turbulence – Softbites

Original paper: Individual-based model of larval transport to coral reefs in turbulent, wave-driven flow: behavioral responses to dissolved settlement inducer

Lost, alone, and buffeted by ocean currents: this is the beginning of life for many oceanic larvae. These tiny organisms, often only 100 micrometers in diameter, must seek a suitable new habitat by searching over length scales thousands of times their own. But searching for something you can’t see while being dragged this way and that by ocean currents can’t be easy. How do these microscopic creatures make sense of the turbulent world around them and find their home?

larvaandslug — Figure 1: The larval form (left) is about 100 micrometers in diameter and swims using the beating hairs on the stumps on at the top of the cell, whereas and the adult sea slug form (right) can grow up to 5 cm in length and stays on the coral. Left figure is taken from Koehl et. al. The right figure is taken from http://www.seaslugforum.net/find/pheslugu.

To answer this question, today’s paper studies a species of sea slug, Phestilla sibogae. These sea slugs have two forms, the baby larval form (Fig. 1 left), which travels through the ocean, and the adult sea slug form (Fig. 1 right), which lives and feeds on their coral prey. After they are born, the young larvae first swim toward light, instinctively leaving their parents’ reef. When they are old enough to settle down and become adults, they must search for a new reef to call home. The metamorphosis from larva to slug is only triggered when the larvae have settled on their coral prey.

The coral prey release a chemical that the sea slug larvae can smell. The chemical acts as an on-off switch for the larva. When there is no chemical, the larva swims in a straight path in a random direction at 170 micrometers per second. Upon encountering a strong enough chemical smell, the larva stops swimming and sinks at 130 micrometers per second. We know how the larva move but how does this movement affect how many and how quickly the larvae make it to the reef? To understand the larvae transport, we need to understand the larval environment.

larvaeinsea — Figure 2: Larva cell (inset) in turbulent waters above the reef. The streaky pattern is from measurements made by the researchers in a wavy flow tank above a reef skeleton. Within the reef skeleton, a fluorescent dye is released. When the fluorescent dye is excited by a laser sheet it emits light. More light means more dye, where the dye represents the coral chemical [1]. The figure is taken from Koehl et. al.

Not only do the larvae have to swim while being buffeted by the wavy turbulent flow, but the waves also affect how the chemical released from the reef spreads. If the coral were in still water, then the amount of chemical would increase smoothly as you travel from the surface waters to the reef due to diffusion. However, the corals live in shallow water, where waves passing over the rough reef surface lead to turbulent waters above the reef. This complicated flow pattern means the chemical smell no longer smoothly increases as you travel towards to reef. Instead, the turbulence creates streaks of very high amounts of chemical and very low amounts of the chemical, as shown in Figure 2.

To investigate the transport of larvae to the reef, Koehl and coauthors build on previous work to create a computer simulation with both the larva swimming behavior and the larva environment based on experimental measurements. To model the background flow environment, they include the net flow, waves, and turbulence. The flow parameters are fit to experimental measurements made in wavy shallow waters in Hawaii [1]. In a similar way, the researchers use experimental measurements to model the swimming behavior of the larvae.

In their simulation, the researchers are able to alter both the environment and the larva swimming behavior and ask what, if any, advantage the on-off swimming behavior brings. The advantage is measured using the steady state larva transport rate, defined as the percentage of the larvae that make it to the reef each minute. With the steady-state approximation, the percentage of larvae that make it to the reef each minute is constant over time.

bothconcpatterns — Figure 3: Turbulent concentration pattern in A shows streaks of high and low concentration while the time-averaged concentration pattern in B smoothly increases towards the bottom of the reef. Figure adapted from Koehl et. al.

First, the researchers investigate whether or not the streakiness of the concentration pattern is an important factor in determining how many larvae reach the coral. When trying to understand how larvae reach the coral, previous researchers made the simplifying assumption that the concentration pattern of the chemical the larvae follow is smooth and uniform over time. As we saw in the streaky pattern in Figure 2, this is not a realistic assumption. But just how wrong is this it? To answer this question, the researchers compare the chemical distribution measured at a single moment in time (Fig. 3A) to the chemical distribution obtained by taking the average of the distribution measured at many different times. This averaging process produces a smoother distribution than would be seen in reality (Fig. 3B). On comparing the two different chemical distributions, the researchers find the larvae transport rates are overestimated by up to 10% in the unrealistic time-averaged environment.

Secondly, because the concentration pattern affects the transport rate, the on-off swimming behavior must affect the transport of the larvae to the reef. In their simulation, the transport rate for naturally swimming and sinking larvae is 45% per minute. The researchers test how the larva behavior affects this transport rate by separately turning off the swimming and sinking behavior of their simulated larva. If a larva sinks but does not swim, the transport rate changes to 20% per minute. If the larva swims but doesn’t sink, the transport rate changes to 25% per minute. Without their on-off switch, the larvae are reliant on the background flow or randomly swimming downwards to be transported to the coral.

From these transport rates, we can understand the relative importance of larval behavior and larval environment. For example, we now know that if the environment was no longer turbulent or if the larvae could no longer swim, the larvae’s rate of transport to the reef would change significantly. This impacts both how many larvae survive to adulthood and where in the ocean the adult sea slugs end up. Building on this work, predictions have also been made for many different species of larvae [2]. From these studies, we not only can get an idea of how local and global populations spread in their natural environments but also how a simple on-off process can help an organism to successfully navigate a complex environment.

[1] See https://academic.oup.com/icb/article/50/4/539/652640, for an overview of how researchers characterize the larva environment.

[2] See https://link.springer.com/article/10.1007%2Fs00227-015-2713-x for more details. Here, the researchers measure both the concentration of chemical and the flow above the reef simultaneously (as described in [1]). With this, they look more generally at the problem of settling on surfaces, investigating a variety of swimming properties and settling sites rather than a specific species.

Original paper: Fluid Dynamics of Bacterial Turbulence

The next time you’re washing your hands, start by turning the water on just a little. Notice how clear is the flow of the water from the tap. There aren’t any bubbles in the water, and when you put your finger in the stream, it smoothly flows around it. This is called laminar flow. Now keep increasing the water flow until it is very fast and rough. The chaotic nature of the flow in this stream is called turbulence, and how a flow turns from being laminar to turbulent is a popular area of research. In general, turbulent flows are very fast and are made from fluids that are not very thick.

In today’s study, Dunkel and his colleagues investigate how bacteria can make flow patterns that look turbulent – chaotic and full of vortices – even though bacteria are tiny and slow. The bacteria push the fluid around as they swim and create vortices, spinning regions in the fluid. The 5 ?m long bacteria create vortices with diameters of 80 ?m by swimming at the speed of 30 ?m/s!

To determine whether a fluid flow is turbulent or laminar, the Reynolds number, a ratio of the strength of the flow (inertial forces) to how much the fluid resists motion (viscous forces) is used and defined as [1]:

$Re = \frac{\rho V D}{\mu}$

It depends on the fluid density ?, flow speed V, the length of an object D (for example, the diameter of a bacterium), and flow viscosity, or thickness, ?. When the Reynolds number of a flow is high, the inertia of the fluid (how powerfully it flows), is much higher than its viscosity (how thick the fluid is). In this case, the resistance of the fluid to small fluctuations in its motion is not enough to prevent the fluctuations from growing and spreading. Before you know it, your previously smooth, easily predictable flow is chaotic and full of vortices – it has transitioned to turbulence.

Typically, flow in a pipe like that in a tap becomes turbulent at a Reynolds number of 2300; the flow of air over an airplane begins transitioning when the Reynolds number is about a million. Bacteria are so tiny and slow that their Reynolds number is very small – on the order of $10^{-5}$ .

The researchers in this study investigate turbulent-like fluid behavior caused by swimming bacteria, B. Subtilis, in a fluid. B. Subtilis is a rod-like bacteria that swims by pushing its surrounding fluid with its flagellum (or tail). The researchers grew the bacteria in a nutritious fluid medium, and they added small, 1 micrometer beads to the fluid to act as tracer particles. When tracer particles are added to a fluid, you can see them being pushed around by its flow. If the particles are small enough, tracking them can be used to show how the fluid is moving. Thus, the experiment consisted of a suspension (a mixture in which particles do not dissolve) of bacteria and particles in the fluid.

The researchers loaded this suspension into sealed microfluidic cylindrical chambers with a 750 ?m radius and 80 ?m height (Figure 1). Since the chamber was sealed, the bacteria grew tired and did not move as much as they ran out of oxygen, so their activity levels decreased during the 10 minutes they spent in the chamber. Thus, by waiting several minutes between sets of measurements, the researchers were able to test how the bacteria affected the fluid at different energy levels.

The researchers took high-speed 2D videos of slices of the suspension containing the bacteria and the particles in the middle of the cylinder (even though the motion of the bacteria was in 3D). They used visible light to illuminate the motion of the bacteria and fluorescent light to view the motion of the tracer particles through a microscope.

setup — Figure 1: Bacteria and tracer particles in the field of view of the camera. Although the entire test chamber is filled with fluid, only the 2D cross-section in the center of the cylinder is in focus.

The researchers used two methods to measure the motion of the suspension of bacteria and particles. First, by monitoring the motion of the bacteria, they obtained vectors showing how the bacteria were moving. Trajectories of bacterial movement were calculated by comparing the position pattern of the bacteria from one image frame to another. In the second approach, they tracked the motion of the particles in the flow by comparing the actual position of the particles from one image frame to the next. The researchers used both methods (monitoring patterns of bacterial motion and tracking individual particle positions) to make sure the flow fields measured from bacteria were an accurate representation of the flow. Since the tracer particles used by the researchers were very small, with diameters of 1 micrometer, they were small enough to be reliably pushed by the flow.

The typical results the researchers got are shown in Figure 2, with visible vortices.

The researchers measured the velocity distributions of the bacteria. They found that the results from the tracer particles and the bacteria had the same distributions – the flow of the bacteria and solvent were very similar, and the flow of the bacteria could be used to represent the fluid flow. However, it is possible that the particles were pushed around by the bacteria, and not by the motion of the fluid itself, which is not mentioned in the paper.

To analyze the average motion of the bacteria, the researchers calculated various properties of the velocity field. They calculated the vorticity, $\omega_z$ , or how much the bacteria rotated the fluid. The average of the square of the vorticity throughout the 2D experimental plane is called the enstrophy, $\Omega_z$ . They then calculated the kinetic energy of the flow, $E_{xy}$ , the energy a fluid has because of its speed, and also calculated its average throughout the space (1). Although the instantaneous kinetic energy and vorticity fluctuated as the bacteria moved, the average kinetic energy and enstrophy over time were approximately constant throughout the 50 seconds of recording.

The researchers then measured two properties of the flow with functions called the VCF (“equal-time spatial velocity autocorrelation function”) and the VACF (“two-time velocity auto-correlation function”). The first function, the VCF, measures how the velocity of the fluid changes throughout the space (the 2D slice of the cylinder). If this function goes from positive, velocities in the same direction, to negative, velocities in different directions, then it indicates that there is a vortex in the fluid (Figure 3a). The results of the VCF are shown in Figure 3b. The researchers calculated that the radius of a vortex in the fluid was $R_v = 40 \mathrm{\mu m}$ from the VCF. The vortex radius did not change with the kinetic energy of the flow.

The average enstrophy was found to be linearly proportional to the time-averaged kinetic energy by about half the vortex radius ( $\Lambda = \mathrm{24 \mu m}$ ) over all the energy scales tested:

$\overline{\Omega_z} =\frac{\overline{E_{xy}}}{\Lambda^2}$

So when bacteria have more energy, the fluid has a higher tendency to form vortices and rotate (but the vortices will be the same size).

The second property the researchers calculated was the “VACF”. The VACF measures how the velocity changes over time. The VACF represents the memory of the fluid. If it decays to 0 slowly, that means the velocities stay similar for a long time; if it decays quickly, the velocity changes in a very short time. The results of the VACF are shown in Figure 3c. The researchers found that at higher energies, the system has a shorter “memory”. The VACF shown in black has the highest energy, and decays much faster than the VACF shown in purple, which has the lowest energy. In bacterial turbulence, bacteria add energy to the system by swimming to make small vortices, which then lead to larger vortices. This is the opposite of how vortices form in a non-active turbulent fluid, where the energy is added to the larger vortices that create smaller vortices because of the fluid’s viscosity.

correlation — Figure 3. (a) Diagram of a vortex. The blue and green vectors are correlated (positive VCF), showing that there is no vortex between them; the blue and red vectors are anticorrelated (negative VCF), showing the presence of a vortex. VCF (b) and VACF (c) of the bacterial motion. The minimum value of the VCF indicates a vortex with a 40-micron radius. The VACF varies with energy when plotted as a function of the time lag. Inset: energy as a function of time; the black spheres show the highest energy and data in purple is at the lowest. Figure modified from the original paper.

Finally, the researchers present a recently developed theory for the flow caused by bacteria. This theory is a continuum model equation – it treats the suspension of fluid, bacteria, and particles as if it were a continuous material, and accounts for the effects of the fluid flow and the forces the bacteria apply to the system.

The equation in the theory can be modeled to predict how the suspension will move. If the parameters of the equation in this theory are chosen to represent flow without bacteria, it simplifies to a theoretical fluid flow model. The researchers chose coefficients in the equation known to represent bacteria that push the fluid, like B. Subtilis in this experiment. They found that the model was accurate to within 10%-15%, making it a good candidate for a quantitative description of bacterial turbulence.

In today’s paper, Dunkel and his colleagues made significant contributions to the understanding of bacterial turbulence. The researchers developed a method to show that the motion of the bacteria swimming in a fluid can be used to measure the motion of the fluid itself. They developed a mathematical model of the motion and tested it with their experimental results to create a method for quantitatively studying how bacterial motion affects fluid flows. The observations they made can be used to compare bacterial turbulence to traditional turbulence in fluid mechanics, and give insight into how other fluids with active particles might behave.

[1] Fox, Robert W., Alan T. McDonald, and Philip J. Pritchard. Chapter 2, Introduction to fluid mechanics. Vol. 5., New York: John Wiley & Sons, 1998.

[2] Dunkel, Jörn, et al. “Fluid dynamics of bacterial turbulence.” Physical review letters 110.22 (2013): 228102.

(1) The vorticity, $\omega_z$ , is calculated by taking the difference between the change in the y-component of the velocity, $v_y$ , in the x-direction and the x-component of the velocity, $v_x$ in the y-direction:

$\omega_z = \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}$

The average kinetic energy (represented by angle brackets is calculated as:

$E_{xy} = \frac{v_x^2+v_y^2}{2}$

As the mass of the bacteria was constant, the energy depends only on the measured velocities of bacteria, and mass was not included in the calculation. As you may remember, a range of energies was tested by varying the bacterial activity.

The enstrophy, $\Omega_z$ , is calculated as the average of the vorticity as:

$\Omega_z = \langle\frac{\omega_z^2}{2}\rangle$

Tag: turbulence

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