Wall Turbulence Research Articles

A Reynolds analogy model for compressible wall turbulence

In the present study, we propose a Reynolds analogy model for compressible wall turbulence. This model is demonstrated to be able to alleviate the defects of the generalized Reynolds analogy model (GRA) (Zhang et al., J. Fluid Mech., vol. 739, 2014, pp. 392–420), and maintains its success in describing the mean velocity–temperature relation. Furthermore, the present model is superior to the GRA in depicting the relationship between their fluctuating fields and also bridges the gap between the phenomenological model and the mathematical representation of the Reynolds analogy. The key points of the present model are validated by analysing the data of compressible wall-bounded turbulence with different Mach numbers, Reynolds numbers and wall thermal conditions.

Unfolding wall turbulence

We present the project FLARE (Unfolding wall turbulence), funded by the Spanish Ministry of Science and FEDER. The main objective of this project is to contribute to understanding the dynamics of wall-bounded turbulence, exploring two techniques: Lie-Symmetry and Machine Learning. Further, we have compiled several high-quality databases of different flows using Direct Numerical Simulations of the Navier-Stokes equations with the code LISO to use these techniques. This data is available to the turbulence community.

Computational Fluid Dynamics (CFD) in Arteriovenous (AV) Graft Implantation Through End-to-Side Anastomosis with Varying Tube Diameters Across Different Vascular Access Locations for Dialysis Treatment.

Background/Objectives: Arteriovenous (AV) graft is a procedure for hemodialysis performed in the arm. Optimizing AV graft design is vital to enhance haemodialytic efficiency in patients with kidney disease. Despite being a standard procedure, making it work optimally is still difficult due to various graft diameters and anastomosis configurations, which have limited studies. This research aims to find the ideal AV graft tube diameter on blood flow and pressure gradients and the ideal body site for AV graft implantation and to study their angles for dialysate flow. Methods: Nine models were designed in Autodesk Fusion 360 with 40°, 50°, and 60° angles each having 2 mm, 5.1 mm, and 14.5 mm diameters, all following specific equations on continuity, momentum (Navier-Stokes Equation)), and the Reynolds Stress Model (RSM). The CFD simulation of these models was performed in ANSYS Fluent with an established parameter of 0.3 m/s inlet velocity and stiff/no-slip graft and artery wall boundary condition. Results: As a result, the design with a diameter of 14.5 mm and a 40° angle was overall the most ideal in terms of minimal wall shear stress and turbulence. Conclusions: Thus the brachiocephalic area or the forearm is calculated to be the most optimal implantation site. Additionally, varying angles do affect dialysate flow, as smaller values cause less stress.

Numerical simulation analysis of turbulent pulsation drag reduction at different intermittent times

The shear stress generated by wall turbulence is the main cause of wall friction resistance in turbulent flow through pipes. This paper investigates the impact of inserting rest periods (regions of constant Reynolds number) within the pulsating operating cycle of velocity on the fully turbulent flow at large time-averaged Reynolds numbers, using the large eddy simulation (LES) method. The study aims to explore the effect of increasing rest periods within pulsations on drag resistance. The dimensionless shear stress and drag reduction rates during different time periods of rest were analyzed. Numerical simulation results indicate that the pulsating velocity operation mode does not necessarily lead to drag reduction; it may even result in increased resistance. Inserting rest periods within the pulsation cycle can achieve drag reduction effects, with the maximum drag reduction rate reaching 21.8%. This paper utilizes large eddy simulation (LES) to compare and validate the feasibility of LES for pipe pulsating operation modes against experimental results and direct numerical simulation (DNS) results from the literature, providing a rationale and proof of concept for further in-depth studies on other pipe pulsating flows.

Numerical study on aerodynamic and aeroacoustic characteristics of sinusoidal wavy square cylinders

This paper numerically investigates the influences of amplitude and wavelength of sinusoidal wavy square cylinders on aerodynamic performance and noise reduction by large eddy simulation along with the Ffowcs Williams–Hawkings equation. The results show that the mean drag, lift fluctuation, and far-field noise of wavy cylinders are all reduced compared to the straight counterpart. The far-field noise of wavy cylinders varies monotonically with amplitude in a specific range but not with wavelength. The case with the largest amplitude demonstrates a significant tonal noise reduction of 47 dB/Hz, while a tonal noise reduction of 23 dB/Hz is observed for the case with the largest wavelength. To explore the mechanisms of noise reduction, the characteristics of a flow field are analyzed. It is found that wavy cylinders attenuate the transverse oscillation of a shear layer and produce more three-dimensional coherent structures in the wake. The wake region is significantly extended due to the delayed vortex shedding, and the mutual interaction between shear layers is remarkably weakened along the entire span. The spanwise coherence is attenuated in a similar way. These lead to the suppression of wall pressure fluctuations and turbulence fluctuations in the wake, which are closely related to far-field noise radiation.

Validation of a Central-Difference Finite Volume Method for Analysing Wall Turbulent Heat Transfer Using Transport Equations for Turbulent Heat Fluxes

Abstract The aim of this study is to validate the finite volume method approach using the transport equations for turbulent heat fluxes in turbulent heat transfer at the wall surface. The finite volume method technique is a numerical analysis based on conservation laws of physical quantities in the flow field, with particular emphasis on the conservation of velocity fluctuation intensity in the flow field. The behaviour of turbulent heat flows is analysed in detail using this method and its validity is tested. In the study, heat transfer in channel turbulence is analysed under a constant heat flux heating condition. The individual terms of the temperature variance equation are analysed and the results are shown to be in agreement with a previous study. The residual is small, confirming the high accuracy of the results. The present study analyses the transport equation for the dissipation of the temperature variance. The transport equations for turbulent heat fluxes in the flow direction and in the vertical wall direction are analysed and show good agreement with those of the previous study. The results show that in the vertical wall direction, non-zero terms such as the pressure diffusion term appear to contribute and reduce the accuracy of the transport equations.

Superposition and Amplitude Modulation Effects in Separated Flow

Abstract This article adopts direct numerical simulation method to study concave cavity flow with high Reynolds number, and explores the influence of outer region structure on inner region turbulence of concave cavity flow, as well as the mechanism of coherent structure in the near wall region of concave cavity flow. This article finds that the superposition and amplitude modulation effects of separated flow are weaker than those in smooth wall turbulence. At different positions within the concave cavity, the superposition effect in the reattachment zone is the strongest, while the superposition effect in the separation zone is the weakest, indicating that flow separation prevents large-scale structures from outer region penetrating into the near wall region. The amplitude modulation effect of the reattachment point is the strongest, which is related to the strong adverse pressure gradient of the reattachment point.

Turbulence from an Observer Perspective

Turbulence is often studied by tracking its spatiotemporal evolution and analyzing the dynamics of its different scales. The dual to this perspective is that of an observer who starts from measurements, or observations, of turbulence and attempts to identify their back-in-time origin, which is the foundation of data assimilation. This back-in-time search must contend with the action of chaos, which obfuscates the interpretation of the observations. When the available measurements satisfy a critical resolution threshold, the influence of chaos can be entirely mitigated and turbulence can be synchronized to the exact state–space trajectory that generated the observations. The critical threshold offers a new interpretation of the Taylor microscale, one that underscores its causal influence. Below the critical threshold, the origin of measurements becomes less definitive in regions where the flow is inconsequential to the observations. In contrast, flow events that influence the measurements, or are within their domain of dependence, are accurately captured. The implications for our understanding of wall turbulence are explored, starting with the highest density of measurements that entirely tame chaos and proceeding all the way to an isolated measurement of wall stress. The article concludes with a discussion of future opportunities and a call to action.

The Response of Turbulent Channel Flow to Standing Wave-like Wall Motion

The effect of wall deformations on wall turbulence is a topic of fundamental importance for the advancement of flow control strategies aimed at reducing the frictional drag. Dynamic wall deformations in the form of wall-normal of in-plane oscillations represents an area that is yet to be fully explored, and that can open a new realm of drag control strategies, as well as provide fresh insights into the structure of wall turbulence. In this study, we present several results from direct numerical simulations aimed at understanding the response of wall turbulence to standing wave-like wall motion, arranged in a checkerboard pattern. More specifically, here we target a low Reynolds number turbulent channel flow, featuring standing wave-like dynamic wall deformations on both bottom and top walls, with parameterizations in terms of the frequency of oscillations, roughness height, and spacing. We quantify the effect that this type of wall motion can have on the skin friction drag, various turbulent statistics, and turbulent flow structures. In addition, taking advantage of the periodicity and the symmetry of the flow, an improved phase-averaging procedure is introduced, which enhances the smoothness of the data. The results show that this type of dynamic wall deflection can have a significant effect on the turbulent flow in proximity to the wall, and that the variation of the spatial wavenumbers of the wall deflections can make a big difference. A slight total drag reduction, in the order of 2%, was observed for some combinations of wavenumbers and frequencies, especially for the highest streamwise wavenumber case.

Uniform momentum zones in turbulent channel flow

The characteristics of uniform momentum zones (UMZs) and UMZ interfaces in turbulent channel flow (TCF) at friction Reynolds number ranging from 600 to 2000 are investigated. The focus is on the Reynolds number effect and the difference among the turbulent channel flow, turbulent pipe flow (TPF), and turbulent boundary layer (TBL). The UMZ is identified using the approach of probability density histograms, a discrete approximation of the probability density function. We find that the average number of UMZs increases with the Reynolds number, similar to the findings in the TBL. The first group of UMZ in the TCF, namely the quiescent core, has a thicker thickness at a higher Reynolds number. Therefore, the UMZ closer to the channel centre significantly influences high Reynolds number wall turbulence by occupying a large portion of the channel. With increasing Reynolds number, the intensity of spanwise vorticity at the UMZ interface increases, while the thickness of the local maximum decreases. The dominance of prograde vorticity at the interface also increases with the Reynolds number. Moreover, the contortion of the UMZ interface is significantly enhanced at high Reynolds numbers than at low Reynolds numbers in both the streamwise and spanwise directions. Furthermore, the number of UMZ for a given snapshot is related to the overall velocity fluctuations in the subregion, increasing with large-scale Q2 events and decreasing with large-scale Q4 events. Comparing the three flow configurations, UMZs show similarities in their trends, along with two notable differences. First, the TPF has more UMZs compared to the TCF and TBL. Second, the innermost group in the TPF is considerably thicker than that of the higher ranks, a pattern absent in the TCF and TBL. The present findings advance the understanding of UMZs in TCF and demonstrate the similarities and differences between the cases of TCF and TBL.

High-flexibility reconstruction of small-scale motions in wall turbulence using a generalized zero-shot learning

This study proposes a novel super-resolution (or SR) framework for generating high-resolution turbulent boundary layer (TBL) flow from low-resolution inputs. The framework combines a super-resolution generative adversarial neural network (SRGAN) with down-sampling modules (DMs), integrating the residual of the continuity equation into the loss function. The DMs selectively filter out components with excessive energy dissipation in low-resolution fields prior to the super-resolution process. The framework iteratively applies the SRGAN and DM procedure to fully capture the energy cascade of multi-scale flow structures, collectively termed the SRGAN-based energy cascade reconstruction framework (EC-SRGAN). Despite being trained solely on turbulent channel flow data (via ‘zero-shot transfer’), EC-SRGAN exhibits remarkable generalization in predicting TBL small-scale velocity fields, accurately reproducing wavenumber spectra compared to direct numerical simulation (DNS) results. Furthermore, a super-resolution core is trained at a specific super-resolution ratio. By leveraging this pretrained super-resolution core, EC-SRGAN efficiently reconstructs TBL fields at multiple super-resolution ratios from various levels of low-resolution inputs, showcasing strong flexibility. By learning turbulent scale invariance, EC-SRGAN demonstrates robustness across different TBL datasets. These results underscore the potential of EC-SRGAN for generating and predicting wall turbulence with high flexibility, offering promising applications in addressing diverse TBL-related challenges.

Utilizing indicator functions with computational data to confirm nature of overlap in normal turbulent stresses: Logarithmic or quarter-power

Indicator functions of the streamwise normal-stress profiles (NSP), based on careful differentiation of some of the best direct numerical simulations (DNS) data from channel and pipe flows, over the range 550<Reτ<16 000, are examined to establish the existence and range in wall distances of either a logarithmic-trend segment or a 1/4-power region. For nine out of 15 cases of DNS data we examined where Reτ<2000, the NSP did not contain either of the proposed trends. As Reτ exceeds around 2000 a 1/4-power, reflecting the “bounded-dissipation” predictions of Chen and Sreenivasan [“Law of bounded dissipation and its consequences in turbulent wall flows,” J. Fluid Mech. 933, A20 (2022); “Reynolds number asymptotics of wall-turbulence fluctuations,” J. Fluid Mech. 976, A21 (2023)] and data analysis of Monkewitz [“Reynolds number scaling and inner-outer overlap of stream-wise Reynoldss stress in wall turbulence,” arXiv:2307.00612 (2023)], develops near y+=1000 and expands with Reynolds numbers extending to 1000<y+<10 000 for Reτ around 15 000. This range of 1/4-power NSP corresponds to a range of outer-scaled Y between around 0.3 and 0.7. The computational database examined did not include the zero-pressure-gradient boundary layer experiments at higher Reynolds numbers where the logarithmic trend in the NSP has been previously reported around y+ of 1000 by Marusic et al. [“Attached eddy model of wall turbulence,” Annu. Rev. Fluid Mech. 51, 49–74 (2019); “The logarithmic variance of streamwise velocity and conundrum in wall turbulence,” J. Fluid Mech. 933, A8 (2022)] according to a “wall-scaled eddy model.”

Wall turbulence perturbed by a bump with organized small-scale roughness: flow statistics

Wall turbulence perturbed by a bump with organized small-scale roughness: flow statistics

URANOS-2.0: Improved performance, enhanced portability, and model extension towards exascale computing of high-speed engineering flows

We present URANOS-2.0, the second major release of our massively parallel, GPU-accelerated solver for compressible wall flow applications. This latest version represents a significant leap forward in our initial tool, which was launched in 2023 (De Vanna et al. [1]), and has been specifically optimized to take full advantage of the opportunities offered by the cutting-edge pre-exascale architectures available within the EuroHPC JU. In particular, URANOS-2.0 emphasizes portability and compatibility improvements with the two top-ranked supercomputing architectures in Europe: LUMI and Leonardo. These systems utilize different GPU architectures, AMD and NVIDIA, respectively, which necessitates extensive efforts to ensure seamless usability across their distinct structures. In pursuit of this objective, the current release adheres to the OpenACC standard. This choice not only facilitates efficient utilization of the full potential inherent in these extensive GPU-based architectures but also upholds the principles of vendor neutrality, a distinctive characteristic of URANOS solvers in the CFD solvers' panorama. However, the URANOS-2.0 version goes beyond the goals of improving usability and portability; it introduces performance enhancements and restructures the most demanding computational kernels. This translates into a 2× speedup over the same architecture. In addition to its enhanced single-GPU performance, the present solver release demonstrates very good scalability in multi-GPU environments. URANOS-2.0, in fact, achieves strong scaling efficiencies of over 80% across 64 compute nodes (256 GPUs) for both LUMI and Leonardo. Furthermore, its weak scaling efficiencies reach approximately 95% and 90% on LUMI and Leonardo, respectively, when up to 256 nodes (1024 GPUs) are considered. These significant performance advancements position URANOS-2.0 as a state-of-the-art supercomputing platform tailored for compressible wall turbulence applications, establishing the solver as an integrated tool for various aerospace and energy engineering applications, which can span from direct numerical simulations, wall-resolved large eddy simulations, up to most recent wall-modeled large eddy simulations. Program summaryProgram title: Unsteady Robust All-around Navier-StOkes Solver (URANOS)CPC Library link to program files:https://doi.org/10.17632/pw5hshn9k6.2Developer's repository link:https://github.com/uranos-gpu/uranos-gpu, https://github.com/uranos-gpu/uranos-gpu/tree/v2.0Licensing provisions: BSD License 2.0Programming language: Modern Fortran, OpenACC, MPINature of problem: Solving the compressible Navier-Stokes equations in a three-dimensional Cartesian framework.Solution method: Convective terms are treated with high-resolution shock-capturing schemes. The system dynamics is advanced in time with a three-stage Runge-Kutta method. Parallelization adopts MPI+OpenACC.

Spatially evolving cascades in wall turbulence with and without interface

Direct numerical simulations of channel flow and temporal boundary layer at a Reynolds number $Re_{\tau } = 1500$ are used to assess the scale-by-scale mechanisms of wall turbulence. From the peak of turbulence production embedded at the small scales of the near-wall region, spatially ascending reverse cascades are generated that move through self-similar eddies growing in size with the wall distance. These fluxes are followed by spatially ascending forward cascades through detached eddies thus reaching sufficiently small scales where eventually scale energy is dissipated. This phenomenology is shared by both boundary layer and channel flow and is recognized as a robust physical feature characterizing wall turbulence in general. Specific features related to the flow configuration are indeed identified in the outer region. In particular, the central region of channels is characterized by a generalized Richardson energy cascade where large scales are in equilibrium with small scales at different wall distances through a combined forward cascade and spatial flux. On the contrary, the interface region of boundary layers is characterized by an almost two-dimensional physics where spatially ascending reverse cascades sustain long and wide interface structures with a forward cascade that survives only in the wall-normal scales. The overall scenario consists in a variety of scale motions that while protruding from the turbulent core towards the external region, squeeze at the interface thus sustaining vertical shear in a thin layer. The observed multidimensional physics sheds light on the complex interactions between outer entrainment and near-wall self-sustaining mechanisms with possible repercussions for theories.

Evaluation of influence of wall roughness on wall turbulence using LES simulation with MSGD model for liquid Li flow inside two-staged contraction nozzle

ABSTRACT Liquid lithium (Li) jet is planned as a beam target in fusion neutron sources (FNSs), such as IFMIF, A-FNS and IFMIF-DONES. For the safety and the efficiency of such FNSs, the high-speed Li jet need to be kept stable. For investigation of the detailed flow pattern inside the Li jet, numerical approaches using CFD simulation have been required to evaluate it because of its opacity. As one of issues about the inner flow structure, the influence of wall roughness on the vortex structure and surface variation is concerned in long-term operation of actual FNSs. In our previous simulation, in order to consider wall roughness in LES simulation, MSGD model was adopted to ANSYS Fluent using User-Defined Function, and CFD simulation in a simple parallel plate flow was conducted to confirm the performance of MSGD model. In this study, firstly, MSGD model was verified by LES simulation of Li flowing on smooth and rough plates. After that, LES simulation with MSGD model for two-staged contraction nozzle, installed into Li loop of Osaka University, was conducted. It was found that MSGD model was valid and spectrum of velocity in roughness wall cases clearly had different behavior from that in smooth wall case.

GPU acceleration of four-way coupled PP-DNS for compressible particle-laden wall turbulence

This paper presents an efficient implementation of the four-way coupled point-particle direct numerical simulation (PP-DNS) for compressible particle-laden wall turbulence, utilizing the open-source finite-difference compressible Navier–Stokes solver, STREAmS. The proposed design integrates a GPU-based two-phase collision detection algorithm known as the spatial subdivision method, along with specialized storage and MPI communication strategies for Lagrangian particles on multi-GPU platforms. Specifically, a ‘page table’ like data structure is designed to store the particle information compactly and to enable highly parallelized packing and unpacking procedures for GPU-GPU data exchange. These advancements significantly reduce the computational cost of four-way coupled particle-laden flow simulations, enabling efficient simulations involving over O(107) particles (an order of magnitude higher than that in the state-of-the-art simulations) on a single NVIDIA A100 GPU. To validate the proposed implementation, we perform simulations of compressible particle-laden wall-bounded turbulence using canonical configurations such as channel flows and zero-pressure-gradient boundary layers. The example results highlight the effects of inter-particle collisions and flow compressibility. Furthermore, we assess single-GPU performance and scalability by employing up to eight NVIDIA GPU devices. Even for four-way coupled simulations, the elapsed time per step scales approximately linearly with the number of particles (when the number of particles is large enough), and a parallel efficiency of 94.1% is achieved on 8 NVIDIA A100 GPUs.

Inter-scale Causality Relations in Wall Turbulence

We investigate the inter-scale energy transfer in the buffer layer between different streamwise and spanwise lengthscales. In a periodic channel, inter-scale transfer of momentum in the periodic directions is only possible through wavenumber interactions in the non-linear advection term. The aim of this study is to map the inter-scale interactions which transfer energy through advection to different lengthscales of interest. The non-linear advection term has nine contributions, which correspond to three different advecting and advected velocity components. Because the flow is anisotropic, each of these contributions can be viewed as portraying a different physical mechanism. Maps of inter-scale transfer are therefore constructed for each of the nine contributions to the advection term in order to understand the relative importance of advecting and advected velocity components in wall turbulence. As a preliminary exploration, two maps of the inter-scale energy transfer in the buffer layer are presented for Reτ = 180, one for transfer to a ‘target’ lengthscale dynamically relevant in the near-wall cycle, λx+=188 , λz+=94 and one for transfer to a ‘target’ lengthscale in the dissipative range, λx+=42 , λz+=21 . In the first case, we observe that the transfer of energy is dominated by transfer into streamwise-velocity (u) by spanwise-elongated structures of spanwise-velocity (w) that advect streamwise-elongated structures of streamwise-velocity (u). This is nothing but the process of streak meandering induced by spanwise-elongated (w) structures. In the second case, the inter-scale transfer can be split into three dominant mechanisms. The first and second mechanisms are observed across many of the nine contributions to the advection term. Velocity structures with size similar or larger than streamwise streaks either advect (mechanism 1) or are advected by (mechanism 2) velocity structures with lengthscales close to the target one. The third mechanism is only significant for streamwise (u) velocity structures advecting streamwise (u) and spanwise (w) velocity, and involves structures with an intermediate scale which is in-between the dissipation scale and the near-wall cycle dynamic lengthscale, but still in the dissipation range. In this mechanism, intermediate-scale streamwise-velocity structures advect intermediate-scale streamwise- and spanwise-velocity structures.

Are the dynamics of wall turbulence in minimal channels and larger domain channels equivalent? A graph-theoretic approach

This work proposes two algorithmic approaches to extract critical dynamical mechanisms in wall-bounded turbulence with minimum human bias. In both approaches, multiple types of coherent structures are spatiotemporally tracked, resulting in a complex multilayer network. Network motif analysis, i.e., extracting dominant non-random elemental patterns within these networks, is used to identify the most dominant dynamical mechanisms. Both approaches, combined with network motif analysis, are used to answer whether the main dynamical mechanisms of a minimal flow unit (MFU) and a larger unconstrained channel flow, labeled a full channel (FC), at Reτ ≈ 180, are equivalent. The first approach tracks traditional coherent structures defined as low- and high-speed streaks, ejections, and sweeps. It is found that the roll-streak pairing, consistent with the current understanding of self-sustaining processes, is the most significant and simplest dynamical mechanism in both flows. However, the MFU has a timescale for this mechanism that is approximately 2.83 times slower than that of the FC. In the second approach, we use semi-Lagrangian wavepackets and define coherent structures from their energetic streak, roll, and small-scale phase space. This method also shows similar motifs for both the MFU and FC. It indicates that, on average, the most dominant phase-space motifs are similar between the two flows, with the significant events taking place approximately 2.21 times slower in the MFU than in the FC. This value is more consistent with the implied timescale ratio of only the slow speed streaks taking part in the roll-streak pairing extracted using the first multi-type spatiotemporal approach, which is approximately 2.17 slower in the MFU than the FC.

Experimental study of the effect of particle–wall interactions on inertial particle dynamics in wall turbulence

Based on Voronoi analysis, the properties related to the near-wall motion of particles in a turbulent boundary layer were experimentally investigated via different release modes, with a friction Reynolds number $Re_\tau =3530$ . For high-inertia sand particles with Stokes number $St^+ \sim O(10^2\unicode{x2013}10^3)$ and a volume fraction $\varPhi _v \sim O(10^{-4})$ , particle image tracking velocimetry was used to determine the particle position and near-wall distribution properties. We established three particle release modes, including top-released, bottom overall-released and bottom partially released sand particles, under the same flow field conditions and calculated the differences in particle near-wall clustering and void properties. It was confirmed that wall effects (including collision and strike-splash) have a great influence on particle clustering and void behaviour near the wall. In the top-released sand particle and locally laid sand particle cases, particles bounced off the smooth walls and re-entered the carrier flow, causing significant clustering and sparsing of particles near the walls. In contrast, in the overall sand-laying case where the bottom wall was completely covered with sand particles, there is no apparent cluster or void phenomenon near the wall $(z/\delta <0.12)$ and the particles are randomly distributed, due to the combined effect of particle impact and splashing. In addition, the clustering and voids of particles become more pronounced with increasing wall-normal distance in the three release modes, and the particle distribution shows some self-similarity at each flow layer. The probability density function of the concentration of cluster particles decreases following a ‘ $-5/3$ ’ power law. However, due to the particle–wall interaction, the probability density function gradually deviates from the ‘ $-5/3$ ’ power law.