Scaling Laws In Turbulence Research Articles

Numerical validation of scaling laws for stratified turbulence

Recent theoretical progress using multiscale asymptotic analysis has revealed various possible regimes of stratified turbulence. Notably, buoyancy transport can either be dominated by advection or diffusion, depending on the effective Péclet number of the flow. Two types of asymptotic models have been proposed, which yield measurably different predictions for the characteristic vertical velocity and length scale of the turbulent eddies in both diffusive and non-diffusive regimes. The first, termed a ‘single-scale model’, is designed to describe flow structures having large horizontal and small vertical scales, while the second, termed a ‘multiscale model’, additionally incorporates flow features with small horizontal scales, and reduces to the single-scale model in their absence. By comparing predicted vertical velocity scaling laws with direct numerical simulation data, we show that the multiscale model correctly captures the properties of strongly stratified turbulence within regions dominated by small-scale isotropic motions, whose volume fraction decreases as the stratification increases. Meanwhile its single-scale reduction accurately describes the more orderly, layer-like, quiescent flow outside those regions.

Unsteady dissipation scaling in static- and active-grid turbulence

A new time-dependent analysis of the global and local fluctuating velocity signals in grid turbulence is conducted to assess the scaling laws for non-equilibrium turbulence. Experimental datasets of static- and active-grid turbulence with different Rossby numbers $R_o({=}U/\varOmega M$ : $U$ is the mean velocity, $\varOmega$ is the mean rotation rate and $M$ is the grid mesh size) are considered. Although the global (long-time-averaged) non-dimensional dissipation rate $C_\varepsilon$ is independent of the Reynolds number $Re_\lambda$ based on the global Taylor microscale, the local (short-time-averaged) non-dimensional dissipation rate $\left \langle C_\varepsilon (t_i) \right \rangle$ ( $t_i$ is the local time) both in the static- and active-grid turbulence clearly show the non-equilibrium scaling $\left \langle C_\varepsilon (t_i)\right \rangle / \sqrt {Re_0} \propto \left \langle Re_\lambda (t_i) \right \rangle ^{-1}$ ( $\left \langle Re_\lambda (t_i) \right \rangle$ and $Re_0$ are the Reynolds numbers based on the local Taylor microscale $\lambda (t_i)$ and the global integral length scale, respectively), which has only been confirmed for global statistics in the near field of grid turbulence. The local value of $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ ( $L(t_i)$ is the local integral length scale) shifts from the equilibrium to non-equilibrium scaling as $\left \langle Re_\lambda (t_i) \right \rangle$ increases, further confirming that the non-equilibrium scalings are recovered for local statistics both in the static- and active-grid turbulence. The local values of $\left \langle C_\varepsilon (t_i) \right \rangle$ and $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ follow the theoretical predictions for global statistics (Bos & Rubinstein, Phys. Rev. Fluids, vol. 2, 2017, 022601).

Kubo combinatorics for turbulence scaling laws

We present an extension to Kolmogorov–Obukhov refined similarity hypotheses for universal fully-developed turbulence. The extension is applied within Z. She and E. Leveque’s multifractal model of inertial range scaling and its generalizations. Our development has relevance to universal fully developed turbulence, a state we describe explicitly under the additional assumptions of the Kolmogorov–Obukhov similarity hypotheses in terms of the coupling between velocity fluctuations and averaged energy dissipation at all orders. This description is unique and leads to a reparametrization of the She–Leveque model that preserves its original forecasts and is infinitely divisible.

Scaling laws of statistics of wall-bounded turbulence at supercritical pressure: Evaluation and mechanism

A growing body of studies support that the real fluid effects related to the abrupt density changes in supercritical fluids significantly affect statistical properties of turbulence, yet developing appropriate scaling laws for wall-bounded turbulence at supercritical state is still difficult. In the present study, we conduct direct numerical simulations on channel flows of supercritical fluids to evaluate the usefulness of classical scaling developed for variable-property flows. We find that the expressions based on semi-local scaling [ϕ=f(y*,Reτ*) and ϕ=f(y*,Reτ*,Pr*)] fail to collapse the statistical profiles at supercritical pressure. We analyze the mechanism of the failure of semi-local scaling by quantifying the modulations of turbulent structures of supercritical fluids due to changes in fluid properties. The intensified ejection and sweep of low-speed streaks destabilize the stream-wise streaks and reduce the stream-wise coherence, changing the statistics and affecting the usefulness of semi-local scaling. To shed light on the scaling laws of fluctuating velocities, we finally examine the hypotheses in Townsend wall-attached eddy theory in the context of flows at a supercritical state. It is found that the attached eddies are self-similar near-wall structures, which result in the logarithmic profiles of stream-wise and span-wise velocity fluctuations; the population density of the attached eddies can be well approximated by an exponential scaling.

Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach.

The calculation of turbulence statistics is considered the key unsolved problem of fluid mechanics, i.e., precisely the computation of arbitrary statistical velocity moments from first principles alone. Using symmetry theory, we derive turbulent scaling laws for moments of arbitrary order in two regions of a turbulent channel flow. Besides the classical scaling symmetries of space and time, the key symmetries for the present work reflect the two well-known characteristics of turbulent flows: non-Gaussianity and intermittency. To validate the new scaling laws we made a new simulation at an unprecedented friction Reynolds number of 10 000, large enough to test the new scaling laws. Two key results appear as an application of symmetry theory, which allowed us to generate symmetry invariant solutions for arbitrary orders of moments for the underlying infinite set of moment equations. First, we show that in the sense of the generalization of the deficit law all moments of the streamwise velocity in the channel center follow a power-law scaling, with exponents depending on the first and second moments alone. Second, we show that the logarithmic law of the mean streamwise velocity in wall-bounded flows is indeed a valid solution of the moment equations, and further, all higher moments in this region follow a power law, where the scaling exponent of the second moment determines all higher moments. With this we give a first complete mathematical framework for all moments in the log region, which was first discovered about 100years ago.

Hydrodynamics of flow through a degraded channel bed

This article presents experimental results of turbulent flow measured in a bimodal degraded channel bed consisting of sand-gravel mixture. Sand and gravel of uniform sizes 0.25 and 3.5 mm were mixed in the same proportions (by weight) to create a bimodal sedimentary bed. A three-dimensional Vectrino velocimeter was employed to collect three-dimensional velocities over bimodal degraded bed under equilibrium condition. The streamwise velocity, Reynolds stresses, turbulent kinetic energy (TKE), and TKE fluxes profiles were compared with the literature. However, the advancement of the existing knowledge was done by exploring the laws of turbulence. To this end, the velocity structure function method was applied. Second and third-order streamwise velocity structure functions followed by mixed third-order velocity structure functions revealed the existence of inertial subrange. The TKE dissipation rate was estimated using Kolmogorov’s and Monin–Yaglom’s scaling laws of turbulence. The anisotropy analysis indicated anisotropic turbulence in the near-bed, whereas above the initial bed-level, the anisotropy tends to follow three-dimensional isotropy. The present study notably enhances the understanding of turbulent flow through a degraded bed by demonstrating the legitimacy of laws of turbulence at different locations over the bed and providing a comprehensible acquaintance in TKE budget and Reynolds stress anisotropy.

Scaling law of mixing layer in cylindrical Rayleigh-Taylor turbulence

The nonlinear evolution of mixing layer in cylindrical Rayleigh-Taylor (RT) turbulence is studied theoretically and numerically. The scaling laws including the hyperbolic cosine growth for outward mixing layer and the cosine growth for inward mixing layer of the cylindrical RT turbulence are proposed for the first time and verified reliably by direct numerical simulation of the Navier-Stokes equations. It is identified that the scaling laws for the cylindrical RT turbulence transcend the classical power law for the planar RT turbulence and can be recovered to the quadratic growth as cylindrical geometry effect vanishes. Further, characteristic time- and length scales are reasonably obtained based on the scaling laws to reveal the self-similar evolution features for the cylindrical RT turbulence.

Empirical scaling laws for wall-bounded turbulence deduced from direct numerical simulations

The dependence of turbulence statistics in wall-bounded flows on the friction Reynolds number $R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$ is complex. Luchini proposed an empirical law for the mean velocity ${U}^{+}$ which has a dependence proportional to $1/R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$ at fixed ${y}^{+}$, and agrees well with direct numerical simulation for channel (Poiseuille), Couette, and pipe flow. We test similar laws for the Reynolds stresses and their budgets. The figure shows the wall-normal Reynolds stress $\overline{{v}^{+}\phantom{\rule{0}{0ex}}{v}^{+}}$ in channel. The two lower curves are results at two Reynolds numbers, and the upper one the extrapolation to infinite $R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$. The symbols show an experiment in pipe at higher $R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$.

Superdiffusion of quantized vortices uncovering scaling laws in quantum turbulence

Generic scaling laws, such as Kolmogorov's 5/3 law, are milestone achievements of turbulence research in classical fluids. For quantum fluids such as atomic Bose-Einstein condensates, superfluid helium, and superfluid neutron stars, turbulence can also exist in the presence of a chaotic tangle of evolving quantized vortex lines. However, due to the lack of suitable experimental tools to directly probe the vortex-tangle motion, so far little is known about possible scaling laws that characterize the velocity correlations and trajectory statistics of the vortices in quantum-fluid turbulence, i.e., quantum turbulence (QT). Acquiring such knowledge could greatly benefit the development of advanced statistical models of QT. Here we report an experiment where a tangle of vortices in superfluid 4He are decorated with solidified deuterium tracer particles. Under experimental conditions where these tracers follow the motion of the vortices, we observed an apparent superdiffusion of the vortices. Our analysis shows that this superdiffusion is not due to Lévy flights, i.e., long-distance hops that are known to be responsible for superdiffusion of random walkers. Instead, a previously unknown power-law scaling of the vortex-velocity temporal correlation is uncovered as the cause. This finding may motivate future research on hidden scaling laws in QT.

Nanoflare Theory Revisited

Abstract At any scale l in the turbulent inertial range, the magnetic field can be divided up into a large-scale component and a small-scale, high spatial frequency component which undergoes magnetic reversals. Such local reconnections, i.e., on any inertial scale l, seem to be an inseparable part of magnetohydrodynamic (MHD) turbulence, whose collective outcome can lead to global reconnection with a rate independent of the small-scale physics dominant at dissipative scales. We show that this picture, known as stochastic reconnection, is intimately related to nanoflare theory, proposed long ago to explain solar coronal heating. We argue that, due to stochastic flux freezing, a generalized version of magnetic flux freezing in turbulence, the field follows the flow in a statistical sense. Turbulence bends and stretches the field, increasing its spatial complexity. Strong magnetic shears associated with such a highly tangled field can trigger local reversals and field annihilations on a wide range of inertial scales which convert magnetic energy into kinetic and thermal energy. The former may efficiently enhance turbulence and the latter heat generation. We support this theoretical picture using scaling laws of MHD turbulence and also recent analytical and numerical studies which suggest a statistical correlation between magnetic spatial complexity and energy dissipation. Finally, using an MHD numerical simulation, we show that the time evolution of the magnetic complexity is statistically correlated with the rate of kinetic energy injection and/or magnetic-to-thermal energy conversion, in agreement with our proposed theoretical picture.

Scaling laws in turbulence.

Following the idea that dissipation in turbulence at high Reynolds number is dominated by singular events in space-time and described by solutions of the inviscid Euler equations, we draw the conclusion that in such flows, scaling laws should depend only on quantities appearing in the Euler equations. This excludes viscosity or a turbulent length as scaling parameters and constrains drastically possible analytical pictures of this limit. We focus on the drag law deduced by Newton for a projectile moving quickly in a fluid at rest. Inspired by this Newton's drag force law (proportional to the square of the speed of the moving object in the limit of large Reynolds numbers), which is well verified in experiments when the location of the detachment of the boundary layer is defined, we propose an explicit relationship between the Reynolds stress in the turbulent wake and quantities depending on the velocity field (averaged in time but depending on space). This model takes the form of an integrodifferential equation for the velocity which is eventually solved for a Poiseuille flow in a circular pipe.

The hierarchy of multi-point probability density functions and characteristic functions in compressible turbulence

The hierarchies of equations for a general multi-point probability density function (PDF) and its characteristic function (CF) are derived for compressible turbulent flows, obeying the ideal gas law. The closure problem of turbulence is clearly exhibited in each of the approaches, with n-point statistics being dependent on the (n + 1)-point statistics and, for some cases, even the (n + 2)-point statistics. When dynamic viscosity and heat conductivity are dependent on temperature as a power-law, the CF hierarchy could contain fractional derivatives if the exponent is a non-integer. The additional conditions satisfied by all the PDFs and CFs in both the hierarchies are also prescribed. The PDF and CF equations derived in this paper, with the unclosed terms explicitly written in terms of higher order PDF/CF, act as a starting point in constructing symmetry-based invariant solutions of compressible turbulence, analogous to the works of Wacławczyk et al. [“Statistical symmetries of the Lundgren–Monin–Novikov hierarchy,” Phys. Rev. E 90, 013022 (2014)] and Oberlack and Rosteck [“New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws,” Discrete Contin. Dyn. Syst. 3, 451–471 (2010)] for incompressible turbulence.

Compressibility Effects on the Scalar Dissipation Rate

ABSTRACT Scalar dissipation rate plays an important role in combustion, especially for fast chemistry in non-premixed flames. Even for finite-rate chemistry, it is a parameter of major importance in models based on flamelets, probability density function methods, and conditional moment closures. Compressibility can influence the behavior of scalar dissipation but existing models do not consider them in the form of large dilatational effects, which are present in combustion in high-speed propulsion systems such as scramjets in hypersonic flight. This paper considers these effects on the scalar dissipation rate, in particular its mean value, spectral density, probability density function, as well the alignment of the scalar gradient with the strain field of the flow. The major result of this study is that the majority of the compressibility effects on inertial-type scales can be taken into account if the classical scaling laws of incompressible turbulence are rescaled by taking only the solenoidal contributions. However, additional effects of compressibility need to be considered at smaller scales.

Symmetries and turbulence modeling

This work applies new insights into turbulent statistics gained by Lie symmetry analysis to the closure problem of turbulence. Founded in the mathematics of partial differential equations, Lie symmetries have helped advances in many fields of modern physics. The main reason for this is their ability to encode important physical principles that are implicitly expressed by governing equations. Newly discovered symmetries of the multi-point correlation equations describing turbulent motion have been shown to encode two central effects of turbulent statistics: intermittency and non-Gaussianity. Moreover, these symmetries play a pivotal role in obtaining turbulent scaling laws such as the logarithmic law of the wall. Evidently, correctly preserving these symmetry properties in a turbulence model would render it capable of accurately predicting important effects of turbulent statistics and turbulent scaling. As these symmetry constraints have so far not been taken into account when devising turbulence models, we present a completely new modeling framework that can yield models fulfilling these conditions. In order to accomplish this, it turns out to be helpful, if not necessary, to introduce an entirely new symmetry-based modeling strategy that allows systematically constructing equations based on symmetry constraints imposed on them. From these considerations, it can be shown that in order to create meaningful turbulence models that fulfill these constraints, it is necessary to introduce a new velocity and pressure field. A possible skeleton of model equations for second moment closure is presented.

The midrange wavenumber spectrum of van Gogh’s Starry Night does not obey a turbulent inertial range scaling law

ABSTRACTThe swirling night sky painted by van Gogh in his Starry Night painting has been interpreted as a representation of turbulence. Indeed, a popular myth is that this painting quantitatively (albeit coincidentally) captures the scaling behaviour of actual turbulence. One of the most familiar scaling laws in turbulence is the Kolmogorov scaling law, whereby the two-point autocorrelation of the velocity field obeys a −5/3 power law dependence in the midrange (‘inertial range') wavenumbers. Compressible turbulence also obeys a similar power law, but with an exponent between −5/3 and −2, depending on Mach number. An examination of the spectral dependence of a two-point autocorrelation of the image intensity in midrange wavenumbers shows that van Gogh’s Starry Night does obey a power law, but instead with an exponent of −1. Examination of various other van Gogh and impressionist artists’ paintings also do not exhibit the correct power law exponent for a turbulent flow. Thus, van Gogh’s Starry Night painting does not accurately embody one of the most fundamental features of turbulent flows, weakening the foundations of the popular and commonly reiterated myth that this painting captures turbulence in a mathematically correct way.

Scaling laws in Hall inertial-range turbulence

Abstract. There is an increasing amount of observational evidence in space plasmas for the breakdown of inertial-range spectra of magnetohydrodynamic (MHD) turbulence on spatial scales smaller than the ion-inertial length. Magnetic energy spectra often exhibit a steepening, which is reminiscent of dissipation of turbulence energy, for example in wave–particle interactions. Electric energy spectra, on the other hand, tend to be flatter than those of MHD turbulence, which is indicative of a dispersive process converting magnetic into electric energy in electromagnetic wave excitation. Here we develop a model of the scaling laws and the power spectra for the Hall inertial range in plasma turbulence. In the present paper we consider a two-dimensional geometry with no wave vector component parallel to the magnetic field as is appropriate in Hall MHD. A phenomenological approach is taken. The Hall electric field attains an electrostatic component when the wave vectors are perpendicular to the mean magnetic field. The power spectra of Hall turbulence are steep for the magnetic field with a slope of -7/3 for compressible magnetic turbulence; they are flatter for the Hall electric field with a slope of -1/3. Our model for the Hall turbulence gives a possible explanation for the steepening of the magnetic energy spectra in the solar wind as an indication of neither the dissipation range nor the dispersive range but as the Hall inertial range. Our model also reproduces the shape of energy spectra in Kelvin–Helmholtz turbulence observed at the Earth's magnetopause.

On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation

A temporally evolving turbulent plane jet is studied both by direct numerical simulation (DNS) and Lie symmetry analysis. The DNS is based on a high-order scheme to solve the Navier–Stokes equations for an incompressible fluid. Computations were conducted at Reynolds number $\mathit{Re}_{0}=8000$, where $\mathit{Re}_{0}$ is defined based on the initial jet thickness, $\unicode[STIX]{x1D6FF}_{0.5}(0)$, and the initial centreline velocity, $\overline{U}_{1}(0)$. A symmetry approach, known as the Lie group, is used to find symmetry transformations, and, in turn, group invariant solutions, which are also denoted as scaling laws in turbulence. This approach, which has been extensively developed to create analytical solutions of differential equations, is presently applied to the mean momentum and two-point correlation equations in a temporally evolving turbulent plane jet. The symmetry analysis of these equations allows us to derive new invariant (self-similar) solutions for the mean flow and higher moments of the velocities in the jet flow. The current DNS validates the consequence of Lie symmetry analysis and therefore confirms the establishment of novel scaling laws in turbulence. It is shown that the classical scaling law for the mean velocity is a specific form of the current scaling (which has a more general form); however, the scaling for the second and higher moments (such as Reynolds stresses) has a completely different structure compared to the classical scaling. While the failure of the classical scaling for the second moments of the fluctuating velocities has been noted from the jet data for many years, the DNS results nicely match with the present self-similar relations derived from Lie symmetry analysis. Key ingredients for the present results, in particular for the scaling laws of the higher moments, are symmetries, which are of a purely statistical nature. i.e. these symmetries are admitted by the moment equations, however, they are not observed by the original Navier–Stokes equations.

Potential landscape and flux field theory for turbulence and nonequilibrium fluid systems

Turbulence is a paradigm for far-from-equilibrium systems without time reversal symmetry. To capture the nonequilibrium irreversible nature of turbulence and investigate its implications, we develop a potential landscape and flux field theory for turbulent flow and more general nonequilibrium fluid systems governed by stochastic Navier–Stokes equations. We find that equilibrium fluid systems with time reversibility are characterized by a detailed balance constraint that quantifies the detailed balance condition. In nonequilibrium fluid systems with nonequilibrium steady states, detailed balance breaking leads directly to a pair of interconnected consequences, namely, the non-Gaussian potential landscape and the irreversible probability flux, forming a ‘nonequilibrium trinity’. The nonequilibrium trinity characterizes the nonequilibrium irreversible essence of fluid systems with intrinsic time irreversibility and is manifested in various aspects of these systems. The nonequilibrium stochastic dynamics of fluid systems including turbulence with detailed balance breaking is shown to be driven by both the non-Gaussian potential landscape gradient and the irreversible probability flux, together with the reversible convective force and the stochastic stirring force. We reveal an underlying connection of the energy flux essential for turbulence energy cascade to the irreversible probability flux and the non-Gaussian potential landscape generated by detailed balance breaking. Using the energy flux as a center of connection, we demonstrate that the four-fifths law in fully developed turbulence is a consequence and reflection of the nonequilibrium trinity. We also show how the nonequilibrium trinity can affect the scaling laws in turbulence.

Scaling laws of wave-cascading superfluid turbulence

Phenomenological model is constructed for superfluid turbulence for two distinct energy cascade scenarios, sound wave cascade and critically-balanced Kelvin wave cascade, using the method for magneto-fluid turbulence theory. Excitations along dispersion relations are used as the primary energy reservoir. The spectral indices in the inertial range are estimated as −3/2 for the long-wavelength sound wave cascade, −3 in the direction to the mean filaments for the Kelvin wave cascade, and −5/3 perpendicular to the filament direction.

The extended law of star formation: the combined role of gas and stars

We present a model for the origin of the extended law of star formation in which the surface density of star formation ($\Sigma_{\rm SFR}$) depends not only on the local surface density of the gas ($\Sigma_{g}$), but also on the stellar surface density ($\Sigma_{*}$), the velocity dispersion of the stars, and on the scaling laws of turbulence in the gas. We compare our model with the spiral, face-on galaxy NGC 628 and show that the dependence of the star formation rate on the entire set of physical quantities for both gas and stars can help explain both the observed general trends in the $\Sigma_{g}-\Sigma_{\rm SFR}$ and $\Sigma_{*}-\Sigma_{\rm SFR}$ relations, but also, and equally important, the scatter in these relations at any value of $\Sigma_{g}$ and $\Sigma_{*}$. Our results point out to the crucial role played by existing stars along with the gaseous component in setting the conditions for large scale gravitational instabilities and star formation in galactic disks.