Performance Analysis of Poppet Valves in Deep-Sea Hydraulic Systems: Considering Viscosity–Pressure Characteristics

Abstract

Deep-sea hydraulic systems, powering a wide range of numerous deep-sea operating equipment, employ many poppet valves to adjust the pressure and flow rate, thereby realizing precise movements of the actuators. With greater depths and ambient pressures, the hydraulic oil viscosity increases exponentially, leading to a significant difference in the performance of the poppet valve compared to on-land usage and across varying depths. Based on the shear stress transport (SST) k-$\omega$ turbulence model and the dynamic mesh method, a computational fluid dynamics (CFD) model of the poppet valve was established. With the viscosity–pressure characteristics considered, the performance of the poppet valve was analyzed under different depths, different inlet flow rates, and different cracking pressures. The results indicate significant performance deterioration in poppet valves at increased depths, characterized by increased pressure loss and extended response rise time. At 11 km underwater, the pressure loss can be 7 MPa larger than the preset cracking pressure of 10 MPa, and the rise time is doubled compared with the land condition. It is recommended to use hydraulic oils with a lower initial viscosity and a slower increase in viscosity with pressure in deep sea conditions.

1. Introduction

Owing to the harshness of the ocean environment, the development and utilization of ocean resources, ocean scientific research, and ocean engineering construction require a variety of advanced deep-sea operation equipment, such as unmanned/manned submersibles [1,2,3], seabed mining vehicles [4], seabed trenchers [5], deep-sea manipulators [6], and so on. Transmission control technology is the key core of deep-sea operation equipment. Only on the basis of the power source provided by transmission control technology, can the deep-sea operation equipment realize the expected designed actions and functions and further complete the deep-sea operation tasks. With the advantages of high load capacity, compact layout, large power-to-weight ratio, fast response speed, easy to compensate for the underwater ambient pressure, etc. [7,8], the hydraulic system is in widespread service for various kinds of deep-sea operation equipment. The Jiaolong manned submersible [9], the Alvin manned submersible [10], the Haima remotely operated vehicle (ROV) [11], the ROSUB 6000 ROV (National Institute of Ocean technology, Chennai, India) [12], as well as ROVs manufactured by Oceaneering International, Inc. (Houston, TX, USA) [13], Forum Energy Technologies, Inc. (Houston, TX, USA) [14], TechnipFMC plc (Houston, TX, USA) [15], and others, are all driven by hydraulic systems.

In a hydraulic system, pressure and flow rate are controlled and adjusted to ensure that an actuator, such as a hydraulic motor or cylinder, is driven with sufficient load capacity and at the expected speed. With the advantages of reliable sealing, simple structure, and fast response [16,17], poppet valves are widely utilized in hydraulic systems for adjusting pressure, controlling flow, and regulating circuit on/off. In order to explore the working performance and flow mechanism of the poppet valve, numerous scholars have carried out in-depth studies on it, and

Table 1. Recent research on poppet valves.

Author (Year)	Research Focus	Main Findings
Yuan et al. (2019) [18]	Flow pattern	Different pressure drops in the poppet valve result in different evolutions of the coherent structure and variations in the flow pattern.
Yuan et al. (2019) [19]	Choked flow	Cavitation occurrence diminishes the pressure influence on flow performance and results in choked flow, with disintegration potentially serving as the governing mechanism.
Yuan et al. (2019) [20]	Cavitation–vortex interaction	Vortical cavitation elongation arises from the vortex shearing effect, while vorticity suppression is driven by cavitation growth in transitional cavitating jet flows.
Jia et al. (2019) [16]	Dynamic characteristics	A higher length-to-diameter ratio of the pre-valve core cylindrical fluid enhances poppet valve stability, with diameter having the most significant influence.
Min et al. (2020) [17]	Unstable vibration and cavitation	Significant cavitation emerges at the valve port as the poppet strikes the valve seat, and it is influenced by the pressure difference and the vibrational amplitude.
Sang et al. (2020) [21]	Dynamic characteristics	Despite an increase in the volume flow rate with increasing set pressure, there is no significant change in the displacement of the poppet due to the sum of the preset spring force and forces caused by back pressure being nearly equal to the lift force.
Burhani and Hos (2021) [22]	Coefficients of force and discharge	Fluid force in the poppet valve diminishes as valve lift increases, while discharge coefficients follow a quadratic relationship, declining with rising lift.
Upadhyay et al. (2021) [23]	Cavitation and structural optimization	Optimizing valve spool structure and streamline direction can reduce cavitation erosion and improve poppet valve performance, enhancing reliability at high pressure differences.
Min et al. (2022) [24]	Discharge coefficient	In low Reynolds number conditions, the pilot poppet valve’s discharge coefficient $C_q$ can still be defined as $C_q=mR{e}^{0.5}$, but the coefficient m is notably lesser than the main stage poppet valve.
HIROSE et al. (2022) [25]	Flow pattern	Five types of flow patterns can be identified in poppet valves, and the relationship between these flow patterns, hysteresis characteristics, and flow forces has been clarified.
Hao et al. (2022) [26]	Flow force and cavitation	A comparative study of three poppet configurations was carried out, one of which achieved up to 44.2% reduction in flow force and 100% reduction in relative vapor volume.
Sang et al. (2023) [27]	Structural optimization	The effects of parameters such as annular clearances, spring stiffnesses and pressure jumps were studied, while the introducing of annular clearances can substantially improve the dynamic performance.

lists the relevant studies in recent years.

The relevant studies in Table 1 were conducted for land-based conditions. In the case of deep-sea hydraulic systems, they are equipped with pressure compensation devices to avoid heavy pressure-resistant structures that are designed to resist ambient pressures that increase with depth [28,29]. As shown in Figure 1, the pressure compensator is connected to the oil tank. Through the displacement or deformation of the pressure-sensitive postion assembly in the pressure compensator, the ambient pressure is introduced into the oil tank. The pump, control valves, and actuators within the hydraulic system are connected to the oil tank via pipes, meaning that ambient pressure is transferred to every location within the hydraulic system. Under ambient pressure, the oil inside the hydraulic system is slightly compressed. The piston assembly of the compensator moves towards the oil chamber to compensate for this volume change. As the ambient pressure changes, the piston assembly moves accordingly in order to dynamically adapt to the change. As a result, the working pressure is established on the basis of the ambient pressure. That is, the absolute pressure on the high-pressure side is the ambient pressure plus the working pressure, while the low-pressure side is the ambient pressure. In this case, the pressure difference in the deep-sea hydraulic system is still only the working pressure, which is consistent with land-based conditions. This also means that the poppet valves work properly.

The flow channel is formed when the hydraulic oil exerts fluid force on the poppet valve core and thus drive it to move. The viscosity of the hydraulic oil is highly dependent on pressure, and extensive experimental testing has shown that the relationship between the viscosity ($\eta$) and the pressure (P) can be fitted by the Barus [30] equation shown below.

$$\eta ={\eta }_0{\mathrm{e}}^{\alpha P},$$

(1)

where ${\eta }_0$ denotes the initial reference viscosity and $\alpha$ denotes the viscosity–pressure coefficient.

For hydraulic oils, the viscosity–pressure coefficient, $\alpha$, is within $1.5\times {10}^{-8}$ Pa⁻¹ to $3.5\times {10}^{-8}$ Pa⁻¹ [31,32,33]. Taking $\alpha$ as $2.2\times {10}^{-8}$ Pa⁻¹ of HFD fire-resistant hydraulic oil [33], and taking the depth of 1 km as corresponding to the ambient pressure of 10 MPa, it can be calculated from Equation (1) that every increase of 1 km in depth will lead to an increase in viscosity by

$$\left[{\displaystyle \frac{{\eta }_0{\mathrm{e}}^{2.2\times {10}^{-8}\mathrm{P}{\mathrm{a}}^{-1}\times \left(P+10\mathrm{MPa}\right)}}{{\eta }_0{\mathrm{e}}^{2.2\times {10}^{-8}\mathrm{P}{\mathrm{a}}^{-1}\times P}}}-1\right]\times 100\%=\left({\mathrm{e}}^{0.022\times 10}-1\right)\times 100\%=24.61\%.$$

(2)

At the bottom of the Mariana Trench, the viscosity of the hydraulic oil could be ${(1+24.61\%)}^{11}=11.25$ times higher than that of the land-based condition, since the ambient pressure is over 110 MPa. Despite consistent working pressure difference, the viscosity change due to the deep-sea environment will lead to changes in the distribution of the flow field in the poppet valve and the forces applied to the valve core, which in turn will make the working performance of the poppet valve quite different from that in the land condition, and it also varies a lot at different depths. In addition, a pressure difference of 10 MPa over the poppet valve is common, meaning that the viscosity of the hydraulic oil can also change by more than 20% during the flowing process.

However, the existing studies treat hydraulic oils as constants and do not consider viscosity changes during the flowing process. At the same time, existing studies are based on land conditions and do not consider the substantial changes due to the deep-sea environment. Whether or not the poppet valve still has good working performance under deep-sea conditions is critical to the deep-sea hydraulic system, the deep-sea operating equipment and ultimately the deep-sea mission. With the consideration of the viscosity–pressure characteristic of the hydraulic oil, this study is an attempt to reveal the performance of the poppet valve under deep sea conditions.

In this study, firstly, the computational fluid dynamics (CFD) model of the poppet valve is developed based on the shear stress transport (SST) k-$\omega$ turbulence model and the dynamic mesh method. Subsequently, the accuracy of the established model is proved by a mesh-independence study, a time-step-independence study and a comparison verification of theoretical values. Finally, based on the established model, the performance of the poppet valve at different depths, different input flow rates, different cracking pressures, and different viscosity–pressure characteristics of hydraulic oil is analyzed and discussed. All the results of this research can provide support and reference for the design, development, and optimization of deep-sea hydraulic poppet valves and deep-sea hydraulic systems.

2. Methodology

2.1. Governing Equation

When CFD simulations are adopted to solve Reynolds-averaged Navier–Stokes equations, mass and momentum conservation should be satisfied, and the governing equations for these two are as follows [34]:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho \overline{u_i}\right)=0,$$

(3)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \overline{u_i}\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho \overline{u_i{u}_j}\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial \rho }{\partial x_j}}\left(\mu {\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}-\rho \overline{{u^{\prime }}_i{u^{\prime }}_j}\right)+F_i,$$

(4)

where $\mu$ and $\rho$ denote the dynamic viscosity and density of the fluid, respectively; $\overline{u_i}$ and $\overline{{u^{\prime }}_i}$ denote the mean components of velocity and pulsation component of velocity, respectively, in the i-directions; p denotes the total pressure; $F_i$ is the external body force components in the i-direction; and $-\rho \overline{{u^{\prime }}_i}\overline{{u^{\prime }}_j}$ denotes Reynolds stress.

2.2. Turbulence Model

In this study, the poppet valve’s opening is typically very small (under all working conditions, the maximum opening does not exceed 0.4 mm). Considering factors like y+ and fluid viscosity, it is extremely challenging to generate boundary layer mesh in such a tiny gap. In addition, the viscosity in this study is pressure-dependent, which on one hand will increase as a whole in response to the ambient pressure change and on the other hand will also change dynamically during the flow process. This further increases the difficulty of meshing. All $\omega$-equations can be integrated over the viscous sublayer, and the laws of the viscous and logarithmic sub-layers are mixed by means of a mixing function, leading to a near-wall treatment that is insensitive to y⁺ and does not require a wall function [35,36]. Therefore, adopting $\omega$-equation can greatly simplify the mesh division.

The SST k-$\omega$ turbulence model takes into account the transport of the turbulence shear stress, allowing it to accurately handle flow separation from smooth surfaces [35,37]. This feature perfectly matches the need to accurately calculate the shear forces acting on the valve core in this study, since this study focuses on the pressure-driven dynamic viscosity change during the flow. The SST k-$\omega$ model given as follows [38,39]:

$$\rho {\displaystyle \frac{\partial k}{\partial t}}+\rho (\overrightarrow{u}·\nabla )k=\nabla ·\left[\left(\mu +{\mu }_T{\sigma }_k\right)\nabla k\right]+P_k-{\beta }^{\ast }\rho \omega k,$$

(5)

$$\rho {\displaystyle \frac{\partial \omega }{\partial t}}+\rho (\overrightarrow{u}·\nabla )\omega =\nabla ·\left[\left(\mu +{\mu }_T{\sigma }_{\omega }\right)\nabla \omega \right]+P_{\omega }-\beta \rho {\omega }^2+2\left(1-F_1\right){\displaystyle \frac{{\sigma }_{\omega 2}\rho }{\omega }}\nabla k\nabla \omega ,$$

(6)

where k and $\omega$ denote the turbulent kinetic energy and the specific energy dissipation rate, respectively; $P_k$ and $P_{\omega }$ denote the effective generation rates of k and $\omega$, respectively; $\beta$, ${\beta }^{\ast }$, ${\theta }_k$, ${\theta }_{\omega }$, and ${\theta }_{\omega 2}$ are empirical coefficients; and $F_1$ is a blending function.

2.3. Dynamic Mesh Method

The axial translational motion of the valve core’s center of gravity can be described by the following equation [35,40]:

$${\dot{v}}_G={\displaystyle \frac{1}{m}}\sum f_G,$$

(7)

where $v_G$ denotes the axial velocity of the center of gravity, m denotes the mass of the valve core, and $f_G$ denotes the axial force applied to the valve core.

By numerically integrating $v_G$, the displacement can be determined and the mesh is then dynamically updated accordingly. In this study, it is planned to generate an all-quadrilateral structured mesh, so that the dynamic layering method can be employed to update the mesh according to the current displacement of the valve core. The meshes at the boundaries will be split to form a new mesh when extended to a height that satisfies Equation (8), and will be destroyed and merged when compressed to a height that satisfies Equation (9).

$$h>\left(1+{\alpha }_s\right)h_{\mathrm{ideal}},$$

(8)

$$h<{\alpha }_c{h}_{\mathrm{ideal}},$$

(9)

where h is the actual height of the current mesh; ${\alpha }_s$ and ${\alpha }_c$ are the layer splitting factor and layer collapse factor, respectively; and $h_{\mathrm{ideal}}$ is the set ideal mesh height, which is set to be consistent with the initially generated mesh in this study.

2.4. Reynolds Number

The structural dimensions at the valve opening in a poppet valve are shown in Figure 2. Taking the distance from the cone surface to the right-angled edge of the valve opening, h, as the characteristic length [24], the Reynolds number can be defined as

$$\mathrm{Re}={\displaystyle \frac{\overline{v}h}{\nu }},$$

(10)

where $\overline{v}$ is the average flow velocity, h is the characteristic length, and $\nu$ is the kinematic viscosity.

The following relationship is satisfied between kinematic viscosity $\nu$ and dynamic viscosity $\eta$:

$$\nu ={\displaystyle \frac{\eta }{\rho }},$$

(11)

where $\rho$ is the density.

According to the geometrical relationship, it can be obtained that

$$h=xsin\alpha ,$$

(12)

where x is the valve core displacement and $\alpha$ is the half-cone angle.

The cross-flow section is a thin circular ring of width h, whose area, A, can be calculated according to the following equation:

$$A=\pi Dh,$$

(13)

where D is the valve seat diameter.

If the volume flow rate is Q, the average flow velocity $\overline{v}$ and area A satisfy the following relationship:

$$Q=\overline{v}A.$$

(14)

By bringing in Equations (11) to (14), Equation (10) can be rewritten as:

$$\mathrm{Re}={\displaystyle \frac{\rho Q}{\pi D\eta }}.$$

(15)

The dynamic viscosity of the oil in this study is not a constant value but varies with pressure according to Equation (1). To facilitate the calculation, the dynamic viscosity corresponding to the outlet pressure of the rear chamber of the poppet valve is taken to calculate the Reynolds number.

3. Model

3.1. Boundary Conditions

The CFD calculations in this study were performed in ANSYS Fluent 2020 R1. Figure 3 illustrates the geometric model of the poppet valve targeted in this study, and

Table 2. Geometric dimensions of the poppet valve.

Symbol	Definition	Value
D	Valve seat diameter	4 mm
$D_1$	Orifice diameter	1 mm
$D_2$	Valve core diameter	5 mm
$D_3$	Rear chamber diameter	12 mm
$L_1$	Orifice length	10 mm
$L_2$	Front chamber length	15 mm
$L_3$	Rear chamber length	40 mm
$L_4$	Cone length	25 mm
$L_5$	Valve core length	8 mm
$\alpha$	Half-cone angle	10°

lists the values of all the corrosponding parameters.

Due to the symmetry of the model, a 2D axisymmetric model was adopted to save computing resources and reduce computation time. The setting of each boundary condition is shown in Figure 4. The outer contour of the valve core is the wall numbered 5 in Figure 4, and its dynamic mesh setting is set to be rigid body motion. Thus, all the forces applied to the valve core can be counted for calculating its displacement according to Equation (7). The dynamic mesh setting for the two interior walls numbered 3 in Figure 4 is stationary. The two interior walls divide the entire flow domain into a total of four regions, A, B, C, and D. Among them, the dynamic mesh setting for region C is a passive rigid body motion. Thus, all the meshes in region C will be fully synchronized with the displacements calculated from the forces on the outer contour of the valve core. The dynamic mesh setting of the two internal walls numbered 3 is stationary with a higher priority. As a result, when the valve core moves to the right, new meshes will be generated at the left internal wall, while the meshes will be compressed and then destroyed at the right internal wall, as shown in Figure 5. In the case of a poppet valve connected to a hydraulic oil tank, its outlet pressure is equal to the pressure in the oil tank, which is approximately the ambient pressure introduced via the pressure compensator [28,29]. To investigate the dynamic movement of the valve core moving to the right, the inlet is set as the volume flow rate jumps from 1 L/min to 1.2 L/min at 2 ms. Since the pressure variations in the study are much smaller than the bulk modulus of the hydraulic fluid, which usually ranges from 1400 to 2000 MPa [41], the volume shrinkage of the oil is negligible, and therefore the density in the study is considered as a constant and takes the value of 872 kg/m³. The viscosity ($\eta$)-pressure (P) characteristics are expressed as Equation (16) [42]. Since ANSYS Fluent does not directly support this exponential viscosity–pressure characteristic, User-Defined Functions (UDF) was used to implement the viscosity–pressure characteristic characterized by Equation (16).

$$\eta =0.04044\times {\mathrm{e}}^{2.252\times 10^{-8}\times P}.$$

(16)

The density of the valve core is 7800 kg/m³. The spring stiffness is 30 kN/mm. Given that the cracking pressure of the poppet valve is 10 MPa, the spring preload, $F_{pre}$, is

$$F_{pre}={\displaystyle \frac{1}{4}}\pi D^2\times 10\mathrm{MPa}=125.66\mathrm{N}.$$

(17)

To ensure the continuity of the fluid calculation domain, a pre-opening of 0.1 mm was set in the model, as shown in Figure 4. Therefore, the spring preload needs to be increased by an additional 3 N and reaches 128.66 N. The effect of gravity is neglected in the model, given that it is much smaller than the fluid forces.

3.2. Mesh Independence Study

The calculation of displacement depends on the accuracy of the total force applied to the contour of the valve core. The calculation of the force involves integrating the pressure against the contour area of the valve core; thus, the fineness of the meshing has a great influence on the calculation accuracy. Then, based on the settings in Section 3.1 and a time step of 1 $\times {10}^{-6}$ s, calculations are carried out under the land condition at different numbers of elements, and the valve core displacements, x, are shown in Figure 6. It can be found from Figure 6 that the steady-state x at 1 L/min and 1.2 L/min, as well as the dynamic processes at the jump, change as the number of elements increases. It should be noted that the x curve for element number 97768 has nearly coincided with that for element number 200,008, indicating that finer meshing has had negligible effect on the results. Eventually, the model in this study was divided into 97,768 elements, as shown in Figure 5. Figure 5 also illustrates the generation and destruction of meshes in the calculation.

3.3. Timestep Independence Study

According to Section 2.3, the calculation of displacement involves integrating against the time; thus, the timestep also has a significant effect on the results. Based on the settings in Section 3.1 and the mesh in Section 3.2, calculations are carried out under the land condition at different timesteps, and the valve core displacements, x, are shown in Figure 7. Since the four selected timesteps are already small, the curves for each x in Figure 7 basically coincide, indicating that the effect of a smaller timestep on the results is already minimal. Ultimately, the timestep in the model is taken as 1 $\times {10}^{-6}$ s.

3.4. Turbulence Model Independence Study

To further confirm the credibility of the developed CFD model, the calculation results using different turbulence models as well as the laminar flow model were also compared, as shown in Figure 8. Figure 8 clearly shows that the calculation results of the SST k-$\omega$ turbulence model used in this study are in good agreement with those of the three major k-$\epsilon$ models as well as the laminar model.

It should be noted that the critical Reynolds number of the poppet valve is in the range of 20–100 [32,43], and the Reynolds number calculated from Equation (15) for each working condition in this study is within this range. This implies a possible laminar flow condition. Encouragingly, Figure 8 shows that the results based on the SST k-$\omega$ model are consistent with that of the laminar flow-based model, which ensures the accuracy of the calculation results.

3.5. Validation

The forces on the valve core are balanced when the poppet valve is in a steady state. At this time, the calculated total force on the valve core, $F_{st}$, the steady-state displacement of the valve core, $x_s$, and the spring preload, $F_{pre}$, should satisfy the following relationship:

$$F_{st}-k{x}_s=F_{pre}=125.66\mathrm{N}.$$

(18)

Table 3. The calculated total force on the valve core and the steady-state displacement of the valve core.

Inlet Volumn Flow Rate (L/min)	${\mathit{F}}_{\mathbf{st}}$ (N)	${\mathit{x}}_{\mathit{s}}$ (mm)	${\mathit{F}}_{\mathbf{st}}-\mathit{k}{\mathit{x}}_{\mathit{s}}$ (N)
0.6	127.93584	0.07587	125.65975
0.8	128.37820	0.09060	125.66029
1	128.79845	0.10460	125.66059
1.2	129.20781	0.11825	125.66037
1.4	129.60358	0.13145	125.66012
1.6	129.99474	0.14447	125.66059
1.8	130.37712	0.15723	125.66017
2	130.75685	0.16986	125.66116

lists the $F_{st}$ and $x_s$ computed by the developed model for different inlet volume flow rates. From the data in Table 3, it can be shown that all the ($F_{st}-k{x}_s$) obtained from the calculation results based on the developed model are about 125.66 N, which is in agreement with the set spring preload. This proves the accuracy of the established CFD model.

4. Results and Discussions

4.1. Effect of Viscosity Variation During Flow Process

The impact of considering viscosity variation during the flow process on the calculation results is analyzed before evaluating the performance of the poppet valve. The displacement curves of the valve core under the land condition in the three cases of considering the viscosity change, constant viscosity at 0 MPa (i.e., viscosity corresponding to outlet pressure or initial viscosity), and constant viscosity at 10 MPa (i.e., viscosity corresponding to inlet pressure or cracking pressure) are shown in Figure 9. Comparing the three curves in Figure 9, it can be found that the displacement of the valve core is smaller than that of the variable viscosity case when calculated with the viscosity corresponding to the outlet pressure of the poppet valve. In contrast, the displacement is larger when calculated with the viscosity corresponding to the cracking pressure of 10 MPa.

The flow field distributions for the variable viscosity case are shown in Figure 10, Figure 11 and Figure 12. At the valve port with smaller through-flow area, the flow velocity increases significantly, as shown in Figure 10. The pressure in the front chamber is about the set cracking pressure, i.e., 10 MPa, while the pressure in the rear chamber is close to 0 MPa, resulting in a huge pressure gradient at the valve port, as shown in Figure 11. The rapid decrease in pressure also causes the oil viscosity at the valve port to decrease rapidly in the direction of flow, as shown in Figure 12.

Thus, both Figure 9 and Figure 12 indicate that calculations based on constant viscosity do not accurately reflect the actual situation. In addition, the ambient pressure varies greatly at different depths in the sea, causing the viscosity of the hydraulic oil to change more than in the flow process. Therefore, it is necessary to consider the viscosity–pressure characteristics of the hydraulic oil.

4.2. Dynamic Performance of the Poppet Valve at Different Flow Rates

Under land-based conditions, the normalized displacement curves (normalized by the steady-state incremental displacement, $x_{si}$) of the valve core for increasing from 1 L/min to different flow rates and for an increase of 0.2 L/min on the basis of different flow rates are plotted in Figure 13 and Figure 14, respectively. The steady-state displacements, $x_s$, at each inlet flow rate are shown in Figure 15. At the same initial flow rate, Figure 13 indicates that the more the flow rate is increased, the larger the displacement overshoot is and the longer the dynamic adjustment time is. According to Equation (15) and the data in Section 3.1, the Reynolds number corresponding to the flow rate from 1.2 L/min to 1.8 L/min decreases from 51 to 34. The smaller the Reynolds number is, the more the viscous force dominance and the damping effect increase, and the more the overshoot of the valve core displacement curves decreases, which also agrees with the change of overshoot of each curve in Figure 13.

While for the same increase in flow rate, Figure 14 indicates that the larger the initial flow is, the larger the displacement overshoot is and the longer the dynamic adjustment time is. Both larger displacement overshoot and longer dynamic adjustment time could bring more pressure and flow fluctuation and reduce the dynamic performance of the poppet valve. It can also be found from Figure 13, Figure 14 and Figure 15 that as the displacement of the valve core increases, i.e., the opening of the valve port becomes larger, the dynamic performance deteriorates.

4.3. Dynamic Performance of the Poppet Valve at Different Underwater Depths

Based on the settings in Section 3, the performance of the poppet valve at different depths was simulated. The steady-state displacement of the valve core at 1.2 L/min, $x_s$, the total pressure drop of the poppet valve, $\Delta P$, the ratio of the viscous force on the valve core ($F_v$) to the total force ($F_t$), and the rise time, $t_r$, are shown in Figure 16. The axial pressure profile at the small orifice and the front chamber is plotted in Figure 17. The normalized displacement curves of the valve core at different depths are shown in Figure 18. Among them, 10 MPa of ambient pressure corresponds to 1 km of underwater depth.

According to Equation (16), the oil viscosity increases with the depth-related ambient pressure. According to Equation (15) and the data in Section 3.1, the corresponding Reynolds number at a flow rate of 1.2 L/min decreases from 34 to 4 when the working depth is increased from 0 to 11 km. It means that viscous forces gradually dominate in the flow field of the poppet valve. Therefore, the percentage of viscous force on the valve core increases, as shown by the yellow curve in Figure 16. In addition, the total pressure drop of the poppet valve, $\Delta P$, increases from close to the set cracking pressure of 10 MPa at shallow depths to a maximum of more than 17 MPa, as shown by the blue curve in Figure 16. The axial pressure curves (Figure 17) indicate that the increased pressure drop comes mainly from the pressure drop when flowing through the small orifice in the poppet valve. And the increase in pressure loss in the small orifice mainly comes from the increase in viscous forces caused by the increase in viscosity. Figure 17 also shows that the pressure in the front chamber, that is, in axial directions greater than 10 mm, increases with depth. According to the poppet valve structure (Figure 3), it can be known that the pressure in the front chamber will drive the valve core to move to the right. The total force, including the increased viscous force, on the valve core will be balanced against the spring preload. Therefore, the steady-state displacement of the valve core, $x_s$, increases with depth, as shown by the red curve in Figure 16. Since $x_s$ increases with depth, the rise time of the displacement response, $t_r$, also increases with depth, as shown by the green curve in Figure 16. In particular, $t_r$ at 11 km reaches 152 $\mathsf{\mu }$s, which is more than a twofold increase from 75 $\mathsf{\mu }$s at 0 km.

According to the conclusion in Section 4.2 and the steady-state displacement, $x_s$, in Figure 16, it can be known that, as the depth and $x_s$ increase, the overshoot of the dynamic process should increase. However, the increased viscosity of the hydraulic oil due to increased depth or ambient pressure will cause higher damping of the poppet valve and reduce the overshoot. The viscosity–pressure characteristic of Equation (16) shows an exponential law, so that the viscosity increases less at shallow depths and more at deeper depths. This makes the reduction in overshoot due to increased viscosity of little effect at shallow depths and significant at deeper depths. Ultimately, the overshoot of the dynamic process presents a law of first increasing and then decreasing with depth due to the combined effect of the poppet valve’s own nonlinearity and the viscosity increase caused by the depth-related ambient pressure, as shown in Figure 18.

To summarize, the poppet valve’s dynamic performance at different underwater depths shows the following pattern:

(1)

The total pressure drop, $\Delta P$, increases with depth;

(2)

The steady-state displacement, $x_s$, increases with depth;

(3)

The rise time of the dynamic process, $t_r$, increases with depth;

(4)

The overshoot of the dynamic process increases and then decreases with depth.

Since the overshoots in Figure 18 are all between 0 and 10%, they cause less impact on pressure and flow fluctuations. An increase in rise time, $t_r$, implies a delayed response, which will affect the pressure and flow rate in the deep-sea hydraulic system, and futher, affect the actuator output accuracy. At the same time, such a large increase in the total pressure drop, $\Delta P$, especially compared to the preset cracking pressure of 10 MPa, will affect the accuracy of the pressure control, energy consumption, and load-driving capacity of the deep-sea hydraulic system.

4.4. Dynamic Performance of the Poppet Valve at Different Cracking Pressures

The normalized displacement curves of the valve core at different depths for preset cracking pressures of Figure 19, Figure 20, Figure 21 and Figure 22 respectively. Figure 23 illustrates the steady-state displacement of the valve core, $x_s$, at different preset cracking pressures and depths.

Figure 19, Figure 20, Figure 21 and Figure 22 all show the same pattern of overshoot increasing first and then decreasing with depth as in Figure 18. The steady-state displacements, $x_s$, at each of the preset cracking pressures in Figure 23 also increase with depth, consistent with the red curves that have been analyzed in Figure 16. The volume flow rate through the poppet valve port satisfies the following equation [24,44]:

$$Q=C_q\pi Dxsin\alpha \sqrt{{\displaystyle \frac{2P_{port}}{\rho }}}.$$

(19)

Then, the displacements, x, can be expressed as follows:

$$x={\displaystyle \frac{Q}{C_q\pi Dsin\alpha \sqrt{\frac{2\Delta P_{port}}{\rho }}}},$$

(20)

where Q, D, $\alpha$, and $\rho$ are the volume flow rate, valve seat diameter, half-cone angle, and hydraulic oil density, respectively; $\Delta P_{port}$ is the pressure difference at the valve port, close to the preset cracking pressure; $C_q$ is the flow coefficient, related to the Reynolds number [24].

At the same depth, the change in viscosity due to the change in preset cracking pressure leads to minor changes in Reynolds number and $C_q$. The remaining parameters in Equation (20) are constants. Therefore, at the same depth, a greater pressure difference at the valve port, which is close to equal to the preset cracking pressure, will result in a smaller steady-state displacement of the valve core, $x_s$, as shown in Figure 23. In particular, at 11 km depth, the $x_s$ is only 0.29496 mm when the cracking pressure is 12 MPa, and it is 0.40827 mm at 6 MPa. Further, the rise time of the valve core displacement is shorter when the preset cracking pressure is greater due to the smaller displacement required to move, as can be seen by comparing Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22.

4.5. Dynamic Performance of the Poppet Valve at Different Viscosity Characteristics

All of the aforementioned performance deterioration of the poppet valve in deep-sea conditions stems from the viscosity increase caused by the depth-related ambient pressure. Therefore, selecting a hydraulic oil with a lower initial viscosity and a slower increase in viscosity with pressure is an effective way to improve the performance of the poppet valve in deep-sea conditions. Based on the viscosity test data of 10# hydraulic oil (10# indicates that the average kinematic viscosity of the hydraulic oil is 10 cSt at 40 °C) conducted by Tian et al. [45], this subsection discusses the performance of the poppet valve when the initial viscosity of the hydraulic oil is lower and the viscosity increases more slowly with pressure.

Figure 24 shows the steady-state displacement of the valve core at 1.2 L/min, $x_s$, the total pressure drop of the poppet valve, $\Delta P$, the ratio of the viscous force on the valve core ($F_v$) to the total force ($F_t$), and the rise time, $t_r$, at different depths when using the 10# hydraulic oil. Figure 25 shows the normalized displacement curves at different depths. As can be known by comparing the range of the axes of each physical quantity in Figure 16 and Figure 24, and by comparing the curves in Figure 18 and Figure 25, a hydraulic oil with a lower initial viscosity and a slower increase in viscosity with pressure attenuates the performance deterioration of the poppet valve in underwater conditions, resulting in a better consistency of its performance at different depths.

5. Conclusions

Based on the viscosity–pressure characteristics of hydraulic oil and computational fluid dynamics simulation, this work analyzed and researched the working performance of poppet valves, which are widely adopted in deep-sea hydraulic systems, under various working conditions, such as different underwater depths, different input flow rates, and different cracking pressures. The main conclusions obtained are as follows:

1.

The poppet valve has nonlinearity, and the larger the displacement of the valve core is, i.e., the larger the opening of the valve port is, the larger the overshoot of the response process is.

2.

As the underwater depth increases, the steady-state displacement, pressure loss, and rise time of the response process all increase dramatically, which will have a great influence on the response speed, load capacity, and energy consumption of the deep-sea hydraulic system. In particular, at 11 km underwater, the total pressure drop increases by 7 MPa compared with the preset cracking pressure of 10 MPa, and the rise time doubles compared with the land case. The overshoot of the response process first increases and then decreases with depth, with little overall change.

3.

At the same depth, the larger the preset cracking pressure, the smaller the displacement of the valve core, i.e., the smaller the opening of the valve port. In particular, at 11 km depth, the steady-state displacement is only 0.29496 mm when the cracking pressure is 12 MPa, and it is 0.40827 mm at 6 MPa. At the same time, a larger cracking pressure results in a smaller rise time, meaning a faster response.

4.

Selecting a hydraulic oil with a lower initial viscosity and a slower increase in viscosity with pressure can significantly slow the performance deterioration of the poppet valves in deep-sea conditions.