Investigation on Secondary Flow of Turbodrill Stator Cascade with Variable Rotary Speed Conditions

Abstract

There are various secondary flow types in turbodrill’s blade cascades, and all kinds of secondary flow have a significant effect on flow loss. In this paper, the stator cascade of φ160 mm turbodrill is taken as the research object, and the CFD method is used to analyze the secondary flow and its evolution. The origin and evolution mechanism of secondary flow is explained from the flow mechanism. The results show that when the working rotary speed is lower than the design rotary speed, the secondary flows are composed of suction surface separation vortex, horseshoe vortex, and passage vortex coexisting. The intensity of secondary flows increases with the decrease of rotary speed. When the working rotary speed is near the design rotary speed, the secondary flows include horseshoe vortex, passage vortex, and corner vortex. When the working rotary speed is higher than the design rotary speed, the secondary flows consist of pressure surface separation vortex and suction surface trailing edge separation vortex. Regardless of rotary speed, secondary flow intensity in the shroud region is greater than the hub region, which has a greater influence on the mainstream. In addition, compared with high rotary speeds, secondary flow intensity is greater at low rotary speeds, resulting in greater flow losses.

1. Introduction

With the development of oil-gas and geological drilling engineering, drilling engineering is constantly moving towards deep-ground, on the one hand, in order to develop deep-ground oil, gas, geothermal, and mineral resources [1], and on the other hand, to promote the development of geoscience [2]. During the process of drilling deep into the ground, drilling engineering faces more and more challenges, such as increasing formation temperature and deteriorating rock drillability. In solving these problems, turbodrill is widely used in deep-ground drilling and high-temperature well drilling due to its advantages of high-temperature resistance and high drilling rate [3]. As an important downhole hydraulic drilling tool (its structure is shown in Figure 1), the key working components of a turbodrill mainly include stators and rotors. The stator converts the pressure energy of drilling fluid into kinetic energy and makes drilling fluid enter the rotor at a certain rate and direction; the rotor converts the kinetic energy and pressure energy of drilling fluid into the mechanical energy of the rotor’s rotation. Therefore, the design of the stator-rotor becomes crucial in the design of turbodrill. Turbodrill’s stator and rotor have axial cascades, and the cascades of stator-rotor have a very low aspect ratio and ultra-small size compared to other turbomachinery. The lower aspect ratio cascade means that the cascade end wall flow is more likely to have a greater influence on the mainstream of cascade and bring greater losses [4], which seriously affects the performance of turbodrill cascades. The ultra-small size means that changes in the flow field can easily affect the characteristics of the flow around the cascade blades.

The secondary flow in the cascade is a flow that is different from the mainstream direction, which can lead to insufficient or excessive fluid steering in the cascade, deviating from the designed flow angle, affecting the distribution of blade load, and causing flow losses. A large number of secondary flow studies have been conducted in turbomachinery, such as gas turbines [6,7,8]. As shown in Figure 2, it is a typical secondary flow structure of a gas turbine cascade [8].

However, in research on turbodrills, the vast majority of researchers focus on the design of turbodrill blades [9,10,11,12]. The research on the internal flow field focuses on the velocity and pressure fields, and its performance is evaluated using macroscopic evaluation indicators [13,14,15] such as output torque, pressure drop, efficiency, etc. However, almost few people have conducted research on the secondary flow structure of turbodrill cascades, although the secondary flow of cascades has a significant influence on the performance of turbodrill. There are significant differences in geometric features of cascade blades between the turbodrill and other turbomachinery, including extremely low aspect ratio, ultra-small blade spanwise height and axial chord length, turbodrill blades with shrouds, and narrow cascade channel, as shown in Figure 3. In order to improve the efficiency of turbodrill cascades and reduce flow losses, it is necessary to conduct research on the secondary flow of turbodrill cascades.

In addition, when turbodrill drives the bottom drill bit to break rock, the change of drilling pressure on the bit and the non-uniformity of the formation’s lithology will cause the loads on turbodrill to change, and its rotary speed will also change as a result. The changes in rotary speed bring about changes in the inflow conditions of the cascade. In velocity triangles, as shown in Figure 4, the black lines represent the inlet and outlet velocity triangles of the turbodrill stator blade under design conditions, the red lines represent the inlet and outlet velocity triangles of the turbodrill stator blade when the rotary speed is higher than its design rotary speed, and the blue lines represent the inlet and outlet velocity triangles of the turbodrill stator blade when the rotary speed is lower than its design rotary speed. From Figure 4, when the rotary speed is higher than the design rotary speed, the inflow will impact the suction surface of the cascade’s blade, while when the rotary speed is lower than the design rotary speed, the inflow will impact the pressure surface of cascade’s blade.

The different inflow conditions make the flow situation of turbodrill cascades more complex. Therefore, it is necessary to analyze the flow process in turbodrill cascade at different rotary speeds and explore the changes of internal secondary flow with variations of rotary speed in order to provide a reference for improving the design of the cascade’s blade.

2. CFD Modeling of Turbodrill Cascade

2.1. Geometric Model

In this paper, the turbodrill with a diameter of φ160 mm, which is commonly used in deep earth drilling, is taken as the research object, and its design point operating parameters are a flow rate of 30 L/s, rotary speed of 3000 RPM, and single stage design torque of 20 N·m. The geometric parameters of the cascade are shown in

Table 1. φ160 mm turbodrill cascade geometric parameters.

Parameter	Value	Parameter	Value
Shroud diameter (mm)	116	Rotor aspect ratio	0.65
Hub diameter (mm)	90	Stator inlet angle (°)	90
Stator blade number	21	Stator outlet angle (°)	31.472
Rotor blade number	20	Stator stagger angle (°)	52.7
Stator chord length (mm)	18.8	Rotor inlet angle (°)	67.267
Rotor chord length (mm)	20	Rotor outlet angle (°)	25.975
Stator aspect ratio	0.69	Rotor stagger angle (°)	41

, and its three-dimensional solid model is shown in Figure 5.

The turbodrill contains multi-stage stators and rotors, and the stator and rotor structures of each stage are identical. In order to avoid the influence of inlet and outlet conditions on the analysis results during calculation, this article selects two-stage turbine stators and rotors to establish a calculation model.

The blade structure of the stator cascade is shown in Figure 6, where 5% Span, 50% Span, and 95% Span are observation surfaces extracted along the blade span.

2.2. Numerical Method

The internal flow of turbodrill cascades is an incompressible three-dimensional viscous flow. The fluid density ρ = const, and the temperature change in its flow field is very small. The variation of internal energy can be completely ignored, so there is no need to solve the energy equation. Therefore, when solving its flow field, only the continuity equation and momentum equation need to be solved. Also, the flow in the blade cascade is in a turbulent state, except for the viscous boundary layer on the wall. In calculation, continuity equations and momentum equations under turbulent conditions are used. The Reynolds Averaged Stress Model (RANS) is used as a turbulence model; therefore, the control equation for calculating the fluid domain of turbodrill blade cascades is the following [16]:

$$\frac{\partial U_i}{\partial x_i}=0$$

(1)

$$\frac{\partial U_i}{\partial t}+\frac{\partial }{\partial x_j}\left(U_i{U}_j\right)=-\frac{1}{\rho }\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}\left(\nu \frac{\partial U_i}{\partial x_j}-\overline{u{’}_i}\overline{u{’}_j}\right)$$

(2)

In the formula, $U$ is the time-averaged velocity, x is the coordinate value variable, t is time, ρ Is the density, P is the time-averaged pressure, $\nu$ is the kinematic viscosity, $\overline{u’}$ is the fluctuating velocity, and i, j is the coordinate axis indicator symbol.

ANSYS CFX was used to solve the Reynolds Averaged Navier–Stokes (RANS) equation. The SST model based on $k-\omega$ was selected as the turbulence model. The SST model based on $k-\omega$ incorporates transport effects into the vortex viscosity formula, taking into account the transfer of turbulent shear stress, overcoming the faultiness of $k-\omega$ and BSL $k-\omega$ models, and can accurately predict flow separation and quantity under inverse pressure gradients. The mathematical expression of SST based on $k-\omega$ the model is as follows [17].

The transport equation for turbulent kinetic energy k is

$$\frac{\partial k}{\partial t}+\overline{u_j}\frac{\partial k}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(v+{\sigma }_k{v}_t\right)\frac{\partial k}{\partial x_j}\right]+P_k-{\beta }^{*}k\omega$$

(3)

The transport equation for the turbulent dissipation rate $\omega$ is

$$\frac{\partial \omega }{\partial t}+\overline{u_j}\frac{\partial \omega }{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(v+{\sigma }_{\omega }{v}_t\right)\frac{\partial \omega }{\partial x_j}\right]+\alpha S^2-\beta {\omega }^2+2(1-F_1){\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}$$

(4)

The turbulent vortex viscosity ${\mu }_{\mathrm{t}}$ is

$${\mu }_{\mathrm{t}}=\frac{{\alpha }_{1}k}{\max ({\alpha }_1\omega ,S{F}_2)}$$

(5)

In Equations (3)–(5), $\nu$ is the kinematic viscosity, and S is the constant term of the shear stress tensor. The expressions for ${\sigma }_{\omega }$, ${\sigma }_k$, $P_k$, $F_1$, and $F_2$ are described as follows.

$${\sigma }_{\omega }=\frac{1}{F_1/{\sigma }_{\omega 1}+(1-F_1)/{\sigma }_{\omega 2}}$$

(6)

$${\sigma }_k=\frac{1}{F_1/{\sigma }_{k1}+(1-F_1)/{\sigma }_{k2}}$$

(7)

$$P_k=\min \left[{\mu }_{\mathrm{t}}\frac{\partial \overline{u_i}}{\partial x_j}\left(\frac{\partial \overline{u_i}}{\partial x_j}+\frac{\partial \overline{u_j}}{\partial x_i}\right),10\rho {\beta }^{*}k\omega \right]$$

(8)

$$F_1=\tanh \left\{{\left\{\min \left[\max \left(\frac{\sqrt{k}}{{\beta }^{*}\omega y},\frac{500v}{\omega y^2}\right),\frac{4\rho {\sigma }_{\omega 2}k}{C{D}_{k\omega }{y}^2}\right]\right\}}^4\right\}$$

(9)

$$F_2=\tanh \left\{\max {\left[\left(\frac{2\sqrt{k}}{{\beta }^{*}\omega y},\frac{500v}{\omega y^2}\right)\right]}^2\right\}$$

(10)

In the equation of $F_1$, the expression of $C{D}_{k\omega }$ is as follows.

$$C{D}_{k\omega }=\max \left(2\rho {\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial k}{\partial x_i},{10}^{-10}\right)$$

(11)

The values of other coefficients in the above equation are as follows:

${\alpha }_1=5/9$, ${\alpha }_2=0.44$, ${\beta }_1=0.075$, ${\beta }_2=0.0828$, ${\beta }^{*}=0.09$, ${\sigma }_{k1}=0.85$, ${\sigma }_{k2}=1$, ${\sigma }_{\omega 1}=0.5$, ${\sigma }_{\omega 2}=0.856$, and y are the minimum distances from the wall.

The superiority of this model has also been demonstrated in much of the literature [18]. The SST turbulence model can provide high-precision predictions of mechanical efficiency and flow field pressure distribution in rotating flow [19], and a large number of researchers have also used SST models to capture complex flows in turbomachinery [20,21]. To accurately simulate the flow of the boundary layer, the wall function adopted automatic near-wall processing, which is a near-wall processing method developed by CFX based on the $k-\omega$ model. The method allows a smooth transition from the low Reynolds number form to the wall function formulation without loss of accuracy [22]. It has good adaptability to the y⁺ of the first layer grid on the wall. When the y⁺ of the first layer grid on the wall is less than 2, the flow field at the wall is directly solved; that is, the wall velocity gradient is linearly related to the distance from the wall normal. When the y⁺ of the first layer grid on the wall is greater than 2, the automatic wall function is called to solve the flow field at the wall [23]. This processing method undoubtedly has great benefits for multi-stage flow turbomachinery and meets the requirements of this paper for analyzing secondary flow.

In the calculation, steady flow calculation is used, and second-order upwind is used for spatial discretization of the control equations.

2.3. Boundary Conditions and Meshing

The purpose of this study is to analyze the influence of cascade geometry on secondary flow. Therefore, the stator was selected as the research object, rather than the rotor, to avoid the influence of centrifugal and Coriolis forces on the fluid in the rotor cascade. Furthermore, due to the absence of guide components at the inlet of the first stage stator, this is different from most stator inlet conditions in turbodrill. In order to make the analysis results representative of the flow situation of the vast majority of stators in turbodrill, two stages of turbines were selected for numerical simulation. The first stage was S1-R1, and the second stage was S2-R2. The stator (S2) of the second stage was selected as the research object in this paper. Due to the periodic repeatability of the flow path in the stator and rotor, a two-stage hydraulic turbine single-channel calculation domain is established, as shown in Figure 7. Each stage of the calculation domain includes the stator domain and rotor domain, and the rotating interface between the stator and rotor was adopted by the Frozen Rotor model. The circumferential interface was set as a periodic interface, and the boundary type is a fluid-fluid type. The inlet boundary condition is the mass flow rate, which is set as the design flow rate of the turbodrill. The outlet boundary condition is the pressure outlet, which is set as 1 atm based on ground experiments. The other solid walls (including shroud, hub, and blades) are set as non-slip wall boundaries.

The meshing was performed using Turbogrid with hexahedral mesh. After the meshing was completed, the number of meshes for the stator and rotor was about 1.34 million, and the total number of meshes in the computational domain of the two-stage turbine was about 5.34 million. The meshing of the near-wall surface is shown in Figure 8.

It is necessary to analyze the impact of fluid domain models under different grid numbers on the calculation results to ensure that the CFD calculation results are independent of the number of grids. This paper selects the output torque of turbodrill to evaluate the impact of grid quantity on CFD calculation results. As shown in Figure 9, when the number of grids in the fluid domain of a single-stage turbine exceeds 1 million, its output torque tends to stabilize. And compared with the grid models in the other literature [14,15] on CFD calculation of turbodrill, the grid number of 1.34 million used in this paper can meet the calculation requirements.

As shown in Figure 10, it is the y⁺ cloud map of the S2 blade, with a maximum value of 9.576 and a minimum value of 0.106. This means that the SST model used its automatic wall function when calculating the blade wall surface.

3. Results and Discussions

3.1. Static Pressure Distribution of Turbodrill Stator Blade

In order to reveal the influence of the end wall in a low aspect ratio blade on the flow field in cascade, the blade static pressure distributions of 5% Span, 50% Span, and 95% Span were taken along the blade spanwise for analysis, as shown in Figure 11 (in Figure 11, PS—pressure surface and SS—suction surface). From Figure 11, the static pressure on the pressure surface shows a general trend of decreasing with increasing rotary speed, while the static pressure values on the suction surface fluctuate within a certain range. Below the design rotary speed (3000 RPM), static pressure values reach a peak near the blade’s leading edge (about 0–10% of the axial chord length) and decrease toward the trailing edge. With increasing rotary speed, this peak gradually moves closer to the blade’s leading edge, but the static pressure values gradually decrease overall. Above the design rotary speed, the static pressure on the pressure surface continues to decrease with increasing rotary speed, and the pressure peak disappears. The static pressure curve tends to flatten, with only certain fluctuations near the leading and trailing edges. The reason for this phenomenon is that, at low rotary speeds, the incoming flow rushes on the pressure surface and forms a stagnation point on the pressure surface where the pressure peak occurs. With the increasing rotary speed, the attack angle of incoming flow gradually decreases (the phenomenon can be obtained from the velocity triangle shown in Figure 4), and the stagnation point on the pressure surface gradually approaches the blade’s leading edge. As the rotary speed continues to increase, the attack angle changes from positive to negative, which leads to the stagnation point moving from the pressure surface past the leading edge to the suction surface of the blade. During the process of changing the attack angle from positive to negative, the angle between the velocity of incoming flow near the leading edge of the pressure surface and the outward normal line of the pressure surface gradually increases, as a result of which the fluid near the pressure surface needs to be turned at a larger angle when flowing along the pressure surface. For the same chord length, the greater the angle at which the fluid turns, the smaller the curvature radius of the streamline and the greater the centrifugal force generated by the fluid. When the centrifugal force of the fluid flowing along the surface of the blade increases, the static pressure of the cascade channel fluid on the blade surface is counteracted more, so the static pressure on the blade decreases as the rotary speed increases. In addition, because of the small radius of the blade’s leading edge, when the static pressure of the cascade channel fluid on the blade surface is counteracted more at negative angles of attack, the fluid flowing at the leading edge of pressure surface does not receive sufficient centripetal force, and flow separation occurs at the leading edge of pressure surface. According to Figure 11, when the rotary speed reaches 5500 RPM, the static pressure on the leading edge of the pressure surface at the 5% Span, 50% Span, and 95% Span fluctuate, indicating a flow separation.

Now, to analyze the distribution law of static pressure on the suction surface. Below the design rotary speed, the suction surface pressure increases sharply from the leading edge to a certain value, then decreases slightly, starts to increase gradually through half of the axial chord length, and peaks near the trailing edge (about 90–100% of the axial chord length). Above the design rotary speed, the suction surface static pressure reaches its maximum at the leading edge, decreases gradually from the leading edge and increases as it approaches the outlet of the cascade. The reason for its formation: When the rotary speed from slightly higher than the design speed (3500 RPM) to the braking condition (0 RPM), the incoming flow stagnation point is located between the pressure surface and the blade’s leading edge. When the incoming flow flows around the leading edge from the stagnation point, due to the small radius of the blade’s leading edge, a large centrifugal force is formed, which counteracts the static pressure formed by the surrounding fluid on the blade’s leading edge, resulting in a small static pressure on the suction surface near the blade’s leading edge. When the fluid circumvents the leading edge and flows around the suction surface, the radius of its flowing increases dramatically, and the static pressure on the suction surface increases sharply. From the blade suction surface profile (as shown in Figure 8), it can be seen that the curvature of the suction surface first increases, then decreases, and then increases from the leading edge to the trailing edge. Therefore, the suction surface static pressure slightly decreases from near the leading edge to the middle of the cascade and gradually increases from the middle to the trailing edge. After passing through the throat of the cascade, a static pressure peak appears near the trailing edge. With the further increase of rotary speed, the incoming stagnation point is gradually transferred to the suction surface. Therefore, the static pressure on the suction surface is the highest at the leading edge. Along the flow direction, the static pressure gradually decreases from the stagnation point at the leading edge and increases near the trailing edge for the same reason as before.

In addition, at rotary speeds below the design rotary speed, the static pressure on both the suction and pressure surfaces is higher than its distribution at high rotary speeds, and it on the pressure surface is much higher than it on the suction surface. At rotary speeds over the design rotary speed, the static pressure on the pressure surface decreases with the increase of rotary speed, and gradually, it on the pressure surface is lower than it on the suction surface, i.e., when the rotary speed is more than 4500 RPM, the static pressure on the suction surface is higher than pressure surface within 0–30% axial chord length. The fundamental reason is that when the rotary speed changes, the angle between the incoming flow velocity and the outward normal line of the pressure surface changes, and the angle between the incoming flow velocity and the outward normal line of the suction surface also changes. When the angle between incoming flow velocity and the outward normal line of the blade profile is acute, the centrifugal force formed by incoming flow is the same as the static pressure of the cascade channel fluid on the blade. At this time, the static pressure acting on the blade profile is the vector sum of incoming flow centrifugal force and the static pressure of the cascade channel fluid on the blade. When the angle between incoming flow velocity and the outward normal line of the blade profile is obtuse, the centrifugal force formed by incoming flow is opposite to the static pressure of the cascade channel fluid on the blade. At this time, the static pressure acting on the blade profile is the vector difference between the incoming flow centrifugal force and the static pressure of the cascade channel fluid on the blade. As the rotary speed changes and attack angle changes, the angles between the incoming flow velocity and outward normal line of both blade pressure surface and suction surface also change, and the magnitude and direction of centrifugal force change. Therefore, the counteraction degree of incoming flow centrifugal force to the fluid static pressure in the cascade channel changes, resulting in the above distribution pattern of blade static pressure.

Comprehensively analyzing the plots (a), (b), and (c) in Figure 11, the differences in static pressure distributions on 5% Span, 50% Span, and 95% Span at the blade are found at any rotary speeds, which suggests that the flow in hub and shroud regions has a significant influence on cascade mainstream. Furthermore, the difference in static pressure distribution between the blade’s 95% Span and 50% Span is much greater than that between the blade’s 5% Span and 50% Span, indicating that the flow in the shroud region has a greater influence on the mainstream, and the mechanism behind this phenomenon needs to be analyzed from the flow field structure.

3.2. Internal Flow Field Structure of Stator Cascade

In order to accurately analyze the flow in the stator channel of turbodrill, this paper analyzes the fluid flow in the stator cascade from the aspects of the hub, 50% Span, shrouded section of the blade, the overall blade, and the cascade channel.

Figure 12 shows the static pressure distribution and limited streamlines of the stator (S2) hub region at different rotary speeds. The incoming flow rushes into the stator cascade, forming a clear stagnation point near the leading edge. Due to the influence of boundary layer viscosity on incoming flow, the fluid velocity near the hub surface is lower than that of the mainstream, resulting in the total pressure of the fluid on the hub surface being lower than that of the mainstream [24]. Therefore, after the incoming flow stagnates at the leading edge, a transverse pressure gradient pointing towards the hub surface will be formed along the blade spanwise, which rolls up the fluid near the hub and pushes it towards the hub surface, forming a vortex. The vortex is divided into two branches under the action of the hub surface, one flowing along the pressure surface and the other flowing along the suction surface, which is the horseshoe vortex (hereinafter referred to as HV) of the hub surface. As shown in Figure 12, at the rotary speed of 0 RPM, the horseshoe vortex is divided into two branches from the high-pressure area near the stagnation point on the pressure surface. One branch flows from the pressure surface to the cascade channel, which is the pressure surface branch of horseshoe vortex (hereinafter referred to as HV_ps); the other branch bypasses the leading edge of the blade and flows towards the suction surface, which is the suction surface branch of horseshoe vortex (hereinafter referred to as HV_ss). When the attack angle of incoming flow changes, the structure of the horseshoe vortex also changes accordingly.

When the attack angle of incoming flow is positive, the stagnation point appears on the pressure surface, and the HV forms near the leading edge of the pressure surface. At the same time, due to the incoming flow velocity being at an acute angle to the outward normal line of the suction surface profile, a large-scale inverse pressure gradient is created near the suction surface, resulting in flow separation and the formation of the suction surface separation vortex (hereinafter referred to as SSV). HV_PS flows into the cascade channel and develops into a passage vortex (hereinafter referred to as PV). After bypassing the leading edge, HV_SS is mixed into the SSV. As the rotary speed increases from 0 rpm, the angle between the incoming flow velocity and the outward normal line of the suction surface profile decreases, the inverse pressure gradient appearing on the suction surface decreases, the SSV intensity gradually weakens, and its influence range in the cascade channel also decreases. At the rotary speed of 2500 RPM, the range of SSV has greatly decreased, and at this time, the HV and its induced PV play a dominant role in hub surface channel, and a corner vortex (hereinafter referred to as CV) is formed in the middle of suction surface.

When the attack angle of incoming flow is negative, the stagnation point appears on the suction surface, and there is no obvious HV on the hub surface (as shown in Figure 12, 5000 and 5500 RPM). The incoming flow separates at the leading edge, forming a pressure surface separation vortex (hereinafter referred to as PSV) in front of the pressure surface. In addition, due to the large curvature variation of the blade suction surface, the angle between the incoming flow velocity and the outward normal line of the suction surface trailing edge is acute, resulting in separation vortices at the suction surface trailing edge (hereinafter referred to as STSV).

The above analysis analyzed the variation process of secondary flow on the hub surface when the attack angle of incoming flow changes from positive to negative. Through comprehensive comparative analysis, it can be found that at low rotary speeds and high attack angles, the secondary flow on the hub surface is mainly composed of SSV, with HV_ps and HV_ss coexisting; as the rotary speed increases while the attack angle decreases, the SSV disappears. At this time, the secondary flow mainly consists of HV_ps and HV_ss, and PV and CV appear; when the attack angle is negative, separation vortices appear on the blade pressure surface and also on the trailing edge of the suction surface. At this time, the secondary flow on the hub surface is mainly composed of separation vortices (including PSV and STSV). Moreover, when the working rotary speed is lower than the design rotary speed of about 1000 RPM (as shown in Figure 12 at 2000 RPM), flow separation begins to occur on the suction surface, while when the working rotary speed is higher than the design rotary speed of more than 2000 RPM (as shown in Figure 12 at 5000 RPM), separation vortices begin to appear on the pressure surface. The influence range of separation vortices at low rotary speeds is greater than that at high rotary speeds. The fundamental reason for this phenomenon is the difference in the angle between incoming flow velocity and the outward normal line of suction and pressure surface profiles.

Figure 13 shows the static pressure distribution and limited streamlines of the stator (S2) shroud region at different rotary speeds. From the figure, it can be seen that the secondary flow vortex structures in the shroud region are similar to that in the hub region, but the secondary flows in the shroud region have a larger range of influence and more intense flow. When the attack angle of incoming flow is positive, the range of SSV increases with the increase of attack angle. Especially when the rotary speed is below 2000 RPM, the SSV gradually affects more than half of the flow on the suction surface, causing a large range of low-pressure areas on it. The HV_ps forms PV in a cascade channel, and HV_ss is mixed into SSV after bypassing the leading edge of the blade. At low rotary speeds with high attack angles, the SSV is the main secondary flow form on the shroud. As the rotary speed increases, the SSV gradually decreases and disappears. When the rotary speed increases to the range of 2500–4000 RPM, the secondary flow in the shroud region is mainly composed of HV, PV, and CV. The lower the rotary speed, the higher the pressure at the stagnation point, the larger the transverse pressure gradient in a cascade channel, and the stronger the strength of HV and PV.

When the attack angle of incoming flow is negative and rotary speed reaches 5000 RPM, flow separation occurs at both the leading edge of the pressure surface and the trailing edge of the suction surface in the shroud region, just like a hub region. As the rotary speed continues to increase, the influence range of PSV gradually occupies half of the cascade channel, and the strength of STSV also increases. Compared with the flow in the hub region, the flow separation situation in the shroud region is more severe. In summary, the secondary flow in the shroud region at negative attack angles is mainly separated vortices.

Figure 14 shows the static pressure distribution and streamlines of the stator (S2) blade at 50% Span with different rotary speeds. From the figure, it can be seen that the secondary flow in the 50% span channel is dominated by the SSV at a positive attack angle. Especially when the rotary speed is below 2000 RPM, the SSV has a greater influence on the flow in the cascade channel, and the lower the rotary speed, the larger the range of influence.

A combined analysis of Figure 12, Figure 13 and Figure 14 reveals that at low rotary speeds, the SSV spans the entire blade span from hub to shroud. The SSV disappears when the rotary speed increases to the design rotary speed. After the rotary speed exceeds 5000 RPM, behind the middle suction surface, the static pressure clearly decreases along the suction surface and increases near the trailing edge. The reason is that the angle between incoming flow velocity and the outward normal line of profile, which is behind the middle suction surface, is acute. The centrifugal force formed by the fluid on the suction surface and the static pressure exerted by the channel fluid on the suction surface counteract each other, resulting in pressure decreasing on the suction surface. However, in the outlet section, the static pressure slightly increases due to a sudden increase in the flow area. On the pressure surface, due to the angle between inflow velocity and the outward normal line of the pressure surface, a low-pressure zone appears in the middle of the pressure surface, but no flow separation occurs. From this, it can be seen that at a rotary speed higher than the design rotary speed, there are no PSV and STSV at 50% span, indicating that these two vortices did not cross the blade span.

Based on the above analysis, it can be found that under various identical rotary speed conditions, the flow intensity of various vortices in the shroud region is greater than that in the hub region. From the pressure cloud map, under the same rotary speed conditions, the high-pressure range on the hub is larger than that on the shroud. The color difference in the pressure cloud map on the shroud is large, indicating a larger pressure gradient and more intense secondary flows. From the perspective of limited streamlining, the influence range of the vortex in the shroud region is greater than that in the hub region, indicating that the flow in the shroud region is more chaotic and severe than in the hub region. Therefore, the flow in the shroud region has a greater effect on the mainstream than the hub region.

In order to more intuitively describe the structure of secondary flow in cascade, the following will be explained from the three-dimensional perspective of the blade. Figure 15 shows the static pressure distribution and limited streamlines of the stator blade pressure surface at different rotary speeds (in the figure, left: hub wall; right: shroud wall). From the figure, it can be seen that the static pressure on the blade pressure surface shows an obvious gradient in the flow direction, and the static pressure value on the pressure surface decreases as the rotary speed increases. When the rotary speed is between 0–3000 RPM, the maximum pressure value is located at the blade’s leading edge and presents a radial pressure gradient from hub to shroud. Under this pressure gradient, the low-energy fluid on the blade pressure surface is pushed to the shroud region, as shown in Figure 15, and the limit streamlines point towards the shroud. This flow makes the boundary layer thickness in the shroud larger than the hub. When the HV_ps rolls up the low-energy fluid, the size of the HV formed in the shroud region is dramatically larger than it is in the hub region. Due to the fact that there are more low-energy fluids mixed into the mainstream in the shroud region, more energy is consumed in the mainstream. This is why the pressure shown in Figure 11c is lower than Figure 11a and also why the shroud region flow has a greater effect on the mainstream than the hub region.

Figure 16 shows the static pressure cloud map and limited streamlines of the blade suction surface at different rotary speeds (in Figure 16, right: hub; left: shroud). As shown in the figure, when the rotary speed is between 0–3000 RPM, the incoming flow rushes on the leading edge of the pressure surface, causing flow separation in the suction surface, which makes the static pressure on the front of the suction surface decrease. The SSV separated from the blade’s leading edge will be re-adsorbed on the blade suction surface by the influence of the transverse pressure of cascade channel fluid. As rotary speed increases with the decreasing attack angle, the scale of SSV decreases, and the low-pressure region formed on the suction surface also decreases. The position of the separation vortex reattachment approaches from the trailing edge to the leading edge.

When the attack angle of incoming flow is negative, it rushes on the leading edge of the suction surface. In the front part of the suction surface, the angle between inflow velocity and the outward normal line of the suction surface is obtuse, and the centrifugal force formed by the fluid near the suction surface is in the same direction as the static pressure of the fluid in cascade on suction surface, resulting in high static pressure in this area. In the middle and rear parts of the suction surface, the angle between inflow velocity and the outward normal line of the suction surface is acute, and the centrifugal force formed by the fluid near the suction surface is in the opposite direction as the static pressure of the fluid in cascade on suction surface, resulting in low static pressure in this area. Overall, when the incoming flow has a negative attack angle, the highest pressure point on the suction surface appears near the leading edge in the shrouded region and forms a pressure gradient from the shroud to the hub. Under this pressure gradient and the incoming flow effect, the low-energy fluid on the suction surface is pushed to the trailing edge of the hub region, which makes the low-energy fluid near the trailing edge increase, and separation occurs.

3.3. Vortex Structure in Stator Cascade Channel

The above mainly analyzed the types of secondary flow in cascade channels based on streamlines and pressure cloud maps. In order to more directly describe the structure of secondary vortices in cascade, the vortex identification method is used to visually describe the vortices. The fluid flow can be divided into translation, rotation, and deformation. The formation of vortices is mainly due to the rotational effect of the fluid. To distinguish between deformation and rotation, the velocity gradient of the flow field is written in the form of symmetric tensor A and anti-symmetric tensor B [25,26]. From the expression, it can be seen that the symmetric tensor represents the deformation, while the anti-symmetric tensor represents the rotation, i.e., [26].

$$\frac{\partial u_i}{\partial x_i}=A+B$$

(12)

Wherein,

$$A=\left[\begin{array}{l}\frac{\partial u}{\partial x}\hspace{1em}\frac{1}{2}(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})\hspace{1em}\frac{1}{2}(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x})\\ \frac{1}{2}(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})\hspace{1em}\frac{\partial y}{\partial y}\hspace{1em}\frac{1}{2}(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})\\ \frac{1}{2}(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x})\hspace{1em}\frac{1}{2}(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})\hspace{1em}\frac{\partial u}{\partial x}\end{array}\right]$$

(13)

$$B=\left[\begin{array}{l}0\hspace{1em}\frac{1}{2}(\frac{\partial u}{\partial y}-\frac{\partial v}{\partial x})\hspace{1em}\frac{1}{2}(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x})\\ -\frac{1}{2}(\frac{\partial u}{\partial y}-\frac{\partial v}{\partial x})\hspace{1em}0\hspace{1em}\frac{1}{2}(\frac{\partial v}{\partial z}-\frac{\partial w}{\partial y})\\ -\frac{1}{2}(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x})\hspace{1em}-\frac{1}{2}(\frac{\partial v}{\partial z}-\frac{\partial w}{\partial y})\hspace{1em}0\end{array}\right]$$

(14)

When the contribution of the ant-symmetric tensor to the velocity gradient is greater than the symmetric tensor, the distribution of the velocity gradient in the flow field is mainly caused by vortices, indicating that vortices exist in the flow field. In this paper, the λ₂ method is chosen as the vortex identification method, and the dimensionless helicity H_n is used for coloring. In the λ₂ method, the pressure in the vortex zone is smaller than the environment [27], i.e.,

$$A^2+B^2=-\nabla (\nabla P)/\rho$$

(15)

In the formula, P is the static pressure, and ρ is the density.

From the equation, it can be seen that when there are two negative eigenvalues for the symmetric tensor A² + B², the pressure is minimum in the plane formed by the eigenvectors corresponding to these two negative eigenvalues. If there are two negative eigenvalues for A² + B², only the second smallest eigenvalues need to be negative [27], i.e., λ₂ < 0, the vortex zone is the area where its eigenvalue is less than 0. In this method, it is influenced by the tensor A that reflects deformation. That is, the vortex structure described by it will be affected by fluid deformation. Therefore, to determine whether the λ₂ isosurface is a vortex structure, dimensionless helicity H_n is used for coloring [28].

Dimensionless helicity [27] H_n:

$$H_n=\frac{\overrightarrow{\xi }\cdot \overrightarrow{v}}{\left|\overrightarrow{\xi }\right|\cdot \left|\overrightarrow{v}\right|}$$

(16)

In the equation, $\overrightarrow{\xi }$ is the vortex vector, and $\overrightarrow{v}$ is the velocity vector. The positive and negative values of H_n represent the rotation direction of the vortex relative to the flow direction; a value of 1 represents a clockwise longitudinal vortex, and −1 represents a counterclockwise longitudinal vortex [29]. From the expression, it can be seen that the physical meaning of dimensionless helicity H_n is the cosine value of the angle between the velocity vector and the vortex vector [30]. In a vortex, the angle between the velocity vector and vortex vector is very small, even 0. Therefore, the value of H_n can respond to whether the identified λ₂ isosurfaces have the characteristics of the vortex. So, when coloring the isosurfaces of λ₂ with H_n, the vortices can be well distinguished from the shear flow.

In this paper, when the value of λ₂ is taken to be −0.02, the vortex structure in the stator cascade channel at different speeds is shown in Figure 17. As shown in Figure 17, the dimensionless helicity values of λ₂ isosurface vortex structure are mainly 1 and −1, indicating that the structure expressed by the isosurface is a vortex structure, and the expressed vortex structure is basically consistent with the vortex characteristics analyzed through the streamline as described above.

At rotary speeds ranging from 0 to 3000 RPM, the HV structures in the hub and shroud regions are clearly visible. The size of HV in the shroud region is larger than the hub region, which proves that the secondary flow in the shroud region is stronger than the hub region once again. In addition, it can be found that the structural size of the HV_ps is larger than HV_ss, and the rotation directions of the two branches are opposite, which is the vortex information that cannot be analyzed using the streamline method. At low speeds (0–2000 RPM), the size of the SSV is larger than the HV, and both types of secondary flow are enhanced with decreasing rotary speed.

When the rotary speed is between 3000~4000 RPM, the secondary flows in the blade cascade are mainly composed of HV and PV, but the HV has been gradually weakened with the increase of rotary speed. At the same time, the CV structure is also very clear in this rotary speed range. Compared with the HV, the size of the CV structure is smaller, and its flow intensity is weaker. When the rotary speed rises to 5000~5500 RPM, the main secondary flows in the cascade are the PSV and STSV, but there is still HV in the hub region.

3.4. Flow Losses in Stator Cascade Channels

In order to evaluate the flow loss caused by secondary flow in cascade, this paper uses the total pressure drop and total pressure loss coefficient to evaluate. Because the turbine stator has no mechanical power output, it is scientific to use the total pressure drop to evaluate the flow loss. The total pressure drop and total pressure loss coefficient are both used to evaluate the loss of mechanical energy in the fluid. The calculation methods are as follows.

Total pressure drop ($\Delta p$):

$$\Delta p=p_{in}-p_{out}$$

(17)

Total pressure loss coefficient [31] ($c_{pt}$):

$$c_{pt}=\frac{p_{in}-p}{0.5\rho v_{out}^2}$$

(18)

In the equation, P_in is inlet total pressure, P_out is outlet total pressure, P is total pressure at each point in the flow field, ρ is fluid density, and v_out is outlet average velocity.

Figure 18 shows the total pressure loss coefficient and total pressure drop of the stator cascade at different rotary speeds. From the graph, it can be seen that the total pressure loss coefficient and total pressure drop both decrease first and then increase with the increase of rotary speed, and the trend of their changes is consistent. When the rotary speed is between 0–3000 RPM, the lower the speed, the higher both the total pressure drop and total pressure loss coefficient. The reason is that the lower the speed, the stronger the SSV and HV, and this has been analyzed in detail earlier. When the rotary speed is between 3000–5000 RPM, both the total pressure loss coefficient and total pressure drop are the smallest, indicating that the optimal working rotary speed range of the turbodrill is between 3000–5000 RPM. When the rotary speed is between 5000–5500 RPM, both the total pressure drop and the total pressure loss coefficient increase, which is caused by an increase in the strength of the PSV and STSV. Overall, at low rotary speeds, the energy loss in the turbine stator cascade is much greater than that at high rotary speeds, indicating that the flow loss caused by SSV and HV is much greater than that caused by PSV and STSV, so special attention should be paid to the adaptability of the stator cascade to low rotary speeds during design.

4. Experimental Verification

A water-flushing experimental platform for turbodrill planar cascades was established to verify the correctness of the above analysis. The experimental system principle is shown in Figure 19, and the main structure of the experimental platform is shown in Figure 20. The experimental process is as follows: The inlet of the platform is connected to the pump outlet, and when the pump starts, the fluid enters the planar cascade platform, then rushes the stator blades after being guided by the guide blades, and then the flow field details of fluid flow around the blade cascade are captured by using a camera. Simulate the flow field changes of the stator cascade at different attack angles (i.e., different speeds) by replacing the guide blades with different stagger angles. The geometric parameters of the stator cascade used in the experiment are shown in Table 1.

In hydraulic machinery, such as water turbines, water pumps, etc., air bubbles are commonly used to display various vortex structures in the cascade, such as inter-blade vortices [32], tip leakage vortices [33], etc. Therefore, in this experiment, air bubbles were also used to demonstrate the vortex structure in the stator cascade.

During the experiment, a small amount of air was added to the pump suction port. After being stirred by the water pump, the bubbles mixed evenly with the water flow, and the bubble size was appropriate, which can display the vortex structure in the flow field. For the experiments, the stator cascade was made transparent to observe the flow field structure in its cascade channel. In order to simulate the flow field structure of the stator cascade at the design rotary speed (3000 RPM), braking (0 RPM), and idling (5500 RPM), the stagger angle of the guide blades are designed to be 0°, 45°, and −45°, respectively.

As shown in Figure 21, the flow field structure of the stator cascade at the rotary speed of 0 RPM with an inlet attack angle of 45° is shown. The vortices’ structure shown in Figure 21 is consistent with the vortices structure shown in Figure 17 at 0 RPM. In the figure, mark 1 indicates the HV structure, and mark 2 indicates the SSV. The HV is divided into two branches at the leading edge of the blade suction: the HVps flows towards the cascade channel, and the HVss mixes with the SSV after it bypasses the leading edge of the blade.

As shown in Figure 22, the flow field structure of the stator cascade at the rotary speed of 5500 RPM with an inlet attack angle of −45° is shown. The vortices’ structure shown in Figure 22 is basically consistent with the vortices structure shown in Figure 17 at 5500 RPM. In the Figure, mark 1 indicates the PSV and mark 2 indicates the STSV. The two kinds of vortex structures are clearly visible, and no HV is found under this operating condition.

As shown in Figure 23, the flow field structure of the stator cascade at the rotary speed of 3000 RPM with an inlet attack angle of 0° is shown. The vortices’ structure shown in Figure 23, is basically consistent with the vortices structure shown in Figure 17 at 3000 RPM. The two branching structures of HV are clearly visible. In the Figure, mark 1 indicates the HVps and mark 2 indicates the HVss.

Based on the comprehensive analysis of Figure 21, Figure 22 and Figure 23, it can be seen that when the attack angle of stator cascade inflow is 45°, the secondary flow intensity in the cascade is the most intense, followed by −45°, and it is the weakest at 0°, which is consistent with the simulation results. Therefore, the above experiments verify the correctness of the simulation calculation in this paper.

5. Conclusions

In this paper, the stator cascade (S2) of a 160 turbodrill was taken as the research object, and the secondary flow structure and its evolution with changes in operating rotary speed were analyzed in the stator cascade channel. The reasons for the formation and evolution mechanism of the secondary flow were explained in terms of the flow mechanism, and the correctness of the simulation calculation in this paper was verified through experimental research. The following conclusions are ultimately drawn:

(1) In stator cascade channel, the main types of secondary flow include HV, PV, CV, SSV, PSV, and STSV. The types of vortices change with the working rotary speed. When the working rotary speed is lower than the design rotary speed, the secondary flow in the cascade channel is mainly composed of SSV, with HV and PV coexisting. When the working rotary speed is near the design rotary speed, the secondary flow in the stator cascade channel mainly includes HV, PV, and CV; When the working rotary speed is higher than the design rotary speed, the secondary flow mainly includes PSV, STAT, and hub surface HV.

(2) Regardless of any working conditions, the secondary flow intensity in the shroud region is greater than that in the hub region, and its influence on the mainstream of the cascade channel is also greater than that in the hub region.

(3) When the working rotary speed is lower than the design rotary speed, the strength of SSV and the HV increases with the decrease of rotary speed, and the action range of SSV extends from the hub region to shroud region, while the HVps is greater than HVss, and the two branches of HV rotate in opposite directions; When the working rotary speed is higher than design rotary speed, the action range of PSV extends from the 50% span of blade to shroud region, and its strength increases with the increase of working rotary speed.

(4) The flow loss in the turbodrill stator cascade first decreases and then increases with the increase of rotary speed, but the flow loss at the working rotary speed is lower than the design rotary speed and is greater than it is at the working rotary speed that is higher than the design rotary speed. The reason for this is that the secondary flow intensity at the working rotary speed is lower than the design rotary speed and is greater than at the working rotary speed, which is higher than the design rotary speed.