The dynamics and timescales of static stall

Airfoil stall plays a central role in the design of safe and efficient lifting surfaces. We typically distinguish between static and dynamic stall based on the unsteady rate of change of an airfoil's angle of attack. Despite the somewhat misleading denotation, the force and flow development of an airfoil undergoing static stall are highly unsteady and the boundary with dynamic stall is not clearly defined. We experimentally investigate the forces acting on a two-dimensional airfoil that is subjected to two manoeuvres leading to static stall: a slow continuous increase in angle of attack with a reduced pitch rate of 1.3e-4 and a step-wise increase in angle of attack from 14.2{\deg} to 14.8{\deg} within 0.04 convective times. We systematically quantify the stall reaction delay for many repetitions of these two manoeuvres. The onset of flow stall is marked by the distinct drop in the lift coefficient. The reaction delay for the slow continuous ramp-up manoeuvre is not influenced by the blade kinematics and its occurrence histogram is normally distributed around 32 convective times. The static reaction delay is compared with dynamic stall delays for dynamic ramp-up motions with reduced pitch rates ranging from 9e-4 to 0.14 and for dynamic sinusoidal pitching motions of different airfoils at higher Reynolds numbers up to 1e6. The stall delays for all conditions follows the same power law decrease from 32 convective times for the most steady case down to an asymptotic value of 3 for reduced pitch rates above 0.04. Static stall is not phenomenologically different than dynamic stall and is merely a typical case of stall for low pitch rates. Based on our results, we suggest that conventional measurements of the static stall angle and the static load curves should be conducted using a continuous and uniform ramp-up motion at a reduced frequency around 1e-4.


Introduction
Flow separation and stall play a central role in the design of lifting surfaces for a wide range of applications such as rotary and fixed wing aircraft, wind turbines, etc. [1,2,3]. Stall is a commonly encountered, mostly undesired, condition that occurs when the angle of attack of an airfoil exceeds a critical angle. We typically distinguish between static and dynamic stall based on the rate of change of the airfoil's angle of attack [4]. The distinction is rather qualitative, as there is no universal criterion to assess whether a motion can be considered either static or dynamic. The denotation of static stall is highly misleading for two reasons: (i) an airfoil can not stall unless it moves past its critical stall angle and (ii) the flow and force development during the transition from an attached to a separated flow state are inherently unsteady. The temporal evolution of aerodynamic loads acting on an airfoil undergoing stall at extremely low pitch rate is often overlooked, as most attention is devoted to dynamic motions.
Literature regarding dynamic stall was initially motivated by helicopter rotor aerodynamics and flutter [5,6,7], and received renewed interest due to problems related to gust interactions [8,9]. The main parameter governing flow unsteadiness related to the kinematics of a pitching airfoil is the reduced pitch rate k defined as: where c is the airfoil chord,α is the pitch rate in radians per second, and U ∞ is the free stream velocity. This parameter represents the ratio of the kinematic to the convective timescales of the flow. The reduced pitch rate can be thought of as a phase lag between the blade kinematics and the surrounding fluid's response, resulting from the inertial effects [10]. For high enough pitch rates, the blade experiences a significant lift overshoot compared to the static case, and stall onset is delayed to an angle of attack beyond the critical stall angle. The additional lift is attributed to the formation, growth and shedding of large-scale dynamic stall vortices [6,11]. The angular delay of flow separation is considered one of the classical hallmarks of dynamic stall [5]. From a timing perspective however, high pitch rates promote flow separation and reduce the blade's reaction time relative to the static case [12,13].
The reaction time is a measure of the time the blade takes to stall after its angle of attack exceeds the critical stall angle. This timespan follows a power law decay for increasing reduced pitch rates, reaching a plateau for reduced pitch rates above 0.04 [14,15]. The minimum value for reaction time is attributed to the vortex formation time. The dynamic stall vortex requires a certain convective time to form before massive flow separation can occur, typically between 3 and 5 convective times for pitching airfoils [16,17].
For extremely low pitch rate values, blade kinematics cease to promote flow separation and the reaction time is expected to reach a maximum value which is not yet well defined. The key differences in the temporal evolution of static and dynamic aerodynamic loads remain to be formulated, and arguably static stall should be regarded as a general case of stall for extremely low values of the pitch rate.
Conventionally, we consider stall to be static when the airfoil's kinematics are slow enough to avoid delaying full flow detachment past the airfoil's critical stall angle. This angle is measured by making subdegree angle of attack increments, allowing the flow around the airfoil to fully develop before further motion is imposed. The last angular position before a loss in lift is observed is considered the critical stall angle.
This value plays a crucial role in characterising the airfoil's performance for dynamic motions and is a key parameter in semi-empirical models for dynamic stall [18,19]. Guidelines that characterise the relationship between reduced pitch rate and the temporal occurrence of stall are relevant to accurately determine the critical stall angle of a given airfoil.
We experimentally investigate the transient development of aerodynamic forces during what is typically considered as conventional static stall. We approach the static stall limit in two different ways: by increasing the angle of attack in small discrete steps and by slowly but continuously increasing the angle of attack.
The systematic acceleration and deceleration related to the stepwise increase of the angle of attack is more likely to disturb the flow than a continuous, extremely slow ramp-up. The time resolved lift response to the two types of quasi-steady motions is compared. Specific focus is directed towards the identification of successive stages in the flow development and the statistical analysis of the timescales associated with the different flow development stages. We quantify the reaction delay between the time when the blade exceeds its static stall angle and the occurrence of stall, determine the limiting values for extremely low and high pitch rates, and compare the results with the stall delays measured for various dynamic motions.
The main objective is to characterise the influence of the reduced pitch rate on the characteristic timescales of an airfoil undergoing stall, and identify qualitative properties that help qualify a motion to be static or dynamic. Output from the load cell was transmitted to a computer using a data acquisition system (National Instruments). Buoyancy forces and load cell factory offset were measured in the water channel without flow for the whole range of angles of attack investigated. These forces were subtracted from the force data to obtain the aerodynamic loads. The force data was filtered using a second-order low-pass filter with the cut-off frequency of 30 Hz. This frequency is multiple orders of magnitude larger than the For the step-wise, full flow reattachment was insured between runs by returning the blade to α = 8°and pitching up to its starting position at α = 14.2°with the same slow pitch rate used for the continuous ramp-up motion. The stall angle was determined based on the data collected with the ramp-up motion and found to be α ss = 14.2°. This information was used to select the start angle for the step-wise motion.

Experimental setup
Ramp-up motions with higher pitch rates ranging from 0.3°/s to 53.3°/s, corresponding to reduced pitch rates ranging from 1 × 10 −3 to 1.4 × 10 −1 , were performed to obtain an overview of the influence of the pitch rate on the delay of stall with respect to the static case. For all experiments, we recorded loads for 5 s or 16.7 convective times prior to the start of the manoeuvre, continuously during the manoeuvre, and for another 5 s after the manoeuvre was completed. The load cell recorded all three components of the force and the moments around all three spatial axes, but we will focus our analysis and discussion on the lift measurements. In most practical applications, the loss of lift due to a transition from attached to fully separated flow raises the most immediate concern. The characteristic timescales that we extract based on the lift response are the result of changes in the flow development which would equally affect other forces and moments, such as the drag or the pitching moment.

Results
The      The formation of stall vortices was found to occur much closer to the blade in the early stages of wake development. Additionally, our experiment is at a relatively low Reynolds number, so viscosity plays a more important role, reducing vortex formation length to near the blade [24,25]. Following this argumentation, the expected vortex shedding frequency f s is:   The stall delay or reaction time decreases with increasing pitch rate following a power law decrease. The additional unsteadiness added to the flow at higher pitch rates promotes flow detachment and the onset of stall. The reaction time decreases rapidly for reduced pitch rates below 0.01 and reaches a plateau at 3 convective times for reduced pitch rates above 0.04. This lower limit is of the order of the vortex formation time of the dynamic stall vortex, which is a classical hallmark of the transition from an attached to a massively separated flow [27,15]. The plateau represents a minimum timescale the blade requires to form a leading edge vortex and reach a fully separated flow condition [17]. Stall onset and vortex formation are characterised by the reaction time and it is the main difference used to distinguish static from dynamic stall [12,15]. The coherent transition between the lowest and higher pitch rates support our hypothesis that the static and dynamic stall responses are phenomenologically the same and their timescales vary continuously as a function of the pitch rate of the underlying motion kinematics.
The standard deviation also decreases rapidly with increasing reduced pitch rate. The airfoil kinematics play a lesser role in the flow development at extremely low pitch rates and do not longer promote the occurrence of stall, resulting in an increasingly random and wide distribution of the reaction time delays.
The reaction time histogram for the lowest pitch rate (figure 5) followed a perfect normal distribution, suggesting this motion can be considered as truly quasi-static.
The universality of these results is challenged by comparing them with measurement from different airfoil geometries, kinematics, and Reynolds number. The timescales of the NACA0018 are compared with those obtained for an OA209 airfoil [12] and for a NACA0015 [26] undergoing sinusoidal pitching motions.
As the pitch rate varies continuously for a sinusoidal motion, we use the instantaneous pitch rate at the time the static stall angle is exceeded as the representative effective pitch rate for the sinusoidal motions [12,14,13]. The effective pitch rates for the sinusoidal motions vary between 0.0035 and 0.02. The measured stall delays or reaction times for the three different airfoils, subjected to different kinematics, at different Reynolds numbers all collapse onto the same power law decay. This suggest that stall onset timescales are universal for airfoils undergoing stall at moderate to high Re where trailing edge stall is most common.
Further investigations are desirable to explore the ranges of validity of this seemingly universal behaviour in terms of Reynolds number and variety of airfoil geometry.
The generality of the variation of the stall onset timescales as a function of the unsteadiness of the pitching motion presented in figure 9 can be used to lay out guidelines for reliably measuring the static stall angle and lift and drag polars. The systematic acceleration and deceleration related to a stepwise increase of the angle of attack is more likely to disturb the flow than a continuous motion. A continuous ramp-up motion with slow uniform pitch rate is thus preferred but how slow is slow enough? To answer that, we first fit a generalised power law decay to our experimental data yielding the following expression: This expression is used to determine the angular accuracy for the measurement of the static stall angle for a given pitch rate determined by the angular increase that occurs during expected stall reaction time: The motion should be slow enough to minimise ∆α. In addition, we want to limit the inertial lift contributions associated with a dynamic motion. The inertial contribution to the lift coefficient for a continuous ramp-up motion can be estimated based on Theodorsen's theory [28] as: The evolution of both the static stall angular accuracy ∆α and the inertial component of the lift coefficient C l,inertial as a function of the reduced pitch rate are presented in figure 10. Overall, the quasi-steady inertial lift contributions are a lesser issue than the stall angle increases. Reduced frequencies of the order of 1 × 10 −4 yield a static stall angle accuracy of ∆α < 1°. The quasi-steady inertial contribution for these pitch rates are negligible. When measuring a static lift response using a continuous slow ramp-up motion, the lift response can be considered a conventional static force response, free of unsteady and quasi-steady influences, and providing a reliable estimate of the critical static stall angle for (αc)/(2U ∞ ) < 1 × 10 −4 .

Conclusion
We The results for the reaction delay from the slow continuous ramp-up motion were compared with results from dynamic ramp-up manoeuvres with reduced pitch rates ranging from 1.3 × 10 −4 to 0.14 and with previously obtained results from dynamic sinusoidal pitching motions with different airfoil geometries at different Reynolds numbers. This comparison revealed a universal power law decay of the stall delays from 32 convective times for the lowest pitch rates to a plateau around 3 convective times for reduced pitch rates above 0.04. The plateau level matches the vortex formation time, which is the minimum time interval required for the boundary layer to roll-up into a coherent stall vortex and separate from the airfoil.
The standard deviation of the observed stall delays across multiple repetitions also rapidly decreased with increasing pitch rate which aids in promoting the occurrence of stall. Static stall is not phenomenologically different than dynamic stall and is merely a typical case of stall for low pitch rates where the onset of flow separation is not promoted by the blade kinematics.
Based on the results, we propose that conventional static stall polars should be measured using a continuous and uniform ramp-up at a reduced frequency < 1 × 10 −4 to minimise the angle of attack variation during the stall delay. The inertial lift contributions at these pitch rates are negligible. A continuous motion is preferred to a stepwise increase as the systematic acceleration and deceleration of a step motion is more likely to cause an unsteady flow response. If a step-wise motion is selected, it is advised to wait at least 30 convective times between the end of the step and the start of the measurements to allow for the flow to respond to the change in the angle of attack.