Sketch of the experiment. The radii (r 1, r 0,1, r 0, r 0,0) and heights (h 1, h 0,1, h 0, h 0,0) are ordered as in the water case (see text, Eqs. (12) and (15)–(17) ). The position of the external wall of height d 0 is variable.
Numerical height profiles ξ(r) for different values of the external depth for type I jumps (lines). Experimental data (circles) obtained from Ref. 11 . Horizontal and vertical scales are normalized with the plate radius L = 37.9 mm and the lowest final depth h = 2.77 mm, respectively. Reproduced by permission from Bohr et al., Physica B 228, 1–10 (1996). Copyright 1996 Elsevier B.V.
Schematic figures of type I and type II hydraulic jumps. A second surfing roll is observed when the outer depth is increased.
Horizontal and vertical coordinates (r, ξ) are normalized with L = 105 mm and h = 4.1 mm, velocities are normalized using a supercritical speed v 1 = 2.14 m/s, given by experimental data. 11 (Left) Free surface (line) and bulk velocity field (arrows). Dark areas denote low velocities. The velocity field shows a separation eddy causing the hydraulic jump. (Right) Surface velocity (line) compared with experimental data upstream (circles) and downstream (squares) the jump. 11 Reproduced by permission from Bohr et al., Physica B 228, 1–10 (1996). Copyright 1996 Elsevier B.V.
The free surface ξ (thick line) and its first three spatial derivatives ξ′ (dotted-dashed line), ξ′′ (dashed line), and ξ′′′ (continuous line). r j is the jump position defined numerically, the jump definition ξ′′ = 0 is marked with a point. Numerical solutions were obtained using Q = 6.32 × 10−1 cm3/s, L = 11.57 cm in the water case. Variables are presented in the non-dimensional notation.
Scalings for the circular hydraulic jump. Experimental data 15 (points) for 3 different viscosities: ν w (water viscosity, circles), ν1 = 15ν w (squares), ν2 = 95ν w (rhomboids) compared with r 0,0 (dotted-dashed line, Eq. (12) ), r 1 (dashed line, solution of Eq. (15) ), r 0 (thick dashed line, Eq. (16) (right), and r 0,1 (thick dotted-dashed line, Eq. (17) ). Hydraulic jump radii are scaled with L = 16.5 cm (plate radius), the fluxes are scaled with Q = 100 cm3/s. Reproduced by permission from Hansen et al., Phys. Rev. E 55, 7048–7061 (1997). Copyright 1997 American Physical Society.
Radii dependence with outer fluid depth for Q = 0.031 l/s, ν = 0.97 × 10−5 m2/s. Experimental results (type I curves (circles) and type II curves (squares)), 11 r 1 (dashed line, solution of Eq. (15) ), r 0 (thick dashed line, Eq. (16) (right)), and r 0,1 (thick dotted-dashed line, Eq. (17) ) compared. Jump radii are normalized with L = 37.9 mm and depths with h = 2.77 mm. Reproduced by permission from Bohr et al., Physica B 228, 1–10 (1996). Copyright 1996 Elsevier B.V.
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