A vortex ring is a flow phenomenon involving a complex toroidal vortex system. When a rotor is in descending flight, it may enter the vortex ring state (VRS), which will affect the rotor’s aerodynamic characteristics and even endanger it. In this paper, to clarify the aerodynamic mechanism by which the vortex ring exerts its effects on an axial descending rotor, an unsteady numerical simulation method for the rotor integrated with structured moving overset grids is proposed and validated using experimental data. This numerical simulation method is then applied to analyze the aerodynamic characteristics of the rotor in descending flight. The variations of the aerodynamic forces and the flow characteristics in the slipstream are analyzed to elucidate the physical mechanisms responsible for the relationships between the aerodynamic loads, flow field, and vortices when the rotor is in the VRS. The effects of the VRS cause a sharp drop in the average aerodynamic forces, which directly affects the safety and reliability of a rotor aircraft. In the slipstream of the descending rotor, a distinct vortex ring forms and moves upward as the velocity of descent increases. The most severe VRS occurs at a nondimensional velocity of descent of 1.13 when the center of the highly unsteady vortex ring is right at the blade tip, and this can be used as an indicator of the VRS. The physical mechanism by which the VRS exerts its effects can be attributed to the low pressure induced by strong vortices. The VRS decreases the pressure on the lower surface of the blade and increases the pressure on the upper surface, resulting in a reduction in the aerodynamic loads. In comparison with the hovering state, the VRS results in a much larger vortex strength and possesses a different vortex structure.

c

Blade chord (m)

Cm

Moment coefficient

Cp

Pressure coefficient

CT

Thrust coefficient

d

Root cut (m)

N

Blade number

Q

Value of Q criterion

R

Rotor radius (m)

r

Spanwise position at rotor (m)

S̃

Strain rate tensor

V

Velocity of descent (m/s)

Vh

Hovering equivalent induced velocity (m/s)

Vi

Nondimensional velocity of descent

x

Chordwise position at blade (m)

z

Axial position below rotor disk (m)

Greek
θc

Collective pitch (deg)

θt

Blade twist (deg)

σ

Blade solidity

Ω

Rotational speed (rad/s)

Ω̃

Rotation rate tensor

The vortex ring is a flow phenomenon that occurs widely in the natural world: dolphins blow vortex ring bubbles in the water,1 dandelions rely on a stable separated vortex ring to enable their seeds to disperse over distances of several miles,2 and birds take advantage of vortex rings around their wings to fly farther.3 The behavior of vortex rings has also been widely exploited in technological applications.4 However, the generation of vortex rings also bring about hazards to helicopters,5 especially when the rotor is in descending flight. The rotating blades rely on the downwash flow to produce thrust. In descending flight, the direction of the inflow is up forward. Particularly when the velocity of descent approaches the induced velocity, the flow around the rotor may exhibit a qualitative change. There will be a considerable loss of thrust in this circumstance, in what is called the vortex ring state (VRS).6 Reference 7 discussed the nature and mechanism of the VRS. The highly unsteady toroidal vortex near the rotor that occurs in VRS is dangerous for helicopters. It will reduce the blades’ angle of attack and result in loss of thrust and stability.

It is therefore essential to obtain a deeper understanding of the VRS as a basis on which to develop methods to increase the safety and reliability of rotor aircraft. In the past, wind-tunnel experiments were the main tools to investigate the VRS.8 The available wind tunnel and flight test data for rotors in the VRS were reviewed by Ref. 9. The unsteady aerodynamic characteristics of a helicopter model operating in the VRS were experimentally investigated by Refs. 10 and 11. Reference 12 used smoke-flow visualization to observe the VRS phenomenon in the neighborhood of a helicopter rotor. Smoke-flow visualizations of VRS were also conducted by Ref. 13, who observed the reverse flow and toroidal vortex structure on rotors. Reference 14 used a cantilever machine to conducted a series of model tests on a rotor in vertical and oblique descent and determined the boundary of the VRS. References 15 and 16 used particle image velocimetry (PIV) to measure the flow field around a rotor in axial descent and observed the flow characteristics when the rotor entered the VRS. Reference 17 also used PIV to observe the changes in thrust during the development and termination of the VRS. Other methods such as high-resolution shadowgraph photography18 have also been used to measure the VRS in descending flight. Experiments at different descent rates were performed with a twin-rotor model, in which the effects of wake interaction on the VRS were analyzed.19 Experiments on an annular jet flow have been performed to simulate the VRS of a rotor system, and it has been found that it is the flow interaction rather than the trailing vortices that drives the VRS.20 

For VRS theoretical analysis, traditional momentum theory and blade element theory will not work, because they ignore viscous and unstable factors.21 Johnson extended momentum theory to establish a dynamical model of the VRS.6 However, this model is not universal, and it gives only rough results in real-time simulations. The vortex particle method has also been used to analyze the VRS on a descending eVTOL rotor.22 Developments in computational fluid dynamics (CFD) have opened up a new path for the simulation of vortex rings.23 CFD has advantages in the simulation of compressible and viscous flows and represents a promising tool for the dynamic modeling of complex descending rotors. References 24 and 25 used CFD to investigate the VRS of a helicopter tail rotor. An unsteady CFD analyses have been carried out to investigate the influence of descent rates on the formation of vortex rings and to determine the conditions under which vortex rings will occur.6,26

Most previous research has focused on the aerodynamic effects of the VRS, and little work has been done to elucidate the mechanism by which the flow field in the VRS affects aerodynamic characteristics. To remedy this deficiency, in this paper, an advanced CFD method integrated with an overset grid technique is developed to shed further light on the detailed flow field features and associated phenomena in the VRS. An unsteady aerodynamic model of a rotor in descending flight is established and then used to analyze the aerodynamic mechanism by which the vortex ring exerts its effects on an axial descending rotor.

The remainder of this paper is organized as follows. The numerical simulation methods is presented in Sec. II and then validated in Sec. III. The results of the simulations of a rotor in descending flight are presented and analyzed in Sec. IV. Several conclusions of this study are summarized in Sec. V.

A Caradonna–Tung rotor associated with reliable test data27 has been chosen for this study. The numerical modeling of the rotor is based on solving the Reynolds-averaged Navier–Stokes equations with the finite volume method. A semi-implicit method for the pressure-linked equations is used in the unsteady simulation. The kω shear-stress transport (SST) and Spalart–Allmaras (SA) turbulence models are compared in the rotor simulation. Figure 1 shows the distributions of the blade chordwise pressure coefficient using these two turbulence models in comparison with experimental data. It can be seen that the simulation based on the k-ω SST turbulence model has higher computational accuracy, especially on the upper surface. Therefore, the k-ω SST model is used in the present modeling.

FIG. 1.

Comparison of blade chordwise pressure coefficient distributions obtained using different turbulence models in comparison with experimental results (θc = 8°, Ω = 1250 rpm, r/R = 0.8).

FIG. 1.

Comparison of blade chordwise pressure coefficient distributions obtained using different turbulence models in comparison with experimental results (θc = 8°, Ω = 1250 rpm, r/R = 0.8).

Close modal

An overset mesh technique is carried out to simulate the unsteady flow of rotating rotor. The overset mesh system is composed of fixed background grids and moving blade overset grids, as shown in Fig. 2. The background grids are divided into two parts, an inner part with dense structured grids, and an outer far-field grid. The rotating blade grids are all structured grids. The wall spacing at the first layer is 0.1 mm. A total of 30 layers are designed for the boundary near the wall. The growth rate of adjacent grids on the boundary layers is set as 1.15. At the airfoil profile, a C-block grid is designed to form boundary layers around the rotor blade, as shown in Fig. 3.

FIG. 2.

Overset mesh system: (a) background grids and component overset grids; (b) blade overset grids.

FIG. 2.

Overset mesh system: (a) background grids and component overset grids; (b) blade overset grids.

Close modal
FIG. 3.

Structured C-block grids for the blade.

FIG. 3.

Structured C-block grids for the blade.

Close modal

Velocity inlet and pressure outlet boundary conditions are set at the outer background domain. The rotating blades are set as wall boundary conditions. Overset boundary conditions are applied to the outside domains of the blade grids. The rotational velocity of the blade overset grids is equal to the real velocity of rotation.

The numerical method is validated using previous experimental data on the Caradonna–Tung rotor from Ref. 27. The main geometric parameters of the Caradonna–Tung rotor are listed in Table I. The blade has constant chord length and zero twist. In the validation, a numerical simulation is performed to solve for the flow field of the Caradonna–Tung rotor in hover.

TABLE I.

Geometric parameters of Caradonna–Tung rotor.

Airfoil NACA0012 
Blade number N 
Rotor radius R 1.143 m 
Blade chord c 0.1905 m 
Root cut d 0.1R 
Blade twist θt 0° 
Blade solidity σ 0.1061 
Airfoil NACA0012 
Blade number N 
Rotor radius R 1.143 m 
Blade chord c 0.1905 m 
Root cut d 0.1R 
Blade twist θt 0° 
Blade solidity σ 0.1061 

Three mesh sizes (2.4 × 106, 4.8 × 106, and 7.2 × 106) are generated to validate mesh independence. The CFD simulation results will be compared with the experimental thrust coefficient CT in hover, which is 0.004 59 at a collective pitch θc of 8° and a rotational speed Ω of 1250 rpm.27 Results for CT and relative calculation errors for the different mesh sizes are listed in Table II.

TABLE II.

Thrust coefficients for different mesh sizes (θc = 8°, Ω = 1250 rpm).

Mesh sizeStep sizeCTError in CT (%)
2.4 × 106 0.0002 0.004 747 3.42 
4.8 × 106 0.0001 0.004 675 1.85 
7.2 × 106 0.000 05 0.004 639 1.10 
Mesh sizeStep sizeCTError in CT (%)
2.4 × 106 0.0002 0.004 747 3.42 
4.8 × 106 0.0001 0.004 675 1.85 
7.2 × 106 0.000 05 0.004 639 1.10 

The calculation errors for CT are 3.42%, 1.85%, and 1.4% for mesh sizes of 2.4 × 106, 4.8 × 106, and 7.2 × 106, respectively. The calculating for the 4.8 × 106 and 7.2 × 106 mesh sizes are both less than 2% and thus within the acceptable tolerance. Therefore, the structured overset mesh with a total cell number of 4.8 × 106 is used in all the following simulations.

Figures 4 and 5 illustrate the blade chordwise pressure coefficient CP distributions at two locations r/R = 0.68 and 0.96. The rotor in Fig. 4 is operating at a rotational speed of 1500 rpm and a collective pitch of 0° and the rotor in at 1250 rpm and 8°. The dots are the experimental data from Ref. 27 and the curves are the CFD results. In Fig. 4, the simulated results are in good agreement with the experimental data. In Fig. 5, when θc = 8°, the CFD predicts the results well at the lower blade surface, an only a few discrepancies appear on the upper blade surface at the leading edge. Thus, the accuracy of the numerical simulation method in calculating the surface pressure distribution is validated.

FIG. 4.

Blade chordwise pressure coefficient distributions (θc = 0°, Ω = 1500 rpm): (a) r/R = 0.68; (b) r/R = 0.96.

FIG. 4.

Blade chordwise pressure coefficient distributions (θc = 0°, Ω = 1500 rpm): (a) r/R = 0.68; (b) r/R = 0.96.

Close modal
FIG. 5.

Blade chordwise pressure coefficient distributions (θc = 8°, Ω = 1250 rpm): (a) r/R = 0.68; (b) r/R = 0.96.

FIG. 5.

Blade chordwise pressure coefficient distributions (θc = 8°, Ω = 1250 rpm): (a) r/R = 0.68; (b) r/R = 0.96.

Close modal

To analyze the combined effects of the downwash flow induced by the rotating rotor and the upward inflow, the numerical simulation must be able to accurately predict the flow in the rotor’s slipstream. To validate the calculations for the slipstream, the numerical simulation method used above is applied to a different, but similar, two-bladed rotor, because there is a lack of experimental data on the slipstream of the Caradonna–Tung rotor. The parameters of this second rotor are given in Table III.28 It runs at a rotational speed of 122.2 rad/s and a blade pitch angle of 11°. Figure 6 compares the predicted dynamic pressure distributions and experimental data along the blade radial direction (r/R) at different axial positions below the rotor disk: z/R = 0.215 and 0.66. On the whole, the simulation results and experimental data agree well. This proves that the proposed CFD method is effective in simulating the rotor slipstream, which provides the basis for the analysis of the flow field around the rotor.

TABLE III.

Parameters of two-bladed rotor.

Airfoil NACA0012 
Blade number N 
Rotor radius R 0.914 m 
Blade chord c 0.1 m 
Root cut d 0.25R 
Blade twist θt 0° 
Collective pitch θc 11° 
Blade solidity σ 0.071 
Airfoil NACA0012 
Blade number N 
Rotor radius R 0.914 m 
Blade chord c 0.1 m 
Root cut d 0.25R 
Blade twist θt 0° 
Collective pitch θc 11° 
Blade solidity σ 0.071 
FIG. 6.

Slipstream dynamic pressure at different axial locations below the rotor disk: (a) z/R = 0.215; (b) z/R = 0.66.

FIG. 6.

Slipstream dynamic pressure at different axial locations below the rotor disk: (a) z/R = 0.215; (b) z/R = 0.66.

Close modal
FIG. 7.

Average thrust and moment coefficients (within one round of convergence) vs nondimensional velocity of descent.

FIG. 7.

Average thrust and moment coefficients (within one round of convergence) vs nondimensional velocity of descent.

Close modal
The CFD method validated above is applied to analyze the rotor aerodynamic characteristics in descending flight. The rotor operates at a rotational speed Ω = 1250 rpm, and a collective pitch θc = 8°. Here, the velocity of descent V is nondimensionalized by the hovering equivalent induced velocity Vh, which is defined as
Vh=ΩRCT2.
(1)

The nondimensional velocity of descent is then defined as Vi = V/Vh. To analyze the aerodynamic characteristics, the rotor is simulated under inflows in the range Vi = 0–2.55, with intervals of 0.28. Figure 7 shows the simulated average thrust and moment coefficients Cm within one round of convergence at different velocities of descent. When Vi < 0.57, the thrust and moment coefficients grow slowly with increasing velocity of descent. However, when Vi approaches 0.85, CT and Cm decrease sharply, and then fall to their lowest values when Vi = 1.13. The sharp drops in aerodynamic forces and moments during the range Vi = 0.8–1.7 result from the occurrence of a VRS. This VRS emerges during the range Vi = 0.8–1.1, reaching its full development during the range Vi = 1.1–1.7. Finally, the rotor recovers from the VRS when Vi > 1.2.

The transition of the flow structure in the rotor slipstream is responsible for the sharp decrease in the aerodynamic loads on the rotor. Figure 8 shows the streamlines around the rotor blades at Vi = 0, 0.57, 1.13, and 2.26. A distinct unsteady vortex ring forms in the slipstream of the rotor and moves upward as Vi increases. A VRS appears when the vortex ring approaches the tip of the blade. When Vi = 1.13, the most severe VRS occurs, with the center of the vortex ring being located right at the blade tip. When Vi = 2.26, the vortex ring moves away from the rotor disk. At that time, the rotor recovers from the VRS and enters a windmill state.

FIG. 8.

Streamlines around rotor blades for different velocities of descent: (a) Vi = 0; (b) Vi = 0.57; (c) Vi = 1.13; (d) Vi = 2.26.

FIG. 8.

Streamlines around rotor blades for different velocities of descent: (a) Vi = 0; (b) Vi = 0.57; (c) Vi = 1.13; (d) Vi = 2.26.

Close modal

Figure 9 presents the definition of the coordinates used in the following discussion. Figure 10 shows the changes in the position of the vortex core relative to the blade disk plane in the inflow direction (Y/R) as the velocity of descent varies. As the velocity of descent increases in the range Vi = 0–0.85, the vortex core moves closer close to the blade disk plane. When the rotor steps enters the VRS at velocities Vi = 0.85–1.7, the relative position Y/R is within in the range of −0.09 to 0.27. As the vortex ring approaches the rotor, it starts to affect the aerodynamic forces on the blades. These effects become most severe when the vortex core moves right up next to the blade.

FIG. 9.

Definition of coordinates.

FIG. 9.

Definition of coordinates.

Close modal
FIG. 10.

Position of vortex core relative to blade disk plane in the inflow direction vs velocity of descent.

FIG. 10.

Position of vortex core relative to blade disk plane in the inflow direction vs velocity of descent.

Close modal

The vortices are characterized by low pressure, especially in the cores, as shown in Fig. 11. The green low-pressure regions below the blades in Figs. 11(a) and 11(b) are induced by the vortices. In Fig. 11(c), the lower surfaces of the blades are surrounded by regions of low pressure induced by the vortex ring, which results in a decrease in aerodynamic loads. In the windmill state, as shown in Fig. 11(d), the low-pressure regions induced by the vortex ring are far above the blades. In that state, the effects of the vortex ring on the blades’ aerodynamic loads disappear.

FIG. 11.

Pressure contours around rotor blades at different velocities of descent: (a) Vi = 0; (b) Vi = 0.57; (c) Vi = 1.13; (d) Vi = 2.26.

FIG. 11.

Pressure contours around rotor blades at different velocities of descent: (a) Vi = 0; (b) Vi = 0.57; (c) Vi = 1.13; (d) Vi = 2.26.

Close modal

To facilitate a deeper analysis of the physical mechanism underlying the relationship between the flow field and the aerodynamic loads, Figs. 12 and 13 show the pressure contours and pressure coefficient distributions at two chordwise slices on the blades: a tip slice at r/R = 0.96 and a middle slice at r/R = 0.68. The aerodynamic lift on the rotor is the resultant of the pressures on the upper and lower surfaces of the blades. In the VRS, the low-pressure region induced by the vortex ring is close to the lower surface of the blade, as shown in Fig. 12(b). The VRS decreases the pressure on the lower surface and increases the pressure on the upper surface, as shown in Fig. 13 (denoted by the orange squares), which results in a reduction in the aerodynamic loads. From examining the pressure distributions on the two slices shown in Figs. 12 and 13, it can be seen that the effects of the VRS on the blade tip (r/R = 0.96) are stronger than those on the inner blade (r/R = 0.68).

FIG. 12.

Pressure contours on the slices r/R = 0.68 and r/R = 0.96 for different velocities of descent.

FIG. 12.

Pressure contours on the slices r/R = 0.68 and r/R = 0.96 for different velocities of descent.

Close modal
FIG. 13.

Blade chordwise pressure distributions at (a) r/R = 0.68 and (b) r/R = 0.96 for different velocities of descent.

FIG. 13.

Blade chordwise pressure distributions at (a) r/R = 0.68 and (b) r/R = 0.96 for different velocities of descent.

Close modal

Figure 14 presents a comparison of the magnitudes of the vortices when the rotor is in the hovering state and the VRS, and Fig. 15 presents a comparison of the corresponding vortex structures. In Fig. 14, the vorticity magnitude contours are shown at spanwise X = 0 and chordwise r/R = 0.96. Most of the region surrounding the blade in the VRS [Figs. 14(b) and 14(d)] has a larger vorticity magnitude than that surrounding the blade in the hovering state [Figs. 14(a) and 14(c)], especially at the blade tip, which is outlined by the black dashed oval in Fig. 14(b).

FIG. 14.

Vorticity magnitude contours at spanwise X = 0 and chordwise r/R = 0.96 when the rotor is in the hovering state and in the VRS: (a) Vi = 0, slice at X = 0; (b) Vi = 1.13, slice at X = 0; (c) Vi = 0, slice at r/R = 0.96; (d) Vi = 1.13, slice at r/R = 0.96.

FIG. 14.

Vorticity magnitude contours at spanwise X = 0 and chordwise r/R = 0.96 when the rotor is in the hovering state and in the VRS: (a) Vi = 0, slice at X = 0; (b) Vi = 1.13, slice at X = 0; (c) Vi = 0, slice at r/R = 0.96; (d) Vi = 1.13, slice at r/R = 0.96.

Close modal
FIG. 15.

Vortex structure according to the Q criterion and velocity contours when the rotor is in the hovering state and in the VRS: (a) Vi = 0, Q = 0.0008; (b) Vi = 1.13, Q = 3000.

FIG. 15.

Vortex structure according to the Q criterion and velocity contours when the rotor is in the hovering state and in the VRS: (a) Vi = 0, Q = 0.0008; (b) Vi = 1.13, Q = 3000.

Close modal
To describe the vortex characteristics in the VRS, the Q criterion29 is used, which is defined as
Q=12Ω̃2S̃2,
(2)
where Ω̃ is the rotation rate tensor and S̃ is the strain rate tensor. Figure 15 compares the vortex structures in the hovering state and the VRS using the Q criterion. As can be seen, the main vortex system induced by the rotating blades includes a blade root vortex, a blade tip vortex, and the main vortex ring. In the hovering state, the vortices are relatively weak, and Q is set as 0.008 to exhibit more vortices, whereas in the VRS, there is a strong vortex ring, and Q is set as 3000 to identify the main vortices. Besides the obvious contrast in vortex strength, the structure of the vortex system differs significantly between the hovering state and the VRS. In the hovering state, as shown in Fig. 15(a), the blade tip vortex and the main vortex ring are below the blade disk plane. In particular, the main vortex ring is far below the blades in this state. In the VRS, as shown in Fig. 15(b), a much stronger main vortex ring is induced, located not far above the blade disk plane.

A numerical simulation method integrated with a moving overset grid technique has been applied to analyze the aerodynamic characteristics of a descending rotor. The results of validation simulations show that the error in calculating CT is 1.85%. The predicted chordwise pressures and slipstream dynamic pressures agree well with experimental data. Rotors with nondimensional velocity of descent in the range Vi = 0–2.55 have been simulated. The variations in the average aerodynamic forces and the detailed flow characteristics have been analyzed. The overall results can be summarized as follows.

  1. The average thrust and moment coefficients exhibit a sharp drop when the nondimensional velocity of descent is in the range Vi = 0.8–1.7, with their lowest values occurring at Vi = 0.8, which is a consequence of the occurrence of the VRS. The most obvious feature induced by the VRS is a sharp drop in the aerodynamic forces on the rotor, which directly affects the safety and reliability of a rotor aircraft.

  2. In the slipstream of the descending rotor, a distinct vortex ring forms and moves upward as the velocity of descent increases. When the rotor enters the VRS, the relative position Y/R is within in the range of −0.09 to 0.27. The most severe VRS occurs at Vi = 1.13 when the center of the vortex ring is right at the blade tip. The location of the vortex ring core can be considered as one of the indicators determining whether the rotor enters the VRS.

  3. The physical mechanism underlying the effects of the VRS can be attributed to the low pressure induced by strong vortices. In the VRS, the low-pressure region induced by the vortex ring is close to the lower surface of the blade, especially at the blade tip. The VRS decreases the pressure on the blade’s lower surface and increases the pressure on the upper surface.

  4. In the VRS, the vortex strength is much greater. The structure of the vortex system differs significantly between the hovering state and the VRS. In the VRS, a much stronger main vortex ring is present immediately above the blade disk plane.

This work was supported by the National Natural Science Foundation of China (Grant No. 12202384) and the Rotor Aerodynamics Key Laboratory Foundation of the China Aerodynamics Research and Development Center (Grant No. RAL202302-2).

The authors have no conflicts to disclose.

Lifang Zeng: Supervision (equal); Writing – original draft (equal). Zeming Gao: Data curation (lead); Formal analysis (lead). Tianyu Xu: Writing – review & editing (lead). Liangquan Wang: Methodology (lead); Validation (lead).

The authors confirm that the data supporting the findings of this study are available within the article.

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