# Numerical Study of Coaxial Main Rotor Aerodynamics in Steep Descent

^{*}

## Abstract

**:**

_{R}= 30–90° and the rate of descent in the range of V

_{y}= 0–30 m/s are considered. The calculations have been carried out under the condition of a fixed time-average thrust of the rotor. The visualization of the rotor wake shapes the flow structures using streamlines, and the flow velocities have been received and analyzed. The VRS boundaries in “V

_{x}–V

_{y}” coordinates have been constructed. The criteria used in this paper are: rotor thrust and torque pulsations, rise of rotor torque and induced velocity relative to the hovering mode. The results of the calculations are compared with the experimental and calculated data of other authors, and a satisfactory match has been obtained. The new results presented in this paper can supplement the existing experience of experimental and numerical research in this field.

## 1. Introduction

_{x}–V

_{y}” has been introduced into the helicopter flight operation manuals, where it is possible for the main rotor to fall into the VRS modes. Thus, the determination of the VRS boundaries in the coordinates “V

_{x}–V

_{y}” is an important task.

_{R}≈ 85° have been performed. Such criteria of the VRS modes as thrust pulsations, growth of torque and induced velocity have been found. In the work by Mohd and Barakos (2017) [26] on the basis of the finite volume method (FVM), using the URANS (Unsteady Reynolds Averaged Navier Stokes Equations), the VRS modes for two model rotors have been studied. This work considers axial flow modes (α

_{R}= 90°). The drop and pulsations in thrust at fixed blade pitch angles have been obtained. In the work by Brown et al. (2002) [27], comparative studies of single-rotor aerodynamics in steep descent modes have been performed using Leishman’s free-vortex model and Brown’s vorticity transport model (VTM). The boundaries of the VRS modes have been found according to the criterion of thrust pulsations. Kinzel et al. (2019) [28] have performed a numerical simulation of the axial descent for the rotors of unmanned aerial vehicles based on the URANS method. The coaxial rotor and equivalent conventional rotor have been considered. In both cases, a decrease in thrust has been obtained, and this indicates the rotor’s falling into the VRS region. It has been shown that, in the case of coaxial rotors, the VRS mode occurs at higher descent speeds, and the thrust drop is less than in case of equivalent conventional rotors.

_{x}–V

_{y}” coordinates. On the other hand, in the VRS modes, the aerodynamic characteristics of the rotor differ in their unsteadiness. Therefore, in order to deeply analyze the rotor aerodynamics in the VRS modes, it is necessary to simulate a large (sometimes up to a hundred or more) number of revolutions of the rotor.

_{R}= 90°). In the proposed study, the simulation of steep descent modes with different angles of attack of the rotor α

_{R}= 70, 50 and 30° has been performed. Together with the results of [29], this made it possible to conduct a comprehensive analysis of the coaxial rotor aerodynamics in the VRS modes and receive their boundaries in the “V

_{x}–V

_{y}” coordinates using various criteria.

## 2. Method and Object of Study

_{L}and the drag force C

_{D}of the blade element are determined at the found angle of attack α and the total flow velocity W based on airfoil steady test data in WT. The system of vortex contours creates a free vortex wake in the form of a grid of longitudinal and transverse vortex segments (see Figure 1). The vortex wake grid is deformed at each calculated step under the influence of external and induced velocity fields.

_{x}–V

_{y}” coordinates, it is necessary to calculate the aerodynamic characteristics of the rotor for a number of operating modes. These modes form a grid in the coordinates of the vertical descent speed V

_{y}and the forward flight speed V

_{x}. Such a grid is shown in Figure 3. The grid includes calculation points for the angles of attack of the rotor α

_{R}= 90, 70, 50 and 30°. Data for α

_{R}= 90° had been previously obtained in work [29]. Calculations for the angles of attack of the rotor α

_{R}= 70, 50 and 30° have been performed in the presented study. In total, more than 70 operating modes have been calculated in this way.

_{T}

_{Σ}= 0.015). This approach requires a lot of time and computational resources compared to the fixed blade pitch angles approach. Meanwhile, it allows obtaining and analyzing a large number of VRS modes criteria including the important criterion of the increase in the required rotor’s power at fixed thrust.

_{Q}

_{UR}= C

_{Q}

_{LR}. In [29], this approach has been used for vertical descent modeling (α

_{R}= 90°). Taking into account the blade pitch angles’ dependencies, presented in Figure 4, it became possible to ensure the balancing of the rotors in most operating modes, with the exception of high descent speed values close to the “autorotation” mode.

## 3. Results and Discussion

#### 3.1. Wake and Flow Structures Analysis of Coaxial Rotor in Step Descent

_{R}= 70, 50 and 30°. Each figure shows the three most characteristic operating modes of the rotor corresponding to different vertical descent speeds V

_{y}. These operating modes have been identified by a comprehensive analysis of the rotor’s aerodynamic characteristics on the whole studied modes set. The first is the most intensive “peak” VRS mode. The second is one of the TWS modes, which follows the VRS modes with an increase in descent speed. The third is the “autorotation” mode, when the rotor rotates due to an external incoming flow without using the power of the power plant. In this case, the total torque of the coaxial rotor is zero C

_{Q}

_{URΣ}= 0.

_{R}= 70°. The following characteristic operating modes are presented: V

_{y}= 10 m/s (VRS mode), V

_{y}= 16 m/s (TWS mode) and V

_{y}= 22 m/s (“autorotation” mode). The characteristic vertical descent rates V

_{y}have less values than for vertical descent (at α

_{R}= 90° characteristic vertical descent rates were V

_{y}= 12, 18 and 23 m/s, respectively [29]). As with α

_{R}= 90°, an “air body” with a circulating flow inside is formed around the rotor in the VRS and TWS modes. However, the flow structure is complicated due to its asymmetry relative to the oY axis. In this case, the flow structure becomes asymmetric not only when viewed from the side, but also when viewed from the top. This asymmetry is especially clearly visible in the vortex wake shapes visualization for V

_{y}= 10 m/s and V

_{y}= 16 m/s (Figure 5a,b). In addition, the front side of the wake maintains a clear structure and the back side is blurred. This is clearly seen in the “vortex ring” mode (Figure 5a, V

_{y}= 10 m/s). Thus, the wake structure in these modes is a “half-vortex-ring”. Another feature is related to the rotor flow structure. Figure 5a,b shows that vortex wake and “air body” are inclined relative to the rotor’s rotational plane and are perpendicular to the incoming flow. This feature is also observed at the other steep descent modes discussed below.

_{R}= 50°. The following vertical descent speeds are presented: V

_{y}= 6 m/s (VRS mode), V

_{y}= 12 m/s (TWS mode) and V

_{y}= 18 m/s (“autorotation” mode). The presented “peak” mode of the VRS observed here at the V

_{y}value is less than half of what it is at α

_{R}= 90° [29]. It can be seen that, at V

_{y}= 6 m/s (Figure 6a), the structure of the vortex wake and the flow structure significantly differ from those observed at the α

_{R}= 90° [29] and α

_{R}= 70°. There is no clear “air body” around the rotor (Figure 6b, V

_{y}= 6 m/s). In the front of the rotor, there is a concentration of the vortex wake in the “half-ring”, located directly in the rotor’s rotational plane (Figure 6a, V

_{y}= 6 m/s, top view). This vortex structure causes a powerful circulating flow in the front side of the rotor (Figure 6b, V

_{y}= 6 m/s). On the back side of the rotor, the vortex wake is blurred. There is no concentrated circulating flow, and the flow from the rotor is directed downward, almost as in the hovering mode [29]. Thus, the front and back sides of the rotor actually work in different flow conditions. At V

_{y}= 12 m/s, even more complex flow structures are observed in the TWS mode (Figure 6a,b, V

_{y}= 12 m/s). A few circulating zones are located near the plane of rotation of the rotors at once. In the “autorotation” mode (V

_{y}= 18 m/s), the shape of the wake and the structure of the flow are significantly simplified. The vortex wake, going up, has a fairly regular helix shape. A flow braking zone appears behind the rotor. The flow around the rotor has a similar structure to the flow around a “flat plate” with a hole in the middle.

_{R}= 30°. Here, are presented the following velocities of descent: V

_{y}= 4 m/s (VRS mode), V

_{y}= 7 m/s (TWS mode) and V

_{y}= 11.5 m/s (“autorotation” mode). It can be noted that the features of the flow and vortex wake structures in these modes mostly repeat the features received for α

_{R}= 50°. At the same time, these characteristic modes have been observed at lower vertical descent velocities. For example, the “peak” of the VRS mode has been registered here at the value of V

_{y}= 4 m/s, which is three times less than at α

_{R}= 90° [29] and almost two times less than at α

_{R}= 70°.

#### 3.2. Coaxial Rotor’s Thrust and Torque Coefficients’ Time-Dependencies in Steep Descent

_{x}–V

_{y}” coordinates.

_{T}= f(n) and C

_{Q}= f(n) for α

_{R}= 70, 50 and 30°. These dependencies are demonstrated for the modes discussed in Section 3.1. For comparison, diagrams in Figure 8, Figure 9, Figure 10 and Figure 11 also include the total thrust and torque coefficients in the hovering mode (red dotted lines). Since the coaxial rotor is considered, thrust and torque dependencies are given both for the UR and LR separately and their sum in total. For the UR and LR, the averaged dependencies of C

_{T}and C

_{Q}are given. For the total curves C

_{T}

_{(UR+LR)}and C

_{Q}

_{(UR+LR)}, in addition, instantaneous pulsations are presented. The instantaneous thrust and torque pulsations at hover and high descent rate V

_{y}(“autorotation” and WBS modes) have a frequency of ≈30 Hz. They are associated with interference between the UR and LR blades [29]. The instantaneous thrust and torque pulsations in the VRS and TWS modes have two–three times greater amplitude than in hovering. They have an irregular character due to the unsteady flow around the rotor and are mainly associated with the rotation of blades inside a complex asymmetric vortex wake structure [29]. Instantaneous thrust and torque pulsations in the VRS modes have a frequency of about ≈15 Hz and, therefore, form continuous regions on the given graphs. Due to the high frequency, instantaneous pulsations form solid areas in the diagrams, colored in light green.

_{T}= f(n) and C

_{Q}= f(n) obtained at α

_{R}= 70° for velocities V

_{y}= 10 m/s (VRS mode), V

_{y}= 16 m/s TWS mode) and V

_{y}= 22 m/s (“autorotation” mode). Unsteady pulsations of averaged thrust and torque coefficients with a period of several rotor’s revolutions as well as high-frequency pulsations of their instantaneous values are observed. These pulsations are the most significant at the VRS (Figure 8a) and TWS (Figure 8b) modes. It can be noted that, in the VRS mode at α

_{R}= 70°, the amplitude of thrust pulsations increases and the pulsation period decreases compared to α

_{R}= 90° [29]. In the “autorotation” mode (Figure 8c), the thrust and torque pulsations are less significant than in the VRS and TWS modes, and the total torque of the coaxial rotor is zero. It can also be noted that, in the VRS (Figure 8a) and TWS (Figure 8b) modes, the torque of the upper and lower rotors are balanced. Moreover, it follows from the diagrams C

_{T}= f(n) in Figure 8 that the time-averaged (averaged over number of n) thrust coefficient values for each of the modes are constant and amount to C

_{T}

_{Σ}≈ 0.015. As it has been mentioned above, this is achieved by selecting the appropriate blade pitch angles (see Figure 4). Figure 9 and Figure 10 demonstrate, similar to the diagrams in Figure 8, dependencies for the angles of attack α

_{R}= 50 and 30°. At α

_{R}= 50° (Figure 9a) and α

_{R}= 30° (Figure 10a), the amplitude of the pulsations continues decreasing and the pulsations become more complex and aperiodic. In the TWS modes, the amplitude of thrust pulsations decreases with a decrease in the angle of attack of the rotor α

_{R}(Figure 9b and Figure 10b). The torque pulsations in the TWS modes for all angles of attack of the rotor (Figure 8b, Figure 9b and Figure 10b) have greater amplitudes than in the VRS modes. At the “autorotation” modes, small pulsations of the rotor thrust and torque are observed at all considered angles of attack of the rotor, except at α

_{R}= 30° (Figure 10c). The increase in the rotor torque observed in the VRS modes (see Figure 8a, Figure 9a and Figure 10a) is relative to the hovering mode at a fixed thrust and has the following values: 111% at α

_{R}= 70°; 108%, at α

_{R}= 50°; 103% for α

_{R}= 30°.

_{T}= f(n) and C

_{Q}= f(n) for α

_{R}= 30–70° in Figure 8, Figure 9 and Figure 10, it follows that, with a decrease in the angle of attack of the rotor α

_{R}, the characteristic indication (criteria) for the VRS modes weakens significantly: the amplitudes of the thrust and torque pulsations decrease; the rise in the rotor torque (required power) in comparison with the hovering mode becomes less.

_{y}and at all considered angles of attack of the rotor (including the data of work [29]) makes it possible to determine the boundaries of the VRS modes in the coordinates “V

_{x}–V

_{y}”.

#### 3.3. Total Aerodynamic Characteristics of the Rotor Analysis

_{T}

_{Σ}≈ 0.015 = constant. This made it possible to estimate the increase in the rotor torque compared to the hovering mode, which is very important and significant for practical results.

_{Q}

_{Σ}/C

_{Q}

_{Σh}= 90, 70, 50 and 30°, the separate dependencies of the torque coefficients of upper C

_{Q}

_{UR}and lower C

_{Q}

_{LR}rotors and total torque C

_{Q}

_{Σ}on the non-dimensional vertical descent speed ${\tilde{V}}_{y}$ are presented. Here, ${\tilde{V}}_{y}$ is the vertical descent speed V

_{y}, related to the induced velocity in the hovering mode: ${\tilde{V}}_{y}={V}_{y}/v$

_{y}. The data used for α

_{R}= 90° (Figure 11a) had been previously obtained in [29]. The data for α

_{R}= 70, 50 and 30° (Figure 11b–d) have been obtained in the presented study. In the VRS modes, a characteristic increase in the rotor’s torque is relative to the hovering mode. It can also be seen from the diagrams in Figure 11 that, with a decrease in the angle of attack of the rotor, the non-dimensional vertical rate of descent ${\tilde{V}}_{y}$ associated with the “autorotation” mode (C

_{Q}

_{Σ}= 0) at first occurs slightly, and then sharply decreases. For α

_{R}= 90° and α

_{R}= 70°, these velocities are close and equal to ${\tilde{V}}_{y}$ ≈ 1.66 and ${\tilde{V}}_{y}$ ≈ 1.6, respectively. At α

_{R}= 50°, ${\tilde{V}}_{y}$ ≈ 1.23 (22% less), and at α

_{R}= 30°, the speed of “autorotation” ${\tilde{V}}_{y}$ ≈ 0.83, which is almost two times less than at α

_{R}= 70°. In addition, from Figure 11a–d, it follows that the use of obtained blade pitch angles laws (see Figure 3) made it possible to balance the coaxial rotor in torque (C

_{Q}

_{UR}= C

_{Q}

_{LR}) almost over the entire range of the V

_{y}speeds except the “autorotation” modes. The analysis of the VRS boundaries according to the criterion of rotor torque (power) growth (C

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1) will be given below.

**Figure 11.**Dependencies of the UR, LR and total (UR + LR) torque coefficients C

_{Q}vs. non-dimensional vertical descent speed ${\tilde{V}}_{y}$ at α

_{R}= 30–90°. (

**a**) α

_{R}= 90°; (

**b**) α

_{R}= 70°; (

**c**) α

_{R}= 50°; (

**d**) α

_{R}= 30°.

_{Q}

_{Σ}/C

_{Q}

_{Σh}on the non-dimensional vertical descent speed ${\tilde{V}}_{y}$, obtained on the basis of data from Figure 11. From Figure 12, it can be seen that, at first, with an increase in the speed ${\tilde{V}}_{y}$ (for all α

_{R}values), a decrease in the relative torque coefficient is observed. This means that the power consumption of the rotor decreases, reaching values of ≈0.92 (at α

_{R}= 90° and ${\tilde{V}}_{y}$ ≈ 0.4) from the power in hovering mode. Further, there is an increase in relative torque coefficient C

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1 in a certain range of vertical descent speed ${\tilde{V}}_{y}$. The region where C

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1 is the region of the VRS modes. With a further increase in vertical descent speed, the relative torque coefficient decreases again. At C

_{Q}

_{Σ}/C

_{Q}

_{Σh}= 0, the rotor enters the “autorotation” mode, and at C

_{Q}

_{Σ}/C

_{Q}

_{Σh}<0, the rotor enters to the WBS modes. It can be seen from the presented diagrams that the region of the VRS modes decreases significantly with a decrease in the angle of attack α

_{R}, and also shifts to lower values of the vertical descent speed ${\tilde{V}}_{y}$. For α

_{R}= 90°, the boundaries of the VRS modes according to the C

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1 criterion observed within ${\tilde{V}}_{y}$ ≈ 0.6–1.1, for α

_{R}= 70° within ${\tilde{V}}_{y}$ ≈ 0.48–0.85, for α

_{R}= 50° within ${\tilde{V}}_{y}$ ≈ 0.34–0.48 and for α

_{R}= 30° within ${\tilde{V}}_{y}$ ≈ 0.26–0.3. The points of the curves C

_{Q}

_{Σ}/C

_{Q}

_{Σh}= f (${\tilde{V}}_{y}$) where the coefficients C

_{Q}

_{Σ}/C

_{Q}

_{Σh}reach maximum values correspond to the “peak” modes of the VRS. These modes have been analyzed in detail in Section 3.1 and Section 3.2 of the article. The maximum increase in the relative torque coefficient reaches 22% (C

_{Q}

_{Σ}/C

_{Q}

_{Σh}≈ 1.22) at α

_{R}= 90° and ${\tilde{V}}_{y}$ ≈ 0.87. At α

_{R}= 70° and α

_{R}= 50°, the increase in torque is 11% (C

_{Q}

_{Σ}/C

_{Q}

_{Σh}≈ 1.11) and 8% (C

_{Q}

_{Σ}/C

_{Q}

_{Σh}≈ 1.08), respectively. At α

_{R}= 30°, the increase in torque is about 3% (C

_{Q}

_{Σ}/C

_{Q}

_{Σh}≈ 1.03). Thus, in the descent modes with angles α

_{R}= 70–90°, the signs of the VRS modes and the associated negative outcomes such as an increase in the rotor power consumptions are manifested most strongly.

_{T}

_{UR}/C

_{T}

_{LR}= f(${\tilde{V}}_{y}$) for α

_{R}= 30–90°. In the hovering mode, the C

_{T}

_{UR}/C

_{T}

_{LR}≈ 1.22 [29]. Further, in a certain range of vertical descent speeds ${\tilde{V}}_{y}$, the value of the C

_{T}

_{UR}/C

_{T}

_{LR}gradually increases to values of 1.3–1.4. With an increase in the speed of ${\tilde{V}}_{y}$, when the rotor falls into the VRS modes area, the value of the C

_{T}

_{UR}/C

_{T}

_{LR}sharply decreases and reaches the value of the C

_{T}

_{UR}/C

_{T}

_{LR}≈ 1 in the “peak” modes of the VRS. After that, with a further increase in the speed ${\tilde{V}}_{y}$, the ratio of the thrust of the UR and LR changes—the thrust of the lower rotor becomes higher than the thrust of the UR. In addition, Figure 13 shows the C

_{T}

_{UR}/C

_{T}

_{LR}curve obtained for α

_{R}= 90° in [7] for C

_{T}

_{Σ}/σ = 0.16. There is a satisfactory agreement with the obtained calculation results.

_{R}= 90° (see Figure 14a) had been previously obtained in [29], and the dependencies for α

_{R}= 70, 50 and 30° (see Figure 14b–d) have been obtained in the presented study. Taking into account the thrust pulsations, this diagram can be represented as an area, as shown in Figure 14. An increase in the induced velocity ${\tilde{\upsilon}}_{y}$ above a threshold value can also serve as one of the criteria for the VRS modes boundary. In addition to the calculated curves ${\tilde{\upsilon}}_{y}$ = f(${\tilde{V}}_{y}$), Figure 14 also shows a number of experimental dependencies obtained by various authors [6,7,10,11,12,16,17]. These are experiments performed for different conditions, different types and sizes of rotors and various experimental rigs. In [7], a model of a coaxial rotor in the WT has been studied, in [6,12], a model of a single rotor has been installed on a moving track and, in [11], rotors with a large blade twist (not typical for helicopter rotors) have been tested. For this reason, the experimental data have significant differences. Nevertheless, for all the angles of attack of the rotor, there is a good qualitative agreement between the calculation and the experimental data. Based on the analysis of the calculated data (Figure 14), we note that, with a decrease in the angle α

_{R}, the maximum time-averaged value ${\tilde{\upsilon}}_{y}$ decreases and shifts to the left. Thus, at α

_{R}= 90°, the non-dimensional induced velocity reaches the value of ${\tilde{\upsilon}}_{y}$ ≈ 2.3 at ${\tilde{V}}_{y}$ ≈ 1 (Figure 14a), at α

_{R}= 70° of the value of ${\tilde{\upsilon}}_{y}$ ≈ 2 at ${\tilde{V}}_{y}$ ≈ 0.7 (Figure 14b), at α

_{R}= 50° of the value of ${\tilde{\upsilon}}_{y}$ ≈ 1.6 at ${\tilde{V}}_{y}$ ≈ 0.45 (Figure 14c), and at α

_{R}= 30° of the value of ${\tilde{\upsilon}}_{y}$ ≈ 1.4 at ${\tilde{V}}_{y}$ ≈ 0.3 (Figure 14d). The boundaries of the VRS modes for the criterion ${\tilde{\upsilon}}_{y}$ > 1.5 observed within the limits are: ${\tilde{V}}_{y}$ ≈ 0.7–1.0 at α

_{R}= 90°, ${\tilde{V}}_{y}$ ≈ 0.7–1.0 at α

_{R}= 70° and ${\tilde{V}}_{y}$ ≈ 0.7–1.0 at α

_{R}= 50°. At α

_{R}= 30°, there is no VRS mode within the selected criterion (Figure 14d).

_{T}= f(n) and the torque coefficients C

_{Q}= f(n), taking into account the pulsations, have been presented in Figure 8, Figure 9 and Figure 10. The analysis of such dependencies allows for estimating the changes in the thrust and torque pulsations for all considered operating modes (Figure 3). The amplitude of the thrust coefficient pulsations is defined as ΔC

_{T}

_{Σ}/2 = (C

_{T}

_{Σmax}− C

_{T}

_{Σmin})/2. It is easy to express this as a percentage of the time-averaged thrust, which is equal for all calculated modes C

_{T}

_{Σ}≈ 0.015 = constant. The amplitude of the torque coefficient pulsations is defined as ΔC

_{Q}

_{Σ}/2 = (C

_{Q}

_{Σmax}− C

_{Q}

_{Σmin})/2. Here, it is expressed as a percentage of the total torque of the rotor in hovering mode.

_{T}

_{Σ}/2 = (${\tilde{V}}_{y}$) for α

_{R}= 30–90°. Here, in addition to the calculation results, some experimental data [11,12,14] are also given. These data have been obtained for various experimental conditions and various model-scale single rotors. In the introduction section, it has been shown that, nowadays, there is no such published experimental or calculated data for full-scale coaxial rotors. However, there is a satisfactory qualitative agreement between the results of calculations and the experimental data presented. The greatest thrust pulsations are observed at α

_{R}= 90° (Figure 15a), and they reach an amplitude of ΔC

_{T}

_{Σ}/2 = 40% at ${\tilde{V}}_{y}$ ≈ 1–1.1. With a decrease in the angle α

_{R}, the amplitude of thrust pulsations decreases significantly and shifts to lower values of ${\tilde{V}}_{y}$. At α

_{R}= 70° (Figure 15b), the maximum amplitude of thrust pulsations reaches the value ΔC

_{T}

_{Σ}/2 ≈ 30% at ${\tilde{V}}_{y}$ ≈ 0.73, at α

_{R}= 50° (Figure 15c) ΔC

_{T}

_{Σ}/2 ≈ 24% at ${\tilde{V}}_{y}$ ≈ 0.5 and ${\tilde{V}}_{y}$ ≈ 0.95 and at α

_{R}= 30° (Figure 15d) ΔC

_{T}

_{Σ}/2 ≈ 20% at ${\tilde{V}}_{y}$ ≈ 0.3. Further, the criterion ΔC

_{T}

_{Σ}/2 > 20% has been taken to determine the boundaries of the VRS modes.

_{Q}

_{Σ}/2 = (${\tilde{V}}_{y}$) for α

_{R}= 30–90°. The largest torque pulsations are observed at α

_{R}= 90° (Figure 16a) and reach an amplitude of ΔC

_{Q}

_{Σ}/2 ≈ 22% at ${\tilde{V}}_{y}$ ≈ 1. At α

_{R}= 70° (Figure 16b), the point of the maximum amplitude of the torque pulsations is shifted to the right by the value of vertical descent speed ${\tilde{V}}_{y}$ ≈ 1.24, and reaches a value of ΔC

_{Q}

_{Σ}/2 ≈ 19%. At α

_{R}= 50° (Figure 16c), the pulsations reach ΔC

_{Q}

_{Σ}/2 ≈ 18% in the range of ${\tilde{V}}_{y}$ ≈ 0.7–1. At α

_{R}= 30° (Figure 16d), the torque pulsations decrease by almost two times their original value and reach ΔC

_{Q}

_{Σ}/2 ≈ 10% at ${\tilde{V}}_{y}$ ≈ 0.44. Further, the criterion ΔC

_{Q}

_{Σ}/2 > 10% has been taken to determine the boundaries of the VRS modes.

#### 3.4. Total Aerodynamic Characteristics of the Rotor Analysis

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1 and C

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1.05; an increase in the non-dimensional induced velocity ${\tilde{\upsilon}}_{y}$ > 1.5; the amplitude of pulsations of the total thrust coefficient ΔC

_{T}

_{Σ}/2 > 20% (compared with the average value of C

_{T}

_{Σ}≈ 0.015 = constant); the amplitude of the pulsations of the total torque coefficient ΔC

_{Q}

_{Σ}/2 > 10% (compared to the total hovering torque). The diagram in Figure 17 shows that approximate areas with different fills are plotted around the calculated points corresponding to the boundaries of the VRS mode region according to various criteria for various angles α

_{R}= 30–90°.

_{R}= 90°). For the steep descent modes (α

_{R}= 50–70°), the calculated boundary of the VRS modes according to the criterion of thrust and torque pulsations is significantly lower than the experimental one, and the calculated boundary of the VRS modes according to the criterion of torque growth is satisfactorily consistent with the experimental curve obtained according to the criterion of the thrust drop. At α

_{R}= 30°, the calculated VRS boundary is located at lower rates of descent.

## 4. Conclusions

_{R}= 70°, 50° and 30° in the range of vertical descent speeds V

_{y}= 0–30 m/s have been considered. The aerodynamic characteristics of the rotor at the fixed time-averaged thrust (C

_{T}

_{Σ}= 0.015 = constant) have been calculated. This approach makes it possible to analyze the growth of the required power of the rotor in the VRS modes area in comparison with the hovering mode. The presented work is a continuation of the studies performed earlier for hovering and vertical descent modes (α

_{R}= 90°). This paper is the main and final part of a comprehensive series of studies.

_{y}and the angles of attack of the rotor α

_{R}. The shapes of the vortex wake and the structure of the rotor flow have been presented and analyzed in a number of the most characteristic modes: the “peak” of VRS, TWS and “autorotation”. A number of features in the forming of the vortex wake and the structure of the flow around the rotor associated with different descent speeds and angles of descent have been revealed. For these modes, the dependencies of the thrust and torque coefficients of the upper and lower rotors, as well as their total values of time (the number of revolutions of the rotor n), taking into account their unsteady pulsations, have been obtained and analyzed.

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1 and C

_{Q}

_{Σ}/C

_{Q}

_{Σh}> 1.05; an increase in the induced velocity in the plane of the rotor disk ${\tilde{\upsilon}}_{y}$ > 1.5; an amplitude of the pulsation of the rotor thrust coefficient ΔC

_{T}

_{Σ}/2 > 20%; an amplitude of the pulsation of the rotor torque coefficient ΔC

_{Q}

_{Σ}/2 > 10%. If necessary, the boundaries of the VRS area can be refined further by changing the selected criteria values.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

ρ | Air density, kg/m^{3} |

t | Time, s |

R | Rotor radius, m |

N_{b} | Number of blades |

c | Blade chord, m |

ωR | Rotational speed of the blade tips, m/s |

n | Number of rotor revolutions |

σ | Rotor solidity, N_{b}·c/πR |

θ_{tw} | Blade twist, degree |

θ | Blade pitch angle, degree |

α_{R} | Angle of rotor attack, degree |

V | Free stream velocity, m/s |

V_{y} | Vertical descent speed, m/s |

T | Rotor thrust, N |

Q | Rotor torque, N·m |

C_{T} | Rotor thrust coefficient, (2·T)/(ρ·(ωR)^{2}·πR^{2}) |

C_{Q} | Rotor torque coefficient, (2·Q)/(ρ·(ωR)^{2}·πR^{3}) |

C_{Q}_{h} | Rotor torque coefficient in hover |

$v$_{y} | Vertical component of the averaged induced velocity in the rotor disc plane, m/s |

$v$_{yh} | Induced velocity in the hover, $0.5\omega R\sqrt{{C}_{T}}$ |

$\tilde{v}$_{y} | Non-dimensional induced velocity ${v}_{y}/v$_{yh} |

${\tilde{V}}_{y}$ | Non-dimensional vertical descent velocity, ${V}_{y}/v$_{yh} |

UR | Upper rotor |

LR | Lower rotor |

VRS | Vortex ring state mode |

TWS | Turbulent wake state mode |

WBS | Windmill brake state mode |

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**Figure 8.**Diagrams of the dependence of the thrust C

_{T}= f(n) and torque C

_{Q}= f(n) coefficients vs. the number of revolutions of the rotor at α

_{R}= 70° for different vertical descent speeds: V

_{y}= 10 m/s (

**a**); V

_{y}= 16 m/s (

**b**); V

_{y}= 22 m/s (

**c**).

**Figure 9.**Diagrams of the dependence of the thrust C

_{T}= f(n) and torque C

_{Q}= f(n) coefficients vs. the number of revolutions of the rotor at α

_{R}= 50° for different vertical descent speeds: V

_{y}= 6 m/s (

**a**); V

_{y}= 12 m/s (

**b**); V

_{y}= 18 m/s (

**c**).

**Figure 10.**Diagrams of the dependence of the thrust C

_{T}= f(n) and torque C

_{Q}= f(n) coefficients vs. the number of revolutions of the rotor at α

_{R}= 30° for different vertical descent speeds: V

_{y}= 4 m/s (

**a**); V

_{y}= 7 m/s (

**b**); V

_{y}= 11.5 m/s (

**c**).

**Figure 12.**Dependencies of the relative torque coefficient C

_{Q}

_{Σ}/C

_{Q}

_{Σh}vs. the non-dimensional vertical descent speed ${\tilde{V}}_{y}$ at α

_{R}= 30–90°.

**Figure 13.**Calculated and experimental [7] dependencies of C

_{T}

_{UR}/C

_{T}

_{LR}= f(${\tilde{V}}_{y}$) at α

_{R}= 30–90°.

**Figure 14.**Dependencies of the UR, LR and total (UR+LR) torque coefficients C

_{Q}vs. non-dimensional vertical descent speed ${\tilde{V}}_{y}$ at α

_{R}= 30–90°. (

**a**) α

_{R}= 90°; (

**b**) α

_{R}= 70°; (

**c**) α

_{R}= 50°; (

**d**) α

_{R}= 30°.

**Figure 16.**Calculated dependencies of torque pulsation amplitude ΔC

_{Q}

_{Σ}/2 = f(${\tilde{V}}_{y}$) at α

_{R}= 30–90°. (

**a**) α

_{R}= 90°; (

**b**) α

_{R}= 70°; (

**c**) α

_{R}= 50°; (

**d**) α

_{R}= 30°.

**Figure 17.**Calculated and experimental [7] VRS boundaries based on various criteria.

Name | Value | Measure Unit |
---|---|---|

Rotor radius, R | 7.95 | m |

Blade tip speed, ωR | 226 | m/s |

UR/LR plane distance | 1.495 | m |

Number of blades, N_{b} | 2 × 3 | – |

Rotor solidity, σ | 0.115 | – |

Blade chord, c | 0.48 | m |

Blade twist, θ_{tw} | −5.8 | degree |

Blade airfoil | NACA 230-12 | – |

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**MDPI and ACS Style**

Makeev, P.; Ignatkin, Y.; Shomov, A.
Numerical Study of Coaxial Main Rotor Aerodynamics in Steep Descent. *Aerospace* **2022**, *9*, 61.
https://doi.org/10.3390/aerospace9020061

**AMA Style**

Makeev P, Ignatkin Y, Shomov A.
Numerical Study of Coaxial Main Rotor Aerodynamics in Steep Descent. *Aerospace*. 2022; 9(2):61.
https://doi.org/10.3390/aerospace9020061

**Chicago/Turabian Style**

Makeev, Pavel, Yuri Ignatkin, and Alexander Shomov.
2022. "Numerical Study of Coaxial Main Rotor Aerodynamics in Steep Descent" *Aerospace* 9, no. 2: 61.
https://doi.org/10.3390/aerospace9020061