Vortex dynamics
H-type VAWT with AR =1.5: volume rendering of the vorticity
magnitude ‖ω‖; the lifting lines are also shown as 3-D
blades.
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The instantaneous wakes of the AR =1.5 machine at the three considered
TSRs are visualized through volume rendering of the vorticity magnitude in
Fig. . They allow several insights into the complex
vortical structure of the wake, which is significantly different from that of
a HAWT. We first consider the design TSR (Fig. b). The
vorticity shed in the wake consists of (i) the blade tip vortices, which
constitute the top and bottom sides of the wake, and (ii) the vortex sheets,
shed due to the time variation of the circulation of the blades, which form
the lateral sides. The tip vortices are the strongest in the vicinity of the
upstream- and downstream-most positions of the blades (around θ=90∘ and 270∘) where the blades operate at their maximum angle
of attack. There, depending on the appearance of stall, or delayed stall
effects, the blade will achieve its maximum circulation and then lose it either
abruptly or progressively, depending on whether the blade is stalled or not.
At the design TSR, the blade exploits the delayed stall to the greatest extent: its
circulation keeps increasing, well past θ=90∘, and then smoothly
decreases. This is explained by two phenomena: (1) the airfoil experiences
the highest delay in the circulation development (beneficial in this case as
it widens the extent of torque production by the blade); (2) the leading-edge
vortex does not introduce a sharp drop in circulation yet (which clearly
happens at lower TSR, see Fig. b). The
dΓ/dt vorticity shedding is maximal when the blades
are close to their lateral positions θ=0∘ (upwind leg) and
180∘ (downwind leg). The corners of the wake, i.e., the intersections
of the two types of vortical structures described above, give rise to the
fastest-growing vortical instabilities, which quickly propagate and cause the
pairing of vortices of unequal circulations. Indeed, the unsteady
aerodynamics have produced vortices with a varying circulation and the shed
vortices will interact with a different section of a preceding/succeeding
vortex. In this kind of event, the stronger vortex distorts the weaker one,
leading to intense stretching, enstrophy production, and the propagation of
disturbances along the vortex cores, therefore bringing an overwhelming
contribution to the transition to turbulence. This mechanism, most visible
in Fig. a and isolated in
Fig. , is well known in vortex dynamics and
has already been identified on aircraft
wakes . As a direct consequence, the
turbulent regions of the wake grow from the corners and the wake only reaches
a fully turbulent state once these regions have merged: the distance to reach
this state will be governed directly by the aspect ratio of the machine.
H-type VAWT with AR =1.5: volume rendering of the vorticity
magnitude ‖ω‖; vortex reconnections are visible on the side of
the wake, here illustrated in the area behind the bottom left corner of the
VAWT, at successive times. The turbine is on the right, and the velocity is
directed to the left.
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The VAWT wake decay is of course also governed by the TSR in a fashion very
similar to that of the HAWT: a high TSR (Fig. c)
induces narrower vortex separations, which directly condition the growth rate
of the instabilities and the time to the reconnection events. This directly,
and very geometrically, translates into an increasing opening angle for the
envelopes of the corner vortical structures, going from
Fig. a to c. Decreasing the
TSR below the design point actually affects the wake even more dramatically.
The stall event on the upstream part of the revolution weakens the upstream
wake contribution (between θ=90 and 180∘), generating a
stopping vortex that will be advected through the rotor
(Fig. a). One can thus expect a two-lobed wake.
Conversely, higher TSRs exhibit weaker vortical structures being advected
through the rotor. As discussed in Sect. , the
dΓ/dt sheets shed on the upstream leg will cross the
rotor and impact the blade aerodynamics on the downstream leg, with an
extreme case being the stall event discussed above.
Effect of aspect ratio at TSR=3.21: contours of the
instantaneous cross-stream vorticity component ωyD/U∞.
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The behaviors of the upstream tip vortices within the rotor are more complex
to apprehend, as they are affected by several factors: the intrinsic roll-up
dynamics of a vortex sheet (with a time-varying strength) and the velocities
induced by the surrounding vortical structures, including the bound vortices
on the blades. To some degree, the latter can be crudely linked to the
overall rotor loading (the Cx values of Table ). For a
highly loaded rotor (TSR=3.21 and 4.28), the generated blockage
effects will push the vortices shed upstream vertically and away from the
downstream blade tips. One only sees the upstream tip vortices impinging upon
the downstream blades at a low rotor loading, as is the case for
TSR=2.14. This observation does not agree with the results of
Scheurich and Brown , which showed upstream vortices
colliding with the blades at high TSRs. A possible explanation might lie in the
relatively short domain and the direct use of the unbounded Biot–Savart law
in their work. One needs to add additional terms to enforce an outflow
condition for this otherwise clipped vorticity field; the present study does
precisely that by enforcing a normal outflow velocity (∂u/∂x=0, v=0, w=0) through its Fourier-based solver .
Other explanations could also be found in a mismatch in the achieved loading
by the VAWT and the use of a curvature correction; these effects should be
further investigated.
Blockage is but one factor, however, and it is a global one. The discussion can be
refined as additional, and less immediate, effects are to be expected from
the machine geometry. The aspect ratio, as indicated by Cx in
Table , has a small effect on the blockage and one can also
expect an influence on the 3-D topology of this blockage effect: a higher AR
thus leads to an increased clearance between the vortices and the blade, as
shown in Fig. . The number of blades also has a
strong influence; the two-bladed machine of Sect. (not
shown here; see also ) exhibits such vortex–blade collisions,
in spite of its high loading Cx=0.874. Finally, beyond the rotor, the
instantaneous vorticity fields of Fig. also offer
some insights into the pairing phenomenon of the tip vortices, the generation
of a turbulent wake, and the recirculation region.
H-type VAWT with AR =1.5: mean streamwise velocity
u¯/U∞ and resolved turbulent kinetic energy
k¯=u′u′‾+v′v′‾+w′w′‾2U∞2 in the y/D=0 plane.
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Average flow statistics
The average behavior of these wakes is studied through the mean axial
velocity u¯ and the turbulent kinetic energy k¯; these
statistics were collected over a period Tavg=30D/U∞.
Figure shows a horizontal slice of these
statistics for the AR =1.5 machine. This averaged wake exhibits several
prominent features that reflect the phenomena identified in the discussion
above. In all the conditions, we observe the generation of turbulent kinetic energy (TKE) on the sides
of the wake and the associated smearing of the velocity deficit. This is
consistent with our discussion of the vortical instabilities in the corner
structures and the subsequent propagation of the turbulent regions. At low
TSR, the averaged velocity field exhibits the expected two-lobed
structure, with a stronger deficit on the side of the
rotor on which the blade is traveling upwind (θ∈[270,90∘]). For the higher TSRs
(Fig. b and c), a backflow region lies inside the
wake at a position that varies with the TSR: it is centered at x/D≃5.5 for TSR =3.21 and at x/D≃4 for TSR =4.28. The location of
this feature clearly coincides with the production of TKE and an accelerated
smearing of the wake velocity deficit; this too agrees with our vortex
dynamics discussion. The topology of the associated recirculation bubbles is
clearly three-dimensional and will not be discussed here.
Finally, the averaged wakes exhibit a slight deviation in this midplane. As
expected, the behaviors of the three TSRs do correlate with the signs and
values of the side forces produced by the rotor (see Cy in
Table ). These side forces also appear in the average
behavior as observed in cross-flow slices
(Fig. ). The deformation of the velocity
deficit clearly suggests the presence of mean streamwise vortices along the
corners of the wake (Fig. a and b), a clear
departure from a HAWT wake with no side slip angle.
H-type VAWT with AR =1.5: mean streamwise velocity
u¯/U∞ and turbulent kinetic energy k¯ in cross-flow
slices.
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The mean streamwise vorticity at three transverse slices is shown in
Fig. . Even though the statistics are converged,
the near-perfect periodicity of the flow leads to a pattern of positive and
negative patches, signatures of the advection of tip vortices shed on the
upstream and downstream parts of the rotation, respectively. The dominant
streamwise vorticity is thus difficult to identify in the near-wake, but large-scale structures can be identified further downstream, also thanks to the
induced deformation of the wake.
H-type VAWT with AR =1.5: mean streamwise vorticity
ω¯/(U∞/L).
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H-type VAWT: dimensionless displacement and momentum surfaces as
functions of the streamwise coordinate.
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Decay diagnostics
We apply classical turbulent wake diagnostics to the characterization of the
wake decay. More specifically, we adapt integral quantities, such as the
displacement and momentum widths, to the present context; the wakes considered do indeed lack symmetry and exhibit strong secondary flow structures,
which makes the definition of a velocity deficit evolution based upon a
single characteristic point unsuitable. Thus, we define dimensionless
displacement and momentum surfaces, respectively, as
S1(x)=1HD∫-∞∞∫-∞∞1-ux(x,y,z)U∞dydzS2(x)=2HD∫-∞∞∫-∞∞1-ux(x,y,z)U∞ux(x,y,z)U∞dydz.
These diagnostics correspond to integrals of flux quantities in cross-stream
sections located at a distance x downstream of the turbine axis; their
practical implementation approximates these integrals through quadrature over
finite square sections [-3D,3D]×[-3D,3D]. Because our
Biot–Savart solver enforces transverse unbounded conditions exactly, it
allows a transverse mass flow due to blockage. As a consequence, S1, shown
in Fig. a, does not vanish (as it would have for a solver
with no through-flow boundaries); it quantifies the blockage effect caused by
the wake on the flow. As a reference, S1 should be compared with the
square of the displacement width (δ2) of an axisymmetric wake for
which classical similarity theory predicts a behavior
S1∼x-1/3 in the far wake. The asymmetry of the wake generator and
its proximity are such that we cannot observe the self-similarity region:
classical results for bluff bodies indicate a development distance of
x/D∼50 to obtain the theoretical far-wake
self-similarity . The decay observed for x/D>5 for most
of the configurations does, however, suggest a power-law-like behavior. For
HAWTs, it has been observed that the decay deviates significantly from the
bluff body behavior in the presence of a turbulence inflow
; similarly, it will be interesting to assess the
sensitivity of VAWT wake decay with respect to the turbulence intensity. Still,
the evolution of S1 does provide a signature of the recirculation region:
the magnitude and the extent of the overshoot S1>1 correlate with the
location and the size of the recirculation bubble for the design and high
TSRs (Fig. b and c). This correspondence also
agrees with the effect of the aspect ratio: an increasing AR pushes both the
recirculation (indicated by the merging of vortical structures in the center
of the wake in Fig. ) and the S1 overshoot
further downstream.
The dimensionless momentum surface S2 is related to the deficit in the
flux of momentum in these planes. In the absence of secondary flows and
pressure gradients, it should in fact correspond to the thrust coefficient
S2≃Cx at large distances behind the VAWT when a factor 2 is used
in the definition of S2, as here in Eq. (). This is confirmed
by our results of Fig. b: after a transition, the curves
tend towards the corresponding Cx values of Table .
Finally, the case TSR=2.14 constitutes an outlier in the
discussions above. This is not unexpected: the instability growth is slower
than for the other cases and does not allow the transition to a well-mixed
fully turbulent wake within the computational domain.