Abstract
Several theoretical and experimental studies have been devoted to the problem of the nonstationary action of the stream behind a shock wave on bodies of varied shape. In particular, in [1], the pressure and density are calculated for flow about bodies of the more typical shapes in the initial stage of the process. The basic relations which accompany the interaction of shock waves are considered in [2, 3]. The analysis of the phenomena of diffraction of shock waves on the sphere, cylinder, and cone is presented in [4]. Problems of unsteady flow about a wing are examined in [5, 6]. A detailed review of the foreign studies on unsteady flow is given in [7].
Of great practical interest is the question of the time for flow formation and the magnitudes of the unsteady loads during this period. Experimental investigations have been made recently [8, 9] in which some criteria are presented for estimating the bow shock formation time for supersonic flow about the sphere and cylinder with flat blunting. However the question of the formation time of the stationary pressure on the body surface is not referred to in these studies and no relationship is shown between the transient position of the reflected wave and the corresponding unsteady pressure on the surface. Moreover, in [8] the dimensionless time criterion is determined very approximately, independently of the Mach number of the shock wave.
The present study was undertaken with the object of determining the basic criteria which characterize unsteady flow about bodies behind a plane shock wave which has time-independent parameters, and clarification of the shock wave reflected from the body and the pressure on the surface of the body during the transient period.
The most typical body shapes were studied: 1) a cylinder with flat face aligned with the stream; 2) a spherically-blunted cylinder; and 3) a cylinder transverse to the stream.
The experiments were conducted in a conventional shock tube using the single-diaphragm scheme. The measurements of the pressure on the models and the velocity of the incident shock wave were made using the technique analogous to that of [10, 11]. A highspeed movie camera was used to record the pattern of the wave diffraction on the body. The Mach number of the incident shock wave varied in the range from M=1.5 to M≈6.0, which corresponded to a range of Mach numbers M∞ of the stream behind the shock wave from 0.6 to 2.1.
The calculations of the required gas dynamic parameters for high temperatures were made with account for equilibrium dissociation of the air on the basis of the data of [10, 12, 13].
The magnitude of the relative maximal shock wave standoff Δ at the stagnation point obtained in the present experiments was compared with the values of Δ from other studies. In the case of the flat-blunted cylinder it was in good agreement with the results of [8–14], and in the case of the spherically-blunted cylinder and the transverse cylinder it was in agreement with the results of [15].
Keywords
Shock Wave Mach Number Shock Tube Unsteady Flow Supersonic FlowNotation
- t
instantaneous time
- t1
time for rarefaction wave to overtake the reflected shock wave
- t2
time for the termination of bow wave formation
- Δ0
value of maximal standoff of reflected shock wave from the stagnation point, corresponding to the steady-state location of the bow wave
- δ0
value of the transient standoff of the reflected shock wave from the stagnation point
- δ1
value of the relative transient standoff of the bow wave, corresponding to the instant of realization of the steady-state pressure at the stagnation point
- p
value of transient pressure on cylinder surface at point in question
- p∘
value of steady-state pressure on cylinder surface at point in question
- L
cylinder length
- R
cylinder radius
- λ
cylinder aspect ratio
- a0
speed of sound behind reflected shock wave for reflection from an infinite plane wall
- W
maximal velocity of reflected wave relative to the cylinder
- τ1
dimensionless time, corresponding to the instant when the rarefaction wave overtakes the reflected shock wave
- τ2
dimensionless time, corresponding to termination of the formation of the stationary bow wave
- τa∘
dimensionless time, corresponding to arrival of the rarefaction wave at the point in question
- P0
- maximal value of the pressure corresponding to the reflection of a wave from a plane wall$$\begin{gathered} p = \frac{r}{R}, \Delta = \frac{{\Delta _0 }}{R}, \delta = \frac{{\delta _0 }}{{\Delta _0 }}, \Delta p = \frac{{p - p^0 }}{{p_0 - p^0 }}, \lambda = \frac{L}{{2R}}, \hfill \\ \tau _\alpha = \frac{{\alpha _0 t}}{{2R}}, \tau _w = \frac{{Wt}}{{\Delta _0 }}, \tau _1 = \frac{{Wt_1 }}{{\Delta _0 }}, \hfill \\ \tau _2 = \frac{{Wt_2 }}{{\Delta _0 }}, \tau _\alpha ^\circ = \frac{{\alpha _0 t_0 }}{{2R}}, \tau = \frac{{\alpha _0 t}}{{R - r}}. \hfill \\ \end{gathered} $$
Preview
Unable to display preview. Download preview PDF.
References
- 1.V. V. Rusanov, “Calculation of the interaction of unsteady shock waves with obstacles,” Zh. vychisl. matem. i matem. fiz.,1, no. 2, 1961.Google Scholar
- 2.G. B. Whitham, “A new approach to problems of shock dynamics, Part I, Two-dimensional problems,” J. Fluid Mech.,5, no. 3, 1959.Google Scholar
- 3.G. B. Whitham, “A new approach to problems of shock dynamics, Part II, Three-dimensional problems,” J. Fluid Mech.,5, no. 3, 1959.Google Scholar
- 4.A. E. Bryson and B. W. F. Gross, “Diffraction of strong shocks by cones, cylinders, and spheres,” J. Fluid Mech.,10, no. 1, 1960.Google Scholar
- 5.A. I. Golubinskii, “Flow of a moving shock wave past a moving plate,” Inzh. zh.,1, no. 2, 1961.Google Scholar
- 6.A. I. Golubinskii, “Lift and moment of a thin profile for arbitrary unsteady flow,” Inzh. zh.,3, no. 3, 1963.Google Scholar
- 7.D. G. Pack, “The reflection and diffraction of shock waves,” J. Fluid Mech.,18, no. 4, 1964.Google Scholar
- 8.F. V. Shugaev, “Interaction of supersonic flow with an obstacle,” PMTF, no. 6, 1963,Google Scholar
- 9.M. P. Syshchikova, M. K. Berezhina, and A. N. Semenov, “Bow wave formation in a shock tube,” ZhTF,34, no. 11, 1964.Google Scholar
- 10.Collection: Shock Tubes [Russian translation], ed. Kh. A. Rakhmatulin and S. S. Semenov, Moscow, Izd. inostr. lit., 1962.Google Scholar
- 11.G. D. Salamandra, T. V. Bazhenova, S. G. Zaitsev, R. I. Soloukhin, I. M. Naboko, and I. K. Sevast'yanova, Some Methods of Studying High-Speed Processes and their Application to Study of the Formation of a Detonation Wave, Moscow, Izd-vo AN SSSR, 1960.Google Scholar
- 12.A. S. Predvoditelev, E. V. Stupochenko, V. P. Ionov, A. S. Pleshanov, I. B. Rozhdestvenskii, and E. V. Samuilov, Thermodynamic Functions of Air for Temperatures from 1000 to 12 000° K and Pressures from 0.001 to 1000 atm [in Russian], Moscow, Izd-vo AN SSSR, 1960.Google Scholar
- 13.V. N. Gusev and M. D. Ladyzhenskii, “Gas dynamic calculation of shock tubes and hypersonic nozzles with equilibrium dissociation and ionization of the air,” Tr. TsAGI, no. 779, 1960.Google Scholar
- 14.H. Serbin, “Supersonic flow around blunt bodies,” J. Aeronaut Sci.,25, no. 1, 1958.Google Scholar
- 15.W. Ambrosio, “Stagnation-point shock-detachment distance for flow around speres and cylinders in air,” J. Aeronaut Sci.,29, no. 7, 1962.Google Scholar
- 16.C. Kim and O. Koichi, “Experimental studies of a plane shock wave with a circular cylinder,” J. Phys. Soc. Japan, vols. 178–180, 1956.Google Scholar