Numerical simulation of the unsteady flow mechanism and dynamic characteristics around cascade

Research on the flow mechanism and dynamic characteristics of a cascade comprising three NACA hydrofoils was conducted under varying cavitation numbers and angles of attack. This study serves as a theoretical foundation for the high-speed miniaturization of fluid machinery. Drawing upon prior findings on individual hydrofoils, comparisons were made regarding the cavitation characteristics of the cascade versus a single hydrofoil, emphasizing pressure and velocity distributions in the flow field. Results suggest that while the cascade consists of three hydrofoils, their interactions profoundly influence the flow field. The cavitation phenomena in the cascade diverge considerably from that of a single hydrofoil. Multiple hydrofoils cause a low-pressure region to appear in the middle of the cascade, making it more susceptible to cavitation flow compared to a single hydrofoil. However, interactions among the hydrofoils constrain the expansion of this low-pressure zone, rendering the cavitation flow in the cascade’s middle layer more stable. As the angle of attack rises, the cavitation flow around the cascade undergoes notable changes, with varying cavitation flows at different cascade positions. The upper layer experiences cavitation flow akin to that of a single hydrofoil. In contrast, the middle layer sustains a thin, weakly periodic cavitation flow.


Introduction
Cavitation describes a multifaceted phase transition process triggered when certain factors cause localized pressure drops within a fluid.This pressure reduction prompts the fluid to shift from a liquid to a gas phase, leading to the formation and subsequent evolution and collapse of bubbles [1].In hydraulic engineering, cavitation emerges as a persistent challenge, dramatically curtailing the operational lifespan of fluid machinery and resulting in significant losses.Underwater fluid machinery operation induces expansive bubble formations on the machinery's surface due to cavitation.These bubbles incessantly form, evolve, and collapse, spawning various adverse consequences, including cavitation erosion, noise, and vibrations.Particularly, cavitation erosion results when collapsing cavitation bubbles repeatedly subject machinery walls to high-pressure impacts.This phenomenon leads to material fatigue wear and potential surface erosion, often referred to as cavitation or, briefly, cavitation erosion.With fluid machinery research progressively steering towards high-speed miniaturization, impeller designs must deliver high loads, tackle cavitation issues, and simultaneously enhance efficiency.
During the 1950s and 60s, extensive experiments sought to address cavitation.In 1985, British engineer Parsons pioneered the first water tunnel apparatus to scrutinize propeller corrosion causes.By increasing the propeller's disk surface, facilitating faster ship speeds.With continuous technological

Numerical simulation method
For the computational analyses presented in this study, the commercial software package Fluent was employed.Our simulation framework predominantly integrates the homogeneous flow model, the Shear Stress Transport (SST) turbulence model, and the Zwart's cavitation model.

Control equations a．Continuity equation equations () 0
(1 ) Where m  is the density of the mixed medium, v  is the density of the gas phase, t is the time, v  is the volume fraction of the gas phase, i u is the velocity component, and i x is the Cartesian coordinate ( i = 1,2,3... ), l  is the density of the liquid phase.b．Momentum equation (1 ) ) Where m p is the pressure,  is the dynamic viscosity of the mixed medium, t  is the turbulent viscosity of the mixed medium, ii  is the Cronet number (when ij = , R is the mass transfer rate between the gas phase and the liquid phase, e R is the steam generation rate, and c R is the steam condensation rate.

Turbulence model
The selection of a turbulence model typically considers two aspects: firstly, the specific flow state of the problem in question; and secondly, the choice of the Reynolds number computation model based on the position of the object's boundary layer grid.The k  − turbulence model can effectively simulate turbulent flow phenomena in the field regions distant from the wall.However, it struggles with boundary issues under pressure gradients.Conversely, the k  − model excels at simulating boundary layer issues under various pressure gradients, but is sensitive to the incoming flow's turbulence parameters.To capitalize on the strengths of both models, Menter [22][23] Where

Cavitation model
Currently, cavitation models can generally be categorized into two types: those based on positive pressure fluid equations of state and those based on mass transport.Zwart et al. [24] introduced a modification to the existing cavitation models proposed by Kubota and Gerber.In their work, Zwart et al. made simplifications by disregarding the surface tension and second-derivative terms associated with cavitation.They assumed that all bubbles within the system were of uniform size.Additionally, they proposed that as the volume fraction of vapor increased during the liquid vaporization process, the density at the nucleation position would decrease accordingly, thus influencing the ultimate cavitation outcome.Consequently, Zwart et al. revised the volume fraction term in the equation for mass cavitation rate to replace the original bubble condensation equation.The resulting formula is as follows: Where nuc  is the cavitation nucleus volume fraction (usually taken as ), cond F is the condensation experience coefficient (usually taken as 0.001), and vap F the gasification experience coefficient (usually taken as 50).This model is simple and widely used, so this paper adopts the Zwart cavitation model.

Numerical setup and description
The cascade used in this study consists of three NACA-0009 hydrofoils.The upper and lower surfaces of this hydrofoil are symmetrical, making the entire cascade a symmetrical structure.The middle hydrofoil of the cascade is influenced by both the upper and lower hydrofoils.Not only does this study compare the differences in cavitation phenomena between the cascade and a single hydrofoil, but it also explores the impact of the entire cascade on the cavitation of the upper hydrofoil.To avoid redundancy, if specific parts of the hydrofoil are not mentioned in the text below, it refers to the middle hydrofoil of the cascade.Figure 1 shows the computational domain of the cascade, surrounded by eight monitoring points to detect changes in pressure and velocity at different positions over time.The chord length C of the hydrofoil is 100mm, the inlet flow velocity  0 is 10m/s, with a turbulence intensity of 1%.Both the hydrofoil surface and the fluid domain boundary adopt adiabatic, no-slip wall conditions, while the boundaries on either side of the domain are symmetrical.The boundary conditions are velocity inlet and pressure outlet, and the cavitation number in the domain can be adjusted by altering the pressure.The main dimensionless parameters used in the article are: Cavitation number:   The specific locations of the eight monitoring points on the cascade are as shown in Table 1.Points A, B, C, and D are located 0.1C away from the leading edge of the cascade, while points E, F, G, and H are positioned 0.9C away from the leading edge.Grid division generally falls into two types: structured grids and unstructured grids.In this study, ANSYS ICEM software was utilized for grid partitioning.Since cavitation phenomena predominantly concentrate on the hydrofoil surface, the grid division process necessitates localized refinement.It is permissible to moderately reduce the grid count in areas unrelated to the flow field, thus conserving computational resources.Figure 2 illustrates the grid partitioning situation for the cascade and the closeup view of the local refinement.During grid division, the computational domain was segmented into two sections, with a circular region established in the middle, facilitating subsequent adjustments for angle of attack calculations.The computational time step was set at 0.00005s, with a total duration of 1s, and a solution accuracy of 10 −5 .To avoid the influence of grid count on the flow field around the cascade, a grid independence check is essential.A lower grid count and quality can significantly degrade the precision and accuracy of computational results, potentially causing model computation failures or errors.Conversely, an excessively high grid count and quality increase the computational load and duration, leading to a reduced computational efficiency.This not only risks wasting computer resources but can also introduce cumulative errors.Taking the cascade with an angle of attack of 0 ° under the operating conditions of a flow velocity of 6m/s as an example, this study conducted a grid independence discussion.As Table 2 indicates: as the grid node count surpasses 300W, the lift coefficient gradually stabilizes.Once the grid count exceeds 400W, the lift coefficient remains largely unchanged.Therefore, the chosen total grid count for the computational domain in this research was approximately 400W.

Cavitation around cascade
In this section, the numerical simulation method proposed above will be used to conduct an in-depth analysis of the unsteady cavitation flow around the 0 angle-of-attack cascade, focusing on the pressure and velocity distribution in the cascade flow field, and explaining the cavitation generation mechanism of the cascade.The cavitation flow field around a single hydrofoil is compared.Figure 3 lists the cavitation evolution process with time around the 0 angle of attack cavitation when the cavitation number is 0.6  = , and the Reynolds number is 5 Re 6.8 10 =.The colored areas are holes, with red representing higher gas content and blue representing lower gas content.The cavitation around the 0 angle-of-attack cascade is weaker, and the cavity area is smaller.0 t will be defined as the initial moment of the cavitation process.At 0 60 t t ms =+ time, cavitation does not occur in the upper and lower layers of the cascade, and cavitation has occurred in the middle layer of the cavitation.As time increases, the cavity area in the middle layer gradually increases.There is still no cavitation in the layer area.At that time, the cavity reached the maximum area, and the shape of the cavity was still elongated, distributed symmetrically, and developed to the tail of the hydrofoil.Furthermore, the tail of the hole begins to collapse, and the area of the hole gradually decreases.At 0 100 t t ms =+ , the area of the hole decreases to the minimum value, but it does not disappear completely, and only a new cycle of growth and collapse process begins.The cavitation flow field around the entire cascade is in the transition stage from sheet-like cavitation to cloud-like cavitation, and the cavitation will show weak periodic characteristics.When the cavitation collapses, the cavitation shrinks from the tail to the hydrofoils head.  .Comparing the pressure cloud diagrams at each time in Figure 4, it can be seen that due to the mutual influence between the three hydrofoils of the cascade, a lowpressure area appears in the middle of the cascade, and it continues to expand, and the area of the cavity also continues to expand.Different from around a single hydrofoil [25], although cavitation flow is difficult to occur around a 0 single hydrofoil at the angle of attack, due to the interaction between the hydrofoils of the cascade, a low-pressure area appears between the hydrofoils of the cascade, and then Cavitation occurs in these regions.It can be seen from Figure 4 that the uppermost and lowermost layers of the cascade are similar to the flow field around a 0 single hydrofoil, and the hydrofoils head pressure is relatively low, not lower than the saturated vapor pressure, and there will be no cavitation.Comparing the velocity cloud images of each frame in Figure 4, it can be seen that the interaction between the hydrofoils of the cascades increases the speed of the water body in the middle of the cascades, and the tail of the hydrofoils between the cascades is also a low-speed area.When 0 40 t t ms =+ , that is, when the cavity area is about to reach the maximum, there is a relatively obvious reverse flow at the tail of the hydrofoil, which is consistent with the research of Zhang Yao [26].Comparing the pressure distribution diagram at this time, it can be seen that there is a large pressure gradient at the tail of the hydrofoil and reverse flow appears.As the pressure field continues to shrink, the reverse jet continues to move toward the head of the hydrofoil.The pressure in the middle of the cascade is low, but the change gradient is small, the area of the cavity is not large, and the reverse jet flow has not been able to affect this area, so the cavity has not collapsed and is basically in a stable state.
In Figure 5, four color lines are used to describe the pressure versus time curves at the cascade A, B, C, and D monitoring points respectively.It is obvious that the pressure change laws of the upper layer and the lower layer are similar, and the pressure change laws of the two monitoring points at the head of the middle hydrofoil are similar.However, the pressure curves of the middle layer and the upper and lower layers are relatively different.The pressure fluctuations of the hydrofoil heads of the upper and lower layers are relatively severe, while the pressure on the hydrofoil heads of the middle layer is relatively stable, and there is no obvious fluctuation with time.The interaction between the wings makes the flow field in the middle of the cascade more stable.The pressure fluctuations at points A and D are periodic, but because the pressure drop at the head of the hydrofoil is limited under the 0 angle of attack, cavitation does not occur, which is also difficult for a 0 single hydrofoil to generate cavitation at the angle of attack reason.The pressure at points B and C is stable, and the pressure value is always higher than that at points A and D, and no cavitation occurs.
The two images in Figure 6 respectively list the pressure and velocity curves at the four detection points at the tail of the cascade with time.Figure 6(a) is the time course of the pressure at the four monitoring points E, F, G and H. Comparing the four curves, it can be seen that the pressure data at the monitoring point at the tail of the cascade is just opposite to that at the head, the pressure fields at the upper and lower points E and H are relatively stable, and the pressure fields at points F and G in the middle layer fluctuate periodically, and the two points are the lowest The pressure is significantly lower than the saturated vapor pressure of the water body, which means that cavitation flow will occur at the tail of the hydrofoil in the middle of the cascade.Comparing the cavitation flow field around the middle hydrofoil of the cascade and the cavitation flow around a single hydrofoil [27], it can be seen that the pressure fluctuation of a single hydrofoil starts from the head and gradually develops towards the tail, while the head of the middle hydrofoil of the cascade Stable, the tail pressure field fluctuates.Combining the pressure cloud diagram in Figure 5 and the pressure curve in Figure 6(a), it can be seen that the state of the pressure field around the hydrofoil in the middle of the cascade is high pressure-low pressure-tail pressure fluctuates continuously.The interaction between the hydrofoils of the cascade makes the pressure field of the cascade tend to be stable, the cavity is collapsed but not completely collapsed, and the head and tail of the cascade are in two completely different states.In conclusion, the cavitation flow around the cascade is different from the flow field around a single hydrofoil.
Figure 6(b) is the time-varying curve of the velocity at four monitoring points E, F, G and H at the tail of the cascade.Comparing the curves in Figure 6(b), it can be seen that with the development of cavitation, the flow velocity of the water body at the tail of the hydrofoil in the middle of the cavitation is also decreasing.direction jet.Compared with the pressure curve pulsation diagram in Figure 6 (a), the pressure and velocity are at the lowest value at the same time, with the same periodicity.However, compared with the single hydrofoil cavitation phenomenon [27], the pressure field in the middle of the cascade is relatively stable, and the duration of the return jet is very short.time, the cavity develops to the maximum extent, and starts to move toward the head, and collapses continuously.At 0 90ms tt =+ time, the hole collapse is complete.When 0.7  = , the flow field around the cascade is more stable, the degree of cavitation is weakened, and the development 0.8  = period is shorter.At 0 20ms tt =+ time, the cavitation around the cascade was weaker and was in the primary cavitation stage: at 0 50ms tt =+ time, cavitation flow began to appear in the middle layer of the cascade, but it was extremely weak; at that time, the cavitation had been fully developed, but the length of the cavitation was much smaller than that of the cavitation 0.6  = length; at 0 80ms tt =+ time, the cavity around the cascade completely collapses and disappears.As the cavitation number increases, the cavitation flow field around the cascade weakens obviously, but the flow field in the middle of the cascade is relatively stable, the cavitation changes are not obvious, and the primary cavitation flow state remains basically stable.As shown on the left side of Figure 8, at a specific time, the flow field around the cascade was transitioning from cloud cavitation to super-cavitation, with a severe degree of cavitation that lacked clear periodicity.When comparing this to the flow field around the NACA66 single hydrofoil, it becomes evident that the critical state only manifests in the flow around the single hydrofoil at a particular point.This suggests that, in comparison to a single hydrofoil, the flow field around the cascade enters the super-cavitation stage more rapidly.However, owing to the mutual influence among the three hydrofoils in the cascade, the flow field between the hydrofoils tends to stabilize.Furthermore, the super-cavitation phenomenon for the hydrofoil in the cascade's middle layer is relatively weak.The right side illustrates the change in dimensionless cavity length over time under varying cavitation numbers for the cascade.At this instance, the cavitation length of the single hydrofoil is zero, so it isn't depicted in the graph.Nonetheless, it is evident that as the cavitation number declines, the cavitation length of   = 0.7 and 0.8.included in the image.From the above discussion, it can be seen that cavitation does not occur at monitoring points A, B, and F, which are only used to discuss pressure changes in the middle of the cascade, and the head and tail of the upper surface are selected for single hydrofoils.

Comparison of different cavitation forms around the cascade
Analysis of the data at the three monitoring points A, B, and F shows that when the cavitation number increases, the pressure at each monitoring point increases significantly, the cavitation period becomes shorter, and the cavitation collapse time moves backward.When 0.8

 =
, the pressure curves of each monitoring point are relatively gentle, and the cavitation is also extremely weak at this time.Different positions of the cavitation cavitation are also affected by the cavitation number.Point B has the least impact, because the hydrofoil interaction makes the watershed in the middle of the cavitation stable.When the cavitation number changes, the middle layer of the cavitation remains relatively stable and does not change much.The pressure at point F is most affected by cavitation.It can be seen from the gas phase cloud image that at this time, the cavitation of the cavitation develops toward the tail of the hydrofoil, and the cavitation area is a low-pressure area, and there will be severe pressure fluctuations with the periodic development of the cavitation.Therefore, The pressure curve at point F fluctuates violently.As the cavitation number decreases, the pressure at point A of the cascade head gradually fluctuates significantly.
Comparing the pressure changes between the single hydrofoil and the cascade at 0.6  = , it can be seen that the upper surface of the upper hydrofoil of the cascade is not affected by other hydrofoils, so its cavitation phenomenon is similar to that of the single hydrofoil.However, it can be seen from monitoring point A that the pressure on the upper part of the cascade will be lower than that of a single hydrofoil, because the overall cascade is larger than that of a single hydrofoil, which will increase the flow velocity in the watershed, thereby making the head pressure lower , so cavitation is more likely to occur at the head of the cascade.From the data of monitoring point F, it can be seen that the pressure at the tail of the cascade is higher than that of a single hydrofoil.The middle structure of the cascade can be regarded as a tapered and divergent pipe, the flow velocity at the head increases, which reduces the pressure, and the flow velocity at the tail slows down, increasing the pressure.The pressure in the middle is the lowest, and when it is lower than the saturated vapor pressure, cavitation will occur, and the cavitation will develop towards the tail and collapse, and the collapse will cause the pressure at the tail to fluctuate violently.Therefore, the cavitation in the middle layer of the cascade is stable, and there will be a weak periodic cavitation collapse process in the tail.

Effect of angle of attack on cavitation around cascade
The angle of attack has a great influence on the degree of cavitation around the cascade, so this paper will discuss the cavitation flow around the cascade at different angles of attack, and compare the differences in the cavitation flow at different positions of the cascade Figure 10 lists the time-varying course of the water vapor nebula flow around the cavitation 0.8  = when the angle of attack is 6 and 8 .t0 is the starting moment of cavitation, which is different from the 0 cavitation flow around the 6 angle-of-attack cascade in Fig7.There will be two stages of cavity growth in the process.The upper layer of the cascade is greatly affected by the angle of attack, flow separation occurs on the suction surface, and a low-pressure area appears at the head, resulting in cavitation flow, which is similar to the cavitation flow of a single hydrofoil [27] .Flow deflection will also occur on the suction surface of the hydrofoil in the middle of the cascade, but because the hydrofoil on the upper layer of the cascade will bear the flow deflection, the pressure drop speed here will slow down, and severe cavitation flow cannot occur.As shown in Figure 10( time, the cavitation in the upper layer and the middle layer of the cascade both began to collapse, and collapsed to the minimum.In the cavitation collapse stage, the flow field in the middle layer of the cascade is stable, the cavitation collapse process is gentle and will not collapse completely, and the cavitation in the upper layer of the cascade collapses to the head of the hydrofoil until it disappears.In short, when the angle of attack is 6°, the cavitation around the cascade is asymmetric, the degree of cavitation is different at different positions, and the periodicity of cavitation is also different. It can be seen from Figure 10(b) that the cavitation flow around the 8 angle-of-attack cascade is basically in the super-cavitation stage, the cavitation flow field fluctuates violently, and the cavitation period is shortened.At 0 tt = time, cavities began to appear in all positions of the cascade, and the cavity area continued to shrink from the upper layer to the lower layer of the cascade.At 0 10 t t ms =+ time, the cavitation flow field fluctuated violently around the hydrofoil on the upper layer of the cascade, reverse jets appeared at the tail of the cavitation, and cavitation also began to appear at the head of the cascade middle layer, but it was still in a slender state.At 0 20 t t ms =+ time, the upper cavitation develops to a maximum extent and begins to collapse.The growth and shedding process around the cascade cavity is obviously accelerated, and the development cycle is greatly shortened.At time, the upper and middle hydrofoils of the cascade were completely enveloped by the cavity, forming a stable super-cavitation flow field structure, and the suction surface of the lower hydrofoil of the cascade was also enveloped by the cavity.Basically in the super-cavitation stage.Comparing the above analysis, it can be seen that the angle of attack has a great influence on the cavitation flow around the cascade.When 0.8  = , the cavitation flow around the cascade at the 0 angle of attack is the primary cavitation stage, and the cavitation flow around the cascade at the 6 angle of attack is the cloud cavitation stage.The cavitation flow in cascade at 8 angle of attack is the stage of super-cavitation.time, the low-pressure area on the upper layer of the cascade rapidly expanded to the tail of the hydrofoil, and cavitation flow appeared, but the low-pressure area around the middle-layer hydrofoil continued to shrink.The analysis shows that due to the mutual influence between the hydrofoils, the low-pressure area cannot expand.There will be no cavitation flow in the region, so the cavitation flow in the middle layer of the cascade only appears at the tail of the hydrofoil.Combined with the distribution of the velocity field, it can be known that the high-speed area around the upper hydrofoil of the cascade continues to expand, the low-speed area at the tail continues to shrink, and the flow velocity gradually decreases, while the middle layer of the cascade has flowed opposite to the direction of the incoming flow, making the cavity around the middle hydrofoil gradually collapsed.At 0 70ms tt =+ time, the low-pressure area around the upper hydrofoil of the cascade developed to the tail of the hydrofoil, and the pressure field gradient in the upper layer of the cascade was relatively large, resulting in a violent jet flow in the opposite direction.Until 0 100ms tt =+ then, the low-pressure area on the upper layer of the cascade was relatively large without significant changes, while the low-pressure area around the middle hydrofoil continued to develop.Combined with the velocity cloud image, it can be seen that the reverse jet around the upper hydrofoil of the cascade continued to expand toward the head of the hydrofoil.At the same time, the strength of the reverse jet is constantly weakening.At 0 120ms tt =+ time, the cavitation in the upper layer of the cascade completely collapsed, and the low-pressure area narrowed toward the head.It can be seen from the velocity diagram that at this time, a stronger reverse jet flow appears on the upper layer of the cascade and develops towards the head of the hydrofoil, where the cavity quickly collapses toward the head of the hydrofoil until it disappears.The cavity around the middle hydrofoil remains basically stable.
It can be seen from Figure 11(c) that the area and development speed of the low-pressure area around the 8 angle-of-attack cascade are significantly larger than the low-pressure area around the 6 angleof-attack cascade, and the low-pressure area merges at the tail of the hydrofoil, gradually enveloping the cascade.It can be seen from Figure 11(d) that the velocity field of the cascade with the 8 angle of attack is more stable, and the intensity of the reverse jet is lower than that of the cascade with the 6 angle of attack.At 0 80ms tt =+ time, the reverse jet flow gradually disappeared.According to analysis, the low-pressure area has completely wrapped the cascade, and the pressure gradient at the tail of the cascade is not obvious, so a strong reverse jet flow cannot be formed.However, part of the reverse flow will be formed at the boundary of the low-pressure area downstream of the cascade.It can be seen that the angle of attack has a great influence on the flow field around the cascade, especially the flow field in the upper layer.The low pressure area at the head develops rapidly to the tail, causing cavitation flow in the upper layer of the cascade, and as the angle of attack increases, the low pressure area The rate of development is faster, making the cavitation cycle faster.However, the flow field in the middle layer of the cascade is relatively stable, increasing the angle of attack reduces the pressure in the middle layer of the cascade, the pressure fluctuation changes are not obvious, the velocity field changes little, and the cavitation periodicity is not prominent., the vortex expands again, and the reverse jet flow becomes more intense, forming a larger vortex area, which continues to expand to the front end of the hydrofoil, and the hydrofoil cavity on the upper layer of the cascade completely collapses.Comparing the images of each frame in Figure 12(b), it can be seen that a large-scale vortex structure quickly appeared at the tail of the upper hydrofoil around the 8 angle-of-attack cascade, and gradually moved to the downstream of the cascade, and super-cavitation flow appeared.When the flow field stabilized, the flow lines around the cascade also  C, the surface pressure of the upper hydrofoil increases sharply, and there will be a large pressure gradient at the end of the cavity, so the length of the cavity can be judged according to the point of steep pressure increase.Combining the surface pressures of the three hydrofoils, it can be seen that from the top to the bottom of the cascade, the pressure-increasing area is constantly approaching the hydrofoil head, and the length of the cavity is constantly shortening.Compared with the cascade around the 6 angle of attack, the middle and lower hydrofoils of the cascade around the 8 angle of attack will have a pressure peak in the middle.The analysis shows that cavitation flow will occur at the head of the hydrofoil around the 8 angle of attack cascade.Head-generated cavity tail.At the same time, the length of the cavity on the upper surface of the cascade around the 8 angle of attack is significantly longer than that of the cascade around the 6 angle of attack.The analysis shows that after the increase of the angle of attack, the development speed of the cavity is accelerated and the cavitation period is shortened; For low pressure, the cavity is generated rapidly, completely covering the upper hydrofoil.The point of steep pressure increase on the surface of the middle and lower hydrofoil moves to the tail, and the cavity is still expanding, but the pressure peak disappears at 0 30 t t ms =+ time.The analysis shows that the two parts of the middle hydrofoil are fused into one, and the 8 degree of cavitation is intensified.At the same time, there is no obvious change in the surface pressure of the hydrofoil around the 6 angle-of-attack cascade, and the cavity development speed is lower than that of the cavitation around the 8 angle-ofattack cascade; The wing is super-cavitated, the hydrofoil is completely covered by the cavity, and the middle and lower layers of the cavity gradually approach the tail of the hydrofoil; at 0 140 t t ms =+ time, the surface pressure of the three hydrofoils around the 8 angle of attack cascade is low pressure, and the cavitation completely covers the cascade , the cascade enters the super-cavitation state.The point of sharp increase in pressure of the upper hydrofoil around the 6 angle-of-attack cascade moves toward the head of the hydrofoil.Combining the above analysis, it can be known that the upper hydrofoil around the 6 angle-of-attack cascade is in the cavitation collapse stage, the low-pressure area is continuously shrinking, and the water in the middle and lower layers The wing cavity is stable, and the surface pressure curve has no obvious change.

Conclusion
This paper deeply explores the unsteady flow mechanism and dynamic characteristics of the cascade around three NACA hydrofoils.By comparing the characteristics of the flow field around the cascade under different cavitation numbers and angles of attack, analyzing the difference between the cascade and a single hydrofoil, and exploring the influence of the cascade structure on the flow field, the following conclusions are obtained: 1) The interaction between the hydrofoils of the cascade makes a low-pressure area appear in the middle of the cascade, so that cavitation flow is more likely to occur at the 0 angle of attack.However, it also makes the flow field in the middle of the cascade more stable, and the cavitation phenomenon is different from that of a single hydrofoil, and the cavitation degree of the cascade is reduced and more stable.
2) Under different cavitation numbers, the cavitation flow around the cavitation is different from that of a single hydrofoil, especially the cloudy cavitation stage will not change drastically with the change of cavitation number, but it will easily enter the super-cavitation stage .
3) The angle of attack has a great influence on the cavitation flow around the cascade.Increasing the angle of attack will intensify the flow separation at the head of the hydrofoil, and the low-pressure area around the cascade will continue to expand, which will intensify the cavitation phenomenon.The hydrofoils are responsible for the flow separation of each other, so that the cavitation flow in the middle layer of the cascade does not change much and is relatively stable.
4) The cavitation flow phenomenon is different at different positions of the cavitation.The cavitation phenomenon in the upper layer of the cascade is similar to that of a single hydrofoil, but the degree of cavitation is more severe than that of a single hydrofoil; the cavitation phenomenon in the middle layer of the cascade is stable, but when the 8 angle of attack increases to , the fusion of the head cavity and the tail cavity greatly intensifies the degree of cavitation, thus making it easier for the cavitation to enter the super-cavitation state.
5) In subsequent sections of this paper, we consider incorporating a Clark-Y airfoil structure within the cascade, which can provide a higher lift, enabling enhanced dynamic characteristics even at reduced cavitation levels.Furthermore, we intend to delve deeper into the time-frequency characteristics during cascade shedding to ensure the safe operation of hydraulic machinery.
introduced the Shear Stress Transport (SST) two-equation model.This model not only retains the reliability of the k  − model in simulating nearwall viscous flow but also upholds the advantages of the k  − model in the shear layer.It achieves this by introducing modifications to the vortex viscosity coefficient, thereby accounting for the transmission effects of turbulent shear stress.Hence, the SST turbulence model can precisely predict a broader range of flows, overcoming the limitations of both k  − and k  − .As a result, this study employs the SST k  − turbulence model.

F
is the lift and drag received by the hydrofoil.l  is the water flow density, s and c represent the extended length and chord length of the hydrofoil, respectively.

Figure 1 .
Figure 1.Computational domain and boundary conditions.

Figure 3 .
Time evolution of the cascade cavitation (

Figure 5 .
Figure 5.Time evolution of monitoring point pressure on cascade head.

Figure 6 .
Figure 6.Time evolution of monitoring point pressure and velocity on cascade tail.

Figure 7
lists the time course of cavitation around the 0 angle of attack cascade when the cavitation number is 0.7 and 0.8.When the cavitation number is 0.7, the cavitation flow field around the cascade is in the sheet cavitation stage: at 0 20ms tt =+ time, the cavitation flow begins to appear in the middle layer of the cascade, and the difference 0.6 =is that the cavitation basically completely collapses and

Figure 7 .
Time evolution of the cascade cavitation under different cavitation numbers.

Figure 8 .
increases.However, due to the mutual interactions among the hydrofoils in the cascade, this increase in cavitation length isn't particularly pronounced.Transition stage from cloud cavitation of cascade and single hydrofoil.

Figure 9
Figure 9 lists the pressure change curves with time at the three monitoring points of cascades A, B and F at the 0 angle of attack when 0.6

Figure 9 .
Figure 9.Time evolution of the cascade pressure at different monitoring point.

Figure 10 .
Time evolution of the cascade cavitation under different attack angle.

Figure 11
Figure 11 lists the time-varying history of the pressure field and velocity field in the flow field around the 0.8  = cascade at the angle of attack 6 and 8 , where Figure 11 (a) and (b) are the 6 flow fields The cloud images of pressure field and velocity field, Figure 11 (c) and (d) are the cloud images of pressure field and velocity field around the 8 angle-of-attack cascade respectively.At 0 tt = time, the head of the cascade around the 6 angle of attack was a low-pressure area, and there was a large lowpressure area in the upper layer of the cascade.The flow separation of the hydrofoils in the middle layer of the cascade restricted the development of the low-pressure area, so there was no cavitation flow at

Figure 11 .
Time evolution of the cascade pressure and velocity under different attack angle.

Figure 12
Figure 12 lists the isolines of the flow function around the 0.8 = cascade at the angle of attack 6 and 8 .It can be seen from Figure12(a) that at 0 60ms tt =+ time, a large-scale vortex appeared at the tail of the upper hydrofoil around the 6 angle-of-attack cascade, and it continued to develop and expand; At 0 80ms tt =+ time, the vortex area reached the maximum value and continued to move toward the head of the hydrofoil.It can be seen from the cloud and pressure cloud images that the first period of cavitation develops to the maximum and begins to collapse at this time; At 0 90ms tt =+ time, the vortex area has decreased, and at the same time the growth of the second cavitation period has begun, although the low-pressure area has not yet shrunk, The reverse jet flow begins to weaken; when 0 100ms tt =+ , the vortex expands again, and the reverse jet flow becomes more intense, forming a larger vortex area, which continues to expand to the front end of the hydrofoil, and the hydrofoil cavity on the upper layer of the cascade completely collapses.Comparing the images of each frame in Figure12(b), it can be seen that a large-scale vortex structure quickly appeared at the tail of the upper hydrofoil around the 8 angle-of-attack cascade, and gradually moved to the downstream of the cascade, and super-cavitation flow appeared.When the flow field stabilized, the flow lines around the cascade also

Figure 12 .
no longer change.Combining the velocity and pressure contours around the cascade around the 8 angle of attack in Figure11(c) and (d), it can be seen that the downstream of the cascade is the boundary of the low-pressure area, and the pressure gradient causes reverse jet flow, which makes the vortex downstream of the cascade.Streamline of cascade under different attack angle.

Figure 13
Figure 13 lists the schematic diagram of the pressure detection positions on the upper surface of the three hydrofoils in the cascade, and Figure 14 lists the changes in the surface pressure of the three hydrofoils around the cascade with the position at different times when 0.8  = , the angle of attack 6 and 8 .At 0 20 t t ms =+ time, among the pressure curves of the three hydrofoils around the 8 angle of attack cascade, the low pressure area of the upper hydrofoil is the largest.When 0.7 x =C, the surface pressure of the upper hydrofoil increases sharply, and there will be a large pressure gradient at the end of the cavity, so the length of the cavity can be judged according to the point of steep pressure increase.Combining the surface pressures of the three hydrofoils, it can be seen that from the top to the bottom of the cascade, the pressure-increasing area is constantly approaching the hydrofoil head, and the length of the cavity is constantly shortening.Compared with the cascade around the 6 angle of attack, the middle and lower hydrofoils of the cascade around the 8 angle of attack will have a pressure peak in the middle.The analysis shows that cavitation flow will occur at the head of the hydrofoil around the 8 angle of attack cascade.Head-generated cavity tail.At the same time, the length of the cavity on the

Figure 13 .
Figure 13.Diagram of upper surface pressure of cascade hydrofoils.

Figure 14 .
Figure 14.Variation of pressure on upper surface of three hydrofoils.

Table 1 .
The specific location of the monitoring point.

Table 2 .
Mesh independence test of the mini cascade.
upper cavity broke into two parts, which developed towards the head and tail of the hydrofoil respectively, and continued to fall off and collapse.The hydrofoil cavities around the middle layer of the cascade are gradually formed and rapidly expanded, and cavitation flows appear at the head and tail of the hydrofoil at the same time, and the two gradually merge to form a more intense