Streamwise Separation Research Articles

Hydrodynamic analysis of fin–fin interactions in two-manta-ray schooling in the vertical plane

This study investigates the interaction of a two-manta-ray school using computational fluid dynamics simulations. The baseline case consists of two in-phase undulating three-dimensional manta models arranged in a stacked configuration. Various vertical stacked and streamwise staggered configurations are studied by altering the locations of the top manta in the upstream and downstream directions. Additionally, phase differences between the two mantas are considered. Simulations are conducted using an in-house developed incompressible flow solver with an immersed boundary method. The results reveal that the follower will significantly benefit from the upstroke vortices (UVs) and downstroke vortices depending on its streamwise separation. We find that placing the top manta 0.5 body length (BL) downstream of the bottom manta optimizes its utilization of UVs from the bottom manta, facilitating the formation of leading-edge vortices (LEVs) on the top manta’s pectoral fins during the downstroke. This LEV strengthening mechanism, in turn, generates a forward suction force on the follower that results in a 72% higher cycle-averaged thrust than a solitary swimmer. This benefit harvested from UVs can be further improved by adjusting the phase of the top follower. By applying a phase difference of π/3 to the top manta, the follower not only benefits from the UVs of the bottom manta but also leverages the auxiliary vortices during the upstroke, leading to stronger tip vortices and a more pronounced forward suction force. The newfound interaction observed in schooling studies offers significant insights that can aid in the development of robot formations inspired by manta rays.

Impact of the numerical domain on turbulent flow statistics: scalings and considerations for canopy flows

Large eddy simulations (LES) are widely used to study the effects of surface morphology on turbulence statistics, exchange processes and turbulence topology in urban canopies. However, as LES are only approximations of reality, special attention is needed for the computational model set-up to ensure an accurate representation of the physical processes of interest. This paper shows that the choice of the numerical domain can significantly affect the accuracy of turbulent flow statistics, potentially causing a mismatch between numerical studies and experimental data. The study examines the influence of cross-stream aspect ratio (YAR), streamwise aspect ratio (XAR) and scale separation (SS) on first- and second-order flow statistics and turbulence topology. It is found that domains with a low YAR underestimate the velocity variance, while those with a low XAR overestimate the variance value. The study proposes a new approach based on the Buckingham Pi theorem to evaluate the effect of SS, as the existing method has major limitations for canopy flows. The results suggest that domains with small SS underpredict the variance value. To minimise the artificial impact of the numerical domain on turbulent flow statistics, the study recommends guidelines for future research, including a YAR of 3 or more, an XAR of 6 or more and an SS of 12 or more. Error tables are presented to allow researchers to select smaller domains than recommended, depending on their research interests in specific parts of the flow.

On low-frequency unsteadiness of single-incident and dual-incident shock waves/turbulent boundary layer interactions with large-scale separations

Interactions between turbulent boundary layer and single-incident or dual-incident shock waves are commonly observed in supersonic inlets. To provide novel insights into the unsteadiness of these interactions, six incident shock wave/turbulent boundary layer interactions (ISWTBLIs), including both single-ISWTBLIs and dual-ISWTBLIs, are experimentally investigated using dynamic wall-pressure measurements at Mach 2.73 flow conditions. Through spectral analysis and correlation analysis, the unsteadiness characteristics of single-ISWTBLIs and dual-ISWTBLIs are comparatively examined, revealing significant influences of shock-wave strength and shock-wave distance on low-frequency unsteadiness. Specifically, both regions near the separation and reattachment points exhibit distinct low-frequency unsteadiness in single-ISWTBLIs and dual-ISWTBLIs with relatively strong separations. An increase in the deflection angle from 10° to 12° in single-ISWTBLIs extends the streamwise separation length, while causing a dramatic decrease in both characteristic frequency and normalized characteristic frequency (Strouhal number) of low-frequency unsteadiness. However, for dual-ISWTBLIs, as the shock-wave distance increases, there is an initial increase followed by a subsequent decrease in the separation length accompanied by a continuously increasing characteristic frequency, thus indicating a different relationship between separation length and characteristic frequency of unsteadiness in dual-ISWTBLIs in comparison with that observed in single-ISWTBLIs. Moreover, correlation analysis revealed that the pressure signals in the reattachment region are of strong negative correlation with those near the separation-shock foot, and the correlation increases with increasing shock-wave strength in single-ISWTBLIs and decreasing shock-wave distance in dual-ISWTBLIs.

Performance analysis of vertical-axis wind turbine clusters: Effect of inter-turbine spacing and turbine rotation

Wind energy has emerged as a viable alternative to fossil fuels, with vertical-axis wind turbines (VAWTs) gaining popularity due to their efficiency and adaptability. Combining the actuator line method (ALM) with large eddy simulation (LES) enables accurate performance evaluations, facilitating the design and optimization of wind turbines. The present study invokes ALM-based methodology to perform calculations for the VAWTs. The results of the LES simulations of the VAWTs have been extensively validated against the available experimental and numerical data. The study further explores a VAWT cluster of three turbines by investigating the influence of turbine spacing (in both inline and staggered configuration) on cluster performance. This study shows that the configuration with a streamwise separation (Xsep) of 0.34 D and a transverse separation (Ysep) of 2.5 D exhibits superior performance to other combinations owing to increased kinetic energy in the wake for the downstream turbines. Furthermore, we have presented the effect of varying the rotation direction (in combinations of Clockwise and Counterclockwise rotation) for the individual turbines in the 3-turbine cluster for the two configurations: inline (Xsep = 0 D, Ysep = 2.5 D) and staggered (Xsep = 0.34 D, Ysep = 2.5 D). Staggered counter-rotating turbine cases show reduced performance compared to co-rotating cases, specifically, the clockwise co-rotating (C-C-C) configuration. In the inline configuration, counter-rotating setups outperform co-rotating ones. Counter-rotation analysis reveals that reducing streamwise separation allows turbines to align in line without sacrificing performance, thereby increasing the power density of the turbine cluster.

Experimental and numerical investigation of shock wave/boundary layer interactions induced by curved back-swept compression ramp

When integrated with supersonic/hypersonic flight vehicles the ramp shock/boundary layer interaction occurs in streamline-traced inward-turning inlets exhibit complex three-dimensional characteristic. Two simplified flow models named the single-sided model and the double-sided model were abstracted to investigate the flow phenomenon in this paper. The single-sided model was designed to focus on curved ramp shock/boundary layer interaction near the corner of the inlet while the double-sided model aimed to expose the coupling mechanism and the evolution process near the symmetry plane of the inlet. The separation region of the single-sided model was observed to develop along the leading edge of the three-dimensional compression ramp using oil flow visualisation. It started at the junction of the ramp and the sidewall, spanned the whole test model, and appeared as a ‘scimitar’ shape with a maximum streamwise length of 0.58δ0. On the other hand, a symmetry separation pattern was observed in the double-sided model. The separation region was ‘crescent-shaped’, originated from the junctions of the ramp and two sidewalls, and coalesced at the symmetry plane, inducing a maximum streamwise separation scale of 0.78δ0 on the symmetry plane. The measured surface pressure distributions showed that the shock wave/boundary layer interactions in the two models behaved similarly around the ramp-sidewall junction.

Supersonic Flow Separation in Front of a Forward-Facing Step

A detailed experimental study has been done to understand the Shock Wave-Boundary Layer Interaction (SWBLI) over a Forward-Facing Step (FFS) of height (h) in a Mach 2.5 flow using Particle Image Velocimetry (PIV). In addition to PIV, unsteady wall pressure measurements, high-speed schlieren and surface oil flow visualization have been studied. Flow separation using surface oil flow visualization was observed around 4.1 step heights upstream of the step face. Shock motions extracted from the schlieren image sequence shows a peak frequency of approximately 1000 Hz, which corresponds to the dominant (low) frequency at which the shock oscillates; this being two orders in magnitude smaller than the characteristics frequency of the incoming boundary layer. Also, spectra of the unsteady wall pressure data shows that the shock oscillates with a peak frequency of approximately 1000 Hz. The effect of shock location (xs) from instantaneous velocity PIV fields in the cross-stream plane due to an upstream influence parameter (line-averaged boundary layer streamwise velocity, ul) and a downstream parameter (separation bubble or recirculation region area, A) is studied. Instantaneous cross-stream PIV measurements show significant variations of the separation shock location, incoming boundary layer velocity fluctuations and recirculation region. Besides, conditionally averaged velocity field based on incoming boundary layer streamwise velocity fluctuations and separation bubble area in the cross-stream plane shows that the shock location (xs) is correlated to the size of the separation bubble and not to boundary layer velocity fluctuations.

Flow-induced vibration of tandem flexible cylinders with a larger-diameter upstream cylinder

A systematic experimental study is conducted for the flow-induced vibration (FIV) of tandem flexible cylinder pairs with diameter ratios (D,u/D,d) ranging from 1 to 2. The diameter of the downstream cylinder (DC), D,d, is fixed at 16 mm, while that of the upstream cylinder (UC), D,u, evenly varies from 16 mm to 32 mm. Cylinders with a total length of 2.7 m and an immersion depth of 1.2 m are allowed to freely oscillate in the cross-flow and in-line directions. Results are examined for Re = 1600–11,200 (based on D,d) and a fixed streamwise separation between the cylinder centers T =6D,d. It is shown that the FIV of the DC (DC-FIV) is strongly sensitive to D,u/D,d, and that increasing D,u/D,d significantly weakens the high-mode and high-frequency response of the DC. A deeper analysis of the dual-cylinder interaction reveals the perturbation mechanism of the UC on the DC-FIV, i.e., the UC forces the DC to oscillate following its dominant frequency and dominant mode through “vibration control”. A map containing eight response patterns is further provided in the D,u/D,d–Vr plane. The Strouhal number rule between DC-FIV and its reduced velocity, which is broken by the UC, is reconstructed with a variation factor.

Measurement of turbulent boundary layer unsteady wall pressures beneath elastomer layers of various thicknesses on a plate

The attenuation of turbulence-inducedwall pressure fluctuations through elastomer layers is studied experimentally and analytically. Wall pressure statistics are measured downstream from a backward facing step, with no elastomer present and beneath 2-, 3- and 4-mm-thick elastomers in a water tunnel facility. In the absence of an elastomer layer, the wall pressure spectra, cross-spectra and velocity statistics measured at the various locations downstream from the backward facing step are in excellent agreement with those reported in the archival literature. The streamwise coherence measured beneath the elastomer layers is higher than that measured in the absence of an elastomer layer, an effect which increases with increasing elastomer thickness. It is speculated that this increase in coherence level is due to the ability of the elastomer to support shear stresses, which effectively increases the area over which an eddy influences the normal stresses measured by the pressure sensors. The high-frequency filtering of the elastomers is also observed in the coherence at the smallest streamwise separation. The attenuation of the turbulent boundary layer wall pressure fluctuations through the elastomer layer using an analytical elastomer transfer function is in excellent agreement with the attenuation measured experimentally through all thicknesses of elastomer and at all the free stream velocities at which the experiments are performed.

Vortex-Generating Shock Control Bumps for Robust Drag Reduction at Transonic Speeds

The 3D contour bump has been integrated with vane-type vortex generators to weaken the shock wave and suppress potential flow separation at higher transonic speeds. The integrated vortex-generating shock control bump is parameterized as a whole, in a manner that the contour bump is designed for shock control and the vortex-generating fins are integrated to the contour bump for separation control. This allows the balance between the two factors affecting the total wing drag. The optimization studies based on the Reynolds-averaged Navier–Stokes equations have been carried out to find the optimal vortex-generating bump designs at the given design conditions. Single-point and multipoint global optimizations are carried out to search the optimum parameters at the corresponding design points. It is found that the vortex-generating bump can further reduce the total drag at the higher transonic Mach numbers due to the alleviation of after-bump streamwise separation by a pair of counter-rotating vortices generated. The inclusion of the vortex generators in the bump design allows for a better balance of the design in controlling both the shock strength and the flow separation after the bump at higher Mach numbers. A multipoint optimization leads to a robust vortex-generating bump design for a range of Mach numbers. The distinctive vortical flow structures induced by the vortex-generating bump are highlighted and discussed.

Numerical study of aerodynamic noise behaviors for a vertically-installed flat strut behind an asymmetrical airfoil

The aerodynamic noise generated from a cooling fan and its support frame is one of the major noise sources of many engineering machines, such as automobile engines and communication base-stations. A generic model composed of a vertically-installed flat strut behind an asymmetrical airfoil is used to investigate numerically the noise characteristics and generation mechanism of a cooling fan by using the SST-SAS turbulence model combining with FW-H formula. The airfoil–strut configuration is varied either by changing the perforation rate of the strut or by adjusting the streamwise gap between the airfoil trailing edge and the strut. The motivation is to investigate the effects of the two parameters on the aerodynamic and noise behaviors of the airfoil–strut​ model and evaluate the influences of noise control methods. The analysis of noise generation mechanism and far-field directivity indicates that in the vertical direction both the unsteady lift forces acting on the airfoil and strut contribute equally to the far-field noise, whereas in the upstream/downstream direction the noise is produced mainly by the unsteady drag on the strut. In addition, the whole model shows the directivity characteristic of an omnidirectional noise source. With the increase of the strut perforation rate, both the unsteady lift and drag on the airfoil and strut are reduced due to the attenuation of the Karman vortex street in the flow field, resulting in a far-field noise reduction. There exists a frequency lock-in (i.e. a resonance) between the airfoil and the strut in which the unsteady characteristic of the airfoil is induced by the downstream strut. However, this lock-in loses its strength with the increase of the streamwise separation between the airfoil and strut, accompanying with a change of far-field noise.

Predictive model of bulk drag coefficient for a nature-based structure exposed to currents

Mangrove vegetation provides natural protection against coastal hazards like flooding and erosion. In spite of their economic and societal value, mangrove forests have experienced a worldwide decline due to human activities. Bamboo structures, formed by poles driven into the soil, are being used to create a sheltered environment for mangrove restoration. The lack of design rules for the structures has led to mixed success rates in their implementation. Improving future designs requires a better understanding of how the bamboo poles affect waves and currents. Currents cause drag forces on the poles, which depend on flow acceleration through the elements (blockage), and the distance from wakes of upstream cylinders (sheltering). We developed a model that predicts the bulk drag coefficient of dense arrays of emergent cylinders in a current, including blockage, sheltering and a balance between turbulence production and dissipation. The model could reproduce measured bulk drag coefficients from the literature within a deviation of 20%. The model also showed that anisotropic structures with small spanwise spacing and large streamwise separation maximize the bulk drag coefficient, and the energy dissipation per pole. The application of the model can guide the design of future mangrove restoration efforts.

Effect of In-Line Tandem Configuration on Performance and Scaling of Pitching Hydrofoils

Simulations are reported on the behavior of two tandem symmetric foils in an in-line configuration undergoing pitching motion. The Reynolds number was varied between 1000 and 4000, whereas the pitching Strouhal number ranged from 0.2 to 0.5 and the amplitude of pitching varied from 4 to 8 deg. The streamwise separation distance between the foils was fixed at one chord length. The performance of the downstream foil was significantly worsened (up to 80% in efficiency) by the tandem effects. The Strouhal number, Reynolds number, and pitching amplitude had substantial impact on the performance of the downstream foil. For the upstream foil, however, the performance improved by up to 147% in efficiency at all ranges of Reynolds number, Strouhal number, and pitching amplitude. The scaling of the individual foil performance metric and the system collective performance metric were investigated for tandem foils. The scaling developed for isolated foils does hold for tandem foils in the in-line configuration with modified empirical coefficients that depend on reduced frequency and Reynolds number in a range of and . However, the leading foil behavior at lower is more scattered compared to higher , which suggests more significant wake interactions that influence the scaling and performance of the leading foil.

Urban Wind Harvesting Using Flow-Induced Vibrations

The growing global interest in sustainable energy has paved the way to the rapid development of large-scale wind farms, consisting of dozens to hundreds of wind turbines. Although these large wind farms can generate enormous amount of power, they are also costly and require large areas of land or water, and thus are not suitable for urban environments. Smaller urban wind turbines have been developed for urban environments, but there are significant challenges to their widespread deployment. One of these challenges are their urban wind flows as they are strongly affected by complex building structures, producing highly turbulent flows. Any urban wind turbine would need to be designed to function efficiently and safely under these flow conditions; however, these unpredictable and turbulent winds can induce undesirable vibrations and cause early failures. Recently, bladeless wind turbines are gaining interest due to their reduced costs compared with conventional wind turbines such as the vertical-axis wind turbine and horizontal-axis wind turbine. These bladeless turbines convert flow wind energy into vibration energy, then converts the vibration energy into electricity. This paper examines the effects of force-induced vibrations on a cantilever beam system through wind tunnel experimentation. When fluid flows around a bluff body, periodic shedding of vortices may occur under the right conditions. The vortex shedding process creates an asymmetric pressure distribution on the body which causes the body to oscillate, known as vortex-induced vibrations. The purpose of the paper is to understand the factors affecting flow-induced vibrations and to improve wind energy harvesting from these vibrations. The first part of the paper focuses on wind tunnel experiments, by utilizing a cantilever beam configuration, conceptualized by previous research. Then, the experimental model was tested in different configurations, to determine the best setup for maximizing vibrations induced on the model. The long-term goal of the project was utilizing the model to optimize the system to improve efficiency of wind energy harvesting. The experimental results showed that the presence of an upstream cylinder will significantly improve the amplitude of vibration for energy harvesting, furthermore, the experiments showed that spacing in different directions also affect the amplitude of the vibrations. A two tandem cylinder system was used in this work, including a fixed rigid upstream cylinder and a downstream cylinder supported by a cantilever beam. Various configurations of these two cylinders in terms of spanwise and streamwise separation distances were studied and their maximum and root mean square displacements are reported for different wind speeds. Results showed that the presence of an upstream cylinder will significantly improve the amplitude of vibrations. This work verified that a wind energy harvester needs to consider the effects of wind speed and separation configuration of the cylinders in order to maximize the harvester’s performance in urban environments. KEYWORDS: Sustainable Energy; Energy Harvesting; Urban Environments; Bladeless Wind Turbines; Flow-Induced Vibrations; Cantilever Beam System; Wind Tunnel; Wake

Numerical study of species separation in rarefied gas mixture flow through micronozzles using DSMC

This work investigates the species separation in the rarefied flow of the argon-helium mixture through convergent-divergent micronozzles. Imposing a molecular mass ratio in the order of 10, the flow of this mixture can lead to the formation of serious nonhomogeneous phenomena such as the species separation. This study is performed in the ranges of 2.0–4.0 for the geometrical expansion ratio, 200–400 K for the wall temperature, and 0.003–1.454 for the inlet Knudsen number. The effects of these parameters are examined on the separative performances of micronozzle. The direct simulation Monte Carlo method is selected as the solution method because it can provide reliable solutions in the current rarefied flow regime study. The current study reveals two important separative effects in the mixture flow through micronozzles. The first effect is the lateral species separation, which results in the enrichment of heavier species near the centerline. The second effect is the streamwise separation, which leads to the enrichment of one species, mostly the lighter one, as the mixture passes through the micronozzle. The current results show that increasing the expansion ratio will enhance the lateral separation monotonically. However, there are specific wall temperature and Knudsen values, which can result in optimum lateral separative effects. In addition, it is observed that the expansion ratio has little effect on the streamwise separation. However, increasing either the wall temperature or the Knudsen number will enhance the streamwise separation, albeit with a limiting value at very high Knudsen numbers.

Exploring wind farms with alternating two- and three-bladed wind turbines

Laboratory experiments and numerical simulations were carried out to explore potential advantages of using alternating rows of 3-bladed and 2-bladed turbines within wind farms. The power output and turbulence around the units were compared with baseline wind farms composed of 3-bladed turbines. The experimental setups consisted of two arrays of aligned turbines with streamwise separation of five and ten rotor diameters d, sharing a spanwise spacing of 2.5d. Complementary Large Eddy Simulations at full-scale Reynolds number of Re≈109 based on d were performed for the 5d setups. Results showed enhanced mean velocity and reduced turbulence levels in the heterogeneous wind farm as compared with the baseline in the 5d layout, which may imply lower dynamic loading; however, this was not reflected in the power fluctuation intensity. A marginal increase in the power of the 3-bladed turbines was offset by the diminished performance of the 2-bladed turbines. For the 10d cases, the benefits associated with the heterogeneous farm were negligible due to longer recovery distances. The performance of the heterogeneous wind farm did not improve over the baseline case; however, it may offer potentially lower costs of manufacturing and transportation, among others.

Low-frequency unsteadiness of swept shock-wave/turbulent-boundary-layer interaction

High-speed turbulent boundary-layer separation can lead to severe wall-pressure fluctuations, often extending over a swept shock region. Having noted the shear layer’s influence within axisymmetric step flows, tests go on to experimentally assess the unsteadiness of a canonical swept separation, caused by a slanted $90^{\circ }$-step discontinuity (with varying azimuthal height) over an axisymmetric turbulent boundary layer. Results document an increase in shock pulsation frequency along the swept separation region ($\unicode[STIX]{x1D6EC}\leqslant 30^{\circ }$ sweep angles) – whereby the recirculation enables downstream feedback via the reverse flow – as the local streamwise separation length is reduced. A link between the spanwise variation in the separation shock’s low-frequency instability and the downstream mass ejection rate, as large shear-layer eddies leave the bubble, is sustained. The local entrainment-recharge dynamics of swept separation are thereby duly evaluated.

Comparison of box-wing and conventional aircraft mission performance using multidisciplinary analysis and optimization

Box-wing aircraft designs have the potential to achieve significant reductions in fuel consumption. Closed non-planar wing designs have been shown to reduce induced drag and the statically indeterminate wing structure can lead to reduced wing weight. In addition, the streamwise separation of the two main wings can provide the moments necessary for static stability and control, eliminating the weight and aerodynamic drag of a horizontal tail.Proper assessment of the disciplinary interactions in box-wing designs is essential to determine any realistic performance benefits arising from the use of such a configuration. This study analyzes both box-wing and conventional aircraft designed for representative regional-jet missions. A preliminary parametric investigation shows a lift-to-drag ratio advantage for box-wing designs, while a more detailed multidisciplinary study indicates that the requirement to carry the mission fuel in the wings leads to an increase of between 5% and 1% in total fuel burn compared to conventional designs. However, the multidisciplinary study identified operating conditions where the box-wing can have superior performance to conventional aircraft despite the fuel volume constraint.

Analytic study of the conditions required for longitudinal stability of dual-wing aircraft

Recent studies of new, fuel-efficient transport aircraft have considered designs, which make use of two principal lifting surfaces to provide the required lift as well as trim and static stability. Such designs include open tandem-wings as well as closed joined and box-wings. As a group, these aircraft can be termed dual-wing designs. This study developed a new analytic model, which takes into account the downwash from the two main wings and is sensitive to three important design variables: the relative areas of each wing, the streamwise separation of the wings, and the center of gravity position. This model was used to better understand trends in the dual-wing geometry on the stability, maneuverability, and lift-to-drag ratio of the aircraft. Dual-wing aircraft have been shown to have reduced the induced drag compared to the conventional designs. In addition, further drag reductions can be realized as the horizontal tail can be removed if the dual-wings have sufficient streamwise stagger to provide the moments necessary for trim and longitudinal stability. As both wings in a dual-wing system carry a significant fraction of the total lift, trends in such designs that led to longitudinal stability can differ from those of the conventional aircraft and have not been the subject of detailed investigation. Results from the analytic model showed that the longitudinal stability required either a reduction of the fore wing area or shifting the center of gravity forward from the midpoint of both wings' aerodynamic centers. In addition, for wing configurations of approximately equal fore and aft wing areas, increasing the separation between the two wings decreased the stability of the aircraft. The source of this unusual behavior was the asymmetric distribution of downwash upstream and downstream of the wing. These relationships between dual-wing geometry and stability will provide initial guidance on the conceptual design of dual-wing aircraft and aid in the understanding of the results of more complex studies of such designs, furthering the development of future transport aircraft.

Aerodynamics of a cylinder in the wake of a V-shaped object

The interaction between two different shaped structures is very important to be understood. Fluid-structure interactions and aerodynamics of a circular cylinder in the wake of a V-shaped cylinder are examined experimentally, including forces, shedding frequencies, lock-in process, etc., with the V-shaped cylinder width d varying from d/D = 0.6 to 2, where D is the circular cylinder diameter. While the streamwise separation between the circular cylinder and V-shaped cylinder was 10D fixed, the transverse distance T between them was varied from T/D = 0 to 1.5. While fluid force and shedding frequency of the circular cylinder were measured using a load cell installed in the circular cylinder, measurement of shedding frequency of the V-shaped cylinder was done by a hotwire. The major findings are: (i) a larger d begets a larger velocity deficit in the wake; (ii) with increase in d/D, the lock-in between the shedding from the two cylinders is centered at d/D = 1.1, occurring at d/D 0.95-1.35 depending on T/D; (iii) at a given T/D, when d/D is increased, the fluctuating lift grows and reaches a maximum before decaying; the d/D corresponding to the maximum fluctuating lift is dependent on T/D, and the relationship between them is linear, expressed as ; that is, a larger d/D corresponds to a greater T/D for the maximum fluctuating lift.

A scaling law for the shear-production range of second-order structure functions

Dimensional analysis suggests that the dissipation length scale ($\ell _{{\it\epsilon}}=u_{\star }^{3}/{\it\epsilon}$) is the appropriate scale for the shear-production range of the second-order streamwise structure function in neutrally stratified turbulent shear flows near solid boundaries, including smooth- and rough-wall boundary layers and shear layers above canopies (e.g. crops, forests and cities). These flows have two major characteristics in common: (i) a single velocity scale, i.e. the friction velocity ($u_{\star }$) and (ii) the presence of large eddies that scale with an external length scale much larger than the local integral length scale. No assumptions are made about the local integral scale, which is shown to be proportional to$\ell _{{\it\epsilon}}$for the scaling analysis to be consistent with Kolmogorov’s result for the inertial subrange. Here${\it\epsilon}$is the rate of dissipation of turbulent kinetic energy (TKE) that represents the rate of energy cascade in the inertial subrange. The scaling yields a log-law dependence of the second-order streamwise structure function on ($r/\ell _{{\it\epsilon}}$), where$r$is the streamwise spatial separation. This scaling law is confirmed by large-eddy simulation (LES) results in the roughness sublayer above a model canopy, where the imbalance between local production and dissipation of TKE is much greater than in the inertial layer of wall turbulence and the local integral scale is affected by two external length scales. Parameters estimated for the log-law dependence on ($r/\ell _{{\it\epsilon}}$) are in reasonable agreement with those reported for the inertial layer of wall turbulence. This leads to two important conclusions. Firstly, the validity of the$\ell _{{\it\epsilon}}$-scaling is extended to shear flows with a much greater imbalance between production and dissipation, indicating possible universality of the shear-production range in flows near solid boundaries. Secondly, from a modelling perspective,$\ell _{{\it\epsilon}}$is the appropriate scale to characterize turbulence in shear flows with multiple externally imposed length scales.