# Mesh Twisting Technique for Swirl Induced Laminar Flow Used to Determine a Desired Blade Shape
## 3 Results and Discussion
### Case Comparison  
Figure 13 shows the results from four different computational cases, with the focus on the swirl number and how it decays along the flow axis. The twist angle is adjusted for all the cases to obtain a swirl number \(S=0.2\) at the swirler exit, except for the (initial) case where the theoretical deviation angle shown in Figure 2 is used directly. It is clear that, because of the relatively few guidance blades and the laminar layer forming along the blades, less than half the swirl number is actually obtained when using the (initial) case. It is also noticeable that Case I overshoots the design value \(S=0.2\) but is otherwise performing relatively well. The decay profiles are examined further in Figure 14, where they are shown as a ratio to the theoretical decay shown in Equation (6), with \(\alpha=0.213179\) obtained from [17]. Case I and Case III (hollow middle) perform similarly, but Case II seems to be superior in terms of keeping the swirling effect along the axis. This confirms that the closer the tangential velocity profile is to the ideal velocity profile, the closer the swirl decay is to the theoretical exponential decay. Figure 11: (**a**) resulting value for parameter \(a\) as a function of radius, needed to obtain a chosen deviation angle (left); and (**b**) the resulting change in twist angle with respect to axial location (right). Figure 12: Swirler design with varying deviation angle at the exit, visualized (**a**) downstream (left); (**b**) upstream (center) and (**c**) under an angle (right). Figure 15 shows the additional pressure drop along the axis when compared to standard Hagen-Poiseuille flow (laminar with no swirling). This is useful to compare the cases as well as observe the difference between added pressure drop because of swirling, and because of the swirler device. It turns out that Case III (hollow) is slightly better than Case II (blades are gathered in the center), but both are superior to Case I where the deviation angle at the exit changes linearly with radius. This is shown more clearly in Table 1 along with the necessary maximum twist angles for the cases. Figure 14: Swirling number decay, expressed as a ratio compared to a theoretical decay, for Cases I–III. Figure 13: Decay in swirl number along the pipe axis (number of diameters downstream of end of the swirler). In the (initial) case, the twist angle is set to \(30^{\circ}\), which corresponds to the theoretical deviation angle for \(S=0.2\). One measure of the performance of different blade setups is how well the axial and tangential velocity profiles conform to the ideal profiles depicted in Figure 2. Figure 16 shows the axial velocity distribution in a cross section half a diameter downsteam of the swirler exit. The results for Cases I and II are very similar; both show the effect of using four blades and thus get an uneven distribution in the angular (tangential) direction. The effect of removing the blades in the centre in Case III is clearly visible, showing a greater axial velocity in the centre of the cross section. Similarly, Figure 17 shows a comparison of the tangential velocities for the three cases. Here, Cases II and III yield similar results and both result in a more uneven velocity distribution in the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case:** & **I** & **II** & **III** \\ \hline Added pressure drop [diameters] & 11.47 & 10.44 & 10.25 \\ Twist angle at swirler exit [\({}^{\circ}\)] & 108.9 & 74.5 & 84.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A comparison of added pressure drop and twist angle at the swirler exit for the three cases. Figure 16: Axial velocity in m/s in a cross section, 0.5 diameters downstream of the swirler exit, (a) Case I (left); (b) Case II (center) and (c) Case III (right). Figure 15: Pressure along the flow direction, expressed as length in number of diameters, but with the normal pressure drop in straight laminar flow subtracted. tangential direction.