Propeller slipstream, or propwash, can significantly affect the aerodynamic characteristics of propeller driven aircraft by providing additional airflow over their aerodynamic and control surfaces. It is therefore essential to have a good knowledge of the induced velocity within the propeller slipstream to determine the aerodynamic forces and moments on slipstream-immersed components. Existing slipstream models based on simple momentum and lifting line theory have limited application since they consider only the acceleration of air within the slipstream and do not take into account the diffusion phenomenon. As such, they yield good results near the propeller where acceleration is dominant but fail to predict induced velocity accurately far behind the propeller where diffusion dominates. This paper presents a slipstream model that takes into account both the acceleration and diffusion phenomena via simple analytical and semi-empirical equations to predict induced velocity accurately up to ~ 8 - 10 propeller diameters downstream of the propeller plane.
螺旋桨滑流或推进洗可以通过在螺旋桨驱动的飞机的气动和控制表面上提供额外的气流来显著影响其气动特性。因此，必须充分了解螺旋桨滑流内的诱导速度，以确定滑流浸没部件上的气动力和力矩，在螺旋桨。基于简单动量和升力线理论的现有滑流模型应用有限，因为它们只考虑滑流中空气的加速度，而没有考虑扩散现象。因此，它们在螺旋桨附近产生了良好的结果附近加速度占主导地位，但在螺旋桨后面扩散占主导地位的地方，它们无法准确预测诱导速度。本文提出了一种滑流模型，该模型通过简单的分析和半经验方程考虑了加速和扩散现象，可以准确预测螺旋桨平面下游约8-10个螺旋桨直径的诱导速度。
I. Introduction
On the other hand, diffusion of propeller slipstream (referred to as propeller jet in marine area) has been researched thoroughly for marine propellers, where the effect of propeller jets on nearby structures including seabed scouring is studied. Researchers in this area are interested in the effects of the propeller jet up to axial distances of ~ 8 propeller diameters from the propeller plane16 and as such, diffusion has to be considered in their works. For this purpose, semi-empirical equations have been developed based on detailed experimental investigation by many researchers. Efforts in this area are summarized in a review by Lam17
另一方面，对船用螺旋桨的螺旋桨滑流扩散（在海洋区域称为螺旋桨射流）进行了深入的研究，研究了螺旋桨射流对附近结构的影响，包括海底冲刷。该领域的研究人员对螺旋桨射流在距离螺旋桨平面约8个螺旋桨直径的轴向距离内的影响感兴趣，因此，他们的工作中必须考虑扩散。为此，许多研究人员在详细的实验研究的基础上开发了半经验方程。Lam17在一篇综述中总结了这方面的努力
This paper presents a simple mathematical model to predict the induced velocity within the propeller slipstream reasonably well up to axial distances of several propeller diameters from the propeller plane. In Section II, a brief discussion is presented on how the propeller slipstream evolves. The underlying phenomena of acceleration and diffusion are discussed alongwith their effect on the slipstream. The developed slipstream model is then discussed in Section III, followed by simulated results in Section IV. Finally the conclusions and future work are presented in Section V.
本文提出了一个简单的数学模型，可以很好地预测螺旋桨滑流中的诱导速度，直到距螺旋桨平面几个螺旋桨直径的轴向距离。在第二节中，简要讨论了螺旋桨滑流是如何演变的。讨论了加速和扩散的基本现象及其对滑流的影响。第三节讨论了开发的滑流模型，第四节讨论了模拟结果。最后，第五节介绍了结论和未来的工作。
II. Evolution of the Propeller Slipstream
Development of an accurate slipstream model requires an understanding of two major phenomena that occur within the slipstream – acceleration and diffusion of the fluid , their cause and effect, and their relative importance in different regions of the slipstream.
开发精确的滑流模型需要了解滑流中发生的两个主要现象——流体的加速和扩散、它们的因果关系以及它们在滑流不同区域的相对重要性。
As air passes through the propeller, it accelerates in the axial, radial and tangential directions due to the pressure force provided by the rotating propeller. This results in an increase in air velocity within the slipstream that must be compensated by a reduction in cross-sectional area of the slipstream for continuity. Therefore, the effect of pressure force is to contract the slipstream. By contrast, diffusion of the slipstream is due to viscosity and turbulence in the air which causes viscous shearing within the slipstream, thereby forming eddies resulting in lateral mixing or diffusion with the ambient flow. As momentum is transferred from the fast-moving slipstream to the slow-moving ambient flow, the air within the slipstream decelerates gradually and is accompanied by an expansion of slipstream for conservation of mass.
当空气通过螺旋桨时，由于旋转螺旋桨提供的压力，它在轴向、径向和切向方向上加速。这导致滑流内的空气速度增加，必须通过减小滑流的横截面积来补偿，以保持连续性。因此，压力的作用是收缩滑流。相比之下，滑流的扩散是由于空气中的粘度和湍流引起的，这会导致滑流内的粘性剪切，从而形成涡流，导致与环境流的横向混合或扩散。当动量从快速移动的滑流转移到缓慢移动的环境流时，滑流中的空气逐渐减速，并伴随着滑流的膨胀以保持质量守恒。
Although both acceleration and diffusion occur simultaneously, their relative importance in different regions of the slipstream determines whether the slipstream will contract or expand. In the vicinity of the propeller, acceleration dominates since the pressure force, due to the rotating propeller, is much larger than the viscous forces and turbulence in this region. Therefore near the propeller, the slipstream contracts. As air moves farther away from the rotating propeller, the pressure force diminishes becoming negligible at some distance behind the propeller. From this point onward, the slipstream no longer contracts but expands under the influence of now relatively larger viscous forces and developed turbulence. The slipstream continues to expand until it becomes weak enough that it cannot be distinguished from the ambient flow.
虽然加速和扩散是同时发生的，但它们在滑流不同区域的相对重要性决定了滑流是收缩还是膨胀。在螺旋桨附近，加速度占主导地位，因为旋转螺旋桨产生的压力远大于该区域的粘性力和湍流。因此，在螺旋桨附近，滑流会收缩。随着空气远离旋转的螺旋桨，压力在螺旋桨后面一定距离处减小到可以忽略不计。从这一点开始，滑流不再收缩，而是在现在相对较大的粘性力和湍流的影响下膨胀。滑流继续膨胀，直到它变得足够弱，无法与环境流区分开来。
III. Propeller Slipstream Model
Based on previous discussion, a good slipstream model can be developed by dividing the slipstream into two regions: a near-field region where diffusion is neglected and only acceleration is considered, and a far-field region where acceleration is neglected and only diffusion is considered as shown in Fig. 2. The two regions are separated by the efflux plane – a transition plane from slipstream contraction to expansion. The induced velocity increases with axial distance from the propeller plane, becomes maximum at the efflux plane, and then decreases. The induced velocity associated with the efflux plane is called the efflux velocity and the corresponsing slipstream radius is called the contracted radius.
基于前面的讨论，可以通过将滑流分为两个区域来开发一个好的滑流模型：一个忽略扩散仅考虑加速度的近场区域，以及一个忽略加速度仅考虑扩散的远场区域，如图2所示。这两个区域由射流平面隔开，射流平面是从滑流收缩到膨胀的过渡平面。诱导速度随着距螺旋桨平面轴向距离的增加而增加，在射流平面处达到最大值，然后减小。与射流平面相关的诱导速度称为射流速度，相应的滑流半径称为收缩半径。
Acceleration in the near-field region causes slipstream contraction which can be analyzed by considering the air to be inviscid and applying first principles of physics such as the momentum theory, Euler’s equations of motion etc. In this way, simple analytical equations have been derived for the induced velocity in the literature18,19. However in the far-field region, where the slipstream expands due to diffusion, one must consider the fluid to be viscous and the flow to be turbulent, making its analysis inherently difficult and therefore no simple analytical equations are found in literature for this case. We must then resort to semi-empirical equations to determine the induced velocity in this region.
近场区域的加速度导致滑流收缩，可以通过考虑空气是无粘性的并应用动量理论、欧拉运动方程等物理学第一原理来分析滑流收缩。通过这种方式，在文献18,19中推导出了诱导速度的简单分析方程。然而，在远场区域，滑流由于扩散而膨胀，必须考虑流体是粘性的，流动是湍流的，这使得其分析本身就很困难，因此文献中没有针对这种情况的简单分析方程。然后，我们必须求助于半经验方程来确定该区域的诱导速度。
As mentioned earlier, air accelerates as it passes through the propeller inducing velocity in all three directions – axial, radial and tangential; but it is well established that the radial and tangential components of the induced velocity are small compared to its axial component18, 20 and are, therefore, neglected in the current work. Further simplification is made by assuming the propeller slipstream to be axisymmetric i.e. the axial component of the induced velocity does not vary in the circumferential direction. Therefore, the main goal of this work is to determine the variation of the axial component of the induced velocity as a function of radial and axial locations within the propeller slipstream.
如前所述，空气在通过螺旋桨时加速，在轴向、径向和切向三个方向上诱导速度；但众所周知，诱导速度的径向和切向分量与其轴向分量18、20相比很小，因此在当前的工作中被忽略了。通过假设螺旋桨滑流是轴对称的，即诱导速度的轴向分量在圆周方向上没有变化，进一步进行了简化。因此，这项工作的主要目标是确定诱导速度的轴向分量随螺旋桨滑流内径向和轴向位置的变化。
A. Induced Velocity at the Propeller Plane
The induced velocity at the propeller plane ( x=0 ) can be found as a function of radial location by applying blade element momentum theory (BEMT) to the propeller. In a companion paper21 on UAV thruster model, two differential thrust equations (one from blade element theory and the other from momentum theory) are equated to yield an expression for the inflow angle :
通过将叶片单元动量理论（BEMT）应用于螺旋桨，可以发现螺旋桨平面（x=0）处的诱导速度是径向位置的函数。在关于无人机推进器模型的配套论文21中，将两个微分推力方程（一个来自叶片单元理论，另一个来自动量理论）等价，得出流入角的表达式：
B. Induced Velocity in the Near-Field Region
As previously alluded, only acceleration of the slipstream needs to be considered in the near-field region for which there are several analytical methods, both simple and detailed, available in literature7-9, 18-20. However, these are not used in the current work to predict induced velocity in the near-field region. Rather, only efflux plane location (i.e. end of near-field region) and contracted radius are found using analytical expressions given in Ref. 19 (done in Section III C). As such, these two quantities are required to evaluate the induced velocity at the efflux plane and in the far-field region (done in Section III D). Once induced velocity is known in all regions except the near field region, interpolation is used to determine the induced velocity in the near-field region. This is discussed in more detail later in Section III E.
如前所述，在近场区域只需要考虑滑流的加速，文献7-9、18-20中提供了几种简单而详细的分析方法。然而，目前的工作中并没有使用这些来预测近场区域的感应速度。相反，使用参考文献19中给出的解析表达式（在第三节C中完成）只能找到射流平面位置（即近场区域的末端）和收缩半径。因此，需要这两个量来评估射流平面和远场区域的诱导速度（在第三节D中完成）。一旦已知除近场区域外的所有区域的诱导速度，就使用插值来确定近场区域的诱导速率。第三节E将对此进行更详细的讨论。
C. Transition from Near-Field to Far-Field Region
In Ref. 19, Euler’s equations of motion for an inviscid fluid are used to obtain expressions for the average induced velocity and the slipstream radius as functions of the axial distance,
在参考文献19中，使用无粘性流体的欧拉运动方程来获得平均诱导速度和滑流半径作为轴向距离的函数的表达式，
The increase in average induced velocity with axial distance x must be accompanied by a contraction in slipstream radius as predicted by Eq. (3). Variation of both induced velocity and slipstream radius with axial distance for a test propeller of diameter 0.254m p D rotating at 6710 rpm is shown in Fig. 3. Evidently, much of the overall velocity increase occurs in the vicinity of the propeller within ~1 propeller diameter from the propeller plane indicating a strong influence of pressure force there. Beyond axial distance of ~1.5 propeller diameter, no appreciable change in induced velocity is seen indicating that the pressure force has diminished to a negligible extent.
平均诱导速度随轴向距离x的增加必须伴随着滑流半径的收缩，如方程式（3）所预测的。图3显示了直径为0.254m p D、以6710 rpm旋转的试验螺旋桨的诱导速度和滑流半径随轴向距离的变化。显然，总体速度的大部分增加发生在距离螺旋桨平面约1个螺旋桨直径范围内的螺旋桨附近，这表明那里的压力有很强的影响。超过螺旋桨直径约1.5的轴向距离，诱导速度没有明显变化，表明压力已经减小到可以忽略不计的程度。
It is proposed that the transition from near field region to far-field region occurs at the location where the average induced velocity (as predicted by Eq. (2)) increases and becomes equal to the efflux velocity (as predicted by semi-empirical equation presented later in Section III D) i.e. , 0 0 ( ) iavg V x V, as shown in Fig. 3. Therefore, with little algebraic rearrangement, the location of efflux plane is given as,
建议从近场区域到远场区域的过渡发生在平均诱导速度（如方程（2）所预测）增加并等于射流速度（如第III D节稍后给出的半经验方程所预测）的位置，即0（）iavg V x V，如图3所示。因此，在代数重排很少的情况下，射流平面的位置被给出为：，
Semi-empirical equation to determine efflux velocity V0 is presented later in Section III D, subsequent to the discussion on propeller jets from ship propellers.
在讨论了船舶螺旋桨的螺旋桨射流后，第三节D将给出确定射流速度V0的半经验方程。
D. Induced Velocity in the Far-Field Region
From the efflux plane onwards, the pressure force is sufficiently small to be neglected. Therefore, only diffusion needs to be considered for the prediction of induced velocity within the far-field region. As noted earlier, this is non trivial due to complex nature of the diffusion process, even with CFD methods. No relevant work is available in the literature pertaining to slipstream diffusion for conventional aircraft; even in recent UAV works, researchers have used simple theories to determine the induced velocity without considering slipstream diffusion. By contrast, significant work has been done for slipstream from ship propellers (referred to as “propeller jets” in marine research) and their effect on structures and seabed in regions that extends up to 8 to 10 propeller diameters from the propeller plane. Many researchers in this area have proposed semi-empirical equations to account for propeller jet diffusion, and these are summarized in a comprehensive review by Lam17.
从流出平面开始，压力足够小，可以忽略不计。因此，在预测远场区域内的诱导速度时，只需要考虑扩散。如前所述，由于扩散过程的复杂性，即使使用CFD方法，这也是不平凡的。文献中没有关于传统飞机滑流扩散的相关工作；即使在最近的无人机研究中，研究人员也使用简单的理论来确定诱导速度，而不考虑滑流扩散。相比之下，对于船舶螺旋桨（在海洋研究中称为“螺旋桨射流”）的滑流及其对螺旋桨平面延伸至8至10个螺旋桨直径区域的结构和海底的影响，已经做了大量工作。该领域的许多研究人员提出了半经验方程来解释螺旋桨射流扩散，这些方程在Lam17的一篇综合综述中进行了总结。
In the current work, slipstream diffusion is also taken into account using the semi-empirical equations established for marine propeller jets. Application of these equations to an aircraft propeller is justified considering that the flow through a small aircraft propeller is also incompressible ( Ma 0.3 ) and that the semi-empirical equations include fluid density as a parameter. However, the equations must be modified slightly to account for the different viscocity of air. The equations can then be used easily for either air or water using appropriate density and viscocity on the condition that the flow is incompressible.
在目前的工作中，还使用为船用螺旋桨喷气机建立的半经验方程考虑了滑流扩散。考虑到通过小型飞机螺旋桨的流动也是不可压缩的（Ma 0.3），并且半经验方程包括流体密度作为参数，将这些方程应用于飞机螺旋桨是合理的。然而，必须对方程进行轻微修改，以考虑空气的不同粘度。然后，在流动不可压缩的条件下，使用适当的密度和粘度，这些方程可以很容易地用于空气或水。
1. Propeller Jets from Ships
Before proceeding with the application of semi-empirical equations developed for ship propellers to aircraft propellers, it is important to have some knowledge of propeller jets from ships. A jet produced from a ship propeller, shown in Fig. 4, is different from a slipstream produced by an aircraft propeller in the sense that a ships propeller jet exhibits little or no contraction and expands continuously starting from the propeller plane, as shown in Fig. 4. The efflux plane is, therefore, almost coincident with the propeller plane.
在将为船舶螺旋桨开发的半经验方程应用于飞机螺旋桨之前，了解船舶螺旋桨射流非常重要。如图4所示，船用螺旋桨产生的射流与飞机螺旋桨产生的滑流不同，因为船用螺旋桨射流几乎不收缩，而是从螺旋桨平面开始连续膨胀，如图4中所示。因此，射流平面几乎与螺旋桨平面重合。
Absence of contraction in a ship’s propeller jet can be attributed to its low rotational speed i.e. around 200-400 rpm, in comparison to a small aircraft propeller which operates around 5000-6000 rpm. At such low rotational speeds, a small pressure force is developed across a ship propeller; thereby producing negligible acceleration. This is evident from the low velocity induced at the ship’s propeller plane which is usually of the order of 1-2 m/s, as compared to an aircraft propeller for which it is of the order of 8-10 m/s at the propeller plane. Thus in a ship’s propeller jet, diffusion is dominant starting from the propeller plane and therefore, the jet exhibits expansion only.
与运行在5000-6000rpm左右的小型飞机螺旋桨相比，船舶螺旋桨喷射器没有收缩可归因于其低转速，即约200-400rpm。在如此低的转速下，船舶螺旋桨上会产生很小的压力；从而产生可忽略的加速度。这从船舶螺旋桨平面处引起的低速中可以明显看出，与螺旋桨平面处的8-10m/s量级的飞机螺旋桨相比，该低速通常为1-2m/s量级。因此，在船的螺旋桨喷气式飞机中，扩散从螺旋桨平面开始占主导地位，因此，喷气式飞机只表现出膨胀。
The propeller jet is divided into 2 zones namely the zone of flow establishment (ZFE) and the zone of established flow (ZEF) as shown in Fig. 4. Within the zone of flow establishment, the induced velocity has two maximum velocity peaks; one on each side of the rotational axis (only one peak is evident in Fig. 4 but the propeller slipstream is symmetric about the rotation axis); and a low velocity core at the center because of the propeller hub. With axial distance, the high velocity fluid moves both inwards towards the low velocity core at the rotation axis and outwards towards the ambient flow. Therefore, the two peaks move gradually toward each other and finally merge into a single peak at the rotation axis. This is when the zone of established flow starts. In this zone, the fluid diffuses outwards only and the single maximum velocity peak at the rotation axis diminishes until it has diffused with the ambient flow completely.
螺旋桨式喷气发动机分为2个区域，即流动建立区（ZFE）和流动已建立区（ZEF），如图4所示。在流动建立区域内，诱导速度有两个最大速度峰值；旋转轴两侧各有一个（图4中只有一个明显的峰值，但螺旋桨滑流关于旋转轴是对称的）；由于螺旋桨轮毂，中心有一个低速核心。随着轴向距离的增加，高速流体既向内朝向旋转轴处的低速核心移动，又向外朝向环境流移动。因此，这两个峰逐渐相互靠近，最终在旋转轴上合并成一个峰。此时，已建立的流量区域开始。在该区域，流体仅向外扩散，旋转轴上的单个最大速度峰值逐渐减小，直到它与环境流完全扩散。
2. Semi-Empirical Equations for Propeller Jets
Semi-empirical equations have been determined from extensive experimental investigation of ship propellers. The earliest work in this field can be attributed to Albertson22 who investigated the induced velocity within a plain water jet using simple momentum theory and characterized the different zones - the zone of flow establishment and zone of established flow, within the propeller jet. Subsequent works by Blaauw and Van de Kaa, Berger, Hamill23, and Stewart24 etc. improved Albertson’s work by modifying the theoretical equations with the help of detailed experiments, thereby developing semi-empirical equations to predict the induced velocity within different zones of the propeller jet. All these semi-empirical equations are based on predicting the efflux velocity first, and then using it to determine the maximum velocity and subsequently the induced velocity distribution at cross-sections along the axial direction.
通过对船舶螺旋桨的广泛实验研究，确定了半经验方程。该领域最早的工作可归因于Albertson22，他使用简单的动量理论研究了平原水射流内的诱导速度，并表征了螺旋桨射流内的不同区域——流动建立区域和已建立流动区域。Blaauw和Van de Kaa、Berger、Hamill 23和Stewart24等人的后续工作改进了Albertson的工作，在详细实验的帮助下修改了理论方程，从而开发了半经验方程来预测螺旋桨射流不同区域内的诱导速度。所有这些半经验方程都是基于首先预测射流速度，然后用它来确定最大速度，然后确定沿轴向横截面的诱导速度分布。
3. Efflux Velocity, Efflux Plane Location and Contracted Radius
射流速度、射流平面位置和收缩半径
4. Induced Velocity at the Efflux plane and in the Far-Field Region
With efflux velocity known, the maximum velocity and subsequently the induced velocity at the efflux plane and in the far-field region may be determined using the semi-empirical equations presented in Ref. 17. These equations are adjusted for an aircraft propeller by:
Replacing the axial distance x with (x-x0 ) in the equations to predict induced velocity starting from the efflux plane ( x=x0 ) rather than from the propeller plane ( x=0 ).
Replacing the propeller diameter p D with the contracted diameter D0=2R0 since at the efflux plane, the slipstream diameter is smaller than the propeller diameter.
Stewart24 proposed that the zone of flow establishment extends up to 3.25 propeller diameters, after which the zone of established flow starts. For an aircraft, the ZFE starts from, and includes, the efflux plane and extends up to 3.25 times the diameter at the efflux plane i.e. the contracted diameter. The axial location of ZFE is thus given as: 
. Beyond the ZFE, the zone of established flow starts and therefore
for ZEF. To predict the maximum velocity decay with axial distance in the two zones, semi-empirical equations developed by Stewart24 are used but with slight modification to account for the different viscocity of air. For marine propeller jets, Stewart’s equations predict a linear decay of the maximum velocity with axial distance; however, for an aircraft’s propeller slipstream, it will decay slower compared to a marine propeller jet owing to relatively higher kinematic visocity of air than water which in turn causes relatively less viscous shearing in a propeller slipstream and consequently less diffusion sideways into the ambient flow. Thus to account for a lower rate of decay with axial distance in an aircraft’s propeller slipstream, a viscocity constant visc K is introduced and associated with the axial distance term 0 x x in all semi-empirical equations developed for marine propeller jets. The modified semi empirical equations are therefore:
Stewart24提出，流动建立区延伸到3.25个螺旋桨直径，之后流动建立区开始。对于飞机，ZFE从射流平面开始，包括射流平面，延伸到射流平面直径的3.25倍，即收缩直径。因此，ZFE的轴向位置如下：。在ZFE之外，已建立的流量区域开始，因此ZEF的流量为。为了预测两个区域中最大速度随轴向距离的衰减，使用了Stewart24开发的半经验方程，但稍作修改以解释空气的不同粘度。对于船用螺旋桨喷气式飞机，斯图尔特方程预测最大速度随轴向距离呈线性衰减；然而，对于飞机的螺旋桨滑流，由于空气的运动粘度相对高于水，因此与船用螺旋桨喷气式飞机相比，它的衰减速度较慢，这反过来又导致螺旋桨滑流中的粘性剪切相对较小，从而减少了侧向扩散到环境流中。因此，为了解释飞机螺旋桨滑流中轴向距离衰减率较低的原因，引入了粘度常数visc K，并将其与为船用螺旋桨喷气式飞机开发的所有半经验方程中的轴向距离项x-x0相关联。因此，修改后的半经验方程为：
E. Complete Propeller Slipstream Model
Eqs. (1) - (10) presented in the preceeding sections constitute a complete model for the propeller slipstream. As stated in section III B, the induced velocity in the near-field region is determined by interpolation once induced velocity is known in the rest of the propeller slipstream (at the propeller plane, efflux plane and in the far-field region). Using the MATLAB griddedInterpolant function, a nonlinear function is constructed from the known induced velocity data. The function is then evaluated at desired locations in the near-field region to complete the propeller slipstream velocity field.
前面章节中给出的方程式（1）-（10）构成了螺旋桨滑流的完整模型。如第三节B所述，一旦已知螺旋桨滑流其余部分（在螺旋桨平面、射流平面和远场区域）的诱导速度，则通过插值确定近场区域的诱导速度。使用MATLAB网格插值函数，根据已知的诱导速度数据构建非线性函数。然后在近场区域的期望位置评估该函数，以完成螺旋桨滑流速度场。
A schematic of the complete propeller slipstream model for small UAVs, with the different regions and zones as well as induced velocity distribution at several cross-sections is shown in Fig. 5. The slipstream shows contraction in the near-field region up to the efflux plane beyond which it expands in the far-field region.
图5显示了小型无人机的完整螺旋桨滑流模型示意图，其中包括不同的区域和地带以及几个横截面上的诱导速度分布。滑流在近场区域显示收缩，直至射流平面，超过该平面后，滑流在远场区域膨胀。
The induced velocity at the propeller plane is determined analytically.
The efflux velocity, efflux plane location, and contracted radius are all determined by coupling analytical equations with semi-empirical equations.
Semi-empirical equations are then used, in conjunction with the previously determined quantities, to predict the induced velocity in the far-field region.
Finally, interpolation is done to determine the induced velocity in the near-field region. The induced velocity is then completely known throughout the slipstream. All relevant equations are summarized in Table I.
螺旋桨平面处的诱导速度是通过分析确定的。
射流速度、射流平面位置和收缩半径都是通过将分析方程与半经验方程耦合来确定的。
然后，结合先前确定的量，使用半经验方程来预测远场区域的诱导速度。
最后，进行插值以确定近场区域的感应速度。在整个滑流中，诱导速度是完全已知的。表I总结了所有相关方程式。
IV. Simulation Results
Simulated induced velocity distribution obtained using the propeller slipstream model is presented in this section for low (1710 rpm) and high (6710 rpm) rotational speeds of a 0.254 m diameter Electrifly 10X4.5 propeller.
本节介绍了使用螺旋桨滑流模型获得的模拟诱导速度分布，适用于直径0.254米的Electrifly 10X4.5螺旋桨的低转速（1710 rpm）和高转速（6710 rpm）。
This particular propeller is selected since experimentally obtained thrust vs. rpm data is readily available for it in Ref. 21, which is needed in the slipstream model. From Ref. 21, the propeller generates ~ 0.54 N and ~ 9.9 N at 1710 rpm and 6710 rpm respectively. Furthermore, following the method described in Ref. 21, the induced velocity distribution at the propeller plane is evaluated for the said rotational speeds and these are plotted in Fig. 6. With this input and setting visc 0.638 K (by assuming 10 K ), the slipstream model was run for each rotational speed at multiple axial and radial locations to generate plots of induced velocity distribution throughout the propeller slipstream. These plots are shown in Fig. 7 and Fig. 8 for low and high rotational speeds respectively.
之所以选择这种特殊的螺旋桨，是因为参考文献21中很容易获得实验获得的推力与转速数据，这是滑流模型所必需的。根据参考文献21，螺旋桨在1710 rpm和6710 rpm时分别产生~0.54 N和~9.9 N。此外，根据参考文献21中描述的方法，对所述转速下螺旋桨平面处的诱导速度分布进行了评估，并将其绘制在图6中。使用此输入和设置visc 0.638 Kɧ（假设为10 K），在多个轴向和径向位置对每个转速运行滑流模型，以生成整个螺旋桨滑流的诱导速度分布图。图7和图8分别显示了低转速和高转速的曲线图。
Increase in induced velocity near the propeller (within 1 propeller diameter) owing to acceleration, is clearly seen in Fig. 7 and 8. After some axial distance, reduction in induced velocity due to slipstream diffusion is also seen in the figures. If, for instance, diffusion of the slipstream had not been considered for the test propeller at 6710 rpm (as is done in exisiting slipstream models7-9), the induced velocity at an axial distance of 3 propeller diameters from the propeller plane would be ~ 17m/s (determined from Fig. 3). This is is atleast twice the value obtained i.e. ~ 9m/s (determined from Fig. 8) if diffusion is considered at that axial location. Furthermore, simple models in the literature such as Ref. 9 only predict an average induced velocity and cannot account for its variation with radial location. Thus, at x = 3Dp, these simple models will estimate the induced velocity to be ~ 17m/s at all radial locations. By contrast, the proposed slipstream model captures the variation of induced velocity with radial distance, as shown in Fig. 7 and 8.
图7和图8清楚地显示了由于加速导致的螺旋桨附近（1个螺旋桨直径内）诱导速度的增加。在一定轴向距离后，图中还可以看到由于滑流扩散导致的诱导速度降低。例如，如果测试螺旋桨在6710 rpm时没有考虑滑流的扩散（如现有滑流模型7-9所示），则距螺旋桨平面3个螺旋桨直径的轴向距离处的诱导速度将为~17m/s（由图3确定）。如果在该轴向位置考虑扩散，则这至少是获得的值的两倍，即~9m/s（由图8确定）。此外，文献中的简单模型，如参考文献9，仅预测平均诱导速度，无法解释其随径向位置的变化。因此，在x=3Dp时，这些简单的模型将估计所有径向位置的诱导速度约为17m/s。相比之下，所提出的滑流模型捕捉到了诱导速度随径向距离的变化，如图7和图8所示。
V. Conclusions
The accurate prediction of induced velocity at any given location is essential for modeling aerodynamics of small UAVs. For example, in the case of small fixed-wing UAVs, the lift on the wings and the tail depends on the square of the total air velocity (sum of relative air velocity due to aircraft motion and the air velocity induced by the propeller ) over these surfaces. Therefore, to determine the aerodynamic forces accurately, one must be able to estimate the value of induced air velocity accurately at components near the propeller such as the wing, as well as at far downstream components such as the tail.
精确预测任何给定位置的诱导速度对于小型无人机的空气动力学建模至关重要。例如，在小型固定翼无人机的情况下，机翼和尾部的升力取决于这些表面上的总空气速度的平方（飞机运动引起的相对空气速度和螺旋桨引起的空气速度之和）。因此，为了准确地确定气动力，必须能够准确地估计螺旋桨附近部件（如机翼）以及远下游部件（如尾部）的诱导空气速度值。
This paper presents a propeller slipstream model for light-weight thruster dominated UAVs which are capable of extreme maneuvers. Unlike other simple models available in literature, the presented slipstream model predicts the axial component of the induced velocity both near and far-aft of the propeller by taking into account both acceleration and diffusion of the slipstream using analytical and semi-empirical equations. Plots showing induced velocity distribution throughout the slipstream as simulated with the slipstream model are shown for both low and high rotational speeds.
本文提出了一种能够进行极端机动的轻型推进器主导无人机的螺旋桨滑流模型。与文献中可用的其他简单模型不同，所提出的滑流模型通过使用分析和半经验方程考虑滑流的加速和扩散，预测了螺旋桨前后诱导速度的轴向分量。显示了低转速和高转速下用滑流模型模拟的整个滑流中诱导速度分布的图。
Future work will be targeted towards the validation of simulated results with experiments at different rotational speeds of the propeller. Experiments using a hot-wire anemometer to measure the induced velocity behind the propeller at a number of predefined locations would be suitable for the validation purpose. Furthermore detailed investigation will be carried out to evaluate the viscosity constant in the semi-empirical equations.
未来的工作将着眼于在螺旋桨的不同转速下通过实验验证模拟结果。使用热线风速计在多个预定义位置测量螺旋桨后面的诱导速度的实验将适用于验证目的。此外，将进行详细的调查，以评估半经验方程中的粘度常数。