3.1. Numerical Simulations for a One Bladed Vertical Axis Turbines
A particular challenge when simulating cross-flow turbines is the relatively high Reynolds number at which these devices operate. Flow regimes for wind turbines are generally turbulent. Numerous studies have computed vertical axis turbines using numerical low order methods, e.g., Ref. [32
] and only very recently Ferrer [19
] has provided results for vertical axis turbines using high order solvers and moving meshes.
In this section, we perform numerical simulations for a one bladed turbine. We compare 3D simulated results issued from the high order Navier–Stokes solver with experimental data and also to fast 2D simulations computed using the commercial finite volume solver Ansys-Fluent-16.2 [38
] This solver is selected to simulate unsteady Reynolds Average (uRANS) Navier–Stokes solutions using a k-omega Shear Stress Transport (SST) turbulence model. These simulations will provide an idea of what to expect from cheap numerical simulations or simple models, so we can assess in upcoming sections the validity of our new reduced order model based on HODMD.
Both high order and low order solvers use sliding meshes and rotate the mesh as time evolves. The high order solver computes 3D simulations (as required by LES type turbulent models) whilst Fluent (using uRANS) allows cheaper 2D simulations. The former involves more expensive simulations, but these are typically more accurate. Numerical details of both solvers are summarised in Table 1
. We show the meshes and a snapshot of the numerical solutions for both solvers in Figure 1
. The turbine has a diameter of radius
m (distance form the center of rotation (x
) = (0, 0) to the airfoil quarter chord. This means that the position of the blades lie within the interval
[−4 m, 4 m] and
[−4 m, 4 m].
Experimental and numerical conditions are detailed in Table 2
for the one bladed turbine, whose blade geometry is characterised by a NACA0015 airfoil. We have maintained the main characteristic numbers
, the tip speed ratio (
We compare the simulated results to experimental tests [39
] for the normal non-dimensional blade forces
represents the fluid density, for a rotating one bladed vertical axis turbine. Figure 2
a compares instantaneous values for the 3D high order DG method to averaged experimental values for four revolutions. We observe comparatively good agreement between 3D high order DG computations (black line) and experiments (orange and blue lines). The experimental data by Oler et al. [39
] includes strain gauge data and pressure data. Whilst the former was reported to be accurate by Oler et al., the latter was obtained by integration of pressure force coefficients (using only five probes) and has been reported to “contain a reasonable amount of error”. Despite the errors, we have included the pressure data to quantify possible experimental errors and to show that the high order simulations are close, in comparative terms, to the gauge data. Additionally, let us note that the experimental forces where filtered using a low pass filter (with frequency 0.6 Hz) whilst our simulations are depicted for a sampling frequency of 1000 Hz, which explains the nosier data of the simulations when compared to the experimental forces.
b compares the low order 2D Fluent solution to the experimental values. We observe good overall agreement but less accuracy than when the high order solver is used.
Previous publications [39
] have compared forces for cheap computational and analytical models to the experiments of Oler et al. [39
] but have showed generally higher discrepancies that our computations. Additionally, these publications have reported non-negligible errors in the experimental measurements for the tangential forces. Consequently, we have preferred to include only the normal forces, which are experimentally accurate and sufficient to show the differences in accuracy between our high and low order simulations.
Returning to the streamwise-velocity snapshots shown in Figure 1
a.2,b.2, we observe a more complex flow with multiple vortices emanating from the trailing edge, when using the high order LES approach and a much smoother flow (with fewer flow details) when using the low order solver with uRANS turbulence. Both the time evolution of the forces and the instantaneous flows suggest that a high order solver with an LES turbulent closure is more accurate than the low order uRANS model. The computational time for the low order model is orders of magnitude lower than when using the high order method, which is attributed the the smaller mesh (2D vs. 3D) and the higher time step enabled by the simpler method and coarser mesh (see Table 2
Finally, we compare velocity components: streamwise (
) and normal (
), on two horizontal lines at
m for a fixed simulation time in Figure 3
. At this instant in time, the blade is at an azimuthal angle of
(measured from the 12:00 a.m. position and rotating anti-clockwise), such that the first line (
m) is at the back of blade trailing edge, whilst the second line (
m) is at the front of the rotating blade. Figure 3
shows that the low order Navier–Stokes solver that uses an uRANS turbulent model provides similar trends that the high order LES simulation, but does not capture all details. In the next sections, we will develop a reduced order model that can capture most of the complex flow physics present in the high order simulation at an affordable computational cost.
3.2. Reduced Order Model for a One Bladed Vertical Axis Turbine
We use the result from the 3D high order numerical simulations for the one bladed turbine, and construct an HODMD-based ROM using 196 snapshots equispaced in time, with
s. Two-dimensional snapshots are extracted from the 3D simulation. These snapshots are collected after one and a half blade revolutions (
s), in order to avoid spurious and transient artifacts of the numerical solution. Consequently, the snapshots are collected during the time interval 80.01 s
s (58.5 s representing
of rotation). Let us remember that a complete rotation of the wind turbine (
) is represented by
s time units (with rotation velocity
rad/s). Figure 4
shows the temporal evolution, which includes the temporal interval retained for this analysis, of two representative points located near the blades rotation region:
with coordinates in meters. As seen in Figure 4
, the flow complexity is very large due to the high level of fluctuations (since the flow is turbulent).
After some calibration, based on the robustness (similar solutions for various type of calculations) of the results [24
], the parameters used for the HODMD analysis are
. It is remarkable that, although the flow is highly complex and the analysis is carried out in an unsteady regime of a numerical simulation (only ∼1 blade revolution is analysed), which limits the values of d
used for the analysis [29
], the results are robust. Nevertheless, the solution obtained would improve when using a larger value for d
, if more snapshots are retained. The reason is that the HODMD expansion, characterized by d
reduces data uncertainty [24
] and helps removing the transient modes, which are treated by the method as noise [13
]). Regarding the tolerances, Figure 4
shows that there are small variations of velocity, of order
(velocity fluctuations), which represent variations of 10% with respect the mean velocity. So, the tolerances
(see Section 2.2
) should be below
to reproduce accurately these fluctuations. In this case, we have set such tolerances to one order of magnitude below to remain conservative. Note that values below
could introduce spurious artifacts in the reduced order model, due to the large complexity of this flow (which is fully turbulent) and should be avoided.
The data complexity and the unfavorable conditions to construct a DMD-based ROM (unsteady turbulent flow and transient data, small time interval to collect the data) suggest that the best option for this case is to combine HODMD with the criterion selection method (HODMDc) that will identify the most relevant modes that will be used to generate the ROM.
shows the growth rates and the amplitudes as function of their corresponding frequencies in the HODMD analysis. As seen, the numerical simulation is still under a transient stage, since the order of magnitude of the growth rates of the DMD modes is defined in the interval (10
s) (transient solution). The method identifies 65 frequencies. HODMDc has been applied to find the most relevant frequencies for the ROM. The prediction of the results has been carried out retaining several number of modes:
in a first case,
. These frequencies are written, for clarity, in Table 3
, including their corresponding amplitudes and growth rates. There is a single mode with null frequency that represents the mean flow. This mode is found in all the analyses, and will not be mentioned again for the sake of brevity. The remaining modes are represented by a complex number (representing the frequency) and its complex conjugate. In other words, when the number of modes retained for the ROM construction is for example
, the DMD expansion (1
) will include 3 modes, one related to the mean flow (zero frequency), one related to the dominant frequency
and its complex conjugate
As seen in the results, HODMDc always captures the mode with highest amplitude, and frequency
). This frequency represents the frequency of rotation of the wind turbine blade (
rad/s). The errors in the calculations of this frequency are attributed to the reduced temporal interval in which the snapshots are collected. Despite this, the method gives a reasonable good approximation of the fundamental frequency. When the number of modes retained is
, the method also captures the harmonic of the fundamental frequency
rad/s. Finally, for values of
, the method captures some harmonics of the fundamental frequency and some high amplitude modes with
. On the one hand, these very high amplitude modes are related to the fluctuations of the wind turbine. Since the value of their growth rate is positive and ∼10
, the influence of these high amplitude modes will grow in time, meaning that, for very large temporal approximations, these modes should be either omitted (as in the ROM proposed in [13
]) or re-calculated at different time instants, but this is far from the scope of this article. On the other hand, although the flow is fully turbulent, it is dominated by a periodic behaviour (geometrical rotation), which is apparent by the modes retained by HODMDc. It is remarkable that the third mode captured by HODMDc is the fourth harmonic of
rad/s, instead of the third harmonic ∼3
rad/s, which is selected in 7th place (
Once the HODMDc has selected the relevant DMD modes, three HODMD-based ROMs are constructed retaining
, 5 and 21 DMD modes and the solution is extrapolated in time ∼160 s time units (the following ∼4 rotations of the wind turbine). In other words, the temporal term of the DMD expansion (1
) is set to
s. Figure 6
shows the original (simulated with the high order DG method) and the predicted instantaneous flow field at time 240 s for the three ROMs constructed. As seen, the results agree favourably to the original data, predicting the regions of larger influence of the turbine blades (
m). The solution is smoother when the number of modes retained is smaller (and somewhat similar to the low order uRANS simulation shown in Section 3.1
), but the fidelity of the model improves significantly when the number of modes is increased. However, it is interesting to note that, using only 3 DMD modes (
), it is possible to represent the main structures of the wind turbine. The RMS error of these predictions is ∼28% , ∼25% and ∼23% for
, 5 and 21, respectively. These errors appear large, having seen the good qualitative agreement of the figures.
In order to understand these errors, Figure 7
and Figure 8
compare the original and approximated spatial evolution of the solution at constant time 240 s and two representative horizontal lines at
m (at a fixed instant in time). The figures show that the three ROMs predict the mean flow field correctly. The predictions with the model with five modes is better than the one of three modes. The model that uses 21 modes is able to approximate some of the high frequency peaks that are related to small scales fluctuations, and related to the modes with very high amplitude. When removing these peaks (i.e., when
m), the HODMD model predicts the solution with a global RMS error smaller than 15%, 13% and 10%, in the models constructed with 3, 5 and 21 modes, respectively. These results reflect that the optimal model is the one retaining five modes, not only because the growth rates of the modes are negative (avoiding that the results diverge at large times), but also because the error for
is similar to the error obtained with
modes, proving that including more modes does not result necessarily in a more accurate model.
3.3. Extensions for Three Bladed Vertical Axis Turbine
In the previous section, we developed and validated a ROM model for a one bladed turbine using snapshots extracted from a 3D high order simulation. In this section, we proceed similarly and generate a ROM model for a three-bladed turbine. The motivation to study this new geometry is three-fold. First, typically, a vertical axis wind turbine has three blades. Second, three blades tend to generate more complex flows (e.g., more vortex-blade interactions [1
], and third, the potential for computational cost savings are greater since three bladed turbines require typically more mesh elements. Furthermore, we will generate the ROM using snapshots exacted during less than one full revolution and show that this is enough to represent the complete turbine behaviour.
The 3D mesh for the high order DG-Fourier solver uses 3676 elements, with polynomial order P = 3 and 64 Fourier planes, given a total number of degrees of freedom of 2.35 millions (2.14 times more that for the one bladed turbine).
As in the previous section, we extract 2D snapshots from the 3D high order simulations and construct an HODMD-based ROM using 196 snapshots equispaced in time. These snapshots have been taken in the time interval
102.416 s (
s, representing ≃
), equispaced in time with
s. With the aim of reducing the computational cost and increasing the efficiency of the ROM methodology, we shift the window for analysis to an an earlier time (than in the previous case). Figure 9
shows the temporal evolution in the interval analyzed of two representative points collected in the blades area of the wind turbine,
meters. As seen, the flow complexity is large due to the velocity fluctuations. In addition, the figure shows that the flow is still evolving in time and consequently the numerical simulation has not reached a periodic state. This will serve to prove the good performance of the method if the snapshots collected do not cover an entire rotation of the turbine blade (but only
). This shortened sampling interval increases the possibility of finding spurious artifacts in the analysis of the simulations, but we show that the proposed method can handle these difficulties. Once the ROM is generated, we use it to predict a solution evolved in time until ∼160 s time units; i.e., the temporal term of expansion (1
) is set for this prediction at
Similarly to the previous case, the parameters used in this analysis are
and the tolerances
. Figure 10
shows the frequencies as function of their amplitudes and growth rates obtained in the analysis. The HODMD method produces 151 modes. Table 4
shows the frequencies, amplitudes and growth rates obtained using HODMDc and retaining 3, 5 and 21 modes (the mode corresponding the mean flow is included in all cases). As for the one bladed turbine, the first frequency retained by HODMDc is the one with highest amplitude
rad/s that represents the frequency of rotation of the turbine blade (
rad/s). The following six frequencies are harmonics of this fundamental frequency and the remaining three frequencies are also high order harmonics of
. In contrast to the previous case, the number of harmonics of
retained by this ROM is larger, representing the major influence of the three blades in the periodic behaviour of the flow. This is also reflected in the amplitude of the DMD modes, since the modes retained by HODMDc correspond to the ones with the highest amplitude. The same analysis has been carried out in different planes of the three-dimensional computational domain to check that the results are very similar and independent of the out-of-plane location (these results are not shown for the sake of brevity).
compares the instantaneous original (high order 3D simulation) and predicted flow fields for the three ROMs constructed with
, 5 and 21 DMD modes, at time
s. As seen, these predictions are in good agreement with the original solution. As expected, small flow structures are better represented when the number of modes retained is larger. However, it is interesting to note that the overall flow structure defining the turbine blade movement can be represented with only three modes. As in the previous case, the RMS error is high in the three cases, being ∼40% (in all the cases). Figure 12
and Figure 13
show that, as already explained, the model cannot predict the peak values of the velocity fluctuations, but that when these peaks are removed (i.e., when
m), the model predicts the original solution with high accuracy (RMS error is ∼15–20%).
Finally, we challenge the HODMD-based ROM by decreasing the number of samples used to construct the ROM. We perform the same analysis but use only 145 snapshots equispaced in time, with s collected in the time interval 25.604 s 68.524 s ( s time units, representing ). Now, the data collected do not represent a complete rotation, making this problem even more challenging than the previous one. In addition, the data set has been taken earlier than before (almost half a rotation earlier). In the first part of the simulation, the data are unstable and are neglected as fully transient (a part of the initialisation of the simulation). This means that these snapshots cannot be used to represent any event in the future (attractor) of the numerical simulation. The parameters used for the HODMD analysis are , (the interval defined for d has decreased with the number of snapshots).
shows the growth rates and the amplitudes as a function of their corresponding frequencies in the HODMD analysis. The method identifies 134 frequencies. Table 5
shows the frequencies, amplitudes and growth rates obtained using HODMDc retaining
, 5 and 21 modes. Again, the first frequency retained by HODMDc is the one with the highest amplitude
rad/s that represents the frequency of rotation of the turbine blades (
rad/s). When the number of modes retained is
or 5, the method captures the harmonic of the fundamental frequency
rad/s. Finally, for values larger than
, the method captures harmonics of the fundamental frequency, but, due to the reduced number of snapshots used in the analysis, the calculations present large errors. Nevertheless, since the amplitude of the remaining frequencies is small (order of magnitude ∼5·10−2
rad/s), the errors induced by these modes in the construction of the ROM are negligible (as shown below). It is noticeable that the third mode retained by HODMDc represents the fourth harmonic of the fundamental frequency
. The last peculiarity was already observed in the one blade turbine case.
Again, the three HODMD-based ROMs are constructed retaining
, 5 and 21 DMD modes and the solution is extrapolated in time to
s. Figure 15
shows the original and the predicted instantaneous flow fields for the three ROMs generated. There is a good agreement between the ROM and the original simulation. The RMS errors for these predictions are ∼40%, ∼41% and ∼44% for
, 5 and 21, respectively. Figure 16
and Figure 17
show that the model cannot predict the peak values of the velocity fluctuations. When these peaks are removed (i.e., when
m), the model predicts the original solution with an RMS error ∼20–25%.