In a recent series of papers, Poll and Schumann have been developing a simple model for estimating fuel burn for turbofan powered, civil transport aircraft for a given mass, Mach number and flight level and in a specified ambient temperature profile for all phases of flight. This paper focuses upon the combination of Mach number and flight level at which an aircraft cruises with the absolute minimum fuel burn. For a given aircraft type, the information necessary to determine these conditions must be specified and this poses a challenge. An initial attempt to obtain these data has been described previously by the first author. In this paper, the optimum conditions are found using a completely different approach. Starting from first principles and using established theory, the equations governing the situation where engine overall efficiency and airframe lift-to-drag ratio both have local maxima at the same flight condition are developed. This special case is termed the “design optimum” condition and, for a specified aircraft mass and a specified atmospheric temperature versus pressure profile, it gives the lowest possible fuel burn for any aircraft and engine combination. The design optimum occurs at a particular Mach number and Reynolds number, and it is a fixed characteristic of the aircraft. The analysis reveals the significance of Reynolds number variations, wave drag, including its derivatives with respect to both lift coefficient and Mach number, and the atmospheric properties. Whilst wave drag is notoriously difficult to determine accurately, it is found that solutions to the equations are not particularly sensitive to the accuracy of this quantity. Consequently, a simple, physically realistic model can give good results. An appropriate model is developed and a complete, approximate solution is obtained. Taking the International Standard Atmosphere as the design atmosphere, results are presented for the 53 aircraft types previously considered by Poll and Schumann. Relative to the design optimum conditions, when Reynolds number is constant and wave drag is zero, compressibility alone reduces L/D by about 5%, reduces lift coefficient by about 1.5% and increases drag coefficient by about 3.5%. Reynolds number variation has little effect upon L/D, but it reduces lift coefficient and drag coefficient by a further 7% and 8% respectively. The reduction in lift coefficient has a significant impact on the optimum cruise flight level. In general, an aircraft’s operating optimum will not coincide with its design optimum, but deviations are expected to be small. Therefore, using the design optimum solution as a reference point, an improved version of the operating optimum estimation method described by Poll and Schumann in previous work is developed. This allows the estimation of the conditions for absolute minimum fuel burn for an aircraft of given mass flying thorough any atmosphere. Updated coefficients for the 53 aircraft types are given.

中文翻译：

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涡扇动力民用运输机绝对最小燃油消耗的条件和简单的波阻模型

在最近的一系列论文中，波尔和舒曼一直在开发一个简单的模型，用于估算涡轮风扇动力民用运输飞机在给定质量、马赫数和飞行高度以及所有飞行阶段的指定环境温度曲线下的燃油消耗。本文重点研究飞机以绝对最低燃油消耗巡航时的马赫数和飞行高度的组合。对于给定的飞机类型，必须指定确定这些条件所需的信息，这构成了挑战。第一作者之前已经描述了获取这些数据的初步尝试。在本文中，使用完全不同的方法找到了最佳条件。从第一原理出发，利用已建立的理论，建立了控制发动机整体效率和机身升阻比在相同飞行条件下均具有局部最大值的情况的方程。这种特殊情况被称为“设计最佳”条件，对于指定的飞机质量和指定的大气温度与压力曲线，它为任何飞机和发动机组合提供了尽可能最低的燃油燃烧。设计最优发生在特定的马赫数和雷诺数，并且它是飞机的固定特性。分析揭示了雷诺数变化、波阻力（包括其关于升力系数和马赫数的导数）以及大气特性的重要性。虽然众所周知，波浪阻力很难准确确定，但人们发现方程的解对该量的精度并不是特别敏感。因此，一个简单的、物理上真实的模型可以给出良好的结果。开发适当的模型并获得完整的近似解。以国际标准大气为设计大气，给出了波尔和舒曼先前考虑的53种机型的结果。相对于设计最佳条件，当雷诺数恒定且波阻为零时，仅压缩性会降低L/D约5%，升力系数减少约1.5%，阻力系数增加约3.5%。雷诺数变化影响不大L/D，但升力系数和阻力系数分别进一步降低了 7% 和 8%。升力系数的降低对最佳巡航飞行高度有显着影响。一般来说，飞机的最佳运行状态与其设计最佳状态不一致，但偏差预计会很小。因此，以设计最优解为参考点，开发了 Poll 和 Schumann 在先前工作中描述的操作最优估计方法的改进版本。这允许估计给定质量的飞机在任何大气层中飞行的绝对最小燃油消耗的条件。给出了 53 种飞机类型的更新系数。