## 2 Vegetation Model  
Logarithmic law (log law) and its modified forms have presented great potential to represent velocity profiles for vegetated flow [14]. To represent velocity distribution across full flow depth, researchers have proposed various two-layer models where velocity has been characterised separately at free water and submerged vegetation layers [4; 15; 16]. For the vegetation-laden layer, the height of vegetation, including the deflection effect, was calculated using the bending moment caused by the vegetation's flexibility [17]. To revise the log law, the roughness factor generated by vegetation has also been considered [4; 15].  
[PERSON] et al. [3] further proposed a three-layer model to more precisely represent submerged rigid vegetation flow. In their proposal, velocity profile modelling was separated  
Figure 1: Schematic diagram for the discussed propeller-induced vegetated flow.  
into a non-vegetation layer, an inner layer, and an outer layer within vegetation. The model starts with the formulation of the momentum equation for steady flow as follows:  
\[\frac{\partial\tau}{\partial z}+\rho_{\rm o}gS_{\rm o}-\frac{1}{2}\rho_{\rm o} C_{\rm D}mDu^{2}=0 \tag{1}\]  
where \(\tau\) is the shear stress, \(\rho_{\rm o}\) is the water density in the unit of kg/m\({}^{3}\), S\({}_{\rm o}\) is the bed slope, C\({}_{\rm D}\) is the drag coefficient, m is the density of vegetation in the unit of stems/m\({}^{2}\), D is the vegetation steam diameter, and u is streamwise flow velocity. The mixing length hypothesis can be utilised to find the shear stress for the inner vegetation region close to the bed [18].  
\[\tau=\rho_{\rm o}l_{\rm m}^{2}\left|\frac{\partial u}{\partial z}\right|\frac {\partial u}{\partial z} \tag{2}\]  
Here, the value of \(l_{\rm m}^{2}\) is the roughness height of the channel due to the boundary layer, and Equation (3) can be obtained by substituting Equation (2) into Equation (1).  
\[l_{\rm m}^{2}\frac{\partial}{\partial z}\left(\left|\frac{\partial u}{ \partial z}\right|\frac{\partial u}{\partial z}\right)+gS_{\rm o}-\frac{1}{2} \rho_{\rm o}C_{\rm D}mDu^{2}=0 \tag{3}\]  
The partial derivative in Equation (3) can be resolved using various numerical or analytical approaches to calculate velocity.  
At the top layer of vegetation, the velocity profile can be represented by  
\[u=\sqrt{2 gS_{\rm o}\{\alpha h_{\rm u}exp(\alpha(z-h_{\rm v}))+1\}/(C_{\rm D}mD )} \tag{4}\]  
where h\({}_{\rm u}\) is the flow depth from the top of vegetation to the free surface, while h\({}_{\rm v}\) is the flow depth from the bed to the top of vegetation. Some Reynolds shear stress assumptions, i.e., ignoring viscosity shear stress and assuming momentum balance in the flow above the vegetated region, have been used to obtain Equation (5).  
\[\tau=\rho_{\rm o}gh_{\rm u}S_{\rm o}exp(\alpha(z-h_{\rm v})) \tag{5}\]  
In the search for a non-vegetated layer velocity profile, the combination of mixing length and vegetation boundary layer was introduced by [PERSON] et al. [3]. Due to the roughness effect of the vegetated zone, the flow flux through vegetation must be considered in this layer. By omitting viscosity shear stress, one can obtain the below equation for Reynolds shear stress.  
\[\overline{u^{\prime}w^{\prime}}=-l^{2}\left(\frac{\partial u}{\partial z} \right)^{2} \tag{6}\]  
Finally, the equation for the velocity profile can be formulated by solving Equation (6) using the mixing length assumption from [18], i.e., Equation (8)  
\[u=u^{*}\left\{\frac{1}{k}ln(1+k(z-h_{\rm v})\ /l_{\rm o}\right\}+C \tag{7}\]  
\[1=[l_{\rm o}+k(z-h_{\rm v})]\sqrt{1-(z-h_{\rm v})/h_{\rm u}} \tag{8}\]  
where \(l_{\rm 0}\) equals the mixing length at the interface of vegetation and flow, and k is the von [PERSON] constant, which is taken as 0.41 [2; 3]. In Equation (7), \(u^{*}\) is the shear velocity, and C is the continuum coefficient at the vegetation and flow interface, which is suggested by [3] to be  
\[C=\sqrt{2(\alpha h_{\rm u}+1)/(C_{\rm D}mDh_{\rm u})} \tag{9}\]