Quasi-one-dimensional numerical study of shock tunnel flows based on cubic equations of state
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摘要: 采用基于立方型气体状态方程的准一维流动数值模拟方法研究了反射式高焓激波风洞的真实气体流动，重点关注了高压真实气体效应对风洞全场流动时空结构和驻室区气流参数的影响，并以理论分析揭示了高压真实气体效应对激波管内流动的作用机理。研究表明：对于以冷高压气体驱动的激波风洞，使用考虑分子体积和分子间作用力的真实气体状态方程能够更准确地描述气体的状态和风洞内的流动状况。高压真实气体效应主要在冷驱动气体中发生作用，其作用效果主要是使当地声速增大，从而使得入射稀疏波和反射稀疏波的传播速度加快；另一方面，高压气体效应在高温气体效应较显著的被驱动气体中作用微弱，且对激波管产生激波的强度和激波后的流动状态影响甚微。稀疏波的加快传播改变了激波管波系的相干时空关系。提前抵达的稀疏波可在一定情况下侵蚀激波风洞的有效试验时间。对于所测试的激波风洞构型，在150 MPa氢气驱动110 kPa氮气的工况下，高压效应导致的有效试验时间缩短约38%。适当加长驱动段长度和采用高温气体驱动均可有效减弱高压真实气体效应的影响。Abstract: The real gas flows of a reflected high-enthalpy shock tunnel are investigated by quasi-one-dimensional numerical simulation based on cubic equations of state, focusing on the influence of high-pressure real gas effect on the spatio-temporal structure of the full-field flow and the flow parameters in the stagnation zone right before the nozzle throat. In the numerical simulations, the thermochemical non-equilibrium due to high temperature caused by shock waves is considered. By embedding cubic equations of state into gas dynamics relations, theoretical analysis is also performed to find out the mechanism of high-pressure effect on the shock tube flow. It is shown that for shock tunnel flows driven by cold and high-pressure gas, the use of a real gas equation of state that takes into account molecular volume and intermolecular forces can more accurately describe the thermodynamic states of gas and the flow conditions in shock tunnels. The high-pressure real gas effect mainly occurs in the cold driving gas, which increases the local sound speed and hence increases the propagation speed of the incident and reflected rarefaction waves; on the other hand, the high-pressure real gas effect plays a weak role in the driven section where the high-temperature gas effect is significant, while it has little effect on the intensity of the shock wave generated by the shock tube as well as the flow state behind the shock. The increase of rarefaction wave speed changes the intersecting time and space of the wave system, which may alter the order of arrival of the reflected rarefaction wave and the contact surface in the stagnation zone. Under this circumstance, the early arrival of rarefaction wave shortens the effective test time of the shock tunnel. For the tested shock tunnel configuration, the high-pressure real gas effect reduces the effective test time by about 38% under the condition of 150 MPa hydrogen driving 110 kPa nitrogen. Both lengthening of the driver section and using high-temperature driver gas can effectively dissolve the influence of the aforementioned high-pressure effect.
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图 2 准一维数值模拟的激波沿程衰减情况与试验结果的对比
Figure 2. Comparison of shock attenuation between quasi-one-dimensional numerical simulations and tests
图 3 不同状态方程下激波风洞驻室压强数值模拟结果与试验结果的对比
Figure 3. Comparison of numerically-simulated pressures in the stagnation zone of a shock tunnel based on different equations of state with the experimental results
图 4 不同混合规则下300 K氢-氮混合气体可压缩性随压强的变化
Figure 4. Variations of compressibility factors with pressure for hydrogen-nitrogen mixture at 300 K under different mixing rules
图 5 不同状态方程300 K氢气可压缩性因子随压强的变化
Figure 5. Variation of compressibility factor with pressure for hydrogen at 300 K with different equations of state
图 6 不同状态方程300 K氢气声速随压强的变化
Figure 6. Variation of sound speed with pressure for hydrogen at 300 K with different equations of state
图 7 激波风洞全场压强x-t云图（理想气体状态方程）
Figure 7. x-t diagram of pressure contour in the shock tunnel with ideal gas state equation
图 8 激波风洞全场温度x-t云图（理想气体状态方程）
Figure 8. x-t diagram of temperature contour in the shock tunnel with ideal gas state equation
图 9 激波风洞全场压强x-t云图（PR状态方程）
Figure 9. x-t diagram of pressure contour in the shock tunnel with PR state equation
图 10 激波风洞全场温度x-t云图（PR状态方程）
Figure 10. x-t diagram of temperature contour in the shock tunnel with PR state equation
图 12 激波风洞驻室参数随时间变化情况
Figure 12. Parameters changing with time in the stagnation zone of the shock tunnel
图 13 分子体积和分子间作用力对氢气声速的影响因子随压强的变化
Figure 13. Influencing factors of molecular volume and intermolecular force on the sound speed of hydrogen changing with pressure
图 14 激波管流动波系和区域示意图
Figure 14. Schematic diagram of wave system and flow regions in the shock tube
图 15 不同状态方程下激波管区域②和区域③气流的p-u曲线
Figure 15. p-u diagrams of the flows in regions ② and ③ of the shock tube for different equations of state
图 16 不同压缩过程高压气体效应的影响程度
Figure 16. Influences of high pressure gas effect in different compression processes
图 17 理想和PR状态方程下不同初始压强配置所产生的入射激波马赫数
Figure 17. Mach numbers of incident shock waves produced under different initial pressure conditions with ideal and PR equations of state
图 18 不同状态方程下激波管区域② 和区域③气流的c-u曲线
Figure 18. c-u diagrams of the flows in regions ② and ③ of the shock tube for different equations of state
图 19 不同状态方程下各个波系到达时刻随总压的变化
Figure 19. Arrival times of waves changing with total pressure under different equations of state
图 20 不同总温、总压条件下高压气体效应对试验时间的影响
Figure 20. Influences of the high-pressure gas effects on test time under different total pressure and temperature conditions
表 1 激波风洞运行典型工况
Table 1. Typical operating conditions of the shock tunnel
管段 压强/Pa 温度/K 初始组分 长度/m 驱动段 5×107 300 氢气 9 被驱动段 1×105 空气 18 喷管 1 空气 7.5 
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表 2 激波风洞运行工况
Table 2. Operating conditions of shock tunnel
管段 压强/Pa 温度/K 初始组分 长度/m 驱动段 15×107 300 氢气 9.0 被驱动段 11×104 氮气 18.0 喷管 10 氮气 7.5
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表 3 各个波系到达驻室入口时刻
Table 3. Arrival time of each wave to stagnation zone
方程 ${t_1}$/ms ${t_2}$/ms ${t_3}$/ms ${t_{{\text{eff}}}}$/ms 理想气体状态方程 6.8 20.6 17.2 10.4 PR状态方程 7.1 13.5 25.5 6.4
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表 4 激波风洞试验气流名义总温对应的初始压比和驱动气体氮气摩尔分数
Table 4. Initial pressure ratios and molar fractions of nitrogen in driver gas corresponding to different nominal total temperatures of test flow in the shock tunnel
总温/K 初始压比 氮气摩尔分数/% 2000 176.5 15.0 3000 444.4 9.5 4000 800.0 5.0 5000 1272.7 1.8 6000 1600.0 0
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