2. The numerical findings of Keeler & Blyth (Reference Keeler and Blyth2024)

In their recent work, Keeler & Blyth (Reference Keeler and Blyth2024) considered the free-surface flow of a 2-D fluid satisfying the nonlinear Euler equations bounded from below by topography. Numerical solutions were obtained with the finite-element method on a truncated domain. Their choice for this topography, $y(x)=a \exp ({-b^2 x^2})$, represents a 2-D ‘bump’ for $a>0$, and a ‘dip’ for $a<0$, both of which are localised near $x=0$. Here, a and b are specified constants. The key result of their work is the discovery of time-periodic solitary waves in this formulation, which emerge from linearly unstable solutions of the steady formulation. An example of this process is shown in figure 1.

Figure 1. Numerical solutions for the flow of an irrotational fluid past indented topography. A steady unstable solution, denoted a hydraulic fall, is shown in (a). The transient behaviour from the instability eventually approaches a time-periodic solution, shown in (b). Figures produced from numerical results of Keeler & Blyth (Reference Keeler and Blyth2024).

Steady free-surface flow over protruding topography, analogous to $a>0$, has been well studied in the potential flow literature for a plethora of geometries and submerged objects, and significant progress has been made through both numerical (Dias & Vanden-Broeck Reference Dias and Vanden-Broeck2002) and asymptotic (Lustri, McCue & Binder Reference Lustri, McCue and Binder2012) approaches. However, fewer works have investigated time-dependent effects beyond the initial disturbances that can arise. In their work, Keeler & Blyth (Reference Keeler and Blyth2024) begin by considering steady solutions of this formulation with $a>0$ and $Fr<1$. They isolate branches of these solutions, confirm with a formal linear stability analysis that these are temporally stable to small perturbations and also show through time evolution that large perturbations result in convergence to the original steady state. Steady solutions for $a>0$ and $Fr>1$ are found to be unstable; perturbations to these solutions induce a solitary wave that travels upstream. Time-evolution plots with $a=0.01$ for both of these examples, demonstrating the nonlinear stability of these solutions, are shown in figure 2(c,d).

Figure 2. Parameter values of steady solutions calculated by Keeler & Blyth (Reference Keeler and Blyth2024) are shown in the $(a,Fr)$ plane. Solutions with $Fr<1$ are hydraulic falls, where the downstream ($x \to \infty$) fluid level is lower than that of the upstream ($x \to -\infty$) level. For the converse case with $Fr>1$, the solutions are hydraulic rises. The qualitative stability properties of these solutions differ depending on the sign of the topography amplitude, $a$. Time-evolution plots for four different solutions are shown in (a–d). The solutions with indented topography ($a<0$) are unstable, and time evolution ultimately reveals that these settle into time-periodic behaviour. Figure adapted from Keeler & Blyth (Reference Keeler and Blyth2024).

However, the time-dependent behaviour for free-surface flows over indented topography has been discovered by Keeler & Blyth (Reference Keeler and Blyth2024) to be significantly different than that for protruding topography. Here, the steady solutions (with $a<0$) are linearly unstable, meaning that small-amplitude perturbations from the steady state initially grow in time. By investigating the long-time behaviour of these perturbations, Keeler & Blyth (Reference Keeler and Blyth2024) discovered that the fluid surface eventually approaches motion which is periodic in time. Time-evolution plots demonstrating this behaviour are shown in figure 2(a,b). Solution (a), with $Fr>1$, contains non-decaying oscillatory waves downstream, whereas solution (b), with $Fr<1$, contains oscillatory behaviour in both the upstream and downstream directions. For this latter case, the oscillatory behaviour is emitted from the central wave crest; large-amplitude travelling waves propagate upstream, and smaller-amplitude waves propagate downstream. The result is a fluid surface that repeats the same unsteady motion after an interval in time, with the oscillatory behaviour extending to spatial infinity. They are denoted generalised solitary waves, since the amplitude of the oscillations does not decay in the far field. These results emphasise the importance of considering solution stability, and the long-time trends of unstable configurations for free-boundary problems.

3. Future directions

The time-periodic surface waves discovered by Keeler & Blyth (Reference Keeler and Blyth2024) in the fluid flow past an indented topography raise a number of interesting questions that warrant additional investigation.

Firstly, it is possible that additional time-periodic solutions exist, which are not associated with the parameter values of the unstable solutions initially calculated by Keeler & Blyth (Reference Keeler and Blyth2024). To investigate this, one can alter the initial value problem with the requirement that the solution returns to its original profile after a period of time. Such an approach has recently been used by Wilkening & Zhao (Reference Wilkening and Zhao2023) to calculate time-periodic water waves with spatially periodic topography. Secondly, analytical results have been obtained for hydraulic-fall solutions with protruding topography by Kamchatnov et al. (Reference Kamchatnov, Kuo, Lin, Horng, Gou, El and Grimshaw2013), who applied Whitham modulation theory to the forced KdV equation which models shallow water flows. An earlier numerical study of this transient phenomenon in the forced KdV equation was performed by Wu (Reference Wu1987). It would be very interesting to see if such an analysis can be adapted to predict the time-periodic solutions uncovered by Keeler & Blyth (Reference Keeler and Blyth2024). Lastly, it is unknown if this time-periodic behaviour can manifest physically, or if some other neglected effect is required in the model, such as the third spatial dimension from which transverse instabilities can emerge. Experimental results exist for hydraulic falls over protruding topography (Tam et al. Reference Tam, Yu, Kelso and Binder2015), confirming the stability of these solutions (analogous to the solution in figure 2d), but not for indented topography. Evidence for the physical existence of these time-periodic solutions would continue to highlight the value of numerical and theoretical investigations into simplified formulations of similar problems across fluid dynamics.