The pump as turbine (PAT), serving as the core energy conversion component under reverse power generation in pumping stations, has internal unsteady flow and pressure fluctuations that are critical to the safe and stable operation of the station. In this study, combined model experiments and numerical simulations were employed to systematically investigate PAT pressure fluctuations and unsteady flow features under both pump and turbine modes. Continuous wavelet transform (CWT) and entropy production theory were introduced to analyse complex flow behaviour, establishing the relationships among pressure fluctuations, flow characteristics, and energy dissipation at key locations. The results reveal that in both modes, pressure fluctuations within the impeller exhibit periodic fluctuations at the shaft frequency (f SF), blade passing frequency (f BPF), and their corresponding harmonics. The rotor–stator interaction induces frequency components that are multiples of the guide vane number (7f SF), particularly evident at the impeller outlet in pump mode and at the impeller inlet in turbine mode. As flow rate increases, harmonic components become more pronounced. Pressure fluctuations in both the inlet and outlet channels are dominated by low-frequency components below 10 Hz with high amplitudes. Entropy production analysis indicates that the impeller serves as the primary energy dissipation region. The coexistence of high turbulent kinetic energy and eddy viscosity zones within the impeller indicates the presence of turbulent fluctuations, which leads to unsteady pressure fluctuations and account for 35%–75% of the total entropy production. These findings provide theoretical insights for addressing resonance-related issues in the engineering application of PAT systems.

Planet formation is inherently linked to the evolution of the protoplanetary disc. Recent developments point towards the possibility that disc evolution results from magnetised winds, rather than turbulent viscosity. This has fundamental implications for planet formation. We investigate planet formation in the context of magnetohydrodynamic (MHD) wind-driven disc evolution under the assumption of accretion being driven in a laminar accretion layer at the disc surface above a disc midplane with low turbulent viscosity. Our study is aimed at testing the global consequences of recent findings from 2D and 3D hydrodynamical simulations regarding inefficient midplane heating and the existence of two sub-regimes of type II migration; namely, slow viscosity-dominated and fast wind-driven migration. To study the global, potentially observable imprints of the physical processes governing planet formation in layered MHD-wind-driven discs, we ran single-embryo planetary population syntheses with varying initial disc conditions (i.e. disc mass, size, and angular momentum transport) and varying embryo starting location. We tested different parametrisations for the accretion layer thickness, Σ_ The extent of type II migration in layered discs depends sensitively on the considered accretion layer thickness. For thin (Σ_ łesssim0.01, or fast (gtrsim12,% sonic velocity) accretion layers, giant planets migrate in the slow viscosity-dominated regime, which strongly limits the extent of type II migration. The fast wind-driven sub-regime nearly never not occurs. For thick (Σ_ ) or slow (łesssim3,% sonic velocity) accretion layers, fast wind-driven type II occurs in contrast frequently, leading to long-range inward migration that sets in once planets reach masses that are sufficiently high to block the accreting layer (typically several 100, ). Disc-limited gas accretion is also strongly affected by deep and early gap opening, limiting maximum giant planet masses. active g/cm^2 active g/cm^2 M_⊕ The existence of two subtypes of type II migration, low type I to type II transition masses and limited runaway gas accretion in layered MHD wind-driven discs strongly influence the final mass–distance diagrams of planets. For thin layers, giant planets form nearly in situ once they have passed into type II migration, which happens already at a few Earth masses. This leads to a bifurcation of the formation tracks where low-mass planets (super-Earths and sub-Neptunes) form closer in while giant planets remain farther out in the disc ≳ 1, . For thick layers, fast wind-driven migration leads in contrast to numerous migrated hot Jupiters. Overall, we find that while the global properties of the emerging planet population are strongly modified relative to classical viscous discs, the key properties of the observed population can be reproduced within this new paradigm. au

This study numerically investigated unsteady tip-leakage flows (TLFs) over a stationary hydrofoil across varying tip-gap heights using delayed detached eddy simulations. The results reveal that multiple secondary vortices form around the tip-leakage vortex (TLV), influencing its structure and pressure characteristics. Secondary vortices originating from the suction-side edge reinforce the TLV, maintaining low pressure near the vortex core, whereas trailing-edge vortices induce flow instabilities and momentary pressure drops. The instantaneous minimum pressure in the TLV is significantly lower than the time-averaged values, particularly downstream of the trailing edge, due to the unsteady shedding vortices. For smaller gaps, the region of higher turbulent viscosity ratio in the boundary layer disperses due to the leakage flow, accelerating TLV diffusion and mitigating the pressure drop. These findings provide insights into TLF-induced cavitation mechanisms, highlighting the role of secondary vortex interactions in sustaining low-pressure regions and influencing cavitation susceptibility in marine hydrofoils.

Direct numerical simulation is performed to study the effects of spanwise curvature on transitioning and turbulent boundary layers. Turbulent transition is induced with an array of resolved cuboids. Spanwise curvature is prescribed using a novel approach with a body force that is applied orthogonally to the bulk flow to curve the mean free-stream streamlines at a set radius. The flows are analysed in a streamline-aligned coordinate system. Although the radius of curvature is large compared with the size of the boundary layer, its effects on the development of the boundary layer are appreciable. The results indicate that spanwise curvature induces a non-uniform mean secondary flow and alters the structure of turbulence within the boundary layer. Analytical expressions for the crossflow are derived in the viscous sublayer and log layer. These alterations are visible as changes in the distribution of the turbulent stresses and alignment of the vortical structures with the mean flow. These modifications are responsible for a misalignment between the Reynolds stress tensor and the velocity gradient tensor, which has important consequences for the validity of the widely used Boussinesq turbulent viscosity hypothesis in Reynolds-averaged Navier–Stokes models. Spanwise curvature was observed to decrease turbulent kinetic energy. These results have important implications on the development of turbulence in general applications, such as the flow over a prolate spheroid.

Abstract Resuspension of fine‐grained bottom sediment under wind‐driven currents and waves is a key process in shaping nearshore environments. Commonly, resuspension is quantified for predicting the dispersion of contaminants and nutrients affecting water quality by numerical modeling and field measurements. Although a large body of research deals with this topic, unique field observations from hypersaline environments coupled with conceptual‐quantitative description of the process are lacking. Here, we present high‐resolution direct measurements of winds, waves, currents, and turbidity conducted along the Dead Sea shores and derivations of an integrated 1D‐numerical model based on mass and momentum conservation laws. Comparing the model predictions and the observations determine, for the first time, that depth‐averaged turbulent viscosity during Dead Sea storms is of order of 10 −3 m 2 s −1 . Resuspension of bottom clay to fine sand is governed primarily by waves inducing shear stress three orders of magnitude larger than current‐induced shear stress, a ratio which is rather constant during Dead Sea storms. The observed spatiotemporal turbidity pattern is reproduced and accounts for the effect of grain‐size distributions on the lake floor. Additionally, we highlight the importance of wave‐induced resuspension as an additional source of sediment involved in the formation of thin, muddy layers that are traditionally interpreted as indicators of inflowing sediment plumes. The novelty of the manuscript lies in the combination of rare observations and modeling, which provides comprehensive physics of the studied processes, an approach that can be used in other nearshore environments of lakes or oceans.

This paper presents a computational-fluid-dynamics-driven machine learning framework to enhance Reynolds-averaged Navier–Stokes turbulence models for complex delta wing flows at high Reynolds numbers and transonic speeds. Building upon the coupled Spalart–Allmaras model framework, this study represents the first application of an interpretable machine learning framework to such complex three-dimensional vortex-dominated external aerodynamic applications. The key contribution lies in applying gene expression programming to learn optimal anisotropy tensor corrections for delta wing flows, where the Boussinesq approximation is extended with nonlinear stress–strain terms and coupled to a redefined production term in the transport equation for turbulent viscosity. Trained on a single-delta wing configuration, the proposed approach is validated on double- and triple-delta wing geometries under challenging transonic flow conditions, demonstrating significant improvements in the prediction of aerodynamic coefficients and showcasing both accuracy and robustness. The use of gene expression programming enables the automated discovery of models, generating explicit, physically interpretable expressions that provide deeper insight into vortex-dominated flow physics and facilitate generalization across related flow regimes. While the present results are specific to delta wing configurations and remain most accurate near the training conditions, they demonstrate the potential of physics-informed machine learning as a promising direction for advancing turbulence modeling in such complex three-dimensional flow regimes.

Context . Modeling how cold giant planets form around M dwarfs remains a challenge, both because their protoplanetary disks can lack sufficient mass and because such planets are expected to migrate inward while interacting with the disk. Moreover, it remains unknown whether inner rocky planets can survive in systems that host a cold giant around very low-mass stars, which could have important implications for the habitability of rocky worlds. Aims . We investigated the conditions required for the formation of giant planets at large orbital distances (1−3 au) around a 0.1 M ⊙ star, and explored the circumstances under which a close-in rocky planet can survive. Methods . We performed N -body simulations in which planetary embryos grow through pebble accretion, followed by gas accretion during the disk lifetime. Assuming a local disk turbulent viscosity (α t ) of 10 −4 , we included planet-disk interactions throughout the disk evolution, using a new prescription that accounts for the onset of outward migration when the planet-to-star mass ratio ( q ) exceeds 0.002. Results . We find that a cold giant planet can form around a late M dwarf, even with an initial pebble mass of only 6 M ⊕ , provided the disk gas mass is 10% of the stellar mass. This outcome requires a compact 20 au disk in which the inner, viscosity-dominated region has a high gas surface density set by a low accretion viscosity (α g =10 −4 ), that planet–planet collisions assemble a ∼ 5 M ⊕ core within 1 Myr, and that the gas disk survives for 10 Myr. In addition, an inner rocky planet can survive in a close-in orbit if it migrates into the inner disk cavity before the outer body grows into a giant. Conclusions . The initial dust mass required for giant planet formation around very low-mass stars does not need to be as extreme as previously thought. A combination of planet–planet collisions, efficient pebble accretion, and a long disk lifetime plays a key role in enabling the formation of cold giant planets with masses between those of Saturn and Jupiter.

The aim of this presentation is to show the evolution of a novel method developed by the authors to provide a reliable and engineering-feasible solution to one of humanity’s greatest enigmas: turbulence. While the first theoretical foundations were laid in 1877, when Boussinesq formulated the turbulent viscosity hypothesis, it was not until 1963 that Smagorinsky proposed the first LES model for CFD applied to meteorology. More than six decades have passed, and the community is still searching for a feasible way to solve this enigma, despite the advantage offered by computational simulation. Assuming that the key lies in solving all the scales present in the Navier-Stokes equations, this has so far been impossible from an analytical point of view and, numerically, only in cases at limited Reynolds numbers. This technique of simulating all scales, called "DNS", from the largest, the integral, to the smallest, the Kolmogorov scale, requires, in the most important cases, grids so fine that they are impossible to process with the available resources. From the 1970s to the present, different methodologies have been presented, most of them focused on modeling certain scales to reduce this cost. Thus, today we have models from the Reynolds Averaged Navier-Stokes (RANS) family, models where part of the scales are simulated and part is modeled, the so-called Large Eddy Simulation (LES), hybrid methods that combine RANS with LES, and, more recently, with the advent of artificial intelligence, models based on Machine Learning. Our proposal aims to not use any model, simulate most scales, and rely on what the subgrid provides. However, instead of incorporating it into the simulation as DNS would do, we propose solving it offline and appropriately building a database (human genome-like) from which the information that the subgrid would provide can be extracted without any correlation, similar to a multiscale calculation. In this way, an accuracy similar to that of DNS is achieved at a cost comparable to RANS. In the presentation, we will show, on a timeline, the developments that have been made, the challenges overcome, and the results obtained, both at the academic level and in complex applications.

ABSTRACT This paper presents the development and application of TURB-3D, a Galerkin finite element – based computational fluid dynamics (CFD) model for simulating turbulent flows in open channels. The model solves the Reynolds-averaged Navier – Stokes (RANS) equations using a penalty method for incompressibility, a nonlinear algebraic stress closure, and the standard k–ε turbulence formulation. The Galerkin finite element method minimizes the residual in a weighted norm, ensuring that the approximate solution is the most accurate representation within the chosen function space. A key motivation for this work arises from the limitations of widely used hydrodynamic solvers such as Delft3D-FLOW and MIKE3. Although often labeled as three-dimensional, these are in fact hydrostatic layered models: vertical momentum is reduced to a hydrostatic balance, while vertical velocity is reconstructed from continuity rather than solved dynamically. Combined with σ- or z-layer coordinate systems, this reduces their ability to capture vertical accelerations, secondary circulations, and non-hydrostatic pressure gradients. Such models are therefore better described as quasi-3D (2.5D) tools, suitable for large-scale shallow-water applications but inadequate for strongly three-dimensional flows. In contrast, most high-resolution 3D CFD solvers, such as FLOW-3D, OpenFOAM, TELEMAC, and ANSYS-CFX, commonly employ isotropic eddy-viscosity turbulence models (e.g., k–ε or k–ω), which represent turbulence in a manner similar to that in confined duct flows. Consequently, secondary currents, turbulence anisotropy near free surfaces, and wall-shear variations are often inadequately captured unless more advanced closures—such as Reynolds Stress Models or anisotropic Large Eddy Simulation (LES) approaches—are utilized. TURB-3D overcomes these limitations by incorporating an anisotropic turbulent viscosity formulation and free-surface proximity functions, which enable realistic prediction of secondary currents, velocity-dip phenomena, and boundary shear stress distributions. Its finite element discretization and penalty scheme provide a significant gain in computational efficiency, making it more accessible for practical engineering use than conventional CFD solvers that demand millions of elements. Validation against laboratory experiments and benchmark CFD studies shows good agreement in velocity profiles, turbulence quantities, and shear velocities, with relative errors in the predicted average shear velocity of −1.8% for AR = 2 and −4.4% for AR = 1. TURB-3D successfully reproduces key three-dimensional flow features, including the depression of velocity maxima, corner-induced vortex structures, and secondary motion cells. The current implementation is limited to steady, fully developed channel flows, and relies on an empirical turbulence closure. Nevertheless, TURB-3D represents a computationally efficient and physically realistic tool for hydraulic engineering, offering a practical framework for channel design, flood analysis, and flow regulation. By bridging the gap between simplified 2D models and resource-intensive full CFD solvers, TURB-3D contributes a novel and robust methodology for simulating turbulent open channel flows.

Background/Objectives: The aberrant subclavian artery (ASA) represents the most common congenital anomaly of the aortic arch, and is frequently associated with a Kommerell diverticulum, an aneurysmal dilation at the anomalous vessel origin. This condition carries a significant risk of rupture and dissection, and growing evidence indicates that local hemodynamic alterations may contribute to its development and progression. Computational Fluid Dynamics (CFD) provides a valuable non-invasive modality to assess biomechanical stresses and elucidate the pathophysiological mechanisms underlying these vascular abnormalities. Methods: In this study, twelve thoracic CT angiography scans were analyzed: six from patients with ASA and six from individuals with normal aortic anatomy. CFD simulations were performed using OpenFOAM, with standardized boundary conditions applied across all cases to isolate the influence of anatomical differences in flow behavior. Four key hemodynamic metrics were evaluated—Wall Shear Stress (WSS), Oscillatory Shear Index (OSI), Drag Forces (DF), and Turbulent Viscosity Ratio (TVR). The aortic arch was subdivided into Ishimaru zones 0–3, with an adapted definition accounting for ASA anatomy. For each region, time- and space-averaged quantities were computed to characterize mean values and oscillatory behavior. Conclusions: The findings demonstrate that patients with ASA exhibit markedly altered hemodynamics in zones 1–3 compared to controls, with consistently elevated WSS, OSI, DF, and TVR. The most pronounced abnormalities occurred in zones 2–3 near the origin of the aberrant vessel, where disturbed flow patterns and off-axis mechanical forces were observed. These features may promote chronic wall stress, endothelial dysfunction, and localized aneurysmal degeneration. Notably, two patients (M1 and M6) displayed particularly elevated drag forces and TVR in the distal arch, correlating with the presence of a distal aneurysm and right-sided arch configuration, respectively. Overall, this work supports the hypothesis that aberrant hemodynamics contribute to Kommerell diverticulum formation and progression, and highlights the CFD’s feasibility for clarifying disease mechanisms, characterizing flow patterns, and informing endovascular planning by identifying hemodynamically favorable landing zones.

This study presents the implementation and systematic comparison of two widely used two-equation turbulence models, k–ϵ and k–ω, within the Smoothed Particle Hydrodynamics (SPH) framework for dam-break flow simulations. The governing equations are discretized using the mesh-free, fully Lagrangian SPH method, which effectively captures highly transient free-surface phenomena characteristic of dam-break events. Separate subroutines were developed to compute turbulent viscosity for each turbulence model, enabling a detailed and consistent evaluation of their performance under identical flow conditions. Two free-surface flow scenarios were examined, namely dam-break flows over a dry bed and over a wet bed with initially quiescent water. Comparative analysis of velocity fields and flow structures revealed significant differences in turbulence closure behaviour, highlighting the specific strengths and limitations of the k–ϵ and k–ω models in representing turbulent free-surface flows within SPH simulations. The findings provide valuable insights into the selection of suitable turbulence models, enhancing the accuracy and reliability of SPH-based predictions in hydraulic and environmental applications.

We investigate a short-wave instability mode recently identified via temporal stability analysis in weakly inclined falling liquid films sheared by a confined turbulent counter-current gas flow (Ishimura et al. J. Fluid Mech. vol. 971, 2023, p. A37). We perform spatio-temporal linear stability calculations based on the Navier–Stokes equations in the liquid film and the Reynolds-averaged Navier–Stokes equations in the gas, and compare these with our own experiments. We find that the short-wave instability mode is always upward-convective. The range of unstable group velocities is very wide and largely coincides with negative values of the wave velocity. Turbulence affects this mode both through the level of gas shear stress imparted and via the shape of the primary-flow gas velocity profile. Beyond a critical value of the counter-current gas flow rate, the short-wave mode merges with the long-wave Kapitza instability mode. The thus obtained merged mode is unstable for group velocities spanning from large negative to large positive values, i.e. it is absolute. The onset of the short-wave mode is precipitated by decreasing the channel height and inclination angle, and by increasing the liquid Reynolds number or the gas-to-liquid dynamic viscosity ratio. For vertically falling liquid films, merging occurs before the short-wave mode can become unstable on its own. Nonetheless, the ability to generate upward-travelling ripples is endowed to the merged mode. Preliminary calculations neglecting the linear perturbation of the turbulent viscosity suggest that three-dimensional perturbations could be more unstable than two-dimensional ones.

Experiments using a rotational rheometer have demonstrated that the apparent viscosity becomes negative under the electric-field-induced turbulent state of conductive nematic liquid crystals [Orihara et al., Phys. Rev. E 99, 012701 (2019)10.1103/PhysRevE.99.012701; F. Kobayashi et al., Phys. Rev. E 101, 022702 (2020)10.1103/PhysRevE.101.022702]. When the upper rotating plate of the rheometer is left free, spontaneous rotation-that is, spontaneous shear flow-has also been observed. In this study, we reproduce these phenomena through three-dimensional simulations based on continuum theory. The simulations reveal characteristic velocity, director, and stress fields in the negative-viscosity state. Furthermore, they clarify the interplay between topological defects (disclinations) and space charges which drive the turbulence.

Effects of turbulent heat diffusion and viscosity on the simulation of turbulent mixing in autonomous PWR subchannel analysis software

The Vortex Particle Method (VPM) represents continuous, incompressible, unsteady, viscous flows by using singular, independent particles represented by small concentrations of vorticity. The VPM is particularly convenient for representing viscous flow fields, which is useful for simulating vortex reconnection in an unbounded domain and flow mixing. This study aims to evaluate the accuracy of the VPM predictions using, among others, the diffusion velocity method to represent the molecular diffusion of vortex structures. The VPM can also represent the viscous diffusion of turbulent flow fields using a turbulent viscosity model. However, such models are defined up to a given constant. The dynamic computation of this constant is required for the turbulence model to encompass the diffusion of all the flow scales. A new fast dynamic technique based on the local loss of enstrophy in the flow is proposed. The coupling of the Diffusion Velocity Method with this dynamic procedure is presented. Finally, the method is validated on several known analytical flow solutions, such as the Lamb–Oseen vortex and isolated and leapfrogging vortex rings.

This study considers the simulation of turbulent two-dimensional vortex-induced vibrations of a spring-mounted cylinder in cross-flow. It focuses on a Reynolds number range of 5×104–105, with spring stiffness 1200 N/m2 and damping coefficients from 28 to 50 Ns/m, for which experimental data are available. Three original outcomes are identified. First, the use of moving overset meshes to track the oscillating cylinder, reduces grid-size requirements and maintains high-quality near-wall resolution even at large oscillation amplitudes. Second, making the turbulent viscosity coefficient of the k−ε model sensitive to the strain rate invariant results in the correct predictions of vibration amplitude and frequency over the available dataset, thus offering a cost-effective approach. Finally, further simulations decouple the effects of the Reynolds number from those of a previously overlooked dimensionless parameter involving the spring constant, and reveal that this second dimensionless parameter exerts a far stronger influence on fluid velocity and flow-induced vibrations than the Reynolds number.

Determining the physical processes driving protoplanetary disc evolution is of paramount importance for understanding planet formation. Our current understanding has crystallised around two possible evolution scenarios: turbulent viscosity and magnetohydrodynamic (MHD) wind-driven. Which of these processes dominates, however, remains unclear. Our aims are twofold. Firstly, we investigate whether a single set of model parameters can reproduce the observational constraints of non-irradiated and irradiated discs. Secondly, we propose a novel approach to break degeneracies between these two scenarios by studying the relation of stellar accretion rate and externally driven wind mass-loss rates, which evolve differently depending on the mechanism of angular momentum transport in the outer disc, and we test this approach using our models. We simulated the evolution of synthetic populations of protoplanetary discs using 1D vertically integrated models for both viscous and MHD wind-driven disc evolution including both internal X-ray and external far ultraviolet (FUV) photoevaporation for both evolution scenarios. We investigated both weak and strong FUV field environments, where the strong FUV field is calculated based on an environment similar to the Cygnus OB2 association. We studied the time evolution of the disc fraction, disc mass--stellar accretion rate relation, the spatial variation of the disc fraction in a highly irradiated cluster, the evolution of disc radii, and the evolution of accretion rates versus wind mass-loss rates. While both evolution scenarios are capable of reproducing observational constraints, our simulations suggest that different parameters are needed for the angular momentum transport to explain disc lifetimes and the disc mass--stellar accretion rate relation in weakly and strongly irradiated regions. We find that the predicted median disc radii are much larger in low FUV environments compared to Cygnus OB2 but also decrease with time. In the viscous scenario, the median disc radius in a low FUV field environment is ∼100 larger than for the MHD wind-driven scenario. We further demonstrate that studying stellar accretion rates and externally driven wind mass-loss rates (provided that they can be isolated from internally driven winds, i.e. MHD wind) is indeed a promising way of disentangling the two evolution scenarios. The fact that a single set of parameters for angular momentum transport is not able to reproduce disc lifetimes in both low and highly irradiated regions at the same time indicates a fundamental difference in the two regions.

This paper presents an experimental and analytical investigation of the turbulent transport and flame geometric characteristics of free turbulent buoyant diffusion flames under different fuel mass fluxes and burner boundary conditions (i.e. with/without a flush floor). The stereo particle image velocimetry technique was utilised to measure the three-dimensional instantaneous velocity fields of the free methane buoyant flames with a burner diameter ( d ) of 0.30 m and dimensionless heat release rates ( $\dot{Q}^{*}$ ) of 0.50–0.90. The results showed that, compared with the configuration without a floor, the time-averaged axial velocity fluctuations squared and the time-averaged radial velocity fluctuations squared decreased, and the peak values of the time-averaged radial velocity, the time-averaged radial velocity fluctuations squared and the time-averaged axial and radial fluctuation product shifted towards the burner centreline in the configuration with a flush floor. Based on the dimensional analysis and the gradient transport assumption, the mean turbulent viscosity within the mean flame height ( $\nu _{t}^{=}$ ) was scaled. Compared with the configuration without a floor of under equal $\dot{Q}^{*}$ , the turbulent viscosity decreased in the configuration with a flush floor, resulting in an increase in mean flame height and a reduction in mean flame width. Based on the concepts of turbulent mixing and equal axial convection and radial diffusion times, semi-physical models were derived for the mean flame height and the mean flame width, respectively. The two correlations agreed well with the experimental data of this work for the two burner configurations with and without a flush floor.

ABSTRACT We theoretically study the line density of quantum vortices in a superfluid turbulent wake based on the conjecture of quantum turbulence of a pure superfluid in the zero temperature limit. The conjecture states that a collection of many quantum vortices can mimic a classical eddy in the continuum approximation. By straightforwardly relating the Kolmogorov's similarity to this conjecture, we can estimate that the effective viscosity of a pure superfluid turbulence is comparable to the circulation quantum , the minimum circulation of quantum vortices. This result naturally leads to a definition of the superfluid Reynolds number, with the characteristic size and velocity , supported by the fact that the equation of motion of a pure superfluid reduces to the Euler equation for high . By using the classical description of the rate of energy injection or dissipation, we quantitatively evaluate a characteristic density of vortex lines in a superfluid turbulent wake and its spatial profile of grid turbulence in terms of the superfluid Reynolds number.

The classical assumption of self-similarity in flow velocities and turbulence statistics has been successfully validated for fully developed flows in open channels, pipes, and boundary layers. However, its application in developing boundary-layer flows in channels with steep slopes and large roughness elements has not yet been thoroughly scrutinized. This study investigates whether turbulence statistics exhibit self-similar behavior when properly scaled in steep-stepped spillways. Specifically, it explores the influence of roughness height (ks)—representing the cavity size of a steep-stepped spillway—on turbulence statistics in the non-aerated skimming flow region. Numerical simulations, extensively validated against experimental data, were conducted for a stepped spillway with a fixed slope angle of 51.34°, using five roughness heights (ks = 6.25, 3.12, 1.56, 0.78 and 0.39 cm), corresponding to step height-to-length ratios of 10:8, 5:4, 2.5:2, 1.25:1 and 0.625:0.5, respectively. The results show that the dimensionless profiles of turbulent kinetic energy (TKE) at the step edges collapse onto a single curve when rescaled by a factor of δ/ksn with n~0.4. Likewise, the dissipation rate of TKE follows a similar collapse with n~0.3. For the turbulent eddy viscosity, an exponent of n~0.5 was adopted based on dimensional analysis, although the values for the smoothest configuration deviate from the curve.