# Mesh Twisting Technique for Swirl Induced Laminar Flow Used to Determine a Desired Blade Shape
## 2 Materials and Methods
### Constructing Guidance Blades  
The general approach in this paper is to use mesh morphing (twisting) to obtain a preferred shape of a guidance blades (vanes) in a swirling device. Thus, the blades, defined as wall patches in OpenFOAM, only need to be specified as straight before the morphing takes place. The aim is to get a distribution of deviation angle as shown in Figure 2, for the flow itself (the flow might not follow the blade's angle immediately). Thus, the deviation angle (\(\phi=\arctan(v_{\theta}/v_{z})\)) indicates a necessary blade shape. Three types of blade setups are considered. The twist angle in the mesh is generally a function of radial and axial location, \(\theta(z,r)\). The cases are described as follows:  
1. Blades where \(\tan(\phi)\) changes linearly from the center of the pipe to the outer wall, assuming that \(\phi\) is the deviation angle. This is equivalent to constructing a single shape of blade, defined by a twist angle \(\theta\) as a function of axial location \(z\) but not affected by radial location \(r\). 2. Blades where the deviation angle changes along the pipe axis, but also follows at the exit the angle defined in Figure 2, in the radial direction. 3. Same blade setup as in Case II, but where 20% of the inner core is removed, thus reducing the flow restriction because of the blade connections in the center. In all cases, the blade's inlet was four diameters from the pipe inlet and all swirlers were one pipe diameter in length. Figure 5: Skewness of the mesh within a given cross section of the pipe. A suitable profile for the twist angle for Case I, where the curvature is zero at the blade exit, is  
\[\theta=\theta_{0}\left(1-\cos\left(\frac{\pi(z-z_{0})}{2(D-z_{0})}\right)\right), \tag{7}\]  
where \(\theta_{0}\) and \(z_{0}\) are the maximum twist angle and inlet location of the swirler, respectively, and \(L\) is the length of the swirler. Figure 6 shows the general blade shape in the axial direction as well as the deviation angle for Case I. In the second case (Case II), we define a function that is the second derivative of the twist angle formula, but with a parameter \(a\) defined,  
\[\theta^{\prime\prime}(\eta)=K\theta_{0}\exp(a\eta)\left(1-\eta^{2}\right), \tag{8}\]  
where \(\eta=\frac{z-D}{D}\) represents dimensionless axial location (\(-1\leq\eta\leq 1\) from inlet to exit of the swirler) and \(K\) is a constant to be determined. This function is closely related to curvature, and it turns out that \(\theta^{\prime\prime}\) is zero at the inlet and exit, as well as \(\theta^{\prime\prime}\) always being positive. Integration where \(K\) is selected in order to get \(\theta(1)=\theta_{0}\) finally gives a blade profile  
\[\theta(\eta)=\theta_{0}\frac{\exp(a\eta)\left(a^{2}(1-\eta^{2})+4a\eta-6 \right)+\exp(-a)\left(4a+6+2a(a+1)(\eta+1)\right)}{\exp(a)\left(4a-6\right)+ \exp(-a)\left(6+4a(a+2)\right)}, \tag{9}\]  
which has the following properties:  
* The curvature at the inlet and exit is zero. * The twist angle is zero at the inlet (\(\eta=-1\)) and \(\theta_{0}\) at the exit (\(\eta=1\)). * The slope with respect to \(\eta\) (deviation) is zero at the inlet, but is controlled by \(a\) at the exit. * The deviation grows monotonically from the inlet to the exit, regardless of the selection of \(a\). Figure 7 illustrates the blade shape in the axial direction for three different end deviation angles as well as the relationship between the parameter \(a\) and the deviation angle at the pipe wall. The wall is selected as an example, since the deviation angle also changes as a function of radius. In addition, note that these are just examples and not optimized shapes as are presented in the Results section. Figure 6: Case I, with four blades and twist angle not affected by radius. (**upper left**) twist angle as a function of axial location; (**lower left**) deviation angle as a function of radial location; (**right**) the blade set up in the pipe. Note that the arrow shows the direction of the flow in front of the inlet.