2.1. Leading-edge suction parameter

Classical thin-aerofoil theory approximates the aerofoil using its camberline with no thickness and, hence, zero leading-edge radius. This assumption requires the flow to turn around the sharp leading edge, giving rise to a theoretically infinite flow velocity ($V_{LE}$) and a force ($\boldsymbol {F}_{LE}$) at the leading edge of a thin aerofoil (figure 1). The perpendicular and parallel components of the leading-edge force with respect to the camberline direction at the leading edge give the normal ($F_{N, LE}$) and suction ($F_{S, LE}$) forces, respectively. While the normal component of the force can either be positive or negative, depending on the location of the stagnation point and direction of the flow around the aerofoil's leading edge, the suction force always acts in the forward (positive) direction on the aerofoil, as illustrated in figures 1(a) and 1(b). When the stagnation point coincides with the leading edge of the camberline, the suction force is zero.

Figure 1. Depiction of flow around, and forces on, the leading edge of a thin aerofoil. ($a$) Stagnation point on the lower surface. ($b$) Stagnation point on the upper surface.

Following the unsteady thin-aerofoil theory of Katz & Plotkin (Reference Katz and Plotkin2001) as implemented in Ramesh et al. (Reference Ramesh, Gopalarathnam, Edwards, Ol and Granlund2013, Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), the chordwise distribution of the time-dependent vortex-sheet strength over the aerofoil, $\gamma (x,t)$, is expressed as a Fourier series as

(2.1)\begin{equation} \gamma(\theta,t)=2U_{ref}\left[A_{0}(t)\frac{1+\cos \theta}{\sin \theta} + \sum_{n=1}^{\infty}A_{n}(t) \sin(n\theta)\right], \end{equation}

where $\theta$ is a variable of transformation related to the chordwise coordinate $x$ as

(2.2)\begin{equation} x=\frac{c}{2}(1-\cos\theta), \end{equation}

and $A_{0}(t), A_{1}(t), \ldots , A_{n}(t)$ are the time-dependent Fourier coefficients, $c$ is the aerofoil chord, $t$ is time and $U_{ref}$ is the reference velocity. This reference velocity is usually invariant with time, and is selected based on the problem. In the current formulation, $U_{ref}$ is set equal to the forward velocity, which is the component of the aerofoil's velocity in the negative $X$ direction. Although recent research efforts (Xia & Mohseni Reference Xia and Mohseni2017; Epps & Greeley Reference Epps and Greeley2018; Epps & Roesler Reference Epps and Roesler2018; Taha & Rezaei Reference Taha and Rezaei2019) have provided updates to the Kutta condition at the trailing edge, the current formulation, which is based on the approach of Katz and Plotkin (Katz & Plotkin Reference Katz and Plotkin2000), assumes zero vortex-sheet strength at the trailing edge, and implicitly enforces the Kutta condition by imposing a zero vortex strength at each time step at the trailing edge. The Fourier coefficients are determined using the instantaneous local downwash on the aerofoil due to all bound and shed vorticity by enforcing the boundary condition that the flow must remain tangential to the aerofoil surface:

(2.3)\begin{equation} A_{0}(t) = -\frac{1}{\rm \pi}\displaystyle \int_{0}^{\rm \pi}\frac{W(x,t)}{U_{ref}}\,\textrm{d} \theta, \end{equation}

(2.4)\begin{equation}A_{n}(t) =\frac{2}{\rm \pi} \displaystyle \int_{0}^{\rm \pi}\frac{W(x,t)}{U_{ref}} \cos (n \theta) \,\textrm{d} \theta. \end{equation}

It has been known for several decades that the onset of separation at the leading edge is governed by criticality of flow parameters at the leading edge. Several researchers (Evans & Mort Reference Evans and Mort1959; Beddoes Reference Beddoes1978; Ekaterinaris & Platzer Reference Ekaterinaris and Platzer1998; Jones & Platzer Reference Jones and Platzer1998; Morris & Rusak Reference Morris and Rusak2013) have correlated leading-edge flow criticality to onset of leading-edge separation and/or static/dynamic stall. The LESP idea of Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), inspired in part by these works, was the result of the search for an appropriate parameter that could be determined as a part of an unsteady aerofoil theoretical calculation.

The LESP is a measure of the suction at the leading edge, which is correlated with the flow velocity around the leading edge, $V_{LE}$. Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) observed that the determining factor for the leading-edge suction and the flow velocity at the leading edge ($V_{LE}$) for an aerofoil is the circulation at the leading edge, $\gamma (0,t)$, which is represented by the first coefficient $A_{0}(t)$. Furthermore, using matched asymptotic expansion, Ramesh (Reference Ramesh2020) showed that $V_{LE}$ is directly proportional to the $A_0$ coefficient in (2.1) as

(2.5)\begin{equation} V_{LE} = \sqrt{ \frac{2}{r} }U_{ref}A_0, \end{equation}

where $r$ is the aerofoil leading-edge radius non-dimensionalized by the chord. It may therefore be argued that the instantaneous $A_0$ value could serve as the LESP. The instantaneous LESP, denoted generally by ${{\mathcal {L}}}$, at any time instant is therefore set equal to the value of $A_{0}(t)$ at that time. Because $A_0$ is defined using the reference velocity, $U_{ref}$, this LESP is based on the reference velocity, and is denoted specifically by $\mathcal {L}^{{ref}}$ (to distinguish it from an updated LESP that is introduced later). We get

(2.6)\begin{equation} \mathcal{L}^{{ref}}(t) = A_{0}(t) \end{equation}

with a positive value of $\mathcal {L}^{{ref}}$ corresponding to when the stagnation point is on the lower surface (figure 1a) and a negative value when the stagnation point is on the upper surface (figure 1b).

2.2. The LESP hypothesis for LEV initiation

As discussed by Katz (Reference Katz1981), real aerofoils have rounded leading edges which can support some suction even when the stagnation point is away from the leading edge. The amount of suction that can be supported is dependent on the aerofoil shape and the operating Reynolds number. Drawing on the observation that the LESP is a measure of the suction/velocity at the leading edge, Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) made the logical choice to develop a correlation for initiation of LEV formation to be based on the LESP. The advantage of this approach is that the instantaneous $\mathcal {L}^{{ref}}$ can be calculated for any unsteady motion from unsteady thin-aerofoil theory.

The LESP hypothesis proposed by Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) is that, for a given aerofoil and Reynolds number combination, the critical LESP, which is the $\mathcal {L}^{{ref}}$ value corresponding to LEV initiation, is independent of motion kinematics. This hypothesis is based in part on the argument that for a rounded leading edge of an aerofoil that is operating at a given chord Reynolds number, attached flow around the leading edge is supported so long as leading-edge suction is below a critical value. When this critical value is exceeded, the flow at the leading edge will separate, resulting in the shedding of vorticity from the leading edge. Because the LESP is a motion-independent, non-dimensional measure of the leading-edge suction, it should be a good predictor of LEV initiation for any motion. The major benefit of this hypothesis being true is that the critical LESP (i.e. $\mathcal {L}^{{ref}}_{crit}$) can be determined for one motion from CFD or experiment and can then be used for prediction of LEV initiation for any other motion.

As mentioned in the work of Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), an important limitation of the LESP hypothesis is that the $\mathcal {L}^{{ref}}$ in that approach is determined using inviscid/attached-flow theory (unsteady thin-aerofoil theory), in which the flow over the aerofoil surfaces is assumed to be attached. Thus, as discussed by Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017), in situations characterized by trailing-edge flow separation, this ‘inviscid LESP’ is unlikely to be a true measure of the leading-edge suction. Furthermore, the $\mathcal {L}^{{ref}}_{crit}$ value derived from unsteady inviscid theory for high pitch-rate motions with negligible trailing-edge boundary-layer separation is not applicable for much lower pitch-rate motions where trailing-edge flow separation is present. The inability of the inviscid LESP to account for the effects of trailing-edge separation was postulated as the reason for the small variation of $\mathcal {L}^{{ref}}_{crit}$ with pitch rate observed in Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017). With a new approach to determine the ‘viscous LESP’ using RANS CFD solutions in the current work, an important aim is to study the viscous $\mathcal {L}^{{ref}}_{crit}$ behaviour to examine if accounting for the effects of trailing-edge separation will collapse the viscous $\mathcal {L}^{{ref}}_{crit}$ values for all motions into a small range, irrespective of whether or not boundary-layer separation is present during initiation of LEV formation.

2.3. Previous hypothesis for the behaviour of LESP after LEV initiation

In the LDVM low-order method algorithm, Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) assumed that once $\mathcal {L}^{{ref}}$ reaches $\mathcal {L}^{{ref}}_{crit}$ and LEV shedding starts, the LESP remains constant at $\mathcal {L}^{{ref}}_{crit}$ until termination of LEV formation. During the LEV shedding process, the strength of the discrete LEV shed during any time step is determined so that the $\mathcal {L}^{{ref}}$ after the discrete LEV is shed is brought to this $\mathcal {L}^{{ref}}_{crit}$ value. LEV shedding is terminated once the instantaneous LESP value drops below $\mathcal {L}^{{ref}}_{crit}$.

The justification to hold the LESP at its predetermined critical value during the shedding process was based in part on the argument proposed by Katz (Reference Katz1981) that leading edges hold a certain amount of suction even when flow is separated. Based on the understanding at that time, it sounded like a reasonable argument that the leading edge had a capacity to hold a certain amount of suction even during LEV shedding, and the excess would be shed as vorticity. Furthermore, there did not seem to be any other guidance on how the suction behaves post-LEV initiation. Finally, and most importantly, when that hypothesis was implemented in the LDVM, the predictions for flow and forces agreed well with CFD and experimental results for all cases (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), which were at low Reynolds numbers of 100 000 and below. One of the objectives of the current work is to carefully test this hypothesis given that the viscous LESP is being determined in the current work using RANS CFD solutions and experimental data.

3.1. CFD method

CFD calculations were performed using North Carolina State University's REACTMB-INS code, which solves the time-dependent incompressible Navier–Stokes equations using a finite-volume method. The governing equations are written in arbitrary Lagrangian/Eulerian form, which enables the motion of a body-fitted computational mesh in accord with prescribed rate laws. Spatial discretization of the inviscid fluxes uses a low-diffusion flux-splitting method valid in the incompressible limit (Cassidy, Edwards & Tian Reference Cassidy, Edwards and Tian2009). This method is extended to higher-order spatial accuracy using piecewise parabolic method interpolations of the primitive variables $[p, u, v, w]^\textrm {T}$ and transported variable for the S-A model, $\tilde {\nu }$. Viscous terms are discretized using second-order central differences. A dual time-stepping method is used to integrate the equations in time. An artificial compressibility technique, discretized in a fully implicit fashion and solved approximately using ILU decomposition, is used to advance the solution in pseudo-time. Typically, eight sub-iterations per physical time step were needed to reduce the residual errors two orders of magnitude. The Spalart–Allmaras model (Spalart & Allmaras Reference Spalart and Allmaras1992), as implemented by Edwards & Chandra (Reference Edwards and Chandra1996), is used for turbulence closure. The geometries for the two aerofoils used in this work, NACA 0012 and SD7003m, are shown in figure 2. The SD7003m is a modified version of the original SD7003 (Selig, Donovan & Fraser Reference Selig, Donovan and Fraser1989), in which the leading-edge radius is twice that of the original SD7003. Two-dimensional body-fitted O-grids for the NACA 0012 and SD7003m aerofoils containing 140 400 and 92 400 cells, respectively, were generated. Two additional high-density grids for the NACA 23012 and SD7003 aerofoils, containing 284 400 and 108 900 cells, respectively, were generated for validation purposes. The wall $y^{+}$ for all the grids generated in the current study was $<5$. Additionally, a grid sensitivity study, carried out to ensure grid convergence, showed that decreasing the $y^{+}$ to a tenth of that used in the final grids does not alter the results noticeably.

Figure 2. ($a$) NACA0012 and ($b$) SD7003m aerofoil geometries.

3.2. CFD validation

Three test computations were performed to establish the accuracy of the numerical code. Steady simulations were performed for the NACA 0012 aerofoil at a free stream Reynolds number ($Re$) of 2.5 million and the computed coefficient of lift ($C_l$) variation with angle of attack ($\alpha$) was compared with experimental data from Carr et al. (Reference Carr, McAlister and McCroskey1977). The results in figure 3 show that computed $C_l$ is within 5 % of experimental data for all pre-stall angles of attack. The CFD overpredicts the stall angle by approximately 1 degree and maximum $C_l$ by approximately 0.08. Next, the NACA 23012 aerofoil pivoted at the quarter chord was simulated for an unsteady sinusoidal motion ($20{{{}^{\circ }}} \pm 10{{{}^{\circ }}} \sin (\varOmega t$), where $\varOmega$ is the angular frequency) at a free stream Reynolds number of 1.5 million and reduced frequency ($k = \varOmega c/(2U$)) of 0.10, where $U$ is the free stream velocity in this situation. The computed $C_l$ vs ${\alpha }$ is compared with experimental data from Leishman (Reference Leishman1990) in figure 4. The results show that LEV initiation, generally associated with the nonlinear increase in $C_l$, occurs at a higher angle of attack in the computations. The computational and experimental results exhibit similar trends after lift stall has occurred. Lastly, an unsteady simulation was performed for the SD7003 aerofoil at a free stream Reynolds number of 30 000 and pivoted about the leading edge at a non-dimensional pitch rate ($K = \dot {\alpha }c/(2U$)) of 0.11, where $\dot {\alpha }$ is the dimensional pitch rate, for a pitch-up-hold-return motion defined using the Eldredge function (Eldredge et al. Reference Eldredge, Wang and Ol2009; Eldredge & Wang Reference Eldredge and Wang2011). Figure 5 compares computed $C_l$ vs $t^{*}$ data with experimental data obtained from the US Air Force Research Laboratory's Horizontal Free-Surface Water Tunnel (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), where $t^{*}$ is the convective time given by $t^{*} = tU/c$. The computed $C_l$ agrees well with experiment in the pitch-up phase but is slightly overpredicted in the hold and return phases of the motion. The computation also correctly captures the time instants and intensities of the spikes due to apparent-mass effects.

Figure 3. Comparison for a stationary aerofoil: steady $C_l$ vs ${\alpha }$ from computations compared with experimental results of Carr et al. (Reference Carr, McAlister and McCroskey1977) for the NACA 0012 aerofoil at $Re = 2.5 \times 10^{6}$.

Figure 4. Comparison for a sinusoidal pitching aerofoil: computed $C_l$ vs ${\alpha }$ compared with the experimental results of Leishman (Reference Leishman1990) for the NACA 23012 aerofoil at $Re = 1.5 \times 10^{6}$, $k = 0.10$, $\alpha (t) = 20{{{}^{\circ }}} \pm 10{{{}^{\circ }}} \sin (\varOmega t)$.

Figure 5. Comparison for a low-$Re$ pitching aerofoil: computed $C_l$ vs $t^{*}$ compared with experimental results in Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) for the SD7003 aerofoil at $Re = 30\,000$, $K = 0.11$, ${\alpha }_{max} = 25{{{}^{\circ }}}$.

3.3. Determination of suction force and LESP from CFD

The determination of the leading-edge suction force and LESP for a thin aerofoil from unsteady theory was discussed in § 2.1. Because the theory assumes attached flow over the aerofoil (i.e. no trailing-edge separation), that LESP is referred to in this work as the ‘inviscid LESP.’ Motivated by the desire to understand the behaviour of the suction force on rounded leading-edge aerofoils in real flows, an approach was developed in the current work to calculate the suction force from RANS CFD solutions. Because the LESP calculated from this CFD-derived suction force accounts for trailing-edge separation and other viscous phenomena, this LESP is referred to as the ‘viscous LESP.’ Gaining an understanding of the behaviour of the viscous LESP would allow for careful evaluation of the LESP-related hypotheses in §§ 2.2 and 2.3, and enable improved modelling of LEV shedding.

When the stagnation point is not at the geometric leading edge of the aerofoil, the flow is forced to travel around the rounded leading edge. This flow curvature gives rise to a low-pressure region near the leading edge. As done in § 2.1, the net force acting on the leading edge is resolved into components acting along and normal to the direction of the camberline at the leading edge. The component of force along the camberline is often a ‘suction’ force acting in the forward direction. The curvature of the streamlines around the leading edge, the resulting suction near the leading edge and the leading-edge forces are illustrated in figure 6. In the current work, the leading-edge force on the aerofoil is calculated from CFD by integrating the surface pressure on the forward portion of the aerofoil from the leading edge to a predefined chordwise location, denoted by $(x/c)_{max}$. This force is then split into its suction and normal components, which when non-dimensionalized by the dynamic pressure ($q_{ref} = \rho U_{ref}^{2}/2$, where $\rho$ is the free stream density) and chord, give the coefficients of suction, $C^{ref}_{s,LE}$, and normal force, $C^{ref}_{n,LE}$, on the leading edge. The superscript ‘ref’ is used here to differentiate these force coefficients from those defined later using a different velocity:

(3.1)\begin{gather} C^{ref}_{s,LE} = F_{S,LE} / {q_{ref} c}, \end{gather}

(3.2)\begin{gather}C^{ref}_{n,LE} = F_{N,LE} / {q_{ref} c}. \end{gather}

Figure 6. Representation of the leading-edge force, $\boldsymbol {F}_{LE}$, resolved into suction force ($F_{S,LE}$) parallel to the camberline, and normal force ($F_{N,LE}$) perpendicular to the camberline at the leading edge.

Figure 7 shows the temporal variation of the coefficient of suction (referred to as $C^{ref}_s$ instead of $C^{ref}_{s,LE}$ from here onward) calculated from the unsteady RANS CFD solution using various values of $(x/c)_{max}$ for a NACA 0012 aerofoil undergoing a pitch-up-return motion with $K = 0.1$ at a Reynolds number of 3 million. We observe that for $(x/c)_{max}\geq 0.30$, the suction force is relatively independent of the selected $(x/c)_{max}$. We note that the $x/c$ location of the maximum thickness of the NACA 0012 aerofoil is at $0.30$. While there is no theoretical guidance on a correct value of $(x/c)_{max}$ to use for integration of CFD pressures to calculate $C^{ref}_s$, we have found that using the $(x/c)_{max}$ corresponding to the maximum thickness location on the chord works well. The calculated value of $C^{ref}_s$ is relatively insensitive to the chosen $(x/c)_{max}$ around this value because the aerofoil surfaces are typically nearly parallel to the chord in the vicinity of the maximum thickness location and the pressure contributions from these portions of the aerofoil to the suction force are small. Thus, the leading-edge suction and normal forces in this work are defined as the appropriate components of the net force acting on the forward portion of the aerofoil from the leading edge to the $x/c$ for the maximum thickness location. Also co-plotted in figure 7 is the time variation of the inviscid $C^{ref}_s$ calculated from the unsteady aerofoil theory for this motion. We see that the CFD-derived $C^{ref}_s$ agrees well with the theoretical $C^{ref}_s$ for small pitch angles at which the flow on the upper surface is fully attached, which shows that the CFD-derived $C^{ref}_s$ is a good equivalence for the theoretical $C^{ref}_s$ in attached-flow conditions. At $t^{*} > \approx 14$, which corresponds to $\alpha > \approx 42$ degrees, when LEV shedding is active, the theoretical $C^{ref}_s$ is seen to be much larger than the CFD-derived $C^{ref}_s$. This large difference is because the theoretical $C^{\rm ref}_s$ does not account for any viscous effects, while the CFD-derived $C^{ref}_s$ is a measure of the suction force from simulations of the real viscous flow.

Figure 7. Time variation of $C_s^{ref}$ for representative motions for various $(x/c)_{max}$ values used for the integration.

The inset in figure 7 shows the surface pressure coefficient ($C^{ref}_p$) variation plotted against $y/c$ for the forward portion of the aerofoil (for $0 \leq x/c \leq 0.30)$ at an example time instant of $t^{*} = 12$ when the flow is fully attached. This pressure coefficient is defined using the reference dynamic pressure as $C_{p}^{ref}=(p-p_{\infty })/q_{ref}$. This plot has the $y/c$ scale on the vertical axis so that the area inside the curve gives a visual representation of the leading-edge suction force acting in the horizontal (chordwise) direction. The vertical distribution of the $C^{ref}_p$ along the $y/c$ in this plot provides information on whether the suction is concentrated on the upper or lower surface of the aerofoil. Such $C_p$ vs $y/c$ plots are used later in this manuscript to discuss the suction behaviour for various situations.

As shown by Ramesh et al. (Reference Ramesh, Gopalarathnam, Edwards, Ol and Granlund2013), aerofoil theory gives the relationship between $C^{ref}_s$ and LESP as

(3.3)\begin{equation} C^{ref}_s = 2 {\rm \pi}(\mathcal{L}^{{ref}})^{2}. \end{equation}

As discussed in § 2.1, the suction force ($F_{S,LE}$) and the suction-force coefficient ($C^{ref}_s$), when calculated from theory, cannot be negative. However, because $C^{ref}_s$ calculated from CFD may not perfectly match up with that from ideal flow, there are a few situations in which, when the theoretical $C^{ref}_s$ value is zero or a small positive value, the CFD-derived $C^{ref}_s$ has a small negative value. To ensure compatibility with the theoretical $C^{ref}_s$ and to avoid problems when calculating the LESP, the CFD-derived LESP is set to zero at the few time instants when the viscous $C^{ref}_s$ has a negative value. Furthermore, the sign of the LESP is set to be the same as that of the $C^{ref}_{n,LE}$, so that positive $C^{ref}_{n,LE}$ is assumed to correspond to a flow with stagnation point on the lower surface, and vice versa. Thus, the CFD-derived viscous LESP is calculated from the CFD-derived $C^{ref}_s$ as

(3.4)\begin{equation} \mathcal{L}^{{ref}} = \begin{cases} {\textrm{sgn}}(C^{ref}_{n,LE}) \sqrt{\frac{C^{ref}_s}{2{\rm \pi}}}, & \textrm{for}\ C^{ref}_s > 0, \\ 0, & \textrm{for}\ C^{ref}_s \leqslant 0. \end{cases} \end{equation}

The time histories of the viscous $C^{ref}_s$ and $C^{ref}_{n,LE}$ calculated from integration of the CFD pressure distribution over the leading-edge region of a NACA 0012 aerofoil at a Reynolds number of 3 million, undergoing sinusoidal pitching ($\alpha (t) = 0{{{}^{\circ }}} \pm 30{{{}^{\circ }}} \sin (\varOmega t$)) with a reduced frequency of $k = \varOmega c/(2U_{ref}) = 0.1$ and pivoted at the half-chord is shown in figure 8(a). The viscous LESP for this case, calculated using (3.4), is shown in figure 8(b). The few flat spots and abrupt jumps when the LESP is close to zero are the result of conversion of viscous $C^{ref}_s$ to viscous LESP using (3.4). These flat spots are inconsequential because the flow events related to leading-edge stall occur only when the LESP is at high values.

Figure 8. Variation of CFD-derived leading-edge quantities for an example sinusoidal motion: ($a$) viscous $C_s^{ref}$ and $C_{n,LE}^{ref}$, and ($b$) viscous $\mathcal {L}^{{ref}}$ vs $t^{*}$.

3.4. Improved determination of LEV initiation from CFD and experiment

An important part of the current work is to examine the events leading to LEV formation for multiple aerofoils, a large set of motion kinematics, and over a range of Reynolds numbers. Although LEV initiation can be qualitatively inferred from CFD flow-field images by marking the time instant at which the first sign of an LEV structure appears during a motion, such a process is subjective and results in noise when comparisons are made for a large number of cases. Furthermore, it is desirable that the approach involve surface quantities and be straightforward to implement so that the data processing can be automated. In earlier work involving low-Reynolds-number unsteady aerofoil flows, a signature in the surface skin-friction coefficient ($C^{ref}_f$, defined using $U_{ref}$) distribution was used for identifying the time instant of LEV initiation (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014; Hirato et al. Reference Hirato, Shen, Gopalarathnam and Edwards2019). An improved approach that works well at both low and high Reynolds numbers is presented in this section. In this illustration, a NACA 0012 aerofoil undergoing a pitch-up motion about the quarter chord at $K = 0.4$ is considered at Reynolds numbers of 30 000 and 3 million. Results from CFD at five time instants are considered to illustrate the events from slightly before to slightly after LEV formation.

For the flow field results from CFD presented in this section and in the remainder of the paper, the colour bars used for all the non-dimensional velocity-magnitude plots (contours of $U_{mag}/U_{\infty }$, where $U_{mag}$ is the velocity magnitude) and non-dimensional vorticity plots (contours of $\omega c /U_{\infty }$, where $\omega$ is the vorticity) are shown in figure 9.

Figure 9. Colour bars for the CFD flow field plots presented in this paper: ($a$) non-dimensional velocity magnitude ($U_{mag}/U_{\infty }$) and ($b$) non-dimensional vorticity ($(\omega c) /U_{\infty }$).

Figure 10 shows the motion history and the time-variation of the inviscid and viscous LESP for the low-Reynolds-number case ($Re = 30\,000$). Only a small time range from slightly before to slightly after initiation of LEV formation is presented here. It is emphasized that the inviscid LESP variation is calculated using attached-flow assumptions, with neither LEV shedding (unlike in the LDVM low-order method (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014)) nor trailing-edge separation (unlike in the unsteady trailing-edge separation method (Narsipur, Gopalarathnam & Edwards Reference Narsipur, Gopalarathnam and Edwards2019)), and is therefore valid only until the onset of LEV shedding even though the LESP variation is plotted for the entire motion. The time instants for the five events are marked in this plot. For each of the five events, figure 11 shows in each row the vorticity contours around the entire aerofoil, a zoomed-in plot of the vorticity variation around the leading-edge region for $0 \leqslant x/c \leqslant 0.25$, the surface-$C^{ref}_f$ for the leading-edge region and a plot of the leading-edge $C^{ref}_p$ vs $y/c$. The events are labelled as follows: ‘A’ for attached flow; ‘O’ for onset of flow reversal on the surface, identifiable by the first occurrence of a small region of negative $C^{\rm ref}_f$ near the leading edge; ‘I’ for the first appearance of an inflection point within the negative-$C^{ref}_f$ region; ‘P’ for the first instant at which the inflection grows to become a positive spike within the negative-$C^{ref}_f$ region; and ‘L’ for the first instant at which the LEV structure is clearly identifiable from the vorticity contour. Mathematically, the inflection point for event I is identified as the first time instant in the motion at which the second derivative of the surface-$C^{ref}_f$, ${\partial ^{2} C^{ref}_f}/{\partial (x/c)^{2}}$, near the leading edge becomes negative. In earlier work (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), the $C^{ref}_f$ signature from CFD results associated with the P event was used as the quantitative criterion to determine the instant of LEV initiation for any given motion. This signature works well for low-Reynolds-number situations.

Figure 10. Events around LEV initiation for the low-$Re$ case: variation of $\mathcal {L}^{{ref}}$ with $t^{*}$ for an unsteady NACA 0012 aerofoil at $Re=30\,000$, $K=0.4$, pivoted at the half-chord.

Figure 11. Events around LEV initiation for the low-$Re$ case: LEV events for an unsteady NACA 0012 aerofoil at $Re=30\,000$, $K=0.4$, pivoted at the half-chord.

Figure 12 shows the motion history and the time variations of inviscid and viscous LESP for the high-Reynolds-number case ($Re = 3$ million). Each row of figure 13 shows the vorticity plot around the aerofoil, vorticity around the leading-edge region ($0 \leqslant x/c \leqslant 0.25$), the surface-$C^{ref}_f$ for the leading-edge region and the $C^{ref}_p$ variation vs $y/c$ for the five events. A noticeable difference between the low-Reynolds-number behaviour (figures 10 and 11) and the high-Reynolds-number behaviour (figures 12 and 13) is that, in the high-Reynolds-number case there is already evidence of an LEV structure in the vorticity plot for event P. This behaviour is consistently seen in all the high-Reynolds-number CFD results from the current work, and indicates that event P is not the best choice for LEV initiation for these cases. Based on these observations, event I – the instant at which an inflection point is first observed within the negative-$C^{ref}_f$ region near the leading edge, is chosen as the event corresponding to LEV initiation for both low- and high-Re cases. It is emphasized that the surface-$C^{ref}_f$ signature corresponding to event I is merely used as a consistent identifier of the time instant of LEV initiation from CFD results for the many cases studied in this work. Although the overall observations presented here are somewhat specific to the RANS CFD method used in this work and earlier related efforts (Ramesh et al. Reference Ramesh, Gopalarathnam, Edwards, Ol and Granlund2013, Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014; Hirato et al. Reference Hirato, Shen, Gopalarathnam and Edwards2019), they are in general agreement with the LEV-formation flow physics discussed by other researchers (Visbal & Shang Reference Visbal and Shang1989; Choudhuri et al. Reference Choudhuri, Knight and Visbal1994; Mulleners & Raffel Reference Mulleners and Raffel2012; Gupta & Ansell Reference Gupta and Ansell2019).

Figure 12. Events around LEV initiation for the high-$Re$ case: variation of $\mathcal {L}^{{ref}}$ with $t^{*}$ for an unsteady NACA 0012 aerofoil at $Re=3\times 10^{6}$, $K=0.4$, pivoted at the half-chord.

Figure 13. Events around LEV initiation for the high-$Re$ case: LEV events for an unsteady NACA 0012 aerofoil at $Re=3\times 10^{6}$, $K=0.4$, pivoted at the half-chord.

Finally it is worth mentioning that, in both low-Re and high-Re cases, between events O and I, there is also the appearance of an inflection point in the $C^{ref}_p$ vs $y/c$ variations. This observation points to the potential use of the inflection in the $C^{ref}_p$ distribution as a signature for LEV initiation from experimental data in which surface shear stress is not measured, but surface-$C^{ref}_p$ distributions are available from a pressure-tapped aerofoil model. However, a reasonably high density of pressure taps would be required to detect the occurrence of the inflection point.

5.1. Low-Reynolds-number cases

The values of viscous LESP at events O and I for the 25 medium and high pitch-rate motions for $Re = 30\,000$ are shown in figure 15, with the results for the NACA 0012 and SD7003m aerofoils shown in figures 15(a) and 15(b), respectively. The data points for event O are shown as open symbols and those for event I are shown as filled symbols. Lines joining the data points for the half-chord pivot cases are included to show the trends of $\mathcal {L}^{{ref}}$ with $\textrm {d}\alpha /\textrm {d}t^{*}$. An error bar is shown to indicate the typical error in $\mathcal {L}^{{ref}}$ of $0.015$ for each event due to the discrete time steps at which the CFD pressure data is available. Also plotted in figure 15(b) are the data points for the critical $\mathcal {L}^{{ref}}$ vs $\textrm {d}\alpha /\textrm {d}t^{*}$ from Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017) for the SD7003 aerofoil undergoing pitch-up motions about a quarter-chord pivot at a Reynolds number of 20 000. These data points from Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017) are the inviscid LESP values for LEV initiation, determined using the LDVM code assuming attached flow, for the time instants corresponding to when the surface-$C_f^{ref}$ signature for event P was observed in the CFD result for pitch-up motions. These data points were also shown in Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017) to qualitatively agree with experimental results from dye-flow visualization of the corresponding unsteady motions in water-tunnel experiments. This experimental confirmation was achieved by showing that, for each motion, there was a formation of distinct LEV structure in the dye-flow visualization just after the time instant at which LEV initiation was observed from the surface-$C_f^{ref}$ signature in the CFD result. Even though the aerofoil, Reynolds number and the event used for identifying the LEV initiation for this data from Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017) are all slightly different from those used in the current work for the SD7003m cases, it is seen that the variation in $\mathcal {L}^{{ref}}$ for event I with instantaneous pitch rate from the current work is very similar to those for the data points from Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017), except for the small offset in $\mathcal {L}^{{ref}}$.

Figure 15. LEV initiation at low Reynolds numbers: variation of $\mathcal {L}^{{ref}}$ at events O and I with instantaneous pitch rate for the ($a$) NACA 0012 and ($b$) SD7003m aerofoils at $Re = 30\,000$. Data points from Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017) are also included in ($b$).

Comparing the results for the two aerofoils from the current effort, it is seen that the values and trends are very similar. The $\mathcal {L}^{{ref}}$ values for the non-half-chord pivot locations for each event are seen to fall within a small range of the corresponding trend line. The two trends lines are seen to have an increase in $\mathcal {L}^{{ref}}$ of approximately $0.10$ for a change in $\textrm {d}\alpha /\textrm {d}t^{*}$ from $0.1$ to $1.0$. The observation that there is a change in viscous $\mathcal {L}^{{ref}}$ with pitch rate from the data in the current effort, and that it is nearly the same as the variation in inviscid $\mathcal {L}^{{ref}}$ from the data of Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017), shows that this effect of pitch rate on $\mathcal {L}^{{ref}}$ is not due to different amounts of trailing-edge separation, as postulated in Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017). Although the critical ${{\mathcal {L}}}^{\rm ref}$ of the trend line is not constant with $\textrm {d}\alpha /\textrm {d}t^{*}$, it is noted that the range of $K$ (and, hence, the range of $\textrm {d}\alpha /\textrm {d}t^{*}$) examined here is very large, with applications spanning from helicopter dynamic stall at the very low values of $K$ to insect and flapping-wing MAV flight at the very high end of the range examined. In practice, within the much smaller range of pitch rates typically used in any single application, the critical LESP can be considered as essentially a constant.

5.2. High-Reynolds-number cases

The values of $\mathcal {L}^{{ref}}$ at events O and I for the higher-$Re$ cases ($Re = 3$ million) are shown in figures 16(a) and 16(b) for the NACA 0012 and SD7003m aerofoils, respectively. It is seen that the $\mathcal {L}^{{ref}}$ values for the high-$Re$ cases are, in general, approximately $0.3$ higher than those for the low-$Re$ cases. The higher $\mathcal {L}^{{ref}}$ values indicate that, for a given motion, the high-$Re$ LEV initiation is delayed to a higher pitch angle compared with the low-$Re$ LEV initiation. This behaviour is to be expected, as the predicted level of boundary-layer separation at higher Reynolds numbers is typically delayed compared with lower Reynolds numbers due to the energizing effects of modelled turbulence. In our cases, the turbulence model was applied without a transition model, meaning that the boundary layer is predicted to be turbulent from the leading edge at the higher Reynolds number. Because the $\mathcal {L}^{{ref}}$ for event I is taken as the critical $\mathcal {L}^{{ref}}$ for modelling initiation of LEV formation in low-order methods like the LDVM (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), it is noted that the $\mathcal {L}^{{ref}}_{crit}$ varies with Reynolds number and needs to be determined using CFD computations at the Reynolds number of interest for the problem at hand.

Figure 16. LEV initiation at high Reynolds numbers: variation of $\mathcal {L}^{{ref}}$ at events O and I with instantaneous pitch rate for the ($a$) NACA 0012 and ($b$) SD7003m aerofoils at $Re = 3$ million.

Compared with the low-$Re$ variations, the trend lines for the half-chord pivot cases in the high-$Re$ results are seen to have much smaller slopes. However, the spread in $\mathcal {L}^{{ref}}$ values for the other pivot locations relative to the corresponding trend lines is significantly larger than those seen in the low-$Re$ cases. This spread in $\mathcal {L}^{{ref}}$ due to pivot location is especially large for the higher-$K$ cases, and reaches a LESP change of almost 0.3 between the leading-edge and trailing-edge pivot locations for the $K = 0.6$ cases. In comparison, the same change for the low-$Re$ case is only 0.05. Furthermore, the variation in $\mathcal {L}^{{ref}}$ with pitch rate is seen to depend on the pivot location. For the leading-edge pivot cases, for instance, $\mathcal {L}^{{ref}}$ is seen to increase with pitch rate, while for the trailing-edge pivot cases, the $\mathcal {L}^{{ref}}$ is seen to decrease with pitch rate. The results show conclusively that the $\mathcal {L}^{{ref}}$, as defined in the earlier works (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014, Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017) and thus far in this paper, fails to have anywhere close to a constant value at either event O or I, negating its utility in low-order prediction of LEV initiation for the high-$Re$ situations.

7.1. Low-Reynolds-number cases

Figures 19(a) and 19(b) show the variation of $\mathcal {L}^{{net}}$ with instantaneous $\textrm {d}\alpha /\textrm {d}t^{*}$ for the low-$Re$ cases of the NACA 0012 and SD7003m aerofoils, respectively. It is seen that the results for the half-chord trend lines for $\mathcal {L}^{{net}}$ are identical to those presented earlier for $\mathcal {L}^{{ref}}$ in figure 15. This is to be expected, as $\mathcal {L}^{{net}}$ and $\mathcal {L}^{{ref}}$ are identical for pitching motions about the half-chord pivot location. In other words, the velocity correction used in the definition of $\mathcal {L}^{{net}}$ has not resulted in any improvement to the variation of LESP with pitch rate. Also seen is that the scatter in $\mathcal {L}^{{net}}$ relative to the trend lines is a little greater than that seen for $\mathcal {L}^{{ref}}$ in figure 15. In particular, for the SD7003m, the $\mathcal {L}^{{net}}$ values for the trailing-edge pivot cases for $K = 0.4$ and $K = 0.6$ and the three-quarter-chord pivot case for $K = 0.6$ are noticeably higher than the corresponding trend lines by approximately $0.05$ to $0.15$. These deviations in $\mathcal {L}^{{net}}$ for the three outliers are discussed in more detail in § 7.3, and can be traced to deviations for these cases in the flow behaviour at LEV formation. Ignoring these three outliers for both the aerofoils, the spread in the $\mathcal {L}^{{net}}$ values from the corresponding trend lines is close to the spread in $\mathcal {L}^{{ref}}$ from the trend lines in figure 15.

Figure 19. LEV initiation at low Reynolds numbers: variation of $\mathcal {L}^{{net}}$ at events O and I with instantaneous pitch rate for the ($a$) NACA 0012 and ($b$) SD7003 aerofoils at $Re = 30\,000$. Data points from Ramesh et al. (Reference Ramesh, Granlund, Ol, Gopalarathnam and Edwards2017), converted to $\mathcal {L}^{{net}}$, are also included in ($b$).

7.2. High-Reynolds-number cases

Figures 20(a) and 20(b) show the variation of $\mathcal {L}^{{net}}$ with instantaneous $\textrm {d}\alpha /\textrm {d}t^{*}$ for high-$Re$ cases of the NACA 0012 and SD7003m aerofoils, respectively. Comparing against the $\mathcal {L}^{{ref}}$ variation in figure 16, we see that the velocity correction to the LESP has resulted in a dramatic improvement. The large scatter of $\mathcal {L}^{{ref}}$ values seen in figure 16 has instead been replaced by $\mathcal {L}^{{net}}$ values that fall in a small band around the two trend lines. This result demonstrates the importance of the velocity scaling used in the updated formulation of the $\mathcal {L}^{{net}}$.

Figure 20. LEV initiation at high Reynolds numbers: variation of $\mathcal {L}^{{net}}$ at events O and I with instantaneous pitch rate for the ($a$) NACA 0012 and ($b$) SD7003m aerofoils at $Re = 3$ million.

It is interesting to note that, while the updated LESP formulation has resulted in substantially different results for the critical LESP for the high-$Re$ cases, it has not resulted in significant change to the results for the low-$Re$ cases. The reason for this difference can be understood by examining the detailed results presented in the appendix. It is seen that, although the time variation of $U_{net}/U_{ref}$ for a given motion is independent of the Reynolds number, the value of $U_{net}/U_{ref}$ at LEV initiation is very much dependent on the Reynolds number. For the high-$K$ motions, LEV initiation occurs at much larger pitch angles at the higher Reynolds number than at the lower Reynolds number. Consequently, the value of $U_{net}/U_{ref}$ at LEV initiation is close to the value of $1$ for the low-Reynolds-number cases, causing $\mathcal {L}^{{net}}$ at LEV initiation to be close to the corresponding $\mathcal {L}^{{ref}}$. For the same motion at the higher Reynolds number, however, $U_{net}/U_{ref}$ at LEV initiation is not close to the value of $1$, causing the $\mathcal {L}^{{net}}$ at LEV initiation to be substantially different from the corresponding $\mathcal {L}^{{ref}}$.

9.1. Cases with LEV formation

Figures 25(a) and 25(b) show the variation with $t^{*}$ of inviscid and viscous LESP, and streamline curvature, $\kappa$, respectively, for a NACA 0012 aerofoil at a Reynolds number of $3$ million, pitching at $K = 0.4$, for the period of time starting from just prior to the initiation of LEV formation to the termination of LEV formation. On these plots are marked eight points corresponding to a sequence of events. The vorticity contours and streamline plots for these eight time instants are shown in figure 26. From these two figures we see that at event (a) the viscous LESP is nearly equal to the inviscid value because the leading-edge flow is attached and the curvature is at a high value. At (b) and (c) we see the early stages of an LEV being formed, resulting in the progressive deviation of the viscous LESP from the inviscid curve, and the beginnings of a sharp drop in streamline curvature at the leading edge. At points (d), (e) and (f) there is a shear layer at the leading edge that feeds a growing LEV structure. As a result of the flow separation at the leading edge, the streamline curvature is close to zero, and the viscous LESP values have large deviations from the corresponding inviscid values. By the time instants corresponding to points (g) and (h), the pitch angle of the aerofoil has started to decrease, resulting in the beginning of the termination process for the LEV. From the flow images (figure 26), it is seen that the streamline curvature has started to increase, which is also reflected in the $\kappa$ variation and in the decreasing difference between the inviscid and viscous LESP values (figure 25a). This example shows that (i) soon after LEV initiation, there is a drop in LESP, and (ii) the reason for this drop is that the separation of the leading-edge flow results in a loss in streamline curvature for the leading-edge flow.

Figure 25. Variation of LESP and curvature with $t^{*}$ for the NACA 0012 aerofoil at $Re = 3$ million, $K= 0.4$, and pivoted at the quarter chord.

Figure 26. Vorticity and streamline plots for the leading-edge region at eight time instants corresponding to figure 25.

Similar inferences regarding the suction behaviour post-LEV initiation can be made for the NACA 0012 aerofoil undergoing a high-$K$ motion with LEV shedding at a low-$Re$ by observing the trends in LESP and $\kappa$ from figure 27, and the vorticity contours and streamline plots from figure 28 for six time instants. Event (a) shows attached flow at the leading edge, negligible difference between viscous LESP from inviscid predictions and a high curvature value. As the LEV starts to initiate at event (b), the deviation between viscous and inviscid LESP starts to increase and a gradual drop in curvature is observed. As the LEV grows (event (c)), figure 27(b) displays a sharp drop in curvature, with the low value being maintained as the shear layer continues to feed into the LEV structure (event (d)). Correspondingly, figure 27(a) shows an increasing difference between inviscid and viscous LESP as the motion progresses from event (c) to (d). As LEV shedding starts to terminate (event (e)) and flow starts to reattach at the leading edge (event (f)), the deviation of viscous LESP from inviscid behaviour reduces and the curvature increases. Overall, this example confirms the correlation between leading-edge suction and streamline curvature, even for the low-$Re$ cases.

Figure 27. Variation of LESP and curvature with $t^{*}$ for the NACA 0012 aerofoil at $Re = 30\,000$, $K=0.4$, and pivoted at the quarter chord.

Figure 28. Vorticity and streamline plots for the leading-edge region at the six time instants corresponding to figure 27.

9.2. Cases without LEV formation

For low pitch-rate cases (of which steady flow ($K = 0$) is a subset) in which LEV shedding does not occur, a drop in LESP similar to that observed for the high-$K$ case in § 9.1 occurs when flow separation reaches the vicinity of the leading edge. Figure 29 shows the variation of (a) inviscid and viscous LESP and (b) leading-edge flow curvature ($\kappa$) for a NACA 0012 aerofoil undergoing a low-$K$ motion. Each row in figure 30 shows the velocity-magnitude contour around the aerofoil, vorticity around the leading-edge region up to $x/c=0.25$, the upper surface-$C_f^{net}$ at the leading edge and the $C_p^{net}$ vs $y/c$ variation up to the quarter chord for the eight time instances marked in figure 29(b). Figures 29 and 30 show that, at events (a) and (b), where there is boundary-layer separation on the aft portion of the upper surface but flow at the leading edge is attached, the difference between inviscid and viscous LESP is negligible and the curvature is high and constant. At event (c), as flow starts to separate at the leading edge, viscous LESP starts to deviate from inviscid behaviour and curvature starts to decrease. Further increase in the angle of attack (event (d)) results in increased deviation of the viscous LESP from inviscid predictions and a corresponding decrease in curvature. At event (e), when the angle of attack is maximum and the degree of flow separation is at its highest for the motion, the deviation in LESP reaches a peak value and the curvature is close to zero and is at its minimum. As the aerofoil pitches down and the angle of attack decreases (events (f) and (g)), a favourable pressure gradient is re-established on the upper surface, leading to flow reattachment at the leading edge. A corresponding decrease in the difference between inviscid and viscous LESP and an increase in flow curvature is observed. Once flow fully reattaches at the leading edge at event (h), viscous LESP reaches a local maximum and proceeds to return to inviscid behaviour with the flow curvature once again reaching its maximum value. Observations from the current low-$K$ case correspond well with the high-$K$ example discussed in § 9.1 in that, similar to cases where LEV shedding occurs, leading-edge flow separation also causes a loss in streamline curvature, which in turn leads to a drop in leading-edge suction. It is noted that, in these cases where there is no LEV formation, the LEV signatures of inflection in the surface-$C_f^{net}$ and the surface-$C_p^{net}$ distributions are also not present, confirming that the absence of those signatures are reliable indicators of the absence of LEV formation.

Figure 29. Variation of LESP and curvature for a low pitch-rate simulation without LEV formation (NACA 0012, $Re = 3\times 10^{6}$, $K = 0.005$, $0{{{}^{\circ }}}-35{{{}^{\circ }}}-0{{{}^{\circ }}}$ pitch-up-return, quarter-chord pivot).

Figure 30. Velocity magnitude, vorticity, skin friction and pressure plots at the eight time instants in figure 29 for the low pitch-rate simulation without LEV (NACA 0012, $Re = 3\times 10^{6}$, $K= 0.005$, $0{{{}^{\circ }}}\text {--}35{{{}^{\circ }}}\text {--}0{{{}^{\circ }}}$ pitch-up-return, quarter-chord pivot).

9.3. Post-leading-edge-separation LESP behaviour for the SD7003m aerofoil

Figure 31 shows the time histories of the LESP for the SD7003m aerofoil undergoing (a) a high-$K$ motion with LEV shedding and (b) a low-$K$ motion without LEV shedding for both low- and high-$Re$ cases.

Figure 31. Post-separation LESP trends for the SD7003m aerofoil: ($a$) high-$K$ motion and ($b$) low-$K$ motion.

As seen with the NACA 0012 aerofoil, the viscous LESP shows a more gradual deviation from inviscid behaviour upon LEV initiation for the low-$Re$ case as compared with the high-$Re$ case for the high-$K$ motion (figure 31a). For the low-$K$ motion without LEV shedding (figure 31b), the peak LESP does not exceed the $\mathcal {L}^{{net}}_{crit}$ values of $0.14$ and $0.41$ (from figures 19b and 20b) for the low- and high-$Re$ cases, respectively, reaffirming the hypothesis that no LEV formation occurs when LESP does not exceed $\mathcal {L}^{{net}}_{crit}$ for the given aerofoil and Reynolds number. Additionally, the behaviour of the LESP after leading-edge separation for the different motions of the SD7003m aerofoil is similar to that observed for equivalent motions of the NACA 0012 aerofoil at both Reynolds numbers.

9.4. Effect of the LESP drop on low-order modelling accuracy

To illustrate the importance of the insight gained here on the drop in LESP after LEV initiation, an example high-Reynolds-number and high-$K$ motion of the SD7003m aerofoil is analysed using a version of the LDVM code (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) with the original and modified implementations of the post-LEV behaviour of the LESP. The results from these low-order implementations are compared with the computational results for this case in figure 32. In the original implementation, labelled LOM1, the LESP is maintained at a constant value of $\mathcal {L}^{{net}}_{crit}$ during LEV shedding. In the modified implementation, labelled LOM2, the LESP drops linearly from LESP$_{crit}$ at LEV initiation to zero using a ${{\textrm {d}}{(\mathcal {L}^{{net}})}}/{{\textrm {d}}{t^{*}}}$ slope that is close to the slope seen in the CFD result. Figure 32(a) compares the LESP variations from the CFD and the two low-order implementations. Figure 32(b) compares the low-order discrete-vortex plot from LOM1 with the vorticity distribution from CFD, showing that the original low-order implementation results in an inaccurate prediction of the location of the LEV structure. With the drop in LESP modelled, as seen in figure 32(c), the location of the LEV structure predicted by the modified low-order method (LOM2) agrees well with that seen in the CFD result. Although the ${{\textrm {d}}{(\mathcal {L}^{{net}})}}/{{\textrm {d}}{t^{*}}}$ used in LOM2 was obtained from the CFD result, this example nevertheless illustrates the importance of correctly modelling the post-separation behaviour of LESP, especially for the high-Reynolds-number cases in which the critical LESP is high (usually $>0.4$), resulting in a significant drop. On the other hand, for low-Reynolds-number cases, because the critical LESP is quite low (${\approx }0.25$ or less), not modelling the small drop did not affect the accuracy of the low-order model noticeably. It is because of this reason that the results in Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) (all low-Reynolds-number cases) agreed well with the experimental and CFD results even with the original implementation of the post-separation LESP behaviour.

Figure 32. Effect of modelling the post-separation LESP drop on low-order results for the SD7003m aerofoil at $Re = 3\times 10^{6}$, pitching about the quarter chord at $K = 0.6$: ($a$) LESP variations from CFD, original model (LOM1) and modified model (LOM2); ($b$) discrete-vortex plot from LOM1 compared with the vorticity plot from CFD at $t^{*} = 12.2$; and ($c$) discrete-vortex plot from LOM2 compared with the vorticity plot from CFD.

9.5. Rate of LESP drop after leading-edge separation

As demonstrated in § 9.4, in order to accurately model the LESP behaviour in low-order methods, the data on the rate of LESP drop is useful. The compiled post-leading-edge-separation ${{\textrm {d}}{(\mathcal {L}^{{net}})}}/{{\textrm {d}}{t^{*}}}$ data for all the cases simulated in the current work (table 1) have been plotted against instantaneous $\textrm {d}\alpha /\textrm {d}t^{*}$ in figure 33. The instantaneous $\textrm {d}\alpha /\textrm {d}t^{*}$ used for the data corresponds to the value at initiation of LEV formation (for cases in which the stall was the result of LEV formation) or the value at which separation reached the leading edge (for cases in which stall occurred without LEV formation). For the symmetric NACA 0012 and cambered SD7003m aerofoils at Reynolds numbers 30 000 and 3 million, the ${{\textrm {d}}{(\mathcal {L}^{{net}})}}/{{\textrm {d}}{t^{*}}}$ ranges from $-0.25$ to $-0.008$ for various motion kinematics. This information can be used in future research to develop extensions of LDVM-like low-order methods (Ramesh et al. Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014; Hirato et al. Reference Hirato, Shen, Gopalarathnam and Edwards2019; Narsipur et al. Reference Narsipur, Gopalarathnam and Edwards2019) in which the post-separation LESP drop is modelled without the need to acquire it from a CFD solution of that problem. An early version of such a low-order method is described in Narsipur (Reference Narsipur2017) and Narsipur, Gopalarathnam & Edwards (Reference Narsipur, Gopalarathnam and Edwards2018).

Figure 33. Scatter plot of ${{\textrm {d}}{({{\mathcal {L}}}^{net})}}/{{\textrm {d}}{t^{*}}}$ after leading-edge separation for all cases.

9.6. Verification of trends in leading-edge suction with experimental data

All the data and resulting analysis presented in the above sections are based on computational results. It is therefore prudent to verify the trends observed in LESP behaviour with experimental data.

Figure 34 compares CFD results with experimental data from Gupta & Ansell (Reference Gupta and Ansell2019) for a NACA 0012 aerofoil undergoing a constant pitch-up motion about the quarter chord at a Reynolds number of 500 000. The LESP and $C_l$ predictions from CFD simulations are compared with experimental data in figures 34(a) and 34(b), respectively. Figure 35 compares the vorticity contour data from CFD simulations with the vorticity fields from time-averaged particle image velocimetry measurements from the experiment at the instant of LEV initiation (event I) and at a point where the LEV is well formed (event L). As explained in § 3.4, the inflection in the $C_p^{net}$ could be used to identify the point of LEV initiation (event I). However, due to the sparsity of pressure ports at the leading edge in the experimental investigation, the $C_p^{net}$ distributions could not be reliably used to identify event I. For the current case, the experimental vorticity field data was used to identify event I.

Figure 34. Comparison of CFD predictions with the experimental results of Gupta & Ansell (Reference Gupta and Ansell2019) for a NACA 0012 aerofoil undergoing a constant pitch-up motion about the quarter chord with $k=0.05$ and $Re = 0.50\times 10^{6}$: ($a$) LESP vs $t^{*}$ and ($b$) $C_l$ vs $t^{*}$.

Figure 35. Flow visualization data at events I and L from experiments (left) and CFD (right) for a NACA 0012 aerofoil undergoing a constant pitch-up motion about the quarter chord with $k=0.05$ and $Re = 0.50\times 10^{6}$.

It is seen that the LESP measurements from the experiment before LEV initiation are higher when compared with CFD predictions. Experimental vorticity measurements (figure 35) indicate the first sign of an LEV at $\alpha = 18.2{{{}^{\circ }}}$ while the signature of event I from the CFD solution was found to occur at a higher angle of attack ($\alpha = 25.2{{{}^{\circ }}}$). After LEV initiation, both CFD and experimental results show a drop in the suction force and a continuous decrease as the LEV grows.

Lift results from the computations compare well with experimental measurements (figure 34b). Both experimental and computational $C_l$ show a linear increase in lift until the point of LEV initiation, after which both methods show a nonlinear rise in lift. Overall, while CFD predicts LEV initiation at a higher angle of attack, trends in LESP, lift and vorticity flow fields compare very well with experimental measurements.

Next we consider experimental results for an S801 aerofoil undergoing a sinusoidal pitching motion ($\alpha _{mean} = 14{{{}^{\circ }}}$, $\alpha _{amplitude} = 10{{{}^{\circ }}}$) about the quarter chord with $k = 0.107$ at a Reynolds number of 750 000 from wind tunnel experiments conducted by Ramsay, Gregorek & Hoffmann (Reference Ramsay, Gregorek and Hoffmann1996). Figure 36 shows (a) the leading edge $C_p^{net}$ vs $y/c$ data for three frames centred about event I, (b) the $C_l$ variation and (c) the $\mathcal {L}^{{net}}$ variation for this case. The availability of a fine distribution of leading-edge pressure data helped in accurately identifying the time instant of event I at $t^{*}=4.44$ from the inflection in the $C_p^{net}$ vs $y/c$ data (shown in figure 36a). The inset in figure 36(a) shows the evolution of the inflection in $C_p^{net}$ at event I by plotting the $C_p^{net}$ vs $y/c$ distribution at time instants immediately before and after said event. An error of ${\rm \Delta} t^{*}=0.74$ is assumed due to the discrete time steps at which the experimental pressure data is available. An additional confirmation of LEV initiation and development is available from figure 36(b) at the point where a nonlinear increase in $C_l$ occurs during the aerofoil's upstroke, a behaviour typically attributed to LEV formation (Leishman Reference Leishman2002), which in the current case lies within the error bounds of the time instant at which the leading edge $C_p^{net}$ distribution indicates event I. At and following event I, figure 36(c) shows a drop in LESP as the LEV initiates and develops, thereby confirming the trends observed in the numerical simulations.

Figure 36. Experimental data for an S801 aerofoil undergoing a sinusoidal pitch motion of $14{{{}^{\circ }}}\pm 10{{{}^{\circ }}}$ about the quarter chord with $k = 0.107$ and $Re = 0.75\times 10^{6}$ (Ramsay et al. Reference Ramsay, Gregorek and Hoffmann1996): ($a$) $C_p^{net}$ vs $y/c$, ($b$) $C_l$ vs $t^{*}$ and ($c$) $\mathcal {L}^{{net}}$ vs $t^{*}$.

In other experimental investigations, Deparday & Mulleners (Reference Deparday and Mulleners2019) and Ansell & Mulleners (Reference Ansell and Mulleners2020) estimated the LESP from surface pressure and time-resolved particle image velocimetry measurements of a sinusoidally pitching OA209 aerofoil ($\alpha = 20{{{}^{\circ }}}\pm 8{{{}^{\circ }}}$, $k=0.05$, quarter-chord pivot) at a Reynolds number of 920 000. The trends from their work are similar to those seen in the current work, providing further experimental confirmation for the observations made in the current work.