## 4 Results and Analysis
### Flow Pressure  
Due to the abrupt change in the channel bed, a stepped spillway features flow separations and significant pressure drop, which may give rise to cavitation damages. Caused by flow acceleration, the pressure fluctuates significantly in the developing flow region [23]. Along the \(y\) direction at the vertex of step no. 10, Figure 4a shows the pressure distributions, \(P/(pgd\text{cos}\varphi)\), at \(\theta\) = 0 and the different \(d_{c}/h\) values, where \(d\) denotes the water depth. For comparison, the results from [PERSON] and [PERSON] [23] are also included. Despite the difference in the step layouts, the numerical results exhibits the same trend as the experimental ones (open symbols), all with a larger \(\partial P/\partial y\) gradient than the hydrostatic near the water surface. Figure 4b compares the effects of \(\theta\) on the \(P/(pgd\text{cos}\varphi)\) distribution. In the near-surface (ideal flow) region, the pressure distributions for all examined angles, except for \(\theta\) = \(-\)20\({}^{\circ}\), exhibit approximately a linear trend. The \(\theta\) = \(-\)20\({}^{\circ}\) layout leads to the lowest \(P/(pgd\text{cos}\varphi)\) values, which is attributable to the smallest step heights. The discrepancy from the hydrostatic distribution is presumably caused by the large velocity gradients. In addition, the flow pressure becomes higher with an increase in \(\theta\). Figure 3: Velocity distributions in the boundary layer at vertices of step no. 10. A half-filled symbol refers to numerical results (this study), and an open one to experimental ones [23]. Along the symmetry plane, Figure 5 presents the pressure contours, \(P/(pgd_{c})\), at steps no. 42 and 43. The sudden change in the bed profile results in flow circulations and substantial pressure drop downstream of the vertical step face. Therefore, the pressures in the step cavities are low; the effects of \(\theta\) on the low-pressure extents are also evident. If the downward inclination is significant (e.g., \(\theta=-20^{\circ}\)), negative pressure occurs on the entire vertical step face. For the downward angles, the low-pressure area in the step cavity becomes smaller if the inclination gets larger. For the upward angles, its size declines with an increase in \(\theta\). The dimensionless pressure, \(P/(pgd_{c})\), near the cavity corner becomes higher with an increasing \(\theta\). On the horizontal step surface, a high-pressure region is also observed, which is associated with the reattachment of the mainstream. Along the step surfaces, both horizontal and vertical, Figure 6 compares the pressure distributions along the symmetry plane, in which \(Y=\) distance along a vertical step surface with its origin at the cavity corner. In the study made by [PERSON] et al. [11], the \(\theta=0\) layout is also numerically examined at \(d_{c}/h=4.8\), and their results are included in the diagram. Obviously, the differences in between are negligibly small. Along the horizontal steps, moderately inclining the steps, either downward or upward, exerts insignificant impacts on the pressure patterns. An S-shaped variation is seen with its max. values at \(X/L=0.7\)-\(0.9\), exclusive of the \(\theta=-20^{\circ}\) layout that has smallest steps close to the bottom. Along the vertical steps, the pressures in all cases demonstrate a descending trend, followed  
Figure 4: Pressure distributions at the step vertex (step no. 10): (**a**) \(\theta=0\), and (**b**) \(d_{c}/h=4.8\). A half-filled symbol refers to numerical results of this study, and an open one to experimental ones by [PERSON] and [PERSON] [23]. Figure 5: Pressure contours (\(P/pgd_{c}\)) at different \(\theta\) values at steps no. 42 and 43 (\(d_{c}/h=4.8\)). by an upward increase. The pressure gets larger as \(\theta\) increases, with its min. occurring approximately at \(Y/H=0.8\) (except for \(\theta=-20^{\circ}\)). On the step surfaces, the min. and max. pressures, denoted as \(P_{\rm min}\) and \(P_{\rm max}\), provide insight into cavitation status and hydraulic loads.