From Bore–Soliton–Splash to a New Wave-to-Wire Wave-Energy Model
Abstract
We explore extreme nonlinear water-wave amplification in a contraction or, analogously, wave amplification in crossing seas. The latter case can lead to extreme or rogue-wave formation at sea. First, amplification of a solitary-water-wave compound running into a contraction is disseminated experimentally in a wave tank. Maximum amplification in our bore–soliton–splash observed is circa tenfold. Subsequently, we summarise some nonlinear and numerical modelling approaches, validated for amplifying, contracting waves. These amplification phenomena observed have led us to develop a novel wave-energy device with wave amplification in a contraction used to enhance wave-activated buoy motion and magnetically induced energy generation. An experimental proof-of-principle shows that our wave-energy device works. Most importantly, we develop a novel wave-to-wire mathematical model of the combined wave hydrodynamics, wave-activated buoy motion and electric power generation by magnetic induction, from first principles, satisfying one grand variational principle in its conservative limit. Wave and buoy dynamics are coupled via a Lagrange multiplier, which boundary value at the waterline is in a subtle way solved explicitly by imposing incompressibility in a weak sense. Dissipative features, such as electrical wire resistance and nonlinear LED loads, are added a posteriori. New is also the intricate and compatible finite-element space–time discretisation of the linearised dynamics, guaranteeing numerical stability and the correct energy transfer between the three subsystems. Preliminary simulations of our simplified and linearised wave-energy model are encouraging and involve a first study of the resonant behaviour and parameter dependence of the device.
Keywords
Water-wave focussing Wave-activated buoy motion Electro-magnetic generator Monolithic variational principle Finite-element modelling1 Introduction
Early September 2010, three applied mathematicians at the University of Twente made requests to create a soliton in a make-shift wave tank for a new “research plaza” opening festivity. Part of that plaza contains a water feature or channel approximately \(45\;\mathrm{m}\) long, \(2\;\mathrm{m}\) wide, and \(1.2\; \mathrm{m}\) deep. Normally filled to its edge with water and harbouring water plants and fish, at the time, it was only partially filled. A soliton is a wave with nonlinearity and dispersion in balance, such that the wave stays coherent and neither disperses nor breaks [13, 33]. Solitons or solitary waves can be generated at the beginning of a rectangular channel with vertical walls: using either a piston moving bespokely, a block lowered at a finite yet fast speed into the water or by a quick sluice-gate removal between a higher rest-water level (\(h_1\)) lock section and a lower rest-water level (\(h_0\)) main section. We have used the latter for solitary-wave generation with an extra channel feature, sketched in Fig. 1 with dimensions in Table 1: a V-shaped channel end with vertical walls.
Details of soliton–splash experiment, including wave tank dimensions
Wave tank length | \(L_y=43.63\pm 0.1\ \mathrm{m}\) |
Wave tank width | \(L_x=2\ \mathrm{m}\) |
Wave tank height | \(L_z=1.2\ \mathrm{m}\) |
Contraction length | \(d=2.7\ \mathrm{m}\) |
Location of sluice gate | \(\ell _{\text {s}}=2.63\ \mathrm{m}\) |
Rest-water level (high) | \(h_1=0.9\ \mathrm{m}\) |
Rest-water level (low) | \(h_0=0.43\ \mathrm{m}\) |
Sluice-gate release speed | \(V_{\text {g}} \approx 2.5\ \mathrm{m/s}\) |
Sluice-gate removal time | \(T_{\text {s}}=h_1/V_{\text {g}} \approx 0.36\ \mathrm{s}\) |
Table with all experimental trials to establish the highest bore–soliton–splash (BSS)
Case | \(h_0\) (m) | \(h_1\) (m) | \(H_s\) | \(H_{rw}\) | Peak | Comments |
---|---|---|---|---|---|---|
\(\pm 0.01\) m | \(\pm 0.01\) m | \(\pm 0.05\) m | \(\pm 0.5\) m | # | ||
1 | 0.32 | 0.67 | – | 0.6 | – | Bore |
2 | 0.38 | 0.74 | – | 2.5 | – | Good splash |
3 | 0.41 | 0.9 | 0.35 | 3.25 | 2\(\mathrm{nd}\) | Thin jet cf. 6 & 8 |
4 | 0.47 | 1.0 | 0.35 | 1 | 2\(\mathrm{nd}\) | Bore and low splash |
5 | 0.41 | 1.02 | 0.40 | 1.5 | 1\(\mathrm{st}\) | Bore and low splash |
6 | 0.41 | 0.9 | 0.35 | 3.5 | 2\(\mathrm{nd}\) | BSS cf. 3 and 8 |
7 | 0.45 | 0.8 | 0.35 | 2.5 | 2\(\mathrm{nd}\) | Good splash |
8 | 0.41 | 0.9 | 0.35 | 3.5 | 2\(\mathrm{nd}\) | BSS and highest splash |
9 | 0.43 | 0.9 | 0.45 | 1.8 | 1\(\mathrm{st}\) | Collapsing into sheets |
Rogue, monster, or extreme waves are anomalously high and rare waves with wave height \(H_{{\text {rw}}}\), generally considered at sea, defined relative to a significant or ambient-wave height \(H_{\text {s}}\) of surrounding, preceding, and following seas. A straightforward definition of rogue waves states that the abnormality index \({\text {AI}}=H_{{\text {rw}}}/H_{\text {s}}>2\), i.e., the rogue-wave height \(H_{{\text {rw}}}\) must be at least twice as high as the ambient-wave height^{2}\(H_{\text {s}}\). Dysthe et al. [14] and Khariff et al. [31] provide more advanced and precise definitions of rogue waves, but this common definition given above suffices here. Rogue waves have a rare, extreme occurrence and are, therefore, difficult to predict, either statistically or deterministically. Understanding their wave height and occurrence is relevant to maritime and coastal engineering given their potential to damage ships, maritime, and coastal structures, including sinking and disappearance of ships; an overview of such disasters is found in Nikolkina and Didulenkova [41]. There are different types of rogue waves, involving linear and nonlinear wave focussing in one horizontal dimension, spatial wave focussing due to coastal or submarine convergences, episodic waves such as tsunamis generated elsewhere, and crossing seas with pyramidal waves [7, 16]. These different rogue-wave types have been (partially) explained within a hierarchy of different models, including, e.g., incompressible Euler equations with a free surface and passive or limited air motion, its potential-flow restriction, and numerous asymptotic approximations of these classical potential-flow water-wave equations such as Benney–Luke equations, Kadomtsev–Petviasvili’s (KP) equation, nonlinear Schrodinger equation(s), and the Korteweg–De-Vries equation [13, 28, 31, 40, 42, 43].
a detailed description of a man-made bore–soliton–splash rogue wave with an abnormality index of \({\text {AI}}\approx 10\);
establishing and employing mathematical and numerical models for experimental cases 8 and 9 (see Table 2), with improved simulations beyond the one in Bokhove and Kalogirou [4]; and,
inspired by the bore–soliton–splash configuration, we invented a novel rogue-wave-energy device, and built and tested a scaled-down version; a first nonlinear mathematical model is developed here, for which we show simulations of its linearised dynamics.
The outline of our paper is as follows. Soliton–splash and bore–soliton–splash experiments are analysed in Sect. 2. Some mathematical and numerical solutions of soliton splashes with Benney–Luke’s model are found in Sect. 3. In Sect. 4, our wave-energy device is introduced with one comprehensive and novel, nonlinear mathematical wave-to-wire model of the hydrodynamics, buoy motion, and power generation. After developing an intricate and novel compatible discretisation of that linearised model, numerical modelling results are presented. We finish with a discussion of open questions and challenges in Sect. 5, also highlighting a splash-inspired artwork.
2 Experimental Set-Up and Results
Our goal in 2010 was to create both a travelling soliton by removing a sluice gate separating two different water levels, initially at rest, and a splash of the highest possible amplitude in a V-shaped contraction. Given time constraints, the only way to determine whether our goal was reachable in practice was to resort to experimentation in two make-shift wave channels: the first one where the Roombeek, a brook, flows onto the University of Twente campus and the second one on the above-mentioned Research Plaza, see Fig. 3. On 20-09-2010, we managed to obtain two soliton splashes with \(h_1\approx 2 h_0\) in the Roombeek convincing us that it was possible to make a larger and reproducible Plaza-channel soliton–splash. Subsequently, seven test cases were completed on 27-09-2010, including six with different rest-water levels \(h_0\) and \(h_1\), and one repeated case with the highest splash to ensure reproducibility on the opening day of the Research Plaza. The optimal case involved \(h_0=0.41\pm 0.01\;\mathrm{m}\) and \(h_1=0.9\pm 0.01\;\mathrm{m}\). These two repeat cases and the general outcomes on 27-09-2010 showed that our experiments to create a bore–soliton–splash were reproducible on the opening day (30-09-2010). All cases are summarised and dated in Table 2 with repeat cases underlined and numbered by 3, 6, and 8. On 30-09-2010, this “optimal” case was successfully repeated, as shown in Fig. 3, followed by a case numbered 9 with the higher water level of \(h_0=0.43\;\mathrm{m}\) set in the main channel by the addition of sluice compartment’s extra water from optimal case 8, while keeping \(h_1=0.9\;\mathrm{m}\); case 9 resulted in a smooth solitary-wave compound without wave breaking and a lower splash. Its evolution in time is displayed in Fig. 4 as bespoke simulations introduced and explained later. Videos of (nearly) all cases are found on Zweers’ YouTube channel [50] and numbered accordingly. Inspection of videos of three repeat cases 3, 6 and 8 reveals that there are some/minor differences, partially commented on in Table 2. We attribute differences to the estimated error of \(\pm 0.01\;\mathrm{m}\) in initial water levels \(h_{0,1}\) and the manner of sluice-gate removal by the excavator, despite training to be as consistent as possible. Case 9 underscores these sensitivities to the initial conditions, because a \(0.02\;\mathrm{m}\) change from \(h_0=0.41\;\mathrm{m}\) (cases 3, 6, and 8) to \(0.43\;\mathrm{m}\), while keeping \(h_1=0.9\;\mathrm{m}\), within measurement error, led to a quite different splash. Note, however, that the outcomes are not chaotic, as three reasonably repeatable cases demonstrate.
3 Mathematical and Numerical Modelling of Soliton Splashes
The bore–soliton–splash involves a series of mathematical and fluid-mechanical ingredients: dispersion, nonlinearity, a turbulent spilling breaker, and collapsing splash. Assuming incompressible fluid flow with a free surface, dispersion in a solitary wave is balanced by nonlinearity due to advection, while the hydraulic bore or spilling breaker highlights that this balance is temporarily and locally broken till turbulent dissipation reduces wave amplitude sufficiently to restore that balance, as we saw in Fig. 3c, d. When the flow is in balance, the soliton compound and splash can be modelled with a single-valued free surface in a singly connected domain till the apex of the splash is reached. Both spilling breaker and collapse of the splash are seen to involve multiply connected domains with bubbles and droplets.
We will start our modelling of the bore–soliton–splash cases for a smooth single-valued free-surface and using potential-flow equations and approximations thereof. Approximations used include a Benney–Luke pair/system of equations. Alternatively, one can explore the single, unidirectional KP equation in two horizontal spatial dimensions. These approximations have the advantage that dispersion is anomalously high which prohibits wave breaking and is, therefore, robust with the disadvantage being that outcomes during wave breaking will be less realistic, as follows. Numerical solutions are required to solve potential-flow and Benney–Luke equations in the actual wave channel, while exact solutions are available for the KP equation in an idealised domain for idealised settings. We will use variational principles and asymptotic theory to enhance numerical stability and robustness: our (novel) numerical techniques are direct, compatible space–time discretisations of relevant variational principles.
Simulations
4 Novel Rogue-Wave-Energy Device
The tapered channel or TapChan device; it consists of a tapered open channel which will enhance the wave amplitude, such that at the channel end the waves overtop a levy and water flows into a reservoir. Elsewhere along the reservoir, water flows down into a turbine to generate electricity via hydropower. A TapChan operated for a couple of years on a Norwegian island, bringing back electricity of \(350\;\mathrm{kW}\) into the Norwegian grid, before it got damaged in a storm.
The IPS wave buoy; it consists of a heaving buoy with a deep-lying piston moving into an anchored vertical shaft with a PTO mechanism to generate energy.
The oscillating water column (OWC); it consists of a tapered channel in which the waves enter one open end of the channel, funnel, and amplify in an enclosed converging section that turns into a vertical blow hole at the top or the top side. Meanwhile, air compresses and decompresses by the rising and sinking wave leading to rapid air flow through the blow hole in which a wells’ turbine is situated and generates electrical power when air flows in either direction.
Our device consists of a contracting channel with a wave buoy constrained to move in only one dimension, either in the vertical by sliding along a guiding mast or along a slightly curved arc pivoting around a horizontal axel at the contraction entrance. Attached to the buoy is either another vertical mast or a curved mast, to which magnets are attached that can move through a series of coils when the buoy is heaving due to the wave motion. An artistic rendering of the second version of the wave-energy device is given in Fig. 6. The latter magnet-and-coil system comprises a magnetic-induction motor, cf., the one in the Faraday shaking light shown in Fig. 7b. Relative to the version with the two vertical masts, one moving and one fixed, it has the advantage that the buoy can be taken out of action in storms and that a rotating axel is mechanically more robust than mast-guide ball bearings. Our device is intended to be part of a breakwater or dock, since waves will be absorbed.
4.1 Wave-to-Wire Monolithic Wave-Energy Model
A comprehensive mathematical model of the new wave-energy device will be developed next within a domain constructed to reproduce an existing small-scale wave tank at the University of Leeds, which is a larger tank than the one used for the proof-of-principle. Both the wave-to-wire formulation of this wave-energy device is novel as well as the compatible and robust discretisation of the linearised model. Such a discretisation is important, because it guarantees numerical stability and the correct two-way feedback between pairs of the three subsystems. The numerical wave tank has a piston wavemaker on its left side, and consists of a channel with a flat bottom at \(z=0\) that ends in a V-shaped contraction at the right end of the channel, cf. Fig. 9, as described in Sect. 3 and Eq. (2). A wave-energy buoy, here constrained to move only in the vertical, resides in the corner of the contraction. The shape of the buoy is described next.
Physical parameters used in three-dimensional numerical calculations for the wave buoy system, including wave tank dimensions, buoy’s mass, and physical properties of water at a room temperature of 25\(\,^{\circ }\mathrm {C}\)
Channel width | \(L_x=0.2\ \mathrm{m}\) |
Channel length | \(L_y=2.0\ \mathrm{m}\) |
Channel height | \(L_z=0.2\ \mathrm{m}\) |
Rest-water depth | \(H_0=0.1\ \mathrm{m}\) |
Buoy mass | \(M=0.05\ \mathrm{kg}\) |
Density of water | \(\rho _{0}=997\ \mathrm{kg/m}^3\) |
Gravity | \(g=9.81\ \mathrm{m/s}^2\) |
4.2 Time Discretisation of the Linearised System
4.3 Space Finite-Element Discretisation of the Linearised System
The model (23) is discretised in a few steps. The first step is to multiply the field equations in (23) by \(C^0\)–test functions, and then integrate over space and by parts. The second step is to expand the fields using (special) \(C^0\)-continuous and compact finite-element basis functions. We will use standard linear and compact Galerkin basis and test functions, which are unity at their home node and zero at neighbouring nodes of the elements connected to the home node. The result is a space-discrete system which will be revealed to be only consistent for certain special choices of the function spaces and expansions. Vice versa, we can first discretise time in a consistent manner, such that again the equations remain consistent as we showed already. Finally, by either discretising the space-discrete system properly in time or the proper time-discrete system in space, we obtain an internally consistent overall space–time discretisation fit for numerical implementation.
4.4 Numerical Results
We have set up a numerical code which simulates the full system as it evolves in time, including the generation/propagation of waves, their impact on the wave-energy buoy, the response of the buoy, and the power output. The numerical results presented next have been obtained using a mesh resolution of \(N_x=10\) and \(N_y=50\), i.e., the total number of elements in the calculations is \(N_{{\text {el}}}=555\) and the total number of nodes is \(N_n=621\). The time step used is \(\varDelta t=0.0028\;\mathrm{s}\). At the start of the simulation the system is at rest and the water depth in the main wave tank is \(H_0=0.1\;\mathrm{m}\). For \(t>0\), waves are generated from the left wall of the tank by a piston wavemaker that follows a periodic motion in time according to \(R(t)=\tfrac{A}{\omega }(1-\cos (\omega t))\), with amplitude \(A=0.0653\;\mathrm{m}\) and frequency \(\omega =\frac{6\pi }{L_y}\sqrt{gH_0}=9.3348\;\mathrm{s}^{-1}\) (which corresponds to a physical frequency of \(\omega /2\pi =1.4857\;\mathrm{Hz}\)). Therefore, on the left wall, \(\partial _y\tilde{\phi }=\dot{R}(t)=A\sin (\omega t)\).
The results of a simulation with the parameters described earlier can be seen in Figs. 11 and 12. The response of the buoy is shown in the top panel of Fig. 11, while the electrical current is shown in the middle panel and the power generated in the bottom panel. Both the power generate (\(\tilde{P}_{\text {g}}(t)\) blue line) and the resistive loss (\(\tilde{P}_{\text {l}}(t)\) orange line) are displayed. Two snapshots of the computed wave height and buoy position in the contraction are displayed in Fig. 12.
Convergence rates n using three different norms (\(\mathcal {L}_1\), \(\mathcal {L}_2\), \(\mathcal {L}_\infty \)) evaluated using the value of the velocity potential \(\tilde{\phi }\) at the final time of the simulation, i.e., at \(t=T\). The time step used is such that the CFL condition is satisfied in similar fashion for each mesh resolution
x-el | y-el | Total | Nodes | Rate | |
---|---|---|---|---|---|
Symbol | \(N_x\) | \(N_y\) | \(N_k\) | \(N_n\) | n |
Mesh 1 | 6 | 30 | 201 | 241 | \(\mathcal {L}_1\): 1.711293 |
Mesh 2 | 12 | 60 | 798 | 877 | \(\mathcal {L}_2\): 1.696554 |
Mesh 3 | 24 | 120 | 3180 | 3337 | \(\mathcal {L}_\infty \): 1.765833 |
Indicative parameter values and units used in three-dimensional numerical calculations for the wave–buoy system. LED: https://en.wikipedia.org/wiki/Shockley_diode_equation
Constant | Value | Unit | Determination |
---|---|---|---|
Magnetic dipole moment | \(m=0.1\) | Am\(^2\) | Estimate |
\(\mu _0=4\pi 10^{-7}\) | N/A\(^2\) | ||
Coil outer radius | \(a=0.04\) | m | Estimate |
\(\alpha _h =0.05,0.2\) | Estimate | ||
\(K=0.53\) | |||
Radius magnet | \(A_{\text {m}} = 0.032\) | m | Estimate |
Coil diameter | \(D=0.2769\) | mm | Coil MW30-9 |
Length coil | \(L =0.08\) | m | Estimate |
Length magnet | \(L_{\text {m}}=0.04\) | m | Estimate |
Winding number | \(N=L/D\) | Calculated | |
Coil induction | \(L_i=\pi a^2\mu _0N^2/L\) | Nm/A\(^2\) | Calculated, see text |
Coil resistance | \(R_{\text {c}}=8aN/({\sigma }D^2)\) | V/A | Calculated, see text |
Circuit resistance | \(R_i=R_{\text {c}}\) | V/A | Estimate |
Shockley | \(n_{\text {q}}=1\) | www | |
Shockley voltage | \(V_{\text {T}}=2.05\) | V | www |
Shockley current | \(I_{{\text {sat}}}=0.02\) | A | www |
Mast length | \(H_{\text {m}}=0.2\) | m | Estimate |
Conductivity | \(\sigma =5.96\times 10^7\) | A/Vm | Copper/wiki |
Mass | \(M=0.08\) | kg | Estimate |
Keel | \(H_k=0.04\) | m | Estimate |
5 Summary and Discussion
In summary, we have reported in detail on the creation of the bore–soliton–splash, summarised modelling of this hydrodynamic splash, and showed how it inspired a novel wave-energy device. We will next provide further context of our work.
Relation to rogue waves at sea The bore–soliton–splash is a nonlinear wave-resonance phenomenon in which a series of travelling solitons reflect in a V-shaped contraction leading to a tenfold resonant amplification of the initial main wave height. Once created in 2010, the phenomenon caught attention of the rogue-wave community. Rogue waves are extreme and rare waves, generally but not exclusively sea waves, at least twice as high as the wave height of the ambient sea. While the original bore–soliton–splash was engineered, it relates to several rogue-wave phenomena at sea, far from and near the coastline. Rogue waves can have several causes and emerge in different situations: rogue-wave emergence in crossing seas, either due to seas with two main directions, e.g., high seas generated by two hurricanes, or one hurricane changing direction. More rare are seas with waves and swell from three different main directions. Our V-shaped contraction walls can, therefore, be re-interpreted as virtual walls concerning two (virtual) waves travelling under two angles \(\pm \varphi \) from the main wave’s direction with two (virtual) planes of no-normal flow leading to a converging point. One difference is that the virtual case supports wave propagation with one splash at one space–time point, cf. [20], while our engineered set-up with solid walls necessarily leads to reflections.
Further modelling of the bore–soliton–splash and related rogue waves We reported (bore–)soliton–splashes including one with smooth solitary waves in which nonlinearity and dispersion are balanced without any wave breaking. The smooth soliton–splash of case 9 was successfully simulated using a compatible, geometric finite-element discretisation of a Benney–Luke model, a bidirectional simplification of the classic potential-flow model for water waves. Case 8 for the maximum bore–soliton–splash was not simulated correctly by the Benney–Luke model. Due to the lack of wave breaking, the case 8 simulation deviated significantly from reality in that the observed resonant interaction was absent. On one hand, it is possible to further explore simplified modelling of rogue waves in crossing seas using the Kadomtsev–Petviashvili (KP) equation, cf. [1, 20, 21, 28, 32]. On the other hand, more advanced modelling of the bore–soliton–splash will either require a full potential-flow model with localised and parameterised wave breaking or the use of models with actual, localised wave breaking while maintaining good dispersion properties. Single-phase or two-phase mixture-theory models, including ones with a Van-der-Waals-type equation of state, may also be good candidates [23, 46].
Optimisation of the wave-energy device Our splash inspired the creation of a novel wave-energy device and we showed a working, experimental proof-of-principle model, but also developed and derived a new and fully nonlinear mathematical model of the combined water-wave dynamics, the wave-activated buoy, and the magnetic-induction power generator. Essential ingredients of this comprehensive model have been captured in one variational principle to which we a posteriori added dissipative effects of the electrical circuit, coils of the actuator, and LEDs used as the loads. The overall model was subsequently linearised and discretised using a finite-element method in space and time. This (linear) algebraic model was made fully compatible with the variational structure in the conservative and continuum limits. Its compatible, novel, and nontrivial discretisation was augmented with the resistances of the electrical circuit and coils of the induction motor as well as the LED loads. Preliminary simulations of the linear model showed promising results including (suboptimal) convergence and energy transfer between the three components. Finally, we investigated the resonant behaviour of the system as function of wave-frequency and load for a long wave-packet of harmonic waves. Nonlinear modelling, optimisation, and control of the wave-energy device require further exploration, and both the geometry of contraction, mass, and wave-buoy shape could be optimised for a given wave climate. We also aim to explore feedback control as function of contraction geometry, the number of coils of the induction motor, and the total load. In addition, higher order and more accurate spatial and temporal discretisation schemes require exploring.
Steel–soliton–splash artwork We finish on an artistic note. Our bore–soliton–splash inspired an artwork, the steel–soliton–splash [49], created by WZ. Snapshots of the video of case 8 were first outlined as silhouettes, two of which formed the basis for a three-dimensional artwork, scaled down by about a factor of three, and welded in stainless steel, see Fig. 18. In 2013, we donated the artwork to the Isaac Newton Institute of Mathematical Sciences in Cambridge, UK.
Footnotes
- 1.
Designer and artist WZ named the “soliton splash” 20-09-2010 video with “bore” indicating intermittent wave breaking.
- 2.
This abnormality index, AI, has been defined and used, e.g., in Didulenkova et al. [10].
- 3.
Movie of 2013 proof-of-principle design/test: https://www.youtube.com/watch?v=SZhe_SOxBWo.
- 4.
Courtesy of Dr. Martin Robinson [45] https://www.youtube.com/watch?v=PRnycO6db1M.
- 5.
Notes
Acknowledgements
Part of this work appeared earlier in [5], in Dutch, as photographs and a first simulation of case 9 in [4], and as illustrative photographs in [17]. Preliminary versions of the wave-energy device and its modelling appeared in conference proceedings [29, 30]. The story of the steel–soliton–splash, first presented in [48], can be found on the website of the Isaac Newton Institute of Mathematical Sciences in Cambridge, UK. It is a pleasure to acknowledge assistance of Dr Paul Steenson and Prof Steve Tobias in formulating the magnetic-induction generator. This is follow-up research grew out of EPSRC Grant EP/L025388/1 for AK and OB. OB also acknowledges participation in workshops (July 2014 and 2017) at the Isaac Newton Institute of Mathematical Sciences in Cambridge, funded under EPSRC Grant EP/R014604/1.
Compliance with ethical standards
Conflict of interest
This manuscript concerns research with no conflict of interest.
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