Role of large-scale advection and small-scale turbulence on vertical migration of gyrotactic swimmers

In this work, we use direct-numerical-simulation-based Eulerian-Lagrangian simulations to investigate the dynamics of small gyrotactic swimmers in free-surface turbulence. We consider open-channel flow turbulence in which bottom-heavy swimmers are dispersed. Swimmers are characterized by different vertical stability, so that some realign to swim upward with a characteristic time smaller than the Kolmogorov timescale, while others possess a reorientation time longer than the Kolmogorov timescale. We cover one order of magnitude in the flow Reynolds number and two orders of magnitude in the stability number, which is a measure of bottom heaviness. We observe that large-scale advection dominates vertical motion when the stability number, scaled on the local Kolmogorov timescale of the flow, is larger than unity: This condition is associated to enhanced migration toward the surface, particularly at low Reynolds number, when swimmers can rise through surface renewal motions that originate directly from the bottom-boundary turbulent bursts. Conversely, small-scale effects become more important when the Kolmogorov-based stability number is below unity: Under this condition, migration toward the surface is hindered, particularly at high Reynolds, when bottom-boundary bursts are less effective in bringing bulk fluid to the surface. In an effort to provide scaling arguments to improve predictions of models for motile microorganisms in turbulent water bodies, we demonstrate that a Kolmogorov-based stability number around unity represents a threshold beyond which swimmer capability to reach the free surface and form clusters saturates.

Figure 1

Sketch of the computational domain with boundary conditions for the fluid.

Figure 2

Gyrotactic microorganisms swim with velocity $v_s$ in the direction given by the orientation vector p, set by a balance of torques: The torque due to cell asymmetry (bottom heaviness: ${\mathbf{T}}_{\text{grav}}$), which tends to align the cell to its preferential orientation along the vertical direction $\mathbf{k}$, and the torque due to the flow ${\mathbf{T}}_{\text{visc}}$.

Figure 3

Fluid velocity statistics. Panels: (a) mean velocity profiles for all Reynolds numbers; (b) fluid velocity RMS at ${\text{Re}}_{\tau }^L=170$; (b) fluid velocity RMS at ${\text{Re}}_{\tau }^I=510$; (c) fluid velocity RMS at ${\text{Re}}_{\tau }^H=1020$.

Figure 4

One-dimensional (streamwise) energy spectra of the streamwise and spanwise surface-parallel velocity fluctuations, $E_x\left(k_x\right)$ (left-hand panels), and $E_y\left(k_x\right)$ (right-hand panels), respectively. Panels: (a), (b) ${\text{Re}}_{\tau }^L=170$; (c), (d) ${\text{Re}}_{\tau }^I=510$; (e), (f) ${\text{Re}}_{\tau }^H=1020$.

Figure 5

Top view of gyrotactic microswimmers distribution on the free surface of a turbulent open-channel flow. Flow is from left to right. Panels: (a), (b) ${\text{Re}}_{\tau }^L=170;$ (c), (d) ${\text{Re}}_{\tau }^I=510;$ (e), (f) ${\text{Re}}_{\tau }^H=1020.$ Left-hand panels: Low gyrotaxis, ${\mathrm{\Psi }}_L^{+}$; right-hand panels: High gyrotaxis, ${\mathrm{\Psi }}_H^{+}$.

Figure 6

Time evolution of the swimmers' correlation dimension at varying gyrotaxis and for different Reynolds numbers: (a) ${\text{Re}}_{\tau }^L=170$; (b) ${\text{Re}}_{\tau }^I=510$; (c) ${\text{Re}}_{\tau }^H=1020$. Note that superscript $+$ in the stability number has been dropped for ease of notation.

Figure 7

Surfacing and accumulation of gyrotactic microswimmers (${\text{Re}}_{\tau }^L=170$). Panels: (a) Wall-normal concentration at varying gyrotaxis in log-linear scale; (b), (c), (d) Evolution of the number of swimmers that have reached the subsurface layer $1<({\text{Re}}_{\tau }-z^{+})/d_p^{+}<5$ for low, intermediate, and high gyrotaxis. The relative contributions due to swimmers in downwellings (${\mathbf{\nabla }}_{\text{2D}}<0$, dash-dotted lines) and upwellings (${\mathbf{\nabla }}_{\text{2D}}>0$, dashed lines) are also shown. In panel (a), the stability numbers are indicated without superscript $+$ for ease of notation. In panels (b)–(d), the gray line represents the evolution of $n/\mathcal{N}$ for the case of swimmers rising in still fluid.

Figure 8

Surfacing and accumulation of gyrotactic microswimmers (${\text{Re}}_{\tau }^I=510$). Panels: (a) Wall-normal concentration at varying gyrotaxis in log-linear scale; (b), (c), (d) Evolution of the number of swimmers that have reached the subsurface layer $1<({\text{Re}}_{\tau }-z^{+})/d_p^{+}<5$ for low, intermediate and high gyrotaxis. The relative contributions due to swimmers in downwellings (${\mathbf{\nabla }}_{\text{2D}}<0$, dash-dotted lines) and upwellings (${\mathbf{\nabla }}_{\text{2D}}>0$, dashed lines) are also shown. In panel (a), the stability numbers are indicated without superscript $+$ for ease of notation. In panels (b)–(d), the gray line represents the evolution of $n/\mathcal{N}$ for the case of swimmers rising in still fluid.

Figure 9

Surfacing and accumulation of gyrotactic microswimmers (${\text{Re}}_{\tau }^H=1020$). Panels: (a) Wall-normal concentration at varying gyrotaxis in log-linear scale; (b), (c), (d) Evolution of the number of swimmers that have reached the subsurface layer $1<({\text{Re}}_{\tau }-z^{+})/d_p^{+}<5$ for low, intermediate and high gyrotaxis. The relative contributions due to swimmers in downwellings (${\mathbf{\nabla }}_{\text{2D}}<0$, dash-dotted lines) and upwellings (${\mathbf{\nabla }}_{\text{2D}}>0$, dashed lines) are also shown. In panel (a), the stability numbers are indicated without superscript $+$ for ease of notation. In panels (b)–(d), the gray line represents the evolution of $n/\mathcal{N}$ for the case of swimmers rising in still fluid.

Figure 10

Peak values of swimmer concentration $C/C_0$ at the free surface in the (${\text{Re}}_{\tau }$, ${\mathrm{\Psi }}^{+}$) parameter space. Two time steps are considered: $t^{+}=2000$ (a) and $t^{+}=4000$ (b). The gray surfaces provide a qualitative rendering of the expected peak value of $C/C_0$ for values of ${\text{Re}}_{\tau }$ and ${\mathrm{\Psi }}^{+}$ within the range considered in this study (as provided by linear interpolation of present concentration data).

Figure 11

Kolmogorov-based stability number, ${\mathrm{\Psi }}_K^{+}$, in the (${\text{Re}}_{\tau }$, ${\mathrm{\Psi }}^{+}$) parameter space. The gray surface provides a qualitative rendering of the expected value of ${\mathrm{\Psi }}_K^{+}$ for values of ${\text{Re}}_{\tau }$ and ${\mathrm{\Psi }}^{+}$ within the range considered in this study (as provided by linear interpolation of the Kolmogorov length scales computed in this study).

Figure 12

Joint probability density function (jPDF) of the instantaneous vertical velocity of the swimmers, $w_p^{+}$, versus surface divergence, ${\mathbf{\nabla }}_{\text{2D}}{|}_p$, conditionally sampled at the swimmer positions for all three Reynolds number. Only swimmers in the region $2<z^{+}<8$ below the free surface are considered. Top-to-bottom panels: (a), (b), (c) ${\text{Re}}_{\tau }^L=170$; (d), (e), (f) ${\text{Re}}_{\tau }^I=510$; (g), (h), (i) ${\text{Re}}_{\tau }^H=1020$. Left-to-right panels: (a), (d), (g) low gyrotaxis, ${\mathrm{\Psi }}_L^{+}$; (b), (e), (h) intermediate gyrotaxis, ${\mathrm{\Psi }}_I^{+}$; (c), (f), (i) high gyrotaxis, ${\mathrm{\Psi }}_H^{+}$. White and black isocontours in each panel correspond to $50\%$ and $10\%$ of the maximum jPDF value.

Figure 13

Mean swimmer orientation, $\langle p_i\rangle$, along the wall-normal direction (left-hand panels) and along the streamwise direction (right-hand panels) at varying gyrotaxis (${\mathrm{\Psi }}_L^{+}$: dotted line; ${\mathrm{\Psi }}_I^{+}$: dashed line; ${\mathrm{\Psi }}_H^{+}$: solid line). The free surface is located at $z^{+}/{\text{Re}}_{\tau }=1$. Top panels: ${\text{Re}}_{\tau }^L=170$; middle panels: ${\text{Re}}_{\tau }^I=510$; bottom panels: ${\text{Re}}_{\tau }^H=1020$.