# Flow Turbulence Characteristics and Mass Transport in the Near-Wake Region of an Aquaculture Cage Net Panel

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Setup and Instrumentation

#### 2.2. Experimental Procedures

^{2}, and the associated resolution was 0.173 mm per pixel. A maximum laser pulse repetition rate of 15 Hz was used, while the pulse duration and time interval between pulses were set at 8 ns and 2700 μs after being optimized for the current PIV measurement. The PLIF images were acquired from the first frame of the double exposures. The instantaneous PIV images were analyzed using the adaptive cross-correlation algorithm in Dynamics Studio, with a final interrogation area (IA) of 32 × 32 pixels with three refinement steps and 33.3% overlap, yielding a 57 × 36 vector field for each instantaneous image. Spurious vectors were removed by range validation and replaced by interpolation. In all cases, the number of replaced vectors was less than 3% of the total number. Pixel-by-pixel calibrations were required for quantitative PLIF measurements before each experiment, in which least-square fitting was performed to relate the grayscale value in the captured images of a number of calibration solutions to the known dye concentration in these solutions. Specifically, a small acrylic tank was placed inside the flume at the location of the FoV with its side walls parallel/perpendicular to the flume side walls. It was filled with uniform Rhodamine B solution with varying concentrations, namely 0 ug/L, 30 ug/L, 60 ug/L, 90 ug/L, and 120 ug/L, and the flume was filled with water to the same level. Once the calibration was completed, the laser and camera settings were kept unchanged throughout the entire experiment. The test section was covered by a black curtain to eliminate the disturbance of background light.

_{0}was established and stabilized along the flume channel, the dye injection was switched on, and simultaneous PIV and PLIF data acquisition were initiated after the dye tracer plume was stabilized at the measurement area. The downstream extent of the measurement region in the various tests was 300 mm, which fell well within the near-wake region (i.e., $x/M\le 20$).

## 3. Results and Discussion

#### 3.1. Time-Averaged Velocity and Concentration Field

_{0}) is commonly used to measure the effects of flow blockage of net structure [4]. The periodicity of the velocity fluctuation behind the net is clearly illustrated by the lateral profiles of the normalized velocity $\overline{U}/{U}_{0}$ in Figure 3a (the gray dashed lines marking the location of the twines), which decays steadily in the downstream direction (i.e., increasing x/M), with the mean velocity becoming more homogeneous and recovering to the incoming velocity (i.e., $\overline{U}/{U}_{0}=1)$. Recirculation (i.e., $\overline{U}/{U}_{0}<0$) was virtually suppressed directly behind the twines at the center-of-mesh measurement plane as observed previously [15].

_{0}) with N2 and N2 with N5 (same U

_{0}but smaller S) in Figure 3b. Moreover, the net solidity of case N1 in the present study was comparable with cases investigated by Bi et al. [9] (residing between reference cases n2 and n3) with a similar incoming velocity, resulting in the associated ratio of velocity reduction also residing between these two reference cases.

#### 3.2. Turbulence Characteristics

_{0}is the virtual origin. This power-law decay was manifested by the straight-line portion in the range of x/M = ~4 to ~10 in the log-log plot shown in Figure 6a. Several previous studies [13,22] have also reported more rapid decay (i.e., larger n in Equation (3)) in the developing (near-field) region than the fully developed (far-field) region of decaying grid turbulence. Furthermore, Isaza et al. [13] observed that the near-field decay region extended to x/M ~ 12. Therefore, the power decay region in this study fell within the near-field decay region. Assuming x

_{0}= −3.5M is used in Equation (3) for the near-field decay region [22], n and a were fitted for the power decay region behind the twine for N1 and N5 in Figure 6b and documented in Table 2. Given the sensitivity of the curve fits to the setting of virtual origin for relatively narrow range of measurement, we also fitted alternative power functions to the same data under the assumption of x

_{0}= 0 [11,13]. Regardless, the range of n values determined in the present study was much smaller than those reported from wind-tunnel experiments [13,22] and a previous water-channel experiment [19]. Although Gan and Krogstad [15] argued that the power-law decay region was not yet attained in their water flume measurements, the turbulence intensity and the slope of decay reported in their study, as well as those of Cardesa et al. [14], were significantly larger than in the current study (Figure 6b). The discrepancy among the various studies is mainly due to the net properties (e.g., mesh size, twine diameter, and rigidity) and working fluid (i.e., air versus water), which dictate the development of the turbulence [11]. Moreover, the elevated turbulence intensity in N1 relative to N5 shown in Figure 6b is also consistent with the finding of Ito et al. [30]. Given the same mesh size, the turbulence intensity increased with grid bar (or net twine) thickness.

#### 3.3. Turbulent Mass Transport

_{t}represents the characteristic size of large-scale eddies that drive the turbulent diffusion of the passive scalar, and can be written in the form

_{y}is the transverse unit vector and r is the separation [15,22]. The longitudinal development of the normalized transverse integral length scale L

_{t}/M for test case N5 is shown in Figure 9. This shows that, while L

_{t}/M at the mesh hole was consistently greater than the corresponding value behind the twine (i.e., at all x/M), the two profiles tended to approach each other at more downstream locations (i.e., x/M > ~5). Corresponding to the power-law decay of the turbulence intensity (Equation (3)), a power-law growth for the transverse integral length scale can also be observed in the range of x/M = ~4 to ~10, which is consistent with Gan and Krogstad [15]. The normalized plume spreading width for test case N5 is also plotted in Figure 9 for comparison purposes. Clearly, this width was initially much smaller than the transverse integral length scale (at the mesh hole), but the difference reduced in the downstream direction (i.e., increasing x/M), indicative of the transition of plume development from the turbulent-convective to turbulent-diffusive regimes.

^{−4}), see Figure 10). These results are generally consistent with previous findings that the gradient-diffusion hypothesis is invalid in the near field region of scalar transport in grid-generated turbulence, where the turbulent flow is still subject to significant anisotropy and inhomogeneity and the plume development has not yet reached the turbulent-diffusive regime [20,32].

## 4. Conclusions

- The wake flow downstream of the model fishing net panel showed a marked reduction and increase in time-averaged streamwise velocity immediately behind the net twine and at the adjacent mesh holes, respectively. The opposite trend was found for the streamwise turbulence intensity. For x > ~3M, the flow field became more homogeneous and entered the turbulence decay region. However, complete recovery of incoming velocity and development of isotropic turbulence was not observed to occur within the downstream extent of experimental measurements (i.e., x → ~15M). Corroborating with these changes in the turbulent flow field, the mean concentration began to decay steadily after x = ~3M.
- Similar to decaying grid-generated turbulence, the turbulence intensity followed a power-law decay over a short range of x = ~4M to 10M. However, the fitted decay exponent was much smaller than reported values for grid turbulence in previous wind-tunnel and water-channel experimental studies. Lateral profiles of the mean scalar concentration and concentration fluctuation exhibited self-similar Gaussian distributions, while those for the transverse turbulent scalar flux were nearly self-similar and antisymmetric about the centerline. These profiles started from a downstream location x = 0.3M, which was much closer to the net panel, with a strongly inhomogeneous wake flow field, than previous findings on scalar plume development downstream of point sources in grid turbulence.
- The presence of the net panel tended to break larger coherent eddies into smaller vortices, leading to a reduction in the transverse integral length scale. In so doing, the panel also accelerated the transition of plume development from the turbulent-convective regime to the more effective turbulent-diffusive regime, with the combined effects leading to slightly enhanced lateral spreading of the scalar plume.
- Direct comparison of the plume spreading width and the transverse integral length scale of the flow indicated that the development of the scalar plume was still transitioning from turbulent-convective regime to turbulent-diffusive regime within the downstream extent of the experimental measurements. This was confirmed by the invalidity of the gradient-diffusion hypothesis in predicting the transverse scalar transport. Nevertheless, the apparent turbulent diffusivity, estimated from the gross plume parameters (i.e., time-averaged velocity and spreading width), appeared to be in reasonable agreement with the Taylor diffusivity calculated as the product of the transverse velocity fluctuation and integral length scale.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | constant associated with the power law turbulence decay |

$\overline{C}$ | mean concentration (µg/L) |

$\overline{{C}_{c}}$ | mean concentration at the plume centerline (µg/L) |

$\overline{{C}_{c0}}$ | mean concentration at the net panels (µg/L) |

c’ | concentration fluctuation (µg/L) |

d | diameter of the net twine (mm) |

D | Taylor (turbulent) diffusivity (m^{2}/s) |

e_{y} | transverse unit vector |

h | water depth in the flume (m) |

K | apparent turbulent diffusivity (m^{2}/s) |

L_{t} | integral length scale of transverse velocity fluctuations (mm) |

M | mesh size of the net/grid (mm) |

n | decay rate (exponent) of the power law turbulence decay |

r | separation in the transverse direction |

Re_{d} | Reynolds number with respect to water depth (=U_{0} h/ν) |

Re_{m} | Reynolds number with respect to net twine (=U_{0} d/ν) |

Re_{w} | Reynolds number with respect to mesh size (=U_{0}M/ν) |

S | net solidity defined as the ratio between the projected area and the total area enclosed by the net panel ($=2d/M-{\left(d/M\right)}^{2}$) |

$\overline{U}$ | mean streamwise velocity behind the net (m/s) |

U_{0} | incoming current velocity in the flume (m/s) |

u’ | velocity fluctuation in the streamwise direction (m/s) |

v’ | velocity fluctuation in the transverse direction (m/s) |

$\overline{{v}^{\prime}{c}^{\prime}}$ | transverse turbulent mass flux (µg/L⋅m/s) |

x,y,z | Cartesian coordinates in the longitudinal, transverse and vertical directions, respectively (mm) |

x_{0} | virtual origin associated with the power law turbulence decay (mm) |

${\sigma}_{y}$ | spreading width of the scalar plume (mm) |

${\sigma}_{y{c}^{\prime}}$ | spreading with of the concentration fluctuations (mm) |

$\eta $ | normalized transverse coordinate ($y/{\sigma}_{y})$ |

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**Figure 2.**Time-averaged (

**a**) velocity field and (

**b**) concentration field downstream of net for the representative case N5.

**Figure 3.**(

**a**) Lateral profiles of normalized velocity $\overline{U}/{U}_{0}$ at different downstream locations for N5; (

**b**) Variation of $\overline{U}/{U}_{0}$ with longitudinal distance along flume centerline.

**Figure 4.**(

**a**) Lateral profiles of the normalized time-averaged concentration $\overline{C}/\overline{{C}_{c}}$ at different downstream locations for N5; (

**b**) Evolution of the plume spreading width ${\sigma}_{y}$ with downstream distance.

**Figure 5.**Longitudinal profiles of the normalized time-averaged concentration $\overline{{C}_{c0}}/\overline{{C}_{c}}$ along the plume centerline.

**Figure 6.**(

**a**) Longitudinal evolution of streamwise turbulence intensity $\overline{{U}^{\prime 2}}/{U}_{0}^{2}$ behind the central twine and the adjacent mesh hole for case N5; (

**b**) Power decay region of $\overline{{U}^{\prime 2}}/{U}_{0}^{2}$ behind the twine for N1 and N5; (

**c**) Lateral profiles of $\overline{{U}^{\prime 2}}/{U}_{0}^{2}$ at different downstream locations for N5; (

**d**) Longitudinal profile of the turbulence anisotropy ratio $a=\sqrt{\overline{{u}^{\prime 2}}}/\sqrt{\overline{{v}^{\prime 2}}}$ behind the twine and mesh hole for N5.

**Figure 7.**(

**a**) Lateral profiles of the root mean square (RMS) of the normalized concentration fluctuation $\sqrt{\overline{{c}^{\prime 2}}}/\sqrt{\overline{{c}_{c}^{\prime 2}}}$; (

**b**) Longitudinal evolution of $\sqrt{\overline{{c}^{\prime 2}}}$ and $\sqrt{\overline{{c}_{c}^{\prime 2}}}/{C}_{c}$ along the plume centerline.

**Figure 8.**Lateral profiles of the normalized transverse turbulent flux $\overline{{v}^{\prime}{c}^{\prime}}/{\overline{{v}^{\prime}{c}^{\prime}}}_{max}$.

**Figure 10.**Longitudinal profiles of turbulent diffusivities ($D\left(x\right)=\sqrt{\overline{{v}^{\prime}{\left(x\right)}^{2}}{L}_{t}}$ and$\text{}K=\frac{\overline{U}}{2}\frac{d{\sigma}_{y}{}^{2}}{dx}$ ).

**Figure 12.**Lateral profiles of (

**a**) transverse turbulent scalar flux $\overline{{v}^{\prime}{c}^{\prime}}$d transverse gradient of mean concentration an $\frac{\partial \overline{C}}{\partial y}$; (

**b**) turbulent diffusivity $D=-\overline{{v}^{\prime}{c}^{\prime}}/\frac{\partial \overline{C}}{\partial y}$.

No. | Current Velocity U_{0} (m/s) | Twine Diameter d (mm) | Mesh Size M (mm) | Net Solidity S (%) | Reynolds Number w.r.t. Water Depth Re _{w} = U_{0}h/ν | Reynolds Number w.r.t. Net Twine Re _{d} = U_{0}d/ν | Reynolds Number w.r.t. Mesh Size Re _{m} = U_{0}M/ν |
---|---|---|---|---|---|---|---|

N0 | 0.290 | / | / | / | 87,000 | / | / |

N1 | 0.142 | 2 | 20 | 19.0 | 42,600 | 284 | 2840 |

N2 | 0.290 | 2 | 20 | 19.0 | 87,000 | 580 | 5800 |

N5 | 0.290 | 1 | 20 | 9.8 | 87,000 | 290 | 5800 |

**Table 2.**Comparison of the fitted parameters of the power decay range of streamwise turbulence intensity.

References | Experimental Conditions | Decay Rate n (n with x_{0} = 0) | Constant a (a with x_{0} = 0) |
---|---|---|---|

Nakamura et al. (1987) [19] | Water channel Regular biplane grids S = 36% Re _{M} = 1480–2970 | 1.45 | 0.0566 |

Isaza et al. (2014) [13] | Wind tunnel Regular biplane grids S = 34% Re _{M} = 42,000 | (1.90) | (0.489) |

Nedic and Tavoularis (2016) [22] | Wind tunnel Steel regular grids S = 25% Re _{M} = 51,000–102,000 | 2.34–2.87 (1.82–1.93) | |

Present study | Water flume Polyethylene knotless square nets S = 9.8–19.0% Re _{M} = 2840–5800 | 0.67–0.85 (0.43–0.55) | 0.023–0.047 (0.011–0.018) |

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**MDPI and ACS Style**

Shao, D.; Huang, L.; Wang, R.-Q.; Gualtieri, C.; Cuthbertson, A.
Flow Turbulence Characteristics and Mass Transport in the Near-Wake Region of an Aquaculture Cage Net Panel. *Water* **2021**, *13*, 294.
https://doi.org/10.3390/w13030294

**AMA Style**

Shao D, Huang L, Wang R-Q, Gualtieri C, Cuthbertson A.
Flow Turbulence Characteristics and Mass Transport in the Near-Wake Region of an Aquaculture Cage Net Panel. *Water*. 2021; 13(3):294.
https://doi.org/10.3390/w13030294

**Chicago/Turabian Style**

Shao, Dongdong, Li Huang, Ruo-Qian Wang, Carlo Gualtieri, and Alan Cuthbertson.
2021. "Flow Turbulence Characteristics and Mass Transport in the Near-Wake Region of an Aquaculture Cage Net Panel" *Water* 13, no. 3: 294.
https://doi.org/10.3390/w13030294