Models for jet power in elliptical galaxies A case for rapidly spinning black holes

高三宇宙奥秘英语阅读理解30题1<背景文章>Black holes are one of the most fascinating and mysterious phenomena in the universe. A black hole is formed when a massive star collapses at the end of its life. The gravitational pull of a black hole is so strong that nothing, not even light, can escape from it.The formation of a black hole begins with the collapse of a massive star. As the star runs out of fuel, it can no longer support its own weight and begins to collapse. The collapse continues until the star reaches a critical density, at which point it becomes a black hole.Black holes have several unique characteristics. One of the most notable is their event horizon, which is the boundary beyond which nothing can escape. Another characteristic is their intense gravitational field, which can distort the space and time around them.Black holes can have a significant impact on the surrounding celestial bodies. They can attract and swallow nearby stars and planets, and their gravitational pull can also affect the orbits of other celestial bodies.Scientists are still working to understand black holes better. They use a variety of tools and techniques, such as telescopes and computer simulations, to study these mysterious objects. Despite significant progressin recent years, there is still much that we don't know about black holes.1. What is a black hole formed by?A. A small star collapsing.B. A massive star collapsing.C. A planet collapsing.D. A moon collapsing.答案：B。

PhysicsofFlaresfromBlackHoleMicroquasarV4641SgrintheRadio,OpticalandX-rays

Soon-WookKim12andShinMineshige3

†

RadioAstronomyDivision,KoreaAstronomyObservatory,Korea†YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Japan

Abstract.Wepresentadiskinstabilitymodelforunusuallyshort,butintenseandchaotic,flaresofafewdaysobservedinablackholemicroquasarV4641Sgr=SAXJ1819.3-2525,amultiplecomponentrelativisticjetsource.Toreproducetheobservedshortdurationoftheflare,aring-like,truncatedKepleriandiskisrequired.Wediscusscausesofsuchatruncatedaccretiondiskbasedonthediskinstabilitymodel,andimplicationtoshort-durationflareeventsobservedinothermicroquasarsandfastX-raytransients.

BLACKHOLEMICROQUASARV4641SAGITTARII=SAXJ1819.3-2525

BlackholemicroquasarV4641SgrwasdiscoveredinFebruary1999inquiescencebyBeppoSAXandRXTE/AllSkyMonitor,anddesignatedasSAXJ1819.3-2525[1][2].OnSeptember1999itsbrightandsuper-Eddington,butunusuallybriefoutburstwasobservedintheradio,opticalandX-raysupto100keV[3][4][5].Ac-companiedsuperluminalradiojetidentifiesthesourceasamicroquasar[3].OpticalspectroscopicobservationsshowthatV4641Sgrisahighmassblackholebinarywiththeblackholeandcompanionmassesof9.6and6.5M,respectively,withanorbitalperiodof2.81days[6].InFigure1,wepresentthe1999FebruarySeptemberX-rayandopticallightcurvesofV4641Sgr.Inthetopandmiddlepanels,X-rayobservationsaredisplayed:publiclyavailabledataofRXTE/PCA(filledandopentriangles),RXTE/ASM4()andBeppoSAX(opensquare)[5].Theopticallightcurveisalsopresented(adottedline)inthetoppanelforacomparison.Inthebot-tompanel,thevisualobservations(opencircle)adoptedfromVSNET5andfrom[7],andCCDdata(filledcircle)[4]arepresented.Intheoptical,theprecursorflaringactivitiesforaoftwoandahalfyears.Inthispaper,wefocusonlyonthe1999outburst.

大射电望远镜馈源支撑系统建模与仿真缪岭,刘玉标(中国科学院力学研究所,北京100190)1引言根据射电天文学发展需要,我国决定利用贵州喀斯特洼地,建设500m 口径球面射电望远镜(Five-hundred-meter Aperture Spherical Telescope ,简称FAST ),建成后FAST 将是世界上最大的单天线射电望远镜,对我国天文学的发展具有重要的意义[1]。

针对如此巨大口径的现实情况,对馈源支撑系统,FAST 采用了一种全新的光机电一体化的创新设计方案[2]。

此设计方案先通过大跨度的悬索对馈源舱进行粗调,然后在馈源舱内采用Stewart 机构进行二次精调,并实时通过激光检测装置检测接收装置的具体位置,与主动反射面配合,反馈给计算机处理进行闭环控制。

由于以往对此工程的机构动力分析一般都建立在有限元的基础上,无法对系统整体进行拖动的实际仿真计算,因此根据上述设计方案,采用多体动力学分析软件MSC .Adams 建立了馈源支撑系统的机械模型及风载模型,并分析了其关键点的响应情况。

2模型的建立按照多体动力学建模方法,对实际机械系统进行抽象,用标准的运动副、驱动约束、力元和外力等要素建立与实际机械系统一致的模型[3]。

悬索系统采用6悬索式悬吊方案。

悬索与馈源舱连接并不交于同一处,而是以相邻两根为一组与馈源舱连接,形成三角形式的连接方案。

悬索模拟采用有限元的方法,将每根悬索分割为多段,以铰接连接,并考虑到悬索的变形,添加了弹簧阻尼元件,使悬索模拟更加符合真实。

馈源舱系统根据分析要求可以分为四部分,分别为三脚架、摆索(阻尼索)、位置器及Stewart 平台,将各部分外形进行简化后,以标准的连接副进行连接。

设S 平台下端动平台中心位置为关键点的位置(此处也为接收器的实际位置)。

设定悬索支点与馈源舱顶部中心水平距离300m ,垂直距离150m ,以主动反射面的底部最低处作为地面,馈源最低点高度约为150m 。

天文上的英语名词解释天文学是一个广阔而神秘的领域，充满了许多令人着迷的现象和概念。

在这篇文章中，我们将解释一些关于天文学的英语名词，帮助读者更好地理解这些术语的含义。

1. 星系 (Galaxy)星系是宇宙中巨大的星际系统。

它由恒星、行星、气体、尘埃和暗物质组成。

以下是几个常见类型的星系：- 棒状螺旋星系 (Barred Spiral Galaxy)：中心有一条棒状结构，围绕着中心旋转的螺旋臂。

- 普通螺旋星系 (Normal Spiral Galaxy)：没有明显的棒状结构，呈现出螺旋状的臂。

- 椭球星系 (Elliptical Galaxy)：呈椭球形状，没有螺旋臂。

- 不规则星系 (Irregular Galaxy)：形状不规则，通常是由于与其他星系的相互作用产生的。

2. 星际尘埃 (Interstellar Dust)星际尘埃是分布在星系间空间中的微小颗粒。

这些尘埃颗粒主要由碳、硅等化学元素组成，它们的存在可以通过散射或吸收光线来观测到。

星际尘埃在星系形成和恒星诞生中起着重要的作用。

3. 星云 (Nebula)星云是由气体和尘埃组成的大型云状物体。

星云通常是通过恒星爆发、超新星爆炸或行星系统形成的。

以下是几种常见的星云类型：- 星际云 (Interstellar Cloud)：主要由氢气和尘埃组成的巨大云状结构。

- 行星状星云 (Planetary Nebula)：恒星快速膨胀和放出外层气体时形成的云状结构，通常呈现出球形或圆盘状。

- 火箭座Cephei-A (RCW 120)是一个著名的恒星形成区，位于南天星云复合体的一部分 (Southern Complex of Star Formation)。

4. 恒星 (Star)恒星是宇宙中辐射出巨大光和热能的球状天体。

它们主要由氢和一小部分氦等元素组成。

恒星通常被分为不同的类型，例如：- 主序星 (Main Sequence Star)：通过核聚变反应将氢转化为氦并产生巨大的能量的恒星。

Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identifies the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great difficulty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR werefirst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs werefirst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.In the next section we attempt to give a mathematical definition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In thefifth section we discuss the excitation and observation of QNMs andfinally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asfinite strings,mem-branes,or cavitiesfilled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedfield.) There is generally an infinite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isfixed at its ends.All such systems can be described by systems of partial differential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of thefiniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the differential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling thefinite string has solutions of various degrees of differentiability. To describe all“realistic situations”,clearly C∞functions should be sufficient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to infinity.The simplest case are waves on an infinite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice differentiable solution.No boundary condition is needed at infinity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of thefields are spatially compact, or–the other extreme–solutions with infinite total energy for which thefields grow at spatial infinity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solu-tions offinite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that wefind a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for infinite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually different to normal modes.To define quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisfies the ordinary differential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and defineG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The definition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which defines a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be defined as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical significance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Thefirst new point concerns the definitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,define the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at infinity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largefinite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for sufficiently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniquefingerprint which would lead to the direct identification of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The difference between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisfies the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate defined byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the effective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testfields as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingfield.For“polar”perturbations the effective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type offluid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving differential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential difficulty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes differentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because infinity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an infinite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。

卫星运行的物理原理是什么卫星运行的物理原理主要包括牛顿力学定律和开普勒三定律。

首先，根据牛顿第一定律，即所有物体都保持静止或以恒定速度直线运动，除非有外力作用于其上。

当卫星处于地球引力的作用下时，牛顿第一定律被满足。

根据牛顿第二定律，物体的加速度与作用在其上的合力成正比，与其质量成反比。

应用到卫星运行中，由于地球对卫星的引力是作用在卫星上的合力，因此卫星会出现加速度。

这一加速度是使卫星保持在轨道上运行的关键。

其次，开普勒三定律也是解释卫星运行的重要原理。

开普勒第一定律称为"椭圆轨道定理"，指出所有行星运动的轨道都是椭圆，太阳位于椭圆的一个焦点上。

同样，卫星绕地球运行的轨道也是一个椭圆，地球位于这个椭圆的一个焦点上。

开普勒第二定律称为"面积定律"，指出在等时间内，行星与太阳连线所扫过的面积是相等的。

对应到卫星运动中，卫星与地球连线扫过的面积也是相等的。

最后，开普勒第三定律称为"调和定律"，指出行星绕太阳的轨道半长轴与轨道周期的平方成正比。

同样的，卫星绕地球的轨道半长轴与轨道周期的平方也成正比。

在应用物理学中，卫星的运行原理可以用开普勒定律和牛顿运动定律的数学公式来描述。

例如，根据开普勒定律可以推导出卫星的轨道半长轴的公式，即a³/T²=G(M+m)/(4π²)，其中a表示卫星的轨道半长轴，T表示卫星绕地球的周期，G表示引力常数，M表示地球的质量，m表示卫星的质量。

这个公式说明了卫星的轨道半长轴与轨道周期的平方成正比。

此外，卫星的运行还需要考虑到其他因素的影响，如地球自转引起的离心力、大气阻力等。

离心力会产生向外的力，而大气阻力则会造成向内的力，这些力都会对卫星的运行产生影响。

为了保持卫星的稳定运行，需要进行轨道控制和姿态控制等操作。

综上所述，卫星运行的物理原理主要包括牛顿力学定律和开普勒三定律。

牛顿力学定律解释了卫星运行中的加速度产生机制，而开普勒三定律则描述了卫星运行轨道的特点和规律。

第９卷㊀第２期２０２４年３月气体物理ＰＨＹＳＩＣＳＯＦＧＡＳＥＳＶｏｌ.９㊀Ｎｏ.２Ｍａｒ.２０２４㊀㊀ＤＯＩ:１０.１９５２７/ｊ.ｃｎｋｉ.２０９６￣１６４２.１０９８Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩的影响章录兴ꎬ㊀王光学ꎬ㊀杜㊀磊ꎬ㊀余发源ꎬ㊀张怀宝(中山大学航空航天学院ꎬ广东深圳５１８１０７)ＥｆｆｅｃｔｓｏｆＭａｃｈＮｕｍｂｅｒａｎｄＷａｌｌＴｅｍｐｅｒａｔｕｒｅｏｎＨｙＴＲＶＢｏｕｎｄａｒｙＬａｙｅｒＴｒａｎｓｉｔｉｏｎＺＨＡＮＧＬｕｘｉｎｇꎬ㊀ＷＡＮＧＧｕａｎｇｘｕｅꎬ㊀ＤＵＬｅｉꎬ㊀ＹＵＦａｙｕａｎꎬ㊀ＺＨＡＮＧＨｕａｉｂａｏ(ＳｃｈｏｏｌｏｆＡｅｒｏｎａｕｔｉｃｓａｎｄＡｓｔｒｏｎａｕｔｉｃｓꎬＳｕｎＹａｔ￣ｓｅｎＵｎｉｖｅｒｓｉｔｙꎬＳｈｅｎｚｈｅｎ５１８１０７ꎬＣｈｉｎａ)摘㊀要:典型的高超声速飞行器流场存在着复杂的转捩现象ꎬ其对飞行器的性能有着显著的影响ꎮ针对ＨｙＴＲＶ这款接近真实高超声速飞行器的升力体模型ꎬ采用数值模拟方法ꎬ研究Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ转捩的影响规律ꎮ采用课题组自研软件开展数值计算ꎬＭａｃｈ数的范围为３~８ꎬ壁面温度的范围为１５０~９００Ｋꎮ首先对γ￣Ｒｅ~θｔ转捩模型和ＳＳＴ湍流模型进行了高超声速修正:将压力梯度系数修正㊁高速横流修正引入到γ￣Ｒｅ~θｔ转捩模型ꎬ并对ＳＳＴ湍流模型闭合系数β∗和β进行可压缩修正ꎻ然后开展了网格无关性验证ꎬ通过与实验结果对比ꎬ确认了修正后的数值方法和软件平台ꎻ最终开展Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩规律的影响研究ꎮ计算结果表明ꎬ转捩区域主要集中在上表面两侧㊁下表面中心线两侧ꎻ增大来流Ｍａｃｈ数ꎬ上下表面转捩起始位置均大幅后移ꎬ湍流区大幅缩小ꎬ但仍会存在ꎬ同时上表面层流区摩阻系数不断增大ꎬ下表面湍流区摩阻系数不断减小ꎻ升高壁面温度ꎬ上下表面转捩起始位置先前移ꎬ然后快速后移ꎬ最终湍流区先后几乎消失ꎮ关键词:转捩ꎻＨｙＴＲＶꎻ摩阻ꎻＭａｃｈ数ꎻ壁面温度㊀㊀㊀收稿日期:２０２３￣１２￣１３ꎻ修回日期:２０２４￣０１￣０２基金项目:国家重大项目(ＧＪＸＭ９２５７９)ꎻ广东省自然科学基金－面上项目(２０２３Ａ１５１５０１００３６)ꎻ中山大学中央高校基本科研业务费专项资金(２２ｑｎｔｄ０７０５)第一作者简介:章录兴(１９９８ )㊀男ꎬ硕士ꎬ主要研究方向为高超声速空气动力学ꎮＥ￣ｍａｉｌ:184****8082＠１６３.ｃｏｍ通信作者简介:张怀宝(１９８５ )㊀男ꎬ副教授ꎬ主要研究方向为空气动力学ꎮＥ￣ｍａｉｌ:ｚｈａｎｇｈｂ２８＠ｍａｉｌ.ｓｙｓｕ.ｅｄｕ.ｃｎ中图分类号:Ｖ２１１ꎻＶ４１１㊀㊀文献标志码:ＡＡｂｓｔｒａｃｔ:Ｔｈｅｒｅｉｓａｃｏｍｐｌｅｘｔｒａｎｓｉｔｉｏｎｐｈｅｎｏｍｅｎｏｎｉｎｔｈｅｆｌｏｗｆｉｅｌｄｏｆａｔｙｐｉｃａｌｈｙｐｅｒｓｏｎｉｃｖｅｈｉｃｌｅꎬｗｈｉｃｈｈａｓａｓｉｇｎｉｆｉ￣ｃａｎｔｉｍｐａｃｔｏｎｔｈｅｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅｖｅｈｉｃｌｅ.ＴｈｅｅｆｆｅｃｔｓｏｆＭａｃｈｎｕｍｂｅｒａｎｄｗａｌｌｔｅｍｐｅｒａｔｕｒｅｏｎｔｈｅｔｒａｎｓｉｔｉｏｎｏｆＨｙＴＲＶｗｅｒｅｓｔｕｄｉｅｄｂｙｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎｍｅｔｈｏｄｓ.Ｔｈｅｓｅｌｆ￣ｄｅｖｅｌｏｐｅｄｓｏｆｔｗａｒｅｏｆｔｈｅｒｅｓｅａｒｃｈｇｒｏｕｐｗａｓｕｓｅｄｔｏｃａｒｒｙｏｕｔｎｕ￣ｍｅｒｉｃａｌｃａｌｃｕｌａｔｉｏｎｓ.ＴｈｅｒａｎｇｅｏｆＭａｃｈｎｕｍｂｅｒｗａｓ３~８ꎬａｎｄｔｈｅｒａｎｇｅｏｆｗａｌｌｔｅｍｐｅｒａｔｕｒｅｗａｓ１５０~９００Ｋ.Ｆｉｒｓｔｌｙꎬｔｈｅｈｙｐｅｒｓｏｎｉｃｃｏｒｒｅｃｔｉｏｎｓｏｆｔｈｅγ￣Ｒｅ~θｔｔｒａｎｓｉｔｉｏｎｍｏｄｅｌａｎｄｔｈｅＳＳＴｔｕｒｂｕｌｅｎｃｅｍｏｄｅｌｗｅｒｅｃａｒｒｉｅｄｏｕｔ.Ｔｈｅｐｒｅｓｓｕｒｅｇｒａｄｉｅｎｔｃｏｅｆｆｉｃｉｅｎｔｃｏｒｒｅｃｔｉｏｎａｎｄｔｈｅｈｉｇｈ￣ｓｐｅｅｄｃｒｏｓｓ￣ｆｌｏｗｃｏｒｒｅｃｔｉｏｎｗｅｒｅｉｎｔｒｏｄｕｃｅｄｉｎｔｏｔｈｅγ￣Ｒｅ~θｔｔｒａｎｓｉｔｉｏｎｍｏｄｅｌꎬａｎｄｔｈｅｃｏｍ￣ｐｒｅｓｓｉｂｉｌｉｔｙｃｏｒｒｅｃｔｉｏｎｓｏｆｔｈｅｃｌｏｓｕｒｅｃｏｅｆｆｉｃｉｅｎｔｓβ∗ａｎｄβｏｆｔｈｅＳＳＴｔｕｒｂｕｌｅｎｃｅｍｏｄｅｌｗｅｒｅｃａｒｒｉｅｄｏｕｔ.Ｔｈｅｎꎬｔｈｅｇｒｉｄｉｎ￣ｄｅｐｅｎｄｅｎｃｅｖｅｒｉｆｉｃａｔｉｏｎｗａｓｃａｒｒｉｅｄｏｕｔꎬａｎｄｔｈｅｍｏｄｉｆｉｅｄｎｕｍｅｒｉｃａｌｍｅｔｈｏｄａｎｄｓｏｆｔｗａｒｅｐｌａｔｆｏｒｍｗｅｒｅｃｏｎｆｉｒｍｅｄｂｙｃｏｍ￣ｐａｒｉｎｇｗｉｔｈｅｘｐｅｒｉｍｅｎｔａｌｒｅｓｕｌｔｓ.ＦｉｎａｌｌｙꎬｔｈｅｅｆｆｅｃｔｓｏｆＭａｃｈｎｕｍｂｅｒａｎｄｗａｌｌｔｅｍｐｅｒａｔｕｒｅｏｎｔｈｅｔｒａｎｓｉｔｉｏｎｌａｗｏｆｔｈｅＨｙＴＲＶｂｏｕｎｄａｒｙｌａｙｅｒｗｅｒｅｓｔｕｄｉｅｄ.Ｔｈｅｒｅｓｕｌｔｓｓｈｏｗｔｈａｔｔｈｅｔｒａｎｓｉｔｉｏｎａｒｅａｉｓｍａｉｎｌｙｃｏｎｃｅｎｔｒａｔｅｄｏｎｂｏｔｈｓｉｄｅｓｏｆｔｈｅｕｐｐｅｒｓｕｒｆａｃｅａｎｄｔｈｅｃｅｎｔｅｒｌｉｎｅｏｆｔｈｅｌｏｗｅｒｓｕｒｆａｃｅ.ＷｉｔｈｔｈｅｉｎｃｒｅａｓｅｏｆｔｈｅｉｎｃｏｍｉｎｇＭａｃｈｎｕｍｂｅｒꎬｔｈｅｓｔａｒｔｉｎｇｐｏｓｉｔｉｏｎｏｆｔｒａｎｓｉｔｉｏｎｏｎｔｈｅｕｐｐｅｒａｎｄｌｏｗｅｒｓｕｒｆａｃｅｓｉｓｇｒｅａｔｌｙｂａｃｋｗａｒｄꎬａｎｄｔｈｅｔｕｒｂｕｌｅｎｔｚｏｎｅｉｓｇｒｅａｔｌｙｒｅｄｕｃｅｄꎬｂｕｔｉｔｓｔｉｌｌｅｘ￣ｉｓｔｓ.Ａｔｔｈｅｓａｍｅｔｉｍｅꎬｔｈｅｆｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｏｆｔｈｅｌａｍｉｎａｒｆｌｏｗｚｏｎｅｏｎｔｈｅｕｐｐｅｒｓｕｒｆａｃｅｉｎｃｒｅａｓｅｓｃｏｎｔｉｎｕｏｕｓｌｙꎬａｎｄｔｈｅｆｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｏｆｔｈｅｔｕｒｂｕｌｅｎｔｚｏｎｅｏｎｔｈｅｌｏｗｅｒｓｕｒｆａｃｅｄｅｃｒｅａｓｅｓ.Ａｓｔｈｅｗａｌｌｔｅｍｐｅｒａｔｕｒｅｉｎｃｒｅａｓｅｓꎬｔｈｅｓｔａｒｔｉｎｇｐｏｓｉ￣ｔｉｏｎｏｆｔｒａｎｓｉｔｉｏｎｏｎｔｈｅｕｐｐｅｒａｎｄｌｏｗｅｒｓｕｒｆａｃｅｓｓｈｉｆｔｓｆｏｒｗａｒｄꎬｔｈｅｎｒａｐｉｄｌｙｓｈｉｆｔｓｂａｃｋｗａｒｄꎬａｎｄｆｉｎａｌｌｙｔｈｅｔｕｒｂｕｌｅｎｔｚｏｎｅａｌｍｏｓｔｄｉｓａｐｐｅａｒｓ.气体物理２０２４年㊀第９卷Ｋｅｙｗｏｒｄｓ:ｔｒａｎｓｉｔｉｏｎꎻＨｙＴＲＶꎻｆｒｉｃｔｉｏｎꎻＭａｃｈｎｕｍｂｅｒꎻｗａｌｌｔｅｍｐｅｒａｔｕｒｅ引㊀言高超声速飞行器具有突防能力强㊁打击范围广㊁响应迅速等显著优势ꎬ正逐渐成为各国空天竞争的热点[１]ꎮ高超声速飞行器边界层转捩是该类飞行器气动设计中的重要问题[２]ꎮ在边界层转捩过程中ꎬ流态由层流转变为湍流ꎬ飞行器的表面摩阻急剧增大到层流时的３~５倍ꎬ严重影响飞行器的气动性能与热防护系统ꎬ转捩还会导致飞行器壁面烧蚀㊁颤振加剧㊁飞行姿态控制难度大等一系列问题ꎬ对飞行器的飞行安全构成严重的威胁[３￣５]ꎬ开展高超声速飞行器边界层转捩研究具有十分重要的意义ꎮ影响边界层转捩的因素很多ꎬ例如ꎬＭａｃｈ数㊁Ｒｅｙｎｏｌｄｓ数㊁湍流强度㊁表面传导热等ꎮ在高超声速流动条件下ꎬ强激波㊁强逆压梯度㊁熵层等高超声速现象及其相互作用ꎬ会使得转捩流动的预测和研究难度进一步增大[６]ꎮ目前高超声速飞行器转捩数值模拟方法主要有直接数值模拟(ＤＮＳ)㊁大涡模拟(ＬＥＳ)和基于Ｒｅｙｎｏｌｄｓ平均Ｎａｖｉｅｒ￣Ｓｔｏｋｅｓ(ＲＡＮＳ)的转捩模型方法ꎬ由于前两种计算量巨大ꎬ难以推广到工程应用ꎬ基于Ｒｅｙｎｏｌｄｓ平均Ｎａｖｉｅｒ￣Ｓｔｏｋｅｓ的转捩模型在工程实践中应用最为广泛ꎬ其中γ￣Ｒｅ~θｔ转捩模型基于局部变量ꎬ与现代ＣＦＤ方法良好兼容ꎬ目前已经有多项研究尝试从一般性的流动问题拓展到高超声速流动转捩模拟[６￣９]ꎮ目前高超声速流动转捩的研究对象主要是结构相对简单的构型ꎮＭｃＤａｎｉｅｌ等[１０]研究了扩口直锥在高超声速流动条件下的转捩现象ꎮＰａｐｐ等[１１]研究了圆锥在高超声速流动条件下的转捩特性ꎮ美国和澳大利亚组织联合实施的ＨＩＦｉＲＥ计划[１２]ꎬ研究了圆锥形状的ＨＩＦｉＲＥ１和椭圆锥形的ＨＩＦｉＲＥ５的转捩问题ꎮ杨云军等[１３]采用数值模拟方法ꎬ分析了椭圆锥的转捩影响机制ꎬ并研究了Ｒｅｙｎｏｌｄｓ数对转捩特性的影响规律ꎮ另外ꎬ袁先旭等[１４]于２０１５年成功实施了圆锥体ＭＦ￣１航天模型飞行试验ꎮ以上对高超声速流动的转捩研究ꎬ都取得了比较理想的结果ꎬ然而所采用的模型都是圆锥㊁椭圆锥等简单几何外形ꎬ这与真实高超声速飞行器有较大差异ꎬ较难反映真实的转捩特性ꎮ为了有效促进对真实高超声速飞行器的转捩问题研究ꎬ中国空气动力研究与发展中心提出并设计了一款接近真实飞行器的升力体模型ꎬ即高超声速转捩研究飞行器(ｈｙｐｅｒｓｏｎｉｃｔｒａｎｓｉｔｉｏｎｒｅｓｅａｒｃｈｖｅｈｉｃｌｅꎬＨｙＴＲＶ)[１５]ꎬ模型详细的参数见参考文献[１６]ꎮＨｙＴＲＶ外形如图１所示ꎬ其整体外形较为复杂ꎬ不同区域发生转捩的情况也不尽相同ꎮ对ＨｙＴＲＶ的转捩问题研究能够显著提高对真实高超声速飞行器转捩特性的认识水平ꎮＬｉｕ等[１７]采用理论分析㊁数值模拟和风洞实验３种方法对ＨｙＴＲＶ的转捩特性进行了研究ꎻ陈坚强等[１５]分析了ＨｙＴＲＶ的边界层失稳特征ꎻＣｈｅｎ等[１８]对ＨｙＴＲＶ进行了多维线性稳定性分析ꎻＱｉ等[１９]在来流Ｍａｃｈ数６㊁攻角０ʎ的条件下对ＨｙＴＲＶ进行了直接数值模拟ꎻ万兵兵等[２０]结合风洞实验与飞行试验ꎬ利用ｅＮ方法预测了ＨｙＴＲＶ升力体横流区的转捩阵面形状ꎮ目前ꎬ相关研究主要集中在ＨｙＴＲＶ的稳定性特征及转捩预测两个方面ꎬ而对若干关键参数ꎬ特别是Ｍａｃｈ数和壁面温度对转捩的影响研究还比较少ꎮ(ａ)Ｆｒｏｎｔｖｉｅｗ(ｂ)Ｓｉｄｅｖｉｅｗ㊀㊀㊀图１㊀ＨｙＴＲＶ外形Ｆｉｇ.１㊀ＳｈａｐｅｏｆＨｙＴＲＶ基于此ꎬ本文采用数值模拟方法ꎬ应用课题组自研软件开展Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ转捩流动的影响规律研究ꎮ１㊀数值方法１.１㊀控制方程和数值方法控制方程为三维可压缩ＲＡＮＳ方程ꎬ采用结构网格技术和有限体积方法ꎬ变量插值方法采用２阶ＭＵＳＣＬ格式ꎬ通量计算采用低耗散的通量向量差分Ｒｏｅ格式ꎬ黏性项离散采用中心格式ꎬ时间推进方法采用ＬＵ￣ＳＧＳ格式ꎮ壁面采用等温㊁无滑移壁面条件ꎬ入口采用Ｒｉｅｍａｎｎ远场边界条件ꎬ出口采用零梯度外推边界条件ꎮ１.２㊀γ￣Ｒｅ~θｔ转捩模型γ￣Ｒｅ~θｔ转捩模型是Ｍｅｎｔｅｒ等[２１ꎬ２２]于２００４年提０１第２期章录兴ꎬ等:Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩的影响出的一种基于拟合公式的间歇因子转捩模型ꎬ在２００９年公布了完整的拟合公式及相关参数[２３]ꎮ许多学者也开发了相应的程序ꎬ并进行了大量的算例验证[２４￣２８]ꎬ证明了该模型具有较好的转捩预测能力ꎬ预测精度较高ꎻ通过合适的标定ꎬγ￣Ｒｅ~θｔ转捩模型可以适用于多种情况下的转捩模拟ꎮ该模型构建了关于间歇因子γ的输运方程和关于转捩动量厚度Ｒｅｙｎｏｌｄｓ数Ｒｅ~θｔ的输运方程ꎮ具体来说ꎬγ表示该位置是湍流流动的概率ꎬ取值范围为０<γ<１ꎮ关于γ的控制方程为Ə(ργ)Əｔ＋Ə(ρｕｊγ)Əｘｊ＝Ｐγ－Ｅγ＋ƏƏｘｊμ＋μｔσｆæèçöø÷ƏγƏｘｊéëêêùûúú其中ꎬＰγ为生成项ꎬＥγ为破坏项ꎮ关于Ｒｅ~θｔ的输运方程为Ə(ρＲｅ~θｔ)Əｔ＋Ə(ρｕｊＲｅ~θｔ)Əｘｊ＝Ｐθｔ＋ƏƏｘｊσθｔ(μ＋μｔ)ƏＲｅ~θｔƏｘｊéëêêùûúú其中ꎬＰθｔ为源项ꎬ其作用是使边界层外部的Ｒｅ~θｔ等于Ｒｅθｔꎬ定义式为Ｐθｔ＝ｃθｔρｔ(Ｒｅθｔ－Ｒｅ~θｔ)(１.０－Ｆθｔ)Ｒｅθｔ采用以下经验公式Ｒｅθｔ＝１１７３.５１－５８９ ４２８Ｔｕ＋０.２１９６Ｔｕ２æèçöø÷Ｆ(λθ)ꎬＴｕɤ０.３Ｒｅθｔ＝３３１.５０(Ｔｕ－０.５６５８)－０.６７１Ｆ(λθ)ꎬＴｕ>０.３ìîíïïïïＦ(λθ)＝１＋(１２.９８６λθ＋１２３.６６λ２θ＋４０５.６８９λ３θ)ｅ－(Ｔｕ１.５)１.５ꎬ㊀λθɤ０Ｆ(λθ)＝１＋０.２７５(１－ｅ－３５.０λθ)ｅ－(Ｔｕ０.５)ꎬλθ>０ìîíïïïï在实际计算中ꎬ通过γ￣Ｒｅ~θｔ转捩模型获得间歇因子ꎬ再通过间歇因子来控制ＳＳＴｋ￣ω湍流模型中湍动能的生成ꎮγ￣Ｒｅ~θｔ转捩模型与ＳＳＴｋ￣ω湍流模型耦合为Ə(ρｋ)Əｔ＋Ə(ρｕｊｋ)Əｘｊ＝γｅｆｆτｉｊƏｕｉƏｘｊ－ｍｉｎ(ｍａｘ(γｅｆｆꎬ０.１)ꎬ１.０)ρβ∗ｋω＋ƏƏｘｊμ＋μｔσｋæèçöø÷ƏｋƏｘｊéëêêùûúúƏ(ρω)Əｔ＋Ə(ρｕｊω)Əｘｊ＝γｖｔτｉｊƏｕｉƏｘｊ－βρω２＋ƏƏｘｊ(μ＋σωμｔ)ƏωƏｘｊéëêêùûúú＋２ρ(１－Ｆ１)σω２１ωƏｋƏｘｊƏωƏｘｊ模型中具体参数定义见文献[２３]ꎮ１.３㊀高超声速修正原始ＳＳＴ湍流模型及γ￣Ｒｅ~θｔ转捩模型都是基于不可压缩流动发展的ꎬ为了更好地预测高超声速流动转捩ꎬ本节引入了３种重要的高超声速修正方法ꎮ１.３.１㊀压力梯度修正压力梯度对边界层转捩的影响较大ꎬ在高Ｍａｃｈ数情况下ꎬ边界层厚度较大ꎬ进而影响压力梯度的大小ꎬ因此在模拟高超声速流动时应该考虑Ｍａｃｈ数对压力梯度的影响ꎮ本文采用张毅峰等[２９]提出的压力梯度修正方法ꎬ具体修正形式如下λᶄθ＝λθ１＋γᶄ－１２Ｍａｅæèçöø÷其中ꎬＭａｅ为边界层外缘Ｍａｃｈ数ꎬγᶄ为比热比ꎮ１.３.２㊀高速横流修正在原始γ￣Ｒｅ~θｔ转捩模型中ꎬ没有考虑横流不稳定性对转捩的影响ꎬ对于横流模态主导的转捩ꎬ原始转捩模型计算的结果并不理想ꎮＬａｎｇｔｒｙ等[３０]在２０１５年对γ￣Ｒｅ~θｔ转捩模型进行了低速横流修正ꎬ向星皓等[９]在Ｌａｎｇｔｒｙ低速横流修正的基础上ꎬ对高超声速椭圆锥转捩ＤＮＳ数据进行了拓展ꎬ提出了高速横流转捩判据ꎬ本文直接采用向星皓提出的高速横流转捩方法ꎮＬａｎｇｔｒｙ将横流强度引入转捩发生动量厚度Ｒｅｙｎｏｌｄｓ数输运方程中Ə(ρＲｅ~θｔ)Əｔ＋Ə(ρｕｊＲｅ~θｔ)Əｘｊ＝Ｐθｔ＋ＤＳＣＦ＋ƏƏｘｊσθｔ(μ＋μｔ)ƏＲｅ~θｔƏｘｊéëêêùûúú式中ꎬＤＳＣＦ为横流源项ꎬＬａｎｇｔｒｙ低速横流修正为ＤＳＣＦ＝ｃθｔρｔｃｃｒｏｓｓｆｌｏｗｍｉｎ(ＲｅＳＣＦ－Ｒｅ~θｔꎬ０.０)Ｆθｔ２其中ꎬＲｅＳＣＦ为低速横流判据ＲｅＳＣＦ＝θｔρＵｌｏｃａｌ０.８２æèçöø÷μ＝－３５.０８８ｌｎｈθｔæèçöø÷＋３１９.５１＋ｆ(＋ΔＨｃｒｏｓｓｆｌｏｗ)－ｆ(－ΔＨｃｒｏｓｓｆｌｏｗ)其中ꎬｈ为壁面粗糙度高度ꎬθｔ为动量厚度ꎬ１１气体物理２０２４年㊀第９卷ΔＨｃｒｏｓｓｆｌｏｗ是横流强度抬升项ꎮ向星皓提出的高速横流转捩判据ꎬ其中高速横流源项ＤＳＣＦ￣Ｈ为ＤＳＣＦ￣Ｈ＝ｃＣＦρｍｉｎ(ＲｅＳＣＦ￣Ｈ－Ｒｅ~θｔꎬ０)ＦθｔＲｅＳＣＦ￣Ｈ＝ＣＣＦ￣１ｌｎｈｌμ＋ＣＣＦ￣２＋(Ｈｃｒｏｓｓｆｌｏｗ)其中ꎬＣＣＦ￣１＝－９.６１８ꎬＣＣＦ￣２＝１２８.３３ꎻｌμ为粗糙度参考高度ꎬｌμ＝１μｍꎻｆ(Ｈｃｒｏｓｓｆｌｏｗ)为抬升函数ｆ(Ｈｃｒｏｓｓｆｌｏｗ)＝６００００.１０６６－ΔＨｃｒｏｓｓｆｌｏｗ＋５００００(０.１０６６－ΔＨｃｒｏｓｓｆｌｏｗ)２其中ꎬΔＨｃｒｏｓｓｆｌｏｗ与Ｌａｎｇｔｒｙ低速横流修正中保持一致ꎮ１.３.３㊀ＳＳＴ可压缩修正高超声速流动具有强可压缩性ꎬ所以在进行高超声速计算时ꎬ应该对湍流模型进行可压缩修正ꎮＳａｒｋａｒ[３１]提出了膨胀耗散修正ꎬ对ＳＳＴ湍流模型中的闭合系数β∗ꎬβ进行了可压缩修正ꎬＷｉｌｃｏｘ[３２]在Ｓａｒｋａｒ修正的基础上考虑了可压缩生成项产生时的延迟效应ꎬ使得可压缩修正在湍流Ｍａｃｈ数较小的近壁面关闭ꎬ在湍流Ｍａｃｈ数较大的自由剪切层打开ꎬ本文采用Ｗｉｌｃｏｘ提出的可压缩性修正β∗＝β∗０[１＋ξ∗Ｆ(Ｍａｔ)]β＝β０－β∗０ξ∗Ｆ(Ｍａｔ)其中ꎬβ０ꎬβ∗均为原始模型中的系数ꎬξ∗＝１.５ꎮＦ(Ｍａｔ)＝[Ｍａｔ－Ｍａｔ０]Ｈ(Ｍａｔ－Ｍａｔ０)Ｍａｔ０＝１/４ꎬＨ(ｘ)＝０ꎬｘɤ０１ꎬｘ>０{其中ꎬＭａｔ＝２ｋ/ａ为湍流Ｍａｃｈ数ꎬａ为当地声速ꎮ２㊀网格无关性验证及数值方法确认２.１㊀网格无关性验证计算采用３套网格ꎬ考虑到ＨｙＴＲＶ的几何对称性ꎬ生成３套半模网格ꎬ第１层网格高度为１ˑ１０－６ｍꎬ确保ｙ＋<１ꎬ流向ˑ法向ˑ周向的网格数分别为:网格１是３０１ˑ２０１ˑ２０１ꎬ网格２是３０１ˑ３０１ˑ２０１ꎬ网格３是４０１ˑ３８１ˑ２８１ꎮ全模下表面如图２所示ꎬ选取ｙ/Ｌ＝０中心线和ｘ/Ｌ＝０.５处ꎬ对比３套网格的表面摩阻系数ꎬ计算结果如图３所示ꎮ采用网格１时ꎬ表面摩阻系数分布与另外两个结果存在明显差异ꎻ而采用网格２和网格３时ꎬ表面摩阻系数曲线基本重合ꎬ表明在流向㊁法向和周向均满足网格无关性ꎬ后续数值计算采用网格２ꎮ图２㊀截取位置示意图Ｆｉｇ.２㊀Ｓｃｈｅｍａｔｉｃｄｉａｇｒａｍｏｆｔｈｅｉｎｔｅｒｃｅｐｔｉｏｎｌｏｃａｔｉｏｎ(ａ)Ｓｕｒｆａｃｅｆｒｉｃｔｉｏｎａｔｙ/Ｌ＝０(ｂ)Ｓｕｒｆａｃｅｆｒｉｃｔｉｏｎａｔｘ/Ｌ＝０.５图３㊀采用３套网格计算得到的摩阻对比Ｆｉｇ.３㊀Ｃｏｍｐａｒｉｓｏｎｏｆｔｈｅｆｒｉｃｔｉｏｎｄｒａｇｃａｌｃｕｌａｔｅｄｕｓｉｎｇｔｈｒｅｅｓｅｔｓｏｆｇｒｉｄｓ２.２㊀数值方法和自研软件的确认采用修正后的转捩模型对ＨｙＴＲＶ开展计算ꎬ计算工况为Ｍａ＝６ꎬ来流温度Ｔɕ＝９７Ｋꎬ单位２１第２期章录兴ꎬ等:Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩的影响Ｒｅｙｎｏｌｄｓ数为Ｒｅ＝１.１ˑ１０７/ｍꎬ攻角α＝０ʎꎬ来流湍流度ＦＳＴＩ＝０.８％ꎬ壁面温度Ｔ＝３００Ｋꎮ为方便对比分析ꎬ计算结果与参考结果均采用上下对称形式布置ꎬ例如ꎬ图４是模型下表面计算结果与实验结果对比:对于下表面两侧转捩的起始位置ꎬ高超声速修正前的转捩位置在ｘ＝０.６８ｍ附近ꎬ高超声速修正后的计算结果与实验结果吻合良好ꎬ均在ｘ＝０.６０ｍ附近ꎬ并且湍流边界层区域形状基本一致ꎬ说明修正后的转捩模型能够较好地预测ＨｙＴＲＶ转捩的位置ꎮ(ａ)Ｃａｌｃｕｌａｔｉｏｎｏｆｔｈｅｆｒｉｃｔｉｏｎｄｉｓｔｒｉｂｕｔｉｏｎ(ｂｅｆｏｒｅｈｙｐｅｒｓｏｎｉｃｃｏｒｒｅｃｔｉｏｎ)(ｂ)Ｃａｌｃｕｌａｔｉｏｎｏｆｔｈｅｆｒｉｃｔｉｏｎｄｉｓｔｒｉｂｕｔｉｏｎ(ａｆｔｅｒｈｙｐｅｒｓｏｎｉｃｃｏｒｒｅｃｔｉｏｎ)(ｃ)Ｅｘｐｅｒｉｍｅｎｔａｌｒｅｓｕｌｔｓｏｆｔｈｅｈｅａｔｆｌｕｘｄｉｓｔｒｉｂｕｔｉｏｎ[１７]图４㊀下表面计算结果和实验结果对比Ｆｉｇ.４㊀Ｃｏｍｐａｒｉｓｏｎｏｆｔｈｅｃａｌｃｕｌａｔｅｄａｎｄｅｘｐｅｒｉｍｅｎｔａｌｒｅｓｕｌｔｓｏｎｔｈｅｌｏｗｅｒｓｕｒｆａｃｅ３㊀ＨｙＴＲＶ转捩的基本流动特性计算工况采用Ｍａ＝６ꎬ攻角α＝０ʎꎬ来流湍流度ＦＳＴＩ＝０.６％ꎬ分析ＨｙＴＲＶ转捩的基本流动特性ꎮ从图５可以看出ꎬ模型两侧和顶端均出现高压区ꎬ高压区之间为低压区ꎬ横截面上存在周向压力梯度ꎬ流动从高压区向低压区汇集ꎬ从而在下表面中心线附近和上表面两侧腰部区域均形成流向涡结构(见图６)ꎬ沿流动方向ꎬ高压区域逐渐扩大ꎬ流向涡结构的影响范围也越大ꎮ在流向涡结构的边缘位置ꎬ壁面附近的低速流体被抬升到外壁面区域ꎬ外壁面区域的高速流体又被带入到近壁面区域ꎬ进而导致流向涡结构边缘处壁面的摩阻显著增加ꎬ最终诱发转捩ꎬ这些流动特征与文献[１５]的结果一致ꎮ图７显示了上下表面摩阻的分布情况ꎬ其中上表面两侧区域在ｘ/Ｌ＝０.８０附近ꎬ摩阻显著增加ꎬ出现明显的转捩现象ꎬ转捩区域分布在两侧边缘位置ꎻ而下表面两侧区域在ｘ/Ｌ＝０.７５附近ꎬ也出现明显的转捩ꎬ转捩区域相对集中在中心线两侧ꎮ图５㊀不同截面位置处的压力云图Ｆｉｇ.５㊀Ｐｒｅｓｓｕｒｅｃｏｎｔｏｕｒｓａｔｄｉｆｆｅｒｅｎｔｃｒｏｓｓ￣ｓｅｃｔｉｏｎｌｏｃａｔｉｏｎｓ图６㊀不同截面位置处的流向速度云图Ｆｉｇ.６㊀Ｓｔｒｅａｍｗｉｓｅｖｅｌｏｃｉｔｙｃｏｎｔｏｕｒｓａｔｄｉｆｆｅｒｅｎｔｃｒｏｓｓ￣ｓｅｃｔｉｏｎｌｏｃａｔｉｏｎｓ３１气体物理２０２４年㊀第９卷(ａ)Ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀(ｂ)Ｌｏｗｅｒｓｕｒｆａｃｅ图７㊀上下表面摩阻分布云图Ｆｉｇ.７㊀Ｆｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｃｏｎｔｏｕｒｓｏｎｔｈｅｕｐｐｅｒａｎｄｌｏｗｅｒｓｕｒｆａｃｅｓ４㊀不同Ｍａｃｈ数对ＨｙＴＲＶ转捩的影响保持来流湍流度ＦＳＴＩ＝０.６％不变ꎬＭａｃｈ数变化范围为３~８ꎮ图８是不同Ｍａｃｈ数条件下ＨｙＴＲＶ上下表面的摩阻分布云图ꎬ从图中可知ꎬ随着Ｍａｃｈ数的增加ꎬ上下表面的湍流区域均逐渐减少ꎬ其中上表面两侧转捩起始位置由ｘ/Ｌ＝０.５６附近后移至ｘ/Ｌ＝０.９２附近ꎬ下表面两侧转捩起始位置由ｘ/Ｌ＝０.４８附近后移至ｘ/Ｌ＝０.９９附近ꎬ上下表面两侧转捩起始位置均大幅后移ꎬ说明Ｍａｃｈ数对ＨｙＴＲＶ转捩的影响很大ꎮｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ａ)Ｍａ＝３ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｂ)Ｍａ＝４ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｃ)Ｍａ＝５４１第２期章录兴ꎬ等:Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩的影响ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｄ)Ｍａ＝６ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｅ)Ｍａ＝７ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｆ)Ｍａ＝８图８㊀不同Ｍａｃｈ数条件下摩阻系数分布云图Ｆｉｇ.８㊀ＦｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｃｏｎｔｏｕｒｓａｔｄｉｆｆｅｒｅｎｔＭａｃｈｎｕｍｂｅｒｓ上表面选取图７中ｚ/Ｌ＝０.１２的位置ꎬ下表面选取ｚ/Ｌ＝０.１０的位置进行分析ꎮ从图９中可以分析出ꎬ随着Ｍａｃｈ数的增加ꎬ上表面转捩起始位置不断后移ꎬ当Ｍａｃｈ数增加到７时ꎬ由于湍流区的缩小ꎬ此处位置不再发生转捩ꎬ此外ꎬＭａｃｈ数越高层流区摩阻系数越大ꎻ下表面转捩起始位置也不断后移ꎬ当Ｍａｃｈ数增加到８时ꎬ此处位置不再发生转捩ꎬ此外ꎬＭａｃｈ数越高ꎬ湍流区的摩阻系数越小ꎬ这些结论与关于来流Ｍａｃｈ数对转捩位置影响的普遍研究结论一致ꎮ(ａ)Ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀(ｂ)Ｌｏｗｅｒｓｕｒｆａｃｅ图９㊀不同位置摩阻系数随Ｍａｃｈ数的变化Ｆｉｇ.９㊀ＶａｒｉａｔｉｏｎｏｆｆｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｗｉｔｈＭａｃｈｎｕｍｂｅｒａｔｄｉｆｆｅｒｅｎｔｌｏｃａｔｉｏｎｓ５１气体物理２０２４年㊀第９卷５㊀不同壁面温度对ＨｙＴＲＶ转捩的影响保持来流湍流度ＦＳＴＩ＝０.６％及Ｍａ＝６不变ꎬ壁面温度的变化范围为１５０~９００Ｋꎮ图１０是不同壁面温度条件下ＨｙＴＲＶ上下表面的摩阻分布云图ꎬ可以看出随着壁面温度的增加ꎬ上表面两侧湍流区域先是缓慢扩大ꎬ在壁面温度为５００Ｋ时湍流区域快速缩小ꎬ增加到９００Ｋ时ꎬ已无明显湍流区域ꎻ下表面两侧湍流区域先是无明显变化ꎬ同样当壁面温度升高到５００Ｋ时ꎬ湍流区域快速缩小ꎬ当壁面温度升高到７００Ｋ时ꎬ两侧已经无明显的湍流区域ꎬ相比上表面两侧湍流区域ꎬ下表面湍流区域消失得更早ꎮ由此可以得出壁面温度对转捩的产生有较大的影响ꎬ壁面温度增加到一定程度将导致ＨｙＴＲＶ没有明显的转捩现象ꎮｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ａ)Ｔ＝１５０Ｋｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｂ)Ｔ＝２００Ｋｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｃ)Ｔ＝３００Ｋｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｄ)Ｔ＝５００Ｋ６１第２期章录兴ꎬ等:Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩的影响ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｅ)Ｔ＝７００Ｋｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀ｌｏｗｅｒｓｕｒｆａｃｅ(ｆ)Ｔ＝９００Ｋ图１０㊀不同壁面温度条件下摩阻系数分布云图Ｆｉｇ.１０㊀Ｆｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｃｏｎｔｏｕｒｓａｔｄｉｆｆｅｒｅｎｔｗａｌｌｔｅｍｐｅｒａｔｕｒｅｃｏｎｄｉｔｉｏｎｓ上表面选取ｚ/Ｌ＝０.１２５的位置ꎬ下表面选取ｚ/Ｌ＝０.１００的位置进行分析ꎮ从图１１中可以分析出ꎬ随着壁面温度的增加ꎬ上表面转捩起始位置先前移ꎬ当壁面温度增加到５００Ｋ时ꎬ转捩起始位置后移ꎬ转捩区长度逐渐增加ꎬ层流区域的摩阻系数逐渐增加ꎬ当壁面温度增加到７００Ｋ时ꎬ该位置已不再出现转捩ꎻ下表面转捩起始位置先小幅后移ꎬ当壁面温度增加到３００Ｋ时ꎬ转捩起始位置开始后移ꎬ当壁面温度增加到７００Ｋ时ꎬ由于湍流区域的减小ꎬ该位置不再发生转捩ꎮ(ａ)Ｕｐｐｅｒｓｕｒｆａｃｅ㊀㊀㊀㊀㊀(ｂ)Ｌｏｗｅｒｓｕｒｆａｃｅ图１１㊀不同位置摩阻系数随壁面温度的变化Ｆｉｇ.１１㊀Ｖａｒｉａｔｉｏｎｏｆｆｒｉｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｗｉｔｈｗａｌｌｔｅｍｐｅｒａｔｕｒｅａｔｄｉｆｆｅｒｅｎｔｌｏｃａｔｉｏｎｓ为进一步分析壁面温度的影响ꎬ本文分别在上下表面湍流区选取一点(０.９ꎬ０.０２９ꎬ０.１４)ꎬ(０.９７ꎬ－０.３４ꎬ０.１２)ꎬ分析边界层湍动能剖面ꎬ结果如图１２所示ꎮ从图中可以看到ꎬ随着壁面温度升高ꎬ边界层厚度先略微变厚ꎬ再变薄ꎬ当壁面温度升高到７００Ｋ时ꎬ边界层厚度迅速降低ꎮ这些结果与转捩位置先前移再后移的结论相符合ꎬ因为边界层厚度会影响不稳定波的时间和空间尺度ꎬ边界层厚度低时ꎬ不稳定波增长速度变慢ꎬ延迟转捩发生ꎮ需要指出的是ꎬ仅采用当前使用的方法ꎬ无法从更深层７１气体物理２０２４年㊀第９卷次揭示转捩反转的流动机理ꎬ而须另外借助稳定性分析方法ꎬ例如ꎬ使用ｅＮ方法开展基于模态的稳定性研究ꎮ文献[３３]采用该手段研究了大掠角平板钝三角翼随壁温比变化出现转捩反转的内在机理:壁温比升高促进横流模态和第１模态扰动增长ꎬ抑制第２模态发展ꎬ在第１㊁２模态联合作用影响下ꎬ出现转捩反转现象ꎮ我们将在后续开展进一步研究ꎮ(ａ)Ｕｐｐｅｒｓｕｒｆａｃｅ(ｂ)Ｌｏｗｅｒｓｕｒｆａｃｅ图１２㊀不同位置湍动能剖面随壁面温度的变化Ｆｉｇ.１２㊀Ｖａｒｉａｔｉｏｎｏｆｔｕｒｂｕｌｅｎｔｋｉｎｅｔｉｃｅｎｅｒｇｙｗｉｔｈｗａｌｌｔｅｍｐｅｒａｔｕｒｅａｔｄｉｆｆｅｒｅｎｔｌｏｃａｔｉｏｎｓ６㊀结论针对ＨｙＴＲＶ转捩问题ꎬ在Ｍａｃｈ数Ｍａ＝３~８ꎬ壁面温度Ｔ＝１５０~９００Ｋ的条件下ꎬ基于课题组自研软件ꎬ对γ￣Ｒｅ~θｔ转捩模型和ＳＳＴ湍流模型进行了高超声速修正ꎬ研究了Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ转捩的影响ꎬ得出以下结论:１)经过高超声速修正后的γ￣Ｒｅ~θｔ转捩模型和ＳＳＴ湍流模型能够较为准确地预测ＨｙＴＲＶ转捩位置ꎬ并且湍流边界层区域形状与实验结果基本一致ꎻＨｙＴＲＶ存在多个不同的转捩区域ꎬ上表面两侧转捩区域分布在两侧边缘位置ꎬ下表面两侧转捩区域分布在中心线两侧ꎮ２)Ｍａｃｈ数的增加会导致上下表面转捩起始位置均大幅后移ꎬ湍流区大幅缩小ꎬ但当Ｍａｃｈ数增加到８时ꎬ湍流区仍然存在ꎬ并没有消失ꎻ上表面层流区摩阻不断增加ꎬ下表面湍流区摩阻不断减小ꎮ３)壁面温度的增加会导致上下表面转捩起始位置先前移ꎬ再后移ꎬ这与边界层厚度变化规律一致ꎬ当壁面温度增加到７００Ｋ时ꎬ下表面湍流区已经基本消失ꎬ当壁面温度增加到９００Ｋ时ꎬ上表面湍流区也基本消失ꎻ上表面在层流区域的摩阻系数逐渐增大ꎬ在湍流区的摩阻系数逐渐减小ꎮ致谢㊀感谢中国空气动力研究与发展中心和空天飞行空气动力科学与技术全国重点实验室提供的ＨｙＴＲＶ模型数据和实验数据ꎮ参考文献(Ｒｅｆｅｒｅｎｃｅｓ)[１]㊀ＯｂｅｒｉｎｇＩＩＩＨꎬＨｅｉｎｒｉｃｈｓＲＬ.Ｍｉｓｓｉｌｅｄｅｆｅｎｓｅｆｏｒｇｒｅａｔｐｏｗｅｒｃｏｎｆｌｉｃｔ:ｏｕｔｍａｎｅｕｖｅｒｉｎｇｔｈｅＣｈｉｎａｔｈｒｅａｔ[Ｊ].Ｓｔｒａ￣ｔｅｇｉｃＳｔｕｄｉｅｓＱｕａｒｔｅｒｌｙꎬ２０１９ꎬ３(４):３７￣５６. [２]ＣｈｅｎｇＣꎬＷｕＪＨꎬＺｈａｎｇＹＬꎬｅｔａｌ.Ａｅｒｏｄｙｎａｍｉｃｓａｎｄｄｙｎａｍｉｃｓｔａｂｉｌｉｔｙｏｆｍｉｃｒｏ￣ａｉｒ￣ｖｅｈｉｃｌｅｗｉｔｈｆｏｕｒｆｌａｐｐｉｎｇｗｉｎｇｓｉｎｈｏｖｅｒｉｎｇｆｌｉｇｈｔ[Ｊ].ＡｄｖａｎｃｅｓｉｎＡｅｒｏｄｙｎａｍｉｃｓꎬ２０２０ꎬ２(３):５.[３]ＢｅｒｔｉｎＪＪꎬＣｕｍｍｉｎｇｓＲＭ.Ｆｉｆｔｙｙｅａｒｓｏｆｈｙｐｅｒｓｏｎｉｃｓ:ｗｈｅｒｅｗｅᶄｖｅｂｅｅｎꎬｗｈｅｒｅｗｅᶄｒｅｇｏｉｎｇ[Ｊ].ＰｒｏｇｒｅｓｓｉｎＡｅｒｏｓｐａｃｅＳｃｉｅｎｃｅｓꎬ２００３ꎬ３９(６/７):５１１￣５３６. [４]陈坚强ꎬ涂国华ꎬ张毅锋ꎬ等.高超声速边界层转捩研究现状与发展趋势[Ｊ].空气动力学学报ꎬ２０１７ꎬ３５(３):３１１￣３３７.ＣｈｅｎＪＱꎬＴｕＧＨꎬＺｈａｎｇＹＦꎬｅｔａｌ.Ｈｙｐｅｒｓｎｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎ:ｗｈａｔｗｅｋｎｏｗꎬｗｈｅｒｅｓｈａｌｌｗｅｇｏ[Ｊ].ＡｃｔａＡｅｒｏｄｙｎａｍｉｃａＳｉｎｉｃａꎬ２０１７ꎬ３５(３):３１１￣３３７(ｉｎＣｈｉｎｅｓｅ).[５]段毅ꎬ姚世勇ꎬ李思怡ꎬ等.高超声速边界层转捩的若干问题及工程应用研究进展综述[Ｊ].空气动力学学报ꎬ２０２０ꎬ３８(２):３９１￣４０３.ＤｕａｎＹꎬＹａｏＳＹꎬＬｉＳＹꎬｅｔａｌ.Ｒｅｖｉｅｗｏｆｐｒｏｇｒｅｓｓｉｎｓｏｍｅｉｓｓｕｅｓａｎｄｅｎｇｉｎｅｅｒｉｎｇａｐｐｌｉｃａｔｉｏｎｏｆｈｙｐｅｒｓｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎ[Ｊ].ＡｃｔａＡｅｒｏｄｙｎａｍｉｃａＳｉｎｉｃａꎬ２０２０ꎬ３８(２):３９１￣４０３(ｉｎＣｈｉｎｅｓｅ).[６]ＺｈａｎｇＹＦꎬＺｈａｎｇＹＲꎬＣｈｅｎＪＱꎬｅｔａｌ.Ｎｕｍｅｒｉｃａｌｓｉ￣８１第２期章录兴ꎬ等:Ｍａｃｈ数和壁面温度对ＨｙＴＲＶ边界层转捩的影响ｍｕｌａｔｉｏｎｓｏｆｈｙｐｅｒｓｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎｂａｓｅｄｏｎｔｈｅｆｌｏｗｓｏｌｖｅｒｃｈａｎｔ２.０[Ｒ].ＡＩＡＡ２０１７￣２４０９ꎬ２０１７. [７]ＫｒａｕｓｅＭꎬＢｅｈｒＭꎬＢａｌｌｍａｎｎＪ.Ｍｏｄｅｌｉｎｇｏｆｔｒａｎｓｉｔｉｏｎｅｆｆｅｃｔｓｉｎｈｙｐｅｒｓｏｎｉｃｉｎｔａｋｅｆｌｏｗｓｕｓｉｎｇａｃｏｒｒｅｌａｔｉｏｎ￣ｂａｓｅｄｉｎｔｅｒｍｉｔｔｅｎｃｙｍｏｄｅｌ[Ｒ].ＡＩＡＡ２００８￣２５９８ꎬ２００８. [８]ＹｉＭＲꎬＺｈａｏＨＹꎬＬｅＪＬ.Ｈｙｐｅｒｓｏｎｉｃｎａｔｕｒａｌａｎｄｆｏｒｃｅｄｔｒａｎｓｉｔｉｏｎｓｉｍｕｌａｔｉｏｎｂｙｃｏｒｒｅｌａｔｉｏｎ￣ｂａｓｅｄｉｎｔｅｒｍｉｔ￣ｔｅｎｃｙ[Ｒ].ＡＩＡＡ２０１７￣２３３７ꎬ２０１７.[９]向星皓ꎬ张毅锋ꎬ袁先旭ꎬ等.Ｃ￣γ￣Ｒｅθ高超声速三维边界层转捩预测模型[Ｊ].航空学报ꎬ２０２１ꎬ４２(９):６２５７１１.ＸｉａｎｇＸＨꎬＺｈａｎｇＹＦꎬＹｕａｎＸＸꎬｅｔａｌ.Ｃ￣γ￣Ｒｅθｍｏｄｅｌｆｏｒｈｙｐｅｒｓｏｎｉｃｔｈｒｅｅ￣ｄｉｍｅｎｓｉｏｎａｌｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎｐｒｅｄｉｃｔｉｏｎ[Ｊ].ＡｃｔａＡｅｒｏｎａｕｔｉｃａｅｔＡｓｔｒｏｎａｕｔｉｃａＳｉｎｉｃａꎬ２０２１ꎬ４２(９):６２５７１１(ｉｎＣｈｉｎｅｓｅ).[１０]ＭｃＤａｎｉｅｌＲＤꎬＮａｎｃｅＲＰꎬＨａｓｓａｎＨＡ.Ｔｒａｎｓｉｔｉｏｎｏｎｓｅｔｐｒｅｄｉｃｔｉｏｎｆｏｒｈｉｇｈ￣ｓｐｅｅｄｆｌｏｗ[Ｊ].ＪｏｕｒｎａｌｏｆＳｐａｃｅｃｒａｆｔａｎｄＲｏｃｋｅｔｓꎬ２０００ꎬ３７(３):３０４￣３０９.[１１]ＰａｐｐＪＬꎬＤａｓｈＳＭ.Ｒａｐｉｄｅｎｇｉｎｅｅｒｉｎｇａｐｐｒｏａｃｈｔｏｍｏｄｅｌｉｎｇｈｙｐｅｒｓｏｎｉｃｌａｍｉｎａｒ￣ｔｏ￣ｔｕｒｂｕｌｅｎｔｔｒａｎｓｉｔｉｏｎａｌｆｌｏｗｓ[Ｊ].ＪｏｕｒｎａｌｏｆＳｐａｃｅｃｒａｆｔａｎｄＲｏｃｋｅｔｓꎬ２００５ꎬ４２(３):４６７￣４７５.[１２]ＪｕｌｉａｎｏＴＪꎬＳｃｈｎｅｉｄｅｒＳＰ.ＩｎｓｔａｂｉｌｉｔｙａｎｄｔｒａｎｓｉｔｉｏｎｏｎｔｈｅＨＩＦｉＲＥ￣５ｉｎａＭａｃｈ￣６ｑｕｉｅｔｔｕｎｎｅｌ[Ｒ].ＡＩＡＡ２０１０￣５００４ꎬ２０１０.[１３]杨云军ꎬ马汉东ꎬ周伟江.高超声速流动转捩的数值研究[Ｊ].宇航学报ꎬ２００６ꎬ２７(１):８５￣８８.ＹａｎｇＹＪꎬＭａＨＤꎬＺｈｏｕＷＪ.Ｎｕｍｅｒｉｃａｌｒｅｓｅａｒｃｈｏｎｓｕｐｅｒｓｏｎｉｃｆｌｏｗｔｒａｎｓｉｔｉｏｎ[Ｊ].ＪｏｕｒｎａｌｏｆＡｓｔｒｏｎａｕｔｉｃｓꎬ２００６ꎬ２７(１):８５￣８８(ｉｎＣｈｉｎｅｓｅ).[１４]袁先旭ꎬ何琨ꎬ陈坚强ꎬ等.ＭＦ￣１模型飞行试验转捩结果初步分析[Ｊ].空气动力学学报ꎬ２０１８ꎬ３６(２):２８６￣２９３.ＹｕａｎＸＸꎬＨｅＫꎬＣｈｅｎＪＱꎬｅｔａｌ.ＰｒｅｌｉｍｉｎａｒｙｔｒａｎｓｉｔｉｏｎｒｅｓｅａｒｃｈａｎａｌｙｓｉｓｏｆＭＦ￣１[Ｊ].ＡｃｔａＡｅｒｏｄｙｎａｍｉｃａＳｉｎｉｃａꎬ２０１８ꎬ３６(２):２８６￣２９３(ｉｎＣｈｉｎｅｓｅ).[１５]陈坚强ꎬ涂国华ꎬ万兵兵ꎬ等.ＨｙＴＲＶ流场特征与边界层稳定性特征分析[Ｊ].航空学报ꎬ２０２１ꎬ４２(６):１２４３１７.ＣｈｅｎＪＱꎬＴｕＧＨꎬＷａｎＢＢꎬｅｔａｌ.Ｃｈａｒａｃｔｅｒｉｓｔｉｃｓｏｆｆｌｏｗｆｉｅｌｄａｎｄｂｏｕｎｄａｒｙ￣ｌａｙｅｒｓｔａｂｉｌｉｔｙｏｆＨｙＴＲＶ[Ｊ].ＡｃｔａＡｅｒｏｎａｕｔｉｃａｅｔＡｓｔｒｏｎａｕｔｉｃａＳｉｎｉｃａꎬ２０２１ꎬ４２(６):１２４３１７(ｉｎＣｈｉｎｅｓｅ).[１６]陈坚强ꎬ刘深深ꎬ刘智勇ꎬ等.用于高超声速边界层转捩研究的标模气动布局及设计方法.中国:１０９９６９３７４Ｂ[Ｐ].２０２１￣０５￣１８.ＣｈｅｎＪＱꎬＬｉｕＳＳꎬＬｉｕＺＹꎬｅｔａｌ.Ｓｔａｎｄａｒｄｍｏｄｅｌａｅｒｏ￣ｄｙｎａｍｉｃｌａｙｏｕｔａｎｄｄｅｓｉｇｎｍｅｔｈｏｄｆｏｒｈｙｐｅｒｓｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎｒｅｓｅａｒｃｈ.ＣＮꎬ１０９９６９３７４Ｂ[Ｐ].２０２１￣０５￣１８(ｉｎＣｈｉｎｅｓｅ).[１７]ＬｉｕＳＳꎬＹｕａｎＸＸꎬＬｉｕＺＹꎬｅｔａｌ.Ｄｅｓｉｇｎａｎｄｔｒａｎｓｉｔｉｏｎｃｈａｒａｃｔｅｒｉｓｔｉｃｓｏｆａｓｔａｎｄａｒｄｍｏｄｅｌｆｏｒｈｙｐｅｒｓｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎｒｅｓｅａｒｃｈ[Ｊ].ＡｃｔａＭｅｃｈａｎｉｃａＳｉｎｉｃａꎬ２０２１ꎬ３７(１１):１６３７￣１６４７.[１８]ＣｈｅｎＸꎬＤｏｎｇＳＷꎬＴｕＧＨꎬｅｔａｌ.Ｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎ￣ｓｉｔｉｏｎａｎｄｌｉｎｅａｒｍｏｄａｌｉｎｓｔａｂｉｌｉｔｉｅｓｏｆｈｙｐｅｒｓｏｎｉｃｆｌｏｗｏｖｅｒａｌｉｆｔｉｎｇｂｏｄｙ[Ｊ].ＪｏｕｒｎａｌｏｆＦｌｕｉｄＭｅｃｈａｎｉｃｓꎬ２０２２ꎬ９３８(４０８):Ａ８.[１９]ＱｉＨꎬＬｉＸＬꎬＹｕＣＰꎬｅｔａｌ.Ｄｉｒｅｃｔｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎｏｆｈｙｐｅｒｓｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎｏｖｅｒａｌｉｆｔｉｎｇ￣ｂｏｄｙｍｏｄｅｌＨｙＴＲＶ[Ｊ].ＡｄｖａｎｃｅｓｉｎＡｅｒｏｄｙｎａｍｉｃｓꎬ２０２１ꎬ３(１):３１.[２０]万兵兵ꎬ陈曦ꎬ陈坚强ꎬ等.三维边界层转捩预测ＨｙＴＥＮ软件在高超声速典型标模中的应用[Ｊ].空天技术ꎬ２０２３(１):１５０￣１５８.ＷａｎＢＢꎬＣｈｅｎＸꎬＣｈｅｎＪＱꎬｅｔａｌ.ＡｐｐｌｉｃａｔｉｏｎｓｏｆＨｙＴＥＮｓｏｆｔｗａｒｅｆｏｒｐｒｅｄｉｃｔｉｎｇｔｈｒｅｅ￣ｄｉｍｅｎｓｉｏｎａｌｂｏｕｎｄａｒｙ￣ｌａｙｅｒｔｒａｎｓｉｔｉｏｎｉｎｔｙｐｉｃａｌｈｙｐｅｒｓｏｎｉｃｍｏｄｅｌｓ[Ｊ].ＡｅｒｏｓｐａｃｅＴｅｃｈｎｏｌｏｇｙꎬ２０２３(１):１５０￣１５８(ｉｎＣｈｉｎｅｓｅ). [２１]ＭｅｎｔｅｒＦＲꎬＬａｎｇｔｒｙＲＢꎬＬｉｋｋｉＳＲꎬｅｔａｌ.Ａｃｏｒｒｅｌａｔｉｏｎ￣ｂａｓｅｄｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｕｓｉｎｇｌｏｃａｌｖａｒｉａｂｌｅｓ ＰａｒｔⅠ:ｍｏｄｅｌｆｏｒｍｕｌａｔｉｏｎ[Ｊ].ＪｏｕｒｎａｌｏｆＴｕｒｂｏｍａｃｈｉｎｅｒｙꎬ２００６ꎬ１２８(３):４１３￣４２２.[２２]ＬａｎｇｔｒｙＲＢꎬＭｅｎｔｅｒＦＲꎬＬｉｋｋｉＳＲꎬｅｔａｌ.Ａｃｏｒｒｅｌａｔｉｏｎ￣ｂａｓｅｄｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｕｓｉｎｇｌｏｃａｌｖａｒｉａｂｌｅｓ ＰａｒｔⅡ:ｔｅｓｔｃａｓｅｓａｎｄｉｎｄｕｓｔｒｉａｌａｐｐｌｉｃａｔｉｏｎｓ[Ｊ].ＪｏｕｒｎａｌｏｆＴｕｒ￣ｂｏｍａｃｈｉｎｅｒｙꎬ２００６ꎬ１２８(３):４２３￣４３４.[２３]ＬａｎｇｔｒｙＲＢꎬＭｅｎｔｅｒＦＲ.Ｃｏｒｒｅｌａｔｉｏｎ￣ｂａｓｅｄｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｉｎｇｆｏｒｕｎｓｔｒｕｃｔｕｒｅｄｐａｒａｌｌｅｌｉｚｅｄｃｏｍｐｕｔａｔｉｏｎａｌｆｌｕｉｄｄｙｎａｍｉｃｓｃｏｄｅｓ[Ｊ].ＡＩＡＡＪｏｕｒｎａｌꎬ２００９ꎬ４７(１２):２８９４￣２９０６.[２４]孟德虹ꎬ张玉伦ꎬ王光学ꎬ等.γ￣Ｒｅθ转捩模型在二维低速问题中的应用[Ｊ].航空学报ꎬ２０１１ꎬ３２(５):７９２￣８０１.ＭｅｎｇＤＨꎬＺｈａｎｇＹＬꎬＷａｎｇＧＸꎬｅｔａｌ.Ａｐｐｌｉｃａｔｉｏｎｏｆγ￣Ｒｅθｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｔｏｔｗｏ￣ｄｉｍｅｎｓｉｏｎａｌｌｏｗｓｐｅｅｄｆｌｏｗｓ[Ｊ].ＡｃｔａＡｅｒｏｎａｕｔｉｃａｅｔＡｓｔｒｏｎａｕｔｉｃａＳｉｎｉｃａꎬ２０１１ꎬ３２(５):７９２￣８０１(ｉｎＣｈｉｎｅｓｅ).[２５]牟斌ꎬ江雄ꎬ肖中云ꎬ等.γ￣Ｒｅθ转捩模型的标定与应用[Ｊ].空气动力学学报ꎬ２０１３ꎬ３１(１):１０３￣１０９.ＭｏｕＢꎬＪｉａｎｇＸꎬＸｉａｏＺＹꎬｅｔａｌ.Ｉｍｐｌｅｍｅｎｔａｔｉｏｎａｎｄｃａｌｉｂｅｒａｔｉｏｎｏｆγ￣Ｒｅθｔｒａｎｓｉｔｉｏｎｍｏｄｅｌ[Ｊ].ＡｃｔａＡｅｒｏｄｙ￣ｎａｍｉｃａＳｉｎｉｃａꎬ２０１３ꎬ３１(１):１０３￣１０９(ｉｎＣｈｉｎｅｓｅ). [２６]郭隽ꎬ刘丽平ꎬ徐晶磊ꎬ等.γ￣Ｒｅ~θｔ转捩模型在跨声速涡轮叶栅中的应用[Ｊ].推进技术ꎬ２０１８ꎬ３９(９):１９９４￣２００１.９１气体物理２０２４年㊀第９卷ＧｕｏＪꎬＬｉｕＬＰꎬＸｕＪＬꎬｅｔａｌ.Ａｐｐｌｉｃａｔｉｏｎｏｆγ￣Ｒｅ~θｔｔｒａｎ￣ｓｉｔｉｏｎｍｏｄｅｌｉｎｔｒａｎｓｏｎｉｃｔｕｒｂｉｎｅｃａｓｃａｄｅｓ[Ｊ].ＪｏｕｒｎａｌｏｆＰｒｏｐｕｌｓｉｏｎＴｅｃｈｎｏｌｏｇｙꎬ２０１８ꎬ３９(９):１９９４￣２００１(ｉｎＣｈｉｎｅｓｅ).[２７]郑赟ꎬ李虹杨ꎬ刘大响.γ￣Ｒｅθ转捩模型在高超声速下的应用及分析[Ｊ].推进技术ꎬ２０１４ꎬ３５(３):２９６￣３０４.ＺｈｅｎｇＹꎬＬｉＨＹꎬＬｉｕＤＸ.Ａｐｐｌｉｃａｔｉｏｎａｎｄａｎａｌｙｓｉｓｏｆγ￣Ｒｅθｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｉｎｈｙｐｅｒｓｏｎｉｃｆｌｏｗ[Ｊ].ＪｏｕｒｎａｌｏｆＰｒｏｐｕｌｓｉｏｎＴｅｃｈｎｏｌｏｇｙꎬ２０１４ꎬ３５(３):２９６￣３０４(ｉｎＣｈｉ￣ｎｅｓｅ).[２８]孔维萱ꎬ阎超ꎬ赵瑞.γ￣Ｒｅθ模式应用于高速边界层转捩的研究[Ｊ].空气动力学学报ꎬ２０１３ꎬ３１(１):１２０￣１２６.ＫｏｎｇＷＸꎬＹａｎＣꎬＺｈａｏＲ.γ￣Ｒｅθｍｏｄｅｌｒｅｓｅａｒｃｈｆｏｒｈｉｇｈ￣ｓｐｅｅｄｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎ[Ｊ].ＡｃｔａＡｅｒｏｄｙ￣ｎａｍｉｃａＳｉｎｉｃａꎬ２０１３ꎬ３１(１):１２０￣１２６(ｉｎＣｈｉｎｅｓｅ). [２９]张毅锋ꎬ何琨ꎬ张益荣ꎬ等.Ｍｅｎｔｅｒ转捩模型在高超声速流动模拟中的改进及验证[Ｊ].宇航学报ꎬ２０１６ꎬ３７(４):３９７￣４０２.ＺｈａｎｇＹＦꎬＨｅＫꎬＺｈａｎｇＹＲꎬｅｔａｌ.ＩｍｐｒｏｖｅｍｅｎｔａｎｄｖａｌｉｄａｔｉｏｎｏｆＭｅｎｔｅｒᶄｓｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｆｏｒｈｙｐｅｒｓｏｎｉｃｆｌｏｗｓｉｍｕｌａｔｉｏｎ[Ｊ].ＪｏｕｒｎａｌｏｆＡｓｔｒｏｎａｕｔｉｃｓꎬ２０１６ꎬ３７(４):３９７￣４０２(ｉｎＣｈｉｎｅｓｅ).[３０]ＬａｎｇｔｒｙＲＢꎬＳｅｎｇｕｐｔａＫꎬＹｅｈＤＴꎬｅｔａｌ.Ｅｘｔｅｎｄｉｎｇｔｈｅγ￣Ｒｅθｔｃｏｒｒｅｌａｔｉｏｎｂａｓｅｄｔｒａｎｓｉｔｉｏｎｍｏｄｅｌｆｏｒｃｒｏｓｓｆｌｏｗｅｆｆｅｃｔｓ(Ｉｎｖｉｔｅｄ)[Ｒ].ＡＩＡＡ２０１５￣２４７４ꎬ２０１５. [３１]ＳａｒｋａｒＳ.Ｔｈｅｐｒｅｓｓｕｒｅ￣ｄｉｌａｔａｔｉｏｎｃｏｒｒｅｌａｔｉｏｎｉｎｃｏｍｐｒｅｓｓｉ￣ｂｌｅｆｌｏｗｓ[Ｊ].ＰｈｙｓｉｃｓｏｆＦｌｕｉｄｓＡꎬ１９９２ꎬ４(１２):２６７４￣２６８２.[３２]ＷｉｌｃｏｘＤＣ.Ｄｉｌａｔａｔｉｏｎ￣ｄｉｓｓｉｐａｔｉｏｎｃｏｒｒｅｃｔｉｏｎｓｆｏｒａｄｖａｎｃｅｄｔｕｒｂｕｌｅｎｃｅｍｏｄｅｌｓ[Ｊ].ＡＩＡＡＪｏｕｒｎａｌꎬ１９９２ꎬ３０(１１):２６３９￣２６４６.[３３]马祎蕾ꎬ余平ꎬ姚世勇.壁温对钝三角翼边界层稳定性及转捩影响[Ｊ].空气动力学学报ꎬ２０２０ꎬ３８(６):１０１７￣１０２６.ＭａＹＬꎬＹｕＰꎬＹａｏＳＹ.Ｅｆｆｅｃｔｏｆｗａｌｌｔｅｍｐｅｒａｔｕｒｅｏｎｓｔａｂｉｌｉｔｙａｎｄｔｒａｎｓｉｔｉｏｎｏｆｈｙｐｅｒｓｏｎｉｃｂｏｕｎｄａｒｙｌａｙｅｒｏｎａｂｌｕｎｔｄｅｌｔａｗｉｎｇ[Ｊ].ＡｃｔａＡｅｒｏｄｙｎａｍｉｃａＳｉｎｉｃａꎬ２０２０ꎬ３８(６):１０１７￣１０２６(ｉｎＣｈｉｎｅｓｅ).０２。