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基于连续性假设,火箭发动机尾喷焰化学反应流动可由可压缩N-S(Navier-Stokes)方程来描述。描述非稳态喷焰流场的方程组包含质量守恒、动量守恒和能量守恒方程。
质量守恒方程:
$$ \frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {x_i}}}(\rho {u_i}) = 0 $$ (1) 动量守恒方程:
$$ \frac{\partial }{{\partial t}}(\rho {u_i}) + \frac{\partial }{{\partial {x_k}}}(\rho {u_i}{u_k}) = - \frac{{\partial p}}{{\partial {x_i}}}{\text{ + }}\frac{{\partial {\tau _{ik}}}}{{\partial {x_k}}} $$ (2) 能量守恒方程:
$$ \frac{{\partial E}}{{\partial t}} + \frac{\partial }{{\partial {x_i}}}(E{u_i} + p{u_i}) = \frac{\partial }{{\partial {x_j}}}({u_j}{\tau _{ij}} + {q_i}) $$ (3) 组分描述方程:
$$ \frac{{\partial (\rho {F_{{s}}})}}{{\partial t}} + \frac{{\partial (\rho {F_s}{u_j})}}{{\partial {x_j}}} = \frac{\partial }{{\partial {x_j}}}\left( {\rho {D_s}\frac{{\partial {F_s}}}{{\partial {x_j}}}} \right) + {\dot \omega _s} $$ (4) 式中:ρ、p、E分别为流体密度、压强和内能;t为时间;u为流体速度;τ为流体粘性应力;F为组分的质量分数;D为组分扩散系数;
$ \dot \omega $ 为复燃化学反应引起的质量变化率;q为热流;在未直接指定的情况下,下标i、j、k表示直角坐标的三个方向;s为组分编号。针对公式(4)中的组分质量变化率,采用有限速率的化学反应动力式进行计算。对于具有
$ {N_r} $ 个基元反应的某反应,其当量表达式可以写为:$$ \sum\limits_{j = 1}^{{N_r}} {{v_{ij}}'{W_j}} \overset {{k_f}_i,{k_b}_i} \longleftrightarrow \sum\limits_{j = 1}^{{N_r}} {{v_{ij}}} ''{W_j}(i = 1,2,\cdots,{N_r}) $$ (5) 式中:
$ {v_{ij}}' $ ,$ {v_{ij}}'' $ 分别为第i个基元反应中组分j的反应物和生成物的当量反应系数;$ {W_j} $ 为反应物组分;$ {k_f}_i $ 、$ {k_{bi}} $ 分别为第i个反应的正、逆反应速率常数。选取9组分10反应的有限速率化学动力模型描述,即采用考虑H2O、CO2、CO、HCl、O、H、H2、OH和N2组分的H2-CO-O2反应体系[11]。为求解上述控制方程,采用应用广泛的RANS(Reynolds-averaged Navier-Stokes)方法和基于涡粘性假设的Realizable
$ k - \varepsilon $ 湍流模型。 -
在连续流域,可认为喷焰流场处于热力学平衡态。喷焰中的高温气体组分发出特定波段的红外光辐射,其主要机制为分子振-转跃迁产生的谱带发射。文中将采用窄谱带模型法计算高温燃气的辐射物性参数。在窄谱带模型中,
$\Delta \eta $ 光谱范围内的平均透过率可由下式[12]表示:$$ \overline \tau = \exp \left\{ { - 2a\left[ {{{\left( {1 + \frac{{\overline \kappa u}}{a}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}} - 1} \right]} \right\} $$ (6) 式中:
$a = \pi {{\overline \gamma } \mathord{\left/ {\vphantom {{\overline \gamma } {\overline d}}} \right. } {\overline d}}$ 为谱带精细结构参数;$\overline \kappa $ 谱带内平均吸收系数;$ u $ 为压力行程长度,前两项参数可由NASA-3080数据库[12]提供。文中针对尾喷焰内H2O、CO2、CO、HCl、NO、OH 6种气体,以NASA-3080光谱数据库为基础,建立了各组分在100~3000 K范围内的窄谱带模型参数库。在进行辐射传输计算时将该数据库读入程序中,并对波数、温度进行插值,而对压力进行修正可得到计算所需的透过率数据。 -
NIPC方法具有坚实的数学基础,是基于PCE方法发展的一种不确定度分析方法。NIPC方法可以将任一随机变量分解为确定和随机两个独立部分,适用于在已知求解器的基础上开展变量灵敏度特性分析。在不确定性系统中,任意随机变量
$ {\alpha ^*} $ (如温度等)可以表示为[13]:$$ {\alpha ^*}(x,\lambda ) \approx \sum\nolimits_{j = 0}^P {{b _j}} (x) {\psi _j}(\lambda ) $$ (7) 式中:
$ {b_j}(x) $ 为确定部分的耦合系数;$ {\psi _j}(\lambda ) $ 为以n维随机变量$ \lambda = ({\lambda _1},{\lambda _2},\cdots,{\lambda _n}) $ 为自变量的j阶随机基函数。未知系数的个数
${N_{{t}}}$ 可由下式计算得到:$$ {N_t} = P + 1 = \frac{{(n + p)!}}{{n!p!}} $$ (8) 式中:n为随机变量的维数;p为PCE中的多项式的最高阶数;P+1为PCE中的多项式项数。
由于NIPC方法不需要对原控制方程进行修改,可采用回归分析的方法对多项式混沌展开系数进行求解,获得新的数学模型或代理模型,可以充分利用现有喷焰红外喷焰数值模拟程序IRSAT (Infrared Signature Analysis Tool)[14],从而大幅减小不确定性分析所需要的工作量。
文中采用随机响应面法[13]来求解PCE系数。首先选取有效的样本点计算未知PCE系数bj(j = 0,1,···,P)。用
$ N $ 表示样本数,采用过采样策略,以两倍于未知PCE系数个数的样本$ N = 2 \times {N_t}$ 可得到相对准确的结果。从设计空间内抽样得到
$ N $ 个有效样本$ \lambda = {\left[ {{\lambda _1},{\lambda _2}, \cdots, {\lambda _i},{\lambda _{i + 1}}, \cdots, {\lambda _N}} \right]^{\rm{T}}} $ ,将其从原随机空间变换到标准随机空间。标准随机空间是指空间中每一维都是标准随机变量。则样本变为$\xi = [ {{\xi _1},{\xi _2}, \cdots,{\xi _i}, {\xi _{i + 1}}, \cdots, {\xi _N}} ]^{\rm{T}}$ 。将抽样所得到的$ N $ 个样本代入原模型,得到各样本处相对应的函数响应值,记为$ G $ ,则:$$ G = {\left[ {g({\lambda _1}),g({\lambda _2}),\cdots g({\lambda _i}),g({\lambda _{i + 1}}),\cdots g({\lambda _N})} \right]^{\rm{T}}} $$ (9) 再将样本和相应函数响应值代入PCE模型得:
$$ \left[ {\begin{array}{*{20}{c}} {{\psi _0}({\xi _1})}&{{\psi _1}({\xi _1})}& \cdots &{{\psi _P}({\xi _1})} \\ {{\psi _0}({\xi _2})}&{{\psi _1}({\xi _2})}& \cdots &{{\psi _P}({\xi _2})} \\ \vdots & \vdots & \ddots & \vdots \\ {{\psi _0}({\xi _N})}&{{\psi _1}({\xi _N})}& \cdots &{{\psi _P}({\xi _N})} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{b_0}} \\ {{b_1}} \\ \vdots \\ {{b_P}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {g({\lambda _1})} \\ {g({\lambda _2})} \\ \vdots \\ {g({\lambda _N})} \end{array}} \right] $$ (10) 多项式混沌展开系数可以根据最小二次回归原理计算求得:
$$b = {({A^{\rm{T}}}A)^{ - 1}}{A^{\rm{T}}}G $$ (11) -
在基于随机响应面法获得多项式混沌展开系数的基础上,总方差可以表示为[13]:
$$ D = {\sum\nolimits_{k = 1}^P {{b_k}} ^2}E\left[ {{\psi _k}^2(\xi )} \right] $$ (12) 总方差可以分解成单个变量的局部方差及各个变量间相互联系、相互作用下的局部方差,即:
$$ \begin{gathered} D = \sum\nolimits_{i = 1}^{i = n} {{D_i}} + \sum\nolimits_{1 \leqslant i < j \leqslant n}^{i = n - 1} {{D_{i,j}}} + \\ {\text{ }}\sum\nolimits_{1 \leqslant i < j < k \leqslant n}^{i = n - 2} {{D_{i,j,k}}} +\cdots + {D_{1,2,\cdots,n}} \\ \end{gathered} $$ (13) 根据混沌多项式理论,局部方差可表示为:
$$ {D_{{i_1}...{i_s}}} = {\sum\nolimits_{\alpha \in {F_{{i_1}...{i_s}}}} {{b_\alpha }} ^2}E\left[ {{\psi _\alpha }^2(\xi )} \right] $$ (14) 其中
$$ {F_{{i_1},\cdots,{i_s}}} = \left\{ {\begin{array}{*{20}{c}} {{\alpha _k} > 0,\forall k = 1,\cdots,n,k \in ({i_1},\cdots,{i_s})} \\ {{\alpha _k} = 0,\forall k = 1,\cdots,n,k \notin ({i_1},\cdots,{i_s})} \end{array}} \right\} $$ (15) 显然,系统的总方差可以用单个变量的局部方差以及各个变量间相互联系、共同作用的局部方差之和来表示。将局部方差与总方差之比定义为Sobol指数,该指数可反映单个变量或各个变量间的共同作用下对模型输出的贡献,即:
$$ {S_{{i_1},\cdots,{i_s}}} = \dfrac{{{D_{{i_1},\cdots,{i_s}}}}}{D} = \dfrac{{{{\displaystyle\sum\nolimits_{\alpha \in {F_{{i_1}\cdots{i_s}}}} {{b_\alpha }} }^2}E\left[ {{\psi _\alpha }^2(\xi )} \right]}}{{{{\displaystyle\sum\nolimits_{k = 1}^P {{b_k}} }^2}E\left[ {{\psi _k}^2(\xi )} \right]}} $$ (16) 式中:
$ {S_i} $ 单个变量$ {X_i} $ 的灵敏度指标,即主Sobol指数。除了单个变量对系统的作用外,该变量和其他变量的共同作用,对系统的总方差也有可能造成很大的影响。此时,需将所有与
$ {X_i} $ 相关的灵敏度指标相加,得到关于变量$ {X_i} $ 总灵敏度指标,称为总Sobol指数,记为${S_i}^{\rm{T}}$ :$$ \begin{gathered} {S_i}^{\rm{T}} = \sum\nolimits_{{\psi _i}} {{S_{{i_1},\cdots,{i_s}}}} \\ {\psi _i} = \left\{ {({i_1},\cdots,{i_s}):\exists k,1 \leqslant k \leqslant s,{i_k} = i} \right\} \\ \end{gathered} $$ (17) 即变量的总灵敏度指数由该变量的各阶灵敏度指数之和表示,
${S_i}^{\rm{T }}$ 的大小表明了变量$ {X_i} $ 本身及其他与变量$ {X_i} $ 共同作用对系统总方差的贡献程度。主Sobol指数$ {S_i} $ 越大,则表示变量$ {X_i} $ 对系统输出的贡献越大,而总Sobol指数${S_i}^{\rm{T}}$ 越大则表示变量$ {X_i} $ 本身及与其他变量的耦合作用对系统输出的贡献越大。 -
考虑攻角下喷焰流场计算域的对称性,选取三维流场计算域的一半进行仿真计算,计算域结构和网格分布如图1所示。图1(a)给出了计算域和边界条件类型,包含来流入口边界、压力远场边界、压力出口边界和多发动机喷口的高速、高温、多组分入口边界。为了捕捉喷焰的流动细节特征和减小计算花费,计算域需要在网格数目和计算精度的折中下合理地离散。图1(b)给出了喷焰流场计算域的结构网格分布示意图,网格在轴向、径向和周向数目分别为350×125×120。此外,在壁面附近和喷焰混合层可能经过的区域进行加密处理,且验证了计算网格的无关性。
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文中以低空(H=15 km)Atlas-IIA火箭发动机欠膨胀喷焰为研究对象。Atlas-IIA采用多喷管煤油/液氧发动机组,其由两个助推发动机和一个主发动机组成,其详细的结构布局和尺寸参考文献[15]。为了简化,多数研究[16-17]采用等效单喷管来简化处理多喷管问题,即依据流量和组分通量等效原则获得单喷管喷口参数。等效喷口参数和来流参数如表1所示。
Parameter Free stream Nozzle U/m·s−1 545.5 2 960 P/Pa 12 112 68 850 T/K 216.7 2 230 H2 0.000 000 0.012 194 O2 0.233 154 0.000 028 H2O 0.000 000 0.272 437 H 0.000 000 0.000 127 OH 0.000 000 0.000 797 N2 0.766 388 0.000 000 CO 0.000 000 0.412 125 CO2 0.000 458 0.302 293 -
文中以来流速度、来流温度、来流压力和攻角为不确定输入参量,对喷焰红外辐射响应的不确定度传播和敏感性分析。在表1中基础来流参数的基础上,假设来流速度U∞服从
$ N(545.5,{55^2}) $ 的正态分布,来流压强P∞服从$ N(12\;112,{1\;200^2}) $ 的正态分布,来流温度T∞服从$ U(211.7,221.7) $ 的均匀分布,攻角ɑ服从$ N(0,{3^2}) $ 的正态分布。NIPC多项式选取3阶,采用拉丁超立方试验设计方法产生70个样本。依据IRSAT代码计算模型喷焰红外辐射响应值,包含2.0~6.0 μm光谱辐射强度、3~5、2.5~3.2、2.8~3.0、4.35~4.65 μm谱带内的积分强度。 -
为了验证喷焰流场计算和红外辐射计算模型的可靠性,将计算结果与美国标准红外计算代码(Standard Infrared Radiation Model, SIRRM)计算的Atlas-IIA 15 km高度下喷焰红外辐射光谱数据进行对照。计算中选取喷焰流场计算域的轴向上游150 m,计算获得2.0~6.0 μm波段的辐射光谱与文献参考数据对照如图2所示。从图中可知,光谱仿真计算值与参考文献[18-19]结果较为接近,验证了喷焰流场和红外辐射计算模型的可靠性。
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通过Ishigami函数来验证NIPC方法预估参数灵敏度的正确性,函数的表达式为:
$$ Y = \sin {X_1} + a \cdot {\sin ^2}{X_2} + b \cdot {X_3}^4 \cdot \sin {X_1} $$ (18) 式中:输入变量
$ {X_i} $ ,$ i = 1,2,3 $ 为均匀分布,取值范围为$ \left[ { - \pi ,\pi } \right] $ ;a和b分别取7和0.1。在参考文献[20]中,该函数作为基准灵敏度的考核方法。Y的方差D和Sobol灵敏度指标可以通过解析计算获得[20]:$$ \begin{array}{l}D=\dfrac{{a}^{2}}{8}+\dfrac{\text{b}\cdot {\pi }^{4}}{5}+\dfrac{{b}^{2}\cdot {\pi }^{8}}{18}+\dfrac{1}{2}\\ {D}_{1}\text=\dfrac{b\cdot {\pi }^{4}}{5}+\dfrac{{b}^{2}\cdot {\pi }^{8}}{50}+\dfrac{1}{2}\\ {D}_{2}=\dfrac{{a}^{2}}{8}\text{;}{D}_{3}=0\text{;}{D}_{12}={D}_{23}=0\text{;}\\ {D}_{13}=\dfrac{8\cdot {b}^{2}\cdot {\pi }^{8}}{225}\text{;}{D}_{123}=0\end{array} $$ (19) 针对公式(18)涉及的3变量系统,采用与文中不确定度量化分析方法一致的模型开展,计算出相应样本的响应值Y对应的均值、方差和Sobol指数,计算结果与理论值吻合,验证了NIPC方法和编制程序的可靠性。
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喷焰红外辐射特性计算需要以喷焰反应流场特性参数为物性底层数据,包含温度、压力和组分含量等。文中以Atlas-IIA发动机为研究对象,分析在基准来流参数一定偏差范围内,输入参量引起系统红外辐射信号的不确定度。在图3(a)~(d)分别给出了H=15 km高度下喷焰流场的温度、组分H2O、CO2和CO的分布云图。可以明显看出,喷焰流场在核心区下游位置(100~200 m范围)出现明显的复燃效应,存在温度和CO2质量分数升高的区域。同理,CO质量分数在复燃区域明显快速降低,这是由于CO为不稳定产物,会与大气中的O2发生反应生成较为稳定的CO2。
基于喷焰流场参数,在考虑1.2节给出的6种气体组分辐射物性参数的基础上,计算喷焰的红外辐射强度。全尺寸喷焰在2.0~6.0 μm波段的光谱曲线在图4中给出。由此可以看出,光谱在2.7 μm和4.3 μm附近存在两个明显的峰值区域。在图4的右上角给出了对应谱带的积分强度,分别包含Band1 (3~5 μm)、Band2 (2.5~3.2 μm)、Band3 (2.8~3.0 μm)、Band4 (4.35~4.65 μm) 4个谱带。可以看出,Band2谱带强度较Band1高出10%左右,Band3和Band4辐射强度接近,但较Band1和Band2低出约2倍。
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为了说明4个来流输入参量对喷焰辐射光谱的影响程度,在图5中给出了光谱平均曲线和不确定度沿波长的分布关系。从图中可以看出。在2.3 μm 波段光谱辐射不确定度较大,但是由于辐射强度较低,产生的影响也较小。在2.7 μm 和4.3 μm 波段辐射强度均值与4.1节基准来流参数下喷焰的辐射强度大致相同,不确定较大。
另外,在表2中给出了喷焰谱带内辐射强度的统计特性,包含平均值、标准差和不确定度。结果表明4个谱带积分强度的标准差与均值成正相关关系,即辐射强度的均值越大标准差越大,其中Band2波段的标准差最大,约为6.77×105 W/sr。然而,辐射强度的不确定度大小则呈现出与辐射强度均值大小相反的规律。其中,Band4 波段的辐射强度不确定度最大,约为9%,较Band2高出约2.5%。显然,来流特性引起的辐射强度不确定度与波段的选取有较高的关联度。
表 2 谱带内辐射强度的统计特性
Table 2. Statistical properties of in-band radiance intensity
Band Mean value/W·sr−1 Standard deviation/W·sr−1 Uncertainty 1 9.83×106 6.62×105 6.70% 2 1.05×107 6.77×105 6.50% 3 3.72×106 2.55×105 6.90% 4 3.85×106 3.50×105 9.10% -
通过Sobol指数Si可表征单个变量对模型输出的贡献,其中下标表示对应的输入参量。图6(a)~(d)分别为来流速度、来流压力、来流温度和攻角对辐射光谱响应量的Sobol指数,其大小表征该参数对于辐射强度的灵敏程度。从图中可以看出,来流速度在除4.3 μm波段外的大部分波段内对光谱辐射强度的灵敏度较高,对应的Sobol指数在0.5以上;来流压力在4.3 μm波段的灵敏度最高,Sobol指数的峰值接近1.0;攻角对光谱辐射强度的Sobol指数在0.4左右,且在多个波段出呈现较高的灵敏度;来流温度对喷焰光谱辐射影响甚微,可以忽略不计。在大部分波段内来流速度对于辐射强度的灵敏程度最高,占主导作用,来流速度和攻角次之,来流温度最小。
图 6 来流参数(a)来流速度、(b)来流压力、(c)来流温度和(d)攻角对光谱辐射强度的主Sobol数
Figure 6. Main Sobol index of free stream parameters to spectral intensity: (a) velocity, (b) pressure, (c) temperature, and (d) angle of attack
为分析变量
$ {X_i} $ 本身与其他变量的耦合作用对系统光谱辐射贡献的大小。计算了任意两个输入参数耦合作用下光谱辐射强度的Sobol指数$ {S_{i,j}} $ 的大小,其中i和j分别表示相互耦合作用的两个参数。图7给出了耦合作用较为明显的参数间Sobol指数分布,其他耦合作用Sobol指数极小未给出。从图7可知,来流速度和来流温度的耦合作用对光谱辐射的贡献最为明显,且在2.0~6.0 μm的大部分波段都有影响;来流速度和来流压强的耦合作用、来流温度与攻角的耦合作用只在某些极窄的谱段内有影响;其他参量间的耦合作用几乎可以忽略。 -
上文从辐射光谱角度分析了各参数和参数间耦合作用对辐射光谱的灵敏度,结果呈现出显著的波段选择特性。在工程应用,除了关注喷焰辐射光谱外,更为关心谱带内的辐射强度特性。因此,有必要对关注的波段内的积分强度进行灵敏度分析。图8(a)~(d)分别给出了2.3节定义的4个波段的主Sobol指数的百分占比。可以看出,来流速度对3~5 μm谱带强度的主Sobol指数最大,占到53.32%,来流压力和攻角的主Sobol指数占比接近。对2.5~3.2 μm波段和2.8~3.0 μm波段而言,来流速度的主Sobol指数占比均在80%左右;其次影响较大的参量为攻角,主Sobol指数占比约在15%左右;来流温度和来流压力的贡献微弱,可忽略不计。然而,来流压强对4.35~4.65 μm谱带积分强度的主Sobol指数最大,约占70%;来流速度的主Sobol指数占比仅占11.45%,攻角占比18.41%。
图 8 来流参数对谱段内光谱辐射强度的Sobol指数:(a) 3~5 μm波段,(b) 2.5~3.2 μm波段,(c) 2.8~3.0 μm波段和(d) 4.35~4.65 μm波段
Figure 8. Sobol index of free stream parameters to in-band spectral intensity: (a) 3-5 μm band, (b) 2.5-3.2 μm band, (c) 2.8-3.0 μm band, and (d) 4.35-4.65 μm band
综上可知,攻角对喷焰谱带内辐射强度的影响虽有一定谱带差异,但灵敏度及波动范围较小,主Sobol指数占比在14%~18%之间。来流速度在4.3 μm波段内的作用效果明显降低,导致对不包含该波段的谱带内辐射强度的贡献占主导,主Sobol指数占比接近80%。在4.3 μm波段附近,来流压力的影响则占主导地位,说明来流速度和来流压强对火箭发动机尾喷焰红外辐射有重要影响。另外,各参数的耦合作用对各谱带内辐射强度的总体影响占比不大,均在4%以下。以上结果表明,喷焰红外辐射信号高保真计算中需依据谱带范围重点关注对应弹道参量的置信水平。
Quantitative analysis of the uncertainty of infrared radiation signature of rocket exhaust plume caused by incoming flow
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摘要: 在火箭发动机喷焰红外辐射预测过程中,由于弹道参数的不确定性因素存在,会影响喷焰红外辐射信号计算结果的置信度。文中以Atlas-IIA低空喷焰为研究对象,以来流速度、来流温度、来流压力和飞行攻角为不确定性输入变量,采用拉丁超立方试验设计样本,开展喷焰反应流场与红外辐射特性计算,获得各样本点对应的喷焰红外辐射响应值,利用非嵌入混沌多项式(No-intrusive Polynomial Chaos, NIPC)方法构建代理模型,采用响应面法求解NIPC多项式系数,基于统计参量研究各参量对红外辐射信号的不确定度和敏感性。结果表明来流特性引起的辐射强度不确定度与波段的选取有较高的关联度。来流速度对光谱辐射强度的灵敏程度最高,来流压力和攻角次之,来流温度的影响可忽略不计;来流速度对2.5~3.2、2.8~3.0 μm谱带内的辐射强度敏感,主Sobol指数占比均在80%左右,来流压力对4.35~4.65 μm波段辐射强度的影响占比为70%;各参数间的耦合作用对喷焰红外辐射的影响不高于4%。该研究可为火箭发动机尾喷焰红外辐射准确预估和置信度评估提供理论支撑。Abstract:
Objective The rocket exhaust plume is one of the key objects of the space-based infrared system (SBIRS) due to its strong infrared radiation characteristics. The infrared radiation signature of the plume is not only related to the motor prototype parameters but also to the flight parameters of the vehicle. In practical applications, the variation characteristics of infrared radiation signature of the rocket exhaust plume can be predicted through numerical methods based on the known motor and flight parameters. However, the flight parameters of non-cooperative targets are often difficult to obtain accurately, which means that there must be some deviation in predicting the infrared radiation signatures of the rocket exhaust plume by numerical methods. Therefore, it is necessary to study the influence of flight parameter disturbance on the infrared radiation of rocket exhaust plumes. For this purpose, the non-intrusive polynomial chaos (NIPC) method is used for the uncertainty quantification and sensitivity analysis of free stream parameters on the infrared radiation signatures of the rocket exhaust plume. Methods The Latin hypercube sampling (LHS) method is used to design the samples of frees tream parameters. The infrared radiation characteristics of the rocket exhaust plume are calculated based on the infrared signature analysis tool (IRSAT), and the corresponding infrared response values of the plume at each sample point are obtained. The regression analysis method is used to solve the polynomial chaotic expansion coefficient and the statistical characteristics of the plume infrared signatures, including the mean value, standard deviation and uncertainty. Based on the Sobol index, the NIPC method can be utilized to quantify the uncertainty and sensitivity of the infrared radiation signature, and analyze the impact of a single variable and multiple variables on the infrared radiation characteristics of the rocket exhaust plume. Results and Discussions The free stream velocity has a high sensitivity to spectral radiation intensity in most spectral bands except for 4.3 μm, and the corresponding Sobol index is above 0.5. The maximum sensitivity of free stream pressure occurs in the 4.3 μm band, and the peak of the Sobol index is close to 1.0. The Sobol index of angle of attack to spectral radiation intensity is about 0.4, and it shows high sensitivity in multiple bands. The influence of the ambient temperature on the radiation spectrum is negligible. The Sobol index of the in-band radiance shows that the free stream velocity is the most sensitive to the radiation intensity in most bands. The free stream pressure and the angle of attack are the second, and the free stream temperature is the smallest. The coupling effect of free stream velocity and ambient temperature has the most obvious contribution to the in-band radiance. The coupling effect of free stream velocity and pressure, and the coupling effect of temperature and angle of attack only affect some extremely narrow spectral bands. The coupling effect among other parameters can be almost ignored. Conclusions The low altitude under-expanded Atlas-IIA plume is taken as the research object, and the uncertainty quantification and sensitivity analysis of infrared radiation signatures of free stream velocity, temperature, pressure and angle of attack are carried out using NIPC method. The uncertainty of radiation intensity caused by the incoming flow has a high correlation with the spectral band. The standard deviation of the in-band radiance is positively correlated with the mean value, and the uncertainty of the radiance is opposite to the mean value of the radiation intensity. In most wavebands, the free stream velocity is the most sensitive to the radiation intensity, followed by the free stream pressure and angle of attack, and the ambient temperature is the least. The coupling effect of velocity and temperature have the most obvious contribution to the radiance. The coupling effect of velocity and pressure, and temperature and angle of attack only have an effect in some very narrow spectral bands. The ratio of angle of attack to the main Sobol index of the radiation intensity is between 15% and 23%. The main Sobol index of the inflow velocity accounts for nearly 80% except for the 4.3 μm band. The impact of inflow pressure in the 4.3 μm band is dominant. The coupling effect of each incoming flow parameter has little influence on the radiation intensity in each spectral band, which is less than 4%. -
Key words:
- free stream parameter /
- rocket exhaust plume /
- NIPC /
- infrared radiation /
- uncertainty
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Parameter Free stream Nozzle U/m·s−1 545.5 2 960 P/Pa 12 112 68 850 T/K 216.7 2 230 H2 0.000 000 0.012 194 O2 0.233 154 0.000 028 H2O 0.000 000 0.272 437 H 0.000 000 0.000 127 OH 0.000 000 0.000 797 N2 0.766 388 0.000 000 CO 0.000 000 0.412 125 CO2 0.000 458 0.302 293 表 2 谱带内辐射强度的统计特性
Table 2. Statistical properties of in-band radiance intensity
Band Mean value/W·sr−1 Standard deviation/W·sr−1 Uncertainty 1 9.83×106 6.62×105 6.70% 2 1.05×107 6.77×105 6.50% 3 3.72×106 2.55×105 6.90% 4 3.85×106 3.50×105 9.10% -
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