Large Eddy Simulation Technique Research Articles

Neural differentiable modeling with diffusion-based super-resolution for two-dimensional spatiotemporal turbulence

Simulating spatiotemporal turbulence with high fidelity remains a cornerstone challenge in computational fluid dynamics (CFD) due to its intricate multiscale nature and prohibitive computational demands. Traditional approaches typically employ closure models, which attempt to represent small-scale features in an unresolved manner. However, these methods often sacrifice accuracy and lose high-frequency/wavenumber information, especially in scenarios involving complex flow physics. In this paper, we introduce an innovative neural differentiable modeling framework designed to enhance the predictability and efficiency of spatiotemporal turbulence simulations. Our approach features differentiable hybrid modeling techniques that seamlessly integrate deep neural networks with numerical PDE solvers within a differentiable programming framework, synergizing deep learning with physics-based CFD modeling. Specifically, a hybrid differentiable neural solver is constructed on a coarser grid to capture large-scale turbulent phenomena, followed by the application of a Bayesian conditional diffusion model that generates small-scale turbulence conditioned on large-scale flow predictions. Two innovative hybrid architecture designs are studied, and their performance is evaluated through comparative analysis against conventional large eddy simulation techniques with physics-based subgrid-scale closures and purely data-driven neural solvers. The findings underscore the potential of the neural differentiable modeling framework to significantly enhance the accuracy and computational efficiency of turbulence simulations. This study not only demonstrates the efficacy of merging deep learning with physics-based numerical solvers but also sets a new precedent for advanced CFD modeling techniques, highlighting the transformative impact of differentiable programming in scientific computing.

Influence of the shock wave-turbulence interaction on the swirl distortion in hypersonic inlet

This study uses the Large Eddy Simulation (LES) technique to conduct an exhaustive analysis of the flow characteristics within the Rectangular-to-Elliptical shape Transition (REST) inlet under Mach 6 conditions. It mainly focuses on investigating the influence of the shock wave-turbulence interaction on the swirl distortion at the inlet exit. At the design condition, characterized by 0° Attack and 0° Sideslip, the incident shock wave at the inlet lip undergoes multiple reflections within the boundary layer of the domain wall, culminating in the formation of turbulent structures. The first reflected shock wave has the highest energy, exerting a significant impact on the boundary layer and the exit swirl distortion. On the contrary, the energy of the incident shock wave is progressively reduced due to repeated reflections, which results in reducing the exit swirl distortion. Under off-design conditions, characterized by 6° Attack and 0° Sideslip as well as 6° Attack with 6° Sideslip, variations in the incoming flow make the incident shock wave move inward, decreasing the frequency of shock wave reflections and even significantly reducing the reflected shock waves under conditions of 6° Attack and 6° Sideslip. However, this results in significantly increasing the exit swirl angle and distortion intensity. The obtained results demonstrate that changes in the incoming flow conditions significantly affect the level of exit swirl distortion by modulating the shock wave-turbulence interaction, especially in terms of the positioning of the incident shock wave and the quantity of reflected shock waves. In addition, this paper studies the wall heat transfer coefficient of the inlet. The obtained results show that the interaction between shock waves and the boundary layer significantly affects the heat transfer coefficient. This study provides a foundation for the comprehension and prediction of the performance of hypersonic inlets across a spectrum of flight conditions, and for the guidance of the design and optimization of such inlets.

Fluid–structure interaction simulations of wind gusts impacting a hyperbolic paraboloid tensile structure

The paper focuses on fluid–structure interactions (FSI) between a turbulent, gusty fluid flow, and a membrane structure. Lightweight structures are particularly vulnerable to wind gusts and can be completely destroyed by them, making it essential to develop and evaluate numerical simulation methods suited for these types of problems. In this study, a thin-walled membrane in the shape of a hyperbolic paraboloid (hypar) is analyzed as a real-scale example. The membrane structure is subjected to discrete wind gusts of varying strength from two different directions. A partitioned FSI approach is employed, utilizing a finite-volume flow solver based on the large-eddy simulation technique and a finite-element solver developed for shell and membrane structures. A recently proposed source-term formulation enables the injection of discrete wind gusts within the fluid domain in front of the structure. In a step-by-step analysis, first the fluid flow around the structure, initially assumed to be rigid, is investigated, including a grid sensitivity analysis. This is followed by examining the two-way coupled FSI system, taking the flexibility of the membrane into account. Finally, the study aims to assess the impact of wind gusts on the resulting deformations and the induced stresses in the tensile material, with a particular focus on the influence of different wind directions.

Effects of serration angle on noise reduction mechanism of trailing edge serration

This study investigates the effects of serration angle on the three-dimensional flow field and noise reduction mechanism of a NACA6512-63 airfoil using large eddy simulation (LES) and spectral proper orthogonal decomposition (SPOD) techniques. The serration angle is defined as the ratio of serration wavelength (λ) to serration amplitude (h). The mean flow field analysis on suction side of serration reveals the outward flow towards the serration edges and the downwash flow between the valleys that generate the counter-rotating vortex pair. This flow alteration changes the serration effective angle of the serration, consequently affecting the noise reduction performance. Additionally, by comparing the general solution of Lyu’s semi-analytical equation with the SPOD mode results, the noise reduction mechanism is examined, and the frequency range in which noise reduction occurs is analyzed. The linearly distributed multipole-like pressure patterns, similar to those in the semi-analytical solution, are observed in the low-frequency range for all three cases, with the λ/h = 0.3 case exhibiting the most effective noise reduction. The findings suggest that the local deviation of the turbulence axis, caused by the strong outward flow near the serration edges, can hinder the formation of the optimal pressure field distribution required for effective noise reduction.

Influence of Time-Varying Freestream Conditions on the Dynamics of Unsteady Boundary-Layer Separation

Unsteady flow separation of a turbulent boundary layer under dynamic pressure gradients is investigated using the Large-Eddy Simulation technique. The unsteadiness is introduced by prescribing an oscillating freestream vertical-velocity profile at the top boundary of the domain. Although previous studies, including Ambrogi et al. (Journal of Fluid Mechanics, Vol. 945, Aug. 2022, p. A10) and Ambrogi et al. (Journal of Fluid Mechanics, Vol. 972, Oct. 2023, p. A36), focused on the kinematics of the flow and the effects of the oscillation frequency on flow separation, the goal of this paper is to analyze the effects of three time-varying freestream-forcing profiles while the oscillation frequency is kept the same for all Cases. Whereas in Case A the freestream-velocity profile changes from suction–blowing to blowing–suction in a complete cycle, Cases B and C are both suction–blowing only and the strength of the adverse pressure gradient is modulated in time. Moreover, the boundary layer in Case B never approaches a zero-pressure gradient condition. A closed separation bubble is formed for all Cases; however, its dimensions change depending on the far-field forcing. The time evolution of turbulent kinetic energy (TKE) reveals an advection mechanism of turbulent structures out of the domain for all Cases. Whereas in Case A and C the high-TKE region, generated in the separated shear layer, is washed out of the domain as a rigid body, in Case B the separation bubble remains present and the advection mechanism of TKE is characterized by a breathing pattern.

Verification and Validation of Spectral Element Code for Supercritical CO2 Flow in Vertical Heated Tubes

The investigation of heat transfer in supercritical CO2 (sCO2) has garnered considerable attention in recent decades, given sCO2’s potential as a promising working fluid for advanced power conversion cycles. Despite previous research efforts, there are still gaps in our understanding of sCO2 heat transfer, particularly in conditions associated with heat transfer deterioration. To delve into sCO2 heat transfer more comprehensively, we propose employing the high-fidelity computational fluid dynamics code NekRS to simulate sCO2 flow using the large eddy simulation technique. Through graphics processing unit acceleration, NekRS achieves a higher computational speed than traditional CPU-based systems. However, before using NekRS in practical applications involving sCO2, it is imperative to perform verification and validation. This paper presents our efforts to verify and validate the NekRS code’s capability for simulating sCO2 using heated vertical tubes, where heat transfer deterioration usually happens. To accommodate the unique properties of sCO2, we have modified the NekRS code by integrating third-party property modules, such as REFPROP and PROPATH. Our simulations are compared with experimental and numerical data from the literature, instilling confidence in leveraging NekRS for future engineering applications. Our simulations also reveal that the accuracy of the property module significantly impacts the results, with REFPROP outperforming PROPATH for sCO2 properties. Additionally, we observed that, depending on the flow direction, buoyancy can either enhance or suppress turbulence in sCO2 flow. In upward flow, under certain conditions, the suppressed turbulence leads to heat transfer deterioration, resulting in elevated wall temperatures.

Wind-Induced Aerodynamic Effects on Set-Back Tall Buildings Using LES

Set-back modification represents a dependable method to reducing the wind effects on high-rise buildings, but the flow mechanism and aerodynamic responses of set-back tall buildings have not been explored systematically. Using the square building as the benchmark model, six set-back building models with varying steps and sizes were chosen for numerical simulation in this study. The results of mean wind pressure and local wind force distribution, power spectral densities of base moment, flow field, and structural response of the proposed building models are compared and analysed using the Large Eddy Simulation (LES) technique. The results demonstrate that the aerodynamic performance of set-back tall buildings dramatically improved, with the actual optimization in the cross-wind direction being more significant. Generally, a single set-back measure can enhance the aerodynamic performance of high-rise buildings more effectively than a double set-back measure. The outcome of this study provides an aerodynamic design guide for set-back tall buildings.

Sub-filter-scale shear stress analysis in hypersonic turbulent Couette flow

Direct numerical simulations of hypersonic turbulent Couette flows are performed for top-wall Mach numbers of 6, 7 and 8, inspired by non-reactive high-enthalpy wind tunnel free-stream conditions, with the goal of analysing the physical processes driving the sub-filter-scale (SFS) stresses to inform development of large-eddy simulation techniques for hypersonic wall-bounded flows. Semi-local scaling laws collapse mean profiles and second-order turbulent statistics well, in spite of the strong wall-normal gradients of temperature and density. On the other hand, the SFS shear stresses exhibit an unexpected profile characterized by a region of pronounced shear stress deficit, which becomes more pronounced for higher Mach numbers. Instantaneous visualizations suggest that the SFS shear stress deficit is induced by the counter-gradient resolved momentum transport driven by residual velocity motions at the interface between high-density low-speed streaks being ejected away from the wall, and low-density high-speed ones replacing the displaced fluid, qualifying this as a compressibility effect. It is shown that the SFS shear stresses are primarily driven by second-order interactions between residual velocities, in spite of their triply nonlinear nature. This, in turn, motivated a statistical quadrant analysis revealing the presence of a SFS shear stress deficit, that is, SFS processes driving momentum transport of the resolved field towards the top wall. Additionally, higher-Reynolds-number simulations reveal that there is an upper limit of spatial filter widths to resolve large-scale structures, and such deficit is observable at any Reynolds number when a reasonable spatial filter is applied.

Comparative numerical investigation of flow-induced noise characteristics of high-speed trains using high-resolution compressible Large Eddy Simulation

The South Korean Ministry of Land, Infrastructure and Transport has developed a “Comprehensive Plan for 400km/h High-Speed Rail” aimed at enhancing the operational speed of the high-speed railways, currently running up to 300km/h. A short-term goal is to elevate the operating speed of high-speed trains to 370 km/h, up from 320 km/h. Because aerodynamic noise proportionally escalates with the 6th powers of the speed, aerodynamic noise becomes more significant at higher speeds. Consequently, there's a pressing need for design solutions that reduce aerodynamic noise in high-speed trains. This study involves an aeroacoustic analysis using real-scale models of the current model and the preliminary design targeting 370 kph operation. Each model's 8-car formation has been simplified to a 5- car setup. A challenge in predicting aerodynamic noise is the generation of detailed sound sources in the near field and precise noise propagation in the acoustic field. For that, a three-dimensional compressible Large Eddy Simulation technique is employed, utilizing high-resolution grids. This allows for concurrent computation of the external flow and acoustic fields for a real-scale, high-speed train in an open environment. The analysis comprehensively examines the aerodynamic and aeroacoustic properties of each train car, including the major contributors to aerodynamic noise in high-speed trains. The radiated noise is predicted using the Ffowcs Williams and Hawkings equation and is further examined in relation to vortex sound sources.

Flow dynamics of a liquid–liquid bi-swirl injector

This work presents detailed two-phase flow simulations of liquid–liquid bi-swirl injectors for engineering applications. Liquid oxygen and kerosene are delivered into the injector, which is initially filled with air. A parametric study is conducted on the recess length of the inner tube to investigate its effect on the flow dynamics and spray pattern. The simulations are performed using an improved multiphase flow solver in OpenFOAM, which employs the volume-of-fluid approach for interface tracking and the large-eddy simulation technique for turbulence modeling. Detailed flow dynamics of both fuel and oxidizer, along with their spray patterns, are examined systematically. A comprehensive account is provided on the impact of the recess length on flow evolution, including the film thickness, air core diameter, and spray cone angle. Spectral and dynamic mode decomposition analyses are performed to further examine the flow structures and dynamics. Results from this study can be effectively used to facilitate the design optimization of injectors for liquid-fueled propulsion systems.

Large Eddy Simulation Combined with System Identification Method to Analyze Non-Premixed Flame Transfer Function

ABSTRACT The heat release response (HRR) of the flame is a pivotal aspect of combustion instability investigation, commonly characterized by the Flame Transfer Function (FTF). This study employed a combination of large eddy simulation (LES) and system identification (SI) techniques to derive the FTF for a methane-air non-premixed swirling burner. The influence of memory time and excitation signals within the LES+SI was investigated as only a few studies have been conducted in this direction so far, and validated against experimental data. Variations in memory time led to notable shifts in frequency and amplitude for gain peaks and valleys, with modest effects on phase delay within the 20–140 Hz range. The number of corner frequencies increased with memory time, whereby smaller memory times flattened the gain curve and underestimated phase delay, and larger memory times intensified curve undulations, distorting finer details. By minimizing the Mean Relative Error (MRE) between LES+SI results and experimental data, the optimized memory time of 0.017 s was identified, which closely aligned with the convection time of 0.0167 s. Consequently, convection time could be referenced in a priori estimation approach for memory time in a non-premixed swirling burner. In comparing FTF obtained from different excitation signals, the Superimposed Sinusoidal Signal (SINE) excelled in the gain curve within the 20–100 Hz range, accurately capturing the 30 Hz gain peak and 80 Hz valley, while the Broadband White Noise (BBWN) excelled in the phase curve within the same range, both yielding a smaller mean relative error (MRE). However, the Discrete Random Binary Signal (DRBS) demonstrated the best consistency with experimental data within the 100–190 Hz range, and boasted an MRE of 38.01% and 28.55% respectively for the gain and phase curve within the entire frequency range. Consequently, we inferred that DRBS functioned as the excitation signal with the most minimal identification error.

The role of the areal parameters on turbulent flow over 2D Gaussian roughness

In the past decades, numerous efforts have been dedicated to establishing a direct correlation between a geometric parameter that represents wall roughness and the corresponding velocity reduction, known as the Roughness Function ΔU+. This reduction is influenced by various statistical measures of roughness height, including average roughness height, peak-to-valley roughness distance, roughness root mean square, and Effective Slope, among others. It has been demonstrated that a singular measure of roughness height cannot sufficiently predict the Roughness Function across all turbulent regimes. Consequently, many studies have concentrated on identifying a universal correlation between roughness geometry and the downward shift of the mean velocity profile. In this study, we investigated the correlation between various geometrical parameters and the Roughness Function using Large Eddy Simulation (LES) techniques in channel flows at a friction Reynolds number of Reτ=400. Given the complexity introduced by the random nature of irregular roughness, we explored specific aspects of the relationship between wall irregularities and the roughness function by studying 2D geometries. This approach allowed us to systematically investigate the impact of geometrical properties on the roughness function and isolate the effects of roughness density and coverage area. Several irregular rough surfaces, characterized by different average oscillations height, different distributions and different densities, were designed throughout 2D Gaussian functions. With the aim to find a universal correlation between the roughness geometry and the Roughness Function, new geometrical quantities were investigated, based on the global area occupied by the roughness A∗. The prediction of the roughness function can be thus obtained using apriori data. To predict the roughness function, we introduced a parameter called Effective Area (EA), which is derived from the correlation between the Effective Slope (ES) and the roughness area A∗. Our findings indicate that a single geometric parameter, whether ES or Area A, is insufficient to predict the roughness effect. Conversely, combining these two parameters enhances predictive accuracy, at least for the proposed roughness model. This improvement can be attributed to the ability of ES to interpret the roughness distribution and height, while the coverage area is effective in predicting roughness density over a flat plate.

Scale Effects Investigation in Physical Modeling of Recirculating Shallow Flow Using Large Eddy Simulation Technique

In this study, the Large Eddy Simulation (LES) model in OpenFOAM was used to investigate the scale effects in the physical modeling of recirculating shallow flow at low Froude numbers. A laboratory test of turbulent flow through a submerged conical island with a Reynolds number of 6,210 was selected. The lab prototype was scaled with factors of 3 and 10 for both undistorted and distorted models. Our study employed the Froude similarity as the gravitational force is more dominant than the others (viscous, drag, and cohesion forces). Because the fluid (water) used for the prototype and model is the same, it is impossible to match the Reynolds, Weber, and Froude numbers simultaneously, resulting in the scale effects. For a scale of 1:1, the LES model could simulate the experimental data by appropriately capturing the vortices behind the conical island. For the undistorted models with scales of 3 and 10, the numerical model captured weaker magnitudes of vortices than the 1:1 scale, indicated by the discrepancies in velocity. In fact, the magnitudes of vortices became weaker with the distorted models. We also observed a significant increment in energy loss behind the conical island (where recirculating flows exist) as the scale increased. However, no significant discrepancies in velocity were observed between the results of the 1:1 scale and the scaled models in front of the conical island, where vortices were absent. These results indicate that the scale effects due to the Froude similarity are quite significant provided that recirculating turbulent flow occurs.

The effects of desk-partition layout on cough droplet dynamics relating to seasonal influenza

The use of physical dividers as control measures for short-range viral transmission in indoor settings has gained increasing attention. However, the understanding regarding their correct usage under different seating arrangements is incomplete. In this study, we focused on assessing the effectiveness of physical dividers in impeding the transient transmission of cough droplets in three representative layouts using the large-eddy simulation technique with the Eulerian–Lagrangian model. We computationally also investigated the effects of ventilation on viral droplet transmission under three representative desk-divider layouts. The ventilation approach was tested using two different ventilation rates (1.0 and 1.8 m/s). A comparative analysis of the ventilation flow fields, topologies, and particle directions has been studied. The findings indicate that the sitting arrangements, ventilation rates, and partition layouts play a significant role in designing effective control measures for indoor infection under the office setup considered. The protected occupied zone ventilation (POV) worked best in situations of low ventilation (1.0 m/s) for cross-partition in protecting the three healthy persons. In addition, the POV performed well for low (1.0 m/s) and high (1.8 m/s) ventilation rates for face-to-face layout. The side-by-side configuration performed poorly for the POV considered in protecting the healthy person seated directly opposite the infected person. The numerical predictions may be used to validate other experimental studies and educate office workers and engineers on indoor airborne infection control.

Numerical simulation of particle flow around the impulsive monopile

In order to explore the particle flow field characteristics around the impulsive monopile, the particle flow field and particle distribution around impulsive monopile were simulated based on the classical impulsive flow around a monopile at medium and low Reynolds number (Re = 550). The Large Eddy Simulation (LES) technique was harnessed to capture the dynamic flow field around the monopile and to facilitate in-depth analysis of vortex evolution. Concurrently, the Lagrangian approach was employed to track and characterize particle trajectories, shedding light on the patterns of particle diffusion and congregation in the vicinity of the monopile under varying Reynolds numbers. Additionally, this investigation unveils the distinctive behavior of particle flow during the impulsive starting and stopping phases adjacent to the monopile. Leveraging these insights, the study elucidates the intricate influence of the unsteady flow field dynamics around the monopile on the progression of particle flow.

Prediction and analysis of flow-induced noise of a real-size high-speed train using compressible Large Eddy Simulation and vortex sound source

As technology advances, high-speed trains have become faster, and the aerodynamic noise generated by their exterior flow fields has become a critical consideration in the design process. Accurately predicting the flow-induced noise of high-speed trains requires high-resolution sound source generation in the near-field and noise propagation in the acoustic field without numerical dissipation. This necessitates the appropriate design of the grids from the point of aerodynamic noise generation and propagation in association with the length scales of the relevant components of an actual train. To address this challenge, this research study employs a three-dimensional compressible Large Eddy Simulation (LES) technique with high-resolution grids to simultaneously calculate the external flow and acoustic fields of a real-size high-speed train consisting of five cars running in open field. Further analysis is carried out to evaluate the contribution of major components, including the cap, pantograph, bogie, intercoach, and HVAC cover, which are known to be significant contributors to high-speed train flow noise. The study also employs the vortex sound source to analyze the generation mechanism of flow-induced noise for each component, providing insights for reducing aerodynamic noise.

Numerical study of flow regimes and force characteristics for cruciform cylinder in oscillatory flow

Oscillatory flow past cruciform cylinder is numerically investigated using the Large-eddy simulation (LES) technique. The flow regimes and force characteristics of the cruciform cylinder are analyzed in detail for β (frequency parameter) numbers ranging from 200 to 500 and KC (Keulegan–Carpenter) number equals 10. Three vortex shedding modes are observed at different β values, which affect the torques exerted on the cruciform cylinder. At sections far from the node, two vortices shed within each cycle. The transverse force curve exhibits a regular pattern and a significant dominant frequency is observed in the transverse force spectra. However, as the node is approached, the vortex shedding becomes increasingly complex, leading to significant changes in the amplitudes and shapes of the transverse forces. Additionally, the sectional force coefficients agree well with those on the isolated cylinder at sections far from the node, whereas they exhibit significant increases at the sections near the node. Finally, the force coefficients of the cruciform cylinder are compared with those of the isolated cylinder for 2 < KC < 30 at β = 500. The results indicate that the cruciform cylinder experiences an increase in force coefficients, attributed to the influence of the node in cruciform cylinder.

Experimental and numerical investigations on the performance and wake characteristics of a tidal turbine under yaw

This paper presents an extensive examination of the wake characteristics and hydrodynamic performance of a 1:60 scale horizontal axis tidal stream turbine under yawed inflow conditions. Experimental investigations were conducted in a laboratory flume, while numerical simulations utilized the actuator line method and large-eddy simulation technique. The turbine was subjected to an inflow turbulence intensity of 6% and was tested at five different yaw angles (5°, 10°, 15°, 20°, and 30°). The findings demonstrate a substantial reduction in the power coefficient as the yaw angle increases, with a notable drop of 47.1% recorded at a yaw angle of 30°. Furthermore, numerical simulations of wake characteristics, including mean velocity and turbulence intensity, were examined and compared with measurements, yielding satisfactory agreement. The yawed wake flow exhibited asymmetrical distributions of mean velocity and deviations from the centerline. Additionally, distinct deformations of the yawed wake in the y-z cross sections were observed, indicating a discernible curled kidney shape that intensified with larger yaw angles. Moreover, interactions between the turbine wake and support structures affected the width and turbulence intensity of the near wake, introducing additional unsteadiness that accelerated wake recovery.

Large-Eddy Simulation of the unsteady supersonic flow around a Mars entry capsule at different angles of attack

Time-evolving high-fidelity simulations of an entry capsule in a supersonic flight regime have been performed at different angles of attack (0∘,5∘,10∘ and 15∘) in Mars' atmosphere. The aim is to contribute to the understanding of the critical unsteady phenomena that arise during the descent phase, gaining insights on the optimal design choices to ensure a safe atmospheric entry and a successful landing procedure. Turbulent structures effectively solved via Implicit Large-Eddy Simulation technique allow us to evaluate the flow properties and the capsule's response both in space and time, placing special attention to the analysis of the turbulent motions responsible for the oscillatory and unsteady dynamics. Ultimately, Proper Orthogonal Decomposition is employed to provide a low-order model of the wake dynamics.

Characterisation of unsteady separation in a turbulent boundary layer: Reynolds stresses and flow dynamics

The large-eddy simulation technique was used to investigate the dynamics of unsteady flow separation on a flat-plate turbulent boundary layer. The unsteadiness was generated by imposing an oscillating, wall-normal velocity profile at the top of the computational domain, and a range of reduced frequencies ( $k$ ), from a very rapid flutter-like motion to a slow quasi-steady oscillation, was studied. Ambrogi et al. (J. Fluid Mech., vol. 945, 2022, A10) showed that the reduced frequency greatly affects the transient separation process, and at a frequency $k=1$ , the separation region became unstable and was advected periodically out of the domain. In this paper, we discuss the causes of the observed advection process and the effects of the unsteadiness on the second moments. The time evolution of turbulent kinetic energy, for instance, reveals that an advection-like phenomenon is also present at a very low reduced frequency, but its dynamic behaviour is completely different from that of the intermediate frequency ( $k=1$ ). At the intermediate frequency the entire recirculation region is advected downstream, keeping its shape. The advected structure is rotational in nature, and moves at constant speed. In contrast, in the low-frequency case the advected fluid originates at the reattachment point, and the structure is shear-dominated. Particle pathlines reflect the fact that the flow at the low frequency is quasi-steady-state, but show peculiar differences at the intermediate frequency, in which the flow response to the freestream forcing depends on the particle positions in the wall-normal direction.