# Hydraulic Performance of Wave-Type Flow at a Sill-Controlled Stilling Basin
## 1 Introduction  
A hydraulic jump is a phenomenon that occurs by converting supercritical to subcritical flow regimens downstream of hydraulic structures. Hydraulic jumps mostly occur in stilling basins with sills or blocks after the sluice gates or weirs. For a sill-controlled stilling basin, the flow passing over the sill changes with the supercritical Froude number, and the relevant hydraulic characteristics (i.e., the hydraulic jump performance) are strongly influenced by the position and height of the sill [1; 2; 3; 4]. In the stilling basin, hydraulic jumps can be classified successively into five types--namely A, B, minimum B, C, and wave-type flow--as the tailswater decreases, as illustrated in Figure 1[5]. The A-jump corresponds practically to a classical jump as the sill position is at the end of the surface roller, and the sill has no effect on the jump (Figure 1a). The B-jump occurs when the tailwater depth decreases, the jump toe moves toward the sill and the deflection of the bottom stream occurs (Figure 1b). Consequently, the minimum B-jump is the formation of a second roller at the downstream of the sill and a C-jump forms when the maximum difference between the flow depth over the sill and the tailwater depth is realized (Figure 1c,d). As the tailwater is gradually reduced, eventually reaching the point of no tailwater, a wave-type flow will occur in the vicinity of the sill, and the resulting downstream flow is characterized by supercritical flow conditions. Distinct from other types of flow, wave-type flows result inexcessive standing waves and highly erosive supercritical flow downstream of the sill; thus, this type of flow should be generally avoided (Figure 1e). In the design of stilling basins or energy dissipators, it is very important to make sure that a hydraulic jump occurs for all possible tailwater depths. Thus, according to the practical engineering design of the sill-controlled stilling basin (i.e., an abrupt bottom rise), wave-type flow inevitably occurs when there is no tailwater [6; 7]. In the literature, much attention has been given to wave formation in different types of hydraulic jumps [8; 9; 10]. Different cases of wave-type flows resulting from abrupt bottom changes have been studied under supercritical downstream conditions. [PERSON] and [PERSON] [11] investigated the wave formation at an abrupt drop and examined various parameters, such as the sequent depth ratio, the maximum height, the location of the plunging point, the length of the downstream jump, and the resultant surface roller. [PERSON] and [PERSON] [5] conducted an analysis on the impact of a continuous, transverse sill on the hydraulic jump in a rectangular channel. The study revealed that the jump controlled by the sill was a perturbed classical jump, as evident from the overall jump pattern. A significant reduction in the energy dissipation of the wave-type flow was observed compared to other patterns. [PERSON] and [PERSON] [12] presented a simple empirical expression for determining the hydraulic characteristics of wave-type flows at abrupt bottom drops. [PERSON] and [PERSON] [13] analyzed the flow pattern and energy dissipation of wave-type flow when the basin bottom both rose and fell. [PERSON] et al. [14] conducted an experimental study on wave characteristics in stilling basins with negative steps. The study describes wave height, period, probability density, and power spectrum in different stilling basins. It establishes relationships between characteristic wave heights and prov  
Figure 1: Sill-controlled flow: (**a**) A-jump; (**b**) B-jump; (**c**) minimum B-jump; (**d**) C-jump; (**e**) wave-type flow. for relative characteristic wave height. [PERSON] et al. [15] studied the energy dissipation of wave-type flow caused by a sill in the stilling basin and compared them with the data of energy dissipation by using a positive step. Moreover, a better agreement with respect to energy dissipation curves can be obtained by considering both upstream and downstream conditions.