymmetric shapes.
We study particles which break parity invariance (chiral particles).
Particles of different chirality may respond differently to the structures of the flow.
Helicoidal-like structures in the flow affect the particles differently depending on the parity of the helicoid as well as on the chirality of the particle.
For flows where one of the two parities of the helicoidal-like structures is more common suspended chiral particles experience different levels on clustering depending on their chirality.
Using analytical methods and direct numerical simulations we investigate the mechanisms of preferential sampling and clustering of chiral particles in flows with local or global breaking of parity invariance.
\end{summary}
\section{Chiral particles}
One of the simplest examples of particles which break parity invariance are \lq isotropic helicoids', first studied by Kelvin~\cite{Kel71}.
Just as spherical particles, isotropic helicoids have diagonal resistance -and moment of inertia tensors.
But unlike spherical particles they have a coupling between the translational and rotational degrees of freedom.
The motion of a small isotropic helicoid with velocity $\ve v$ and angular velocity $\ve\omega$ is governed by the force $\ve F$ and torque $\ve\tau$~\cite{Bre83}
\begin{eqnarray}
\ve F=m\dot{\ve v}\!\!\!\!&=&\!\!\!\!C^{{\rm tt}}(\ve u-\ve v)+C^{{\rm tr}}(\ve\Omega-\ve\omega)\nonumber\\
\ve\tau=I_0\dot{\ve\omega}\!\!\!\!&=&\!\!\!\!C^{{\rm tr}}(\ve u-\ve v)+C^{\rm rr}(\ve\Omega-\ve\omega)\,.
\label{eq:eqm}
\end{eqnarray}
Here $m$ and $I_0$ are the mass and moment of inertia of the particle, $\ve u$ is the flow velocity at the particle position and $\ve\Omega\equiv\ve\nabla\wedge\ve u/2$ is half the vorticity of the flow.
The diagonal components of the resistance tensors, $C^{{\rm tt}}$ and $C^{\rm rr}$, and the parity-breaking coupling, $C^{{\rm tr}}$, are all scalar parameters.
When $C^{{\rm tr}}=0$, Eqs.~(\ref{eq:eqm}) simplify to Stokes equations for a spherical particle.
When $C^{{\rm tr}}\ne 0$, the equations are no longer invariant under parity transformations: if the spatial coordinates change sign, the terms proportional to $C^{{\rm tr}}\ne 0$ change sign relative to all other terms.
A second example of a parity-breaking particle is a rigid asymmetric particle consisting of four beads, as illustrated in Fig.~\ref{fig:four_bead}{\bf a} and~{\bf b}.
The equations of motion for the four-bead particle can be obtained using the methods in Ref.~\cite{Car99}.
Unlike spherical particles and isotropic helicoids, the center of mass of the asymmetric four-bead particle does not follow the streamlines of the flow in the limit of no particle inertia.
\begin{figure}[b]
\begin{center}
\includegraphics[width=8cm]{four_bead}
\begin{picture}(0,0)
\put(-130,38){x}
\put(-95,30){y}
\put(-109,74){z}
\end{picture}
\hspace{-1cm}\raisebox{-0.4cm}{\includegraphics[width=4cm]{helicoid}}
\begin{picture}(0,0)
\put(-240,0){{\bf a}}
\put(-153,0){{\bf b}}
\put(-58,0){{\bf c}}
\end{picture}
\end{center}
\caption{{\bf a}, {\bf b}: Asymmetric particles consisting of four well-separated beads with two different chiralities. The relative positions and orientations of the beads are kept fixed by infinitesimally thin rods. {\bf c}: Inertialess motion of the particles in a simple helicoid, $\ve u=(-5y,5x,1)$. One streamline of the flow $\ve u$ is shown as a blue line. Even though the particles lack inertia, their center of mass can not follow the streamlines. The particle trajectories are shown as green for particle {\bf a} and red for particle {\bf b}.}
\label{fig:four_bead}
\end{figure}
%Helicoidal flow: \cite{Ari13}
%Shear: \cite{}
The local helicity $H(\ve r)$ of a fluid flow can be expressed in terms of the alignment between the fluid velocity, $\ve u$, and the vorticity of the flow, $2\ve\Omega$, i.e. $H\equiv 2\ve u\cdot \ve\Omega$.
It is expected that particles of different chiralities (different sign of $C^{{\rm tr}}$ in Eq.~(\ref{eq:eqm}), or the mirror images in Fig.~\ref{fig:four_bead}{\bf a} and~{\bf b}) behave differently depending on the sign and magnitude of $H$.
To illustrate the asymmetry between the particles in Fig.~\ref{fig:four_bead}{\bf a} and~{\bf b}, their trajectories are plotted in Fig.~\ref{fig:four_bead}{\bf c} for a helicoidal flow, $\ve u=(-5y,5x,1)$ and $\ve\Omega=(0,0,5)$.
If the helicoidal flow in Fig.~\ref{fig:four_bead}{\bf c} is mirrored in the origin, the role of the red and green particles is interchanged.
\section{Flows with local or global breaking of parity invariance}
Particles of different chirality move differently through the flow, allowing them to sample different flow regions.
If the flow is reflection invariant, it is as likely to find a specific helical structure as it is to find its mirror image.
This implies that there is no major difference between trajectories of particles with different chiralities: the only difference is a reflection of the flow.
Homogeneous isotropic turbulence is reflection invariant, and the magnitude of clustering and preferential sampling is thus, on average, independent of the particle chirality.
Due to boundary conditions, or due to the nature of the forcing of the turbulence, reflection invariance is often broken in systems encountered in nature and in applications.
In flows with broken parity invariance helical structures with one sign of the local helicity $H$ is more common than the other sign.
As a net effect, particles may show different statistical properties depending on their chiralities.
We study the motion of chiral particles in DNS of turbulent flows, as well as in model flows in which one sign of the helicity is more likely.
One example of such flow is the Arnold-Beltrami-Childress (ABC) flow~\cite{Arn65}
\begin{eqnarray}
\ve u=(C\cos y+A\sin z,A\cos z+B\sin x,B\cos x+C\sin y)\,.
\end{eqnarray}
This flow has $\ve\Omega=\ve u/2$ which implies a positive helicity $H=|\ve u|^2$.
A second method to construct a reflection-breaking model flow is to project out and remove the modes of a flow which contributes to the undesired sign of $H$~\cite{Bif13}.
We address the questions on which regions of the flow are preferentially sampled by chiral particles and how the chiral particles cluster.
We discuss the similarities and differences between isotropic helicoids (\ref{eq:eqm}) and the four-bead model (Fig.~\ref{fig:four_bead}) for both reflection invariant flows and the reflection-breaking flows mentioned above.
%Very different equations of motion (four-bead model detach from flow in inertialess limit): Same clustering mechanism or not? (i.e. is there a universal clustering mechanism for chiral particles, or does clustering depend on other properties of the particle as well)
We compare our results to previous results on clustering of spherical particles in reflection-invariant flows, see for instance Ref.~\cite{Gus15a}, and in rotating flows~\cite{Elp98}.
\\
{\em Acknowledgements}.
Partially supported by ERC AdG NewTURB n. 339032.
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\bibliography{../../latex/shared/biblio}
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12:00
15 mins
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Multiscale Statistics of Lagrangian and Eulerian Acceleration in Turbulent Stratified Shear Flows
Frank Jacobitz, Kai Schneider, Marie Farge
Abstract: Direct numerical simulation data of homogeneous turbulence with shear and stratification are analyzed to study the Lagrangian and Eulerian acceleration statistics. Richardson numbers from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear flow, are considered. The scale dependence of the acceleration statistics is studied using a wavelet-based approach. The probability density functions (pdfs) of both Lagrangian and Eulerian accelerations exhibit a strong and similar influence on $Ri$. The extreme values for Lagrangian acceleration are weaker than those observed for the Eulerian acceleration. Similarly, the Lagrangian time-rate of change of fluctuating density is observed to have smaller extreme values than that of the Eulerian time-rate of change. Thus the time-rate of change of fluctuating density obtained at a fixed location is mainly due to advection of fluctuating density through this location. In contrast the time-rate of change of fluctuating density following a fluid particle is substantially smaller, and due to production and dissipation of fluctuating density.
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12:15
15 mins
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Sweeping has no effect on renormalized turbulent viscosity
Mahendra K. Verma, Abhishek Kumar
Abstract: We perform renormalization group analysis (RG) of the Navier-Stokes equation in the presence of constant mean velocity field $\mathbf U_0$, and show that the renormalized viscosity is unaffected by $\mathbf U_0$, thus negating the ``sweeping effect", proposed by Kraichnan [Phys. Fluids {\bf 7}, 1723 (1964)] using random Galilean invariance. Using direct numerical simulation, we show that the correlation functions $\langle {\mathbf u} ({\mathbf k}, t){\mathbf u}({\mathbf k}, t+\tau) \rangle$ for $\mathbf U_0 =0$ and $\mathbf U_0 \ne 0$ differ from each other, but the renormalized viscosity for the two cases are the same. Our numerical results are consistent with the RG calculations.
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