Wall-bounded Flows Research Articles

Outer-layer universality of the mean velocity profile in turbulent wall-bounded flows

This paper deals with discerning outer-layer universality in the mean velocity profiles of turbulent boundary layers. For that purpose we derive an objective criterion which generalizes the classical wall scaling, as well as the Rotta-Clauser and Zagarola-Smits scaling. Outer-layer universality in the present scaling clearly emerges at displacement thickness Reynolds number exceeding about 3000. We propose universal velocity distributions based on simple patched logarithmic/parabolic fitting functions, which are accurate to within less than 1%

Decomposition of the wall-heat flux of compressible boundary layers

We use the method developed by Elnahhas and Johnson [“On the enhancement of boundary layer skin friction by turbulence: An angular momentum approach,” J. Fluid Mech. 940, A36 (2022)] and Xu et al. [“Decomposition of the skin-friction coefficient of compressible boundary layers,” Phys. Fluids 35, 035107 (2023)] for the decomposition of the skin-friction coefficient to integrate the mean temperature equation for high-Reynolds-number compressible boundary layers and arrive at an identity for the decomposition of the wall-heat flux. The physical interpretation of the identity and the limitations of this approach are discussed. We perform an integration on the mean temperature equation to obtain an identity that is the heat-transfer analog to the compressible von Kármán momentum integral equation for the skin-friction coefficient. This identity is applied to numerical data for laminar and turbulent compressible boundary layers, revealing that the mean-flow dissipation and production of turbulent kinetic energy given by the Favre–Reynolds stresses dominate the thermal-energy balance. The term related to the growth of the turbulent boundary layer opposes the wall cooling. Other identities for the wall-heat flux, inspired by the method of Fukagata et al. [“Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows,” Phys. Fluids 14(11), L73–L76 (2002)], are studied numerically and by asymptotic methods. The terms of these identities depend spuriously on the upper integration bound because this bound is a mathematical quantity used in the derivation. When the bound is asymptotically large, the integral identities simplify to the heat-transfer analog to the von Kármán momentum equation. We also prove that an existing multiple-integration identity reduces to the definition of the wall-heat flux when the number of integrations is asymptotically large. No information about the wall-heat transfer is extracted because the impact of the integration number is nonphysical.

Homoclinic bifurcation and switching of edge state in plane Couette flow.

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.

Drag model of finite-sized particle in turbulent wall-bound flow over sediment bed

Drag force acting on a particle is vital for the accurate simulation of turbulent multiphase flows, but the robust drag model is still an open issue. Fully resolved direct numerical simulation (DNS) with an immersed boundary method is performed to investigate the drag force on saltating particles in wall turbulence over a sediment bed. Results show that, for saltating particles, the drag force along the particle trajectories cannot be estimated accurately by traditional drag models originally developed for an isolated particle that depends on the particle-wall separation distance or local volume fraction in addition to the particle Reynolds number. The errors between the models and DNS are especially clear during the descending phase of the particles. Through simple theoretical analysis and DNS data fitting, we present a corrected factor using the classical, particle Reynolds number dependent drag force model as the benchmark model. The new drag model, which takes the particle vertical velocity into account, can reasonably predict the mean drag force obtained by DNS along a particle trajectory.

Natural grid stretching for DNS of compressible wall-bounded flows

We propose a physics-driven stretching function for direct numerical simulation (DNS) of compressible turbulent wall-bounded flows, which blends uniform near-wall spacing with uniform resolution in terms of semi-local Kolmogorov units in the outer wall layer. Given target Mach number, Reynolds number and wall temperature, our procedure yields a well-defined prescription for the number of grid points and their distribution which guarantee at the same time numerical accuracy and judicious exploitation of computational resources. DNS of high-speed turbulent boundary layers are used to evaluate the quality of the proposed stretching function, which show that one can achieve identical results as with general-purpose stretching functions, however with substantially higher efficiency. A Python script is provided to facilitate implementation of the proposed grid stretching.

Patterns in transitional shear turbulence. Part 2. Emergence and optimal wavelength

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of oblique, spatially intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatio-temporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent–laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds-number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and nonlinear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar–turbulent interfaces. This optimal behaviour is due primarily to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.

Patterns in transitional shear turbulence. Part 1. Energy transfer and mean-flow interaction

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of spatially intermittent turbulent structures. In plane shear flows, these appear as a regular pattern of alternating turbulent and quasi-laminar flow. Both the physical and the spectral energy balance of a turbulent–laminar pattern in plane Couette flow are computed and compared to those of uniform turbulence. In the patterned state, the mean flow is strongly modulated and is fuelled by two mechanisms: primarily, the nonlinear self-interaction of the mean flow (via mean advection), and secondly, the extraction of energy from turbulent fluctuations (via negative spectral production, associated with an energy transfer from small to large scales). Negative production at large scales is also found in the uniformly turbulent state. Important features of the energy budgets are surveyed as a function of$Re$through the transition between uniform turbulence and turbulent–laminar patterns.

The turbulence development of a vertical natural convection boundary layer

Results from direct numerical simulations of a vertical natural convection boundary layer (NCBL) with $Pr=0.71$ reveal that the turbulence development of such a thermally driven convective flow has two distinct stages: at relatively low Grashof number, the bulk flow is turbulent while the near-wall region is laminar-like or weakly turbulent; at sufficiently high Grashof number, the entire flow becomes turbulent in the sense of von Kármán (cf. Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56, for the ultimate turbulent regime). Investigations on the turbulence statistics show that the near-wall Reynolds shear stress is negligible in the weakly turbulent regime but will grow in magnitude as the flow transitions to the ultimate regime at higher Grashof number. Similar behaviour is also seen in the streamwise turbulence intensity, where it develops from a mono-peak profile into a dual-peak structure as the Grashof number increases. At higher Grashof number, the near-wall energetic site is shown to have an energy distribution similar to that of a canonical wall-bounded turbulence (e.g. Hutchins & Marusic, Phil. Trans. R. Soc.A, vol. 365, 2007, pp. 647–664), with a peak centred at fixed location and wavelength ( $y^+=18$ and $\lambda ^+_x=1000$ ) in viscous coordinates. Investigation on the spanwise spectra also suggests that the turbulent near-wall streaks emerge only at sufficiently high Grashof number, with constant spacing $\lambda _z^+\approx 130$ . The extent of the weakly turbulent regime is identified using the maximum velocity location $\delta_m$ and a laminar length scale $\delta_u$ . The development of near-wall turbulence is also investigated by examining the turbulence kinetic energy budget. In the weakly turbulent regime, the near-wall turbulence is sustained predominantly by the pressure transport in addition to the shear production. At higher Grashof number, the flow becomes fully turbulent, and both turbulent transport and shear production become stronger, while the pressure transport is decreased. These results also reveal that the production–dissipation ratio $P/\varepsilon$ of the NCBL would follow a fundamentally different trend to the canonical wall-bounded flows, which further supports that the near-wall turbulence generation is affected by the bulk flow.

Interaction of a turbulent flame with the very-large-scale structures in a channel flow

Combustion processes taking place in large-scale burners may need to consider the impact of the recently identified very-large-scale motions (VLSM) appearing in wall-bounded turbulent flows. While VLSM in non-reacting, cold flows have been found to be important and noticeably impact turbulence statistics, the possible influence of VLSM on turbulent premixed flames has not been considered up to now to the best of our knowledge. The present study investigates the interaction of a turbulent flame with VLSM in a channel flow using direct numerical simulations (DNS). Detailed chemical kinetics have been used to compute a H2 flame. The turbulent channel flow corresponds to Reτ= 280, while the premixed flame is associated to a Damköhler number Da = 0.167. First, VLSM have been identified for the non-reacting flow by proper orthogonal decomposition (POD). Interestingly, the propagation of the turbulent flame is found to have a negligible effect on the VLSM, highlighting their resilience. At the same time, the VLSM do not modulate strongly the flame structure, while it is noticeably affected by small-scale vortices. This decoupling might be explained by the very different length scales of both processes, the VLSM covering the full channel length, while the turbulent flame is characterized by small-scale structures. In the same manner, the characteristic time scales also differ widely, the flame time scale being much faster than the characteristic time of the VLSM. The flame curvature is positively correlated with the fluctuating streamwise velocity when the flame is still far from the wall. When approaching the wall, flame quenching takes place in a complex process. Overall, it appears for the present conditions that VLSM do not impact directly the turbulent flame propagation, but there is still an indirect impact through the modifications of the turbulence statistics.

Scaling of secondary flows with surface parameters: A linear approach

Secondary flows are generated when a lateral variation of the topography, such as streamwise aligned ridges, is imposed upon a turbulent wall-bounded flow. In this case, the time-averaged flow field is characterized by streamwise vortices known as Prandtl’s vortices of the second kind (Prandtl, 1952). As demonstrated in previous experimental and numerical works, the strength and flow organization of these vortices depend on the ridge shape. In this paper, the effect of the ridge geometry on the generation of secondary flows is investigated using the linearized RANS-based model proposed by Zampino et al. (2022). The model is derived from the assumption that the ridges are shallow, with height smaller than any other length scale, e.g. the viscous length scale. Symmetric channels with rectangular, triangular, elliptical and trapezoidal ridges are studied. The model predicts that the strength of secondary flows can be scaled with the mean ridge height, regardless of the ridge shape, when the ridges are narrower than the half channel height and isolated, i.e. when the lateral separation between the ridges is much larger than the ridge width. Finally, the appearance of tertiary flows and the effect of the ridge shape on the flow organization is studied in detail for trapezoidal geometries. It is observed that tertiary flows emerge for ridge configurations where the scaling behaviour does not hold.

A Database for Reduced-Complexity Modeling of Fluid Flows

We present a publicly accessible database specifically designed to aid in the conception, training, demonstration, evaluation, and comparison of reduced-complexity models for fluid mechanics. Availability of high-quality flow data is essential for all of these aspects of model development for both data-driven and physics-based methods. The current database is unique in that it has been curated with this need in mind. The database contains time-resolved data for six distinct datasets: a large eddy simulation of a turbulent jet, direct numerical simulations of a zero-pressure-gradient turbulent boundary layer, particle-image-velocimetry measurements for the same boundary layer at several Reynolds numbers, direct numerical simulations of laminar stationary and pitching flat-plate airfoils, particle-image-velocimetry and force measurements of an airfoil encountering a gust, and a large eddy simulation of the separated, turbulent flow over an airfoil. These six cases span several key flow categories: laminar and turbulent, statistically stationary and transient, tonal and broadband spectral content, canonical and application-oriented, wall-bounded and free-shear flow, and simulation and experimental measurements. For each dataset, we describe the flow setup and computational/experimental methods, catalog the data available in the database, and provide examples of how these data can be used for reduced-complexity modeling. All data can be downloaded using a browser interface or Globus. Our vision is that the common testbed provided by this database will aid the fluid mechanics community in clarifying the distinct capabilities of new and existing methods.

Coherent structures at the origin of time irreversibility in wall turbulence

Time irreversibility is a distinctive feature of non-equilibrium phenomena such as turbulent flows, where irreversibility is mainly associated with an energy cascade process. The connection between time irreversibility and coherent motions in wall turbulence, however, has not been investigated yet. An Eulerian, multiscale analysis of time irreversibility in wall-bounded turbulence is proposed in this study, which differs from previous works relying on a Lagrangian approach and mainly focusing on homogeneous turbulence. Outcomes reveal a strong connection between irreversibility levels and coherent structures in both turbulent channel and boundary layer flows. In the near-wall region, irreversibility is directly related to the inner spectral peak originating from small-scale turbulent structures in the buffer layer. Conversely, stronger irreversibility is found in correspondence to the outer spectral peak originating from larger turbulent flow scales far from the wall. Our results represent a first effort to characterize Eulerian TI in wall-bounded turbulent flows, thus paving the way for further developments in wall-turbulence modeling and control accounting for broken temporal symmetry.

Phononic-subsurface flow stabilization by subwavelength locally resonant metamaterials

The interactions between a solid surface and a fluid flow underlie dynamical processes relevant to air, sea, and land vehicle performance and numerous other technologies. Key among these processes are unstable flow disturbances that contribute to fundamental transformations in the flow field. Precise control of these disturbances is possible by introducing a phononic subsurface (PSub). This comprises locally attaching a finite phononic structure nominally perpendicular to an elastic surface exposed to the flowing fluid. This structure experiences ongoing excitation by an unstable flow mode, or more than one mode, traveling in conjunction with the mean flow. The excitation generates small deformations at the surface that trigger elastic wave propagation within the structure, traveling away from the flow and reflecting at the end of the structure to return to the fluid-structure interface and back into the flow. By targeted tuning of the unit-cell and finite-structure characteristics of the PSub, the returning waves may be devised to resonate and reenter the flow out of phase, leading to significant destructive interference of the continuously incoming flow waves near the surface and subsequently to their attenuation over the spatial extent of the control region. This entire mechanism is passive, responsive, and engineered offline without needing coupled fluid-structure simulations; only the flow instability’s frequency, wavelength, and overall modal characteristics must be known. Disturbance stabilization in a wall-bounded transitional flow leads to delay in laminar-to-turbulent transition and reduction in skin-friction drag. Destabilization is also possible by alternatively designing the PSub to induce constructive interference, which is beneficial for delaying flow separation and enhancing chemical mixing and combustion. In this paper, we present a PSub in the form of a locally resonant elastic metamaterial, designed to operate in the elastic subwavelength regime and hence being significantly shorter in length compared to a phononic-crystal-based PSub. This is enabled by utilizing a sub-hybridization resonance. Using direct numerical simulations of channel flows, both types of PSubs are investigated, and their controlled spatial and energetic influence on the wall-bounded flow behavior is demonstrated and analyzed. We show that the PSub’s effect is spatially localized as intended, with a rapidly diminishing streamwise influence away from its location in the subsurface.

Influence of channel bend angle on the turbulent statistics in sharply bent channel flows

Turbulence in wall-bounded flows shows a wide range of regimes, where the interaction between scales significantly occur. Reynolds number is used to characterize the dynamics of fluid corresponding to single phase channel flows. Meanwhile, different flow behavior exists in curved channels even at fixed Reynolds number, where the curvature varies as shown by Brethouwer [“Turbulent flow in curved channels,” J. Fluid Mech. 931, A21 (2022)]. In the present study, we show how wall-bounded turbulent flow behaves on sharply bending the channel by investigating the time averaged velocity profiles at the straight section, at the bend, and in the inclined section. The well-known logarithmic behavior of the time averaged normalized velocity profile is retained, where the von Kármán and the additive constants assume altered values depending on the sharp bend inclination. The near-wall fluctuations at the bend are enhanced, which is due to diffusion of counter-rotating vortices leading to increased turbulent activity. In terms of spatial structure of the random fluctuating field, the two-point correlation statistics suggest that multiple high speed and low speed streak pairs are generated, and there are multiple streamwise vortices of different sizes when bend inclination is increased. Budgets of turbulent kinetic energy are presented for various inclinations of the bend at different sections of the bend channel, which depicts that turbulent kinetic energy is modulated at the bend.

A wall model learned from the periodic hill data and the law of the wall

Toward data-driven wall-modeled large-eddy simulations of different wall-bounded turbulent flows, a wall model is learned in this work using the wall-resolved large-eddy simulation (WRLES) data of the flow over periodic hills (PH) and the law of the wall (LoW). The feedforward neural network (FNN) is employed to construct the model. The obtained FNN_PH-LoW model is successfully tested using the direct numerical simulation data of turbulent channel flows and the WRLES data of PH cases, and applied to turbulent channel flows for a wide range of Reynolds numbers.

Revisiting “bursts” in wall-bounded turbulent flows

In this paper, we demonstrate that the problem of turbulent bursts can be tackled through a complex systems approach. Specifically, by considering both duration and intensity of the bursting events, one can incorporate the effect of bursts on the turbulence statistics at any specified scale of the flow, thereby allowing one to connect the origin of bursts to the presence of organized eddy motions in turbulent flows called coherent structures. Through our approach we discover a particular aspect of universality associated with turbulent bursts and contribute toward the development of next-generation turbulence models by creating a union between complex systems science and fluid mechanics.

Direct numerical simulation of flow and heat transport in a closely-spaced bare rod bundle

Direct Numerical Simulation (DNS) of fully-developed flow and heat transfer is performed for a bare rod bundle with a low pitch-to-diameter ratio using the highly-scalabale spectral element code Nek5000. The simulation is performed at a Reynolds number, Reh, based on the bulk velocity and hydraulic diameter, of 9800, with iso-temperature and iso-flux thermal boundary conditions and Prandtl numbers 0.025, 1.0 and 2.0. The mean velocity and temperature statistics are first validated against the experimental measurements of Hooper et al. (1983) and complementary DNS results of Lai et al. (2019) which are reported in the literature at a higher Reh of 22,600. The mean flow and temperature are demonstrated a similar behaviour at the present low Renolds number and the higher Reynolds number of Lai et al. (2019), indicating the Reynolds number scaling of the mean flow. The anisotropy of the flow is illustrated using distribution of normal and Reynolds stresses, and the Lumley invariant map. It is shown that the turbulence in the subchannel region is characteristic of wall-bounded turbulent flows such as channel flows. In the narrow gap, however, turbulence is suppressed, only enhanced in the streamwise direction due to flow pulsations. Further, temperature statistics with both thermal boundary conditions are presented, demonstrating the effect of the Prandtl number on the mean and fluctuating quantities. The temperature statistics with iso-temperature wall boundary conditions, not reported in literature before, are shown to behave differently in the interstitial subchannel region and the narrow gap between two rods. The distribution of turbulent heat flux is also presented. The streamwise component of turbulent heat flux is again shown to be enhanced, in comparison to the same in the subchannel. The pulsating motion of flow across the narrow gap is discussed further. A frequency analysis of the velocity in the narrow gap elucidates the frequency of flow pulsations. Unlike mean flow statistics, the flow pulsations are shown to exhibit a low Reynolds number effect. The reference DNS data presented in the present work, complementary to that of Lai et al. (2019), serves as a benchmark against which lower order turbulence models may be further developed or improved. In particular, the newly presented anisotropy data and the turbulence heat flux data are essential for further development, improvement and validation of turbulent heat transfer models at different Prandtl numbers.

Robust Eddy Viscosity Turbulence Modeling with Elliptic Relaxation and Compound Wall Treatment

This paper presents a holistic Reynolds-averaged Navier–Stokes (RANS) turbulence modeling framework for the computational fluid dynamics (CFD) simulations of complex wall-bounded turbulent flows. Based on the elliptic relaxation idea, the deployed eddy viscosity turbulence model reconstructs the near-wall stress anisotropy and nonviscous effects. The appropriate selection of the turbulent quantities that are being solved for, together with the zero value wall boundary condition for the related turbulent quantities, renders the model less sensitive to the near-wall grid nonuniformities and resolution. The unified near-wall velocity profile, obtained based on the boundary layer theory, is used to devise the compound near-wall treatment that ensures the robustness of the numerical simulation. The proposed turbulence modeling framework is implemented into the general-purpose open-source CFD code and validated against the generic test cases with satisfactory agreement.

Efficient spectral implementation of ODE wall model and the extension of integral wall model to unstructured LES solvers

While wall modeling has become an indispensable tool for scale-resolving simulation of high Reynolds numbers wall-bounded turbulent flows, discussion of efficient implementation of wall models in parallel unstructured-grid solvers is lacking. In this paper, we discuss physical and numerical improvements to two zonal wall models based on integral form of the boundary layer differential equations within an unstructured-grid cell-centered finite-volume LES solver. The first model is a novel implementation of the ODE equilibrium wall model, where the velocity profile is expressed in the integral form using the constant shear-stress layer assumption and the integral is evaluated using a spectral quadrature method, resulting in a local and algebraic (grid-free) formulation. The second model is the modified form of the integral wall model of Yang et al. (2015) [14], which is based on the vertically-integrated thin-boundary-layer PDE along with a prescribed composite velocity profile in the wall-modeled region. A modification to the inner layer velocity profile which fixes the coordinate invariance issue of the original model is proposed. Additionally, several numerical challenges unique to the implementation of these integral models in unstructured mesh environments, such as the exchange of wall quantities between wall faces and LES cells, and the computation of surface gradients, are identified and possible remedies are proposed. The computational performance of the wall models is assessed both in a priori and a posteriori settings against the traditional finite-volume based ODE equilibrium wall model, showing a comparable computational cost for the integral wall model, and superior cost scaling for the spectral implementation over the finite-volume based approach. Load imbalance among the processors in parallel simulations seems to severely degrade the parallel efficiency of finite-volume based ODE wall model, whereas the spectral implementation is remarkably agnostic to these effects. The integral wall model, however, is even more prone to parallel efficiency degradation than the finite-volume based ODE wall model due to communication overhead among processors in highly parallelized settings.

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

This work details the assessment of the performance of the drag-porosity model implemented in ARPS (Advanced Regional Prediction System) atmospheric Large-Eddy Simulation (LES) solver for simulating the atmospheric boundary layer developing over the urban canopy with comparison with literature. The flow within and above an idealized urban canopy consisting of a staggered array of cubes with various packing densities modeled with the drag-porosity approach immersed into a neutral, Coriolis-free atmospheric boundary layer at high Reynolds is investigated. Besides one-points statistics, particular interest was given to the ability of the model to reproduce the turbulent coherent structures and their characteristic scales. A detailed analysis of one-point statistics, one- and two-dimensional spectra and two-point correlation functions revealed the presence of typical structures and features found in wall-bounded turbulent flows (two-scale behavior in the roughness sublayer, ejections, sweeps, self-similar wall-attached large scale streaky motions, canopy-independent very large scale motions). Further investigation to identify the interaction mechanisms between large and small scales based on spectral filtering highlighted an interaction mechanism that resembles an amplitude modulation process, as observed in literature on wall-bounded flows. These findings therefore show that the proposed approach is able to reproduce all the key features of the flow developing over urban terrain.