A novel hydraulic balanced turbine for stability control and application in drilling tools

ABSTRACT Most turbine designs are based on the continuous rotating turbine structure. In this paper, a novel hydraulic balanced turbine (HBT) with a symmetrical blade structure that can be used in a hydro-environment is proposed. The turbine, which uses the hydraulic impact of the nozzle jet and balanced turbine blade design in stability control, could be used to control stability in certain applications, e.g. in drilling engineering. This is a completely new way of thinking about turbine design. Because of the special blade design of the HBT, when the nozzle deviates from the balance position, the HBT will be restored to the balance position under the action of external moment. The follow-up performance of the HBT is directly related to the rotational speed of the nozzle. The theoretical and numerical solutions in this paper have good consistency. Modern computational resources allow computational fluid dynamics to be an integral part of turbine design. The simulation results proved that this innovative design is feasible and has potential application prospects. The HBT realizes the ‘soft’ connection between the controlled stable platform and the actuator in the vertical drilling tool, which can significantly improve the control precision of mechanical vertical drilling.

Nozzle distance from the center of the turbine l 2 Nozzle length a Nozzle angle r 1 Outer radius

Introduction
In general, most of the turbine designs (Tengs et al., 2018) in the reviewed literature are based on the continuously rotating turbine structure. Modern computational resources allow computational fluid dynamics (CFD) to be an integral part of turbine design. A vast amount of research has been carried out on the numerical simulation of hydraulic turbines. Position control during rotary motion (Moghadam et al., 2019) is not difficult for electromechanical systems. Using a rotary position sensor (Bajic et al., 2014), which is an essential component, and a stepper motor or servo-hydraulic rotary actuator (Sadeghieh et al., 2012), we can achieve precise control. However, in some special applications, such as petroleum drilling (Jones et al., 2016), the nuclear industry  and lunar drilling (Koemle et al., 2008), electronic components are not adaptable because the control system is usually required to work at temperatures exceeding 200°C, which exceed the limits of most electronic components. In general, we need a completely mechanical way to achieve simple closed-loop control. Not only is the actuator of the control implemented by a mechanical structure, but the function of the sensor is also implemented mechanically. However, it is much more difficult to achieve the above-mentioned stable balance control. Before the arrival of the electronic age, turbine engine control was achieved through purely mechanical systems (Dmitriev & Snegov, 2001;Hu et al., 2003;Ong, 1972;Shammas et al., 2005); advanced control methods, such as differential/integral control and PID control, were implemented through the mechanical structure of hydraulic oil, pistons, connecting rods, etc. Electronic sensors and computers make these controls easier, more reliable and more accurate. But even so, this does not mean that the purely mechanical control system is completely useless. In oil drilling engineering, the mechanical automatic vertical drilling system (Jones et al., 2016;Xue et al., 2019), which is a purely mechanical control system that can be used in environments over 200°C, brings tens of millions in profits to oil companies every year.
In a purely mechanical structure, fluid pressure (Xue et al., 2016) is often used as a source of power; compared to the motor drive system, hydraulic system has higher power density (Amundson et al., 2006). The fluid power technology can achieve high power output with low system mass compared to other common powertrain technologies (Seok et al., 2013). A hydraulic turbine (Dorji & Ghomashchi, 2014;Liu et al., 2019;Powell et al., 2018) is a common energy conversion device that converts hydraulic pressure into rotary motion, and is commonly used in wind energy, hydraulic turbines, hydraulic motors, etc. (Eker, 2004;Giosio et al., 2017;Li et al., 2016). Ge et al. (2016) studied the Reynolds number of airfoil aerodynamic performance and its influence on the optimal design of wind turbine rotors, and discussed the maximum power coefficient; Hansen (2018) developed and tested an airfoil optimization method for wind turbine applications, which can control the performance loss caused by leading edge contamination. In view of the failure of traditional fluid machinery in supersonic flow, Paniagua et al. (2014) designed and analyzed advanced high-sonic axial flow turbines. Pereiras et al. (2014) calculated the efficiency of the twin-turbine structure based on the results of the numerical model, and paid attention to the influence of the turbine's torque and flow rate under the reverse mode conditions. Kwon et al. (2014) used the Rayleigh-Ritz hypothesis model method to analyze a turbine generator, determined the resonant frequency and intensity of the rotating multi-package blade system excited by multiple nozzle forces, and studied the influence of different parameters, such as the number of nozzles and damping coefficient, on the system response. Muller et al. (2009) analyzed the wind energy converter, and discussed the modern transformation of this traction-type energy converter for building integration. Lanzafame and Messina (2007) improved the blade element momentum theory and established a mathematical model for the design of hydrodynamic wind turbines at low wind speed, which they used to optimize the performance of the rotor. The axial-flow Wells turbine is widely operated in most energy stations. The principle of operation of the Wells turbine is based on aerodynamics, and several variants of the Wells turbine have been proposed and developed (Nazeryan & Lakzian, 2018;Shaaban, 2012). The performance and shape of Wells turbines have been optimized by numerical (Halder & Samad, 2015) and experimental (Setoguchi et al., 2003) methods for each of the sea states (Halder & Samad, 2016). A novel twin-rotor radial-inflow air turbine was recently proposed (Falcao et al., 2015), which has two sets of rotor blades equipped on a common shaft that are axially offset from each other. Another new topology that uses twin unidirectional turbines has been proposed and studied (Jayashankar et al., 2009).
However, all of the turbine designs mentioned above are based on the continuous rotating turbine structure. In this paper, a novel hydraulic balanced turbine (HBT) with a symmetrical blade structure that can be used in the hydro-environment is proposed. The turbine, which uses the hydraulic impact of the nozzle jet and balanced turbine blade design in stability control, could be used to control stability in certain applications. This is a completely new way of thinking about turbine design.
The behavioral modeling of hydraulic turbines using computational fluid dynamic (CFD) techniques (Esfahanizadeh et al., 2022) is known to be extremely beneficial, especially in applications where experimental analysis is difficult or even impossible. For instance, Ghalandari et al. (2019) conducted a CFD simulation of the nano-fluids inside a root canal and obtained beneficial information and an appropriate insight into the flow characteristics during irrigation in the root canal.
Automatic vertical drilling technology adopts downhole closed-loop control (Elshafei & Al-majed, 2015), which can realize active deviation prevention and correction. When the drilling tool is inclined, the gravity inclination measurement and control mechanism (hereinafter referred to as the bias platform) turn to the low side of the hole under the action of gravity, which drives the upper disc valve to stabilize in the expected position. In this paper, a hydraulic balancing turbine is installed at position 'A', as shown in Figure 2. The balance moment generated by the turbine is used to stabilize the upper disc valve at a predetermined position. The platform is only controlled by the liquid flow nozzle to drive the novel HBT. The 'soft' connection between the bias platform and the upper disc valve is realized, so that the bias platform is not affected by the friction resistance of the upper and lower disc valves, and the abnormal swing of the bias platform will not be transmitted to the upper disc valve immediately, which makes the guiding concentrated force of the wing rib more stable. The HBT realizes the soft connection between the controlled stable platform and the actuator in the vertical drilling tool, which can significantly improve the control precision of the mechanical vertical drilling.
The remainder of this article is organized as follows. The conceptual design and geometric features of the new turbine are presented in Section 2. The mechanical model of the HBT and the mathematical methods of simulation based on finite volume methods are presented in Section 3, including the start-up performance of the HBT, stable  performance of the HBT and dynamics of the nozzle. Next, the main results of the paper are given in Section 4, where numerical simulations using the models introduced in Section 3 are presented. Finally, conclusions are presented in Section 5.

The new design of a hydraulic balanced turbine
The HBT shown in Figure 1 can produce the balance moment, which can significantly improve the control accuracy and reliability of mechanical control mechanisms when applied to mechanical automatic vertical drilling tools. Its mechanical modulation structure is shown in Figure 2.
A schematic diagram of the HBT is shown in Figure 1(a). The fluid flows through a nozzle into the control housing, then the output torque can be generated by the turbine. In this paper, however, the turbine does not produce continuous torque, as shown in Figure 1(b) and (c); in most cases, the turbine is working in a balanced state and the thrust F provided by the turbine is zero. The nozzles are located in the middle of the forward and reverse turbines. The turbine torques on both sides are balanced, and the turbine is stable. If the turbine deviates from the equilibrium position owing to load torque fluctuations, the balance between forward and reverse torque will be broken. As shown in Figure 1(d) and (e), the hydraulic shock generates recovery torque to restore the turbine to the equilibrium position. In this model, the nozzle structure, turbine blade shape, nozzle installation height h 0 , turbine blade installation angle γ , blade number n and separation angle β, designed to prevent positive and negative turbine interaction, form a functional relationship with the output torque T o : T o = f (h 0 , γ , n, β). This mathematical model of torque can evaluate the balance performance of the turbine. If the control position changes, the control housing rotates, which drives the nozzle to rotate, and the turbine rotates, which reflects the dynamic follow-up performance of the turbine.
The HBT achieves the soft connection between the control platform and the control actuator, optimizing the performance of the control mechanism. The advantages are reflected in two aspects: first, it effectively avoids the influence of the frictional resistance of the upper and lower disc valves, as well as the influence of stick-slip vibration of the bottom drilling tools on the biased platform; and secondly, the use of the unique follow-up and hysteresis characteristics of the HBT can cushion the vibration of the bias block, effectively reduce the influence of the swing of the bias block on the upper plate valve, and improve the stability control.
The design criteria of the proposed device were tested using three-dimensional (3D) CFD, which allows quick simulation of a large number of different geometries. CFD is a powerful method for modeling various physical systems to predict the evolution of the governing state variables. This approach to solving engineering problems has recently gained importance owing to its effectiveness and applicability (Ramezanizadeh et al., 2019).

Mechanical model of the HBT
To verify the feasibility of the new design in this paper, the 3D analysis model of HBT is established in SolidWorks software. Since conventional turbine design is complex (Caboni et al., 2016;Luo et al., 2018;Rehman et al., 2018), to make the design of the turbine simple and efficient, a projection modeling method is used for the blade. The blade profile is controlled only by the outer angle of the blade (Dixon & Hall, 2013). The design parameters of the HBT, as shown in Figure 3, include the blade angle γ , blade number n, blade thickness w, blade length l 1 and turbine height h.
The blade element momentum theory is used to analyze the torque force on the turbine (Hodges, 1980;MacNeill & Verstraete, 2017;Suzuki & Mahfuz, 2018). The blade element theory divides the blade along the blade extension direction into continuous microsegments with independent fluid performance. Each micro-segment is called a blade element. The force on the blade can be obtained by integrating the force on each blade element along the radial direction. The fluid velocity vector and blade element force are shown in Figure 4.  To calculate the turbine torque, first, each fluid speed is obtained: where v is the absolute velocity of the nozzle fluid, v r1 , v r2 is the velocity of the fluid relative to the turbine, u is the rotational speed of the blade element, ω is the turbine angular velocity, and l is the distance from the blade element to the center of the turbine. From the fluid momentum equation: Therefore, the tangential force dF h and impact force dF v of the blade per unit length can be expressed as follows: where α is the blade element angle and dA is the projection area of the nozzle on the blade element. Turbine blade angle change can be expressed as: where L is the distance between the blade outer edge and turbine center, and l is the distance from the blade element to the turbine center. Substituting Equation (4) into Equation (3) and integrating, the impact force and impact torque of the turbine can be expressed as: where r 0 is the nozzle radius, and k is the distance from the nozzle to the center of the turbine. Turbine rotation is affected by fluid resistance, including frictional resistance and shape resistance. Friction resistance is caused by viscosity, which can be calculated by boundary layer theory. Shape resistance, also known as differential pressure resistance, depends on the pressure difference before and after the flow around the object, which is related to the shape of the object, especially the shape of the back half of the object. At present, the size of shape resistance is generally determined by experiments. When the turbine rotates, it is impacted not only by the nozzle, but also by the fluid resistance (mainly the shape resistance), and the resistance dF d and lift dF l of the blade element are as follows: where ρ is the fluid density, ds is the area of the blade element, C d is the drag coefficient of the blade element, When the turbine works in the stable state, the lift of the turbine is zero, the turbine is only subjected to the rotational resistance generated by the fluid, and the rotational resistance torque T d of the turbine is the integral of the resistance of the blade, which can be expressed as follows: where n is the number of blades. The total torque T t of the turbine when it rotates is: When the turbine starts rotating, assuming that T e is the load torque, the torque acting on the turbine must satisfy T d > T e .

Numerical simulation model
A flowing fluid follows the law of conservation of mass, momentum and energy. However, it is very difficult to solve the velocity field and pressure field in complex flows accurately. The CFD method can be used to find the approximate solution to satisfy the requirements of the engineering application. The accuracy of CFD methods has been verified in many fields (Bhutta et al., 2012;Halder et al., 2017;Wang et al., 2017). This paper used numerical simulation tools to demonstrate the results of the theoretical analysis, and to study the variation of geometric parameters. The steady operating performance of the turbine was evaluated using a set of dimensionless coefficients, which are characterized in terms of the torque coefficient C t , input power coefficient f i , turbine efficiency η and flow coefficient ϕ. The definitions related to these parameters are expressed as follows: where P is the total pressure drop between the inlet and outlet of the turbine; T o is the output torque; Q and ρ denote the air volumetric flow rate and liquid density, respectively; s and ω represent the number of rotor blades and the angular velocity of the turbine, respectively; v r and u r are mean axial flow velocity and circumferential velocity, respectively; and n indicates the area of each annular blade. A flow simulation was carried out using ANSYS Fluent 19.0 software, which uses the finite volume numerical method for solving the Reynolds-averaged Navier-Stokes equations by means of a pressure-based solver. The whole 3D geometry model of the turbine was established using ANSYS Workbench (specific parameter values can be obtained in Table 1), and grids were generated using the preprocessing software ICEM-CFD.
The k-reliable model is chosen as the fluid analysis model. To model the near-wall region, the enhanced wall function is used; this can save substantial computational resources, because the viscosity-affected nearwall region, in which the solution variables change most rapidly, does not need to be resolved. The second-order upwind model is used for discretization, the simple solution algorithm is selected and the PRESTO pressure discretization scheme is chosen. Because of the complexity of the turbine model, the hexahedral mesh and tetrahedral mesh are used. The value of y + , which depends on the wall function, is between 3 and 10. The CFD simulation analysis model and meshing are shown in Figure 5. We observe that the y + values are always lower than 300, which satisfies the upper limit for the renormalization group (RNG) k-turbulence model combined with a scalable wall function (Maduka & Li, 2021). With the exception of the blades that are not hit by the flow, all grid nodes also satisfy the lower limit given by 30 ≤ y + (Maduka & Li, 2021). The impact torque and impact force of the turbine directly affect the start-up performance of the turbine. The magnitude of the hydraulic torque determines whether the turbine can overcome the system load torque and the friction torque. Therefore, when the turbine is working, it is not necessary to consider the hydraulic efficiency of the blades, and only the torque and impact force of the HBT are important. There are many factors affecting turbine performance. This paper uses CFD software to simulate and analyze the system of the nozzle and the HBT to explore the influence of different factors on the turbine force. For the start-up performance analysis, the flow-field analysis area is entirely static and the nozzle position is fixed. We chose the basic parameters of the HBT in the actual field drilling project, because the focus of this article is to study the working characteristics of the HBT; we will describe its application in another paper. The basic dimensions of the turbine are determined: the inner diameter is 44.5 mm and the blade thickness is 2.75 mm.
The grid independence was proven by comparing the simulation results (the fluid stable pressure) with different numbers of grid elements from the coarsest to the finest, and the results are shown in Table 1. it is shown that the grid-independence analysis can be performed satisfactorily by increasing the total number of grid cells, but it is not necessary to increase the number of mesh cells of the impeller fluid domain.
Turbine and nozzle parameters are designed as shown in Table 2, where 'No.' represents the system parameters corresponding to different simulation tests.
An important factor in the design of the HBT is the flow angle β. Near the flow angle, the external force acting on the turbine changes dynamically. Therefore, transient analysis is used to study the change in the force acting on the turbine when the nozzle is near the symmetrical position and under different flow angles. Currently, the nozzle is in motion and moves slowly.
Both the follow-up process and the dynamic balancing process of the turbine are dynamic. In this paper, the sliding mesh method is used to analyze the dynamic process of the turbine.
With respect to dynamic meshes, the integral form of the conservation equation for a general scalar, φ, on an arbitrary control volume, V, with a moving boundary, can be written as: where ρ is the fluid density, u is the flow velocity vector, u g is the mesh velocity of the moving mesh, is the diffusion coefficient, and S φ is the source term of φ.
Here, ∂V is used to represent the boundary of the control volume V. The time derivative can be written as: In the case of sliding mesh, the motion of moving zones is tracked relative to the stationary frame. Therefore, no moving reference frames are attached to the computational domain, simplifying the flux transfers across the interfaces.
To study the influence of different operating modes of the control platform on the follow-up performance of the turbine, two different modes of motion of the nozzle are defined, as shown Equation (16): uniform nozzle motion and sinusoidal nozzle motion. The nozzle is initially located at the balance position and then moves in these two different modes. The nozzle is set as the rotating region and rotates around the central axis at different angular velocities.
Using CFD software to monitor changes in the torques, impact forces and rotary speed of the turbine, the state changes during the turbine motion can be obtained. There is no doubt that other parameters also affect the performance of the turbine; however, this is outside the scope of this paper.

Theoretical model validation
Under the condition of the basic size parameters listed in Table 1, the position of the turbine is at ± 90°, as shown in Figure 1(b). As shown in Figure 6, the computer simulation results of the torque coefficient C t basically match the  theoretical solutions in the change of the turbine blade rotational angle γ . If the turbine blade angle is set too small or too large, the relative velocity of the liquid flow at the outlet of the flow channel is no longer parallel to the blade plane. This means that under the conditions of the optimal blade rotation angle, the eddy current and impact loss of the liquid flow are the smallest; however, this factor is not considered in the theoretical analysis. As a result, the solution calculated by the theoretical formula is slightly larger than the simulated value.

Stability of the HBT
As shown in Figure 7, the velocity distribution of the fluid is uniform when it just leaves the nozzle. After flowing a certain distance in the Y direction, the boundary of the jet becomes wider and wider, while the velocity of the jet decreases gradually, because the jet pumps and entrains a large amount of surrounding fluid. The jet boundary layer expands to both sides as the distance of the exit increases in the Y direction. When the turbine is in a steady state, the final stop position of the turbine is as shown in Figure 8(a), just like the 0°position as shown in Figure 1(b). At this time, the nozzle is in the balance position, and the turbine is balanced and stationary. However, considering the effect of load disturbance, the turbine will not stop at the balance position. When the turbine finally stops, the fluid resistance moment is zero and the impact torque is equal to the friction resistance moment, the turbine will stay in the static balance region. Furthermore, Figure 8(b) shows the unstable state of the HBT, when the nozzle is in the 180°position of the steady-state position of the turbine. This is the most easily disturbed state; the fluid deflects to both sides and produces two relative torques, like a ball placed on the top of a mountain. Once a small disturbance breaks the equilibrium state, the system will eventually return to the state shown in Figure 8 The key design parameter for the symmetrical position of the turbine is the flow angle β. As shown in Figure 9, the nozzle position changes from left 90 to right 90 in symmetrical position 'A' (as shown in Figure 3). When the flow angle increases from 0 to 40, the change in the torque of the turbine is basically the same. The impact force on the turbine decreases with the increase in the flow angle. With the increase in the flow angle, the impediment of the turbine to the flow of the fluid decreases. Reducing the impact force is beneficial to reducing the friction of the whole balanced turbine control mechanism. When the flow angle is too large, the distance between the left-handed blade and the righthanded blade is larger than the diameter of the nozzle, and the torque of the turbine is zero in a certain range.

Parameter optimization
The effects of different parameters on the turbine rotation torque T v and vertical force F d (Figure 3) are shown in Figure 10. The fluid ejected from the nozzle acts on the turbine. The impediment of the flow caused by the turbine changes the momentum of the fluid. Different factors have different effects on the change in turbine momentum, so the forces acting on the fluid are different.
The fluid impact force on a turbine varies with the shape of the turbine and the size of the nozzle. Under the condition that other factors remain unchanged (as shown in Table 1), it can be seen from Figure 10 that turbine blade angle γ , nozzle diameter d 0 , blade thickness w, fluid velocity v and nozzle distance h 0 from the center of the turbine have a greater impact on the torque of the turbine than the other factors. As shown in Figure 10(a), we  obtained the maximum torque value with a blade angle of 36°. With the increase in nozzle diameter d 0 , fluid velocity v and distance k from the nozzle to the turbine center, the absolute value of the torque and impact force on the turbine increase (Figure 10(d)). With the increase in blade thickness (Figure 10(e)), the changes in turbine torque and impact force are opposite to one another. The increase in blade thickness leads to a decrease in the absolute value of turbine torque and an increase in the absolute value of impact force.
As shown in Figure 10, the influence of nozzle height (b), blade spacing angle (c), blade length (g) and nozzle length (j) on the turbine torque is small. When the submerged jet flows, it draws and entrains a large amount of surrounding fluid, making the boundary of the jet wider and wider, while the velocity of the jet is gradually reduced. Therefore, as the height of the nozzle increases (Figure 10(b)), the jet spreads continuously and the torque decreases, but the impact force increases. In HBT design, the nozzle jet should be fully impacted on the blade, so the blade spacing angle should not be too large. The increase in blade length (Figure 10(g)) leads to an increase in torque, but at the same time, the distance between the blades increases and the blade shape changes under the nozzle projection area. Therefore, the increase in blade length leads to a variety of factors, so there seems to be no obvious law relating the increase in blade length to the turbine torque. With the increase in turbine height (Figure 10(i)), the torque of the turbine fluctuates and the impact force decreases, so it is advisable to consider reducing the turbine height in turbine design. With the increase in nozzle length (Figure 10(j)), the force of the turbine is basically unchanged, mainly because the crosssectional area of the cylindrical nozzle is unchanged, and the friction force on the nozzle wall has little influence on the fluid flow, resulting in the nozzle ejection velocity remaining basically unchanged.
The turbine start-up performance is very important to the turbine. Therefore, the parameter design of the turbine should be determined according to the influence of different factors on the start-up performance of the turbine. The turbine torque directly determines whether the turbine can overcome the friction torque. When designing the turbine blade, it is necessary to consider this comprehensively to achieve the optimal relationship between the torque and the impact force. In principle, the greater the torque of the turbine the better, and the smaller the vertical force the better. However, considering the strength and durability of the turbine, the turbine blade thickness was set to 4 mm and the fluid velocity was 20 m/s. Considering the coupling relationship between the nozzle diameter and the distance of the nozzle from the center, when the nozzle diameter is 40 mm, the preferable distance between the nozzle and the center of the turbine is 88 mm. Through the results of the above CFD simulation, the recommended parameters of the turbine are shown in Table 3. We chose the basic parameters of the HBT in the actual field drilling project, because the focus of this article is to study the working characteristics of the HBT. We will describe its application in another paper.

Dynamic follow-up performance of the HBT
Under the impact of the nozzle jet (Jeffers et al., 2016), the turbine overcomes the frictional resistance torque and then moves. The ability of the hydraulic turbine to overcome the frictional resistance torque is called the start-up performance of the turbine. When the nozzle moves, driven by the control mechanism, the turbine will move along with the nozzle, as shown in Figure 11. We call this characteristic of the turbine the dynamic follow-up performance, which describes the dynamics of the HBT when the upper part of the nozzle moves. Figure 12 shows the force variations of the turbine when the nozzle rotates for one circle at the speed of 0.0000167, 1, 2, 3, 4 and 5 rev/s when the turbine is fixed. When the nozzle is located on the right-handed blade, the torque of the turbine is negative; when the nozzle is located on the left-handed blade, the torque of the turbine is positive; when the nozzle is stabilized at the symmetrical position A or B, as shown in Figure 2, the torque of the turbine is zero.   Turbine torque variations can be divided into two cases. Near the symmetrical positions A and B, the torque varies approximately linearly, while at other positions, the torque fluctuates regularly. This is because when the nozzle is near the symmetrical position, the fluid ejected from the nozzle acts simultaneously on the blades with different rotating directions, and the two blades are subjected to the opposite direction of torque. When the nozzle moves counter-clockwise crossing the symmetrical position A, the torque of the right-handed blade decreases and that of the left-handed blade increases, so the absolute value of the torque decreases first and then increases, and vice versa. At other locations, the torque fluctuates regularly owing to the change in the surface area of the turbine blade impacted by the nozzle fluid.

Nozzle movement at constant speed
The velocity of the nozzle jet is actually the combined velocity of nozzle rotation speed and fluid nozzle speed. The change in nozzle speed affects the change in velocity of the fluid relative to the turbine. Therefore, the force of the turbine varies with different nozzle speeds. At the same time, because of the movement of the nozzle, the velocity of the fluid is not vertical downward, but there is an angle with the vertical direction. Therefore, when the rotating speed of the nozzle is different and the torque of the turbine is zero, the nozzle is not located at the symmetrical positions A and B, but deviates by a certain angle. As far as the speed given in this paper is concerned, as the nozzle speed increases, the torque and impact force of the turbine tend to become larger. In fact, as the nozzle speed increases, the direction of the jet velocity approaches the horizontal direction, and the attack angle between the jet and the blade changes, resulting in a complex variation of the turbine force.
The vertical impact force on the turbine is always negative, and it fluctuates greatly. Because of the symmetrical structure of the turbine, at the symmetrical position A, the fluid tends to concentrate and the fluid momentum changes greatly, while at the symmetrical position B, the fluid flow is dispersed and the momentum changes little. When the nozzle velocity is 0.0000167 rev/s, the influence of the nozzle speed on the nozzle outlet fluid velocity can be neglected. When the nozzle is in a position near the symmetrical positions A and B, the impact force takes the minimum value and the maximum value, respectively.

Nozzle sine oscillation
When the nozzle is driven by the control mechanism, it rotates at the speed of ω = π rad/s and ω = π 2 /2 * sin (π * t). The angular velocity, rotation angle and torque of the turbine vary with time, as shown in Figures 13-15. The left-hand figures are the variation curves of the nozzle in uniform motion and the right-hand figures are the variation curves of the nozzle in sinusoidal motion. The amplitude of the turbine rotational speed decreases continuously and finally it is basically consistent with the nozzle speed, as shown in Figure 13(a). As shown in Figure 14(a), the turbine position appears to be ahead of the nozzle or lags behind the nozzle within t < 1 s, but the difference in the rotation angle between the nozzle and the turbine tends to be stable over time.
The total moment acting on the turbine changes sinusoidally with time, and finally stabilizes near zero, while the impact force fluctuates in the initial stage and finally tends to stabilize, as shown in Figure 15(a). When the nozzle moves away from the balance position at a constant speed driven by the control mechanism, the torque balance of the turbine is broken. Under the impact moment produced by the nozzle, the turbine starts to move, and the speed gradually increases from zero. When the turbine speed is the same as the nozzle speed for the first time, the nozzle deviates farthest from the balance position; when the turbine speed increases to the extreme value, the nozzle and the turbine are in the balance position. Then, as the turbine continues to rotate and the nozzle crosses the balance position, the impact torque on the turbine is in the opposite direction, as shown in Figure 15(a), and the turbine begins to decelerate. The extreme point of the turbine speed curve corresponds to the balance position of the turbine. When the nozzle crosses the balance position, the direction of the turbine torque changes and the direction of the turbine acceleration changes. When the nozzle stops moving, the turbine will continue to rotate until it gradually stabilizes to its balance position.
When the nozzle moves at sine oscillation, the speed of the turbine still fluctuates up and down based on the nozzle speed, but the fluctuation range is smaller than that of the nozzle rotating at a constant speed. The overall trend of the turbine speed is consistent with the sinusoidal variation of the nozzle speed, as shown in Figure 13(b). The variation curve of the nozzle position basically coincides with that of the turbine, which proves that the angle of nozzle deviation from the balance position is very small, as shown in Figure 14(b). The torque variation range of the turbine is smaller than the torque variation range when the nozzle rotates at a constant speed, and the impact force variation range is not very different, as shown in Figure 15(b), which also proves that the angle range of the nozzle deviating from the balance position when the nozzle moves at a constant speed is larger than that when the nozzle moves at sinusoidal speed. When the nozzle speed changes sinusoidally (the nozzle speed changes from zero), the turbine speed is basically the same as the nozzle speed, but fluctuates only slightly. When t = 1 s, the nozzle velocity decays to zero and stops moving, and the turbine speed approaches zero. However, the impact moment of the turbine is not zero, and the turbine will continue to rotate until it gradually stabilizes to its balance position.

Application in vertical drilling tools
Automatic vertical drilling technology adopts downhole closed-loop control (Wang et al., 2021), which is programmed on the surface to automatically keeps the wellbore being drilled vertical. When the drilling tool is inclined, the gravity inclination measurement and control mechanism turn to the lower side of the hole under the action of gravity, which drives the upper disc valve to stabilize in the expected position, as shown in Figure 2. The lower disc valve rotates with the lower hole drilling tool, and frictional forces between the pads and the borehole wall will reduce the instantaneous rotational speed of the drill bit. The pads of the tools in the vertical drilling system constantly push against the borehole wall, making the bottom hole by a cycle of nonlinear damping forces, which leads to the chaotic and disordered motion of the bottom drilling tool (Xue et al., 2019). The main factors affecting the accuracy of deviation control are due to the friction resistance of the upper and lower disc valves, and the bias platform cannot be stabilized at the lower side of the well.
A new type of HBT is presented and applied to the vertical drilling system, which is used to drive the upper disc valve to reduce the frictional resistance and the influence of vibration of the bottom drilling tool. This can achieve a soft connection between the eccentric platform and the upper valves, eliminate the effect of the friction between the upper valve and the lower valve, and optimize the control ability. After HBT optimization, the stable angle is closer to the low side, which can make the control of the vertical drilling system more accurate.  the turbine position appears to be ahead of the nozzle or lags behind the nozzle within t < 1 s, but the difference in the rotation angle between the nozzle and the turbine tends to be stable over time; (b) sinusoidal rotation of the nozzle: the turbine position basically coincides with that of the nozzle, which proves that the angle of nozzle deviation from the balance position is very small. Figure 15. Impact force and impact torque variations of the turbine with different nozzle motion modes. (a) Uniform rotation of the nozzle: the total moment acting on the turbine changes sinusoidally with time, and finally stabilizes near zero, while the force fluctuates in the initial stage and finally tends to stabilize; (b) sinusoidal rotation of the nozzle: the torque variation range of the turbine is smaller than the torque variation range when the nozzle rotates at a constant speed, and the impact force variation range is not very different. As shown in Figure 2, in the new design, the eccentric platform controls the upper disc valve with a nozzle to drive the HBT in a stable manner in the low direction of the wellbore. Therefore, the vibration of the tool caused by the pads will not affect the stability of the eccentric platform. The soft connection between the eccentric platform and the upper disc valve is realized.
When the bottom drill string does not rotate, the eccentric block rotates downward from 90°from the lower side. Figure 16 shows the angular velocityφ curve for an eccentric block before and after optimization; after optimization of the HBT, the eccentric block approaches the steady state more quickly.
The objective of this research was to improve the performance of a mechanical vertical drilling tool (i.e. a tool designed to keep the hole vertical in the face of deflection forces). The main contribution of this paper is the addition of a novel turbine to the internal mechanism of the vertical drilling tool. This turbine should allow the tool to be more effective; that is, to keep the hole more closely aligned to vertical. Our experiment can be carried out on the entire drilling tool to verify the effect of the procedure with or without HBT optimization.

Conclusions
Most turbine designs are based on the continuous rotating turbine structure. In this paper, a novel HBT with a symmetrical blade structure that can be used in the hydro environment is proposed. The turbine, which uses the hydraulic impact of the nozzle jet and balanced turbine blade design in stability control, could be used to control stability in special applications. The performance of the turbine is studied using CFD software. Finally, the following conclusions can be reached.
(1) The torque and impact force acting on the HBT are the most important parameters affecting the performance of the turbine. Turbine blade angle, nozzle diameter, blade thickness, fluid velocity and distance from the center of the nozzle to that of the turbine have a great influence on the turbine torque, which is the main factor to be considered in HBT design. Recommended parameters of the turbine are shown in Table 3. We chose the basic parameters of the HBT in the actual field drilling project.
(2) Because of the special blade design of the HBT, when the nozzle deviates from the balance position, the HBT will be restored to the balance position under the action of external moment. Under the action of dynamic friction, the turbine stability time is less than is the case without friction. Considering the static friction force, the position where the turbine stops moving is not at the balance position, but at an angle to the balance position. (3) The follow-up performance of the HBT is directly related to the rotational speed of the nozzle. When the nozzle moves in sinusoidal oscillation, the turbine has better follow-up performance. When t = 1 s, the nozzle velocity decays to zero and stops moving, and the turbine speed approaches zero. However, the impact moment of the turbine is not zero, and the turbine will continue to rotate until it gradually stabilizes to its balance position.
The simulation results show that the change in nozzle velocity directly affects the change in turbine speed in the process of the turbine following the nozzle. When the nozzle deviates from its balance position at a certain speed, the turbine will move with the movement of the nozzle. During the follow-up process of the nozzle, the external torque tends to make the turbine speed consistent with the nozzle speed. When the nozzle stops, owing to the inertia of the turbine, the latter will continue to rotate and eventually return to its balance position under the action of external torque. To reduce the stability time of the turbine, it is hoped that when the nozzle movement stops, the turbine will have a low speed and the angle of the nozzle deviating from the balance position will be small. From the above analysis, when the velocity of the nozzle changes sinusoidally and continuously, the angle of deviation of the nozzle from the balance position is smaller and the follow-up performance is better, which is also beneficial to the stabilization process of the turbine.
The HBT is used to drive the upper disc valve to reduce the frictional resistance and the influence of the bottom drilling tool vibration on the bias platform, to improve the control accuracy. This research has revealed the optimization mechanism by which the HBT controls the mechanical performance of the mechanical vertical drilling disc valve mechanism. The present turbine is only a preliminary design scheme and more parameters could be modified using optimization techniques. Further improvements to the turbine and modeling experiments will be carried out in future work.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by the National Natural Science Foundation of China [grant number 42072341].