Direct numerical simulations are conducted in this study to investigate fully developed turbulent flows in spanwise rotating ducts with different aspect ratios. The global friction Reynolds number Re-tau is fixed at 180, while the global friction rotation number Ro(tau) varies from 0 to 10. The duct aspect ratios (AR) considered include 0.5, 1, and 2. The objective of this study is to explore in depth the kinematic and dynamic characteristics of secondary flows under different rotation numbers and aspect ratios, as well as their impact on the flow physics of wall-bounded turbulence. The results show that as the rotation rate increases, the secondary flow structures in both square (AR = 1) and AR = 2 ducts follow an evolution path from corner vortices, through additional vortices, to large-scale circulations dominated by the Coriolis force, although differences exist in the generation, development, and decay processes of each structure. In contrast, the AR = 0.5 duct directly evolves from corner vortices to Coriolis-dominated circulations. Analysis of the secondary flow intensity indicates that the expansion of the spanwise scale allows the secondary flow to fully develop, and the secondary flow intensity in the duct is significantly enhanced. The study further confirms that the secondary flow in rotating ducts is mainly driven by the Ekman layer, and the Ekman layers in ducts with different aspect ratios exhibit similarities. Turbulence statistics and transport analysis further reveal that the square duct with a smaller AR shows higher sensitivity of mean velocity and turbulence fluctuations to changes in rotation number, with more pronounced suppression effects by rotation, while the secondary flow significantly enhances the local turbulence intensity and transport capacity.
Numerical simulations of axial flow in a rectangular channel with two parallel fins were conducted using the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations coupled with a Reynolds Stress Model (RSM). The primary objective of this study is to investigate the formation of quasi-periodic coherent structures and their significant role in fluid transport across the inter-fin region. Computational Fluid Dynamics (CFD) results were validated against experimental data available in the literature, specifically for Strouhal numbers and the spacings of coherent structures in the five studied test sections, with a ratio of fin height to inter-fin spacing ranging from 2.5 to 7.5. The Reynolds number, based on the hydraulic diameter of the channel and the bulk velocity, was held constant at 40 & times; 10(4). Velocity profiles and contours were analysed to capture and describe the presence of coherent structures within this family of finned rectangular channels.
We derive a two-equation model for the energy balance in statistically homogeneous turbulence. The present formulation is expressed in terms of the energy flux, unlike the classical approach, where the dissipation rate appears in the kinetic energy equation. This enables a unified description of both forced and decaying turbulence with a single set of model constants. The model also captures the time lag between the evolution of the kinetic energy and the dissipation rate. Reformulating the system as an equation for the dissipation rate further clarifies how non-equilibrium effects can be incorporated into existing turbulence models.
Direct numerical simulations are used to investigate the impact of compressibility and compositional variable-density (VD) effects on the growth of instability in buoyancy-driven shear layers (BSL), which exhibit features common to both Rayleigh-Taylor (RTI) and Kelvin-Helmholtz (KHI) instabilities. The two-dimensional simulations are performed in a periodic domain with alternating high- and low-density fluid columns under neutral background stratification. This study expands on previous work by Gat et al. [Incompressible variable-density turbulence in an external acceleration field. J Fluid Mech. 2017;827:506-535. doi: 10.1017/jfm.2017.490] conducted in the incompressible regime. The Atwood number, A, and the isothermal Mach number, Ma, are varied to isolate VD and compressibility effects. Two flow regimes are identified. At early times, shear dominates and the evolution resembles KHI with minimal sensitivity to A or Ma. Later, buoyancy becomes dominant and the flow exhibits RTI-like behaviour; increasing A accelerates this transition. Compressibility suppresses the growth of the mixing-layer, reduces bubble and spike penetration, and amplifies the VD-induced mixing asymmetry. Enhanced molecular mixing weakens the baroclinic torque, shifting vorticity production toward dissipative scales and eventually suppressing mixing-layer growth at sufficiently large Ma. The suppressive effect of compressibility strengthens with increasing A, indicating that VD and compressibility are coupled yet competing influences.
Adverse bed-slope transitions can strongly reorganise open-channel turbulence, but their impact on anisotropy, intermittency, and nonlinear dynamics is not yet well quantified. This study experimentally investigates turbulence over an adverse sloping bed in an open channel, aiming to elucidate how bed-induced geometric transitions shape turbulence characteristics. High-frequency three-dimensional velocity measurements were acquired at multiple streamwise stations and depths, and analysed using Reynolds-stress anisotropy diagnostics, intermittency indicators based on quadrant joint-PDFs, velocity spectra, and chaos-information theoretic measures. The results show that turbulence anisotropy evolves gradually along the bed transition. The flow shifts from predominantly bi-directional states upstream toward more mixed states over the slope, followed by partial recovery downstream. Anisotropy is strongest near the bed, exhibits peaks just below mid-depth, and becomes more vertically self-similar with increasing current magnitude. Positionally, no anomalous tendency is observed under stronger currents. In contrast, intermittency remains strongly localised, with slope-transition stations acting as hotspots where joint-PDF contours broaden toward ejection-sweep dominance and spectra steepen. This is consistent with sporadic bursting and sharper departures from local trends, whereas non-transitional stations retain tighter contours and flatter spectra indicative of steadier coherence. Chaos and information analyses further indicate increasing trajectory divergence and reduced memory with height as slope-driven upwellings are present. Collectively, these observations suggest that anisotropy trends and intermittency hotspots can exhibit distinct spatial signatures. Therefore, anisotropy should be interpreted alongside intermittency and memory metrics for position-aware turbulence assessment over adverse slopes.
Vortex research is critical for environmental and engineering applications. Quantitative evaluation on vortex properties were centred on two-dimensional (2D) contours (based on 2D or three-dimensional (3D) flow fields) in previous studies. In this study we proposed a 3D isosurface-based method with higher ability and accuracy for vortex extraction to characterise 3D vortices. The key lies in the usage of a fast triangle-triangle intersection method together with a triangular cutting plane. Vortices were extracted as polygons from intersection lines between the 3D isosurface and the triangular cutting plane. Vortex population density, morphological descriptors (circularity, radius, convex solidity, and axis ratio), and orientation for different thresholds and wall distances were analysed. As the threshold increases, the number of extracted vortices decreases logarithmically, and vortices become smaller, more circular and convex. Similar to the threshold case, a logarithmic relationship between vortex density and wall distance exists. And as the wall distance increases, vortices also become more circular and convex, and inclination angle increases and stabilises in the wake layer. Compared with findings based on 2D contours in previous studies, the isosurface-based method in this study represented 3D vortices in a clearer perspective. Due to the usage of threshold varied with the wall distance, vortex radius increased with the increasing wall distance in previous studies. However, results based on 3D isosurface show that in the fully developed channel flow vortex size is strongly correlated with the velocity gradient, and vortex size only increases in the near-wall region, beyond which it remains nearly constant. The proposed method may help to delve into turbulent structures along with their interactions and evolutionary mechanisms.
In this study, we assess the capabilities of large-eddy simulation (LES) to predict the features of shock-wave turbulent boundary layer interaction (STBLI) and examine the effects of wall temperature on such features. We consider a $ 24<^>{\circ } $ 24 degrees compression corner configuration at the free-stream Mach number of 2.9 and freestream unit Reynolds number of 5581.4 mm $ <^>{-1} $ -1, respectively. First, we assess LES capabilities in terms of the role of various subgrid models in the prediction of statistics of STBLI. Specifically, we consider four models, namely, the no-model approach, the dynamic Lagrangian model for eddy viscosity, the wall-adapted local eddy-viscosity (WALE) model, and the locally dynamic kinetic energy model. Overall, the WALE model showed a better agreement with reference results compared to other models. Next, we examine the effects of wall temperature on the features of STBLI by considering three cases with increasing ratio of the bottom wall temperature to the recovery temperature with $ T_{{\rm w}}/T_{{\rm r}} = 0.6 $ Tw/Tr=0.6, 1.14, and 1.4. With an increase in $ T_{{\rm w}} $ Tw, we observe that all features of STBLI get affected, such as a decrease in the coherence of the quasi-streamwise vortical structures, a decrease in the energy content of the small scales, an earlier separation, an increase in the characteristic frequency associated with the low-frequency unsteadiness, etc. Additionally, the streamwise variation of skin friction and heat transfer coefficients and the wall pressure and its fluctuation intensity show a strong dependence on the wall temperature.
This study presents a novel rotationally invariant data-driven subgrid-scale (SGS) models for large-eddy simulation (LES) of wall-bounded turbulent flows. Previous studies have typically imposed invariance via scalar invariants or the canonical form (e.g. based on its eigenframe of rate-of-strain tensor), whereas this study introduces a modified deep neural network (DNN) architecture that inherently respects rotational invariance. Building upon the multiscale convolutional neural network SGS model (MSC model) developed by the authors, which outputs SGS stress tensors ( $ \tau _{ij} $ tau ij), the DNN architecture is modified to satisfy the principle of material objectivity by removing bias terms and batch normalisation layers while incorporating a spatial transformer network algorithm. The proposed data-driven SGS models were trained on a turbulent channel flow at $ {\rm Re}_\tau = 180 $ Re tau=180 and evaluated under both non-rotated and rotated input conditions. The models accurately predicted $ \tau _{ij} $ tau ij and key turbulence statistics, including SGS dissipation, backscatter, and SGS transport, for non-rotated inputs. In addition, in the case of rotated inputs, they significantly outperformed the baseline MSC model, reducing the mean absolute error (MAE) of the $ \tau _{12} $ tau 12 predictions from 0.402 to below 0.047 and achieving up to two orders of magnitude lower MAE in the turbulence statistics. Moreover, the models effectively generalise to unseen rotated inputs, accurately predicting $ \tau _{ij} $ tau ij despite the input configurations not being encountered during the training, indicating the general applicability of the proposed model. These findings highlight that the proposed data-driven SGS models address the key limitations of common data-driven SGS approaches, particularly their sensitivity to rotated input conditions. It also marks an important advancement in data-driven SGS modelling for LES, particularly in flow configurations where rotational effects are non-negligible.
The Wilcox $ k-\omega $ k-omega turbulence model predicts turbulent boundary layers well, both fully-developed channel flows and flat-plate boundary layers. However, it predicts too low a turbulent kinetic energy. This is a feature it shares with most other two-equation turbulence models. When comparing the terms in the k equations with DNS data it is found that the production and dissipation terms are well predicted but the turbulent diffusion is not. In the present work the poor modelling of the turbulent diffusion is improved using Physics Informed Neural Network (PINN) and Neural Network (NN). The k equation is turned into an ordinary differential equation for the turbulent viscosity in the k equation, $ u _{t,PINN} $ nu t,PINN, which is solved using PINN. A new turbulent Prandtl number is then computed as $ \sigma _{k,PINN} = u _{t}/ u _{t,PINN} $ sigma k,PINN=nu t/nu t,PINN where $ u _t = k/\omega $ nu t=k/omega. Hence, the turbulent Prandtl number, $ \sigma _{k,PINN} $ sigma k,PINN, is determined using PINN, followed by the use of DNS data for estimating $ C_{k,PINN} $ Ck,PINN and $ C_{\omega 2,PINN} $ C omega 2,PINN which appear in the destruction terms in the k and omega equation, respectively. Neural networks are then used to generalise these results and thus construct a turbulence model. All Python PINN, NN and pySR scripts as well as the Python CFD code can be downloaded [Davidson. Using physical informed neural network (PINN) and neural network (NN) to improve a $ k-\omega $ k-omega turbulence model: python CFD code and PINN script. In: Division of fluid dynamics. Gothenburg: Dept. of Mechanics and Maritime Sciences, Chalmers University of Technology; 2025].
We present a methodology for the study of the dispersion of trajectories of stochastic processes in reconstructed phase spaces from observed data. The methodology allows to find ensembles of analog states, i.e. states that are close in the phase space. Once these states are found, we focus on the characterisation of their dispersion in function of (1) the time and (2) their initial separation. We study an experimental turbulent velocity measurement and two scale-invariant stochastic processes: a regularised fractional Brownian motion and a regularised multifractal random walk. Both stochastic processes are synthesised to have the same covariance structure as the experimental turbulent velocity, but only the regularised multifractal random walk mimics the intermittency of turbulent velocity. We illustrate that while the covariance structure of the processes governs the time dependence of the dispersion of the analog states, the intermittency phenomenon is responsible for the impact of the initial separation of the analogs on their dispersion.
Direct numerical simulations have been performed by applying a control method of producing large-scale vortices along the streamwise direction to reduce skin-friction drag in supersonic turbulent channel flows. The effectiveness of large-scale control for supersonic flows at bulk Mach number $ Ma_b = 1.5 $ Mab=1.5 and bulk Reynolds number $ Re_{b} = 3000 $ Reb=3000 is studied by varying input parameters, i.e. body force amplitude A and spanwise wavelength $ \Lambda <^>+ $ Lambda+ of large-scale control. Several different spanwise wavelengths and forcing amplitudes have been studied since the combination of the above two affects the strength of the vortices. The highest drag reduction (DR) of approximately $ 14.4\% $ 14.4% with a net power saving of 13.81% and the highest drag increase (DI) of $ 10.4\% $ 10.4% are observed for different strengths of large-scale vortices. The variation of coherent and random parts of Reynolds stresses is analysed and subsequently their contribution to skin-friction coefficient using Fukagata-Iwamoto-Kasagi (FIK) identity. The redistribution of turbulent kinetic energy (TKE) in the cross-flow plane points to the spatial distribution of high and low turbulence concentration regions in the channel. The strength of the vortices dictates wall-normal momentum transfer, by the upwash and downwash of quasi-streamwise vortical structures. The quadrant analysis suggests enhanced sweeps and ejections for high-strength vortices, and reduction in those events for low-strength vortices. The effect of bulk Mach number variation is also studied for $ Ma_b $ Mab = 0.3 and 2.5, and the results reveal that the efficacy of the control fades for higher $ Ma_b $ Mab flows, and a much smaller skin-friction drag reduction of 2.3% with a relatively low net power saving of 1.7% is obtained.
This study presents the implementation and systematic comparison of two widely used two-equation turbulence models, k-& varepsilon; and k-omega, 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-& varepsilon; and k-omega 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 develop hybrid RANS-LES strategies within the spectral element code Nek5000 based on the k-tau SST turbulence model. We chose airfoil sections with chord-based Reynolds number on the order of 10(5)-10(6), in both attached and stalled conditions, as our target problem to comprehensively test the solver accuracy and performance. Verification and validation of the k-tau SST model are performed for two reference cases: for the zero-pressure gradient boundary layer developing on a flat plate and for mild adverse-pressure gradient boundary layers developing on suction side of NACA0012. The k-tau SST model shows good grid convergence characteristics, at par or better in comparison to existing reference results. The results also show good corroboration with existing experimental and numerical datasets for low incoming flow angles. A small discrepancy appears at higher angle in comparison with the experiments, which is in line with our expectations from an RANS formulation. Building on this foundation, we construct a hybrid RANS-LES framework based on the Delayed Detached-Eddy Simulation (DDES) approach. DDES captures both the attached and separated flow dynamics well when compared with available numerical datasets. We demonstrate that for the hybrid approach a high-order spectral element discretization converges faster (i.e. with less resolution) and captures the flow dynamics more accurately than representative low-order approaches. We also revise some of the guidelines on sample size requirements for statistics convergence for massively separated flow within the current numerical framework. Finally, we analyse some of the observed discrepancies of our unconfined DDES at higher angles with the experiments by evaluating the 'blocking' effect of wind tunnel walls. We carry out additional simulations for confined domains and assess the observed differences as a function of Reynolds number.
This study presents a hybrid CFD-GRNN framework for predicting wind forces on tall buildings. The framework leverages OpenFOAM, using a steady RANS standard k-& varepsilon; turbulence model, to generate high-fidelity CFD data. This data includes aerodynamic coefficients (CFx, CFy, CMz) and wind pressure coefficients (Cp), which are used to train a Gradient-Regularized Neural Network (GRNN). The GRNN utilises a custom loss function that penalises the spatial gradient of the predicted Cp, ensuring physically plausible and smooth pressure distributions. The study investigates wind interference from an interfering building with varying heights and orientations from 0 degrees to 180 degrees in 5 degrees increments. The framework exhibits high predictive efficiency, accurately forecasting Cp for unsimulated configurations. Key findings include a persistent negative drag force coefficient (CFx) up to approximately 40 degrees for a height ratio (Hi/Hp) of 1.5. Additionally, significant variations in the across-wind force coefficient (CFy) are noted at orientations of 45 degrees, 60 degrees, and 70 degrees for Hi/Hp ratios of 0.6, 1.0, and 1.5, respectively.
A deterministic forcing scheme, formulated in physical space, is proposed to generate turbulence from a quiescent initial flow field, by referencing the Taylor-Green vortex. The scheme is analytically defined, ensuring isotropy and divergence-free characteristics. A series of direct numerical simulations (DNS) of forced homogeneous isotropic turbulence across varying Reynolds numbers is performed to benchmark the forcing method. The resulting turbulence is fully developed and statistically isotropic. The forcing scheme is further applied to DNS of compressible turbulence at both low and high Mach numbers. At low Mach numbers, the simulations reach statistically steady states consistent with the incompressible turbulence, while at high Mach numbers, strong compressibility effects are present, characterised by significant dilation and the formation of shocklets. Additionally, correlations between turbulence statistics and forcing parameters are established, providing further guidance for the application of such a forcing scheme to generate isotropic turbulence.
Spanwise homogeneous turbulent boundary layers convect groups of flow structures that move with similar momentum, often called coherent structures. Large-scale motions on the order of 20 delta, termed superstructures, are conjectured to be present throughout the outer region of the boundary layer. Their presence in close vicinity of the wall makes them a strong candidate to be the prime contributors to low-wavenumber pressure fluctuations in smooth walls. This study presents experimental evidence of superstructures in the logarithmic layer using time-resolved Particle Image Velocimetry and wall-pressure measurements acquired at varying mean pressure gradients and freestream Reynolds numbers to explore the relationship between the two physical quantities. Turbulence statistics are shown to collapse on outer boundary layer parameters, and impacts of external pressure gradients and Reynolds number variation on turbulence length scales are presented. Although synchronous velocity-pressure measurements were not obtained, comparative spectral features suggest a strong connection between superstructures and sub-convective wall-pressure fluctuations.
The central Indian region routinely experiences intense heat waves from April to June, peaking in May, with air temperatures frequently surpassing 40 $ <^>{\circ } $ degrees C. This study employs a high-resolution regional numerical weather prediction model, Consortium for small-scale modelling (COSMO), to explore the role of atmospheric boundary layer (ABL) processes in the sustenance of these extreme events. Using hourly data of surface-layer radiation and meteorological parameters from 26 inland stations, we assess the contribution of turbulent fluxes to ABL evolution. A detailed examination is carried out for four representative stations: Jodhpur and Jalore, characterised by intense heat wave conditions, and Jagdalpur and Gopalpur, characterised by mild heat wave conditions.The time-altitude structure of eddy diffusivity for heat (Kh) highlights stark contrasts in ABL dynamics between these sites. Over Jodhpur and Jalore, the ABL deepens up to 1.65 km and maintains this depth for extended durations during extreme heat wave episodes. In contrast, at Jagdalpur and Gopalpur, the ABL fails to sustain its growth, collapsing from 1.60 km to 485 m well before sunset. The influence of dense vegetation over Jagadalpur and Gopalpur limits surface heating and suppress boundary-layer growth during mild heat wave events. These findings underscore the critical role of ABL processes in modulating the severity and persistence of heat waves over the Indian subcontinent.
The drag force induced by roughness on the surrounding fluid depends heavily on surface statistics and flow conditions. The drag forces obtained from Direct Numerical Simulation data in turbulent channel flows over k-type rough surfaces are examined to find a relevant scaling for the drag force. The database explores the impact of the most influential parameters for this type of rough surfaces. Within the framework of the double-averaged Navier-Stokes equations, which include slice-by-slice spatial averaging in addition to the Reynolds average, several drag force models have been developed over the years. However, they all share a common weakness linked to the absence of interdependence between the slices. A dimensionless number, constructed from the local Reynolds number $ Re_d $ Red based on the equivalent roughness diameter d, is shown to group all the drag force coefficients together. A scaling law was then derived, significantly improving the prediction of drag force distributions compared with previous modelling.
The dependence of turbulence statistics and wall friction on Reynolds number in fully developed turbulent pipe flow remains a fundamental subject in fluid mechanics. This paper cross-validates experimental and numerical results, focusing on the scaling of turbulence statistics at the pipe centerline and across the inner-outer flow region. Pipe flow experiments were reviewed for friction Reynolds numbers $ 810 \le {\rm Re}_{\tau} \le 55 \times 10<^>3 $ 810 <= Re tau <= 55x103, where $ {\rm Re}_{\tau} = u_{\tau} R/\nu $ Re tau=u tau R/nu, $ u_\tau $ u tau is the wall friction velocity, $ R $ R the pipe radius, and $ \nu $ nu the kinematic viscosity. Complementary DNS data for $ 180 \le {\rm Re}_\tau \le 2880 $ 180 <= Re tau <= 2880 provide detailed insight into near-wall turbulence. A novel friction correlation, $ \rm {Re}_\tau = 0.048\,{\rm {Re}}_c<^>{0.923} $ Re tau=0.048Rec0.923 is introduced, predicting pipe-wall friction across a wide range of $ {\rm Re}_c $ Rec with accuracy better than $ \pm 2.06\% $ +/- 2.06%, where $ {\rm Re}_c $ Rec is the Reynolds number based on the centerline streamwise mean velocity component $ \overline{U}_{zc} $ Uzc. This correlation enables reliable friction estimates from centerline single-point measurements or DNS data without requiring near-wall or streamwise pressure-gradient information and is validated by consistent agreement with both experiments and DNS. The monotonic decrease in centerline turbulence intensity $ {\langle{u_z'}<^>{2}\rangle}<^>{1/2}/{\overline{{U}}_{zc}} $ < uz ' 2 > 1/2/Uzc with increasing $ {\rm Re}_c $ Rec is explained using the streamwise mean momentum equation. Finally, azimuthal spatial filtering of DNS data highlights the limitations of hot-wire resolution near the wall. For $ {\rm Re}_{\tau} \ge 2880 $ Re tau >= 2880, higher-order experimental statistics agree well with DNS for $ y<^>+ \ge 30 $ y+>= 30 and into the logarithmic region, with both datasets equally well described by logarithmic or power-law correlations, while near-wall discrepancies remain due to resolution limits.
The Taylor-Couette flow case is a turbulent benchmark that assesses the capability of numerical methods to simulate problems with curved boundaries and boundary layers. In this study, we consider the case with a rotating inner cylinder and a stationary outer cylinder at a Reynolds number $ \mathrm {Re} = 4000 $ Re=4000. This allows us to assess the accuracy of our continuous Galerkin finite element solver, which uses an implicit Large-Eddy Simulation (LES) approach and SUPG/PSPG stabilisation techniques. We perform numerical experiments using different polynomial orders p = 1, 2, 3 with up to 6M cells and 716M degrees of freedom. We compare enstrophy and kinetic energy profiles, along with vorticity and Q criterion distributions. Moreover, we compute the numerical dissipation of the implicit LES approach using the energy equation. The results show that the prediction of enstrophy is more accurate with increasing order p and refinement of the mesh. The energy balance analysis allows us to show that high-order elements can obtain less numerical dissipation with a lower number of degrees of freedom and with fewer computational resources. The work raises the question of what is an acceptable amount of numerical dissipation and provides valuable data for future users of this benchmark.
