frequency on the turbine. The model developed and validated in Chapter3 demonstrates that the averaged flow field of the combined pitch/surge flow including the dynamic stall phenomenon can be captured using simply the first two harmonics of the pitch/surge frequency. While these primary and secondary separation modes model the physics of the flow, including leading edge vortex formation, their relationship determines the phase in the airfoil motion where separation and reattachment take place. LEV separation and stall occur before the change in direction of the maximum angle of attack and thus an additional physical mechanism, beyond the pitching/surging motion, determines the modal phase relationship that results in the point of dynamic stall.
0 10 20 29 20 10 0 0
0.1 0.2 0.3 0.4 0.5
Angle of attackα UΓ c¯U
A Φ B Ψ C
(a)
0 10 20 29 20 10 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Angle of attackα
A Φ B Ψ C
Vortexdiameter[c]
(b)
0 10 20 29 20 10 0
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2
X(vortex)[c]
Angle of attackα
A Φ B Ψ C
(c)
0 10 20 29 20 10 0
0 0.1 0.2 0.3 0.4 0.5 0.6
Angle of attackα
A Φ B Ψ C
Y(vortex)[c]
(d)
Figure 4.1: Leading edge vortex circulation, diameter, and position over half a pitch up/down cycle α≥0.
0 50 100 150 200 250 300 350
−30
−20
−10 0 10 20 30
α
0 50 100 150 200 250 300 350
0.5 1 1.5
x 105
Turbine Angle θ
R e
A
B
C Ψ
Φ
Figure 4.2: Location of points A-A’ from figure 3.1 in pitch/surge cycle. Angle of attack α (top) Reynolds number (bottom).
Figure 4.3: Γ2= 2.2/πcontour of identified leading edge vortex colored by vorticity. Positive angle of attack half of pitch/surge period shown. Time shown from right to left to provide best angle to view vortex structure.
at increasing angle of attack. As the airfoil passesα= 10◦, however, the size and strength of this region decrease as flow rotation at the leading edge begins to dominate.
4.2.2 Leading edge vortex development
Atα∼15◦, point Φ, the leading edge vortex begins to dominate the rotation of the flow around the airfoil and is identified by the Γ criteria. This vortex can be seen from above in figure4.3between points Φ and B, as well as from below in figure4.5. In this regime the core of this vortex remains in a constant position within 5% of the leading edge inxand about 0.2-.25c above the airfoil centerline in y. Initially the vortex is nearly circular just above the leading edge as seen atα+= 19◦in figure4.4 and at the beginning (left) of figure4.5. In this regime this strong nearly circular core that contains most of the vorticity remains at the leading edge. As time progresses, however, the vortex includes the flow that rotates around the airfoil behind this core with weaker vorticity. This can be seen from above as the blue structure growing toward the trailing edge in figure4.3, and as a thin layer extending from the top of the vortex core visible from below in figure4.5. In this region the leading edge vortex circulation uΓ and equivalent diameter grow linearly as more and more circulation is entrained in the LEV (figures4.1(a) and 4.1(b)) and the weaker rotational layer behind the LEV grows toward the trailing edge.
Figure 4.4: Vorticity contour with velocity vector plot near the beginning of LEV formation (α= 19◦). Γ2= 2/πcontour in white at the leading edge of the airfoil.
Figure 4.5: Isocontour of Γ2 = 2.2/π colored by vorticity seen from below indicating the leading edge vortex during the development regime beginning at point Φ, at the left side of the isocontour and ending at B on the right side of the contour. From this angle the primary core of the vortex can be seen as a circular structure. This structure remains just above the leading edge of the airfoil, which has been removed from the image, so the shape of the vorticity isocontour can be observed.
4.2.3 Leading edge vortex separation
At point B the strong circular core of vorticity at the leading edge that characterizes the LEV in the development regime begins to extend backward toward the trailing edge. This results in a strong vorticity signature appearing in the Γ2= 2.2/πisocontour behind the leading edge (figure4.3). This corresponds to the center of the vortex beginning to move toward the trailing edge and up away from the airfoil in they direction (figures4.1(c)and4.1(d)). This rearward motion of LEV toward the trailing edge indicates moment stall, as defined byCarr et al.(1977). Since the vortex has begun to move back at this point and is no longer attached to the leading edge of the airfoil, point B is defined as the time when the leading edge vortex is shed.
After the vortex has shed the diameter continues to grow at the same rate as during LEV formation (figure 4.1(b)); however, the rate of circulation growth increases rapidly as the vortex convects into the shear layer (figure4.1(a)). This apparent increase in circulation is observed because after the LEV is shed, the Γ2≥2.2/π includes the vortex as well as the developing separated shear layer behind it. This increase in total circulation likely corresponds to the continuing lift increase as the LEV convects over the airfoil surface before lift stall, as defined by Carr et al.(1977). The LEV separation regime ends when the center of the vortex reachesxv= 0.5c(figure 4.1(c)) and the vortex extends all the way back to the trailing edge at point Ψ.
4.2.4 Stalled flow
Finally, after Ψ the leading edge vortex structure leaves the airfoil behind, followed by the trailing edge vortex seen in figure3.4 and investigated byRival et al.(2009). At this point the circulation of the vortex is no longer bound to the airfoil, and lift stall occurs (Carr et al., 1977). After lift stall, the flow is fully separated, and the Γ criteria pick up vortices in the separated shear layer.
As such, the isocontour in figure4.3 no longer shows a single coherent vortex but instead appears rough, indicating multiple smaller vortices. As the LEV convects away from the airfoil and decays, circulation plateaus and eventually drops, while the size of the rotational region increases to include large portions of the separated shear layer.
4.2.5 Non-phase averaged results
A comparison of the phase averaged data with that of a single measurement in the front field of view, during vortex development, and after vortex shedding (figure4.6) show similar general behavior, but smaller structure in the instantaneous measurement that is removed with phase averaging. This effect
(a)α+=27◦.(b)α=30◦.
Figure 4.6: Vorticity contours from phase-averaged realization (left) and instantaneous (right) from the front field of view. Red and blue contours indicate positive and negative vorticity, respectively.
The green area indicates the PIV laser shadow.
is similar to the measurements performed bySim˜ao Ferreira et al.(2009) who found smaller vortical structure in the instantaneous measurements, within the phase-averaged LEV contour. Their results found a very slightly (∼5%) stronger LEV circulation values in the instantaneous measurement due to small amounts of negative vorticity being included in the phase averaged LEV contour. These slight differences however do not change the results or conclusions presented in this section.
4.2.6 Vortex formation time
The development of the leading edge vortex can be further studied by considering the variation of the vortex circulation with time. The circulation of the LEV from α= 0◦ through separation at point Ψ from figure4.1is plotted against the airfoil convective time scale ˆT =RT
0 U(t)
c dtin figure4.7, whereU(t) is the time dependent freestream velocity relative to the airfoil. The LEV development phase between Φ and B occurs in ˆT ∼4. This is consistent with the universal vortex formation time ofGharib et al.(1998), with unsteady conditions as developed byDabiri and Gharib(2005). This result suggests that at such low reduced frequencies (k= 0.12), the vortex formation is dominated by quasi-steady dynamics, and the vortex is able to grow to its maximum strength before convecting downstream at moment stall. Furthermore during this regime the rotational velocity of the airfoil
Figure 4.7: LEV circulation to formation time ˆT over vortex growth period.
˙
α is small (and decreasing as it approaches maximum angle of attack), suggesting that the flow devlopment during vortex formation is not strongly affected by the increasing angle of attack. As such, the leading edge vortex development and shedding that leads to dynamic stall are dominated by the optimal vortex formation time. Therefore, while LEV formation is linked to the motion of the airfoil and appears in the phase-averaged results, the timescale responsible for the phase at which LEV develops and sheds, and thus the phase between the separation modes in Chapter3, is associated with vortex formation while the arirfoil motion period determines the beginning of the vortex formation, (i.e. point Φ). This result is consistent with experiments byRival et al.(2009) on plunging airfoils at slightly higher reduced frequencies.