As depicted in Figure 4.6, the C-H mesh has better agreement near the tip than the O mesh, but the results are similar for the rest of the blade area. For further simulations, this particular setup, employing the C-H mesh and DDES-SST turbulence model, is applied unless stated otherwise. For the MEXICO rotor, all evaluations of the 3D flows are carried out using Δt= 1°.
4.3 Root Flow Characteristics
hub connection, except at r/R = 0.35 where the axial and tangential forces are overestimated. Atθ= 0°, it can be seen that the axial flow acceleration in the root area is concentrated from the center of rotation up to the blade end section (r/R≈ 0.12). The root flow acceleration seems to occupy a larger area successively atθ = -45° and θ = -90°, especially close to the rotor plane at X/R = 0. Akay et al.
[150, 151] and Sherryet al. [152] observed also a similar phenomenon. Despite that, in their studies the increase of U was caused by the radial flow that is directed towards the center of rotation as a result from the nacelle flow interaction. In fact, if the blade was properly connected to the nacelle as in their studies, the velocity increase was not as high as the present observation. The magnitude of U/U∞ is slightly higher than unity which is significantly above the expected axial velocities based on the momentum theory.
As can be seen in Figure 4.13a, the lower side of the blade is characterized by a negative azimuthal velocity component. This is caused by the local motion of the fluid flow around the blade from the stagnation point towards the leading edge of the blade, and its direction is the same as the rotor motion. In contrary, this particular fluid flow opposes the rotor direction on the suction side and its magnitude is increased by the displacement effect. Furthermore, it is shown that positive magnitude of the velocity component dominates the flow field downstream of the rotor atθ= 0°. This observation shows that the inviscid part of the flow field rotates opposite to the rotor, in agreement with the momentum theory. A similar observation was also given in [150, 151].
Some regions downstream of the rotor (X/R > 0) are locally characterized by negative circumferential velocity fields. These are observed atX/R≈0.4 and 0.8 forθ
= 0°,X/R≈0.1 and 0.5 forθ= -45°andX/R≈0.3 and 0.7 forθ= -90°, marking the location of the nearby vortices in the wake area along the blade radius. The origin of this phenomenon is expected to stem from two main factors: (1) the change sign of the trailing vortices due to the positive/negative gradient of the bound circulation along the blade; and (2) the interaction of the current and previous blade passages as discussed in [150–152]. Regarding to the latter effect, it is shown that the strength of the velocity interaction between the blade passages decreases with increasing streamwise distance.
This effect starts just after the boundary layer materials shed into the wake area and are dragged together with the rotor motion [150, 151]. As no hub is introduced, this phenomenon becomes remarkably strong near the root area at X/R ≈ 0.27. It is expected that the local root flow circulation in the wake is stronger than if the blade is properly attached on a nacelle. Furthermore, it is shown that the radial expansion of the most inboard root vortices seems very weak due to the nozzle effect.
The third component of the near wake flow,i.e., the radial flow, for the isolated MEXICO rotor is presented in Figure 4.14. Similar to the observation of the circumferential velocity, the root vortices are clearly observed for this velocity component. The location of the vortices is marked by the change of sign of the radial velocity. In the inboard area, a distinct root vortex is observed approximately atr/R
= 0.15. It can be seen that the root vortices move in helical fashion as the blade rotates, as can be depicted from the velocity contour planes atθ = 0°, -45°and -90°
(a)θ= 0° (b)θ= -45°
(c)θ= -90°
Figure 4.12:Dimensionless axial velocity distributions for three different azimuth angles, 0°(4.12a), -45°(4.12b) and -90°(4.12c). Inflow is in +X/Rdirection. Black line indicates the position of the blade. The contour line indicates the magnitude ofU/U∞ = 1. A small discontinuity of the contour in Figure 4.12b atr/R= 0.25 andX/R= 0.1 is caused by errors in the Chimera interpolation within the overlapping meshes.Figures 4.12a and 4.12b are taken from [146].
4.3 Root Flow Characteristics
(a)θ= 0° (b)θ= -45°
(c)θ= -90°
Figure 4.13: Dimensionless circumferential velocity distributions for three different azimuth angles, 0°(4.13a), -45°(4.13b) and -90°(4.13c). Inflow is in +X/Rdirection.
Black line indicates the position of the blade. A small discontinuity of the contour in Figure 4.13b atr/R= 0.25 andX/R= 0.1 is caused by errors in the Chimera interpolation within the overlapping meshes.Figures 4.13a and 4.13b are taken from [146].
(a)θ= 0° (b)θ= -45°
(c)θ= -90°
Figure 4.14:Dimensionless radial velocity distributions for three different azimuth angles, 0°(4.14a), -45°(4.14b) and -90°(4.14c). Inflow is in +X/Rdirection. Black line indicates the position of the blade. A small discontinuity of the contour in Figure 4.14b atr/R= 0.25 andX/R= 0.1 is caused by errors in the Chimera interpolation within the overlapping meshes.Figures 4.14a and 4.14b are taken from [146].
4.3 Root Flow Characteristics
Figure 4.15: Iso-surface visualizing the vortical structures downstream of the rotor by λ2= -413 s−2.Figure is taken from [146].
and from the iso-surface in Figure 4.15. Atθ= 0°, a powerful outboard motion of the fluid flow is observed adjacent to the suction side of the blade surface, dominating up to r/R ≈ 0.2. This is caused by the root effect of the isolated rotor. A similar phenomenon was observed to occur near the tip area, but in opposite direction. In contrast, in the range ofr/R= 0.375-0.42 and aroundr/R= 0.6, the flow motion is directed inboard and connected with the local induction effect of the blade loads,i.e., it is related to flow separation. In the normal force distribution presented in Figure 4.6c, roughly representing the bound circulation, it is shown that the force locally decreases at the aforementioned locations causing a local change of the bound vortex direction and induction effect.
Figure 4.16 shows the velocity distribution extracted from Figures 4.12, 4.13 and 4.14 forθ= 0°. It can be seen that the axial velocity increases up to around 1.075U∞ at the edge of the blade root atr/R= 0.12 and around 1.01U∞in the near tip region.
It is interesting to note that the axial velocity increase near the root is even higher than near the tip region. In the blade middle stations, between 0.2< r/R < 0.8, the magnitude of this velocity component reduces as it enters the wake region. This indicates that the axial flow component is less reduced near the blade, which may be related to the displacement effect. For the circumferential velocity component in Figure 4.16b,Vθincreases significantly around the blade and the influence becomes smaller for largerX/R, which clearly indicates that the effect is caused by the fluid flow around the blade due to displacement effect. Interestingly, it seems that the circumferential velocity augmentation becomes stronger approaching the center of rotation along the blade. The velocity suddenly drops atr/R= 0.12 where the edge of the blade root is located. In contrast, the radial flow increases significantly at this area shown in Figure 4.16c. The increased circumferential velocity component is in good agreement with the increased suction peak with decreasing blade radius as the angle of attack becomes larger. The pressure distributions for different blade stations can be seen in Figure 4.19 in Section 4.4. Unlike the axial and circumferential velocities, no considerable change
of the radial velocity component is observed along the blade, especially further away from the rotor plane. This confirms preceding studies of the 3D effects that the strong radial flow is only prominent within the separated boundary layer area.
Figure 4.17 presents the spanwise flow (vz) in the rotating frame of reference,i.e., the rotational speed of the blade (Ω) is taken into account, at several radial positions, namely 0.25R, 0.35R, 0.6R, 0.82Rand 0.92R. This flow component is the same asVr
only atθ= 0°. The termvzis presented instead ofVrbecause the spanwise flow defines the chordwise component of the Coriolis force that is important for separation delay.
The velocity component is non-dimensionalized by the local kinematic velocity. The red domains in Figure 4.17 represent the regions characterized by a strong spanwise flow. The spanwise velocity on the pressure side is small compared to the kinematic
r/R [-]
U / U [-] c [m]
0 0.2 0.4 0.6 0.8 1 1.2 0.7
0.8 0.9 1 1.1 1.2
0 0.2 0.4 0.6 0.8 1 1.2 X/R = 0.1 1.4 X/R = 0.15 X/R = 0.2 Chord
(a)Axial component.
r/R [-]
V / U [-] c [m]
0 0.2 0.4 0.6 0.8 1 1.2 0
0.2 0.4 0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
(b)Circumferential component.
r/R [-]
V
r/ U [-] c [m]
0 0.2 0.4 0.6 0.8 1 1.2 -0.2
-0.1 0 0.1 0.2 0.3 0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
(c)Radial component.
Figure 4.16: Dimensionless velocity components forθ= 0°at several axial distances, X/R= 0.1, 0.15 and 0.2.Figure is taken from [146].
4.3 Root Flow Characteristics
(a)c/r= 0.4. (b)c/r= 0.24.
(c)c/r= 0.1. (d)c/r= 0.06.
(e)c/r= 0.05.
Figure 4.17:Time averaged dimensionless spanwise velocity in rotating frame of reference (vz) at several radial stations,r/R= 0.25, 0.35, 0.6, 0.82 and 0.92. The contour level below than 0.2 is omitted for clarity purpose.Figures 4.17b-4.17e are taken from [146].
velocity. Furthermore, the flow is attached on the pressure side of the blade. On the other hand, it can be seen that the spanwise flow is remarkable on the blade suction side where separation occurs especially in the inboard area. It extends up tor/R
= 0.6 but the magnitude significantly reduces even though the nature of the flow is massively separated, indicating that the spanwise flow is less dominant forc/r0.1.
By means of this information, the 3D correction of the 2D polar for engineering codes like BEM can be applied forc/r0.1, and pure 2D polars can be used in the other region. Additionally, it is noted that the spanwise flow occurs only within the separated boundary layer supporting the studies for the 3D effects [1–4, 27, 29, 57].
4.4 3D Effects on Aerodynamic Coefficients
In this section, the influence of the 3D effects on the aerodynamic coefficients is discussed. Firstly, the angle of attack was extracted from the DDES results using the reduced axial velocity method [60, 126, 127] described in Section 2.4. Then, the resulting angle of attack was used to calculate the lift (CL) and drag (CD) coefficients from the CFD simulations, defined as:
(4.2) CL = L
0.5ρ Vrel2 c and CD= D 0.5ρ Vrel2 c,
whereLandDare the lift and drag forces, respectively, andVrelis the relative velocity including the induction effect. There is no information ofα in the new MEXICO measurements [119]. Thus, the same angle of attack (as the extracted CFD results) was assumed to calculateCLandCDfor the experimental data.
To demonstrate the 3D effects, the 3D rotating blade results need to be compared to the 2D data with consistent inflow conditions like the angle of attack and Reynolds number. For this purpose, two-dimensional simulations of the blade sections were performed. In these computations, the same angle of attack and Reynolds number as the 3D rotor were applied. The 2D simulations make use of the URANS approach employing the Menter SST [102] turbulence model. The DDES approach is not applied for the 2D case due to the 3D nature of the eddies. Similar to the 3D rotor computations, no laminar-turbulent transition was considered.
It shall be noted that two-dimensional measurement data for airfoils constructing the MEXICO rotor is available, documented in Ref. [153]. Despite that, it is worthwhile to emphasize that the 3D rotor blade operates at differentRethan the 2D experiment.
Furthermore, boundary layer tripping was applied in the 2D measurement while the 3D rotor simulations assume a fully turbulent flow. This indicates that there may be uncertainty induced if the 3D rotor results are directly compared to the 2D experimental data. In order to asses the accuracy of 2D CFD simulations, additional two-dimensional simulations of these airfoils were performed at consistent inflow conditions as in the experiment.
To sum up, in total, five different cases are employed in the present studies to assess the three-dimensional effects:
4.4 3D Effects on Aerodynamic Coefficients
• 3D Exp.:3D experiment obtained from Ref. [119].
• 2D Exp.:2D experiment obtained from Ref. [153].
• 3D DDES:3D rotor simulationsemploying the DDES-SST turbulence model.
• 2D URANS-A:2D rotor computationsemploying the URANS-SST turbulence model at the same inflow conditions as the3D rotor simulations.
• 2D URANS-B:2D rotor computationsemploying the URANS-SST turbulence model at the same inflow conditions as the2D experiment.
Details of the inflow conditions employed can be seen in Table 4.2. The angle of attack shown is obtained from the analysis described in the preceding paragraph. In the following paragraphs, it shall be noted that the 2D CFD accuracy is assessed by comparing2D URANS-Bto2D Exp., 3D CFD accuracy by comparing3D DDESto 3D Exp., and the three-dimensional effects are evaluated by comparing3D DDES/3D Exp.to2D URANS-A/2D Exp..
Figure 4.18 presents theCLandCD distributions along the blade radius. In the most inboard position, at r/R= 0.25, the 2D and 3D CFD simulations accurately predict the measured aerodynamic coefficients, except for 2DCD where the data is overestimated. It is clearly shown that the 3D effects causeCLaugmentation up to 2 times compared to the 2D situation. In Figure 4.19a, it can be seen that the distribution ofCpatr/R= 0.25 agrees well with the behaviour ofCLin Figure 4.18. The suction peak ofCpremarkably enhances from -4.8 in the 2D case to -8.85 in the 3D case, but Table 4.2: EmployedReandαfor the 2D and 3D cases. SS represents the suction side and PS is for the pressure side. The angle of attack is obtained from the 3D rotor computations employing the RAV approach [60, 126, 127].
r/R[-] Airfoil Re[-] α[°] Tripping [-]
2D Experiment
0.25R DU91-W2-250 0.7x106 22.8 0.05 (SS) and 0.1 (PS) 0.35R DU91-W2-250 0.7x106 19.7 0.05 (SS) and 0.1 (PS) 0.6R RISØ-A1-21 1.6x106 16.9 0.05 (SS) and 0.1 (PS) 0.82R NACA 64-418 0.7x106 13.8 0.05 (SS) and 0.1 (PS) 0.82R NACA 64-418 0.7x106 13.7 0.05 (SS) and 0.1 (PS)
3D Rotor Simulations
0.25R DU91-W2-250 0.50x106 22.8 fully turbulent 0.35R DU91-W2-250 0.54x106 19.7 fully turbulent 0.6R RISØ-A1-21 0.61x106 16.9 fully turbulent 0.82R NACA 64-418 0.64x106 13.8 fully turbulent 0.82R NACA 64-418 0.62x106 13.7 fully turbulent
r/R [-]
CL [-]
0.2 0.4 0.6 0.8 1
0.5 1.0 1.5 2.0 2.5
3D Exp.
3D DDES 2D Exp.
2D URANS-A 2D URANS-B
r/R [-]
CD [-]
0.2 0.4 0.6 0.8 1
0.0 0.1 0.2 0.3
3D Exp.
3D DDES 2D Exp.
2D URANS-A 2D URANS-B
Figure 4.18:Lift (left) and drag (right) coefficients along the blade radius from the time averaged CFD results and experiments. Lift increases in the blade inner part while drag decreases.Measurement data for the 3D and 2D cases are obtained from [119] and [153], respectively.
reduces compared to the 2D inviscidCplevel of -17.65. This effect is caused by the delay of separation fromx/c≈0.2 in the 2D case tox/c≈0.4 in the 3D case, which stems from mixed influences of the inviscid and viscous rotational effects described in Chapter 3. It worthwhile to mention again that the inviscid effect tends to reduce the suction peak and, consequently, the pressure gradient due to the influence of the root vortices.
In contrast, the root vortices become weaker if the viscous effect is considered, and, as the radial flow presents, the Himmelskamp effect takes place delaying the occurrence of separation. However, it was shown in Chapter 3 that the pressure distributions for both the inviscid and the viscous cases do not much differ, indicating that the viscous rotational effect is driven by the inviscid characteristics of the blade. It is suggested to refer back to Chapter 3 for detailed discussion of the phenomena.
It is shown, atr/R= 0.25, that the 3D CFD results are similar to the measurement data although the nacelle was not modelled in CFD, see Figure 4.18. This may indicate that the root vortices have weaker influence to the 3D effects than the observation made in Chapter 3 despite the fact that a distinct hub vortex is observed in Figure 4.15. The reason seems to be related to the airfoil thickness. In the MEXICO rotor, the inboard region employs the DU91-W2-250 airfoil (25% relative thickness) which is characterized by trailing edge separation. This is in contrast to the NACA 0010 airfoil (10% relative thickness) employed in Chapter 3 which shows leading edge separation. Furthermore, the cylindrical portion of the most inboard blade area reduces the tendency of the distinct root vortex in affecting the blade sections further outboard.
A similar behaviour is shown at a larger radial position atr/R= 0.35. Surprisingly, the 2D URANS computations accurately predict 2DCL in the inboard region even
4.4 3D Effects on Aerodynamic Coefficients
x/c [-]
Cp [-]
0 0.2 0.4 0.6 0.8 1
-10 -8 -6 -4 -2 0
3D 2D 2D Inv 0.25R
(a)c/r= 0.4.
x/c [-]
Cp [-]
0 0.2 0.4 0.6 0.8 1
-10 -8 -6 -4 -2 0
3D 2D 2D Inv 0.35R
(b)c/r= 0.24.
x/c [-]
Cp [-]
0 0.2 0.4 0.6 0.8 1
-5 -4 -3 -2 -1 0 1
3D 2D 2D Inv 0.6R
(c)c/r= 0.1.
x/c [-]
Cp [-]
0 0.2 0.4 0.6 0.8 1
-5 -4 -3 -2 -1 0 1
3D 2D 2D Inv 0.82R
(d)c/r= 0.06.
x/c [-]
Cp [-]
0 0.2 0.4 0.6 0.8 1
-5 -4 -3 -2 -1 0 1
3D 2D 2D Inv 0.92R
(e)c/r= 0.05.
Figure 4.19:Time averagedCpdistributions of the blade section and 2D airfoil data for several radial stations obtained from3D DDESand2D URANS-A, respectively.
though the angle of attack is large involving massive flow separation,e.g.,atr/R= 0.25 the angle of attack is 22.8°. However, in the 3D CFD results, it can be seen
thatCL is overpredicted andCD is underpredicted. This agrees with the behaviour for the prediction ofFy presented in Section 4.2. It was documented by Weihing et al. [78] that a proper modelling of the hub-nacelle geometry can fix the issue. Thus, it is quite reasonable to expect the cause of the phenomena based on the influence of the root vortex induction effect. Despite that, this shall not be confused with the root vortex influence for the inviscid flows described above. It was documented in Chapter 3 that the roll-up motion of the root vortex causes the vortex-induced centrifugal force which produce streamtube widening and reduces the suction peak. In this particular radial station, the root vortex creates the induction effect, which increases the local wind speed and, consequently, the local angle of attack perceived by the blade section.
Furthermore, it is shown in Figure 4.17b that the relative spanwise flow component is stronger compared to the smaller radial station of 0.25R, which seems strange because the values ofαandc/rare smaller than those at 0.25R. It is expected that the root vortices are responsible for this. As a consequence, the Himmelskamp effect within the blade boundary layer is overestimated and the rotational augmentation becomes stronger.
The enhancement of lift is observed up tor/R = 0.6, corresponding to thec/r ratio of 0.1 which is defined as the limit where the 3D effects are prominent in Chapter 3 and Section 4.3. These are observed in both the measurement data and the CFD simulations with good agreement. It is interesting to note that 3DCLis comparable to the 2D case in the middle blade regions, namely at 0.6Rand 0.82R, even though strong separation occurs, see Figure 4.17. It shows that as long asvz/Vkinis small, the airfoil section generally behaves like in the two-dimensional situation.
Particularly atr/R = 0.6, the locations of separation in the 2D and 3D cases are similar (Figure 4.19c), but the 3D lift coefficient in Figure 4.18 and the suction peak value in Figure 4.19c for the corresponding section are slightly higher than in the 2D case. The small lift augmentation is likely to be caused by the reduced pressure effect rather than the separation delay for this particular section. The centrifugal force seems to be responsible for this effect since the main mechanism of this force is on the boundary layer thinning. Similarly, Sicotet al.[154] also observed a lift enhancement for a small turbine rotor without the evidence of separation delay.
In the near tip region, the 3D DDES computations also deliver accurate results compared to the measurement data. On the other hand, the 2D URANS results for the outboard NACA 64-418 airfoil atr/R = 0.82 and 0.92 clearly overestimate the lift coefficient, but are accurate for the drag coefficient. It is already well known that URANS often inaccurately predicts the maximum lift coefficient of airfoils [68, 69, 104, 147, 155]. Considering the maximum lift for this airfoil is atα= 15°,CLoverestimations for the radial stations of 82% and 92% become reasonable, because the airfoil sections operate at α = 13.8°and 13.7°, respectively. Note that the 2D prediction for the inboard airfoil (DU91-W2-250) has a better agreement with the experimental data although the angle of attack is significantly larger. This discrepancy can be attributed to the airfoil-type (Table 4.2) dependency influence. Furthermore, it will be shown in Chapter 5 - Figure 5.6a (for differentRe) that the prediction of the DU91-W2-250