Hydraulic jump in one-dimensional flow

Abstract

Abstract. In the presence of viscosity the hydraulic jump in one dimension is seen to be a first-order transition. A scaling relation for the position of the jump has been determined by applying an averaging technique on the stationary hydrodynamic equations. This gives a linear height profile before the jump, as well as a clear dependence of the magnitude of the jump on the outer boundary condition. The importance of viscosity in the jump formation has been convincingly established, and its physical basis has been understood by a time-dependent analysis of the flow equations. In doing so, a very close correspondence has been revealed between a perturbation equation for the flow rate and the metric of an acoustic white hole. We finally provide experimental support for our heuristically developed theory.

Application of Finite Difference Schemes to 1D St. Venant for Simulating Weir Overflow

“…Computed water profile compared and examined to the result from previous experimental research as shown in Figure 2 and Figure 3 for MacCormack scheme, and Figure 4 and The close agreement between the computed results and the experiment for hydraulic jump location, supercritical region and upstream water depth shows that the proposed method is comparatively accurate, which value of error shown in Table 2. However, both models fail to compute length of hydraulic jump since 1D SWE gives a linear height profile before the jump as stated in Singha et al [10]. In addition, models are not valid for submerged hydraulic jump condition and there is significantly error on the upstream as roller effect influence flow significantly, which is 2D phenomenon.…”

Implications of nonlinearity for spherically symmetric accretion

“…Looking at equation (18), we realize that H (φ,φ) = Aφ + Bφ and V(φ) = C(φ 2 /2) + ǫD(φ 3 /3), with the constant coefficients, A, B, C and D having to be read from equations (19). To investigate the properties of the equilibrium points resulting from equation (20), we need to decompose this second-order differential equation into a coupled first-order dynamical system.…”

“…A most striking feature of this equation is that even on accommodating nonlinearity in full order, it conforms to the structure of the metric equation of a scalar field in Lorentzian geometry (Section III). This fluid analogue (an "acoustic black hole"), emulating many features of a general relativistic black hole, is a matter of continuing interest in fluid mechanics from diverse points of view [8,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Then we apply our nonlinear equation of the perturbation to study the stability of globally subsonic stationary solutions under large-amplitude time-dependent perturbations.…”

The hydraulic jump and the shallow-water equations

“…Singha et al [8] introduced the viscous BLSWE (3) for a planar geometry (also see [7]), as [6] did before them, for a circular geometry. The present article addresses the planar jump, primarily, where the BLSWE with corresponding simplified boundary-conditions, and the local and global continuity equations are respectively…”

“…An understanding of it has been sought at least since the time of Rayleigh [3,4]. There exists a large body of literature on standing hydraulic jumps and this problem has been intensely studied since the last 100 years using various approaches like vertical averaging [5][6][7][8] of the boundary-layer shallow-water equations (BLSWE), numerically using Navier-Stokes simulations [9,10] and using higher-order boundary-layer analyses [11][12][13]. The interested reader is also referred to [14] where a comprehensive literature survey of the problem spanning last 100 years has been presented.…”