The use of hydropower is rapidly increasing worldwide, with different-range-scale hydropower plants being utilized to meet the growing demand for energy while taking advantage of the numerous benefits offered by this technology 
. These benefits include sustainability, reliable storage, environmental friendliness, regulation flexibility, and high efficiency. However, achieving maximum efficiency in hydraulic turbines requires operating at the best efficiency point (QBEP
), which is rarely feasible because flexible regulation and adjustments to accommodate changing power grid loads are needed.
Turbine operating conditions are divided into ranges based on flow rate with respect to QBEP
. Under part load conditions, where the flow rate is lower than QBEP
, the tangential velocity component is codirectional to the peripheral runner revolution velocity. The resulting tangential velocity cannot be fully transformed into runner torque, and flow in the draft tube remains swirled. Under high-load conditions, where the flow rate is higher than QBEP
, the tangential velocity component is oppositely directed to runner rotation, resulting in excessive swirl and uneven velocity distribution. During partial-load conditions, an excess of swirling can result in flow separation. This can cause a stationary area, flow reversal, and formation of a helical rotating vortex rope (RVR) in the center of the draft tube 
. RVR is helical vortex breakdown 
, which is also called a precessing vortex core (PVC) 
. The RVR rotates in the same direction as the runner, and its precessing frequency is roughly 30% of the runner’s rotational frequency. When operating at high loads, the swirling flow would rotate in the opposite direction to the runner, forming an axisymmetric vortex rope that resembles a torch and precesses in the opposite direction of the runner’s rotation. The high load vortex produces synchronous pressure pulsations associated with cavity volume oscillations, while the part load vortex is a source of asynchronous pulsations associated with rotation of the helix around the axis. The resulting pressure pulsations and, in general, the existing vibrations have a negative impact on hydropower equipment 
The problem of pulsation mitigation under non-optimal operating conditions associated with the presence of a vortex rope started to be addressed 
almost as soon as the nature of flow surge was identified 
. By mitigating the vortex phenomenon, one can reduce pressure pulsation amplitudes and therefore expand the range of stable operation of the hydroturbine. The relevance of developing flow control in hydroturbines is evidenced by numerous recent reviews 
The most capacious requirements for the control of vortex structures in hydropower equipment were formulated by Prof. Susan-Resiga 
. Addressing the primary source of excitation is of utmost importance for ensuring optimal turbine performance. The vortex rope, a phenomenon of vortex breakdown in the swirling flow in the draft tube behind the runner, has been identified as the root cause of excitation. Therefore, it is imperative to modify its precessing frequency or entirely eliminate the unsteady vortex breakdown to prevent the detrimental effects on the turbine. In addition to addressing the primary source of excitation, it is crucial to ensure that the control device does not interfere with the turbine operation when it is not required. This is particularly important when operating at or near the optimal efficiency. Per se, the vortex rope control should be switchable on/off or continuously adjusted based on the operating regime to prevent any adverse effects on turbine efficiency.
It is also essential to note that any control method implemented must not significantly reduce turbine efficiency while reducing or eliminating pressure fluctuations. Therefore, it is necessary to strike a balance between controlling the vortex rope and maintaining optimal turbine performance. Finally, to ensure easy acceptance and realization in industrial practice, the practical implementation of the control method should be simple and robust. Any modifications to the current turbine design and costs should be kept minimal to ensure widespread adoption. With these considerations in mind, efficient control over the vortex rope can be achieved, leading to improved turbine performance and reduced maintenance costs.
Nowadays, there are two basic approaches to surge mitigation, which are based on the operation turbine principle. The first one involves modifying the existing geometry through geometric optimization 
, seeking to improve performance over a wider range of operations by changing parametric dimensions. This approach mainly involves replacing the draft tube and runner to counteract draft tube surge. Other solutions designed to hold the flow steady by weakening the swirl content include runner cone extensions, modifying the draft tube cone using J-grooves 
, finning the inner surface of the draft tube 
, crosspieces 
, runner crown extension 
, and the maximum power point tracking control strategy 
. These devices reduce tangential momentum, i.e., the swirl intensity. In their recent study, Joy et al. 
described that the vortex rope was significantly suppressed by inserting an “adjustable” special guide vane into the draft tube cone.
The second approach aims to reduce flow swirl and break up stagnant zones by injecting water or air into the flow. The mechanism of impact is different in this case. The air jet, having several orders of magnitude lower density, virtually does not introduce additional momentum to the flow but rather redistributes the low-pressure area, making it more uniform. On the other hand, the water jet can directly affect the rotational momentum of the flow.
Each countermeasure has its own unique benefits and side effects. The most commonly recurring problems include an inability to regulate the frequency of instabilities, which increases the risk of resonance. Additionally, passive elements can contribute to secondary instabilities. Jet injections require additional energy and may be accompanied by hydraulic losses.
2. Air Addition
2.1. Experimental Works
Injecting air into hydraulic turbines to reduce pressure pulsations has a history dating back to the 1950s when it was first introduced in real power plants (refer to 
for an overview of early work). Initially, the primary objective was to increase the amount of dissolved oxygen at the inlet for environmental reasons and to maintain the required oxygen level at the tailwater depth. However, it was discovered that hydroturbine aeration had other beneficial effects, including reduced vibration and increased efficiency of the turbine per se. Although some studies were performed quite a long time ago, some of them are still of particular interest.
conducted experiments using a simplified model of a draft tube and a stationary guide vane system to test different levels of swirl. The experiments revealed that at certain levels of swirl, there were powerful pressure pulsations and vibrations in the draft tube. Air was injected through a pipe, 10 mm in diameter, into the center of the draft tube, with the amount of air varied from 0 to 6%. As the ratio between circulating velocity and axial velocity of water increased, more air was required to prevent vibration. When the amount of air was insufficient, vibrations in the draft tube increased, but the draft action was improved. It was found that only 3% or more air injected could suppress pressure pulsation. Although the model was dissimilar to the geometry of a hydraulic turbine, the results of this study were among the first findings demonstrating that the vortex rope can be controlled by air injection.
Further progress has been made by Nakanishi and Ueda 
in designing and applying air injection to prevent hydraulic turbine surge. Their study examined various parameters that could have the most significant impact on pressure pulsations under different operating conditions. In addition to gas content, they also investigated the optimal position and number of air feed pipes. When the tail race level at a power station is raised to consider runner cavitation characteristics, the static pressure in the draft tube increases, thus impeding natural air admission. Therefore, it is best to find a position of minimal pressure for easy air intake and place the air pipe opening accordingly. However, in some cases, providing enough air to reduce surging may be challenging, and forced air supply from a jet pump may be required.
Tests were conducted for three types of Francis turbines with different specific speeds up to n = 220 mkW for the high-head cavitation test rig. Two opposing tubes were primarily used to supply air. When air was supplied into the draft tube, air foam gathered in the low-pressure interior of the center cone, breaking the forced vortex core and making it larger. However, the center cone may sometimes tremble more, and the amplitude of water pressure fluctuation may rise. Supplying more air than needed increases draft tube pressure and worsens turbine efficiency. Therefore, it is crucial to obtain the minimal amount of air supply required to stabilize the center cone. Air content changes somewhat when the guide vane opening angle varies, but no noticeable correlation between these values is observed. The optimum airflow rate ranges from 1.5% to 2.5%. When traversing the air pipe opening radially to the draft tube, the optimal air value becomes higher, up to 3–4%, if air is fed from near the draft tube wall. The simple idea is that if the injection site is far enough from the vortex position in terms of radius and the region of low pressure, some air will not reach the vortex core and will be lost. Therefore, air injection from the axis is most promising.
The outcomes of this study remain pertinent to date, since no principally new systems that would be more efficient have been proposed since then. Moreover, the numerous subsequent studies have neglected a crucial air content parameter. Of course, due to the lack of information about the flow structure, no conclusions could be made about the mechanism of pulsation suppression, except for the fact that the vortex rope is transformed into a symmetric vapor–gas cavity. It was not until much later that this issue was considered in studies employing modern methods of computational fluid dynamics. Obviously, research into the prototype turbines is of greatest interest, but simplified experimental rigs will also be considered.
In 2010, García and Viveros 
conducted field tests at a real hydropower plant equipped with several 95 MW Francis turbines, each with a 3.8 m runner diameter. In order to determine the crucial conditions for the structure, the researchers gauged the pressure within the draft tube and strain exerted on its external wall at different power levels. At partial load between 30 and 50 MW, they observed that the average pressure inside the draft tube and its amplitude increased significantly, leading to elevated stress on the wall, with pressure pulsation exhibiting a low frequency, which was likely due to the vortex rope.
Additionally, they carried out investigations under partial load conditions by infusing air through the stay vanes from a storage tank containing compressed air at 5.45 MPa. The researchers indirectly estimated the amount of air supplied from the pressure in the pressure tank and calculated that air flow was 0.207 m3
/s at a water discharge of 60.8 m3
/s, which corresponded to gas content of 0.34%. Air flow rate as low as 0.34% with respect to water discharge proved to be sufficient for enhancing hydraulic stability and reducing pressure pulsation and wall stress. However, the maximum gas content studied was significantly lower than the threshold values of 2–3% found in previous works by Murakami 
as well as Nakanishi and Ueda 
. This difference could be due to inaccurate estimation of the air flow rate, since only pressure in the pressure tank was taken into account, whereas the pressure drop was unknown and could decrease under non-optimal turbine conditions when a large-scale area with vortex rope was formed. Additionally, multiphase effects were notoriously difficult to scale up with increasing turbine size and resulting Reynolds number, and smaller gas contents could have an effect.
Papillon et al. 
have contributed by developing three different aeration systems: through the runner cone, the peripheral discharge ring aeration system, and aeration by the wicket gate trailing edge. These methods affect turbine performance in terms of flow rate, power output, and efficiency. The central aeration alternatives had a greater impact on efficiency compared to the peripheral aeration system. The introduction of air bubbles into the water changes the fluid density and alters water velocities. Moreover, addition of air affects local pressure, thus reducing the “internal” head observed by the runner. This results in a shift in the runner operating point and drop-in efficiency and output as the internal head decreases. Therefore, air admission alters the entire performance curve of the turbine.
Experimental data obtained from a real hydroelectric power plant are crucial in assessing the efficiency of an air injection system. Türkmenoğlu 
studied the vibration effects of a vortex in high-head Francis turbines at the Darca-1 hydroelectric power plant in Ordu Province, Turkey. The vortex, caused by oxygen undissolved in water and parallel to the alternator load, resulted in significant vibration within the alternator and turbine bearings, negatively impacting their capacity. To address this issue, an air admission system was added, increasing power production from 44 MW to 49.5 MW (an increase by 11.11%). The air admission rate can be manually adjusted through a valve located on top of the system, which was fully open at Darca-1. After installing the air admission system, the generator can produce power within the safe vibration range at full load. Lowering the vibration amplitude of turbine bearings improves their operating time and safety. At maximum air injection rates, relative radial vibration decreased by up to 40%, indicating promising results for turbine safety.
Muntean et al. 
went beyond the regimes with the highest turbine efficiency and considered seven different operating conditions in the range Q = 0.29 QBEP
, where QBEP
is the optimal flow rate. The test case corresponds to a medium specific speed Francis turbine with dimensionless specific speed n
In order to register pressure pulsations at different points in the conical part of the draft tube, several pressure transducers were mounted; the signal from them was analyzed using the Fourier spectra in order to identify the fundamental frequency and the associated amplitude. Three operating ranges in which air injection has fundamentally different effects can be distinguished. At high flow rates (>0.81 QBEP), a small influence of air injection is revealed; when the turbine operates between 0.29 QBEP and 0.53 QBEP with air injection, the dynamic behavior is deteriorated. In particular, turbine operation near Q = 0.42 QBEP air injection can lead to mechanical problems. The most positive impact was found near the operating point Q = 0.69 QBEP at which air injection significantly improved the dynamic behavior.
Nakashima et al. 
studied the impact of air injection on turbine surge and performance using the Francis turbine model test facility. Forced aeration at Q = 0, 0.5, and 1% was performed downstream of the diffuser inlet at a distance of 0.5 draft tube throat diameter. Pressure pulsations were measured in the cross-section using several transducers so that one could differentiate the frequency associated with cavitation and vortices from the frequency of the unsteady vortex by signal analysis. The aeration suppresses the vortex rope behavior and the cavitation surge by half at Q = 0.5% and fully suppresses at Q = 1%, but the aeration increases the loss in the diffuser and decreases the efficiency of the water turbine by several percent.
Sometimes studies are conducted not only on the prototypes or Francis turbine models but also on very simplified geometries in order to better focus on one effect and exclude the influence of other factors. The results of experimental modeling of the swirling flow in the simplified Turbine-99 draft tube model are presented by Skripkin et al. 
. The research was carried out on a single swirl parameter corresponding to the part-load regime provided by a stationary swirler. Dimensional vortex frequency was significantly changed when the gas content was varied from 0 to 5%. Although no data were presented for vortex rope suppression and pressure pulsations, these results may be useful in terms of flexible control of an unsteady vortex if the precession frequency needs to be changed under the resonance conditions.
Data on vortex rope control in simplified geometry can be found in the study by Tănasă et al. 
. The hydraulic system was utilized to produce a flow pattern that mimicked the one experienced by a Francis runner operating at partial discharge. On the other hand, the swirl generator, unlike the turbine runner, featured a stationary and rotating blade annular section generating a swirling flow. The air system was utilized to introduce air into the inlet of the conical diffuser via a nozzle. In order to examine the impact of air admission, the authors installed eight pressure sensors in four different cross-sections. This allowed them to decompose the signal from pressure transducers into two distinct components: the plunging/synchronous and rotating/asynchronous components. The interaction between the helical vortex and the draft tube bend led to asynchronous fluctuation. The amplitude of the rotating component linked to the vortex rope gradually decreased until it reached Qair
/Q ratio = 1% and almost disappeared at Qair
/Q = 1.7%. The synchronous component behaved in a similar manner but did not disappear completely even at high gas contents. It can be assumed that the frequency jump above 1% in the plot at Qair
/Q was related to low-frequency oscillations of the gas cavity rather than to vortex rotation, since the synchronous component should have already been completely suppressed.
Unterluggauer et al. 
, as well as, previously, Türkmenoğlu 
, studied an important parameter such as turbine bearing vibration via the acceleration probes in the Francis turbine prototype with a speed factor ned
= 0.39. It was assumed that the unusual location would be advantageous in mitigating the erosive action and reducing the dynamic impact on the entire runner. Operating conditions at p
= 63% of the nominal turbine power were chosen for air impact testing. By using the acceleration sensors in two planar (x, y) directions, one can observe the effect of damping and vibration elimination at the hydraulic turbine bearing. However, the z-directional vibrations of the runner were not significantly impacted by air admission. Reduction of radial bearing vibration could well be due to suppression of the vortex rope whose influence in the radial direction due to its rotation was higher than that in the z direction. Unfortunately, no information about the amount of air supplied was provided.
The findings reported by Platonov et al. 
demonstrated a significant effect of air injection on the intensity of pressure fluctuations obtained when operating a medium-scale hydrodynamic test bench with a Francis turbine. It was found that when air was injected at operating conditions close to the optimal regimes, a substantial increase in pressure pulsations was observed, while when air was injected under conditions with the maximum pulsations, it could significantly reduce the surge. Experimental results have confirmed that reduction of pressure pulsations was directly related to the amount of air injected into the system. The ideal flow rate of air required to efficiently minimize draft tube pressure pulsations was 1.0% when operating at a guide vane opening of 50% of the optimal value. It was found that this particular flow rate did not affect turbine efficiency, while significantly reducing pressure pulsations in the draft tube.
Bucur et al. 
and Bunea et al. 
proposed an innovative aeration device in a small Francis turbine. Although the main aim of their work was to develop an aeration device to increase the DO level for environmental reasons, the effect of air injection on the turbine’s energy and vibration performance was also considered. The development of the aeration system was preceded by numerical calculation to determine the optimal location for air injection, so that air supply would not lead to additional economic costs. Depending on the operating regime of the turbine, aeration can be either natural (without associated energy consumption) or forced (with compressed air). Although no reduction in turbine efficiency was observed when air was supplied, this method is inefficient in terms of suppressing the pulsations associated with the vortex. Air bubbles had little or no interaction with the vortex core that had already been formed and did not converge in the central dead water zone.
2.2. Numerical Work
Due to the high cost of experimental tests and their inability to provide detailed information on internal flow characteristics, numerical simulations are often utilized to gain a better understanding of the flow mechanisms within turbines. However, research into application of air injection to suppress unsteady vortices in hydraulic turbines using computational fluid dynamics started to emerge later, after a certain level of computational power had been achieved and numerical models describing two-phase flow with acceptable accuracy had been developed. For instance, one can refer to the work by Qian et al. 
who studied the three-dimensional unsteady multiphase flow in the Francis hydraulic turbine. They observed that the air was directed towards the center of the draft tube cone through a hollow shaft ending at the outlet of the runner cone. The computational results revealed that when air was admitted through the spindle hole, it reduced the pressure difference in the horizontal section of the draft tube, thus decreasing the amplitude of low-frequency pressure pulsation. However, the rotor–stator interaction between the air inlet and the runner increased the blade-frequency pressure pulsation in front of the runner. The researchers also found that air in the draft tube rotated with the vortex rope and its fraction was high in the low-pressure areas. Although air admission did not change the dominating frequency in the draft tube, it significantly decreased its amplitude. The lowest amplitude was observed when the air discharge was 0.5%.
Chirkov et al. 
analyzed stability issues in electric power stations that were arising when a cavity was formed behind the runner, resulting in self-excited oscillations throughout the flow duct. They found that a symmetrical cavity can appear when a hydropower station operates at high loads, specifically at power equal to 1.175 times its nominal power. Their numerical model considered air as an incompressible fluid with constant density and solved the basic governing equations of the three-phase model using their CADRUN solver. By varying the gas flow rate Qair
from 0.1 to 0.4% of the liquid discharge rate Q at the center of the runner hub, they observed that at a maximum air flow rate (0.4%), the recirculation zone at the draft tube center expanded to reach the runner fairing, eliminating cavitation in the flow. The air was accumulated below the runner fairing, resulting in minimal pressure pulsations and reducing cavitation phenomena below the runner. Due to its buoyancy, the air also changed the flow pattern, expanding the flow stagnation zone at the draft tube center, leading to flow stabilization. However, their results differed from the experimental data reported by Muntean et al. 
, which showed that air supply had almost no effect on pressure pulsations under full load conditions. In their subsequent paper, Chirkov et al. 
tested their numerical model in the part-load regime by comparing the calculated data with the experimental measurements performed at the Laboratory of Hydraulic Turbines, Leningrad Metal Plant. They observed that their numerical calculations agreed acceptably with the experiment in single-phase flow but failed to adequately describe the unsteady properties of the vortex rope with air injection, making them unreliable to draw any conclusions about the effect of air effect on its frequency and amplitude.
The research by Mohammadi et al. 
, unlike most studies, paid attention to important parameters such as the injection point, number of injection points, and appropriate nozzle diameter in addition to varying the air flow rate. They also considered a combined configuration with an additional water jet from the runner cone. However, the authors neither investigated the influence of injection on unsteady vortex structures nor explained the mechanism of changes in performance. The flow from the spiral case to the end of the draft tube was simulated using the SST k-ω turbulence and two-phase models, which showed the best results. After conducting a parameter search, they selected 72 nozzles with a diameter of 1 cm for 1.5% air injection from the draft tube wall. This configuration increased turbine efficiency up to 4.3% under certain operating conditions and may also be successful in suppressing vortex ropes.
Luo et al. 
and Zhu et al. 
made significant progress in suppressing vortex ropes in Francis turbines with a specific speed of 125 mkW. In order to gather empirical information regarding the hydraulic performance and pressure vibration at customary monitoring locations, a series of model trials were carried out on a test apparatus at Harbin Electric Company Ltd. located in China, Harbin. To simulate the unsteady flow, the Reynolds averaged Navier–Stokes (RANS) procedure was employed in combination with the k-ω SST turbulence model and a homogeneous cavitation model. After conducting mesh independence examination, the computational domain was composed of 3,310,293 mesh elements. Comparisons between the simulation and experiment for the averaged hydraulic performance and instantaneous pressure pulsation showed acceptable agreement 
Initially, air was supplied through fins on the draft tube wall to reduce pressure pulsations on the draft tube wall downstream by 26.3% with 2% air supply. However, upstream pulsations were not affected by this method. Injecting air through the ventilation hole at the runner hub produced the best results. Zhu et al. 
noted that homogeneous distributions of pressure and pressure gradient due to air admission efficiently suppressed pressure vibration in the draft tube. The ventilation rate Q = 0.04 was found to be very efficient in suppressing pressure vibration due to changes in vortex rope geometry from helical to cylindrical ones, while Q = 0.01 had little effect.
The velocity distribution along the diameter shows swirl intensity in the draft tube. The findings suggest that injecting air into the draft tube makes the circumferential velocity distribution more symmetrical and reduces its magnitude, especially at high air flow rates. As aeration volume increases, the backflow area expands and the backflow core moves towards the section center. Since air has lower density than water, the momentum and kinetic energy of the vortex rope decrease rapidly with large aeration volume at the backflow core, as is observed in the case of Q = 4%. It remarkably decreases pressure gradients, and pressure at the vortex core is recovered for the largest aeration volume. Additionally, high-pressure gas contributes to pressure increase in the vortex core and decreases pressure gradients, reducing draft tube wall pulsations in general.
Kim et al. 
conducted a study using the Francis turbine model with Qed
= 0.33 and ned
= 0.48 to investigate vortex rope suppression using an air supply system from the runner cone and anti-swirl fins. Unsteady-state calculations were performed using the ANSYS CFX software, employing the unsteady-state RANS equations supported by the SAS-SST turbulence model. The working fluid consisted of three-phase flows of water, vapor, and air at 298 K. Part-load conditions at Q = 0.78 QBEP
with well-developed vortex rope were considered.
The results showed that an injection of 0.1% Q resulted in a 14% higher magnitude than that obtained with anti-swirl fins alone. The low flow rate of air injection affected the generation of long vortex rope with high swirl number distribution along the flow direction, leading to highly unsteady pressure characteristics. The magnitude decreased significantly by 55% when injection of 0.5% Q was employed.
One of the more recent numerical studies in which the authors tried to explain the mechanism of vortex rope mitigation and pressure pulsation suppression can be found in the paper by Sun et al. 
. Previously, Sun et al. 
reported the successful use of air injection to mitigate inter-blade vortices. They determined that an air volume fraction of 0.7 provided the best balance between reducing the vortex and minimizing energy consumption. By injecting air into water, formation of inter-blade vortices was significantly reduced. This was due to the presence of air cavities in low-pressure flow regions, which disrupted the flow and altered the distribution of streamlines in the runner.
To investigate the influence of air on the unsteady vortex rope behavior, they chose an operating mode with parameters Q11 = 92.49% and n11 = 117.78% that produced 70% rated output power within the partial-load zone. The calculation involved two stages: the first stage dealt with a single-phase problem and the second stage introduced an additional dispersed air volume with a mean diameter of 0.0001 m into the homogeneous multiphase model from the first stage. This allowed the researchers to examine the impact of air admission on the vortex structure at different air volume fractions, ranging from 0 to 3%.
Injecting 1.0% air into the runner cone maintained the visibility and strength of the helical vortex rope, similar to conditions without air. Increasing the air injection to 2.0% caused some disordered and distinct vortex structures in the elbow section, but the helical vortex rope in the cone section remained intact. At 3.0% air injection, the vortex rope disappeared in the draft tube, leaving an umbrella-shaped vortex structure near the inlet of the cone section and a free vortex in the elbow section. As a result, the researchers observed significant drops in pressure amplitude at air volume fractions of 1.0% and 2.0%, with amplitudes decreasing by 41.1% and 49.3%, respectively. No significant pulsations were detected at an air volume fraction of 3.0%.
Overall, the authors concluded that further research into pressure redistribution mechanisms would be needed to improve air supply methods for turbines. They only varied the airflow rate, while keeping injection site and operating conditions unchanged. Nonetheless, this work brings us closer to understanding the complex mechanisms of interaction between unsteady vortices and the multiphase swirling flow.
In contrast to numerical works, analytical approaches for describing two-phase swirling flows in the presence of large-scale vortex structures are extremely rare. This is primarily due to the complexity of constructing models that must account for the interaction between air bubbles of different scales and the vortex core. Furthermore, the lack of quantitative experimental data containing reliable information on the instantaneous velocity and pressure fields for both phases has hindered the development of analytical approaches that could shed light on the complex vortex phenomena. Simple analytical models 
, such as one-dimensional models, typically describe the pulsations associated with changes in the volume of the cavity per se, which is more typical of full-load turbine conditions. These models allow one to consider how filling the cavity with air changes the natural frequency and cavitation compliance, but they do not account for the precession frequency of the vortex rope. Among the few three-dimensional analytical models worth mentioning is the one presented in 
. Kuibin et al. 
utilized the single-phase vortex model 
as a foundation for their two-phase model, which has demonstrated adequate accuracy in describing single-phase flows 
. The frequency of a vortex is a complex parameter that is influenced by various factors. These factors include the vortex curvature, the vortex torsion, the impact of the tube wall, translational motion, and the uneven vorticity distribution within the vortex core. When the air vortex core is filled, changes occur that affect the curvature and vorticity distribution within the core. This shift from a circular to an annular structure has a significant impact on the precession frequency of the vortex.
Geometrical relationships are considered to analyze the impact of the gas phase on the precession frequency of the vortex. It is suggested that the observed decrease in precession frequency during the transition from a pure liquid to a two-phase condition can only be explained by a non-monotonic dependence of the helical vortex pitch on the gas cavity size. This decrease is followed by an increase in frequency as the gas flow rate increases.
Without delving into the mathematical calculations presented by Kuibin et al. 
, which predicts the behavior of the precession frequency as a function of the volume flow rate. Despite several simplifications and assumptions, the gas–liquid vortex model adequately describes the frequency response according to the experimental data reported by Shtork et al. 
Further development of the model requires extensive experimental data on the vortex structure, such as the vortex core size, precession radius velocity at the vortex axis, and pitch of the helical structure in different engineering applications and flow configurations. However, for the vortex control purpose, precession frequency is rarely studied due to its small variation in experimental works falling within the range of 0–3% gas content, beyond which pressure pulsations associated with the vortex rope are typically suppressed.