Boundary layer transition experiments with embedded streamwise vortices

Extensive hot-wire measurements have been made to determine the structure of the large eddy in a synthejc turbulent boundary layer on a flat-plate model. The experiments were carried out in a wind tunnel at a nominal free-stream velocity of 12 m/s. The synthetic turbulent boundary layer had a hexagonal pattern of eddies and a ratio of streamwise scale to spanwise scale of 3.2:1. The measured celerity of the large eddy was 84.2 percent of the free-stream velocity. There was some loss of coherence, but very little distortion, as the eddies moved downstream. Several mean properties of the synthetic boundary layer were found to agree quite well with the mean properties of a natural turbulent boundary layer at the same Reynolds number. The large eddy is composed of a pair of primary counter-rotating vortices about five δ long in the streamwise direction and about one δ apart in the spanwise direction, where δ is the mean boundary-layer thickness. The sense of the primary pair is such as to pump fluid away from the wall in the region between the vortices. A secondary pair of counter-rotating streamwise vortices, having a sense opposite to that of the primary pair, is observed outside of and slightly downstream from the primary vortices. Both pairs of vortices extend across the full thickness of the boundary layer and are inclined at a shallow angle to the surface of the flat plate. The data show that the mean vorticity vectors are not tangential to the large-eddy vortices. In fact, the streamwise and normal vorticity components that signal the presence of the eddy are of the same order of magnitude. Definite signatures are obtained in terms of the mean skin-friction coefficient and the mean wake parameter averaged at constant phase. Velocities induced by the vortices are partly responsible for entrainment of irrotational fluid, for transport of momentum, for generation of Reynolds stresses, and for maintenance of streamwise and normal vorticity in the outer flow. A stretching mechanism is important in matching spanwise vorticity close to the wall to variations in turbulent shearing stress. Regions where the stretching term is large coincide with regions of large wall shearing stress and large turbulence production.

Three-dimensional velocity distributions and corresponding wall stresses are measured concurrently in the inner part of a turbulent boundary layer over a smooth wall using digital holographic microscopy. The measurements are performed in a square duct channel flow atReδ= 50000 andReτ= 1470. A spatial resolution of 3–8 wall units (δυ= μm) in streamwise and spanwise directions and 1 wall unit in the wall-normal direction are sufficient for resolving buffer layer structures and for measuring the instantaneous wall shear stresses from velocity gradients in the viscous sublayer. Mean velocity and Reynolds stress profiles agree well with previous publications. Rudimentary observations classify the buffer layer three-dimensional flow into (i) a pair of counter-rotating inclined vortices, (ii) multiple streamwise vortices, some of them powerful, and (iii) no apparent buffer layer structures. Each appears in about one third of the realizations. Conditional sampling based on local wall shear stress maxima and minima reveals two types of three-dimensional buffer layer structures that generate extreme stress events. The first structure develops as spanwise vorticity lifts from the wall abruptly and within a short distance of about 10 wall units, creating initially a vertical arch. Its only precursors are a slight velocity deficit that does not involve an inflection point and low levels of vertical vorticity. This arch is subsequently stretched vertically and in the streamwise direction, culminating in formation of a pair of inclined, counter-rotating vortices with similar strength and inclination angle exceeding 45°. A wall stress minimum exists under the point of initial lifting. A pair of stress maxima develops 35δυdownstream, on the outer (downflow) sides of the vortex pair and is displaced laterally by 35–40δυfrom the minimum. This flow structure exists not only in the conditionally averaged field but in the instantaneous measurement as well and appears in 16.4% of the realizations. Most of the streamwise velocity deficit generated by this phenomenon develops during this initial lifting, but it persists between the pair of vortices. Distribution of velocity fluctuations shows that spanwise transport of streamwise momentum plays a dominant role and that vertical transport is small under the vortices. In other regions, e.g. during initial lifting, and between the vortices, vertical transport dominates. The characteristics of this structure are compared to early experimental findings, highlighting similarities and differences. Abundance of pairs of streamwise vortices with similar strength is inconsistent with conclusions of several studies based on analysis of direct numerical simulation (DNS) data. The second buffer layer structure generating high wall stresses is a single, predominantly streamwise vortex, with characteristic diameter of 20–40δυand inclination angle of 12°. It generates an elongated, strong stress maximum on one side and a weak minimum on the other and has been observed in 20.4% of the realizations. Except for a limited region of sweep above the high-stress region, this low-lying vortex mostly induces spanwise momentum transport. This structure appears to be similar to those observed in several numerical studies.

Blasius’s flat plate boundary layer equation has always been a model for a better understanding of the boundary layer concept and a didactic example of the exact solution for a particular case of Navier Stokes equations. However, considering problems in the equation deduction, the meaning of some related parameters, such as displacement thickness, \(\delta\)I, and the momentum thickness, \(\delta\)I, and the existence of a unique value of the similarity parameter at infinity, i.e., \(\eta\)\(\infty\), set valid for the entire plate, may change such a reputation, turning it unreliable. These issues have been calling the attention of researchers for more than a century, who incorporated comments, critics, techniques, arguments, and suggestions to improve the classical theory and its results. Unfortunately, most of these contributions have not succeeded in explaining the common doubts related to an incompressible fluid advancing over an ideal flat plate. In fact, they hampered the determination of a model capable of describing this physical phenomenon. This work explains how it occurs and presents new equations compatible with Prandtl’s concept of boundary layer used to describe the flat plate boundary layer. The proposition of a new equation and solution requires that the ordinary third-order differential equation be solved with just three boundary conditions, as mathematically recommended; preserves the original flow design, and velocity gradients for a chosen station, x, determined in terms of positions situated all along the boundary layer thickness.

Numerical investigation of a jet and vortex actuator in a cross flow boundary layer

The linear stability analysis has been performed at Reτ=180 for a turbulent-channel-type base flow with a periodic undulation in the spanwise direction in order to elucidate the generation mechanism of streamwise vorticity through the instability of streaks in near-wall turbulence. It is found that there appear three different instability modes depending on the spanwise wavenumber of streaks. In the case of the streak with around 100 wall-unit spanwise wavelength its critical velocity amplitude lies at 〓U+c〓3, above which streaky flow is unstable to an infinitesimal sinuous disturbance, i.e.a bending mode along the streamwise direction. The instability is identified to originate from inflection points, i.e.wake-like instability, in the spanwise variation of the streaky flow. In this case, unstable eignmodes take a form that is inclined towards the streamwise direction from the wall-normal direction, and they directly induce the streamwise vorticity on low-and high-speed streaks. In addition, the streamwise vorticity is secondarily produced pricipally through tilting of the wall-normal disturbance vorticity by the base flos whear across the wall-normal direction.

This study investigates flow past a sphere immersed in a flat-plate turbulent boundary layer by using direct numerical simulations, with the objective of clarifying the effects of a wall-proximity sphere on turbulent coherent structures and turbulence statistics. Three cases are evaluated with gap ratios (G/D) of 0.1, 0.25, and 0.5 at a Reynolds number of ReD=2500, where D is the diameter of the sphere and G is the gap width between the bottom of the sphere and the flat plate. The results show that the wake of the sphere plays an important role in the streamwise region 0<x/D<10. The near-wall streaks break into small-scale point-like or patch-like structures in the near-wake region, with the most significant effect at G/D=0.1. This can be attributed to the interactions between the shedding vortex behind the sphere and vortical structures within the flat-plate boundary layer. Detail analysis of turbulence statistics indicates that the flat-plate boundary layer thickness is increased at x/D<−0.5 owing to the blockage effect of the sphere, whereas it is decreased at x/D>0.5 because of the high-speed jet flow around it. In addition, the presence of a wall-proximity sphere significantly affects the skin friction coefficient. The budgets of the turbulent kinetic energy show that turbulence production and viscous dissipation are augmented due to the formation of small-scale vortices and interactions among them in the near-wake region, especially at G/D=0.1.

The particle image velocimetry (PIV) and hydrogen-bubble visualization technique are used to investigate the flat-plate boundary layer transition induced by an array of roughness elements. The streamwise evolutions of the mean and fluctuation velocity are analyzed, and the critical Reynolds number Rek,c is determined between 339 and 443 under current experimental setup. The hairpin vortices shed from supercritical roughness elements are visualized by swirling strength, in which two pairs of counter-rotating vortices can be observed: one vortex pair is the manifestation of the neck of the hairpin vortices shed from the top of the roughness; the other vortex pair, which originates from the lower part of the roughness, comes from the streamwise vortices that are constantly perturbed by the hairpin vortex shedding.

Detailed three-dimensional measurements of the first vortex pairing of a large plane mixing layer reveal excitation of several three-dimensional instability modes. Time evolution in three-dimensional space (x, y, z, t) shows how the two-dimensional rollers become three-dimensional as they approach each other and that the linear growth of at least two instability waves leads to a spanwise periodic pairing. The results are based on phase-locked measurements made in three-dimensional spatial grids, with a mesh spacing of 8.5% of the fundamental instability wavelength. Spanwise-uniform, periodic acoustic excitation stabilizes the most probable two-dimensional natural features – roll-up and first pairing. The second subharmonic is added to study the effect of alternate streamwise pairing locations on the three-dimensional characteristics of vortex pairing. Velocities are measured using hot-wire anemometry, and the coherent structures are reconstructed from the ensemble-averaged vorticity field.Vortex pairing is shown to initiate through local ‘bridging’ at the maxima of periodic spanwise undulations. The undulations result from linear amplification of various instability modes on pairing rollers having different strengths. Bridging results from the change of the relative phase between the spanwise undulations of the pairing rollers from in-phase (due to the initial translative mode) to out-of-phase (due to the amplification of bulging-like and non-axisymmetric modes). It is found that when pairing occurs sufficiently far upstream, only axisymmetric waves are amplified and the evolution results in axisymmetric merging. In contrast, when pairing occurs sufficiently far downstream, both axisymmetric and non-axisymmetric waves are amplified and the evolution results in non-axisymmetric merging.The results indicate that vortex pairing is accompanied by the counter-rotating pairs of secondary structures (‘streamwise vortices’ or ‘ribs’) located in the mixing-layer braids and residing in the valleys of the spanwise-roller waves. Time evolution of these secondary structures shows that they move in the transverse direction, following the rollers.

Experimental and computational study on the linear instability region downstream of a two-dimensional roughness element(a square rod)placed in a laminar boundary layer on a flat plate is described. The numerical solutions of the two-dimensional N-S equations were obtained by the MAC(marker-and-cell)method. Hot wire measurements were conducted in the boundary layer on a flat plate in a low-turbulence wind tunnel. A sinusoidal fluctuation, which grows exponentially, is observed behind the roughness element. The characteristics of sinusoidal fluctuation(linear region)depend on κ / δ*κ, where κ is the height of the roughness element and δ*κ is the displacement thickness at the roughness element. At κ / δ*κ=1.0, the predominant frequency of naturally excited sinusoidal fluctuation lies in the unstable zone, which is predicted using the theoretical calculation based on the Orr-Sommerfeld equation applied for Lin's profile. The response characteristics of the linear instability region to artificial external excitation were investigated in detail using sound as an exciting agent. Also, flow fields with forced time-dependent disturbances are studied by the numerical method. The effects of the artificial external excitation become maximum when the frequency of disturbance coincides with the natural frequency. The numerical results for the linear instability region(frequency, phase velocity and amplification rates)agree with the experimental measurements

Machine Learning (ML) is becoming increasingly popular in fluid dynamics. Powerful ML algorithms such as neural networks or ensemble methods are notoriously difficult to interpret. Here, we introduce the novel Shapley additive explanations (SHAP) algorithm (Lundberg & Lee, Advances in Neural Information Processing Systems, 2017, pp. 4765–4774), a game-theoretic approach that explains the output of a given ML model in the fluid dynamics context. We give a proof of concept concerning SHAP as an explainable artificial intelligence method providing useful and human-interpretable insight for fluid dynamics. To show that the feature importance ranking provided by SHAP can be interpreted physically, we first consider data from an established low-dimensional model based on the self-sustaining process (SSP) in wall-bounded shear flows, where each data feature has a clear physical and dynamical interpretation in terms of known representative features of the near-wall dynamics, i.e. streamwise vortices, streaks and linear streak instabilities. SHAP determines consistently that only the laminar profile, the streamwise vortex and a specific streak instability play a major role in the prediction. We demonstrate that the method can be applied to larger fluid dynamics datasets by a SHAP evaluation on plane Couette flow in a minimal flow unit focussing on the relevance of streaks and their instabilities for the prediction of relaminarisation events. Here, we find that the prediction is based on proxies for streak modulations corresponding to linear streak instabilities within the SSP. That is, the SHAP analysis suggests that the break-up of the self-sustaining cycle is connected with a suppression of streak instabilities.

Using direct numerical simulations of turbulent channel flow, we present new insight into the formation mechanism of near-wall longitudinal vortices. Instability of lifted, vortex-free low-speed streaks is shown to generate, upon nonlinear saturation, new streamwise vortices, which dominate near-wall turbulence production, drag, and heat transfer. The instability requires sufficiently strong streaks (y circulation per unit x > 7.6) and is inviscid in nature, despite the proximity of the no-slip wall. Streamwise vortex formation (collapse) is dominated by stretching, rather than rollup, of instability generated (ωx sheets. In turn, direct stretching results from the positive ∂u/∂x (i.e. positive VISA) associated with streak waviness in the (x,z) plane, generated upon finite-amplitude evolution of the sinuous instability mode. Significantly, the 3D features of the (instantaneous) instability-generated vortices agree well with the coherent structures educed (i.e. ensemble averaged) from fully turbulent flow, suggesting the prevalence of this instability mechanism. Fundamental differences in the regeneration dynamics of minimal channel and Couette flows are revealed regarding (nonlinear) streak instability, vortex formation and evolution, and wall shear behavior.KeywordsWall Shear StressCouette FlowVortex FormationStreamwise VortexVortex RegenerationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Using direct numerical simulations of turbulent channel flow, we present new insight into the formation mechanism of near-wall longitudinal vortices. Instability of lifted, vortex-free low-speed streaks is shown to generate, upon nonlinear saturation, new streamwise vortices, which dominate near-wall turbulence production, drag, and heat transfer. The instability requires sufficiently strong streaks (y circulation per unit x > 7.6) and is inviscid in nature, despite the proximity of the no-slip wall. Streamwise vortex formation (collapse) is dominated by stretching, rather than rollup, of instability generated (ωx sheets. In turn, direct stretching results from the positive ∂u/∂x (i.e. positive VISA) associated with streak waviness in the (x,z) plane, generated upon finite-amplitude evolution of the sinuous instability mode. Significantly, the 3D features of the (instantaneous) instability-generated vortices agree well with the coherent structures educed (i.e. ensemble averaged) from fully turbulent flow, suggesting the prevalence of this instability mechanism. Fundamental differences in the regeneration dynamics of minimal channel and Couette flows are revealed regarding (nonlinear) streak instability, vortex formation and evolution, and wall shear behavior.

Lateral growth of turbulent wedge is examined experimentally in a flat-plate boundary layer at low Reynolds numbers. Turbulent wedge is generated by exciting hairpin vortices through a small hole at a location where the Reynolds number is subcritical. The lateral growth of turbulent wedge occurs at the momentum thickness Reynolds numbers above 170, with successive growth of low-speed streaks in the ambient laminar flow region. Beyond the critical Reynolds number for the lateral turbulent contamination, the lateral spacing of low-speed streaks newly generated is 45 in wall units using the friction velocity of Blasius flow at the stage of their first appearance and increases to that of the developed wall turbulence as the instability and breakdown of low-speed streaks develop.

The three-dimensional behavior of flow transition in circular and 6-chevron jets at Re = 5000 is investigated with experiments conducted on a free water jet by time-resolved tomographic particle image velocimetry. The emphasis is on the unsteady organization of coherent flow structures, which play a role in the generation of acoustic noise. Shedding and pairing of vortices are the most pronounced phenomena observed in the near field of the circular jet. The first and second pairing amplify the axial pulsatile motion in the jet column and lead to the growth of azimuthal waves culminating in the breakup of the vortex ring. Streamwise vortices of axial and radial vorticity are observed in the outer region and move inward and outward under the effect of the vortex rings. In the jet with chevrons, the axisymmetric ring-like coherence of the circular jet is not encountered. Instead, streamwise flow structures of azimuthal vorticity emanate from the chevron apices, and counter-rotating streamwise vortices of axial and radial vorticity develop from the chevron notches. The decay of streamwise vortices is accompanied by the formation of C-shaped structures. The three-dimensional analysis allows quantifying the vortex stretching and tilting activity, which, for the circular jet exit, is related to the azimuthal instabilities and the streamwise vortices connecting the vortex rings. In the chevron jet, stretching and tilting peak during the formation of C-structures. Following Powell’s aeroacoustic analogy, the spatial distribution of the source term is mapped, evaluating the temporal derivative of the Lamb vector. The spatio-temporal evolution of such source term is visualized revealing that the events of highest activity are associated with the processes of vortex-ring pairing and vortex-ring disruption for the circular jet, and with the decay of streamwise instabilities and the formation of C-shaped structures for the chevron case.

Stereoscopic particle image velocimetry is used to investigate the effects of micro-ramp sub-boundary layer vortex generators, on an incident shock wave/boundary layer interaction at Mach 1.84. Single- and double-row arrangements of micro-ramps are considered. The micro-ramps have a height of 20% of the unperturbed boundary layer thickness and the measurement planes are located 0.1 and 0.6 boundary layer thicknesses from the wall. The micro-ramps generate packets of individual vortex pairs downstream of their vertices, which produce counter-rotating longitudinal streamwise vortex pairs in a time-averaged view. These structures induce a pronounced spanwise variation of the flow properties, namely the mixing across the boundary layer interface. The probability of reversed-flow occurrence is decreased by 20 and 30% for the single- and double-row configurations, respectively. Both configurations of micro-ramps stabilize the shock motion by reducing the length of its motion by about 20% in the lower measurement plane. The results are summarized by a conceptual model describing the boundary layer’s and interaction’s flow pattern under the effect of the micro-ramps.