## 3 Results  
In this section, the results of all simulations are presented. First, the case with 0 degree azimuthal rotation are considered. Figure 4 shows the center-line profiles along the \(z\)-axis. The true theoretical profiles are shown in red. The blue curve is the estimated mean profile (\(\bar{v}\)), and the dashed black curves shows the relative standard deviation (\(\bar{\sigma}\)). The top row shows the case with 90 degrees beam-to-flow angle. Since the beamforming was performed along the \(x\), \(y\) or \(z\) direction, the method is expected to perform better when the flow is parallel to one of these directions. When the flow deviates from the cardinal directions, the beamformed signals start to decorrelate, which negatively affects the method's precision. It is visible that the profile for the \(v_{x}\) component (blue curve) of the flow matches the theoretical profile well (red line). Further the middle and bottom rows show the cases for beam-to-flow angles of 75 and 60 degrees, respectively. These cases both have a \(z\) component of the flow as well. It can be seen that the profiles matches the theoretical profiles well. It is seen that the bias and standard deviation increase with decreasing beam-to-flow angle. Figure 5 presents relative standard deviation (RSD) and relative bias (RB) for the entire center plane of the cases above to quantify the performance in the volume rather than along a single line. RSD is less than (1.288%, 0.934%, 0.434%) for the first case (90 degrees beam-to-flow angle) for all three velocity components (\(v_{x}\), \(v_{y}\), \(v_{z}\)). The RB varies from \(-2.64\%\) to \(-0.22\%\). At a 75 degree beam-to-flow angle, RSD increases to 2.88% for \(v_{x}\) component. RB is also growing and varies from \(-0.57\%\) to \(-14.25\%\). In the case with a 60 degrees beam-to-flow angle all parameters are increasing again. The RSD is (5.53%, 5.31%, 0.82%) for (\(v_{x}\), \(v_{y}\), \(v_{z}\)), respectively. The RB is (\(-14.1\%\), \(2.33\%\), \(-5.38\%\)) for (\(v_{x}\), \(v_{y}\), \(v_{z}\)). Figure 6 shows the profiles along the center lines for a 45 degree azimuthal rotation and the three beam-to-flow angles. Figure 4: Estimated center profiles from simulated data for \(\alpha=(60,75,90)^{\circ}\), \(\beta=0^{\circ}\). Standard settings from Table 1 were used. The red line is theoretical profile, the blue curve is estimated mean profile, the dashed black curves are the relative standard deviation. Figure 5: RSD and RB for simulations with (90, 75, 60) degrees beam-to-flow angles and 0 degrees rotation angle. The dashed lines show the relative standard deviation, the solid lines show the relative bias. The red color is \(v_{x}\), the black is \(v_{y}\) and the blue is \(v_{z}\) components of the flow. The top row presents the case with a 90 degrees beam-to-flow angle, the middle row for 75 degrees, and the bottom row presents for 60 degrees. For all these cases the flow will also have a \(y\)-component. Good correspondence is seen between estimated and true profiles, although with increasing deviations when the beam-to-flow angle is decreased. Comparing Figures 4 and 6, it is seen that changes to the beam-to-flow angle have greater effect than azimuthal rotation. This is supported by the RSD and RB in Figures 5 and 7. Figure 7 shows the RSD and RB for three cases from Figure 6, again calculated from the entire center plane. One can see that the RSD is (1.15%, 1.10%, 0.39%) for 90 degrees beam-to-flow angle, the RB is (9.81%, 3.16%, 4.45%). The RSD for 75 degrees beam-to-flow angle (2.36%, 3.07%, 0.49%) and the RB is (\(-13.23\%,\)\(-1.54\%,\) 6.08%). The RSD for 60 degrees beam-to-flow angle is (4.67%, 4.99%, 0.96%) and the RB is (\(-9.78\%,\) 1.82%, 14.65%). It is visible that the RSD is the same for cases without a rotation angle and for cases with 45 degrees rotation angle as stated above, but the RB shows larger variations. Figure 6: Estimated center profiles from simulated data for \(\alpha=(60,75,90)^{\circ}\), \(\beta=45^{\circ}\). Standard settings from Table 1 were used.