Self-similarity Hypothesis Research Articles

Using the method of self-similar functions for estimating the normal boiling temperatures of homologous series

An equation for estimating the normal boiling points of homologous series with the general formula R1(CH2)nR2 has been developed. The equation is based on the hypothesis of functional self-similarity. The parameters of the equation have been determined using a current experimental database for 10 homologous series. The equation allows predicting the normal boiling temperature of any member of any homologous series of the type R1(CH2)nR2 if the normal boiling temperature is known for one compound which belongs to this series. The proposed method gives an absolute average percentage deviation from the experimental values of the normal boiling point of 0.72 % for all the homologous series studied. A thermodynamically consistent model for estimating the critical temperature, critical pressure, and normal boiling point of n-alkanes has been developed.

Evidence of entropy cascade in collisionless magnetized plasma turbulence

The turbulence of collisionless magnetized plasmas, as observed in space, astrophysical, and magnetically confined fusion plasmas, has attracted considerable interest for a long-time. The entropy cascade in collisionless magnetized plasmas is a theoretically proposed dynamics comparable to the Kolmogorov energy cascade in fluid turbulence. Here, we present evidence of an entropy cascade in laboratory plasmas by direct visualization of the entropy distribution in the phase space of turbulence in laboratory experiments. This measurement confirms the scaling laws predicted by the gyrokinetic theory with the dual self-similarity hypothesis, which reflects the interplay between the position and velocity of ions by perpendicular nonlinear phase mixing. This verification contributes to our understanding of turbulent heating in the solar corona, accretion disks, and magnetically confined fusion plasmas.

Numerical simulation of a highly turbulent wall jet in a confined basin

ABSTRACT Wall jets are important hydraulic flows that occur in relation to many civil engineering structures. Researchers have made efforts to study wall jet hydraulics in an unconfined field that can be described by the self-similarity theory. Herein, a highly turbulent submerged wall jet entering a confined basin was considered to investigate numerically. Attempts were made to give details of the flow structure in the confined basin using OpenFOAM, employing five turbulence closures: standard k–ε, realizable k–ε, RNG k–ε, standard k–ω, and k–ω SST. The simulated vertical profiles of velocity were compared with laboratorial measurements available in the literature to determine the best performance. It was found that, while dealing with cases of the wall jet entering a confined basin, the self-similarity hypothesis, which is widely used to describe wall jets in unconfined domain, failed to predict the evolution of the wall jet further downstream. In the regions near the jet entering, in which self-similarity dominated, all the turbulence models simulated the wall jet successfully and computed the same results. On the other hand, in the transition stage, which shows the evolution of wall jet profile to uniform profile, the k–ω SST turbulence model predictions had better agreement with experimental data.

Self-similar cosmological solutions in f(R,T) gravity theory

We study the [Formula: see text] cosmological models under the self-similarity hypothesis. We determine the exact form that each physical and geometrical quantity may take in order that the field equations (FE) admit exact self-similar (SS) solutions through the matter collineation approach. We study two models: the case [Formula: see text] and the case [Formula: see text]. In each case, we state general theorems which determine completely the form of the unknown functions [Formula: see text] such that the FE admit SS solutions. We also state some corollaries as limiting cases. These results are quite general and valid for any homogeneous SS metric[Formula: see text] In this way, we are able to generate new cosmological scenarios. As examples, we study two cases by finding exact solutions to these particular models.

Response of Reynolds stresses and scaling behavior of high-order structure functions to a water-worked gravel-bed surface and its implication on sediment transport

The aim of this experimental study is to investigate the interaction between turbulent flow and a gravel-bed that mimics the actual roughness structures of a natural bed and its implication on sediment transport. In particular, the response of the Reynolds stresses and the role of intermittency to the bed roughness is the primary focus of the current study. To this end, the flow field, measured with an Acoustic–Doppler Velocimeter (ADV), is thoroughly examined, considering the conditional Reynolds shear stresses and the high-order structure functions of velocity. The study results show that the features and the roughness structure of the water-worked gravel-bed (WWGB) have a strong impact on bursting events. The reason for this is attributed to the flow layer above the crest in the WWGBs being primarily affected by sweep events rather than ejection events, although the latter events are prevalent in this flow layer in an immobile, man-made, gravel-bed. These coherent structures, which are primarily responsible for sediment transport, also are the key cause of the presence of an intermittency excess that breaks the Kolmogorov self-similarity hypothesis, leading to multi-fractal behavior of the velocity structure functions.

Method of self-similar functions for estimating the critical properties of homologous series

The coefficients of equations for estimating the critical temperatures, pressures, and densities of homologous series with the general formula R1(CH2)nR2 have been calculated using the current experimental database. The equations are based on the hypothesis of functional self-similarity and the scaling behaviour of the critical constants of long-chain molecules (Nikitin et al. Fluid Phase Equilib., 2005, 235, 1). The equations allow predicting the critical constants of any member of any homologous series of this type if these quantities are known for one substance which belongs to this series. The method works well for compounds with a rather large number of CH2 groups in a molecule (n≥4). The results of calculating the critical temperatures, pressures, and densities of 16 homologous series by the proposed method have been compared with the critical properties estimated by the Constantinou/Gani, Marrero/Gani, and Hukkerikar et al. techniques.

Wavelet analysis of shearless turbulent mixing layer

The intermittency and scaling exponents of structure functions are experimentally studied in a shearless turbulent mixing layer. Motivated by previous studies on the anomalous scaling in homogeneous/inhomogeneous turbulent flows, this study aims to investigate the effect of strong intermittency caused by turbulent kinetic energy diffusion without energy production by mean shear. We applied an orthonormal wavelet transformation to time series data of streamwise velocity fluctuations measured by hot-wire anemometry. Intermittent fluctuations are extracted by a conditional method with the local intermittency measure, and the scaling exponents of strong and weak intermittent fluctuations are calculated based on the extended self-similarity. The results show that the intermittency is stronger in the mixing layer region than in the quasi-homogeneous isotropic turbulent regions, especially at small scales. The deviation of higher-order scaling exponents from Kolmogorov's self-similarity hypothesis is significant in the mixing layer region, and the large deviation is caused by strong, intermittent fluctuations even without mean shear. The total intermittent energy ratio is also different in the mixing layer region, suggesting that the total intermittent energy ratio is not universal but depends on turbulent flows. The scaling exponents of weak fluctuations with a wavelet coefficient flatness corresponding to the Gaussian distribution value of 3 follow the Kolmogorov theory up to fifth order. However, the sixth order scaling exponent is still affected by these weak fluctuations.

A self-similar solution of a fluid with spherical distribution in general relativity

The purpose of this article is to use an algorithm to obtain a non-static solution to the Einstein–Euler equations of a spherically symmetric matter distribution of a perfect fluid. The equations obtained make up a nonlinear system of partial derivative equations (PDE), which in the vast majority of cases is quite difficult to solve, either analytically or numerically. In this context, the spherical symmetry of space-time is very useful since it allows to establish the self-similarity of the solutions. This assumption reduces the PDEs to a set of ordinary differential equations (ODE). Therefore, the self-similarity hypothesis is very powerful in the search for non-static analytical or numerical solutions. The ODEs thus established allow defining a set of variables adjusted to the boundary conditions of the spherical distribution of matter, together with an equation of state (EOS). In this work, Tolman’s V solution has been selected as EOS. Once the numerical integration has been carried out, the variables established in the ODE and some other physical variables determined in the algorithm (density, pressure, radiation, etc.) can present damped oscillations (rebounds), depending on the initial values. This behavior is similar to that found near phase transitions in condensed matter physics, but now the mass distribution plays the role of an order parameter. This result has been obtained in other simulations of numerical relativity, where the PDE obtained from the gravitational field equations are integrated and the expression of an ultrarelativistic fluid $$\left( P=\kappa \,\rho \right) $$ . It is important to note that there is the possibility of carrying out simulations with other EOS (with electrical charge, anisotropy, etc.) with this algorithm. Many of the calculations to obtain the field equations, such as the conservation equations, were performed using the GRTensorIII computational algebra package, running on Maple 2017; as well as some Maple routines that have been used for these types of problems.

Experimental study on triangular central baffle flume

In this paper the results of the experiments performed to study the flow through a Triangular Central Baffle Flume (TCBF) are reported. The investigated flume consists of a triangular baffle of the apex angle of 75° with a given base width. The theoretical stage-discharge formula was deduced by applying the Buckingham's Theorem and incomplete self-similarity hypothesis and was calibrated using the laboratory measurements carried out in this investigation. The proposed stage-discharge formula is characterized by a mean absolute relative error of 7.4% and 72% of the data points are in an error range of ±5%. The results indicate that TCBF flume is characterized by a flow capacity higher than that of a typical central baffle flume. Experimental observations show that the contraction ratio is a key parameter to distinguish between free and submerged flow regimes through a TCBF. Finally, to identify the flow condition, submergence threshold condition was formulated.

A comparative study of some cosmological models with time varying G and Λ: A symmetry approach

We study how the constants G and Λ may vary in four different theoretical models: general relativity with time-varying constants (Y.-K. Lau. Aust. J. Phys. 38, 547 (1985). doi: 10.1071/PH850547 ), the model proposed by Lu et al. (Phys Rev D, 89, 063526 (2014). doi: 10.1103/PhysRevD.89.063526 ), the model proposed by Bonanno et al. (Class. Quant. Grav. 24, 1443 (2007). doi: 10.1088/0264-9381/24/6/005 ), and the Brans–Dicke model with Λ([Formula: see text]) [ 25 ]. To carry out this study, we work under the self-similar hypothesis and we assume the same metric, a flat Friedmann–Robertson–Walker metric, and the same matter source, a perfect fluid. We put special emphasis on mathematical and formal aspects, which allows us to calculate exact power-law solutions through symmetry methods, matter collineation, and Noether symmetries. This enables us to compare the solutions of each model and in the same way to contrast the results with some observational data.

Assessing flow resistance law in vegetated channels by dimensional analysis and self-similarity

In this paper experimental data collected by Kouwen et al., Wilson and Horrit, Raffaelli et al. and Carollo et al., using straight flumes having a bed covered by grass-like vegetation with different stem concentrations, were used to analyze flow resistance for flexible submerged elements. At first, the dimensional analysis and the incomplete self-similarity hypothesis was applied to deduce the flow velocity distribution and the resulting theoretical expression of the Darcy-Weisbach friction factor. Then, a relationship between the Γ function of the velocity profile and the biomechanical characteristics of vegetation, the channel slope, the Reynolds number and the flow Froude number was empirically deduced by the available measurements. This relationship was established distinguishing between the low stem concentration values and the highest values for which a quasi-smooth skimming flow occurs. Finally, the analysis showed that, in comparison with previous results, a more accurate Darcy-Weisbach friction factor estimate can be obtained by the theoretical approach based on a power-velocity profile.

Discussion of “Applying Hypothesis of Self-Similarity for Flow-Resistance Law in Calabrian Gravel-Bed Rivers” by Vito Ferro and Paolo Porto

Discussion of “Applying Hypothesis of Self-Similarity for Flow-Resistance Law in Calabrian Gravel-Bed Rivers” by Vito Ferro and Paolo Porto

Closure to “Applying Hypothesis of Self-Similarity for Flow-Resistance Law in Calabrian Gravel-Bed Rivers” by Vito Ferro and Paolo Porto

Closure to “Applying Hypothesis of Self-Similarity for Flow-Resistance Law in Calabrian Gravel-Bed Rivers” by Vito Ferro and Paolo Porto

Regularities of crustal faulting and tectonophysical indicators of fault metastability

The problem of forecasting seismic hazards is discussed. The stress state data characterizing various aspects of brittle failure are reviewed in detail. It is shown that the most convenient tool for analyzing such data is the Mohr stress diagram and the Coulomb criterion. Noted is the role of a fluid in not only reducing the normal stresses responsible for brittle failure, but also predetermining the major processes in fault zones. In each fault body, a node can be distinguished as a fault part wherein the main structural and material transformations take place. The node contains narrow elongated zones of modification of mylonites, from protomylonites to ultramylonites and blastomylonites, that are related to the localization of continuous and discontinuous shear deformation. Due to the metamorphic processes, fault zones are less strong than the surrounding consolidated blocks of the crust. A theoretical analysis of the mechanism of displacements along the discontinuities of different scale ranks shows differences in their manifestation. Tectonic and seismic displacements along the rupture occupy the entire area at once, while displacements along the fault zone occur in stages along its extent and follow the ‘rolling-carpet’ principle that is also typical of intra-crystal dislocations. The stress state in the vicinity of ruptures and faults has different characteristic features. Based on the seismological and tectonophysical data on earthquake focal parameters and discontinuities, it is possible to identify two or three ranks of stresses, which differ in the laws predetermining their mutual relationships. Actually, this conclusion contradicts the hypothesis of self-similarity of discontinuities in their continuous range, from a dislocation to a fault zone, which length amounts to tens of kilometers. Besides, it imposes a restriction on the use of statistical analysis of seismic data. The seismic data show that in the source of a large earthquake, displacement develops as a running band (‘rolling-carpet’ principle). In the source of a weak earthquake, it occupies the entire earthquake focal area at once. The differences in the types of shearing in the sources of weak and strong earthquakes are related to the relationships between three dynamic parameters of the medium: velocity of seismic wave propagation, rate of rupture propagation, and displacement rate of the sides of the fracture. Using tectonophysical methods, the stress state was reconstructed for the seismically active regions of the planet and the sources of the mega-earthquakes of the 21st century. Based on the reconstructions, the mean strength and stress values were calculated, and the specific features of the stress fields were revealed. It is established that the strongest regional earthquakes ‘avoided’ the areas with increased effective isotropic pressure. The sizes of the sources of the strongest earthquakes were controlled by the size of the region with decreased effective pressure. The sites, wherefrom the earthquake were initiated, were often located in the zones of the highest stress gradients. These regularities support the term “metastability of the state of fault zone” (introduced to seismology from the physics of the states of matter) and justify it by a specific distribution pattern of stress values prior to the mega-earthquake. Based on the tectonophysical definition of the metastable state of faults, the important role is outlined for a stress gradient zone that represents a location wherein a trigger earthquake occurs. The ‘maturity’ of the zone with increased stress gradient values is, in essence, a characteristic of the time interval of metastability of the fault zone.

Intermittency enhancement in quantum turbulence in superfluidHe4

Intermittency is a hallmark of turbulence, which exists not only in turbulent flows of classical viscous fluids but also in flows of quantum fluids such as superfluid $^4$He. Despite the established similarity between turbulence in classical fluids and quasi-classical turbulence in superfluid $^4$He, it has been predicted that intermittency in superfluid $^4$He is temperature dependent and enhanced for certain temperatures, which strikingly contrasts the nearly flow-independent intermittency in classical turbulence. Experimental verification of this theoretical prediction is challenging since it requires well-controlled generation of quantum turbulence in $^4$He and flow measurement tools with high spatial and temporal resolution. Here, we report an experimental study of quantum turbulence generated by towing a grid through a stationary sample of superfluid $^4$He. The decaying turbulent quantum flow is probed by combining a recently developed He$^*_2$ molecular tracer-line tagging velocimetry technique and a traditional second sound attenuation method. We observe quasi-classical decays of turbulent kinetic energy in the normal fluid and of vortex line density in the superfluid component. For several time instants during the decay, we calculate the transverse velocity structure functions. Their scaling exponents, deduced using the extended self-similarity hypothesis, display non-monotonic temperature-dependent intermittency enhancement, in excellent agreement with recent theoretical/numerical study of Biferale et al. [Phys. Rev. Fluids 3, 024605 (2018)].

Applying Hypothesis of Self-Similarity for Flow-Resistance Law in Calabrian Gravel-Bed Rivers

AbstractIn this paper, the results of an investigation carried out to test the applicability of a flow-resistance law on gravel-bed rivers in southern Italy (fiumare) are reported. First, dimension...

Statistics of turbulence in the energy-containing range of Taylor–Couette compared to canonical wall-bounded flows

Considering structure functions of the streamwise velocity component in a framework akin to the extended self-similarity hypothesis (ESS), de Silva et al. (J. Fluid Mech., vol. 823, 2017, pp. 498–510) observed that remarkably the large-scale (energy-containing range) statistics in canonical wall-bounded flows exhibit universal behaviour. In the present study, we extend this universality, which was seen to encompass also flows at moderate Reynolds number, to Taylor–Couette flow. In doing so, we find that also the transversal structure function of the spanwise velocity component exhibits the same universal behaviour across all flow types considered. We further demonstrate that these observations are consistent with predictions developed based on an attached-eddy hypothesis. These considerations also yield a possible explanation for the efficacy of the ESS framework by showing that it relaxes the self-similarity assumption for the attached-eddy contributions. By taking the effect of streamwise alignment into account, the attached-eddy model predicts different behaviour for structure functions in the streamwise and in the spanwise directions and that this effect cancels in the ESS framework – both consistent with the data. Moreover, it is demonstrated here that also the additive constants, which were previously believed to be flow dependent, are indeed universal at least in turbulent boundary layers and pipe flow where high Reynolds number data are currently available.

A new analytical scaling for turbulent wind-bent plumes: Comparison of scaling laws with analog experiments and a new database of eruptive conditions for predicting the height of volcanic plumes

Various scaling relationships relate the height of volcanic plumes to eruptive source conditions, atmospheric density stratification, turbulent entrainment, and wind stresses. However, observational, analog, and numerical studies used to test these scalings capture only a narrow range of natural eruptive conditions. In particular, existing analytical scalings are not appropriate for the wind stress conditions typical of the majority of volcanic eruptions. Accordingly, we develop a new analytical scaling for the height of buoyant plumes rising in density-stratified uniform crossflows. We compare this scaling to existing analytical scalings (Morton et al. 1956; Hewett et al. 1971) as well as “functional” scalings (i.e., parameterizations expressed as a function of the Morton et al. (1956) scaling and a regime parameter related to wind stress, Degruyter and Bonadonna (2012), Woodhouse et al. (2013), Carazzo et al. (2014)) using the extensive experimental dataset from Carazzo et al. (2014) along with natural events from a new database including 94 eruptive phases. Our proposed scaling best predicts the height of experimental plumes, which enables us to constrain the ratio of the wind to radial entrainment coefficients. For natural eruptions, the Woodhouse et al. (2013) and Carazzo et al. (2014) scalings, which account explicitly for wind gradient, best predicts plume heights. We show that accounting for atmospheric stratification and wind improves empirical relationships between mass eruption rates and plume heights. For tested scalings, analyze of residual heights for natural eruption supports the hypothesis that volcanic plumes rise higher in a wetter atmosphere. Finally, for analog plumes rising under moderate to high wind stresses, we show that plume shapes evolve over the plume height, violating the self-similarity assumption on which all scalings and integral model results rely. We discuss consequences for relaxing the self-similarity hypothesis as well as potential improvements for standard integral plume models, in turn.

Universality of the energy-containing structures in wall-bounded turbulence

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

New Flow-Resistance Law for Steep Mountain Streams Based on Velocity Profile

AbstractThis paper deduces a new flow-resistance equation for open-channel flow applying dimensional analysis and self-similarity theory. The incomplete self-similarity hypothesis is used to establ...