Hydraulic jumps on adverse slopes

Hydraulic jumps on rough and smooth beds: aggregate approach for horizontal and adverse-sloped beds

“…The rough bed height values were measured both transversally and longitudinally, in a mesh 1 cm × 1 cm, using a point gauge with ± 0.1 mm precision and then the obtained data were used to determine ET and PT. Figure 1 shows a sketch of a hydraulic jump occurring on an adversesloped rough bed, and illustrates key geometric parameters, i.e., the effective upstream and downstream depths y 1 and y 2 , the roller length of the jump L r (McCorquodale & Mohamed, 1994;Pagliara & Peruginelli, 2000), and the bed slope (negative) i. Figure 2 is a schematic sketch of the flume illustrating PT and ET elevations as well as the longitudinal and vertical coordinate system (x and y, respectively).…”

“…Assuming Y = y 2 /y 1 (sequent depth ratio) and considering Eqs. (2a) and (2b), the following general expression can be easily derived (McCorquodale & Mohamed, 1994;Pagliara & Peruginelli, 2000):…”

“…Note that in the case of a hydraulic jump occurring in a flow over an adverse-sloped bed, without a sill, the length of the hydraulic jump can be assumed to be equal to the length of the roller (McCorquodale & Mohamed, 1994;Pagliara & Peruginelli, 2000). The parameter G 1 is termed the "adverse jump parameter" and G 1 = F 1 for α = 0 (i.e., i = 0).…”

“…In the following, the coefficient K is assumed to be equal to 1 (McCorquodale & Mohamed, 1994;Pagliara & Peruginelli, 2000). Carollo et al (2007) stated that F τ (integrated shear stress per unit width) can be expressed as follows:…”

“…In particular, hydraulic jump characteristics in a flow on a smooth adverse-sloped bed were analysed by McCorquodale & Mohamed (1994), Okada & Aki (1955), Rajaratnam (1966Rajaratnam ( , 1967 and Stevens (1942). McCorquodale & Mohamed (1994) proposed a theoretical approach based on conservation of momentum principles to estimate hydraulic jump characteristics and found that the stability of a hydraulic jump on an adverse slope is also dependent on the upstream Froude number. The presence of a sill increases hydraulic jump stability for both horizontal (Hager & Bretz, 1988;Othsu, 1981;Ohtsu, Yasuda, & Yamanaka, 1991) and adverse-sloped beds (Pagliara & Peruginelli, 2000).…”