To aid in prediction of turbulent boundary layer flows over rough surfaces, a new model is proposed to estimate hydrodynamic roughness based solely on geometric surface information. The model is based on a fluid-mechanics motivated geometric parameter called the wind-shade factor. Sheltering is included using a rapid algorithm adapted from the landscape shadow literature, while local pressure drag is estimated using a piecewise potential flow approximation. Similarly to evaluating traditional surface parameters such as skewness or average slope magnitude, the wind-shade factor is purely geometric and can be evaluated efficiently from knowing the surface elevation map and the mean flow direction. The wind-shade roughness model is applied to over 100 different surfaces available in a public roughness database and some others, and the predicted sandgrain-roughness heights are compared with measured values. Effects of various model ingredients are analysed, and transitionally rough surfaces are treated by adding a term representing the viscous stress component.

Losses due to wake interactions between wind turbines can significantly reduce the power output of wind farms. The possibility of active flow control by wake deflection downstream of yawed horizontal-axis wind turbines has motivated research on the fluid mechanics involved. We summarize the findings of a wind tunnel study (Bastankhah & Porté-Agel, J. Fluid Mech., vol. 806, 2016, pp. 506–541) of the flow associated with a yawed model wind turbine, and the insights and modelling developments that have followed this important study.

Two common definitions of the spatially local rate of kinetic energy cascade at some scale $\ell$ in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’ $R>0$, $Q>0$ quadrant, where $R$ and $Q$ are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when $R$ is large.

Recent advances in manufacturing techniques are providing porous media with both high permeability, necessary for effective passive flow control, and high structural strength, essential for engineering applications. We therefore examine the predictive accuracy of the standard Darcy–Forchheimer (DF) law, which is often used to model porous media flows, for inclusion Reynolds numbers (Re) ranging from the low linear regime to the high nonlinear regime where unsteady effects such as vortex shedding become evident. We consider two different inclusion shapes, square and circular, and three different arrangements of the inclusions – inline, staggered and random. The numerical simulations show that the DF law performs well for low-Re flows, irrespective of the inclusion configuration. For intermediate/high-Re flows, the DF law is adequate only when the arrangement is highly random. However, for the regularly arranged topologies or less random geometries at intermediate/high-Re flows, the DF-law performance diminishes significantly due to flow sheltering and redirection (‘inertial flow deflection’) effects that arise from flow inertia, separation and vortex shedding in the wake of the inclusions. It is shown that the standard DF law, in which the nonlinear permeability tensor is independent of orientation, does not capture such effects. We modified the DF law to capture flow redirection effects by allowing the Forchheimer permeability tensor to depend on the flow orientation with respect to the principal geometrical directions of the porous geometry, and examined this extended DF law for these flows.

The recent Lagrangian relaxation towards equilibrium (LaRTE) approach (Fowler et al., J. Fluid Mech., vol. 934, 2022, A44) is a wall model for large-eddy simulations (LES) that isolates quasi-equilibrium wall-stress dynamics from non-equilibrium responses to time-varying LES inputs. Non-equilibrium physics can then be modelled separately, such as the laminar Stokes layers that form in the viscous region and generate rapid wall-stress responses to fast changes in the pressure gradient. To capture additional wall-stress contributions due to near-wall turbulent eddies, a model term motivated by the attached eddy hypothesis is proposed. The total modelled wall stress thus includes contributions from various processes operating at different time scales (i.e. the LaRTE quasi-equilibrium plus laminar and turbulent non-equilibrium wall stresses) and is called the multi-time-scale (MTS) wall model. It is applied in LES of turbulent channel flow subject to a wide range of unsteady conditions from quasi-equilibrium to non-equilibrium. Flows considered include pulsating and linearly accelerating channel flow for several forcing frequencies and acceleration rates, respectively. We also revisit the sudden spanwise pressure gradient flow (considered in Fowler et al., J. Fluid Mech., vol. 934, 2022, A44) to review how the newly introduced model features affect this flow. Results obtained with the MTS wall model show good agreement with direct numerical simulation data over a vast range of conditions in these various non-equilibrium channel flows. To further understand the MTS model, we also describe and test the instantaneous-equilibrium limit of the MTS wall model. In this limit, good wall-stress predictions are obtained with reduced model complexity but providing less complete information about the wall-stress physics.

Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number $Re_\tau =1000$. The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.

Based on a generalized local Kolmogorov–Hill equation expressing the evolution of kinetic energy integrated over spheres of size $\ell$ in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at scale $\ell$ is shown to follow exponential behaviour with the entropy generation rate if the latter is defined by including an appropriately defined notion of ‘temperature of turbulence’ proportional to the kinetic energy at scale $\ell$.

A large eddy simulation wall model is developed based on a formal interpretation of quasi-equilibrium that governs the momentum balance integrated in the wall-normal direction. The model substitutes the law-of-the-wall velocity profile for a smooth surface into the wall-normal integrated momentum balance, leading to a Lagrangian relaxation towards equilibrium (LaRTE) transport equation for the friction–velocity vector ${\boldsymbol u}_\tau (x,z,t)$. This partial differential equation includes a relaxation time scale governing the rate at which the wall stress can respond to imposed fluctuations due to the inertia of the fluid layer from the wall to the wall-model height. A priori tests based on channel flow direct numerical simulation (DNS) data show that the identified relaxation time scale ensures self-consistency with assumed quasi-equilibrium conditions. The new approach enables us to formally distinguish quasi-equilibrium from additional, non-equilibrium contributions to the wall stress. A particular model for non-equilibrium contributions is derived, motivated by laminar Stokes layer dynamics in the viscous sublayer when applying fast-varying pressure gradients. The new wall modelling approach is first tested in standard equilibrium channel flow in order to document various properties of the approach. The model is then applied in large eddy simulation of channel flow with a suddenly applied spanwise pressure gradient (SSPG). The resulting mean wall-stress evolution is compared with DNS with good agreement. At the onset of the SSPG, the laminar Stokes layer develops rapidly while the LaRTE portion of the stress has a delayed response due to its inherent relaxation dynamics. Results also highlight open challenges such as modelling the response of near-wall turbulence occurring above the viscous sublayer and at time scales faster than quasi-equilibrium conditions.

Aiming to study the rough-wall turbulent boundary layer structure over differently arranged roughness elements, an experimental study was conducted on flows with regular and random roughness. Varying planform densities of truncated cone roughness elements in a square staggered pattern were investigated. The same planform densities were also investigated in random arrangements. Velocity statistics were measured via two-component laser Doppler velocimetry and stereoscopic particle image velocimetry. Friction velocity, thickness, roughness length and zero-plane displacement, determined from spatially averaged flow statistics, showed only minor differences between the regular and random arrangements at the same density. Recent a priori morphometric and statistical drag prediction methods were evaluated against experimentally determined roughness length. Observed differences between regular and random surface flow parameters were due to the presence of secondary flows which manifest as high-momentum pathways and low-momentum pathways in the streamwise velocity. Contrary to expectation, these secondary flows were present over the random surfaces and not discernible over the regular surfaces. Previously identified streamwise-coherent spanwise roughness heterogeneity does not seem to be present, suggesting that such roughness heterogeneity is not necessary to sustain secondary flows. Evidence suggests that the observed secondary flows were initiated at the front edge of the roughness and sustained over irregular roughness. Due to the secondary flows, local turbulent boundary layer profiles do not scale with local wall shear stress but appear to scale with local turbulent shear stress above the roughness canopy. Additionally, quadrant analysis shows distinct changes in the populations of ejection and sweep events.

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions.

By analysing the Karman–Howarth equation for filtered-velocity fields in turbulent flows, we show that the two-point correlation between the filtered strain-rate and subfilter stress tensors plays a central role in the evolution of filtered-velocity correlation functions. Two-point correlation-based statistical a priori tests thus enable rigorous and physically meaningful studies of turbulence models. Using data from direct numerical simulations of isotropic and channel flow turbulence, we show that local eddy-viscosity models fail to exhibit the long tails observed in the real subfilter stress–strain-rate correlation functions. Stronger non-local correlations may be achieved by defining the eddy-viscosity model based on fractional gradients of order $0<\alpha <1$ (where $\alpha$ is the fractional gradient order) rather than the classical gradient corresponding to $\alpha =1$. Analyses of such correlation functions are presented for various orders of the fractional-gradient operators. It is found that in isotropic turbulence fractional derivative order $\alpha \sim 0.5$ yields best results, while for channel flow $\alpha \sim 0.2$ yields better results for the correlations in the streamwise direction, even well into the core channel region. In the spanwise direction, channel flow results show significantly more local interactions. The overall results confirm strong non-locality in the interactions between subfilter stresses and resolved-scale fluid deformation rates, but with non-trivial directional dependencies in non-isotropic flows. Hence, non-local operators thus exhibit interesting modelling capabilities and potential for large-eddy simulations although more developments are required, both on the theoretical and computational implementation fronts.

A quantitative understanding of the dominant mechanisms that govern the generation and decay of the counter-rotating vortex pair (CVP) produced by yawed wind turbines is needed to fully realize the potential of yawing for wind farm power maximization and regulation. Observations from large eddy simulations (LES) of yawed wind turbines in the turbulent atmospheric boundary layer and concepts from the aircraft trailing vortex literature inform a model for the shed vorticity and circulation. The model is formed through analytical integration of simplified forms of the vorticity transport equation. Based on an eddy viscosity approach, it uses the boundary-layer friction velocity as the velocity scale and the width of the vorticity distribution itself as the length scale. As with the widely used Jensen model for wake deficit evolution in wind farms, our analytical expressions do not require costly numerical integration of differential equations. The predicted downstream decay of maximum vorticity and total circulation agree well with LES results. We also show that the vorticity length scale grows linearly with downstream distance and find several power laws for the decay of maximum vorticity. These results support the notion that the decay of the CVP is dominated by gradual cancellation of the vorticity at the line of symmetry of the wake through cross-diffusion.

The spatio-temporal dynamics of separation bubbles induced to form in a fully developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate is studied via direct numerical simulations. Two different separation bubbles are examined: one induced by a suction–blowing velocity profile on the top boundary and the other by a suction-only velocity profile. The latter condition allows reattachment to occur without an externally imposed favourable pressure gradient and leads to a separation bubble more representative of those occurring over airfoils and in diffusers. The suction-only separation bubble exhibits a range of clearly distinguishable modes, including a high-frequency mode and a low-frequency ‘breathing’ mode that has been observed in some previous experiments. The high-frequency mode is well characterized by classical frequency scalings for a plane mixing layer and is associated with the formation and shedding of spanwise-oriented vortex rollers. The topology associated with the low-frequency motion is revealed by applying dynamic mode decomposition to the data from the simulations and is shown to be dominated by highly elongated structures in the streamwise direction. The possibility of Görtler instability induced by the streamwise curvature on the upstream end of the separation bubble as the underlying mechanism for these structures and the associated low frequency is explored.

Motivated by the need for accurate determination of wall shear stress from profile measurements in turbulent boundary layer flows, the total shear stress balance is analysed and reformulated using several well-established semi-empirical relations. The analysis highlights the significant effect that small pressure gradients can have on parameters deduced from data even in nominally zero pressure gradient boundary layers. Using the comprehensive shear stress balance together with the log-law equation, it is shown that friction velocity, roughness length and zero-plane displacement can be determined with only velocity and turbulent shear stress profile measurements at a single streamwise location for nominally zero pressure gradient turbulent boundary layers. Application of the proposed analysis to turbulent smooth- and rough-wall experimental data shows that the friction velocity is determined with accuracy comparable to force balances (approximately 1 %–4 %). Additionally, application to boundary layer data from previous studies provides clear evidence that the often cited discrepancy between directly measured friction velocities (e.g. using force balances) and those derived from traditional total shear stress methods is likely due to the small favourable pressure gradient imposed by a fixed cross-section facility. The proposed comprehensive shear stress analysis can account for these small pressure gradients and allows more accurate boundary layer wall shear stress or friction velocity determination using commonly available mean velocity and shear stress profile data from a single streamwise location.

Lifting line theory describes the cumulative effect of shed vorticity from finite span lifting surfaces. In this work, the theory is reformulated to improve the accuracy of the actuator line model (ALM). This model is a computational tool used to represent lifting surfaces, such as wind-turbine blades in computational fluid dynamics. In ALM, blade segments are represented by means of a Gaussian body force distribution with a prescribed kernel size. Prior analysis has shown that a representation of the blade using an optimal kernel width $\unicode[STIX]{x1D716}^{opt}$ of approximately one quarter of the chord size results in accurate predictions of the velocity field and loads along the blades. Also, simulations have shown that use of the optimal kernel size yields accurate representation of the tip-vortex size and the associated downwash resulting in accurate predictions of the tip losses. In this work, we address the issue of how to represent the effects of finite span wings and tip vortices when using Gaussian body forces with a kernel size larger than the optimal value. This question is relevant in the context of coarse-scale large-eddy simulations that cannot afford the fine resolutions required to resolve the optimal kernel size. For this purpose, we present a filtered lifting line theory for a Gaussian force distribution. Based on the streamwise component of the vorticity transport equation, we develop an analytical model for the induced velocity resulting from the spanwise changes in lift force for an arbitrary kernel scale. The results are used to derive a subfilter-scale velocity model that is used to correct the velocity along the blade when using kernel sizes larger than $\unicode[STIX]{x1D716}^{opt}$. Tests are performed in large-eddy simulation of flow over fixed wings with constant and elliptic chord distributions using various kernel sizes. Results show that by using the proposed subfilter velocity model, kernel-size independent predictions of lift coefficient and total lift forces agree with those obtained with the optimal kernel size.

This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.

Yawing wind turbines has emerged as an appealing method for wake deflection. However, the associated flow properties, including the magnitude of the transverse velocity associated with yawed turbines, are not fully understood. In this paper, we view a yawed turbine as a lifting surface with an elliptic distribution of transverse lift. Prandtl’s lifting line theory provides predictions for the transverse velocity and magnitude of the shed counter-rotating vortex pair known to form downstream of the yawed turbine. The streamwise velocity deficit behind the turbine can then be obtained using classical momentum theory. This new model for the near-disk inviscid region of the flow is compared to numerical simulations and found to yield more accurate predictions of the initial transverse velocity and wake skewness angle than existing models. We use these predictions as initial conditions in a wake model of the downstream evolution of the turbulent wake flow and compare predicted wake deflection with measurements from wind tunnel experiments.

The detailed dynamics of small-scale turbulence are not directly accessible in large-eddy simulations (LES), posing a modelling challenge, because many micro-physical processes such as deformation of aggregates, drops, bubbles and polymers dynamics depend strongly on the velocity gradient tensor, which is dominated by the turbulence structure in the viscous range. In this paper, we introduce a method for coupling existing stochastic models for the Lagrangian evolution of the velocity gradient tensor with coarse-grained fluid simulations to recover small-scale physics without resorting to direct numerical simulations (DNS). The proposed approach is implemented in LES of turbulent channel flow and detailed comparisons with DNS are carried out. An application to modelling the fate of deformable, small (sub-Kolmogorov) droplets at negligible Stokes number and low volume fraction with one-way coupling is carried out and results are again compared to DNS results. Results illustrate the ability of the proposed model to predict the influence of small-scale turbulence on droplet micro-physics in the context of LES.

The fluctuations in power output from wind farms display significantly reduced spectra compared to single wind turbines due to power smoothing and averaging. In order to better understand these spectral features and to relate them to properties of turbulent boundary layers, we perform a wind tunnel experiment in which we measure spatio-temporal characteristics of an experimental surrogate of the power output from a micro wind farm with 100 porous disk models. The experimental results show that the frequency spectrum of the total wind farm power follows a power law with a slope between $-5/3$ and $-2$, and up to lower frequencies than seen for any individual turbine model. In agreement with previous studies in the literature, peaks in the spectrum are observed at frequencies corresponding to the mean flow convection time between consecutive turbines. In the current work we interpret the sum of power extraction from an array of turbines as a discrete spatial filtering of a turbulent boundary layer and derive the associated transfer function. We apply it to an existing model for the wavenumber–frequency spectrum of turbulent boundary layers. This approach allows us to verify the individual roles of Doppler shift and broadening of frequencies on the resulting spatially sampled frequency spectrum. Comparison with the wind tunnel data confirms that the approach captures and explains the main features in the spectrum, indicating the crucial role of the interaction between the spatial sampling and the space–time correlations inherently present in the flow. The frequency spectrum of the aggregated power from a wind farm thus depends on both the spectrum of the incoming turbulence and its modulation by the spatial distribution of turbines in the boundary layer flow.