Time-averaged and time-resolved volumetric velocimetry measurements of a laminar separation bubble on an airfoil

European Journal of Mechanics B/Fluids 41 (2013) 46–59

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European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu

Time-averaged and time-resolved volumetric velocimetry measurements of a laminar separation bubble on an airfoil Redha Wahidi a,∗ , Wing Lai b , James P. Hubner a , Amy Lang a a

Department of Aerospace Engineering and Mechanics, The University of Alabama, Tuscaloosa, AL, 35487, USA

b

TSI Incorporated, Fluid Mechanics Division, Shoreview, MN, 55126, USA

highlights • • • •

Performed volumetric 3-component measurements of low Re number flow over an airfoil. Related 3D Reynolds stresses to transition and reattachment of transitional bubble. Reynolds stresses indicate growth of 3D disturbances before breakdown to turbulence. Organized structures are recognized between vortices and ejection and sweep events.

article

info

Article history: Received 15 October 2012 Received in revised form 16 February 2013 Accepted 15 April 2013 Available online 2 May 2013 Keywords: Separation Transition Reattachment Vorticity Volumetric measurements

abstract Measurements of a laminar separation bubble on an NACA4412 airfoil were performed using volumetric three-component velocimetry (V3V) at a Reynolds number of 50,000 and different angles of attack. The time-averaged V3V results were analyzed to estimate the locations of transition onset and reattachment by interpreting the relationships of the different Reynolds normal and shear stresses to these locations. More specifically, the locations of separation and mean reattachment are defined as the beginning and end of the reverse-flow region, respectively. The onset of transition is located by finding the point where the growth rates of the Reynolds shear stresses changes significantly. The results show that the spanwise Reynolds normal stresses and the Reynolds shear stresses grow in the transition and reattachment regions, respectively, suggesting that three-dimensional disturbances grow before the complete breakdown to turbulence. Counter-rotating streamwise and wall-normal vorticity exist in the reattachment region. Organized relations are found between the streamwise vorticity, vortex cores, and ejection and sweep events. © 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction Laminar separation bubbles (LSB) occur on airfoils operating at low Reynolds numbers (50,000–1,000,000) [1,2]. The existence of an LSB increases the pressure drag and, in some cases, (depending on the Reynolds number, angle of attack, and airfoil geometry) also decreases the lift. The increase in drag and decrease in lift adversely affect the performance of airfoils operating in the low Reynolds number regime such as wings of sail planes, unmanned air vehicles (UAVs), micro air vehicles (MAVs), wind turbine blades, and low pressure turbine blades. Moreover, the unsteady nature of the LSB, in general, and its reattachment, in particular, creates an additional challenge to the design of small UAVs and MAVs. Therefore, there is increased interest to further understand the dynamics of laminar separation bubbles especially if passive and

∗

Corresponding author. Tel.: +1 662 617 4541. E-mail addresses: [email protected], [email protected] (R. Wahidi).

0997-7546/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechflu.2013.04.003

active flow control techniques are to be developed. In addition to these practical reasons for the increasing interest in this flow phenomenon, LSBs attract academic interests since they involve strong interactions between laminar separation, transition and turbulent reattachment occurring over a relatively small distance. Low-Reynolds-number laminar flow over an airfoil can separate when it encounters an adverse pressure gradient. The laminar separated shear layer then transitions due to the growing disturbances in the flow. If the Reynolds number is high enough, or if the angle of attack is low enough, the separated shear layer reattaches to the surface of the airfoil as a turbulent boundary layer downstream of a reverse-flow vortex. The region enclosed between the locations of separation and reattachment is referred to as the laminar separation bubble. Two types of LSBs are classified by a criterion introduced by [3], based on the displacement thickness Reynolds number at separation (Reδs∗ ). This classification states that for short and long bubbles, Reδs∗ > 450 and Reδs∗ < 400, respectively. The existence of the long bubble significantly alters the surface pressure distribution with a much smaller suction peak

R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

resulting in lower lift at a given angle of attack. Normally at low Reynolds number flows, the long bubble is a transformation of a short bubble to a long one as the angle of attack is increased. This transformation appears as a ‘‘kink’’ in the lift curve [4]. The long bubble smoothly grows as the angle of attack increases until it fails to reattach to the airfoil surface and extends into the wake causing a major decrease in lift and increase in drag. The short bubble, on the other hand, slightly decreases the magnitude of the suction peak of the pressure distribution, and the peak continues to increase with increasing the angle of attack. The short bubble appears as a perturbation to the surface pressure distribution (Cp ) in the form of a constant pressure region. Several investigations [5–9] have examined the details of the transition process in the separated shear layer of the LSB. These investigations indicated that the Tollmien–Schlichting (TS) waves in the attached boundary layer increase in magnitude and the inviscid Kelvin–Helmholtz (KH) instabilities amplify in the separated shear layer resulting in the development of three-dimensional vortices and transition to turbulence. The contributions of linear and secondary instabilities and threedimensional disturbances to the transition process in laminar separation bubbles have been investigated [10]. The results showed that the initial instability is of TS-type, and it was suggested that only in larger bubbles do the KH instabilities contribute to the transition process. In support of the suggestion in [10,11] it was found that the separated shear layer of the long bubble is governed by KH instabilities, and related this observation to the stronger velocity gradient normal to the airfoil surface. These KH instabilities, rather than small-scale instabilities, cause a quasi-periodic roll up of the shear layer and the formation of a large vortex in the reattachment region which separates from the recirculation region and travels downstream. Due to the recent advancement in flow measurement techniques such as Tomo-PIV and volumetric three-component velocimetry (V3V), researchers are revisiting this flow phenomenon in an effort to uncover more details about separation-induced transition and the subsequent reattachment. These laser-based techniques enable three-dimensional flow measurements to assist in the better understanding of the LSB structure. The scanning PIV measurements of [11–13] delivered insight about the three-dimensional vortical structure in the reattachment region of a long laminar separation bubble. Only more recently have instantaneous three-dimensional, three-component measurement techniques become available. Measurements of laminar separation bubbles were carried out by [14,15]; however, details of the Reynolds normal and shear stresses, reverse-flow region and vorticity and vortical structures were not presented in detail in these studies. The purpose of this investigation is to measure the laminar separation bubble on an airfoil at a low Reynolds number to correlate separation, transition and reattachment with Reynolds normal and shear stresses in all three dimensions. Also, since the V3V enables instantaneous snapshots of the three components of the velocity vectors in a three-dimensional volume, details of the streamwise, spanwise and wall-normal vorticity, and vortical structures are presented and discussed. The LSB measured in this investigation, based on computationally-determined characteristics of the pressure and shear stress distribution and the experimentally-determined characteristics of separation and reattachment, is of the short type. 2. Experimental facility and procedure The experiments were carried out on an NACA4412 airfoil in a water tunnel facility at The University of Alabama. The length, width and height of the test section of the tunnel are 2.743 m

47

Fig. 1. Schematic of the experimental setup (laser volume is normal to the page).

(108 in.), 0.381 m (15 in.) and 0.762 m (30 in.), respectively. The tunnel is capable of delivering freestream velocities of 0.5 m/s (1.64 ft/s) with a manufacturer-reported turbulence level of 0.4% at a freestream velocity of 0.051 m/s (2 in./s). A two-dimensional NACA4412 airfoil is made from clear plexiglass, painted black and sanded to ensure smoothness of the surface. The airfoil model has a chord (c) of 0.3048 m (12 in.) and a span of 0.610 m (24 in.). Fig. 1 shows a schematic of the experimental setup where the model is vertically placed in the water tunnel using a mounting bracket at the top and a bushing at the bottom. This connection is made by running a steel rod through the airfoil’s aerodynamic center. This mounting system ensures that the airfoil is held securely in the water tunnel and allows the angle of attack to be changed. To minimize the effects of the tunnel walls, the airfoil’s chord line is positioned at the midpoint of the test section width. The solid blockage is estimated to be 3.8% and 4.5% for the airfoil at α = 4° and 6°, respectively. The airfoil extends 0.102 m (4 in.) above the water free surface. A bushing connected to the bottom end of the airfoil creates a gap of 0.0254 m (1 in.) between the spanwise end of the airfoil and the bottom wall of the water tunnel. The center of the measurement volume is located at the airfoil’s midspan location. The flow is measured using volumetric three-component velocimetry (V3V). The flow is seeded with 50 µm polymer particles, and the volume of interest is illuminated with a 425 mJ/pulse, dual-cavity Nd:YAG laser. These pulses are spread into laser volumes and directed into the water tunnel through the bottom window. Two cylindrical lenses are used expand the laser beam into a laser cone and to control the size of the illumination volume. The boundaries of the illumination volume are further set by masking the region away from the airfoil surface. This is done to concentrate the illumination to the region of interest which results in higher vector concentration. The V3V camera has three apertures, where each aperture contains a 4 million pixel CCD sensor. The three CCD sensors are arranged in a triangular pattern which allows the search and identification of the location of the seed particles in the 3D volume, using a triplet search algorithm. Similar to image capture in PIV, two laser straddled captures with the camera probe are acquired and synchronized with two laser pulses as one realization. After acquiring the images, the Insight V3V 3GTM software scans each image for particles which are identified and matched among the other two apertures according to a calibration map to determine the threedimensional particle locations at the two instances of time. The particles at the two captures are tracked in 3D space using the relaxation method [16,17] and interpolated onto a regular grid to yield instantaneous volumes of approximate size of 140 mm by 140 mm by 100 mm in the streamwise, spanwise and wallnormal directions, respectively. A typical single capture yields

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R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59 Table 1 Estimated values of separation (S), onset of transition (T ) and mean reattachment (R) obtained from XFLR5 software.

a

α

S (X /C )

T (X /C )

R (X /C )

4° 6°

0.42 0.37

0.64 0.57

0.70 0.61

b Fig. 3. Spanwise slice (x–z plane) colored by averaged U /U ∞ ; α = 6°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3. Results

Fig. 2. Results obtained from XFLR5 (a) Cp curve and (b) Cf curve.

12,000 independent randomly-spaced velocity vectors. The vectors are interpolated onto a rectangular grid by defining grid nodes throughout the volume spaced at 3.15 mm and vectors within 4.5 mm of each grid node are averaged using Gaussian-weighting based on distance from the grid node. The uncertainty in the velocity measurements is estimated to be 1% in the x and y directions (U and V velocity components) and 4% in the z direction (W velocity component). The pulse separation between the two laser pulses, 1t, is set to allow the particles to move 4–6 pixels. It must be noted here that the particles in each image represent the particles in the entire volume. This means that some particles are closer to the surface and others are in the freestream. Therefore, several raw images must be examined carefully by the experimenter before 1t is selected since the movement of the particles varies significantly within the measurement volume. Three overlapping measurement volumes are needed to cover the entire chordwise distance of the airfoil. The center of each measurement volume is aligned with the midspan location. At each measurement volume, five-hundred statistically-independent instantaneous realizations (images) are captured at 7.25 Hz. The streamwise, spanwise and wall-normal coordinates used in this investigation are x, y and z, respectively. The corresponding velocity components are U , V and W for the streamwise, spanwise and wall-normal directions, respectively. Planar fields in all figures are defined (labeled) by the unit vector direction normal to the plane. The measurements are conducted at a Reynolds number (Rec ) of 50,000 based on the chord length and freestream velocity. Two angles of attack are investigated which are 4° and 6°.

Since the structure of long and short laminar separation bubbles may differ [1–3], a distinction between them in relation to this study is necessary. As mentioned above, a short laminar separation bubble appears as a perturbation to the surface pressure distribution (Cp ) in the form of a constant pressure region. The beginning of this pressure plateau indicates the location of separation (S), and its downstream end signifies the location of the onset of transition (T ). The location of the time-averaged reattachment normally corresponds to the location where the slopes of the viscous (actual) and inviscid (or fully turbulent) Cp curves match [1,18–21]. Surface pressure data obtained from XFLR5 software, a free version of XFOIL, are shown in Fig. 2(a). The Cp curves indicate that the laminar separation bubble at this Reynolds number and angles of attack are of the ‘‘short’’ type since they only appear as a perturbation to the Cp curves. The skin friction coefficient (Cf ) data (Fig. 2(b)) are also obtained from the XFLR5 software to estimate the separation and reattachment locations. Separation is where Cf turns negative and reattachment is where Cf turns positive again in the mean. The location of onset of transition can be directly obtained from the XFLR5 software which utilizes the eN transition prediction method. An N value of 9 is recommended for a wind tunnel with a turbulence intensity of 0.1%–0.2%. However, since the turbulence intensity in the current facility is around 0.4%, an N value of 5 is selected to represent the higher turbulence intensity in the current water tunnel. The higher N value is determined based on Eq. (1) as proposed by [22], N = −8.43 − 2.4 ln(TI )

(1)

where TI is the turbulence intensity. The predicted locations of separation (S), onset of transition (T ) and time-averaged reattachment (R) obtained from XFLR5 are summarized in Table 1. 3.1. Time-averaged results Fig. 3 shows a slice in the spanwise direction (x–z plane) flooded by the time-averaged streamwise velocity (U /U∞ ), with the flow from left to right. The separated shear layer and the recirculation region (colored in gray) are apparent and well resolved. The timeaveraged reverse-flow regions are presented in spanwise slices in Fig. 4. The reverse-flow regions start at x/c = 0.45 and 0.42 for the cases at 4° and 6°, respectively, which are slightly downstream of the predicted XFLR5 separation locations at x/c = 0.42 and

R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

a

a

b

b

Fig. 4. Spanwise slice (x–z plane) colored by U /U ∞ < 0 (a) α = 4°, (b) α = 6°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.37. The reverse-flow region for the case at α = 6° begins upstream of that at α = 4° suggesting an earlier separation at the higher angle of attack. For both angles of attack, the reverse flow moves faster in the rear portion of the bubble reaching velocities larger than 15% and 20% of the freestream velocity for the cases at 4° and 6°, respectively. The location at which the reverse flow becomes stronger corresponds with the turbulent portion of the bubble (distance between the locations of onset of transition and reattachment). The recirculation region extends downstream of the numerically-predicted location of reattachment and scattered regions of recirculating flow extend to a location slightly upstream of the trailing edge of the airfoil (scattered regions vary in the spanwise direction and can only be seen from a top-view image). The existence of these scattered regions of reverse flow could be attributed to unsteadiness of the reattachment region. Previous results [5] showed that regions of backflow exist intermittently downstream of the mean reattachment location. Therefore, for the time-averaged resultant to show regions of scattered large backflow, these regions must occur at low frequency, close to the sampling rate, with stronger upstream flow accompanied with variation in the spanwise direction. A backflow coefficient (BFC) is introduced here to obtain insight on the intermittency within the recirculation region. The BFC is defined as the percentage of time where the flow moves in the upstream direction. Fig. 5 shows the BFC on a spanwise slice (x–z plane) at y/c = 0. The BFC reaches values near 80% between the locations of separation and transition and downstream of reattachment, and decreases between (T ) and (R). DNS results [5] showed that the percent of time where the flow is reverse starts upstream of separation and increases to 90% near reattachment after which it continues to decrease to zero after several bubble lengths downstream of reattachment. A similar behavior is observed here as the BFC values of 40% appear upstream of separation. Additionally, previous DNS results [5] showed a region between separation and transition of a sudden decrease in the percentage of time where the flow was reverse which is also seen in Fig. 5; however, a discussion of this observation was not provided. A reviewer of this paper explains this behavior as an indication of the demarcation between the dead-air zone downstream of separation and the much stronger reverse-flow vortex upstream of reattachment. In the same study, they performed a probability density function (PDF) analysis along the bubble at different streamwise locations. Their results showed

49

Fig. 5. Spanwise slice (x–z plane) colored by BFC (a) α = 4°, (b) α = 6°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

that the locations of separation and reattachment based on the negative Cf values correspond to the locations where the flow moves in the upstream direction 50% of the time. The BFC value in the current study is approximately 85% near the experimentallydetermined location of reattachment with larger gradient of the BFC values in the wall-normal direction than in the streamwise direction. Figs. 6 and 7 show iso-surfaces defined where the timeaveraged streamwise velocity (U) is −0.04U∞ and −0.1U∞ , respectively, and colored by the BFC. The BFC value at U /U∞ = −0.04 is more than 75% in the forward portion of the bubble whereas it is less than 40% in the aft portion of the bubble. This decrease in BFC is gradual along the streamwise direction of the bubble. Fig. 7 shows that only in the aft portion of the bubble the reverse flow reaches −0.1U∞ . The corresponding backflow coefficient in this region reaches 70%. It is noted that there is more variance in the BFC values at U /U∞ = −0.1 with areas of BFC reaching 90%. Comparing the results in Figs. 6 and 7, it can be concluded that the backflow in the aft portion of the bubble is stronger than in the forward portion of the bubble and that the slower backflow dominates the forward portion of the bubble. The scattered regions of reverse flow downstream of reattachment mentioned above are clear in these figures and they are more obvious for the case at 4°. It is also clear in the figures that there is a discontinuity between these regions and the bubble. These regions have strong reverse flow reaching 10% of U∞ with large BFC values of up to 70%. Turbulence statistics are investigated to define criteria for estimating the locations of separation, transition onset and reattachment. The turbulence statistics are performed on a sample of 500 realizations. The convergence of data with 500 samples (realizations) is confirmed by investigating the Reynolds normal and shear stresses at different sample sizes. Fig. 8 shows the maximum values of Reynolds normal and shear stresses averaged over the spanwise direction. The V3V data are structured in a 3D x, y and z grid. The maximum values of the Reynolds stresses are determined at each x–y location. This results in multiple values at each streamwise location corresponding to the number of spanwise (y) grid points. The spanwise values are then averaged at each streamwise location. The Reynolds normal  stresses in the streamwise and spanwise direction ( ⟨u′2 ⟩/U∞ and



⟨v ′2 ⟩/U∞ ) clearly converge for a sample size of 250 realizations. However, the Reynolds normal stresses in the wall-normal

50

R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

a

a

b

b

Fig. 6. Iso-surfaces defined where U /U ∞ = −0.04 and flooded with BFC (a) α = 4°, (b) α = 6°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Iso-surfaces defined where U /U ∞ = −0.1 and flooded with BFC (a) α = 4°, (b) α = 6°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2 direction ( ⟨w ′2 ⟩/U∞ ) and Reynolds shear stress (⟨u′ w ′ ⟩/U∞ ) fully converge at 400 samples. This provides confidence in presenting the turbulence statistics obtained from averaging 500 samples. The larger sample required for the convergence of the data involving w ′ is probably due the larger uncertainty in this measurement. The uncertainty of the V3V measurements is much higher in the wall-normal direction (z) than in the streamwise and spanwise directions. It must be noted here that the Reynolds stresses involving w ′ (Fig. 8(c) and (d)) show variations in the region between x/c = 0.8 and 0.9. These variations still converge for a sample of 500 realizations as evident by the identical values at samples of 400 and 500 realizations. However, the reason for these variations at these locations remains unknown, and further investigation of the measurement technique is required. The streamwise, spanwise and wall-normal Reynolds normal    stresses ( ⟨u′2 ⟩/U∞ , ⟨v ′2 ⟩/U∞ and ⟨w ′2 ⟩/U∞ , respectively) are shown in Fig. 9 on slices in the spanwise direction (x–z plane). Hot-wire data [23,24], showed that the rms (root mean square) of the fluctuating streamwise velocity increases near the location of separation and further increases near the location of transition onset with a peak occurring near (U /U∞ = 0.5) and eventually exhibits a wider and larger peak in the  reattachment region. Fig. 9 displays that the larger values of ( ⟨u′2 ⟩/U∞ ) take place in the region of the separated shear layer and that the peak in the streamwise Reynolds stress widens in the region of reattachment.

These observations are in agreement with hot-wire data [23,24]. However, the slight increase near separation (S) followed by gradual increase until the transition onset (T ) and reattachment (R) locations seen in the hot-wire data is absent in the sense that large values are observed near (S)without further increase near (T ) and (R). Instead, larger values of ⟨u′2 ⟩/U∞ are observed between (S) and (T ), where they decrease near (T ) and then continue to increase at some distance downstream of (T ). The reason for the larger values of the streamwise Reynolds normal stresses near (S) could be attributed to the higher uncertainty in the measurement. Another possible reason is the reflection of the airfoil surface on the data in the proximity of the wall. The peaks of these Reynolds normal stresses occur approximately in the separated shear layer. The shear layer near (S) is very close to the surface of the airfoil where the uncertainty of the measurement is expected to be higher and the quality of seeding to be less. The spanwise and wall-normal Reynolds stresses increase in the streamwise direction and their peaks widen significantly in the reattachment region. These stresses considerably magnify downstream of transition onset and continue to increase further downstream. The same high values of wall-normal Reynolds normal stress have been shown in [11] and it was explained that these high values are generated by the vortices in the reattachment region. The growth of the spanwise and wall-normal Reynolds stresses in the reattachment region strongly suggests the evolution



R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

a

b

c

d

51

Fig. 8. Maximum Reynolds normal and shear stresses for different sample sizes; α = 4°.

a

b

Fig. 9. Spanwise slices (x–z plane) flooded with ( ⟨u′2 ⟩/U∞ ), ( ⟨v ′2 ⟩/U∞ ) and ( ⟨w ′2 ⟩/U∞ ) (a) α = 4°, (b) α = 6°.



of three-dimensional disturbances manifesting themselves in three-dimensional vortices. The results of the spanwise and wallnormal Reynolds stresses also show that the rate of change of these quantities can be used to estimate the location of reattachment. To obtain insight of the changes of Reynolds normal and shear stresses in the shear layer where they are expected to grow, topviews of the iso-surfaces of U /U∞ = 0.5 flooded by the Reynolds normal and shear stresses are presented in Fig. 10. The streamwise Reynolds normal stresses fluctuate (Fig. 10(a)) over the separated shear layer (i.e. U /U∞ =0.5). This behavior can also be seen in the maximum values of ( ⟨u′2 ⟩/U∞ ) shown in Fig. 8(a). The wallnormal Reynolds normal stresses (Fig. 10(c)) and the turbulent kinetic energy (not shown here) also fluctuate on the separated shear layer, whereas these fluctuations are not observed in, the spanwise Reynolds normal stresses (Fig. 10(b)). The reason for this behavior could be related to the measurement technique and procedure. However, it can also be speculated that the fluctuations of the





streamwise and wall-normal Reynolds normal stresses are due to the flapping of the separated shear layer. Since the flapping of the shear layer is a low-frequency phenomenon [25] and the sampling rate (i.e. 7.25 Hz) is low, it is possible that a complete period of the flapping motion is not captured even with 500 samples. This can be examined if the shear layer has a different wall-normal distance along the chord. Creating an iso-surface of U /U∞ = 0.5 (representing the shear layer) shows that the distance of the shear layer varies slightly along the chord. This variation ends downstream of reattachment as expected since the iso surface is no longer represent the shear layer but a region in the boundary layer. 2 The Reynolds shear stresses (⟨u′ w ′ ⟩/U∞ ), shown in Fig. 10(d), gradually increase in the shear layer and significantly increase at x/c = 0.7 followed by a further gradual increase. Negative values 2 of ⟨u′ v ′ ⟩/U∞ (Fig. 10(e)) grow near the transition onset (x/c = 0.7 and 0.68 for the cases at 4° and 6°, respectively) and then regions of large positive and negative values appear downstream

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R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

a

d

b

e

c

f

2 2 2 Fig. 10. Iso-surfaces of U /U ∞ = 0.5 flooded with (a) ( ⟨u′2 ⟩/U∞ ), (b) ( ⟨v ′2 ⟩/U∞ ), (c) ( ⟨w ′2 ⟩/U∞ ), (d) (⟨u′ w ′ ⟩/U∞ ), (e) (⟨u′ v ′ ⟩/U∞ ), (f) (⟨v ′ w ′ ⟩/U∞ ); α = 4°.





of the end of the reverse-flow region at x/c = 0.85. This behavior exists on iso-surfaces covering the region 0.1 < U /U∞ < 0.7 2 with the same absolute values of ⟨u′ v ′ ⟩/U∞ and continues in the region 0.7 < U /U∞ < 0.9 but with smaller absolute values of 2 2 ⟨u′ v ′ ⟩/U∞ . Regions of positive and negative values of ⟨v ′ w ′ ⟩/U∞ (Fig. 10(g)) also grow downstream of the reverse-flow region. These regions of positive and negative values of the Reynolds shear stresses confirm the existence of three-dimensional disturbances in the reattachment region. Fig. 11 presents a slice in the x–y plane flooded with the instantaneous spanwise velocity (V /U ∞ ). Regions of large spanwise velocities appear near (T ) and become stronger and appear more frequently near (R). This explains the observation



2 2 of the positive and negative regions of ⟨u′ v ′ ⟩/U∞ and ⟨v ′ w ′ ⟩/U∞ which appear in the reattachment region. Also, the growth of the 2 2 and ⟨v ′ w ′ ⟩/U∞ suggests the Reynolds shear stresses ⟨u′ v ′ ⟩/U∞ presence of wall-normal and streamwise vorticity, respectively, as will be discussed later. A criterion was suggested for locating the onset of transition 2 when ⟨u′ w ′ ⟩/U∞ < −0.001 [26]. Reasonable agreement is found between this criterion and the numerically-predicted location of onset of transition (T ). However, the threshold criterion is more inclined to be affected by measurement errors. Instead, the location of transition onset is better estimated as the location where the rate of change of the Reynolds shear stress doubles [11].

R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

53

Table 2 Locations of the increased growth rates of the Reynolds normal and shear stresses.

α



4° 6°

0.64 0.66

⟨v ′2 ⟩/U∞



⟨w′2 ⟩/U∞

0.68 0.64

2 Minimum ⟨u′ w ′ ⟩/U∞

2 Maximum ⟨u′ v ′ ⟩/U∞

2 Minimum ⟨u′ v ′ ⟩/U∞

2 Maximum ⟨v ′ w ′ ⟩/U∞

2 Minimum ⟨v ′ w ′ ⟩/U∞

0.71 0.68

0.82 0.80

0.80 0.81

0.79 0.79

0.80 0.78

Table 3 Comparison of the locations of separation (S), onset of transition (T ) and mean reattachment (R) obtained from experiments and XFLR5 software.

Fig. 11. A wall-normal slice (X –Y plane) flooded with instantaneous spanwise velocity (V /U∞ ); α = 4°.

A similar criterion is used here for estimating the location of transition onset where it is defined as the location where a significant change in the rate of growth of the Reynolds shear 2 stresses (⟨u′ w ′ ⟩/U∞ ) is observed. The significant change referred to here is when the growth rate (slope of line) approximately doubles. However, an exact value, such as double, is not preferred to be used as a criterion since it could also be affected by the uncertainty of the data. Fig. 12(a) shows the variation of the minimum values (smallest absolute values) of the Reynolds shear 2 stresses (⟨u′ w ′ ⟩/U∞ ) with the streamwise direction. The lines in the figure represent the rate of change of the Reynolds shear 2 stresses. The growth rate of (⟨u′ w ′ ⟩/U∞ ) increase by factors of 3 and 8 at x/c = 0.71 and 0.68 for the cases at 4° and 6°, respectively. These positions are downstream of the numericallypredicted transition onset locations at x/c = 0.64 and 0.57 for the cases at 4° and 6°, respectively. Fig. 12(b) and (c) show the spanwise and wall-normal Reynolds normal stresses and the rate of change of these quantities. Significant changes in the rate of growth of the Reynolds normal stresses are clear. Although the spanwise Reynolds stresses are lower in magnitude than the wallnormal stresses, they also show a significant change in their rate of growth. The growth rates increase by factors of approximately 4 and 7 for the spanwise and wall-normal Reynolds normal stresses, respectively. The increase of the growth rates of the Reynolds normal and shear stresses are larger for the case at 6°. The locations where the growth rates of the Reynolds normal stresses increase 2 occur slightly upstream of the locations where (⟨u′ w ′ ⟩/U∞ ) change their rate of growth. The rate of growth does not change downstream of these locations and continue to increase linearly in the reattachment region. It is expected to observe a large growth in the wall-normal Reynolds stresses in the transition area due to the momentum transport of the fluid. However, the significant growth of the spanwise Reynolds normal stresses suggests that three-dimensional disturbances grow in the transition region and prior to the breakdown to turbulence. Fig. 12(d) and (e) show the

α

S

S (XFLR5)

T

T (XFLR5)

R

R (XFLR5)

4° 6°

0.45 0.42

0.42 0.37

0.71 0.68

0.64 0.57

0.85 0.80

0.70 0.61

maximum and minimum values (maximum absolute values) of the 2 2 Reynolds shear stresses ⟨u′ v ′ ⟩/U∞ and ⟨v ′ w ′ ⟩/U∞ , respectively. The scatter in these Reynolds stresses is large; however, some trends can be observed. These Reynolds stresses are much smaller 2 2 in magnitude than ⟨u′ w ′ ⟩/U∞ , whereas ⟨u′ v ′ ⟩/U∞ is slightly ′ ′ 2 smaller than ⟨v w ⟩/U∞ . The maximum and minimum values of these Reynolds shear stresses change their rate of growth in the reattachment region. The growth rates of these Reynolds stresses are gradual until reaching locations slightly upstream of reattachment where significant increases in the growth rates are observed. Since the data points used to determine the slopes of the growth rates are selected arbitrarily, the locations of the significant increase cannot be determined with higher accuracy. This results in difficulty in distinguishing any effects of the angle of attack. In general, however, the growth rates approximately increase by a factor of 2. Table 2 summarizes the locations of the changes in the growth rates of the Reynolds normal and shear stresses as seen in Fig. 12. The reattachment location is defined where the reverse-flow region ends. Fig. 4 shows that these locations are at x/c = 0.85 and 0.8 for the cases at 4° and 6°, respectively. These locations are slightly downstream of the locations where the Reynolds shear stresses significantly increase as seen in Figs. 10 and 12. The increase in the growth rates of the Reynolds shear 2 2 stresses ⟨u′ v ′ ⟩/U∞ and ⟨v ′ w ′ ⟩/U∞ occur between transition onset and reattachment. It can, therefore, be speculated that the increase of the growth rates occur at the end of the transition process where the shear layer if fully turbulent. Table 3 compares the locations of separation (S), transition onset (T ) and reattachment (R) determined from the experimental results and XFLR5. The experimentally-estimated locations of separation are within 8% of the numerically-predicted locations. The difference between these values can be attributed to the uncertainty in the velocity measurements and in determining the streamwise locations. Also, the reverse-flow region is thin near the location of separation which decreases the certainty in determining the actual beginning of the reverse-flow region. The experimentallydetermined locations of transition onset and reattachment are downstream of those predicted by XFLR5. The difference between the locations determined by the experiments and XFLR5 varies between 10% and 30%. 3.2. Time-resolved results The vortex shedding from the separated shear layer is illustrated in Fig. 13 where values of λ2 are shown on slices in the spanwise direction (x–z plane) at different instants of time. The vortex identification method used here is based on the negative second eigenvalues of the velocity gradient tensor known as the λ2 -criterion [27]. The velocity vectors colored by (U /U ∞ ) are also shown in the figure to indicate that valid vectors exist in the region of the vortices. The distances between the vortices at this spanwise

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a

b

c

d

e

2 2 Fig. 12. (a) Minimum (⟨u′ w ′ ⟩/U∞ ), (b) Maximum ( ⟨v ′2 ⟩/U∞ ), (c) Maximum ( ⟨w ′2 ⟩/U∞ ), (d) Maximum and Minimum (⟨u′ v ′ ⟩/U∞ ), (e) Maximum and Minimum 2 (⟨v ′ w ′ ⟩/U∞ ).

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location, i.e. y/c = 0, are determined by locating the centers of the negative λ2 regions. By investigating several instantaneous snapshots, it is found that the distances between the vortex cores range from 0.062c to 0.082c and are on average 0.075c (22.8 mm) with the most common distance between the vortices being 0.07c. Further investigation of the vortex cores shows that some of the vortices move slightly in the spanwise direction, as will be shown later, which makes them disappear from the plane at y/c = 0. This may explain the discrepancies found in the streamwise distances between the vortices. Calculations of the convective velocity (uc ) of the vortices based on the average distance between the vortices (0.075c) and the time interval (1/7.25 Hz = 0.138 s) show that the vortices travel with the freestream velocity. This result is unexpected since the KH instabilities convect at a lower velocity than the freestream. The reason for this unexpected result is the assumption that the vortices always exist on a particular spanwise plane. As an alternative method for tracking the vortices, they can be traced on a wall-normal plane (x–y plane). Since the



vortices change their location in the wall-normal direction as well, this approach will also result in an inaccurate estimate of the convective velocity. Therefore, a three-dimensional presentation of the vortices is better suited for their tracking thereby calculating their convective velocity as will be discussed later. Fig. 14 shows the instantaneous wall-normal velocity (W /U ∞ ) presented on the same spanwise slices as in Fig. 13. Additionally, grayscale contour lines of λ2 are presented in the figure to demonstrate the relationship between the wall-normal velocity and the vortices. The pairs of positive and negative wall-normal velocities are induced by the vortices and they indicate a swirling motion about the y-axis (spanwise direction). The relationship between the swirling motion and the wall-normal velocity is seen where the centers of the vortex cores are bounded between the positive and negative wall-normal velocities. These pairs indicate a clockwise rotation of the vortex where W /U ∞ > 0 and W /U ∞ < 0 is upstream and downstream of the vortex, respectively. The negative values of W /U ∞ extend farther from the separated shear

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Fig. 13. Spanwise slices (x–z plane) flooded with λ2 and also shown the vector field colored by U /U ∞ ; α = 4°. (a) at t0 , (b) t0 + 1t, (c) t0 + 21t, (d) t0 + 31t. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

a

b

Fig. 14. Spanwise slices (x–z plane) flooded with the wall-normal velocity (W /U ∞ ) and lines colored by λ2 ; α = 4° (a) at t0 , (b) t0 + 71t. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

layer and the surface of the airfoil indicating the fluid deflection towards the surface of the airfoil. To visualize the structure of the vortices on the separated shear layer and downstream of reattachment, instantaneous isosurfaces of negative λ2 are presented in Fig. 15 and flooded with the streamwise velocity (U /U ∞ ), streamwise vorticity (ωx c /U∞ ) and

wall-normal vorticity (ωz c /U∞ ). In general, the vortices (Fig. 15(a)) have large aspect ratio and are aligned with their primary axis in the spanwise direction. Their thickness (wall-normal dimension) is 0.035c and their streamwise dimension is 0.04c. However, no common spanwise dimension is recognized due to the interaction between the vortex cores. Previous V3V results showed that the average dimensions of the vortices in the reattachment region are 0.04c, 0.06c and 0.04c in the streamwise, spanwise and wall-normal directions, respectively [14]. Thus, there is agreement between the streamwise and wall-normal dimensions presented in this study and by Ref. [14]. The vortices found in this study, however, have much larger spanwise dimension. The streamwise distances between the vortical structures are determined by estimating the centers of these structures of several snapshots and their average is found to be 0.06c. This streamwise distance is slightly smaller than the average distance found on a spanwise plane of 0.075c as discussed above. The vortices (Fig. 15(a)) extend in the wall-normal direction over a distance where U /U ∞ ranges from 0.1 to 1.3. The vortical structures are flooded by the streamwise and wall-normal vorticity to investigate the distribution of the vorticity in the vortical structures. The streamwise vorticity is distributed asymmetrically over the vortices (Fig. 15(c)) whereas the distribution of the wall-normal vorticity is not always clear (Fig. 15(d)). Asymmetric distribution of both the streamwise and wall-normal vorticity was previously reported [14] which indicates the three-dimensionality of the vortical structures. Nonetheless, more vortices at different snapshots show similar distribution as those found by [14]. The structure of the vortices found by them is more defined than the vortices of the current study, but when the threshold of the λ2 values is increased (more negative), the vortical structure becomes more recognizable. Also, the distances between the vortex cores and the chessboard pattern found in their study are not clearly identified in the current experiments. It should be noted that some of the iso-surfaces of the vortical structures shown in Fig. 15 appear incomplete where only portions of the iso-surface are shown. These structures appear incomplete because the regions

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a

b

c

d

Fig. 15. Iso-surfaces defined at negative λ2 and flooded by (a) streamwise velocity (U /U ∞ ), interpolated vectors unmasked, (b) streamwise velocity (U /U ∞ ), interpolated vectors masked, (c) streamwise vorticity (ωx c /U∞ ) and (d) wall-normal vorticity (ωz c /U∞ ); α = 4°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of interpolated velocity vectors are removed, and only the results based on the valid vectors are shown. Fig. 15(a) and (b) illustrate the effect of removing the interpolated vectors. Since some of the vortical structures exist close to the airfoil surface where the seeding quality decreases, their portion near the surface is based on interpolated vectors. The velocity vectors on planes closer to the surface are also shown in Fig. 15(a) and (b) and colored by the streamwise velocity. The vortices exist in regions of backflow or slower flow as can be seen from the vector fields. Similar observation was noted by [11,14]. Therefore, although the vortices exist in the vicinity of the separated shear layer and farther from the wall, they still project foot prints near the surface of the airfoil. Visual inspection of the vortices suggests that they travel in the streamwise, spanwise and wall-normal directions while changing their strength and orientation. Therefore, the discussion about the structure and tracking of the vortices is only qualitative and only serves as a general view of this complex behavior. Fig. 16 shows the vortex cores colored by the wall-normal distance and velocity vectors colored by the streamwise velocity, and illustrates the tracking of randomly selected vortex cores. Vortex cores and their centers are visually identified and the distances traveled by their centers at four instants of time are determined, and these distances are then averaged. This procedure is applied to different sets of images to ensure that the results are consistent. The average distance traveled of the labeled vortices in the four snapshots is 0.046c which results in a vortex drift velocity of 0.62U∞ for the case at α = 4°. This result is in general agreement with those of [11,14] who reported drift velocities ranging from 0.58 to 0.69 of the freestream velocity depending on the angle of attack and Reynolds number. The general agreement between the drift velocities of these studies and the current investigation supports

the above statement that estimating the vortex convective velocity based on the vortices visualized on a spanwise plane can be misleading. Fig. 17 shows iso-surfaces of the instantaneous streamwise vorticity (ωx c /U∞ ) and wall-normal vorticity (ωz c /U∞ ). The streamwise vorticity is oriented in the streamwise direction and their streamwise dimension is larger than their spanwise dimension. The vorticity exists in counter-rotating pairs, and its spanwise dimension increases downstream of reattachment (i.e., becomes thicker). The structure of the wall-normal vorticity is generally similar to the streamwise vorticity as they are both oriented in the streamwise direction with larger streamwise dimension. Additionally, the wall-normal vorticity exists in a counter-rotating pairs, but unlike the streamwise vorticity, its spanwise dimension does not increase downstream of reattachment. The distances between the counter-rotating pairs of the streamwise and wall-normal vorticity decrease in the reattachment region. The vorticity and their counter-rotating behavior explain the increase in the growth rate 2 2 of the Reynolds shear stresses ⟨u′ v ′ ⟩/U∞ and ⟨v ′ w ′ ⟩/U∞ . The connectivity of the counter-rotating streamwise vorticity to the vortex cores is illustrated in Fig. 18. The direction of rotation of the counter-rotating vorticity switches between the upstream and downstream ends of the vortex. This is illustrated by noticing the orientation of the counter-rotating vorticity connected to vortex cores labeled A, C and D as compared to B and E. However, for the most part, the streamwise vorticity is connected to the vortex core from the upstream end (similar to structures B and E). Moreover, the vorticity pairs connected to the vortex cores (rollers) from the upstream end are closer to the surface of the airfoil than the vortex cores (structure E). The vorticity pairs connected to the downstream end of the vortex cores are at the same level or a little further away from the wall than the vortex cores (structure D).

R. Wahidi et al. / European Journal of Mechanics B/Fluids 41 (2013) 46–59

a

c

b

d

57

Fig. 16. Iso-surfaces defined at λ2 < −35 and flooded by the wall-normal dimensions (z /c ); α = 4° (a) at t0 , (b) t0 + 1t, (c) t0 + 21t, (d) t0 + 31t. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

a

c

b

d

Fig. 17. Iso-surfaces defined at ωx c /U∞ > 10 and ωx c /U∞ < −10—at (a) t0 , and (b) t0 + 21t; ωz c /U∞ > 15 and ωz c /U∞ < −15—at (c) t0 , and (d) t0 + 21t; α = 4° – positive (red) – negative (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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By investigating a number of snapshots, similar relationship can be observed between the counter rotating wall-normal vorticity and these events, but these events do not occur as frequently to provide confidence that it is an organized structure. Iso-surfaces of Q2 and Q4 are shown in Fig. 19(b) with iso-surfaces of large λ2 (gray) to illustrate the relationship between the vortex cores and the ejection and sweep events. The ejection events occur upstream of the vortex cores, whereas the sweep events occur downstream of the vortex cores. The vortex cores are farther away from the wall than the ejection events and approximately at the same distance as the sweep events. This is an opposite behavior when compared to the streamwise vorticity where this vorticity is closer to the surface than the Q2 and Q4 events. Also, the sweep events are, in general, farther away from the wall than the ejection events.

Fig. 18. Iso-surfaces defined at ωx c /U∞ > 10 (red) and ωx c /U∞ < −10 (blue) and negative λ2 (gray) at t0 ; α = 4°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The vortical structures found in the reattachment region are investigated in relation to the quadrants of the Reynolds shear stress (u′ w ′ ). The second quadrant denoted Q2 represents the ejections of low-momentum fluid from the region near the wall to regions of higher momentum further away from the wall. The fourth quadrant (Q4 ) represents sweeps of higher momentum fluid moving towards regions of lower momentum fluids near the wall. The second and fourth quadrants of the Reynolds shear stresses are defined where u′ < 0 and w ′ > 0, and u′ > 0 and w ′ < 0, respectively. The relationship between the vortical structures and these events, i.e. ejections and sweeps, is shown in Fig. 19. Fig. 19(a) shows a top-view of the iso-surfaces defined at large positive ωx c /U∞ (gray), large negative ωx c /U∞ (black), Q2 (red) and Q4 (blue). The threshold value for the Q2 and Q4 iso2 surfaces is −u′ w ′ /U∞ = 0.025. The system presented in box A shows that ejection events occur between the counter-rotating pairs of streamwise vorticity. The pair shown in box B has an opposite direction of rotation than the pair in system B. In system B, sweep event occurs between the positive and negative vorticity. This organized structure can be seen in all the snapshots and it suggests that the pairs of counter-rotating streamwise vorticity contribute to the production of turbulence in the reattachment region of the transitional bubble. It is also noted that the vorticity exists closer to the surface than the ejection and sweep events. The relationship between the wall-normal vorticity and the Q2 and Q4 events is not as organized as the streamwise vorticity.

a

4. Conclusion Volumetric three-component measurements were carried out on an NACA4412 airfoil at low Reynolds number where a laminar separation bubble exists. The time-averaged results show that the reverse flow moves faster in the turbulent portion of the bubble reaching velocities of more than 10% of the freestream velocity with a back-flow coefficient of 70%. The instantaneous backflow in the aft portion of the bubble is stronger than in the forward portion of the bubble, whereas the forward portion of the bubble is dominated by slower reverse flow. The increasing growth rate of the spanwise and wall-normal Reynolds normal stresses in the transitional and reattachment regions indicate the evolution of three-dimensional disturbances. This is also supported by the appearance of regions of positive and negative Reynolds shear stresses which also indicate that presence of streamwise and wallnormal vorticity. Three-dimensional vortical structures appear in the reattachment region. Although these vortices occur mainly in the shear layer, they project footprints near the surface of the airfoil as regions of slow positive or reverse flows. Also, counter-rotating pairs of streamwise and wall-normal vorticity are observed in the reattachment region. The counter-rotating pairs of streamwise vorticity connect to the vortex cores from the upstream and downstream ends with opposite direction of rotation. Also, the results show some organized relations between the streamwise vorticity, vortex cores, and ejection and sweep events. Depending on the direction of rotation of the streamwise vorticity, ejection or sweep events occur between the counter-rotating pairs. The vortex cores are connected to ejection events from their upstream end and to sweep events from their downstream ends.

b

Fig. 19. Iso-surfaces defined at Q2 (red), Q4 (blue) and (a) ωx c /U∞ > 0 (gray) and ωx c /U∞ < 0 (black) and (b) negative λ2 (gray); α = 4°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. 0958668. The support of The University of Alabama is also acknowledged through the Research Stimulation Post-Doctoral Fellow Program. The authors also acknowledge the assistance and discussions with Mr. Triston Cambonie at ESPCI Paris regarding the computing schemes used in handling the V3V data. References [1] B.H. Carmichael, Low Reynolds number airfoil survey, volume I, NASA Contractor Report 165803, 1981. [2] W. Shyy, Y. Lina, J. Tang, D. Viieru, H. Liu, Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, New York, 2008. [3] P.R. Owen, L. Klanfer, On the Laminar boundary layer separation from the leading edge of a thin airfoil, ARC Current Paper No 220, 1953. [4] L.F. Crabtree, The formation of regions of separated flow on wing surfaces, ARC R&M 3122, 1959. [5] M. Alam, N.D. Sandham, Direct numerical simulation of short laminar separation bubbles with turbulent reattachment, J. Fluid Mech. 410 (2000) 1–28. [6] R. Hain, C.J. Kähler, R. Radespiel, Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils, J. Fluid Mech. 630 (2009) 129–153. [7] M. Lang, U. Rist, S. Wagner, Investigations on disturbance amplification in a laminar separation bubble by means of LDA and PIV, Exp. Fluids 36 (2004) 43–52. [8] O. Marxen, U. Rist, S. Wagner, The effect of spanwise-modulated disturbances on transition in 2-D separated boundary layer, AIAA J. 42 (2004) 937–944. [9] J.H. Watmuff, Evolution of a wave packet into vortex loop in a laminar separation bubble, J. Fluid Mech. 397 (1999) 119–169. [10] U. Rist, On instability and transition in laminar separation bubbles, in: Proceedings of the CEAS Aerospace Aerodynamics Research Conference, UK, June 10–12 2002. [11] S. Burgmann, J. Dannemann, W. Schröder, Time-resolved and volumetric PIV measurements of a transitional separation bubble on a SD7003 airfoil, Exp. Fluids 44 (2008) 609–622. [12] S. Burgmann, C. Brücker, W. Schröder, Scanning PIV measurements of a laminar separation bubble, Exp. Fluids 41 (2006) 319–326.

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