Towards Micro-vortices Generated by Liquid Water’s Structural Heterogeneity

Towards Micro-vortices Generated by Liquid Water’s Structural Heterogeneity | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Towards Micro-vortices Generated by Liquid Water’s Structural Heterogeneity Arturo Tozzi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3845315/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Turbulence is a widespread phenomenon detectable in physical and biological systems. Examining a theoretical model of liquid water flowing in a cylinder at different Raleigh numbers, we propose a novel approach to elucidate the first stages of turbulent flows. The weakly bonded molecular assemblies of liquid distilled water form a fluctuating branched polymer in which every micro-cluster displays different density. Against the common view of liquid water as an incompressible and continuous fluid, we consider it as a non-homogeneous, compressible medium characterised by density differences. We suggest that the occurrence of transient local aggregates in liquid water could produce the vortices and eddies that are the hallmarks of turbulence. As in a two-fluid model, lighter fluid interacts with heavier fluid as if one of the two were an obstacle. Micro-assemblies of such obstacles might justify the presence of micro-vortices and hence of turbulence. We quantify the local changes in velocity, diameter and density required to engender obstacles to the average flow. Then, we explain how these microstructures, equipped with different Raleigh numbers and characterized by high percolation index, could generate boundary layers that contribute to micro-vortices production. We explore the theoretical possibility that three-dimensional turbulence might originate from micro-vortices, contrary to the common view that three-dimensional turbulence is caused by energy cascades from larger to smaller vortices. We conclude that the genesis of turbulence cannot be assessed in terms of collective phenomena, rather is sustained, among many other factors, by the underrated microscopic inhomogeneities of fluids like liquid water. high-density water molecular dynamics water model percolation theory Figures Figure 1 INTRODUCTION Turbulence is a dissipative phenomenon characterized by mathematically untreatable fluctuating instabilities. Various interacting factors, such as linear dimension, inertial forces, density gradients and mass/heat/momentum random transport provide turbulent flows with intrinsic unpredictability. One of the major difficulties in forecasting turbulent flows is the breakdown of the continuum flow assumption that is a hallmark of the Navier-Stokes equations (Barber and Emerson, 2002 ). The most studied example of turbulence consists of incompressible fluids with constant density impacting solid surfaces, such as airfoils placed in wind tunnels. The more the flow approaches the cylindrical surface, the slower the velocity. The separation of the fluid’s boundary layer from the solid surface leads to unsteady flow conditions and onset of one of the hallmarks of turbulence, i.e., the occurrence of vortices, eddies and swings at different length scales (Sturm et al., 2012 ). Overflowed surfaces are subject to counter-rotating foci, separation and saddle points, extinction and even inversion of the velocity, leading to turbulent flows’ production, redistribution and dissipation (Ma et al., 2020 ). Transition from laminar flow to turbulence produces distinct stages of expanding fluctuations regions (Cerbus et al., 2019 ) where large vortices break up to form smaller ones, locally transferring the kinetic energy in a cascading waterfall devoid of long-range transfers (Kalmár-Nagy and Bak, 2019 ; Ortiz-Suslow and Wang, 2019 ). The common tenet, based on the Kolmogorov’s mean field theory (Kolmogorov 1941 ), states that turbulence’s multiscale properties are governed by fluid viscosity and by the average cascade of kinetic energy transferred from large spatial vortices to small ones (Iyer et al., 2021 ). Though Kolmogorov’s theory predicts that the decaying fluctuations’ statistical properties vary with scale as power laws whose exponents are universal, evidence suggests systematic departures from power-law behaviour calling for theoretical understanding (Küchler et al., 2023 ). Here we point towards a novel fluid dynamic approach to quantify the occurrence of micro-vortices in turbulent flows. Concerning the fluid to be discussed, our choice falls on liquid distilled water, because it consists of a transient dynamical network of fluctuating non-covalent, hydrogen-bonded links (Al-Hamdani and Tkatchenko 2019; Lodish et al., 2000 ). Contrary to the common approach that regards liquid water as an incompressible, isotropic and homogeneous fluid assessable through continuous models (Nakayama 2017 ; Gao et al., 2021 ), we regard liquid water as a compressible fluid. Indeed, water consists of a mixture of low-density water (LDW) and high-density water (HDW) transient assemblies (Muthachikavil et al., 2022 ) that generate patchy network inhomogeneities and micro-variations in local density which can be assessed just through discretized models. Focusing on a simple model of distilled liquid water flowing in a cylinder, we provide numerical simulations that illustrate how the everchanging, amorphous network configuration of liquid water could contribute to generate micro-vortices. In sum, our theoretical account suggests that that the occurrence of transient networks fluctuating between reversible LDW and HDW assemblies might explain crucial features of turbulent flows in liquid water. LIQUID WATER’S TRANSIENT MICRO-ASSEMBLIES AND ENSUING EFFECTS ON TURBULENT FLOWS The first step is to describe and quantify the structural and dynamical properties of liquid water, focusing on the possible occurrence of transient micro-assemblies. Premise: the structural heterogeneity of liquid water . Since hydrogen-bonds are continuously assembled and disassembled, various geometric manifolds have been proposed to describe the water’s branched polymer (Shiotari and Sugimoto, 2017 ). Water polymorphisms have been generally studied in extreme settings (Mariedahl et al., 2019 ) such as supercritical water (Skarmoutsos and Samios, 2016 ), high-pressure crystals in supercooled water (Kim et al., 2009 ; Lin et al., 2018 ), frozen water confined in nanometric slit pores (Koga et al., 2020), nanochannels formed of cubic crystalline phases ( Das et al., 2019 ). Nevertheless, a few studies focused on the micro-structure of liquid water at ambient temperature and pressure. Every water molecule forms a maximum of four hydrogen bonds with the surrounding water molecules, producing a tetrahedral structure (Fanetti et al., 2014 ; Liu et al. 2017 ; Milovanović et al., 2020 ). Thaomola et al. (2017) proposed “short-live” and “long-live” exchange periods with fluctuations in hydrogen bonds’ number from 2 to 6, with the nearest neighbors either “loosely” or “tightly” bound to a central water molecule. According to Shiotari and Sugimoto ( 2017 ) and Formanek and Martelli ( 2020 ), liquid water is composed of assemblies of pentagonal and hexagonal rings, while Liu et al. ( 2017 ) pointed towards a dynamical mixture of tetrahedral molecules and ring-and-chain-like structures that produce a densely connected, spherical core of ≈ 140 water molecules surrounded by a fuzzy zone of ≈ 1800 loosely connected molecules. According to Naserifar et al. (2019), strong hydrogen bonds at room temperature form multibranched polymers consisting of 151 H 2 O molecules per chain, while Ansari et al. ( 2018 ) proposed that density fluctuations in liquid water create regions of empty spaces in the shape of spherical or fractal-like voids. In turn, dos Santos et al. (2002) described at ambient conditions the existence of a giant cluster percolating the whole system and Jedlovszky et al. ( 2007 ) noticed that tree structure of the largest water cluster is dominated by a linear, chain-like arrangement. The so called “two-liquids scenario” theorizes the occurrence in water of two competing local molecular structures characterized by low (LDW) and high local density (HDW). Differences in densities, easier to detect in extreme settings, have been described even in liquid water at ambient conditions (Cheng et al. 2019). HDW is regarded as a high-entropy unstructured state, while LDW is believed to display ordered gaps between the first and second molecular shell (de Oca et al., 2019 ). Camisasca et al. ( 2019 ) suggested that HDW could be formed by chains, while LDW by fused dodecahedra working as templates for tetrahedral fluctuations. It is believed that HDW patches tend to be more tetrahedral (Ansari et al., 2018 ) and to display higher connectivity than LDW patches (Faccio et al., 2022 ). Cheng et al. (2019) identified also a third type of local structure in liquid water, characterized by ultra-high density and more stable hydrogen bonds. In our account, we regard LDW and HDW assemblies as unnoticed “impurities” in the ever-changing three-dimensional structure of liquid water. We conjecture that these impurities might lead to tiny variations in density able to generate micro-surfaces and ensuing local modifications of the average macroscopic flow. It is well-known that the addition of impurities induces strong fluctuations in stable and laminar fluid flows and that tiny amounts of long-chain flexible polymers dissolved in turbulent fluids can drastically change flow properties (Zhang et al., 2021 ). In sum we hypothesize the occurrence in liquid water of tiny micro-assemblies (henceforward MA) able to generate micro-vortices in turbulent flows. Characterized by density different from the average density of the whole liquid, MA could deliver slight variations in fluid velocity compared with the average velocity of the whole liquid. Experimental setting . We aim to theoretically evaluate whether microscopic, weak, non-covalent interactions taking spontaneously place in liquid water might produce macroscopic turbulent flows. We are required to build a simulated system to assess and quantify the structural and dynamical properties of microscopic liquid water. We suggest a system consisting of distilled water at temperature = 283.15 K and pressure = 1 Atm, flowing inside a cylinder of diameter L = 1 meter. Water flow is induced by a force constantly exerted at the proximal cylindrical end. The best available quantitative approach to tackle the complexity of turbulent flows consists of the dimensionless Reynolds number (henceforward Re), which describes the ratio of inertial/viscous forces: Re = \(\frac{{\rho } \text{・}\text{L} \text{・}\text{u}}{{\mu }}\) where ρ is the fluid density (kg/m 3 ), L is the characteristic linear dimension (in this case, the cylinder’s diameter in m), u is the fluid velocity (m/s) and µ is the dynamic (kg/m・s). Re differentiates between laminar (Re 3,000). The cylinder must be smooth and carefully aligned so that turbulent slugs appear naturally at Re > ∼3,000 (Wygnanski and Champagne, 2006 ). Simulations can be performed through freely available calculators, such as, e.g., https://www.omnicalculator.com/physics/reynolds-number . It is well-known that transition to turbulence could be initiated at low Re by introducing modifications in the physical parameters ρ and/or u. In our suggested simulation, the dynamic viscosity (= 0.001308 Kg/m s) is held constant, while the average fluid density (= 999.7 kg/m 3 ), the average fluid velocity (= 0.1) and the cylinder diameter L can be modified ad libitum. Some theoretical results are illustrated in Figure . Summarizing, we suggest that appropriate changes in the parameters L, u, ρ might lead to the formation of MA, i.e., local assemblies of liquid water characterized by physical features such as the Reynolds number that are different when compared with the average flow. Micro-assemblies in liquid water? To assess whether the local features of liquid water could generate MA, a comparison is required between our simulations and the real physical parameters of liquid water. The size of a water molecule is ∼2 Å (magnitude: 1 x 10 − 10 meters), while the size of LDW and HDW patches encompasses short (0.3–0.5 nm) as well as long (1–2 nm) assemblies (magnitude: ∼1 x10 − 9 meters) (Ansari et al., 2018 ). Note that the shorter is the MA diameter L, the more laminar is the flow. This means that slight changes in u and ρ in confined zones with low L may cause local modifications in the Reynolds number ( Figure ). In confined liquid water near ambient temperature and pressure, the average density is LDW = 0.78 g/cm 3 , while HDW = 1.08 g/cm 3 (Nomura et al., 2017 ). Therefore, microscopic patches of different density and length can form tiny islands of laminar flows inside turbulent flows. Even though multiple noncovalent bonds in liquid water at room temperature have an existence briefer than 200 femtoseconds (200 x 10 − 15 seconds) (Lodish et al., 2000 ; Bakó et al., 2013 ; Naserifar et al, 2019), LDW has been found to last ∼half a second at 160 K (Lin et al., 2018 ). Therefore, despite their short life (Camisasca et al., 2019 ), HDW and LDW could produce local changes in density affecting upon local environment’s dynamics (Skarmoutsos and Samios, 2016 ) and leading to macroscopic modifications of chemical and biophysical processes (Fanetti et al., 2014 ). Another factor must be considered when assessing turbulent flows, i.e., the average fluid velocity. Since our framework does not assume that the flow is constant and uniform, every micro-volume could theoretically display different velocity. Summarizing, transient assemblies might occur in liquid water. The physical micro-structure of liquid water displays magnitude parameters that allow the formation of MA. In the sequel, we will see how MA might contribute to modify the mainstream flow, leading to the onset of micro-turbulences. CAN MICROSCOPIC FLUID INHOMOGENEITIES IN WATER SUSTAIN MICRO-VORTICES? Once attained that MA might exist in liquid water, the next step is to assess their potential ability to generate micro-vortices in turbulent flows. We suggest that MA stand for “impurities” that provide micro-obstacles to the main flow and therefore generate vortices. The occurrence of MA’s liquid layers gives rise to boundary layers, i.e., liquid water layers in which the fluid velocity is close to zero in the vicinity of MA. The breakdown pattern in boundary-layer flow bears connection to laminar instability and can be reconstructed using the transition zone’s macroscopic properties like persistence times and transitional intermittency (Vinod and Govindarajan, 2004 ). Changes in drag between the boundary layer and the undisturbed flow might lead to micro-forces acting opposite to the average motion of the fluid. This might produce micro-vortices that can be analytically assessed through the available vorticity equations. When a uniform flow approaches the boundary of a solid body, the fluid particles closest to the body surface describe two simultaneous paths: A path along the x axis, parallel to the body surface. Another path along the y axis, normal to the body surface. When the flow impacts the object along the x axis in the same direction of the current, vorticity equations describe the fluid’s infinitesimal bidimensional surface dx \(\varDelta\) y. The subsequent potential flow is characterized by two stagnation points: an anterior stagnation point r. A posterior stagnation point r + \(\varDelta\) r, where backflow takes place just downstream from the separation front. The vorticity is: Where ω is the total vorticity in a point of the body’s profile surface, \(\varDelta\) r is the local thickness of the fluid layer close to the body surface and is the curvilinear abscissa along the flow profile. At low values of Re, the first slight modifications towards instability take place in the vicinity of the interface. Then, turbulent slugs progressively fill the entire cylindrical cross-section as they proceed downstream, growing in length and decreasing in velocity (Wygnanski and Champagne, 2006 ). Flow evolution is characterized by a phase space trajectory containing many recurrence patterns that convey the interactions among turbulent eddies (Wygnanski and Champagne, 2006 ). The number of self-crossings in each recurrent loop reflects the temporal complexity: a significant number of simpler trajectories have just a few self-crossings, while a small number of complex trajectories contain > 100 self-crossings (Wu 2020 ). In touch with our suggestion that MA could generate macroscopic behaviour, it has been demonstrated that local micro-defects in water layers growing on metal surfaces deeply modify the wetting processes (Gao et al., 2021 ). Could micro-aggregates modify turbulent flows in liquid water? The occurrence of laminar MA inside turbulent flows is easier to achieve than the occurrence of turbulent MA inside laminar flows. As stated above, the smaller the dimension L, the more the flow is laminar. Therefore, very small MAs might be regarded as islands of laminar flow. If the MA dimension is not far from to the average free path of the fluid molecules, the fluid can no longer be regarded as being in thermodynamic equilibrium. This leads to a non-continuum regime that influences velocity profiles, mass flow rates and boundary shear stresses (Barber and Emerson, 2002 ). Therefore, when small obstacles such as MAs are introduced into the main flow, turbulences could be observed far downstream. To affect the formation of macroscopic turbulences, short-lived clusters are required to form a wide network able to interact with the average flow. It is known that large connectivity in high-density assemblies leads to percolation inside three-dimensional water lattices (Timonin 2018 ). Does percolation really occur in water? The answer is positive (Bernabei and Ricci, 2008 ; Strong et al., 2018 ). For instance, percolation transition of hydrogen bond networks has been demonstrated in supercritical water, preferentially at high molecular densities (Jedlovszky et al., 2007 ). Percolation thresholds and cluster size distribution follow a universal power law rule, such that percolation transition occurs when the fractal dimension of the largest cluster reaches the value of 2.53 (Galam and Mauger, 1996 ; Jedlovszky et al., 2007 ). Simulations for liquid water suggest that initially disconnected clusters suddenly produce at cut-off values a large space-filling percolating network, in which just a few disconnected fragments/polygonal closures can be found (Geiger 1978). Bearing in mind that 18,01528 grams of water encompass 6,02214076×10 23 water particles, the number of percolating molecules is very high. The probability of finding a cluster that spans the three-dimensional system reaches 0.65 in liquid water (Oleinikova et al., 2002 ). The occurrence of percolation provides an obstacle to the average flow, considering that the bond energies of liquid water assemblies are done in such way as to hinder the main flow and cause turbulence. The energy of a water’s hydrogen bond is 1–5 kcal/mol, much lower than the energy of ≈ 110 kcal/mol required to break a single covalent bond (Lodish et al., 2000 ). Being the molecular average kinetic energy of ∼0.6 kcal/mol at room temperature, many molecules have enough energy to break the noncovalent bonds (Lodish et al., 2000 ). This means that liquid water flows can be influenced by obstacles made by hydrogen bonds. Summarizing, simultaneous variations of different physical parameters might generate transient micro-zones of laminar flow inside turbulent flows. When these small obstacles are introduced into the main flow, turbulences might be observed far downstream. CONCLUSIONS Turbulence is a widespread phenomenon that can be found in unexpected contexts too, such as, e.g., the intracytoplasmic medium (Fan et al., 2011 ; Kalmár-Nagy and Bak, 2019 ; Beppu et al., 2021 ) and the EEG waves (Sheremet et al., 2019 ). Occurring more rapidly than molecular diffusion, turbulent flows are crucial for rapid mixing and transport in systems dealing with combustion, pollutant/contaminant reduction, etc. Still, studies concerning turbulent flows are plagued by lack of knowledge of the subtending mathematics. We suggest that the occurrence of turbulent flows can be sustained, among other factors, by the underrated, transient microscopic “impurities” occurring in almost all the fluids. Flows are usually deemed incompressible when the Mach number (the ratio of the flow velocity past a boundary to the local speed of the sound) is smaller than 0.3. Nevertheless, we suggest that the fluid in turbulent systems should be considered, contrary to the common belief, inhomogeneous and compressible. Most of the fluids in which turbulent flows arise are not isotropic and homogeneous, rather encompass microscopic, scattered “singularities”, “impurities” or “holes” behaving like seeds that contribute to the generation of turbulence. The assembly of large clusters can be methodologically used to discretize continuous liquids. The Knudsen number determines whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used to model a system (Barber and Emerson, 2002 ). The Knudsen number (Kn) is a dimensionless number defined as follows: Kn = \(\frac{{\lambda }}{\text{L}}\) Where λ is the average free path i.e., the average distance over which a fluid particle travels before changing its direction/energy as a result of collisions with an obstacle. When the Knudsen number is ≥ 1, the average free path is comparable to the length scale of the problem and the continuum assumption of fluid mechanics is no longer a useful approximation. In such cases, statistical methods should be used. Among the countless fluids, here we focus of liquid water, which is chemically characterized by intrinsic inhomogeneity. Liquid water could be regarded as a three-dimensional structure where microscopic local changes in density take place. We hypothesize that local micro-defects in the water’s polymer might contribute to the making and fuelling of turbulent flows. We suggest that crucial features of turbulence in liquid water, i.e., the vortices, might be explained by the occurrence of transient networks of LDW and HDW assemblies. This work has some limitations. The phase spaces where water’s transient assemblies are believed to occur are difficult to explore, leading to the upsetting concept of “water’s no-man’s land” (Lin et al., 2018 ). Despite liquid water can be tackled in terms of a dynamically evolving, fluctuating, branched polymer (Naserifar et al, 2019), a full understanding of its dynamical and structural properties is still lacking (Fanetti et al., 2014 ) due to technical difficulties in gaining experimental information on ultrafast interplay (Tamtögl et al., 2020 ). Weak, non-covalent interactions have been studied just in small molecular complexes, falling short of the macroscopic structural properties (Al-Hamdani and Tkatchenko, 2019) that are typical of complex soft materials such as, e.g., supramolecular aggregates. To make things more complicated, totally different networks topologies and physical interpretations have been provided, depending on how rings have been counted (Das et al., 2019 ; Formanek and Martelli, 2020 ). The heterogeneity of water is generally approached through simulation of molecular dynamics, such as, e.g., conventional QM/MM scheme and ONIOM-XS methods (Thaomola et al, 2017), second-order Møller–Plesset perturbation theory (Liu et al. 2017 ), quantum Monte Carlo, non-canonical coupled cluster theory (Al-Hamdani and Tkatchenko, 2019), modified Louvain algorithm of graph community (Gao et al., 2021 ), topological local (clustering coefficient, path length and degree distribution) and global (spectral analysis) properties (dos Santos et al., 2002; Carreras et al., 2008 ; Steinberg et al., 2019 ), persistent homology methods (Wu 2020 ) and so on. It is unclear whether the water in the liquid state displays randomness or long-range interactions. For instance, dos Santos et al. (2002) suggested that the water network’s behaviour at room temperature is very similar to a Poisson distribution compatible with a random graph. On the contrary, some authors are opposed to purely random arrangements of water assemblies, pointing towards medium- to long-range order (Faccio et al. ( 2022 ). To provide a few examples, Tamtögl et al. ( 2020 ) suggested that the motion of water at the surface of a topological insulator displays signatures of correlated motion instead of Brownian motion, while Ansari et al. ( 2018 ) and Gao et al. ( 2021 ) observed collective translational fluctuations of hydrogen-bonded rings and clusters of water molecules. Concerning percolation approaches to liquid water, the limit is that the percolation threshold cannot be accurately located through the cluster size distribution (Jedlovszky et al., 2007 ). Furthermore, in the assessment of LDW and HDW, the temperature must be taken into account. Since the lower the temperature, the higher the energy linking the hydrogen bonds, increases in temperature upon melting lead to quick drops in the average number of assemblies (Gao et al., 2021 ) and broader ring size distribution (Bakó et al., 2013 ; Naserifar et al, 2019). Therefore, microscopic assemblies in water are not easily indistinguishable beyond the isochore end point of 292 K (Nomura et al., 2017 ). In conclusion, we suggest on a theoretical basis a mechanism explaining the onset of turbulence in liquids, with an emphasis on realistic water. We consider water as a non-homogeneous, compressible medium, characterised by density differences. As in a two-fluid model, lighter fluid interacts with heavier fluid as if one of the two were an obstacle. Micro-assemblies of such obstacles might justify the presence of micro-vortices and hence of turbulence. Declarations Ethics approval and consent to participate: This research does not contain any studies with human participants or animals performed by the Author. Consent for publication: The Author transfers all copyright ownership, in the event the work is published. The undersigned author warrants that the article is original, does not infringe on any copyright or other proprietary right of any third part, is not under consideration by another journal, and has not been previously published. Availability of data and materials: all data and materials generated or analyzed during this study are included in the manuscript. The Author had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. Competing interests: The Author do not have any known or potential conflict of interest including any financial, personal or other relationships with other people or organizations within three years of beginning the submitted work that could inappropriately influence, or be perceived to influence, their work. Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Authors' contributions: The Author performed study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical, and material support, study supervision. Acknowledgements: none. References Al-Hamdani Ys, Tkatchenko A. 2019. Understanding non-covalent interactions in larger molecular complexes from first principles featured. J. Chem. 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Milovanović MR, Živković JM, Ninković DB, Stanković IM, Zarić SD. 2020. How flexible is the water molecule structure? Analysis of crystal structures and the potential energy surface. Phys. Chem. Chem. Phys., 2020,22, 4138-4143. https://doi.org/10.1039/C9CP07042G. Muthachikavil AV, Kontogeorgis GM, Liang X, Lei Q, Peng B. 2022. Structural characteristics of low-density environments in liquid water. Phys. Rev. E 105, 034604. Nakayama K. 2017. Topological features and properties associated with development/decay of vortices in isotropic homogeneous turbulence. Phys. Rev. Fluids 2, 014701. Naserifar S, Goddard III WA. 2019. Liquid water is a dynamic polydisperse branched polymer. PNAS. 116 (6) 1998-2003. https://doi.org/10.1073/pnas.1817383116 Nomura K, Kaneko T, Bai J et al. 2017. Evidence of low-density and high-density liquid phases and isochore end point for water confined to carbon nanotube. PNAS. 114 (16) 4066-4071. https://doi.org/10.1073/pnas.1701609114. Oleinikova A, Brovchenko IV, Geiger A, Guillot B. 2002. Percolation of water in aqueous solution and liquid-liquid immiscibility. The Journal of Chemical Physics 117(7):3296-3304. DOI: 10.1063/1.1493183. Ortiz-Suslow DG, Wang Q. 2019. An Evaluation of Kolmogorov's −5/3 Power Law Observed Within the Turbulent Airflow Above the Ocean. Geophysical Research letters. https://doi.org/10.1029/2019GL085083. Pusztai L. 2000. Comparison between the structures of liquid water and (high- and low-density) amorphous ice. Physical Chemistry Chemical Physics. Issue 12. Sheremet A, Qin Y, Kennedy JP, Zhou Y, Maurer AP. 2019. Are EEG flows turbulent? Front Syst Neurosci. 2018; 12: 62. doi: 10.3389/fnsys.2018.00062. Shiotari A, Sugimoto Y. 2017. Ultrahigh-resolution imaging of water networks by atomic force microscopy. Nat Commun 8, 14313 (2017). https://doi.org/10.1038/ncomms14313ù Skarmoutsos I, Samios J. 2016. Local Density Inhomogeneities and Dynamics in Supercritical Water: A Molecular Dynamics Simulation Approach. J. Phys. Chem. B 2006, 110, 43, 21931–21937. https://doi.org/10.1021/jp060955p. Steinberg L, Russo J, Frey J. 2019. A new topological descriptor for water network structure. J Cheminform 11, 48 (2019). https://doi.org/10.1186/s13321-019-0369-0. Strong SE, Shi L, Skinner JL. 2018. Percolation in supercritical water: Do the Widom and percolation lines coincide? J. Chem. Phys. 149, 084504 (2018); https://doi.org/10.1063/1.5042556. Sturm H, Dumstorff G, Busche P, Westermann D, Lang W. 2012. Boundary Layer Separation and Reattachment Detection on Airfoils by Thermal Flow Sensors. Sensors (Basel, Switzerland). DOI:10.3390/s121114292. Tamtögl A., Sacchi M., Avidor N, Calvo-Almazán I, Townsend PSM, et al. 2020. Nanoscopic diffusion of water on a topological insulator. Nat Commun 11, 278. https://doi.org/10.1038/s41467-019-14064-7 Thaomola S. Tongraar A, Kerdcharoen T. 2012. Insights into the structure and dynamics of liquid water: A comparative study of conventional QM/MM and ONIOM-XS MD simulations. Journal of Molecular Liquids. Volume 174, October 2012, Pages 26-33. Timonin PN. 2018. Statistical mechanics of high-density bond percolation. Phys. Rev. E 97, 052119. Vinod N, Govindarajan R. 2004. Pattern of Breakdown of Laminar Flow into Turbulent Spots. Phys. Rev. Lett. 93, 114501. Wu, H. 2020. The topological features of a fully developed turbulent wake flow. APS Division of Fluid Dynamics (Fall) 2020, abstract id.S09.007. Wygnanski IJ, Champagne FH. 2006. On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. Journal of Fluid Mechanics. Volume 59, Issue 2. On transition in a pipe. Part 1. The origin of puffs. Yang C, Zhang C, Ye F, Zhou X. 2019. Ultra-high-density local structure in liquid water. Chinese Physics B. 28(11): 116104. Zhang Y-B, Bodenschatzhait E, Xu A.Xi H-D. 2021. Experimental observation of the elastic range scaling in turbulent flow with polymer additives. Science Advances. Vol. 7, No. 14. DOI: 10.1126/sciadv.abd3525. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3845315","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":265943349,"identity":"290da725-0ac5-4e2a-aba5-dd322552c28d","order_by":0,"name":"Arturo Tozzi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3ElEQVRIie3PuwrCMBSA4ZRCuwRdI4i+QkTQ0Vc5odBZcHEQDQh2c9a3iRQ6BWfBIoWCc1zEQcTU6iKS6uaQfzkZ8uWCkM32r0E5nCyj4rESpt34SYgeLoUXMRn8nAXxCLzON5GBH+UqmxymdB8nYximrX7EnViZbsECKCQjQrdhuAN67DalqHgYAQHgAaES9zSJ2ap4npkwLuBWkqEmsy9I4HC2KAnSBEglkYmL2BIaa+kFRP+ls8IbLqSB+FGUny5nqNeku1HqmraJP4/V2EA+5fAfgc1ms9neuwN84VMmjm0+/gAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Arturo","middleName":"","lastName":"Tozzi","suffix":""}],"badges":[],"createdAt":"2024-01-08 12:02:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3845315/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3845315/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":49431225,"identity":"2a2442eb-a9af-417e-a37e-4f3dac6d74c0","added_by":"auto","created_at":"2024-01-10 17:31:41","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":622972,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3845315/v1/feb95425a0588dcef235bb1e.png"},{"id":50306810,"identity":"29373e4b-4130-4f62-ab79-ec9adc0eb0e1","added_by":"auto","created_at":"2024-01-29 13:30:32","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":626285,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3845315/v1/cd7f0f9e-9e07-460f-8af8-d9503bda7e97.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eTowards Micro-vortices Generated by Liquid Water’s Structural Heterogeneity\u003c/p\u003e","fulltext":[{"header":"INTRODUCTION","content":"\u003cp\u003eTurbulence is a dissipative phenomenon characterized by mathematically untreatable fluctuating instabilities. Various interacting factors, such as linear dimension, inertial forces, density gradients and mass/heat/momentum random transport provide turbulent flows with intrinsic unpredictability. One of the major difficulties in forecasting turbulent flows is the breakdown of the continuum flow assumption that is a hallmark of the Navier-Stokes equations (Barber and Emerson, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). The most studied example of turbulence consists of incompressible fluids with constant density impacting solid surfaces, such as airfoils placed in wind tunnels. The more the flow approaches the cylindrical surface, the slower the velocity. The separation of the fluid\u0026rsquo;s boundary layer from the solid surface leads to unsteady flow conditions and onset of one of the hallmarks of turbulence, i.e., the occurrence of vortices, eddies and swings at different length scales (Sturm et al., \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). Overflowed surfaces are subject to counter-rotating foci, separation and saddle points, extinction and even inversion of the velocity, leading to turbulent flows\u0026rsquo; production, redistribution and dissipation (Ma et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Transition from laminar flow to turbulence produces distinct stages of expanding fluctuations regions (Cerbus et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) where large vortices break up to form smaller ones, locally transferring the kinetic energy in a cascading waterfall devoid of long-range transfers (Kalm\u0026aacute;r-Nagy and Bak, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Ortiz-Suslow and Wang, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). The common tenet, based on the Kolmogorov\u0026rsquo;s mean field theory (Kolmogorov \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1941\u003c/span\u003e), states that turbulence\u0026rsquo;s multiscale properties are governed by fluid viscosity and by the average cascade of kinetic energy transferred from large spatial vortices to small ones (Iyer et al., \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Though Kolmogorov\u0026rsquo;s theory predicts that the decaying fluctuations\u0026rsquo; statistical properties vary with scale as power laws whose exponents are universal, evidence suggests systematic departures from power-law behaviour calling for theoretical understanding (K\u0026uuml;chler et al., \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eHere we point towards a novel fluid dynamic approach to quantify the occurrence of micro-vortices in turbulent flows. Concerning the fluid to be discussed, our choice falls on liquid distilled water, because it consists of a transient dynamical network of fluctuating non-covalent, hydrogen-bonded links (Al-Hamdani and Tkatchenko 2019; Lodish et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). Contrary to the common approach that regards liquid water as an incompressible, isotropic and homogeneous fluid assessable through continuous models (Nakayama \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Gao et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), we regard liquid water as a compressible fluid. Indeed, water consists of a mixture of low-density water (LDW) and high-density water (HDW) transient assemblies (Muthachikavil et al., \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) that generate patchy network inhomogeneities and micro-variations in local density which can be assessed just through discretized models. Focusing on a simple model of distilled liquid water flowing in a cylinder, we provide numerical simulations that illustrate how the everchanging, amorphous network configuration of liquid water could contribute to generate micro-vortices. In sum, our theoretical account suggests that that the occurrence of transient networks fluctuating between reversible LDW and HDW assemblies might explain crucial features of turbulent flows in liquid water.\u003c/p\u003e"},{"header":"LIQUID WATER’S TRANSIENT MICRO-ASSEMBLIES AND ENSUING EFFECTS ON TURBULENT FLOWS","content":"\u003cp\u003eThe first step is to describe and quantify the structural and dynamical properties of liquid water, focusing on the possible occurrence of transient micro-assemblies.\u003c/p\u003e \u003cp\u003e \u003cb\u003ePremise: the structural heterogeneity of liquid water\u003c/b\u003e. Since hydrogen-bonds are continuously assembled and disassembled, various geometric manifolds have been proposed to describe the water’s branched polymer (Shiotari and Sugimoto, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Water polymorphisms have been generally studied in extreme settings (Mariedahl et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) such as supercritical water (Skarmoutsos and Samios, \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), high-pressure crystals in supercooled water (Kim et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Lin et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), frozen water confined in nanometric slit pores (Koga et al., 2020), nanochannels formed of cubic crystalline phases \u003cb\u003e(\u003c/b\u003eDas et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Nevertheless, a few studies focused on the micro-structure of liquid water at ambient temperature and pressure. Every water molecule forms a maximum of four hydrogen bonds with the surrounding water molecules, producing a tetrahedral structure (Fanetti et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Milovanović et al., \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Thaomola et al. (2017) proposed “short-live” and “long-live” exchange periods with fluctuations in hydrogen bonds’ number from 2 to 6, with the nearest neighbors either “loosely” or “tightly” bound to a central water molecule. According to Shiotari and Sugimoto (\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) and Formanek and Martelli (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), liquid water is composed of assemblies of pentagonal and hexagonal rings, while Liu et al. (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) pointed towards a dynamical mixture of tetrahedral molecules and ring-and-chain-like structures that produce a densely connected, spherical core of ≈ 140 water molecules surrounded by a fuzzy zone of ≈ 1800 loosely connected molecules. According to Naserifar et al. (2019), strong hydrogen bonds at room temperature form multibranched polymers consisting of 151 H\u003csub\u003e2\u003c/sub\u003eO molecules per chain, while Ansari et al. (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) proposed that density fluctuations in liquid water create regions of empty spaces in the shape of spherical or fractal-like voids. In turn, dos Santos et al. (2002) described at ambient conditions the existence of a giant cluster percolating the whole system and Jedlovszky et al. (\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) noticed that tree structure of the largest water cluster is dominated by a linear, chain-like arrangement.\u003c/p\u003e \u003cp\u003eThe so called “two-liquids scenario” theorizes the occurrence in water of two competing local molecular structures characterized by low (LDW) and high local density (HDW). Differences in densities, easier to detect in extreme settings, have been described even in liquid water at ambient conditions (Cheng et al. 2019). HDW is regarded as a high-entropy unstructured state, while LDW is believed to display ordered gaps between the first and second molecular shell (de Oca et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Camisasca et al. (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) suggested that HDW could be formed by chains, while LDW by fused dodecahedra working as templates for tetrahedral fluctuations. It is believed that HDW patches tend to be more tetrahedral (Ansari et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) and to display higher connectivity than LDW patches (Faccio et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Cheng et al. (2019) identified also a third type of local structure in liquid water, characterized by ultra-high density and more stable hydrogen bonds.\u003c/p\u003e \u003cp\u003eIn our account, we regard LDW and HDW assemblies as unnoticed “impurities” in the ever-changing three-dimensional structure of liquid water. We conjecture that these impurities might lead to tiny variations in density able to generate micro-surfaces and ensuing local modifications of the average macroscopic flow. It is well-known that the addition of impurities induces strong fluctuations in stable and laminar fluid flows and that tiny amounts of long-chain flexible polymers dissolved in turbulent fluids can drastically change flow properties (Zhang et al., \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In sum we hypothesize the occurrence in liquid water of tiny micro-assemblies (henceforward MA) able to generate micro-vortices in turbulent flows. Characterized by density different from the average density of the whole liquid, MA could deliver slight variations in fluid velocity compared with the average velocity of the whole liquid.\u003c/p\u003e \u003cp\u003e \u003cb\u003eExperimental setting\u003c/b\u003e. We aim to theoretically evaluate whether microscopic, weak, non-covalent interactions taking spontaneously place in liquid water might produce macroscopic turbulent flows. We are required to build a simulated system to assess and quantify the structural and dynamical properties of microscopic liquid water. We suggest a system consisting of distilled water at temperature = 283.15 K and pressure = 1 Atm, flowing inside a cylinder of diameter L = 1 meter. Water flow is induced by a force constantly exerted at the proximal cylindrical end. The best available quantitative approach to tackle the complexity of turbulent flows consists of the dimensionless Reynolds number (henceforward Re), which describes the ratio of inertial/viscous forces:\u003c/p\u003e \u003cp\u003eRe =\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\rho } \\text{・}\\text{L} \\text{・}\\text{u}}{{\\mu }}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003ewhere ρ is the fluid density (kg/m\u003csup\u003e3\u003c/sup\u003e), L is the characteristic linear dimension (in this case, the cylinder’s diameter in m), u is the fluid velocity (m/s) and µ is the dynamic (kg/m・s). Re differentiates between laminar (Re \u0026lt; 2,100) and turbulent flows (Re \u0026gt; 3,000). The cylinder must be smooth and carefully aligned so that turbulent slugs appear naturally at Re \u0026gt; ∼3,000 (Wygnanski and Champagne, \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). Simulations can be performed through freely available calculators, such as, e.g., \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.omnicalculator.com/physics/reynolds-number\u003c/span\u003e\u003cspan address=\"https://www.omnicalculator.com/physics/reynolds-number\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. It is well-known that transition to turbulence could be initiated at low Re by introducing modifications in the physical parameters ρ and/or u. In our suggested simulation, the dynamic viscosity (= 0.001308 Kg/m s) is held constant, while the average fluid density (= 999.7 kg/m\u003csup\u003e3\u003c/sup\u003e), the average fluid velocity (= 0.1) and the cylinder diameter L can be modified ad libitum. Some theoretical results are illustrated in \u003cb\u003eFigure\u003c/b\u003e.\u003c/p\u003e \u003cp\u003eSummarizing, we suggest that appropriate changes in the parameters L, u, ρ might lead to the formation of MA, i.e., local assemblies of liquid water characterized by physical features such as the Reynolds number that are different when compared with the average flow.\u003c/p\u003e \u003cp\u003e \u003cb\u003eMicro-assemblies in liquid water?\u003c/b\u003e To assess whether the local features of liquid water could generate MA, a comparison is required between our simulations and the real physical parameters of liquid water. The size of a water molecule is ∼2 Å (magnitude: 1 x 10\u003csup\u003e− 10\u003c/sup\u003e meters), while the size of LDW and HDW patches encompasses short (0.3–0.5 nm) as well as long (1–2 nm) assemblies (magnitude: ∼1 x10\u003csup\u003e− 9\u003c/sup\u003e meters) (Ansari et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Note that the shorter is the MA diameter L, the more laminar is the flow. This means that slight changes in u and ρ in confined zones with low L may cause local modifications in the Reynolds number (\u003cb\u003eFigure\u003c/b\u003e).\u003c/p\u003e \u003cp\u003eIn confined liquid water near ambient temperature and pressure, the average density is LDW = 0.78 g/cm\u003csup\u003e3\u003c/sup\u003e, while HDW = 1.08 g/cm\u003csup\u003e3\u003c/sup\u003e (Nomura et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Therefore, microscopic patches of different density and length can form tiny islands of laminar flows inside turbulent flows.\u003c/p\u003e \u003cp\u003eEven though multiple noncovalent bonds in liquid water at room temperature have an existence briefer than 200 femtoseconds (200 x 10\u003csup\u003e− 15\u003c/sup\u003e seconds) (Lodish et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Bakó et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Naserifar et al, 2019), LDW has been found to last ∼half a second at 160 K (Lin et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Therefore, despite their short life (Camisasca et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), HDW and LDW could produce local changes in density affecting upon local environment’s dynamics (Skarmoutsos and Samios, \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) and leading to macroscopic modifications of chemical and biophysical processes (Fanetti et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Another factor must be considered when assessing turbulent flows, i.e., the average fluid velocity. Since our framework does not assume that the flow is constant and uniform, every micro-volume could theoretically display different velocity.\u003c/p\u003e \u003cp\u003eSummarizing, transient assemblies might occur in liquid water. The physical micro-structure of liquid water displays magnitude parameters that allow the formation of MA. In the sequel, we will see how MA might contribute to modify the mainstream flow, leading to the onset of micro-turbulences.\u003c/p\u003e \u003c/div\u003e"},{"header":"CAN MICROSCOPIC FLUID INHOMOGENEITIES IN WATER SUSTAIN MICRO-VORTICES?","content":"\u003cp\u003eOnce attained that MA might exist in liquid water, the next step is to assess their potential ability to generate micro-vortices in turbulent flows. We suggest that MA stand for “impurities” that provide micro-obstacles to the main flow and therefore generate vortices. The occurrence of MA’s liquid layers gives rise to boundary layers, i.e., liquid water layers in which the fluid velocity is close to zero in the vicinity of MA. The breakdown pattern in boundary-layer flow bears connection to laminar instability and can be reconstructed using the transition zone’s macroscopic properties like persistence times and transitional intermittency (Vinod and Govindarajan, \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). Changes in drag between the boundary layer and the undisturbed flow might lead to micro-forces acting opposite to the average motion of the fluid. This might produce micro-vortices that can be analytically assessed through the available vorticity equations. When a uniform flow approaches the boundary of a solid body, the fluid particles closest to the body surface describe two simultaneous paths:\u003c/p\u003e\u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eA path along the x axis, parallel to the body surface.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eAnother path along the y axis, normal to the body surface.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eWhen the flow impacts the object along the x axis in the same direction of the current, vorticity equations describe the fluid’s infinitesimal bidimensional surface dx \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varDelta\\)\u003c/span\u003e\u003c/span\u003ey. The subsequent potential flow is characterized by two stagnation points:\u003c/p\u003e\u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003ean anterior stagnation point r.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eA posterior stagnation point r + \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varDelta\\)\u003c/span\u003e\u003c/span\u003er, where backflow takes place just downstream from the separation front.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e\u003cp\u003eThe vorticity is:\u003c/p\u003e\u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/122228_c8a1650c59388082/122228_custom_files/img1704907175.png\"\u003e\u003cbr\u003e\u003c/p\u003e\n\u003ch3\u003e\u003c/h3\u003e\n\u003cp\u003eWhere ω is the total vorticity in a point of the body’s profile surface, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varDelta\\)\u003c/span\u003e\u003c/span\u003er is the local thickness of the fluid layer close to the body surface and is the curvilinear abscissa along the flow profile.\u003c/p\u003e\u003cp\u003eAt low values of Re, the first slight modifications towards instability take place in the vicinity of the interface. Then, turbulent slugs progressively fill the entire cylindrical cross-section as they proceed downstream, growing in length and decreasing in velocity (Wygnanski and Champagne, \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). Flow evolution is characterized by a phase space trajectory containing many recurrence patterns that convey the interactions among turbulent eddies (Wygnanski and Champagne, \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). The number of self-crossings in each recurrent loop reflects the temporal complexity: a significant number of simpler trajectories have just a few self-crossings, while a small number of complex trajectories contain \u0026gt; 100 self-crossings (Wu \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In touch with our suggestion that MA could generate macroscopic behaviour, it has been demonstrated that local micro-defects in water layers growing on metal surfaces deeply modify the wetting processes (Gao et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e \u003cb\u003eCould micro-aggregates modify turbulent flows in liquid water?\u003c/b\u003e The occurrence of laminar MA inside turbulent flows is easier to achieve than the occurrence of turbulent MA inside laminar flows. As stated above, the smaller the dimension L, the more the flow is laminar. Therefore, very small MAs might be regarded as islands of laminar flow. If the MA dimension is not far from to the average free path of the fluid molecules, the fluid can no longer be regarded as being in thermodynamic equilibrium. This leads to a non-continuum regime that influences velocity profiles, mass flow rates and boundary shear stresses (Barber and Emerson, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). Therefore, when small obstacles such as MAs are introduced into the main flow, turbulences could be observed far downstream.\u003c/p\u003e\u003cp\u003eTo affect the formation of macroscopic turbulences, short-lived clusters are required to form a wide network able to interact with the average flow. It is known that large connectivity in high-density assemblies leads to percolation inside three-dimensional water lattices (Timonin \u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Does percolation really occur in water? The answer is positive (Bernabei and Ricci, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Strong et al., \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). For instance, percolation transition of hydrogen bond networks has been demonstrated in supercritical water, preferentially at high molecular densities (Jedlovszky et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Percolation thresholds and cluster size distribution follow a universal power law rule, such that percolation transition occurs when the fractal dimension of the largest cluster reaches the value of 2.53 (Galam and Mauger, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Jedlovszky et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Simulations for liquid water suggest that initially disconnected clusters suddenly produce at cut-off values a large space-filling percolating network, in which just a few disconnected fragments/polygonal closures can be found (Geiger 1978). Bearing in mind that 18,01528 grams of water encompass 6,02214076×10\u003csup\u003e23\u003c/sup\u003e water particles, the number of percolating molecules is very high. The probability of finding a cluster that spans the three-dimensional system reaches 0.65 in liquid water (Oleinikova et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). The occurrence of percolation provides an obstacle to the average flow, considering that the bond energies of liquid water assemblies are done in such way as to hinder the main flow and cause turbulence. The energy of a water’s hydrogen bond is 1–5 kcal/mol, much lower than the energy of ≈ 110 kcal/mol required to break a single covalent bond (Lodish et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). Being the molecular average kinetic energy of ∼0.6 kcal/mol at room temperature, many molecules have enough energy to break the noncovalent bonds (Lodish et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). This means that liquid water flows can be influenced by obstacles made by hydrogen bonds.\u003c/p\u003e\u003cp\u003eSummarizing, simultaneous variations of different physical parameters might generate transient micro-zones of laminar flow inside turbulent flows. When these small obstacles are introduced into the main flow, turbulences might be observed far downstream.\u003c/p\u003e"},{"header":"CONCLUSIONS","content":"\u003cp\u003eTurbulence is a widespread phenomenon that can be found in unexpected contexts too, such as, e.g., the intracytoplasmic medium (Fan et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Kalm\u0026aacute;r-Nagy and Bak, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Beppu et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) and the EEG waves (Sheremet et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Occurring more rapidly than molecular diffusion, turbulent flows are crucial for rapid mixing and transport in systems dealing with combustion, pollutant/contaminant reduction, etc. Still, studies concerning turbulent flows are plagued by lack of knowledge of the subtending mathematics. We suggest that the occurrence of turbulent flows can be sustained, among other factors, by the underrated, transient microscopic \u0026ldquo;impurities\u0026rdquo; occurring in almost all the fluids. Flows are usually deemed incompressible when the Mach number (the ratio of the flow velocity past a boundary to the local speed of the sound) is smaller than 0.3. Nevertheless, we suggest that the fluid in turbulent systems should be considered, contrary to the common belief, inhomogeneous and compressible. Most of the fluids in which turbulent flows arise are not isotropic and homogeneous, rather encompass microscopic, scattered \u0026ldquo;singularities\u0026rdquo;, \u0026ldquo;impurities\u0026rdquo; or \u0026ldquo;holes\u0026rdquo; behaving like seeds that contribute to the generation of turbulence.\u003c/p\u003e \u003cp\u003eThe assembly of large clusters can be methodologically used to discretize continuous liquids. The Knudsen number determines whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used to model a system (Barber and Emerson, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). The Knudsen number (Kn) is a dimensionless number defined as follows:\u003c/p\u003e \u003cp\u003eKn =\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\lambda }}{\\text{L}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eWhere λ is the average free path i.e., the average distance over which a fluid particle travels before changing its direction/energy as a result of collisions with an obstacle. When the Knudsen number is \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u0026ge;\u003c/span\u003e\u0026thinsp;1, the average free path is comparable to the length scale of the problem and the continuum assumption of fluid mechanics is no longer a useful approximation. In such cases, statistical methods should be used.\u003c/p\u003e \u003cp\u003eAmong the countless fluids, here we focus of liquid water, which is chemically characterized by intrinsic inhomogeneity. Liquid water could be regarded as a three-dimensional structure where microscopic local changes in density take place. We hypothesize that local micro-defects in the water\u0026rsquo;s polymer might contribute to the making and fuelling of turbulent flows. We suggest that crucial features of turbulence in liquid water, i.e., the vortices, might be explained by the occurrence of transient networks of LDW and HDW assemblies.\u003c/p\u003e \u003cp\u003eThis work has some limitations. The phase spaces where water\u0026rsquo;s transient assemblies are believed to occur are difficult to explore, leading to the upsetting concept of \u0026ldquo;water\u0026rsquo;s no-man\u0026rsquo;s land\u0026rdquo; (Lin et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Despite liquid water can be tackled in terms of a dynamically evolving, fluctuating, branched polymer (Naserifar et al, 2019), a full understanding of its dynamical and structural properties is still lacking (Fanetti et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) due to technical difficulties in gaining experimental information on ultrafast interplay (Tamt\u0026ouml;gl et al., \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Weak, non-covalent interactions have been studied just in small molecular complexes, falling short of the macroscopic structural properties (Al-Hamdani and Tkatchenko, 2019) that are typical of complex soft materials such as, e.g., supramolecular aggregates. To make things more complicated, totally different networks topologies and physical interpretations have been provided, depending on how rings have been counted (Das et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Formanek and Martelli, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). The heterogeneity of water is generally approached through simulation of molecular dynamics, such as, e.g., conventional QM/MM scheme and ONIOM-XS methods (Thaomola et al, 2017), second-order M\u0026oslash;ller\u0026ndash;Plesset perturbation theory (Liu et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), quantum Monte Carlo, non-canonical coupled cluster theory (Al-Hamdani and Tkatchenko, 2019), modified Louvain algorithm of graph community (Gao et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), topological local (clustering coefficient, path length and degree distribution) and global (spectral analysis) properties (dos Santos et al., 2002; Carreras et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Steinberg et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), persistent homology methods (Wu \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) and so on.\u003c/p\u003e \u003cp\u003eIt is unclear whether the water in the liquid state displays randomness or long-range interactions. For instance, dos Santos et al. (2002) suggested that the water network\u0026rsquo;s behaviour at room temperature is very similar to a Poisson distribution compatible with a random graph. On the contrary, some authors are opposed to purely random arrangements of water assemblies, pointing towards medium- to long-range order (Faccio et al. (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). To provide a few examples, Tamt\u0026ouml;gl et al. (\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) suggested that the motion of water at the surface of a topological insulator displays signatures of correlated motion instead of Brownian motion, while Ansari et al. (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) and Gao et al. (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) observed collective translational fluctuations of hydrogen-bonded rings and clusters of water molecules. Concerning percolation approaches to liquid water, the limit is that the percolation threshold cannot be accurately located through the cluster size distribution (Jedlovszky et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Furthermore, in the assessment of LDW and HDW, the temperature must be taken into account. Since the lower the temperature, the higher the energy linking the hydrogen bonds, increases in temperature upon melting lead to quick drops in the average number of assemblies (Gao et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) and broader ring size distribution (Bak\u0026oacute; et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Naserifar et al, 2019). Therefore, microscopic assemblies in water are not easily indistinguishable beyond the isochore end point of 292 K (Nomura et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2017\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eIn conclusion, we suggest on a theoretical basis a mechanism explaining the onset of turbulence in liquids, with an emphasis on realistic water. We consider water as a non-homogeneous, compressible medium, characterised by density differences. As in a two-fluid model, lighter fluid interacts with heavier fluid as if one of the two were an obstacle. Micro-assemblies of such obstacles might justify the presence of micro-vortices and hence of turbulence.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate:\u0026nbsp;\u003c/strong\u003eThis research does not contain any studies with human participants or animals performed by the Author.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication:\u0026nbsp;\u003c/strong\u003eThe Author transfers all copyright ownership, in the event the work is published. \u0026nbsp;The undersigned author warrants that the article is original, does not infringe on any copyright or other proprietary right of any third part, is not under consideration by another journal, and has not been previously published.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials:\u0026nbsp;\u003c/strong\u003eall data and materials generated or analyzed during this study are included in the manuscript. \u0026nbsp;The Author had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests:\u0026nbsp;\u003c/strong\u003eThe Author do not have any known or potential conflict of interest including any financial, personal or other relationships with other people or organizations within three years of beginning the submitted work that could inappropriately influence, or be perceived to influence, their work.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u0026nbsp;\u003c/strong\u003eThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors\u0026apos; contributions:\u0026nbsp;\u003c/strong\u003eThe Author performed study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical, and material support, study supervision.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements:\u0026nbsp;\u003c/strong\u003enone.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAl-Hamdani Ys, Tkatchenko A. 2019. 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DOI: 10.1126/sciadv.abd3525. \u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"high-density water, molecular dynamics, water model, percolation theory","lastPublishedDoi":"10.21203/rs.3.rs-3845315/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3845315/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTurbulence is a widespread phenomenon detectable in physical and biological systems. Examining a theoretical model of liquid water flowing in a cylinder at different Raleigh numbers, we propose a novel approach to elucidate the first stages of turbulent flows. The weakly bonded molecular assemblies of liquid distilled water form a fluctuating branched polymer in which every micro-cluster displays different density. Against the common view of liquid water as an incompressible and continuous fluid, we consider it as a non-homogeneous, compressible medium characterised by density differences. We suggest that the occurrence of transient local aggregates in liquid water could produce the vortices and eddies that are the hallmarks of turbulence. As in a two-fluid model, lighter fluid interacts with heavier fluid as if one of the two were an obstacle. Micro-assemblies of such obstacles might justify the presence of micro-vortices and hence of turbulence. We quantify the local changes in velocity, diameter and density required to engender obstacles to the average flow. Then, we explain how these microstructures, equipped with different Raleigh numbers and characterized by high percolation index, could generate boundary layers that contribute to micro-vortices production. We explore the theoretical possibility that three-dimensional turbulence might originate from micro-vortices, contrary to the common view that three-dimensional turbulence is caused by energy cascades from larger to smaller vortices. We conclude that the genesis of turbulence cannot be assessed in terms of collective phenomena, rather is sustained, among many other factors, by the underrated microscopic inhomogeneities of fluids like liquid water.\u003c/p\u003e","manuscriptTitle":"Towards Micro-vortices Generated by Liquid Water’s Structural Heterogeneity","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-10 17:31:36","doi":"10.21203/rs.3.rs-3845315/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6c1a9893-087c-4729-827d-1fd5096e8d96","owner":[],"postedDate":"January 10th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-01-29T13:30:19+00:00","versionOfRecord":[],"versionCreatedAt":"2024-01-10 17:31:36","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3845315","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3845315","identity":"rs-3845315","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}