Role of gravity and capillary waves in the origin of circular hydraulic jumps

In almost all of the studies on the circular hydraulic jump (CHJ), gravity had been considered as a significant variable that affects the formation of the jump. Most recently, gravity was deprived of being important in the origin of the CHJ, which challenged researchers in this field of fluid mechanics. This study addresses in detail the physical concepts behind this intriguing phenomenon occurring in the radial outspreading of a vertically downward free-surface liquid impinging jet upon a horizontal plate. The aim is to find out whether gravity plays any role in the origin of the CHJ. Accordingly, the jump evolution is investigated in two cases: first, the initial formation of the CHJ in which the subcritical flow downstream from the jump is approaching the outlet boundary (developing jump). Second, the final evolution of the CHJ in which a steady-state flow is circumventing an obstacle at the edge of the impinged plate and falling uniformly down from the outlet boundary (developed jump). The results indicate the existence of two different flow regimes in the jump formation: gravity- and capillary-dominant flow regimes. In general, the role of gravity in the formation of developing or developed jumps cannot be eliminated; however, its importance lies in the fact of which regime dominates the flow. Intensification of gravitational effects is observed when capillary waves are dampened by increasing viscosity, density, or volume flow rate as well as by decreasing surface tension. Finally, a generalized scaling relation for the jump radius is obtained considering both capillary and gravitational effects in the critical flow condition. In contrast to the previous results, this generalized scaling relation predicts more accurately the radius of both a developing and a developed jump.

Figure 1

Formation of a CHJ before and after the arrival of the flow at the outlet boundary. (a) Developing CHJ. (b) Developed CHJ.

Figure 2

(a) Computational domain with the boundary conditions; dashed lines schematically show an initial formation of the jump before the flow reaches the outlet boundary (a developing jump) and the light blue background segment illustrates a steady-state flow dropping down from the outlet (a developed jump) (b) Qualitative grid cells distribution and the dimensionless length scales based on the nozzle diameter.

Figure 3

Independence of the interface profile from the grid resolution: (a) Flow approaching the outlet boundary at $\tau =8\times {10}^{-2}$; (b) final evolution of the flow over the obstacle.

Figure 4

(a) Independence of the wall gradient from the grid resolution. (b) Comparison with the experimental study by Duchesne et al. [37].

Figure 5

Formation of a CHJ in gravity-dominant (left column) and in capillary-dominant (right column) regimes. (a) Initial formation of a CHJ showing a constant position for the jump, before the flow reaches the outlet boundary. (b) Distributions of the inverse Weber number and the Froude number squared, and that of ${\text{Fr}}^{-2}+{\text{We}}^{-1}$ before (supercritical flow) and after (subcritical flow) the jump position at $\tau =7\times {10}^{-2}$. (c) Jump position (${\xi }_j=R_j/D$) based on the conditions $\text{Fr}=1,\text{We}=1,{\text{Fr}}^{-2}+{\text{We}}^{-1}=1$ or where $dh/dr$ has its maximum. (d) Gravity $\left(\sqrt{gh}\right)$, capillary $\left(\sqrt{\sigma /\rho h}\right)$, and capillary-gravity $(\sqrt{gh+\sigma /\rho h}$ waves velocities before and after the jump at $\tau =7\times {10}^{-2}$.

Figure 6

(a) Possible deviations of the scaling relation of Bhagat et al. [2], Eq. (4), from the jump position. (b) Dispersion relation of undamped capillary-gravity waves in water. The different chromatic regions represent the approximate boundaries for pure capillary waves (small wavelengths, $\lambda <0.7$ cm) and gravity waves (long wavelengths, $\lambda >4$ cm). The region in the middle has mixed capillary-gravity waves. The solid line represents the relatively sharp boundary between the shallow- and deep-water limits [18].

Figure 7

Influence of viscosity on developing (left column) and developed (right column) CHJs: (a) interface geometry; (b) and (c) variation of capillary and gravity effects in the jump region.

Figure 8

Influence of volume flow rate on developing (left column) and developed (right column) CHJs: (a) interface geometry; (b) and (c) variation of capillary and gravity effects in the jump region.

Figure 9

Surface tension effects on the formation of CHJs: interface geometry for a developing (a1) and a developed (a2 and b) CHJ; (c) local variation of the inverse Froude number squared, the inverse Weber number, and that of ${\text{Fr}}^{-2}+{\text{We}}^{-1}$ in capillary- ($\sigma =70$ mN ${\mathrm{m}}^{-1}$) and gravity-dominated ($\sigma =10$ mN ${\mathrm{m}}^{-1}$) flow regimes for a developed jump; (d) gravity waves ($\sqrt{gh}$), capillary waves ($\sqrt{\sigma /\rho h}$), gravity-capillary waves ($\sqrt{gh+\sigma /\rho h}$), and the mean flow ($\overline{u}$) velocities for a developed jump.

Figure 10

Gravity effects on the formation of a developing (left column) and a developed (right column) CHJ: (a) and (b) interface geometry; (c) and (d) local variation of the inverse Froude number squared, the inverse Weber number, and that of ${\text{Fr}}^{-2}+{\text{We}}^{-1}$; (e) gravity waves ($\sqrt{gh}$), capillary waves ($\sqrt{\sigma /\rho h}$), gravity-capillary waves ($\sqrt{gh+\sigma /\rho h}$), and the mean flow ($\overline{u}$) velocities.

Figure 11

Influence of density on the formation of a developing (left column) and a developed (right column) CHJ: (a) and (b) interface geometry; (c) and (d) local variation of the inverse Froude number squared, the inverse Weber, and that of ${\text{Fr}}^{-2}+{\text{We}}^{-1}$; (e) gravity waves ($\sqrt{gh}$), capillary waves ($\sqrt{\sigma /\rho h}$), gravity-capillary waves ($\sqrt{gh+\sigma /\rho h}$), and the mean flow ($\overline{u}$) velocities.

Figure 12

Comparisons between the jump positions predicted by $R_j=c{\left(q^5{\nu }^{-3}{g}^{-1}\right)}^{1/8}$ [Eq. (2)] and $R_j=c{\left(q^3\rho {\nu }^{-1}{\sigma }^{-1}\right)}^{1/4}$ [Eq. (4)] and Eq. (13) in terms of viscosity (a), volume flow rate (b), and density (c) for the developing (left column) and developed (right column) CHJs.

Figure 13

Comparisons between the jump positions predicted by $R_j=c{\left(q^5{\nu }^{-3}{g}^{-1}\right)}^{1/8}$ [Eqs. (2)] and $R_j=c{\left(q^3\rho {\nu }^{-1}{\sigma }^{-1}\right)}^{1/4}$ [Eq. (4)] and Eq. (13) in terms of surface tension (a) and gravity (b) for the developing (left column) and developed (right column) CHJs.

Figure 14

(a) Influence of the obstacle height on the jump position and on the scaling relation of Eq. (13); (b) evaluation of the constant coefficient of Eq. (13) for the developing (b1) and developed (b2) jumps.