Flow Regime Map Research Articles

Analysis of two-phase flow regimes and parameter distribution in 5 x 5 rod bundle using image processing techniques

With the recent developments in image processing and analysis, this paper presents bubble characteristics distribution in adiabatic air-water two-phase flow through a 5 × 5 rod bundle. The experiment covered water superficial velocities (Jl = 0.012 m/s – 0.421 m/s) and air superficial velocities (Jg = 0.042 m/s – 0.987 m/s) in which three distinct flow regimes were identified. The flow regime map was compared with existing flow regime transition criteria for vertical rod bundles. Distinct features from the two-phase flow images were extracted to train a classifier model to distinguish between regimes from a separate experiment. The model distinguishes between the bubbly flow regime and others accurately. The void fraction and velocity distributions were also extracted from the R–CCN masked images. Bubble-induced turbulence that was dominant in the subchannel at (Jl = 0.28 m/s) shifted to the outer subchannels and gaps when the flow rate increased (Jl = 0.42 m/s). These methods over-predicted the void faction around the surfaces of the inner rods.

A flow map for core/shell microdroplet formation in the co-flow Microchannel using ternary phase-field numerical model

Core/shell microdroplets formation with uniform size is investigated numerically in the co-flow microchannel. The interface and volume fraction contour between three immiscible fluids are captured using a ternary phase-field model. Previous research has shown that the effective parameters of microdroplet size are the physical properties and velocity of the three phases. By adjusting these variables, five main flow patterns are observed in numerical simulations. A core/shell dripping/slug regime is observed when the inertia of the continuous phase breaks the flow of the core and shell phases and makes a droplet. In the slug regime, the continuous phase has less inertia, and the droplets that form are surrounded by the channel walls, while in the dripping regime, the shell phase fluid is surrounded by the continuous phase. An increase in continuous-fluid or shell-fluid flow rate leads to dripping to a jetting transition. When three immiscible liquids flow continuously and parallel to one another without dispersing, this is known as laminar flow. In the tubing regime, the core phase flows continuously in the channel's central region, the shell phase flows in the annulus formed by the core phase's central region, and the continuous phase flows between the shell phase fluid and channel walls. In order to discriminate between the aforementioned flow patterns using Weber and Capillary numbers and establish regime transition criteria based on these two dimensionless variables, a flow regime map is provided. Finally, a correlation for shell thickness using shell-to-core phase velocity ratio and conducting 51 CFD simulations was proposed.

Applying a microfluidic device to improve the Ca2+ separation performance of the liquid–liquid extraction process

This study investigates the application of extraction solvent in a new microfluidic apparatus to separate calcium ions (Ca2+). Indeed, a serpentine microfluidic device has been utilized to separate calcium ions. The flow regime map shows that it is possible to completely separate organic and aqueous phases using the serpentine microfluidic device. The suggested microfluidic device reaches the extraction efficiency of 24.59% at 4.2 s of the residence time. This research also employs the Box–Behnken design (BBD) strategy in the response surface methodology (RSM) for performing the modeling and optimization of the suggested extraction process using the recorded experimental data. Flow rate and pH of the aquatic phase as well as Dicyclohexano-18-crown-6 (DC18C6) concentration are those independent features engaged in the model derivation task. The optimum values of pH 6.34, the DC18C6 concentration of 0.015 M, and the flow rate = 20 µl/min have been achieved for the aquatic phase. The results indicated that the extraction efficiency of Ca2+ is 63.6%, and microfluidic extraction is 24.59% in this optimum condition. It is also observed that the microfluidic extraction percentage and experimental efficiency achieved by the suggested serpentine microchannel are higher than the previous separation ranges reported in the literature.

Microgravity two-phase flow research in the context of vapor compression cycle experiments on parabolic flights

Research on two-phase channel flow in microgravity has extensively investigated flow regimes, pressure drop, and heat transfer during evaporation and condensation processes over several decades. This literature has been primarily motivated by the goal of enabling future space applications for two-phase thermal management systems, among them vapor compression refrigeration cycles. However, relative to the number of two-phase channel flow experiments, research on vapor compression cycles in microgravity has been almost non-existent. This paper reviews two-phase flow research and compares key outcomes with those from system-level measurements obtained from recent testing of a vapor compression cycle at varying inclination angles and on parabolic flights. Previous two-phase channel flow experiments have resulted in several microgravity-specific flow regime maps and have also generally found reduced condensation heat transfer coefficients in microgravity as compared to normal gravity. The flow pattern map of Jayawardena et al. (1997) and generally reduced condensation heat transfer are confirmed by the vapor compression cycle experiments. Regarding evaporation heat transfer coefficients and friction factors, there is not a consensus in the literature on the effects of microgravity in two-phase channel flows. Similarly, the system-level experiments do not show repeatable increases or decreases for those parameters. Lastly, the dependence of system-level behavior on orientation with respect to gravity was tested by performing dynamic inclination experiments, where the inclination angle was changed every 2 min by a 90-degree increment. These tests reveal an increased relative dependence of the vapor compression cycle to orientation with decreasing mass fluxes, in line with two-phase channel flow research that has generally shown that increasing the flow inertia acts to mitigate various flow instabilities and dependence on orientation. However, inclination testing with longer locking times per angle showed a constant relative dependence of the vapor compression cycle to orientation changes.

Gas–liquid two-phase flow patterns and pressure drop of decaying swirling flow inside a horizontal pipe

Gas–liquid two phase swirling flow has been widely used in industry. Its flow pattern is fundamental to investigate the two-phase flow. In this paper, two-phase air–water flow patterns and pressure drop of decaying swirling flow generated by vane-type swirler inside a horizontal pipe have been systematically investigated by visualization experiment. The range of superficial gas phase Reynolds number is 17–23902, while superficial liquid phase Reynolds number is 12–37871, and gauge pressure is 0.003–0.17 MPa. The five typical flow patterns in a horizontal swirling flow were characterized based on flow visualizations and fluctuation characteristics of pressure drop signals. Then swirling flow regime map nearby the swirler outlet was proposed. As flow patterns altered along the streamwise direction due to swirl decay, flow pattern maps at different positions were proposed, which indicated the swirling flow patterns gradually transformed to non-swirling flow patterns. Finally, a new model to predict frictional pressure drop of air–water swirling flow inside a horizontal pipe containing vane-type swirler was proposed. This correlation performed well on predicting the frictional pressure drop.

Experimental Investigation of Flow Inside the Back-Sidewall Gaps of Centrifugal Pump and its Impact on Axial Thrust

Despite the sidewall gaps of centrifugal pumps are tiny narrow spaces, the flow inside them can have a great impact on the flow field in the whole machine. Regardless of its influence, there is no full understanding of the flow in those regions. A theory exists for the simplified cylindrical cavity with a rotating disk, whose application to real hydraulic machines is questionable. To investigate the flow in the sidewall gaps of centrifugal pumps, a test rig including a real impeller has been built up. It enables observation of the flow in the back-sidewall gap by optical methods and evaluation of axial thrust and torque. The measurements were performed for a range of rotational speeds in order to map possible flow regimes and patterns. It was found out, that the basic theory and map of the flow regimes obtained for a simple rotating cavity is inapplicable for the back-sidewall gaps of real hydraulic machines. The flow field is significantly influenced by the impeller blades, and consequently, the map of the regimes is shifted compared to the one derived for the simple cylindrical cavity with a rotating disk. The discrepancies were described and a new map of flow regimes was introduced.

The study on the partition method of two-phase wall drag in the one-dimensional two-fluid model

The interfacial area transport equation(IATE) is being investigated to replace the flow regime map in the one-dimensional two-fluid model (1-D TFM). The partition of two-phase wall drag is a part of flow regime-dependent constitutive relations in the code using the 1-D TFM such as RELAP5. Hence, to develop the partition method for two-phase wall drag that is consistent with 1-D TFM using the IATE, the mechanism for the partition was investigated and clarified in this work. And the partition method for two-phase wall drag that can capture the dynamic change of two-phase flow structure was developed with the clarified mechanism. Furthermore, the method was preliminarily validated with the experimental data, which shows that the two-phase wall drag computed by the method can accurately capture the variation of flow structure and relative velocity in transient two-phase flow. Accordingly, the developed method is able to take advantage of the IATE.

Local solids flow structure in a countercurrent liquid-upward and solids-downward fluidized bed (CCLSFB) system

The local solids holdup and local particle velocity in a Countercurrent Liquid-upward and Solids-downward Fluidized Bed (CCLSFB) were investigated in details using optical fiber probes with two different models in a Plexiglas column of 1.5 m in height and 7.0 cm in inner diameter. A new flow regime map including fluidized bed, transition, and flooding regimes was established. The axial solids holdup distribution is almost uniform at low liquid velocity and/or solids flowrate and becomes less uniform with higher solids holdup at the top of the column after the operating liquid velocity is reaching the flooding velocity. The radial solids holdup profile is also nearly flat with a slightly lower solids holdup in the near-wall region at low liquid velocity and solids flowrate but becomes nonuniform as the operating liquid velocity approaches the flooding velocity. Two equations were also proposed to correlate radial local solids holdups. The descending particle velocity in CCLSFB increases with the decrease of the liquid velocity and the increase of the solids flowrate. A generally uniform particle velocity distribution was found in the axial direction, as well as in the radial direction except for a small decrease near the wall. These results on the local solids flow structure would provide basic information and theoretical supports for the design and industrial application of CCLSFB.

Performance intensification and anti-fouling of the two-phase flow enhanced direct contact membrane distillation for seawater desalination

Membrane distillation is a promising approach to the fresh water crisis due to the low operating temperature and high rejection of non-volatile components. However, it suffers from the polarization effect and membrane fouling. This paper aims at developing a novel two-phase flow enhanced direct contact membrane distillation (TP-DCMD) system and investigating the effect of two-phase flow behaviors on the heat and mass transfer and membrane fouling comprehensively. First, a novel experimental system of TP-DCMD is developed and well validated using the mass and energy conservations. Then, the effect of flow parameters on the transmembrane permeate flux is discussed. The introduction of gas could improve the membrane distillation performance by as high as 27 %. The relationship between the performance enhancement and two-phase flow regimes is discussed. Both permeate flux and multiplier clouds are proposed based on two-phase flow regime map. The slug flow demonstrates superior enhancing effect to the other flow regimes due to the low possibility of gas penetration. Finally, the effect of two-phase flows on the membrane fouling is investigated. After 420 min of accelerating fouling test, the transmembrane permeate flux drops by 38.2 % in the conventional DCMD system, while it just decreases by 6.6 % in TP-DCMD system.

Generalized flow regime map for two-phase mini/micro-channel flows

This study concerns the development of a generalized approach for predicting two-phase flow regimes in mini/micro-channel flows. To achieve this, a total of 7391 two-phase flow regime data points for mini/micro-channels are amassed from 25 sources, including the micro-channel experimental data of the authors and data from previous studies on two-phase flow patterns in mini/micro-channels in the literature. The consolidated database encompasses 16 different working fluid data for both single-component and two/multi-component mixtures with hydraulic diameters of 0.2–4.0 mm, superficial liquid Reynolds numbers of 4.6–918,938, superficial gas Reynolds numbers of 0.4–261,017, mass velocities of 2.4–6028 kg/m2·s, flow qualities of 0–0.99, and reduced pressures of 0.036–0.78. Based on the flow regime definition of the present study, the database was classified into two main categories (non-annular and annular flows) and seven subcategories (bubbly, intermittent, transition, multiple, wavy-annular, smooth-annular, and dispersed-annular flows). Predictions of previous generalized flow regime maps are compared to four subsets of the consolidated database, revealing significant deviations in the transition boundaries between the two. Accordingly, a new generalized flow regime map with flow regime boundary correlations was proposed for two-phase mini/micro-channel flows. The new flow regime map was demonstrated to be capable of providing excellent predictions against the entire consolidated database, with evenly good predictions for different flow regimes, working fluids, and over a broad range of hydraulic diameters.

Flow pattern transition and void fraction prediction of gas–liquid flow in helically coiled tubes

• Flow patterns in helically coiled tubes (HCTs) are classified by visual observation. • Flow-regime map is proposed and mechanisms of flow transitions are presented. • Void fraction for the entire tube cross-section is measured by WMS technique. • New correlations of distribution parameter and drift velocity are proposed. Helically coiled tubes (HCTs) are widely used in industries owing to their compact structure and excellent heat and mass transfer efficiency. Due to the complexity of three-dimensional flow in HCTs, profound knowledge on the flow pattern characteristics is not well documented in existing literature. In this paper, we employ the wire-mesh sensor (WMS) and image tomography technique to measure a series of essential parameters (e.g. flow structures, phase fraction) to explore the flow pattern transition mechanism. According to our observation, bubbly, plug, slug, wavy and annular flow are recognized, and the mechanisms of flow pattern transitions are carefully discussed based on the analysis of the forces. In addition, the variation of void fraction in both temporal and spatial dimensions is discussed in detail. By introducing the drift flux analysis, new correlations of distribution parameter and drift velocity are proposed taking both HCTs structure and flow condition into account.

Biogas bioconversion into poly(3-hydroxybutyrate) by a mixed microbial culture in a novel Taylor flow bioreactor

Biogas-based biopolymer production represents an alternative biogas valorization route with potential to cut down plastic pollution and greenhouse gas emissions. This study investigated for the first time the continuous bioconversion of methane, contained in biogas, into poly(3-hydroxybutyrate) (PHB) by a mixed methanotrophic culture using an innovative high mass-transfer Taylor flow bioreactor. Following a hydrodynamic flow regime mapping, the influence of the gas residence time and the internal gas recirculation on CH4 abatement was assessed under non nutrient limiting conditions. Under optimal operational conditions (gas residence time of 60 min and internal gas recycling ratio of 17), the bioreactor was able to support a CH4 removal efficiency of 63.3%, a robust CH4 elimination capacity (17.2 g-CH4 m-3h-1) and a stable biomass concentration (1.0 g L-1). The simultaneous CH4 abatement and PHB synthesis was investigated under 24-h:24-h nitrogen feast/famine continuous operation. The cyclic nitrogen starvation and the Taylor flow imposed in the bioreactor resulted in a relatively constant biomass concentration of 0.6 g L-1 with PHB contents ranging from 11 to 32% w w-1 (on a dry weight basis), entailing an average PHB productivity of 5.9 g-PHB m-3 d-1 with an associated PHB yield of 19.8 mg-PHB g-CH4-1. Finally, the molecular analysis of the microbial population structure indicated that type II methanotrophs outcompeted non-PHB accumulating type I methanotrophs, with a heterotrophic-methanotrophic consortium enriched in Methylocystis, Hyphomicrobium, Rubinisphaeraceae SH PL14 and Pseudonocardia.

Development of CFD based model for sodium flow boiling in narrow channel akin to SFR fuel subchannel towards high void fraction regimes

The Sodium cooled fast reactor (SFR) is one of the Gen IV reactor type meant for the highest nuclear safety standards. Understanding of flow boiling heat transfer characteristics of liquid sodium is essential under certain postulated events of SFR. This study is intended towards the development of computational fluid dynamics (CFD) based multiphase flow model for the sodium boiling in SFR subchannel using the finite volume method. The numerical simulations have been performed for the basic two-dimensional domain that has a resemblance with the flow subchannel of SFR fuel subassembly. The sodium boiling inception, the nucleate flow boiling regime, and further the forced convective vaporization regime characteristics are numerically predicted by the Eulerian-Eulerian model (two-fluid model) developed using flow regime based modeling within the Algebraic Interfacial Area Density (AIAD) framework for the interfacial closure correlations. The improved non-equilibrium wall boiling model is implemented for wall heat flux partitioning. The liquid turbulence is modeled using a Realizable k-ε model with standard wall function. The CFD simulations are performed for the reference experimental cases of liquid sodium flow boiling in narrow annuli and a tubular channel till high void fraction regimes using the model developed. The predictions of void growth behavior along the axial and in the radial direction depict the dominance of annular flow boiling pattern as observed in sodium flow boiling experiments based on the literature. The numerical predictions for the sodium two-phase flow pressure drop are compared with the experimental results and also with the existing empirical correlations. The mechanistic model-based predictions for boiling heat transfer coefficient are obtained from the wall heat balance model and compared with the available empirical correlations. Also based on the flow parameters predicted the possible flow patterns are identified using the available flow regime maps.

Experimental investigation of two-phase flow regimes in rectangular micro-channel with two mixer types

This study primarily aimed to explore the two-phase flow characteristics in a rectangular channel with a hydraulic diameter of 1.33 mm. High-speed video imaging techniques were used to capture two-phase flow behavior corresponding to different superficial gas and liquid velocities. To examine the effect of the gas–liquid mixer configuration on two-phase flow patterns, experiments were conducted using two types of mixers. The results show that the mixer configuration can affect the flow structure only in the slug flow regime. The present study provides a more detailed classification of flow regimes than previous studies. The flow regimes observed were classified into seven categories: slug, aerated-slug, transition, multiple, wavy-annular, smooth-annular, and dispersed-annular flows. Moreover, the quantitative characterization of flow regimes and transition boundaries in micro-channel flows using the standard deviation of differential pressure fluctuations was attempted. This study comprehensively reviews and assesses previous two-phase flow regime maps for both macro-channels and mini/micro-channels. It is shown that the two-phase flow pattern in mini/micro-channels is influenced by various factors, such as the channel hydraulic diameter, the channel cross-sectional shape, the gas–liquid mixer configuration, the difference in the hydraulic diameters between the mixer outlet and test channel, and the thermophysical properties of the working fluids. The comparison between the present experimental data and predictions of the previous flow regime maps demonstrates the necessity of developing a new flow regime map to accurately predict two-phase flow regimes in micro-channels.

A computational fluid dynamics study on rimming flow in a rotating cylinder

Extensive computational fluid dynamics (CFD) simulations were conducted to study “rimming” flow in a partially filled horizontally rotating cylinder. These flows are encountered in aero-engine bearing chambers, which often exhibit complex two-phase flow scenarios as well as in multiple other engineering applications. In this study, a robust numerical scheme to model two-phase rimming flow has been adopted and validated against analytical expression and experimental data obtained from the literature. Additionally, a vast parametric study of the flow conditions has been performed. We used the volume of fluid method to solve the system of multi-phase flow governing equations and track the interface of rimming flow. The time-dependent gas–liquid interface was resolved, and the liquid-film thickness was determined. First, we performed our simulations within small to moderate ranges of Reynolds and Bond numbers and compared our results with previously reported analytical and experimental investigations. The present CFD results were found to be in very good agreement with previously reported data, both in identifying different regimes reported in the literature for rimming flow and in liquid-film thickness predictions. We also performed several additional simulations at much larger and practical ranges of Reynolds and Bond numbers, beyond the limitations imposed in previous analytical and experimental investigations on thin-film flows. We showed that three different flow regimes—shear-dominated, transitional, and gravitational-dominated—are attainable for the rimming flow for different combinations of Reynolds, Bond, and gravitational numbers. The present numerical results led us to propose a new map of rimming flow regimes by introducing functions of the Froude number and capillary number, which successfully identify and separate these regimes for a significant number of flow conditions.

Gas-solid hydrodynamics in a stator-rotor vortex chamber reactor

The gas–solid vortex reactor (GSVR) has enormous process intensification potential. However the huge gas consumption can be a serious disadvantage for the GSVR in some applications such as fast pyrolysis. In this work, we demonstrate a recent novel design, where a stator-rotor vortex chamber (STARVOC) is driven by the fluid’s kinetic energy, to decouple the solids bed rotation and gas. Gas-solid fluidization by using air and monosized aluminum balls was performed to investigate the hydrodynamics. A constructed fluidization flow regime map for a fixed solids loading of 100 g shows that the bed can only be fluidized for a rotation speed between 200 and 400 RPM. Below 200 RPM, particles settle down on the bottom plate and cannot form a stable bed due to inertia and friction. Above 400 RPM, the bed cannot be fluidized with superficial velocities up to 1.8 m/s (air flow rate of 90 Nm3/h). The bed thickness shows some non-uniformities, being smaller at the top of the bed than at the bottom counterpart. However by increasing the air flow rate or rotation speed the axial nonuniformity can be resolved. The bed pressure drop first increases with increasing gas flow rate and then levels off, showing similar characteristics as conventional fluidized beds. Theoretical pressure drops calculated from mathematical models such as Kao et al. model agree well with experimental measurements. Particle velocity discrepancies between the top and bottom particles reveal that the impact of gravity cannot be completely neglected. Design guidelines and possible applications for further development of STARVOC concept are proposed based on fundamental data provided in this work.

Recognition of gas-liquid flow regimes in helically coiled tube using wire-mesh sensor and KNN algorithm

• Flow patterns in helically coiled tube are classified by visual observation. • Void fraction for entire cross-section is measured by wire-mesh sensor. • k -Nearest Neighbors algorithm is utilized to identify flow patterns objectively. • Influence of centrifugal force and secondary flow on phase distribution and flow transition is revealed. Helically coiled tube (HCT) is used extensively in compact types of heat exchangers. Complete knowledge of gas-liquid flow behavior in HCT is essential in designing the heat transfer equipment involving phase change. Although the gas-liquid flow patterns in straight tubes have been widely studied, investigations on flow pattern recognition in HCT are still scarce. In this paper, based on the backlight imaging tomography and self-designed wire-mesh sensor, air-water experiments are carried out to study the characteristics of gas-liquid flow in a transparent HCT. Bubble, plug, slug, wavy and annular flow are observed, and their void fraction at the cross-section of the tube are measured and analyzed using the statistic parameters of time-trace, PDF and CPDF. Owing to the simplicity of implementation, the void fraction data are subsequently utilized along with the k -Nearest Neighbors algorithm (KNN algorithm) to identify the flow patterns objectively. Finally, a flow regime map is proposed and compared with the previous straight flow map and HCT map in the literatures. Accordingly, the influence of curvature-induced centrifugal force and secondary flow on the phase distribution and flow-regime transition is revealed in the present study.

Segregation and intermixing in polydisperse liquid–solid fluidized beds: A multifluid model validation study

AbstractMultifluid model (MFM) simulations have been carried out on liquid–solid fluidized beds (LSFB) consisting of binary and higher‐order polydisperse particle mixtures. The role of particle–particle interactions was found to be as crucial as the drag force under laminar and homogenous LSFB flow regimes. The commonly used particle–particle closure models are designed for turbulent and heterogeneous gas–solid flow regimes and thus exhibit limited to no success when implemented for LSFB operating under laminar and homogenous conditions. A need is perceived to carry out direct numerical simulations of liquid–solid flows and extract data from them to develop rational closure terms to account for the physics of LSFB. Finally, a recommendation flow regime map signifying the performance of the MFM has been proposed. This map will act as a potential guideline to identify whether or not the bed expansion characteristics of a given polydisperse LSFB can be correctly simulated using MFM closures tested.

Characterization of gelatin microparticle production in a flow focusing microfluidic system

In this study, gelatin microdroplets were generated using a PDMS square microchannel (100 ×100 µm). The flow regime map was experimentally determined, and the regimes in which is possible to obtain microdroplets, dripping and jetting, were studied in detail. The regime map was compared with maps published in the literature and it was found that the regime map depends significantly on the viscosity ratio. For each regime, a correlation to calculate the normalized size of the droplet was obtained by fitting the experimental data. The smaller microdroplets obtained had a size of 53 µm and 60 µm, for dripping and jetting regimes, respectively. The droplets generated in this microfluidic device were collected and gelled. We concluded that occurred a decrease in the size of the final microparticles during the curing process. A coefficient of variation (CV - %) between 2% and 4% was found for the microdroplets and microparticles, which demonstrates that this method can be used to produce monodispersed gelatin microparticles.

Investigations into the ventilated cavities around a surface-piercing hydrofoil at high Froude numbers

This study investigates the ventilated cavities around a surface-piercing hydrofoil, aiming to extend previous studies by an in-depth understanding of the vaporous cavity behaviors and the flow-regime transition at high Froude numbers. An experiment is carried out in a constrained-launching water tank with a vertically cantilevered hydrofoil piercing a still water surface. The cavity is recorded using high-speed photography, and flow-regime maps are summarized over a broad range of Froude number and yaw angle at different immersed aspect ratios. In addition to the well-known steady flow regimes (i.e., fully wetted flow and fully ventilated flow), an unsteady vaporous cavitating flow is revealed at a very high Froude number with a small yaw angle, which exhibits cavitation shedding dynamics behaviors, including the cavity growth, destabilization, and collapse. The transition from the fully wetted flow to the fully ventilated flow is attributed to the vapor-cavitation-induced ventilation besides the tip-vortex-induced ventilation. Vaporous cavitation promotes ventilation formation, but it has to meet the criterion that air should enter the sub-atmospheric cavity through the tip-vortex path before the cavity length reaches the maximum. Moreover, an improved lifting-line model is developed with considering the effects of free surface and finite aspect ratio. Both analytical modeling and experimental measurements reveal that the vaporous cavity length follows a power relation against the cavitation parameter. Such knowledge lays a foundation for the design optimization and control strategy of high-speed hydrofoils.