Results are presented from an experimental investigation of the motion of a shallow layer of water in a square tank that was oscillated horizontally with small amplitude at frequencies close to the natural frequency of the layer. The aim was to assess the validity of certain theoretical predictions relating to nonlinear resonance in shallow layers based on asymptotic analysis. These concern the hysteretic nature of the resonance profile and the existence of non-trivial oscillatory components associated with shock-like discontinuities in the derived solution for the free surface configuration. The experimental results support the theoretical predictions and show a coincidence in parameter space between the regions of hysteresis and oscillatory transition.
Considerable attention has been given to the role of nonlinear mechanisms in fluid flow problems. Much of this effort has been directed towards understanding the behaviour of closed flows such as Rayleigh–Bénard convection (Bodenschatz et al. 2000) and Taylor–Couette flow (Tagg 1994), where the intrinsic patterned nature of the flow gives rise to a finite number of nonlinearly interacting fluid elements. There has also been interest in the possibility of low-dimensional nonlinear mechanisms in open flows that undergo spatial transition due to convective instability (Huerre & Monkewitz 1990). However, in the present study the particular area of fluid dynamics that is of interest from the point of view of nonlinear dynamics is that of free surface flow. In particular, since a fluid surface acts as a natural oscillator when displaced from equilibrium, it is pertinent to consider how such a system responds when subjected to some form of external periodic forcing (Ockendon & Ockendon 2001).
Of the studies to date that have chosen to look at the problem of forced free surfaces, the vast majority have been for the case of vertical forcing in which the free surface is oscillated in the direction perpendicular to its equilibrium orientation. Vertical vibration gives rise to standing waves on the free surface that are known as Faraday waves (Miles & Henderson 1990). These act to segment the free surface into a pattern of nonlinearly interacting elements, much as in the closed flow problems mentioned above. As such, it is perhaps not surprising that low-dimensional dynamics have been found under such conditions. However, there is an alternative forcing regime that is arguably more generic and certainly holds much more validity when considering practical problems in which free surfaces are involved. This is the case of horizontal forcing, where the free surface is perturbed in a direction parallel to its equilibrium orientation. Under such circumstances, there is no fundamental instability from a trivial state as in the case of the Faraday instability. The problem is now one of a forced nonlinear oscillator with resonant responses when the frequency of the applied forcing coincides with any of the natural frequencies of the free surface.
The first systematic theoretical study of the nonlinear resonant characteristics of a contained free surface was carried out by Moiseyev (1958) for the case of relatively deep layers. The problem was also looked at independently by Chester (1968), who established an appropriate description of the resonance in terms of a nonlinear ordinary differential equation derived from the Laplace equation for the two-dimensional velocity potential. The work of Chester was consolidated by Ockendon & Ockendon (1973), who applied rigorous asymptotic methods to the solution of the governing equation with particular emphasis on the regime where the depth of the fluid layer is small in comparison to the horizontal tank dimension. The most significant findings of Chester and Ockendon & Ockendon were that the direction of resonant hysteresis reverses as the depth passes through a critical value, and that there is a range of forcing parameters in which the solutions representing the configuration of the free surface develop discontinuities in the form of compressive shocks. Experimental evidence for this was offered by Chester & Bones (1968) by way of observations of undular bores that developed on the free surface in addition to the basic oscillation. The original asymptotic analysis was subsequently extended by Ockendon et al. (1986) to investigate further the emergence of the shock-like solutions and their development and proliferation. This led to predictions that there should be multiple secondary oscillations in response to the applied forcing, and also that there should be bursts of rapidly decaying high-frequency oscillations following the shocks for certain parameter values. These predictions were subsequently extended by Byatt-Smith (1988) using asymptotic and numerical methods. Grimshaw & Tian (1994) showed by means of a Melnikov analysis that these multiple oscillations could in fact form sequences of subharmonic bifurcations leading ultimately to chaotic behaviour. Similar predictions have been made by Mackey & Cox (2003) for a forced two-layer configuration. Recently the three-dimensional behaviour that arises at large response amplitudes in square tanks containing relatively deep layers has been considered by Faltinsen et al. (2003, 2005).
The main aim of the present study was to provide further experimental data for comparison with theoretical predictions relating to the particular case of shallow two-dimensional layers that are forced with small amplitudes at frequencies close to fundamental resonance. This was achieved by means of a mechanical rig that was driven by computer-controlled stepper motors. The rig could accommodate a variety of liquid-filled tanks, but for the present purpose attention was restricted to a square geometry. The qualitative aspects of the free surface displacement were determined by eye and video recording. More detailed quantitative measurements were performed non-invasively using an image processing technique that allowed time series of the free surface displacement at a particular location to be measured and recorded. For a given layer depth and forcing amplitude, the free surface response grew and then diminished as the forcing frequency passed through the fundamental resonance. Results will be presented that confirm established theoretical predictions relating to the hysteretic nature of this resonant response and the reversal of the direction of hysteresis as the layer depth is varied through a critical value. Furthermore, a connection between the resonant hysteresis and the critical conditions for the existence of secondary oscillations will be shown for the first time.
2. Theoretical background
Following the procedure outlined by Ockendon et al. (1986), it is pertinent to consider the case of two-dimensional motion in a tank of width πl where the depth of the undisturbed liquid layer is hl. The tank oscillates horizontally with frequency ω and amplitude a. Taking the length scale of the problem as l leads to dimensionless parameters for the forcing amplitude and frequency that are ϵ=a/l and δ=−1+(lω2 coth h)/g, respectively. The parameter δ is a detuning relative to the fundamental resonant frequency of the layer. By restricting attention to forcing amplitudes such that a≪l and to forcing frequencies close to the fundamental frequency, both ϵ and δ can be considered as small parameters. If attention is further restricted to the case of shallow fluid layers then h is also a small parameter of the problem.
The parameters ϵ, δ and h are appropriately related in terms of the Korteweg–de Vries scalings, in which κ=hϵ−1/4 and λ=δϵ−1/2 are both O(1) (Ockendon & Ockendon 1973). This is strictly only true for the h=O(1) case studied by Moiseyev (1958), but Ockendon et al. (1986) show that the vertical spatial coordinate, the velocity potential and the surface elevation can all be rescaled to allow the same scalings as above to hold true in the shallow-layer case under consideration here.
The assumption is made that the fluid motion in response to the forcing is both inviscid and irrotational. Thus solutions for the velocity potential satisfying the Laplace equation can be sought in terms of appropriate asymptotic expansions involving the various small parameters of the problem. The details of this procedure are given by Ockendon et al. (1986) and references therein. The pertinent result as far as the present study is concerned is that the dimensionless surface elevation ϵη of the fluid layer relative to the undisturbed level is given by(2.1)The function G in (2.1) is a 2π-periodic solution of the equation(2.2)which emerges from the asymptotic analysis. The conditions that pertain in (2.2) are that(2.3)(2.4)One particular regime in which solutions to (2.2) can be obtained without undue difficulty is the case of zero dispersion when κ=0. This corresponds to the shallow-layer limit in which h→0. Equation (2.2) then reduces to(2.5)The interpretation of (2.5) was considered in detail by Chester (1964) in the context of resonant oscillations in closed tubes of gas, and again by Chester (1968) in the present context of horizontally forced surface waves. The trivial case is when c≥1, which yields regular oscillatory solutions that are continuous (in the sense that G remains real) for all values of t. It can be shown from (2.3) that c≥1 corresponds to the condition that . Thus a prediction is that for values of the detuning parameter δ=ϵ1/2λ sufficiently above or below resonance (δ=0), the motion of the free surface should correspond to regular standing wave behaviour. This is shown schematically in figure 1 for regions of parameter space above and below the central region.
The case of c<1 (and hence ) in (2.5) is highly non-trivial, as was shown by Chester (1964, 1968). A full discussion of the appropriate interpretation in this range is beyond the scope of this paper. In essence, as summarized by Ockendon et al. (1986), the solutions now involve compressive shocks that serve as the mechanism to overcome what would otherwise be a breakdown in continuity of (2.5) for certain values of t. The birth of these shocks can be glimpsed by means of an asymptotic analysis when |λ| is close to its critical value. This involves introducing a small dissipative term to balance the effect of forcing, whereupon the variation of G with t is seen to develop large values of G′ in proximity to where continuity would otherwise break down. These shock solutions are shown in figure 1 within the central delineated region.
The experimental results presented in §4 will be used to illustrate the crucial qualitative differences in the motion of the oscillating free surface depending on the value of the parameter λ. In order to compare the theoretical predictions described above with the experimental configuration used in the present study, it is necessary to translate certain parameters and results into equivalent terms based on measurable experimental quantities. Consider a rectangular tank of dimensional width L that contains a fluid layer of dimensional depth H in the quiescent state. The tank is forced horizontally in the direction of the width L by a displacement that is given by A cos Ωt. Then the dimensionless parameters ϵ and δ for forcing amplitude and detuning respectively are given by ϵ=πA/L and δ=−1+(Ω/Ω0)2, where Ω0 is the measured natural frequency of free oscillations of the fluid layer of depth H.
The detuning scaling λ=δϵ−1/2 used in the theoretical analysis is obtained by substituting the above expressions for ϵ and δ. The critical values of λ that separate continuous and discontinuous solutions of (2.5) are those that satisfy the condition(2.6)In terms of the experimental parameters, (2.6) translates into a locus of critical points in the parameter space of the experiment that is given by(2.7)Part of this locus is shown in figure 1 for 0≤A/L≤0.016 (0≤ϵ≤0.05). Solutions of (2.5) remain regular outside the quasi-parabolic region bounded by (2.7) but develop shock-like discontinuities for parameter values inside this region. In the experiment, these discontinuities will be smoothed out by the action of viscosity and surface tension to become solitary waves of elevation propagating on the free surface in the form of undular bores, as discussed in §1 and illustrated in §4.
3. Experimental details
The experimental configuration is shown in figure 2. A cubic perspex tank of interior length 200.0±0.5 mm sat on a table that could move horizontally on linear bearings. Oscillatory motion of the table was driven by a Scotch yoke mechanism rotated by a stepper motor via a 10 : 1 reduction gearbox. The table in fact consisted of two translating decks that could move perpendicular to each other. The lower deck incorporated a crank mechanism to allow perturbed sinusoidal forcing if required (see figure 2). However, in this study only the pure sinusoidal motion of the upper deck was used; the additional perpendicular component remains available for future experimental investigations using this apparatus.
The frequency of the sinusoidal translation of the table could be set in increments of 0.005 Hz in the range between 0 and 2 Hz with a stability of better than 0.2%. The amplitude of translation was set manually by means of a lead screw running through a radial offset bearing. One turn of the screw corresponded to a change of forcing amplitude of approximately 0.35 mm. The stroke amplitude was preset before each experiment and was measured using a dial test indicator to an accuracy of 0.01 mm in the range between 0 and 15 mm.
The working fluid that was used in all the experiments reported here was water at room temperature. A small quantity (approx. 0.001% per volume) of wetting agent was added to reduce surface tension and to increase the mobility of the contact lines on the inner surfaces of the tank. A further quantity (approx. 0.1% per volume) of water-based food dye was added to improve the visual contrast between the water layer and the ambient background.
The fluid motion was recorded using a video camera that operated at a rate of 25 fps. The camera was mounted in the laboratory frame of reference in order to avoid the additional weight that would have been incurred in mounting the camera rigidly on the oscillating table. The maximum stroke length used in the experiments was approximately 5 mm, which resulted in negligible changes of viewing perspective. A diffuse light source was positioned on the opposite side of the tank from the camera, giving strong contrast in the field of visualization. Qualitative observations of the fluid motion were made from both the front and the sides of the tank. Additionally, quantitative recordings were made of the advancing and retreating contact line on the front face of the tank. This was done using a frame grabber card in conjunction with a computer. A single vertical column of image pixels was extracted on a frame-by-frame basis from a section of the recording, and for each frame the location of the contrast change between the relatively dark fluid region and the light background was noted. The spatial resolution achieved in the vertical direction using this method was approximately 0.2 mm and the temporal resolution was 0.04 s.
(a) Linear oscillations
A test of the experimental set-up was performed by investigating the extent of agreement between the fluid motion at small oscillation amplitudes and the predictions of linear theory with regard to the natural frequency of oscillation of the fundamental planar oscillatory mode. This was done by setting the fluid layer in oscillation with an impulsive motion of the oscillating table, which was then stationary subsequently. After the initial transient response of the fluid layer was judged to have decayed, the time taken for 20 oscillations of the forward contact line was measured using a handheld stopwatch. This was repeated three times for dimensional depths between 10 and 120 mm in intervals of 5 mm. The undisturbed depth was measured from a transparent grid scale graduated in 1 mm intervals that was attached to the forward side of the tank.
The results of this configuration test are shown in figure 3. The points are the experimental values of frequency recorded at each of the layer depths; experimental errors are within the radial extent of the plotting symbol. The solid line is the prediction of linear theory based on the standard relationship between angular frequency Ω0 and layer depth H for a wave of wavelength 2π/k under gravitational acceleration g. The fundamental oscillatory mode has a wavelength of 2L, where L is the horizontal tank dimension equal to 200 mm in this case. The excellent agreement between theory and experiment as displayed in figure 3 gives support to the assumption that the experimental arrangement as described above is a good and proper realization of the shallow-layer fluid oscillator whose nonlinear resonant properties are to be investigated.
(b) Resonant hysteresis
Before considering the nonlinear effects that occur in the case of the shallowest layers for which , it is instructive to first look at the behaviour of deeper layers to ascertain the nature of the resonant response for a fixed forcing amplitude as the forcing frequency is varied through the fundamental resonance.
The amplitude response of layers of depth H/L=0.1, 0.2, 0.3 and 0.4 was investigated experimentally in the present system. The tank was forced to move with amplitude A/L=0.0026 in the frequency range Ω/Ω0=0.8–1.2. The frequency was changed in incremental steps and, after each change, the system was allowed to come to equilibrium before the peak-to-peak amplitude W of oscillation of one of the spanwise contact lines was measured using the video technique as described in §3. The measuring location was at the midpoint across the tank. The results of these experiments are plotted in figure 4 in terms of the oscillation amplitude relative to the layer depth. Measurements were made with both increasing and decreasing increments of the forcing frequency to capture any hysteresis that existed in the response curves. Measurements were made only if the free surface remained two dimensional as evidenced by the contact line remaining flat across the width of the tank. As will be discussed in §4e, there were parameter regimes in which the free surface developed three-dimensional patterns of oscillation. This would then have meant that the oscillation amplitude measured using the present technique would have become dependent on the spanwise measuring position.
The theoretical prediction (Ockendon & Ockendon 1973; Waterhouse 1994) is indeed for a change in resonant response from ‘hard-spring’ to ‘soft-spring’ behaviour as the layer depth is increased. This terminology relates to the hysteresis that is frequently encountered in the graph of amplitude response against forcing frequency for oscillatory mechanical systems. For shallower layers, the theoretical prediction is for the amplitude response to increase monotonically as the forcing frequency is increased towards the fundamental resonance. A hysteretic response should then be encountered over a frequency range whose values are greater than the natural frequency. Beyond a certain critical depth this behaviour reverses, and hysteresis is encountered at values less than the natural frequency. The critical depth H* is predicted as H*/L=0.33741. At this depth the resonance curve becomes symmetric about the natural frequency, just as in the case of resonance in simple harmonic systems.
The resonance plots in figure 4 confirm the predicted change in hysteresis behaviour with increasing layer depth. Although the datasets for H/L>0.1 are not complete due to the onset of three-dimensional effects, it is still possible to observe the change in orientation about the fundamental frequency. The results suggest that the critical depth occurs at H*/L≈0.3, which is in line with the value predicted theoretically.
(c) Two-dimensional secondary oscillations
In order to assess the effect on the two-dimensional free surface motion of the discontinuous solutions that are predicted theoretically, a series of experiments was performed in which the forcing amplitude was fixed and the forcing frequency was varied in stages across the delineated region of parameter space shown in figure 1. Initially, a sequence of still images taken from video recordings of the fluid motion is shown in figure 5. The fluid layer was viewed perpendicularly to the direction of oscillation and in each case the forcing amplitude was fixed at A/L=0.01. The forcing frequencies were chosen to form a vertical set of equally spaced sample points in the parameter space of the system as represented by figure 1. The first and last images in figure 5 were recorded at points just outside the quasi-parabolic region that corresponds to shocks in the solution of (2.5). The other images were recorded at points inside this region of parameter space. The forcing frequency increases from figure 5a–g.
Starting at figure 5a, the observed motion of the fluid layer was a simple oscillation that was in phase with the applied oscillatory forcing. The contact lines at either end of the tank rose and fell in a regular sinusoidal manner with a relative phase difference of π, while the midpoint of the free surface remained at the ambient depth level. When the frequency was increased to the value corresponding to figure 5b a definite qualitative change in the fluid motion was observed to occur. Although the overall sinusoidal oscillation that had been observed previously was still evident, there was an additional feature that resembled ripples propagating on the free surface. Both the amplitude and wavelength of this new motion were very much smaller than those of the basic oscillation, and hence it is difficult to discern in figure 5b. The difference was most pronounced at the left and right contact lines where a rapid ‘flutter’ could be seen in the motion, especially during the maximal phase of the basic oscillation. The next increase in frequency resulted in both the amplitude and the wavelength of this secondary oscillatory component increasing, as can be seen in figure 5c. The free surface then resembled an undular bore consisting of three small local maxima that travelled at the same speed across the tank and which were reflected from the lateral walls. The subsequent progression with increasing forcing frequency was to a reflected bore with first two (figure 5d) and then one (figure 5e) local maximum travelling along the free surface. Once the frequency had been increased to the penultimate value close to the upper limit of the delineated region in figure 1, the motion of the free surface had the definite appearance of an individual solitary wave travelling backwards and forwards across the tank, as shown in figure 5f. The final increase in forcing frequency took the system out of the quasi-parabolic region in figure 1, and the free surface then reverted to the regular sinusoidal motion of the first frequency in the sequence discussed above. The only difference was that the fluid oscillation was then out of phase with the forcing oscillation by a factor of π.
The development of the non-trivial secondary oscillations described above is further illustrated by the sequence of time-series plots shown in figure 6. These were recorded at the centre of one of the horizontal spanwise contact lines using the video digitization method as described in §3. The time series that are shown were recorded at the same forcing amplitude and frequencies that were used to obtain figure 5b–f. Time series of the regular oscillatory behaviour associated with figure 5a,g are trivial and are omitted here. The progression of the secondary component from relatively small ripple-like motion to large soliton-like behaviour can be seen again in figure 6. What can also be seen very clearly is the synchronization between the underlying oscillation and the secondary component. The two behaviours are slaved to the imposed forcing, but their relative contributions to the overall fluid motion change dramatically as the forcing frequency is varied across the delineated region of parameter space.
The above observations of a developing sequence of secondary oscillations superposed on the basic fundamental mode of the liquid layer offer experimental support for the theoretical prediction of Ockendon et al. (1986) that for forced shallow layers in the limit of vanishing dispersion there should be multiple oscillations triggered periodically by the shock-like discontinuities in (2.5) for . Furthermore, the time series shown in figure 6 bear a striking resemblance to some of those calculated numerically by Grimshaw & Tian (1994), while investigating periodic forcing of a modified form of the Korteweg–de Vries equation. In that case, it was shown that the system was in fact undergoing a sequence of bifurcations leading to higher harmonic modes and eventually to chaotic dynamics. The present observations could indicate that a similar mechanism is present in the experiment, although more investigation is required before any further conclusions can be drawn. Most likely the emergence of any chaotic behaviour would be complicated by three-dimensional effects (see §4e).
(d) Parameter space results
The above qualitative observations of the onset and termination of secondary oscillations associated with parametric variation relative to the locus shown in figure 1 raise the question of the extent of quantitative agreement between the theoretical predictions of critical parameter values and those measured experimentally. This aspect was investigated in the present study using layers of depth H/L=0.05, 0.075 and 0.1. The results are plotted in figure 7. The experiments were performed by first setting the forcing amplitude A to a particular value, and then increasing the forcing frequency Ω gradually from 0. The motion of one of the spanwise contact lines was observed until the onset of the ripple-like behaviour described above was detected. The forcing frequency was then gradually decreased until the secondary oscillation was judged to have disappeared. This alternating increase and decrease of the forcing frequency was repeated and successively refined until the critical value for the first appearance of secondary ripples could be estimated with a relative error of no more than 2%. This was the practical limit of measurement of this particular transition given the very subtle distinction between the oscillatory behaviour of the fluid layer before and after the change occurred.
Once the lower frequency limit for secondary oscillations had been located, the forcing frequency was increased further until the well-developed soliton-like behaviour similar to that shown in figure 5f became established. The frequency was then increased in small relative increments until the motion of the free surface was seen to revert back to the regular small-amplitude oscillatory behaviour similar to figure 5g. The frequency was then reduced incrementally until the soliton-like feature returned, and as before the process was repeated iteratively until a chosen accuracy was obtained. In this case, the significant qualitative differences between the oscillatory behaviour before and after the transition allowed the critical forcing frequency to be located with a relative error of approximately 0.5%. The improvement in accuracy was also facilitated by the fact that there was generally a significant directional hysteresis in the transition with variation of forcing frequency. This was to be expected given the orientation of the resonance curve shown in figure 4a for H/L=0.1. Values for the upper and lower frequencies of the transition were measured in each case.
The experimental results plotted in figure 7 are generally in very good agreement with the theoretical prediction given by (2.7). The lower branch of the secondary region is tracked initially for all three layer depths as the forcing amplitude is increased, with the eventual departure occurring at higher amplitudes as the layer is made deeper. The upper branch now assumes the role of predicting not the specific value of reverse transition but rather the mean value based on the upper and lower limits of stability of the nonlinear resonant hysteresis. Here too the agreement becomes better over a wider range of forcing amplitude as the depth of the layer is increased.
(e) Three-dimensional secondary oscillations
An example of the three-dimensional oscillatory behaviour that was encountered during the present study is shown in figure 8. An initially undisturbed layer of depth H/L=0.2 is shown in figure 8a, where the perspective is parallel with the direction of forcing. The amplitude was set at A/L=0.0026 and the forcing frequency was increased gradually from 0 to Ω/Ω0=1.037. This corresponds to the system travelling along the left-hand branch of the resonance curve shown in figure 4b. The behaviour that was observed when the system was configured in this manner is shown in figure 8b,c. These images were taken at successive times of closest proximity between the tank and the camera, when the front contact line had risen up the presenting wall of the tank. The contact line was no longer flat across the width of the tank, but had developed a pronounced spiked pattern whose phase was found to alternate regularly between successive forcing cycles as shown. This behaviour was stable and persisted for as long as the forcing was maintained. The corresponding patterns were also seen on the opposite contact line at the apogee of the receding forcing stroke. During this phase, the front contact line was relatively flat but its motion was characterized by a burst of very rapid oscillation of small and strongly decaying amplitude. This may well be an example of the behaviour hypothesized by Ockendon et al. (1986) as discussed in §1.
The present study has demonstrated once again the extent of interesting and challenging phenomena associated with nonlinear resonance in fluid dynamics. Even the relatively simple situation considered here of a uniform layer of water in a square container that oscillates sinusoidally along an axis parallel with two sides of the tank is rich in behavioural detail. The experiments that were carried out to investigate the nonlinear resonant response were distinct in the application of a non-invasive measuring technique to obtain quantitative information describing the motion of the free surface. This is particularly significant given the emphasis on shallow layers, where the flow is especially susceptible to disturbances that would otherwise be introduced by invasive measuring devices.
Of the results that have been presented, the most significant is the experimental determination of the locus in parameter space that separates the regular oscillatory regime from that of nonlinear secondary oscillations. What is now clear is that although the theoretical prediction of this locus is indeed accurate, the upper branch does not correspond to a simple reversible transition between the two behaviours but rather to the location of a region of hysteretic exchange. The results also indicate that the agreement between theory and experiment extends to higher forcing amplitudes as the depth of the relatively shallow layer is increased. This is somewhat surprising given that the theoretical prediction is obtained from an asymptotic analysis that invokes the smallness of the forcing amplitude and layer depth. It suggests that further investigation in this regime is called for.
The fact remains that the majority of unresolved issues relating to nonlinearly resonant fluid layers centre around the emergence and development of three-dimensional oscillatory patterns on the free surface. It is hoped that the results of the present study will serve as indicators to future investigations of this important aspect of the problem.
The author is grateful to H. Ockendon and J. R. Ockendon for their extremely useful discussions. This work was funded by the Royal Society under its University Research Fellowship programme.
One contribution of 6 to a Theme Issue ‘Experimental nonlinear dynamics II. Fluids’.
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