the complex nonlinear shock wave phenomena in literature.It has long been studied experimentally, analytically as well as numerically. Takayama in 1987 ...
ISSN 0567 - 7718
ACTA MECHANICA SINICA,Vol.7, No.4, November 1991 Science Press, Beijing, China Allerton Press, INC., New York, U.S.A.
UNSTEADY INTERACTION OF SHOCK WAVE DIFFRACTING AROUND A CIRCULAR CYLINDER IN AIR* H u a n g Wensheng ( ~ 3 ~ )
(Peking University,Beo'ing 100871, China )
O. O n o d e r a and K. Takayama
( Tohoku University, Japan )
The reflectionand diffractionof a planar shock wavearound a circularcylinderare a typicalproblemof the complexnonlinear shockwavephenomena in literature.Ithas long been studied experimentally,analyticallyas well as numerically.Takayama in 1987obtained clear experimentalpictures of isopycnicsin shock tube under the condition that the impingingshock wavepropagates as far as 3 diametersaway from the cylinder.To know more completely the whole unsteady process, it is desirable to get experimentalresults in a region which is more than 10 diameters away from the cylinder.This is what has been done in this paper by using the pulsed laser holographic intefferometry for severalshock Mach numbers of the impinging shock. Results for several moments are shown, giving more knowledge about the whole unsteady flow field. This is usefulfor a reliable and complete understanding of the changing force acting on the cylinder,and providesinterestingdata to check the performanceof many recentlydeveloped high resolutionnumericalmethods for unsteadyshock wavecalculation. ABSTRACT:
shock wave diffraction, large flow field around a cylinder, pulsed laser holographic interferometry, isopycnics,complexinteraction of curved shocks
KEY W O R D S :
I , INTRODUCTION The reflection of a planar shock wave impinging upon a wedge has been studied and analysed experimentlly and theoretically by Smith ( 1945 ), White (1952), W h i t h a m ( 1957 ), Henderson & Lozzi (1975) as well as Ben-Dor & Glass (1978)et al. since 1945. It is now well established that the reflection patterns can be divided into four types according to the incident shockwave Mach numberMi, wedge angle 0w and gas properties. The four types are regular reflection ( R R ) , single-Mach reflection ( S M R ) , complex-Mach reflection ( C M R ) a n d double-Mach reflection ( D M R ) , respectively.Several criteria have been proposed to analytically predict the transition boundaries between reflection types. However the case of a planar shock wave reflected on the circular cylinder does not fall within the above-mentioned types. This is a complete unsteady flow process and its flow pattern is very complex. Specially, when the flow M a c h n u m b e r Mi behind incident shock wave exceeds Me 9 (the critical M a c h n u m b e r ) , a supersonic flow region will appear around the circular cylinder and a new shock wave can be produced. Owing to the interaction between the new shock wave and the b o u n d a r y layer, the unsteady separated flow region will be formed resulting in a rapid change of the unsteady aerodynamic force. It should be pointed out that the circular cylinder surface is the simplest curved surface. For this reason, the reflection and diffraction of a planar shock wave on circular cylinder become a typical problem of the complex nonlinear shock wave propagation. Bryson & Gross (1961), Heilig (1969) as well as Iton ( 1 9 8 1 ) e t al. have studied the unsteady interaction of shock wave diffracted a r o u n d a circular cylinder by means of the shadowgraphic and schlieren method. Then they clearly pointed out that critical angle ~Oc9 of R R - M R transition does not depend on the cylinder radius, but only depends on the Mach n u m b e r of incident shock wave.Their results are qualitative and only the position and shape of incident shock wave and reflected shock wave are provided. In order to understand the development and change of complete unsteady flow process, the shape and flow pattern of a planar shock wave that has propagated far away from circular cylinder must be obtained.The paper Received 6 April 1990, revised 4 January 1991 * The project suported partially by National Natural Science Foundation of China
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reports results in a region of about 10 - - 1 2 diameters away from the cylinder. These results are very important and useful for the theoretical development and practice. II. EXPERIMENT A schematic diagram of the shock tube and its optical arrangement are shown in Fig. 1. Optical experiment of this paper was conducted on a transonic shock tube at the Institute of Fluid Science, Tohoku University. The transonic shock tube consists of a 1.5m long high pressure chamber of 230 mm in diameter and an 8.0m long low pressure channel with a cross section of 60 mm x 150 mm. The test section has a field of view of 150mm x 290mm. The model of the optical experiment is a circular cylinder with a diameter of 10 mm. This model is supported between the observation windows which are made of plastic glass. The test gas is drived air at 1.2--101.3 kPa and the driver gas is nitrogen at 344.4 kPa and helium at 273.5kPa. A mylar film 0. lmm thick is used as the diaphragm. The shock speed is measured by two pressure transducers (Kistler 603A) placed 250mm apart in front of the test section. The incident shock Mach numbers thus obtained are 1.40 + 0.01 and 1.93 -I-0.03, respectively. A good reproducibility is obtained. Paraboloidal mirror D = 300mm
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Test section
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Fig.l Schematicdiagram of shocktube and optical arrangement The present optical system consists of a pair ofparaboloidal schlieren mirrors 300mm in diameter and with a 3m focal length, a 6 : 4 beam splitter, an image lens system for the object beam and mirrors and lenses for the reference beam. As a light source, a pulsed ruby laser of 694.3nm wavelength and 20 nsec pulse duration (Apollo Lasers Inc. 22 HD ) is used, Agfa 10 E75 4 u x 5 /f sheet films are used for constructing holograms and Neopan SS 4 /~ x 5 1, sheet films are used for reconstruction. The holographic film is placed flat on the film holder. For the reconstruction an Argon-Ion laser of 514.5 nm wavelength is used. The distortion of reconstructed images due to the difference ofwavelength between the ruby laser and the Argon-Ion laser is found to be negligible and can be easily eliminated ifa computer aided data processing system is applied. In order to construct a double-exposure holographic interferogram, two exposures are required. The first one is conducted before the event and the second exposure is triggered by the event. Consequently the interference fringes, are obtained through the difference in the phase recorded on the hologram between these two exposures. These fringe distributions in an infinite interferogram correspond to the isopycnics. Therefore, if the ionization effect can be ignored, the interference fringe shift is related to the density variation as follows
Ap=N)~ / KL where N is the fringe number ( N = 1, 2, 3 .... )
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K :Gladstone-Dale constant (K = 0.225 c m 3 / g ) L : t h e width of the test section ( L = 6 0 m m ) 2: the wavelength of the ruby laser (2 = 694.3nm ) The density of static gas is p = 1.098 x 1 0 - 3 g / c m 3 . So that one fringe shift corresponds to the density variation of 4.68 % for air at the standard condition. It should be noted here that since this value changes in inverse proportion to the shock tube w i d t h , a wider shock tube is advantageous in order to obtain better accuracy so far as two-dimensional phenomena are concerned. IlI. RESULTS A N D DISCUSSION In order to simplify the analysis, it is necessary to make flow parameters dimentionless. W e can take the front stagnation point as the origin of coordinates and take the diameter D of the circular cylinder as the characteristic length. Thus, the abscissa and the ordinate are _,Y = x / D , Y=y/D, respectively. 1. The Reflection of a Planar ~aock Wave on the Circular Cylinder Surface
( 1 ) F r o m the interferograms obtained we can see that the initial reflection is a regular reflection ( R R ) . As time goes on, the regular reflection is changed to Mach reflection as shown in Fig. 2. When the incident shock wave Mach number M i = 1.40 and 1.93, the critical transition angle (Ocr = 5 2 o and 48 ~ respectively. These results are in agreement with the result of detachment criterion. It is noted that the use of optical observations alone may confuse the regular reflection with Mach reflections, when M ach stem is very short. Strictly speaking, these obtained values of o~c 9 are only approximate. However, there can be no doubt that the fall in transition angle for concave cylinder surface is genuine. There is a danger of failing to detect the first appearance of a triple point on convex surface, but this should only affect the accuracy of the measurements.
(a) Regularreflection
(b) Mach reflection Fig. 2 The reflectionon circularcylinder(M i= 1.40 )
(a) pattern of waves
(c) Mach reflection
(b) photograph of the intefferogram
Fig. 3 The propogationapart from the back stagnationpoint ( 1)
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(2) A slipstream line appears when a shock wave reaches the apex of cylinder. The head of the slipstream line takes the form of vortex and it is apart from the surface of circular cylinder. Because the stagnation pressure in front of the slipstream line is larger than the back pressure, therefore,with the triple point as a base, the slipstream line and Mach stem become curve, as shown in Fig. 2 (c). 2. A Planar Shock Wave Propagates on the Back of Circular Cylinder (1)As a shock wave propagates forward, the upper and the lower two Mach stems should intersect at the back stagnation point G, then, these Mach stems reflect at G and the reflected Mach stems RMS are produced. These reflected Mach stems move upsteam along the surface of circular cylinder as shown in Fig. 3 (a) and 3 (b). (2) When reflected Mach stems (RMS)move further upsteam, the boundary layer becomes thicker and the flow within boundary layer begins to separate. In the separated region, the free vortex appears and the flow within the separated region becomes extremely complex, as shown in Fig. 4. (3)As time goes on, the free vortex continuously increases and moves forward. When Mi = 1.40 and 1.93, respectively, the major slipstream line disappears at about X = 5 . 5 and 11.0, as shown in Fig.5. Then the free vortex is away from circular cylinder surface and becomes an indepen-
T MS
~S
(a) pattern of waves (b) photograph of the interferogram Fig .4 The propogation apart from the back stagnation point (2)
Fig. 5
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dent vortex, the front of incident shock wave coincides with Mach stem and becomes a straight line near points X = 18.5 and 16.5, respectively, as shown in Fig .6.
Fig. 6
3. The Triple Point Trajectory and the Angle co between Incident and Reflect Shock Wave The triple point trajectory obtained by the present experiment is shown in Fig. 7 . It will be seen from Fig. 7 that two triple trajectories tend to coincide, when Mi = 1.40 and 1.93, respectively. These results show that the triple point trajectory is independent of the distance X, when the incident shock Mach number M i < 2. The present optical experimental results agree with the results of Heilig and others. The angle co between incident and reflect shock waves becomes a monotonically decreasing curve,when Mi = 1.40 and 1.93, respectively. When the distance X reaches a certain value, the value of the co will be a constant.
8 Y,
6 o'g" y.~ 4
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9 M, = 1.40
Y, M~ = 1.93
2
o,- N
t 2
L
I 4
I
1 6
t
L 8
I
I 10
I
I 12
F i g . 7 The triple point trajectory
Acknowledgement : The authors wish to express their thanks to Mr. H. Ojima, technician of the Institute of Fluid Science, Tohoku University, for his assistance in conducting the present experiment. REFERENCES [ 1] [ 2] [ 3] l 4] [ 5] [ 6] [ 7] [ 8] [ 9]
Smith, W . R . , OSRD (1945),Rep. No.6271. White, D. R., Proc. 2nd Midwest Conf. on Fluid Mech. (1952), 21 No. 3,252-- 263. Bryson, A. E and Gross, R. W. F., J. Fluid Mech., 10 (1961). Heilig, W. H . , Phys. Fluid. Suppl. 1, 12, pt. 2 (1969), 154-- 157. Iton, S. and Itaya, M., Proc. 12th Int. Syrup. on Shock Tubes and Waves (1980). Iton, S.,Okazaki, N. andltaya, M., J. Fluid Mech., 108 (1981), 383-- 400. Ben-Dor, G., Takayama, K. andKawauchi, T., J. Fluid Mech., 100(1980), 147-- 160. Iton, K. and Takayama, K., Theoretical and Applied Mechanics., 35 (1987), 2 7 - 36. Huang Wensheng, Liu Ziqiang, Hu Yongsheng., Acta Mechanica Sinica, 19, Sup 9 July (1987), 73 - - 78 (in Chinese).
