Experimental study on wave attenuation characteristics of different types of vegetation under regular waves
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摘要: 为探究规则波作用下刚性、柔性以及刚柔组合型植被消波特性差异，利用实验室水槽开展了一系列物理模型试验，定量分析了刚性、柔性及刚柔组合型植被对波浪的衰减作用，确定不同类型植被拖曳力系数CD与雷诺数Re、邱卡数KC和厄塞尔数Ur的关系。研究表明，3种植被配置均能引起波高沿程递减；随着入射波周期或植被淹没度的增加，各植被类型的消波效果均减弱；在波高影响方面，刚性植被的消波效果随波高增大持续显著增强，柔性植被则呈现先增强后减弱的非线性趋势，而刚柔组合型植被兼具两者优势，其消波效果亦随波高增大而增强。此外，3种植被的CD可采用统一的理论公式表达，其主要差异在于反映植被摆动对波高衰减的影响因子γ不同。CD与Re、KC以及Ur之间均存在显著统计关系，并可用统一经验公式描述。本研究结果可为海岸生态防护工程中植被的优化配置提供理论依据与设计参考。Abstract: To investigate the differences in wave attenuation characteristics among rigid, flexible, and rigid-flexible composite vegetation under regular waves, a series of physical model tests were conducted in a laboratory flume. The wave attenuation effects of these three vegetation types were quantitatively analyzed, and the relationships between the drag coefficient (CD) and Reynolds number (Re), Keulegan-Carpenter number (KC), and Ursell number (Ur) were determined. Results show that all three configurations induce a progressive along-flume reduction in wave height. Increasing incident wave period or vegetation submergence ratio consistently weakens wave dissipation for all vegetation types. The response to wave height differs by configuration: dissipation by rigid vegetation increases markedly and continuously with wave height, whereas flexible vegetation exhibits a nonlinear behavior, strengthening at first and then weakening as wave height further increases. The rigid-flexible combined configuration integrates these advantages and also shows enhanced dissipation with increasing wave height. Moreover, CD for the three vegetation types can be represented using a unified theoretical expression; the primary distinction among configurations is the value of the influence factor γ, which accounts for the effect of vegetation swaying on wave-height attenuation. Statistically significant dependencies of CD on Re, KC, and Ur are observed and can be parameterized by a unified empirical formulation. These results provide a theoretical basis and design reference for optimizing vegetation configurations in coastal ecological protection and restoration engineering.
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图 2 试验中的植被模型
a. 刚性；b. 刚柔组合；c. 柔性
Fig. 2 Vegetation models in the experiment
a. Rigid; b. rigid–flexible composite vegetation; c. flexible
图 3 试验中植被排布方式、株距与行距
空心蓝圈代表柔性植株，实心绿圈代表刚性圆杆
Fig. 3 Arrangement, plant spacing and row spacing of the vegetation in the experiment
Hollow blue circles represent flexible plants and solid green circles represent rigid circular rods
图 4 入射波高对不同类型植被消波影响（T = 1.2 s, h = 0.30 m）
Fig. 4 Effect of incident wave height on wave dissipation of different vegetation types (T = 1.2 s, h = 0.30 m)
图 5 入射波周期对不同类型植被消波影响（H = 0.06 m, h = 0.30 m）
Fig. 5 Effect of incident wave period on wave dissipation of different vegetation types (H = 0.06 m, h = 0.30 m)
图 6 水深对不同类型植被消波影响（H = 0.05 m, T = 1.2 s）
Fig. 6 Effect of water depth on wave dissipation of different vegetation types (H = 0.05 m, T = 1.2 s)
图 7 淹没度和相对波高对不同类型植被的消波影响
Fig. 7 Effect of submergence degree and relative wave height on wave dissipation of different types of vegetation
图 8 刚柔组合型植被模型两种拟合方案对比
Fig. 8 Comparison of two fitting schemes of rigid-flexible combined vegetation model
图 10 拖曳力系数CD与KC、Ur、Re的关系
Fig. 10 Relationships between drag coefficient CD and KC, Ur, Re
表 1 植被模型参数
Tab. 1 Vegetation model parameters
模型编号 树型 长度/m 植被高度hv /cm 密度/(株·m–2) 排布方式 M1 刚性 1 50 400 方形 M2 柔+刚 0.5 + 0.5 26 + 50 761 + 400 方形 + 交错 M3 柔性 1 26 761 交错 
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表 2 试验工况
Tab. 2 Experimental conditions
组次 试验水深h/cm 入射波波高H/cm 入射波周期T/s 1 20 5 1.2 2 25 5 1.2 3 30 4/5/6/7/8 1.2 4 30 6 0.8/1.0/1.2/1.4/1.6 5 35 5 1.2 6 40 5 1.2
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表 3 CD与KC、Ur或Re关系综述
Tab. 3 Review on the relationships between CD and KC, Ur, or Re
研究 波浪条件 植被类型 公式 Veelen等[31] 规则波 刚性 CD = (81/KC)0.36 (53 ≤ KC ≤ 133) R2 = 0.54 柔性 CD = 0.26+(43/KC)5.3 (53 ≤ KC ≤ 133) R2 = 0.54 Reis等[32] 规则波 刚性 CD = 0.83+(14.8/KC)1.24 (13 ≤ KC ≤ 68) R2 = 0.71 CD = 0.79+( 1014 /Re)1.14 (895 ≤ Re ≤3615 )R2 = 0.69 柔性 CD = 1.11+(22.4/KC)4.1 (22 ≤ KC ≤ 60) R2 = 0.56 CD = ( 5265 /Re)0.33 (1520 ≤ Re ≤3025 )R2 = 0.25 Gong等[18] 规则波 柔性 CD = 0.163+(4.37/KC)2.07 (12 ≤ KC ≤ 45) R2 = 0.65 CD = 0.095+ ( 1516 /Re)1.74 (850 ≤ Re ≤9800 )R2 = 0.66 目前的研究 规则波 刚性 CD = 5/KC0.9+1 (10 ≤ KC ≤ 24) R2 = 0.73 CD = 19.8/Ur 2.3+1.1 (2 ≤ Ur ≤ 14) R2 = 0.76 CD = 4556 /Re−0.6 (1800 ≤ Re ≤3400 )R2 = 0.75 刚柔组合 CD = 9/KC0.9+0.5 (14 ≤ KC ≤ 53) R2 = 0.76 CD = 10.7/Ur 2.3+0.8 (2 ≤ Ur ≤ 14) R2 = 0.92 CD = 2800 /Re−0.35 (1650 ≤ Re ≤3300 )R2 = 0.83 柔性 CD = 8.6/KC0.9+0.1 (18 ≤ KC ≤ 56) R2 = 0.82 CD = 9.8/Ur2.8+0.4 (2 ≤ Ur ≤ 14) R2 = 0.88 CD = 1500 /Re−0.15 (1800 ≤ Re ≤3150 )R2 = 0.79
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