This section begins with a discussion on the key role played by the flow attack angle in determining the energy conversion efficiency. Then the power coefficients are analysed for all the simulated case studies. Finally, three practical issues that must be considered to choose the diffuser geometry are discussed: the mitigation of torque ripple, the mitigation of the overall drag, and the wake characteristics if a multi-device arrangement is planned.
Free stream velocity and turbulence are 2.25 m/s and 5%. TSR
is set to the optimal value, that is 1.7 for bare turbines [33
] and, due to the flow acceleration inside the diffuser [26
], 1.9 for ducted turbines (as it resulted from a preliminary investigation, not reported here for brevity). The blade angular position (ϑ
) and the yaw angle of the flow (φ
) are defined in Figure 6
, together with an illustration of the upwind and downwind paths of the blade during one revolution. Irrespective of the (anticlockwise or clockwise) rotation sense the upwind path (0° ≤ ϑ
≤ 180°) starts with the blade chord parallel to the current, whereas the downwind path covers the range 180° ≤ ϑ
≤ 360°. The rotation verse of the turbine pair is set according to [30
]. In case of yawed flow, the “first” and “second” turbines are the rotor that is first and second approached by the flow (in the figure, they correspond to the lower and to the upper rotors, respectively).
4.1. Importance of the Attack Angle
In CFTs most of the torque is generated upwind, since downwind the kinetic energy of the current is much lower due to upwind blade passage. The tangential force (Ft) that generates torque coincides with the tangential projection of the overall hydrodynamic force (sum of lift and drag) exerted by the fluid on the blade. Two parameters determine Ft, they are the flow apparent velocity (W) and the attack angle (α).
As depicted in Figure 7
results from the vector composition of the absolute flow velocity (U
) and the blade speed (V
), whereas α
is the angle between W
and the chord. Since during one revolution the blade angular position changes, both α
vary in a cyclic manner. W
is maximum at ϑ
= 0° (where the U
module is added to the V
module), and is minimum at ϑ
= 180° (where U
is subtracted from V
). The highest values of α
occur at ~90° and ~270°, and the lowest at ~0° and ~180°. The lift (Fl
) and the drag (Fd
) forces are responsible for the torque generated by the blade. However, Fd
, which always opposes the blade motion, is relatively much smaller than Fl
and thus it is here neglected for simplicity of treatment. Ft
is greatly influenced by the instantaneous value of the attack angle. Indeed, the lift force on a wing of unitary span is
The above two relations are only valid in the pre-stall region, and for small values of α, for which sin(α) can be approximated whit α (in radians). Thus, a few degrees of increasing in α lead to a significant enhancing of the instantaneous torque. This would be helpful at the beginning and at the end of upwind, where α is very small not only because the blade chord is almost aligned to the free stream direction (x-axis), but also because the flow approaching the blade is x-directed only in the central part of upwind, while in early and late upwind it diverges laterally. This phenomenon (and the drag effects) justify why the blade of a single bare turbine starts to generate torque only at ϑ~30°.
gives a qualitative comparison of the velocity triangles and hydrodynamic forces when the blade is in early upwind (ϑ
~40°) for three typical situations: a bare turbine, a turbine belonging to a closely spaced pair, a ducted turbine.
In the case of the turbine pair it can be observed that the presence of the upper turbine prevents the flow from diverging to the inner sides of the configuration, and therefore U
-aligned, causing a significant increase of α
], yet, the best torque is expected to be achieved by adopting a diffuser with a convergent section, since the diffuser wall direct U
in a way that implies an even more great increasing of α
]. One could object that the W
vector is unfavourably shortened by any changing in the U
direction that leads to a more favourable α
. However, as can be seen looking at the velocity triangles of Figure 7
, the W
reduction appears very smaller respect to the α
increasing. The reader can verify this by means of the following example. Let’s assuming ϑ
= 30°, TSR
= 1.9, U
(in reality, U
is significantly smaller than U∞
since the turbine slows down the flow as an obstacle partially permeable to the flow would do). Reasonable U
angles with respect to the x-axis could be: 20° (flow divergent to the turbine side) for the bare turbine; 0° for the turbine pair; −20° (flow convergent to the turbine axis) for the ducted turbine. From the trigonometry laws it can be easily found that α
is 2.77° for the single, 8.21° for the paired turbine and 13.29° for the ducted turbine, whereas W
is 13.8 m/s, 13.5 m/s an 12.8 m/s, exhibiting a much lower sensitivity to the direction of the flow approaching the blade.
4.2. CP of the Turbine Pairs
A difficulty encountered by the designers of high performance diffusers is that the behaviour of an empty diffuser is completely different from that of a diffuser with a rotor working inside. Thus, numerical tools adopted to predict a new diffuser performance (CFD, or low order methods [29
]) should also consider the turbine presence. Preliminary simulations of our diffusers (without turbines) predicted a slight flow separation in the final part of the inner walls for the hydrofoil shaped diffuser, and a massive separation for the symmetrical one leading to a 24% lower flow rate. Nevertheless, when turbines were inserted, neither the hydrofoil shaped nor the symmetrical diffuser showed any separation of the flow, as can be seen in Figure 8
a, depicting details of the velocity map for the original diffuser. Scientists agree to attribute this trend to the effect of the high momentum jet establishing in the gap between the rotor and the duct wall when the turbine works [11
]. This jet energizes the boundary layer in the adverse pressure gradient flow through the diffuser, suppressing trailing edge separation and therefore enhancing the turbine performance through a mechanism similar to circumferential blowing [52
Another high momentum region occurs in the narrow aisle between the two rotors at both the sides of the central separation wall. For completeness, Figure 8
b shows the turbulent kinetic energy map. Two high turbulence regions are visible, one being due to the vortices naturally shed by the blade at the beginning of downwind, the other being induced by the blunt head of the separation wall (see Figure 3
b). However, only the second perturbs the torque production, as will be clarified in the following.
tries to justify the diffuser effect on the local flow speed (U
) of the flow approaching the blade when its position varies during the revolution. At this purpose, the path-lines passing through the grid interface lines are shown for the bare turbine pair and for the ducted turbines. Despite only the symmetrical diffuser is treated, the following observations are valid in general. The path-lines set can roughly be considered the “stream tube” of the rotor pair. The stream tube’s width at a sufficient distance upstream the rotors is indicative of the flow rate processed by the turbines. Strictly speaking, the width measurement must be taken at the tube “source” (i.e., where free stream conditions occur).
By comparing the widths of the upstream part of the tubes of Figure 9
a it is evident that ducted turbines are processing a much greater flow rate than bare turbines. Since the flow rate is constant along the stream tube for the continuity law, the local speed is inversely proportional to the stream tube cross-section. Thus, at the diffuser inlet, where the section appears very large, a flow speed even lower than in case of bare turbines is expected (in Figure 9
a we indicated some relatively low speed zones with blue marks), and the same could happen after the diffuser exit. Focusing on the blade path, the highest U
increasing respect to the bare turbines happens during downwind since local sections are equal (at ϑ
= 180° and 360°) or even slightly smaller of the stream tube sections of the bare turbines (we indicated relatively high speed zones with red marks). In upwind the (relative) accelerating effect of the diffuser is much less intense and only relegated to the early and late upwind where stream tube sections are just slightly larger than in case of bare turbines. At half upwind, since the blade distance from the rotor axis is the highest and therefore the stream tube section can be significantly larger, the flow speed can result similar, or even lower, than in case of bare turbines. The velocity magnitude maps of Figure 9
d confirm the above interpretation. In conclusion, the beneficial U
increasing due to the diffuser action is much more effective in downwind than in upwind.
allows a qualitative comparison of the behaviour of the two diffuser types at different yaw angles.
Considering that the widths of the upstream stream tubes are representative of the flow rate passing through the turbines, the following general features are noticeable:
Taking into account the rotation direction of each turbine (shown in Figure 6
) it is evident that the effect of the converging walls of the symmetrical diffuser occurs mainly during the final part of the blade upwind path. As demonstrated by the direction of the path-lines, this effect consists of an increase in the attack angle, whose consequences on lifting and torque have already been illustrated in Figure 7
. It happens for both the rotors if φ
= 0°, only for the “second turbine” (that is the upper one) if φ
> 0°. The relative extent of this favourable effect (i.e., in comparison to what happens in the original diffuser) is greater the greater the yaw angle is.
For φ = 0°, the stream tube of the symmetrical diffuser appears larger (a measurement done at the domain inlet boundary revealed +3.5% respect to the original diffuser) indicating a better flow concentration capability.
For the hydrofoil shaped diffuser the stream tube width slightly increases at φ = 15° but appears drastically narrowed at φ = 30°, while for the symmetrical diffuser the width continues surprisingly to grow up to φ = 30°.
In summary, a diffuser acts modifying α and U. It is worth noting that the effect marked with Equation (1) only concerns the fluid dynamics internal to the diffuser, whereas effects marked with Equations (2) and (3) need to be justified by accounting for the viscous phenomena occurring in the external flow, as will be better explained later in the section.
Despite the fact that for all the cases that have been investigated, the CP
was higher the higher the stream tube width was measured, the diffuser accelerating capability alone cannot justify the formidable power output improving that have been calculated with CFD, especially for yawed flows. As shown by Castelli et al. [53
] an effective way to asses and justify the energy performance of a straight-bladed CFT is to graphically represent the evolution of both the rotor torque (or CP(ϑ)
) and blade angle of attack. Then, to deeply understand how the diffuser presence affects the key mechanisms that determine torque generation we analyse the polar diagrams of Ux
) and α
), and then the one-blade CP
) predicted during one revolution, shown in Figure 11
It must be specified that α
) is determined from the x
-components of the flow absolute velocity, Ux(ϑ)
, that are recorded along the blade path in a point moving simultaneously with the blade, and set at a proper distance ahead [33
]. The best distance rises from a compromise, as the flow velocity is perturbed by both the bound circulation around the blade and by its wake. It was set at 1.5c
on the base of a preliminary CFD investigation.
The graph of normalised U
) confirms the aforementioned flow speed increasing in early and late upwind, and during the entire downwind, for both the diffusers in comparison to the single bare turbine. It is worthy of note that a U
improving already occurs in the bare paired configuration in early upwind and late downwind [33
The attack angle (Figure 11
b) is high in the early upwind region for all the paired configurations, including the bare turbines [33
], yet the diffusers also allow high α
in the late upwind zone, especially in case of the symmetrical one (as already observed comparing the path-lines inclination in Figure 10
a,d). It can be noticed that the diffusers also produce α
augmentation (in absolute value) in downwind, however this is a consequence of the U
increasing that modifies the velocity triangle.
The features of the one-blade CP
graph (Figure 11
c) follow those of the α
graph: with both diffusers the torque generation begins earlier and lasts longer in upwind, and is significantly improved during the whole downwind. The detail that only appears in the CP
graph, and therefore cannot be related to α
, is a performance drop that occurs in the final 25° of downwind due to the high turbulence level generated by vortices released by the central wall (visible in Figure 8
b). With respect to the single bare turbine, the average CP
enhancing results of 30.9% for the bare paired rotors, and 63.7% and 70.3% for the hydrofoil shaped and the symmetrical diffusers, respectively.
Before analysing the polar graphs of yawed flow cases, it is useful to have a look at the characteristics of the flow field, focusing on the behaviour of the flow external to diffusers. Path-lines and static pressure maps are depicted in Figure 12
for the symmetrical and in Figure 13
for the original diffuser in case of φ
= 0°, 15° and 30°.
A vortex formation like the Karman vortex street is seen downstream the trailing edges of the symmetrical diffuser at φ
= 0°. As the yaw angle increases, the vortex shedding becomes asymmetric: it almost expires at the upstream side of the diffuser, and conversely becomes stronger and stronger at the downstream side, entailing a very low pressure zone behind the diffuser. The vortex shedding from the original diffuser is much less important, being significant only for φ
= 15° and 30°, yet the low pressure zones behind the diffuser are less extended and modest in absolute values. In our opinion, these differences in the external fluid dynamics could explain the different flow acceleration attitudes, and therefore flow rate passing through turbines, exhibited by the two diffusers. The literature on ducted turbines gives evidence of how viscous interactions between internal and external flows can be advantageously exploited to achieve a remarkable increasing of the flow concentration capability in diffusers equipped with deflectors [26
] or flanges [54
] at the exit periphery. The mechanism of flow acceleration was explained as follows [55
]: a large vortex is generated behind a flange and the backpressure of the diffuser becomes lower than that of a diffuser without a flange. Therefore, the mass flow drawn into the diffuser increases and the flow speed through the turbine becomes high. Although we don’t have a flange, strong vortices at the diffuser sides occur thanks to the concave external shape of the symmetrical diffuser. These vortices boost the suction in the turbine wakes, as suggested by the rear stagnation points of the diffuser (i.e., where internal and external flows meet) that are moved forward, farther from the diffuser axis, as if virtually the aspect ratio of diffuser was enlarged. This can be realized comparing the rear stagnation point positions of Figure 12
Let’s now analyse the polar graphs of yawed cases in Figure 14
, limiting to ϕ
= 15° an 22.5° for brevity. About U
, at φ
= 15° both the diffusers benefit from viscous interactions between internal and external flows (exhibiting a flow rate slightly greater than in case of φ
= 0°), whereas at φ
= 22.5° this only happens for the symmetrical diffuser.
About α, it is considerably smaller during late upwind for the second turbine of the original diffuser, and this deficiency is higher the higher the yaw angle is. This is because the original diffuser is not equipped with a converging wall capable of holding back the flow from the exit towards the device side and orient it to the rotor with a favourable entry angle. As a result, CP are similar at φ = 15°, yet very different at φ = 22.5°.
The arguments presented so far are useful to understand the results summarised in Figure 15
, which shows the performance of each rotor of the pair in terms of diffuser efficiency (CP
normalised by the bare turbine CP
). It is seen that, at any yaw angle, the symmetrical diffuser has a performance higher than that of the original diffuser. In case of symmetrical diffuser, the performance of the second rotor overcomes that of the first rotor, and CP
grows up to φ
= 30° (exhibiting a drastic drop at φ
= 37.5°). Whereas, for the hydrofoil shaped diffuser, the first rotor works better than the second, and CP
improves up to φ
= 15°, then it starts to drop. It is interesting to notice that the enhancing of the diffuser performance with φ
was never found in case of ducted horizontal axis turbines, which at least exhibit a constant CP
(or just a small drop) until a certain φ
4.4. Torque Ripple
The periodic fluctuation of torque and power is a characteristic of any CFT that should be mitigated as much as possible. To quantify the fluctuation a parameter is used, named Torque Ripple Factor (TRF
), defined as:
are the average, the lowest and the highest instantaneous torque in the period. For bare turbines the period is one revolution, yet for ducted turbines the vortex shedding can entail much longer periods (of the order of 10 revolutions). When this happens (in our study, only with symmetrical diffusers) it is useful to distinguish the contributions of turbine and diffuser to the overall TRF
. We calculated the former on two consecutive pick and trough, and the latter as the remaining part of the overall TRF
. The TRF
of ducted pairs is shown in Figure 18
for the original and the symmetrical diffusers. Results indicate that the symmetrical diffuser, despite the contribution of the diffuser to the ripple, implies lower (with the exception of φ
= 0°) and almost constant TRF
when yaw angle changes. However, since the denominator of TRF
grows with φ
, also absolute fluctuations rise (with very high values at φ
= 22.5° and 30°).
Thus, any modification to the diffuser shape reducing both absolute and relative fluctuations would be useful. At this purpose we now compare the torque ripple behaviour of symmetrical diffusers A, B and C (Figure 19
). It is seen that A leads to much higher TRF
than B and C, and that B results the best shape, since it is able to maintain TRF
al low values until φ
≤ 22.5°, as shown in Figure 20
, which allows to compare the power fluctuation behaviour of the A and B diffusers for yaw angles of 0° and 22.5°.
explains the reasons for the TRF
differences found for A and B at φ
= 0° and 22.5°. In one revolution six peaks occur in the CP
curve, this is because the blades are three and each one produces torque in upwind and downwind. One-blade CP
of each blade are also reported (dashed lines). CP
at a certain ϑ
is the sum of the three one-blade values. In general, to suppress TRF
it is desirable to level peaks and troughs, bringing them closer to the average torque. The main reasons for the drastic TRF
reduction exhibited by B is a great rising of troughs and, only in case of φ
= 0°, a reduction of peaks. Figure 21
c,f reveal why the “smoothing out” of the turbine CP
curve can occur. The straight throat inserted in B causes a reduction of α
in early and late upwind and an Ux
increasing in downwind (due to the cross section narrowing).
As a consequence, at φ
= 0° less torque is generated in early and late upwind (leading to a reduction of peaks) and more torque is generated at half downwind (leading to an increasing of troughs). However, the huge rising of troughs found at φ
= 22.5° is also due to a much higher flow rate of B respect to A (noticeable in Figure 17
b), that improves the one-blade torque at ϑ
= 120° and 290°.