Numerical investigation the shape of groove on tip leakage vortex suppression of a NACA0009 hydrofoil

To investigate the effect of different bionic grooves on the tip leakage vortex, four bionic grooves with different shapes were designed, which are semicircular, rectangular, triangular and trapezoid. The streamline, pressure, vortex strength and turbulence kinetic energy were analyzed to clarify the suppression mechanism. When the groove is arranged, the velocity of vortex center and the area of high-intensity vortex area decrease. The vortex strength in the triangular groove is higher than other groove shapes. The pressure in the center of the tip leakage vortex core increases. The low pressure area reduces obviously, which is the smallest in the trapezoidal groove condition. The turbulent kinetic energy in the main flow field increases, but decreases in the tip clearance. The high turbulent kinetic energy region is divided into small pieces by the bionic groove, which is the smallest in the rectangular groove condition. The bionic groove leads to the decrease in lift-to-drag ratio, which is the largest in the semicircular groove and smallest in the rectangular groove condition.


Introduction
Semi-open centrifugal pump is an important tool for fluid conveying in engineering.It is widely used in agriculture and chemical industry because of the simple structure, convenient processing and cleaning [1] [2].However, due to the particularity of the structure, there is a certain gap between the blade tip and the shroud.The tip leakage flow in the tip clearance incudes complex vortex structures such as tip leakage vortex, clearance separation vortex and induced vortex, which increase the hydraulic losses and decreases the energy performance of the unit [3] [4].Therefore, the researchers focus on investigate the flow mechanism and inhibition methods of tip leakage flow.
Li et al. [5] experimentally investigated the effect of tip clearance size and flow rate on the flow rate around the tip of an axial turbomachine rotor and found that the generation position and movement trajectory of the clearance vortex system change depending on the clearance width, which in turn affects the clearance turbulence losses.Through numerical simulation, Wu et al. [6] found that the C-shaped groove can effectively reduce the flow rate of tip leakage flow, thereby inhibiting the generation of high-speed leakage flow.Gao et al. [7] studied the influence of centrifugal compressor performance under different tip clearances, and found that zero clearance is not the best working condition, and the flow loss is minimized when the tip clearance is 0.9% of the impeller outlet height.Feng et al. [8] studied the influence of changing the tip clearance on clearance flow and leakage vortex at large angle of attack by numerical simulation method, and found that the development of leakage vortex along the hydrofoil with small clearance is more complicated and unstable.Liu et al. [9] studied the influence of counter-rotating compressor under different tip clearance based on numerical methods, and the results showed that the total pressure ratio and efficiency of the compressor were inversely proportional to the width of the tip clearance, and there was an optimal clearance combination for the counter-rotating compressor, that is, the tip clearance of rotors R1 and R2 was 1.0τ and 0.5τ (τ represents the design clearance), respectively.Under this condition, the peak efficiency and surge margin increase by about 0.62% and 6.9% respectively compared with the design clearance.Jia et al. [10] used numerical simulation to study the aerodynamic performance.By changing the clearance, he found that the blade with certain clearance was more efficient than the blade without clearance.By comparing the semi-open centrifugal pumps with low rotation speeds, Jia et al. [11] and Boitel et al. [12] found that the larger the tip clearance, the faster the lift and efficiency will decrease, but the hump phenomenon at low flow conditions is improved.Wang et al. [13] carried out experimental analysis on the flow field of compressor cascade with tip clearance.The results show that the relative motion state of top wall and side wall has no significant effect on the leakage flow of the clearance and the flow characteristics of the hydrofoil tail flow field.Cheng et al. [14] and Wu et al. [15] analyzed the shape of secondary leakage vortex under different tip clearances, and analyzed the influence of tip clearance leakage on the flow structure of centrifugal pump.
To study the effect of different groove shapes on the tip leakage vortex, four bionic grooves with different cross section shapes were designed by numerical simulation.The effect of different cross section shape bionic grooves on tip leakage was compared by comparing the flow line distribution, pressure coefficient distribution, vortex intensity and turbulent kinetic energy of hydrofoil, which provided theoretical data for the study of tip leakage vortex.

Physical model
In this paper, the same geometry of winged NACA0009 as in the visualization experiments of Dreyer et al. [16][17] was used to perform experiments on NACA0009 in a rectangular flume, as shown in Figure 1.The length L of the flume is 750 mm, the height b and the width a are both 150 mm, the chord length c of the hydrofoil is 100 mm, the blade spread height h is 150 mm, the maximum thickness tm is 9.9 mm, the angle of attack α is 10 degrees, the distance between the front edge of the hydrofoil and the inlet is 200 mm, Define the dimensionless tip clearance dimension τ as τ= /tm, and τ is selected as 0.2 for follow-up research

Geometric parameters of different bionic grooves
The structure and position of the impeller are shown in Figure 2. The bionic grooves are distributed on the clearance wall above the wing.The bionic grooves have a length x of 140 mm and a width y of 58 mm, which are 1.4 times and 0.58 times of the chord length c of the wing, respectively.The cross sections of the bionic grooves are rectangular, semicircular, isosceles triangular and isosceles trapezoid, the widths s of the four models are all equal to 2 mm, including the diameter of the semicircular, the bottom of the trapezoid and the bottom of the triangular, the height h and the radius r of the model are all equal to 1 mm, the upper bottom t of the trapezoid is 1 mm, and the distance k between each bionic groove and the width s are equal to 2 mm, as shown in Figure 2.

Numerical method
In this paper, ANSYSCFX16.0 software is used for numerical simulation.According to the method of Smirnov et al. [18], not only the local rotation characteristics and curvature effects are considered, but also the turbulent flow model modified by SST-CC curvature can be used to accurately predict the flow separation phenomenon under the pressure gradient, and the turbulent kinetic energy generation term in the turbulent flow model is modified, and the sensitivity of the convection curve curvature and the rotation motion is enhanced, so that more accurate leakage flow can be captured.
In the formula, k and ω are the turbulent kinetic energy and the turbulent dissipation frequency respectively, F 1 and F 2 are mixed functions, P k is the turbulent kinetic energy generation term, α、β、 The setting of boundary conditions is the same as that of Dreyer's experiment, that is, the inlet is set as a uniform speed inlet, the W  size is 10 m/s, the turbulence strength is set as 1%, the outlet is set as a static pressure outlet, ensure that the inlet pressure P  is 1 atm, and all wall surfaces are set as nonslip boundary conditions.

Simulation verification
According to the experimental results of NACA0009 hydrofoil by Dreyer et al. [16][17], the results of numerical simulation are compared with the experimental results.The vortex structure is defined by Q criterion, where Q=5.0×10 6 s -2 .It can be seen that the numerical simulation results and the experimental results have a high degree of consistency, the trajectory of the leakage vortex is similar to the numerical simulation results, and the magnitude of the induced vortex is basically the same, and the angle between the vortex and the hydrofoil is basically the same as the experiment.Figure 4 shows the comparison between the axial velocity cloud map of the hydrofoil z/c=1 section obtained by the experiment and the axial velocity cloud map of the same section obtained by numerical simulation.The numerical simulation results of the mainstream area are close to those of the experiment, the position of the low-speed area is basically the same, the size and position of the low-speed area are Bionic groove position bionic groove section geometry similar, the comparison results are basically similar, and the numerical simulation results are credible.

Streamline distribution
In order to investigate the effect of bionic grooves of different section shapes on tip leakage vortex, the flow line distribution of different models is shown in Figure 5. Firstly, it is obvious that the velocity of the original working condition leakage vortex is larger at the initial position, and as the leakage vortex develops downstream, the leakage vortex gradually moves away from the hydrofoil suction surface and the velocity gradually decreases.Compared with the other four with bionic grooves, the angle between the leaked vortex and the wing shape is significantly reduced, the disturbance area is reduced, and the speed of the vortex center is reduced.The flow field is affected by the four bionic grooves.Compared with the other three, the flow line shape of the triangular grooves is closer to the original working condition, and the arc of the flow line is smaller and smoother; The flow line of the semicircular groove has larger radian.

Vortex intensity distribution
For further analysis, Figure 6 shows a cloud view of the vortex intensity in 10 sections from the leading edge to the trailing edge of the hydrofoil.The area of the high-intensity vortex area is obviously reduced after adding the bionic groove, and is particularly obvious by comparing the sections 2,3 and 10 ; Compared with section 4, it is found that the vortex intensity of the induced vortex area increases obviously after adding the bionic groove, the vortex intensity of the induced vortex in the semicircular groove is the largest, and the vortex intensity of the induced vortex in the triangular groove is the smallest.After the bionic groove is added, a high-strength vortex appears in the bionic groove, the strength of the semicircular groove is the largest, the strength of the triangular groove is the smallest, and the section area of the bionic groove is proportional to that of the bionic groove.At sections 1 and 2, the initial position of the leakage vortex, the leakage vortex trajectory is basically the same under five conditions.Starting from section 3, after adding different bionic grooves, the y-axis coordinate of the leakage vortex trajectory is obviously reduced, that is, the leakage vortex is closer to the wing-shaped suction surface, and the influence of different bionic grooves on the leakage vortex trajectory is basically the same.In section 5, it can be seen that the vortex strength in the center of the triangular groove vortex is higher than that of the others.

Pressure coefficient distribution
The pressure distribution of different sections under the original working condition and the bionic groove working condition are selected for analysis.The definition of the pressure coefficient is defined as follows: where   represents the pressure at the mesh node and  ∞ represents the outlet pressure.As shown in Figure 7, the center pressure coefficients of the separation vortex and the leakage vortex are relatively low.This low pressure characteristic will induce the phenomenon of vortex cavitation.As the axial position increases downstream of the leakage vortex, the pressure coefficient Original Semicircular groove Rectangular groove Triangular groove Trapezoidal groove of the vortex core gradually increases.Compared with the original working condition, the pressure coefficient of the center of the leaked vortex core is improved by the bionic groove, but the area of the same pressure coefficient is reduced, and the section of 10 is the most obvious.In section 7, the trapezoidal and rectangular grooves have a smaller high-pressure area.low-pressure area is obviously reduced, and the area of the triangular groove and the trapezoidal groove is not significantly different.The area of the rectangular groove is slightly larger, and the area of the low-pressure area of the semicircular groove is the largest, indicating that the inhibition effect of the triangular groove and the trapezoidal groove is obvious, and the inhibition effect of the semicircular groove is the worst.

Turbulent kinetic energy distribution
Figure 9 shows the distribution of turbulent kinetic energy in 10 sections of the hydrofoil.It can be seen that under the action of blade tip clearance, tip leakage vortex occurs on the suction surface of the hydrofoil, and the turbulent kinetic energy increases.By comparing the 10 sections, it can be found that the triangular groove has the largest turbulent kinetic energy, while the rectangular groove has the smallest.However, in sections 3 and 4, it can be seen that the turbulent kinetic energy in the tip clearance decreases after the bionic groove is added.10, it can be seen that the turbulent kinetic energy on the tip surface of the original working condition blade starts from the leading edge and spreads along the hydrofoil in a fan shape, with the strongest turbulent kinetic energy in the middle of the hydrofoil and then gradually decreases, and there is a small area in the middle of the beginning part where the turbulent kinetic energy is lower than the surrounding area.After adding the bionic groove, it can be clearly seen that the turbulent kinetic energy is reduced, the area is reduced, and the turbulent kinetic energy generated by the tail wing has a tendency to be dispersed into two parts; and the high-energy areas of the semicircular groove and the rectangular groove are dispersed into two parts; and the trapezoidal

Comparison of lift-to-drag ratio
As can be seen from Figure 11, after the bionic groove is added, the lift force decreases, while the drag increases, and the lift-drag ratio also decreases, indicating that the addition of the bionic groove will reduce the lift-drag ratio to a certain extent, thereby reducing the influence of leakage vortex.The lift and drag of the triangular groove after increasing the bionic groove are the largest.The semicircular groove has the least drag and the rectangular groove has the least lift.By contrast, the rectangular groove has the smallest lift-to-drag ratio, with a decrease of 3.10% compared with the original working condition, and the semicircular groove has the largest lift-to-drag ratio, which is 2.75% smaller than the original working condition.

Conclusion
The influence of groove shapes on the tip leakage vortex is studied in this paper.Through qualitative and quantitative analysis of the pressure, velocity and vortex structure, the detailed conclusions are as follows: (1) The bionic groove can decrease the velocity of the vortex center and the area of the highintensity vortex region.The vortex strength in the triangular groove is higher than that in other groove shapes.
(2) The pressure in the center of the tip leakage vortex core increases when the bionic groove was arranged.The low pressure area near the leading edge of the hydrofoil reduces which is the smallest in the trapezoidal groove and largest in the semi-circular groove condition.
(3) The high turbulent kinetic energy region near the tip clearance is divided into small pieces by the bionic groove, which is the smallest in the rectangular condition.
(4) The bionic groove leads to the decrease in lift-to-drag ratio, which decrease greatly in the semicircular groove condition.

Figure 2 .
Figure 2. Schematic diagram of the position and structure of the bionic groove.

Figure 3 .
Figure 3.Comparison of experimental and numerical simulation of vortex structure.

Figure 4 .
Figure 4. Comparison of axial velocity of z/c=1 section.

Figure 5 .
Figure 5.Comparison of flow lines under different working conditions.

Figure 6 .
Figure 6.Vortex intensity clouds of different cross-sections.

Figure 7 .
Figure 7. Pressure clouds of different sections.

Figure 8 .
Figure 8. Distribution of C  on tipwall.Figure8shows the pressure coefficient distribution on the tip surface of the hydrofoil.It can be seen that the original working condition has a low-pressure area at the front edge of the hydrofoil and spreads in two directions along the axial direction.After installing the bionic groove, the area of the

Figure 9 .
Figure 9. Distribution of turbulent kinetic energy in different sections.According to Figure10, it can be seen that the turbulent kinetic energy on the tip surface of the original working condition blade starts from the leading edge and spreads along the hydrofoil in a fan shape, with the strongest turbulent kinetic energy in the middle of the hydrofoil and then gradually decreases, and there is a small area in the middle of the beginning part where the turbulent kinetic energy is lower than the surrounding area.After adding the bionic groove, it can be clearly seen that the turbulent kinetic energy is reduced, the area is reduced, and the turbulent kinetic energy generated by the tail wing has a tendency to be dispersed into two parts; and the high-energy areas of the semicircular groove and the rectangular groove are dispersed into two parts; and the trapezoidal

Figure 10 .
Figure 10.Distribution of turbulent kinetic energy on the surface section of blade tip.