Spanwise Gradient Research Articles

The effect of Reynolds number on the separated flow over a low-aspect-ratio wing

At high incidence, low-aspect-ratio wings present a unique set of aerodynamic characteristics, including flow separation, vortex shedding and unsteady force production. Furthermore, low-aspect-ratio wings exhibit a highly impactful tip vortex, which introduces strong spanwise gradients into an already complex flow. In this work, we explore the interaction between leading-edge flow separation and a strong, persistent tip vortex over a Reynolds number range of $600 \leq Re \leq 10{\,}000$ . In performing this study, we aim to bridge the insight gained from existing low-Reynolds-number studies of separated flow on finite wings ( $Re \approx 10^2$ ) and turbulent flows at higher Reynolds numbers ( $Re \approx 10^4$ ). Our study suggests two primary effects of the Reynolds number. First, we observe a break from periodicity, along with a dramatic increase in the intensity and concentration of small-scale eddies, as we shift from $Re = 600$ to $Re = 2500$ . Second, we observe that many of our flow diagnostics, including the time-averaged aerodynamic force, exhibit reduced sensitivity to Reynolds number beyond $Re = 2500$ , an observation attributed to the stabilising impact of the wing tip vortex. This latter point illustrates the manner by which the tip vortex drives flow over low-aspect-ratio wings, and provides insight into how our existing understanding of this flow field may be adjusted for higher-Reynolds-number applications.

Numerical analysis of sediment transport and depth averaged flow velocity in non-prismatic compound channels

The flow velocity in compound open channels is greatly affected by the geometry of both the main channel (MC) and floodplain (FP). In this study, a numerical investigation of the joint effect between the FP divergence angle and the size of suspended sediment particles on sediment transport and flow depth-averaged velocity (DAV) in non-prismatic diverging compound channels was conducted. In addition, the influence of relative flow depth on the DAV in different flow directions (i.e., stream-wise and span-wise) was examined. Three divergence angles of 4.0, 6.3 and 11.3°, four sediment particle sizes of 1, 5, 50 and 500 µm and two relative depths of the FP with respect to the MC of 0.20 and 0.40 were considered during simulations. A Computational Fluid Dynamics (CFD) software ANSYS-Fluent was used to perform the required simulation scenarios and the DAV was expressed in terms of a dimensionless velocity (i.e., relative depth-averaged velocity; RDAV) to generalize the use of the obtained results. Results revealed that the span-wise RDAV gradient between the MC and FP at the middle and the end sections of the divergence reach is much larger in the case of higher divergence angle. Moreover, the presence of coarse sediment particles in the sediment-laden flow resulted in an augmentation in the fluctuations of span-wise and stream-wise RDAVs as compared to those obtained in the case of fine sediment particles. Results also showed that the span-wise RDAV gradient between the two subsections (MC and FP) increases with decreasing the relative flow depth. Sediment particle size had a considerable impact on the span-wise RDAV values, especially in the case of lower relative depth. As the sediment particle size increases, the distortion in the span-wise RDAV distribution curves increases. However, the stream-wise RDAV along the MC and FP decreases, as the sediment particle size decreases, especially in the case of higher relative depth. Based on the results, it can be concluded that the span-wise DAV gradient was greatly affected by the FP divergence angle and sediment particle size. The fluctuations in the span-wise DAV gradient were more pronounced in the case of higher divergence angles and coarser sediment particles. The stream-wise DAV distribution along the MC and FP was mainly influenced by the relative depth and sediment particle size. The stream-wise DAV dip along the MC and FP was more explicit in the case of higher relative depths and finer sediment particles.

Multiscale analysis of some shear layers in a fully developed turbulent channel flow

Multiscale edge detection-wavelet analysis is applied to the velocity field emanating from direct numerical simulations of a fully developed turbulent channel flow. The direct numerical simulations are realized in large computational domains at four different Karman numbers up to 1100. The shear layers (SL) related to the streamwise and spanwise velocity fields involving streamwise and spanwise gradients are analyzed scale-by-scale, over the homogeneous planes. The SL involving the spanwise gradient, in particular the wall vorticity layers surrounding the quasi-streamwise vortices (QSV) are organized into long streamwise streaks, whose spanwise spacing and thickness are determined versus the wavelet-wavenumber. The shear layers related to the streamwise gradient are further analyzed by using the Rice decomposition of the wavelet coefficients into a local phase and a local amplitude. Large zones of constant phase with slowly varying local amplitude have been identified at different scales. It is found that the constant phase-locked regions coincide with the wakes of the buffer layer QSV. The arrival of the quasi-streamwise vortices cause phase jumps and large intermittency in the local amplitude. The analogy between these peculiarities and the imperfect chaos synchronization phenomena together with the consequences in term of wall turbulence control are discussed. These results confirm globally our previous investigation realized through a low Reynolds number channel flow DNS in a small computational box and clarify some points that could not then elucidated.

Experimental and numerical study of bending-induced buckling of stiffened composite plate assemblies

Despite their importance in benchmarking numerical simulations, buckling tests still feature compromises between component-level and high-fidelity large-scale tests. For example, compression-induced buckling tests cannot capture the through-thickness or span-wise stress gradients in wing skins. Consequently, the results obtained often require careful interpretation and conservative considerations before applying to a structure. Alternatively, a system-level large-scale test can be used, yet at considerably increased time and expense. There has been little progression towards capturing system-level behaviour in a simplified test. Herein, for buckling behaviour assessment, a three-point bending test is used, which is quick, simple to implement, and cost-effective compared to existing conventional methods. The proposed method relies on subjecting a panel with auxiliary stiffeners to bending to introduce compression-induced buckling in the skin. The three-point bend test is used, because it provides readily controllable loading and boundary conditions. The location of the neutral plane can be tailored via design of the stiffeners, thus allowing for control of the through-thickness stress gradient induced in the skin. This method is applicable to buckling of stiffened structures subject to bending (e.g., aircraft wingboxes). Numerical models are used to explore the limits of the proposed method and comparing it against traditional coupon and full-scale structural level tests. The test method is experimentally demonstrated for capturing the buckling behaviour of a thermoplastic composite panel made via automated fibre placement. The proposed approach is shown to reliably capture the buckling behaviour of a large-scale test using a simpler and more time and cost-efficient setup than conventional methods.

2列マイクロフォンアレイを用いた乱流境界層における壁面圧力勾配変動の測定

It is known that wall pressure fluctuations in a turbulent boundary layer are closely related to the turbulent flow structures. In this study, a two-rows 60ch microphone array has been developed to evaluate the wall pressure fluctuations and its spatial gradients. The measurements have been conducted in a turbulent boundary layer. The probability density functions (PDFs) of the wall pressure fluctuations exhibit intermittent characteristics and clear departure from the Gaussian distribution. In the PDFs of the wall pressure fluctuation gradient, the spanwise gradient indicates the symmetric profile as it is expected from the geometrical symmetry of the flow. On the other hand, although the streamwise gradient is considered to show the negative skewness because of the effect of the mean velocity gradient on the fluctuations of the spanwise vorticity flux, the PDFs of the present data show rather symmetric distributions.

Analysis of momentum recovery within the near wake of a cross-flow turbine using Large Eddy Simulation

Momentum recovery within the near wake of a cross-flow turbine is studied, focusing on the different terms of the momentum balance equation. The flow is computed via Large-Eddy Simulation, coupled with an Immersed-Boundary method. Results demonstrate the dominant role of the spanwise flows (oriented along the axis of the turbine) in contributing to momentum recovery within the near wake, especially via advection in the vicinity of its spanwise boundaries and via turbulent transport at inner locations, closer to the mid span. Large streamwise-oriented vortices on the windward side promote momentum recovery into the wake core from its spanwise boundaries, moving the lower-momentum fluid downstream of the turbine from the leeward side towards the windward side, increasing this way the asymmetry of the wake system. Away from the turbine the importance of the spanwise flows declines, compared to that of the cross-stream ones (orthogonal to the axis of the turbine), due to a faster depletion of the spanwise gradients within the wake. As a consequence, the relative importance of the cross-stream flows on further momentum recovery becomes more significant. Meanwhile, as mean gradients decay away from the turbine, turbulent transport contributes more than advection to complete the process of momentum replenishment within the wake.

Energetic structures in the turbulent boundary layer over a spanwise-heterogeneous converging/diverging riblets wall

Time-invariant (or mean) secondary flows, in terms of large-scale counter-rotating roll modes filling boundary layer, have been observed in turbulent boundary layer (TBL) flows over various spanwise-heterogeneous rough walls. Recent studies show that these mean secondary flows are inherently connected with instantaneous large-scale structures in TBL flows over such spanwise-heterogeneous rough walls. In this work, the technique of proper orthogonal decomposition (POD) is used to extract dominant energetic (and thus large-scale) structures from TBL flows over the surface with a spanwise-periodic converging/diverging riblets pattern, one of the spanwise-heterogeneous rough walls adopted previously. POD analyses are conducted on the three fluctuating components of velocities, measured by stereoscopic particle image velocimetry at Reθ = 13 000, in a cross-stream plane of the TBL flows over a spanwise section of the converging riblets pattern, where low-momentum eruptions consistently occur. It is found that first two POD modes with large temporal coefficients are linked to large-scale structures oscillating vigorously in the transverse direction within one section of the converging riblets. Superposition of these instantaneous large-scale structures reveals the observed pattern of mean secondary flows. Furthermore, these large-scale structures are associated with enhanced streamwise vortices and spanwise gradients of the streamwise velocity, compared with that in the smooth wall flow, thus rendering profound effects on the spatial correlations of velocities as well as the distributions of Reynolds stresses.

Simulation and flow physics of a shocked and reshocked high-energy-density mixing layer

Abstract

Computational Analysis of Unstart in Variable-Geometry Inlet

Computational fluid dynamics was used to study the flow through a scaled, mixed-compression, high-speed inlet with a rotating cowl at Mach 4.0 conditions. First, steady Reynolds-averaged Navier–Stokes computations were undertaken with a range of popular turbulence models including the Spalart–Allmaras model, the realizable model, the cubic model, and the Menter shear stress transport (SST) model to assess the impact on the inlet operating state. It was found that two models, the Spalart–Allmaras model and the Menter SST model, predicted the inlet to become unstarted at a cowl angle where the experimental data indicated the inlet remained started. Next, steady-state flow structures were studied at three discrete cowl positions, identifying highly three-dimensional flow features including regions of separated flow and spanwise gradients that became stronger as the cowl opened. Finally, the study culminated with the development of the new transient model, which allowed for time-accurate investigation into the unstart, restart, and hysteresis. The evolution of the separation bubbles was shown to be a major factor in the hysteresis, causing the inlet to restart at an angle different from where it unstarted. The utility of unsteady Reynolds-averaged Navier–Stokes computations to capture the complex time-dependent details of such flows was demonstrated.

The wake flow downstream of a propeller-rudder system

We report wall-resolved, large-eddy simulations for the case of a propeller operating upstream of a hydrofoil, mimicking a rudder. Our primary objective is the identification of wake features that are unique to this coupled system, when compared to open-water cases, which are usually the focus of experiments and computations in the literature. We were able to achieve unprecedented levels of numerical resolution, which capture the dynamics of all energetic eddies in the flow by using a scalable, conservative, structured solver in cylindrical coordinates. The boundary conditions on the rotating propeller and hydrofoil were enforced via an immersed boundary formulation. The largest values of turbulent stresses in the wake of the hydrofoil are achieved outwards from the radial coordinate of the tip of the propeller blades. This is due to spanwise gradients across the hydrofoil (in the direction parallel to the span of the hydrofoil), producing a displacement of the pressure side legs of the tip vortices towards outer coordinates, where they experience shear with the wake of the hydrofoil. The evolution of turbulence is non-monotonic across the streamwise direction. This is a consequence of the growing shear resulting from the complex interactions involving the shear layers from the trailing edge, the tip vortices and the two branches of the hub vortex coming from the two sides of the hydrofoil. Such a shear is reinforced by the spanwise velocities developed by the two branches of the propeller wake across the hydrofoil. Compared to an isolated propeller, these phenomena enhance turbulence production. The present results highlight that a downstream hydrofoil, typical of surface ships, is able to significantly intensify the wake signature of a propeller.

Homogenization of streaks in a laminar boundary layer

The present work, based on experimental, numerical and theoretical investigations, introduces a method to homogenize streaks in the laminar boundary layer. The streaks are created by a spanwise array of roughness elements on the surface of a flat plate. A homogenization body in the form of a horizontal bar is added at a downstream location away from the roughness array to homogenize the velocity differences of the streaks in the laminar boundary layer. Measurements are done with hot-film anemometry and supported by numerical simulations and linear stability theory. The streak amplitude can be significantly reduced with the proposed homogenization body. Furthermore, the reduction in spanwise gradients of the mean velocity leads to a significant reduction in the sinuous instability of the streaky flow. The effects of the homogenization body on the displacement thickness and the observation of flow unsteadiness downstream of the homogenization body are discussed. The present work thus proposes and explores a passive technique to control undesired streaks in the laminar boundary layer.Graphic abstract

LES investigation of the influence of cavitation on flow patterns in a confined tip-leakage flow

In the present paper, Large Eddy Simulations combined with the Zwart-Gerber-Belamri cavitation model are conducted to study the flow characteristics in a tip-leakage flow. A reasonable agreement is obtained between the numerical and experimental data. With the numerical results, the influence of cavitation in a tip-leakage flow on the gross features of tip-leakage vortex (TLV) and tip-separation vortex (TSV), the behavior of tip-leakage jet, the time-averaged vorticity field and turbulent kinetic energy (TKE) distributions are discussed in detailed. Our results demonstrate that more complex induced vortexes are observed in the non-cavitation case when compared with the cavitation case. Moreover, the results indicate that cavitation promotes the fusion of TLV and TSV. The narrower passageway in the gap when cavitation occurs increases the strength of tip-leakage jet, which leaves significant influence on local vorticity and turbulent kinetic energy. Cavitation trends to suppress the production of spanwise and pitchwise vorticity and promotes the TKE production. Further analysis shows that the spanwise and pitchwise flow patterns are mainly responsible for the TKE production. The fluctuations of v and w velocity and the corresponding spanwise and pitchwise velocity gradient components need to be controlled if a decrease in the viscous losses is desired.

Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices

This study investigates the stability and transition of Görtler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Görtler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Görtler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Görtler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

Granular surface avalanching induced by drainage from a narrow silo

Using theory and experiments, we investigate granular surface avalanching due to material outflow from a narrow silo. The assumed silo geometry is a deep rectangular box, of moderate spanwise width and small gap thickness between smooth front and back walls. A small orifice deep below the free surface lets grains drain out at a constant rate. The resulting granular flows can therefore be assumed quasi-two-dimensional and quasi-steady over most of the surface descent history. To model these flows, we couple a kinematic model of deep granular flow with a dynamic model of shallow surface avalanching. We then compare the calculated flow fields with detailed particle tracking measurements, letting the silo ascend relative to the high-speed camera to increase spatial resolution. The results show that the avalanching surface shape and near-surface flow are controlled by the spanwise gradient in subsidence velocity, and how this gradient is in turn controlled by the height above orifice and the gap thickness. Whereas the deep flow pattern is rate independent, shallow avalanching is paced by the granular rheology.

Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution

Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.

Secondary crossflow instability through global analysis of measured base flows

A combined experimental and numerical approach to the analysis of the secondary stability of realistic swept-wing boundary layers is presented. Global linear stability theory is applied to experimentally measured base flows. These base flows are three-dimensional laminar boundary layers subject to spanwise distortion due to the presence of primary stationary crossflow vortices. A full three-dimensional description of these flows is accessed through the use of tomographic particle image velocimetry (PIV). The stability analysis solves for the secondary high-frequency modes of type I and type II, ultimately responsible for turbulent breakdown. Several pertinent parameters arising from the application of the proposed methodology are investigated, including the mean flow ensemble size and the measurement domain extent. Extensive use is made of the decomposition of the eigensolutions into the terms of the Reynolds–Orr equation, allowing insight into the production and/or destruction of perturbations from various base flow features. Stability results demonstrate satisfactory convergence with respect to the mean flow ensemble size and are independent of the handling of the exterior of the measurement domain. The Reynolds–Orr analysis reveals a close relationship between the type I and type II instability modes with spanwise and wall-normal gradients of the base flow, respectively. The structural role of the in-plane velocity components in the perturbation growth, topology and sensitivity is identified. Using the developed framework, further insight is gained into the linear growth mechanisms and later stages of transition via the primary and secondary crossflow instabilities. Furthermore, the proposed methodology enables the extension and enhancement of the experimental measurement data to the pertinent instability eigenmodes. The present work is the first demonstration of the use of a measured base flow for stability analysis applied to the swept-wing boundary layer, directly avoiding the modelling of the primary vortices receptivity processes.

Streamwise Vortices and Velocity Streaks in a Locally Drag-Reduced Turbulent Boundary Layer

This work aims to understand the changes associated with the near-wall streaky structures in a turbulent boundary layer (TBL) where the local skin-friction drag is substantially reduced. The Reynolds number is R e 𝜃 = 1000 based on the momentum thickness or R e τ = 440 based on the friction velocity of the uncontrolled flow. The TBL is perturbed via a local surface oscillation produced by an array of spanwise-aligned piezo-ceramic (PZT) actuators and measurements are made in two orthogonal planes using particle image velocimetry (PIV). Data analyses are conducted using the vortex detection, streaky structure identification, spatial correlation and proper orthogonal decomposition (POD) techniques. It is found that the streaky structures are greatly modified in the near-wall region. Firstly, the near-wall streamwise vortices are increased in number and swirling strength but decreased in size, and are associated with greatly altered velocity correlations. Secondly, the velocity streaks grow in number and strength but contract in width and spacing, exhibiting a regular spatial arrangement. Other aspects of the streaky structures are also characterized; they include the spanwise gradient of the longitudinal fluctuating velocity and both streamwise and spanwise integral length scales. The POD analysis indicates that the turbulent kinetic energy of the streaky structures is reduced. When possible, our results are compared with those obtained by other control techniques such as a spanwise-wall oscillation, a spanwise oscillatory Lorentz force and a transverse traveling wave.

Flow separation on flapping and rotating profiles with spanwise gradients

The growth of leading-edge vortices (LEV) on analogous flapping and rotating profiles has been investigated experimentally. Three time-varying cases were considered: a two-dimensional reference case with a spanwise-uniform angle-of-attack variation α; a case with increasing α towards the profile tip (similar to flapping flyers); and a case with increasing α towards the profile root (similar to rotor blades experiencing an axial gust). It has been shown that the time-varying spanwise angle-of-attack gradient produces a vorticity gradient, which, in combination with spanwise flow, results in a redistribution of circulation along the profile. Specifically, when replicating the angle-of-attack gradient characteristic of a rotor experiencing an axial gust, the spanwise-vorticity gradient is aligned such that circulation increases within the measurement domain. This in turn increases the local LEV growth rate, which is suggestive of force augmentation on the blade. Reversing the relative alignment of the spanwise-vorticity gradient and spanwise flow, thereby replicating that arrangement found in a flapping flyer, was found to reduce local circulation. From this, we can conclude that spanwise flow can be arranged to vary LEV growth to prolong lift augmentation and reduce the unsteadiness of cyclic loads.

Toward robust prediction of crossflow-wave instability in hypersonic boundary layers

Abstract Traveling and stationary crossflow-wave instabilities in a laminar Mach 6 boundary layer are investigated on a 38.1% scale model of the Fifth Hypersonic International Flight Research Experiment (HIFiRE-5) elliptic cone at zero angle of attack and yaw. The Langley Stability and Transition Analysis Code (LASTRAC) was used to analyze the crossflow dominated boundary layer in the mid-span region near the downstream end of the model. Disturbance growth rates, wave angles, and phase speeds are computed with LASTRAC using quasi-parallel Linear Stability Theory (LST), Linear Parabolized StabilityEquations (LPSE), and two-plane or surface marching LPSE (2pLPSE). The predicted wave angles and phase speeds are validated using experimental data, and are found to be in better agreement than previous computations. Further numerical analysis is conducted using the Spatial BiGlobal technique (SBG), which simultaneously accounts for wall-normal and spanwise gradients in the mean boundary layer at a particular axial station. For the first time in the literature, a comparison is made between crossflow wave growth rates computed using LST, LPSE, and 2pLPSE and those computed using SBG, accounting for curvature and geometric divergence of the elliptic cone. The agreement between LST, LPSE, and SBG is fair at best, but excellent agreement is realized between 2pLPSE and SBG. This result constitutes a co-verification of the LASTRAC and SBG stability codes, and provides evidence that 2pLPSE accurately models the physics of the traveling-crossflow instability.

A two-element high-lift airfoil in disturbed flow conditions

This contribution presents experimental results of a two-element high-lift airfoil in a disturbed inflow. Permanent disturbances are generated by the wakes of two static airfoils located upstream of the research airfoil, one of which is a planar wake by an infinite airfoil and the other is a longitudinal vortex emanating from a finite wing. The disturbances induce spanwise gradients into the flow. The disturbed flow field is measured by Particle Image Velocimetry. The interaction of the disturbed flow field with the high-lift airfoil is investigated by means of static surface pressure close to the airfoil’s leading- and trailing-edge, as well as surface hotfilm measurements and oil flow visualizations on the high-lift flap. For small and moderate incidences the airfoil is mainly influenced by the circulation induced by the disturbances, which affect the effective flow angles. Local effects that result from the turbulence in the airfoil-wake and the induced transverse velocity of the disturbances are likewise considered. At high angle of attack, the prevailing stall conditions with strong variations in spanwise direction are discussed.