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We have performed direct numerical simulation of the turbulent flow of a polymer solution in a square duct, with the FENE-P model used to simulate the presence of polymers. First, a simulation at a fixed moderate Reynolds number is performed and its results compared with those of a Newtonian fluid to understand the mechanism of drag reduction and how the secondary motion, typical of the turbulent flow in non-axisymmetric ducts, is affected by polymer additives. Our study shows that the Prandtl’s secondary flow is modified by the polymers: the circulation of the streamwise main vortices increases and the location of the maximum vorticity moves towards the centre of the duct. In-plane fluctuations are reduced while the streamwise ones are enhanced in the centre of the duct and dumped in the corners due to a substantial modification of the quasi-streamwise vortices and the associated near-wall low- and high-speed streaks; these grow in size and depart from the walls, their streamwise coherence increasing. Finally, we investigated the effect of the parameters defining the viscoelastic behaviour of the flow and found that the Weissenberg number strongly influences the flow, with the cross-stream vortical structures growing in size and the in-plane velocity fluctuations reducing for increasing flow elasticity.
Direct numerical simulation of incompressible turbulence in a straight square duct finds the post-transition flow evolving substantially, and for Reynolds numbers based on the friction velocity and duct hydraulic radius greater than 600 a two-structure secondary flow regime has been established, suggesting the coexistence of two distinct sources of mean streamwise vorticity. The nominal source terms in the equation for the mean streamwise vorticity involve turbulent variables only, that allow us to identify the dominant dynamical process that marks and/or sustains the transverse mean flow. Close to the corner a mean profile instability is dominant, while farther away turbulent streamwise vorticity intensification is broadly distributed near the duct walls. The instability-driven secondary velocity maximum on the duct diagonals scales with the friction velocity. There is limited scaling of turbulent intensities on the wall bisectors.
We present a new approach – the entropy-viscosity method (EVM) – for numerical modelling of high Reynolds number flows and investigate its potential by simulating fully developed incompressible turbulent flow, first in a stationary pipe and subsequently in a flexible pipe. This method, which was first proposed by Guermond et al. (J. Comput. Phys., vol. 230 (11), 2011, pp. 4248–4267), introduces the concept of entropy viscosity, computed based on the nonlinear localized residual obtained from the energy equation. Specifically, this nonlinear viscosity based on the local size of entropy production is added to the spectral element discretization employed in our work for stabilization at insufficient resolution. Unlike its original formulation, which includes an ad hoc tuneable parameter $\unicode[STIX]{x1D6FC}$ , here, we determine the value of $\unicode[STIX]{x1D6FC}$ by assuming that the entropy viscosity is analogous to the eddy viscosity of the Smagorinsky model. However, the overall approach has the flavour of the implicit large eddy simulation (ILES) instead of the standard large eddy simulation (LES). Given the empiricism of our approach, we have performed systematic studies of homogeneous isotropic turbulence for validation (see appendix A). We have also carried out a more complete numerical simulation study to investigate incompressible turbulent flow in a stationary pipe at $Re_{D}=5300$ and $Re_{D}=44\,000$ , following the work of Wu & Moin (J. Fluid Mech., vol. 608, 2008, pp. 81–112) who performed very accurate direct numerical simulations (DNS) of these two cases. We found that the mean flow, turbulence fluctuations, and two-point correlations of the EVM-based LES are in good agreement with the DNS of Wu & Moin despite the fact that we employed grids with resolution two orders of magnitude smaller. If we instead use the standard Smagorinsky model in our simulations, the computations become unstable due to insufficient resolution of the smaller scales. Another important difference is that the entropy-viscosity model scales with the cube of the distance from the wall and approaches zero at the wall, which is theoretically correct, as shown by our a posteriori tests. Based on the validated EVM approach, we then simulated fully developed turbulent flow at $Re_{D}=5300$ in a flexible pipe subject to prescribed vibrations in the cross-flow plane corresponding to a standing wave of amplitude $A$ and wavelength $\unicode[STIX]{x1D706}=3D$ , where $D=2R$ is the pipe diameter and $R$ is the radius. We have simulated 11 cases corresponding to increasing values of wave steepness $s_{o}=2A/\unicode[STIX]{x1D706}$ , with $s_{o}\in [0,0.067]$ . We found a quadratic dependence of the friction factor on $s_{o}$ with the minimum at approximately $s_{o}\approx 0.01$ , so, surprisingly, we have a slight decrease in drag at first and then a substantial increase compared to the stationary pipe. To obtain the turbulence statistics, we averaged the simulated flow over twenty time periods at the nodes and anti-nodes separately. We found substantial changes in the mean velocity profile at distances $(1-r)^{+}>5$ while the peaks of turbulent intensities were amplified by 50 % in the axial direction and by 200 % in the normal and azimuthal directions at $s_{o}=0.067$ . The peak shear stress at the node increased by more than 200 % whereas at the anti-node it attained negative values. Turbulent budgets revealed large changes close to the wall at $(1-r)^{+}<50$ while flow visualizations showed that many more strong worm-like vortices were generated in the near-wall regions compared to the stationary pipe. We have also computed various spatio-temporal correlations that show that the pressure fluctuations are very sensitive to the pipe vibration and scale quadratically with $s_{o}$ . Both pressure and velocity correlations exhibit cellular patterns consistent with the standing-wave pipe motion.
Wall turbulence is a ubiquitous phenomenon in nature and engineering applications, yet predicting such turbulence is difficult due to its complexity. High-Reynolds-number turbulence arises in most practical flows, and is particularly complicated because of its wide range of scales. Although the attached-eddy hypothesis postulated by Townsend can be used to predict turbulence intensities and serves as a unified theory for the asymptotic behaviours of turbulence, the presence of coherent structures that contribute to the logarithmic behaviours has not been observed in instantaneous flow fields. Here, we demonstrate the logarithmic region of the turbulence intensity by identifying wall-attached structures of the velocity fluctuations ( $u_{i}$ ) through the direct numerical simulation of a moderate-Reynolds-number boundary layer ( $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ ). The wall-attached structures are self-similar with respect to their heights ( $l_{y}$ ), and in particular the population density of the streamwise component ( $u$ ) scales inversely with $l_{y}$ , reminiscent of the hierarchy of attached eddies. The turbulence intensities contained within the wall-parallel components ( $u$ and $w$ ) exhibit the logarithmic behaviour. The tall attached structures ( $l_{y}^{+}>100$ ) of $u$ are composed of multiple uniform momentum zones (UMZs) with long streamwise extents, whereas those of the cross-stream components ( $v$ and $w$ ) are relatively short with a comparable width, suggesting the presence of tall vortical structures associated with multiple UMZs. The magnitude of the near-wall peak observed in the streamwise turbulent intensity increases with increasing $l_{y}$ , reflecting the nested hierarchies of the attached $u$ structures. These findings suggest that the identified structures are prime candidates for Townsend’s attached-eddy hypothesis and that they can serve as cornerstones for understanding the multiscale phenomena of high-Reynolds-number boundary layers.
We have performed a particle-resolved direct numerical simulation of a turbulent channel flow past a moving dilute array of spherical particles. The flow shares important features with dilute vertical gas solid flow at high Stokes number, such as significant attenuation of the turbulence kinetic energy (TKE) at low particle volume fraction. The flow has been simulated by means of an overset grid method, using spherical grids around each particle overset on a background non-uniform Cartesian grid. The main focus of the present paper is on the TKE budget, which is analysed both in the fixed channel frame of reference and in the moving particle frame of reference. The overall (domain-integrated) TKE and turbulence production due to mean shear are reduced compared to unladen flow. In the fixed frame, the interfacial term, which represents production due to relative (slip) velocity, accounts for approximately 40 % of the total turbulence production in the channel. As a consequence, the total turbulence production and the overall turbulence dissipation rate remain approximately the same as in the unladen flow. However, a comparison with laminar flow past the same particle configuration reveals that significant parts of various fixed-frame statistics are due to non-turbulent structures, spatial variations that are steady in the moving particle frame. In order to obtain a clearer picture of the modification of the true turbulence and in order to reveal the rich three-dimensional (3-D) statistical structure of turbulence interacting with particles, time averaging in the moving frame of reference of the particle is used to extract the fluctuations entirely due to true turbulence. In the moving frame, the turbulence production is positive near the sides and in the wake, but negative in a region near the front of the particle. The turbulence dissipation rate and even more the dissipation rate of the 3-D mean flow attain very large values on a large part of the particle surface, up to approximately 400 and 4000 times the local turbulence dissipation rate of the unladen flow, respectively. Very close to the particle, viscous diffusion is the dominant transport term, but somewhat further away, in particular near the front and sides of the particle, pressure diffusion and also convection provide large and positive transport contributions to the moving-frame budget. A radial analysis shows that the regions around the particles draw energy from the regions further away via the surprising dominance of the pressure diffusion flux over a large range of radii. Spectra show that (very) far away from the particles all scales of the (true) turbulence are reduced. Near the particles enhancement of small scale turbulence is observed, for the streamwise component of the velocity fluctuation more than for the other components. The most important reason for turbulence reduction and anisotropy increase appears to be particle-induced non-uniformity of the mean driving force of the flow.
We present wall-resolved large-eddy simulation (LES) of flow with free-stream velocity $\boldsymbol{U}_{\infty }$ over a cylinder of diameter $D$ rotating at constant angular velocity $\unicode[STIX]{x1D6FA}$ , with the focus on the lift crisis, which takes place at relatively high Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. Two sets of LES are performed within the ( $Re_{D}$ , $\unicode[STIX]{x1D6FC}$ )-plane with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FA}D/(2U_{\infty })$ the dimensionless cylinder rotation speed. One set, at $Re_{D}=5000$ , is used as a reference flow and does not exhibit a lift crisis. Our main LES varies $\unicode[STIX]{x1D6FC}$ in $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 2.0$ at fixed $Re_{D}=6\times 10^{4}$ . For $\unicode[STIX]{x1D6FC}$ in the range $\unicode[STIX]{x1D6FC}=0.48{-}0.6$ we find a lift crisis. This range is in agreement with experiment although the LES shows a deeper local minimum in the lift coefficient than the measured value. Diagnostics that include instantaneous surface portraits of the surface skin-friction vector field $\boldsymbol{C}_{\boldsymbol{f}}$ , spanwise-averaged flow-streamline plots, and a statistical analysis of local, near-surface flow reversal show that, on the leeward-bottom cylinder surface, the flow experiences large-scale reorganization as $\unicode[STIX]{x1D6FC}$ increases through the lift crisis. At $\unicode[STIX]{x1D6FC}=0.48$ the primary-flow features comprise a shear layer separating from that side of the cylinder that moves with the free stream and a pattern of oscillatory but largely attached flow zones surrounded by scattered patches of local flow separation/reattachment on the lee and underside of the cylinder surface. Large-scale, unsteady vortex shedding is observed. At $\unicode[STIX]{x1D6FC}=0.6$ the flow has transitioned to a more ordered state where the small-scale separation/reattachment cells concentrate into a relatively narrow zone with largely attached flow elsewhere. This induces a low-pressure region which produces a sudden decrease in lift and hence the lift crisis. Through this process, the boundary layer does not show classical turbulence behaviour. As $\unicode[STIX]{x1D6FC}$ is further increased at constant $Re_{D}$ , the localized separation zone dissipates with corresponding attached flow on most of the cylinder surface. The lift coefficient then resumes its increasing trend. A logarithmic region is found within the boundary layer at $\unicode[STIX]{x1D6FC}=1.0$ .
Wall-resolved large-eddy simulation (LES) is used to simulate flow over an axisymmetric body of revolution at a Reynolds number, $Re=1.1\times 10^{6}$ , based on the free-stream velocity and the length of the body. The geometry used in the present work is an idealized submarine hull (DARPA SUBOFF without appendages) at zero angle of pitch and yaw. The computational domain is chosen to avoid confinement effects and capture the wake up to fifteen diameters downstream of the body. The unstructured computational grid is designed to capture the fine near-wall flow structures as well as the wake evolution. LES results show good agreement with the available experimental data. The axisymmetric turbulent boundary layer has higher skin friction and higher radial decay of turbulence away from the wall, compared to a planar turbulent boundary layer under similar conditions. The mean streamwise velocity exhibits self-similarity, but the turbulent intensities are not self-similar over the length of the simulated wake, consistent with previous studies reported in the literature. The axisymmetric wake shifts from high- $Re$ to low- $Re$ equilibrium self-similar solutions, which were only observed for axisymmetric wakes of bluff bodies in the past.
We present numerical simulations of laminar and turbulent channel flow of an elastoviscoplastic fluid. The non-Newtonian flow is simulated by solving the full incompressible Navier–Stokes equations coupled with the evolution equation for the elastoviscoplastic stress tensor. The laminar simulations are carried out for a wide range of Reynolds numbers, Bingham numbers and ratios of the fluid and total viscosity, while the turbulent flow simulations are performed at a fixed bulk Reynolds number equal to 2800 and weak elasticity. We show that in the laminar flow regime the friction factor increases monotonically with the Bingham number (yield stress) and decreases with the viscosity ratio, while in the turbulent regime the friction factor is almost independent of the viscosity ratio and decreases with the Bingham number, until the flow eventually returns to a fully laminar condition for large enough yield stresses. Three main regimes are found in the turbulent case, depending on the Bingham number: for low values, the friction Reynolds number and the turbulent flow statistics only slightly differ from those of a Newtonian fluid; for intermediate values of the Bingham number, the fluctuations increase and the inertial equilibrium range is lost. Finally, for higher values the flow completely laminarizes. These different behaviours are associated with a progressive increases of the volume where the fluid is not yielded, growing from the centreline towards the walls as the Bingham number increases. The unyielded region interacts with the near-wall structures, forming preferentially above the high-speed streaks. In particular, the near-wall streaks and the associated quasi-streamwise vortices are strongly enhanced in an highly elastoviscoplastic fluid and the flow becomes more correlated in the streamwise direction.
Unconfined three-dimensional gravity currents generated by lock exchange using a small dividing gate in a sufficiently large tank are investigated by means of large eddy simulations under the Boussinesq approximation, with Grashof numbers varying over five orders of magnitudes. The study shows that, after an initial transient, the flow can be separated into an axisymmetric expansion and a globally translating motion. In particular, the circular frontline spreads like a constant-flow-rate, axially symmetric gravity current about a virtual source translating along the symmetry axis. The flow is characterised by the presence of lobe and cleft instabilities and hydrodynamic shocks. Depending on the Grashof number, the shocks can either be isolated or produced continuously. In the latter case a typical ring structure is visible in the density and velocity fields. The analysis of the frontal spreading of the axisymmetric part of the current indicates the presence of three regimes, namely, a slumping phase, an inertial–buoyancy equilibrium regime and a viscous–buoyancy equilibrium regime. The viscous–buoyancy phase is in good agreement with the model of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), while the inertial phase is consistent with the experiments of Britter (Atmos. Environ., vol. 13, 1979, pp. 1241–1247), conducted for purely axially symmetric, constant inflow, gravity currents. The adoption of the slumping model of Huppert & Simpson (J. Fluid Mech., vol. 99 (04), 1980, pp. 785–799), which is here extended to the case of constant-flow-rate cylindrical currents, allows reconciling of the different theories about the initial radial spreading in the context of different asymptotic regimes. As expected, the slumping phase is governed by the Froude number at the lock’s gate, whereas the transition to the viscous phase depends on both the Froude number at the gate and the Grashof number. The identification of the inertial–buoyancy regime in the presence of hydrodynamic shocks for this class of flows is important, due to the lack of analytical solutions for the similarity problem in the framework of shallow water theory. This fact has considerably slowed the research on variable-flow-rate axisymmetric gravity currents, as opposed to the rapid development of the knowledge about cylindrical constant-volume and planar gravity currents, despite their own environmental relevance.
In this paper we study the wall pressure and vorticity fields of the Stokes boundary layer in the intermittently turbulent regime through direct numerical simulation (DNS). The DNS results are compared to experimental measurements and a good agreement is found for the mean and fluctuating velocity fields. We observe maxima of the turbulent kinetic energy and wall shear stress in the early deceleration stage and minima in the late acceleration stage. The wall pressure field is characterized by large fluctuations with respect to the root mean square level, while the skewness and kurtosis of the wall pressure show significant deviations from their Gaussian values. The wall vorticity components show different behaviours during the cycle: for the streamwise component, positive and negative fluctuations have the same probability of occurrence throughout the cycle while the spanwise fluctuations favour negative extrema in the acceleration stage and positive extrema in the deceleration stage. The wall vorticity flux is a function of the wall pressure gradients. Vorticity creation at the wall reaches a maximum at the beginning of the deceleration stage due to the increase of uncorrelated wall pressure signals. The spanwise vorticity component is the most affected by the oscillations of the outer flow. These findings have consequences for the design of wave energy converters. In extreme seas, wave induced fluid velocities can be very high and extreme wall pressure fluctuations may occur. Moreover, the spanwise vortical fields oscillate violently in a wave cycle, inducing strong interactions between vortices and the device that can enhance the device motion.
Direct numerical simulations (DNSs) are performed to analyse the secondary flow of Prandtl’s second kind in fully developed spanwise-periodic channels with in-plane sinusoidal walls. The secondary flow is characterized for different combinations of wave parameters defining the wall geometry at $Re_{h}=2500$ and 5000, where $h$ is the half-height of the channel. The total cross-flow rate in the channel $Q_{yz}$ is defined along with a theoretical model to predict its behaviour. Interaction between the secondary flows from opposite walls is observed if $\unicode[STIX]{x1D706}\simeq h\simeq A$ , where $A$ and $\unicode[STIX]{x1D706}$ are the amplitude and wavelength of the sinusoidal function defining the wall geometry. As the outer-scaled wavelength ( $\unicode[STIX]{x1D706}/h$ ) is reduced, the secondary vortices become smaller and faster, increasing the total cross-flow rate per wall. However, if the inner-scaled wavelength ( $\unicode[STIX]{x1D706}^{+}$ ) is below 130 viscous units, the cross-flow decays for smaller wavelengths. By analysing cases in which the wavelength of the wall is much smaller than the half-height of the channel $\unicode[STIX]{x1D706}\ll h$ , we show that the cross-flow distribution depends almost entirely on the separation between the scales of the instantaneous vortices, where the upper and lower bounds are determined by $\unicode[STIX]{x1D706}/h$ and $\unicode[STIX]{x1D706}^{+}$ , respectively. Therefore, the distribution of the secondary flow relative to the size of the wave at a given $Re_{h}$ can be replicated at higher $Re_{h}$ by decreasing $\unicode[STIX]{x1D706}/h$ and keeping $\unicode[STIX]{x1D706}^{+}$ constant. The mechanisms that contribute to the mean cross-flow are analysed in terms of the Reynolds stresses and using quadrant analysis to evaluate the probability density function of the bursting events. These events are further classified with respect to the sign of their instantaneous spanwise velocities. Sweeping events and ejections are preferentially located in the valleys and peaks of the wall, respectively. The sweeps direct the instantaneous cross-flow from the core of the channel towards the wall, turning in the wall-tangent direction towards the peaks. The ejections drive the instantaneous cross-flow from the near-wall region towards the core. This preferential behaviour is identified as one of the main contributors to the secondary flow.
We study how the properties of forcing and dissipation affect the scaling behaviour of the vortex population in the two-dimensional turbulent inverse energy cascade. When the flow is forced at scales intermediate between the domain and dissipation scales, the growth rates of the largest vortex area and the spectral peak length scale are robust to all simulation parameters. For white-in-time forcing the number density distribution of vortex areas follows the scaling theory predictions of Burgess & Scott (J. Fluid Mech., vol. 811, 2017, pp. 742–756) and shows little sensitivity either to the forcing bandwidth or to the nature of the small-scale dissipation: both narrowband and broadband forcing generate nearly identical vortex populations, as do Laplacian diffusion and hyperdiffusion. The greatest differences arise in comparing simulations with correlated forcing to those with white-in-time forcing: in flows with correlated forcing the intermediate range in the vortex number density steepens significantly past the predicted scale-invariant $A^{-1}$ scaling. We also study the impact of the forcing Reynolds number $Re_{f}$ , a measure of the relative importance of nonlinear terms and dissipation at the forcing scale, on vortex formation and the scaling of the number density. As $Re_{f}$ decreases, the flow changes from one dominated by intense circular vortices surrounded by filaments to a less structured flow in which vortex formation becomes progressively more suppressed and the filamentary nature of the surrounding vorticity field is lost. However, even at very small $Re_{f}$ , and in the absence of intense coherent vortex formation, regions of anomalously high vorticity merge and grow in area as predicted by the scaling theory, generating a three-part number density similar to that found at higher $Re_{f}$ . At late enough stages the aggregation process results in the formation of long-lived circular vortices, demonstrating a strong tendency to vortex formation, and via a route distinct from the axisymmetrization of forcing extrema seen at higher $Re_{f}$ . Our results establish coherent vortices as a robust feature of the two-dimensional inverse energy cascade, and provide clues as to the dynamical mechanisms shaping their statistics.
Direct numerical simulations are used to examine large-scale motions with a streamwise length $2\sim 4h$ ( $h$ denotes the channel half-width) in the logarithmic and outer regions of a turbulent channel flow. We test a minimal ‘streamwise’ flow unit (Toh & Itano, J. Fluid Mech., vol. 524, 2005, pp. 249–262) (or MSU) for larger Kármán numbers ( $h^{+}=395$ and 1020) than in the original work. This flow unit consists of a sufficiently long ( ${L_{x}}^{+}\approx 400$ ) streamwise domain to maintain near-wall turbulence (Jiménez & Moin, J. Fluid Mech., vol. 225, 1991, pp. 213–240) and a spanwise domain which is large enough to represent the spanwise behaviour of inner and outer structures correctly; as $h^{+}$ increases, the streamwise extent of the MSU domain decreases with respect to $h$ . Particular attention is given to whether the spanwise organization of the large-scale structures may be represented properly in this simplified system at sufficiently large $h^{+}$ and how these structures are associated with the mean streamwise velocity $\overline{U}$ . It is shown that, in the MSU, the large-scale structures become approximately two-dimensional at $h^{+}=1020$ . In this case, the streamwise velocity fluctuation $u$ is energized, whereas the spanwise velocity fluctuation $w$ is weakened significantly. Indeed, there is a reduced energy redistribution arising from the impaired global nature of the pressure, which is linked to the reduced linear–nonlinear interaction in the Poisson equation (i.e. the rapid pressure). The logarithmic dependence of $\overline{ww}$ is also more evident due to the reduced large-scale spanwise meandering. On the other hand, the spanwise organization of the large-scale $u$ structures is essentially identical for the MSU and large streamwise domain (LSD). One discernible difference, relative to the LSD, is that the large-scale structures in the MSU are more energized in the outer region due to a reduced turbulent diffusion. In this region, there is a tight coupling between neighbouring structures, which yields antisymmetric pairs (with respect to centreline) of large-scale structures with a spanwise spacing of approximately $3h$ ; this is intrinsically identical with the outer energetic mode in the optimal transient growth of perturbations (del Álamo & Jiménez, J. Fluid Mech., vol. 561, 2006, pp. 329–358).
We study wind turbulence over breaking waves based on direct numerical simulation (DNS) of two-fluid flows. In the DNS, the air and water are simulated as a coherent system, with the interface captured using the coupled level-set and volume-of-fluid method. Because the wave breaking is an unsteady process, we use ensemble averaging over 100 runs to define turbulence statistics. We focus on analysing the turbulence statistics of the airflow over breaking waves. The effects of wave age and wave steepness are investigated. It is found that before wave breaking, the turbulence statistics are largely influenced by the wave age. The vertical gradient of mean streamwise velocity is positive at small and intermediate wave ages, but it becomes negative near the wave surface at large wave age as the pressure force changes from drag to thrust. Furthermore, wave-coherent motions make increasingly important contributions to the momentum flux and kinetic energy of velocity fluctuations (KE-F) as the wave age increases. During the wave breaking process, spilling breakers do not influence the wind field significantly; in contrast, plunging breakers alter the structures of wind turbulence near the wave surface drastically. It is observed from the DNS results that during wave plunging, a high pressure region occurs ahead of the wave front, which further accelerates the wind in the downstream direction. Meanwhile, a large spanwise vortex is generated, which greatly disturbs the airflow around it, resulting in large magnitudes of Reynolds stress and turbulence kinetic energy (TKE) below the wave crest. Above the crest, the magnitude of KE-F is enhanced during wave plunging at small and large wave ages, but at intermediate wave age, the transient enhancement of KE-F is absent. The effect of wave breaking on the magnitude of KE-F is further investigated through the analysis of the KE-F production. It is discovered that at small wave age, the transient enhancement of KE-F is caused by the appearance of a local maximum in the profile of total momentum flux; but at large wave age, it results from the change in the sign of the KE-F production from negative to positive, due to the sign change in the wave-coherent momentum flux. At intermediate wave age, neither of these two processes is present, and the transient growth of KE-F does not take place.
The values of the highest Lyapunov exponent (HLE) $\unicode[STIX]{x1D706}_{1}$ for turbulent flow in a plane channel at Reynolds numbers up to $Re_{\unicode[STIX]{x1D70F}}=586$ are determined. The instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are investigated. The LLV is calculated by numerical solution of the Navier–Stokes equations linearized about the non-stationary base solution corresponding to the developed turbulent flow. The base turbulent flow is calculated in parallel with the calculation of the evolution of the perturbations. For arbitrary initial conditions, the regime of exponential growth ${\sim}\exp (\unicode[STIX]{x1D706}_{1}t)$ which corresponds to the approaching of the perturbation to the LLV is achieved already at $t^{+}<50$ . It is found that the HLE increases with increasing Reynolds number from $\unicode[STIX]{x1D706}_{1}^{+}\approx 0.021$ at $Re_{\unicode[STIX]{x1D70F}}=180$ to $\unicode[STIX]{x1D706}_{1}^{+}\approx 0.026$ at $Re_{\unicode[STIX]{x1D70F}}=586$ . The LLV structures are concentrated mainly in a region of the buffer layer and are manifested in the form of spots of increased fluctuation intensity localized both in time and space. The root-mean-square (r.m.s.) profiles of the velocity and vorticity intensities in the LLV are qualitatively close to the corresponding profiles in the base flow with artificially removed near-wall streaks. The difference is the larger concentration of LLV perturbations in the vicinity of the buffer layer and a relatively larger (by approximately 80 %) amplitude of the vorticity pulsations. Based on the energy spectra of velocity and vorticity pulsations, the integral spatial scales of the LLV structures are determined. It is found that LLV structures are on average twice narrower and twice shorter than the corresponding structures of the base flow. The contribution of each of the terms entering into the expression for the production of the perturbation kinetic energy is determined. It is shown that the process of perturbation development is essentially dictated by the inhomogeneity of the base flow, as well as by the presence of transversal motion in it. Neglecting of these factors leads to a significant underestimation of the perturbation growth rate. The presence of near-wall streaks in the base flow, on the contrary, does not play a significant role in the development of the LLV perturbations. Artificial removal of streaks from the base flow does not change the character of the perturbation growth.
The irreversible mixing efficiency is studied using large-eddy simulations (LES) of stratified turbulence, where three different subgrid-scale (SGS) parameterizations are employed. For comparison, direct numerical simulations (DNS) and hyperviscosity simulations are also performed. In the regime of stratified turbulence where $Fr_{v}\sim 1$ , the irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ in LES scales like $1/(1+2Pr_{t})$ , where $Fr_{v}$ and $Pr_{t}$ are the vertical Froude number and turbulent Prandtl number, respectively. Assuming a unit scaling coefficient and $Pr_{t}=1$ , $\unicode[STIX]{x1D6FE}_{i}$ goes to a constant value $1/3$ , in agreement with DNS results. In addition, our results show that the irreversible mixing efficiency in LES, while consistent with this prediction, depends on SGS parameterizations and the grid spacing $\unicode[STIX]{x1D6E5}$ . Overall, the LES approach can reproduce mixing efficiency results similar to those from the DNS approach if $\unicode[STIX]{x1D6E5}\lesssim L_{o}$ , where $L_{o}$ is the Ozmidov scale. In this situation, the computational costs of numerical simulations are significantly reduced because LES runs require much smaller computational resources in comparison with expensive DNS runs.
We propose a body-fitted mesh approach based on a semi-Lagrangian streaming step combined with an entropy-based collision model. After determining the order of convergence of the method, we analyse the flow past a circular cylinder in the lower subcritical regime, at a Reynolds number $Re=3900$ , in order to assess the numerical performances for wall-bounded turbulence. The results are compared to experimental and numerical data available in the literature. Overall, the agreement is satisfactory. By adopting an efficient local refinement strategy together with the enhanced stability features of the entropic model, this method extends the range of applicability of the lattice Boltzmann approach to the solution of realistic fluid dynamics problems, at high Reynolds numbers, involving complex geometries.
A three-dimensional wavelet multi-resolution analysis of direct numerical simulations of a turbulent premixed flame is performed in order to investigate the spatially localized spectral transfer of kinetic energy across scales in the vicinity of the flame front. A formulation is developed that addresses the compressible spectral dynamics of the kinetic energy in wavelet space. The wavelet basis enables the examination of local energy spectra, along with inter-scale and subfilter-scale (SFS) cumulative energy fluxes across a scale cutoff, all quantities being available either unconditioned or conditioned on the local instantaneous value of the progress variable across the flame brush. The results include the quantification of mean spectral values and associated spatial variabilities. The energy spectra undergo, in most locations in the flame brush, a precipitous drop that starts at scales of the same order as the characteristic flame scale and continues to smaller scales, even though the corresponding decrease of the mean spectra is much more gradual. The mean convective inter-scale flux indicates that convection increases the energy of small scales, although it does so in a non-conservative manner due to the high aspect ratio of the grid, which limits the maximum scale level that can be used in the wavelet transform, and to the non-periodic boundary conditions, which exchange energy through surface forces, as explicitly elucidated by the formulation. The mean pressure-gradient inter-scale flux extracts energy from intermediate scales of the same order as the characteristic flame scale, and injects energy in the smaller and larger scales. The local SFS-cumulative contribution of the convective and pressure-gradient mechanisms of energy transfer across a given cutoff scale imposed by a wavelet filter is analysed. The local SFS-cumulative energy flux is such that the subfilter scales upstream from the flame always receive energy on average. Conversely, within the flame brush, energy is drained on average from the subfilter scales by convective and pressure-gradient effects most intensely when the filter cutoff is larger than the characteristic flame scale.
Direct numerical simulation of a turbulent boundary layer (TBL) subjected to a moderate adverse pressure gradient (APG, $\unicode[STIX]{x1D6FD}=1.45$ ) is performed to explore the contribution of large scales to the skin friction, where $\unicode[STIX]{x1D6FD}$ is the Clauser pressure gradient parameter. The Reynolds number based on the momentum thickness develops from $Re_{\unicode[STIX]{x1D703}}\approx 110$ to 6000 with an equilibrium region in $Re_{\unicode[STIX]{x1D703}}=4000$ –5500. The spanwise wavelength ( $\unicode[STIX]{x1D706}_{z}$ ) spectra of the streamwise and spanwise velocity fluctuations show that the large-scale energy is significantly enhanced throughout the boundary layer. We quantify the superposition and amplitude modulation effects of these enhanced large scales on the skin friction coefficient ( $C_{f}$ ) by employing two approaches: (i) spanwise co-spectra of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ ; (ii) conditionally averaged $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ . The velocity–vorticity correlations $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ are related to the advective transport and the vortex stretching, respectively. Although $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ negatively contributes to $C_{f}$ , the positive contribution of the large scales ( $\unicode[STIX]{x1D706}_{z}>0.5\unicode[STIX]{x1D6FF}$ ) is observed in the co-spectra of weighted $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ . For the co-spectra of weighted $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ , we observe an outer peak at $\unicode[STIX]{x1D706}_{z}\approx 0.75\unicode[STIX]{x1D6FF}$ and the superposition of the large scales in the buffer region, leading to the enhancement of $C_{f}$ . The magnitude of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ depends on the large-scale streamwise velocity fluctuations ( $u_{L}$ ). In particular, the negative- $u_{L}$ events amplify $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ in the outer region, and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ is enhanced by the positive- $u_{L}$ events. As a result, the skin friction induced by $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ increases in the present APG TBL.
In this study, macro-rough flows over beds with different permeability values are simulated using the large-eddy simulation, and the results are analysed by applying the double-averaging (DA) methodology. Spheres of different sizes and arrangements were used to form the beds, which are deemed to be permeable granular beds. The influence of bed permeability on the turbulence dynamics and structure is investigated. It was observed that the scales of the spanwise vortical structures over more permeable beds are larger than those over less permeable beds. This is attributed to large-scale spanwise-alternate strips of varying Reynolds shear stress (RSS), emerging from the surface of macro-rough elements for the permeable beds. The DA stress balance suggests that the time-averaged spanwise vortical structure leads to a damping in DA RSS and an unusual peak of the form-induced stress in the main flow. In the streamwise direction, both large turbulent structures that originate from the Kelvin–Helmholtz-type instability and small turbulent structures that are associated with the turbulent transport across the gaps of the roughness elements are more prevalent over highly permeable beds. Near the bed, the relative magnitude of turbulent events shows a transition from a ejections-dominating to sweeps-dominating zone with vertical distance. Further, several hydrodynamic characteristics normalized by inner scales (kinematic viscosity to shear velocity ratio) show a greater dependency on permeability Reynolds number than those normalized by sediment size. The study provides an insight into the mechanism of mass transfer near the fluid–particle interface, which is vital to benthic and aquatic ecology.
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