張 森，席德科，劉治斌，陳寶峰

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基于雙橢圓弧型中弧線的系列翼型設(shè)計(jì)方法

張 森1，席德科1※，劉治斌2，陳寶峰2

（1. 西北工業(yè)大學(xué)航空學(xué)院，西安 710072； 2. 山西安瑞風(fēng)機(jī)電氣股份有限公司，運(yùn)城 044402）

為了能夠方便快捷的設(shè)計(jì)和修改翼型，采用兩段橢圓弧來(lái)構(gòu)造翼型的中弧線，并推導(dǎo)了描述中弧線的方程式。用該方法構(gòu)造的中弧線光滑連續(xù)，且不存在拐點(diǎn)。選用現(xiàn)有翼型的厚度分布，與中弧線分布函數(shù)進(jìn)行疊加，并引入厚度比例因子來(lái)實(shí)現(xiàn)對(duì)厚度的調(diào)整，最終得到了一種基于雙橢圓弧型中弧線的翼型設(shè)計(jì)方法，稱之為DEA（double ellipse arcs）翼型。選用Clark-Y翼型作為基礎(chǔ)翼型，設(shè)計(jì)了多款DEA翼型，并利用X-foil軟件對(duì)翼型氣動(dòng)性能進(jìn)行求解，分別研究了最大相對(duì)彎度、最大彎度相對(duì)位置、最大相對(duì)厚度以及翼型中弧線的形狀因子對(duì)翼型氣動(dòng)性能的影響。研究表明：增加最大相對(duì)彎度，可以提高翼型的升力系數(shù)，同時(shí)使翼型的升阻特性得到一定的改善；最大彎度位置前移，可以提高翼型在小攻角下的升力系數(shù)，同時(shí)增加翼型高效升阻比的攻角范圍；增加最大相對(duì)厚度可以提高翼型的最大升力系數(shù)，以及增大失速攻角，同時(shí)，高效升阻比的攻角范圍也隨著翼型最大相對(duì)厚度的增大而增加；中弧線前、后緣形狀因子對(duì)翼型氣動(dòng)性能的影響相對(duì)較小。

設(shè)計(jì)；翼型；數(shù)值分析；中弧線；橢圓弧

0 引 言

翼型作為航空技術(shù)發(fā)展的產(chǎn)物，被廣泛的應(yīng)用于流體機(jī)械產(chǎn)品的設(shè)計(jì)中，如壓縮機(jī)、軸流風(fēng)機(jī)、軸流泵、風(fēng)力機(jī)等，翼型的氣動(dòng)特性是決定流體機(jī)械性能優(yōu)劣的一個(gè)關(guān)鍵因素[1-5]。

當(dāng)選擇已有的翼型尚不能滿足工程要求時(shí)，需要重新設(shè)計(jì)或者對(duì)原有翼型進(jìn)行修型。目前此項(xiàng)工作主要依靠計(jì)算空氣動(dòng)力學(xué)的方法設(shè)計(jì)，以減少甚至取代翼型風(fēng)洞試驗(yàn)，具體實(shí)施方法有直接法和逆設(shè)計(jì)[6-8]。直接法基本過(guò)程：首先確定目標(biāo)，然后人工修改翼型型線或數(shù)據(jù)，計(jì)算氣動(dòng)特性并與設(shè)計(jì)要求比較分析，重復(fù)進(jìn)行修正，直到滿足要求，該方法要求設(shè)計(jì)者有較深厚的專業(yè)知識(shí)和豐富的設(shè)計(jì)經(jīng)驗(yàn)。逆設(shè)計(jì)基本過(guò)程：給定壓力分布目標(biāo)函數(shù)和約束條件，通過(guò)智能優(yōu)化方法自動(dòng)修改翼型外形，經(jīng)過(guò)多次迭代，達(dá)到收斂條件為止，該方法可以進(jìn)行多點(diǎn)/多目標(biāo)的設(shè)計(jì)，但計(jì)算量大，特別是設(shè)計(jì)變量較多時(shí)尤為突出[9-15]。

翼型的型線決定了繞翼型流場(chǎng)特性參數(shù)的分布，即翼型的氣動(dòng)性能，是翼型分析的基礎(chǔ)和關(guān)鍵。翼型的型線表達(dá)方法主要有：外形參數(shù)化方法[16]、形函數(shù)擾動(dòng)法[17]、解析函數(shù)法[18-19]。Ray T等[20]采用翼型的特征參數(shù)對(duì)翼型外形參數(shù)化，設(shè)計(jì)變量幾何意義明確，但未能給出解析表達(dá)式。形函數(shù)擾動(dòng)法是由原始翼型和擾動(dòng)形函數(shù)的線性疊加實(shí)現(xiàn)[21-22]。解析函數(shù)法就是用一個(gè)解析函數(shù)直接表示翼型形狀，例如早期用多項(xiàng)式表達(dá)的NACA的4位數(shù)、5位數(shù)系列翼型，此外也有研究用級(jí)數(shù)表達(dá)翼型的方法[23]。

Mark[24]于1989年開(kāi)發(fā)了X-foil翼型分析與設(shè)計(jì)系統(tǒng)，該軟件對(duì)于黏性流體采用了面元法和邊界層理論，由于其計(jì)算簡(jiǎn)單方便，適合于低速翼型的快速分析和設(shè)計(jì)，因此得到了廣泛應(yīng)用[25-29]。Ashok G等[30]對(duì)比了NLF(1)-0416和NLF(1)-0215F翼型的X-foil計(jì)算結(jié)果與試驗(yàn)數(shù)據(jù)，指出X-foil程序可以用于優(yōu)化設(shè)計(jì)中計(jì)算不同外型的翼型氣動(dòng)性能。鄧?yán)诘萚31]研究并發(fā)展了一套進(jìn)行高升阻比自然層流翼型多設(shè)計(jì)點(diǎn)/多設(shè)計(jì)目標(biāo)的優(yōu)化設(shè)計(jì)方法。為減少設(shè)計(jì)中的計(jì)算量，使用了X-foil程序進(jìn)行流場(chǎng)計(jì)算，進(jìn)一步驗(yàn)證了該軟件的可靠性。

本文采用兩段橢圓弧來(lái)構(gòu)造翼型的中弧線，并推導(dǎo)了中弧線的方程式，最終得到了一種基于雙橢圓弧型中弧線的翼型設(shè)計(jì)方法，利用X-foil軟件分析了翼型幾何特征參數(shù)對(duì)該系列翼型氣動(dòng)性能的影響規(guī)律，并給出翼型設(shè)計(jì)指導(dǎo)準(zhǔn)則。

1 翼型幾何特征參數(shù)

圖1中給出了翼型的幾何特征參數(shù)。

注：c表示翼型的弦長(zhǎng)，mm，將c設(shè)定為1；t表示翼型的最大相對(duì)厚度；f表示翼型的最大相對(duì)彎度；xt表示翼型最大厚度相對(duì)弦長(zhǎng)的位置；xf表示翼型最大彎度相對(duì)弦長(zhǎng)的位置。

2 雙橢圓弧型中弧線函數(shù)的構(gòu)造

2.1 雙橢圓弧型中弧線函數(shù)的求解

采用頂點(diǎn)重合的2個(gè)橢圓上截得的橢圓弧來(lái)構(gòu)造翼型中弧線，具體方法如圖2所示，為了方便描述，將翼型的弦長(zhǎng)定義為1。

圖2 雙橢圓弧型中弧線示意圖

圖2中，點(diǎn)(0,0)與(1,0)分別為翼型的前緣點(diǎn)和后緣點(diǎn)，點(diǎn)(x,)為橢圓的頂點(diǎn)。中弧線型線方程y如式（1）所示。

式中1為翼型中弧線前緣形狀因子，?1<1≤0；2為翼型中弧線后緣形狀因子，?1<2≤0。

2.2 中弧線形狀因子k1和k2對(duì)中弧線形狀的影響

當(dāng)翼型中弧線的最大相對(duì)彎度和最大彎度相對(duì)位置x一定時(shí)，通過(guò)調(diào)整方程（1）中形狀因子1和2的大小可以局部調(diào)整中弧線的形狀。

為了研究形狀因子1和2對(duì)中弧線形狀的影響，在圖3中給出了最大相對(duì)彎度=3.43%，最大彎度相對(duì)位置x=40%時(shí)，中弧線前緣形狀因子1=?0.99，中弧線后緣形狀因子2分別為0、?0.2、?0.5、?0.7、?0.99，和中弧線后緣形狀因子2=?0.99，中弧線前緣形狀因子1分別為0、?0.2、?0.5、?0.7、?0.99時(shí)的型線圖。

圖3 不同k1和k2時(shí)的中弧線型線

由圖3可知，當(dāng)中弧線前緣形狀因子一定而中弧線后緣形狀因子變化時(shí)，前緣點(diǎn)至最大彎度處翼型中弧線型線保持不變，翼型后緣氣流出口角隨著中弧線后緣形狀因子的增大而增大，當(dāng)中弧線后緣形狀因子2=0時(shí)，達(dá)到最大值。同樣，當(dāng)中弧線后緣形狀因子一定而中弧線前緣形狀因子變化時(shí)，最大彎度至后緣點(diǎn)翼型中弧線型線保持不變，翼型前緣氣流出口角隨著中弧線前緣形狀因子的增大而增大，當(dāng)中弧線前緣形狀因子1=0時(shí)，達(dá)到最大值。結(jié)果表明，當(dāng)中弧線的最大相對(duì)彎度和最大彎度相對(duì)位置一定時(shí)，隨著形狀因子1與2增大，翼型中弧線型線變得飽滿，對(duì)應(yīng)的氣流角也逐漸增大，且形狀因子1與2對(duì)型線的影響以最大彎度位置為分界點(diǎn)，相互獨(dú)立。

3 系列翼型設(shè)計(jì)方法

翼型的型面函數(shù)可以表示為中弧線分布函數(shù)與厚度分布函數(shù)的疊加，如式（2）所示。

式中y、y分別表示翼型的上表面型面函數(shù)和下表面型面函數(shù)；y表示翼型的厚度分布函數(shù)。

式（2）中的中弧線分布函數(shù)y使用本文推導(dǎo)的式（1），厚度分布函數(shù)則選用現(xiàn)有翼型的厚度分布。為了能夠調(diào)節(jié)翼型最大相對(duì)厚度的大小，引入了厚度比例因子，其表達(dá)式為

式中des表示設(shè)計(jì)翼型的最大相對(duì)厚度；ori表示原始翼型的最大相對(duì)厚度。

因此，可根據(jù)實(shí)際需要，通過(guò)調(diào)整厚度比例因子的大小得到理想的最大相對(duì)厚度。在式（2）的厚度項(xiàng)y前乘以系數(shù)，得到

式（4）給出了最終的翼型型面函數(shù)表達(dá)式。本文將該方法構(gòu)造的翼型稱之為DEA（double ellipse arcs）翼型。

4 結(jié)果與分析

選用Clark-Y翼型作為基礎(chǔ)翼型，運(yùn)用上述DEA翼型設(shè)計(jì)方法進(jìn)行翼型設(shè)計(jì)。Clark-Y翼型的幾何特征參數(shù)如下：最大相對(duì)厚度=11.71%，最大厚度相對(duì)位置x=28%，最大相對(duì)彎度=3.43%，最大彎度相對(duì)位置x=42%。在與Clark-Y翼型具有相同幾何特征參數(shù)的條件下，取中弧線形狀因子1=?0.8，2=?0.99，設(shè)計(jì)了一款DEA翼型，如圖4所示。

圖4 Clark-Y翼型與DEA翼型的型線對(duì)比

Clark-Y翼型與DEA翼型的型線對(duì)比結(jié)果顯示，兩者的型線有較好的重合度，采用DEA翼型設(shè)計(jì)方法能夠便捷的對(duì)現(xiàn)有翼型進(jìn)行參數(shù)化。

為了研究幾何特征參數(shù)對(duì)DEA翼型氣動(dòng)性能的影響，本文利用X-foil軟件對(duì)翼型氣動(dòng)性能進(jìn)行求解，取雷諾數(shù)=1.0×105。

4.1 最大相對(duì)彎度f(wàn)對(duì)DEA翼型氣動(dòng)性能的影響

在最大相對(duì)厚度=11.71%、最大厚度相對(duì)位置x=28%和最大彎度相對(duì)位置x=42%的情況下，取中弧線形狀因子1=?0.8，2=?0.99，分別設(shè)計(jì)了最大相對(duì)彎度為3.42%、4%、5%、6%時(shí)的翼型，氣動(dòng)性能計(jì)算結(jié)果如圖5所示。

由圖5可看出，在最大相對(duì)厚度、最大厚度相對(duì)位置和最大彎度相對(duì)位置相同的情況下，基礎(chǔ)翼型Clark-Y與最大相對(duì)彎度=3.43%時(shí)的DEA翼型的性能曲線有較高的重合度，這里進(jìn)一步驗(yàn)證了用DEA翼型參數(shù)化設(shè)計(jì)方法設(shè)計(jì)翼型的可靠性。當(dāng)只改變最大相對(duì)彎度時(shí)，翼型的升力系數(shù)曲線在小攻角范圍內(nèi)基本保持平行，在同一攻角下，隨著最大相對(duì)彎度的增大，升力系數(shù)也隨之增大。此外，隨著最大相對(duì)彎度的增大，翼型的升阻比曲線也發(fā)生了變化，當(dāng)最大相對(duì)彎度=6%時(shí)，在0°、6°和12°攻角處，翼型的升阻比相對(duì)于Clark-Y翼型分別提高了6.88%、1.59%和24.49%。研究表明，在翼型厚度、最大厚度位置和最大彎度位置不變的情況下，增大相對(duì)彎度，可以提高DEA翼型的升力系數(shù)，同時(shí)使翼型的升阻特性得到一定的改善。

注：表示翼型攻角，(o)；Cl表示翼型的升力系數(shù)；Cd表示翼型的阻力系數(shù)；Cl/Cd表示翼型的升阻比。

4.2 最大彎度相對(duì)位置xf對(duì)DEA翼型氣動(dòng)性能的影響

為了研究最大彎度相對(duì)位置x對(duì)DEA翼型氣動(dòng)特性的影響，在最大相對(duì)厚度=11.71%、最大厚度相對(duì)位置x=28%和最大相對(duì)彎度=3.43%的條件下，取中弧線形狀因子1=?0.8，2=?0.99，分別設(shè)計(jì)了最大彎度相對(duì)位置x為25%、42%、50%、60%時(shí)的翼型，并對(duì)其氣動(dòng)性能進(jìn)行了計(jì)算，計(jì)算結(jié)果如圖6所示。

圖6 不同xf下的DEA翼型氣動(dòng)性能曲線

4.3 最大相對(duì)厚度t對(duì)DEA翼型氣動(dòng)性能的影響

由薄翼理論可知，對(duì)于理想不可壓縮流體的翼型繞流，如果氣流繞翼型的迎角、翼型厚度、翼型彎度都很小，則繞流場(chǎng)是一個(gè)小擾動(dòng)的勢(shì)流場(chǎng)。這時(shí)，翼面上的邊界條件和壓強(qiáng)系數(shù)可以線性化為厚度、彎度、迎角三者影響的疊加。因此翼型的厚度對(duì)氣動(dòng)性能也起到至關(guān)重要的影響。圖7中給出了最大彎度相對(duì)位置x=25%、最大厚度相對(duì)位置x=28%、最大相對(duì)彎度=3.43%，最大相對(duì)厚度分別為6%、8%、10%和11.71%時(shí)的DEA翼型氣動(dòng)特性曲線，中弧線形狀因子取1=?0.8，2=?0.99。

圖7 不同t下的DEA翼型氣動(dòng)性能曲線

4.4 中弧線形狀因子k1與k2對(duì)DEA翼型氣動(dòng)性能的影響

為了研究中弧線形狀因子1和2對(duì)DEA翼型氣動(dòng)特性的影響，分別設(shè)計(jì)了最大相對(duì)厚度=11.71%、最大彎度相對(duì)位置x=42%、最大厚度相對(duì)位置x=28%、最大相對(duì)彎度=3.43%時(shí)，中弧線形狀因子1=?0.2、?0.5、?0.7、?0.99，2=?0.99和1=?0.99，2=?0.2、?0.5、?0.7、?0.99的翼型，性能計(jì)算結(jié)果如圖8和圖9所示。

圖8 不同k1下的DEA翼型氣動(dòng)性能曲線

圖9 不同k2下的DEA翼型氣動(dòng)性能曲線

5 結(jié) 論

1）采用兩段橢圓弧來(lái)構(gòu)造翼型中弧線，并推導(dǎo)了中弧線方程式，該方程通過(guò)改變最大彎度和最大彎度相對(duì)位置對(duì)中弧線形狀進(jìn)行控制，并由翼型中弧線形狀因子實(shí)現(xiàn)局部微調(diào)。用該方法構(gòu)造的中弧線光滑連續(xù)，且不存在拐點(diǎn)。

2）選用現(xiàn)有翼型的厚度分布，與中弧線分布函數(shù)進(jìn)行疊加，并引入厚度比例因子來(lái)實(shí)現(xiàn)對(duì)厚度的控制，最終得到了一種基于雙橢圓弧中弧線的翼型設(shè)計(jì)方法，本文將該方法構(gòu)造的翼型稱之為DEA（double ellipse arcs）翼型。

3）選用Clark-Y翼型作為基礎(chǔ)翼型，設(shè)計(jì)了多款DEA翼型，并利用X-foil軟件對(duì)翼型氣動(dòng)性能進(jìn)行求解，分別研究了最大相對(duì)彎度、最大彎度相對(duì)位置、最大相對(duì)厚度，以及翼型中弧線的形狀因子對(duì)翼型氣動(dòng)性能的影響。研究表明：增加最大相對(duì)彎度，可以提高翼型的升力系數(shù)，同時(shí)使翼型的升阻特性得到一定的改善；最大彎度位置前移，可以提高翼型在小攻角下的升力系數(shù)，同時(shí)增加翼型高效升阻比的攻角區(qū)間，但最優(yōu)升阻比會(huì)逐漸減??；增加最大相對(duì)厚度可以提高翼型的最大升力系數(shù)，以及增大失速攻角，同時(shí)，高效升阻比的攻角區(qū)間也隨著翼型厚度的增大而增加，但最優(yōu)升阻比會(huì)逐漸減小；中弧線形狀因子對(duì)翼型氣動(dòng)性能的影響較小。

依據(jù)上述參數(shù)的調(diào)整變化原則，可設(shè)計(jì)出新的翼型或者對(duì)原翼型進(jìn)行修型以滿足目標(biāo)任務(wù)的需要。

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Design method for series airfoil based on mean camber line consisting of double ellipse arcs

Zhang Sen, Xi Deke, Liu Zhibin, Chen Baofeng

Institute of Aeronautics, Northwestern Polytechnical University, Xi’an, ChinaShanxi Anrui Fan Electric Co., Ltd, Yuncheng, China

Airfoil, as a product of aviation technology, has been widely used in the design of fluid machinery products. The aerodynamic characteristics of airfoils are a key factor in determining the performance of fluid machinery. When the existing airfoils are not able to meet the engineering requirements, it is necessary to redesign or trim the original airfoils. In this research, 2 ellipse arcs were used to form the mean camber line of the airfoil, and the corresponding equation was deduced. This equation controls the shape of the mean camber line by changing the maximum camber and the relative position of maximum camber, and adjusts the local shape by changing the 2 shape factors of the mean camber line. The mean camber line constructed by this method is smooth and continuous, and there is no knee point. Then the thickness distribution of the existing airfoil was superposed with the distribution function of mean camber line, and a thickness scale factor was introduced to adjust the thickness distribution. Ultimately, the design method for a series of airfoils based on mean camber line of double ellipse arcs is achieved, which is called DEA (double ellipse arcs) airfoil. The airfoil profile function constructed by this method has definite physical meaning, simple and reliable, and it is easy to realize serialization. In order to study the influence of airfoil characteristic parameters on aerodynamic performance of the DEA airfoil, the Clark-Y airfoil was taken as the basic airfoil, and a number of DEA airfoils were designed using the thickness distribution of the Clark-Y airfoil. Then the aerodynamic characteristics of the designed airfoils were solved by the X-foil software to study the influence of the maximum camber, the relative position of the maximum camber, the maximum thickness and the shape factors of the mean camber line on the DEA airfoil aerodynamic performance. There are 5 characteristic parameters in all that influence the shape of the DEA airfoil. We selected one of the 5 characteristic parameters as variable and fixed the other 4 characteristic parameters to design different DEA airfoils. And the aerodynamic characteristics were achieved at Reynolds number of 1.0×105. The calculation results of the 4 DEA airfoils with different values of maximum camber show that the increase of the maximum camber can improve the lift coefficient and ameliorate the characteristics of the lift-drag ratio. The calculation results of the 4 DEA airfoils with different values of relative position of the maximum camber show that as the relative position of the maximum camber moves forward, the lift coefficient under small angles of attack is improved, and the range of efficient lift-drag ratio gets broadened. The calculation results of the 4 DEA airfoils with different values of maximum thickness show that the increase of the maximum thickness can increase the maximum lift coefficient and the stall angle. At the same time, with the increase of thickness, the range of efficient lift-drag ratio also gets broadened. The calculation results of the DEA airfoils with different shape factors of the mean camber line also were achieved. At small attack angle, the change of the leading shape factors of the mean camber line has little influence on lift coefficient. With the decrease of the leading shape factors of the mean camber line, the interval of efficient lift-drag ratio has a tendency to move to high attack angle range. With the decrease of the trailing shape factors of the mean camber line, the lift coefficient and lift-drag ratio decrease gradually. Moreover, the interval of efficient lift-drag ratio also decreases and the decrease is mainly at the range of small attack angle. According to the adjustment principle of the above parameters, a new airfoil can be designed or modified to meet the needs of the target task.

design; airfoils; numerical analysis; mean camber line; double ellipse arcs

10.11975/j.issn.1002-6819.2018.02.006

TB126/TB21

A

1002-6819(2018)-02-0040-07

2017-07-07

2017-12-12

國(guó)家自然科學(xué)基金（11172243）；陜西省科技統(tǒng)籌創(chuàng)新工程計(jì)劃（2011KTCQ01-02）

張 森，河南新鄉(xiāng)人，博士生，研究方向?yàn)楹娇崭呖萍架娹D(zhuǎn)民技術(shù)及流體機(jī)械設(shè)計(jì)。Email：sen96@mail.nwpu.edu.cn

席德科，教授，博士生導(dǎo)師，研究方向?yàn)轱L(fēng)洞設(shè)計(jì)及流體機(jī)械設(shè)計(jì)。Email：xideke@nwpu.edu.cn

張 森，席德科，劉治斌，陳寶峰. 基于雙橢圓弧型中弧線的系列翼型設(shè)計(jì)方法[J]. 農(nóng)業(yè)工程學(xué)報(bào)，2018，34(2)：40－46. doi：10.11975/j.issn.1002-6819.2018.02.006 http://www.tcsae.org

Zhang Sen, Xi Deke, Liu Zhibin, Chen Baofeng. Design method for series airfoil based on mean camber line consisting of double ellipse arcs[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2018, 34(2): 40－46. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2018.02.006 http://www.tcsae.org