**Research article**
06 Sep 2021

**Research article** | 06 Sep 2021

# A study on the drag coefficient in wave attenuation by vegetation

Zhilin Zhang Bensheng Huang Chao Tan and Xiangju Cheng

^{1,2,3,4,5},

^{1,2,3,4},

^{1,2,3,4},

^{5}

**Zhilin Zhang et al.**Zhilin Zhang Bensheng Huang Chao Tan and Xiangju Cheng

^{1,2,3,4,5},

^{1,2,3,4},

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^{5}

^{1}Guangdong Research Institute of Water Resources and Hydropower, Guangzhou 510630, China^{2}State and Local Joint Engineering Laboratory of Estuarine and Hydraulic Technology, Guangzhou 510630, China^{3}Guangdong Provincial Science and Technology Collaborative Innovation Center for Water Safety, Guangzhou 510630, China^{4}Guangdong Key Laboratory of Hydrodynamic Research, Guangzhou 510630, China^{5}School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, China

^{1}Guangdong Research Institute of Water Resources and Hydropower, Guangzhou 510630, China^{2}State and Local Joint Engineering Laboratory of Estuarine and Hydraulic Technology, Guangzhou 510630, China^{3}Guangdong Provincial Science and Technology Collaborative Innovation Center for Water Safety, Guangzhou 510630, China^{4}Guangdong Key Laboratory of Hydrodynamic Research, Guangzhou 510630, China^{5}School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, China

**Correspondence**: Zhilin Zhang (zhilin_zhang@outlook.com)

**Correspondence**: Zhilin Zhang (zhilin_zhang@outlook.com)

Received: 02 Apr 2021 – Discussion started: 15 Apr 2021 – Revised: 27 Jul 2021 – Accepted: 14 Aug 2021 – Published: 06 Sep 2021

Vegetation in wetlands is a large-scale nature-based resource providing a myriad of services for human beings and the environment, such as dissipating incoming wave energy and protecting coastal areas. For understanding wave height attenuation by vegetation, there are two main traditional calibration approaches to the drag effect acting on the vegetation. One of them is based on the rule that wave height decays through the vegetated area by a reciprocal function and another by an exponential function. In both functions, the local wave height reduces with distance from the beginning of the vegetation depending on damping factors. These two damping factors, which are usually obtained from calibration by measured local wave height, are linked to the drag coefficient and measurable parameters, respectively. So the drag coefficient that quantifies the effect of the vegetation can be calculated by different methods, followed by connecting this coefficient to hydraulic parameters to make it predictable. In this study, two relations between these two damping factors and methods to calculate the drag coefficient have been investigated by 99 laboratory experiments. Finally, relations between the drag coefficient and relevant hydraulic parameters were analyzed. The results show that emergent conditions of the vegetation should be considered when studying the drag coefficient; traditional methods which had overlooked this condition cannot perform well when the vegetation was emerged. The new method based on the relation between these two damping factors performed as well as the well-recognized method for emerged and submerged vegetation. Additionally, the Keulegan–Carpenter number can be a suitable hydraulic parameter to predict the drag coefficient and only the experimental setup, especially the densities of the vegetation, can affect the prediction equations.

To meet the current wave prevention requirements, it is practical to construct ecological safety barriers with wetland vegetation based on natural conditions. Vegetation in wetlands can enhance the toughness of the coast and save construction investment effectively by dissipating incoming wave energy (Reguero et al., 2018). Practice also has proved that vegetation in wetlands can provide services such as enhancing the coastal ecosystem and biodiversity, enhancing fisheries and forestry production, increasing bank stability, and promoting tourism economy, whereas the vegetated area occupies land resources in the floodplain (Schaubroeck, 2017; Keesstra, 2018). Hence, it is necessary to better understand the mechanism of wave attenuation to promote the efficiency of the nature-based solution.

Wave attenuation by vegetation is mainly induced by the drag force provided
by the vegetation acting on water motion, as investigated in different
research topics such as numerical modeling (e.g., Wu et al., 2016; Suzuki et al.,
2019), laboratory experiment (e.g., Hu et al., 2014; Wu and Cox, 2015,
2016), or field study (e.g., Danielsen et al., 2005; Quartel et al., 2007).
The drag force is closely related to the drag coefficient *C*_{D} which
quantifies the drag or resistance of vegetation in water (Chen et al.,
2018). This coefficient is one of the most uncertain parameters in the
complicated interaction between the vegetated area and water, because the
drag effect can be fairly different on various time and space scales.

The calibration method for the drag coefficient is based on the perspective
of wave energy dissipation and wave height reduction which will be discussed
in Sect. 2, while Dean (1979) and Kobayashi et al. (1993) proposed that
local wave height decaying through the vegetated area follows a reciprocal
function and exponential function, respectively. These two calibration
functions describe local wave height with a distance from the beginning of
vegetation and a factor reflecting the damping, so the corresponding factor
can be calibrated based on measured wave height through the vegetated area.
The damping factor *α*^{′} from the reciprocal function and the
exponential damping factor *k*^{′} from the exponential function are often
linked to the drag coefficient *C*_{D} as well as measurable parameters such as
water depth and density of stems. For instance, Dean (1979) proposed an
equation to calculate *C*_{D} based on the damping factor, and the model has
been developed by researcher teams such as Knutson et al. (1982), Dalrymple et
al. (1984), and Losada et al. (2016). Overall, the drag coefficient can be
calculated by calibrating *α*^{′} or *k*^{′} using measured local wave
height; then the researchers built nonlinear relations between *C*_{D} and
hydraulic parameters such as the Reynolds number (e.g., Hu et al., 2014; He
et al., 2019). In this way, the drag of vegetation in water becomes
predictable based on the nonlinear relations and the values of these
hydraulic parameters under different operating conditions.

Zhang et al. (2021) compared these two calibration approaches by these
two featured functions directly and yielded a connection between *α*^{′} and *k*^{′}; then a new equation to calculate the drag coefficient has
been revealed. However, Zhang et al. (2021) overlooked the relation between
*k*^{′} and *C*_{D} by Kobayashi et al. (1993) and only used the relation
between *α*^{′} and *C*_{D} by Dean (1979). In this article, using the
well-documented relation between the damping factor *α*^{′} and the drag
coefficient *C*_{D} by Dalrymple et al. (1984), as well as the mentioned
relation by Kobayashi et al. (1993), these two traditional approaches have
been compared from another perspective, and the second connection between
*α*^{′} and *k*^{′} has been revealed.

Hence, there are two relations between the damping factor and the
exponential damping factor from the two perspectives, and they have been analyzed
by 99 cases from collected data and experiments in this study.
Additionally, in normal tidal conditions and at the initial stage of storm
surge, vegetation in wetlands can be emerged, while, by storm surge,
vegetation is submerged or nearly submerged. Existing methods to calculate the
drag coefficient have been compared considering these emergence conditions.
Finally, relations between *C*_{D} and hydraulic parameters, e.g.,
the Reynolds number (*R**e*), the Keulegan–Carpenter number (*K**C*), and the
Ursell number (*U**r*), have been studied.

Typically, the drag coefficient *C*_{D} is determined from the perspective
of wave energy dissipation, represented by the decay of wave height. Dean
(1979) proposed one of the first models for wave attenuation by vegetation
in which wave height throughout the vegetated area can be expressed as a
reciprocal function:

where *K*_{X} (–) is the relative wave height at a distance *X* (m) through
the vegetation field from the beginning of vegetation, *H*(*X*) (m) is the
local wave height, *H*_{0} (m) is the incident wave height, and *α*^{′}
(m^{−1}) is the damping factor.

Based on empirical estimates of fluid drag forces acting on vertical, rigid cylinders, Dean (1979) found that

where *d* (m) is the diameter of the circular vegetation cylinder, *h* (m)
is the water depth, and *N* (stems m^{−2}) is the average number of stems
per unit area.

Then Dalrymple et al. (1984) formulated an algebraic dissipation equation
practicing linear theory and conservation of wave energy where *α*^{′}
can be expressed as

where *d*_{v} (m) is the vegetated area per unit height of plant normal to
wave direction, *k*_{w} (rad m^{−1}) is the wave number, and *l*_{s} (m)
is the submerged stem height.

On the other hand, Kobayashi et al. (1993) published that the local wave height decays exponentially through submerged artificial kelp:

where *k*^{′} (m^{−1}) is the exponential damping factor. Based on linear
wave theory and the conservation equation of energy, *k*^{′} is expressed as the following
(Kobayashi et al., 1993):

If we compare these relations between the (exponential) damping factor and
the drag coefficient (Eqs. 3 and 5), a relation between the damping
factor *α*^{′} and the exponential damping factor *k*^{′} can be derived:

Recently, Zhang et al. (2021) presented a relation between *α*^{′} and
*k*^{′}, looking at these featured functions (Eqs. 1 and 4) directly.
This method firstly scaled the distance *X*:

and

where *α* ($={\mathit{\alpha}}^{\prime}L)$ (–) is the scaled damping factor, *L* (m) is
the length of vegetated area, *x* ($=X/L$) (–) is the scaled distance
through the vegetation field, *k* ($={k}^{\prime}L$) (–) is the scaled exponential
damping factor, and *F*(*x*) and *G*(*x*) represent functions.

Then, by using the Taylor expansion, when the scaled distance *x* equals
half, the following equations were derived:

and

where *R*_{1}(*x*) and *R*_{2}(*x*) are the residual terms. The relative
magnitude of each term in Eqs. (9) and (10) has been analyzed by Zhang et
al. (2021), and they revealed that the first two terms on the right-hand side
of these equations are relatively large compared to other terms. Hence,
considering only these two terms in Eqs. (9) and (10), the proportionality
$\frac{\mathrm{2}}{\mathit{\alpha}+\mathrm{2}}/\frac{\mathrm{1}}{{e}^{k/\mathrm{2}}}=\frac{\mathrm{4}\mathit{\alpha}}{(\mathit{\alpha}+\mathrm{2}{)}^{\mathrm{2}}}(x-\mathrm{1}/\mathrm{2})/\frac{k}{{e}^{k/\mathrm{2}}}(x-\mathrm{1}/\mathrm{2})$ results in

which equals

Equations (6) and (12) have built bridges between the exponential function and reciprocal function, verifying that these two functions are reliable and capable to describe the wave height attenuation by vegetation satisfactorily. The rule of the attenuation is then limited by two functions, which can increase the reliability of the calibration.

However, application of Eq. (6) in Eq. (12) results in ${k}^{\prime}L\cong \mathrm{0}$, which is not appropriate when there is vegetation in the wetlands. Hence, it is worth it to further study the relation between these two damping factors to help us better understand the drag coefficient and wave attenuation by vegetation.

In addition, we had studied the relation between *C*_{D} and three relevant
hydraulic parameters, which are frequently used to model *C*_{D}, including
(1) the Reynolds number, *R**e* ($={u}_{\mathrm{max}}{d}_{\mathrm{v}}/\mathit{\nu})$, where
*ν* $(=\mathrm{1.011}\times {\mathrm{10}}^{-\mathrm{6}}$ m^{2} s^{−1}) is the
kinematic viscosity of water and ${u}_{\mathrm{max}}(=\mathrm{2}\mathit{\pi}{H}_{\mathrm{0}}/\mathrm{2}T\mathrm{tanh}{k}_{\mathrm{w}}h)$ is the maximum horizontal wave velocity from
linear wave theory, where *T* (s) is the wave period; (2) the
Keulegan–Carpenter number, *K**C* ($={u}_{\mathrm{max}}T/{d}_{\mathrm{v}})$, representing
oscillatory flow around cylinders; and (3) the Ursell number, *U**r* ($={\mathit{\lambda}}^{\mathrm{2}}{H}_{\mathrm{0}}/{h}^{\mathrm{3}}$), characterizing the balance between wave steepness and the
relative water depth, where *λ* (m) is the wavelength. Researchers
have reported several formulas between *C*_{D} and *R**e*. For instance, Wu et al. (2011) obtained the following empirical equation:

Besides this, He et al. (2019) revealed that

Hence, the following two formulas are most possible solutions to study the
nonlinear relation between *C*_{D} and these parameters:

where $\stackrel{\mathrm{\u203e}}{X}$ can be *R**e*, *K**C*, or *U**r*, and *a*, *b*, and *c* are the factors.
Values of these factors can be obtained by the regression of *C*_{D} by
calibrated *α*^{′} or *k*^{′} (Eqs. 3 or 5) and these parameters, and
in this way, *C*_{D} becomes predictable under different operation
conditions. We have obtained the values of the factors and the corresponding
adjusted *R*^{2} as in Sect. 5.4 by both equations, and it is hard to
tell the difference between these results from Eqs. (15) and (16). The
former was at last chosen because it contains less factors and is
simpler than the latter.

The experiments were conducted in a wave flume in Guangdong Key Laboratory of Hydrodynamic Research at Guangdong Research Institute of Water Resources and Hydropower, China. The wave flume is 80.0 m long, 1.8 m wide, and 2.6 m deep (schematized in Fig. 1a, unit: meters). The wave was generated by a wave generator at one end and absorbed at the opposite end.

The start of the vegetated area was located 52.7 m from the wave generator.
The uniform vegetation was constructed by putting mimic plants (Fig. 1b) in
holes drilled in the bottom. These two heights of mimic plants (*l*_{v})
were 0.3 and 0.5 m, and *d*_{v} of the mimic plants was 0.057 m, considering average
diameters of the stem and leaves, while the height ratio of them was about
0.5 (Fig. 1b). The three horizontal lengths of the vegetated area (*L*) were
4, 5, and 6 m, and two mimic stem densities (*N*) were 25 and 50 stems
m^{−2} (marked as N1 and N2; see Fig. 1c and d). The two water levels of
the flume were 0.8 and 1.0 m, so the corresponding water depths of the
floodplain (*h*) were 0.3 and 0.5 m.

The original wave height (*H*_{ori}) of each designed regular wave
was calibrated at 30 m from the wave generator before these tests. In this study,
seven wave gages (G1 to G7) were used to measure the wave height time
series, which were placed 1 m apart from each other from the beginning of
the vegetated area (Fig. 1a), and the measurement at G1 was used as the
incident wave height (*H*_{0}) (Wu and Cox, 2015).

Control tests were carried out with no mimic plants to reduce the influence of flume bed and sidewalls. As listed in Table 1, 16 cases were conducted including various conditions. Data of each test were collected during more than 200 s, and each case was repeated for three times.

Besides experiments in this study, observations in published literature have been collected from Hu et al. (2014), Wu et al. (2011), and Wu and Cox (2015, 2016) as Zhang et al. (2021) presented. The summarized experimental setup is shown in Table 2. Overall, different laboratory experiments with different operation conditions have been conducted by the research studies.

## 5.1 Reduction of wave height

Wave height along the vegetated area is a significant index for wave
attenuation by vegetation. The calibrated reductions of wave height by three
equations demonstrating two examples (cases 13 and 16) are shown in Fig. 2.
It is clear that Eqs. (7) and (8) were reliable relations between the scaled
distance and the relative wave height. Additionally, with the calibrated *k*
value from Eq. (8), we calculated the value of *α* according to Eq. (11). Applying the calculated *α* in Eq. (7), the calculated relative
wave height, which was named by Eq. (11) in Fig. 2, was applicable to fit the
measurements, which suggested that Eq. (11) is valid. Results also show that
the larger the value of the scaled damping factors, the stronger the wave
attenuates.

## 5.2 Relation between *α* and *k*

The relation between calibrated values of *α* and *k* by 99 cases from
this study and collected data is shown in Fig. 3. In the study of Wu et al. (2011), Hu et al. (2014), and this research, both submerged and emerged
cases have been conducted, and in the study of Wu and Cox (2015, 2016) the
vegetation was emerged. The emerged and submerged cases have been separated
for studying the influence of the emergent condition (emerged or submerged).
Figure 3 shows that there is an obvious relation between *α* and *k*
for all cases. However, Eq. (6), which was obtained by comparing these
relations between the (exponential) damping factor and the drag coefficient
by Dalrymple et al. (1984) and Kobayashi et al. (1993), worked well only
when values of *α* and *k* were smaller than around 0.4. Equation
(11), on the other hand, seemed a possible solution for the relation of
these two factors, and the relation between *α* and *k* is not
strongly affected by the emergent condition even though these values are
indeed relatively small when the vegetation is submerged (0.04*<**α**<*0.56) than when it is emerged (0.12*<**α**<*1.43). Notably, the analytical solution by Kobayashi et al. (1993), i.e., Eq. (5), was obtained and conducted using deeply submerged
artificial kelp, and *H*(*X*)^{3}≅*H*_{0}*H*(*X*)^{2} was assumed, which
can only be valid when wave height reduces slightly through submerged
vegetated areas and the exponential damping factor is small. This is why Eq. (6) can only be useful for submerged vegetation.

Equation (11) also revealed that $\mathit{\alpha}-k={k}^{\mathrm{2}}/(\mathrm{2}-k)>\mathrm{0}$ since *k* is
smaller than 2 (Fig. 3). When the vegetation is deeply submerged, the
calibrated *k* is close to zero, and *α* is larger than but approximately equal to
*k* (Eq. 6); when the vegetation becomes emerged, *α* and *k* become
relatively large and the difference between them enlarges, which can be seen
in Figs. 2 and 3. That is to say, Fig. 3 shows that Eq. (11) works well and
it includes Eq. (6) to some extent.

## 5.3 Calculating *C*_{D} by different methods

### 5.3.1 Calculating *C*_{D} by Dean (1979)

Several studies paid attention to the emergent condition of the vegetation
recently. This condition (e.g., by *l*_{s}) has been included in Eq. (3) by
Dalrymple et al. (1984), while it had not been considered in Eq. (2) by Dean
(1979). Both methods by Dean (1979) and Dalrymple et al. (1984) consider
wave height decaying by the reciprocal function, in which the damping factor
can be obtained by fitting the local wave height by Eq. (7). In this case,
the value of the drag coefficient can be calculated using Eqs. (2) or (3), and the comparison of results by these two equations is shown in Fig. 4. The result shows that these 99 cases obviously can be divided into two
categories, and they can be fitted by two linear lines. Both the values of
the adjusted *R*^{2} of the linear fit of emerged category and submerged
category are 0.97, which means the results by these two equations are
comparable. However, the slope of the former is about twice as large as the
latter, so the emergent condition is necessary to consider when
calculating the drag coefficient in wave attenuation by vegetation.
Additionally, the linear fit of the submerged category is close to the 1:1
line, which means both equations are reliable and applicable for this
category, while one of them is not suitable for emerged category considering
the slope of the linear line. Since Eq. (3) pays attention to the
emergent condition, it is then regarded as a more satisfactory solution to
calculate the drag coefficient for different conditions, while for emerged
cases Eq. (2) can lead to larger values.

### 5.3.2 Calculating *C*_{D} by Kobayashi et al. (1993)

Equation (5) by Kobayashi et al. (1993) also considered the emergent
condition, and it was obtained by using local wave height decaying
exponentially. Hence, in this part, the comparison of the values of the drag
coefficient by Eqs. (3) and (5) was studied to learn the influence of
different decaying functions, and the result is shown in Fig. 5. The value of
*C*_{D} by Kobayashi et al. (1993) was obtained by calculating *C*_{D} using
Eq. (5) on the basis of the calibrated exponential damping factor by fitting
the local wave height using Eq. (8). Figure 5 reveals that *C*_{D} by Eq. (5) is always smaller than *C*_{D} by Eq. (3). Also, cases can be divided
into two categories. For submerged cases, the drag coefficient by Eq. (5) is
close to but slightly smaller than that by Eq. (3), with a slope of 0.96 in
Fig. 5; for emerged cased, the former is smaller than the latter when
the drag coefficient is larger. This is consistent with the conclusion in
Sect. 5.2 since *C*_{D} has positive correlation with *α* and *k*.
In short, for calculating the drag coefficient in wave attenuation by
submerged vegetation, both Eqs. (3) and (5) can be the solution. However,
for emerged cases, Eq. (5) can lead to underestimated values of the
calibrated *C*_{D}.

Additionally, although the regression of data should not be linear since
$k/\mathit{\alpha}=(\mathrm{2}-k)/\mathrm{2}<\mathrm{1}$, which is not a constant, if we obtain *C*_{D} by
calibrating the exponential function for emerged cases, we have a rapid
assessment that the value will be approximately 77 % of the needed value.
Moreover, the result reveals that ${k}^{\prime}/{\mathit{\alpha}}^{\prime}\approx \mathrm{0.77}$. Combining Eq. (12), $k{}^{\prime}L=k$ approximates to 0.46, then *K*_{X}≈0.63 at the end
of the vegetation according to Eqs. (4) and (8). This means that the reduction
rate ($=\mathrm{1}-{K}_{X}$) of the wave height for the emerged cases is about
37 %. Furthermore, if we apply *k*≈0.46 in Eq. (11), *α* is
about 0.60; then *K*_{X}≈0.63 according to Eqs. (1) and (7). Values of
*K*_{X} which are close to *α* and *k* can be used to assess the wave
attenuation by emerged vegetation in a very preliminary way.

Of course, several parameters can affect the drag effect. In this case,
certain cases should be considered separately instead of using the result
from a regression by all the cases with different operating conditions; then
the slope of the comparison between the calculated *C*_{D} by Eqs. (3) and
(5) will be different so the calculated relative wave height will be
different.

### 5.3.3 Calculating *C*_{D} by a new method

The new method obtains the damping factor *α*^{′} by using
the calibrated *k*^{′} based on measured wave height and Eq. (12),
so the drag coefficient *C*_{D} can be calculated by Eq. (3). The Eq. (12)-based method used the rule that the local wave height decays
exponentially and the classic relation between the damping factor and
*C*_{D} by Dalrymple et al. (1984). The comparison of the calculated values
of *C*_{D} by Eq. (3) and the new method is shown in Fig. 6. The result
shows that there is a strong linear relationship among the calculated values
in 99 cases from different research studies. The slope of the linear fit is about
unitary, and the adjusted *R*^{2} equals 0.99. The result is inspiring and
shows that the new method can lead to comparable results to the method by
Dalrymple et al. (1984) for the drag coefficient. It is revealed that Eq. (12) is satisfactory and can be a bridge between the damping factor and the
exponential damping factor. Based on the results in Figs. 5 and 6, the
exponential damping factor *k*^{′} can be used to calculate *C*_{D} while it
needs to be converted to *α*^{′} based on Eq. (12) instead to be used
directly in Eq. (5) for emerged cases; while for submerged cases, it can be
a solution to calculate *C*_{D} directly.

## 5.4 Relating *C*_{D} to *R**e*, *K**C*, and *U**r*

### 5.4.1 Relating *C*_{D} to *R**e*

Relating the calculated *C*_{D} by calibration method to *R**e*, *K**C*, or
*U**r* is a common method to predict *C*_{D}. The relation between *R**e* and
the calibrated *C*_{D} by the new method and the nonlinear fit by Eq. (15)
are shown in Fig. 7. In the study by Hu et al. (2014) and this research,
cases were grouped by different densities. The values of *R**e* ranged from
370 to 38 000, and the solid line following different groups of symbols can
basically fit. Results reveal that separating cases from different
densities is necessary for studying this relation, while the effect of the
emergent condition can be ignorable. Equation (15) was utilized to study
this relation, and the outcomes of the factors from nonlinear fit between
*R**e* and *C*_{D} by the new method and by Eq. (3) are shown in Table 3.
Results show that values for a certain factor (*a* or *b*) based on the new
method and Eq. (3) are close to each other, especially for cases from Hu et
al. (2014), supporting the fact that the new method is comparable to Dalrymple et al. (1984). Moreover, values of factors can be quite different in various groups
in Table 3; hence, laboratory setup could play an important role in the
relation between the drag coefficient and the Reynolds number. Hence, this
relation is not universal for different cases. For example, the calculated
line by Eq. (13) published by Wu et al. (2011) is not very suitable for
other groups of measurements. Hence, for engineering applications, case
studies are needed for certain issues.

### 5.4.2 Relating *C*_{D} to *K**C*

The relation between *K**C* and *C*_{D} by the new method is shown in Fig. 8.
The values of *K**C* ranged from 9 to 130, and the range is much smaller than
that of *R**e* in Fig. 7. Similarly, Eq. (15) was utilized to study the
relation between *K**C* and *C*_{D}, and outcomes of the factors are shown in
Table 4. Results show that these fit lines are closer to each other than
those in Fig. 7. The adjusted *R*^{2} values in Table 4 are overall larger
than the corresponding numbers in Table 3. In addition, values for a certain
factor based on these two methods are closer to each other than the results
in Table 3. From these studied cases, the Keulegan–Carpenter number can be a
better parameter to describe the drag coefficient than *R**e*. Besides, for
predicting *C*_{D} by *K**C*, factors in Eq. (15) can be different for
different densities of vegetation and operation conditions, but the emergent
condition will not affect the result.

### 5.4.3 Relating *C*_{D} to *U**r*

The relation between *C*_{D} and the Ursell number *U**r* have also been
studied (Fig. 9). The values of *U**r* ranged from 1 to 68. However, the
nonlinear fit by Eqs. (15) is unsatisfactory for all groups since the
relation of these data is not so strong. Results show that comparing to
*R**e* and *K**C*, *U**r* is not a well-performed parameter for studying the
drag coefficient in wave attenuation by vegetation.

Wave attenuation by vegetation in wetlands is a large-scale nature-based
solution providing a myriad of services for human beings. For understanding
wave attenuation, two main traditional calibration approaches to the drag
effect acting on the vegetation have been established, based on local wave
height decaying by a reciprocal function or exponential function. These two
reliable calibration methods by Dean (1979) and Kobayashi et al. (1993) can
be combined from two perspectives: one by combining these featured functions
directly (Eqs. 1 and 4) and another by the relations between the
(exponential) damping factor and the drag coefficient (Eqs. 3 and 5).
So, two relations between the damping factor *α*^{′} and the
exponential damping factor *k*^{′} have been derived (Eqs. 6 and 12). Then,
the relation between *α*^{′} and *k*^{′} and the drag coefficient in wave
attenuation were analyzed by 99 laboratory experiments. Furthermore,
relations between *C*_{D} and important hydraulic parameters (*R**e*, *K**C*,
and *U**r*) were analyzed to make *C*_{D} predictable under certain
conditions.

The results showed that the reduction of wave height can be well described
by both reciprocal and exponential functions. For submerged vegetation,
which reduces wave height relatively slightly, the damping factor
approximately equalled the exponential damping factor, and Eq. (6) was
applied. However, Eq. (12) was applicable no matter how submerged the
vegetation was, which is a satisfactory result. Besides, for submerged
vegetation, values of *C*_{D} calculated with Eq. (2) by Dean (1979) and with
Eq. (5) by Kobayashi et al. (1993) were consistent with the well-recognized
Eq. (3) by Dalrymple et al. (1984). However, when the vegetation was
emerged, Eqs. (2) and (5) were not in line with Eq. (3). On the other hand,
the calculated *C*_{D} values by the new method by Zhang et al. (2021) in
combination with Eq. (3) were almost the same as the results from the method
by Dalrymple et al. (1984). Additionally, it is appeared that *K**C* performed
best to predict *C*_{D}, better than *R**e* and *U**r*, although the factors
were different in different groups of laboratory observations. Therefore,
further studies are needed in a variety of laboratory experiments.

Building a bridge between the two reliable methods by Dean (1979) and Kobayashi et al. (1993) is helpful. In this way, the reduction of wave height is limited by two functions, so experimental outliers can be distinguished. Also, emergent conditions and densities are very significant aspects to study the drag coefficient by vegetation. This method for the drag coefficient have been validated by a great amount of data under different laboratory conditions; however, the interaction between the vegetation and flow field is complicated, and laboratory errors may affect the result, so verification and/or calibration are needed further for predicting the drag coefficient.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

ZZ, BH, and CT did the conceptualization and methodology. ZZ and TC did the data curation and formal analysis. ZZ did validation and visualization. BH and CT did the funding acquisition and project administration. BH and XC did the supervision. All the authors contributed to writing and editing of the article.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors especially thank Zhan Hu and Wei-Cheng Wu for sharing laboratory data.

This work has supported by Guangzhou Science and Technology Program key projects (grant no. 201806010143), the Soft Science Research Program of Guangdong (grant no. 2018B020207004), and the National Key Research and Development Program of China (grant no. 2016YFC0402607).

This paper was edited by Hubert H. G. Savenije and reviewed by two anonymous referees.

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