1. Introduction
Positive surges are frequently observed in both artificial and natural channels. In channels used for irrigation and water energy production, a positive surge may be created by a partial or complete closure of a gate, causing a sudden increase in flow depth [
1,
2]. In natural channels, tidal bores are characteristic of the rivers and estuaries of Europe, China, Australia, and South America, such as the Seine, Garonne, Severn, Elbe, Qiantang, and Amazon, while Tsunami-induced bores were also observed [
3]. The waves associated with a tidal bore propagate upstream, rapidly increasing the free-surface profile as the tidal flow turns to rising. Bores are formed when the tidal range exceeds 4.5 to 6 m, and the tidal wave is amplified by the funnel shape of the river mouth and the lower estuarine zone [
3]. Despite the fact that they are sometimes seen as a tourist attraction, tidal bores can be very dangerous as they can adversely impact the local natural ecosystems (large sediment resuspension, impairment of aquatic organisms and fish reproduction and development), severely damage local infrastructures (bridges, roads, levees, etc.), and more generally hinder the development of local resources [
3].
In laboratory flumes, positive surges are studied upon the Froude similitude, because field and laboratory data demonstrated that the characteristics of a surge are related to its inflow Froude number
Fr1. If the Froude number ranges from 1 to approximately 1.5, the surge is followed by a train of quasi-periodic secondary waves and is called
undular [
3]. On the other hand, a maximum in wave amplitude and steepness was found for
Fr1 = 1.4 to 1.5, which was associated with some breaking and air entrainment at the first wave crest. If the Froude number is larger, the surge has a breaking front with a roller, and hence it is termed a
breaking or
weak surge [
3].
After the first pioneering studies [
4,
5,
6], several researchers investigated the positive surges and tidal bores [
7,
8,
9,
10,
11]. While older studies were limited to visual observations and sometimes free-surface measurements [
12,
13,
14], in the last two decades unsteady turbulence, air entrainment, and sediment transport characteristics were measured using particle image velocimetry (PIV) and acoustic Doppler velocimetry (ADV) in the laboratory [
15,
16,
17] and in the field [
18,
19]. Finally, numerical studies on tidal bores, both with Large Eddy Simulation (LES) and Smoothed Particles Hydrodynamics (SPH) approaches, focused on the prediction of unsteady flow free-surface dynamics and velocity distribution, where the requirement for high-quality datasets for the validation of the numerical results remains imperative [
20,
21,
22].
Turbulent flows are defined and characterized by irregularity, diffusivity, large Reynolds numbers, 3D vorticity fluctuations, and dissipation [
23]. Irregularity or randomness cannot be measured using a deterministic approach, but it can only be reliably measured using statistical methods [
23]. Complexity measures were previously used for the extraction of information, such as environmental time series (cosmic rays, solar and UV radiation) [
24], biomedical signals [
25,
26], testing of random number generators, etc., from data. In the last decade, such measures, namely the Kolmogorov complexity, were applied to study the randomness of turbulent environmental fluid mechanic (EFM) flows [
27], such as the flow series and regimes [
28,
29] and different types of open channel flows [
30,
31,
32].
This paper presents the results of the application of Kolmogorov complexity (KC) and Kolmogorov complexity spectrum (KCS) to the experimental velocity data of both an undular and breaking surge. The study aims at identifying how those two metrics of complexity can improve our current knowledge about positive surges. The manuscript is structured as follows. After some remarks about randomness, the Kolmogorov complexity and spectrum, as well as the concept of Information, are presented. Second, a short description of the laboratory experiments carried out to collect the velocity data and the procedure used to post-process such data to derive KC and KCS are presented. Then, some basic observations about the flow field during the surge passage are provided. After that, the results from this study in terms of KC and KCS are presented and discussed to identify some novel insights and limitations of the present research, and, finally, some conclusions are proposed.
2. Some Remarks about Randomness, Kolmogorov Complexity and Information
While a comprehensive mechanical theory of turbulence is still missing [
27], the term
randomness that is broadly used in different sciences dealing with fluids and randomness is also one of the fundamental characteristics of a turbulent flow.
While Ichimiya and Nakamura reviewed several definitions of randomness [
33], Khrennikov [
34] further developed the original definition of randomness proposed by Kolmogorov. He outlined three different interpretations of randomness: (i) randomness as unpredictability; (ii) randomness as typicality; and (iii) randomness as complexity. He noted that: “As we have seen, none of the three basic mathematical approaches to the notion of randomness (based on unpredictability, typicality, and algorithmic complexity) led to a consistent and commonly accepted theory of randomness”. Among these interpretations, only that from Kolmogorov is based upon the concept of complexity, but this viewpoint is not always accepted in our understanding of probability. Furthermore, Kolmogorov complexity is not even computable. Therefore, randomness is either a subjective measure or an objective measure that is non-computable. Following this discussion, we might conclude that randomness is not a mathematical notion, but rather a physical notion. Namely, it is the physical procedure where the true randomness is hidden. Therefore, mathematical methods might not be sufficient to theoretically establish the concept of randomness itself. Notably, in Kolmogorov’s approach there is “no room” in which the
level of randomness can be placed.
Many scientific analyses use the term “algorithmic” randomness, which is directly related to the definition of complexity proposed by Kolmogorov [
35]. Such metrics can be quantified by their algorithmic complexity, which is a measure of how long an algorithm would take to complete given an input of size
n. This time bound should be finite and practical even for large values of
n. Hence, complexity is calculated asymptotically as
n approaches infinity as a measure of randomness.
It should be noted that in physical and engineering sciences, scientists often apply a
heuristic technique to choose a model. To formulate a heuristic, we consider any approach to problem solving that uses a practical method that is not guaranteed to be optimal, but it is sufficient either to rapidly reach a goal or until a better approach is developed [
36].
Complexity measures, such as Kolmogorov complexity and its derivatives, are
information measures, i.e., they stem from algorithmic
information theory, where
information is broadly defined as the
pattern of organization of mass and energy [
37]. This definition relates the concept of information to any
process of transfer of mass, momentum, or energy in any fluid and across any environmental interface [
27]. Inherently, the concept of information includes all patterns of organization of matter and energy in the brains and bodies of human beings and animals. This information comes up from their genetic heritage and is further created by their interaction with the external and inner worlds and later recorded in their sensory, nervous, and biochemical systems. Thus, our subjective understanding of the world, which is embedded in our minds and feelings, can be regarded from the outer as a body of information as having that pattern of organization [
37]. Interestingly, peripheral nerve fibers and neural pathways dedicated to conveying information from the body’s interior to the brain end in their own dedicated region, the insular cortex, whose activity patterns are perturbed by emotions [
38].
6. Discussion
The analysis of KC and KCS was addressed on six specific questions: (1) Does KC present a recognizable pattern over the depth in both SS and US flow conditions? (2) Is there any difference in the pattern between the undular surge and the breaking surge? (3) Is there any difference in the pattern for the longitudinal velocity Vx and the transverse velocity Vy? (4) Is there any difference between the undular and the breaking surge in terms of KCS distribution? (5) If compared with Reynolds normal stresses, does the KC vertical distribution show a similar trend? (6) Do the vertical distributions of Reynolds stresses and the KC provide different, complementary insights about the flow structure of a surge? These questions lead to a more general and broad question: which new knowledge about positive surges could be derived from the calculation of those two metrics of complexity?
To address Questions nn. 1-2-3, the vertical distributions of KC for both the surges and both the velocity components Vx and Vy were analyzed in two flow conditions: before the formation of the surge (steady-state-SS) and during the passage of the surge (unsteady-state-US).
The vertical pattern observed for
Vx, that is that the KC was largest at the bed and decreased as the distance from the bed increases, was found in SS and US for the breaking surge but not for the undular surge, where the KC was almost constant over the depth and generally smaller than that in the breaking surge. This difference in the KC values for the US may be related to the different patterns observed for
Vx in the undular and breaking surge (
Figure 4 and
Figure 5). While in the former
Vx followed an undular pattern (
Figure 4), in the latter
Vx had a sudden decrease to a value which was almost constant but related to the elevation from the bed and also negative for
z/d0 = 0.060 (
Figure 5a,b). Hence, the undular pattern of
Vx resulted in a lower degree of randomness that was unaffected from the distance from the bed (
Figure 8a). On the other hand, in the breaking surge, randomness was found to decrease as the elevation from the bed increased (
Figure 9a). Such a pattern can be explained considering that approaching the interface, turbulent motions become increasingly damped, and only small eddies can develop [
41]. Hence, close to the bed, the flow is dominated by small eddies being random [
42] and contributing as expected to the higher randomness observed at the bed. Far from the bed, the eddies are larger and coherently organized, so they cannot introduce more randomness in the flow. Interestingly, such a decaying pattern of KC with the distance from the bed is consistent with the observations by Mihailović et al. [
30] for the KC in an open channel flow with canopies of different density but only for the water depth among the cylinders (Figure 10 in [
30]).
Afterall, from this analysis, it seems that the Kolmogorov complexity for Vx could be related to both the eddy sizes and to the temporal patterns of the longitudinal velocity (undular or not). While in SS the distance from the bed is the main factor controlling KC vertical distribution, in US the undular pattern, the longitudinal velocity Vx associated with the undular surge resulted in a lower and constant degree of randomness that was unaffected by the distance from the bed. On the other side, the almost constant velocity associated with the passage of the breaking surge resulted in a KC vertical distribution close to that in SS. Hence, comparatively, the vertical distribution of KC highlighted a difference between the undular surge and the breaking surge for Vx (Question n.2).
To address Question n.3, it should be noted that the vertical distribution of the KC for the transverse component Vy had no clear trend, while KC values for the SS both for the undular surge and the breaking surge were generally larger than those in US. At any elevation from the bed, in both SS and US conditions and for both surges, the KC values for Vy were also generally larger than those for Vx.
To address Question n.4, KCS data presented the distribution of KC over the velocity values observed during the experiments. The analysis of the KCS data showed that at each elevation KC peaked for both surges at about the same value of Vx, 0.72 m/s at the bed and 0.86-0.83 m/s about the free surface, but the peak for the breaking surge was larger than that for the undular surge. Hence, the distribution of the KC values for Vx was related both to the elevation from the bed for the KC peak being larger at the free surface and to the surge type (undular vs. breaking). This is consistent with the larger randomness associated with the breaking surge previously pointed out (Question n.2). On the other side, the KC values for Vy peaked and centered about 0.00 m/s, while the largest peak was observed at the bed for both surges. Comparatively, for both surges, while transverse velocity was seen to peak at the bed, longitudinal velocity had the largest KC at z′ = 100.00 mm, while transverse velocity had at the bed a peak in KC larger than that of the longitudinal velocity. In the end, the analysis of KCS revealed some differences between the undular and the breaking surge as well as between the longitudinal and transverse velocity that were not identified through the classical metrics previously applied to a surge.
To address Questions nn.5–6, the vertical distribution of the normal Reynolds stresses and KC were compared. In SS, the normal Reynolds stresses for both surges and for both
Vx and
Vy were seen to peak at the bed and, after a rapid decay, to remain almost constant as the distance from the bed increased. In US, while for the undular surge the normal Reynolds stresses were much larger than those in SS, for the breaking surge such large difference was not observed. Furthermore, while in the undular surge the normal Reynolds stresses increased with the elevation from the bed, as already found in previous studies [
3], in the breaking surge those stresses seemed not to be affected from the distance from the bed. Comparatively, the vertical distribution of normal Reynolds stresses revealed a significant difference between the two surges. However, the vertical patterns observed for the normal Reynolds stresses were different from those related to the Kolmogorov Complexity. Interestingly, while the vertical distribution of both the normal Reynolds stresses and KC for both
Vx and
Vy seemed to not be affected by the passage of the breaking surge, it was the passage of the undular surge that largely modified the vertical distribution of both the normal Reynolds stresses and of KC for
Vx only. Furthermore, the KC vertical distribution provided information about eddy size structures that could not be revealed by the distribution of the normal Reynolds stresses. The larger KC observed at the bed could be related to a flow structure dominated by small eddies being random [
42], while, far from the bed, the eddies are larger and coherently organized, so they cannot introduce more randomness into the flow. During the surge passage, while for the breaking surge the KC vertical distribution was unchanged (
Figure 8a), for the undular surge the KC was almost constant (and lower) over the depth suggesting that the passage of the undular surge modifies the vertical distribution of eddy size. In the end, the comparison between the vertical distribution of the normal Reynolds stresses and KC demonstrated that these metrics could provide different and complementary information about the flow structure of a surge.
In a previous study [
30], the vertical distribution of KC was associated with the integral length scale of turbulence, defined as a measure of the longest correlation distance between the velocity at two points of the flow field. Both parameters, i.e., KC and integral length scale, were observed to peak at the bed and to decay as the distance from the bed increased (Table 3 and Figure 10 in [
30]).
Some characteristic turbulent time scales were derived in the literature from the instantaneous velocity data for positive surges and tidal bores. Chanson and Toi [
43] found that, in the laboratory, for both undular and breaking surges, the dimensionless integral time scales
Tv (
g/
d0)
1/2 for the horizontal and transverse velocity components were similar, probably reflecting some turbulence anisotropy in the order of 0.2–0.25, while for the vertical velocity component they were 0.08–0.1. Notably, the approximate ratio of 2 between the longitudinal and vertical velocity time scales was consistent with the analytical relationship for isotropic turbulence [
44]. Further experiments suggested that, in a breaking surge, integral turbulent time and length scales decreased as the vertical elevations from the bed increased [
45]. For field data collected from the Sélune River, it was found that the dimensionless integral time scales were about 0.1–0.12 for the horizontal velocity component and between 0.04 and 0.06 for the transverse and vertical components of the velocity [
43]. Afterall, the analysis of the turbulent integral scales indicated that the propagation of a surge was an anisotropic process, where the vortical structures had sizes and lifespans in the vertical direction longer than those in the horizontal directions [
45].
Finally, it should be acknowledged that the analysis presented herein is based upon a single repetition of the experiments, and this might represent a limitation of this study. Past studies on unsteady flows and positive surges have demonstrated that the statistical analysis of transient flows is not an easy task [
16,
46], because in highly unsteady transient flows, such as the leading edge of surges, the time scale of the physical processes is often very short, even at a prototype scale [
46]. Hence, Chanson [
47] argued that in laboratory experiments, while a single experiment can properly provide information on qualitative patterns and instantaneous quantities, to derive robust statistical data, the repetition of the experiments and the ensemble statistics are the most reliable approach. It was suggested that a selection of 25 repeats could provide independence in terms of free-surface properties, longitudinal velocity, and average tangential Reynolds stress in monophase flows [
47]. On the other side, a larger number of repeated experiments might be required if more advanced parameters, such as triple correlations, extreme pressure values, and air–water flow characteristics, are investigated [
47]. Chanson [
47] suggested that instantaneous ensemble data are best analyzed using instantaneous medians, quartiles, and percentiles of the data ensemble, which are robust parameters that are insensitive to the presence of outliers. On the opposite hand, ensemble-averaged properties, including root mean square errors, are not robust estimators because they may be biased by outliers and extreme values within small data samples [
47]. Finally, in the field experiments of transient flows, that in most situations cannot be repeated under well-controlled conditions, a Fourier component approach may therefore be the most appropriate statistical analysis [
19,
47].
In the end, as the analysis presented herein was based upon a single repetition of the experiments, future studies on the application of Kolmogorov complexity and its spectrum related to unsteady flows should include a larger number of repetitions.
7. Conclusions
Positive surges are commonly observed both in artificial and natural channels. In rivers and estuaries, a common type of positive surge is the tidal bore. Tidal bores can have relevant and negative impacts on a range of natural and socio-economic resources, such local infrastructures, sedimentary processes in the upper estuary, aquatic organism and native fish species reproduction and development, and more generally on the sustainability of those aquatic systems [
3].
This paper presented the application of Kolmogorov complexity to the 2D (longitudinal and transversal) velocity data collected during the laboratory investigation of a positive surge. Two types of surges were considered: an undular surge and a breaking surge. In both cases, the velocity data were collected during the unsteady flow condition (US) associated with the passage of the surge as well as during the preceding steady-state flow condition (SS). For both surges, the Kolmogorov complexity (KC) and the Kolmogorov complexity spectrum (KCS) were calculated in both flow conditions to identify whether those complexity metrics can provide new knowledge about positive surges.
The analysis of the vertical distribution of the KC in both flow conditions highlighted some interesting features. First, the results showed that the vertical distribution of KC for
Vx in SS is dominated by the distance from the bed as the KC is the largest at the bed and the lowest at the free surface. This trend is consistent with that observed in a past application of KC to a canopy open channel flow [
30] and, even with past literature studies on integral length scales of turbulence in positive surges [
43], suggesting that the relationship between KC and those scales should be further explored. Second, only the passage of the undular surge was able to drastically modify such a vertical distribution of KC resulting in a lower and constant randomness throughout the water depth, while the KC vertical distribution was virtually unaffected by the passage of the breaking surge. Hence the vertical distribution of KC highlighted a difference between the undular surge and the breaking surge for
Vx. Third, while these findings were identified for the longitudinal velocity, the vertical distribution of KC for
Vy had no clear trend, but, for both surges, the KC values in SS were generally larger than those in US. Fourth, the KCS were found to peak at each elevation for both surges at about the same value of
Vx, but the peak for the breaking surge was larger than that for the undular surge. This confirms the larger randomness associated with the breaking surge. Furthermore, the distribution of the KC values for
Vx was related both to the elevation from the bed and to the surge type. On the other hand, the KC values for
Vy peaked at and were centered about 0.00 m/s, while the largest peak was observed at the bed for both surges. Finally, the analysis of the vertical distribution of the normal Reynolds stresses revealed that the passage of the undular surge was able to significantly modify such a distribution, which instead was not affected by the passage of the breaking surge.
The application of the Kolmogorov complexity measures to a positive surge identified a clear difference between undular and breaking surges in terms of the randomness vertical distribution during the passage of the surge providing some novel findings to characterize such intricate hydrodynamic process. In addition, the comparative analysis of the vertical distribution of the normal Reynolds stresses and of the Kolmogorov Complexity demonstrated that both metrics were significantly affected by the passage of the undular surge only, and that they are providing different and, in some sense, complementary information about the flow structure of a surge and, more generally, about the changes observed in the flow as it moves from a steady state to an unsteady state.
However, it is acknowledged that the present study is based upon one single experiment, for both the undular and the breaking surges. As past studies have demonstrated that the analysis of laboratory data from highly transient flows, such as positive surges should be carried out through an ensemble-averaging approach based on 25 repeated experiments [
47]. It is advisable that future studies on the application of Kolmogorov complexity and its spectrum to unsteady flows should be based on repeated experiments.