WANG Zhipeng, HAN Zhijun, WANG Longfei. Dynamic Instability of Composite Plate under Stress Wave Based on Galerkin Method[J]. Chinese Journal of High Pressure Physics, 2021, 35(5): 054204. doi: 10.11858/gywlxb.20210705
Citation: HE Xin, TIAN Hui, WANG Jian, CHEN Wanlei, WEI Zhaoxuan, LIU Jincheng, QI Dongli, SHEN Longhai. Density Generalized Function Theory Study on New MAX Phase M2SeC (M=Zr, Hf) under High Pressure[J]. Chinese Journal of High Pressure Physics, 2023, 37(4): 041102. doi: 10.11858/gywlxb.20230644

Density Generalized Function Theory Study on New MAX Phase M2SeC (M=Zr, Hf) under High Pressure

doi: 10.11858/gywlxb.20230644
  • Received Date: 20 Apr 2023
  • Rev Recd Date: 11 May 2023
  • Accepted Date: 07 Jun 2023
  • Available Online: 13 Sep 2023
  • Issue Publish Date: 01 Sep 2023
  • The effects of pressure on the crystal structure, elasticity, electronic and thermodynamic properties of the new MAX phases Zr2SeC and Hf2SeC were investigated by employing the first principle of density generalized function theory. Elastic constants and phonon calculations show that both compounds have stable structure in the pressure range of 0–40 GPa. Unlike most MAX phases, Zr2SeC and Hf2SeC are more easily compressed along the a-axis than along the c-axis, and the effect of external pressure on the crystal structure of Zr2SeC is more significant than Hf2SeC. Electronic structure calculations show that Zr2SeC and Hf2SeC have metallic properties, and the electronic density of states at the Fermi energy level decrease gradually with increasing pressure, thus improving the stability of Zr2SeC and Hf2SeC. In addition, the elastic modulus, the Poisson’s ratio and the anisotropy index show an enhancement with increasing pressure. In the pressure range of 0–40 GPa, the elastic modulus of Hf2SeC is greater than that of Zr2SeC at the same pressure, indicating that Hf2SeC has stronger resistance to fracture and deformation than Zr2SeC at high pressure. Thermodynamic property calculations show that Zr2SeC and Hf2SeC have higher melting temperatures in the pressure range of 0–40 GPa.

     

  • 激波与两种不同密度流体界面的相互作用包括激波的反射、透射和绕射等复杂的激波动力学过程,同时由于密度梯度方向与压力梯度方向不一致导致界面上沉积斜压涡量,引起Richtmyer-Meshkov(RM)不稳定性[1-2],使界面上的微小扰动被放大,甚至出现湍流混合。激波与气柱的相互作用是研究此类问题最为简单的构型,对于深入认识惯性约束聚变、高超声速燃烧、爆燃转爆轰[3]等复杂实际应用中RM不稳定性的产生、发展及作用具有重要的工程价值。

    良好的界面形成方法和精密的诊断技术对实验研究RM不稳定性至关重要。Haas等人[4]通过硝化纤维薄膜形成气柱界面,采用纹影技术观察到界面在激波冲击后演化成涡对结构。Jacobs[5]利用层流射流方法生成无膜气柱,采用可以精确捕捉界面的平面激光诱导荧光(Planar Laser-Induced Fluorescence,PLIF)技术,发现涡对结构主导了气柱界面的非线性演化阶段,演化后期出现诸如Kelvin-Helmholtz等二次不稳定性。Jacobs提出的无膜气柱方法克服了有膜气柱方法中支架和破碎薄膜干扰流场的弊端,随后被广泛采用。近年来,激波冲击气柱的RM不稳定性得到了广泛而深入的研究,从混合区宽度等大尺度结构的描述[6]到对小尺度结构的关注[7],从演化早期[8]延伸到后期混合[9],从单圆气柱拓扑到多气柱[10]、椭圆气柱[11],从平面激波加载到汇聚激波加载[12],诊断技术也从传统的纹影技术发展到能够精细定量描述流场信息的粒子图像测速(Particle Image Velocimetry, PIV)和PLIF[13]等片光技术。最近,陈模军等人[14]提出采用环约束方法形成气柱,不但流场干扰小,而且可以减小界面处的气体扩散。

    在超新星动力学[15]、惯性约束聚变[16]等涉及RM不稳定性的实际应用中,界面往往经历激波的二次甚至多次冲击。激波的二次冲击为演化中的界面提供了新的能量,引入了二次压力扰动,并再次产生斜压涡量。王显圣等[17]、Zhai等[18]和何惠琴等[19]对激波二次冲击处于演化早中期的气柱界面进行了数值模拟和实验研究,通过高速摄影结合激光片光技术探讨了界面变形规律,发现激波二次冲击后界面形成二次涡对。张赋等人[20]在此基础上利用PIV算法处理高速摄影图像,获得了界面演化的速度场信息,并通过测量涡对速度估算了环量。本研究基于此前对重气体气柱RM不稳定性的高速摄影研究[21],采用双曝光PIV技术,对激波在不同时刻二次冲击处于演化中后期的气柱界面进行拓展研究,定量表征流场的速度场和涡量场,并由速度场直接计算环量,以期为理论模型和数值程序的校验提供有益的参考。

    激波两次冲击重气体SF6(密度约为空气的5倍)气柱实验在横截面为100 mm×100 mm的水平激波管中进行,侧视图如图 1所示。初始时驱动段中的高压氮气与被驱动段中的常压空气由塑料薄膜隔开,实验时通过瞬时释放的大电流加热电阻丝熔化薄膜,产生从左向右传播的平面入射激波。这种破膜方式具有膜片安装简便、激波平面度高、实验可重复性好等优点。入射激波第一次冲击气柱后将发生反射、透射和绕射,一段时间后将在试验段尾端的固壁处反射,形成从右向左传播的反射激波,并二次冲击演化中的气柱界面。将试验段尾端的固壁设计成可移动反射端壁,通过调节反射端壁与气柱初始位置的距离实现反射激波在不同时刻二次冲击气柱界面。定义反射距离(L0)为初始气柱中心至反射端壁的距离,如图 1所示。

    图  1  实验装置示意图
    Figure  1.  Schematic illustration of experimental devices

    圆形重气柱的生成采用层流射流方法[11, 22]。Rightley等[23]和Prestridge等[24]的研究表明,乙二醇烟雾具有很好的流场跟随性,是比较理想的示踪粒子,为此选择乙二醇作为示踪粒子。把气箱置于试验段上方,首先充入SF6气体,利用烟雾发生器把液体乙二醇汽化为白色乙二醇烟雾(粒子的平均直径约为0.5 μm),经气箱底部充入并与SF6充分混合。在重力的作用下,混合气体自上而下流动,经过安装在试验段上方的圆形喷嘴(直径D0=5 mm)后在试验段中形成垂直方向的圆形气柱。通过调节气箱阀门调整混合气体的流速,从而控制气柱的稳定性。由于SF6气柱的流速(约0.1 m/s)远小于激波速度(约420 m/s)和波后气流速度(约110 m/s),气柱界面在激波作用下的演化可认为是准二维的,因此可以采用激光片光技术对流场进行观测。

    利用双曝光PIV技术诊断气柱界面演化过程中的速度信息。PIV技术的操作流程为:先在流场中加入示踪粒子,然后以很短的时间间隔至少两次照亮流场,记录流场中示踪粒子的散射光,最后进行PIV图像的后处理。本实验采用两台独立的Nd:YAG脉冲激光器作为光源分别出光照射流场,激光器发出的激光脉冲波长均为532 nm,最大能量均为200 mJ,历时8 ns,因此两次照射流场的时间间隔在理论上可以为零,非常适合高速流场的诊断。两台激光器发出的激光经过特定光路调节为共轴,通过片光透镜后形成厚度约1 mm的片光,透过反射端壁上的K9玻璃后照射流场。片光平面距圆形喷嘴出口约20 mm。将HiSense 4M高速相机(2 048像素×2 048像素,最小跨帧为200 ns)安装在试验段正上方,相机沿垂直于片光平面方向记录两次激光出光时示踪粒子的散射光,分别存储于不同帧。PIV相机聚焦的流场区域大小为51 mm×51 mm。

    后处理采用标准的双帧互相关算法[25],由PIV图像计算流场的速度场。查询窗口大小设为16像素×16像素,要求每个查询窗口至少包含10个示踪粒子。由互相关算法获得原始的速度场后,再利用PIV后处理中相关的通用验证算法和替换算法[25]剔除无效矢量和伪矢量,并根据周围的矢量进行替换。

    本实验考察了3种反射距离(L0),分别为100、150和210 mm。对于每种反射距离,利用PIV技术分别测量气柱在4个不同演化时刻的速度场,包括激波二次冲击前1个时刻和激波二次冲击后3个时刻,如表 1所示,其中反射激波二次冲击气柱时间由廖深飞等人[21]的高速摄影结果估算而得。入射激波马赫数Ma=1.22±0.01。为方便描述,界面中靠近高压段部分称为左界面,靠近反射端壁部分称为右界面。

    表  1  3种反射距离下PIV的测量时刻
    Table  1.  Times of PIV measurements for 3 endwall distances
    L0/(mm)Reshock time/(μs)Times of PIV measurements/(μs)
    100450420, 650, 750, 1 050
    150670640, 870, 970, 1 270
    210900870, 1 100, 1 200, 1 500
    下载: 导出CSV 
    | 显示表格

    双曝光PIV技术中两次出光的时间间隔可以很小,非常适合诊断高速非定常流场,但是本实验所采用的高能量激光器的出光频率偏低,每次实验只能诊断一个或几个时刻的流场信息,因此流场信息的演化过程只能依靠多次实验,对实验的可重复性提出了更高的要求。为此,本研究针对每个演化时刻均进行多次实验,确保实验的可重复性。图 2给出了3种反射距离条件下采用PIV技术拍摄的图像与高速摄影图像的比较。可以看出:两者对于气柱演化过程中大尺度结构的诊断较为吻合;激波首次冲击气柱后,气柱界面的演化由一对涡对结构主导;反射激波二次冲击气柱后,当反射距离较小(L0=100 mm)时,气柱的右界面衍生出一对二次涡对结构,当反射距离较大(L0=210 mm)时,则没有出现明显的二次涡对结构。因此,可以采用本实验的PIV技术定量表征气柱在激波二次冲击下的速度、涡量等流场信息的发展。

    图  2  3种反射距离下气柱界面形态的演化
    Figure  2.  Morphological evolution of the gas cylinder for 3 endwall distances

    3种反射距离下气柱瞬时速度场和涡量场的演化如图 3所示,其中速度场已减去气柱的流向平均速度,相当于在随气柱平动的坐标系中观察气柱的运动,涡量的正方向定义为垂直纸面向外。从图 3可以看到,气柱速度场中某些区域(如图 3(a)中650 μs时刻的速度场)的速度矢量形成了封闭的圆或椭圆,同时涡量也聚集在这些区域,标志着涡结构的形成。根据Jeong等人[26]提出的用于辨识涡结构的λ2准则(λ2为应变率张量与涡张量组合张量的特征值),即负的λ2的局部最小值标志着涡核,由速度场所得λ2等值线图,进一步证实了涡结构的存在,如图 4(a)所示。

    图  3  3种反射距离下气柱速度场和涡量场的演化
    Figure  3.  Evolutions of the velocity and vorticity fields of the gas cylinder for 3 endwall distances
    图  4  速度场和λ2的等值线图(图 3中实线方框标示区域)
    Figure  4.  Velocity field and contour of λ2 (Area marked by the solid line box in Fig. 3)

    速度场和涡量场清晰地揭示了在激波二次冲击前气柱界面的演化由初始涡对(Primary Vortex Pair,PVP)主导,两个涡的旋转方向相反,强度相当。廖深飞等人[21]采用高速摄影技术发现激波二次冲击前气柱的左界面以恒速运动,根据PIV技术获得的速度场可统计出3种反射距离下气柱左界面的平均速度为(90.3±4.4)m/s,略高于一维气体动力学理论预测结果76.4 m/s,与Rudinger等人[27]预测的涡对速度(99.5 m/s)接近。PIV实验结果略高于一维气体动力学理论预测结果的原因可能是实验中无膜气柱与环境气体之间存在扩散,界面并不是理想的密度间断面,而是具有有限厚度的密度过渡层,导致界面处的密度梯度小于理想情况。

    激波二次冲击后,当反射距离较小(L0=100 mm)时:气柱的右界面衍生出二次涡对(Secondary Vortex Pair,SVP),且二次涡对的旋转方向与初始涡对相反,正如Zhai等人[18]所指出的,二次涡对的出现是由斜压机制引起;二次涡对涡核处涡量的绝对值明显小于初始涡对涡核处涡量的绝对值,从涡量分布上看二次涡对的强度也小于初始涡对,原因可能是气体的扩散和混合导致激波二次冲击气柱时界面两侧的密度梯度小于激波首次冲击气柱时界面两侧的密度梯度。当反射距离较大(L0=210 mm)时,激波二次冲击后气柱没有衍生出二次涡对,说明激波与气柱的相互作用与气柱的界面形状密切相关。当L0=150 mm时,激波二次冲击后,气柱右界面也衍生出二次涡对,如图 4(b)所示,该现象在图 2所示的高速摄影图像中很难分辨,说明PIV技术具有比传统高速摄影技术更好的空间分辨能力。

    二次涡对的出现对流场的演化具有重要的影响。二次涡对的旋转方向与初始涡对的旋转方向相反,因此两者诱导的流向速度方向相反(如图 3(a)中650 μs时刻图像),促进了混合区在流向上的快速增长,最终二次涡对与初始涡对分离(如图 3(a)中1 050 μs时刻图像),从而解释了廖深飞等人[21]观察到的现象,即气柱界面流向宽度的增长率随反射距离的减小而增大。另外,二次涡对和初始涡对都是稳定的大尺度结构,持续时间较长,这些大尺度结构在流场的质量、动量和能量输运以及流体混合中占主导地位,从图 2图 3中可以明显地看到质量由初始涡对向二次涡对转移。图 3中的涡量场还显示,在演化后期,无论是初始涡对还是二次涡对,这些大尺度结构会逐渐分解成小尺度结构,涡量分布由聚集状态逐渐过渡到分散状态。

    涡结构是激波冲击气柱后RM不稳定性诱导的典型结构,其演化主要由激波与气柱相互作用过程中斜压涡量的产生和分布所主导[5];而环量则表征沉积在界面上的斜压涡量值,反映了涡的强度,因此是流场演化的一个重要物理量,其实验测量对于深入认识RM不稳定性的演化机理及校验流体动力学程序具有重要意义。环量Γ定义为速度U沿某一封闭周线的线积分

    Γ=CUds
    (1)

    图 3所示的涡量场可以看到,在大尺度结构分解之前,涡量主要聚集在有限大小的涡核附近,类似于Rankine涡,因此定义包含涡核的长方形作为线积分路径计算环量(如图 3(a)中红色长方形所示),类似于文献[28]。积分路径上某一微元处的速度矢量由速度场中该微元的4个周围矢量结合双线性插值方法计算。计算每个时刻涡对中两个涡的环量,并将两个涡环量的绝对值的平均值作为涡对的环量。

    Samtaney等人[29]基于激波极线理论对平面激波单次冲击圆形界面进行了分析,提出并建立了理论模型(简称SZ模型)以预测由斜压机制而沉积在界面上的环量,即

    ΓSZ=r0(1+π2)(1η1/2)[1+2Ma(Ma+1)]v
    (2)

    式中:r0为圆形界面半径,η为圆形界面内气体与环境气体的密度比,v为波后气流速度。鉴于SZ模型的预测结果与数值模拟结果吻合很好[29],本研究采用SZ模型预测初始涡对的环量,其中气体的初始参数采用文献[30]中的参数。相比于单次冲击,激波二次冲击演化中的界面更加复杂,从激波与界面的相互作用过程出发预测环量更加困难。为此,Haehn等人[31]将与激波冲击历史无关的Kelvin环量模型(简称K模型)推广到激波二次冲击气泡中,发现K模型的预测结果与PIV的实验测量结果吻合很好。为此,本研究采用K模型预测二次涡对的环量

    ΓK=4πvdefR(ln8Ra14)1
    (3)

    式中:vdef为涡对相对于流场的速度,由于反射激波的波后流场速度为零,因此本研究将二次涡对右界面的速度近似为涡对速度;R为涡对中涡核中心间距的1/2,a为涡核半径,如图 5(a)所示。

    图  5  K模型参数定义示意图(a)及环量随时间的变化(b)(l0=L0/D0, t=0为反射激波二次冲击气柱的时刻)
    Figure  5.  (a) Schematic of parameter definition for K model; (b) Variations of circulation with time (l0=L0/D0; t=0 is the time when the gas cylinder is reshocked by a reflected shock wave)

    图 5(b)给出了PIV所得环量和理论模型预测环量。可见,实验结果与理论模型的预测结果大体上吻合,尤其是二次涡对的环量,说明本实验所采用的PIV技术可以较好地定量表征气柱演化。对于实验结果与理论模型预测结果的偏差:一方面来自于理论模型的理想假定与实验中实际情况的偏差,如SZ模型中的密度间断面假定、激波绕射圆界面时压力梯度方向与密度梯度方向垂直的假定等;另一方面来自于PIV技术本身的误差,如示踪粒子只播撒在气柱气体中,PIV后处理时界面附近速度的不连续会带来一定的误差。

    PIV所得环量随时间演化的结果表明,激波二次冲击气柱前(420~870 μs),初始涡对的环量变化很小,与Orlicz等人[28]观察到的激波冲击气帘时初始涡对环量的演化相似。激波二次冲击气柱后,初始涡对的环量随时间逐渐减小,说明能量逐渐由流场中的大尺度结构转移至小尺度结构,加剧了流体之间的混合。初始涡对分解成小尺度结构之前,其强度显著大于二次涡对的强度,Haehn等人[31]在激波二次冲击气泡时也观察到类似现象。由一维气体动力学可知,激波二次冲击气柱时的压力梯度略大于激波首次冲击气柱时的压力梯度,说明界面两侧的密度梯度和界面形状对二次涡对的形成和发展影响显著。

    采用双曝光PIV技术实验研究了激波两次冲击下重气柱界面的RM不稳定性,定量表征了界面演化的速度场和涡量场。根据速度矢量分布和涡辨识准则,研究发现:当反射距离较小时,激波二次冲击后,界面衍生出二次涡对,其旋转方向与初始涡对相反,强度显著小于初始涡对;当反射距离较大时,不会衍生大尺度涡结构。相比于传统的高速摄影技术,PIV技术具有更好的空间分辨力。由PIV所得的初始涡对和二次涡对的环量与理论模型的预测结果吻合较好,在一定程度上验证了实验中所采用的PIV技术在定量表征气柱演化方面的可靠性。激波二次冲击气柱后,初始涡对的环量随时间逐渐减小,说明能量逐渐由流场中的大尺度结构转移至小尺度结构。实验结果表明,界面两侧的密度梯度和界面形状对二次涡对的形成和发展影响显著。

    本实验中示踪粒子只播撒在气柱气体中,PIV后处理时界面附近的速度不连续给结果带来一定的误差,后续研究将考虑在气柱气体和环境气体中同时播撒示踪粒子,以提高PIV结果的精度。

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