Reynolds number influence on the formation of vortical structures on a pitching flat plate

Abstract

The impact of chord-based Reynolds number on the formation of leading-edge vortices (LEVs) on unsteady pitching flat plates is investigated. The influence of secondary flow structures on the shear layer feeding the LEV and the subsequent topological change at the leading edge as the result of viscous processes are demonstrated. Time-resolved velocity fields are measured using particle image velocimetry simultaneously in two fields of view to correlate local and global flow phenomena in order to identify unsteady boundary-layer separation and the subsequent flow structures. Finally, the Reynolds number is identified as a parameter that is responsible for the transition in mechanisms leading to LEV detachment from an aerofoil, as it determines the viscous response of the boundary layer in the vortex–wall interaction.

1. Introduction

Leading-edge vortices (LEVs) play an important role in unsteady aerodynamics. They increase the achievable lift beyond static conditions through the delayed stall effect [1,2]. In biological flapping flight or technical applications (e.g. micro air vehicles (MAVs)), these effects are used to generate the necessary lift, which, according to Jones et al. [3], would be insufficient using fixed wings at low Reynolds numbers. LEVs may also be used to maximize the efficiency of energy harvesting devices [4]. The range of Reynolds numbers, in which these unsteady effects are encountered and a strong impact on the aerodynamics is observed, reaches from very low Reynolds numbers (Re ≈ 10³ for insects and MAVs) to high Reynolds numbers (Re ≈ 10⁶ for helicopters). The existence of these vortices results in large moments at the root of wind turbine blades and may promote mechanical fatigue. According to Leishman [5], knowledge about unsteady aerodynamic loads is also crucial for the design of helicopter rotor blades. Many models to capture the effects of LEVs exist (e.g. [5,6]), but as they rely on empirically determined coefficients (stemming mostly from steady experiments) instead of physical mechanisms, their application is limited to very specific conditions.

The unsteady force acting due to the vortex evolution may be expressed by the change of vortical impulse, as formulated by Wu et al. [7]:

where ω is the vorticity field, S the area covered by the vortex, ρ the fluid density and the circulation . After differentiation , the vortical force is split into two terms. The first is proportional to the growth rate of the circulation , and the second one is proportional to the convection of the circulation . Jones & Babinsky [8] show that the overshoot in unsteady lift and drag persists as long as the LEV grows and is attached to the wing at a Reynolds number of 60 000. As soon as the LEV detaches and convects downstream, the force drops immediately. Therefore, the vortex detachment mechanisms determine the maximum achievable circulation, the achievable unsteady lift and the instant of force breakdown. To predict vortex dynamics and unsteady lift, these detachment mechanisms have to be understood. Crucial for the optimization of all these applications is the prediction of maximal vortex size and the timing of vortex detachment. The mechanisms limiting the LEV growth and causing its detachment are addressed in this study.

The most commonly used concept for the prediction of vortex growth in biological flight is the concept of optimal vortex formation, which was introduced by Gharib et al. [9]. A piston moves through a cylinder over a distance l with average velocity and pushes fluid through an orifice of diameter d into a plenum. The shear layers at the edges of the orifice roll up into a vortex ring. A non-dimensional number describing the vortex formation time was defined, in which t describes time. The termination of vortex growth occurred at T* ≈ 4 for a wide range of experimental parameters, indicating universal vortex behaviour. The limiting mechanism of the vortex growth was associated with topological changes, preventing more vorticity-carrying fluid from being entrained into the primary vortex ring. Dabiri [10] extended this concept to flapping flight, which, according to Taylor et al. [11] and Triantafyllou et al. [12], usually lies within the Strouhal number range and a reduced frequency of , where h is the plunging height of a moving aerofoil, U the free stream velocity, T the motion period of an unsteady aerofoil and c its chord length. In this flight regime, a formation number T* ≈ 4 is found for a wide range of biological swimmers and flappers.

Rival et al. [13,14] show that the chord length is an appropriate length scale for the limitation of the LEV circulation, at least for the typical flapping flight regime.

Nevertheless, the appropriateness of chord length as a characteristic length scale can be questioned, especially because the LEV growth is most immediately influenced by the feeding shear layer at the leading edge. A more encompassing interpretation has been introduced by Doligalski et al. [15], in which the interaction of the LEV with the viscous boundary layer on the plate is examined. In this study, two possibilities are identified by which LEV growth can be terminated. Both are schematically shown in figure 1, using the flow topology terms introduced in [17]: nodes are denoted by N, saddle points by S and half-saddles attached to a rigid wall by S′. A description of flow topology and a definition of the flow structures are provided in §2.1. The role of nodes and saddles in the LEV detachment process is described below. Figure 1a shows an attached LEV during its growth period; the vortex, with the centre N₁, is bound to the aerofoil surface by two saddle points.

Figure 1.

Topological representation of the leading edge vortex: N, node; S, saddle; S′, half-saddle; ω, vorticity. Adapted from Widmann & Tropea [16]. (a) Flow topology during the formation of the LEV. (b) LEV detachment according to the bluff body detachment mechanism, with a full saddle point and flow reversal at the trailing edge. (c) Flow topology after boundary-layer eruption.

The first of these two underlying LEV detachment mechanisms involves flow reversal at the trailing edge and is pictured in figure 1b. This mechanism is referred to as bluff body detachment, as a similar topology would result for a bluff body, e.g. a plate placed vertically in the flow. The interaction of initially separated shear layers of opposite sign vorticity at the leading and the trailing edge leads to the limitation of LEV growth. If the LEV grows sufficiently, then the rear half-saddle moves beyond the trailing edge and forms a full saddle S₁ in the wake and flow reversal at the trailing edge results. The LEV entrains a portion of flow carrying opposite signed vorticity (ω⁺) from the trailing edge and convects it towards the leading-edge shear layer ω⁻. Both layers interact, and the feeding of the LEV through the leading-edge shear layer is inhibited. Owing to the flow reversal at the trailing edge, another vortex is formed there. This mechanism is analogous to the vortex-shedding process behind bluff bodies, as described by Gerrard [18]. These observations suggest that the chord length is a universal length scale for LEV formation.

Although in biological flight the concept of universal vortex formation may be applied, there are experiments (e.g. [8,13,19,20]) where deviations from this concept are observed and the LEVs detach without flow reversal at the trailing edge. The second of the two principal mechanisms for inhibiting LEV growth accounts for such cases and is shown in figure 1c. This mechanism involves the viscous/inviscid interaction of the primary LEV with the boundary layer and a subsequent separation of the boundary layer, as explained by Doligalski et al. [15]. The separation (or boundary-layer eruption) occurs as a viscous response of the boundary layer to the externally applied pressure gradient by the LEV and is inherent to any vortex wall configuration. This eruption marks the onset of the formation of secondary vortical structures and causes a change of flow topology at the leading edge. Through the boundary-layer eruption, two secondary vortices (N₂, ω⁺ and N₃, ω⁻) and a full saddle point S₁ are formed. The saddle point redistributes the fluid emerging at the leading edge into the secondary vortex N₃; the primary vortex N₁ is cut off from its feeding layer and stops growing. The impact of these secondary vortical structures (which occur independently of the trailing edge) on the LEV detachment process is emphasized by Eslam Panah et al. [21], Wojcik & Buchholz [22] and Widmann & Tropea [16]. As this mechanism involves a viscous/inviscid interaction, it can be assumed that an enhancement of the viscous forces (controlled by a decrease of the Reynolds number) facilitates a transition between both detachment mechanisms.

The aim of this study is to investigate the hypothesis that the role of viscosity as an isolated parameter causes a transition between the two vortex detachment mechanisms. For Reynolds numbers larger than a certain value, the bluff body mechanism causes LEV detachment; for lower values, secondary structures appear independently of the trailing edge and cause the LEV to detach earlier at a lower non-dimensional circulation. It is not the goal to identify a threshold in the Reynolds number that distinguishes both vortex detachment mechanisms, as besides the viscosity, many effects, such as the aerofoil kinematics, the motion frequency, the aerofoil shape or three dimensionality, also have a strong influence on these mechanisms.

In this study, the flow around an aerodynamic profile pitching from a geometrical angle of attack of 0° to 30° and kept fixed at that end position is investigated by means of particle image velocimetry (PIV). A variation of the Reynolds number Re is conducted, while keeping all other parameters, such as reduced frequency k or Strouhal number St, constant, to distinguish regimes in which one or the other mechanism of detachment is dominant. The reduced frequency was fixed at k = 0.25; the Reynolds number was varied between Re = 10 000 and Re = 80 000. A reduced frequency of k = 0.25 was chosen, as it represents a typical number in unsteady flight and is sufficiently high for LEV formation. The Reynolds number range of Re = 10 000 to Re = 80 000 was chosen as the widest possible band in which all other non-dimensional parameters could be kept constant due to constraints of the experimental facility. The velocity fields are analysed by vortex identification methods to identify secondary flow structures and estimate their effect on the LEV circulation.

2. Experimental set-up

Time-resolved PIV is used to investigate the flow field generated by a pitching flat plate in the open return wind tunnel at the Technische Universität Darmstadt. A flat plate with sharp leading and trailing edges and a chord length of c = 120 mm and a thickness of d_p = 5 mm was used; the flow separation point was thereby fixed at the leading edge. Flat plates were chosen to exclude geometry-caused pressure gradients from affecting the results. The flat plate was pitched down from a geometric angle of attack of 0° to 30° during half a motion period T (with the leading edge as the pivoting point) and was then fixed at its bottom dead position to avoid upstroke effects in the formation of the LEV and the secondary structures. Linear actuators with a streamwise spacing of s = 80 mm are used to create the pitching motion. While the upstream actuator remains fixed, the downstream actuator executes a half-cycle of a sinusoidal vertical motion to lower the trailing edge and yield a geometric angle of attack. The geometric angle of attack variation over dimensionless time is shown in figure 2.

Figure 2.

Variation of geometric angle of attack over dimensionless time.

The free stream velocity U_∞ was adjusted in order to vary the Reynolds number between Re = 10 000 and Re = 80 000. The reduced frequency , based on a full motion period T, was kept constant by adjusting the pitch velocity. The value for the reduced frequency and the angle of attack history have been chosen to resemble the kinematics prevailing in efficient forward flight and to produce a distinct LEV. The experimental parameters of the present study are listed in table 1.

Table 1.

Experimental parameters of the four test cases.

test case	motion period T (ms)	freestream velocity
Re = 10 000	2 × 566	1.33
Re = 35 000	2 × 162	4.66
Re = 60 000	2 × 94	8.00
Re = 80 000	2 × 71	10.66

Linear actuators with a streamwise spacing of s = 80 mm are used to create the pitching motion. While the upstream actuator remains fixed, the downstream actuator executes a half-cycle of a sinusoidal vertical motion to lower the trailing edge and yield a geometric angle of attack. The set-up in the test section is shown in figure 3.

Figure 3.

Schematic view of the experimental set-up using two cameras to observe the global flow field (FOV 1) and the flow field near the leading edge (FOV 2) simultaneously. The incoming flow is denoted by (a), the linear actuators below the test section are denoted by (b). The flat plate (c) is mounted on the actuators with connecting rods. The cameras (d) recording FOV 1 and FOV 2 are placed outside the wind tunnel, next to the mirror (e). The light sheet (green area) is introduced to the test section by the light sheet optics (f). The global field of view (FOV 1) is indicated by the solid black frame and denoted by (g); the local leading-edge field of view (FOV 2) is indicated by a dashed black frame and denoted by (h).

High-speed PIV measurements were conducted to obtain time-resolved flow fields at an acquisition rate of 1 or 2 kHz. Two Phantom v12 (12 bit monochrome image sensor, 1200 × 800 pixels resolution) cameras have been triggered to simultaneously capture two different fields of view. A Litron LDY-303 laser with a wavelength of λ = 527 nm was used to illuminate the test section. Light sheet optics was set downstream of the aerofoil. Optical filters with the corresponding wavelength were used to reduce background noise. The output energy was 18 mJ for an acquisition frequency of 2 kHz. Diethylhexyl sebacate (DEHS) was used as seeding particles and introduced to the settling chamber. The diameter of these particles is specified by the manufacturer with d_DEHS = 1 µm; its density is ρ_DEHS = 910 kg m⁻³. The raw images were correlated with Dynamic studio 3.4 using a multi-grid approach with a stepwise refinement of the interrogation area size from 128 × 128 to 16 × 16 pixels and a 50% overlap. Velocity vectors with a difference six times larger than the local standard deviation were removed using a 3 × 3 neighbourhood filter. The overall LEV evolution is recorded in a global field of view (FOV 1); the secondary structures appear in a local field of view (FOV 2) near the leading edge. The different fields of view are schematically shown in figure 4. In order to prevent both cameras from concealing each other's field of view, a mirror has been mounted to observe FOV 2. A Nikon 60 mmf/2.8 lens was used to capture the flow in FOV 1. For FOV 2, a Nikon 110 mmf/2.8 lens was used. With these optical devices, FOV 1 covers an area of in streamwise and in vertical direction and FOV 2 covers an area of in streamwise and in vertical direction.

Figure 4.

Schematic view of the experimental set-up using two cameras to observe the global flow field (FOV 1) and the flow field near the leading edge (FOV 2) simultaneously. The right-hand side depicts the side view of the set-up, and the left-hand side shows the wind tunnel cross section in free stream direction. The flow topology near the leading edge including secondary flow structures and critical points is depicted schematically.

2.1. Methods

For each Reynolds number, five independent test runs were recorded. The results are presented by averaged values and error bars correspond to 1 s.d. The flow fields are shown for non-averaged time-resolved data.

In figure 1, the impact of secondary structures and the LEV detachment has been schematically illustrated. This section describes how different methods based on the flow topology are used to describe the boundary-layer separation due to the viscous/inviscid interaction and the subsequent topological changes.

2.1.1. Flow topology and vortex identification

Foss [17] connects the Euler characteristic of a flow domain with the flow features inside this domain. Each flow feature, such as vortices and stagnation points, may be represented by critical points (vortices as nodes N, stagnation points as saddle points S; saddle points directly at a solid wall are denoted as half-saddles S′). A strict topological constraint is that the sum of their Poincaré indices corresponds to the Euler characteristic. If a new flow feature forms, then a set of flow features must appear to fulfil these topological constraints. An example for the topological features occurring in the present set-up is given in figure 5.

Figure 5.

Topological features near the leading edge (FOV 2) after the boundary-layer eruption. The black contour lines mark the regions where the vortex identification method found vortical structures. Vortex centres correspond to the nodes N₁ and N₂. The saddle point S₁ illustrates the redistribution of vorticity-carrying fluid emerging from the shear layer. Individual vortical structures are marked by a black contour. Note that only every second velocity vector is shown to improve clarity.

The boundary-layer eruption results in the formation of a secondary vortex (figure 5: N₂ is of opposite rotation with respect to the primary LEV N₁) below the LEV. To balance this flow feature, another secondary vortex (figure 5: N₃ is of same rotation with respect to the LEV N₁) forms directly at the shear layer; a full saddle (S₂) and a set of half-saddles ( and ) appear. The full saddle redistributes the vorticity-carrying fluid emerging from the shear layer. The shear layer does not feed the LEV exclusively into the primary LEV, but into the secondary vortex, until the primary LEV stops growing. The interaction of the LEV feeding shear layer with secondary structures is amplified, and LEV detachment is accelerated for lower Reynolds numbers with a higher impact of viscosity.

From the velocity fields obtained in both fields of view, different quantities are extracted to characterize the flow. For each velocity field, the vortex identification method proposed by Graftieaux et al. [23] has been performed; the vortex centres and the rear reattachment point of the LEV have been tracked automatically.

The integral vortex identification method uses the measured velocity field data to obtain two scalar fields Γ₁ and Γ₂, which characterize vortices. Γ₁ and Γ₂ define the circular motion around discrete control points in the flow field.

At control points, where these scalars exceed the thresholds of |Γ₁| > 0.9 a vortex centre is identified and where |Γ₂| > 0.6 this point is defined as part of a vortex. Note that Γ₂ is invariant of the flow velocity. For Γ₁, a laboratory frame is used to ensure the conformity of the vortex centre with a critical point, which by definition exhibits zero flow velocity. The mathematical definition of both scalars is provided in formula (2.1); figure 6 illustrates the method further. S denotes the integration area with centre point P including N control points M. PM is the distance between control point and centre point. The average of the angle θ_M between the velocity U_M at M and PM gives Γ₁, which represents the circular fluid motion around P. Thus, the value of N corresponds to a spatial filter.

Figure 6.

Schematic of the vortex identification method by Graftieaux et al. [23] (adapted from [24]).

The difference between the calculation of Γ₁ and Γ₂ is that instead of the absolute flow velocity U_M the velocity is used. The calculated angle is, therefore, determined by velocity relative to the averaged convection speed in S, which is given by .

2.1

In FOV 1, the global flow features are extracted, while in FOV 2, the topology near the leading edge can be investigated due to the higher magnification. Therefore, the LEV circulation, the TEV (trailing edge vortex) circulation, the overall circulation and the position of the rear reattachment point are calculated from FOV 1, while the secondary vortex circulation and the leading-edge topology are obtained from FOV 2.

To obtain the position of the rear stagnation point behind the LEV, critical points in the flow field are calculated according to the method introduced by Cormier et al. [25]. The rear stagnation point X_RSP is defined as the critical point with a Poincaré Index of –1, which is positioned farthest downstream.

Coherent regions which exceed the threshold for Γ₂ = 0.6, as suggested by Graftieaux et al. [23], are identified as vortical structures. The primary LEV is identified as the vortical structure in FOV 1 carrying the highest negative circulation. Its centre in chordwise direction x_LEV and in wall normal direction y_LEV is defined as the spatial point with the highest value Γ₁. The TEV is represented by the vortical structure carrying the highest amount of positive circulation in FOV 1. The overall negative circulation Γ⁻ is calculated by integrating all negative vorticity contained in vortical structures. The eruption vortex is calculated by integrating all positive vorticity contained in vortical structures in FOV 2. The occurrences of the secondary structures (secondary LEV and eruption vortex) are connected to the flow topology by the critical points. The vortex centres are represented by nodes from a topological point of view. The situation is schematically depicted in figure 5. The individual vortical structures are marked by the black contour.

If the boundary-layer eruption causes a vortex to form (node N₂) by separation of the boundary layer from the aerofoil (half-saddle S′₃), a saddle point (S₁) must necessarily arise. To fulfil the topological constraints, the shear layer must roll up into a secondary vortex (node N₁). Therefore, the formation of the eruption vortex and the secondary vortex indicate a topological change near the leading edge. Through the formation of the secondary LEV, the saddle point redistributes the fluid emerging from the shear layer. The primary LEV is no longer fed, rather the shear layer feeds the secondary LEV, which in turn starts growing and thereby limits the primary LEV size.

LEV detachment is defined according to Fage & Johansen [26] as the instant in the stroke cycle when the vortex circulation reaches a peak value.

The quantities Γ^LEV, Γ⁻, Γ^TEV, Γ^Eruption (N₂ in figure 5), x_LEV, y_LEV and x_RSP are recorded for each run and then averaged over all five runs.

To determine whether the advancement of the rear reattachment point is a necessary condition for the formation of the secondary vortical structures leading to the separation of the LEV from its feeding shear layer, several quantities have been measured and correlated. The overall circulation Γ⁻ and the LEV circulation Γ^LEV have been measured for each Reynolds number. To verify the hypothesis made in the introduction (§1) that the role of viscosity as an isolated parameter causes a transition between the two vortex detachment mechanisms, the following approach was chosen: the coincidence of distinct points in Γ^LEV and Γ⁻ with the formation of an eruption vortex of the circulation Γ^Eruption and a secondary vortex of the circulation Γ^secLEV have to be compared. The instant in the motion cycle when the rear reattachment point x_RP reaches the trailing edge has been marked in all following diagrams. It shows whether the formation of certain flow structures is a result of the reversed flow at the trailing edge (due to ). The maximal standard deviation for the instant in time when the rear half-saddle exceeds the trailing edge is 5.9% of the aerofoil chord length.

3. Results

The results of the LEV evolution and detachment for four different Reynolds numbers are presented in the following sections. The first two sections show the LEV development (figure 7) in the form of time-resolved velocity data for the global (FOV 1) and the local (FOV 2) field of view as normalized vorticity contours with overlaid streamlines. Positive vorticity is red; negative vorticity is blue.

Figure 7.

Time-resolved global flow field (FOV 1) development in terms of normalized vorticity and streamlines for each experimental case at . (a) Re = 10 000, (b) Re = 35 000, (c) Re = 60 000 and (d) Re = 80 000.

It is hypothesized that viscous effects are pronounced at lower Reynolds numbers; therefore, the development of viscosity-driven flow structures such as the boundary-layer eruption is also pronounced. Larger and further developed secondary structures are expected at the leading edge. For large Reynolds numbers, a boundary-layer separation and a subsequent topological change with the LEV feeding shear layer are not expected. Their occurrence may be attributed to the flow reversal at the trailing edge and the upstream transport of fluid to the leading edge.

3.1. Global flow evolution

The time-resolved global flow fields (FOV 1) at the four different Reynolds numbers are shown in figure 7. The evolution of an LEV beginning with the shear layer separation (early in the motion cycle) until LEV detachment is shown. It is apparent that the LEV evolution, including boundary-layer separation and vortex roll-up, is shifted towards earlier times in the motion cycle for the lowest Reynolds number. At higher Reynolds number (Re > 35 000), the LEV evolution in each case is synchronous with respect to the motion period T. The LEV detaches from the plate surface much earlier in the motion cycle t/T, while for Re > 35 000, the LEV still continues to grow.

3.2. Local flow evolution

A detailed view of the leading-edge region for the same instants in the motion period is shown in figure 8. In FOV 2, the evolution of secondary structures can be identified. Data for FOV 2 show that the secondary structures due to topological changes at the leading edge appear earlier and are much more pronounced the lower the Reynolds number is. Distinct secondary vortices develop at the leading edge for Re = 10 000, already while the LEV is still growing and prior to LEV detachment and flow reversal at the trailing edge. For higher Reynolds numbers than Re = 10 000, a layer of opposite signed vorticity with respect to the LEV forms between the primary vortex and the aerofoil surface. But in contrast with the lowest Reynolds number, no distinct secondary vortical features near the leading edge have developed.

Figure 8.

Time-resolved development of the flow field in the leading-edge region (FOV 2) in terms of normalized vorticity and velocity vectors for each experimental case at . (a) Re = 10 000, (b) Re = 35 000, (c) Re = 60 000 and (d) Re = 80 000.

3.3. Derived quantities

To check the validity of the hypotheses made in §1, several quantities are derived from the velocity fields. The normalized quantities of the LEV circulation , the TEV circulation , the eruption vortex circulation and the LEV trajectory are shown in figures 9–12 for the Reynolds numbers 10 000, 35 000, 60 000 and 80 000.

Figure 9.

Temporal evolution of the averaged LEV circulation for all investigated Reynolds numbers. One standard deviation computed from five experimental runs is given as an error bar. The second peak for Re = 10 000 indicates the growth of a second LEV during 0 < t/T < 0.75; the vertical dashed lines indicate the onset of flow reversal at the trailing edge. It should be noted that the circulation increase towards the end for the Re = 10 000 case is attributed to the formation of a subsequent LEV.

Figure 10.

Temporal evolution of the averaged normalized vortex centre in streamwise direction . The vertical dashed lines indicate the onset of flow reversal at the trailing edge.

Figure 11.

Temporal evolution of the averaged TEV circulation for all investigated Reynolds numbers. One standard deviation computed from five experimental runs is given as an error bar. The vertical dashed lines indicate the onset of flow reversal at the trailing edge.

Figure 12.

Temporal evolution of the averaged eruption vortex circulation for all investigated Reynolds numbers. One standard deviation computed from five experimental runs is given as an error bar. The vertical dashed lines indicate the onset of flow reversal at the trailing edge.

Figure 9 shows the temporal evolution of the normalized LEV circulation ; the vertical dashed lines indicate when the rear reattachment point exceeds the trailing edge and flow reversal is initiated. While the normalized LEV circulation curves nearly collapse for Re ≥ 35 000 and exhibit similar peak values and detachment times around t/T = 0.45, the evolution and the detachment of the LEV are shifted towards earlier stages and lower peak circulation values for Re = 10 000. This difference may be explained by the formation of secondary structures arising from the boundary-layer eruption.

In figure 10, the normalized position of the vortex centre in the streamwise direction is plotted, together with the onset of flow reversal (marked by dashed vertical lines). The normalized LEV circulation is added as dash-dotted lines to accentuate the temporal relation with the streamwise position of the vortex centre. It can be seen that as long as the vortex grows and has not reached its peak circulation, the vortex centre remains close to the leading edge. As the LEV circulation nears its peak value, the vortex centre begins to move rapidly downstream. The LEV detaches and begins to convect away at a high normalized convection velocity of for the respective Reynolds numbers after the peak circulation has been reached. While the LEV centre acceleration occurs after flow reversal at the trailing edge for higher Reynolds numbers, it is independent of the flow reversal for Re = 10 000.

The TEV grows as a result of the flow reversal at the trailing edge; its normalized circulation is shown in figure 11 over the motion cycle t/T. The normalized peak circulation decreases with increasing Reynolds number. During a period of , the LEV circulation and the TEV circulation increase simultaneously, which indicates that the LEV detachment follows the flow reversal at the trailing edge.

The formation of secondary vortical structures, which is initiated by either a viscous vortex–wall interaction (viscous mechanism) or the transport of fluid with vorticity-carrying fluid from the trailing edge to the leading edge (analogous to bluff body vortex shedding) or their interplay, leads to the formation of a secondary vortex (here, eruption vortex) with rotation opposite to that of the primary LEV rotation. This vortex corresponds to the node N₂ in figure 5. This topological change eventually results in the separation of an LEV from its feeding layer. To indicate the onset of this topological change, the evolution of the eruption vortex has been measured and correlated with the events occurring in the LEV evolution history.

Figure 12 shows the normalized circulation of the eruption vortex . While such a vortex can be identified already before fluid from the trailing edge can be transported upstream at low Reynolds numbers (Re = 10 000 and Re = 35 000), for higher Reynolds numbers this vortex is identified after flow reversal . The peak value of decreases with increasing Reynolds numbers and is shifted towards later stages in the motion period. It is noteworthy that the peaks of Γ^ErupcU_∞ and coincide for lower Reynolds numbers, while they occur only after the LEV has separated from its shear layer at higher Reynolds numbers. The peaks may rather be a result of LEV detachment than its cause. The role of the viscosity is made clear: for low Reynolds numbers the eruption vortex is a result of viscous effects, for higher Reynolds numbers a result of reversed flow at the trailing edge. The role of viscosity diminishes at higher Reynolds numbers.

4. Discussion

Section 3 shows that viscous processes may have a significant influence on the formation and detachment of LEVs for low Reynolds numbers. For a reduced frequency of k = 0.25 and high Reynolds numbers, the vortex formation time agrees with the value predicted by the concept of optimal vortex formation. The results agree with Jones & Babinsky [27], who investigate the LEV evolution of accelerated aerofoils at a constant angle of attack in a water tunnel at different Reynolds numbers. They find an accelerated LEV evolution for lower Reynolds numbers and a collapse of normalized circulation at higher Reynolds numbers.

Both LEV detachment mechanisms (bluff body and boundary-layer eruption) introduced in §1 can lead to topological changes and the subsequent development of secondary vortical structures at the leading edge. If the Reynolds number is decreased and viscous effects become more dominant, the boundary-layer eruption altering the leading-edge topology can occur decoupled from the inviscid global flow topology.

Because the peak LEV circulation coincides with an acceleration in its trajectory, agreement is found between the definition of LEV detachment provided by Fage & Johansen [26] and the experimental observations.

These observations confirm some points related to the hypotheses made in §1 that the role of viscosity as an isolated parameter causes a transition between the two vortex detachment mechanisms, summarized as follows:

• — As the effect of viscosity is increased with lower Reynolds numbers, the viscous response of the boundary layer and the subsequent formation of secondary structures occur earlier in the stroke cycle. The circulation of these secondary structures is stronger.

• — For higher Reynolds numbers, these secondary structures occur later in the stroke and only after a result of the flow reversal, with the chord length being characteristic for that process.

• — For sufficiently high Reynolds numbers, the normalized circulation remains unchanged, because due to hampered viscous effects the LEV detaches as a result of the flow reversal at the trailing edge. The chord length is the characteristic length scale and leads to a common normalized LEV circulation.

• — For low Reynolds numbers, LEV detachment occurs earlier in the stroke cycle due to viscous effects, without flow reversal at the trailing edge. The LEV growth is diminished and the normalized circulation is reduced; the chord length cannot be considered a characteristic length scale.

Therefore, the Reynolds number may facilitate a transition in the LEV detachment mechanism, as proposed in the hypotheses given in §1 that the role of viscosity as an isolated parameter causes a transition between the two vortex detachment mechanisms. The topological changes leading to LEV detachment are not necessarily coupled with the inviscid global flow topology and, therefore, the flow reversal at the trailing edge is not a universal LEV mechanism in the process of limiting LEV growth.

With a changing Reynolds number, the properties of the LEV feeding shear layer also change. For a decreasing Reynolds number, the shear layer thickness decreases; for increasing Reynolds number, the shear layer thickness decreases and converges towards a common value. The shear layer thickness determines the distribution of vorticity in the primary LEV. For LEVs at higher Reynolds numbers, the vorticity distribution is homogeneous, due to the thin shear layer, while the inhomogeneous vorticity distribution for LEVs at low Reynolds numbers causes the boundary-layer eruption. This leads to a premature LEV detachment and lower peak LEV circulation values. For higher Reynolds numbers, these effects are negligible, as the global flow topology, in particular the position of the rear reattachment point, determines the onset of LEV detachment and peak values of the eruption vortex circulation are reached. The LEV circulation has already reached its maximum.

5. Conclusion and outlook

The separation of an LEV from its feeding shear layer and the subsequent detachment from an aerofoil can be attributed to topological changes near the leading edge. A saddle point redistributes the vorticity-carrying fluid emerging from the shear layer into a secondary vortex, whose growth halts the increase in circulation of the primary LEV. Depending on the role of viscosity (and therefore Reynolds number), this topological change can be provoked by two independent and different mechanisms. If the influence of viscosity is high (the Reynolds number is sufficiently low), then the boundary layer below the LEV erupts and a secondary vortex appears (in topological terms: a new node). If the influence of the viscosity is low, then no viscous response can be observed; the node altering the flow topology develops by the accumulation of fluid transported upstream from the trailing edge. This mechanism depends on the global inviscid flow topology and can only be observed if the rear reattachment of the LEV moves beyond the trailing edge and enables reverse flow. This mechanism corresponds to the events leading to bluff body vortex shedding. Boundary-layer eruption and reversed flow are not coupled, which means that the formation of secondary structures (leading to LEV detachment) is not attributed to the transport of fluid from the trailing edge upstream to the leading edge, but relies on local viscous mechanisms. For the cases when LEV detachment occurs independently of the position of the rear stagnation point, the chord length is not characteristic for the LEV behaviour and it cannot be universally applied.

In the investigated cases, the transition between both LEV detachment mechanisms occurs in the range 10 000 < Re < 35 000. However, a fixed transition boundary between both experiments is not expected in the range 10 000 < Re < 35 000. In the transition range, the simultaneous occurrence of the boundary-layer eruption and the flow reversal at the trailing edge leads to a mixture of both mechanisms.

A second effect is the accelerated vortex dynamics (with respect to the aerofoil kinematics) at lower Reynolds numbers. The flow separation and the subsequent LEV roll-up at the trailing edge are shifted to earlier motion stages with sufficiently low Reynolds numbers. Therefore, besides the transition in the detachment mechanism, the vortex evolution stages are shifted towards lower angles of attack.

As a result, in the range 10 000 < Re < 35 000, an interplay of various effects can be attributed to the change in viscosity, not only a transition in LEV detachment mechanisms. To clarify the influence of the shifted vortex dynamics on LEV detachment, experiments at Reynolds numbers below Re = 10 000 are necessary. To narrow down the range in which the transition of LEV detachment mechanisms occurs for the given kinematics, more measurements need to be performed in the range where a mixture of both detachment mechanisms is expected. The experiments have been carried out under two-dimensional conditions; no spanwise flow was present over the aerofoil. It is unclear how the boundary-layer separation and the viscous effects behave under three-dimensional conditions with strong spanwise flow or tip vortices, as the pressure and the circulation distribution along the span are altered by these effects. For instance, the spanwise flow may balance the pressure gradient imposed on the boundary layer and inhibit the formation of secondary structures. If so, the flow reversal at the trailing edge as a limiting mechanism would extend to lower Reynolds number regions.

These observations directly suggest a way to manipulate the LEV detachment behaviour in order to delay its separation from its shear layer, i.e. prolong its growth to higher circulation values and achieve higher lift values. The development of the secondary structures has to be suppressed for such a manipulation. As the critical points determining the flow topology have low velocities by definition, it can be expected that already small body forces may be sufficient to achieve this goal, if the force is introduced at appropriate locations of low flow velocities and over appropriate times. Gursul et al. [28] apply devices for the manipulation of secondary structures (suction and blowing of air or plasma actuators) near the leading edge of a non-slender delta wing in order to stabilize the main LEV and increase lift. Plasma actuators may be a useful tool to suppress the boundary-layer separation leading to the eruption, as demonstrated in numerous flow control experiments summarized by Gursul et al. [28]. Little et al. [29], for example, use plasma actuators for the control of trailing edge separation; Visbal & Gaitonde [30] delay the vortex breakdown on delta wings using plasma actuators. Depending on the detachment mechanisms, the actuator may either be placed at the trailing edge to prevent flow reversal or near the leading edge to prevent the boundary-layer eruption. For the latter case, the appropriate position may be a function of the Reynolds number, as the eruption is caused by viscous effects. A first investigation exploring this possibility has been presented in [31]. The present experiment shows how it is possible to enhance LEV growth with only small power input into the flow, if suitable locations are known. The manipulation of LEV growth, therefore, requires detailed knowledge about the detachment mechanisms.

Authors' contributions

This work was performed in the framework of A.W.'s PhD. C.T. assisted in formulating the manuscript.

Competing interests

We declare we have no competing interests.

Funding

The authors acknowledge funding from the Deutsche Forschungsgemeinschaft under grant no. TR 194/52.