Modeling of Lagrangian particles in turbulence boundary layer considering attached eddies: particle trajectories and concentration profiles

<p>     Sediment particles in flow not only follow the mean drift, but also diffuse randomly due to turbulence. Owing to this property, Lagrangian particle trajectory is regarded as a stochastic process in this study. The proposed model based on Lagrangian methods will combine physical mechanisms and stochastic methods to simulate the particle motion, and uses the Brownian motion to describe the diffusion affected by turbulence. In turbulence boundary layer, there are eddies with different length and velocity scales. Eddies affect the motion of a particle, like the occurrences of ejection and sweep events. Among others, those extended to the wall, named attached eddies, are primarily responsible for most of the turbulent kinetic energy and Reynolds shear stresses. Perry & Marušić (1995) further divided the attached eddies into two types, those directly attached to the wall are called Type-A eddies while others not directly attached to the wall in the wake region are called Type-B eddies. The scales of Type-B eddies are affected by the distance away from the wall. Therefore, this study will combine the above-mentioned theory and the stochastic diffusion particle tracking model (SD-PTM) to simulate the Lagrangian sediment particles in turbulence boundary layer considering the effects of attached eddies.<br>     The SD-PTM which has been built on the Lagrangian scheme and derived from the Langevin equation has two main parts – the mean drift term and the turbulence term. The proposed model will separate the turbulence term into the effects by Type-A eddies and the effects by Type-B eddies, respectively. In the simulation results of sediment concentration in Tsai & Huang (2019), it can be found that when only Type-A eddies are considered, there were some discrepancies except for the near wall region within about 20% of the thickness of turbulence boundary layer. Hence, after taking into account for the effects of Type-B eddies in the proposed model, it can be expected that accuracy of the simulation results of Lagrangian particle trajectories and sediment concentrations can be improved throughout the whole boundary layer.</p><p>Keywords: Lagrangian methods, stochastic particle tracking model, attached eddies, Brownian motion, particle trajectories</p>

Observations of Nearbed Turbulence over Mobile Bedforms in Combined, Collinear Wave-Current Flows

Collinear wave-current shear interactions are often assumed to be the same for currents following or opposing the direction of regular wave propagation; with momentum and mass exchanges restricted to the thin oscillating boundary layer (zero-flux condition) and enhanced but equal wave-averaged bed shear stresses. To examine these assumptions, a prototype-scale experiment investigated the nature of turbulent exchanges in flows with currents aligned to, and opposing, wave propagation over a mobile sandy bed. Estimated mean and maximum stresses from measurements above the bed exceeded predictions by models of bed shear stress subscribing to the assumptions above, suggesting the combined boundary layer is larger than predicted by theory. The core flow experiences upward turbulent fluxes in aligned flows, coupled with sediment entrainment by vortex shedding at flow reversal, whilst downward fluxes of eddies generated by the core flow, and strong adverse shear can enhance near-bed mass transport, in opposing currents. Current-aligned coherent structures contribute significantly to the stress and energy dissipation, and display characteristics of wall-attached eddies formed by the pairing of counter-rotating vortices. These preliminary findings suggest a notable difference in wave-following and wave-opposing wave-current interactions, and highlight the need to account for intermittent momentum-exchanges in predicting stress, boundary layer thickness and sediment transport.

Identity of attached eddies in turbulent channel flows with bidimensional empirical mode decomposition

Bidimensional empirical mode decomposition (BEMD) is used to identify attached eddies in turbulent channel flows and quantify their relationship with the mean skin-friction drag generation. BEMD is an adaptive, non-intrusive, data-driven method for mode decomposition of multiscale signals especially suitable for non-stationary and nonlinear processes such as those encountered in turbulent flows. In the present study, we decompose the velocity fluctuations obtained by direct numerical simulation of channel flows into BEMD modes characterized by specific length scales. Unlike previous works (e.g. Flores & Jiménez, Phys. Fluids, vol. 22(7), 2010, 071704; Hwang, J. Fluid Mech., vol. 767, 2015, pp. 254–289), the current approach employs naturally evolving wall-bounded turbulence without modifications of the Navier–Stokes equations to maintain the inherent turbulent dynamics, and minimize artificial numerical enforcement or truncation. We show that modes identified by BEMD exhibit a self-similar behaviour, and that single attached eddies are mainly composed of streaky structures carrying intense streamwise velocity fluctuations and vortex packets permeating in all velocity components. Our findings are consistent with the existence of attached eddies in actual wall-bounded flows, and show that BEMD modes are tenable candidates to represent Townsend attached eddies. Finally, we evaluate the turbulent-drag generation from the perspective of attached eddies with the aid of the Fukagata–Iwamoto–Kasagi identity (Fukagata et al., Phys. Fluids, vol. 14(11), 2002, pp. L73–L76) by splitting the Reynolds shear stress into four different terms related to the length scale of the attached eddies.

Characteristic scales of Townsend’s wall-attached eddies

Townsend (The Structure of Turbulent Shear Flow, 1976, Cambridge University Press) proposed a structural model for the logarithmic layer (log layer) of wall turbulence at high Reynolds numbers, where the dominant momentum-carrying motions are organised into a multiscale population of eddies attached to the wall. In the attached-eddy framework, the relevant length and velocity scales of the wall-attached eddies are the friction velocity and the distance to the wall. In the present work, we hypothesise that the momentum-carrying eddies are controlled by the mean momentum flux and mean shear with no explicit reference to the distance to the wall and propose new characteristic velocity, length and time scales consistent with this argument. Our hypothesis is supported by direct numerical simulation of turbulent channel flows driven by non-uniform body forces and modified mean velocity profiles, where the resulting outer-layer flow structures are substantially altered to accommodate the new mean momentum transfer. The proposed scaling is further corroborated by simulations where the no-slip wall is replaced by a Robin boundary condition for the three velocity components, allowing for substantial wall-normal transpiration at all length scales. We show that the outer-layer one-point statistics and spectra of this channel with transpiration agree quantitatively with those of its wall-bounded counterpart. The results reveal that the wall-parallel no-slip condition is not required to recover classic wall-bounded turbulence far from the wall and, more importantly, neither is the impermeability condition at the wall.

Wall-attached structures of velocity fluctuations in a turbulent boundary layer

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.

Implications of Nonlocal Transport and Conditionally Averaged Statistics on Monin–Obukhov Similarity Theory and Townsend’s Attached Eddy Hypothesis

According to Townsend’s hypothesis, so-called wall-attached eddies are the main contributors to turbulent transport in the atmospheric surface layer (ASL). This is also one of the main assumptions of Monin–Obukhov similarity theory (MOST). However, previous evidence seems to indicate that outer-scale eddies can impact the ASL, resulting in deviations from the classic MOST scaling. We conduct large-eddy simulations and direct numerical simulations of a dry convective boundary layer to investigate the impact of coherent structures on the ASL. A height-dependent passive tracer enables coherent structure detection and conditional analysis based on updrafts and subsidence. The MOST similarity functions computed from the simulation results indicate a larger deviation of the momentum similarity function ϕ m from classical scaling relationships compared to the temperature similarity function ϕ h. The conditional-averaged ϕ m for updrafts and subsidence are similar, indicating strong interactions between the inner and outer layers. However, ϕ h conditioned on subsidence follows the mixed-layer scaling, while its updraft counterpart is well predicted by MOST. Updrafts are the dominant contributors to the transport of momentum and temperature. Subsidence, which comprises eddies that originate from the outer layer, contributes increasingly to the transport of temperature with increasing instability. However, u′ of different signs are distributed symmetrically in subsidence unlike the predominantly negative θ′ as instability increases. Thus, the spatial patterns of u′ w′ differ compared to θ′ w′ in regions of subsidence. These results depict the mechanisms for departure from the MOST scaling, which is related to the stronger role of subsidence.