Turbulent Wall Flows Research Articles

Proposal of innovative arc-shaped wall-jet film cooling: A numerical investigation

Numerous methods are conceived to pick up the heat from hot plats, where, amidst all, film cooling methods possess several benefits and have always been considered. However, increasing the cooling effectiveness of this method while reducing the coolant mass flow rate has always been considered one of the concerns, and much literature has yet to be presented to surmount this problem. In this study, three different novel jet configurations including simple, semi-mushroom, and semi-oval jet types are proposed to boost the effectiveness of the film cooling method while the mass rate is drastically lower compared to the traditional jet types. Instead of using the traditional film jet that usually blows the coolant flow at a 30-90-degree angle, it can be changed so that the coolant is wholly blown in the mainstream path, simultaneously reducing the mixing ratio and increasing the diffusion of coolant film. Meantime, the arc-shaped jet design can increase the surface coverage by the coolant fluid, especially in the transverse direction. This innovative concept is also reckoned for in this study by presenting computational simulation using the k − ω − SST turbulence model, which has some superiority for turbulent near wall flows. The results showed that at BR=0.5 and x/d=5, the proposed novel jet (simple type) achieved a 17.9% improvement in averaged cooling effectiveness compared to regular jets, while utilizing a coolant mass flow rate ten times lower. Also, it found that contrary to the results related to regular jets, the averaged cooling effectiveness increases with the increment of blowing ratio in innovative proposed jet types.

Molecular Mechanisms Underlying Intracranial Aneurysm Rupture

Intracranial aneurysms, a major cause of subarachnoid hemorrhage(SAH), pose a significant social burden due to their poor patient outcomes. Recent studies, including those using animal models, have shed light on a new disease concept: intracranial aneurysms as a chronic inflammatory disease. This process is triggered by abnormal hemodynamic forces and mediated by immune cells like macrophages and neutrophils. The initiation of intracranial aneurysms is a two-step process. First, high wall shear stress and mechanical stretch work together to promote macrophage infiltration into the arterial walls. This infiltration is facilitated by endothelial cells and fibroblasts, which are activated to produce chemoattractants. Once the lesions enlarge, low wall shear stress and turbulent flow take over, maintaining macrophage infiltration. As the disease progresses towards rupture, infiltration creates hypoxic conditions that exacerbate the situation. These conditions, in turn, induce the formation of neovessels at the weakest point of the aneurysm and promote specific inflammatory microenvironments rich in neutrophils. The excessive tissue destruction caused by neutrophil-mediated inflammation ultimately leads to lesion rupture. Therefore, intracranial aneurysm rupture requires not only structural changes but also qualitative alterations within the chronic inflammatory environment. This suggests that factors mediating chronic inflammation could be potential targets for predicting or preventing aneurysm rupture.

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.”

On the relevance of lift force modelling in turbulent wall flows with small inertial particles

On the relevance of lift force modelling in turbulent wall flows with small inertial particles

Pressure drag reduction via imposition of spanwise wall oscillations on a rough wall

The present study tests the efficacy of the well-known viscous drag reduction strategy of imposing spanwise wall oscillations to reduce pressure drag contributions in transitional and fully rough turbulent wall flow. This is achieved by conducting a series of direct numerical simulations of a turbulent flow over two-dimensional (spanwise-aligned) semi-cylindrical rods, placed periodically along the streamwise direction with varying streamwise spacing. Surface oscillations, imposed at fixed viscous-scaled actuation parameters optimum for smooth wall drag reduction, are found to yield substantial drag reduction ( $\gtrsim$ 25 %) for all the rough wall cases, maintained at matched roughness Reynolds numbers. While the total drag reduction is due to a drop in both viscous and pressure drag in the case of transitionally rough flow (i.e. with large inter-rod spacing), it is associated solely with pressure drag reduction for the fully rough cases (i.e. with small inter-rod spacing), with the latter being reported for the first time. The study finds that pressure drag reduction in all cases is caused by the attenuation of the vortex shedding activity in the roughness wake, in response to wall oscillation frequencies that are of the same order as the vortex shedding frequencies. Contrary to speculations in the literature, this study confirms that the mechanism behind pressure drag reduction, achieved via imposition of spanwise oscillations, is independent of the viscous drag reduction. This mechanism is responsible for weakening of the Reynolds stresses and increase in base pressure in the roughness wake, explaining the pressure drag reduction observed by past studies, across varying roughness heights and geometries.

Passive scalar transport in a turbulent spot within a poiseuille flow

A pair of counter-rotating vortices are introduced in a Poiseuille flow to trigger the by-pass transition. The accent is put on the passive scalar transport within the resulting transitional spot. The latter is identified and tracked in time and space using the fluctuating vorticity and temperature fields it induces. The velocity and temperature statistics are determined strictly within the spot. They are scaled by the inner variables, related to the shear velocity and the wall temperature flux induced by the spot. The statistics are compared with those of the corresponding fully developed turbulent channel flow, in order to determine the degree of similarity between the internal structure of the spot and the fully developed wall turbulence. The analysis includes the fine structure of the fluctuating temperature field, its transport equations and the effect of the coherent structures within the spot. It is found that the turbulent part of the spot is closely similar to the turbulent wall flow, both for the fluctuating flow, and the passive scalar fields. The root-mean-square of the temperature fluctuations determined in the entire spot is nearly twice larger than that in the fully developed wall turbulence in the outer part of the flow. The transport is enhanced and the turbulent fluxes increase compared to the fully developed turbulent channel flow. This is due to the wave area of the spot that contains, in their great majority, the structures coming from the breakdown process of waves. The latter is responsible for the increase of the small-scale mixing and of the large peripheral temperature fluctuations, which is rather good news from an application point of view.

Particle transport in a turbulent pipe flow: direct numerical simulations, phenomenological modelling and physical mechanisms

In particle-laden turbulent wall flows, transport of particles towards solid walls is phenomenologically thought to be governed by the wall-normal turbulence intensity supporting the underlying particle–eddy interactions that are usually modelled by a combination of turbophoresis and turbulent diffusion. We estimate the turbophoretic and turbulent diffusive coefficients as a function of wall-normal coordinate directly from a generated direct numerical simulation (DNS) database of low volume fraction point particles in a turbulent pipe flow. These coefficients are then used in an advection–diffusion equation to estimate the particle concentration as a function of wall-normal distance and time, with favourable comparison against DNS for smaller Stokes number ( $St^+$ ) particles suggesting a limitation of the common gradient diffusion hypothesis for larger $St^+$ particles. Using DNS we explore the non-trivial effects of $St^+$ , pipe wall condition (particle absorbing or elastic) as well as the influence of drag and lift force on the velocity and particle statistics giving rise to different particle concentrations. We then appraise various Eulerian-based models of turbophoretic and turbulent diffusive coefficients and, finally, use physical insights from Lagrangian correlation times, conditional quadrant analysis and flow topology to shed further light on the particle transport as a function of various parameters and the limits of gradient diffusion hypothesis.

RANS thermal modelling of a natural convection boundary layer at low Prandtl number

The present work evaluates the performance of different RANS turbulence models in a natural convection boundary layer at three different Prandtl numbers: 0.71, 0.2 and 0.025. An assessment on the prediction of the boundary layer is presented in function of the wall fluxes, mean and turbulent flow fields. It is observed that the chosen turbulence model to close the momentum equation is relevant in order to capture the flow field at low Prandtl numbers. This is due to the strong interaction between the momentum and thermal fields in natural convection. From the thermal point of view, the analogy between the turbulent momentum and turbulent thermal diffusivities presents serious issues to correctly predict the flow field independently of the Prandtl number. Advanced models that employ algebraic or transport equations of the turbulent heat fluxes introduces the needed physics to correctly predict the natural convection boundary layer.

Bounded dissipation predicts finite asymptotic state of near-wall turbulence

Enormous efforts have been devoted to the prediction and control of turbulent wall flows. A primary consideration here is the Reynolds number scaling problem in which non-dimensional representations are sought that render the normalized variables of interest unchanged for varying Reynolds number. Relative to this objective and in contrast to the mean velocity, there remains a considerable lack of clarity associated with the apparent failure of inner normalization (i.e. using the friction velocity and kinematic viscosity) when applied to the statistical profiles of fluctuating quantities in the near-wall region. In their present work Chen & Sreenivasan (J. Fluid Mech., vol. 933, 2022, A20) generalize their earlier effort, Chen & Sreenivasan (J. Fluid Mech., vol. 908, 2021, R3), and present a rational framework for characterizing and describing the evolution of turbulence quantities that either attain a near-wall peak or have non-zero wall values. Their analysis enjoys considerable empirical support. Physically, the asymptotic boundedness of the inner-normalized dissipation is used to reason that there is a limiting state of near-wall turbulence at asymptotically large Reynolds numbers. The law of bounded dissipation arguments put forth by Chen and Sreenivasan prescribe the recovery of inner scaling and suggest new possibilities regarding the physics of how wall turbulence matures to its asymptotic state.

Computational Study of Abdominal Aortic Aneurysms with Severely Angulated Neck Based on Transient Hemodynamics Using an Idealized Model

An abdominal aortic aneurysm (AAA) is an enlargement of the abdominal aorta that can become a life-threatening disease. The pulsatile blood flow exhibits intricate laminar patterns in the abdominal portion of the human aorta under normal resting conditions, whereas secondary flows are caused by adjacent branches and abnormal vessel geometries. If a pathological disorder (e.g., aneurysm) alters the structural composition of the artery wall, the flow dynamics become more complex. In this study, we analyzed the hemodynamics of pulsatile blood flow in three-dimensional AAA models. Computational predictions of hemodynamic changes were performed considering idealized models for four severe proximal neck angulations of symmetric aneurysms assuming conditions of laminar flow and a rigid artery wall. The predictions were based on computational fluid dynamics throughout the cardiac cycle. Postprocessing was used to visualize the numerical findings. The hemodynamic changes in factors such as velocity, flow streamline, pressure, and wall shear stress were obtained and visualized. The resulting blood flow through the severely angulated proximal neck of the abdominal aorta caused strong turbulence and asymmetric flow inside the aneurysm sac, leading to blood recirculation, especially during diastole. The simulation results showed the formation of regions with high and low wall shear stress, turbulent flow, and recirculation in the aneurysm sac depending on the angulation, which could have led to aortic wall weakness.

Law of bounded dissipation and its consequences in turbulent wall flows

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number $Re_\tau$ . This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of $Re_\tau$ , owing to the natural constraint that the dissipation rate is bounded. Specifically, $\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$ where $\varPhi$ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript $\infty$ denotes the bounded asymptotic value of $\varPhi$ , and the coefficient $C_\varPhi$ depends on $\varPhi$ but not on $Re_\tau$ . Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form $\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$ , where $\varphi$ represents the streamwise or spanwise velocity fluctuation, and $\alpha _q$ and $\beta _q$ are independent of $Re_\tau$ . Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear $q$ -norm Gaussian’, is proposed to explain the observed linear dependence of $\alpha _q$ on $q$ .

A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

Integral measures of the zero pressure gradient boundary layer over the Reynolds number range ≤Rτ<∞

A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ>≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.

Statistical properties of streamline geometry in turbulent wall-flows

This study investigates the characteristics of streamline curvature and torsion in wall-bounded turbulence using DNS data from channel and boundary layer flows. The analysis includes characterizing the curvature of probability distributions as a function of the wall-normal distance. Far from the wall, the probability density function of the curvature decays with a -4 power dependence for the extreme events that are comparable with the Kolmogorov length scale. Consistent with previous studies, these large values of curvature are hypothesized to reflect the flow in the vicinity of stagnation points in the fluctuating field.

High-order conservative formulation of viscous terms for variable viscosity flows

The work presents a general strategy to design high-order conservative co-located finite-difference approximations of viscous/diffusion terms for flows featuring extreme variations of diffusive properties. The proposed scheme becomes equivalent to central finite-difference derivatives with corresponding order in the case of uniform flow properties, while in variable viscosity/diffusion conditions it grants a strong preservation and a proper telescoping of viscous/diffusion terms. Presented tests show that standard co-located discretisation of the viscous terms is not able to describe the flow when the viscosity field experiences substantial variations, while the proposed method always reproduces the correct behaviour. Thus, the process is recommended for such flows whose viscosity field highly varies, in both laminar and turbulent conditions, relying on a more robust approximation of diffuse terms in any situation. Hence, the proposed discretisation should be used in all these cases and, for example, in large eddy simulations of turbulent wall flows where the eddy viscosity abruptly changes in the near-wall region.

Theory-based Reynolds-averaged Navier–Stokes equations with large eddy simulation capability for separated turbulent flow simulations

The prediction and investigation of very high Reynolds number turbulent wall flows pose a significant challenge: experimental studies and large eddy simulation (LES) are often inapplicable to these flows, and Reynolds-averaged Navier–Stokes (RANS) methods often fail to characterize the essential flow characteristics, in particular, for separated flows. These facts explain the need for the development of hybrid RANS-LES methods. The predominant approach to deal with this question is the combination of RANS and LES equation elements. This often implies shortcomings in simulations: the lack of control of modeled and resolved motions, which are involved in hybrid simulations, can lead to inconsistencies and imbalances. A novel approach based on a theoretical solution to the latter problem (referred to as continuous eddy simulation method) is investigated here via simulations of periodic hill flows (involving flow separation and reattachment) for a range of very high Reynolds numbers. We study the mechanism and simulation performance of these new hybrid methods. The results presented demonstrate their excellent performance and advantages to differently designed hybrid methods. We also consider the reliability of flow predictions for which data for model validation are unavailable. Criteria for the reliability of such hybrid simulations are suggested. It is shown that the new hybrid method satisfy these criteria for reliable flow predictions. The results indicate the existence of an asymptotic flow regime far above Reynolds numbers that can be realized in experimental studies and resolved LES.

Multiscale interaction of inertial particles with turbulent motions in open channel flow

A spatial filtering strategy is used to investigate inertial particles interacting with certain features of wall turbulent flow, thus isolating large-scale motions (LSMs) and very large-scale motions (VLSMs). In the one-way coupled limit, low-inertia particle transport is dictated both by LSMs and VLSMs, while high-inertia particles are more influenced by VLSMs. In the two-way coupled limit, particle-flow interaction is dictated by interactions with LSMs. However, the effects of VLSMs still cannot be ignored.

(Invited) Thoughts on the Viscous Sublayer

Years ago people postulated that there is a laminar sublayer near a solid wall in turbulent flow. Murphree (1932) and Levich (1942, 1944) postulated that the turbulence eddies decay only gradually as the solid wall is approached. There are two schools of thought: one says that the eddy diffusivity is proportional to y 3 near the wall; the other says y 4. Which is right? Perhaps there is some truth in both views. It may be that y 4 prevails near the outer limit of the viscous sublayer, adjacent to the outer turbulent flow, but that y3 prevails very close to the wall. It has been thought that this was important only for mass transfer in turbulent flow at high Schmidt numbers, that is, for diffusion coefficients much smaller than the kinematic viscosity. However, it is shown that Nikuradse’s data (1932) for pressure drops (or friction factors) in turbulent pipe flow do not agree with his very careful measurements of profiles of the eddy viscosity. In other words, the viscous sublayer cannot be ignored even in fluid flow. The profile of predicted eddy diffusivity of Wasan, Tien, and Wilke (1963) show regions of y 3 dependence very close to the wall and y dependence in the outer turbulent flow with perhaps a y 4 dependence in the intermediate region. On this diagram, one can identify the thickness of the diffusion layer as a function of the Schmidt number. As is well known, the diffusion layer gets very thin at high Schmidt numbers. Using this profile, one can show that the y 4 dependence will not show up clearly in mass-transfer data at any Schmidt number. Any region of y 4 dependence of the eddy diffusivity is outside the diffusion layer where the concentration does not change (at high Schmidt numbers). These thoughts inspire a method of calculating the profiles of eddy diffusivity and eddy viscosity from consideration of fluctuations themselves. Hopefully this can be a subject of future work.

A comparative study of the velocity and vorticity structure in pipes and boundary layers at friction Reynolds numbers up to

This study presents findings from a first-of-its-kind measurement campaign that includes simultaneous measurements of the full velocity and vorticity vectors in both pipe and boundary layer flows under matched spatial resolution and Reynolds number conditions. Comparison of canonical turbulent flows offers insight into the role(s) played by features that are unique to one or the other. Pipe and zero pressure gradient boundary layer flows are often compared with the goal of elucidating the roles of geometry and a free boundary condition on turbulent wall flows. Prior experimental efforts towards this end have focused primarily on the streamwise component of velocity, while direct numerical simulations are at relatively low Reynolds numbers. In contrast, this study presents experimental measurements of all three components of both velocity and vorticity for friction Reynolds numbers$Re_{\unicode[STIX]{x1D70F}}$ranging from 5000 to 10 000. Differences in the two transverse Reynolds normal stresses are shown to exist throughout the log layer and wake layer at Reynolds numbers that exceed those of existing numerical data sets. The turbulence enstrophy profiles are also shown to exhibit differences spanning from the outer edge of the log layer to the outer flow boundary. Skewness and kurtosis profiles of the velocity and vorticity components imply the existence of a ‘quiescent core’ in pipe flow, as described by Kwonet al. (J. Fluid Mech., vol. 751, 2014, pp. 228–254) for channel flow at lower$Re_{\unicode[STIX]{x1D70F}}$, and characterize the extent of its influence in the pipe. Observed differences between statistical profiles of velocity and vorticity are then discussed in the context of a structural difference between free-stream intermittency in the boundary layer and ‘quiescent core’ intermittency in the pipe that is detectable to wall distances as small as 5 % of the layer thickness.

Nonlocal Turbophoresis of Particles in the Logarithmic Layer of Wall Turbulence

The dynamics of high-inertia particles in the logarithmic layer of a wall turbulent flow has been studied using Lagrangian stochastic simulation. Because of the linear dependence of the turbulent time scale on the distance to the wall in the logarithmic layer, turbophoresis, i.e., the migration of particles toward the wall, is observed. It has a nonlocal character in the wall region because of a significant contribution of ballistic motion of particles. The profiles of the concentration and second moment of particle velocities have been obtained for an arbitrary inertia parameter of a particle and different values of the restitution coefficient of the particle velocity at rebound from the wall. It has been shown that the probability density distribution of particle velocities has “heavy tails” in the wall region. These tails exhibit a power-law behavior in the inelastic collapse regime for particle trajectories, which is observed at restitution coefficients below a critical value of 0.163 in agreement with the results reported in Belan et al., Phys. Rev. E 93, 052206 (2016).