Aerodynamic Department, Airbus Operations S.A.S.,
空中客车运营公司空气动力学部，
306 Route de Bayonne,  巴约讷路线 306 号，
Toulouse Cedex 9 31000, France
Toulouse Cedex 9 31000， 法国
e-mail: benjamin.francois@cerfacs.fr
电子邮件： benjamin.francois@cerfacs.fr

Michel Costes  米歇尔·科斯特斯
Onera, The French Aerospace Lab,
Onera，法国航空航天实验室，
8 Rue des Vertugadins,  8 Rue des Vertugadins，
Meudon F-92190, France  Meudon F-92190， 法国
e-mail: michel.costes@onera.fr
电子邮件： michel.costes@onera.fr

In the context of increasing costs for fuel, the development of new aircraft designs is mainly driven by the need to reduce fuel burn. To reach this end, new engine concepts such as contrarotating open rotors appear to be one suitable option for the single aisle segment, currently dominated by the Airbus A320 and Boeing 737. This concept was the focus of a large research effort led by NASA and US industry in the late 1970s and 1980s, motivated by the high fuel costs arising from the 1973 oil crisis [1]. Significant advances were achieved, but due to the decrease in oil prices, the interest in bringing those engines to market waned. Presently, the CROR concept appears again to be one promising option for powering the new generation of short-range aircraft.
在燃料成本增加的背景下，新飞机设计的开发主要是出于减少燃料消耗的需求。为了达到这一目的，新的发动机概念，如对转开式旋翼，似乎是目前由空客 A320 和波音 737 主导的单通道细分市场的一个合适选择。这个概念是 NASA 和美国工业界在 1970 年代末和 1980 年代领导的一项大型研究工作的重点，其动机是 1973 年石油危机导致的高燃料成本 [1]。取得了重大进展，但由于油价下跌，将这些发动机推向市场的兴趣减弱了。目前，CROR 概念似乎再次成为为新一代短程飞机提供动力的一个有前途的选择。

This new concept raises major challenges for aircraft manufacturers. One of them is the impact of forces transverse to the rotation axis of the propellers, commonly named in-plane forces ${}^1$${}^1$^(1){ }^{1}, which are caused by a nonhomogeneous inflow velocity flow field in the propeller plane. Such conditions are encountered when the
这一新概念给飞机制造商带来了重大挑战。其中之一是横向螺旋桨旋转轴的力的冲击，通常称为面内力 ${}^1$${}^1$^(1){ }^{1} ，这是由螺旋桨平面中的非均匀流入速度流场引起的。当

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the Journal of Turbomachinery. Manuscript received January 17, 2013; final manuscript received December 11, 2013; published online January 31, 2014. Assoc. Editor: Alok Sinha.
由 ASME 的国际燃气轮机研究所 （IGTI） 投稿，发表在 Journal of Turbomachinery.手稿接收于 2013 年 1 月 17 日;2013 年 12 月 11 日收到最终手稿;2014 年 1 月 31 日在线发布。副编辑：Alok Sinha。
${}^1$${}^1$^(1){ }^{1} The expression $1P$$1P$1P1 P-forces (1P stands for once-per-revolution) can also be found in literature to define the in-plane forces on a propeller. However, the $1P$$1P$1P1 P-forces expression will not be used in this work.
${}^1$${}^1$^(1){ }^{1} 表达式 $1P$$1P$1P1 P -forces（1P 代表每转一次）也可以在文献中找到，用于定义螺旋桨上的面内力。但是， $1P$$1P$1P1 P -forces 表达式不会在这项工作中使用。
far-field inflow has an angle of attack with respect to the rotation axis (incidence, sideslip) or for an installed propeller configuration. Therefore, it is essential to predict these forces for the structural design of the installed engine system. In-plane forces contribute also to the sizing of HTP and VTP because these forces need to be counterbalanced to meet handling quality requirements. Then, in-plane forces can initiate a flutter phenomenon on the blades or on the whole engine system. Finally, these forces are unsteady and may lead to vibrations on the whole aircraft that may degrade the comfort of the passengers and lead to structural fatigue.
远场流入具有相对于旋转轴（入射、侧滑）或已安装螺旋桨配置的攻角。因此，预测这些力对于已安装的发动机系统的结构设计至关重要。平面内力也有助于 HTP 和 VTP 的尺寸，因为需要平衡这些力以满足处理质量要求。然后，面内力会在叶片或整个发动机系统上引发颤振现象。最后，这些力是不稳定的，可能会导致整个飞机的振动，从而降低乘客的舒适度并导致结构疲劳。

In order to predict accurately the in-plane forces on open rotors, a lot of effort is devoted to the development and the validation of methods and tools for high-fidelity aerodynamic simulations and to wind tunnel test campaigns on open rotor configurations. The work presented in this paper focuses on the assessment and the understanding of in-plane forces on an isolated open rotor configuration at high-speed (conditions in which in-plane force magnitude can reach the same order of the thrust level) at nonzero angle of attack.
为了准确预测开式转子上的面内力，投入了大量精力来开发和验证用于高保真空气动力学模拟的方法和工具，以及在开式转子配置上进行风洞测试活动。本文介绍的工作侧重于在非零攻角下高速（面内力大小可以达到推力水平的相同数量级）评估和理解孤立开式转子配置上的面内力。

Numerous works have been done in the past to predict and understand the origin of the in-plane forces around propellers. The in-plane forces on propellers at nonzero angle of attack were pointed out for the first time in 1909 by Lanchester [2]. A few years later, the first basic theories emerged with Harris [3] and Glauert [4] who proposed an analogy with a fin: A propeller at
过去已经做了大量工作来预测和理解螺旋桨周围面内力的来源。Lanchester 于 1909 年首次指出了非零攻角螺旋桨的面内力 [2]。几年后，Harris [3] 和 Glauert [4] 提出了第一个基本理论，他们提出了一个带有鳍的类比：螺旋桨在
nonzero angle of attack develops a normal force in the orthogonal direction of the incoming velocity vector as a fin. Forces and moments arising were expressed with an analytical expression using the thrust and torque of an uninclined propeller. Forces applied on propellers were split into the forces along the rotation axis, commonly called thrust, and the force normal to the rotation axis. In 1935, Glauert [5] proposed a widely used analytical expression of the normal force. For an inclined propeller, this force is a function of the angle of inclination, the advance ratio $J$$J$JJ, the power coefficient $C_p$$C_p$C_(p)C_{p}, and the thrust distribution along the blade. Glauert’s definition considered only the component contained in the vertical plane for an inclined propeller. Lateral force was not taken into account. Comparison of Glauert’s theory to experimental results [6] showed little discrepancies, but this validation was limited up to blade settings angles of 45 deg and advance ratio value of 2.0. Later, this theory was exploited by Ribner [7] who extended it to contra-rotating propellers. From the 1950s to the 1980s, fewer articles about propellers were published, probably due to a decreasing interest in propellers and the emergence of the turbofan technology. These analytical models were rapid methods for the improvement of propellers aerodynamics but were not valid for compressible flows.
非零攻角 （Nonzero Point of Attack of Force） 在传入速度矢量的正交方向上产生法向力作为鳍片。产生的力和力矩使用不倾斜螺旋桨的推力和扭矩用解析表达式表示。施加在螺旋桨上的力分为沿旋转轴的力（通常称为推力）和垂直于旋转轴的力。1935 年，Glauert [5] 提出了一种广泛使用的法向力解析表达式。对于倾斜的螺旋桨，该力是倾斜角、前进比 $J$$J$JJ 、功率系数 $C_p$$C_p$C_(p)C_{p} 和沿叶片的推力分布的函数。Glauert 的定义仅考虑了倾斜螺旋桨的垂直平面中包含的组件。未考虑侧向力。将 Glauert 的理论与实验结果进行比较 [6] 显示差异不大，但这种验证仅限于 45 度的叶片设置角度和 2.0 的提前比值。后来，Ribner [7] 利用了这一理论，并将其扩展到对转螺旋桨。从 1950 年代到 1980 年代，发表的关于螺旋桨的文章较少，这可能是由于对螺旋桨的兴趣下降和涡轮风扇技术的出现。这些分析模型是改善螺旋桨空气动力学的快速方法，但对可压缩流无效。

After the oil crisis, many studies on propellers reemerged in the 1980s with the first three-dimensional steady Euler computations at high speed with the work of Bober et al. [8]. The move to computational procedures for solving the fluid dynamic equations was motivated by two main reasons. First, analytical models were only valid at low-speed and, thus, unfit for transonic flows. Second, Euler computations were accurate means of determining the aerodynamic characteristics of a complex blade for which the threedimensional geometry cannot be handled by existing analytical models. The computations of Bober et al. were performed on isolated propellers at zero angle of attack. The computational domain was reduced to a single blade passage with periodicity boundary conditions. The flow was solved with a steady approach in the rotating frame. Comparison with experiments showed that the power coefficient is overpredicted, but the variation regarding its blade angle was well captured. According to Bober et al., these discrepancies could be attributed to the viscous effects. These methods were extended to contra-rotating propellers by Wong et al. [9] and Nicoud et al. [10]. The rotor-rotor interface was modeled with a mixing-plane condition [11]. First aerodynamic simulations of propellers at nonzero angle of attack at high-speed (Mach number from 0.6 to 0.8 ) were achieved by Nallasamy [12] in 1994. All the blades have to be accounted for in the computation because there are no flow periodicity relations. The flow was solved with an unsteady approach because the inflow seen by the blade varies depending on its azimuthal position. Pressure transducers measurements over a rotation for different radii were compared to the numerical simulation results and showed acceptable discrepancies. Nonetheless, nonlinear variations of the measured pressure were not reproduced by the numerical procedure. Then, unsteady Euler computations of high-speed propellers with aircraft were simulated by Bousquet and Gardarein [13] and were compared to wind tunnel in-plane forces measurements at Mach number 0.7. The comparison showed that normal and lateral forces were underpredicted by $15-20\mathrm{\%}$$15-20\mathrm{\%}$15-20%15-20 \%.
石油危机后，许多关于螺旋桨的研究在 1980 年代重新出现，Bober 等人 [8] 首次进行了高速三维稳态欧拉计算。转向求解流体动力学方程的计算程序有两个主要原因。首先，解析模型仅在低速时有效，因此不适用于跨音速流。其次，欧拉计算是确定复杂叶片空气动力学特性的准确方法，现有分析模型无法处理其三维几何形状。Bober 等人的计算是在零迎角的孤立螺旋桨上进行的。计算域被简化为具有周期性边界条件的单个叶片通道。流动是通过在旋转框架中稳定地解决的。与实验的比较表明，功率系数被高估了，但其叶片角度的变化被很好地捕捉到了。根据 Bober 等人的说法，这些差异可归因于粘性效应。Wong 等[9]和 Nicoud 等[10]将这些方法扩展到对转螺旋桨。转子-转子界面是用混合面条件建模的 [11]。1994 年，Nallasamy [12]首次实现了螺旋桨在高速下非零攻角（马赫数从 0.6 到 0.8）的空气动力学模拟。在计算中必须考虑所有叶片，因为没有流周期关系。该流是用非稳态方法求解的，因为叶片看到的流入量根据其方位角位置而变化。 将不同半径旋转的压力传感器测量值与数值模拟结果进行比较，并显示出可接受的差异。尽管如此，数值程序并未再现测量压力的非线性变化。然后，Bousquet 和 Gardarein [13] 模拟了飞机高速螺旋桨的非定常欧拉计算，并与马赫数为 0.7 的风洞面内力测量进行了比较。比较表明，法向力和侧向力的预测值低。 $15-20\mathrm{\%}$$15-20\mathrm{\%}$15-20%15-20 \%

Improvements in numerical simulations made it possible to account for the viscous effects using a Navier-Stokes solver to study the propellers’ aerodynamics. First, Stuermer [14] achieved advanced three-dimensional Navier-Stokes simulations on isolated open rotors focusing on the aerodynamic performance and in-plane forces at low-speed and high-speed. Zachariadis and Hall [15] focused on the prediction of the rotor performance by investigating the best numerical settings (mesh strategy, boundary conditions) and by comparing them with wind tunnel measurements. These enhancements in the computational approach allowed focus on the acoustic prediction $[16,17]$$[16,17]$[16,17][16,17] and on the prediction of performance and in-plane forces on installed open rotor configurations [18,19]. An important contribution for the simulation of
数值仿真的改进使得使用 Navier-Stokes 求解器来研究螺旋桨的空气动力学来解释粘性效应成为可能。首先，Stuermer [14] 在孤立的开式转子上实现了先进的三维 Navier-Stokes 仿真，重点关注低速和高速下的空气动力学性能和面内力。Zachariadis 和 Hall [15] 通过研究最佳数值设置（网格策略、边界条件）并将其与风洞测量值进行比较，专注于转子性能的预测。计算方法的这些改进使人们可以专注于声学预测 $[16,17]$$[16,17]$[16,17][16,17] 以及对已安装的开式转子配置的性能和面内力的预测[18,19]。对模拟的重要贡献
open-rotors at nonzero angle of attack is the work of Brandvik et al. [20]. It focuses on the interaction of the front rotor wake and tip vortex with the rear rotor for acoustic purposes. However, to the authors’ knowledge, no numerical prediction of the in-plane forces on an open rotor was compared to experimental data. Ortun et al. [21] performed such a comparison but only on an isolated single propeller. In-plane forces results matched quite well with experimental measurements at low-speed but presented larger discrepancies at high-speed, which are not fully understood. In the same work, Ortun et al. [21] also presented an analysis of the origin of the normal and lateral component of in-plane forces applied on a single propeller. Such an analysis was never performed before and enables us to understand which aerodynamic phenomena are at stake. The use of the lifting-line technique coupled with an unsteady wake model (gathered in the HOST code [22]) enable us to deepen and separate the different in-plane forces contributions. However, this comparison and this analysis have never been applied to contra-rotating open rotors.
非零迎角的开转子是 Brandvik 等人 [20] 的工作。它侧重于前转子尾流和尖端涡流与后转子的相互作用，以实现声学目的。然而，据作者所知，没有将开式转子上面内力的数值预测与实验数据进行比较。Ortun 等[21]进行了这样的比较，但仅限于一个孤立的单螺旋桨。面内力结果与低速时的实验测量结果非常吻合，但在高速时呈现出更大的差异，这一点尚不完全清楚。在同一研究中，Ortun 等[21]还分析了施加在单个螺旋桨上的面内力的法向和横向分量的来源。这样的分析以前从未进行过，使我们能够了解哪些空气动力学现象处于危险之中。使用提升线技术与非定常尾流模型（收集在 HOST 代码 [22] 中）相结合，使我们能够加深和分离不同的面内力贡献。然而，这种比较和分析从未应用于对转开式转子。
In this context, the primary objective of this contribution is to provide a physical analysis of the aerodynamics of in-plane forces for a CROR in high speed at nonzero angle of attack using unsteady simulations. High-speed conditions are selected, as they are one of the most critical for an aircraft with respect to in-plane forces because their magnitude can reach the same order as the thrust level in these conditions. As no validation data are available, confidence in the numerical results is built through a code-to-code comparison. This is a first step in the verification process of in-plane forces prediction, consolidating the reliability of the CFD results. Thus, two computational processes for URANS simulations of an isolated open rotor at nonzero angle of attack are compared: computational strategy, open rotor meshing, and aerodynamic results. The comparisons focus on rotor forces, blades thrust, and pressure distributions.
在这种情况下，本贡献的主要目标是使用非定常仿真为高速非零攻角的 CROR 提供面内力空气动力学的物理分析。选择高速条件，因为它们是飞机相对于平面内力最关键的条件之一，因为它们的大小可以达到与这些条件下的推力水平相同的数量级。由于没有可用的验证数据，因此通过代码到代码的比较来建立对数值结果的信心。这是面内力预测验证过程的第一步，巩固了 CFD 结果的可靠性。因此，比较了非零迎角下孤立开式转子的 URANS 仿真的两种计算过程：计算策略、开式转子网格划分和空气动力学结果。比较的重点是转子力、叶片推力和压力分布。
In the second part of this work, an in-depth analysis of the mechanisms contributing to the in-plane forces is proposed. The results of the simulations are used to identify and explain the key aerodynamic phenomena leading to in-plane forces for each stage. Their contributions to the different components (modulus, direction) are discussed.
在这项工作的第二部分，提出了对导致面内力的机制的深入分析。仿真结果用于识别和解释导致每个阶段的面内力的关键空气动力学现象。讨论了它们对不同分量（模量、方向）的贡献。

Two approaches for the unsteady aerodynamic computations of an isolated CROR operating at high-speed conditions $({\mathrm{M}}_{\mathrm{\infty }}=0.73$$\left({\mathrm{M}}_{\mathrm{\infty }}=0.73\right.$(M_(oo)=0.73:}\left(\mathrm{M}_{\infty}=0.73\right., alt $=10,668\mathrm{m}(35,000\mathrm{ft}))$$\left.=10,668\mathrm{m}(35,000\mathrm{ft})\right)${:=10,668(m)(35,000ft))\left.=10,668 \mathrm{~m}(35,000 \mathrm{ft})\right) at angle of attack of 1 deg are presented, both solving the unsteady Reynolds-averaged Navier-Stokes equations.
提出了两种在高速条件下 $({\mathrm{M}}_{\mathrm{\infty }}=0.73$$\left({\mathrm{M}}_{\mathrm{\infty }}=0.73\right.$(M_(oo)=0.73:}\left(\mathrm{M}_{\infty}=0.73\right. 运行的孤立 CROR 的非定常空气动力学计算方法，alt $=10,668\mathrm{m}(35,000\mathrm{ft}))$$\left.=10,668\mathrm{m}(35,000\mathrm{ft})\right)${:=10,668(m)(35,000ft))\left.=10,668 \mathrm{~m}(35,000 \mathrm{ft})\right) 攻角为 1 度，都求解了非定常雷诺平均纳维-斯托克斯方程。
2.1 CFD Solvers. The first CFD solver used for the computations is the elsA [23] code, which solves the compressible RANS equations on multiblock structured grids using a finite volume method. The elsA code has been developed by ONERA since 1997 and codeveloped by CERFACS since 2001. It has been extensively used for turbomachinery, helicopter, and aircraft applications and is the production code of several aeronautical companies (Safran, Airbus, Eurocopter, etc.).
2.1 CFD 求解器。第一个用于计算的 CFD 求解器是 elsA [23] 代码，它使用有限体积法在多块结构网格上求解可压缩的 RANS 方程。elsA 代码自 1997 年起由 ONERA 开发，自 2001 年起由 CERFACS 共同开发。它已广泛用于涡轮机械、直升机和飞机应用，是多家航空公司（赛峰集团、空客、欧洲直升机公司等）的生产规范。

The second CFD solver used for the computations is the ENSOLV code [24,25]. ENSOLV uses a finite volume formulation and multiblock boundary-conforming structured grids. The ENSOLV code was developed since 1988 as a collaboration between NLR, CIRA, and ALENIA. The code has been used for aircraft, helicopter, launcher, ship, and turbomachinery applications as well as acoustic wave propagation problems.
用于计算的第二个 CFD 求解器是 ENSOLV 代码 [24\u201225]。ENSOLV 使用有限体积公式和符合边界的多块结构化网格。ENSOLV 代码自 1988 年以来由 NLR、CIRA 和 ALENIA 合作开发。该代码已用于飞机、直升机、发射器、船舶和涡轮机械应用以及声波传播问题。
2.2 Numerical Setup. For the elsA computations, a centered Jameson scheme [26] with artificial viscosity is used for the spatial discretization. The time integration of the governing equations is based on a dual time stepping (DTS) [27] approach. The scheme for the physical time is a second-order Gear scheme, and the one
2.2 数值设置。对于 elsA 计算，使用具有人工粘度的中心 Jameson 方案 [26] 进行空间离散化。控制方程的时间积分基于双时间步长 （DTS） [27] 方法。物理时间的方案是二阶 Gear 方案，而

Fig. 1 AI-PX7 geometry  图 1 AI-PX7 几何图形

for the fictive time is a first-order backward Euler scheme. The Spalart-Allmaras one-equation turbulence model [28] is used for closure of the RANS equations. A time step convergence study was performed by comparing results obtained with time steps of 0.25 deg and 0.5 deg and showed that the in-plane forces modulus and angle results present discrepancies lower than $0.1\mathrm{\%}$$0.1\mathrm{\%}$0.1%0.1 \%. Thus, the simulations are performed with a time step equivalent to a propeller rotation of 0.5 deg . For the prediction of in-plane forces, the computation is considered converged when the moving average and the root mean square ( rms ) rotor forces vary respectively by less than $0.1\mathrm{\%}$$0.1\mathrm{\%}$0.1%0.1 \% and $1\mathrm{\%}$$1\mathrm{\%}$1%1 \% during two consecutive rotations. To fulfill this criterion, six rotations are performed. For the DTS scheme inner loop, ten subiterations are used and enable us to reach the convergence of the aerodynamic forces, as simulations with 30 subiterations show identical forces. For the implicit scheme, the lower upper symmetric successive over relation (LUSSOR) scheme developed by Yoon and Jameson is used [29]. The computations are initialized with a uniform flow-field.
for the fictive time 是一阶向后欧拉方案。Spalart-Allmaras 单方程湍流模型 [28] 用于 RANS 方程的闭合。通过比较 0.25 度和 0.5 度的时间步长获得的结果，进行了时间步长收敛研究，结果表明面内力模量和角度结果的差异小于 $0.1\mathrm{\%}$$0.1\mathrm{\%}$0.1%0.1 \% 。因此，模拟以相当于螺旋桨旋转 0.5 度 的时间步长执行。对于面内力的预测，当移动平均和均方根 （ rms ） 转子力分别变化小于 $0.1\mathrm{\%}$$0.1\mathrm{\%}$0.1%0.1 \% 和 $1\mathrm{\%}$$1\mathrm{\%}$1%1 \% 连续两次旋转时，计算被认为是收敛的。为了满足此标准，将执行六次旋转。对于 DTS 方案内循环，使用了 10 个子迭代，使我们能够达到空气动力的收敛，因为具有 30 个子迭代的模拟显示了相同的力。对于隐式方案，使用了 Yoon 和 Jameson 开发的下上对称连续关系 （LUSSOR） 方案 [29]。计算使用均匀流场进行初始化。

For the ENSOLV computations, the flow equations are solved using cell-centered finite-volume schemes. The time integration of the governing equations is based on a DTS [27] approach. The scheme for the physical time is a second-order Gear scheme, and the one for the fictive time is a first-order backward Euler scheme. Kok’s $k-\omega$$k-\omega$k-omegak-\omega model [24] is used for the turbulence closure. For the current application, a fourth-order accurate finite volume scheme [25] is used. This scheme is dispersion-relation and symmetry preserving, resulting in low numerical dispersion and dissipation. This property ensures the accurate capturing of propeller slipstreams,
对于 ENSOLV 计算，流动方程使用单元心有限体积方案求解。控制方程的时间积分基于 DTS [27] 方法。物理时间的方案是二阶 Gear 方案，虚构时间的方案是一阶向后欧拉方案。Kok $k-\omega$$k-\omega$k-omegak-\omega 模型 [24] 用于湍流闭合。对于当前的应用，使用了四阶精确有限体积方案 [25]。该方案保持色散关系和对称性，从而产生较低的数值色散和耗散。此特性可确保准确捕获螺旋桨滑流，
propeller tip vortices, and acoustic waves. The convergence is reached when the thrust blade force from two consecutive rotations match. Thus, five rotor rotations are performed with 40 subiterations (simulations with 60 subiterations present identical forces). The simulations are performed with a time step equivalent to a propeller rotation of 0.5 deg . For the implicit scheme, the implicit residual averaging developed by Jameson and Yoon [30] is used. The computations are initialized with a uniform flow field.
螺旋桨尖端涡流和声波。当来自两个连续旋转的推力叶片力匹配时，达到收敛。因此，用 40 个子迭代执行 5 个转子旋转（用 60 个子迭代的模拟呈现相同的力）。模拟以相当于螺旋桨旋转 0.5 度 的时间步长执行。对于隐式方案，使用了 Jameson 和 Yoon [30] 开发的隐式残差平均。计算使用均匀流场进行初始化。
2.3 Computational Domain and Boundary Conditions. Unsteady simulations of propellers at nonzero angles of attack require the use of a full annulus computational domain. No periodicity can be established between flow features of neighboring blades because each blade has a different inflow depending on its azimuthal position. The computational domain is bounded by a cylindrical box.
2.3 计算域和边界条件。在非零迎角下对螺旋桨进行非稳态仿真需要使用完整的环形计算域。无法在相邻叶片的流动特征之间建立周期性，因为每个叶片都有不同的流入，具体取决于其方位角位置。计算域由一个圆柱形框包围。

For elsA computations, its dimensions are 28 times the radius of the front rotor in the axial direction and 13 times the radius of the front rotor in the radial direction. The external boundaries are modeled with a nonreflective condition, which prevents the acoustic waves from reflecting on the external boundaries. The methodology is based on a characteristic relation approach using a gradient technique for the determination of the wave propagation direction (see Couaillier [31]). All the walls (blades and nacelle) are modeled with an adiabatic condition of viscous wall.
对于 elsA 计算，其尺寸是前转子轴向半径的 28 倍，径向前转子半径的 13 倍。外部边界使用非反射条件进行建模，以防止声波在外部边界上反射。该方法基于特征关系法，使用梯度技术来确定波的传播方向（参见 Couaillier [31]）。所有壁（叶片和机舱）都使用粘性壁的绝热条件进行建模。

For ENSOLV computations, the dimensions of the computational domain are 14 times the radius of the front rotor in the axial direction and four times the radius of the front rotor in the radial direction. At the external boundaries, the freestream state vector is imposed. Large cells neighboring the external boundaries enable damping of the reflection of acoustic waves. Walls on blades are modeled with an adiabatic condition of viscous wall while the nacelle is modeled with wall slip condition.
对于 ENSOLV 计算，计算域的尺寸是前转子轴向半径的 14 倍，径向前转子半径的 4 倍。在外部边界处，施加了 freestream 状态向量。与外部边界相邻的大单元可以抑制声波的反射。叶片上的壁是用粘性壁的绝热条件建模的，而机舱是用壁滑移条件建模的。

The dimensions of the computational domain for both codes are consistent with the standards reported in the literature for CROR simulations $[15,16,32]$$[15,16,32]$[15,16,32][15,16,32] : (i) a radial length of the computational domain of at least three times the rotor diameter and (ii) an axial extent of at least seven to eight diameters.
两种代码的计算域尺寸与 CROR 仿真文献中报道的标准一致 $[15,16,32]$$[15,16,32]$[15,16,32][15,16,32] ：（i） 计算域的径向长度至少为转子直径的三倍，以及 （ii） 轴向范围至少为 7 到 8 个直径。

The open rotor test case is the AI-PX7 configuration, pictured in Fig. 1. This is a generic open rotor designed by Airbus [33] and used to test, validate, and develop numerical approaches, within the Clean Sky JTI-SFWA European project in terms of CFD techniques and mesh requirements, and enhance the understanding of the complex aerodynamics around CROR. The geometry is an $11\times 9$$11\times 9$11 xx911 \times 9 bladed pusher configuration with a rotor diameter of $D=4.2672\mathrm{m}(14\mathrm{ft})$$D=4.2672\mathrm{m}(14\mathrm{ft})$D=4.2672m(14ft)D=4.2672 \mathrm{~m}(14 \mathrm{ft}). Inlet and exhaust are not modeled in the nacelle shape. Table 1 gives a short overview of the AI-PX7 configuration features in high-speed conditions.
开式转子测试案例是 AI-PX7 配置，如图 1 所示。这是由空客公司 [33] 设计的通用开式旋翼，用于在 Clean Sky JTI-SFWA 欧洲项目中测试、验证和开发数值方法，包括 CFD 技术和网格要求，并增强对 CROR 周围复杂空气动力学的理解。几何形状是转子直径为 $D=4.2672\mathrm{m}(14\mathrm{ft})$$D=4.2672\mathrm{m}(14\mathrm{ft})$D=4.2672m(14ft)D=4.2672 \mathrm{~m}(14 \mathrm{ft}) 的 $11\times 9$$11\times 9$11 xx911 \times 9 叶片式推杆配置。入口和排气口未以机舱形状建模。表 1 简要概述了高速条件下的 AI-PX7 配置功能。

The blade geometry used in this paper is a blade with a sweep function varying from negative value at low radius to positive values up to the tip, as shown in Fig. 4. The transonic blades need to be swept to decrease the magnitude of the shock structure. The blade has a low thickness-to-chord ratio, similar to current transonic fan blades, and low camber throughout its span. The rear rotor diameter is reduced by $10\mathrm{\%}$$10\mathrm{\%}$10%10 \% relative to the front blade while the rear blade chord is increased in order to generate a thrust of equivalent magnitude for both rotors. The rear rotor cropping enables us to decrease the impact of tip vortices convected from the front rotor to the rear rotor.
本文中使用的叶片几何形状是具有扫掠函数的叶片，从小半径的负值到尖端的正值不等，如图 4 所示。需要扫掠跨音速叶片以减小激波结构的幅度。该叶片具有较低的厚度与弦比，类似于当前的跨音速风扇叶片，并且在整个跨度内具有低外倾角。后转子直径 $10\mathrm{\%}$$10\mathrm{\%}$10%10 \% 相对于前叶片减小，而后叶片弦增加，以便为两个转子产生同等大小的推力。后转子裁剪使我们能够减少从前转子到后转子的尖端涡流的影响。

The nacelle design corresponds to a transonic nacelle design with the objective of minimizing as much as possible the local Mach number seen by the blades. The flow is, thus, only slightly accelerated by the front part of the nacelle. In high-speed conditions, Mach number increases from 0.73 in the freestream to 0.75 just upstream of the front rotor. The nacelle designs used in elsA and ENSOLV simulations for the comparison are slightly different (see Fig. 2). Considering that the maximum cross section of both
机舱设计对应于跨音速机舱设计，其目的是尽可能减少叶片看到的局部马赫数。因此，机舱的前部仅略微加速了流动。在高速条件下，马赫数从自由流中的 0.73 增加到前转子上游的 0.75。elsA 和 ENSOLV 仿真中用于比较的机舱设计略有不同（见图 2）。考虑到两者的最大横截面