Fhdd Dynamics, ~91 35. No. l. 2000
TYPE OF R E F L E C T I O N OF A NEAR-WALL S H O C K WAVE IN THE PROCESS OF D I F F R A C T I O N F R O M A SQUARE C H A N N E L T. V. Bazhenova, T. A. Bormotova, V. V. Golub, A. M. Shurmeister, and C. B. Shcherbak
UDC 533.6.011.72
The results of an experimental and numerical investigation of the process of diffraction of shock waves from a square channel at a ninetydegree convex corner are presented for various incident shock wave Mach numbers M,, (1.4 < M~ < 7). The type of reflection of the near-wall fragment of the diffracting shock wave from the wall and the wave velocity are determined as functions of M~. direction, and time.
With increase in the angle of a wedge along which a shock wave propagates the type of reflection changes from Machtype to regular, the pressure on the wedge increasing sharply [1]. After emerging from the channel the shock wave propagates along the surface near the outlet (as in the emergence o f a shock wave from a mine shaft or a w i n d o w in an explosion). The type o f reflection of the diffracting shock wave from the wall depends on the angle o f inclination o f the shock to the wall and the shock wave Mach number near the wall. In its turn, these parameters are determined both by the angle between the wall and the direction of propagation o f the shock front and the incident shock wave Mach number. W h e n a shock wave is diffracted at a convex corner, the velocity of the near-wall fragment of the shock wave decreases due to increase in the gas volume captured b y the shock. The velocity (and, as a result, the shape) vary in the part o f the shock wave which interacts with the centered rarefaction wave departing from the vertex o f the corner and with the reversed flow beneath the separation line. In the self-similar case the velocity of the near-wall fragment o f the shock wave does not vary with time. The velocity o f the near-wall fragment of the shock wave determines the m a x i m u m pressure at the wall. In [2] the available experimental data [3-5] on the dependence o f the type of reflection o f a diffracting shock wave on the wedge angle and the incident shock wave Mach number M o were classified for the self-similar case (Fig. 1). The diffracting shock wave profile can be divided into two parts: the part o f the shock wave in the neighborhood o f the surface of the convex corner and the part of the curved main wave front which interacts with the rarefaction wave fan departing from the vertex o f the corner. The velocity o f the near-wall wave is less than the velocity o f the main part of the wave and depends on the gas flow parameters beneath the separation line. The near-wall wave is j o i n e d with the main shock wave. The reflection is called normal when the diffracting shock wave slips along the wall and the wave front is directed at right angles to it (N-type, Fig. l a ) . As Mo increases, the near-wall part lags behind the main part and a kink develops on the diffracting shock (K-type, Fig. lb). In this case the second derivative o f the wave profile is discontinuous. With further increase in M 0 the near-wall part of the wave, as before, is perpendicular to the wall but the direction o f the main wave is such that a smooth junction is not possible. The kink is transformed into a triple point and in the neighborhood of the wall a three-shock Mach-type configuration develops (M-type, Fig. lc) with a reflected wave departing from the triple point. In this case the first and second derivatives of the wave profile are discontinuous. On the range o f higher M 0 number triple point trajectory decreases and regular reflection (R-type, Fig. ld) m a y develop. In this case the diffracting wave is reflected at an acute angle directly from the wall. In the plane self-similar case when the diffraction takes place at a 90 ~ corner normal reflection goes over into reflection with a kink at the M a c h number M0=3 and this type of reflection does not change with time. The angle between the diffracting wave and the wall (angle o f incidence) is close to 90 ~ The angle of incidence o~1 decreases monotonically from 90 ~ to 65 ~ with increase in M 0 from 3 to 4.3. At M0=4.3 a kink is observed on the near-wall part o f the wave. At M0=7.5 (r ~ the three-shock configuration develops. For 7.5 < M 0 < 11 the angle o f incidence decreases to 50 ~ and regular reflection with an oblique reflected shock on the wall is observed. The results o f numerical calculations [6] for the plane self-similar case also fit within the above classification. In the three-dimensional case the shape o f the shock wave varies with time and with direction in space [7]. In the process of diffraction of a shock wave with M0=3.0 from a square duct the type of reflection changes in time and space and
Moscow. Translated from Izvestiya Rossiiskoi A k a d e m i i Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 145-151, J a n u a r y - F e b r u a r y , 2000. Original article submitted M a y 15, 1998. 0015-4628/00/3501-0117525.00 9 2000 K l u w e r A c a d e m i c / P l e n u m Publishers
117
a
M~
~j
b
*M H
c
Mo
rl
*M n
d
Fig. 1. Types of reflection of a diffracting shock wave from a wall: type N (a), type K (b), type M (c), and type R (d).
goes over from N-type to K-type. In a spherical explosion, in the neighborhood of a plane surface the dimensionless pressure distribution in time and space is determined by the type of reflection of the shock wave from the surface [8]. The type of reflection changes as the explosion wave propagates along the surface as a consequence o f variation in the angle of incidence and the wave intensity.
1.
EXPERIMENTAL APPARATUS
In order to investigate three-dimensional shock wave diffraction an apparatus consisting o f a shock tube connected to a cylindrical vacuum chamber 80 m m in diameter and 120 m m in length was used. At the end o f the shock tube we installed a flange with a channel located inside the tube. The shock tube had a 4 0 x 4 0 m m cross-section and the channel was 100 m m in length and had a 20 x 20 m m square cross-section. The input channel edge facing the direction of motion of the shock wave had a 10 ~ taper on the outside. As a result, only sonic disturbance entered the channel. The time of conservation of constant parameters at the inlet to the insert channel consists of the time of propagation of the incident shock wave to the end of the shock tube and time of propagation of the reflected shock wave to the inlet o f the nozzle. For all the incident shock wave Mach numbers investigated this time was greater than 160 ps, while the diffracting shock wave observation time did not exceed 150 ps. The equality of the Mach numbers of the shock waves measured in the tube and in the nozzle is confirmed by measurement of the shock wave velocity on the basis of a series of shadowgraphs at the initial moment of diffraction. The end of the shock tube with the flange, whose surface formed a 90 ~ angle with the shock tube wall, was located between the plane-parallel optic windows of the vacuum chamber. By turning the flange relative to the axis of the tube, we could transilluminate the flow behind the tube outlet in different directions. The flow pattern was visualized by means of an IAB-451 shadow instrument. To investigate the development of the time-dependent gasdynamic processes a BSK-5 high-speed camera was used. Since this camera was not intended for working with the IAB-451, it was modifies by replacing the standard lens, installing an additional lenses, and reconstructing the electrodynamic shutter. This made it possible to obtain 72 16 x 22 m m frames with an interval of 4 - 7 ps in a single run. Tair-3S and Jupiter 37 lenses located in series between the slit plane and the BSK5 lens were used as an additional optical system in conjunction with the BSK-5 and the standard lens. In order to investigate the shadowgraph details with a large frame visualization was carried out with a pulsed light source of 1 ps duration. The shadow pattern was recorded on high-speed photographic film with a 60 x 60 m m frame. At a given instant the light source was switched on by a synchronization unit with a controlled delay. The unit was switched on by the signal from a piezoelectric pressure transducer located in the wall of the shock tube in the neighborhood of the end face. The accuracy o f the time measurements was 1 /as. The low-pressure chamber of the shock tube and the vacuum chamber were filled with air. The pressm'e o f the air in the low-pressure chamber and the air, nitrogen or helium in the high-pressure chamber and the thread depth o f the copper diaphragms were so chosen that an incident shock wave with a given Mach number M 0 developed in the shock tube. The velocity of the incident shock wave was measured correct to 1% by the base method using pressure transducer signals.
118
Fig. 2. Successive shadowgraphs of the flow pattern for a shock wave diffracting from a square channel obtained by transillumination. Figures (a) and (b) correspond to the projections in the direction of the side of the square and (c) and (d) the diagonal of the square, respectively. D 1 is the diffracting shock wave, K is the contact surface, and R is the rarefaction wave fan. Types of reflection: regular (a), Mach-type (b), normal (c), with kink (d). 2.
NUMERICAL SIMULATION
The problem of non-self-similar diffraction was numerically simulated by solving the Euler equations by means of the second-order Godunov method. In the outlet of the channel installed at the end of the shock tube we specified the flow parameters calculated for a given incident shock wave Mach number. The flow field for an ideal gas in space and time was calculated as a function of the incident shock wave Mach number and the specific heat ratio of the outflowing gas.
119
8
X, cm a
2
4 J
b
1
..g
0
80
0
80
t, ItS 160
Fig. 3. Time dependence of the position of the near-wall part of a diffracting shock wave: in the direction of the side of the square (a) for M0=1.62, 1.95, 2.7, and 4.65 (data 1-4); in the direction of the diagonal of the square (b) for M0=1.54, 3.05 and 5.4 (data 1-3).
The three-dimensional problem was calculated on a rectangular grid consisting of (60 x 60 x 60) and (120 x 120 x 120) cells. When the number of calculation cells was increased by 8 times, the difference in the values of the gas parameters was within 5%. The calculation results were nondimensionalized using the parameters of the undisturbed gas in the vacuum chamber: the density p(, and pressure P0. Distances were divided by the tube diameter or the side of the square d. The dimensionless time z is related to dimensional time t as follows: "c=(t/d)(pJpo) m. To compare the results of the numerical calculations with experiment we constructed the front trajectories in the direction of the side and the diagonal of the square. For this purpose from a numerical array corresponding to the flow parameter distribution we chose the coordinates of the shock front on the pressure curve (points with the maximum pressure gradient) in a given cross-section at a given instant of time. At this point the pressure amplitude amounts to 0.9 of the maximum. The results of the calculations of the trajectory of the diffracting wave front along the wail and in the direction of the channel axis coincided with the experimental data. The shape of the diffracting shock front in various directions in a plane passing through the tube axis was also determined from the image of the isobar field obtained numerically. However, the trajectories of the front could not be determined from the sequence of isobar fields because the pressure distribution inside the shock wave could not be resolved. The velocity of the leading front of the shock wave determined from the image of the isobar field obtained numerically turned out to be less than that measured experimentally.
3.
E X P E R I M E N T A L RESULTS
Series of shadowgraphs of the development of the process of diffraction of a shock wave from a square channel at a 90 ~ convex corner were obtained for various Mach numbers (M0=1,4-7) in the direction of the side and the diagonal of the square. In Fig. 2 we have reproduced shadowgraphs of the flow pattern in the diffraction process. The photos show all the basic elements of the structure investigated in the self-similar case: the diffracting shock wave D~, the contact surface K, and the wave rarefaction fan R. The projection of the diffracting shock front depends on the direction of transillumination and varies differently with time. In the photos reproduced we can also trace the change in the type of reflection from normal to Mach type. In Fig. 3 we have plotted graphs of the time dependence of the location of the near-wall part of the diffracting wave along the direction of the side and the diagonal of the square for various M 0. The approximating curves for the position of the shock front as a function of the initial Mach number were constructed by least squares. The mean square deviation of the experimental points from the constructed curves is 1%. The wave velocity was determined as the derivative of the dependence of the wave front coordinate on time. The mean value of the front coordinate was 3.5 cm and the mean time from 70 to 150 ps depending on the initial Mach numbel: The accuracy of determination of the time and coordinate was +1 ps and +1 mm, respectively. The relative error in determining the diffracting shock Mach number was 6%. The trajectories were approximated by second-order polynomials. On the time interval investigated (up to 150 ps) the front velocity in the direction of the side and the diagonal of the square depends only slightly on time; therefore, to investigate the dependence of the near-wall wave Mach number on M 0 we used a linear approximation. In Fig. 4 we have plotted the graph of the Mach number M H of the near-wall part of the shock wave in the direction of the side of the square
120
x , cm
2.41
_
2
MH
/
....x
1.6
--x
S/
+j
~Xo ON e,R
0.8
3
5
40
M0
Fig. 4
t, ~s
80
Fig. 5
Fig. 4. Relation between the Mach numbers of the near-wall (M H) and the incident (M 0) shock waves. In the direction of the side of the square: experiment (1) and calculation (2); in the direction of the diagonal of the square: experiment (approximation) (3), calculation (4), and plane self-similar case (experiment [1]). Fig. 5. Time dependence of the position of the near-wall part of a diffracting shock wave in the direction of the side of the square for M0=1.62, 2.6, and 2.72 (curves 1-3). Types of reflection: normal (N) and regular (R).
as a function of Mo and the analogous straight line for the plane self-similar case [1] approximated by the relation MH=I + (1 - sin cqJ2)(iVl~- 1), where 0% is the wedge angle. On the basis of these experimental data, on the observation interval (150 ps) the Mach numbers of the near-wall part of the shock can be approximated by the following relations in the direction of the side and the diagonal: Mu=0.3M0 + 0.7 and Mn=0.22Mo- 0.01Mo + 0.83. The first dependence is close to that obtained in the self-similar case [1]. In the direction of the channel z axis the velocity of the shock wave increases with the M0 number and decreases more strongly with time, for example, at M0=3 the diffracting wave Mach number can be approximated by the relation M.=2.92 - 1.08t
(t=0, z = d )
These approximations of the experimental data satisfactorily describe the result of the numerical calculation of the front velocity within the accuracy of the measurements (Fig. 4). 4.
DISCUSSION OF T H E RESULTS
An analysis of the series of diffraction shadowgraphs makes it possible to establish the laws of change in the type of reflection of a diffracting wave emerging from a square channel. The shape of the diffracting shock wave varies with time since the velocity of propagation of the front in different directions depends on time in different ways. Accordingly, the angle of inclination of the diffracting wave to the wall varies in time and space. This affects the type of reflection. In the present paper we establish the limits of transition from normal reflection of the diffracting shock wave at a 90 ~ comer to reflection with a kink on the Mach number interval 1.4 < M 0 < 7 in the three-dimensional case. For Mach numbers 1.5 < M 0 < 2 the type of reflection in the direction of the side of the square is always normal. On the interval of incident wave Mach numbers 2 < M 0 < 2.7 the type of reflection changes in the process of propagation of the shock wave from Ntype to R-type and the instant of change in the type of reflection comes earlier with increase in the Mach number (Fig. 5). The velocity of the near-wall part of the shock wave does not change when the type of reflection changes. When M 0 is greater than 2.7, the initially normal reflection goes over into K-type reflection with a kink and into Mach reflection at M 0 > 4.7. For M0=7 Mach reflection is observed from the onset. The nature of the change in the type of reflection is identical in the direction of the side and the diagonal of the square.
121
60 1
88
V
72
0
40
80
t, ~ts
Fig. 6. Angle o)~ between the diflYacting shock wave and the wall at Mo=3 in the direction o f the diagonal of the square at various instants o f time.
The variation o f the type o f reflection with time determined from the color image o f the numerical isobar field is in agreement with that observed experimentally. At the limit of transition to K-type reflection (M0=3) transition instability observed, with changes in transition from one type to another and back. In this case the angle eo~ in the direction o f the diagonal oscillates b et w een 72 ~ and 90 ~ (Fig. 6), MH=I.3. For the plane self-similar problem normal reflection should be observed for these angles o)j and M H < 2. During the observation period the state o f the flow at the channel outlet did not vary, the accuracy o f measurement o f the angle being +1 ~ Oscillations of the angle of incidence of the near-wall part o f the shock wave were not observed at larger and smaller M 0. The restructuring o f the flow at the limit o f transition from one type of reflection to another is characterized by instability of the angle o f incidence, S u m m a r y . Our experiments and calculations showed that in the process o f diffraction from a square channel the dependence o f the M a c h number of the near-wall wave in the direction of the side o f the square on the incident shock M a c h number M 0 is close to that obtained for the self-similar case. In the direction of the diagonal the velocity o f the shock w a v e is lower and on the entire interval o f M 0 numbers investigated a low-pressure region develops. The dependence o f the type o f reflection on the M 0 number in the process o f diffraction at a 90 ~ corner o f a shock w a v e from a square channel differs from the results of investigating the self-similar diffraction. On the interval 1.4 < M0 < 7 the type of reflection changes in time and space in accordance with a law that depends on the incident wave Mach number. The work was carried out with partial support fi'om the Russian Foundation for Basic Research (project No. 96-0216170a).
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T.V. Bazhenova and L. G. Gvozdeva, "Unsteady Interaction of Shock Waves [in Russian], Nauka, Moscow (1977). K. Matsuo, T. Aoki, and H. Kashimura. "Diffraction of a shock wave around a convex corner," in: Y. M. Kim (Ed.), Proc. 17th Intern. Symp. on Shock Waves and Shock Tubes. Amer. Inst. Phys., N.Y. (1990), E 252. B.W. Skews, "The shape of a diffracting shock wave," J. Fhdd Mech., 29, 297 (1967). T.V. Bazhenova. L. G. Gvozdeva, V. S. Komarov, and B. G. Sukhov, "Investigation of strong shock wave diffraction at a convex corner," lzv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 4, 122 (1973). T.V. Bazhenova, L. G. Gvozdeva. and Yu. V. Zhilin, "Variation of shock wave intensity in the process of diffraction at a convex corner," Teplofiz. Vysok. Temp., 14, 436 (1976). R. Hillier, "Numerical modelling of shock wave diffraction," in: Shock Waves @ Marseille IV, Physico-Chemical Plvcesses and Nonequilibrium Flows, Proc. 19th Intern. Syrup. on Shock Waves, Springer. Berlin (1995), P.17. T.V. Bazhenova, T. A. Bormotova, V. V. Golub et al., "Diffraction of a shock wave from a square channel at a ninety degree convex corner." lzv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 114 (1999). J. Brossard, C. Desrosier, H. Purnomo, and J. Renard, "Pressure loads on a plane surface submitted to an explosion," in: Shock Waves @ Marseille IV, Physico-Chemical Processes aml Nonequilibrium Flows, Pivc. 19th Intern. Syrup. on Shock Waves, Springer. Berlin (1995), E 387.
