Characteristics of flow structure of free-surface flow in a partly obstructed open channel with vegetation patch

Abstract

Free-surface flows over patchy vegetation are common in aquatic environments. In this study, the hydrodynamics of free-surface flow in a rectangular channel with a bed of rigid vegetation-like cylinders occupying half of the channel bed was investigated and interpreted by means of laboratory experiments and numerical simulations. The channel configurations have low width-to-depth aspect ratio (1.235 and 2.153). Experimental results show that the adjustment length for the flow to be fully developed through the vegetation patch in the present study is shorter than observed for large-aspect-ratio channels in other studies. Outside the lateral edge of the vegetation patch, negative velocity gradient ($\partial \overline{u}/\partial z < 0$) and a local velocity maximum are observed in the vertical profile of the longitudinal velocity in the near-bed region, corresponding to the negative Reynolds stress ($- \overline{{u^{\prime}w^{\prime}}} < 0$) at the same location. Assuming coherent vortices to be the dominant factor influencing the mean flow field, an improved Spalart–Allmaras turbulence model is developed. The model improvement is based on an enhanced turbulence length scale accounting for coherent vortices due to the effect of the porous vegetation canopy and channel bed. This particular flow characteristic is more profound in the case of high vegetation density due to the stronger momentum exchange of horizontal coherent vortices. Numerical simulations confirmed the local maximum velocity and negative gradient in the velocity profile due to the presence of vegetation and bed friction. This in turn supports the physical interpretation of the flow processes in the partly obstructed channel with vegetation patch. In addition, the vertical profile of the longitudinal velocity can also be explained by the vertical behavior of the horizontal coherent vortices based on a theoretical argument.

Appendix

On the nonvegetated side, the bed shear stress is balanced by the Reynolds stress. On the vegetated side, the bed shear stress is balanced by the Reynolds stress and the stem drag force.

1.1 Nonvegetated side

$$u_{*}^{2} = - \overline{{u^{\prime}}{w^{\prime}}}+ \frac{{\partial \overline{{u^{\prime}}{v^{\prime}}}}}{\partial y}\Delta z.$$

Assuming the mixing length theory to be valid,

$$u_{*}^{2} = \kappa^{2} z^{2} \left({\frac{\partial u}{\partial z}} \right)^{2} + \frac{{\partial \overline{{u^{\prime}}{v^{\prime}}}}} {\partial y}\Delta z.$$

With algebraic manipulations and integrations,

$$\begin{aligned} \frac{\partial u}{\partial z} = \frac{{\sqrt {u_{*}^{2} - \frac{{\partial \overline{{u^{\prime}v^{\prime}}}}}{\partial y}\Delta z}}}{\kappa z} \hfill \\ u = \int_{{z_{\text{s}}}}^{z} {\frac{{\sqrt {u_{*}^{2} - \frac{{\partial \overline{{u^{\prime}v^{\prime}}}}}{\partial y}\Delta z}}}{\kappa z}} \; {\rm d}z, \hfill \\ \end{aligned}$$

where z _s is the viscous sublayer thickness.

At the near-bed region, horizontal coherent vortices are absent due to the bed friction, and we have $- \partial \overline{{u^{\prime}v^{\prime}}}/\partial y \approx 0.$ Therefore,

$$u = \int_{{z_{\text{s}}}}^{z} {\frac{{u_{*}}}{\kappa z}} \, {\rm d}z = \underbrace {{\frac{{u_{*}}}{\kappa}\ln \frac{z}{{z_{\text{s}}}}}}_{\text{Log law}}.$$

As the distance from the bed increases, the horizontal coherent vortices are triggered by the flow shear, and we have $- \partial \overline{{u^{\prime}v^{\prime}}}/\partial y < 0$. Therefore,

$$u = \int_{{z_{\text{s}}}}^{z} {\frac{{\sqrt {u_{*}^{2} - \frac{{\partial \overline{{u^{\prime}v^{\prime}}}}}{\partial y}\Delta z}}}{\kappa z}}\; {\rm d}z < \int_{{z_{\text{s}}}}^{z} {\frac{{u_{*}}}{\kappa z}}\; {\rm d}z = \underbrace {{\frac{{u_{*}}}{\kappa}\ln \frac{z}{{z_{\text{s}}}}}}_{\text{Log law}}.$$

Nonvegetated side

$$u_{*}^{2} = - \overline{{u^{\prime}w^{\prime}}} + \frac{{\partial \overline{u^{\prime}v^{\prime}}}}{\partial y}\Delta z - C_{d} \Delta z.$$

Assuming the mixing length theory to be valid,

$$u_{*}^{2} = \kappa^{2} z^{2} \left({\frac{\partial u}{\partial z}} \right)^{2} + \frac{{\partial \overline{u^{\prime}v^{\prime}}}}{\partial y}\Delta z - C_{d} \Delta z.$$

With algebraic manipulations and integrations,

$$\begin{aligned} \frac{\partial u}{\partial z} = \frac{{\sqrt {u_{*}^{2} - \frac{{\partial \overline{{u^{\prime}v^{\prime}}}}}{\partial y}\Delta z + C_{d} \Delta z}}}{\kappa z} \hfill \\ u = \int_{{z_{\text{s}}}}^{z} {\frac{{\sqrt {u_{*}^{2} - \frac{{\partial \overline{u^{\prime}v^{\prime}}}}{\partial y}\Delta z + C_{d} \Delta z}}}{\kappa z}} {\rm d}z. \hfill \\ \end{aligned}$$

At the near-bed region, the horizontal coherent vortices are absent due to bed friction, and we have $- \partial \overline{{u^{\prime}v^{\prime}}}/\partial y \approx 0$. Therefore,

$$u = \int_{{z_{\text{s}}}}^{z} {\frac{{\sqrt {u_{*}^{2} + C_{d} \Delta z}}}{\kappa z}} {\rm d}z = \underbrace {{\frac{{\sqrt {u_{*}^{2} + C_{d} \Delta z}}}{\kappa}\ln \frac{z}{{z_{\text{s}}}}}}_{\text{Log law}}.$$

As the distance from the bed increases, the horizontal coherent vortices are triggered by the flow shear, and we have $- \partial \overline{{u^{\prime}v^{\prime}}}/\partial y > 0$. Therefore,

$$u = \int_{{z_{\text{s}}}}^{z} {\frac{{\sqrt {u_{*}^{2} - \frac{{\partial \overline{{u^{\prime}v^{\prime}}}}}{\partial y}\Delta z + C_{d} \Delta z}}}{\kappa z}} {\rm d}z > \int_{{z_{\text{s}}}}^{z} {\frac{{\sqrt {u_{*}^{2} + C_{d} \Delta z}}}{\kappa z}} {\rm d}z = \underbrace {{\frac{{\sqrt {u_{*}^{2} + C_{d} \Delta z}}}{\kappa}\ln \frac{z}{{z_{\text{s}}}}}}_{\text{Log law}}.$$