# Mass Transport and Turbulent Statistics within Two Branching Coral Colonies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Immersed Boundary Method with Large Eddy Simulation

#### 2.2. Coral Geometries

## 3. Results

#### 3.1. Validation of the Simulation Results

#### 3.2. Comparison of Flow Profiles and Transport Mechanism between Pocillopra Geometries

#### 3.3. Turbulent Momentum Flux

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic of flow domain used in the simulation (not to scale). $x=0$ indicates the beginning of the flow domain, while ${x}^{\prime}=0$ represents the beginning (front edge) of the coral in the flow domain.

**Figure 3.**(

**A**,

**B**) Rendered stereolithography files of the P. meandrina and P. eydouxi colony geometries used in the simulations. (

**C**,

**D**) Mean branch diameter at normalized height from the base of the Pocillopora colonies. Horizontal bars indicate the variance.

**Figure 4.**Comparison of (

**A**) the normalized streamwise velocity and (

**B**) Reynolds stress over a single P. meandrina colony from the current simulation with the experimental results over 81 P. meandrina colonies at $Re=23,513$ by Asher et al. [20]. The dotted line represents the top of the colony. The experimental data were obtained using GRABIT in MATLAB [52] and reproduced with permission from Asher et al. Vertical variations of coral reef drag forces. Journal of Geophysical Research: Oceans

**2016**, 121, 3549–3563. For both data sets, the mean velocity and Reynolds stress were normalized by the friction velocity.

**Figure 5.**Top view of the instantaneous velocity vector fields at two different heights ($z=0.1H$ and $0.4H$, where H is the height of the colony) inside the P. meandrina colony. These vector fields show the detailed interaction of the oncoming flow with the individual branches of the colony. The lower right figure shows a zoomed-in view of the velocity vectors at the frontal section of the colony at $z=0.1H$. The difference in vector fields at the front and rear halves of the densely branched coral, P. menadrina is clearly observable from the figure.

**Figure 6.**Top view of the instantaneous velocity vector fields at two different heights ($z=0.1H$ and $0.4H$, where H is the height of the colony) inside the P. eydouxi colony. In comparison to P. meandrina, the velocity vector field demonstrates relatively higher flow penetration into the P. eydouxi colony because of the openness of the structure. The formation of wakes can be observed behind individual branches.

**Figure 7.**Front-view streamline sections plotted at ${x}^{\prime}=0.5$ L for both Pocillopora colonies. The formation of vortices can clearly be seen in the inter-branch velocity field of P. meandrina in (

**A**), whereas relatively few vortices are formed around the branches of P. eydouxi, as seen in (

**B**).

**Figure 8.**The Stanton number calculated in the interior of the P. meandrina colony based on the mean flow magnitude and advection time scale at Reynolds number 15,000. Here, the advection time scale was calculated based one the mean vortex diameter at different sections of P. meandrina perpendicular to the flow direction, and the mean velocity value in those sections.

**Figure 9.**Mean velocity profile and internal cross-sectional area (y-z plane) along the length of (

**A**) the densely branched P. meandrina colony and (

**B**) the loosely branched P. eydouxi colony. Here, the area and velocity are normalized by the maximum cross-sectional area and oncoming velocity magnitude, respectively.

**Figure 10.**Side view of instantaneous mean (

**left**) and fluctuating (

**right**) velocity vector components above P. eydouxi at $y=0.5W$, where W is the width of the colony. Here, the flow is moving from right to left. The instantaneous mean vector field (

**left**) shows a higher concentration of velocity vectors above the colony as most of the flow is diverted to the top and sides of the colony. The fluctuating vector field was obtained by subtracting mean velocity from the instantaneous velocity field. These fluctuating velocity components also appear denser at the top surface of colony and contribute, to the Reynolds stress which controls mass transport to and from the top surface of the colony.

**Figure 11.**Mean Reynolds stress plotted as a function of normalized height above the top surface of the colony for (

**A**) P. meandrina and (

**B**) P. eydouxi at a Reynolds number of 15,000. Similar Reynolds stresses were observed above both colonies, despite their differences in branching patterns.

**Figure 12.**(

**A**,

**B**) streamwise, and (

**C**,

**D**), vertical velocity variance computed over P. meandrina and P. eydouxi, respectively. In both cases, the magnitude of the variance was found to be higher for P. meandrina where the inter-branch distance was smaller than for P. eydouxi.

Colony | Length (m) | Width (m) | Height (m) | Volume/Surface Area | Interbranch Distance/Diameter |
---|---|---|---|---|---|

P. meandrina | $0.172$ | $0.172$ | $0.100$ | $2.060$ | $2.8$ to $4.0$ |

P. eydouxi | $0.120$ | $0.110$ | $0.100$ | $2.090$ | 8 |

**Table 2.**Stanton number calculated along the length of the P. meandrina colony in the flow direction using two different methods (top and middle values) and the Stanton number calculated along the length of the P. eydouxi colony using the first method only (bottom values). Key: * The mean branch diameter was used as a length scale to calculate these values. † The mean diameter of the vortices was used as a length scale to calculate these values.

Colony | ${\mathit{x}}^{\prime}$ (m) | $\overline{\mathit{u}}$ (m/s) | Diameter (m) | ${\mathit{\tau}}_{\mathit{a}\mathit{d}\mathit{v}}$ (s) | St | Increase (%) |
---|---|---|---|---|---|---|

P. meandrina (*) | ${x}^{\prime}=0.2$ L | $0.142$ | $0.013$ | $0.091$ | $5.92\times {10}^{-5}$ | |

P. meandrina (†) | ${x}^{\prime}=0.4$ L | $0.084$ | $0.004$ | $0.048$ | $1.89\times {10}^{-4}$ | $219\%$ |

${x}^{\prime}=0.5$ L | $0.0675$ | $0.0045$ | $0.066$ | $1.70\times {10}^{-4}$ | $187\%$ | |

${x}^{\prime}=0.6$ L | $0.057$ | $0.0045$ | $0.079$ | $1.54\times {10}^{-4}$ | $160\%$ | |

${x}^{\prime}=0.8$ L | $0.066$ | $0.0050$ | $0.075$ | $1.71\times {10}^{-4}$ | $189\%$ | |

P. eydouxi (*) | ${x}^{\prime}=0.2$ L | $0.138$ | $0.00585$ | $0.0423$ | $2.84\times {10}^{-4}$ |

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Hossain, M.M.; Staples, A.E.
Mass Transport and Turbulent Statistics within Two Branching Coral Colonies. *Fluids* **2020**, *5*, 153.
https://doi.org/10.3390/fluids5030153

**AMA Style**

Hossain MM, Staples AE.
Mass Transport and Turbulent Statistics within Two Branching Coral Colonies. *Fluids*. 2020; 5(3):153.
https://doi.org/10.3390/fluids5030153

**Chicago/Turabian Style**

Hossain, Md Monir, and Anne E. Staples.
2020. "Mass Transport and Turbulent Statistics within Two Branching Coral Colonies" *Fluids* 5, no. 3: 153.
https://doi.org/10.3390/fluids5030153