Hydraulic Jumps on Rough Beds

Analysis of Turbulent Hydraulic Jump over a Transitional Rough Bed of a Rectangular Channel: Universal Relations

“…Furthermore, a simple linear relation (57) which is also shown in Figure 3(a) compares slightly better with the experimental data for k/h 1 ≤ 1. The relation proposed by Carollo et al (2007) (58) is also shown in Figure 3(a), and compares well with Equation (56). The bed roughness drag coefficient λ against alternate roughness parameter k/h c is shown in Figure 2(b) along with the proposed relations (59) (60) that also compare well with the experimental data.…”

“…The proposed theory for rough bed rectangular channels predicted universal solution depth profile, sequent depth ratio, jump length, and roller lengths in terms of effective upstream Froude number F S1 and are explicitly independent of bed roughness. The experimental data of Hughes and Flack (1984), Carollo et al (2007Carollo et al ( , 2009, and Ead and Rajaratnam (2002) support the proposed universal theory. Thus, the results for rough bed channels can be directly deduced from the classical smooth bed hydraulic jump theory, provided the upstream Froude number F 1 may be replaced by the effective upstream Froude number F S1 .…”

“…To test this linear proposition, the same experimental data is shown in Figure 9(a); within the scatter, the data fit well with coefficient 8ϵ R /3= 4.2. The role of the effective upstream Froude number F S1 on roller length L R /h 1 from the experimental data of Carollo et al (2007) and Ead and Rajaratnam (2002) is shown in Figure 9(b), which also provides strong support for this study's universal relations, explicitly independent of bed roughness. The leading term perturbation solution in parameter F S1 → 1 from roller length Equation (44) in terms of L R /h 1 is also shown in Figure 9(b) and is in good agreement with the experimental data.…”

“…In rectangular channels the nondimensional roller length L R /h 2 and L R /h 1 against the upstream Froude number F 1 are shown in Figures 8(a) and 8(b), respectively, from the rough bed data of Hughes and Flack (1984), Ead and Rajaratnam (2002), and Carollo et al (2007). For large values of upstream Froude numbers F 1 → ∞, the roller length L R /h 2 approaches a constant universal value that depends on the bed roughness.…”

“…Gill (1980) predicted the effects of bed roughness upon sequent depth h 2 /h 1 as a first perturbation over smooth surface sequent depth α -1 * = (h 2 /h 1 ) e=0 in a rectangular channel as ( 4 ) which compares with experimental data for F 1 ≤ 10. Carollo et al (2007) considered M 2 ≠ 0 and proposed sequent depth Equation (1), where -e = -λ was bed roughness effects correlated with bed roughness k/h 1 . Later, Carollo et al (2009) adopted k/h c as bed roughness and proposed an empirical expression: ( 5 ) where h c = critical water depth, instead of h 1 used earlier by Carollo et al (2007).…”