Hydrodynamic Turbulence Research Articles

Modeling oil dispersion under breaking waves. Part I: Wave hydrodynamics

The dispersion (transport and breakup) of oil droplets under breaking waves is essential for evaluating the impact of oil spills on the environment, and for designing countermeasures. In this part of the work (Part I), the hydrodynamics of a wave breaker in a 1.0 m deep tank obtained experimentally by Rapp and Melville (Philos Trans R Soc Lond A Math Phys Eng Sci 331(1622):735–800, 1990) using the dispersive focusing method was investigated numerically using Reynold-averaged Navier–Stokes (RANS) within the CFD code ANSYS Fluent. The renormalization group (RNG) k–e turbulence closure model was adopted to simulate wave turbulence, and the transient water–air interface was captured using volume of fluid (VOF) method. The simulated surface excursion and velocity fields matched closely the experimental observations during wave breaking. The energy dissipation of the wave crest showed good agreement with recent laboratory work and numerical simulations through evaluating the breaking parameter. The RANS approach was able to reproduce the turbulent kinetic energy field engendered by the breakers and simulated the residual turbulence within about two wave periods after the passage of the wave train. The VOF scheme Compressive provided a better agreement with the observation than the Geo-reconstruct scheme. The approach herein suggests that RANS method coupled with VOF in ANSYS Fluent is capable of capturing the major hydrodynamic forces and turbulence, and thus could be used to predict environmental processes within the breaking waves such as oil droplet formation and transport.

On the Closure Problem of the Coarse-Grained Hydrodynamics of Turbulent Superfluids

The coarse-grained hydrodynamics of turbulent superfluid fluids describes the fluid flow in terms of two equations for the averaged normal and superfluid velocities. These two equations are coupled via the mutual friction term, which contains the quantity $${\mathcal {L}}(r,t)$$ –the vortex line density (VLD) of the vortex tangle. The question of how to treat the quantity $${\mathcal {L}}(r,t)$$ –the so-called closure procedure is crucial for the correct description of flow of turbulent superfluids. The article provides a critical analysis of several approaches to the closure procedure. The first one, which is usually referred to as HVBK method suggests to express the quantity $${\mathcal {L}}(r,t)$$ via coarse-grained vorticity $$\nabla \times {\mathbf {v}}_{s}$$ using the famous Feynman rule. This method and idea on the vortex bundle structure, which justifies the use of the HVBK approach, are analyzed and discussed in detail. Another approach that has been popular before, but is still used sometimes, is called the Gorter–Mellink relation. This method suggests that the VLD $${\mathcal {L}}(r,t)$$ is proportional to the squared relative velocity between normal and superfluid components $$\propto (v_{n}-v_{s})^{2}$$ . One more variant of the closure procedure, discussed in the paper, is based on the method in which the vortex line density $${\mathcal {L}}(r,t)$$ is not expressed directly via the velocity (and/or vorticity) field, but is an independent equipollent variable, controlled by a separate equation. The latter approach is called as hydrodynamics of superfluid turbulence (HST). The advantages and disadvantages of each method are discussed.

Hydrodynamical turbulence in eccentric circumbinary discs and its impact on the in situ formation of circumbinary planets

ABSTRACT Eccentric gaseous discs are unstable to a parametric instability involving the resonant interaction between inertial-gravity waves and the eccentric mode in the disc. We present three-dimensional global hydrodynamical simulations of inviscid circumbinary discs that form an inner cavity and become eccentric through interaction with the central binary. The parametric instability grows and generates turbulence that transports angular momentum with stress parameter α ∼ 5 × 10−3 at distances ≲ 7 abin, where abin is the binary semimajor axis. Vertical turbulent diffusion occurs at a rate corresponding to αdiff ∼ 1–2 × 10−3. We examine the impact of turbulent diffusion on the vertical settling of pebbles, and on the rate of pebble accretion by embedded planets. In steady state, dust particles with Stokes numbers St ≲ 0.1 form a layer of finite thickness Hd ≳ 0.1H, where H is the gas scale height. Pebble accretion efficiency is then reduced by a factor racc/Hd, where racc is the accretion radius, compared to the rate in a laminar disc. For accreting core masses with mp ≲ 0.1 M⊕, pebble accretion for particles with St ≳ 0.5 is also reduced because of velocity kicks induced by the turbulence. These effects combine to make the time needed by a Ceres mass object to grow to the pebble isolation mass, when significant gas accretion can occur, longer than typical disc lifetimes. Hence, the origins of circumbinary planets orbiting close to their central binary systems, as discovered by the Kepler mission, are difficult to explain using an in situ model that invokes a combination of the streaming instability and pebble accretion.

Vortex line density and thermal pulse dynamics in superfluid helium

The propagation of intense rectangular thermal pulses in superfluid helium is studied numerically using superfluid turbulence hydrodynamics with various equations for vortex line density dynamics. It is shown that the experimental data are most adequately described by the Vinen equation. The impact of the background vortex line density on the dynamics of the pulses at different temperatures of the unperturbed fluid is studied by using the Vinen equation that describes the vortex line density dynamics. It is found that a high background vortex line density leads to a significant change in the pulse shape and overheating of the fluid. At the same values of the heat flux supplied to the heater and the background vortex line density, the temperature perturbations decrease with the increasing temperature of the unperturbed fluid. This is due to changes in the thermodynamic properties of the fluid and the vortex line density dynamics. As the temperature increases, the increase in the vortex line density slows down, as a result of which the fluid temperature perturbations decrease significantly.

Production and excitation of molecules by dissipation of two-dimensional turbulence

ABSTRACT The interstellar medium (ISM) is typically a hostile environment: cold, dilute and irradiated. Nevertheless, it appears very fertile for molecules. The localized heating resulting from turbulence dissipation is a possible channel to produce and excite molecules. However, large-scale simulations cannot resolve the dissipative scales of the ISM. Here, we present two-dimensional small-scale simulations of decaying hydrodynamic turbulence using the chemses code, with fully resolved viscous dissipation, time-dependent heating, cooling, chemistry and excitation of a few rotational levels of H2. We show that molecules are produced and excited in the wake of strong dissipation ridges. We carefully identify shocks and we assess their statistics and contribution to the molecular yields and excitation. We find that the formation of molecules is strongly linked to increased density as a result of shock compression and to the opening of endothermic chemical routes because of higher temperatures. We identify a new channel for molecule production via H2 excitation, illustrated by CH+ yields in our simulations. Despite low temperatures and the absence of magnetic fields (favouring CH+ production through ion-neutral velocity drifts), the excitation of the first few rotational levels of H2 shrinks the energy gap to form CH+. The present study demonstrates how dissipative chemistry can be modelled by statistical collections of one-dimensional steady-state shocks. Thus, the excitation of higher J levels of H2 is likely to be a direct signature of turbulence dissipation, and an indirect probe for molecule formation. We hope these results will help to bring new tools and ideas for the interpretation of current observations of H2 rotational lines carried out using the Stratospheric Observatory for Infrared Astronomy (SOFIA), and pave the way for a better understanding of the high-resolution mapping of H2 emission by future instruments, such as theJames Webb Space Telescope and the Space Infrared Telescope for Cosmology and Astrophysics.

Inverse cascade of energy in helical turbulence

Using direct numerical simulation of hydrodynamic turbulence with helicity forcing applied at all scales, a near-maximum helical turbulent state is obtained, with an inverse energy cascade at scales larger than the energy forcing scale and a forward helicity cascade at scales smaller than the energy forcing scale. In contrast to previous studies using decimated triads, our simulations contain all possible triads. By computing the shell-to-shell energy fluxes, we show that the inverse energy cascade results from weakly non-local interactions among homochiral triads. Varying the helicity injection range of scales leads to necessary conditions to obtain an inverse energy cascade.

Scaling exponents saturate in three-dimensional isotropic turbulence

From a database of direct numerical simulations of homogeneous and isotropic turbulence, generated in periodic boxes of various sizes, we extract the spherically symmetric part of moments of velocity increments and first verify the following (somewhat contested) results: the $4/5$-ths law holds in an intermediate range of scales and that the second order exponent over the same range of scales is {\it{anomalous}}, departing from the self-similar value of $2/3$ and approaching a constant of $0.72$ at high Reynolds numbers. We compare with some typical theories the dependence of longitudinal exponents as well as their derivatives with respect to the moment order $n$, and estimate the most probable value of the H\"older exponent. We demonstrate that the transverse scaling exponents saturate for large $n$, and trace this trend to the presence of large localized jumps in the signal. The saturation value of about $2$ at the highest Reynolds number suggests, when interpreted in the spirit of fractals, the presence of vortex sheets rather than more complex singularities. In general, the scaling concept in hydrodynamic turbulence appears to be more complex than even the multifractal description.

Nonlinear hydrodynamic instability and turbulence in pulsatile flow

Pulsating flows through tubular geometries are laminar provided that velocities are moderate. This in particular is also believed to apply to cardiovascular flows where inertial forces are typically too low to sustain turbulence. On the other hand, flow instabilities and fluctuating shear stresses are held responsible for a variety of cardiovascular diseases. Here we report a nonlinear instability mechanism for pulsating pipe flow that gives rise to bursts of turbulence at low flow rates. Geometrical distortions of small, yet finite, amplitude are found to excite a state consisting of helical vortices during flow deceleration. The resulting flow pattern grows rapidly in magnitude, breaks down into turbulence, and eventually returns to laminar when the flow accelerates. This scenario causes shear stress fluctuations and flow reversal during each pulsation cycle. Such unsteady conditions can adversely affect blood vessels and have been shown to promote inflammation and dysfunction of the shear stress-sensitive endothelial cell layer.

Turbulent drag reduction in magnetohydrodynamic and quasi-static magnetohydrodynamic turbulence

In hydrodynamic turbulence, the kinetic energy injected at large scales cascades to the inertial range, leading to a constant kinetic energy flux. In contrast, in magnetohydrodynamic (MHD) turbulence, a fraction of kinetic energy is transferred to the magnetic energy. Consequently, for the same kinetic energy injection rate, the kinetic energy flux in MHD turbulence is reduced compared to its hydrodynamic counterpart. This leads to relative weakening of the nonlinear term ⟨|(u·∇)u|⟩, (where u is the velocity field) and turbulent drag, but strengthening of the velocity field in MHD turbulence. We verify the above using shell model simulations of hydrodynamic and MHD turbulence. Quasi-static MHD turbulence too exhibits turbulent drag reduction similar to MHD turbulence.

Probability Density Functions in Homogeneous and Isotropic Magneto-Hydrodynamic Turbulence

We derive a hierarchy of evolution equations for multi-point probability density functions in magneto-hydrodynamic (MHD) turbulence. We discuss the relation to the moment hierarchy in MHD turbulence formulated by Chandrasekhar (S. Chandrasekhar, Proc. R. Soc. Lond. A 1951, 204, 435–449) and derive a functional equation for a joint characteristic functional, which can be considered as the analogon to the Hopf functional in hydrodynamic turbulence. Furthermore, we develop a closure method for the evolution equation of the single-point magnetic field probability density function, which is based on a joint Gaussian assumption for unclosed terms. It is explicitly shown that this closure, together with the assumptions of homogeneity and isotropy, leads to vanishing nonlinear terms. We discuss the implications of this finding for magnetic field generation and give a brief outlook on an axisymmetric theory, which includes a mean magnetic field.

Generation of Solenoidal Modes and Magnetic Fields in Turbulence Driven by Compressive Driving

Abstract We perform numerical simulations of hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence driven by compressive driving, to study the generation of solenoidal velocity components and the small-scale magnetic field. We mainly focus on the effects of mean magnetic field (B 0) and the sonic Mach number (M s ). The dependence of solenoidal ratio (i.e., ratio of solenoidal to kinetic energies) and magnetic energy density on M s in compressively driven turbulence is already established, but that on B 0 is not yet. We also consider two different driving schemes in terms of the correlation timescale of forcing vectors: a finite-correlated driving and a delta-correlated driving. Our findings are as follows. First, when we fix the value of B 0, the solenoidal ratio after saturation increases as increases. A similar trend is observed for generation of magnetic field when B 0 is small. Second, when we fix the value of , HD and MHD simulations result in similar solenoidal ratios when B 0 is not strong (say, M A ≳ 5, where M A is Alfvén Mach number). However, the ratio increases when M A ≲ 5. Roughly speaking, the magnetic energy density after saturation is a linearly increasing function of B 0 irrespective of M s . Third, generation of the solenoidal velocity component is not sensitive to numerical resolution, but that of magnetic energy density is mildly sensitive. Finally, when initial conditions are same, the finite-correlated driving always produces more solenoidal velocity and small-scale magnetic field components than the delta-correlated driving. We additionally analyze the vorticity equation to understand why higher and B 0 yield a larger quantity of the solenoidal velocity component.

Green corrosion inhibitor for oilfield application II: The time–evolution effect on the sweet corrosion of API X60 steel in synthetic brine and the inhibition performance of 2-(2-pyridyl) benzimidazole under turbulent hydrodynamics

Time–evolution study of 2-(2-pyridyl) benzimidazole (2PB) as CO2 corrosion inhibitor for API X60 steel under hydrodynamic conditions in a synthetic brine solution has been undertaken. Electrochemical impedance spectroscopy (EIS) and potentiodynamic polarization (PDP) techniques were employed for this study. The results indicate that 2.56 mM of 2PB could provide up to 88.3 % inhibition efficiency after 12 h at 4000 rpm in the synthetic brine solution. The addition of 0.14 mM thiobarbituric acid (TBA) significantly enhances 2PB performance to 95.6 % using PDP. ATR-IR, SEM and optical image characterizations confirm that 2PB adsorption suppresses the localized corrosion of the steel.

Convex integration and phenomenologies in turbulence

In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier–Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash’s fundamental ideas on C^1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration , have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [51,54,55,200]. Mathematical problems arising in turbulence(such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative C^{0+}_{x,t} weak solutions of the Euler equations, first proven by De Lellis–Székelyhidi Jr. [52,53]. Second, we present Isett’s [108] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [21] of De Lellis–Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C^{1/3-}_{x,t} are constructed, attaining any energy profile. Third, we give a concise proof of the authors’ recent result [23], which proves the existence of infinitely many weak solutions of the Navier–Stokes in the regularity class C^0_t L^{2+}_x \cap C^0_t W^{1,1+}_x . We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

Scale-free distributions of waiting times for earthquakes

We investigate through detailed statistical analyses the dynamics of earthquakes with epicenters in Romania, Japan, California (USA), and Italy, and show that the distribution of waiting times, defined here akin to econophysics, as to implicitly include information about the magnitude of earthquakes, is very similar to the scale-free distributions observed in hydrodynamic turbulence and stock market dynamics. Our results show that the shape of the observed distributions depends on the size of the magnitude threshold δ, with a cut-off at small waiting times, and is sensitive with respect to the sign of the aforementioned threshold. In the case of earthquakes originating in Romania we also show that the distributions of waiting times for depths have the same power-law behaviour. Moreover, we show that the distributions of released daily energies calculated for Romania and California (USA) have a prominent scale-free nature, which reinforce the idea that seismic zones can be seen as critical systems.

Knudsen number effects on the nonlinear acoustic spectral energy cascade.

We present a numerical investigation of the effects of gas rarefaction on the energy dynamics of resonating planar nonlinear acoustic waves. The problem setup is a gas-filled, adiabatic tube, excited from one end by a piston oscillating at the fundamental resonant frequency of the tube and closed at the other end; nonlinear wave steepening occurs until a limit cycle is reached, resulting in shock formation for sufficiently high densities. The Knudsen number, defined here as the ratio of the characteristic molecular collision timescale to the resonance period, is varied in the range Kn=10^{-1}-10^{-5}, from rarefied to dense regime, by changing the base density of the gas. The working fluid is Argon. A numerical solution of the Boltzmann equation, closed with the Bhatnagar-Gross-Krook model, is used to simulate cases for Kn≥0.01. The fully compressible one-dimensional Navier-Stokes equations are used for Kn<0.01 with adaptive mesh refinement to resolve the resonating weak shocks, reaching wave Mach numbers up to 1.01. Nonlinear wave steepening and shock formation are associated with spectral broadening of the acoustic energy in the wavenumber-frequency domain; the latter is defined based on the exact energy corollary for second-order nonlinear acoustics derived by Gupta and Scalo [Phys. Rev. E 98, 033117 (2018)2470-004510.1103/PhysRevE.98.033117], representing the Lyapunov function of the system. At the limit cycle, the acoustic energy spectra exhibit an equilibrium energy cascade with a -2 slope in the inertial range, also observed in freely decaying nonlinear acoustic waves by the same authors. In the present system, energy is introduced externally via a piston at low wavenumbers or frequencies and balanced by thermoviscous dissipation at high wavenumbers or frequencies, responsible for the base temperature increase in the system. The thermoviscous dissipation rate is shown to scale as Kn^{2} for fixed Reynolds number based on the maximum velocity amplitude, i.e., increasing with the degree of flow rarefaction; consistently, the smallest length scale of the steepened waves at the limit cycle, corresponding to the thickness of the shock (when present) also increases with Kn. For a given fixed piston velocity amplitude, the bandwidth of the inertial range of the spectral energy cascade decreases with increasing Knudsen numbers, resulting in a reduced resonant response of the system. By exploiting dimensionless scaling laws borrowed by Kolmogorov's theory of hydrodynamic turbulence, it is shown that an inertial range for spectral energy transfer can be expected for acoustic Reynolds numbers Re_{U_{max}}>100, based on the maximum acoustic velocity amplitude in the domain.

On the possible distributions of temperature in nonequilibrium steady states

Superstatistics is a framework in nonequilibrium statistical mechanics that successfully describes a wide variety of complex systems, including hydrodynamic turbulence, weakly-collisional plasmas, cosmic rays, power grid fluctuations, among several others. In this work we analyze the class of nonequilibrium steady-state systems consisting of a subsystem and its environment, and where the subsystem is described by the superstatistical framework. In this case we provide an answer to the mechanism by which a broad distribution of temperature arises, namely due to correlation between subsystem and environment. We prove that there is a unique microscopic definition of inverse temperature compatible with superstatistics, in the sense that all moments of and coincide. The function however, cannot depend on the degrees of freedom of the system itself, only on the environment, in full agreement with our previous impossibility theorem (Davis and Gutiérrez 2018 Physica A 505 864–70). The present results also constrain the possible joint ensembles of system and environment compatible with superstatistics.

Generation of a large-scale vorticity in a fast-rotating density-stratified turbulence or turbulent convection.

We find an instability resulting in generation of large-scale vorticity in a fast-rotating small-scale turbulence or turbulent convection with inhomogeneous fluid density along the rotational axis in anelastic approximation. The large-scale instability causes excitation of two modes: (i) the mode with dominant vertical vorticity and with the mean velocity being independent of the vertical coordinate; (ii) the mode with dominant horizontal vorticity and with the mean momentum being independent of the vertical coordinate. The mode with the dominant vertical vorticity can be excited in a fast-rotating density-stratified hydrodynamic turbulence or turbulent convection. For this mode, the mean entropy is depleted inside the cyclonic vortices, while it is enhanced inside the anticyclonic vortices. The mode with the dominant horizontal vorticity can be excited only in a fast-rotating density-stratified turbulent convection. The developed theory may be relevant for explanation of an origin of large spots observed as immense storms in great planets, e.g., the Great Red Spot in Jupiter and large spots in Saturn. It may be also useful for explanation of an origin of high-latitude spots in rapidly rotating late-type stars.

Development of modeling and simulation of bubble‐liquid hydrodynamics in bubble column

AbstractIn the present work, the investigation on strategies to alleviate the cell death in terms of bubble‐liquid two‐phase hydrodynamics in bubble column was performed using the modeling and numerical simulation approaches. A multifluid bubble‐liquid turbulent model based on the second‐order moment algorithm was developed to predict turbulent hydrodynamics. In this model, the anisotropic characteristics of Reynolds stresses and the interactions between bubble and liquid phases are considered by means of the defined bubble‐liquid two‐phase fluctuation velocities correlation. Hydrodynamic parameters such as turbulent energy dissipation rate, the turbulent kinetic energy, the distributions of bubble normal and shear stresses, and the operation conditions of gas entrance velocity and height to diameter of column are accurately simulated. A proposed correlation named turbulent energy production rate is revealed successfully the higher cell death rate at the gas inlet regions that cannot be verified by the rest of the two‐phase hydrodynamics. Superficial gas velocity is the predominant influence on this value distribution as smaller the ratio of height to diameter of bubble column, which reduced by two order of magnitude under decreased by one level of gas velocity. Near inlet region with large height‐to‐diameter ratio, this value is always larger than those of small one.

Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions

The aim of this work is to prove an existence and uniqueness result of Kato–Fujita type for the Navier–Stokes equations, in vorticity form, in 2D and 3D, perturbed by a gradient-type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by Barbu and Rockner (J Differ Equ 263:5395–5411, 2017) that treats the stochastic 3D Navier–Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in Barbu and Rockner (2017), but the transformation is different and adapted to our gradient-type noise. Then, global unique existence results are proved for the transformed equation, while for the original stochastic Navier–Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time.

Similarities between the structure functions of thermal convection and hydrodynamic turbulence

In this paper, we analyze the scaling of velocity structure functions of turbulent thermal convection. Using high-resolution numerical simulations, we show that the structure functions scale similar to those of hydrodynamic turbulence, with the scaling exponents in agreement with the predictions of She and Leveque [“Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. 72, 336–339 (1994)]. The probability distribution functions of velocity increments are non-Gaussian with wide tails in the dissipative scales and become close to Gaussian in the inertial range. The tails of the probability distribution follow a stretched exponential. We also show that in thermal convection, the energy flux in the inertial range is less than the viscous dissipation rate. This is unlike in hydrodynamic turbulence where the energy flux and the dissipation rate are equal.