5.2.3. Comparison of *h<sup>m</sup>* and *h<sup>τ</sup>*

Table 3 presents the locations (*h<sup>m</sup>* and *hτ*) where the maximum velocity and zero shear stress occur for all cases. These two locations were not consistent and had distinct difference, i.e., calculated *hm*/*H* > *hτ*/*H*. The location of zero shear stress is closer to the channel bed than that of maximum velocity [13,29,38]. By contrast, the locations of the maximum velocity and zero shear stress for the symmetry and open channel flows were the same. Specifically, the location of maximum velocity and zero shear stress for the symmetry flows was at the middle water depth, and the location for the open channel flows was at the free water surface.

Considering the locations of the maximum velocity and zero shear stress for cases 1, 3 and 5, when the ratio of *n<sup>i</sup>* to *n<sup>b</sup>* increased, *hm*/*H* decreased, so the location of the maximum velocity approached the channel bed, and the vertical inhomogeneity of the velocity profile was strengthened. Meanwhile, *τi*/*τ<sup>b</sup>* increased with the increase in *ni*/*n<sup>b</sup>* ; then, *hτ*/*H* decreased, which indicates the location of the zero shear stress gets closer to the channel bed.

#### 5.2.4. Empirical Constants *D<sup>b</sup>* , *E<sup>b</sup>* , *D<sup>i</sup>* and *E<sup>i</sup>*

The empirical constants of turbulence intensity for all cases are listed in Table 4. No remarkable changes of *D<sup>b</sup>* and *D<sup>i</sup>* were observed in any case. The mean *D<sup>b</sup>* was 2.22 with a standard deviation of 0.05, and the mean *D<sup>i</sup>* was 2.16 with a standard deviation of 0.06. Hence, it is reasonable to consider that *D<sup>b</sup>* = *D<sup>i</sup>* . The difference between *E<sup>b</sup>* and *E<sup>i</sup>* was not negligible, similar to the results of Li et al. (2020) [29].


**Table 4.** Parameters to predict the turbulence intensity in each case.

### **6. Conclusions**

The existence of ice cover dramatically changed the flow velocity and turbulence structure. We here proposed theoretical models to describe the vertical distribution of longitudinal velocity, shear stress and turbulence intensity. By dividing the ice-covered flow into an ice-affected layer and a channel bed-affected layer, a two-power-law function was adopted to predict the vertical profile of velocity. The calculated velocity distribution presents that the maximum velocity occurred near the middle of the water depth close to the channel bed with smooth boundary. Theoretical analysis shows that the shear stress had a linear distribution form in the vertical direction, with the positive values in the lower bed layer and negative values in the upper ice layer. Moreover, the Manning's roughness coefficient of the ice cover was larger than that of the channel bed. The two exponents *m<sup>b</sup>* and *m<sup>i</sup>* were influenced by the roughness coefficients of the channel bed and ice cover. The location of zero shear stress was not the same as that of maximum velocity and was closer to the smooth fixed boundary than the plane of maximum velocity, namely, *h<sup>m</sup>* > *hτ*. A comparison of the analytical and experimental velocities, the Reynolds stress and turbulence intensity displays that the theoretical models can provide satisfied predictions of the vertical distribution of these flow characteristics. This study expands our understanding of the effects of ice cover on the hydraulic characteristics in the open channels. However, we still need to do more research to explore the application of the proposed models in other conditions, like compound channels or confluence channels and we will involve comprehensive experiments to reveal detailed flow characteristics, such as vortex structure.

**Author Contributions:** Conceptualization, J.Z. and W.W.; methodology, J.Z.; software, Z.X., Y.Z.; resources, Q.L., Y.Z.; data curation, H.Q.; writing—original draft preparation, J.Z.; writing—review and editing, J.Z.; visualization, H.Q.; supervision, W.W.; project administration, Z.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Key R&D Program of China (2017YFC0504704), the National Natural Science Foundation of China (Grant No. 51609198), the Science and Technology Project Funded by Shaanxi Provincial Department of Water Resources (2020slkj-10) and the Technology Project Funded by Clean Energy and Ecological Water Conservancy Engineering Research Center (QNZX-2019-03).

**Conflicts of Interest:** The authors declare no conflict of interest.
