ZHANG Xiong-xing, WANG Wei*, LIU Guang-hai, GUO Zi-long, HU Rui

(1.School of Optoelectronics Engineering,Xi′an Technological University,Xi′an 710021,China;2.Sichuan Physcience Optics and Fine Mechanics Co.,Ltd.,Mianyang 621900,China)

Abstract: In order to study temperature field measurement techniques of Schlieren systems, the principle of Schlieren quantitative measuring techniques is expounded. Based on the relationship between the grayness of a Schlieren image, along with the area of a light source, we propose an algorithm that is dependent on Schlieren to calculate flow field temperature. Firstly, a transmission Schlieren system is built on an optical platform and a hot plate is placed in the test area. Schlieren images are then captured by using a CCD camera, which are uploaded and stored on a computer for image processing. Finally, the algorithm is used to get the measured values of the temperature field, which then are compared with those by thermocouple measurement. The experimental results show that, when the temperature of the hot plate is set at 50 ℃ or 90 ℃, the relative errors of the flow field temperature measurement are less than 10%, the reliability of the algorithm is therefore proved and the quantitative measurement of temperature fields based on Schlieren techniques is realized.

Key words: schlieren;quantitative measurement;flow field;image gray magnitude

1 Introduction

引 言

In modern flow field measurement, as the demand for non-contact flow field temperature measurement grows, traditional instruments are failing to satisfy observation requirements. Schlieren measuring techniques through optics has been attracting much attention for its multiple outstanding characteristics, among which are its informative demonstration of flow fields, its non-invasive but high level of quantitative precision, and its low cost. This is especially true for the measurement of temperature in flow fields when compared to traditional methods using probes. Using Schlieren measuring techniques, one can measure the temperature of flow fields without affecting the flow field itself[1-6].

在現(xiàn)代流場(chǎng)測(cè)量中，對(duì)于非接觸式流場(chǎng)溫度定量測(cè)量需求逐步增強(qiáng)，傳統(tǒng)流場(chǎng)顯示設(shè)備己很難滿足觀測(cè)需求。以紋影定量測(cè)量技術(shù)為代表的光學(xué)顯示技術(shù)備受關(guān)注，其顯著特點(diǎn)是流場(chǎng)信息量大、非侵入定量測(cè)量精度高、成本低。特別是對(duì)于流場(chǎng)的溫度測(cè)量，相比傳統(tǒng)的探針測(cè)量方式，利用紋影定量技術(shù)可以在不影響流場(chǎng)本身特性的前提下，滿足溫度場(chǎng)測(cè)量需求[1-6]。

Since year 2000, Schlieren measurement technology has seen rapid development. In 2008, C. Alvarezherreraetal.[7]performed reconstruction using an orthogonal decomposition method and observed the combustion temperature of two flames from fuels mixed partially and fully with air. It was discovered that there were structural differences in the flame but it didn't find the flame′s relevant temperature ranges. In 2009, M.K.Campbelletal.[8]set a reflective Schlieren measuring device above a metal hot plate, selected two temperatures, and observed their differences using a thermocouple with reasonable consistency. However, since the temperature range used was somewhat narrow, there is no way to tell how the results will vary under more extreme temperatures. In 2012, A. Martínezgonzálezetal.[9]used traditional Schlieren measuring tools to simultaneously take measurements of speed and temperature, but did so with a relatively large margin of error for a series of different temperature values. In 2016, A.Martínezgonzálezetal.[10]analyzed lab results using two pictures taken at different temperatures and discovered that changes in the camera′s exposure allowed measurements at different temperature ranges. However, the published results failed to fully analyze the phenomenon. Domestic research of Schlieren systems is progressively increasing in importance. In 2013, Jifei Yeetal.[11-12]used a multi-coloured Schlieren method to measure the density of axis-symmetric flow fields. However, the fabrication of light colour filters is relatively difficult. In 2015, Sheng Meng,etal., from Zhejiang University of Technology.[13-14], used a Z-shaped Schlieren system and standard spectrophotometry to quantify flame temperature. That same year, China Aerodynamics and Research Development Center′s Jun Zhangetal.[15]used background Schlieren techniques with flow field density and temperature distribution to perform measurements, but background Schlieren techniques are more appropriately used with large-scale measurements.

自2000年以后，紋影定量技術(shù)得到了迅猛發(fā)展。2008年，Alvarezherrera C等人[7]利用一種正交分解的方法對(duì)充分和部分兩種空氣混合的燃料燃燒的火焰溫度場(chǎng)進(jìn)行重建，找出兩種火焰結(jié)構(gòu)上的差異，但是并沒(méi)有測(cè)量出火焰的溫度值；2009年，Campbell M K等人[8]利用反射式紋影儀測(cè)量出金屬加熱平臺(tái)上方的空氣溫度，選取兩個(gè)溫度值，與熱電偶所測(cè)量的溫度值進(jìn)行比較，得到較好的一致性。由于加熱平臺(tái)選取溫度值較為單一，無(wú)法比較低溫度值和高溫度值范圍內(nèi)測(cè)量誤差；2012年，Martínezgonzález A等人[9]利用傳統(tǒng)紋影儀實(shí)現(xiàn)了溫度和速度同時(shí)測(cè)量，其對(duì)可控加熱平臺(tái)設(shè)定一系列溫度值進(jìn)行測(cè)量，但是測(cè)量誤差偏大；2016年，Martínezgonzález A等人[10]通過(guò)實(shí)驗(yàn)對(duì)不同溫度值的圖片進(jìn)行分析，發(fā)現(xiàn)通過(guò)改變相機(jī)的曝光時(shí)間，可以測(cè)量不同范圍的溫度值，但是沒(méi)有提出具體的理論分析。國(guó)內(nèi)對(duì)紋影定量測(cè)量技術(shù)的研究也逐漸重視，2013年，葉繼飛等人[11-12]采用彩虹紋影法對(duì)軸對(duì)稱(chēng)流場(chǎng)的密度場(chǎng)進(jìn)行了定量測(cè)量，但彩色濾光片的制作較難；2015年，浙江工業(yè)大學(xué)孟昇等人[13-14]利用“Z”字形紋影系統(tǒng)，采用標(biāo)準(zhǔn)光度法對(duì)火焰溫度進(jìn)行了定量化；同年，中國(guó)空氣動(dòng)力研究與發(fā)展中心的張俊等人[15]利用背景紋影技術(shù)對(duì)流場(chǎng)的密度和溫度分布進(jìn)行了測(cè)量，但背景紋影技術(shù)適用于大視場(chǎng)的定量測(cè)量。

From the above, one can see that there is no standard method of calculating the temperature of Schlieren systems or verifying tests in this field. Using Schlieren measurement principles as a foundation, the relationship between degrees of grayness in Schlieren images and the amount of source light obstruction is first analyzed, a method of calculating flow field temperature is proposed, an appropriate measurement scope is meticulously analyzed, and the results of a laboratory test on temperature field measurements to verify the aforementioned algorithm is described.

綜上可以看出，紋影系統(tǒng)的溫度定量測(cè)量技術(shù)在計(jì)算方法上并沒(méi)有一個(gè)完整的理論推導(dǎo)和試驗(yàn)驗(yàn)證。而本文的創(chuàng)新點(diǎn)在于，以紋影法的定量測(cè)量原理為基礎(chǔ)，分析了紋影圖像灰度值大小與未被遮擋的光源像面積的關(guān)系，提出了一套完整的流場(chǎng)溫度定量計(jì)算方法，詳細(xì)分析了該方法的測(cè)量范圍，并通過(guò)溫度場(chǎng)定量測(cè)量試驗(yàn)，驗(yàn)證了這套方法的可行性。

2 Measurement Principles

測(cè)量原理

Light rays whose indexes of refraction are inconsistent are dispersed unevenly. As seen in Fig.1, the three different coloured regions depict that the internal positions of one flow field are adjacent.

光線通過(guò)折射率不均勻的流場(chǎng)時(shí)會(huì)發(fā)生偏折，如圖1所示，圖中3塊不同顏色的區(qū)域表示了一個(gè)流場(chǎng)內(nèi)部3個(gè)相鄰區(qū)域。

Fig.1 Transmission of light beam in media 圖1 光在介質(zhì)中的傳播

Light follows the Huygens principle when passing through a transparent medium. Because of changes in the index of refraction, the velocity of the light wave can change, along with its direction. This leads to the following relationship:

光在透明介質(zhì)中傳播時(shí)遵循惠更斯原理，而折射率的改變會(huì)導(dǎo)致波前光線速度發(fā)生改變化，光束的方向也會(huì)有所不同，由此即可獲得如下關(guān)系：

(1)

While working with formula 1, because the neighbouring flow fields are very close, the following formula 2 is created：

在式(1)中,由于流場(chǎng)相鄰區(qū)域的折射率非常接近，經(jīng)過(guò)化簡(jiǎn)后得到公式(2)：

(2)

Becauseθis a very small angle, they-axis can be written as:

因?yàn)棣仁且粋€(gè)非常小的角度，以y軸方向?yàn)槔治觯?/p>

(3)

Combining this with formula (2) and integrating with respect to thex-axis yields

將其帶入公式(2),并沿z軸方向做一次積分得到式(4)：

(4)

After light passes through a flow field it is refracted toward a blade, as shown in Fig.2.

在光線通過(guò)流場(chǎng)區(qū)域后偏折角與刀口的偏移量如圖2所示。

Fig.2 Relationship between deflection angle θ and offset a 圖2 偏折角θ與偏移量a關(guān)系圖

From Fig.2, one can see that the angle of lightθcan be refracted by offset a at the blade, depending on the nature of the focal plane:

從圖2中可看出，光線的偏移角θ會(huì)在刀口處產(chǎn)生一定的偏移量a，根據(jù)焦平面的性質(zhì)則有：

a=f2tanθ≈f2θ.

(5)

The following flow field density formula, formula (6), is obtained by combining formulas (4) and (5) with the Gladstone-Dale law:

結(jié)合公式(4)、(5)和格拉斯通-戴爾(Gladstone-Dale)定律可得到流場(chǎng)密度梯度公式(6)：

(6)

In Equation (6),Krepresents the Gladstone-Dale constant,Lrepresents the length of the flow field along the optical path, andf2represents the focal length of Schlieren lens 2, all of which are constant. After simplifying formula 6, the density value of the flow field with respect to they-axis can be obtained, as shown in formula 7.

式中，K表示Gladstone-Dale常數(shù)，L表示流場(chǎng)沿光路方向的長(zhǎng)度，f2表示紋影系統(tǒng)透鏡2的焦距，以上3個(gè)值均為常量。對(duì)公式(6)進(jìn)行化簡(jiǎn)后即可得到流場(chǎng)關(guān)于y軸方向的密度值，如式(7)所示:

(7)

In equation (7),ρyrepresents the density to be measured,ρ0represents the reference density,ξ0represents the reference edge position, andξ1represents the offset of the light. Combined with the ideal gas state equation, the temperature of the gas flow field is obtained as follows:

式中，ρy表示待測(cè)密度，ρ0表示基準(zhǔn)密度，ξ0表示參考刀口位置，ξ1表示光線的偏移量。再結(jié)合理想氣體狀態(tài)方程，可得到氣體流場(chǎng)的溫度如下：

(8)

WhereT0is surrounding temperature,ρ0is the surrounding air density,Tyis the temperature to be measured, andρyis the density to be measured. It can be seen from formula 8 that the integral with respect to the amount of refracted light is key to calculating gas flow field temperature.

式中，T0為周?chē)h(huán)境溫度，ρ0為周?chē)h(huán)境空氣密度，Ty為待測(cè)溫度，ρy為待測(cè)密度。由公式(8)可以看出，其中關(guān)于光線偏折量的一次積分是計(jì)算氣體流場(chǎng)溫度的關(guān)鍵所在。

3 Measurement Methods and Data Processing

測(cè)量流程與數(shù)據(jù)處理方法

By analyzing of the principles of quantitative measurement of Schlieren, it can be obtained that:Due to the different densities in the flow field, the light will undergo different degrees of offset when passing through the flow field, and different degrees of light refraction will cause different gray values on the Schlieren image. Therefore, the main idea of the algorithm is to establish the relationship between the gray level of the Schlieren image and the refraction of the light. This requires calibration between the offset of the light and the gray value of the Schlieren image.

經(jīng)過(guò)對(duì)紋影定量測(cè)量原理的分析可得：由于流場(chǎng)內(nèi)密度不同導(dǎo)致光線在穿過(guò)流場(chǎng)時(shí)，會(huì)發(fā)生不同程度的偏移，而不同程度的光線偏移會(huì)造成紋影圖像上灰度值的不同，故紋影儀流場(chǎng)溫度測(cè)量的主體思想為構(gòu)建紋影圖像灰度大小與光線偏移量這二者之間的關(guān)系。這就需要對(duì)兩者作出標(biāo)定。

The calibration method used herein is the calibration Schlieren method, which is to convert the relationship between the offset of the light relative to the blade and the gray scale of the Schlieren image into the relationship between the relative light offset of the blade and the grayness of the Schlieren image. As shown in Fig.3, the two relative motions are actually identical in terms of the area of the occluded light from the source.

本文所采用的標(biāo)定方法為定標(biāo)紋影法，就是將光線相對(duì)刀口的偏移量與紋影圖像灰度變化的關(guān)系轉(zhuǎn)化為刀口相對(duì)光線偏移量與紋影圖像灰度變化的關(guān)系，如圖3所示，兩種相對(duì)運(yùn)動(dòng)實(shí)際對(duì)遮擋光源像面積上是一致的。

Fig.3 Relationship between the position of the blade and light source image 圖3 刀口切割量與光源像的位置關(guān)系示意圖

The specific method of establishing the relationship between the grayness of the Schlieren image and the refraction of light is:start with a measurement area with no flow fields, place the blade in a position in which the light is not being obscured, then slowly and successively move the blade by Δauntil it completely covers the area. Use a CCD camera to record all of the obscured light and stop when the entire image has been recorded. Set the longitudinal coordinate of any point on the grain image toi, the latitudinal coordinate toj, and use the superscriptsto indicate the amount of light obstruction applied by the blade.Hs[i,j] indicates the grayness of the horizontal and vertical coordinates of the Schlieren picture atiandjwhen no fluid is added. The grayness of a point, the degree of occlusion and position of the blade edge is determined by the point in which source light is first obstructed.

紋影圖像灰度與刀口裝置遮擋量曲線關(guān)系建立的具體方法是：在測(cè)量區(qū)域內(nèi)不添加任何流場(chǎng)，將刀口從不遮擋光源像的位置開(kāi)始，以固定Δa逐步遮擋光源像，CCD相機(jī)記錄刀口逐步遮擋所捕獲的紋影圖像，當(dāng)?shù)犊趯⒐庠聪裢耆趽鯐r(shí)停止。設(shè)紋影圖像上任意點(diǎn)的橫坐標(biāo)為i、縱向坐標(biāo)為j，上標(biāo)s表示刀口遮擋量，Hs[i,j]表示沒(méi)有流體加入時(shí)紋影圖像上橫縱坐標(biāo)分別為i、j點(diǎn)的灰度值；刀口遮擋位置程度由光源像剛被遮擋開(kāi)始到剛好全被遮擋結(jié)束來(lái)決定。

Fig.4 Curve of relationship between image grayness and blade edge occlusion 圖4 圖像灰度與刀口遮擋量關(guān)系曲線

Since the integral of the denominator in equation (8) is related to the obstruction of light, the obstruction of light then determines the size of the image of the unobstructed light source. This then affects the camera′s recorded grayness from the Schlieren image. The principle behind the proposed calculation method is to record the changes in the obstructed light from beginning to end. If the blade′s edge is regarded as a line on the circular light source image, the positional relationship between the blade edge and the light source image can be equivalent to the three different positional relationships shown in Fig.5.

由于式(8)中分母上的一次積分與光線的偏移量有關(guān)，而光線偏移量又決定了未被遮擋光源像面積的大小，進(jìn)而影響紋影圖像在相機(jī)中的灰度大小。故該計(jì)算方法的核心理念是找出光源像面積被遮擋前后的的變化量。若將刀口看作圓形光源像上的弦，刀口和光源像的位置關(guān)系可以等價(jià)為圖5表示的3種不同位置關(guān)系。

Fig.5 Area change schematic diagram of the source image 圖5 光源像面積變化示意圖

The black solid line indicates the reference position of the blade′s edge, and the red solid line indicates the position after the blade is moved, that is, the position of the blade after the light is moved. The three anglesα,β, andθrepresent the triangle′s apex angle, formed by the light source′s center and perimeter. Generally, these angles are decided by the degree to which the light fans outward. Because the size of the image is inversely proportional to the amount of light obstructed by the blade, the change in grayness is similar to that in Fig.5, where it is equal to the change in the area between the black and red lines. These 3 instances are calculated as follows:

圖中實(shí)線1表示刀口的參考位置，實(shí)線2表示刀口移動(dòng)后的位置，即等價(jià)于光線移動(dòng)后的刀口位置，α、β、θ三者表示光源像圓心和刀口形成的三角形頂角的角度，一般這個(gè)角度等于所應(yīng)對(duì)的扇形角角度。由于圖像灰度值的大小與刀口遮擋量成反比，因此灰度值的變化相當(dāng)于圖5當(dāng)中實(shí)線1與實(shí)線2之間的光源像面積變化。對(duì)于3種情況的計(jì)算如下：

When a line is on the center of the circle:

當(dāng)一弦在圓心上時(shí)：

(9)

When the two lines are on the same side：

當(dāng)兩弦在圓心同側(cè)時(shí)：

(10)

When the two lines are on either side of the circle′s center：

當(dāng)兩弦在圓心兩側(cè)時(shí):

(11)

In these equations,ris the radius of the light source in the image, which can be calculated from the displacement of the blade′s edge in the calibration curve, and ΔSis the change in area between the solid black line and the solid red line, denoted as bothS1andS2in equations 9 and 10. TheS0,S1andS2in equation (11) are the three parts between the red and black strings, that is, the area of the two circular areas, the black triangular area and the red triangular area. With this, formula (8) can modified to:

其中，r為光源像的半徑，可由標(biāo)定曲線中刀口位移量計(jì)算得出，ΔS為實(shí)線1與實(shí)線2之間區(qū)域面積變化，式(9)和式(10)中S1與S2均為弦1與弦2分別與圓形成的下方閉合部分的面積；式(11)中S0、S1和S2為紅黑弦之間的3個(gè)部分即兩個(gè)扇形區(qū)域面積、上部三角區(qū)域面積和下部三角區(qū)域面積。此時(shí)公式(8)可變化為：

(12)

To summarize, the process for quantitatively calculating flow field temperature is：

①After adjusting the Schlieren′s light path toward the observation area, determine the calibration curve and the radius of the image of the source light,r.

②After adding the flow field, adjust the blade′s position such that the image′s contrast is appropriate, then record the blade′s position.

③Capture the image of the Schlieren when there is a flow field, record the grayness of the observation area on the image, then refer to the calibration curve using the blade′s positiona2.

④Analyse the positional relationship betweena1anda2, then use either of the formulas (9), (10), and (11) to calculate the area difference ΔS.

⑤Use the calculated ΔSin formula (12)，from which the corresponding air temperature can be discovered.

綜上所述，紋影流場(chǎng)的溫度定量計(jì)算流程為：

①完成紋影光路的調(diào)節(jié)后，對(duì)需要測(cè)量的區(qū)域，先作出標(biāo)定曲線，并計(jì)算出光源像半徑r；

②加入流場(chǎng)后調(diào)節(jié)刀口位置，以圖像對(duì)比度較好為準(zhǔn)，并記下此時(shí)刀口的位置a1；

③捕獲有流場(chǎng)時(shí)紋影圖像，將圖像上測(cè)量區(qū)域的灰度值，通過(guò)標(biāo)定曲線找到對(duì)應(yīng)的刀口位置a2；

④通過(guò)分析a1與a2的位置關(guān)系，找出相應(yīng)情況的計(jì)算公式(9)、(10)及(11)，計(jì)算面積變化量ΔS；

⑤再將ΔS計(jì)算帶入公式(12)，即可得到相應(yīng)的空氣流場(chǎng)溫度信息。

4 Experimental Process

實(shí)驗(yàn)過(guò)程

The experiment makes use of traditional Schlieren transmissions. The entire setup is shown in Fig.6. The relevant equipment includes: a 520 nm LED monochromatic light source, a 50.8 mm diameter double-bonded lens group, a blade setup, a CCD camera, and an electronically controlled hot plate.

實(shí)驗(yàn)采用傳統(tǒng)透射式紋影光路，所搭的試驗(yàn)平臺(tái)如圖6所示。主要設(shè)備包括：520 nm的LED單色光源，直徑為50.8 mm的雙膠合透鏡組，刀口裝置，CCD相機(jī)和電控加熱平臺(tái)。

Fig.6 Experimental platform 圖6 實(shí)驗(yàn)平臺(tái)

In order to control the thickness of the flow of light during calculation, the object to be measured is the heated air above the rectangular hot plate, which is set to 50 ℃ and 90 ℃. The results of the experiment are compared with the measurements of a thermocouple in order to verify the validity of the algorithm. The experiment was conducted in a closed environment where the temperature was 20 ℃(293 K) and the air density was 1.29 g/cm3. It is known from the Gladstone-Dale formula that the relationship between the density and refractive index when the flow field is a gas is as follows:

為了控制該計(jì)算方法中流場(chǎng)沿光路方向的厚度這一變量，被測(cè)對(duì)象為矩形電控加熱平臺(tái)上方的受熱空氣，加熱平臺(tái)的溫度分別設(shè)定50 ℃與90 ℃。通過(guò)比較熱電偶的測(cè)量值與實(shí)驗(yàn)結(jié)果相進(jìn)行驗(yàn)證。測(cè)量實(shí)驗(yàn)在20 ℃(293 K)下的封閉環(huán)境中進(jìn)行，環(huán)境密度為1.29 g/cm3，由Gladstone-Dale公式可知，流場(chǎng)為氣體時(shí)，其密度與折射率之間的關(guān)系如下：

n-1=ρK,

(13)

In formula (13),nis the refractive index of the fluid,ρis the density of the gas, andKis the Gladstone-Dale constant, which is generally determined by the gas′s composition, among other characteristics. Furthermore, the relationship of a wavelength of a wave of light as it passes through a gas and the Gladstone-Dale constant is as follows:

公式(13)中，n為流體的折射率，ρ為氣體的密度，K為Gladstone-Dale常數(shù)，而這個(gè)常數(shù)一般由氣體的組分等特性決定，并且受光波長(zhǎng)的影響，當(dāng)氣體為空氣時(shí)，光波波長(zhǎng)與Gladstone-Dale常數(shù)的關(guān)系如下：

(14)

In which，λ, or the wavelength, is 520 nm， whileKis 2.261×10-4m3/kg.

式中，λ為光波波長(zhǎng)，其值為520 nm，K值約為2.261×10-4m3/kg。

Using to the position of the blade in the calibration curve, it is not difficult to obtain the radius of the circular light source image using the blade′s positiond2, where no source light is blocked, and the blade positiond1, where it is blocked completely, as demonstrated in formula (15):

根據(jù)標(biāo)定曲線中刀口從完全不遮擋光源像的位置d2到完全遮擋光源像的位置d1，不難得出圓形光源像的半徑大小為式(15):

0.845 mm .

(15)

The hot plate was added to the observation area and set to the appropriate temperature(50 ℃ and 90 ℃). When the hot plate reached the target temperature and stabilized, the Schlieren images were captured, as shown in Fig.7. In the figure, there are obvious fluctuations in the grayness of the image. From the darker areas, it is clear how the temperature is changing against the background of the image. Lighter areas rise on they-axis while darker areas fall.

將加熱平臺(tái)植入待測(cè)區(qū)域并設(shè)定好溫度值(50 ℃與90 ℃)，待其平臺(tái)表面穩(wěn)定并達(dá)到設(shè)定值后開(kāi)始捕獲紋影圖像，結(jié)果如圖7所示。在圖中有明顯的亮暗變化，從亮暗區(qū)域的分界就可以看出溫度在該分界處的變化方向，比背景圖亮的區(qū)域，溫度變化方向沿y軸向上；比背景圖暗的區(qū)域，溫度變化方向沿y軸向下。

Fig.7 Schlieren image on the heating platform at different temperatures 圖7 不同溫度下加熱平臺(tái)上方紋影圖像

The red squares in Fig.7 highlight the observation area of the image. The grayness of any point in the Schlieren image is found on the calibration curve. Given the position of the knife edge and the change of the grayness of the point, one can analyze the change in obstructed light of the image area. Using the appropriate formula, ΔSis calculated, and the final temperature value of the point is then found using formula (8).

圖7中方塊區(qū)域?yàn)闇y(cè)量區(qū)域，查詢紋影圖像中某點(diǎn)灰度值的標(biāo)定曲線，找到該點(diǎn)灰度值變化量對(duì)應(yīng)的刀口位置，分析刀口切割光源像面積的變化關(guān)系，并帶入相應(yīng)公式，計(jì)算出ΔS，最后通過(guò)公式(8)算出該點(diǎn)的溫度值。

5 Data Analysis

數(shù)據(jù)分析

In order to demonstrate the feasibility of the formula, the results of the experiment are compared with the measurement of a thermocouple on the hot plate, which measures in the same direction of the flow field at distances of 0.0 cm, 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, 2.5 cm, 3.0 cm, 3.5 cm and 4.0 cm from the surface of the hot plate. The average temperature was measured 100 times at each height, allowing the actual temperatures to be compared with the calculated temperatures. Tab.1 and 2 show the measured and calculated temperatures at each of the 9 heights. Fig.8 shows a graph of the temperature at different heights, given by both the calculation and the thermocouple.

為了表明該計(jì)算方法的可行性，試驗(yàn)利用電熱偶探針由平臺(tái)表面，沿著空氣流場(chǎng)流動(dòng)的方向依次測(cè)量，每次測(cè)量間距離平臺(tái)為0.0、0.5、1.0、1.5、2.0、2.5、3.0、3.5、4.0 cm，并且在每個(gè)高度測(cè)量100次取平均溫度值。再與所計(jì)算出的溫度值相比較。表1和表2為這9個(gè)高度位置上，由算法計(jì)算的溫度值與電熱偶測(cè)量值對(duì)比表。圖8為算法得出的受熱空氣的溫度值與測(cè)溫儀得出的溫度值對(duì)比曲線圖。

Tab.1 ComparisonTable of measurement values at 50℃ 表1 50 ℃測(cè)量值對(duì)比表

Tab.2 ComparisonTable of measurement values at 90 ℃ 表2 90 ℃下測(cè)量值對(duì)比表

Fig.8 Schlieren image on the heating platform at different temperatures 圖8 定量算法計(jì)算值與測(cè)溫儀測(cè)量值比較圖(上)50 ℃和(下)90 ℃

From the Fig.8, it is clear that there is a big drop in temperature, after which the measured temperature slowly and stably declines. This is because the closer the air is to the platform, the stronger the heating effect and the weaker the influence of the surrounding environment. Contrarily, as the heat from the hot plate is weaker, the influence from the environment is stronger. Tabl.1 and 2 reflect this fast-to-slow phenomenon, where both the measured result from the thermocouple and the result of the calculation reflect this consistently. However, at the surface of the hot plate, one can sea a huge discrepancy between the results of the measurement and calculation at 90 ℃, with consistency returning thereafter. It is believed that this is because the heat exceeds the measurement range at this point. The method of analyzing this measurement range follows.

從圖8中不難看出，距離加熱平臺(tái)表面越遠(yuǎn)，期間測(cè)量數(shù)值有一個(gè)相當(dāng)大的陡變時(shí)期，經(jīng)過(guò)這個(gè)陡變期后，測(cè)量數(shù)值基本保持穩(wěn)中有降的趨勢(shì)。這是因?yàn)榫嚯x平臺(tái)越近空氣的受加熱作用越強(qiáng)且受到周?chē)h(huán)境的影響越弱，反之受熱作用越弱且受到周?chē)h(huán)境的影響越強(qiáng)。由表1和表2中也能夠看出同樣的溫度變化趨勢(shì)，用測(cè)溫儀測(cè)量的數(shù)據(jù)與紋影算法計(jì)算的數(shù)值相吻合。但是在圖8下圖和表2中，可以明顯看出，溫度在90 ℃時(shí)，該計(jì)算方法對(duì)于緊貼平臺(tái)表面位置的測(cè)量有一個(gè)非常大的誤差，考慮之后計(jì)算出的溫度與測(cè)量值相差無(wú)幾，初步斷定此時(shí)已經(jīng)超出該計(jì)算方法其量程，以下是具體分析過(guò)程。

5.1 Algorithm Measurement Range

算法測(cè)量量程

The core of the calculation is to measure the change in the light that′s being occluded. From formula (12) and the results of the experiment, it can be seen that the proposed calculation method is limited by its scope in temperature. This range is determined by the reference temperatureT0(equivalent reference densityρ0). The main reasons for this are as follows:

由于算法的核心理念是找出光源像面積被遮擋前后的變化量。從公式(12)和試驗(yàn)結(jié)果中不難看出，該算法對(duì)溫度測(cè)量范圍有所限制，測(cè)量范圍由基準(zhǔn)溫度T0(等同基準(zhǔn)密度ρ0)決定，其主要原因有以下兩點(diǎn)：

(1)The scope of grayness of the camera. When making the calibration curve using the blade and image, the 8-bit grayscale camera will be unable to produce shades that fit on the calibration curve whenever a shade of light surpasses it′s 0-255 shade range. Therefore, this method is only valid within the range set by the calibration curve, and that range is restricted to a camera′s 0-255 shades.

(1)相機(jī)的灰度范圍。在作圖像灰度與刀口切割量的標(biāo)定曲線時(shí)，相機(jī)的8位灰度值范圍(0～255)會(huì)對(duì)測(cè)量范圍造成影響，因?yàn)楫?dāng)灰度值一旦超過(guò)255或者低于0時(shí)，其光線的偏折是無(wú)法通過(guò)標(biāo)定曲線來(lái)取值的。故該方法只有在標(biāo)定曲線的線性范圍內(nèi)有效且標(biāo)定曲線的線性范圍不能超過(guò)灰度范圍0～255；

(2)The area of the light source on the image. If the measured object is the same medium and the focal length of the second Schlieren lens remains constant，and ifKandf2in formula (8) are constant values, then the measurement scope is the formula′s ΔS, which is directly influenced by the knife′s position and the size of the light source on the image. Generally, the blade′s position should be such that it cuts the light source by half, but in reality this is affected by different Schlieren measuring devices and the size of the Schlieren points on the light source's image. It may seem as though using a larger light source will make a greater measurement range but this actually causes the sensitivity to Schlieren systems and flow fields to fall drastically. Researchers planning to perform this experiment should take this into consideration when selecting a light source.

(2)光源像面積大小。如果測(cè)量對(duì)象為同一介質(zhì)且第二個(gè)紋影透鏡的焦距保持不變，即公式(8)中的K值與f2為定值，那么限制測(cè)量范圍的就是ΔS，而ΔS受刀口參考位置和光源像面積大小二者影響，一般來(lái)說(shuō)，刀口的參考位置理論上為切割光源像一半，但實(shí)際由不同紋影設(shè)備的成像效果而定；而光源像面積受到所使用的紋影點(diǎn)光源的影響，雖然表面看似如果用發(fā)光面積較大的光源會(huì)讓測(cè)量范圍變大，但是發(fā)光面積過(guò)大的會(huì)使整個(gè)紋影測(cè)量系統(tǒng)對(duì)流場(chǎng)的靈敏程度下降很多，這就需要試驗(yàn)人員在挑選紋影光源的時(shí)候做出平衡選擇。

(3)The standard temperature(or standard density). From formula (7), it is apparent that the highest and lowest measurement of the algorithm′s temperature is manipulated byρ0. The formula itself produces the change in the temperature, but only does so using formula (12), which adds to the standard temperature in order to calculate the actual temperature.

(3)基準(zhǔn)溫度(或基準(zhǔn)密度)。從式(7)可以看出，該計(jì)算方法測(cè)量溫度的最大值與最小值與公式中的ρ0有關(guān)，該方法計(jì)算出的本來(lái)就是一個(gè)溫度變化量，只不過(guò)是通過(guò)式(12)將這個(gè)溫度變化量附加到基準(zhǔn)溫度上，進(jìn)而得出實(shí)際溫度。

In summary, the grayness of calibration curve used in this experiment has a scope of 0-180, which did not exceed the camera′s most extreme values. Also, the value of ΔSis dependent on the relationship between the position of the blade and the light source. In the experiment, the radius of the light source′s image was 0.845 mm, which causes ΔSto have a scope of 0-1.569 mm2. Therefore, according to formula (12), it can be deduced that the temperature range for the experiment′s temperature range is -8～55.3 ℃. Because of this, when the hot plate′s temperature is set to 90 ℃, the calculation is no longer appropriate for use. As the temperature was out of range, the resulting temperature from the calculation had a large error.

綜上所述，本實(shí)驗(yàn)標(biāo)定曲線的灰度范圍為0～180，未超過(guò)相機(jī)的最大灰度范圍；而ΔS值一般根據(jù)刀口與光源像位置關(guān)系來(lái)決定。實(shí)驗(yàn)中光源像半徑為0.845 mm，故ΔS的范圍為0～1.569 mm2，那么根據(jù)式(12)可以得出本實(shí)驗(yàn)的測(cè)量范圍為-8～55.3 ℃。因此在加熱平臺(tái)為90 ℃時(shí)，該計(jì)算方法已經(jīng)不能適用，實(shí)際溫度已超出其量程，進(jìn)而導(dǎo)致出現(xiàn)非常大的相對(duì)誤差。

5.2 Error Analysis

誤差分析

According to the analysis of the algorithm′s range set out in 5.1, the error analysis for measurements that fall within the range are as follows:the measured temperature is expressed by formula (8), whoseρyvalue has a significant influence on the results. Furthermore, the algorithm′s error comes from ΔS, whose formula of error transfer is:

根據(jù)5.1算法量程分析結(jié)果，對(duì)其在量程范圍內(nèi)的測(cè)量數(shù)據(jù)進(jìn)行誤差分析：測(cè)量溫度的表達(dá)式(8)，其中分母ρy為主要影響參數(shù)，而該參數(shù)的誤差主要來(lái)源于ΔS，其誤差傳遞公式為:

(16)

The algorithm′s Δ(ΔS) is the difference between the actual amount of change of the unoccluded light source image area and the measured unoccluded light source image area change. The actual measured area of the change in source light comes from two points:The first is the researcher′s measurement of the radius of the light source′s image. The second is the researcher′s judgement of the position of the blade in relation to the light source. Both of these points are a result of the fact that it is impossible to obtain a measurement of the obstruction of light when making the calibration curve, which in turn is because the camera demands a minimum difference in the level of obstruction as the knife is being shifted. Therefore, the measured diameter of the light source′s image and the position of the measured light source will have some discrepancy. This form of error, the adjustment of the used equipment and the equipment itself all lead to the systematic error.

式中，Δ(ΔS)為實(shí)際的未遮擋光源像面積的變化量與實(shí)際測(cè)量的未遮擋光源像面積變化量之間的差值。實(shí)際測(cè)量的未遮擋光源像面積變化量主要來(lái)源于兩點(diǎn)：一是試驗(yàn)人員對(duì)光源像半徑大小的讀數(shù)存在誤差；二是試驗(yàn)人員對(duì)刀口遮擋光源像位置的判定誤差。由于相機(jī)對(duì)連續(xù)的刀口遮擋時(shí)有一個(gè)最小遮擋量的圖像區(qū)別度，其一定要明顯，而在做標(biāo)定曲線時(shí)，無(wú)法獲得連續(xù)的刀口遮擋量。故在光源像半徑讀取和光源像位置的判定上會(huì)有一定的偏差。這類(lèi)誤差與試驗(yàn)所用的儀器本身誤差和調(diào)節(jié)儀器時(shí)產(chǎn)生的誤差，均屬于系統(tǒng)誤差。

By comparing and analyzing Fig.8 and Tab.1 and Tab.2 with regards to their measured and calculated temperatures, it is apparent that:calculating the temperature and measuring the temperature of flow fields in Schlieren give a relative error of less than 10%; algorithmic calculation of the temperature of Schlieren produces reasonably accurate results and; such method is applicable to measuring Schlieren flow fields.

從對(duì)比圖8、表1和表2中測(cè)量值與算法計(jì)算值之間的差異，再結(jié)合上述分析，不難看出：在紋影儀空氣流場(chǎng)溫度定量測(cè)量試驗(yàn)中，算法的相對(duì)誤差總體小于10%，說(shuō)明紋影儀溫度定量算法具有較好的準(zhǔn)確性，對(duì)紋影法測(cè)量空氣氣體流場(chǎng)具備一定借鑒意義。

6 Conclusion

結(jié) 論

This paper built Schlieren models and used their principles of measurement in order to analyze the relationship between a light source′s image and the position of a blade. From this, a relationship between the blade and the grayness of an image was obtained, which in turn provided a method of measuring Schlieren temperature. This paper used laboratory measurements and calculations to carefully analyze and provide detail about the shortcomings of the algorithm. The results are:when measuring the temperature of Schlieren flow fields, the proposed algorithmic method has relatively accurate results, providing error that falls under 10% compared to methods using tools of measurement.

本文從紋影定量測(cè)量原理的基礎(chǔ)上，以典型的紋影系統(tǒng)為基礎(chǔ)，通過(guò)分析刀口與光源像的位置關(guān)系，以得到刀口位置與灰度值的對(duì)應(yīng)關(guān)系，進(jìn)而給出一種紋影定量計(jì)算方法，并對(duì)其測(cè)量范圍和實(shí)驗(yàn)結(jié)果作了詳細(xì)分析，彌補(bǔ)了在紋影定量計(jì)算推導(dǎo)中的不足。結(jié)果表明：在紋影儀空氣流場(chǎng)溫度定量測(cè)量試驗(yàn)中，紋影儀溫度定量算法具有較好的準(zhǔn)確性，該計(jì)算方法的相對(duì)誤差小于10%，對(duì)紋影法測(cè)量氣體流場(chǎng)溫度具備一定的借鑒意義。