Gyrotactic Trapping in Laminar and Turbulent shear flow

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1 Gyrotactic Trapping in Laminar and Turbulent shear flow Filippo De Lillo, Guido Boffetta, Francesco Santamaria Dipartimento di Fisica and INFN, Università di Torino Massimo Cencini ISC-CNR Rome Gyrotactic trapping in laminar and turbulent Kolmogorov flow Santamaria et al., Phys Fluids 6, (014). Lisbon, December 014

2 Model gyrotactic algae - bottom heavy swimmers, e.g. Chlamydomonas - dilute -> no interaction, no feed-back on the fluid - Translational and rotational diffusion negligible - Modulus of the swimming velocity constant GRAVITAXIS fluid vorticity kinematic viscosity gravitational acceleration JO Kessler (Nature 1985) T Pedley and JO Kessler (Annu. Rev. Fluid. Mech. 199)

3 Gyrotactic trapping in shear flows W.M. Durham, J.O. Kessler and R. Stocker, Science 33, 1067 (009) Thin layers of high phytoplankton concentration. Vertical thickness cm to m, horizontal size up to km, persistence up to days. Trapping in a shear Thin layers of Heterosigma akashiwo near Shannon Point (WA) Equation for the angle z Bω < 1 z Bω < 1 u(z) u(z) equilibrium direction no equilibrium: tumbling motion

4 Gyrotactic trapping in shear flows W.M. Durham, J.O. Kessler and R. Stocker, Science 33, 1067 (009) For a shear trajectories of swimming cells are trapped by tumbling Swimming cells concentrate in regions of high shear Chlamydomonas rehinartii Heterosigma akashivo

5 Kolmogorov flow Navier-Stokes equations for incompressible velocity field u! Kolmogorov body force: Stationary solution:! For For the laminar solution is linearly unstable the flow becomes turbulent: DNS are necessary. Why this flow? because it is the simplest periodic shear flow because the mean profile of the turbulent flow is still a cosine G. I. Sivashinsky, Physica D (1985) Musacchio and Boffetta, Physical Review E (014)

6 Swimmers in Laminar Kolmogorov flow Equations of motion Ẋ = cos Z + Ż = Ẏ = p z p y p x ṗ x = 1 ṗ y = 1 ṗ z = 1 p x p z 1 sin Z p z p y p z (1 p z)+ 1 sin Z p x X and Y do not enter the dynamics ( actually makes the dynamics three-dimensional) p =1 swimming number stability number Ż = p z 1 ṗ x = 1 ṗ y = ṗ z = 1 p x p z 1 sin Z p z p y p z (1 p z)+ 1 sin Z p x However, in this laminar case we can say much more. In analogy with the constant shear case we could expect < 1 > 1 quasi-equilibrium solutions with ṗ =0exist for all Z, swimmers can escape for some Z rotation due to shear dominates, swimmers are trapped

7 Swimmers in Laminar Kolmogorov flow Ż = p z ṗ x = 1 ṗ y = 1 ṗ z = 1 p x p z 1 sin Z p z p y p z (1 p z)+ 1 sin Z p x If a swimmer is not trapped, Z is not limited, p y 0 The system becomes D. Two constants of motion! the system is integrable = G(Z)@ Z H Ż = G(Z) =e G(Z)@ H Z G(Z):inverse integrating factor A Zöttl, H Stark, PRL (01) find similar results for prolate cells in Poiseuille flow

8 Swimmers in Laminar Kolmogorov flow H Conservation of implies that for large Z (untrapped swimmers) p x = ( cos Z sin Z) trapping partial trapping This defines a c = 1 solutions for < c apple c = 1 4 1/ no trapping > c solutions for any swimming number stability number

9 trapping partial trapping no trapping

10 Swimmers in Turbulent Kolmogorov flow laminar turbulent Effective diffusion due to turbulence makes trapping transient Technical note: in turbulent Kolmogorov flow the relative intensity of fluctuations is constant. We change it by decomposing the velocity field! From now on we consider only conditions that would give trapping in a laminar flow

11 Density Profiles π Ψ=1.1 Ψ=1.5 3π/ z π γ=0.01 γ=0.05 γ=0. π/ ρ(z) ρ(z) ρ(z) Turbulence intensity

12 Deviation From Homogeneity =0.05 =1.1 =0. Circles Squares

13 How long do layers last? Exit time: time to swim half a period upwards Exit time distribution compared with an inverse gaussian (i.e. the shape for a pure diffusion with drift) =0.05 If typical parameters for the ocean are used hours < Tp < days =0. Circles Squares ht i /hv z i T p

14 Conclusions - We considered the formation of thin layers of gyrotactic algae in a laminar and turbulent shear. - Analytical conditions for the formation of TLs in laminar Kolmogorov flow can be derived - Turbulence causes layers to dissolve in a finite time. We discussed some estimates of the lifetime of layers, with the correct orders of magnitude.! On the paper you can also find a discussion of the effects of rotational diffusion. Gyrotactic trapping in laminar and turbulent Kolmogorov flow Santamaria et al., Phys Fluids 6, (014). Muito obrigado!

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16 Model gyrotactic algae - bottom heavy swimmers, e.g. Chlamydomonas - dilute -> no interaction, no feed-back on the fluid - Translational and rotational diffusion negligible - Modulus of the swimming velocity constant Gyrotactic focusing in pipe flows JO Kessler, Nature 313, 18 (1985) JO Kessler (Nature 1985) T Pedley and JO Kessler (Annu. Rev. Fluid. Mech. 199)

17 Model gyrotactic algae - bottom heavy swimmers, e.g. Chlamydomonas - dilute no interaction, no feed-back on the fluid - Translational and rotational diffusion negligible - Modulus of the swimming velocity constant Gyrotactic focusing in pipe flows JO Kessler, Nature 313, 18 (1985) JO Kessler (Nature 1985) T Pedley and JO Kessler (Annu. Rev. Fluid. Mech. 199)

18 A turbulence primer... What we mean by turbulence (your neighbour might give a different definition): -a solution of the Navier-Stokes equation at large Re!! - more than chaotic : many active scales E(k) dissipative scale (Kolmogorov s scale) typical velocity at scale Energy flux across scales constant in the inertial range typical eddy-turn-over time at scale INJECTION DISSIPATION Important for what follows: small scales are determined by and only! k