# Applying a New Force–Velocity Synchronizing Algorithm to Derive Drag Coefficients of Rigid Vegetation in Oscillatory Flows

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## Abstract

**:**

_{D}) is of great importance to the quantification of vegetation-induced wave dissipation. Recently, a direct measuring approach has been developed to derive vegetation drag coefficient more accurately compared to the conventional calibration approach. However, as this approach requires perfectly in-phase force and velocity signals, there are two difficulties associated with it. The first difficulty is the availability of a suitable force sensor to compose synchronized force–velocity measuring systems. The second difficulty is related to realigning the obtained timeseries of force and velocity data. This technical note develops a new synchronized force–velocity measuring system by using standard force sensors and an acoustic doppler velocimeter (ADV). This system is applied together with an automatic realignment algorithm to ensure in-phase data for C

_{D}deviation. The algorithm reduces the phase shift between force–velocity signals from ca. 0.26 s to 0.003 s. Both time-varying and period-averaged C

_{D}can be obtained using this method. The derived C

_{D}can be used to accurately reproduce the measured maximum total acting force on vegetation (R

^{2}= 0.759), which shows the reliability of the automatic alignment algorithm. The newly-developed synchronized force–velocity measuring system and alignment algorithm are expected to be useful in future experiments on vegetation–wave interactions with various hydrodynamic and vegetation settings.

## 1. Introduction

_{D}) and inertia force (F

_{M}) [23]. For normal field conditions, the drag force is the dominant force, and most relevant for wave energy dissipation. In the Morison equation, F

_{D}is proportional to the square of impact velocity on vegetation stems. When the velocity scale is determined, the magnitude of F

_{D}varies linearly with vegetation drag coefficients (C

_{D}). In oscillatory (wavy) flows, the C

_{D}values have a large range of variations (i.e., 0.1 to 100) [24]. The C

_{D}values depend on canopy density, hydrodynamic conditions, as well as the morphology of the individual canopy elements. Thus, choosing appropriate C

_{D}values are important for accurate simulation of F

_{D}, and the resultant wave dampening in many modelling studies [18,25,26,27,28,29,30].

_{D}: the calibration method and the direct measurement method. The calibration method is a convectional method developed in the 1990s [31,32], and has been widely used since [33,34,35]. It derives C

_{D}by calibrating its values to obtain the best fit between modelled and measured wave height evolution over vegetation fields. The direct measurement method is a new method, which has been developed since the 2010s [36,37,38]. This method directly applies the Morrison equation and measured in-phase force and velocity data to determine C

_{D}. The main differences between the calibration and the direct measurement method are: (1) the calibration method can only provide period-averaged C

_{D}, but the direct measurement method can derive both period-averaged and time-varying C

_{D}; (2) the direct measurement method can eliminate the potential errors often associated with the calibration method, and lead to C

_{D}–Re (Reynolds number) relations with better fits, which are desirable for model applications [37].

_{D}derivation [37]. The data alignment is critical for the direct measurement method, as there are time lags between original force data and velocity data signals (ca. 0.2 s), which may lead to large errors in the derived C

_{D}. These time lags may originate from small misalignments between force sensors and velocity measurement [37]. They may also be induced by intrinsic time shifts in instrument recordings. The maximum wave energy dissipation occurs at the peak wave orbital velocity in phase with the peak drag force when C

_{D}values matter the most. Thus, in order to obtain accurate C

_{D}values, it is important to minimize the time lags. Previous studies firstly set an intrinsic time lag between the force and velocity data, and then started iterations to reduce the time lag. Note that this intrinsic time lag varies with different instrument set-ups. This intrinsic time lag needs to be carefully tuned to obtain in-phase data. It is, however, preferable to have an automatic algorithm that can provide generic solutions to the alignment problem.

_{D}derivation. The new force–velocity measuring systems are applied in a flume at four locations in a mimicked mangrove canopy, which were tested with various simulated wave conditions. The automatic alignment algorithm was then applied to reduce the time lags between force–velocity signals. The processed data were subsequently used to derive both time-varying and period-averaged C

_{D}. To evaluate the accuracy of the derived C

_{D}(and also the alignment algorithm), we used the derived C

_{D}to reproduce the total acting force on mimicked vegetation, and compared it with the measurements.

## 2. Materials and Methods

#### 2.1. The Direct Measuring Method for C_{D} Derivation

_{D}is drag force, and F

_{M}is inertia force. $\rho $ is the density of the fluid. C

_{D}and C

_{M}are the drag and inertia coefficients, respectively. h

_{v}is the height of vegetation in water, and b

_{v}is the diameter of circular cylinder. U is the depth-averaged flow velocity. When linear wave theory is applied, the U varies as a function of sine:

_{w}is the amplitude of horizontal wave orbital velocity. Following linear wave theory, U

_{w}can be expressed as

_{0}is the vertical position of the considered point, which is 0 at the still wave level, and −h at the sea bed. Equation (3) was used to estimate U

_{w}when the velocity measurement is unavailable. C

_{M}is often assumed to be equal to 2 for cylinders (e.g., [39]). To derive the time-varying C

_{D}, we can apply the following equation:

_{M}can be derived based on $\frac{\partial U}{\partial t}$ using the timeseries of velocity data, and other parameters (i.e., $\rho $, b

_{v}, h

_{v}, C

_{M}) in F

_{M}are known. Thus, time-varying C

_{D}can be obtained readily when in-phase force and velocity data is obtained.

_{D}is relevant to vegetation-induced wave dissipation. It was not computed as the temporal mean of the time-varying C

_{D}. Time-varying C

_{D}has great variability over one wave period [37]. Specifically, its value is infinite when the velocity is close to zero. However, those C

_{D}values are not relevant for vegetation-induced wave energy dissipation, as dissipation is highest at the velocity peaks. Thus, C

_{D}values at high velocity matter the most. To obtain relevant period-averaged C

_{D}values, the direct measurement method applies the technique of quantifying the power and work done by the acting force ($\epsilon $) [37]. The time-varying power of F

_{D}and F

_{M}is evaluated as follows:

_{D}and W

_{M}is the work done by F

_{D}and F

_{M}over a full period, respectively. As U is a sine function (Equation (2)), the work done by ${F}_{M}$ over a full wave period (i.e., second term on the right) is zero. Thus, the work done by $F$ is equal to the work done by ${F}_{D}$:

_{D}can be derived based on the above equation:

_{D}values. As W is proportional to ${U}^{3}$ (Equation (9)), the integration of FU over a period is largely contributed to by the moments with relatively high velocity, and to a very limited extent, by the moments with low velocity. Thus, deriving period-averaged C

_{D}via the technique of quantifying $\epsilon $ can automatically assign large weight to the moments with high velocities in a wave period, resulting in most relevant C

_{D}values for wave dissipation analysis. To check the validity of the direct measuring method, we used the derived period-averaged C

_{D}values to reproduce the total acting forcing (F

_{rep}) using Equation (1), and compare it with the actual measurement. Additionally, another reproduced total force F

_{rep}’ is included by assuming C

_{D}= 1. It is used as a reference for the F

_{rep}.

#### 2.2. Synchronized Force–Velocity Measuring System

^{2}. It was built on top of a false bottom in order to elevate the canopy so that the force sensors can be mounted underneath it. The mimicked vegetation canopy was submerged in water (water depth = 0.25 m), and it was subjected to various wave conditions. The tested wave height varied from 0.03 to 0.09 m, and the tested wave period varied from 0.6 to 1.2 s. A space-averaged C

_{D}can be obtained by taking the mean C

_{D}values of the four measuring spots in each test.

_{v}= 1.25) is small [24,37], and it can significantly reduce the labor involved. Two of the ADVs were made by Nortek (Vectrino see http://www.nortekusa.com/usa/products/acoustic-doppler-velocimeters/vectrino-1), and the other two were made by SonTek (MicroADV, see https://www.sontek.com/argonaut-adv). These four ADVs are common instruments in fluid mechanics labs. Their basis measurement technology is coherent Doppler processing. They measure 3D water velocity of a small cylinder (i.e., within 1 cm

^{3}) that is a few centimeters away from measuring probes in the water. They can measure at frequencies as high as 64 Hz, which are desirable for the direct measuring method. In our experiment, the ADV data acquisition followed their respective user manuals. The measuring frequency was set as 40 Hz in order to accommodate the measuring frequency of the force sensor (i.e., 20 Hz). The obtained data are filtered through a low-pass filter to remove high frequency spikes following a similar method described in Strom and Papanicolaou [40].

#### 2.3. Automatic Alignment Algorithm

_{D}) should be in phase, which can be used to evaluate the time-lag between velocity and force measurement. A flow chart for the data realignment is shown in Figure 2. The inputs are the timeseries of force (F) and velocity (U). As we can assume that C

_{M}= 2 [39], the inertia force can be calculated based on the velocity. Then, the drag force (F

_{D}) can be computed by subtracting inertia force (F

_{M}) from the total force (F). Subsequently, we can determine the phase shift (∆t) between the velocity and drag force peaks. Lastly, this phase shift (∆t) will be recorded and used to adjust the velocity timeseries, aiming to obtain more in-phase velocity and force data. The obtained new velocity and force data will be used as input in the same loop. This loop continues 30 times, and we chose the minimum phase shift (∆t) and the resultant velocity and force timeseries as outputs for C

_{D}derivation. The automatic alignment algorithm is provided in the Appendix A as a MATLAB script. To verify the 30 loop count criterion, a sensitivity analysis is conducted by changing the loop count to 10, 20, 30, and 50. The resulting phase shifts with those loop counts are subsequently compared.

## 3. Results

#### 3.1. Velocity Profiles in the Vegetation Canopy

_{w}measured at the half water depth is 0.049 m/s. The difference between these two is small. The velocity profiles in wave0709 have greater vertical gradient, i.e., higher velocity at the top and lower velocity near the bottom. Overall, the difference between U

_{w}measured at the half water depth (0.106 m/s) and the amplitude of depth-averaged in-canopy velocity (0.115 m/s) is small. Therefore, it is acceptable to use U

_{w}measured at the half water depth as a representative value of the depth-averaged in-canopy velocity.

#### 3.2. Wave Height and Wave Orbital Velocity in the Mimicked Vegetation Canopy

_{w}) through mimicked vegetation canopy can be observed in Figure 4. The wave height reduces continuously from the canopy front to the end. The final wave height reduction rate was 55% (Figure 4a). The shown wave orbital velocity is obtained by ADV measurement at location 1–3. The ADV measurement at location 4 failed during the experiment. The shown U

_{w}is obtained by using Equation (3) based on an average wave height between x = 6–8 m in Figure 4a. With the reduced wave height, the magnitude of wave orbital velocity also reduces from 0.155 m/s to 0.095 m/s from the beginning to the end of the canopy. The reduced wave orbital velocity leads to variations in acting force on vegetation stem as well as in vegetation drag coefficient (C

_{D}), which are shown in the following sections.

#### 3.3. Data Alignment and Time-Varying C_{D}

_{D}and U), which lead to errors in C

_{D}.

_{D}and F

_{M}can be estimated as shown in Figure 5e. However, it should be noted that as the total force and velocity data are not in phase. The derived F

_{D}and F

_{M}are the first estimation. The derived peaks of the F

_{D}are even higher than the total force, which is not possible. This result highlights the necessity of obtaining in-phase data. Judging from the peaks between U and estimated F

_{D}peaks, the time lag between those two signals is 0.26 s.

_{D}is largely reduced (Figure 6a,b), but it cannot be completely eliminated, as small shifts in signal peaks and troughs still exist. Based on the phase shifts of the two peaks and two troughs (as indicated by the red double arrow lines), the time shift of the realigned U and F

_{D}is significantly reduced to 0.003 s, which is only 1% of the original time shift before the realignment. Note that this optimized time shift is the mean shift of the tested two wave periods (i.e., shifts of two peaks and two troughs). Apart from the reduced time shift, the magnitude of F

_{D}is also reduced to be lower than the total force (F), which is in line with the original Morison equation. It is noted that the peak drag force before alignment is about twice as large as the peak drag force after the alignment. Thus, if the original F timeseries were used, the derived drag coefficient would be considerably overestimated.

_{D}can be derived (Figure 6e). The time-varying C

_{D}values vary periodically with the changing velocity signals. The values are small when the velocity is large, but they reach to infinity when the velocity is close to zero. The reason for the unrealistically large C

_{D}values is because the F

_{D}are divided by very small velocity values in Equation (4). These unrealistically large C

_{D}values are not useful in modelling wave dissipation by vegetation, since they are associated with period with very low velocity when the energy dissipation is very limited.

#### 3.4. Deriving Period-Averaged C_{D}

_{D}by quantifying the power and work done, F

_{D}and F

_{M}. It is clear that the time-varying power of F

_{D}is always positive, and its magnitude varies in phase with the velocity magnitude (Figure 8). For ideal sinusoidal velocity signals, P

_{D}at the velocity peaks should be equal to the troughs. However, in our test case (and in real field conditions), the wave orbital velocity is asymmetrical: higher in the positive direction and lower in the negative direction. Thus, P

_{D}is larger near the wave peaks and smaller near the wave trough. The difference between peaks and troughs are much more apparent in P

_{D}compared to the difference in velocity. It is because that P

_{D}is to the third power of velocity. Small asymmetry in velocity will be greatly magnified in P

_{D}. The variation of P

_{M}is different from that of the P

_{D}. It is clear that P

_{M}varies between positive and negative values. The zero-crossings in P

_{M}occur when velocity is zero or when velocity is at its maximum in both directions, i.e., when F

_{M}is zero in Equation (5). Note that there are small fluctuations at the peaks of the P

_{D}. These fluctuations may be induced by the small phase shifts between the F

_{D}and U timeseries.

_{D}and P

_{W}over one wave period, we can obtain the work done by drag force (W

_{D}) and inertia force (W

_{M}) as shown in Equation (8). The averaged W

_{D}and W

_{M}over the two periods in Figure 8 is $2.0\text{}\times \text{}{10}^{-3}J$ and $-2.0\text{}\times \text{}{10}^{-4}J$, respectively. In case of ideal sinusoidal velocity signals, W

_{M}should be exactly zero. Due to the asymmetric wave velocity, the W

_{M}is not zero, but the magnitude of W

_{M}is fairly small, i.e., one tenth of the W

_{D}. Since the magnitude of W

_{M}is considerably small compared to W

_{D}, the assumption that W

_{M}can be ignored in the deriving period-averaged C

_{D}(in Equation (8)) is still valid, and the period-averaged C

_{D}in the shown case is derived as 3.25.

_{D}at three functional measuring locations of all the tested cases are listed in Table 2. It shows that relatively large deviations in C

_{D}values exist among different measuring locations. The C

_{D}values generally increase from locations from 1 to 3. Previous studies have shown that C

_{D}values increase with the reduced velocity (i.e., Reynolds number) [24,37]. The obtained increase of C

_{D}values may be related to the reduction of wave orbital velocity from the front to the end of the vegetation canopy, as shown in Figure 4b Thus, the spatial variation in C

_{D}values is in-line with previous studies. Additionally, it is noted that the cases with larger wave height and wave period (i.e., higher wave orbital velocity) generally lead to smaller spatially averaged C

_{D}, which is also in agreement with previous studies [24,37].

#### 3.5. Assessing the Derived C_{D} by Reproducing Acting Force

_{D}, we used the derived values to reproduce the total force from the velocity signals using Equation (1). The reproduced total force (F

_{rep}) is subsequently compared with the measured actual total force (Figure 9). The reproduced total force (F

_{rep}’) using C

_{D}= 1 is also included as reference. It is clear that F

_{rep}is in good agreement with the measured force over the shown two wave periods, although small differences exist between them. Notably, the measured maximum force is well captured in F

_{rep}near x = 0.5 π, which is important as the maximum force is critical not only for energy dissipation but also for assessing the stem strength to wave loading. As a comparison, the difference between F

_{rep}’ and the measurement is large, which shows the validity of using period-averaged C

_{D}to reproduce the total force.

_{rep}and maximum measured total force obtained at all three functional measuring locations in all test cases. In general, the reproduced maximum force is well in-line with the measurement, as most of the data points are fairly close to the 1:1 reference line. The R

^{2}value is 0.759 for data of all three functional measuring locations in all test cases. This result indicates the C

_{D}deriving procedure is valid, and the intrinsic errors associated with this procedure are acceptable.

## 4. Discussion

#### 4.1. Advantages of the Current Measuring System and Alignment Algorithm

_{D}(Figure 5 and Table 2). Thus, it is important to have synchronized force–velocity measurement at multiple locations by a number of force–velocity measuring systems. The selected standard force sensors are small enough to be installed at multiple locations in wave flumes. Additionally, these sensors are designed with built-in tapped holes, which facilitate testing various vegetation mimics, e.g., rigid cylinders, flexible stripes, and real vegetation stems.

_{D}values (Figure 6 and Figure 7). In our experiment, we aligned the instruments as good as possible (please see Figure 1d of the manuscript), but it is inevitable to have small misalignment to cause the delay. The main source causing the delay may be the inherent difference in instruments’ speed of recording and receiving data, as the force and velocity measurements have their separate data acquisition systems. Thus, the automatic synchronizing algorithm is necessary and valuable in the current study. By using this algorithm, the time shift between two signals can be reduced to 0.003 s, which is only 1% of the original time shift before the realignment. The obtained time shifts are believed to be acceptable when comparing to normal wave periods (1–2 s) tested in our lab flume. The time shifts are merely 0.3% to the tested wave period.

_{D}. Furthermore, this algorithm can run very efficiently, which is desirable when processing large data sets from multiple measuring locations. Lastly, this alignment algorithm is applied to process the velocity data from ADVs, but it is worth noticing that this algorithm is also applicable for other velocity measuring technologies, e.g., EMF (electromagnetic flow manufacture meter) and PIV (particle image velocimetry) [41,42,43].

#### 4.2. Current Limitations and Future Applications

_{D}values. This may partly explain the difference between the maximum measured force and the reproduced force. In order to improve the velocity measuring accuracy and reduce the labor involved, PIV system can be applied in future experiments. The PIV system can provide detailed velocity information of velocity field [41,42]. By applying such a system, it is also possible to obtain the relative velocity between water motion and the motion of flexible vegetation stems. Thus, the developed technics in the current study can be further applied in flexible vegetation canopies, e.g., saltmarshes and seagrasses, which is interesting to both coastal engineers and ecologists.

## 5. Conclusions

_{D}) following the direct measuring method [36,37,38]. Different from previous studies, standard force sensors are applied to compose four synchronized force–velocity measuring systems in the current experiment. These standard force sensors are robust and suitable for flume applications. The composed force–velocity measuring systems can provide synchronized force–velocity measurement. Although one of the ADV instruments failed, the other three ADVs functioned well during the experiment. Importantly, an automatic algorithm was developed to realign the obtained force and velocity signals for direct C

_{D}deviation. This algorithm is expected to be able to accommodate a variety of velocity measuring techniques, providing possibilities to extend current application range. The developed force–velocity measuring systems and the automatic realignment algorithm may assist future experiments on vegetation–wave interactions for better understanding and prediction of vegetation-induced wave dissipation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. MATLAB Code for Force and Velocity Data Realignment

_{D}derivations.

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**Figure 1.**(

**a**) Flume experiment set-up. The numbers 1–4 indicate the locations of the in-phase force–velocity measurement; (

**b**) A force sensor connected to a PVC pipe; (

**c**) Mimicked vegetation canopy constructed by PVC pipes; (

**d**) Instrument deployments in the flume without mimicked vegetation. “WG” stands for wave gauge, “ADV” stands for acoustic doppler velocimeter, and “FS” stands for force sensor. The dashed line indicates the ADV and force sensor are placed at the same cross-section to ensure in-phase measurements.

**Figure 2.**Flow chart to realign velocity and force data signals. The algorithm is provided in the Appendix A as a MATLAB script.

**Figure 3.**(

**a**) Measured velocity profile for case wave0312 with 3 cm wave height and 1.2 s wave period, U

_{min}is the highest wave orbital velocity in negative direction (oppose to wave propagation), U

_{max}is the highest wave orbital velocity in positive direction (same as wave propagation), and U

_{w}is the amplitude of wave orbital velocity; (

**b**) Measured velocity profile for case wave0709 with 7 cm wave height and 0.9 s wave period.

**Figure 4.**(

**a**) Spatial variations of wave height (H) through the vegetation canopy (i.e., x = 0–8 m) indicated as green bars; (

**b**) Spatial variations of the magnitude of wave orbital velocity (U

_{w}) through the vegetation canopy. The first 3 U

_{w}data points (x = 0–6 m) are obtained from ADV measurement, whereas the last data point at x = 7 m is obtained by using Equation (3) based on an average wave height between x = 6 m and 8 m in panel (

**a**). The shown test case is wave0712 with 7 cm wave height and 1.2 s wave period.

**Figure 5.**(

**a**–

**d**) Raw velocity and total force data measured at four locations (1–4) in the order of the wave propagation in the mimicked vegetation canopy; (

**e**) The shaded area in panel (

**c**) is blown up in panel (

**e**) for detailed analysis, where inertia force and drag force were derived based on non-synchronized data. The time shift (∆t) between the F

_{D}and U is 0.26 s. The shown test case is wave0712 with 7 cm wave height and 1.2 s wave period. The velocity data in panel (

**d**) is not included, due to the ADV measurement failure at location 4.

**Figure 6.**(

**a**–

**d**) Time-varying U, F

_{D}, F

_{M}, and F data over two wave periods. The gray lines are before realignment, and the red ones are after realignment. The vertical double-arrowed lines in panel (a) and (

**b**) indicate the synchronization status before and after the realignment; (

**e**) Time-varying C

_{D}derived based on realigned U and F

_{D}data. The shown test case is wave0712 with 7 cm wave height and 1.2 s wave period.

**Figure 7.**Sensitivity analysis of the number of loops used in the realignment algorithm. Case 1 is the case wave0712 with 7 cm wave height and 1.2 s wave period, and case 2 is the case wave0512 with 5 cm wave height and 1.2 s wave period.

**Figure 8.**Time-varying U, P

_{D}, and P

_{M}over two wave periods. The shown test case is wave0712, with 7 cm wave height and 1.2 s wave period. The averaged work done by drag force (W

_{D}) and inertia force (W

_{M}) over the shown two periods is $2.0\text{}\times \text{}{10}^{-3}J$ and $-2.0\text{}\times \text{}{10}^{-4}J$, respectively. The derived period-averaged C

_{D}is 3.25.

**Figure 9.**Comparison between reproduced total acting force and measured total force. The red solid line is the measured total force; The black dash line is quantified by using derived period-averaged C

_{D}in Equation (1) (i.e., F

_{rep}); The blue dash line is quantified by using C

_{D}= 1 in Equation (1) for reference (i.e., F

_{rep}’). The shown test case is wave0712, with 7 cm wave height and 1.2 s wave period.

**Figure 10.**Comparison between the maximum reproduced total acting force (F

_{rep}) and the measured maximum total force at three functional measuring locations in all the tested cases.

Direction | Known Weights (g) | 1st Reading ^{a} (g) | 2nd Reading ^{b} (g) | 3rd Reading ^{c} (g) | Mean Reading (g) | Absolute Error (g) | Relative Error |
---|---|---|---|---|---|---|---|

+ | 5 g | 5.10 | 5.00 | 5.00 | 5.03 | 0.03 | 0.60% |

10 g | 10.10 | 10.10 | 10.00 | 10.07 | 0.07 | 0.70% | |

20 g | 20.10 | 20.00 | 20.00 | 20.03 | 0.03 | 0.15% | |

50 g | 49.90 | 49.90 | 49.90 | 49.90 | 0.10 | 0.02% | |

− | 5 g | −5.00 | −5.00 | −5.10 | −5.03 | −0.03 | −0.60% |

10 g | −10.00 | −10.10 | −10.10 | −10.07 | −0.07 | −0.70% | |

20 g | −20.00 | −20.00 | −20.00 | −20.00 | 0 | 0% | |

50 g | −50.0 | −49.9 | −49.9 | −49.93 | 0.07 | 0.14% |

^{a,b,c}The 1st, 2nd, and 3rd time readings were taking when the weights were put at the bottom, middle, and tip of the testing pipe, respectively.

Test Number | Wave Height (m) | Wave Period (s) | C_{D} at Location 1 | C_{D} at Location 2 | C_{D} at Location 3 | Space-Mean C_{D} | Standard Deviation |
---|---|---|---|---|---|---|---|

1 | 0.03 | 0.6 | 5.41 | 10.02 | 10.14 | 8.52 | 7.28 |

2 | 0.03 | 0.9 | 3.19 | 4.88 | 7.09 | 5.05 | 3.82 |

3 | 0.03 | 1.2 | 3.60 | 2.79 | 5.25 | 3.88 | 1.56 |

4 | 0.05 | 0.6 | 7.61 | 5.83 | 5.96 | 6.46 | 0.98 |

5 | 0.05 | 0.9 | 3.82 | 2.36 | 3.62 | 3.27 | 0.62 |

6 | 0.05 | 1.2 | 2.84 | 3.03 | 3.97 | 3.28 | 0.37 |

7 | 0.07 | 0.6 | 3.43 | 3.01 | 7.04 | 4.49 | 4.90 |

8 | 0.07 | 0.9 | 1.97 | 1.83 | 2.94 | 2.25 | 0.37 |

9 | 0.07 | 1.2 | 1.77 | 2.77 | 3.51 | 2.68 | 0.76 |

10 | 0.09 | 0.6 | 3.02 | 5.76 | 6.10 | 4.96 | 2.86 |

11 | 0.09 | 0.9 | 1.26 | 1.79 | 2.64 | 1.89 | 0.49 |

12 | 0.09 | 1.2 | 1.44 | 2.54 | 3.00 | 2.33 | 0.64 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yao, P.; Chen, H.; Huang, B.; Tan, C.; Hu, Z.; Ren, L.; Yang, Q.
Applying a New Force–Velocity Synchronizing Algorithm to Derive Drag Coefficients of Rigid Vegetation in Oscillatory Flows. *Water* **2018**, *10*, 906.
https://doi.org/10.3390/w10070906

**AMA Style**

Yao P, Chen H, Huang B, Tan C, Hu Z, Ren L, Yang Q.
Applying a New Force–Velocity Synchronizing Algorithm to Derive Drag Coefficients of Rigid Vegetation in Oscillatory Flows. *Water*. 2018; 10(7):906.
https://doi.org/10.3390/w10070906

**Chicago/Turabian Style**

Yao, Peng, Hui Chen, Bensheng Huang, Chao Tan, Zhan Hu, Lei Ren, and Qingshu Yang.
2018. "Applying a New Force–Velocity Synchronizing Algorithm to Derive Drag Coefficients of Rigid Vegetation in Oscillatory Flows" *Water* 10, no. 7: 906.
https://doi.org/10.3390/w10070906