Phenomenology of transition to quantum turbulence in flows of superfluid helium.

Transition from laminar to turbulent states of classical viscous fluids is complex and incompletely understood. Transition to quantum turbulence (QT), by which we mean the turbulent motion of quantum fluids such as helium II, whose physical properties depend on quantum physics in some crucial respects, is naturally more complex. This increased complexity arises from superfluidity, quantization of circulation, and, at finite temperatures below the critical, the two-fluid behavior. Transition to QT could involve, as an initial step, the transition of the classical component, or the intrinsic or extrinsic nucleation of quantized vortices in the superfluid component, or a simultaneous occurrence of both scenarios-and the subsequent interconnected evolution. In spite of the multiplicity of scenarios, aspects of transition to QT can be understood at a phenomenological level on the basis of some general principles, and compared meaningfully with transition in classical flows.

Forecasting small-scale dynamics of fluid turbulence using deep neural networks.

Turbulence in fluid flows is characterized by a wide range of interacting scales. Since the scale range increases as some power of the flow Reynolds number, a faithful simulation of the entire scale range is prohibitively expensive at high Reynolds numbers. The most expensive aspect concerns the small-scale motions; thus, major emphasis is placed on understanding and modeling them, taking advantage of their putative universality. In this work, using physics-informed deep learning methods, we present a modeling framework to capture and predict the small-scale dynamics of turbulence, via the velocity gradient tensor. The model is based on obtaining functional closures for the pressure Hessian and viscous Laplacian contributions as functions of velocity gradient tensor. This task is accomplished using deep neural networks that are consistent with physical constraints and explicitly incorporate Reynolds number dependence to account for small-scale intermittency. We then utilize a massive direct numerical simulation database, spanning two orders of magnitude in the large-scale Reynolds number, for training and validation. The model learns from low to moderate Reynolds numbers and successfully predicts velocity gradient statistics at both seen and higher (unseen) Reynolds numbers. The success of our present approach demonstrates the viability of deep learning over traditional modeling approaches in capturing and predicting small-scale features of turbulence.

Hybrid quantum algorithms for flow problems.

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with end-to-end practical applications-in this instance, fluid dynamics. We debut here a high-performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well-known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates [Formula: see text] (a parameter crucial to Hamiltonian simulations) to the condition number [Formula: see text] of the simulation matrix and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for a) the functional form or sparse quantum state preparation and b) in situ quantum postprocessing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be [Formula: see text], where [Formula: see text] is the size of the linear system of equations, [Formula: see text] is the solution error, and [Formula: see text] is the error in postprocessing. This work suggests a path toward quantum simulation of fluid flows and highlights the special considerations needed at the gate-level implementation of QC.

Self-similar Rayleigh-Taylor mixing with accelerations varying in time and space.

As a ubiquitous paradigm of instabilities and mixing that occur in instances as diverse as supernovae, plasma fusion, oil recovery, and nanofabrication, the Rayleigh-Taylor (RT) problem is rightly regarded as important. The acceleration of the fluid medium in these instances often depends on time and space, whereas most past studies assume it to be constant or impulsive. Here, we analyze the symmetries of RT mixing for variable accelerations and obtain the scaling of correlations and spectra for classes of self-similar dynamics. RT mixing is shown to retain the memory of deterministic conditions for all accelerations, with the dynamics ranging from superballistic to subdiffusive. These results contribute to our understanding and control of the RT phenomena and reveal specific conditions under which Kolmogorov turbulence might be realized in RT mixing.

The area rule for circulation in three-dimensional turbulence.

An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called area rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the area rule, for both planar and nonplanar loops, using high-resolution data from direct numerical simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is farther from unity and when the moment order increases. The differences do not exceed about 5% for any condition examined here. The aspect ratio dependence observed for the second-order moment is indistinguishable from results for the Gaussian random field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller ( < 2%) but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence.

Phenomenology of quantum turbulence in superfluid helium.

Quantum turbulence-the stochastic motion of quantum fluids such as 4He and 3He-B, which display pure superfluidity at zero temperature and two-fluid behavior at finite but low temperatures-has been a subject of intense experimental, theoretical, and numerical studies over the last half a century. Yet, there does not exist a satisfactory phenomenological framework that captures the rich variety of experimental observations, physical properties, and characteristic features, at the same level of detail as incompressible turbulence in conventional viscous fluids. Here we present such a phenomenology that captures in simple terms many known features and regimes of quantum turbulence, in both the limit of zero temperature and the temperature range of two-fluid behavior.

Saturation and Multifractality of Lagrangian and Eulerian Scaling Exponents in Three-Dimensional Turbulence.

Inertial-range scaling exponents for both Lagrangian and Eulerian structure functions are obtained from direct numerical simulations of isotropic turbulence in triply periodic domains at Taylor-scale Reynolds number up to 1300. We reaffirm that transverse Eulerian scaling exponents saturate at ≈2.1 for moment orders p≥10, significantly differing from the longitudinal exponents (which are predicted to saturate at ≈7.3 for p≥30 from a recent theory). The Lagrangian scaling exponents likewise saturate at ≈2 for p≥8. The saturation of Lagrangian exponents and transverse Eulerian exponents is related by the same multifractal spectrum by utilizing the well-known frozen hypothesis to relate spatial and temporal scales. Furthermore, this spectrum is different from the known spectra for Eulerian longitudinal exponents, suggesting that Lagrangian intermittency is characterized solely by transverse Eulerian intermittency. We discuss possible implications of this outlook when extending multifractal predictions to the dissipation range, especially for Lagrangian acceleration.

Classical 1/3 scaling of convection holds up to Ra = 1015.

The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh number [Formula: see text]-the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells for [Formula: see text] have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh-Bénard convection flows in a slender cylindrical cell of aspect ratio [Formula: see text], that the Nusselt number-the dimensionless measure of heat transport-follows the classical power law of [Formula: see text] up to [Formula: see text] Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at all [Formula: see text] considered here, increasing with increasing [Formula: see text], and suggest that an abrupt transition of the boundary layer to turbulence does not take place.

Scaling of Acceleration Statistics in High Reynolds Number Turbulence.

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 an 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 scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

Laws of turbulence decay from direct numerical simulations.

Inspection of available data on the decay exponent for the kinetic energy of homogeneous and isotropic turbulence (HIT) shows that it varies by as much as 100%. Measurements and simulations often show no correspondence with theoretical arguments, which are themselves varied. This situation is unsatisfactory given that HIT is a building block of turbulence theory and modelling. We take recourse to a large base of direct numerical simulations and study decaying HIT for a variety of initial conditions. We show that the Kolmogorov decay exponent and the Birkhoff-Saffman decay are both observed, albeit approximately, for long periods of time if the initial conditions are appropriately arranged. We also present, for both cases, other turbulent statistics such as the velocity derivative skewness, energy spectra and dissipation, and show that the decay and growth laws are approximately as expected theoretically, though the wavenumber spectrum near the origin begins to change relatively quickly, suggesting that the invariants do not strictly exist. We comment briefly on why the decay exponent has varied so widely in past experiments and simulations. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Turbulent mixing: A perspective.

Mixing of initially distinct substances plays an important role in our daily lives as well as in ecological and technological worlds. From the continuum point of view, which we adopt here, mixing is complete when the substances come together across smallest flow scales determined in part by molecular mechanisms, but important stages of the process occur via the advection of substances by an underlying flow. We know how smooth flows enable mixing but less well the manner in which a turbulent flow influences it; but the latter is the more common occurrence on Earth and in the universe. We focus here on turbulent mixing, with more attention paid to the postmixing state than to the transient process of initiation. In particular, we examine turbulent mixing when the substance is a scalar (i.e., characterized only by the scalar property of its concentration), and the mixing process does not influence the flow itself (i.e., the scalar is "passive"). This is the simplest paradigm of turbulent mixing. Within this paradigm, we discuss how a turbulently mixed state depends on the flow Reynolds number and the Schmidt number of the scalar (the ratio of fluid viscosity to the scalar diffusivity), point out some fundamental aspects of turbulent mixing that render it difficult to be addressed quantitatively, and summarize a set of ideas that help us appreciate its physics in diverse circumstances. We consider the so-called universal and anomalous features and summarize a few model studies that help us understand them both.

Reconnection scaling in quantum fluids.

Fundamental to classical and quantum vortices, superconductors, magnetic flux tubes, liquid crystals, cosmic strings, and DNA is the phenomenon of reconnection of line-like singularities. We visualize reconnection of quantum vortices in superfluid 4He, using submicrometer frozen air tracers. Compared with previous work, the fluid was almost at rest, leading to fewer, straighter, and slower-moving vortices. For distances that are large compared with vortex diameter but small compared with those from other nonparticipating vortices and solid boundaries (called here the intermediate asymptotic region), we find a robust 1/2-power scaling of the intervortex separation with time and characterize the influence of the intervortex angle on the evolution of the recoiling vortices. The agreement of the experimental data with the analytical and numerical models suggests that the dynamics of reconnection of long straight vortices can be described by self-similar solutions of the local induction approximation or Biot-Savart equations. Reconnection dynamics for straight vortices in the intermediate asymptotic region are substantially different from those in a vortex tangle or on distances of the order of the vortex diameter.

Deep learning in turbulent convection networks.

We explore heat transport properties of turbulent Rayleigh-Bénard convection in horizontally extended systems by using deep-learning algorithms that greatly reduce the number of degrees of freedom. Particular attention is paid to the slowly evolving turbulent superstructures-so called because they are larger in extent than the height of the convection layer-which appear as temporal patterns of ridges of hot upwelling and cold downwelling fluid, including defects where the ridges merge or end. The machine-learning algorithm trains a deep convolutional neural network (CNN) with U-shaped architecture, consisting of a contraction and a subsequent expansion branch, to reduce the complex 3D turbulent superstructure to a temporal planar network in the midplane of the layer. This results in a data compression by more than five orders of magnitude at the highest Rayleigh number, and its application yields a discrete transport network with dynamically varying defect points, including points of locally enhanced heat flux or "hot spots." One conclusion is that the fraction of heat transport by the superstructure decreases as the Rayleigh number increases (although they might remain individually strong), correspondingly implying the increased importance of small-scale background turbulence.

Turbulence is an Ineffective Mixer when Schmidt Numbers Are Large.

We solve the advection-diffusion equation for a stochastically stationary passive scalar θ, in conjunction with forced 3D Navier-Stokes equations, using direct numerical simulations in periodic domains of various sizes, the largest being 8192^{3}. The Taylor-scale Reynolds number varies in the range 140-650 and the Schmidt number Sc≡ν/D in the range 1-512, where ν is the kinematic viscosity of the fluid and D is the molecular diffusivity of θ. Our results show that turbulence becomes an ineffective mixer when Sc is large. First, the mean scalar dissipation rate ⟨χ⟩=2D⟨|∇θ|^{2}⟩, when suitably nondimensionalized, decreases as 1/logSc. Second, 1D cuts through the scalar field indicate increasing density of sharp fronts on larger scales, oscillating with large excursions leading to reduced mixing, and additionally suggesting weakening of scalar variance flux across the scales. The scaling exponents of the scalar structure functions in the inertial-convective range appear to saturate with respect to the moment order and the saturation exponent approaches unity as Sc increases, qualitatively consistent with 1D cuts of the scalar.

Small-Scale Isotropy and Ramp-Cliff Structures in Scalar Turbulence.

Passive scalars advected by three-dimensional Navier-Stokes turbulence exhibit a fundamental anomaly in odd-order moments because of the characteristic ramp-cliff structures, violating small-scale isotropy. We use data from direct numerical simulations with grid resolution of up to 8192^{3} at high Péclet numbers to understand this anomaly as the scalar diffusivity, D, diminishes, or as the Schmidt number, Sc=ν/D, increases; here ν is the kinematic viscosity of the fluid. The microscale Reynolds number varies from 140 to 650 and Sc varies from 1 to 512. A simple model for the ramp-cliff structures is developed and shown to characterize the scalar derivative statistics very well. It accurately captures how the small-scale isotropy is restored in the large-Sc limit, and additionally suggests a possible correction to the Batchelor length scale as the relevant smallest scale in the scalar field.

Oscillations Modulating Power Law Exponents in Isotropic Turbulence: Comparison of Experiments with Simulations.

Inertial-range features of turbulence are investigated using data from experimental measurements of grid turbulence and direct numerical simulations of isotropic turbulence simulated in a periodic box, both at the Taylor-scale Reynolds number R_{λ}∼1000. In particular, oscillations modulating the power-law scaling in the inertial range are examined for structure functions up to sixth-order moments. The oscillations in exponent ratios decrease with increasing sample size in simulations, although in experiments they survive at a low value of 4 parts in 1000 even after massive averaging. The two datasets are consistent in their intermittent character but differ in small but observable respects. Neither the scaling exponents themselves nor all the viscous effects are consistently reproduced by existing models of intermittency.

Solenoidal Scaling Laws for Compressible Mixing.

Mixing of passive scalars in compressible turbulence does not obey the same classical Reynolds number scaling as its incompressible counterpart. We first show from a large database of direct numerical simulations that even the solenoidal part of the velocity field fails to follow the classical incompressible scaling when the forcing includes a substantial dilatational component. Though the dilatational effects on the flow remain significant, our main results are that both the solenoidal energy spectrum and the passive scalar spectrum assume incompressible forms, and that the scalar gradient essentially aligns with the most compressive eigenvalue of the solenoidal part, provided that only the solenoidal components are consistently used for scaling. A slight refinement of this statement is also pointed out.

Steep Cliffs and Saturated Exponents in Three-Dimensional Scalar Turbulence.

The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of 650 is studied using direct numerical simulations on a 4096^{3} grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to 1.2 for moment orders beyond about 12, indicating that scalar intermittency is dominated by the most singular shocklike cliffs in the scalar field. We show that the fractal dimension of the spatial support of steep cliffs is about 1.8, whose sum with the saturation exponent value of 1.2 adds up to the space dimension of 3, thus demonstrating a deep connection between the geometry and statistics in turbulent scalar mixing. The anomaly for the fourth and sixth order moments is comparable to that in the Kraichnan model for the roughness exponent of 4/3.

Extreme events in computational turbulence.

We have performed direct numerical simulations of homogeneous and isotropic turbulence in a periodic box with 8,192(3) grid points. These are the largest simulations performed, to date, aimed at improving our understanding of turbulence small-scale structure. We present some basic statistical results and focus on "extreme" events (whose magnitudes are several tens of thousands the mean value). The structure of these extreme events is quite different from that of moderately large events (of the order of 10 times the mean value). In particular, intense vorticity occurs primarily in the form of tubes for moderately large events whereas it is much more "chunky" for extreme events (though probably overlaid on the traditional vortex tubes). We track the temporal evolution of extreme events and find that they are generally short-lived. Extreme magnitudes of energy dissipation rate and enstrophy occur simultaneously in space and remain nearly colocated during their evolution.