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1. 中国空气动力研究与发展中心 高速空气动力研究所, 绵阳 621000;
2. 中国空气动力研究与发展中心 空气动力学国家重点实验室, 绵阳 621000

Measurement technique optimization of turbulence level in compressible fluid by changing overheat ratio of hot wire anemometer
DU Yufeng1, LIN Jun1, WANG Xunnian2, XIONG Neng1
1. High Speed Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China;
2. State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China
Abstract: In order to increase the measurement precision of a compressible flow, an optimization of turbulence level measurement technique is studied. By changing the overheat ratio of hot-wire anemometer and neglecting the pressure fluctuation terms in the governing equations, the turbulence level is solved. In order to evaluate the turbulence level in high speed flow more precisely, the algorithm for turbulence level based on response function of constant temperature hot-wire anemometer in compressible fluid is theoretically optimized by introducing pressure fluctuation. Turbulence level measurement experiments are carried out with the Mach number varied from 0.3 to 0.7 and the experimental data is processed by using algorithm of turbulence level before and after optimization. The results indicate that the magnitudes of turbulence level from the two methods are similar, but the variation tendency of turbulence level with Mach number obtained by using the optimized method is in accordance with the objective physical law. The uncertainty of turbulence level is obtained using Monte Carlo simulation, and the magnitude of the uncertainty is much smaller than that of the turbulence level. The results suggest that the turbulence level obtained using the optimized method could be regarded as the true values. The results proved the correctness of the turbulence level measurement technique after optimization and the feasibility of application of constant temperature hot-wire anemometer into turbulence level measurement in high speed wind tunnels.
Keywords: compressible fluid     turbulence level     overheat ratio     Monte Carlo simulation     uncertainty

1 湍流度求解方法优化 1.1 湍流度求解方法

 $\frac{{\Delta E}}{E} = {F_{{\rm{CTA}}}}\frac{{\Delta m}}{m} - {G_{{\rm{CTA}}}}\frac{{\Delta {T_0}}}{{{T_0}}}$ （1）

 $\theta = \frac{{\Delta m}}{m}r - \frac{{\Delta {T_0}}}{{{T_0}}}$ （2）

 ${\left( {\sqrt {\overline {{\theta ^2}} } } \right)^2} = \overline {{{\left( {\frac{{\Delta m}}{m}} \right)}^2}} {r^2} - 2\overline {\left( {\frac{{\Delta m}}{m}} \right)\left( {\frac{{\Delta {T_0}}}{{{T_0}}}} \right)} r + \overline {{{\left( {\frac{{\Delta {T_0}}}{{{T_0}}}} \right)}^2}}$ （3）

1.2 湍流度求解方法优化的理论推导

 $\frac{{\Delta m}}{m} = \frac{{\Delta \left( {\rho U} \right)}}{{\rho U}} = \frac{{\Delta \rho }}{\rho } + \frac{{\Delta U}}{U}$ （4）

 ${T_0} = T\left( {1 + \frac{{\gamma - 1}}{2}M{a^2}} \right)$ （5）

 $\begin{array}{*{20}{c}} {\frac{{\partial {T_0}}}{{{T_0}}} = \frac{{\partial T}}{T} + \frac{{\partial \left( {1 + \frac{{\gamma - 1}}{2}M{a^2}} \right)}}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}} = }\\ {\frac{{\partial T}}{T} + \frac{{\left( {\gamma - 1} \right)Ma}}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}}\partial Ma} \end{array}$ （6）

 $Ma = \frac{U}{{\sqrt {\gamma RT} }}$ （7）

 $\begin{array}{l} \partial Ma = \partial \left( {\frac{U}{{\sqrt {\gamma RT} }}} \right) = \frac{{\sqrt {\gamma RT} \partial U - U\frac{{\sqrt {\gamma R} }}{{2\sqrt T }}\partial T}}{{\gamma RT}} = \\ \;\;\;\;\;\;\frac{U}{{\sqrt {\gamma RT} }}\left( {\frac{{\partial U}}{U} - \frac{1}{2} \cdot \frac{{\partial T}}{T}} \right) = Ma\left( {\frac{{\partial U}}{U} - \frac{1}{2} \cdot \frac{{\partial T}}{T}} \right) \end{array}$ （8）

 $\begin{array}{l} \frac{{\partial {T_0}}}{{{T_0}}} = \frac{{\partial T}}{T} + \frac{{\left( {\gamma - 1} \right)M{a^2}}}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}}\left( {\frac{{\partial U}}{U} - \frac{1}{2} \cdot \frac{{\partial T}}{T}} \right) = \\ \;\;\;\;\;\;\frac{1}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}} \cdot \frac{{\partial T}}{T} + \frac{{\left( {\gamma - 1} \right)M{a^2}}}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}} \cdot \frac{{\partial U}}{U} \end{array}$ （9）

 $\frac{{\Delta {T_0}}}{{{T_0}}} = \alpha \frac{{\Delta T}}{T} + \beta \frac{{\Delta U}}{U}$ （10）

 $\alpha = \frac{1}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}}$ （11）
 $\beta = \frac{{\left( {\gamma - 1} \right)M{a^2}}}{{1 + \frac{{\gamma - 1}}{2}M{a^2}}}$ （12）

 $\theta = \left( {\frac{{\Delta \rho }}{\rho } + \frac{{\Delta U}}{U}} \right)r - \left( {\alpha \frac{{\Delta T}}{T} + \beta \frac{{\Delta U}}{U}} \right)$ （13）

 $p = C{\rho ^\gamma }$ （14）

 $\frac{{\Delta \rho }}{\rho } = \frac{1}{\gamma } \cdot \frac{{\Delta p}}{p}$ （15）

 $p = \rho RT$ （16）

 $\frac{{\Delta p}}{p} = \frac{{\Delta \rho }}{\rho } + \frac{{\Delta T}}{T}$ （17）

 $\frac{{\Delta T}}{T} = \frac{{\gamma - 1}}{\gamma } \cdot \frac{{\Delta p}}{p}$ （18）

 $\begin{array}{l} \theta = \left( {\frac{1}{\gamma } \cdot \frac{{\Delta p}}{p} + \frac{{\Delta U}}{U}} \right)r - \left( {\alpha \frac{{\gamma - 1}}{\gamma } \cdot \frac{{\Delta p}}{p} + \beta \frac{{\Delta U}}{U}} \right) = \\ \;\;\;\;\;\left( {\frac{1}{\gamma } \cdot \frac{{\Delta p}}{p} + \frac{{\Delta U}}{U}} \right)r - \beta \left( {\frac{1}{{\gamma M{a^2}}} \cdot \frac{{\Delta p}}{p} + \frac{{\Delta U}}{U}} \right) \end{array}$ （19）

 $\begin{array}{l} {\left( {\sqrt {\overline {{\theta ^2}} } } \right)^2} = \\ \;\;\;\;\;\left[ {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} + \frac{2}{\gamma }\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} + \frac{1}{{{\gamma ^2}}}\overline {{{\left( {\frac{{\Delta p}}{p}} \right)}^2}} } \right]{r^2} - \\ \;\;\;\;\;2\beta r\left[ {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} + \frac{1}{\gamma }\left( {1 + \frac{1}{{M{a^2}}}} \right)\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} + } \right.\\ \;\;\;\;\;\left. {\frac{1}{{{\gamma ^2}M{a^2}}}\overline {{{\left( {\frac{{\Delta p}}{p}} \right)}^2}} } \right] + {\beta ^2}\left[ {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} + } \right.\\ \;\;\;\;\;\left. {\frac{2}{{\gamma M{a^2}}}\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} + \frac{1}{{{\gamma ^2}M{a^4}}}{{\left( {\frac{{\Delta p}}{p}} \right)}^2}} \right] \end{array}$ （20）

 $\overline {{{\left( {\frac{{\Delta m}}{m}} \right)}^2}} = \overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} + \frac{2}{\gamma }\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} + \frac{1}{{{\gamma ^2}}}\overline {{{\left( {\frac{{\Delta p}}{p}} \right)}^2}}$ （21）
 $\begin{array}{l} \overline {\frac{{\Delta m}}{m} \cdot \frac{{\Delta {T_0}}}{{{T_0}}}} = \beta \left[ {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} + } \right.\\ \;\;\;\;\;\;\left. {\frac{1}{\gamma }\left( {1 + \frac{1}{{M{a^2}}}} \right)\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} + \frac{1}{{{\gamma ^2}M{a^2}}}\overline {{{\left( {\frac{{\Delta p}}{p}} \right)}^2}} } \right] \end{array}$ （22）
 $\begin{array}{l} \overline {{{\left( {\frac{{\Delta {T_0}}}{{{T_0}}}} \right)}^2}} = {\beta ^2}\left[ {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} + } \right.\\ \;\;\;\;\;\left. {\frac{2}{{\gamma M{a^2}}}\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} + \frac{1}{{{\gamma ^2}M{a^4}}}\overline {{{\left( {\frac{{\Delta p}}{p}} \right)}^2}} } \right] \end{array}$ （23）

 $\left[ {\begin{array}{*{20}{c}} {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} }\\ {\overline {\frac{{\Delta U}}{U} \cdot \frac{{\Delta p}}{p}} }\\ {\overline {{{\left( {\frac{{\Delta p}}{p}} \right)}^2}} } \end{array}} \right] = \mathit{\boldsymbol{D}} \cdot \mathit{\boldsymbol{X}}$ （24）

 $\mathit{\boldsymbol{D}} = {\left[ {\begin{array}{*{20}{c}} 1&{\frac{2}{\gamma }}&{\frac{1}{{{\gamma ^2}}}}\\ \beta &{\frac{\beta }{\gamma }\left( {1 + \frac{1}{{M{a^2}}}} \right)}&{\frac{\beta }{{{\gamma ^2}M{a^2}}}}\\ {{\beta ^2}}&{\frac{{2{\beta ^2}}}{{\gamma M{a^2}}}}&{\frac{{{\beta ^2}}}{{{\gamma ^2}M{a^4}}}} \end{array}} \right]^{ - 1}}$ （25）
 $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {\overline {{{\left( {\frac{{\Delta m}}{m}} \right)}^2}} }\\ {\overline {\frac{{\Delta m}}{m} \cdot \frac{{\Delta {T_0}}}{{{T_0}}}} }\\ {\overline {{{\left( {\frac{{\Delta {T_0}}}{{{T_0}}}} \right)}^2}} } \end{array}} \right]$ （26）

 $Tu = \sqrt {\overline {{{\left( {\frac{{\Delta U}}{U}} \right)}^2}} }$ （27）
2 湍流度测量试验及结果分析 2.1 风洞及测量仪器

 技术参数 数值 试验段喷管出口截面尺寸Ø/mm 50 马赫数调节范围 0.05~1 总压调节范围/MPa 0.05~0.25 总温调节范围/K 278~330 迎角调节范围/(°) -30~30
 图 1 探针校准风洞示意图 Fig. 1 Schematic of probe calibration wind tunnel
2.2 试验结果分析

 Ma=0.330 Ma=0.420 Ma=0.525 Ma=0.627 Ma=0.719 r $\sqrt{\theta^{2}} / 10^{-4}$ r $\sqrt{\theta^{2}} / 10^{-4}$ r $\sqrt{\theta^{2}} / 10^{-4}$ r $\sqrt{\theta^{2}} / 10^{-4}$ r $\sqrt{\theta^{2}} / 10^{-4}$ 0.034 7.53 0.034 5.95 0.034 6.23 0.034 5.59 0.033 7.56 0.053 6.57 0.053 5.43 0.052 5.27 0.052 4.72 0.051 4.55 0.073 4.49 0.073 4.64 0.073 3.61 0.073 3.81 0.071 3.68 0.095 4.25 0.096 3.66 0.119 4.13 0.119 3.68 0.116 3.97 0.119 5.60 0.121 5.04 0.146 4.37 0.146 3.57 0.142 4.21 0.146 5.19 0.148 4.81 0.176 5.17 0.177 3.84 0.171 4.83 0.177 5.88 0.179 5.78 0.210 5.90 0.212 4.36 0.205 5.32 0.211 6.33 0.215 6.73 0.250 7.07 0.252 5.16 0.243 6.27 0.251 7.66 0.257 7.65 0.638 15.3 0.659 12.5 0.603 12.2 0.640 22.5 0.681 21.2 1.290 32.4 1.399 27.5 1.165 24.5

 图 2 双曲线拟合结果 Fig. 2 Results of hyperbola fitting

 Ma 拟合优度 Tu/% 优化后 优化前 0.330 0.997 4 0.355 0.469 0.420 0.998 6 0.357 0.365 0.525 0.999 5 0.329 0.269 0.627 0.999 5 0.306 0.208 0.719 0.996 0 0.425 0.221

 图 3 优化前后湍流度对比 Fig. 3 Contrast of turbulence level before and after optimization
3 湍流度不确定度的评估

1) 构造概率统计模型。在各个马赫数下，利用双曲线进行拟合时，可利用计算机进行仿真试验生成大量待拟合的散点数据，待拟合的散点数据到湍流度测量试验结果散点数据(后文简称“已知散点”)的纵向距离符合以已知散点的纵向位置为均值μ、所有已知散点到拟合曲线的纵向距离均值的1/3为标准差σ的正态分布N(μ, σ2)(即符合“3σ”原则)[30]

2) 模型的随机抽样。利用MATLAB软件生成符合正态分布N(μ, σ2)的1 000组随机数，将已知散点与生成的1 000组随机数叠加，即为待拟合的散点数据。

3) 确定评估值。利用双曲线拟合方法对生成的1 000组待拟合的散点数据进行拟合，得到1 000组湍流度拟合结果，并对湍流度的不确定度进行求解，即

 $u = \sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {{x_i} - \bar x} \right)}^2}} }$ （28）

 Ma 湍流度均值/% 不确定度/% 0.330 0.355 0.003 0.420 0.357 0.002 0.525 0.329 0.005 0.627 0.306 0.006 0.719 0.425 0.011

4 结论

1) 针对在研的变热线过热比湍流度测量方法忽略压力脉动项的问题进行了完善，从理论上优化了利用恒温热线风速仪对可压缩流湍流度进行测量的方法。

2) 在马赫数Ma=0.3~0.7进行了湍流度测量试验，利用优化后的湍流度求解方法对湍流度进行了求解，并与优化前的求解结果进行了对比。结果表明优化后的湍流度求解方法所得的湍流度更加符合客观物理规律及部分文献测试结果，验证了优化后方法的有效性。

3) 利用蒙特卡洛方法对湍流度的不确定度进行了求解，不确定度量值远小于湍流度量值，验证了湍流度测量结果的稳定性。

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http://dx.doi.org/10.7527/S1000-6893.2019.23067

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#### 文章信息

DU Yufeng, LIN Jun, WANG Xunnian, XIONG Neng

Measurement technique optimization of turbulence level in compressible fluid by changing overheat ratio of hot wire anemometer

Acta Aeronautica et Astronautica Sinica, 2019, 40(12): 123067.
http://dx.doi.org/10.7527/S1000-6893.2019.23067