The dynamics and timescales of static stall
TThe dynamics and timescales of static stall
S´ebastien Le Fouest a , Julien Deparday a,1 , Karen Mulleners a, ∗ a Institute of Mechanical Engineering, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Abstract
Airfoil stall plays a central role in the design of safe and efficient lifting surfaces. We typically distinguishbetween static and dynamic stall based on the unsteady rate of change of an airfoil’s angle of attack. Despitethe somewhat misleading denotation, the force and flow development of an airfoil undergoing static stall arehighly unsteady and the boundary with dynamic stall is not clearly defined. We experimentally investigatethe forces acting on a two-dimensional airfoil that is subjected to two manoeuvres leading to static stall: aslow continuous increase in angle of attack with a reduced pitch rate of 1.3 × − and a step-wise increasein angle of attack from 14.2 ° to 14.8 ° within 0.04 convective times. We systematically quantify the stallreaction delay, or the timespan between the moment the blade exceeds its critical static stall angle and theonset of stall, for many repetitions of these two manoeuvres. The onset of flow stall is marked by the distinctdrop in the lift coefficient. The reaction delay for the slow continuous ramp-up manoeuvre is not influencedby 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 reducedpitch rates ranging from 9 × − to 0.14 and for dynamic sinusoidal pitching motions of different airfoilsat higher Reynolds numbers up to 1 × . The stall delays for all conditions follows the same power lawdecrease from 32 convective times for the most steady case down to an asymptotic value of 3 convectivetimes for reduced pitch rates above 0.04. Static stall is not phenomenologically different than dynamic stalland is merely a typical case of stall for low pitch rates where the onset of flow separation is not promotedby the blade kinematics. Based on our results, we suggest that conventional measurements of the staticstall angle and the static load curves should be conducted using a continuous and uniform ramp-up motionat a reduced frequency around 1 × − . Keywords: static stall, dynamic stall, stall delay, NACA0018
1. Introduction
Flow separation and stall play a central role in the design of lifting surfaces for a wide range of applica-tions such as rotary and fixed wing aircraft, wind turbines, etc. [1, 2, 3]. Stall is a commonly encountered, ∗ Corresponding author
Email address: [email protected] (Karen Mulleners) Present affiliation: Institute for Energy Technology, Eastern Switzerland University of Applied Sciences (OST), CH-8645Rapperswil, Switzerland
Preprint submitted to Journal of Fluids and Structures February 10, 2021 a r X i v : . [ phy s i c s . f l u - dyn ] F e b ostly undesired, condition that occurs when the angle of attack of an airfoil exceeds a critical angle. Wetypically distinguish between static and dynamic stall based on the rate of change of the airfoil’s angleof attack [4]. The distinction is rather qualitative, as there is no universal criterion to assess whether amotion can be considered either static or dynamic. The denotation of static stall is highly misleading fortwo reasons: (i) an airfoil can not stall unless it moves past its critical stall angle and (ii) the flow and forcedevelopment during the transition from an attached to a separated flow state are inherently unsteady. Thetemporal evolution of aerodynamic loads acting on an airfoil undergoing stall at extremely low pitch rateis 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 mainparameter governing flow unsteadiness related to the kinematics of a pitching airfoil is the reduced pitchrate k defined as: k = c ˙ α ∞ , (1)where c is the airfoil chord, ˙ α is the pitch rate in radians per second, and U ∞ is the free stream velocity. Thisparameter represents the ratio of the kinematic to the convective timescales of the flow. The reduced pitchrate 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 liftovershoot compared to the static case, and stall onset is delayed to an angle of attack beyond the criticalstall angle. The additional lift is attributed to the formation, growth and shedding of large-scale dynamicstall vortices [6, 11]. The angular delay of flow separation is considered one of the classical hallmarks ofdynamic stall [5]. From a timing perspective however, high pitch rates promote flow separation and reducethe 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 thecritical stall angle. This timespan follows a power law decay for increasing reduced pitch rates, reachinga plateau for reduced pitch rates above 0.04 [14, 15]. The minimum value for reaction time is attributedto the vortex formation time. The dynamic stall vortex requires a certain convective time to form beforemassive 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 reactiontime is expected to reach a maximum value which is not yet well defined. The key differences in thetemporal evolution of static and dynamic aerodynamic loads remain to be formulated, and arguably staticstall 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 avoiddelaying full flow detachment past the airfoil’s critical stall angle. This angle is measured by making sub-degree angle of attack increments, allowing the flow around the airfoil to fully develop before further motionis imposed. The last angular position before a loss in lift is observed is considered the critical stall angle.2his value plays a crucial role in characterising the airfoil’s performance for dynamic motions and is a keyparameter in semi-empirical models for dynamic stall [18, 19]. Guidelines that characterise the relationshipbetween reduced pitch rate and the temporal occurrence of stall are relevant to accurately determine thecritical stall angle of a given airfoil.We experimentally investigate the transient development of aerodynamic forces during what is typicallyconsidered as conventional static stall. We approach the static stall limit in two different ways: by increasingthe 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 morelikely to disturb the flow than a continuous, extremely slow ramp-up. The time resolved lift response tothe two types of quasi-steady motions is compared. Specific focus is directed towards the identificationof successive stages in the flow development and the statistical analysis of the timescales associated withthe different flow development stages. We quantify the reaction delay between the time when the bladeexceeds its static stall angle and the occurrence of stall, determine the limiting values for extremely lowand 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 timescalesof an airfoil undergoing stall, and identify qualitative properties that help qualify a motion to be static ordynamic.
2. Experimental setup
A NACA0018 profile is vertically suspended in a recirculating water channel with fully-transparentacrylic windows and a test section with dimensions 0.6 m × × ∞ = 0.50 m/s, resulting in a Reynolds number of Re = 7.5 × . Forces are measured with a sixdegrees of freedom load cell (ATI Nano 25) placed at the interface between the shaft of the airfoil and themotor shaft. Force data were recorded with a sampling frequency of 1000 Hz, a sensing range of 125 N anda resolution of 0.02 N. Output from the load cell was transmitted to a computer using a data acquisitionsystem (National Instruments). Buoyancy forces and load cell factory offset were measured in the waterchannel without flow for the whole range of angles of attack investigated. These forces were subtracted fromthe force data to obtain the aerodynamic loads. The force data was filtered using a second-order low-passfilter with the cut-off frequency of 30 Hz. This frequency is multiple orders of magnitude larger than the3 igure 1: Schematic representation of the two-dimensional NACA0018 blade mounted vertically in a free surface recirculatingwater channel with a cross section of 0.6 m × pitching frequency and 48 times larger than the expected post-stall vortex shedding frequency based on achord-based Strouhal number of 0.2 [19].Two quasi-steady manoeuvres that lead to the occurrence of what can be considered conventional staticstall are investigated: a slow continuous ramp-up motion and a step-wise increase. For the first type ofmanoeuvre, we continuously increased the angle of attack from α = 8 ° up to α = 18 ° with a constant pitchrate of 0.05 ° /s, which corresponds to a reduced pitch rate of 1 . × − . For the second type of manoeuvre,we performed a step-wise increase in the angle of attack from 14.2 ° to 14.8 ° within 0.04 convective timesat 50 ° /s, which corresponds to a reduced pitch rate of k = 1 .
3. The airfoil was rotated around its quarter-chord axis using a stepper motor with a 1 . ° step angle and 1 : 25 planetary gearbox reduction, resultingin a position accuracy of 0 . ° . The initial blade position was calibrated using a small angle of attacksweep between − ° and 5 ° to find the angle that resulted in zero net lift. The pitch rate for the continuousramp-up motion was selected by progressively reducing the pitch rate until we systematically measured nosignificant lift overshoot at the onset of stall. Each motion was repeated for approximately 100 runs to allowfor a statistical analysis of the transient aerodynamic load fluctuations occurring during stall development.For the step-wise, full flow reattachment was insured between runs by returning the blade to α = 8 ° andpitching up to its starting position at α = 14 . ° with the same slow pitch rate used for the continuousramp-up motion. The stall angle was determined based on the data collected with the ramp-up motionand found to be α ss = 14 . ° . This information was used to select the start angle for the step-wise motion.Ramp-up motions with higher pitch rates ranging from 0.3 ° /s to 53.3 ° /s, corresponding to reduced pitchrates ranging from 1 × − to 1.4 × − , were performed to obtain an overview of the influence of thepitch rate on the delay of stall with respect to the static case. For all experiments, we recorded loads for4 s or 16.7 convective times prior to the start of the manoeuvre, continuously during the manoeuvre, andfor another 5 s after the manoeuvre was completed. The load cell recorded all three components of theforce and the moments around all three spatial axes, but we will focus our analysis and discussion on thelift measurements. In most practical applications, the loss of lift due to a transition from attached to fullyseparated flow raises the most immediate concern. The characteristic timescales that we extract based onthe lift response are the result of changes in the flow development which would equally affect other forcesand moments, such as the drag or the pitching moment.
3. Results
The temporal evolution of the lift coefficient in response to the slow continuous ramp-up and the stepmanoeuvre are shown in figure 2. The timing is indicated in terms of convective time defined as t c = t U ∞ /c ,relative to t c = 0 when the blade’s angle of attack exceeds 14.2 ° . The value of 14.2 ° is considered the criticalangle of attack above which the blade will always stall even if no further motion is imposed. Across theentire range of experiments conducted with this blade under the given flow conditions, 14.2 ° was the lowerbound at which stall would consistently occur. Results are insensitive to the exact choice of critical angle,as moving it up or down one sub-degree would result in a homogeneous shift in timing for all experimentalruns. A close-up view of the lift drop during stall for the continuous ramp-up in figure 2d facilitates thecomparison between the continuous ramp-up and the step manoeuvre. Each line in figure 2b,d,f representsa different run. A single run is highlighted in black for both experiments to show the characteristics of theresponse.In the continuous ramp-up experiment (figure 2a-d), the lift does not immediately drop once the bladeexceeds the critical angle and remains at an approximately constant value for a certain time before dropping.The time delay between the moment at which the critical angle is exceeded and the moment at which thelift starts to drop differs for each run. We call this time delay the reaction delay and refer to the periodcovering the reaction delay as the holding stage. Although the blade’s angle of attack can increase by asmuch as 0.6 ° during the holding stage for some runs, the blade does not produce additional circulation andthe lift coefficient fluctuates steadily around 1 during this stage. For the highlighted run, the holding stagelasts about 33 convective times, followed by the drop stage where the lift coefficient falls from C L =1.05 to0.6 in 9 convective times. A third characteristic time instant is identified as the first local lift minimumbelow the average post-stall level. At this point, the flow is considered to be in a fully developed stalledstate. The fully developed post-stall stage is dominated by periodic load fluctuations due to characteristicbluff-body-like vortex-shedding [20, 21].For the step-wise increase in angle of attack (figure 2e,f), the blade is static for the first 5 s or 16.7convective times of the load recording during which the lift coefficient fluctuates around 1.05. The bladeis then subjected to a step-wise increase of 0.6 ° beyond the critical angle of attack. The origin on the5 igure 2: (a) Variation of the angle of attack during the continuous ramp-up manoeuvre. (b) The corresponding lift responsesfor 94 repetitions. (c,d) Close up views of the transient region corresponding to the shaded areas in the left plots. (e) Variationof the angle of attack during the step manoeuvre. (f) The lift response to the step-wise increase in the angle of attack for 100repetitions. Convective time t U ∞ /c = 0 corresponds to the time when the airfoil’s angle of attack exceeds the critical angle of14.2 ° . The thicker black lines in the bottow row show the lift coefficient for a single run to highlight the characteristics of theindividual responses. Pre-stall and post-stall reference levels are indicated by the red dashed lines. The green dashed linesindicate the upper and lower threshold values used to identify the timing of the stall stage. The start of the lift drop is t c,pre isdefined as the time at which the lift has dropped more than 4 % of the total lift drop ∆ C l . The end of the lift drop is t c,post isdefined as the time at which the lift has dropped more than 96 % of the total lift drop. convective timescale marks again the time at which the blade exceeds the critical stall angle. The loads forall runs show a highly repeatable inertial response to the step manoeuvre characterised by lift fluctuationsat 4 Hz over 5.5 convective times. The fluctuations progressively decay and the lift returns to the same levelas before the step manoeuvre. Similarly to the response to the continuous ramp-up motion, the lift doesnot collapse immediately after the angle is increased beyond the critical angle but remains at its pre-stallvalue for a time duration that varies for all runs. For the highlighted run, the lift coefficient has a last localpeak above the pre-stall level around 14 convective times after the blade manoeuvre is conducted, followedby a drop from 1.05 to 0.6 in 9 convective times. About 32 convective time after the step manoeuvre, theflow reaches a post-stall stage dominated by large periodic load fluctuations.For both manoeuvres, we distinguish three flow stages that characterise the transient flow developmentfrom fully attached to a fully stalled condition: a holding stage, a drop stage, and a relaxation stage. Thetime delays associated with these stages are indicated in figure 2 and are defined as follows:1. the reaction delay defined as ∆ t c,reaction = t c,pre − t c,ss , where t c,pre is the time when the lift drops below anupper threshold limit, and t c,ss is when the blade exceeds its critical stall angle of 14.2 ° ,2. the drop time delay ∆ t c,drop = t c,post − t c,pre where t c,post is the time when the lift coefficient drops below a6 igure 3: Drop time ∆ t c,drop occurrence histogram for all runs of (a) the slow continuous ramp-up and (b) the step manoeuvrebased on threshold levels of 4 % and 96 % to identify the start and end of the lift drop. The solid line represents the fittednormal distribution corresponding to the histogram. The dashed and dotted lines indicate how the normal distributions shiftwhen varying the threshold values. (c) Temporal evolution of the the lift coefficient for all the runs of both the slow ramp-upand step-wise experiment shifted along the time axis with respect to the middle of the drop delay. The shaded area indicatesthe overall average drop delay of 8 convective times. lower threshold limit,3. the relaxation delay ∆ t c,relax = t c,min − t c,post , where t c,min is the time when the lift reaches its first localpost-stall minimum.The upper and lower threshold levels are reference values that are determined with respect to the averagepre-stall and post-stall limits and are fractions of total lift drop. Using thresholds at each boundary ofthe drop section, instead of using local extrema values, reduces the sensitivity of timescales to the largefluctuations occurring in the reaction and relaxation stages. Here, the upper threshold level was set 4 % ofthe total lift drop ∆ C l below the average pre-stall value C l,pre-stall . The lower threshold level was set 96 % ofthe total lift drop below the average pre-stall value C l,pre-stall or 4 % of the total lift drop above the averagepost-stall value C l,post-stall . The percentages of the lift drop for the threshold limits were selected followingan iterative procedure that aimed at maximising the drop stage length while avoiding outliers issuing fromfluctuations at the edges of the drop stage. The drop stage analysis was completed following the IEEEstandards of a negative-going transition [22]. A general idea of the influence of the choice of the thresholdvalues on the results is provided by displaying results obtained with lower and higher threshold limits inthe following figures.The distributions of the extracted drop times ∆ t c,drop for both manoeuvres are compared in figure 3a,b.Both cases display a normal distribution centred around 8 convective times. The lift drop rate is constantfor both experiments and yields a highly repeatable portion of the stall transient. The selected upper limitend lower limit threshold values represent a trade off between spanning the largest possible region for thedrop stage, while limiting the spread of the reaction time occurrence histograms. Narrowing the threshold7 igure 4: Relaxation delay ∆ t c,relax occurrence histogram for all runs of (a) the slow continuous ramp-up and (b) the stepmanoeuvre. percentages (6 % to 94 %) reduces the spread but will artificial increase the repeatability of the reaction andrelaxation stages by accounting for a greater portion of the repeatable drop stage in the neighbouring stages.Widening the threshold percentages (2 % to 98 %) increases the sensitivity of the drop stage duration tofluctuations occurring during the end of the reaction and the beginning of the relaxation stage, resulting inan increased standard deviation of the distribution. The numerical values depend slightly on the selectedthresholds for the identification of the lift drop start and end, but the distributions of the drop timesconsistently show a normal distribution and the lift evolution during the drop stage is highly repeatableacross all repetitions. The self-similarity of the lift response during the drop time is clearly visualised infigure 3 where all the lift responses for both cases are shown on top of each other shifted in time withrespect to the middle of the drop time.The distributions of the relaxation times ∆ t c,relax are compared for both manoeuvres in figure 4. Therelaxation time ∆ t c,relax shows similar characteristics for both motions investigated: a skewed distributionwith a mean value around 5. The lift coefficient has a standard deviation close to 10 % of its mean valuein the post-stall stage. These significant fluctuations complicate the identification of the actual relaxationtime, as several local minima could be considered as the first post-stall minimum. Further analysis ofthe influence of fluctuations on flow detachment will clarify the role that instabilities play in post-stallrelaxation.The distributions of the reaction times ∆ t c,reaction are compared for both manoeuvres in figure 5. Thereaction time follows a standard normal distribution with an average of 32 convective times and a standarddeviation of 8 convective times for the slow continuous ramp-up manoeuvre (figure 5a). Periodic peaksspaced by approximately 4 to 5 convective times suggest that pre-stall fluctuations cause periodicallyreturning conditions that are favourable for the boundary layer to start separating. The reaction timefor the step manoeuvre (figure 5b) follows a skewed normal distribution with a mean value of 14 convective8 igure 5: Reaction delay ∆ t c,reaction occurrence histogram for all runs of (a) the slow continuous ramp-up and (b) the stepmanoeuvre. times and a standard deviation of 4 convective times. The skewness, shorter time delay, and narrow spreadrelative to the continuous ramp up motion are attributed to the step angle of attack increase to beyondthe static stall angle. The sudden and fast motion disturbs the surrounding flow, yielding increased loadfluctuations compared to the slow continuous ramp-up manoeuvre. This unsteadiness promotes full flowdetachment and creates a bias in the reaction time occurrence, increasing the repeatability between runs.To estimate the dominant frequencies of instabilities and assess their role in the occurrence of stall,we calculated a time-frequency plot for both manoeuvres. The dominating frequencies of the load fluctua-tions were quantified by completing a fast Fourier transform (FFT) on the load coefficient. The FFT wascalculated using a sliding time window over the whole time domain for each individual repetition of bothmanoeuvres. The window width was set to 5.5 convective times, which corresponds to the time durationof the load transient that followed the step manoeuvre. The ensemble-averaged temporal evolution of theamplitude spectra are presented in figure 6 for both manoeuvres. The ensemble-averaged temporal evolu-tion of lift was included to facilitate the comparison between time-scales, load fluctuations, and dominantfrequencies. The timing is indicated in terms of convective time relative to t c = 0 when the blade’s angleof attack exceeds 14.2 ° . The blade undergoes load fluctuations with a frequency of 4.2 Hz immediatelyafter the step around t c = 0, as highlighted in figure 6b. This response to a single-point excitation on theblade apparatus is equivalent to a modal test. The highest energy peak in the amplitude spectrum, around4.2 Hz, corresponds to the natural frequency of the system. Further peaks in the vicinity of 4.2 Hz areassumed to correspond to structural vibrations.High energy peaks are observed around 0.65 Hz in the transient lift drop region at the time whenlift peaks for both manoeuvres: near 32 convective times for the slow continuous ramp-up and near 18convective times for the step manoeuvre. The expected vortex shedding frequency is calculated based ona Strouhal number St = 0 .
20 found experimentally for a NACA0018 operating at a Reynolds number of9 igure 6: Ensemble-averaged temporal evolution of lift (top) and of the amplitude spectrum of lift (bottom) for (a,c) thetransient portion of the slow continuous ramp-up manoeuvre and (b,d) the step manoeuvre. Re = 7 . × [23]. The chord was chosen as characteristic length scale for vortex shedding. The projectedchord length is more significant to characterise the interplay between vortices in a fully developed wake.The formation of stall vortices was found to occur much closer to the blade in the early stages of wakedevelopment. Additionally, our experiment is at a relatively low Reynolds number, so viscosity plays a moreimportant role, reducing vortex formation length to near the blade [24, 25]. Following this argumentation,the expected vortex shedding frequency f s is: f s = U ∞ Stc = 0 .
67 Hz . (2)This frequency corresponds to the dominating load fluctuation frequencies observed around 0.65 Hzin the stall transient and post-stall stages for both manoeuvres figure 6. A load fluctuation frequency of0.65 Hz corresponds to temporally spaced lift peaks of about 5 convective times. This value also correspondsto the spread in the drop time histogram (figure 3). The temporal spacing between the peaks of the reactiontime histogram for the slow continuous ramp-up manoeuvre (figure 5a) was also around around 5 convectivetimes. The onset of stall occurs at periodically returning conditions after a randomly distributed number ofcycles when the flow is no longer influenced by the blade kinematics. The amplitude peaks around 4.2 Hz inthe stall transient and post-stall stages highlight structural vibrations. This analysis highlights the fact thatflow unsteadiness plays a central role in the timing of the transient lift drop at stall. The frequency of loadfluctuations explains the periodicity of the reaction time distribution and confirms the high repeatabilityof the drop time. 10 igure 7: Reaction time ∆ t c,reaction against load fluctuations for all runs of (a) the slow continuous ramp-up manoeuvre and (b)the step manoeuvre. Error bars represent the difference between results obtained with threshold levels 6 % to 94 % and 2 %to 98 % to identify the start and end of the drop stage. The reaction and relaxation delays varied significantly between individual runs for both manoeuvres.To assess the correlation between load fluctuation and the reaction and relaxation times, the magnitude ofthe load fluctuations was quantified. To obtain the magnitude of load fluctuations, the mean lift coefficientwas computed on a sliding window with a 5 convective time width over the full time domain. The standarddeviation was systematically calculated relative to the local mean C L . The lift fluctuation magnitude in theholding stage C (cid:48) L,pre-stall is computed as the average of the local standard deviations: C (cid:48) L,pre-stall = 1 N pre-stall N (cid:88) n =1 (cid:0) C L,n − C L,n (cid:1) (3)where N is the number of points in the holding stage, C L,n is the local lift coefficient and C L,n is the meanlift coefficient across the local window. The lift fluctuation magnitude during the drop stage C (cid:48) L,drop wascalculated with an analogous expression, replacing N pre-stall by N drop .The reaction delay ∆ t c,reaction is presented against load fluctuations in the holding stage for both manoeu-vres in figure 7. Load fluctuations for the slow continuous ramp-up manoeuvre are confined between 0.09and 0.013, which is around 1 % of the local mean lift coefficient. There is no apparent correlation betweenthe stall onset timing and the load fluctuation for the low levels of unsteadiness in the slow continuousramp-up manoeuvre, suggesting that the motion is truly quasi-steady and does not influence the onsetof flow detachment. The mean reaction time of 32 convective times observed for this manoeuvre can beconsidered to be a lower bound for the waiting time between subsequent angle of attack steps in staticstall measurements. For the step manoeuvre, we observe a wider range of fluctuation levels in the holdingstage, reaching up to 3 % of local mean lift coefficient (figure 7b). The reaction time linearly decays withload fluctuations, supporting the fact that flow unsteadiness promotes the onset of stall. The error barsrepresent the difference between results obtained with the narrower (6 % to 94 %) and the wider threshold11 igure 8: Relaxation time ∆ t c,relax against load fluctuations for all runs of (a) the slow continuous ramp-up manoeuvre and (b)the step manoeuvre. Error bars represent the difference between results obtained with threshold levels 6 % to 94 % and 2 %to 98 % to identify the start and end of the drop stage. limits (2 % to 98 %) identifying the start and end of the drop stages. The error bars are small and indicatethat these observations are not sensitive to the drop stage threshold selection for either kinematic.The relaxation time ∆ t c,relax is compared to the load fluctuations in the drop stage in figure 9. Bothexperiments show similar fluctuation levels and a clear decreasing linear trend suggesting load fluctuationsoccurring during the lift drop promote full flow detachment. The decrease happens at the same rate for bothmanoeuvres. This suggests that the onset of vortex shedding is independent of kinematics for static motions.This timescale shows a much greater sensitivity to the drop stage threshold selection for some cases. Theincreased sensitivity is due to the greater fluctuations that occur in the post-stall regime compared tothe pre-stall regime. The decay in relaxation time with increasing load fluctuations in the drop stage isnevertheless apparent.The influence of pitch rate on the reaction time was investigated by comparing results of the slowcontinuous ramp-up manoeuvre with higher constant pitch rate manoeuvres. Data was collected for reducedpitch rates ranging from 1 × − to 0.14 with 5 repetitions for each pitch rate. We systematically computethe reaction time ∆ t c,reaction = t c,pre − t c,ss , where t c,pre is the time where the lift coefficient starts to drop and t c,ss is when the blade exceeds its critical stall angle of 14.2 ° . The average reaction time calculated over the 5repetitions is presented against reduced pitch rate in figure 9. The error bars show the standard deviationand indicate the spread or the width of the distribution of the measured reaction times at a given reducedpitch rate.The stall delay or reaction time decreases with increasing pitch rate following a power law decrease. Theadditional unsteadiness added to the flow at higher pitch rates promotes flow detachment and the onset ofstall. The reaction time decreases rapidly for reduced pitch rates below 0.01 and reaches a plateau at 3convective times for reduced pitch rates above 0.04. This lower limit is of the order of the vortex formation12 igure 9: Average reaction time delay ∆ t c,reaction as a function of reduced pitch rate for three different airfoils. Results from thecontinuous ramp-up motion of our NACA0018 are compared with previous results from a sinusoidally pitching OA209 airfoilat Re =9.2 × [12] and a sinusoidally pitching NACA0015 airfoil at Re =9.2 × [26]. time of the dynamic stall vortex, which is a classical hallmark of the transition from an attached to amassively separated flow [27, 15]. The plateau represents a minimum timescale the blade requires to forma leading edge vortex and reach a fully separated flow condition [17]. Stall onset and vortex formation arecharacterised 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 thestatic and dynamic stall responses are phenomenologically the same and their timescales vary continuouslyas a function of the pitch rate of the underlying motion kinematics.The standard deviation also decreases rapidly with increasing reduced pitch rate. The airfoil kinematicsplay a lesser role in the flow development at extremely low pitch rates and do not longer promote theoccurrence 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 differentairfoil geometries, kinematics, and Reynolds number. The timescales of the NACA0018 are compared withthose 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 thetime 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 measuredstall delays or reaction times for the three different airfoils, subjected to different kinematics, at differentReynolds numbers all collapse onto the same power law decay. This suggest that stall onset timescales areuniversal 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 behaviour13 igure 10: Estimated static stall accuracy ∆ α and inertial component of the lift coefficient C L,inertial against reduced frequencyfor static stall measurements performed with a continuous and uniform ramp-up motion. 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 thepitching motion presented in figure 9 can be used to lay out guidelines for reliably measuring the static stallangle and lift and drag polars. The systematic acceleration and deceleration related to a stepwise increaseof the angle of attack is more likely to disturb the flow than a continuous motion. A continuous ramp-upmotion with slow uniform pitch rate is thus preferred but how slow is slow enough? To answer that, wefirst fit a generalised power law decay to our experimental data yielding the following expression:∆ t c,reaction = 1 . (cid:18) ˙ αc ∞ (cid:19) − . . (4)This expression is used to determine the angular accuracy for the measurement of the static stall angle fora given pitch rate determined by the angular increase that occurs during expected stall reaction time:∆ α = ˙ αc U ∞ t c,reaction = 2 . (cid:18) ˙ αc ∞ (cid:19) . . (5)The motion should be slow enough to minimise ∆ α . In addition, we want to limit the inertial lift contri-butions associated with a dynamic motion. The inertial contribution to the lift coefficient for a continuousramp-up motion can be estimated based on Theodorsen’s theory [28] as: C l,inertial = π ˙ αc ∞ . (6)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 inertiallift contributions are a lesser issue than the stall angle increases. Reduced frequencies of the order of1 × − yield a static stall angle accuracy of ∆ α < ° . The quasi-steady inertial contribution for thesepitch rates are negligible. When measuring a static lift response using a continuous slow ramp-up motion,14he lift response can be considered a conventional static force response, free of unsteady and quasi-steadyinfluences, and providing a reliable estimate of the critical static stall angle for ( ˙ αc ) / (2 U ∞ ) < × − .
4. Conclusion
We investigated the dynamic load variations and timescales of static stall by measuring loads acting on ablade undergoing two quasi-steady manoeuvres: (i) a slow continuous ramp-up motion from angles of attack8 ° to 18 ° at a constant reduced pitch rate of 1.3 × − and (ii) a step-wise increase in angle of attack from14.2 ° to 14.8 ° within 0.04 convective times. We defined three characteristic time delays associated with thetransient flow development from attached to fully separated in response to the two types of manoeuvres: areaction delay, a drop delay, and a relaxation delay.The reaction delay is the time delay between the moment when the blade exceeds its stall angle of 14.2 ° and the moment when the lift collapses. This timescale characterises the duration of stall onset, whichplays a central role in the distinction between static and dynamic stall. The reaction time is not influencedby pre-stall fluctuations for the slowest continuous ramp-up motion, and its occurrence histogram follows anormal distribution centred around 32 convective times. The unsteadiness induced by the step manoeuvreleads to pre-stall load fluctuations that are three times larger than those induced by the slow continuousramp-up manoeuvre and do promote the onset of stall. The reaction delay linearly decreases with increasingfluctuations in the step manoeuvre.The results for the reaction delay from the slow continuous ramp-up motion were compared with resultsfrom dynamic ramp-up manoeuvres with reduced pitch rates ranging from 1.3 × − to 0.14 and withpreviously obtained results from dynamic sinusoidal pitching motions with different airfoil geometries atdifferent Reynolds numbers. This comparison revealed a universal power law decay of the stall delaysfrom 32 convective times for the lowest pitch rates to a plateau around 3 convective times for reducedpitch rates above 0.04. The plateau level matches the vortex formation time, which is the minimum timeinterval 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 withincreasing pitch rate which aids in promoting the occurrence of stall. Static stall is not phenomenologicallydifferent than dynamic stall and is merely a typical case of stall for low pitch rates where the onset of flowseparation is not promoted by the blade kinematics.Based on the results, we propose that conventional static stall polars should be measured using acontinuous and uniform ramp-up at a reduced frequency < × − to minimise the angle of attack variationduring the stall delay. The inertial lift contributions at these pitch rates are negligible. A continuous motionis preferred to a stepwise increase as the systematic acceleration and deceleration of a step motion is morelikely to cause an unsteady flow response. If a step-wise motion is selected, it is advised to wait at least30 convective times between the end of the step and the start of the measurements to allow for the flow to15espond to the change in the angle of attack. Acknowledgements
This work was supported by the Swiss national science foundation under grant number PYAPP2 173652.
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