4.1 Initial condition at $\unicode[STIX]{x1D6FC}=8^{\circ }$

To begin, a statistically stationary solution at $\unicode[STIX]{x1D6FC}=8^{\circ }$ is computed. The grid is oriented such that the airfoil is horizontal (as shown in figure 1 b) with the initial $\unicode[STIX]{x1D6FC}$ being set in the free stream. At this condition, the airfoil boundary layer is characterised by a small LSB near the leading edge that is responsible for the transition of the upper surface boundary layer to turbulence. The signature of the bubble is shown in figure 2 in both the pressure and skin-friction coefficients. As can be inferred from the inset in figure 2, the boundary layer separates at $x/c=0.018$ and reattaches at $x/c=0.05$ , giving a bubble length (in surface distance) of $L/c=0.034$ . The pressure surface boundary layer is laminar from the stagnation point to the trailing edge.

Figure 2.

Time-mean wall pressure coefficient for the static solution at $\unicode[STIX]{x1D6FC}=8^{\circ }$ . The inset shows the wall skin-friction coefficient in the vicinity of the laminar separation bubble.

This condition of $\unicode[STIX]{x1D6FC}=8^{\circ }$ is chosen to minimise computational requirements for the simulation of the onset of dynamic stall at computationally demanding Reynolds numbers of $O(10^{6})$ . The angle is high enough to ensure a laminar boundary layer on the pressure surface for the entirety of the computation, but low enough such that the constant pitch rate can be achieved well below the point at which the airfoil pitches above the static stall angle, $\unicode[STIX]{x1D6FC}=12^{\circ }{-}14^{\circ }$ (Critzos, Heyson & Boswinkle Jr Reference Critzos, Heyson and Boswinkle1955; Sheldahl & Klimas Reference Sheldahl and Klimas1981).

4.2 Overview of dynamic stall development

A general overview of the dynamic stall process is first given before a more in-depth discussion of the specific boundary-layer and vortex interactions are presented. Beginning at $\unicode[STIX]{x1D6FC}=8^{\circ }$ , a short acceleration period is specified. This transitions the airfoil motion to the constant pitch rate of $\unicode[STIX]{x1D6FA}_{0}=0.05$ , which is then maintained to the point at which the simulation is terminated. All visualisations of the flow solution are presented with the grid pitched relative to the initial condition of $\unicode[STIX]{x1D6FC}=8^{\circ }$ .

The force history for the manoeuvre is shown in figure 3 with shaded regions highlighting the various stages of boundary-layer and vortex development that describe the onset of dynamic stall. Each of these stages is described in detail in the following sections. Many of the trends in the force history are classic in dynamic stall research namely the high normal force coefficient ( $C_{n}$ ) above the static stall angle with a peak that corresponds to vortex development. Note that when the simulation is terminated ( $\unicode[STIX]{x1D6FC}\approx 30^{\circ }$ ), the DSV has propagated to $x/c=0.6$ and $C_{n}$ would be expected to drop suddenly once the vortex has convected past the trailing edge. At approximately $\unicode[STIX]{x1D6FC}=20^{\circ }$ there is a sudden rise in the drag coefficient ( $C_{d}$ ) as well as a roll off in the quarter-chord moment coefficient ( $C_{m,c/4}$ ), both of which are due to the effective thickening of the airfoil upper surface due to the reversed flow within the turbulent boundary layer. A brief reversal of the trend in moment coefficient is noted at the point of vortex development. This transitions to a rapid increase in the negative moment once the DSV begins to convect downstream.

Figure 3.

Force coefficients as a function of $\unicode[STIX]{x1D6FC}$ showing the dynamic stall process.

For $Re_{c}=1.0\times 10^{6}$ the stages of the onset of dynamic stall listed and highlighted in figure 3 are visualised in figures 4(c) and 5. In figure 4, a spatio-temporal contour of skin-friction coefficient ( $C_{f}$ ) illustrates the contraction of the small region of reverse wall shear associated with the leading-edge LSB, which occurs for each $Re_{c}$ . At $\unicode[STIX]{x1D6FC}\approx 15^{\circ }$ , turbulent separation (TS) appears near the trailing edge. This is marked with a dashed line and is shown to propagate upstream. Figure 4(a,b) show similar LSB contraction and upstream TS propagation at the lower Reynolds numbers of $Re_{c}=0.2\times 10^{6}$ and $0.5\times 10^{6}$ . At each lower $Re_{c}$ shown here, dynamic stall develops from the bursting of the leading-edge LSB (Visbal Reference Visbal2014b ; Visbal & Garmann Reference Visbal and Garmann2018). It should be noted that these data at lower $Re_{c}$ are from new computations beginning with $\unicode[STIX]{x1D6FC}_{0}=8^{\circ }$ to allow for direct comparison to $Re_{c}=1\times 10^{6}$ . Comparison with the previous results (which began at $\unicode[STIX]{x1D6FC}_{0}=4^{\circ }$ ) from Visbal (Reference Visbal2014b ) and Visbal & Garmann (Reference Visbal and Garmann2018) are excellent. With increasing $Re_{c}$ the LSB is more stable to the bursting process which allows for further upstream propagation of TS before the DSV develops.

The case of $Re_{c}=1.0\times 10^{6}$ represents a transition to a new form of dynamic stall within which the LSB exists but is stable to the bursting process and TS is able to propagate upstream to the LSB. The development of the vortex structures from these boundary-layer interactions is illustrated using select instantaneous mid-span contours of entropy in figure 5 (a corresponding animation is made available in the supplementary material at https://doi.org/10.1017/jfm.2018.939). At $\unicode[STIX]{x1D6FC}=24^{\circ }$ the TS point has propagated upstream and meets the turbulent reattachment point of the LSB (event 1 in figure 5). This causes a bursting event that results in the development of a small leading-edge vortex (LEV) structure. Meanwhile, the turbulent boundary layer is separated over nearly 98 % of the airfoil suction surface and has begun to roll up to form a second vortex structure termed the turbulent-separation vortex (TSV) in this work (event 2). These structures merge as the LEV wraps up the TSV, pulling it slightly upstream (event 3). This final vortex structure is referred to as the DSV, in the classical sense used in the literature. The DSV then propagates downstream as is typically observed. After formation of the DSV, the simulation is terminated as it is known that the interaction between the DSV and the trailing edge, is highly dependent on full wing geometry and longer-wavelength instabilities not captured in the current spanwise-periodic slice.

Figure 4.

Spatio-temporal ( $x{-}\unicode[STIX]{x1D6FC}$ ) contours of span-averaged wall skin-friction coefficient on the airfoil suction surface. Data at the lower Reynolds numbers of $Re_{c}=0.2\times 10^{6}$ (a) and $0.5\times 10^{6}$  (b) are included for comparison.

Figure 5.

Instantaneous contours of entropy at the mid-span showing overall process of dynamic stall. Numbers refer to the list of events in figure 3. An animation of this contour is included in the supplementary material.

The following sections describe these events in further detail with an emphasis on the unsteady boundary-layer mechanics that result in the onset of dynamic stall. At $Re_{c}$ of $O(10^{6})$ the boundary layer becomes increasingly difficult to resolve experimentally due to its decreasing thickness and the unsteady motion of the airfoil surface. This indicates that high-fidelity computations, such as the current study, are particularly well suited to study these events.

4.3 Laminar separation bubble development

A primary conclusion of the current study is that, despite a relatively high airfoil $Re_{c}$ and a very small LSB, the LSB is crucial in the overall development of the DSV and ultimately sets the inception point of the DSV. Accurately accounting for the effect of the LSB is key to locating the DSV and predicting the pressure loads during the unsteady stall process.

As described for the static condition at $\unicode[STIX]{x1D6FC}=8^{\circ }$ the LSB is located near the leading edge with an initial length of $L/c=0.034$ . As the airfoil pitches up, the flow acceleration around the leading edge increases, resulting in a higher peak suction as well as a stronger adverse pressure gradient across the LSB. Plots of LSB length and suction-surface minimum pressure coefficient ( $C_{p,min}$ ) are included in figure 6. For reference, similar data at lower $Re_{c}$ are included. At each of the lower $Re_{c}$ a sharp increase in the LSB length occurs associated with the onset of pressure-gradient-induced LSB bursting. This does not occur at $Re_{c}=1.0\times 10^{6}$ , where the LSB breaks down due to interaction with the turbulent separation. With increasing $Re_{c}$ there is a strong reduction in the LSB length, corresponding to the strong sensitivity of the dynamic stall process to $Re_{c}$ within the range presented here. For $\unicode[STIX]{x1D6FC}=8^{\circ }$ the length of the LSB as a function of $Re_{c}$ is shown to be proportional to $Re_{c}^{-1/2}$ as might be expected. This suggests (preliminary computations at $Re_{c}=1.5\times 10^{6}$ confirm this) that the sensitivity of the dynamic stall process for  $Re_{c}>1.0\times 10^{6}$ is likely reduced as suggested by Carr et al. (Reference Carr, McAlister and McCroskey1977) and Leishman (Reference Leishman1990).

Figure 6.

LSB length (a) and span-averaged minimum suction surface $C_{p}$  (b) each as a function of  $\unicode[STIX]{x1D6FC}$ . Data at lower $Re_{c}$ are included for comparison.

Before vortex development, the leading-edge flow structure is characterised by a contraction of the LSB from $L/c=0.034$ to 0.02. Simultaneously, peak suction increases from $C_{p}=-4$ to $-15$ suggesting flow acceleration up to four times $U_{\infty }$ resulting in a local Mach number as high as $M_{\infty }=0.5$ , despite the low free-stream Mach number of 0.1. The fact that under this high adverse pressure gradient, no pressure-gradient-induced bursting is observed is an interesting conclusion. Preliminary attempts to apply bursting criteria from the literature (Tani Reference Tani1964; Diwan, Chetan & Ramesh Reference Diwan, Chetan, Ramesh and Govindarajan2006) were made, but with little success. It is likely that a successful bursting criteria in the current situation must take into account the effects of the unsteady motion. This is an area requiring further study.

4.4 Trailing-edge separation movement

For $Re_{c}=1.0\times 10^{6}$ , turbulent separation begins to appear at the trailing edge as the airfoil pitches past $\unicode[STIX]{x1D6FC}\approx 15^{\circ }$ . Turbulent separation is marked by a black dashed line in figure 4 with a grey shaded region that represents two standard deviations of the recorded instances of zero wall shear. This relative unsteadiness of the separation location decreases as the TS point moves upstream with increasing  $\unicode[STIX]{x1D6FC}$ . After appearing at the trailing edge, the TS propagates upstream at a rate of $0.3U_{\infty }$ with a slight acceleration to $0.5U_{\infty }$ by the time separation reaches mid-chord. After this point, upstream propagation decelerates and continues at a mostly constant rate of $0.4U_{\infty }$ toward the leading edge. From approximately $x/c=0.06$ to the reattachment point of the LSB the TS movement accelerates rapidly, as shown in the inset figure in figure 4(c) and discussed further in the next section. This is the precursor to LEV development.

Figure 7.

Turbulent-separation location (a) and pitch rate comparison against Carr et al. (Reference Carr, McAlister and McCroskey1977) (b) each as a function of  $\unicode[STIX]{x1D6FC}$ . Data at lower $Re_{c}$ are included for comparison.

As an implicit validation of the current simulation, as well as a demonstration of the applicability of the current ramp-type motion to understanding sinusoidal pitching, the location of the TS is compared against data presented by Carr et al. (Reference Carr, McAlister and McCroskey1977) at the same $Re_{c}$ . This comparison is made as a function of $\unicode[STIX]{x1D6FC}$ in both figures 4(c) and 7(a). A comparison of the pitch rate as a function of $\unicode[STIX]{x1D6FC}$ is included in figure 7(b), where a thin dashed line shows the sinusoidal motion prescribed experimentally and black squares highlight the points in the motion where the location of the TS had been recorded. The comparison between the two motions is facilitated by the generally similar pitch rates (within the range $10^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 20^{\circ }$ ) and the observation made experimentally (Carr Reference Carr1988) that favourable comparisons of dynamic stall behaviour are observed when the pitch rate at the static stall angle is consistent. Given the highly unsteady location of the TS and the somewhat large experimental uncertainty documented by Carr et al. (Reference Carr, McAlister and McCroskey1977) this agreement is excellent. This comparison suggests that the turbulent boundary layer is properly resolved in order to predict the large momentum containing structures that determine averaged features such as turbulent separation.

4.5 Bursting due to LSB/TS interaction

Figure 8.

Regions of reverse flow ( $U_{x}<0$ , black) in the airfoil frame of reference for select instances leading to LEV development. Note in the expanded figures on the left, the $y$ -direction is scaled by a factor of two for clarity.

Figure 9.

Spatio-temporal contour of span-averaged skin-friction coefficient (a) alongside instantaneous skin-friction profiles (b) and instantaneous contours of spanwise vorticity (c) each illustrating the development of the LEV from the leading-edge LSB. Carets mark $x/c$ locations in increments of $0.01c$ in (c).

At the point at which the TS reaches the turbulent reattachment point of the LSB a complex series of events are initiated. Regions of span-averaged reverse flow ( $U_{x}<0$ , airfoil frame of reference) are highlighted in figure 8. As described by Carr et al. (Reference Carr, McAlister and McCroskey1977) a thin region of reverse flow develops over the majority of the airfoil suction surface. Periodic undulations in the outer boundary indicate low-frequency large structures in the separated turbulent boundary layer, a feature that was also noted using smoke visualisations in Carr et al. (Reference Carr, McAlister and McCroskey1977). The expanded views on the left of figure 8 show the upstream extent of the TS point and the small region associated with the recirculating flow within the LSB. At $\unicode[STIX]{x1D6FC}=24.08^{\circ }$ the two regions meet and by $\unicode[STIX]{x1D6FC}=24.36^{\circ }$ a distinct feature associated with the LEV is apparent. This interaction is expanded further in figure 9 to illustrate the development of the LEV directly from the LSB. The region of strong reverse flow, which terminates the chordwise extent of the LSB, develops continuously into the reverse flow associated with the LEV. This transition is shown through instantaneous snapshots of the skin-friction profile in the leading-edge region as well as contours of instantaneous vorticity in figures 9(b) and 9(c), respectively. During the formation of the LEV (illustrated in figure 9 c) vorticity is fed into this vortex structure directly from the leading-edge shear layer as well as the roll up of the near-stagnant turbulent boundary layer vorticity just downstream of the LSB.

Figure 10.

Contours of entropy showing rapid development of LEV.

A series of instantaneous mid-span contours of entropy are included in figure 10 to illustrate the development of the LEV and TSV structures. Corresponding span-averaged wall pressure coefficient distributions are included in figure 11. The development of the LEV structure is very rapid, occurring in the span of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}=0.2^{\circ }$ ( $\unicode[STIX]{x0394}t=0.07$ , figure 9). Experimental evidence for the LEV structure is mixed, given the difficulty in capturing such a small, rapidly developing flow feature. However, it is alluded to in the classical sketches in Carr et al. (Reference Carr, McAlister and McCroskey1977) through the depiction of a small vortex near the leading edge following the upstream spread of turbulent separation. Furthermore, at a similar Reynolds number on the OA209 airfoil, Mulleners & Raffel (Reference Mulleners and Raffel2013) described a ‘primary instability stage’ whereby, after reverse flow spread over much of the airfoil chord, the shear layer broke down into a system of substructures that mutually engulf each other to form a single DSV structure. Although no specific inference was made to an LSB in their work, a small vortex emanating from the leading edge is observed which convects downstream at a rate much quicker than the other similarly sized vortices. Previous research on this airfoil (Richez et al. Reference Richez, Mary, Gleize and Basdevant2008; Le Pape et al. Reference Le Pape, Costes, Joubert, David and Deluc2012), spanning the same flow conditions, specifically focused on the presence and behaviour of a small leading-edge LSB, indicating a similar boundary-layer topology as that described in the current computations. It should be noted that of the various pitching motions analysed in Mulleners & Raffel (Reference Mulleners and Raffel2013), the maximum pitch rate is a factor of 2–3 less than that used in the current study. At the same pitch rate and ramp-type motion as that of the current study, Gupta & Ansell (Reference Gupta and Ansell2018), using particle image velocimetry, show both leading-edge and turbulent-separation vortices which interact to form the DSV.

Figure 11.

Suction-surface pressure coefficient showing suction collapse, LEV development and TSV development. Time instances approximately correspond to the contours in figure 10.

A distinct suction peak associated with the LEV is apparent starting at $\unicode[STIX]{x1D6FC}=24.2^{\circ }$ at $x/c=0.03$ in figure 11. Up to $\unicode[STIX]{x1D6FC}=24.4^{\circ }$ the magnitude of the suction peak grows in strength and then begins to weaken and widen as the LEV convects downstream and departs from the airfoil surface. For a small region of the flow just downstream of the LSB ( $0.025\leqslant x/c\leqslant 0.10$ ) a distinct ‘spike’ or ‘suction transient’ is observed in the wall pressure coefficient as a function of $\unicode[STIX]{x1D6FC}$ (see figure 12, marked ‘LEV’). Skin-friction coefficient time histories in figure 12 show that this event is associated with strong reverse flow corresponding to the passing of the LEV. The histories at $x/c=0.03$ are initially downstream of the LSB at the point at which the bursting occurs. Tracking the extrema of these events shows that the LEV initially convects downstream at a velocity of $0.48U_{\infty }$ . Recent experimental data have captured this pressure transient in a high-frequency wall pressure signal (Gupta & Ansell Reference Gupta and Ansell2018) and it is also noted in the work of Leishman (Reference Leishman1990) who provided wall pressure data for a single cycle of the pitching motion. This sharp transient associated with the LEV is most easily visualised just downstream of the LSB reattachment as the LEV develops, where the slow roll off in suction is distinct from the rapid suction as the LEV passes (figure 12, $x/c=0.03{-}0.05$ ). The brief pressure minimum is also visible in figure 6(b) in the plots of $C_{p,min}$ where the LEV development temporarily increases the peak suction level after the collapse of leading-edge suction. Also visible in figure 12 at $x/c=0.01$ is a period of rapid pressure fluctuation prior to suction collapse. This is noted in the experiments of Lorber & Carta (Reference Lorber and Carta1988) at $Re_{c}=2.0\times 10^{6}$ and has been shown to be associated with increased acoustic radiation emanating from the LSB which may be considered as a precursor to the onset of dynamic stall (Visbal & Garmann Reference Visbal and Garmann2018).

Figure 12.

Span-averaged wall pressure coefficient (upper) and skin-friction coefficient (lower) at $x/c=0.01$ , 0.03 and 0.05 (left to right) as a function of  $\unicode[STIX]{x1D6FC}$ . A moving mean filter with a time window of 0.02 is applied to the skin-friction coefficient to better visualise sharp transitions.

A second feature in figure 10 is the TSV structure that develops over approximately $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}=0.5^{\circ }$ ( $\unicode[STIX]{x0394}t=0.17$ ). This structure develops due to the roll up of the turbulent separated boundary layer. Starting at $\unicode[STIX]{x1D6FC}=24.5^{\circ }$ , when the suction peak is beginning to decay (figure 6 b), the turbulent shear layer appears to break up into a system of separate structures. At least four structures are noted clearly in the contours of entropy. These structures then wrap up together to form the TSV structure ( $\unicode[STIX]{x1D6FC}=24.8^{\circ }$ , figure 10). This agrees well with the so-called ‘primary instability’ mechanism captured by Mulleners & Raffel (Reference Mulleners and Raffel2013). Figure 11 shows a distinct suction peak associated with the TSV starting at $\unicode[STIX]{x1D6FC}=24.5^{\circ }$ . At approximately $x/c=0.35$ this peak only slightly moves downstream as it develops in strength.

Figure 13.

Comparison of contours of vorticity (upper) compared to entropy (lower) highlighting strength and coherence of LEV compared to TSV.

Figure 13 shows a comparison of vorticity contours contrasted against entropy contours for select stages within the LEV and TSV development process. This depiction highlights the increased strength and coherence of the LEV as compared to the TSV, something which is somewhat obscured through the entropy visualisations. This imbalance in strength sets the dynamics that occurs during the merging process.

4.6 Merging of LEV and TSV

The merging of the LEV and TSV structures is described next. This interaction is key in setting the initial position and strength of the DSV prior to its convection downstream, which is important for the prediction of the force histories throughout the unsteady motion.

The origin of both the TSV and LEV structures is the generation of strong clockwise vorticity in the favourable pressure gradient region of the boundary layer between the stagnation point and the location of peak suction. Until the development of the LEV from the LSB breakdown, the separated turbulent shear layer is continuously fed by the leading-edge vorticity source. This continues as the TSV begins to develop, however at the point at which LEV development occurs, this source of vorticity no longer feeds the TSV resulting in a saturation of total circulation of this structure. Instead leading-edge vorticity feeds the LEV structure which grows in strength to a point at which its total circulation is of similar order to the TSV before it detaches to move downstream.

As the LEV begins to move downstream, the TSV structure is pulled upstream and swept underneath the LEV structure, causing the LEV to lift further from the airfoil surface. This is illustrated from $\unicode[STIX]{x1D6FC}=24.9^{\circ }$ to $\unicode[STIX]{x1D6FC}=25.5^{\circ }$ in figure 14. Over this same time period figures 15 and 16 show the merging of the local suction peaks associated with these turbulent structures. At $\unicode[STIX]{x1D6FC}=24.9^{\circ }$ , the suction peak associated with the LEV is positioned at $x/c=0.12$ and the TSV at $x/c=0.4$ . Through the merging process the TSV suction peak is pulled upstream and the two peaks merge at $x/c=0.25$ . At this point ( $\unicode[STIX]{x1D6FC}=25.5^{\circ }$ ) the two vortex cores (visualised using entropy, figure 14) have yet to merge, but are simply stacked on top of each other. As the cores merge, the suction peak associated with the DSV decreases in strength, widens slightly and begins to convect downstream. A convection speed of $0.25U_{\infty }$ was estimated for the DSV based upon $x$ -location of minimum $C_{p}$ over the time window $6.6\leqslant t\leqslant 8.0$ .

Figure 14.

Contours of entropy showing merging process of the LEV and the TSV.

Figure 15.

Suction-surface pressure coefficient showing merging process between LEV and TSV. Time instances correspond to the contours in figure 14.

Figure 16.

Spatio-temporal contour of span-averaged wall pressure coefficient showing the merging process of the LEV and TSV structures.

This process, which sets the initial location of the DSV at $x/c=0.2{-}0.3$ , occurs over the span $24^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 26^{\circ }$ in which the average location of the wall suction related to the vortex structures is primarily fixed. The effects of this interaction are observed in the force histories in figure 3, where the roll off of the moment coefficient is briefly reversed and a high normal force coefficient is maintained. As the airfoil continues to pitch above $\unicode[STIX]{x1D6FC}=26^{\circ }$ the DSV begins to convect downstream, shifting the location of peak suction. Accurate prediction of these events leading to the initial DSV location is important to predict the later time series of the forces in which the strength and position of the DSV is paramount. A similar observation of an extended time period of vortex amalgamation prior to downstream convection is included in Pruski & Bowersox (Reference Pruski and Bowersox2013). For the NACA 0012 airfoil at $Re_{c}=1.0\times 10^{6}$ , Pruski & Bowersox (Reference Pruski and Bowersox2013) discuss observations of a seemingly ‘suspended’ DSV that ‘forms and gathers strength’.

Figure 17.

Span-averaged wall pressure coefficient as a function of $\unicode[STIX]{x1D6FC}$ at $x/c=0.3$ , 0.4 and 0.5 (a–c) showing suction peaks associated with the TSV and DSV structures.

The span-averaged wall pressure coefficient versus $\unicode[STIX]{x1D6FC}$ is plotted in figure 17 over the range $x/c=0.3{-}0.5$ . It is important to point out that the separate development of the TSV and the fact that it is pulled upstream creates a situation where single-point time histories of wall pressure are difficult to interpret. At $x/c=0.4$ pressure decreases in association with the development of the TSV. As described previously, this structure is pulled upstream by the LEV and merges to form the DSV. The pressure rises when the TSV moves upstream and then drops again as the DSV passes. The high temporal and spatial resolution of these computations allow for these two events to be fully characterised.