The inlet and outlet radii are important design parameters that directly determine the internal/external characteristics and cavitation characteristics of the torque converter (TC). The stator and turbine are the main areas of cavitation in TCs. Based on this, the outlet radius of the stator and turbine is taken as the research object of this paper. In this study, computational fluid dynamics (CFD) models of different turbine and stator outlet radii are established, and the influence of stator/turbine outlet radius design parameters on the performance of TCs is revealed by comparing the internal/external characteristics and cavitation characteristics. The results show that reducing the outlet radius of the stator/increasing the outlet radius of the turbine will cause the stator and the turbine blade to be shorter, increase the area of the vaneless region between the impellers, and reduce the risk of cavitation in TCs. However, it will also lead to a decrease in the external characteristics of the low-speed ratio (*SR*) condition and an increase in the external characteristics of the high-*SR* condition. With the decrease in the stator outlet radius/the increase in the turbine outlet radius, the mass flow rate of TCs will decrease, and the mass flow loss caused by cavitation will decrease from the original 42.51 to 6.95 and 21.95 kg/s, respectively. The suppression rates of the stator/turbine outlet radius on TC cavitation volume are 58.894% and 52.359%, respectively. The research results of this study can provide practical engineering guidance for the design of high-performance TCs and cavitation suppression.

## I. INTRODUCTION

The torque converter (TC) is widely used in loaders, mine trucks, and automobiles because of its advantages of vibration isolation and vibration reduction, stepless speed change, and torque multiplication. The most basic TC consists of a rotating pump, a rotating turbine, and a fixed stator. Cavitation is the most common phenomenon in TCs, which seriously affects the performance of TCs. In severe cases, it can cause problems such as blade fatigue damage, flow passage blockage, vibration, and noise. Therefore, it is necessary to study the cavitation mechanism and its suppression method inside the TC.

The cavitation phenomenon inside the TC will undergo a continuous quasi-periodic change process of onset, development, shedding, and collapse. Many researchers have studied the cavitation phenomenon inside the TC by means of visual experimental measurement technology and numerical simulation. Watanabe *et al.* made a transparent prototype of the TC and tested and observed it, and they found a large area of attached cavitation near the leading edge of the stator near the suction surface.^{1,2} Mekkes *et al.* and Guo *et al.* placed a dynamic pressure sensor near the suction surface of the stator blade of the TC, and it was found through experimental tests that there was a large low-pressure area near the leading edge of the stator blade suction surface, demonstrating that this position was the main region where TC cavitation occurred.^{3,4} Chai *et al.*, Ju *et al.*, and Tsutsumi *et al.* conducted a numerical simulation study on cavitation of stamping welded blades and flat TCs. The results show that cavitation will cause the flow passage of the TC to be seriously blocked under low-speed ratio operating conditions.^{5–7} Zhao *et al.*, Ran *et al.*, and Zhang *et al.* conducted a comprehensive numerical analysis of the cavitation bubble breakup behavior inside the TC, and the research shows that the cavitation movement is an unstable process.^{8–10}

Cavitation will deteriorate the performance of the torque converter and restrict its development toward high power density. Therefore, it is necessary to seek reliable technical means to achieve effective suppression of cavitation. Both the external boundary and the internal transmission medium properties have a significant effect on the cavitation of the TC. Guo *et al.* studied the effects of rotational speed, thermal properties of the transmission medium, and inlet temperature on cavitation, and the results show that reducing the inlet temperature can effectively suppress the cavitation of the TC.^{11–13} Liu *et al.* studied the effects of charging pressure and charging position on cavitation, and the results show that increasing the charging pressure and using the turbine–stator vaneless region for oil charging can effectively suppress the cavitation of the TC.^{14} Guo *et al.* and Chai *et al.* studied the influence of the blade outlet angle and inlet deflection angle on the performance of the TC.^{15–17} Wang *et al.* and Liu *et al.* studied the influence of the stator blade angle and blade profile on cavitation. Increasing the inlet angle of the stator blade or reducing the inclination angle of the stator blade can effectively suppress stator cavitation.^{18,19} Xiong *et al.* used a Joukowsky airfoil to construct the blade of the TC, which can effectively suppress the cavitation of the stator.^{20} Liu *et al.* slotted the blade of the stator to effectively suppress the cavitation of the stator.^{21}

It can be seen from previous research that most of the research focuses on the suppression analysis of blade parameters, external boundary, and thermal properties of the internal transmission medium on the cavitation of the TC. A few researchers have studied the influence of torus parameters on the cavitation performance of TCs. Parameters such as the radius of the inlet and outlet of the torus are important because this parameter directly determines the length of the blade and the area of the vaneless region. The influence of the radius of the inlet and outlet on the cavitation of the TC is not revealed. At the same time, through previous studies, it is found that the turbine and the stator are the main domains where the cavitation of the TC occurs. Therefore, this paper takes the outlet radius of the stator and the turbine as the research object to study its influence on the cavitation of the TC. In order to facilitate the parametric modification of the blade outlet radius, this paper first uses the spline curve to parametrically reconstruct the TC baseline blade. Then the computational fluid dynamics (CFD) results of the parametrically reconstructed original blade with/without cavitation conditions are compared with the experimental results. Then, this paper analyzes the influence of different stator and turbine outlet radii on the external characteristics and internal flow field, and it reveals the influence of stator/turbine outlet radius on the cavitation performance of the TC. The research of this paper can provide guidance for the design of high-performance and low cavitation TCs.

## II. RESEARCH OBJECTS AND PROBLEM DESCRIPTION

### A. Torus and cascade system of torque converter

In order to facilitate the parametric modification of the inlet and outlet radius of the TC in this paper, the spline curve is used to parametrically reconstruct the baseline blade, and the parameterized reconstructed blade shape is used as the original blade shape in this paper. The specific method of parametric reconstruction of the baseline blade in this paper can be seen in Ref. 22. The effective diameter of the TC in this paper is 375 mm, and the number of blades of the pump, turbine, and stator is 29, 24, and 22, respectively. Figure 1(a) defines the torus and the inlet and outlet radius of the TC. The torus consists of a core streamline (inner ring streamline), a shell streamline (outer ring streamline), a middle streamline, and six inlet and outlet edges intersecting with the middle streamline. The distance from the intersection of the inlet and outlet edges and the middle streamline to the rotation axis is called the inlet and outlet radius. The inlet and outlet radius determines the length of the blade in the streamwise direction and the width of the vaneless region between the impellers.^{23} The stator and the turbine are the main domains where the TC cavitation occurs. Therefore, the outlet radius of the turbine and the stator is taken as the research object of this paper to study its influence on TC cavitation. Figure 1(b) shows three different turbine outlet radii and stator outlet radii designed in the study.

In Fig. 1, *R*_{B1} denotes the pump inlet radius (the initial value is 115.8476 mm), *R*_{B2} denotes the pump outlet radius (the initial value is 176.0436 mm), *R*_{T1} denotes the turbine inlet radius (the initial value is 176.0436 mm), *R*_{T2} denotes the turbine outlet radius (also known as *R*_{T2-Ori}; the initial value is 116.6712 mm), *R*_{S1} denotes the stator inlet radius (the initial value is 107.6944 mm), and *R*_{S2} denotes the stator outlet radius (also known as *R*_{S2-Ori}; the initial value is 107.6125 mm). Figure 2 shows the three-dimensional (3D) impeller, cascade, and flow passage model and six outlet radius design schemes of the TC.

Table I shows the design scheme of the six inlet and outlet radii of the TC in this paper. It can be seen from the table that the scheme of cases 1–3 only modifies the turbine outlet radius, and the scheme of cases 4–5 only modifies the stator outlet radius to study the influence of the turbine and stator outlet radius on TC cavitation. The value of *R*_{T2-1} is 131.3506 mm. The value of *R*_{T2-2} is 123.7108 mm. The value of *R*_{S2-1} is 103.6016 mm. The value of *R*_{S2-2} is 104.8840 mm.

. | . | . | . | R_{T2}
. | . | R_{S2}
. | ||||
---|---|---|---|---|---|---|---|---|---|---|

Case Radius . | R_{B1}
. | R_{B2}
. | R_{T1}
. | 1 . | 2 . | Ori . | R_{S1}
. | 1 . | 2 . | Ori . |

1 | ||||||||||

2 | ||||||||||

3 | ||||||||||

4 | ||||||||||

5 | ||||||||||

6 |

. | . | . | . | R_{T2}
. | . | R_{S2}
. | ||||
---|---|---|---|---|---|---|---|---|---|---|

Case Radius . | R_{B1}
. | R_{B2}
. | R_{T1}
. | 1 . | 2 . | Ori . | R_{S1}
. | 1 . | 2 . | Ori . |

1 | ||||||||||

2 | ||||||||||

3 | ||||||||||

4 | ||||||||||

5 | ||||||||||

6 |

In order to facilitate the labeling in the following content, the study uses *R*_{T2-1} to identify case 1, which represents the design scheme of the cascade system with the first set of the turbine outlet radius. Case 2 is identified by *R*_{T2-2}, which represents the design scheme of the cascade system with the second set of the turbine outlet radius. Case 3 is identified by *R*_{T2-Ori}, which represents the design scheme of the cascade system with the third set of the turbine outlet radius, which is consistent with the design parameters of the original torque converter. *R*_{S2-1} is used to identify case 4, which represents the design scheme of the cascade system with the first set of the stator outlet radius. *R*_{S2-2} is used to identify case 5, which represents the design scheme of the cascade system with the second set of the stator outlet radius. Case 6 is identified by *R*_{S2-Ori}, which represents the design scheme of the cascade system with the third set of the stator outlet radius, which is consistent with the design parameters of the original TC.

### B. CFD model and grid independence analysis

*CF*) under stall conditions is used to verify the grid independence under different grid densities. The definition of

*ɛ*(

*CF*) is shown in Eq. (1). When

*ɛ*(

*CF*) is less than 3%, it can be considered that the calculation result of the TC is independent of the grid size.

^{24}When the number of grids is 4.5 × 10

^{6}, the change rate of the

*CF*is 2.35%, less than 3%, and the calculation time is 5.76 h. At this time, if we continue to reduce the grid density and increase the number of grids, it will lead to a rapid increase in computational costs,

*y*+ is an important parameter for the wall cavitation capture of the TC. In this paper, Eq. (2) was used to estimate the first layer grid height of different impeller blades of the TC so as to ensure that the

*y*+ value of the blade surface does not exceed 2 and to ensure the capture accuracy of the cavitation of the blade wall of the TC,

*C*

_{f}denotes the wall friction factor,

*τ*

_{w}denotes the wall friction stress,

*u*

_{*}denotes the wall friction velocity,

*y*denotes the height of the first layer of the wall grid,

*μ*represents the fluid viscosity,

*ρ*represents the fluid density,

*U*

_{freestream}represents the far-field velocity, and

*L*represents the characteristic length (the length of the impeller blade is selected as the characteristic length in this paper). For the calculation of

*C*

_{f}, this paper uses the method defined in Ref. 25.

In order to ensure that the *y*+ value is less than 2, the center point of the first layer grid is placed in the viscous sublayer of the boundary layer, the first layer of the grid height near the blade wall is 0.015 mm, the thickness of the encryption layer is 1 mm, the grid growth rate is 1.2, and the grid has 12 layers. As shown in Fig. 3(d), the maximum *y*+ value of the pump blade is 1.4, which appears at the inlet of the pump blade, and the minimum *y*+ value of the pump blade is 0.1. The maximum *y*+ value of the turbine blade is 2.0, which appears at the inlet of the turbine blade, and the minimum *y*+ value of the turbine blade is 0.2. The maximum *y*+ value of the stator blade is 1.5, which appears at the inlet of the stator blade, and the minimum *y*+ value of the stator blade is 0.1. The *y*+ values on the surface of the three impeller blades are all less than 2, indicating that the boundary layer meshing meets the analytical requirements of the TC cavitation flow field.

## III. NUMERICAL CALCULATION

### A. Turbulence model

*x*

_{i}is an index in the form of a tensor and Δ

*x*

_{i}is the component of the grid scale in the

*i*direction.

*u*

_{i}and

*u*

_{j}are the time-averaged velocities in the tensor form,

*μ*

_{m}is the viscosity of the mixed phase, and the variables with the upper line are the filtered field variables.

*τ*

_{ij}is an anisotropic sub-grid stress, and its calculation equation is as follows:

*μ*

_{t}is the turbulent viscosity,

*τ*

_{kk}is the isotropic sub-grid stress, and

*S*

_{ij}is the stress tensor ratio. In this paper, the algebraic wall-modeled large eddy simulation (WMLES) model is used to solve

*μ*

_{t}. In the WMLES sub-grid model,

*μ*

_{t}is defined as follows:

*d*

_{w}represents the distance between the calculated position and the wall,

*κ*= 0.41, and

*C*

_{Smag}= 0.2.

### B. Cavitation model

*R*

_{e}represents the evaporation rate,

*R*

_{c}represents the condensation rate, and

*R*

_{B}represents the bubble diameter.

*ρ*

_{l},

*ρ*

_{v}, and

*ρ*

_{m}represent the densities of the liquid phase, vapor phase, and mixed phase, respectively.

*α*

_{v}represents the vapor volume fraction,

*p*represents the pressure in the flow field, and

*p*

_{v}represents the saturated vapor pressure, which is 110 Pa. The density of the mixed phase

*ρ*

_{m}, the viscosity of the mixed phase

*μ*

_{m}, and the cavitation bubble radius

*R*

_{B}are defined as follows:

### C. Boundary conditions and simulation parameter settings

^{26}For cavitation conditions, this paper uses ten-time steps to solve the flow field of the pump across a blade,

^{27}

*n*

_{B}denotes the pump rotational speed,

*C*

_{B}denotes the number of pump blades, Δ

*t*

_{noc}is the time step without the cavitation condition, and Δ

*t*

_{cavi}is the time step with the cavitation condition.

The boundary conditions and simulation parameters of the TC CFD calculation are shown in Table II. The transient model was used to simulate the non-cavitation condition and cavitation condition. The numerical simulation of the TC without cavitation condition only involves single-phase calculation, and the convergence is good. The time step is set to 0.0005 s/step, and the number of steps is 240. The numerical simulation of the TC with cavitation conditions involves multiphase flow calculation. In order to facilitate the convergence of the calculation, the time step is set to 0.0001 s/step, and the number of steps is 1200.

Parameters . | No cavitation condition . | Cavitation condition . |
---|---|---|

Analysis type | Transient state | Transient state |

Pressure–velocity coupling | SIMPLE | SIMPLE |

Advection scheme | Second order upwind | Second order upwind |

Multiphase model | None | Mixture |

Oil density (kg/m^{3}) | 860 | 860 |

Oil viscosity (Pa·s) | 0.0258 | 0.0258 |

Vapor density (kg/m^{3}) | None | 2.1 |

Vapor viscosity (Pa s) | None | 1.2 × 10^{−5} |

Saturated vapor pressure (Pa) | None | 110 |

Turbulence model | WMLES | WMLES |

Cavitation model | None | Schnerr–Sauer |

Pump state | 2000 rpm | 2000 rpm |

Turbine state | 0–1600 rpm | 0–1600 rpm |

Stator state | Stationary | Stationary |

Inlet condition | 0.8 MPa | 0.8 MPa |

Outlet condition | 0.4 MPa | 0.4 MPa |

Convergence target | 1 × 10^{−5} | 1 × 10^{−5} |

Parameters . | No cavitation condition . | Cavitation condition . |
---|---|---|

Analysis type | Transient state | Transient state |

Pressure–velocity coupling | SIMPLE | SIMPLE |

Advection scheme | Second order upwind | Second order upwind |

Multiphase model | None | Mixture |

Oil density (kg/m^{3}) | 860 | 860 |

Oil viscosity (Pa·s) | 0.0258 | 0.0258 |

Vapor density (kg/m^{3}) | None | 2.1 |

Vapor viscosity (Pa s) | None | 1.2 × 10^{−5} |

Saturated vapor pressure (Pa) | None | 110 |

Turbulence model | WMLES | WMLES |

Cavitation model | None | Schnerr–Sauer |

Pump state | 2000 rpm | 2000 rpm |

Turbine state | 0–1600 rpm | 0–1600 rpm |

Stator state | Stationary | Stationary |

Inlet condition | 0.8 MPa | 0.8 MPa |

Outlet condition | 0.4 MPa | 0.4 MPa |

Convergence target | 1 × 10^{−5} | 1 × 10^{−5} |

## IV. EXPERIMENTAL VERIFICATION AND RESULT ANALYSIS

### A. External characteristic test

Figure 4 shows the test rig system of the TC. The TC test rig system consists of six parts: drive motor (maximum power 500 kW), load motor (maximum power 500 kW), torque/speed sensor, TC assembly box, signal acquisition system, and pump station and its cooling system. The oil supply pressure of the torque converter is 0.8 MPa, and the back pressure is 0.4 MPa. The inlet oil temperature of the TC is 80 °C. During the test, the driving motor is kept at 2000 rpm, and the switching of different speed ratio conditions of the TC is realized by adjusting the load motor. Then the torque of the pump and turbine under different speed ratio conditions is extracted by the signal acquisition system, and finally, the external characteristic curve of the TC is drawn.

*TR*), efficiency (

*η*), and

*CF*. The external characteristic parameters of the TC are defined as follows:

*T*

_{B}and

*T*

_{T}represent the pump and turbine torque, respectively;

*SR*represents the speed ratio (

*SR*=

*n*

_{T}/

*n*

_{B});

*CF*represents the capacity factor (10

^{−6}min

^{2}r

^{−2}m

^{−1}), which characterizes the ability of the pump to absorb engine power; and

*TR*is the torque ratio, which characterizes the torque conversion capability of the TC.

### B. Analysis of simulation and experimental results

Figure 5 shows the comparison of simulation and test results of the TC under different *SR* operating conditions. It can be seen from the *CF* curve in the figure that when *SR* < 0.4, the *CF* prediction results without cavitation conditions will cause a maximum error of 12%. After considering the cavitation conditions, the prediction error of the *CF* is greatly reduced, and the maximum error is less than 4%. When *SR* ≥ 0.4, the predicted values of the non-cavitation condition and the cavitation condition are consistent, which are greater than the experimental values by about 2%. This may be due to the lack of consideration of the internal leakage of the TC and the fitting error caused by the parametric reconstruction of the baseline blade. For *TR* and *η* curves, when *SR* < 0.4, the calculation results considering cavitation are lower than those without cavitation, which are closer to the experimental results. When *SR* ≥ 0.4, the predicted values of non-cavitation conditions and cavitation conditions are consistent, both of which are greater than the experimental values by about 2%. From the comparison between the CFD calculation results and the experimental results of the TC with/without cavitation, it can be seen that the non-cavitation condition will cause the prediction error to be larger under the low-*SR* condition, and the cavitation condition will be closer to the real test condition, which demonstrates the accuracy of the CFD calculation model of the TC cavitation in this paper.

### C. Analysis of performance prediction results

In Sec. IV B, it is concluded that considering cavitation conditions can improve the prediction accuracy of the external characteristics of the TC. Therefore, the influence of cavitation effect is considered in the numerical calculation of different design schemes of stator and turbine outlet radius.

#### 1. The influence of stator outlet radius on the external characteristics and cavitation characteristics of the TC

Figure 6 shows the comparison of the external characteristics and cavitation volume of the TC with different stator outlet radii. From the external characteristic curve in Fig. 6, it can be seen that as the outlet radius of the stator decreases, the stator blade becomes shorter. At *SR* ≤ 0.4, *TR* and *η* decrease, and the *CF* will be lower. When *SR* > 0.4, the *CF* will become larger. It can be seen from the cavitation volume curve in Fig. 6 that the cavitation volume of the TC gradually decreases with the decrease in the stator outlet radius. When *SR* > 0.4, the cavitation volume inside the TC is reduced to 0. As a whole, it can be seen that the *TR*, *η*, and *CF* of the low-*SR* of the TC will decrease with the decrease in the stator outlet radius, and the cavitation inside the TC will gradually decrease.

#### 2. The influence of turbine outlet radius on the external characteristics and cavitation characteristics of TC

Figure 7 shows the comparison of the external characteristics and cavitation volume of the TC with different turbine outlet radii. It can be seen from the external characteristic curve in Fig. 7 that as the turbine outlet radius increases, the turbine blade becomes shorter, and *TR* and *η* decrease. When *SR* < 0.3, the *CF* will be lower. When *SR* ≥ 0.3, the *CF* becomes larger. It can be seen from the cavitation volume curve in Fig. 7 that the cavitation volume of the TC gradually decreases with the increase in the turbine outlet radius. When *SR* > 0.4, the cavitation volume inside the TC is reduced to 0. As a whole, it can be seen that as the turbine outlet radius increases, the *TR*, *η*, and *CF* of the low-*SR* of the TC will decrease, and the cavitation inside the TC will gradually decrease.

## V. FLOW FIELD ANALYSIS

### A. Two-dimensional flow field analysis

Figure 8 shows the streamwise velocity distribution of different stator/turbine outlet radii. It can be seen from Fig. 8(a) that as the outlet radius of the stator decreases, the stator blade becomes shorter, and the area of the vaneless region between the stator and the pump becomes larger. The flow separation phenomenon on the suction surface of the stator blade becomes more advanced, resulting in a lower efficiency of the TC under low *SR* conditions. With the decrease in the outlet radius of the stator, the stator blade becomes shorter and smaller, the flow velocity of the suction surface of the stator blade decreases, the local pressure increases, and the risk of cavitation of the stator blade is reduced. This explains the reason why the cavitation volume of the TC decreases with the decrease in the outlet radius of the stator, as shown in Fig. 6. As the outlet radius of the stator decreases, the flow velocity at the outlet of the pump decreases significantly, and the velocity circulation from the inlet to the outlet of the pump also decreases, which eventually leads to the decrease in the *CF* under stall conditions. It can be seen from Fig. 8(b) that as the turbine outlet radius increases, the turbine blade becomes shorter, and the flow velocity at the pump outlet decreases significantly. The velocity circulation increase from the inlet to the outlet of the pump decreases, resulting in a decrease in the *CF* under stall conditions. As the turbine blade becomes shorter, the area of the turbine–stator vaneless region becomes larger, and the flow velocity of the fluid impacting the suction surface of the stator blade is significantly reduced. It can be known from the Bernoulli equation that the lower the flow rate and the higher the pressure, the lower the risk of cavitation in the stator blade of the TC, which explains the reason why the cavitation volume of the TC becomes smaller as the turbine outlet radius becomes larger, as shown in Fig. 7.

Figure 9 shows the streamwise pressure distribution of different stator/turbine outlet radii. From Fig. 9(a), it can be seen that with the decrease in the outlet radius of the stator, the stator blade becomes shorter, the low-pressure area of the suction surface of the stator blade becomes smaller, and the risk of cavitation becomes smaller. At the same time, the area of the high-pressure area of the turbine blade becomes smaller, indicating that the TC’s torque conversion ability becomes smaller, which eventually leads to a decrease in *TR*. It can be seen from Fig. 9(b) that as the turbine outlet radius becomes larger, the turbine blade becomes shorter, and the area of the turbine–stator vaneless region becomes larger, resulting in a decrease in the area of the low-pressure zone on the suction surface of the stator blade, which realizes the effective suppression of the stator cavitation. The increase in the turbine outlet radius will also lead to the decrease in turbine blade pressure, which will reduce the torque conversion ability of the TC and ultimately lead to the decrease in *TR*.

Figure 10 shows the comparison of meridional pressure distribution with different stator/turbine outlet radii. It can be seen from the figure that as the stator outlet radius decreases/the turbine outlet radius increases, the pressure in the turbine blade region will decrease significantly, which means that the TC’s torque capability will decrease and the *TR* will decrease significantly. At the same time, the high-pressure area of the pump decreases, and the *CF* of the torque converter becomes smaller. With the decrease in the outlet radius of the stator/the increase in the outlet radius of the turbine, the stator blade becomes shorter/the turbine blade becomes shorter, the area of the low-pressure region on the meridional surface of the stator gradually becomes smaller, and the cavitation risk of the TC will become smaller.

### B. Comparison of cavitation vapor distribution and 3D vortex structure

Figure 11 shows the comparison of 3D cavitation vapor distribution of different stator/turbine outlet radii. It can be seen from the figure that as the outlet radius of the stator decreases, the stator blade becomes shorter and smaller, and the cavitation bubbles attached to the surface of the stator blade gradually decrease. The cavitation bubble near the suction surface of the turbine blade is from the original attachment to the whole span to only the attachment span to about 1/2 of the shell (out ring) of the stator, and the cavitation bubble attachment area in the streamwise direction of the blade is also reduced. As the turbine outlet radius increases, the turbine blades become shorter and smaller, and the area of the vaneless region between the turbine and the stator increases. The cavitation near the leading edge of the stator and turbine is effectively suppressed, and the volume of cavitation bubbles is significantly reduced.

*Q*criterion is used to characterize the vortex coherent structure of the TC. The calculation formula of the

*Q*criterion is as follows:

^{28}

*B*and

*A*denote the anti-symmetric tensor (vortex tensor) and symmetric tensor (strain rate tensor) of the velocity gradient, respectively, ∇

*V*denotes the velocity gradient, $BF2$ represents the square of the matrix norm, and the

*Q*criterion determines the part of

*Q*> 0 as the vortex region.

Figure 12 shows the comparison of 3D vortex structures with different stator/turbine outlet radii. It can be seen from the figure that as the stator outlet radius decreases, the separation vortex phenomenon on the suction surface of the stator near the middle position becomes more prominent, which will directly lead to more flow rate losses in the TC and reduce the efficiency of the TC at low *SR*. In addition, the stator blade becomes shorter, and a large separation vortex appears at the leading and trailing edge of the stator blade, which is consistent with the flow separation phenomenon on the suction surface of the stator blade, as shown in Fig. 8(a). As the turbine outlet radius increases, the turbine blade becomes shorter, and the vorticity in the turbine domain changes from a short vortex to a long-shaped vortex attached to the surface of the turbine blade, which will increase the flow loss caused by greater velocity disturbance, thereby reducing the efficiency under low *SR* conditions. The area of the vaneless region between the turbine and the stator becomes larger, and the flow velocity near the suction leading edge of the turbine and the stator is significantly reduced.

### C. Comparison of cavitation performance and mass flow loss

*σ*) is an important parameter for cavitation prediction of the TC. The cavitation number inside the TC can be calculated by the following formula:

*p*

_{ref}denotes reference pressure,

*p*

_{v}denotes the saturated vapor pressure of oil,

*ρ*

_{l}denotes the density of oil,

*v*

_{ref}denotes the reference velocity (this article takes the flow velocity of the stator inlet as the reference velocity),

*A*is the cross-sectional area of the flow, and

*MF*is the mass flow rate.

Figure 13 shows the comparison of the cavitation number of different stator/turbine outlet radii. It can be seen from Fig. 13(a) that as the outlet radius of the stator decreases, the cavitation number of the TC becomes larger, which means that the risk of cavitation of the TC becomes lower. Although the cavitation number under the high *SR* condition shows the opposite trend, the cavitation of the TC under the high *SR* condition disappeared, so the cavitation number under the high *SR* condition has lost the significance of cavitation prediction. It can be seen from Fig. 13(b) that as the turbine outlet radius becomes larger, the cavitation number of the TC becomes larger under the low *SR* condition, and the risk of cavitation of the TC becomes lower.

Figure 14 shows the comparison of mass flow and mass flow loss of different stator/turbine outlet radii. It can be seen from Figs. 14(a) and 14(b) that as the outlet radius of the stator becomes smaller, the mass flow rate of the TC decreases, resulting in a decrease in the *CF* of the TC. The mass flow loss caused by cavitation inside the TC is reduced from 42.51 to 6.95 kg/s (*SR* = 0), indicating that the cavitation inside the TC is effectively suppressed. It can be seen from Figs. 14(c) and 14(d) that as the turbine outlet radius increases, the mass flow rate of the TC decreases significantly, and the mass flow loss caused by the cavitation of the TC also decreases significantly. It is worth noting that when the turbine outlet radius increases to a certain extent, the mass flow rate inside the TC will increase first and then decrease with the increase in *SR*, which will lead to the same trend of the *CF* with the change in *SR*. The mass loss caused by cavitation inside the TC is reduced from 42.51 to 21.59 kg/s (*SR* = 0), indicating that the cavitation inside the TC is effectively suppressed. When *SR* > 0.4, the internal mass flow loss of the TC with different stator/turbine outlet radii becomes 0, indicating that the cavitation of the TC has completely disappeared. In general, the decrease in the stator outlet radius and the increase in the turbine outlet radius will reduce the external characteristics of the TC under low *SR* conditions. At the same time, the cavitation suppression effect of the TC is better, and the mass flow loss caused by cavitation is smaller.

## VI. CONCLUSIONS

As an important parameter in the design of TCs, the inlet and outlet radii determine the distribution of impeller blades on the meridional surface, which will directly affect the external characteristics, internal flow field, and cavitation characteristics of the TC. The stator and turbine are the main regions where TC cavitation occurs. Based on this, the numerical study of the cavitation characteristics and hydrodynamic characteristics of the TC with different stator and turbine outlet radii is carried out in this study. Three primary conclusions were drawn from the results:

The decrease in the outlet radius of the stator will cause the stator blade to become shorter and smaller, and the flow separation phenomenon on the suction surface of the stator becomes obvious, resulting in a decrease in the efficiency of TC under low

*SR*operating conditions. The flow rate at the outlet of the pump is reduced, and the increase in velocity circulation from the inlet to the outlet of the pump is reduced, which eventually leads to the decrease in the*CF.*The pressure on the meridional surface and the streamwise surface of the turbine decreases, resulting in a decrease in the torque conversion capability of the TC, which ultimately reduces the*TR.*The decrease in the stator outlet radius will make the low-pressure area of the stator and the turbine suction surface smaller, which reduces the risk of cavitation of the TC.The increase in the turbine outlet radius will lead to the increase in the area of the turbine–stator vaneless region and the decrease in the flow velocity near the stator and the turbine suction surface, which will increase the fluid pressure and reduce the risk of cavitation inside the TC. The increase in the outlet radius of the turbine will lead to a decrease in the flow rate at the pump outlet, resulting in a decrease in the velocity circulation increase from the pump inlet to the outlet, ultimately decreasing the

*CF.*The increase in the turbine outlet radius will cause the pressure of the turbine meridional surface and the streamwise surface to decrease, which will eventually lead to a decrease in*TR.*The decrease in the stator outlet radius and the increase in the turbine outlet radius will reduce the mass flow rate of the TC. The mass flow loss caused by the cavitation of the TC will be reduced from the original 42.51 to 6.95 and 21.95 kg/s, respectively. The cavitation volume inside the TC will be reduced from the original 2.0394 × 10

^{−6}to 8.3832 × 10^{−7}m^{3}(cavitation suppression rate is 58.894%) and 9.7159 × 10^{−7}m^{3}(cavitation suppression rate is 52.359%), respectively. It is worth noting that the increase in the turbine outlet radius to a certain extent will cause the mass flow rate and*CF*of the TC to increase first and then decrease with the*SR.*

In general, the influence of the stator outlet radius on the cavitation of the TC is greater than that of the turbine outlet radius. The reasonable design of the stator and turbine outlet radius can effectively suppress the cavitation inside the TC, thereby reducing the mass flow loss inside the TC. The research in this paper can provide guidance for the cavitation suppression and high-performance design of the TC.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. 52075212), the Chongqing Jiaotong University Research Start-up Funding Project (Grant No. F1240031), the Natural Science Foundation of Chongqing (Grant No. CSTB2023NSCQ-MSX0321) and the Exploration Foundation of National Key Laboratory of Automotive Chassis Integration and Bionics (Grant No. ascl-zytsxm-202010). The authors thank the Postdoctoral Research Funding Project of Zhejiang Province (Grant No. ZJ2024164).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Zilin Ran**: Funding acquisition (equal); Writing – original draft (equal). **Huanhui Zhou**: Validation (equal); Writing – review & editing (equal). **Weida Yang**: Data curation (equal); Methodology (equal). **Shuoshuo Lu**: Data curation (equal). **Xianwei Chen**: Data curation (equal). **Bosen Chai**: Conceptualization (equal); Funding acquisition (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Boundary Layer Theory*