# Design Approach and Mechanism Analysis for Cavitation-Tolerant Torque Converter Blades
## 3 Application of Suppression Slot in Stator Cascade
### Analysis of the Flow Field Mechanism with Different Slot Designs  
There is no uniform criterion for vortex structure identification in academia. [PERSON] and [PERSON] first proposed a new method for vortex structure identification, the \(\lambda_{2}\) criterion [27]. This method makes up for the defect of the Q criterion in identifying near-wall vortex flow. The calculation formula of the \(\lambda_{2}\) criterion can be obtained through the inviscid incompressible assumption of the transport equation. The coherent vortex structure in the fluid is calculated by the \(\lambda_{2}\) criterion. Firstly, it is necessary to calculate \(S^{2}+\Omega^{2}\) (\(S\) represents the symmetric part of the velocity gradient tensor, namely the strain rate tensor; \(\Omega\) represents the antisymmetric part of the velocity gradient tensor, i.e., the vorticity magnitude). Then, the characteristic root is calculated, and the second eigenvalue is selected according to the order of magnitude as the criterion of the vortex. Generally, a surface with \(\lambda_{2}<0\) is chosen as the vortex. They compared the difference between the \(\lambda_{2}\) criterion and the other three commonly used criteria (Q criterion, \(\Delta\) criterion, \(|\omega|\) criterion) in their paper, and they proved that this criterion was obviously superior to the other three criteria through experimental tests and discussions. In particular, for an axisymmetric axial vortex  
Figure 9: Vapor distribution in various suppression slot stators under stall operating condition in 3D view (1—attached cavity, 2—slot cavity, 3—trailing cavity, 4—scatter cavity). within a vortex ring and a conically symmetric vortex with axial velocity, the correctness of the vortex structure results of the Q criterion is less than that of the \(\lambda_{2}\) criterion. In addition, [PERSON] et al. used the \(\lambda_{2}\) criterion to well capture the vortex structure near the groove structure [28]. The impellers of the torque converter belong to the rotated symmetric fluid machinery. Therefore, this paper adopts the \(\lambda_{2}\) criterion to capture the vortex structure near the stator suppression slots. The \(\lambda_{2}\) criterion is derived from the matrix \(S^{2}+\Omega^{2}\). The strain rate tensor is  
\[S=\frac{1}{2}(\
abla V+\
abla V^{T}) \tag{13}\]  
The vorticity magnitude is  
\[\Omega=\frac{1}{2}(\
abla V-\
abla V^{T}) \tag{14}\]  
The velocity gradient tensor is  
\[\
abla V=\left( \begin{array}{ccc}\frac{\partial UI}{\partial x}&\frac{ \partial V}{\partial y}&\frac{\partial W}{\partial y}\\ \frac{\partial U}{\partial y}&\frac{\partial V}{\partial y}&\frac{\partial W }{\partial y}\\ \frac{\partial U}{\partial z}&\frac{\partial V}{\partial z}&\frac{\partial W }{\partial z}\\ \end{array} \right)=\left( \begin{array}{ccc}U_{x}&V_{x}&W_{x}\\ U_{y}&V_{y}&W_{y}\\ U_{z}&V_{z}&W_{z}\\ \end{array} \right) \tag{15}\]  
In the formula, \(U\), \(V\), and \(W\) represent the velocity components of the velocity in \(x\), \(y\), and \(z\) directions, respectively. The matrix \(S^{2}+\Omega^{2}\) can be simplified as:  
\[\left(\begin{array}{ccc}U_{x}{}^{2}+U_{y}V_{x}+U_{z}W_{x}&\frac{1}{2}((U_{x} +V_{y})(U_{y}+V_{x})+V_{z}W_{x}+U_{z}W_{y})&\frac{1}{4}(U_{y}V_{z}+(U_{x}+W_{ z})(U_{z}+W_{x})+V_{x}W_{y})\\ \frac{1}{2}((U_{x}+V_{y})(U_{y}+V_{x})+V_{z}W_{x}+U_{z}W_{y})&V_{y}{}^{2}+U_{y }V_{x}+V_{z}W_{y}&\frac{1}{2}(U_{x}V_{x}+U_{y}W_{x}+(V_{y}+W_{z})(V_{z}+W_Figure 11 shows the distribution of velocity streamlines in the 0.5 span for different suppression slots. Compared with the flow field of the stator without slot structure, the flow field of the stator with the four slot designs has a high level of eddy flow, which is the reason for the decline in the hydrodynamic performance of the torque converter caused by the slot design. Plan 1 forms a high-speed jet in the slot, and a vortex flow is formed on both sides of the slot outlet. The flow velocity at the suction leading edge of the blade does not decrease to a great extent.