In this paper,focusing of a toroidal shock wave propagating from a shock tube of an- nular cross-section into a cylindrical chamber was investigated numerically with the dispersion- controlled scheme. For CFD validation, the numerical code was rst applied to calculate both viscous and inviscid ows at a low Mach number of 1.5, which was compared with the experi- ment results and got better consistency. Then the validated code was used to calculate several cases for high Mach numbers. From the result, several major factors that in uent the ow, such as the Mach number and the viscosity, were analyzed detailedly and along with the high Mach number some unusual ow structure was observed and explained theoreticall

Han Z

Hossein SHR.

Jiang ZL(姜宗林)

Takayama K

Teng HH(滕宏辉)

滕宏辉

Institute Of Mechanics,Chinese Academy of Sciences

Numerical investigation of toroidal shock wavesfocusingH.H. Teng", Z. Jiang?, Z.Y. Han", S.H.R. Hosseini", and K. Takayama''1 LHD, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China2 Dept. of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui230026, China3 Shock Wave Research Center, Institute of Fluid Science, Tohoku University, Sendai980-8577, JapanAbstract. In this paper, focusing of a toroidal shock wave propagating from a shock tube of an-nular cross-section into a cylindrical chamber was investigated numerically with th e dispersion-controlled scheme. For CFD validation, the numerical code was first applied to calculate bothviscous and inviscid flows at a low Mach number of 1.5, which was compared with the experi-ment results and got better consistency. Then the validated code was used to calculate severalcases for high Mach numbers. From the result, several major factors that influent the flow, suchas the Mach number and the viscosity, were analyzed detailedly and along with the high Machnumber some unusual flow structure was observed and explained in detail.1 IntroductionShock wave focusing has been a fundamental topic in shock wave research for manyyears. Especially, in axisymmetric cases, the cylindrical shock wave focuses at a point,instead of a planar shock focusing in a plane. The phenomenon results in a small region ofhigh temperature and pressure around a certain point, which leads to vortex generation,turbulence mixing and new pattern of shock wave reflection. So, the relevant research isnot only meaningful for fundamental study, but also has wide applications in industry andmedicine. Sod [1) studied the focusing of cylindrical shock waves by numerical simulationas early as in 1977. Takayama et al. [2] investigated the stability of cylindrical shock wavesin air experimentally in 1987. In 1989, Groenig [3) had reviewed shock waves focusingand pointed out the focusing of cylindrical shock waves to be one of the research topicsworth trying in future . Jiang et al. [4) discussed the focusing of the toroidal shock wavedischarged from a coaxial annual shock tube. Hamid et al. [5) investigated the focusingof the toroidal shock wave in the shock tube experimentally with Mach number 1.5.Because of the difficulty of generating high quality toroidal shock waves, there are noexperimental results of higher Mach number focusing published, which is more complexand practical.This paper is devoted to numerical simulation of the toroidal shock wave focusingin the cylindrical shock tube, including both viscous and inviscid flow. For numericalvalidation, the dispersion-controlled scheme [6) is used to solve both Euler equation andNavier-Stokes equation for inviscid and viscous axisymmetric flow respectively at a lowMach number 1.5, which the experiment has been accomplished. Then some cases for highMach numbers are simulated and through the analysis of numerical results, the focusingprocess is investigated and also the influence of the Mach number and the viscosity arediscussed.474 H.H.Teng et al.2 Problem descriptionThe model investigated here is a toroidal shock wave propagating from a shock tube ofannular cross-section into a cylindrical chamber, as shown in Fig 1. The shock wave willdiffract over the bent corner first and then implode toward the axis of symmetry, andfinally focus at a certain point on it. The focusing point is singular point and withoutregard to the viscosity pressure and temperature on it will become infinite. In practicethe effect of the viscosity and real gas decreases the value of pressure and temperatureto being definite . When carrying on numerical simulation, because the grid can not beinfinitesimal, the results is also definite, which is usually lower than actual results. But theresults of numerical simulation, which show the process of the focusing and interactionof complex wave system, still play an important role in research and application of shockwaves focusing.The axi s o f symme t ryT~Tor oidal shock waveFig. 1. Computation domain with D:d=5:43 Numerical methodsIn this numerical investigation, Euler equation and Navier-Stokes equation are solvedseparately in the axisymmetric flow field. In the inviscid flow, the dispersion-controlledscheme is used to solve the Euler equation, which has been demonstrated to be capable ofpredicting complex flow fields, including shock wave interactions, without non-physicaloscillations and any need for artificial viscosity. For the viscous flow, the dissipationterms, which introduce the effect of the viscosity, are discretized with the simple centraldifference. As the initial condition, the toroidal shock wave is moving and up to enterthe cylindrical shock tube from the annular shock tube. As the boundary condition thereflection and slip boundary conditions are applied to the solid wall when the Eulerequation is solved, while no-slip and adiabatic conditions are used on the solid wall whenthe Navier-Stokes equation is solved.4 Solution verificationFor the numerical validation, the density contour is integrated into the 'numerical holo-graphic interferogram' [7] and compared with the experimental holographic interferogramdirectly. In Fig. 2, the experimental result(a) [5], viscous numerical result(b) and invis-cid numerical result(c) are shown when focusing. It can be found out that not only thenumber of fringes but also the distribution of individual fringe are very close, so theNumerical investigation of toroidal shock waves focusing 475a b cFig. 2. Holographic interferograms when focusing for M. = 1.5calculation results are credible and the code can be used to investigate the shock wavefocusing. Through comparing the viscous and inviscid numerical result , it 's found thatnear the wall the inviscid result is not so accurate . If taking into account of the viscosity,the departure will be improved.Fig. 3. Isobars(lower half) and isopycnics (upper half) for viscous and inviscid flow (M. = 2.0)476 H.H.Teng et al,ahFig. 4. Isobars(lower half) and isopycnics (upper half) for viscous and inviscid flow (M. = 3.0)5 Results and discussion5.1 Case 1: M 8 = 2.0The case for a shock Mach number of M. = 2.0 is simulated and shown in Fig. 3, whichviscous results are on the left column from Fig.3a to c and inviscid ones on the right. Inevery frame of the figure, isobars are shown on the lower half and isopycnics are shownon the upper half . The same display is also used in 'other cases. The toroidal shockwave, which is discharged from a annular shock tube to a co-axial cylindrical shock tube,diffracts and implodes toward the axis of symmetry. At first the reflecting pattern on theaxis is regular reflection, with the lapse of time it quickly transits to Mach reflection asshow in Fig.3a and the pressure behind the shock wave decreases rapidly. From Fig.3ato b, the Mach stem propagates forward and the triple point moves away from the axisof symmetry. In the axisymmetric case, the triple point is actually a ring and becomesbigger and bigger. So in Fig.3c, the shock wave surface gradually becomes a plane whichis mainly the Mach stem . From the figure, the slip line derived from the Mach stem canbe observed by comparing the isobars and isopycnics. Through comparing the left columnand the right , differences between viscous and inviscid flow can be found out easily. Asshown in Fig. 3, the flow structure taking into account the effect of viscosity on the leftis similar to that inviscid one on the right approximately. But near the wall of the shocktube and the entry plane of the toroidal shock wave, the shock wave and contact surfacebecome illegible.5.2 Case 2: M 8 = 3.0In this case, the flow changes induced by better focusing effect on the axis of symmetry.As shown in Fig. 4, instead of decaying back to plane like the case M. = 2.0, it can beNumerical investigation of toroidal shock waves focusing 477found that the Mach stem is further accelerated near the axis so that the curved Machstem develops. Behind the curved Mach stem, vortex and the second contact surface canbe observed, which is peculiar for higher Mach number . To investigate these phenomenonsdeeply, higher Mach number cases are carried out in the next section.Fig. 5. Isobars(lower half) and isopycnics (upper half) for viscous and inviscid flow (M. = 4.0)5.3 Case 3: M B = 4.0In this case, numerical results for M, = 4.0 are shown in Fig. 5. The peculiar phe-nomenons along with the higher Mach number become more clear. The Mach stem iscurved forward nearly to be spherical around the axis of symmetry, and the second triplepoint and contact surface are generating at a certain point of the first one. So the shockwave front surface is divided into three sectors: the outside original diffracting shockwave, the middle first Mach stem and the inside second Mach stem. This kind of reflec-tion , with the second Mach stem spherical, is called spherical double Mach reflection.Behind the inside Mach stem , there are very complex interactions between shock wave,vortex, contact surface and perhaps jet. From the Fig. 5 vortex and another short shockwave generation can be observed .The pressure value of every point on the axis of symmetry is checked for each timestep and the maximum ones are recorded , as shown in Fig. 6. It can be found out thatthe maximum pressure of the focusing point, which is also the peak pressure of the axis ofsymmetry, mainly depends on the incident Mach number and the focusing effect can beimproved rapidly with the incident shock wave Mach number increasing. The viscosityinfluences the result slightly, which decrease the focusing effect . Interestingly, we find478 H.H.Teng et al.- YISCOUI:zoo1_" 1IOeo 'eo20 (\,..,eo(\701\,~ ) \.. ' '0'20I i50\' 00fl!:.0 fl!:.0 44 ~ fQ. '0 \eoi \\Q.\ 30 .. \20 40<,; '0 20--·'0+-_ - --.--- _ -,o 2X Axio(M,03.0).00+-_ - -.-- _ -.o 2X Axio(M,04 .0)Fig. 6. Historic maximum pressure of every point on the centerlinethere are two peak values. One is on the back wall and the other is not the first arrivingpoint of diffracting shock waves but after it .6 ConclusionsFrom the numerical investigation, some results are summarized as follows: Being differentfrom low Mach number focusing, in which shock wave front decays back to planar afterfocusing, the shock wave front surface around the axis of symmetry is curved forwardand generate spherical double Mach reflection. High incident Mach number also inducessome complicated phenomenons and generate two peak focusing pressure. When takingthe flow viscosity into account, the flow structure appears similar to the inviscid onebut the dissipation induced by viscous terms weakens the effect of shock wave focusingslightly.References1. G.A. Sod: A numerical study of a converging cylindrical shock. J .Fliud Mech. 83,795 (1977)2. K. Takayama, H. Kieine and G. Groenig: An experimental investigation of the stabilitycylindrical shock wave in air. Exp . in Fluid . 5, 315 (1987)3. H. Groenig: 'Past, present and future of shock wave focusing research'. In: Proceeding of theInt . Workshop on Shock Wave Focusing,Sendai,Japan. 1, (1989)4. Z. Jiang and K. Takayama: Reflection and focusing of toroidal shock waves from coaxialannular shock tubes. Computers Fluids. 27, 553 (1998)5. S. Hamid, R. Hosseini and K. Takayama: 'Study of shock wave focusing and reflection oversymmetrical axis of a compact vertical co-axial diaphragmless shock tube'. In: ISSW23,(2001)6. Z. Jiang, K. Takayama and Y.S. Chen: Dispersion conditions for on-oscillatory shock cap-turing schemes and its applications. CFD Journal. 4, 137 (1995)7. Z. Jiang and K. Takayama: An Investigation into the Validation of Numerical Solution ofComplex Flowfields. J .Comput.Phys. 151, 479 (1999)

Numerical Investigation of Toroidal Shock Wave Focusing

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Numerical Investigation of Toroidal Shock Wave Focusing

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Institute Of Mechanics,Chinese Academy of Sciences