The Kelvin-Helmholtz theorem on conservation of circulations is supposed to hold for ideal inviscid fluids and is believed to be play a crucial role in turbulent phenomena, such as production of dissipation by vortex line-stretching. However, this expectation does not take into account singularities in turbulent velocity fields at infinite Reynolds number. We present evidence from numerical simulations for the breakdown of the classical Kelvin theorem in the three-dimensional turbulent energy cascade. Although violated in individual realizations, we find that circulations are still conserved in some average sense. For comparison, we show that Kelvin's theorem holds for individual realizations in the two-dimensional enstrophy cascade, in agreement with theory. The turbulent ``cascade of circulations'' is shown to be a classical analogue of phase-slip due to quantized vortices in superfluids and various applications in geophysics and astrophysics are outlined.
The effect of Prandtl number on the evolution of Kelvin-Helmholtz (KH) billows and the amount of mixing they generate is studied through direct numerical simulation (DNS). The results indicate that the time evolution of the rate of mixing through different stages of the life-cycle of KH flow is significantly influenced by the Prandtl number. As the Prandtl number increases, the final amount of mixing increases for Reynolds that are too low to support active three-dimensional motions. This trend is the opposite in sufficiently high Reynolds number KH flows that can overcome viscous effects, and…
We present evidence of Kelvin excitations in space-time resolved spectra of numerical simulations of quantum turbulence. Kelvin waves are transverse and circularly polarized waves that propagate along quantized vortices, for which the restitutive force is the tension of the vortex line, and which play an important role in theories of superfluid turbulence. We use the Gross-Pitaevskii equation to model quantum flows, letting an initial array of well-organized vortices develop into a turbulent bundle of intertwined vortex filaments. By achieving high spatial and temporal resolution we are able t…
Kelvin-Helmholtz {instability induced} turbulence is one promising mechanism by which loops in the solar corona can be heated by MHD waves. In this paper we present an analytical model of the dissipation rate of {Kelvin-Helmholtz instability induced} turbulence $\varepsilon_{\rm D}$, finding it scales as the wave amplitude ($d$) to the third power ($\varepsilon_{\rm D}\propto d^3$). Based on the concept of steady-state turbulence, we expect the turbulence heating throughout the volume of {the} loop to match the total energy injected through its footpoints. In situations where this holds, the w…
Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length-scale separating a quasi-inerti…
We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magneto-hydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two- and three-dimensional MHD turbulence. In particular, rescaling relations between longitudinal and transverse structure functions are derived and utilized in order to compare different scaling behavior in the inertial range. It is found that there are no substantial differences between longitudinal and transverse structure function…
We performed numerical simulations of decaying quantum turbulence by using a generalized Gross-Pitaevskii equation, that includes a beyond mean field correction and a nonlocal interaction potential. The nonlocal potential is chosen in order to mimic He II by introducing a roton minimum in the excitation spectrum. We observe that at large scales the statistical behavior of the flow is independent of the interaction potential, but at scales smaller than the intervortex distance a Kelvin wave cascade is enhanced in the generalized model. In this range, the incompressible kinetic energy spectrum o…
A hallmark of fluid turbulence theory is the universal power law scaling of the velocity difference statistics between two points in space in the inertial range between the large energy injection scale and the small energy dissipation scale. Even at the highest Reynolds numbers available, laboratory and natural flows such universal power laws have not been convincingly demonstrated. Here we show for the decaying active grid turbulence of the Max Planck Variable Density Turbulence Tunnel that the velocity difference statistics at high Reynolds numbers do not exhibit a power law, but have a univ…
The effects of Reynolds number and Stokes number on particle-pair relative velocity (RV) were investigated systematically using a recently developed planar four-frame particle tracking technique in a novel homogeneous and isotropic turbulence chamber.
Kelvin's Theorem on conservation of circulations is an essential ingredient of G. I. Taylor's theory of turbulent energy dissipation by the process of vortex-line stretching. In previous work, we have proposed a nonlinear mechanism for the breakdown of Kelvin's Theorem in ideal turbulence at infinite Reynolds number. We develop here a detailed physical theory of this ``cascade of circulations''. Our analysis is based upon an effective equation for large-scale ``coarse-grained'' velocity, which contains a turbulent-induced ``vortex-force'' that can violate Kelvin's Theorem. We show that singula…
We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids, and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at near-zero temperatures. In this paper, we show analytically that the solution proposed in Ref. \cite{10LN} enjoys existence, uniqueness and regularity of the pre-factor. Furthermore, we present numerical results of the dynamical …
Using the Lagrangian transport of momentum, the Reynolds shear stress can be expressed in terms of basic turbulence parameters. In this view, the Reynolds stress gradient represents the lateral transport of streamwise momentum, balanced by the u2 transport, pressure and shear force terms in the momentum equation. We extend this formalism to other turbulence parameters such as diagonal components of the Reynolds stress, and also to more complex flows (boundary layer flows with adverse pressure gradients and pipe flows with swirl). Data from direct numerical simulations (DNS) are used to validat…
We present a unique method for solving for the Reynolds stress in turbulent canonical flows, based on the momentum balance for a control volume moving at the local mean velocity. A differential transform converts this momentum balance to a solvable form. Comparisons with experimental and computational data in simple geometries show quite good agreements. An alternate picture for the turbulence momentum transport is offered, as verified with data, where the turbulence momentum is transported by the mean velocity while being dissipated by viscosity. The net momentum transport is the Reynolds str…
The Reynolds number dependence of the dimensionless dissipation rate is derived directly from the Karman-Howarth equation in terms of the inverse of the integral scale Reynolds number and a coefficient which depends on the second- and third-order structure functions. Fitting this form to the results from a direct numerical simulation indicates that the coefficient is effectively constant.
Iterative coarse-graining procedure based on Wyld's perturbation expansion is applied to the problem of Navier-Stokes turbulence. It is shown that the low-order calculation gives the fixed-point Reynolds number $ Re_{fp}$ (coupling constant) almost identical to the Reynolds number of the recently discovered transition to anomalous scaling of the moments of {\bf "velocity derivatives"}. Using this result as a dynamic constraint, it is argued that in the vicinity of the fixed point (integral scale) the high-order non-linearities, generated by the procedure, are irrelevant. The infra-red divergen…
Using the Lagrangian transport of momentum, the Reynolds stress can be expressed in terms of basic turbulence parameters. The Reynolds stress gradient represents the lateral transport of stream-wise momentum, balanced by the u2 transport, pressure and shear force terms in the momentum equation. Data from direct numerical simulations (DNS) have been used to validate this approach, with a good degree of consistency and agreement. This expression for the Reynolds stress gradient adds a transport equation to the Reynolds-averaged Navier-Stokes (RANS) equation, so that the turbulence structure can …
The way the increment statistics of turbulent velocity fluctuations scale with the increment size is a centerpiece of turbulence theories. We report data on decaying turbulence in the Max Planck Variable Density Turbulence Tunnel (VDTT), which show an approach of the inertial range statistics toward a nontrivial shape at small scales. By correcting for the contributions of energy decay to the large-scale statistics with a model, we find the scaling exponent of the second-order velocity increment statistics to be independent of the Reynolds number and equal to $0.693\pm0.003$ for $2000\lesssim …
Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the EDQNM model and compared to results of the multifractal formalism. At the highest Reynolds number available in windtunnel experiments, $R_λ=2500$, both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For EDQNM, this correction is a finite Reynolds number effect, whereas for the multifractal formalism it is an intermittency correction that persists at any high Reynolds number. Furthermore, the …
Turbulent motions in a fluid relax at a certain rate once stirring has stopped. The role of the most basic parameter in fluid mechanics, the Reynolds number, in setting the relaxation rate is not generally known. This paper concerns the high-Reynolds-number limit of the process. In a classical grid-turbulence wind-tunnel experiment that both reached higher Reynolds numbers than ever before and covered a wide range of them ($10^4 < Re = UM/ν< 5\times10^6$), we measured the relaxation rate with the unprecedented precision of about 2\%. Here $U$ is the mean speed of the flow, $M$ the forcing scal…
Numerical simulations of turbulent round jets, using explicit-filtered LES technique, are performed, for three different Reynolds numbers(Re = 3600, 88000, 400000), to understand the Reynolds number effect on subsonic jets with Mach number 0.9. Eigth-order compact schemes are used for spatial derivative estimation. Second order Runge-Kutta method is used for Time evolution of flow. One-parameter fourth-order compact filter is used for low-pass explicit filtering of transported variables. Decreased centreline axial velocity decay rate and reduced turbulence intensities are observed as jet Reyno…
The small scales of turbulence in a high-Reynolds-number jet ($Re_λ \approx 350$) are investigated with a \textmu PIV setup to overcome the optical limitations of conventional tomographic PIV setups. With the aim of validating the performances of tomographic long-distance \textmu PIV, analyses are carried out involving statistical aspects of the small scales of turbulence. The technique is assessed and the data are bench-marked to be applied to the analysis of any three dimensional small-scale phenomena in large-scale flow domains.
There exists continuous demand of improved turbulence models for the closure of Reynolds Averaged Navier-Stokes (RANS) simulations. Machine Learning (ML) offers effective tools for establishing advanced empirical Reynolds stress closures on the basis of high fidelity simulation data. This paper presents a turbulence model based on the Deep Neural Network(DNN) which takes into account the non-linear relationship between the Reynolds stress anisotropy tensor and the local mean velocity gradient as well as the near-wall effect. The construction and the tuning of the DNN-turbulence model are detai…
The elementary structures of turbulence, i.e., vortex tubes, are studied using velocity data obtained in laboratory experiments for boundary layers and duct flows at microscale Reynolds numbers 332-1934. While past experimental studies focused on intense vortex tubes, the present study focuses on all vortex tubes with various intensities. We obtain the mean velocity profile. The radius scales with the Kolmogorov length. The circulation velocity scales with the Kolmogorov velocity, in contrast to the case of intense vortex tubes alone where the circulation velocity scales with the rms velocity …
Quantum turbulence (QT) generated by the Rayleigh-Taylor instability in binary immiscible ultracold 87Rb atoms at zero temperature is studied theoretically. We show that the quantum vortex tangle is qualitatively different from previously considered superfluids, which reveals deep relations between QT and classical turbulence. The present QT may be generated at arbitrarily small Mach numbers, which is a unique property not found in previously studied superfluids. By numerical solution of the coupled Gross-Pitaevskii equations we find that the Kolmogorov scaling law holds for the incompressible…
Direct numerical simulation is used to investigate the decay exponent of isotropic homogeneous turbulence over a range of Reynolds numbers sufficient to display both high and low Re number decay behavior. The initial turbulence is generated by the stirring action of the flow past many small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low-wavenumber and large scale behavior of the turbulent spectrum is generated by the fluid and is not imposed. It is shown that the decay exponent in the resulting turbulence closely matches the theoretical predictio…
In this paper, a turbulence model based on deep neural network is developed for turbulent flow around airfoil at high Reynolds numbers. According to the data got from the Spalart-Allmaras (SA) turbulence model, we build a neural network model that maps flow features to eddy viscosity. The model is then used to replace the SA turbulence model to mutually couple with the CFD solver. We build this suitable data-driven turbulence model mainly from the inputs, outputs features and loss function of the model. A feature selection method based on feature importance is also implemented. The results sho…
In experimental study of very high Reynolds number turbulence, we found evidences that there are distinguished vortex structures in the intermediate range, that is, between the Kolmogorov and Taylor microscales, where they are indeed expected to be present. These structures are responsible for the intermittency, and, in the same time, they contribute into asymmetry of turbulent statistics, the latter following from the Kolmogorov law.
New scaling relations for the mean velocity and Reynolds shear stress in viscous sublayer were proposed based on the application of matched asymptotic expansion method to the mean momentum balance. It was shown that the new parameter $Λ_+$ is relevant for the wall turbulence in addition to the friction velocity. From the proposed new scaling of the viscous sublayer the power law for the mean velocity in the overlap region was derived. The complete or incomplete similarity of the parameter $Λ_+$ was shown to be the key factor, which determines whether the mean velocity obeys the power law or lo…
Using exact relations between velocity structure functions (Hill, Hill and Boratav, and Yakhot) and neglecting pressure contributions in a first approximation, we obtain a closed system and derive simple order-dependent rescaling relationships between longitudinal and transverse structure functions. By means of numerical data with turbulent Reynolds numbers ranging from $\Re_λ=320$ to $\Re_λ=730$, we establish a clear correspondence between their respective scaling range, while confirming that their scaling exponents do differ. This difference does not seem to depend on Reynolds number. Making…
A symbolic regression-based implicit algebraic stress turbulence model: incorporating the production of non-dimensional Reynolds stress deviatoric tensor
The modified Townsend-Perry attached eddy model of Vassilicos et al (2015) combines the outer peak/plateau behaviour of rms streamwise turbulence velocity profiles and the Townsend-Perry log-decay of these profiles at higher distances from the wall. This model was validated by these authors for high Reynolds number tur- bulent pipe flow data and is shown here to describe equally well and with about the same parameter values turbulent boundary layer flow data from four different facilities and a wide range of Reynolds numbers. The model has predictive value as, when extrapolated to the extremel…
Short term unpredictability is discovered numerically for high Reynolds number fluid flows under periodic boundary conditions. Furthermore, the abundance of the short term unpredictability is also discovered. These discoveries support our theory that fully developed turbulence is constantly driven by such short term unpredictability.
We present measurements of fluid particle accelerations in turbulent water flows between counter-rotating disks using three-dimensional Lagrangian particle tracking. By simultaneously following multiple particles with sub-Kolmogorov-time-scale temporal resolution, we measured the spatial correlation of fluid particle acceleration at Taylor microscale Reynolds numbers between 200 and 690. We also obtained indirect, non-intrusive measurements of the Eulerian pressure structure functions by integrating the acceleration correlations. Our experimental data provide strong support to the theoretical …
Turbulence in fluids is an ubiquitous phenomenon, characterized by spontaneous transition of a smooth, laminar flow to rapidly changing, chaotic dynamics. In 1883, Reynolds experimentally demonstrated that, in an initially laminar flow of water, turbulent motions emerge without any measurable external disturbance. To this day, turbulence remains a major unresolved phenomenon in fluid mechanics; in particular, there is a lack of a mathematical model where turbulent dynamics emerge naturally from a laminar flow. Recently, we proposed a new theory of turbulence in gases, according to which turbul…
In this work we address the open problem of high Reynolds number limit in hydrodynamic turbulence, which we modify by considering a vanishing random (instead of deterministic) viscosity. In this formulation, a small-scale noise propagates to large scales in an inverse cascade, which can be described using qualitative arguments of the Kolmogorov-Obukhov theory. We conjecture that the limit of the resulting probability distribution exists as $\mathrm{Re} \to \infty$, and the limiting flow at finite time remains stochastic even if forcing, initial and boundary conditions are deterministic. This c…
The radial relative velocity between particles suspended in turbulent flow plays a critical role in droplet collision and growth. We present a simple and accurate approach to RV measurement in isotropic turbulence - planar 4-frame particle tracking velocimetry - using routine PIV hardware. This study demonstrates the feasibility of accurately measuring RV using routine hardware and verifies, for the first time, the path-history and inertial filtering effects on particle-pair RV at large particle separations experimentally.
The streamwise and transverse velocities are measured simultaneously in isotropic grid turbulence at relatively high Reynolds numbers, Re(lambda) = 110-330. Using a conditional averaging technique, we extract typical intermittency patterns, which are consistent with velocity profiles of a model for a vortex tube, i.e., Burgers vortex. The radii of the vortex tubes are several of the Kolmogorov length regardless of the Reynolds number. Using the distribution of an interval between successive enhancements of a small-scale velocity increment, we study the spatial distribution of vortex tubes. The…
For low-Reynolds number shear-flows of neutrally-buoyant suspensions, the shear stress is often modeled using an effective viscosity that depends only on the solid fraction. As the Reynolds number ($Re$) is increased and inertia becomes important, the effective viscosity also depends on the Reynolds number itself. The current experiments measure the torque for flows of neutrally-buoyant particles in a coaxial-cylinder rheometer for solid fractions, $ϕ$, from 10\% to 50\% and Reynolds numbers based on particle diameter from 2 to 1,000. For experiments for Reynolds of O(10) and solid fractions l…
The most elementary structures of turbulence, i.e., vortex tubes, are studied using velocity data obtained in a laboratory experiment for boundary layers with microscale Reynolds numbers 295-1258. We conduct conditional averaging for enhancements of a small-scale velocity increment and obtain the typical velocity profile for vortex tubes. Their radii are of the order of the Kolmogorov length. Their circulation velocities are of the order of the root-mean-square velocity fluctuation. We also obtain the distribution of the interval between successive enhancements of the velocity increment as the…
The evolution of buoyancy-driven homogeneous variable-density turbulence (HVDT) at Atwood numbers up to 0.75 and large Reynolds numbers is studied by using high-resolution Direct Numerical Simulations. To help understand the highly non-equilibrium nature of buoyancy-driven HVDT, the flow evolution is divided into four different regimes based on the behavior of turbulent kinetic energy derivatives. The results show that each regime has a unique type of dependency on both Atwood and Reynolds numbers. It is found that the local statistics of the flow based on the flow composition are more sensiti…
Data-driven turbulence modeling studies have reached such a stage that the fundamental framework is basically settled, but several essential issues remain that strongly affect the performance, including accuracy, smoothness, and generalization capacity. Two problems are studied in the current research: (1) the processing of the Reynolds stress tensor and (2) the coupling method between the machine learning turbulence model and flow solver. The first determines the form of predicting targets and the resulting physical completeness and interpretability. The second determines the training process…
The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as a naive extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the sca…
Direct numerical simulations (DNS) of fully-developed turbulent channel flows for very low Reynolds numbers have been performed with a larger computational box sizes than those of existing DNS. The friction Reynolds number was decreased down to 60, where the friction Reynolds number is based on the friction velocity and the channel half width. When the Reynolds number was decreased to 60 with small computational box size, the flow became laminar. Using a large box, we found that a localized turbulence was observed to sustain in the form of periodic oblique band. This type of locally disordered…
We study the fractal scaling of iso-levels sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The Schmidt number is unity. A fractal box-counting dimension $D_F$ can be obtained for iso-levels below about $3$ standard deviations of the scalar fluctuation on either side of its mean value. The dimension varies systematically with the iso-level, with a maximum of about $8/3$ for the iso-level at the mean; this maximum dimension also follows as an upper bound from the geometric measure theory. We interpret this result to mean that mix…
Developing data-driven subgrid-scale (SGS) models for large eddy simulations (LES) has received substantial attention recently. Despite some success, particularly in a priori (offline) tests, challenges have been identified that include numerical instabilities in a posteriori (online) tests and generalization (i.e., extrapolation) of trained data-driven SGS models, for example to higher Reynolds numbers. Here, using the stochastically forced Burgers turbulence as the test-bed, we show that deep neural networks trained using properly pre-conditioned (augmented) data yield stable and accurate a …
This paper examines the behavior of the dimensionless dissipation rate $C_{\varepsilon}$ for stationary and nonstationary magnetohydrodynamic (MHD) turbulence in presence of external forces. By combining with previous studies for freely decaying MHD turbulence, we obtain here both the most general model equation for $C_{\varepsilon}$ applicable to homogeneous MHD turbulence and a comprehensive numerical study of the Reynolds number dependence of the dimensionless total energy dissipation rate at unity magnetic Prandtl number. We carry out a series of medium to high resolution direct numerical …
Spatial and spectral energy exchanges associated with the turbulence kinetic energy per unit mass, or half-trace of the velocity covariance tensor, are studied in an anisothermal low Mach number turbulent channel flow. The temperatures of the two channel walls are 293 K and 586 K. This generates a strong temperature gradient in the wall-normal direction. The effect of the temperature gradient on the energy exchanges is investigated using two direct numerical simulations of the channel, at the mean friction Reynolds numbers 180 and 395. The temperature gradient creates an asymmetry between the …
Numerical simulations based on Reynolds-Averaged Navier--Stokes (RANS) equations are widely used in engineering design and analysis involving turbulent flows. However, RANS simulations are known to be unreliable in many flows of engineering relevance, which is largely caused by model-form uncertainties associated with the Reynolds stresses. Recently, a machine-learning approach has been proposed to assist RANS modeling by building a functional mapping from mean flow features to discrepancies in RANS modeled Reynolds stresses as compared to high-fidelity data. However, it remains a challenge to…
Very recently, a defect model which depicts the growth tendency of the near-wall peak of the streamwise turbulence intensity has been developed (Chen $\&$ Sreenivasan, J. Fluid Mech. (2021), vol.908, R3). Based on the finiteness of the near-wall turbulence production, this model predicts that the magnitude of the peak will approach a finite limit as the Reynolds number increases. In the present study, we revisit the basic hypotheses of the model, such as the balance between the turbulence production and the wall dissipation in the region of peak production, the negligible effects of the logari…
The turbulence problem at the level of scaling exponents is hard in part because of the multifractal scaling of small scales, which demands that each moment order be treated and understood independently. This conclusion derives from studies of velocity structure functions, energy dissipation, enstrophy density (that is, square of vorticity), etc. However, it is likely that there exist other physically pertinent quantities with uncomplicated structure in the inertial range, potentially resulting in huge simplifications in the turbulence theory. We show that velocity circulation around closed lo…
Large-scale magnetic fields thread through the electrically conducting matter of the interplanetary and interstellar medium, stellar interiors, and other astrophysical plasmas, producing anisotropic flows with regions of high-Reynolds-number turbulence. It is common to encounter turbulent flows structured by a magnetic field with a strength approximately equal to the root-mean-square magnetic fluctuations. In this work, direct numerical simulations of anisotropic magnetohydrodynamic (MHD) turbulence influenced by such a magnetic field are conducted for a series of cases that have identical res…
Extreme dissipation events in turbulent flows are rare, but they can be orders of magnitude stronger than the mean dissipation rate. Despite its importance in many small-scale physical processes, there is presently no accurate theory or model for predicting the extrema as a function of the Reynolds number. Here, we introduce a new model for the dissipation PDF based on the concept of significant shear layers, which are thin regions of elevated local mean dissipation. At very high Reynolds numbers these significant shear layers develop layered substructures. The flow domain is divided into the …
We experimentally investigate the flow of a viscoelastic fluid in a parallel shear geometry at low Reynolds number. As the flow becomes unstable via a nonlinear subcritical instability, velocimetry measurements show non-periodic fluctuations over a broad range of frequencies and wavelengths, consistent with the main features of elastic turbulence. Using the same experimental setup, we compare these features to those in the flow around cylinders, which is upstream to the parallel shear region; we find significant differences in power spectra scaling, intermittency statistics, and flow structure…
