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Computational Simulations of Pulsating Soft Corals

By Lina Sorg

Soft corals in the family Xeniidae are sessile organisms that pulsate. They exhibit active motion but do not use this motion to locomote. Because such behavior is very uncommon, biologists are interested in exploring its potential benefits. During a minisymposium presentation at the 2020 SIAM Conference on Mathematics of Planet Earth, which took place virtually earlier this month, Shilpa Khatri of the University of California, Merced presented computational simulations in the context of fluid flow to explore the competitive advantages of coral pulsations.

Khatri began with three hypotheses about the potential benefits of pulsating motion in Xeniidae corals. She suggested that the pulses might (i) remove oxygen to increase photosynthesis in the corals’ symbiotic algae; (ii) prevent refiltration by neighboring coral, thus benefitting the colony as a whole; and (iii) increase the uptake of dissolved matter in the water. To explore these theories, Khatri traveled to the Red Sea and went snorkeling. She observed Heteroxenia fuscescens, a species of soft coral in the Xeniidae family that lives in large colonies. Each individual polyp comprises a stalk with a ring of eight large, feathery tentacles.

Khatri explained that researchers can use an underwater particle image velocimetry (PIV) system at depths of five to 10 meters to gather in situ data and measure flow fields directly above Heteroxenia fuscescens. A resulting video revealed a strong background flow. According to a 2013 PNAS paper, if an animal is at rest (not pulsating) but still exhibits a strong background flow, no measurable turbulence exists. But if an animal is pulsating and the background flow is weak, turbulence is present. These findings indicate that a pulsating motion is quite different from background flow. 

Khatri decided to conduct numerical simulations on Heteroxenia fuscescens to understand the fluid flow associated with the coral’s mixing efforts. She was specifically interested in the animal’s ability to successfully create enough mixing to ensure the pulsing motion’s effectiveness. To model this phenomenon, Khatri employed a fluid-structure interaction with elastic body and flow and prescribed a motion for the coral. She used the Navier-Stokes equations to solve for fluid flow, accounting for the velocity, pressure, position, time, and the coral’s force on the fluid. In these simulations, the Reynolds number (\(Re\)) for Heteroxenia fuscescens was about \(10\). Khatri utilized the immersed boundary method to compute the simulations. The Navier-Stokes equations coupled the Eulerian mesh (on which the fluid equations are solved) and the Lagrangian mesh (which represents the coral).

Next, Khatri relied on a predefined integral to advance one timestep. This process involved advancing the coral location, computing the corals’ force on the fluid grid, and solving for the fluid flow using the forces from the coral. Khatri conducted these simulations with the IBAMR library. “This library is adaptive in both time and space, which makes three-dimensional (3D) simulations feasible in a somewhat reasonable time,” she said.

A laboratory video provided details about the kinematic motion of Heteroxenia fuscescens. Khatri tracked six different points along one tentacle, fit a polynomial at each frame, computed the arclenghs using the polynomials, nondimensionalized to ensure that she could average over some number of pulses and some number of animals, and fit polynomials of arclength at each frame. Khatri averaged over five polyps with five pulses each. Then she observed the behavior of one tentacle. Over the course of one pulse, the animal remains at rest 34 percent of the time, contracts for 29 percent, and expands for 37 percent. Pulsation lasts roughly two seconds, which yields a \(Re\) of \(10.66\). 

Khatri presented a 3D simulation with 10 pulses on an adaptive grid (see Figure 1). One can see that the vorticity ring forms but does not fully separate. The animal pulls in fluid from the side as it contracts and pushes the fluid outward. It brings the nearby fluid downward as it relaxes back open, resulting in a mixing region. Overall, minimal backflow occurs. And though it is not visible in the simulation, a continuous upward jet is present — even when the animal is relaxed.

Figure 1.Three-dimensional simulations on an adaptive grid.

To test the simulation’s accuracy, Khatri compared it to experimental data using PIV. A strong upward jet is present in the velocity field as the animal closes; when it opens, the continuous upward jet persists and a return flow is present. Backflow, which occurs close to the animal, generates mixing to remove oxygen waste product that is emitted by symbiotic algae. In short, the experimental data agrees with the simulation.

Khatri then conducted quantitative analysis and presented both experimental and simulation-based results. She measured the average horizontal and vertical velocity and found that the maximum vertical velocity is roughly the same for both PIV data and the simulation, though more decay is visible in the latter. Although horizontal velocity trends for both instances are fairly similar as well, some differences are present. Khatri surmised that these discrepancies occurred because she and her collaborators treat the whole animal as a solid plate; Heteroxenia fuscescens tentacles are actually feathery and thus necessitate the addition of some coarse material to the simulation.

Khatri next examined the animal at different Reynolds numbers. Heteroxenia fuscescens lives at \(Re=10\), but Khatri also depicted a simulation of \(Re=0.5\) and \(Re=80\). When \(Re\) is less than or equal to \(5\), the coral cannot maintain a continuous jet; therefore, the continuous upward jet is not at all apparent for \(Re=0.5\). A high \(Re\) boasts a full separation of the vortices with minimal backward flow towards the animal. \(Re=10\) is thus a “sweet spot” at which both an upward jet and healthy backflow are present.

To finish her presentation and test her initial hypotheses, Khatri summarized Matea Santiago’s (UC Merced) work, which couples the hypotheses with an oxygen-limited model to ultimately generate a photosynthesis model. This model measures the amount of oxygen that the tentacle creates at time \(t\).

In conclusion, Khatri has developed computational tools that allow her to explore the flows that result from pulsating corals. She has already begun conducting some preliminary mixing analysis that utilizes dynamical systems techniques and is also working to measure the amount of mixing that occurs at different Reynolds numbers. In the future, she hopes to perform more accurate modeling of solutes and particles, incorporate muscle mechanics, and account for interactions between multiple polyps.

 Lina Sorg is the managing editor of SIAM News.
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