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A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation

Zhilin Zhang Bensheng Huang Hongxiang Ji Xin Tian Jing Qiu Chao Tan Xiangju Cheng

Zhilin Zhang, Bensheng Huang, Hongxiang Ji, Xin Tian, Jing Qiu, Chao Tan, Xiangju Cheng. A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-019-0000-0
Citation: Zhilin Zhang, Bensheng Huang, Hongxiang Ji, Xin Tian, Jing Qiu, Chao Tan, Xiangju Cheng. A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-019-0000-0

doi: 10.1007/s13131-019-0000-0

A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation

Funds: The National Key Research and Development Program of China under contract No. 2016YFC0402607; Key Research and Development Projects in Guangdong Province under contract No. 2019B111101002; 2018 Guangzhou Science and Technology Project under contract No. 201806010143; Water Resource Science and Technology Innovation Program of Guangdong Province under contract No. 2017-17.
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出版历程

A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation

doi: 10.1007/s13131-019-0000-0
    基金项目:  The National Key Research and Development Program of China under contract No. 2016YFC0402607; Key Research and Development Projects in Guangdong Province under contract No. 2019B111101002; 2018 Guangzhou Science and Technology Project under contract No. 201806010143; Water Resource Science and Technology Innovation Program of Guangdong Province under contract No. 2017-17.
    通讯作者: E-mail: bensheng@21cn.com

English Abstract

Zhilin Zhang, Bensheng Huang, Hongxiang Ji, Xin Tian, Jing Qiu, Chao Tan, Xiangju Cheng. A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-019-0000-0
Citation: Zhilin Zhang, Bensheng Huang, Hongxiang Ji, Xin Tian, Jing Qiu, Chao Tan, Xiangju Cheng. A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-019-0000-0
    • The estuarine and coastal area is an ecological habitat, commonly with abundant resources. This area provides water, food, energy, minerals, fishery and other resources for human beings, serving as an important place for leisure, recreation, and transportation. However, coastal areas, especially at low altitudes, have long been at risk of flooding from hurricanes and other extreme storm events, and the water security is not guaranteed; sea-level rise and climate change could also lead to increased frequency and intensity of storms, exacerbating the threat (Anderson et al., 2011). Most of the existing single seawalls are difficult to meet the current wave prevention requirements, hence, it is of practical significance to consider the natural conditions and construct the ecological safety barrier with wetland vegetation, which can enhance the toughness of the coast and save construction investment effectively (e.g., Reguero et al., 2018). Practice has proved that wetland vegetation in China (e.g., Liu et al., 2019), Vietnam (e.g., Mazda et al., 1997), Malaysia (e.g., Ghazali et al., 2016), Indonesia (e.g., Yanagisawa et al., 2010), Sri Lanka (e.g., Tanaka et al., 2007), India (e.g., Danielsen et al., 2005), Thailand (e.g., Thampanya et al., 2006) and other places have effectively protected the coast by dissipating incoming wave energy. Vegetation in wetlands as a large-scale nature-based solution can also provide services such as increase bank stability, enhance coastal ecosystem and biodiversity, enhance fisheries and forestry production, and promote tourism economy, whereas the vegetation occupies floodplain resources (Schaubroeck, 2017; Keesstra et al., 2018).

      Therefore, it is necessary to understand the mechanism of wave attenuation, fueling the efficiency of the nature-based solution. It is assumed that the influence of vegetation on the flow field can be expressed by the resistance acting on the vegetation (Kobayashi et al., 1993). No matter which kind of approach is used, such as numerical modeling (e.g., Wu et al., 2016; Suzuki et al., 2019), laboratory experiment (e.g., Houser et al., 2015; Peruzzo et al., 2018), or field study (e.g., Quartel et al., 2007), wave attenuation by vegetation is mainly induced by the drag force ($ {F}_{D} $) provided by the vegetation acting on the water motion. The drag force is closely related to the drag coefficient ($ {C}_{D} $) which is used to quantify the drag or resistance of vegetation in water. Since the drag coefficient can be quite different on different time and space scales, this parameter is one of the most uncertain parameters in the complicate interaction between vegetation and water. Parameterizing $ {C}_{D} $ (reviewed by e.g., Chen et al., 2018) was mainly by two methods, calibration or direct measurement. The calibration method comes from the perspective of wave energy dissipation and will be presented in detail in Section 2. The direct measurement method is based on the acting force on the vegetation, and the periodic averaged drag coefficient is obtained by calculating the work done by the drag force over one period. It is crucial to connect these two methods to reduce the uncertainties related to the drag coefficient, however, Chen et al. (2018) had considered the combined current-wave flow condition following Losada et al. (2016) which can be expanded to other cases.

      This article aims to reveal an effective method for rapidly assessing wave attenuation by vegetation following the conventional calibration approaches, and comparing the predicted value from the new method to the historical observation either by calibration method or direct measurement method. This eliminates the need for different formulas for different operating conditions and to form a unified, simplified and rapid assessment technology.

    • The calibration method determines the drag coefficient ($ {C}_{D} $) from the perspective of wave energy dissipation, represented by the decay of wave height. Following one of the first hydrodynamic models for wave attenuation proposed by Dean (1979), wave height decay can be expressed as:

      $$ \frac{H\left(X\right)}{{H}_{0}}=\frac{1}{1+\alpha {'}X}{,} $$ (1)

      where $ H\left(X\right) $ (m) is the wave height at a distance $ X $ (m) through the vegetation field; $ {H}_{0} $ (m) is the initial wave height measured at the start of the vegetation field; and $ \alpha {'} $ (m−1) is the damping factor.

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

      $$ \alpha {'}=\frac{{C}_{D}d}{6\pi h}{NH}_{0}{,} $$ (2)

      where $ d $ (m) is the diameter of circular vegetation cylinder; $ h $ (m) is the water depth; and $ N $ (stems/m2) is the average number of stems per unit area.

      Then, Knutson et al. (1982) reformed Eq. (2) by incorporating an empirical plant drag coefficient, $ {C}_{P} $, to account for the difference between rigid cylinders and plants:

      $$ \alpha {'}=\frac{{{C}_{P}C}_{D}d}{6\pi h}{NH}_{0}{.} $$ (3)

      Further, Dalrymple et al. (1984) formulated an algebraic dissipation equation practicing linear wave theory and conservation of wave energy by considering a vegetation bed as an array of rigid, vertical cylinders. The time-averaged energy dissipation is the product of the drag force per unit volume and the fluid velocity due to wave motion, and instead of Eq. (2), the solution was given by:

      $$ \begin{split} \alpha {'}=& \frac{1}{3\pi }{C}_{D}Nd\left({\rm{sinh}}^{3}{k}_{w}{l}_{s}+3{\rm{sinh}}{k}_{w}{l}_{s}\right)\\ & \left[\frac{4{k}_{w}}{3{\rm{sinh}}{k}_{w}h({\rm{sinh}}2{k}_{w}h+2{k}_{w}h)}\right]{H}_{0}{,} \end{split} $$ (4)

      where $ {k}_{w} $ (rad/m) is the wave number and $ {l}_{s} $ (m) is the average stem length.

      Moreover, in combined current-wave flow, Losada et al. (2016) modified the analytical formulation of Dalrymple et al. (1984):

      $$ \begin{split} \alpha =& \left[ {\frac{{16}}{{3\pi }}{C_D}N{h_v}d{{\left( {\frac{{g{k_w}}}{{2{\sigma _{wc}}}}} \right)}^3}\frac{{{\rm{sin}}{{\rm{h}}^3}{k_w}{h_v} + 3{\rm{sinh}}{k_w}{h_v}}}{{3{k_w}{\rm{cos}}{{\rm{h}}^3}{k_w}h}}{H_0}} \right]\bigg/\\ & \left[ {g\left( {1 + \frac{{2{k_w}h}}{{{\rm{sinh}}2{k_w}h}}} \right){{\left( {\frac{g}{{{k_w}}}{\rm{tanh}}{k_w}h} \right)}^{\frac{1}{2}}} + } \right.\\ & \left. {gu\left( {3 + \frac{{4{k_w}h}}{{{\rm{sinh}}2{k_w}h}}} \right) + 3k{u^2}{{\left( {\frac{g}{k}{\rm{coth}}{k_w}h} \right)}^{1/2}}} \right], \end{split} $$ (5)

      where $ {h}_{v} $ (m) is the height of vegetation in water; $ {\sigma }_{wc} $ (rad/s) is the angular frequency associated with combined waves and currents ($ {\sigma }_{wc}=\sigma -{u}_{0}{k}_{w} $); $ \sigma $ (rad/s) is angular frequency; and $ u $ (m/s) is unidirectional current velocity.

      Overall, different equations for the damping factor ($ \alpha $) can be obtained under different operating conditions, and the researchers all connected $ \alpha $ to the drag coefficient ($ {C}_{D} $) and other easily measurable parameters, such as the water depth, the average stem density, the average stem diameter, and the initial wave height. When spatial wave height is available, the damping factor can be obtained by calibrating the wave attenuation Eq. (1), then $ {C}_{D} $ can be obtained by the above published equations for $ \alpha $. Finally, relations between $ {C}_{D} $ and Reynolds number ($ {R}_{e} $) and/or Keuglan-Carpenter number ($ KC $) can be obtained by regressions.

    • On one hand, Dean (1979) based on empirical estimates of fluid drag forces acting on vertical, rigid cylinders, the model for the damping of incident wave height ($ {H}_{0} $) by coastal plants:

      $$ {K}_{v}=\frac{H}{{H}_{0}}=\frac{1}{1+\alpha {'}X}=\frac{1}{1+\alpha x}=F\left(x\right){,} $$ (6)

      in which

      $$ {\alpha }^{{'}}=\frac{{C}_{D}d}{6\pi h}{NH}_{0}(0\leqslant x=X/L\leqslant 1){,} $$ (7)

      where $ {K}_{v} $ (–) is the scaled wave height, $ L $ (m) is the length of vegetation area, $ \alpha $ ($ =\alpha {'}L $) (–) is the scaled damping factor, and $ x $ (–) is the scaled distance through the vegetation field.

      On the other hands, Kobayashi et al. (1993) linearized the horizontal drag force as a function of fluid particle velocity. The local wave height was assumed to decay exponentially with propagation through a vegetation bed according to the following form:

      $$ \frac{H}{{H}_{0}}={\exp}\left(-k{'}X\right)={\exp}\left(-kx\right)=G\left(x\right){,} $$ (8)

      where $ k{'} $ (m–1) is an exponential damping factor, indicating a slighter decrease in a lower value. $ k $ ($ =k{'}L $) (–) is the scaled exponential damping factor.

      Based on reliable calibration methods, these two expressions are linked. Using the Taylor expansion, when the scaled distance $ x $ equals half, the following equations are derived:

      $$ \begin{split} F\left(x\right)=& \frac{2}{\alpha +2}-\frac{4\alpha }{{\left(\alpha +2\right)}^{2}}\left(x-\frac{1}{2}\right)+\frac{8{\alpha }^{2}}{{\left(\alpha +2\right)}^{3}}{\left(x-\frac{1}{2}\right)}^{2}-\\ & \frac{16{\alpha }^{3}}{{\left(\alpha +2\right)}^{4}}{\left(x-\frac{1}{2}\right)}^{3}+{R}_{1}\left(x\right), \end{split}$$ (9)

      and

      $$ \begin{split} G\left(x\right)=& \frac{1}{{{\rm{e}}}^{k/2}}-\frac{k}{{{\rm{e}}}^{k/2}}(x-1/2)+\frac{{k}^{2}}{2{{\rm{e}}}^{k/2}}{(x-1/2)}^{2}-\\ & \frac{{k}^{3}}{6{{\rm{e}}}^{k/2}}{(x-1/2)}^{3}+{R}_{2}\left(x\right){,} \end{split}$$ (10)

      where $ {R}_{1}\left(x\right) $ and $ {R}_{2}\left(x\right) $ are the residual terms. To analyze the importance of each term in Eq. (9), these terms are represented:

      $$ {f}_{1}=\frac{2}{\alpha +2}{,} $$ (11)
      $$ {f}_{2}=-\frac{4\alpha }{{\left(\alpha +2\right)}^{2}}\left(x-1/2\right){,} $$ (12)
      $$ {f}_{3}=\frac{8{\alpha }^{2}}{{(\alpha +2)}^{3}}{(x-1/2)}^{2}{,} $$ (13)
      $$ {f}_{4}=-\frac{16{\alpha }^{3}}{{(\alpha +2)}^{4}}{(x-1/2)}^{3}{.} $$ (14)

      In these above Eqs (11)–(14), $ \alpha $ is larger than zero due to the fact of wave attenuation. Since $ x $ is in the range of zero to unit, Eq. (12) can obtain its largest value when $ x $ equals zero, and in this case,

      $$ {f}_{2,{\rm{max}}}=\frac{2\alpha }{{\left(\alpha +2\right)}^{2}}{.} $$ (15)

      Similarly, Eq. (13) has the largest value when $ x $ equals zero or unit:

      $$ {f}_{3,{\rm{max}}}=\frac{2{\alpha }^{2}}{{(\alpha +2)}^{3}}{.} $$ (16)

      And Eq. (14) can obtain the largest value in the case of $ x=0 $:

      $$ {f}_{4,{\rm{max}}}=\frac{2{\alpha }^{3}}{{(\alpha +2)}^{4}}{.} $$ (17)

      To evaluate the relative magnitudes of the different terms of Eq. (9), Fig. 1 presents the factors. The result demonstrates that the first two terms can play the most significant roles.

      Figure 1.  Comparison between the factors in Eq. (9) as a function of the damping factor $ \alpha $.

      Comparably, the importance of each term in Eq. (10) is analyzed and the following expressions are obtained:

      $$ {g}_{1}=\frac{1}{{e}^{k/2}}{,} $$ (18)
      $$ {g}_{2,{\rm{max}}}=\frac{k}{{e}^{k/2}}{,} $$ (19)
      $$ {g}_{3,{\rm{max}}}=\frac{{k}^{2}}{8{e}^{k/2}}{,} $$ (20)
      $$ {g}_{4,{\rm{max}}}=\frac{{k}^{3}}{48{e}^{k/2}}{.} $$ (21)

      Hence, Fig. 2 shows the comparison between these equations as a function of the exponential damping factor. Based on experiences, the value of $ k $ is always in the range of zero to two (see for instance, Table 1 in Section 4.2). Under this circumstance, it is obvious that the first two terms are the key ingredients to Eq. (10) and the lower the value of $ k $, which means the slower the wave attenuate, the more important the first term can be.

      Figure 2.  Comparison between the factors in Eq. (10) as a function of the exponential damping factor $ k $.

      Table 1.  Values of parameters from references and the calibrated $ k $.

      ReferenceType of
      vegetation
      CasePlant
      height
      /m
      Plant
      diameter
      /m
      Plant
      densitystem
      /m2
      Incident
      wave
      period/s
      Incident
      wave
      height/m
      Collected
      $ {C}_{D} $/–
      Calibrated
      $ k $/–
      Wu et al. (2011)birch dowels124363010.63 0.009 4 3501.20.0852.550.47
      124350010.48 0.009 4 3501.20.0841.710.33
      126363010.63 0.009 4 6231.20.0832.740.72
      Wu and Cox (2015)plastic strips5a0.120.0052 1001.60.0162.230.43
      5b0.120.0052 1001.60.0242.120.55
      5c0.120.0052 1001.60.0331.950.66
      5d0.120.0052 1001.60.0411.620.76
      Wu and Cox (2016)plastic stripsCase20.120.0051 6180.6 0.018 73.740.43
      Case50.120.0051 6181.2 0.031 32.210.38
      Yao et al. (2018)PVC pipeswave07120.2 0.02 1391.20.07 2.680.58

      Therefore, consider only the first two terms, instead of distance-dependent terms, in Eqs (9)–(10), the proportionality results in:

      $$ \alpha =\frac{2k}{2-k}{.} $$ (22)

      Then, a new expression for the drag coefficient is derived based on Eq. (2):

      $$ {C}_{D}=\frac{12\pi kh}{(2-k)dN{H}_{0}L}{.} $$ (23)

      Therefore, by measuring local wave height $ H\left(x\right) $, the exponential damping rate ($ k $) can be calibrated easily, for instance, by Microsoft EXCEL, instead of more professional numerical tools, and $ {C}_{D} $ can be obtained by Eq. (23).

      To look upon Eq. (23) further, a derivative of this equation with respect to $ k $ is taken and the result is obtained:

      $$ {{\left({C}_{D}\right)}^{{'}}}_{k}=\frac{4}{{\left(2-k\right)}^{2}}\frac{12\pi h}{dN{H}_{0}L}{.} $$ (24)

      It is obvious that Eq. (24) is larger than zero when $ k $ is unequal to 2 since all these parameters are positive. Then $ {C}_{D} $ is a monotone function of the exponential damping factor. In other word, the more considerably the wave attenuate, the larger the drag coefficient. Besides, the length of the vegetation field is introduced for the bulk drag coefficient comparing to traditional Eq. (2), and it has an inverse correlation with $ {C}_{D} $ which is reasonable: A wider vegetated area drives a larger (exponential) damping factor and these parameters compensate with each other.

      Furthermore, the force $ F $ can be expressed as:

      $$ F=\frac{1}{2}\rho {C}_{D}NdU\left|U\right|=\rho \frac{6\pi kh}{(2-k){H}_{0}L}U\left|U\right|{.} $$ (25)
    • To test the new expression Eq. (23) for the bulk drag coefficient, measured data in the published literature has been collected. The researchers in the previous studies showed the values of $ {C}_{D} $ based on either the calibration method or the direct measurement method.

      The laboratory experiments by Hu et al. (2014) were conducted in a wave flume, with a 6 m long vegetation mimic canopy. The mimics were constructed by stiff wooden rods with a height of 0.36 m and a diameter of 0.01 m. Three mimic stem densities (62, 139, and 556 stems/m2) were constructed and control tests with no stems were measured to reduce the effect of friction of flume bed and sidewalls. Meanwhile, two water depths (0.25 m and 0.50 m) were used to study the emergent and submerged conditions. Instantaneous force on stems and horizontal velocity were measured by force transducers and electromagnetic flow manufacture meters. Then the periodic averaged drag coefficient was obtained by the measurements directly.

      In addition, Wu et al. (2011) reported a series of experiments in laboratory with a 3.6 m long vegetation field in a sloping beach. The rigid vegetation mimicked by 9.4 mm diameter birch dowels were studied by three stem densities (156, 350, and 623 stems/m2) and two stem heights (0.63 m and 0.48 m). Water surface displacement along the vegetated area were measured by wave gages and video imaging, then the wave height was calculated and the drag coefficient was obtained by fitting the wave attenuation model.

      Besides, the laboratory experiments by Wu and Cox (2015) were conducted in a wave flume with a water depth of 12 cm and the 1.8 m long vegetated area was modeled by plastic strips, 5 mm wide by 1 mm thick. The length of the strips was 14 cm and the density was 2 100 stems/ m2. Acoustic wave gages were used to record the free water surface.

      Moreover, data in publications by, for instance, Wu and Cox (2016) and Yao et al. (2018), are also collected.

    • Wave height along the vegetated area is a significant index for the rapid assessment technique, and two examples were taken as Fig. 3 shown. Revisiting the attenuation from Wu and Cox (2015) and Wu et al. (2011), it is clear that Eq. (8) is a reliable relation between the scaled distance and the relative wave height. Results show that the larger the value of $ k $, the faster the wave attenuates. The values of the bulk drag coefficient ($ {C}_{D} $) and the calibrated $ k $, besides the data collected from Hu et al. (2014), are listed in Table 1.

      Figure 3.  Wave attenuation for data from Wu and Cox (2015) (a) and Wu et al. (2011) (b). Solid symbols indicate measurements and lines represent the curves fitted by Eq. (8). Names of cases follow the references.

    • To validate the new expression for the drag coefficient, a myriad of measured data by previous researchers introduced in Section 4.1 have been collected. The predicted values of the drag coefficient by the new rapid assessment technique are obtained by applying Eq. (23) then. Comparing the outcomes with former published data, the results are displayed in Fig. 4. Based on Hu et al. (2014), which contained a bunch of laboratory studies, operating conditions are represented by VD1, VD2, and VD3 for those three mimic stem densities. The results show that the predicted values are in a range of 0.7 to 10.6 and mostly concentrated on values less than 5.1. More importantly, the predicted values have a strong relation with the 36 historical observations by different researchers. The value of R2 is 0.90 by regression in Fig. 4, which is quite promising and satisfactory.

      Figure 4.  Comparison of the values of the drag coefficient. The predicted values are obtained by using Eq. (23) and the published data of the drag coefficient are collected from references. Different symbols represent different publications.

      The slope of the trendline which equals 0.25 is not troublesome according to Knutson et al. (1982), who has introduced an empirical plant drag coefficient as Eq. (3) shown, with a calibrated value of $ {C}_{P} $ = 5 by minimizing the error between predicted and measured wave heights for data collected in S. alterniflora marshes of Chesapeake Bay. Figure 4 then reveals a predicted equation for the drag coefficient:

      $$ {C}_{D}=\frac{3\pi kh}{(2-k)dN{H}_{0}L}+0.95{.} $$ (26)

      In analogical, if Eq. (3) was utilized instead of Eq. (2) in the derivation process above, $ {C}_{P} $ equals 4 for Eq. (23), which is close to the solution by Knutson et al. (1982).

    • Wave attenuation by vegetation in wetlands is a large-scale nature-based solution providing different services for humans. To understand wave attenuation, there are two main traditional calibration approaches to the drag effect acting on the vegetation. On one hand, previous researchers found that the decrease of the wave height fits the exponential function well, however, this simple equation has not been applied to the drag coefficient ($ {C}_{D} $), one of the most crucial parameters in wave attenuation by vegetation. On the other hand, following the research by Dean (1979), there are several equations based on fitting wave height attenuation for the damping factor $ \alpha $, which is derived as a function of $ {C}_{D} $. It is possible that the more complicated the equation, the more precise the equation could be, while it is also necessary to derive a simple approach which can be applicable to different circumstances. By combining these two reliable calibration methods, a simple and visualized relation (Eq. (23)) between the drag coefficient and measurable parameters has been derived. The predicted values yielded from this relation then were compared with a myriad of measured data in the published literature. These 36 published observations come from either the calibration method from the perspective of wave energy dissipation or direct measurement method based on the acting force on the vegetation. Result showed that there is a strong linear relationship between the predicted and published values, and the linear regression revealed a promising R2 value of 0.90. Moreover, a calibrated value of 4 for the empirical plant drag coefficient $ {C}_{P} $ introduced by Knutson et al. (1982) has been obtained, which is close to the previous outcome. Finally, a new equation (Eq. (26)) has been obtained for predicting the drag coefficient in wave attenuation by vegetation.

      This rapid assessment method for wave attenuation is useful. Firstly, the new method verified that both exponential function and inverse proportional function are reliable and capable for describing the wave attenuation by vegetation satisfactorily, but the new method has built a bridge between them. The exponential damping rate ($ k $), however, can be obtained much easier than the damping rate ($ \alpha $), and no professional numerical tools is needed. In addition, it is promising that the new technique can be applied under different circumstances and there is no need to consider which equation should be used for the damping factor $ \left(\alpha \right) $ by the calibration method. Besides, a bridge between the calibration and direct measurement methods has been built. Based on different mechanisms, the calibration and direct measurement methods were utilized together to validate the new technique for wave attenuation. The new rapid assessment for the drag coefficient has been validated by a great amount of data under different laboratory conditions, however, the interaction between the vegetation and flow filed is complicated so verification and/or calibration are needed before applying the result to certain cases. Lastly, more data are needed to test the applicability of the new rapid assessment method for the drag coefficient in future studies.

    • We especially thank Hu Zhan for sharing laboratory data.

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