Standing and travelling waves in the shallow-water circular hydraulic jump

Abstract

A wave equation for a time-dependent perturbation about the steady shallow-water solution emulates the metric an acoustic white hole, even upon the incorporation of nonlinearity in the lowest order. A standing wave in the sub-critical region of the flow is stabilised by viscosity, and the resulting time scale for the amplitude decay helps in providing a scaling argument for the formation of the hydraulic jump. A standing wave in the super-critical region, on the other hand, displays an unstable character, which, although somewhat mitigated by viscosity, needs nonlinear effects to be saturated. A travelling wave moving upstream from the sub-critical region, destabilises the flow in the vicinity of the jump, for which experimental support has been given.

Implications of nonlinearity for spherically symmetric accretion

“…nonlinearity higher than the second. Studies of the hydraulic jump phenomenon have reported similar saturation of a nonlinearly growing instability in the vicinity of the critical point of the flow [22,24].…”

“…For all that, a most remarkable fact that has emerged in consequence of including nonlinearity in the perturbative analysis, is that notwithstanding the order of nonlinearity that we may adopt, the symmetric form of the metric equation will remain unchanged, as shown very clearly by equation (9). Precedence of the survival of this symmetry under nonlinear conditions, can be found in the fluid problems of the hydraulic jump [24] and spherically symmetric outflows of nuclear matter [30].…”

“…This state of affairs has been depicted in the right side of the plot in Figure 1, showing a continuously diverging phase solution. Given that the eigenvalues, ω, have been yielded on using solutions of the type, exp(ωt), the e-folding time scale of this growing mode of the perturbation is ω −1 , with ω having to be read from equation (24). With C = 1 and D = −1, the coordinate of the second fixed point (a saddle point) is set at (1, 0).…”

“…Since a saddle point cannot be eliminated by the inclusion of higher orders of nonlinearity [34], the best that we may hope for is that the instability may grow in time till it reaches a saturation level imposed by the higher nonlinear orders (but the instability will never be decayed down). This type of instability has a precedence in the laboratory fluid problem of the hydraulic jump [22,24]. While the discussion so far holds forth on the perturbative perspective, its crux lies in the far-reaching implications of the saddle point for the non-perturbative evolutionary dynamics.…”

“…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.…”