﻿ 针栓式喷注单元雾化角模型分析
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1. 西安航天动力研究所 液体火箭发动机技术重点实验室, 西安 710100;
2. 中国船舶工业集团有限公司, 北京 100044

Analysis on spray angle model for pintle injector element
WANG Kai1, LEI Fanpei2, ZHANG Botao1, YANG Anlong1, ZHOU Lixin1
1. Key Laboratory for Liquid Rocket Engine Technology, Xi'an Aerospace Propulsion Institute, Xi'an 710100, China;
2. China State Shipbuilding Corporation Limited, Beijing 100044, China
Abstract: In order to accurately predict the spray angle of pintle injector of different radial orifice shapes, the theoretical models of the spray angles of one liquid sheet impinging on another liquid sheet and a liquid sheet impinging on a liquid jet are modified based on the momentum conservation equation. For the injection unit of one liquid sheet impacting another liquid sheet, two deformation factors are introduced in the model by theoretical derivation, which associates the geometric deformation effect of the impact with spray angle. For a liquid sheet impact a liquid jet, the effective impinging momentum ratio is defined by introducing the blocking rate. At the same time, the influence of the inlet hole shape of the liquid jet is implicitly considered in the deformation factors. Finally, according to the results of high-speed photography test and the numerical simulation results, the corresponding combination coefficients of deformation factors are obtained, which makes the new spray angle model more adaptable and more accurate than the original model. The results show that the predicted values of the theoretical model with the introduction of deformation factors and blocking rate are in good agreement with the experimental and numerical simulation results. For one liquid sheet impinging on another liquid sheet, the deformation factors are basically maintained at 0.9~1.1, and the recommended values of the deformation factors are C1=0.99 and C2=1.06, according to the experimental results and simulation results. For a liquid sheet impinging on a liquid jet, the recommended values of the deformation factor are C1=0.75 and C2=1.25. This model calculates the spray angle according to the actual axial momentum and the actual synthetic total momentum at the exit, which implicitly considers the influence of the impact effect. Compared with the common model that calculates the spray angle based on the axial momentum and the synthetic total momentum at the entrance before the impact, the accuracy of the new model is significantly improved, which provides an important reference for the theoretical research and engineering design of pintle injectors.
Keywords: injector    spray angle model    deformation factor    blocking rate    effective impinging momentum ratio

 图 1 针栓式喷注器原理图 Fig. 1 Schematic diagram of pintle injector

1 雾化角理论模型分析 1.1 液膜/液膜撞击雾化角模型 1.1.1 基本定义

 ${\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta = \frac{1}{{{C_1} + {C_2} \cdot {C_{{\rm{TMR}}}}}}$ （8）

1.2 液膜/液束撞击雾化角模型

 $\begin{array}{l} {C_{{\rm{MReff }}}} = \frac{{{{\bar \rho }_2}u_{{\rm{2 in }}}^2{A_{{\rm{2 in }}}}}}{{{{\bar \rho }_1}u_{{\rm{1in }}}^2{A_{{\rm{1in }}}}}} = \frac{{{{\bar \rho }_2}u_{{\rm{2in }}}^2{A_{{\rm{2in }}}}}}{{{{\bar \rho }_1}u_{{\rm{1in }}}^2{A_{{\rm{1in}}{\kern 1pt} {\kern 1pt} {\rm{total }}}} \cdot \frac{w}{L}}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{C_{{\rm{TMR}}}}}}{{{C_{{\rm{BF}}}}}} \end{array}$ （9）

 $\begin{array}{*{20}{c}} {{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta = \frac{1}{{{C_1} + {C_2}{C_{{\rm{ MReff }}}}}} = }\\ {\frac{1}{{{C_1} + {C_2}\frac{{{C_{{\rm{ TMR }}}}}}{{{C_{{\rm{ BF }}}}}}}}} \end{array}$ （10）
 图 4 液膜/液束撞击单元 Fig. 4 Injection element of liquid sheet impacting liquid jet

1.3 模型分析讨论

1) 由于推导过程中以守恒量ρV2整体进行积分变换，后面又引入了平均密度，因此式(8)既适用于不可压的液体膜及气体膜撞击，也适用于可压缩的气体膜撞击。

2) 当认为两路液体撞击后出口截面面积与各自入口截面面积相等时，式(8)中2个变形因子均等于1，因此雾化角公式简化为

 ${\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta = \frac{1}{{1 + {C_{{\rm{TMR}}}}}}$ （11）

3) 从式(5)或式(6)不难看出，该模型的雾化角余弦值等于轴向入口动量(即控制体1的入口截面动量)与撞击后出口总动量(即控制体1和2的出口截面的动量和)之比，是根据实际的出口轴向动量(等于入口轴向动量)和出口合成总动量来计算雾化角，隐含考虑了撞击造成的影响，也适合引入撞击液膜与液束的几何形状因素。该模型仅认为轴向动量守恒，径向动量不守恒。而相比于另一种常用的雾化角模型：tan θ=CTMR，即

 $\begin{array}{*{20}{l}} {{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta = \frac{1}{{\sqrt {1 + C_{{\rm{TMR}}}^2} }} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{{\bar \rho }_1}u_{{\rm{1in}}}^2{A_{{\rm{1in}}}}}}{{\sqrt {{{({{\bar \rho }_1}u_{{\rm{1in}}}^2{A_{{\rm{1in}}}})}^2} + {{({{\bar \rho }_2}u_{{\rm{2in}}}^2{A_{{\rm{2in}}}})}^2}} }}} \end{array}$ （12）

2 数值物理模型 2.1 数值方法 2.1.1 控制方程

 $\left\{ \begin{array}{l} \rho \left( {\frac{{\partial \mathit{\boldsymbol{V}}}}{{\partial t}} + \mathit{\boldsymbol{V}} \cdot \mathit{\boldsymbol{\nabla}} \mathit{\boldsymbol{V}}} \right) = - \mathit{\boldsymbol{\nabla}} p + \mathit{\boldsymbol{\nabla}} \cdot (2\mu \mathit{\boldsymbol{D}}) + \sigma \kappa {\delta _{\rm{s}}}\mathit{\boldsymbol{n}}\\ \begin{array}{*{20}{l}} {\frac{{\partial \rho }}{{\partial t}} + \mathit{\boldsymbol{\nabla}} \cdot (\rho \mathit{\boldsymbol{V}}) = 0}\\ {\mathit{\boldsymbol{\nabla}} \cdot \mathit{\boldsymbol{V}} = 0} \end{array} \end{array} \right.$ （13）

 $\left\{ {\begin{array}{*{20}{l}} {\rho ({c_1},{c_2}) = {c_1}{\rho _{{\rm{l1}}}} + {c_2}{\rho _{{\rm{l2}}}} + (1 - {c_1} - {c_2}){\rho _{\rm{g}}}}\\ {\mu ({c_1},{c_2}) = {c_1}{\mu _{{\rm{l1}}}} + {c_2}{\mu _{{\rm{l2}}}} + (1 - {c_1} - {c_2}){\mu _{\rm{g}}}} \end{array}} \right.$ （14）

 图 5 基于分段线性VOF几何重构方法的自适应网格加密 Fig. 5 Adaptive mesh refinement based on a piecewise-linear geometrical VOF
2.1.2 气液相界面捕捉

VOF方法通过定义体积分数函数c来描述界面(含有运动界面的网格满足0 < c < 1，充满目标流体的网格c=1，不含目标流体的网格c=0)，通过求解函数c的输运方程跟踪界面运动[40]。优点是可以方便地计算复杂的相界面变化过程，捕捉的相界面锐利程度高，对计算内存的要求较低，具有守恒的特性。

 图 6 分段线性的VOF界面重构方法 Fig. 6 Piecewise-linear VOF interface reconstruction method
2.2 计算模型

 图 7 计算域示意图 Fig. 7 Sketch map of computation zone

 h1/mm h2 /mm 0.25 0.25 0.4 0.25 0.65 0.25, 0.4

 Axial velocity/(m·s-1) Radial velocity/(m·s-1) 10 5, 7.5, 10, 15, 20, 25 15 20, 25, 30, 35 20 25, 35, 45

 h1/mm d/mm 0.25 0.5，0.8, 1.0 0.45 0.8 0.65 0.8

 Axial velocity/(m·s-1) Radial velocity/(m·s-1) 10 5, 7.5, 10, 12.5, 15, 17.5, 20 20 12.5, 17.5, 22.5, 27.5 15 12.5 25 12.5
 图 8 数值仿真获得的雾化角 Fig. 8 Spray angle of numerical simulation
3 喷注单元试验装置及试验测量系统 3.1 喷注单元试验装置

 图 9 液液平面针栓式喷注单元试验件结构 Fig. 9 Structure of liquid-liquid plane pintle injector element

 h1/mm d/mm 0.25 0.5，0.8，1.0 0.45 0.5，0.8，1.0，1.2 0.65 1.0 0.85 1.0
3.2 试验测量系统

 图 10 喷嘴雾化试验系统 Fig. 10 Spray experiment system for injectors
3.3 试验图像处理方法

 图 11 试验结果图像处理过程 Fig. 11 Image processing of experimental results
 图 12 试验测得的雾化角 Fig. 12 Spray angle of experimental result
4 结果讨论与分析 4.1 液膜/液膜撞击雾化角结果分析

 图 13 液膜撞击液膜流场结构[33] Fig. 13 Flow field structure of liquid sheet impinging on liquid sheet[33]
 图 14 面积比的计算结果 Fig. 14 Calculation results of area ratio
 图 15 雾化角拟合获得的面积比 Fig. 15 Fitting curve of area ratio according to spray angle

 图 16 不同动量比下的雾化角结果对比(液膜撞击液膜) Fig. 16 Comparison of spray angles at various momentum ratios (liquid sheet impinging on liquid sheet)
4.2 液膜/液束撞击雾化角结果分析

 图 17 不同动量比下的雾化角结果对比(液膜撞击液束) Fig. 17 Comparison of spray angles at various momentum ratios (liquid sheet impinging on liquid jet)

 图 18 液膜绕液束的流动与变形过程 Fig. 18 Flow and deformation process of liquid sheet around liquid jet
 图 19 液膜与液束撞击形成的“Ω”形雾扇 Fig. 19 "Ω" shape spray fan formed by liquid sheet impinging on liquid jet

5 结论

1) 基于轴向动量守恒和径向动量不守恒的假设，通过理论推导引入了2个变形因子，将撞击的几何变形效应与雾化角关联，推导建立了液膜撞击液膜的雾化角新模型。引入变形因子的理论模型预测值与试验及数值仿真结果吻合很好；变形因子基本维持在0.9~1.1，根据试验结果及仿真结果的推荐值为C1=0.99和C2=1.06。

2) 基于液膜撞击液膜的雾化角分析模型，通过引入阻塞率定义有效撞击动量比，同时将液膜液束变形的影响隐含考虑在变形因子中，推导建立了液膜撞击液束的雾化角新模型。引入变形因子和阻塞率的理论模型预测值与试验及数值仿真结果吻合很好，理论模型和数值仿真模型均得到了有效验证；根据试验结果及仿真结果获得的变形因子推荐值为C1=0.75和C2=1.25。

3) 建立的雾化角新模型根据实际出口的轴向动量(等于入口轴向动量)和合成总动量计算雾化角，隐含考虑了撞击变形作用的影响，较根据撞击前入口的轴向动量和合成总动量计算雾化角的常用tan模型预测值准确度显著提高，这是2种雾化角模型之间的本质区别，为针栓式喷注器的理论研究和工程设计提供了重要参考。

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http://dx.doi.org/10.7527/S1000-6893.2019.23622

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#### 文章信息

WANG Kai, LEI Fanpei, ZHANG Botao, YANG Anlong, ZHOU Lixin

Analysis on spray angle model for pintle injector element

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 123622.
http://dx.doi.org/10.7527/S1000-6893.2019.23622