English

The near-wall, subgrid-scale (SGS) model [Chung and Pullin, Large-eddy simulation and wall-modeling of turbulent channel flow, J. Fluid Mech. 631, 281--309 (2009)] is used to perform large-eddy simulations (LES) of the incompressible developing, smooth-wall, flat-plate turbulent boundary layer. In this model, the stretched-vortex, SGS closure is utilized in conjunction with a tailored, near-wall model designed to incorporate anisotropic vorticity scales in the presence of the wall. The composite SGS-wall model is presently incorporated into a computer code suitable for the LES of developing flat-plate boundary layers. This is then used to study several aspects of zero- and adverse-pressure gradient turbulent boundary layers. First, LES of the zero-pressure gradient turbulent boundary layer are performed at Reynolds numbers Reθ based on the free-stream velocity and the momentum thickness in the range Reθ = 103 - 1012. Results include the inverse skin friction coefficient, √2/Cf, velocity profiles, the shape factor H, the Karman constant, and the Coles wake factor as functions of Reθ. Comparisons with some direct numerical simulation (DNS) and experiment are made, including turbulent intensity data from atmospheric-layer measurements at Reθ = O(106. At extremely large Reθ, the empirical Coles-Fernholz relation for skin-friction coefficient provides a reasonable representation of the LES predictions. While the present LES methodology cannot of itself probe the structure of the near-wall region, the present results show turbulence intensities that scale on the wall-friction velocity and on the Clauser length scale over almost all of the outer boundary layer. It is argued that the LES is suggestive of the asymptotic, infinite Reynolds-number limit for the smooth-wall turbulent boundary layer and different ways in which this limit can be approached are discussed. The maximum Reθ of the present simulations appears to be limited by machine precision and it is speculated, but not demonstrated, that even larger Reθ could be achieved with quad- or higher-precision arithmetic. Second, the time series velocity signals obtained from LES within the logarithmic region of the zero-pressure gradient turbulent boundary layer are used in combination with an empirical, predictive inner--outer wall model [Marusic et al., Predictive model for wall-bounded turbulent flow, Science 329, 193 (2010)] to calculate the statistics of the fluctuating streamwise velocity in the inner region of the zero-pressure gradient turbulent boundary layer. Results, including spectra and moments up to fourth order, are compared with equivalent predictions using experimental time series, as well as with direct experimental measurements at Reynolds numbers Reτ based on the friction velocity and the boundary layer thickness, Reτ =7,300, 13,600 and 19,000. LES combined with the wall model are then used to extend the inner-layer predictions to Reynolds numbers Reτ =62,000, 100,000 and 200,000 that lie within a gap in log(Reτ) space between laboratory measurements and surface-layer, atmospheric experiments. The present results support a log-like increase in the near-wall peak of the streamwise turbulence intensities with Reτ and also provide a means of extending LES results at large Reynolds numbers to the near-wall region of wall-bounded turbulent flows. Finally, we apply the wall model to LES of a turbulent boundary layer subject to an adverse pressure gradient. Computed statistics are found to be consistent with recent experiments and some Reynolds number similarity is observed over a range of two orders of magnitude.

This paper revisits the classical logarithmic velocity profile of wall-bounded turbulent shear flows, with particular emphasis on the persistent scatter in its defining parameters—the von Kármán constant (κ) and the additive constant (B). A combined experimental–theoretical framework is developed to extract these parameters across the three canonical flows: plane channel, circular pipe, and zero-pressure-gradient (ZPG) boundary layer. The authors' derivations show that three different partial differential equations governing these flows in the fully developed state behave similarly as Reτ→∞, particularly within the inertial sublayer, thereby supporting the universality of the log-law, where Reτ denotes the friction Reynolds number. The theoretical approach, based on the slope of the mean velocity profile in logarithmic coordinates, is validated using new hot-wire measurements together with literature data, accounting for wall-curvature effects in pipe flow and flow development in ZPG boundary layers. The analysis indicates that, at sufficiently high Reynolds numbers, the inertial sublayer is governed by the log-law, with an estimated von Kármán constant of κ≈1/e≈0.367879⋯. This value is supported by renormalization group theory and exponential self-similarity arguments, reinforcing that κ is not merely a fitting parameter but a consequence of turbulence dynamics. Furthermore, the present study demonstrates that κ and B cannot be specified independently, as multiple κ–B pairs yield equally valid log-law predictions. A linear κB–B relation, κB = 0.4582 B − 0.3134, is established, reducing data scatter and clarifying the cross-dependence of the two parameters. The present findings highlight the robustness of the logarithmic law and the ongoing challenge of defining its universal constants in wall-bounded turbulent flows, while providing a physical framework for developing and calibrating turbulence models.

Wall-modeling is required to make large-eddy simulations of high-Reynolds number wall-bounded turbulent flows feasible in terms of computational cost. Here, an extension of the integral wall-model for large-eddy simulations (iWMLESs) for incompressible flows developed by Yang et al. [“Integral wall model for large eddy simulations of wall-bounded turbulent flows,” Phys. Fluids 27(2), 025112 (2015)] to compressible and isothermal flows is proposed and assessed. The iWMLES approach is analogous to the von Kármán-Pohlhausen integral method for laminar flows: the velocity profile is parameterized, and unknown coefficients are determined by matching boundary conditions obeying the integral boundary layer momentum equation. It allows non-equilibrium effects such as pressure gradient and convection to be included at a computing cost similar to analytical wall-models. To take into account density variations and temperature gradients, the temperature profile is also parameterized and the integral compressible boundary layer energy equation is considered. Parameterized profiles are based on the usual logarithmic wall functions with corrective terms to extend their range of validity. Instead of solving a set of differential equations as wall-models based on the thin boundary layer equation approach, a simple linear system is solved. The proposed wall-model is implemented in a finite-volume cell-centered structured grid solver and assessed on adiabatic and isothermal plane channel flows at several friction Reynolds and Mach numbers. For low Mach number cases, mean profiles, wall fluxes, and turbulent fluctuations are in agreement with those of Direct Numerical Simulation (DNS). For supersonic flows, the results are in good agreement with the DNS data, especially the mean velocity quantities and the wall friction, while standard analytical wall-models show their limits.

The extraction of spatiotemporal coherence in high-dimensional, chaotic, nonlinear dynamical systems, such as turbulent flows, remains a fundamental challenge in physics, mathematics, and engineering. In this work, we employ Shannon transfer entropy (TE) to identify causally coherent motions in a zero-pressure-gradient turbulent boundary layer (TBL). This causality metric, rooted in information theory, enables the identification of sources and targets in dynamical systems using the corresponding time series. However, TE requires sophisticated tuning of various hyperparameters, such as the Markovian order of the source (m), which can spatially vary in wall-bounded turbulent flow. Here, we present an adaptive tuning and discuss the influence of m across different TBLs. We introduce the concept of causally coherent structures (CCS), i.e., coherent structures interpreted as spatiotemporal patterns of causality. Moreover, the net transfer entropy flux is also utilized to identify boundary layer locations acting either as sources or targets. The standard viscous, logarithmic, and outer layers are characterized by information fluxes, highlighting, for example, dominant top-down interactions between the inner and outer layers, analogously to the classical energy cascade. This work extends techniques previously employed in the literature, such as correlation and spectral analysis, and presents an approach that is inherently general and applicable to a wide range of chaotic dynamical systems, with applications in cognitive sciences, systems biology and finance.

We summarize the results of our recent work that show the logarithmic region of the streamwise turbulent intensity in wall-bounded turbulent flows. We extract the three-dimensional structures of the intense streamwise velocity fluctuations in the instantaneous flow field obtained from direct numerical simulation data of turbulent boundary layer at \(Re_\tau \approx 1000\). The identified structures are physically attached to the wall, and their height (\(l_y\)) spans from the buffer layer to the boundary layer thickness. Their length and width scale with \(l_y\) and the population density function of the tall attached structures (\(290< l_y^+ < 550\)) is inversely proportional to \(l_y\), reflecting Townsend’s attached-eddy hypothesis. We conditionally average the streamwise turbulent intensity associated with the tall attached structures and show the existence of the logarithmic region. This logarithmic behavior is further verified by examining the diagnostic function.

A new formulation for the streamwise turbulence intensity distribution in wall-bounded turbulent flows

The excellent drag reduction effect of the bubble drag reduction technique has been proved through many experiments since it was proposed. In this paper, the authors investigate the bubble-turbulence interaction and the corresponding drag reduction effect with a two-way coupled Euler–Lagrange code. The liquid phase is simulated by using a large eddy simulation method with the immersed bubbles treated using a nonlinear collision model to accurately simulate the bubble–wall interaction. A Gaussian distributed method is adopted to obtain the void fraction and interphase forces in the two-way coupled algorithm. Two typical wall-bounded turbulent flow problems (turbulent channel flow and boundary layer flow) are simulated to validate the accuracy and stability in bubbly flows and investigate the drag reduction mechanism. First, the effect of bubbles on the turbulent flow is studied in the channel flow cases in which the bubbles are observed attaching to the upper plate and swaying in the spanwise direction. In this case, Reynolds stress near the wall is decreased, which contributes to the drag reduction. Moreover, drag reduction of a turbulent boundary layer flow with bubble injection is studied in which the drag reduction under different air flow rates is in good agreement with experimental results. The contribution of turbulence and different liquid forces to the migration of bubbles away from the wall is investigated. The bubble trajectory in the turbulent boundary layer is divided into three distinct stages and discussed in detail finally.

Most of the flow problems encountered in practical engineering are wall-bounded turbulent flows at high Reynolds numbers. Wall-modeled large eddy simulation (WMLES) is one of the most viable approaches for predicting these realistic flows. Immersed boundary (IB) approach is an efficient computational technique to solve flow problems involving complex and/or moving geometries. This work extends a sharp-interface IB method, named the local domain-free discretion (DFD), to WMLES of compressible flows at high Reynolds numbers. An equilibrium wall model based on solving the simplified compressible turbulent boundary layer equations is utilized to alleviate the requirement of high near-wall mesh resolution. In conjunction with the approximate boundary conditions prescribed by the modeled wall shear stress and wall heat flux, the tangential velocity and temperature at an exterior dependent node are evaluated. Then, the closure of the discrete form of governing equations at an interior node in the immediate vicinity of the immersed wall is accomplished. A simple non-equilibrium correction of the wall shear stress provided by the equilibrium wall model is introduced explicitly. The WMLES/DFD method is applied to a supersonic zero-pressure-gradient turbulent boundary layer flow, a shock wave/flat-plate boundary layer interaction, a supersonic compression ramp flow and high-speed turbulent Couette flows with various thermal boundary conditions. The influence of grid resolution is investigated in the simulation of zero-pressure-gradient turbulent boundary layer flow. By comparing the computed results with the referenced experimental data and/or numerical results, the accuracy and ability of the WMLES/DFD method to simulate compressible turbulent flows are verified.

The logarithmic law for the mean velocity in turbulent boundary layers has long provided a valuable and robust reference for comparison with theories, models and large-eddy simulations (LES) of wall-bounded turbulence. More recently, analysis of high-Reynolds-number experimental boundary-layer data has shown that also the variance and higher-order moments of the streamwise velocity fluctuations $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}u^{\prime +}$ display logarithmic laws. Such experimental observations motivate the question whether LES can accurately reproduce the variance and the higher-order moments, in particular their logarithmic dependency on distance to the wall. In this study we perform LES of very high-Reynolds-number wall-modelled channel flow and focus on profiles of variance and higher-order moments of the streamwise velocity fluctuations. In agreement with the experimental data, we observe an approximately logarithmic law for the variance in the LES, with a ‘Townsend–Perry’ constant of $A_1\approx 1.25$. The LES also yields approximate logarithmic laws for the higher-order moments of the streamwise velocity. Good agreement is found between $A_p$, the generalized ‘Townsend–Perry’ constants for moments of order $2p$, from experiments and simulations. Both are indicative of sub-Gaussian behaviour of the streamwise velocity fluctuations. The near-wall behaviour of the variance, the ranges of validity of the logarithmic law and in particular possible dependencies on characteristic length scales such as the roughness length $z_0$, the LES grid scale $\Delta $, and subgrid scale mixing length $C_s\Delta $ are examined. We also present LES results on moments of spanwise and wall-normal fluctuations of velocity.

The trinity of so-called "canonical" wall-bounded turbulent flows, comprising the zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL, turbulent pipe flow and channel/duct flows has continued to receive intense attention as new and more reliable experimental data have become available. Nevertheless, the debate on whether the logarithmic part of the mean velocity profile, in particular the K\'arm\'an constant $\kappa$, is identical for these three canonical flows or flow-dependent is still ongoing. In this paper, which expands upon Monkewitz and Nagib (24th ICTAM Conf., Montreal, 2016), the asymptotic matching requirement of equal $\kappa$ in the log-law and in the expression for the centerline/free-stream velocity is reiterated and shown to preclude a single universal log-law in the three canonical flows or at least make it very unlikely. The current re-analysis of high quality mean velocity profiles in ZPG TBL's, the Princeton "Superpipe" and in channels and ducts leads to a coherent description of (almost) all seemingly contradictory data interpretations in terms of TWO logarithmic regions in pipes and channels: A universal interior, near-wall logarithmic region with the same parameters as in the ZPG TBL, in particular $\kappa_{\mathrm{wall}} \cong 0.384$, but only extending from around $150$ to around $10^3$ wall units, and shrinking with increasing pressure gradient, followed by an exterior logarithmic region with a flow specific $\kappa$ matching the logarithmic slope of the respective free-stream or centerline velocity. The log-law parameters of the exterior logarithmic region in channels and pipes are shown to depend monotonically on the pressure gradient.

Large-eddy simulation of the urban boundary layer using drag-porosity modeling

Energy integral equation for premixed flame-wall interaction in turbulent boundary layers and its application to turbulent burning velocity and wall flux evaluations

High-order methods are well-suited for the numerical simulation of complex compressible turbulent flows, but require additional stabilization techniques to capture instabilities arising from the underlying non-linear hyperbolic equations. This paper provides a detailed comparison of the effectiveness of entropy stable discontinuous Galerkin methods for the stabilization of compressible (wall-bounded) turbulent flows. For this investigation, an entropy stable discontinuous Galerkin spectral element method is applied on Gauss–Legendre and Gauss–Lobatto nodes. In the compressible regime, an additional stabilization technique for shock capturing based on a convex blending of a low-order finite volume with the high-order discontinuous Galerkin operator is utilized. The present investigation provides a systematic study from convergence tests, to the Taylor–Green vortex and finally to a more intricate turbulent wall-bounded 3D diffuser flow, encompassing both weakly compressible and compressible regimes. The comparison demonstrates that the DGSEM on Gauss–Lobatto nodes is generally less accurate for an equal amount of degrees of freedom. Conversely, it is faster than the DGSEM on Gauss—Legendre nodes due to a less severe time step restriction and simpler numerical operator. A performance comparison reveals that the DGSEM on Gauss–Lobatto nodes generally outperforms the DGSEM on Gauss nodes for underresolved turbulence in the subsonic regime on a periodic domain. Conversely, the opposite effect can be observed for wall-bounded flows as well as the supersonic regime, the latter depending of course on the chosen shock-capturing scheme. To the author’s knowledge, this is the first time for which a comparison of entropy stable DGSEM on Gauss–Lobatto and Gauss–Legendre has been performed for compressible, wall-bounded turbulent flows with separation.

A stochastic method in simulating particles transport and deposition in wall-bounded turbulent flow

Flow in the juncture of surface mounted stream-lined slabs of various aspect ratios was studied in a low speed wind tunnel for laminar as well as turbulent approach boundary layer conditions for Reynolds number of 70,000, 90,000 and 120,000. For a laminar approach boundary layer a wavy structure along the primary separation line was observed for an aspect ratio of 1 while at higher aspect ratios and in case of the turbulent boundary layer for all aspect ratios this pattern was not visible. Mean pressures measured along the plane of symmetry on the surface exhibited influence of aspect ratio as well as boundary layer condition. While the surface pressure gradient for the turbulent case continuously increased in the direction of the slab, the laminar boundary layer case had a distinct change in slope, the location of which was aspect ratio dependent. Additionally the pressure gradient for the turbulent boundary layer was lower in comparison to the laminar boundary layer. The pressure differential at the mean vortex location for the turbulent boundary layer was higher than the laminar boundary layer case and reduced with increasing aspect ratio for both cases. The mean vortex location moved away from the slab leading edge with increasing aspect ratio for the laminar case while movement in case of the turbulent boundary layer was not discernable. The mean vortex locations for the turbulent boundary layer were further upstream as compared to the laminar boundary layer cases