Reliability of the eN Method Applied to Hypersonic Blunt Cone Boundary Layers for Transition Prediction
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摘要: eN方法基于扰动在边界层中线性演化过程中的幅值增长程度来预测转捩。以来流Mach数为6、不同壁面温度条件下不同钝度圆锥为研究对象,结合直接数值模拟和抛物化稳定性方程,从eN方法是否能够准确描述扰动在上述边界层中线性增长的角度,分析了该方法预测转捩的可靠性。研究结果表明,在小钝度或高壁面温度情况下,扰动在向下游的演化过程中从第1模态转变为第2模态,基于线性稳定性理论的eN方法变得不再可靠。壁面温度相同,头部钝度越大,eN方法越可靠;同等钝度下,壁面温度越低,eN方法越可靠。由于存在模态转换时,线性稳定性理论总是低估扰动的增长,因而对于给定的转捩判据NT(可由某一工况实验标定给出),若钝度减小或壁面温度增加到一定程度,eN方法给出的转捩位置比实际情况更靠后。重新标定转捩判据时,钝度越小,壁面温度越高,NT的修正程度就越大。Abstract: The eN method predicts transition based on the level of the linear amplitude amplification of the disturbances in the boundary layer. Boundary layers over cones at Mach 6 were investigated with different nose bluntness and under different wall temperature conditions. Combined with direct numerical simulation (DNS) and parabolized stability equations (PSE), from the viewpoint whether the eN method is able to accurately describe the amplification of the disturbances in the above boundary layers, its reliability for transition prediction was interrogated. Results show that in the cases of small bluntness or high wall temperature, the disturbances undergo the intermodal exchange from the first mode to the second mode when travelling downstream in the boundary layer, so that the eN method based on linear stability theory becomes less reliable. Under the same wall temperature condition, as the nose bluntness increases, the eN method is more reliable. For the same nose bluntness, as the wall temperature decreases, the eN method is more reliable. Since linear stability theory always underestimates the amplification of the disturbances when there is an intermodal exchange, for the given transition criterion NT, which could be calibrated by a certain case, as the bluntness decreases or the wall temperature increases to some extent, the eN method tends to produce a further downstream transition location than reality. To recalibrate the transition criterion, the smaller the nose bluntness or the higher the wall temperature, the larger the modification of NT factor.
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Key words:
- hypersonic /
- boundary layer /
- linear stability theory /
- transition prediction /
- eN method
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表 1 PSE和eN方法给出的转捩位置的比较
Table 1. Comparison of predicted transition locations obtained by PSE and eN method
Tw/K Rn/mm NT=4.5 NT=10 PSE/m eN/m ΔξT/m PSE/m eN/m ΔξT/m adiabatic 0.025 4 0.09 0.16 0.07 0.28 0.51 0.23 adiabatic 0.79 0.23 0.24 0.01 0.49 0.60 0.11 adiabatic 2.54 0.60 0.60 0.00 0.95 0.95 0.00 300 0.025 4 0.08 0.11 0.03 0.26 0.37 0.11 300 0.79 0.19 0.19 0.00 0.39 0.40 0.01 300 2.54 0.51 0.51 0.00 0.75 0.75 0.00 表 2 绝热壁下eN方法用于不同钝度工况所需的N值修正
Table 2. Modification of N factor in eN method for adiabatic cones with different nose bluntness
Rn/mm NT=4.5 NT=10 PSE: ξT/m eN: ΔN PSE: ξT/m eN: ΔN 0.025 4 0.09 2.25 0.28 3.95 0.79 0.23 0.35 0.49 1.25 2.54 0.60 0.00 0.95 0.00 表 3 不同壁面温度下eN方法所需的N值修正(Rn=0.79 mm)
Table 3. Modification of N factor in eN method for cases with different wall temperatures (Rn=0.79 mm)
Tw/K NT=4.5 NT=10 PSE: ξT/m eN: ΔN PSE: ξT/m eN: ΔN 300 0.19 0.00 0.39 0.25 400 0.22 0.10 0.47 0.80 adiabatic 0.23 0.35 0.49 1.25 500 0.25 0.39 0.50 1.85 -
[1] Bertin J J, Cummings R M. Fifty years of hypersonics: where we've been, where we're going[J]. Progress in Aerospace Sciences, 2003, 39(6/7): 511-536. [2] 叶友达, 张涵信, 蒋勤学, 等. 近空间高超声速飞行器气动特性研究的若干关键问题[J]. 力学学报, 2018, 50(6): 1292-1310. https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201806004.htmYe Y D, Zhang H X, Jiang Q X, et al. Some key problems in the study of aerodynamic characteristics of near-space hypersonic vehicles[J]. Chinese Journal of Theoretical and Applied Mechanic, 2018, 50(6): 1292-1310 (in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201806004.htm [3] Reshotko E. Is Reθ/Me a meaningful transition criterion?[J]. AIAA Journal, 2007, 45(7): 1441-1443. doi: 10.2514/1.29952 [4] Langtry R B, Menter F R. Transition modeling for general CFD applications in aeronautics[R]. AIAA 2005-522, 2005. [5] Menter F R, Langtry R B, Likki S R, et al. A correlation-based transition model using local variables-Part Ⅰ: model formulation[J]. Journal of Turbomachinery, 2006, 128(3): 413-422. doi: 10.1115/1.2184352 [6] Langtry R B, Menter F R, Likki S R, et al. A correlation-based transition model using local variables-Part Ⅱ: test cases and industrial applications[J]. Journal of Turbomachinery, 2006, 128(3): 423-434. doi: 10.1115/1.2184353 [7] Smith A M, Gamberoni N. Transition, pressure gradient and stability theory[R]. Douglas Aircraft Report ES-26388, 1956. [8] van Ingen J L. A suggested semi-empirical method for the calculation of the boundary layer transition region[R]. Technical Report VTH-74, 1956. [9] Fischer J S, Soemarwoto B I, van der Weide, et al. Automatic transition prediction in a Navier-Stokes solver using linear stability theory[J]. AIAA Journal, 2021, 59(7): 2409-2426. doi: 10.2514/1.J059910 [10] 罗纪生. 高超声速边界层的转捩及预测[J]. 航空学报, 2015, 36(1): 357-372. https://www.cnki.com.cn/Article/CJFDTOTAL-HKXB201501027.htmLuo J S. Transition and prediction for hypersonic boundary layers[J]. Acta Aeronautica et Astronautica Sinica, 2015, 36(1): 357-372 (in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-HKXB201501027.htm [11] Stock H W, Haase W. Navier-Stokes airfoil computations with eN transition prediction including transitional flow regions[J]. AIAA Journal, 2000, 38(11): 2059-2066. doi: 10.2514/2.893 [12] 朱震, 宋文萍, 韩忠华. 基于双eN方法的翼身组合体流动转捩自动判断[J]. 航空学报, 2018, 39(2): 123-134. https://www.cnki.com.cn/Article/CJFDTOTAL-HKXB201802012.htmZhu Z, Song W P, Han Z H. Automatic transition prediction for wing-body configurations using dual eN method[J]. Acta Aeronautica et Astronautica Sinica, 2018, 39(2): 123-134 (in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-HKXB201802012.htm [13] Crouch J D, Crouch I W, Ng L L. Transition prediction for three-dimensional boundary layers in computational fluid dynamics applications[J]. AIAA Journal, 2002, 40(8): 1536-1541. doi: 10.2514/2.1850 [14] 安复兴, 李磊, 苏伟, 等. 高超声速飞行器气动设计中的若干关键问题[J]. 中国科学: 物理学、力学、天文学, 2021, 51(10): 104702. https://www.cnki.com.cn/Article/CJFDTOTAL-JGXK202110003.htmAn F X, Li L, Su W, et al. Key issues in hypersonic vehicle aerodynamic design[J]. Scientia Sinica Physica, Mechanica & Astronomica, 2021, 51(10): 104702 (in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-JGXK202110003.htm [15] Jaffe N A, Okamura T T, Smith A M. Determination of spatial amplification factors and their application to predicting transition[J]. AIAA Journal, 1970, 8(2): 301-308. doi: 10.2514/3.5660 [16] Malik M. Prediction and control of transition in hypersonic boundary layers[C]. 19th AIAA, Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Honolulu: AIAA, 1987. [17] Chen F J, Malik M R, Beckwith I E. Boundary-layer transition on a cone and flat plate at Mach 3.5[J]. AIAA Journal, 1989, 27(6): 687-693. doi: 10.2514/3.10166 [18] Horvath T J, Berry S A, Hollis B R, et al. Boundary layer transition on slender cones in conventional and low disturbance Mach 6 wind tunnels[R]. AIAA 2002-2743, 2002. [19] Marineau E C, Moraru C G, Lewis D R, et al. Mach 10 boundary layer transition experiments on sharp and blunted cones[R]. AIAA 2014-3108, 2014. [20] Goparaju H, Unnikrishnan S, Gaitonde D V. Effects of nose bluntness on hypersonic boundary-layer receptivity and stability[J]. Journal of Spacecraft and Rockets, 2021, 58(3): 668-684. doi: 10.2514/1.A34829 [21] Liang X, Li X L, Fu D X, et al. Effects of wall temperature on boundary layer stability over a blunt cone at Mach 7.99[J]. Computers & Fluids, 2010, 39(2): 359-371. [22] Balakumar P. Receptivity of hypersonic boundary layers to acoustic and vortical disturbances (invited)[R]. AIAA 2015-2473, 2015. [23] Wan B B, Luo J S, Su C H. Response of a hypersonic blunt cone boundary layer to slow acoustic waves with assessment of various routes of receptivity[J]. Applied Mathematics and Mechanics, 2018, 39(11): 1643-1660. doi: 10.1007/s10483-018-2391-6 [24] Zhang Y M, Zhou H. Verification of parabolized stability equations for its application to compressible boundary layers[J]. Applied Mathematics and Mechanics, 2007, 28(8): 987-998. doi: 10.1007/s10483-007-0801-3 [25] Zhang Y M, Zhou H. PSE as applied to problems of transition in compressible boundary layers[J]. Applied Mathematics and Mechanics, 2008, 29(7): 833-840. doi: 10.1007/s10483-008-0701-8 [26] Kara K, Balakumar P, Kandil O A. Effects of nose bluntness on hypersonic boundary-layer receptivity and stability over cones[J]. AIAA Journal, 2011, 49(12): 2593-2606. doi: 10.2514/1.J050032 [27] Klaimon J H. Bow shock correlation for slightly blunted cones[J]. AIAA Journal, 1963, 1(2): 490-491. doi: 10.2514/3.1581 [28] 万兵兵, 罗纪生. 超声速绕钝板熵层不稳定性的研究[J]. 空气动力学学报, 2018, 36(2): 247-253. https://www.cnki.com.cn/Article/CJFDTOTAL-KQDX201802009.htmWan B B, Luo J S. Entropy-layer instability over a blunt plate in supersonic flow[J]. Acta Aerodynamica Sinica, 2018, 36(2): 247-253 (in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-KQDX201802009.htm [29] 苏彩虹. 高超声速边界层转捩预测中的关键科学问题——感受性、扰动演化及转捩判据研究进展[J]. 空气动力学学报, 2020, 38(2): 355-367. https://www.cnki.com.cn/Article/CJFDTOTAL-KQDX202002019.htmSu C H. Progress in key scientific problems of hypersonic boundary-layer transition prediction: receptivity, evolution of disturbances and transition criterion[J]. Acta Aerodynamica Sinica, 2020, 38(2): 355-367 (in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-KQDX202002019.htm [30] Su C H, Zhou H. Stability analysis and transition prediction of hypersonic boundary layer over a blunt cone with small nose bluntness at zero angle of attack[J]. Applied Mathematics and Mechanics, 2007, 28(5): 563-572. doi: 10.1007/s10483-007-0501-1