Study on the flow field and structural stress field of the primary retarder considering the throttling characteristics of the valve

Abstract

The valve plays an important role in the control of the filling rate (q_fill) and braking characteristics of the hydrodynamic retarder (HR). However, the dynamic throttling characteristics of the valve were rarely considered in previous studies. Therefore, this paper considers the influence of the external valve system (retarder valve) of HR on its flow field, structural field and braking characteristics. Firstly, numerical calculation and experimental verification are carried out on the flow passage model without inlet and outlet (Plan A), with inlet and outlet (Plan B) and with retarder valve (Plan C). After considering the throttling characteristics of the retarder valve, the maximum errors of the impeller rotational speed (n), q_fill and braking torque (T) characteristic curves are 2.678% and 3.517%, respectively, and the average errors are 4.526% and 5.012%, respectively. Compared with the other two schemes, the average error and maximum error of HR have been greatly reduced. Then, this paper compares the differences between the three schemes from the perspective of flow field and structural field, and expounds the influence of retarder valve on the numerical calculation results of retarder from the internal mechanism. Finally, this paper discusses the influence of n, q_fill and retarder valve opening on the calculation results of Scheme C. The results show that the opening of the valve port will seriously affect the flow rate at the inlet of HR, and the flow rate in the retarder valve will have a hysteresis effect on the circulation flow in the mainstream region of HR. The research method of this paper can provide reference for the numerical study of turbomachinery.

Keywords

Hydrodynamic retarder valve system Fluid-Solid Interaction throttling characteristics braking characteristics

Introduction

Hydrodynamic retarder (HR) is widely used in loaders, mine trucks and rail vehicles because of its low noise, smooth braking, and small size (Figure 1). HR is composed of a rotor and a stator. The rotating rotor throws out the fluid to impact the stator to realize the energy conversion of mechanical energy-fluid kinetic energy-heat energy, and finally realizes the continuous braking of the vehicle. The research object of this paper is the primary retarder. The primary retarder is a retarder arranged in front of the hydrodynamic torque converter (HTC). Compared with the traditional HR, the primary retarder has three obvious advantages. It has a more compact axial space arrangement, more convenient for oil circuit arrangement, greater braking power density.

Figure 1.

Application field of HR and its composition and control components.

Many researchers have made a lot of efforts in the numerical calculation of HR. Mu et al. proposed a new type of plunger-type spoiler structure. He compared the influence of installing spoiler structure and not installing spoiler structure on the idling loss of HR.¹ Mu et al. established the parametric equation of the blade top arc of the dual torus HR, and optimized the cascade system to improve the braking performance of HR.² Wei et al. studied the influence of the blade orientation angle of the dual torus HR on the braking performance, and improved the braking performance of HR by optimization.³ Li et al. used large eddy simulation and sliding grid method to numerically analyze the HR, and used optimization algorithm to optimize the blade inclination angle and wedge angle, which improved the braking performance of HR.⁴ Lei et al. conducted a systematic study on the internal flow field flow mechanism, partially filling characteristics and thermal management system of the water medium HR.^5–6 Yuan et al. conducted a Thermal-Fluid-Solid Interaction (TFSI) analysis on the closed HR. The results show that the maximum equivalent stress at the root of the outer ring of the blade is as high as 326.13 MPa.⁷ Xu et al. and Bu et al. studied the influence of variable density and viscosity transmission medium on the internal flow field and external characteristics of HR.^8–10 Wang et al. proposed to process the stripe structure on the surface of the HR blade, which could increase the braking torque (T) of the retarder by 14.36%.¹¹ Wang et al. studied the internal flow field and external characteristics of the double-cavity HR. After bench test, the accuracy of the numerical calculation of the double-cavity HR was verified.¹² Based on Rankine vortex dynamics, Huang et al. established a new blade design theory of HR.¹³ Hur et al. used VOF and sliding mesh methods to study the flow mechanism of the internal flow field and predict the external characteristics of the water medium HR under partial filling ratio and different rotational speed (n).^14,15

In the aspect of valve system and hydromechatronics integrated control of HR, Kong et al. studied the influence of cartridge valve on the control of filling rate of HR. The research shows that the throttling characteristics of cartridge valve can cause great fluctuation of T and circulating mass flow rate (C_mf) of HR.¹⁶ Wei studied the influence of the charging valve system and the non-charging valve system model on the charging characteristics of HR. The research shows that the charging valve system has obvious hysteresis effect on the transient T.^17,18 Mu et al. built a dynamic oil filling and discharging control system for a dual torus HR on the AMEsim platform. He developed a control algorithm to achieve constant-speed downhill and constant-torque downhill control of HR.¹⁹ Xue et al. built the dynamic oil filling and discharging system of the pneumatic HR, and carried out simulation analysis and experimental verification. The filling time of HR is 2.43 s.²⁰ Ma et al. established the hydromechatronics integration simulation model of HR, and simulated the HR under the fixed slope ramp and the variable slope ramp.^21,22 Wang et al. built a dynamic filling and discharging system of HR on the AMEsim platform, and compared the numerical and experimental results. The filling time of the HR is 2.3 s, and the discharging time is 0.9 s.²³ Based on the transmission control protocol, a co-simulation model combining one-dimensional (1D) hydraulic system and three-dimensional (3D) computational fluid dynamics (CFD) technology was constructed by Zhang et al. Through bench test verification, the prediction error of the model for the dynamic T of HR does not exceed 7%.^24,25 It can be seen from the above literature that the control of HR is closely related to the valve system. The accuracy of the valve system model will directly affect the accuracy of numerical calculation.

In addition, it can be seen from the above research that most researchers mainly focus on the study of the middle and rear retarders, and there is little research on the primary retarder with higher braking power density. Most researchers rarely consider the influence of the throttling characteristics of the valve on the HR when studying the flow field in the HR, cascade system optimization and multi-physical field coupling. During the actual test and loading operation of HR, the valve system on the HR housing shell will have a great influence on the filling of HR. The opening of the valve port will affect the filling rate (q_fill) of HR, and the filling and discharging of HR is a dynamic process, so the flow field calculation and multi-physics coupling analysis of HR must also be a dynamic process. In the study of HR, the researchers pay little attention to the influence of the throttling characteristics of the valve on the internal flow field and external characteristics of HR. Obviously, this analysis result is flawed.

Based on this, this paper studies the flow field and structural field in the HR considering the throttling characteristics of the valve. The second section of this paper mainly introduces the Fluid-Solid Interaction (FSI) theory of HR. In the third section, FSI numerical calculation is carried out. In the fourth section, the flow field and structural field of three schemes without inlet and outlet, with inlet and outlet and with retarder valve are compared. In the fifth section of this paper, the different operating conditions of the valve retarder are discussed. The influence of n, q_fill and valve opening on the flow field and structural field of HR is mainly considered, and the influence of valve throttling characteristics on the HR is expounded. In the study, considering the influence of valve throttling characteristics on the flow field and structural field in the HR, it can provide reference for the numerical research of other turbomachinery.

Fluid-Solid Interaction theory of primary retarder

Hydrodynamic retarder theory

HR is developed based on 1D beam theory. The 1D beam theory clarifies the relationship between T and q_fill, n (Tq_filln characteristic surface), the relationship between circulating mass flow rate (C_mf) and q_fill, n (C_mfq_filln characteristic surface), and the relationship between T and C_mf.

According to the 1D beam theory, the T of HR has the following relationship with q_fill and n:

T = q_{fill} λ_{B} ρ g n^{2} D^{5}

(1)

where ρ is the fluid density, λ_B is the capacity coefficient, and T is the braking torque. q_fill and D represent the filling rate and nominal diameter of HR, respectively.

When the n is fixed, there is a linear relationship between T and q_fill.^8,10 When the q_fill is fixed, there is a quadratic function relationship between T and n.

According to the 1D beam theory, when the q_fill of HR is fixed, the C_mf satisfies the law of linear increase with the increase of the n, when the n is fixed, the C_mf and the q_fill also meet the linear increasing relationship. Therefore, the relationship between the C_mf of HR and the q_fill and the n is as follows:

C_{mf} = kn q_{fill}

(2)

where k is a constant.

The calculation formula of the circulating mass flow rate of the HR is as follows²⁶:

{\begin{matrix} C_{mf} = v_{m} A = 2 π r_{i} b_{i} ψ ω_{B} r_{B} \sqrt{\frac{1 - a^{2}}{C_{f}}} \\ a = \frac{r_{B 1}}{r_{B 2}} \end{matrix}

(3)

where v_m denotes the meridional velocity of HR, A denotes the cross-sectional area of the flow passage, r_i denotes the radius of rotation, and b_i denotes the width of the meridional flow passage of the retarder, a represents the ratio of the inlet radius r_B1 to the outlet radius r_B2 of the rotor, ω_B represents the rotational angular velocity of the rotor, C_f represents the total friction resistance coefficient in the curved rotating pipe of HR, and ψ denotes the crowding coefficient of the blade.

There is a linear relationship between the T and the C_mf, and the calculation formula is

T = ρ C_{mf} (v_{u 2} r_{B 2} - v_{u 1} r_{B 1})

(4)

where v_u1 and v_u2 are the circumferential velocities at the inlet and outlet of the retarder, respectively.

Internal structure and working principle of retarder valve

There are two states of the retarder valve : (1) State 1 (the retarder valve is not connected to the air source pressure): the inlet and outlet oil circuits of HR are cut off, the outlet of HR and the oil sump are connected, the oil inside the HR is emptied, and HR is in not working state; (2) State 2 (retarder valve is connected to the air source pressure): the inlet and outlet oil circuits of HR are connected, the outlet of HR and the oil sump is disconnected, the oil chamber of HR is continuously filled with liquid, and HR is in working state. By controlling different air source pressures, the valve spool of retarder valve produces different displacements, thereby controlling the overcurrent cross-sectional area of the inlet and outlet of HR, realizing the control of the filling rate of HR, and finally realizing the control of the braking torque of HR. The 3D model and 2D internal structure of retarder valve are shown in Figure 2.

Figure 2.

3D model and 2D internal structure of retarder valve: (a) Top view of 3D model of retarder valve (1 – outlet oil circuit of HTC; 2 – inlet oil circuit of HTC; 3 – air source pressure inlet; 4 – lubricating oil circuit of transmission; 5 – temperature sensor; 6 – spring; 7 – valve spool; 8 – piston). (b) Internal structure of the retarder valve in working State 1 (A&E: heat exchanger inlet (valve outlet); B: oil sump; C: HR inlet; D: inlet oil circuit of transmission; F: heat exchanger outlet (valve inlet); G: HTC outlet; P: HR outlet). (c) Internal structure of the retarder valve in working State 2.

The force balance equation of the valve spool of retarder valve is as follows:

p_{c} A - (F_{cmd} + k_{s} x_{s}) - \sum_{i = 1}^{8} F_{jet i} = M \frac{\partial^{2} x_{s}}{\partial t^{2}} + R \frac{\partial x_{s}}{\partial t}

(5)

where p_c represents the air source pressure, A represents the overflow area of the valve spool, and F_cmd represents the preload of the spring; k_s represents the spring stiffness, x_s represents the displacement of the valve spool, and M represents the valve spool mass of the retarder valve; R represents damping, and F_jeti represents the steady-state hydrodynamic force of each throttling edge of the valve spool of the retarder valve.

Governing equations

The multi-physics coupling of HR mainly involves flow field calculation and solid field calculation. The fluid field is calculated by Finite Volume Method (FVM), and the solid field is solved by Finite Element Method (FEM).

(1) Fluid control equations. CFD numerical calculation method is a full 3D solution method, which has been widely used in the numerical calculation of HR. The numerical calculation inside the HR mainly involves the continuity equation, momentum equation and energy equation. In this paper, the large eddy simulation (LES) model is used to solve the turbulence of HR. The LES uses the filter function to filter the Navier-Stokes (N-S) equations. The turbulence structure larger than the grid scale is solved by the LES model, and the turbulence structure smaller than the grid scale is modeled. The filtering functions in the LES model are as follows:

G (| x - x' |) = {\begin{matrix} \frac{1}{Δ x_{1} Δ x_{2} Δ x_{3}}, | {x'}_{i} - x_{i} | \leq \frac{Δ x_{1}}{2}, i = 1, 2, 3 \\ 0, | {x'}_{i} - x_{i} | > \frac{Δ x_{1}}{2}, i = 1, 2, 3 \end{matrix}

(6)

where x_i is the index of tensor form. Δx_i is the component of the grid scale in the i direction

The N-S equation filtered by the filter function is as follows:

{\begin{matrix} \frac{\partial ρ_{m}}{\partial t} + \frac{\partial (ρ_{m} {\bar{u}}_{j})}{\partial x_{j}} = 0 \\ \frac{\partial}{\partial t} (ρ_{m} {\bar{u}}_{i}) + \frac{\partial}{\partial x_{j}} (ρ_{m} {\bar{u}}_{i} {\bar{u}}_{j}) = - \frac{\partial \bar{p}}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (μ_{m} \frac{\partial {\bar{u}}_{i}}{\partial x_{j}}) - \frac{\partial τ_{ij}}{\partial x_{j}} \end{matrix}

(7)

where ρ_m and μ_m represent the density and viscosity of the mixed phase, respectively. The definitions of ρ_m and μ_m are as follows:

{\begin{matrix} ρ_{m} = α_{v} ρ_{v} + (1 - α_{v}) ρ_{l} \\ μ_{m} = α_{v} μ_{v} + (1 - α_{v}) μ_{l} \end{matrix}

(8)

where α_v denotes the volume fraction of vapor, ρ_v and ρ_l denote the densities of vapor and liquid phases, respectively, and μ_v and μ_l denote the viscosities of vapor and liquid phases, respectively.

u represents the time-averaged velocity in tensor form, p represents the pressure in the flow field, and the overlined variables represent the filtered field variables. τ_ij represents the anisotropic sub-grid stress. The definition of τ_ij is as follows:

{\begin{matrix} τ_{ij} = ρ_{m} (\bar{u_{i} u_{j}} - \bar{u_{i}} \bar{u_{j}}) = \frac{1}{3} τ_{kk} δ_{ij} - 2 μ_{t} \bar{S_{ij}} \\ \bar{S_{ij}} = \frac{1}{2} (\frac{\partial {\bar{u}}_{i}}{\partial x_{j}} + \frac{\partial {\bar{u}}_{j}}{\partial x_{i}}) \end{matrix}

(9)

where τ_kk denotes isotropic sub-grid stress, S_ij represents the stress tensor ratio, δ_ij represents the Kronecker function, and μ_t represents the turbulent viscosity. In this paper, μ_t is calculated by the dynamic kinetic energy transport (KET) sub-grid model in the LES model. In the KET sub-grid model, the calculation formula for μ_t is as follows:

{\begin{matrix} μ_{t} = C_{k} ρ_{m} k_{sgs}^{1 / 2} V^{1 / 3} \\ ρ \frac{\partial \bar{k_{sgs}}}{\partial t} + ρ \frac{\partial \bar{u_{j}} \bar{k_{sgs}}}{\partial x_{j}} = (2 C_{k} ρ k_{sgs}^{1 / 2} V^{1 / 3} \bar{S_{ij}} - \frac{2}{3} ρ k_{sgs} δ_{ij}) \frac{\partial \bar{u_{i}}}{\partial x_{j}} \\ - C_{ε} ρ k_{sgs}^{3 / 2} V^{- 1 / 3} + \frac{\partial}{\partial x_{j}} (C_{k} ρ V^{1 / 3} k_{sgs}^{1 / 2} \frac{\partial k_{sgs}}{\partial x_{j}}) \end{matrix}

(10)

where k_sgs represents the turbulent kinetic energy of the sub-grid model, V represents the volume of the calculation unit, C_k represents the model coefficient of the KET sub-grid model, and C_ε represents the dynamic variable.

energy equation:

\frac{\partial (ρ_{m} E)}{\partial t} + \frac{\partial}{\partial x_{j}} [u_{i} (ρ_{m} E + p)] = \frac{\partial}{\partial x} (λ \frac{\partial T_{t}}{\partial x_{j}} - \sum_{j} h_{j} J_{j} + u_{j} τ_{ij}) + S_{h}

(11)

where E represents the energy of the unit mass, λ and h_j represent the heat transfer coefficient and enthalpy, S_h represents the viscous dissipation term, and T_t represents the temperature.

(2) Solid control equations. The conservation equation of the solid part can be derived from Newton’s second law:

ρ_{s} \frac{\partial^{2} d_{s}}{\partial t^{2}} = \nabla \cdot σ_{s} + f_{s}

(12)

where ρ_s is the solid density, σ_s is the Cauchy stress tensor, and f_s is the body force vector, d_s is the displacement in the solid domain.

(3) FSI equations. FSI follows the most basic conservation principle, so at the interface of solid-liquid coupling, it should satisfy the equality or conservation of variables such as fluid and solid stress ( $τ$ ), displacement (d), heat flow rate (h_f), and temperature (T_t). i.e. the following equation is satisfied:

{\begin{matrix} τ_{l} \cdot n_{s} = τ_{l} \cdot n_{s} \\ d_{l} = d_{s} \\ h_{fl} = h_{fs} \\ T_{tl} = T_{ts} \end{matrix}

(13)

Fluid-Solid Interaction model

The specific process of FSI of the primary retarder is shown in Figure 3. In this paper, the parameters of the torus and cascade system are extracted according to the impellers model of the primary retarder. Then, the physical boundary is defined according to the flow passage model, and the mesh is divided. The pressure distribution on the blade surface is obtained by CFD calculation. The pressure distribution is loaded onto the primary retarder impellers. The boundary constraints and motion of the HR impellers are defined. The stress, strain and deformation of the impellers (solid domain) are obtained by ANSYS Workbench. By changing the n and q_fill of the fluid domain to adjust the working condition of HR, the parameters such as pressure, velocity and braking torque of the fluid domain and the parameters such as stress, strain and deformation of the solid domain can be obtained, and the FSI calculation of the primary retarder can be realized.

Figure 3.

FSI principle of primary retarder.

Numerical calculation of Fluid-Solid Interaction of primary retarder

Physical model

In this paper, the parameters of the torus and cascade system are extracted from the impellers model of the primary retarder, and the key parameters of the primary retarder can be obtained as shown in Table 1. In this paper, the nominal diameter of the torus of the primary retarder is 377 mm, and the blade inclination angle and wedge angle are 60° and 30°.

Table 1.

Primary retarder torus and cascade system parameters.

D (mm)	d ₀ (mm)	B (mm)	h _R (mm)	h _S1(mm)	h _S2(mm)	Z _R	Z _S
377	319	29.5	3.00	4.42	13.00	49	40

In Table 1, D is the nominal diameter of the torus of the primary retarder, d₀ is the minimum diameter of the torus, and B is the width of the torus. h_R is the thickness of rotor blades, h_S1 is the thickness of adjustable stator blades, h_S2 is the thickness of fixed stator blades, Z_R is the number of rotor blades, Z_S is the number of stator blades. Figure 4 is the assembly model of the primary retarder impellers and valve. The primary retarder consists of a rotor (yellow color), a stator (adjustable stator-green color, fixed stator-red color) and a retarder valve (cyan color). The retarder valve controls the flow cross-section area of the retarder inlet through the spool displacement, thereby controlling the filling rate of HR and generating different braking torques to meet the braking performance requirements of the vehicle on different slopes.

Figure 4.

Impeller and valve assembly model of primary retarder.

In this paper, three schemes are considered to numerically simulate the primary retarder: (1) Plan A: the model of primary retarder model without inlet and outlet (simplified model); (2) Plan B: the model of primary retarder model with inlet and outlet (simplified model); (3) Plan C: the model of primary retarder with valve (accurate model). In this paper, the differences between the three schemes are discussed, and the flow field and structural stress field are comprehensively evaluated. As is shown in Figure 5, the flow passage model of Plan A is composed of rotor flow passage and stator flow passage, ignoring the influence of the inlet and outlet of HR. The HR flow passage model of Plan B consists of five parts: rotor flow passage, stator flow passage, outer ring flow passage, inlet flow passage (inlet flow passage is arranged on the stator) and outlet flow passage (outlet flow passage is arranged outside the outer ring flow passage). The retarder flow passage model of Plan C is composed of Plan B flow passage model and retarder valve flow passage model (the retarder valve flow passage model is composed of valve inlet flow passage and valve outlet flow passage). The HR flow passage model of Plan C considers not only the internal circulation of HR, but also the throttling effect of the valve.

Figure 5.

Primary retarder flow passage models of different schemes.

Mesh model and mesh independence verification

In this paper, the mesh independence verification of HR flow passage model of Plan C is carried out under the condition of full filling rate of HR and 2000 rpm of the impeller rotational speed.^10,12 A large number of literature studies on HRs have shown that the smaller the global grid size, the more the number of grids, the better the calculation accuracy.^20,25,27,28 However, as the global grid size decreases, the number of grids increases sharply, resulting in a rapid increase in computing time. Therefore, in order to balance the calculation cost and calculation accuracy, it is necessary to carry out the grid independence verification. This paper refers to the research of relevant literature,^20,25,27,28 the change rate of impeller torque ε(T) is used as the calculation index when the grid independence verification of the retarder is carried out. The calculation formula of ε(T) is as follows:

ε (T) = \frac{T (n) - T (n - 1)}{T (n)} \times 100 %

(14)

Table 2 is the statistics of the number of meshes and the calculation results under different mesh sizes. The mesh model of the primary retarder and the verification results of mesh independence are shown in Figure 6. In this paper, the mesh size is 1.550 mm and the number of meshes is 4,007,565 elements. At this time, the torque change rate is 2.856%, which is lower than 3%.²⁹ At the same time, the calculation time is 7.923 h. It can be considered that the calculation result is independent of the mesh size. At this time, reducing the mesh size will greatly increase the number of meshes and increase the computational cost.

Table 2.

Number of meshes with different sizes.

Mesh size	Rotor	Stator	Out ring	Total	ε(T)/%	Computation time (h)
2.500 mm	367,842	404,793	235,662	1,008,297	6.967	1.720
1.975 mm	721,209	824,191	461,433	2,006,833	5.824	3.435
1.719 mm	1,095,519	1,206,711	705,477	3,007,707	3.835	5.597
1.550 mm	1,407,417	1,639,005	961,143	4,007,565	2.856	7.923
1.438 mm	1,830,126	2,004,965	1,169,361	5,004,452	2.548	10.158
1.350 mm	2,175,667	2,419,874	1,412,256	6,007,797	2.153	12.626
1.275 mm	2,596,468	2,752,527	1,654,491	7,003,486	2.042	15.173

Note. The bold font is the number of grids selected for the CFD calculation of HR in this paper.

Figure 6.

Mesh model and mesh independence verification of primary retarder: (a) mesh model and (b) mesh independence results.

CFD calculation and experimental verification

The boundary conditions of the fluid domain of primary retarder are defined as shown in Table 3. The table clarifies the interaction between the rotor fluid domain, the stator fluid domain and the valve fluid domain and the type of fluid boundary.

Table 3.

Boundary conditions of fluid domain of primary retarder.

Item	Fluid boundary conditions	Description
R-blades	Moving wall	Blade boundary of rotor fluid domain
R-inter1	Interface	Rotor and stator fluid domain interface
R-shell	Moving wall	Shell boundary of rotor fluid domain
S-blades	Stationary wall	Blade boundary of stator fluid domain
S-inter	Interface	Stator and rotor fluid domain interface
S-shell	Stationary wall	Shell boundary of stator fluid domain
Out ring-shell	Stationary wall	Shell boundary of out ring fluid domain
Our ring-inter	Interface	Out ring and rotor fluid domain interface
R-inter2	Interface	Rotor and out ring fluid domain interface
Inlet-wall	Stationary wall	Shell boundary of HR inlet
Outlet-wall	Stationary wall	Shell boundary of HR outlet
Valve inlet1	Velocity inlet	The inlet 1 boundary of retarder valve
Valve inlet2	Velocity inlet	The inlet 2 boundary of retarder valve
Valve outlet	Outflow	The outlet boundary of retarder valve
valve-shell	Stationary wall	Shell boundary of valve fluid domain

In the working process of HR, the Rotor-stator interaction (RSI) effect between the rotor and stator of HR is very prominent. In order to ensure the computational efficiency and flow field capture accuracy of HR, this paper uses equation (15) to determine the time step of the CFD calculation of HR. For the fast CFD calculation of the external characteristics of HR, a time step is used to solve the flow field of the rotor across a blade. For the high-precision analysis of the flow field in HR, the strong RSI effect between the rotor and the stator is captured, and 5-time steps are used to solve the flow field of the rotor passing through a blade.

{\begin{matrix} Δ t_{fast} = \frac{60}{n} \frac{1}{Z_{R}} \geq 0.0005 s \\ Δ t_{accurate} = \frac{60}{n} \frac{1}{Z_{R}} \frac{1}{5} \geq 0.0001 s \end{matrix}

(15)

where n represents the impeller rotational speed (2000 rpm is selected as the calculation rotational speed in the time step calculation in this paper, and 2000 rpm is a common speed operation condition of HR^10,12). Z_R represents the number of rotor blades, which is 49 in this paper. Δt_fast represents the time step of the fast CFD calculation of the external characteristics of HR. Δt_accurate represents the time step of the accurate analysis of the flow field of HR.

The detailed parameter settings for the CFD calculation of the primary retarder in this paper are shown in Table 4. The calculation step is set to 240 steps (0.0005 s/step), and the impeller rotates four times. At this time, the flow field calculation result of HR is stable. By setting different n and q_fill, the primary retarder is simulated to work in different operating conditions. By calculating the CFD calculation results under different operating conditions, the T under different n and q_fill is calculated, and the braking characteristic curve of the primary retarder is obtained.

Table 4.

Detailed parameter settings for the CFD calculation of the primary retarder.

Boundary conditions/parameters	Condition/values
Analysis type	Transient state
Solver type	Pressure-Based
Multiphase model	Mixture
Interaction	Sliding mesh
Pressure-velocity coupling	SIMPLEC
Momentum	Bounded central differential
Transient formula	Second order implicit
Other term spatial discretization	Second order upwind
Turbulence model	LES model
Sub-grid scale models	KET model
Oil density and viscosity	860 kg/m³, 0.0258 Pa·s
Air density and viscosity	1.225 kg/m³, 1.789 × 10⁻⁵ Pa·s
Rotor status	Varied from 800–2400 rpm
Stator status	Stationary
Filling rate (q)	Varied from 0.2 to 1
Temperature	343.15 K
Inlet oil flow	115 L/min
Inlet velocity	0.5 m/s
Outlet pressure	Outflow
Boundary details	No-slip and smooth wall
Solve for time step	0.0005 s/step (fast CFD)/0.0001 s/step (accurate CFD)
Convergence target	1 × 10⁻⁵

The integrated control system of the primary retarder is shown in Figure 7(a). The braking torque of HR is determined by the joint control of the retarder valve and the motor. The retarder valve is controlled by a variable air source pressure (an adjustable pressure air storage tank with valve control). The air source pressure can drive the retarder valve spool, and the valve spool of the retarder valve is balanced under the action of air source pressure and spring force (equation (5)). The pressure of the variable air source can be changed infinitely between 0 and 8 bar. The effective air source pressure of the retarder valve is 3bar, which is used to overcome the pre-load of the oil circuit and make the spool completely pass through the State 1 position of the retarder valve. When the air source pressure is 4–8 bar, the opening degree at the inlet of HR is 0.2–1 (the opening degree is affected by the displacement of the valve spool), and the filling rate of HR chamber is 0.2–1. In Figure 2(c), when the air source pressure becomes larger, the cross-sectional area of the flow from valve inlet (F) to HR inlet (C) and HR outlet (P) to valve outlet (A&E) increases from small to large, and the oil entering the oil chamber of HR increases, so as to control the filling rate of the oil chamber of HR by controlling the spool displacement of the retarder valve, and finally realize the control of the braking torque of HR. The entity model and internal structure of the retarder valve are shown in Figure 7(b) and Figure 2, respectively.

Figure 7.

Integrated control system of primary retarder and its test bench layout: (a) Integrated control system of primary retarder (1 – outlet oil circuit of HTC; 2 – inlet oil circuit of HTC; 3 – air source pressure inlet; 4 – lubricating oil circuit of transmission; 5 – temperature sensor; valve inlet-outlet oil circuit of heat exchanger; valve outlet-inlet oil circuit of heat exchanger). (b) Internal structure of retarder valve (A&E: heat exchanger inlet (valve outlet); B: oil sump; C: HR inlet; D: inlet oil circuit of transmission; F: heat exchanger outlet (valve inlet); G: HTC outlet; P: HR outlet; T: HTC inlet (not shown in the figure)). (c) Schematic diagram of HR test bench.

The HR adjusts the speed of the rotor through a computer-controlled motor, and simulates the velocity of the vehicle wheel during the long downhill process (the HR is in the long downhill braking process, and the power transmission route is: vehicle wheel-transmission-HR rotor). The external oil circuit control of HR is controlled by the ECU module outside the transmission. The on-off/opening degree control of HR chamber and the external oil circuit is controlled by the retarder valve (Figure 7(c)).

In this paper, the impellers torque at different rotational speeds (800-2400rpm) under full filling rate conditions of HR is tested by the built HR test bench. The torque of HR impellers with different filling rate is also tested at 2000 rpm. In order to reduce the error of test data acquisition, the impellers torque under one working condition is measured many times and averaged during the test.

The CFD results of the three schemes are compared with the experimental results (Figure 8). From the T curves at different impeller rotational speeds in Figure 8, it can be seen that Plan C has the highest prediction accuracy, with an average error of 2.678%, less than 5%, and a maximum error of 4.526%, less than 5%, appearing at 800 rpm. The prediction accuracy of Plan A is the lowest, with an average error of 7.467% and a maximum error of 9.845%, which is close to 10%, appearing at 800 rpm. From the T curves under different filling rates in Figure 8, it can be seen that Plan C has the highest prediction accuracy, with an average error of 3.517%, less than 5%, and a maximum error of 5.012%, close to 5%, which appears at q = 0.2. The prediction accuracy of Plan A is the lowest, with an average error of 7.268% and a maximum error of 9.562%, which appears at q = 0.4. In general, the numerical prediction error of HR is relatively large at low n and low q_fill. Plan C has significantly improved the prediction accuracy of its overall external characteristics compared with other schemes because it considers the influence of valve throttling characteristics.

Figure 8.

Comparison of external characteristics of three calculation schemes.

Structural field calculation

After the flow field calculation of the flow passage model of the primary retarder, the pressure load on the blade surface is obtained. According to the mesh size used in the FSI analysis of HTC impellers in Wang et al.,³⁰ the global size of the solid mesh of the primary retarder is set to 2 mm, and the overall mesh number of the impellers is 1102,336 elements. Before the solid domain calculation, it is necessary to constrain and define the motion of the impeller to achieve synchronous transmission with the fluid domain and accurate pressure loading. Figure 9 is the constraint definition of the primary retarder impeller. For the stator, the fixed constraint is used in this paper. For the rotor, the displacement limit and the rotation motion are defined for the inner ring to ensure the synchronous motion and accurate pressure loading with the rotation of the fluid domain.

Figure 9.

Motion and constraint definition of retarder impellers: (a) definition of stator constraints, (b) definition of rotor motion, (c) impellers model with constraint definition.

Figure 10 shows the stress loading of the primary retarder impellers. The pressure data of 3D space points are extracted from the fluid domain and then applied to the solid domain interface of the corresponding space points. According to the principle that the variables such as stress (τ), displacement (d), heat flow (h_f), and temperature (T_t) on the interface between fluid and solid are equal, the pressure loading on the surface of rotor and stator is completed.

Figure 10.

Pressure loading of primary retarder impellers: (a) pressure distribution on the rotor blades, (b) pressure loading of rotor blades, (c) pressure distribution on the stator blades, and (d) pressure loading of stator blades.

Analysis of flow field and structure field of primary retarder

Comparison of pressure field

In the previous sections, the differences in the calculation results of the external characteristics of the three schemes are compared in detail, and it is necessary to explain them from the mechanism. The internal flow field of the retarder determines its external characteristics, so this section mainly evaluates the differences between the three schemes from the internal flow field information such as pressure, velocity, vorticity, and structural field information such as stress and deformation. The torque of the impeller is formed by the integral of the product of pressure and radius on the blade surface. Figure 11 is the comparison of the pressure contours of the three schemes. It can be seen from the figure that the high-pressure area of Plan A is larger than that of Plan B and Plan C, and the distribution area of high-pressure area of Plan C is the smallest, which explains the reason why the torque of Plan C impellers is lower than that of other schemes in the three schemes. The reason for the uneven circumferential pressure distribution of the rotor and stator in Figure 11 is that the inlet flow passage on the stator of HR occupies five blades of the stator. The number of blades in the rotor is 49, which is centrosymmetric in the circumferential direction of the impellers. The number of blades in the stator is 40, which is non-centrosymmetric in the circumferential direction of the impellers, resulting in the non-centrosymmetric flow passage of the stator blade, which enlarges the Rotor-stator interaction (RSI) and non-uniform effect between the rotor and the stator.

Figure 11.

Pressure contour comparison of three schemes.

The blade load curves of the three schemes are shown in Figure 12. The total pressure distribution curve of the blade cross section is shown in Figure 12(a). It can be seen from the figure that the total pressure of the blade cross section of the three schemes is: Plan A > Plan B > Plan C, which demonstrates the reason for the maximum braking torque of Plan A in Figure 8. The pressure surface and suction surface of the blade are in a relatively high-pressure state, and the blade wedge angle is in a low-pressure state, which creates favorable conditions for the cavitation of HR. The distribution trend of the blade load curve of the blade section in this paper is consistent with the Yang et al.²⁵

Figure 12.

Comparison of blade load curves of three schemes: (a) comparison of total pressure at sectional blade (α: wedge angle of blades; β: incline angle of blades; 0: root position of blade pressure surface; 0.5: position of blade wedge angle; 1: root position of blade suction surface). (b) the static pressure and total pressure distribution on the blade streamwise direction.

From the static pressure distribution of the blade in the flow direction in Figure 12(b), it can be seen that the inlet and outlet positions of the blade are in a low-pressure state during the fluid flow process, and the middle position of the blade is in a high-pressure state.⁹ From the total pressure distribution of the blade in Figure 12(b), it can be seen that under the action of the centrifugal force of the impeller, the fluid flows from the inlet of the rotor blade to the outlet, realizing the conversion of mechanical energy to fluid kinetic energy. The total pressure at the outlet of the blade is higher than the total pressure at the inlet of the blade. In general, the static pressure and total pressure of Plan A in the blade streamwise direction are higher than those of the other two schemes, which will lead to excessive prediction of braking performance.

Comparison of velocity contour and streamline

The 1D beam theory indicates that the impellers torque is equal to the outlet velocity circulation of HR impellers minus its inlet velocity circulation (equation (4)). Figure 13 is the comparison of 2D streamwise velocity contours of the three schemes. It can be seen from the figure that the velocity at the interface between the rotor and the stator of the Plan A model is significantly greater than the other two schemes, indicating that the velocity circulation at the outlet of the Plan A model is the largest, which explains the reason why the T of Plan A is greater than the other two schemes. The total velocity is composed of tangential velocity, radial velocity and circumferential velocity. It can be seen from Figure 13 that most of the contribution of the total velocity inside the HR comes from its circumferential velocity. The product of circumferential velocity and radius is the velocity circulation. The circumferential velocity of Plan A is significantly larger than that of the other two schemes, and the circumferential velocity of Plan C is the lowest. It also demonstrates that Plan A has the largest T and Plan C has the lowest T. This shows that Plan A ignores the influence of inlet and outlet on the flow of HR, which will cause the calculation result of the velocity in the flow field to be higher, resulting in the prediction accuracy of its external characteristics to be too low.

Figure 13.

Comparison of 2D streamwise velocity contours of the three schemes (R = 184 mm).

Figure 14(a) shows the 3D velocity streamline comparison of the three schemes. The high n causes the velocity of the mainstream region of HR to be greater than the flow velocity at the inlet and outlet and inside the retarder valve, but the fluid velocity at the inlet and outlet is perpendicular to the flow velocity of the mainstream region, which will have a significant impact on the velocity of the mainstream region. From Figure 14(a), it can be seen that the model of Plan A does not consider the influence of inlet and outlet, and the flow velocity is too large. Plan B and Plan C are affected by the inlet and outlet and valve throttling characteristics respectively. The flow velocity in the mainstream region is disturbed, and the flow velocity is significantly reduced.

Figure 14.

Comparison of 3D velocity streamlines and vorticity of three schemes: (a) 3D velocity streamlines and (b) 3D vorticity.

Comparison of three-dimensional vorticity

According to the 1D beam design theory of HR, due to the influence of the centrifugal force of the HR impellers, a circular flow will be formed on its meridional surface, forming a relative velocity between the blade and the fluid. The motion of the impellers will form a traction motion velocity in its circumferential direction, and the relative motion velocity and the traction motion velocity form a 3D circumferential vortex of HR. Figure 14(b) shows the 3D vorticity comparison of the three schemes. It can be seen from the figure that the vorticity of HR presents the characteristics of the circumferential vortex, which is consistent with the theoretical analysis results and demonstrates the effectiveness of the numerical calculation results. The results of 3D vorticity and 3D velocity streamline analysis are consistent. The influence of inlet and outlet and valve throttling characteristics will hinder the mainstream region of the primary retarder.

Analysis of stress and deformation

Figure 15 shows the comparison of stress contour of the three schemes. It can be seen from Figure 15(a) that the maximum stress of the rotor reaches 484.59 MPa. Because the impellers are produced by nodular cast-iron after strengthening treatment, its tensile strength is above 900 MPa, and the strength of HR impellers in this paper meets the requirements. The high stress area of the three schemes is Plan A > Plan B > Plan C. It can be seen that the stress prediction of Plan A is too large, and the actual stress distribution should be closer to the results of Plan C scheme. The maximum stress of the stator is 123.21 MPa, about 1/4 of the maximum stress of the rotor. The thickness of the rotor blade is about 1/4 of the stator blade (Table 1). The thickness of the rotor blade is relatively thin, and the rotor blade is more dangerous.

Figure 15.

Comparison of stress contour of three schemes: (a) the stress contour of the rotor and (b) the stress contour of the stator.

Figure 16 is the comparison of the deformation contour of the three schemes. The maximum deformation of the rotor is located in the outer ring of the impeller, reaching a deformation of 1.23 mm, and the maximum deformation of the stator is located in the inner ring of the impeller, reaching 0.051 mm (the thickness of the stator blade is greater than that of the rotor blade, and its stiffness is greater and its anti-deformation ability is stronger). The impellers deformation of Plan A is the largest of the three schemes, and the overall deformation is more uniform in the circumferential direction, which is the result of not considering the disturbance of inlet and outlet and valve throttling characteristics. Plan C has the smallest deformation. The arrangement of the retarder valve on one side of HR will lead to the uneven distribution of the impeller deformation in the circumferential direction, which is closer to the actual working condition of HR.

Figure 16.

Comparison of deformation contour of three schemes: (a) the deformation contour of the rotor and (b) the deformation contour of the stator.

Result and discussion

The influence of rotational speed on the temperature field of HR

The average temperature curve of the inlet and outlet of HR with time is shown in Figure 17. It can be seen from the figure that with the increase of time, the average temperature of HR outlet tends to be stable, and the heat balance state is reached inside the HR at about 0.15 s. The average temperature at the inlet of HR is maintained at about 333.15K, and the average temperature at the outlet of HR is finally maintained at 367.6038K, and the temperature is increased by 34.4538K. The change trend of temperature with time calculated in this paper is consistent with that in Liu et al.⁹ (because the cavity shape of HR in Liu et al.⁹ is larger, the braking performance at the same rotational speed is higher than that in this paper, so the average temperature at the outlet of HR is relatively higher).

Figure 17.

Average temperature of the inlet and outlet of HR varies with time. (n = 2400 rpm).

The relationship between the average outlet temperature of HR and time is as follows:

\begin{matrix} T_{t outlet} = 332.031 + 511.725 t - 1742.491 t^{2} - 5992.465 t^{3} \\ + 46581.641 t^{4} - 72689.315 t^{5} R^{2} = 0.97721 \end{matrix}

(16)

The influence of rotational speed on the temperature field distribution and the average temperature of HR outlet is shown in Figure 18. It can be seen from Figure 18(a) that the inlet of HR is in a low temperature state. As the impellers rotates, the fluid is thrown out from the rotor outlet under the action of centrifugal force to impact the stator, realizing the conversion of flow kinetic energy to thermal energy. The interior of HR is in a high temperature state, and the heat is circulated to the heat exchanger through the outlet of HR to realize the external circulation of heat. As the rotational speed increases, the heat exchange inside the HR becomes more intense, and the temperature on the HR impellers continues to increase. The highest local temperature of HR flow field can reach 372.4438 K (n = 2000 rpm).

Figure 18.

Relationship between the temperature field distribution, the average temperature of HR outlet and the rotational speed (q = 1): (a) Temperature distribution on the surface of the impellers at different rotational speeds and (b) Average temperature at the outlet of HR and the maximum temperature of the flow field change with the rotational speed curve.

It can be seen from Figure 18(b) that with the increase of rotational speed, the average temperature of HR outlet and the maximum temperature of the flow field will increase. The average outlet temperature of HR increases from 338.2433 K to 367.6038 K, and the maximum temperature of the flow field increases from 342.2506 K to 399.6924 K (126.5424°C).

The influence of rotational speed on flow field and structure field

In order to better reveal the flow mechanism of the primary retarder considering the throttling characteristics of the valve, this paper discusses the influence of n, q_fill and valve opening on the flow field and structural field in the primary retarder.

Figure 19 shows the relationship between n and maximum static pressure of flow field. As is shown in Figure 19(a), with the increase of n, the distribution of high-pressure region on the impellers surface is more uniform. As is shown in Figure 19(b), there is a quadratic function relationship between the n and the maximum static pressure of the flow field (equation (17)).

{\begin{matrix} P_{Rmax} = 0.7148 - 0.826 (\frac{n}{400}) + 0.2853 {(\frac{n}{400})}^{2} R^{2} = 0.94129 \\ P_{Smax} = 0.54052 - 0.63498 (\frac{n}{400}) + 0.25656 {(\frac{n}{400})}^{2} R^{2} = 0.9705 \end{matrix}

(17)

where P_Rmax represents the maximum static pressure of the rotor flow field, and P_Smax represents the maximum static pressure of the stator flow field.

Figure 19.

The influence of rotational speed on pressure field (q = 1): (a) the relationship between n and pressure distribution and (b) the relationship between n and maximum pressure.

Figure 20 shows the influence of n on the velocity field. With the increase of n, the velocity of the flow field increases, and the difference between the outlet and inlet circulation of the primary retarder, that is, the impeller torque increases. It can be seen from Figure 20(b) that there is a linear relationship between the n and the maximum velocity of the flow field, and there is a quadratic function relationship between the n and the maximum dynamic pressure of the flow field (equation (18)). The formula for calculating the dynamic pressure of the flow field is 1/2ρv². It can be seen that the numerical calculation and theoretical results in this paper are consistent.

{\begin{matrix} v_{\max} = - 6.743 + 1.34085 (\frac{n}{20}) R^{2} = 0.99957 \\ P_{dmax} = - 0.43093 + 0.31777 (\frac{n}{400}) + 0.23563 {(\frac{n}{400})}^{2} \\ R^{2} = 0.99825 \end{matrix}

(18)

where v_max represents the maximum velocity in the flow field of HR, and P_dmax represents the maximum dynamic pressure in the flow field of HR.

Figure 20.

The influence of n on velocity field (q = 1): (a) the relationship between n and flow field velocity distribution (R = 184 mm). and (b) the relationship between n and maximum velocity.

Figure 21 shows the influence of n on the structural field. The increase of n will significantly lead to the increase of impellers stress, and also lead to the increase of impellers deformation. It can be seen from Figure 21(b) that there is a significant linear relationship between n and maximum stress and maximum deformation (equation (19)).

{\begin{matrix} τ_{\max} = - 301.702 + 1.90252 (\frac{n}{5}) R^{2} = 0.96687 \\ d_{\max} = - 0.75427 + 1.94054 (\frac{n}{2000}) R^{2} = 0.97856 \end{matrix}

(19)

where τ_max represents the maximum stress and d_max represents the maximum deformation.

Figure 21.

The influence of n on structural field (q = 1): (a) the relationship between the n of the rotor and the stress and deformation contour and (b) the relationship between n and maximum stress, maximum deformation.

The influence of filling rate on flow field and structure field

Figure 22 shows the effect of q_fill on the pressure field. With the increase of q_fill, the maximum pressure region is more evenly distributed on the impellers. Furthermore, there is a significant quadratic function relationship between the q_fill and the maximum pressure of the impellers (Equation (20)).

{\begin{matrix} P_{Rmax} = 0.48245 - 2.08503 q_{fill} + 5.5497 q_{fill}^{2} R^{2} = 0.99239 \\ P_{Smax} = 0.12462 + 0.60413 q_{fill} + 3.23021 q_{fill}^{2} R^{2} = 0.99217 \end{matrix}

(20)

Figure 22.

The effect of q_fill on pressure field (n = 2000 rpm): (a) the relationship between q_fill and impeller pressure distribution (2000 rpm) and (b) the relationship between q_fill and maximum pressure of flow field.

Figure 23 shows the effect of q_fill on the velocity field. It can be seen from the figure that there is a positive correlation between the filling rate and the flow field velocity. Furthermore, there is a linear function and a quadratic function relationship between the q_fill and the maximum velocity and maximum dynamic pressure of the flow field, respectively (equation (21)).

{\begin{matrix} v_{\max} = 18.7897 + 106.697 q_{fill} R^{2} = 0.99735 \\ P_{dmax} = 0.26348 + 1.18545 q_{fill} R^{2} = 0.99729 \end{matrix}

(21)

Figure 23.

The effect of q_fill on velocity field (n = 2000 rpm): (a) the relationship between q_fill and velocity distribution of flow field (R = 184 mm) and (b) the relationship between q_fill and maximum velocity of flow field.

Figure 24 shows the effect of q_fill on the structure field. It can be seen from the figure that the q_fill will lead to a significant increase in the maximum stress and maximum deformation. Furthermore, there is an approximate linear relationship between the q_fill and the maximum stress and maximum deformation (equation (22)).

{\begin{matrix} τ_{\max} = - 123.7079 + 609.532 q_{fill} R^{2} = 0.9648 \\ d_{\max} = - 0.31797 + 1.57738 q_{fill} R^{2} = 0.95636 \end{matrix}

(22)

Figure 24.

The effect of q_fill on the structure field (n = 2000 rpm): (a) the relationship between q_fill and stress, deformation contour and (b) the relationship between q_fill and maximum stress, maximum deformation.

Valve orifice throttling velocity contour and streamline analysis

Different displacements of the spool of the retarder valve will form different throttling areas to control different filling rates. When the retarder brakes on the long downhill, the driver will control the T of HR by controlling the spool displacement of the retarder valve, so it is necessary to analyze the influence of the opening of the retarder valve on the internal flow field. Figure 25(a) is the inlet and outlet throttling velocity contour of the retarder valve with different valve openings. As the opening of the retarder valve port increases, the fluid velocity entering the HR will increase, thereby increasing the q_fill inside the HR cavity. When the valve opening is 1, the velocity of the flow field will increase sharply, and the fluid flow through the valve will be more complicated, resulting in more turbulence.

Figure 25.

Retarder valve inlet and outlet throttling velocity contour and velocity streamlines of different valve openings: (a) velocity contour; (b) velocity streamlines (n = 2000 rpm).

Figure 25(b) shows the inlet and outlet throttling velocity streamlines of the retarder valve with different valve openings. It can be seen from the figure that when the valve opening is relatively small, the flow is relatively stable when the fluid passes through the throttle area of the valve. However, it can be seen that the fluid will produce a large amount of vortex flow through the valve orifice, which will bring serious flow loss to the HR, increase the flow resistance, and then affect the braking performance of HR. When the valve opening becomes 1, because the valve orifice becomes larger, the rapid increase of the flow through the valve will form a high velocity and disordered turbulent flow inside the valve, and the flow resistance will also increase significantly. In general, considering the throttling characteristics of the valve can simulate the filling process of HR more realistically, which is closer to the real operating condition of HR, and can improve the prediction accuracy of the external characteristics and internal flow field of HR.

Conclusion

In the study, three numerical simulation schemes (Plan A: the flow passage of model without inlet and outlet; Plan B: the flow passage of model with inlet and outlet; Plan C: the flow passage of model with retarder valve) of the primary retarder are designed. The differences between the three schemes are compared from the perspectives of internal flow field and external characteristics. Furthermore, the effects of n, q_fill and valve opening on the internal flow field and external characteristics of HR are discussed in detail.

The main conclusions of this paper are as follows:

(1) The HR model without inlet and outlet will cause excessive prediction of impeller torque, flow velocity, fluid pressure in the internal flow field and stress and deformation in the structural field. On the contrary, the retarder considering the throttling characteristics of the valve is closer to the actual working environment of HR. The throttling characteristics of the valve will directly hinder the fluid flow in the mainstream area of HR, resulting in a reduction in impeller torque and will not cause excessive prediction of impeller torque.

(2) There is a significant quadratic function relationship between n and maximum pressure of flow field. There is a significant linear function relationship between the n and the maximum velocity of the flow field, the maximum stress and the maximum deformation of the structural field. There is also a significant quadratic function relationship between the q_fill and the maximum pressure of the flow field. There is a significant linear function relationship between the q_fill and the maximum velocity of the flow field, the maximum stress and the maximum deformation of the structural field.

(3) As the opening of the retarder valve port increases, the fluid velocity entering the HR will increase, thereby increasing the q_fill inside the HR cavity. When the valve opening is relatively small, the flow of the fluid through the valve throttling area is relatively stable. However, it can still be seen that the fluid will produce a large amount of vortex flow through the valve orifice, which will bring serious flow loss to the HR, and then affect the braking performance of HR. When the valve opening becomes 1, because the valve orifice becomes larger, the sharp increase in the flow through the valve will form a high-velocity and chaotic turbulent flow inside the valve, and the flow resistance is also greater.

In general, considering the throttling characteristics of the valve, the actual filling process of HR can be more realistically simulated, which is closer to the actual working condition of HR. It can improve the prediction accuracy of the external characteristics and internal flow field of HR, and can provide reference for the numerical simulation of related turbomachinery.

Footnotes

Handling Editor: Sharmili Pandian

Author contributions

Huanhui Zhou wrote and editing the manuscript; Zilin Ran contributed to the conception of the study and provided funding acquisition; Hu Dai contributed designed and performed the experiments; Weida Yang performed the data acquisition and data analysis; Xianwei Chen helped perform the analysis with constructive discussions and simulation calculation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Postdoctoral Research Project of Zhejiang Province (Grant No. ZJ2024164) and Chongqing Jiaotong University Research Start-up Funding Project (Grant No. F1240031).

ORCID iD

Zilin Ran

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