^{1,a)}, M. Argentina

^{2}and E. Tirapegui

^{3}

### Abstract

This document presents theoretical and numerical results of the circular hydraulic jump derived from the inertial lubrication theory[N. O. Rojas, M. Argentina, E. Cerda, and E. Tirapegui, Phys. Rev. Lett.104, 187801–1187801–4 (Year: 2010)]10.1103/PhysRevLett.104.187801. In particular, a correction for the hydraulic jump scaling is obtained. The results depend on subcritical depth, density, and surface tension, in agreement with experimental data at low Reynolds numbers[T. Bohr, C. Ellegaard, A. E. Hansen, and A. Haaning, Physica B228, 1–10 (Year: 1996)10.1016/S0921-4526(96)00373-0; S. H. Hansen, S. Horluck, D. Zauner, P. Dimon, C. Ellegaard, and S. C. Creagh, Phys. Rev. E55, 7048–7061 (Year: 1997)]10.1103/PhysRevE.55.7048.

The authors thank Tomas Bohr and Steen Hansen for providing permissions and useful experimental data. E.T. thanks project Fondecyt 1120329. N.R. acknowledges the support of CONICYT PAI/ACADEMIA 79112030.

I. INTRODUCTION

II. GENERAL FORMULATION

III. RESULTS

A. Free surface and velocity field

B. Scaling laws

1. First approach

2. Second approach

### Key Topics

- Hydraulics
- 17.0
- Viscosity
- 16.0
- Surface tension
- 13.0
- Fluid equations
- 12.0
- Numerical solutions
- 7.0

## Figures

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.

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.

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.

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.

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.

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.

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.

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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