UDC 533.6.011.72

HYDROMECHANICS

K. E. BOGOSLOVSKII

UNSTEADY INTERACTION OF BLUNTED BODIES WITH A SHOCK WAVE

(Presented by Academician G. I. Petrov on July 6, 1965)

A number of works of both theoretical and experimental character have been devoted to the unsteady interaction of a flow moving behind a shock wave with bodies of various shapes. In particular, in work (¹), by the method of through computation, the pressure and density fields in the flow around the most typical bodies were determined. In works (², ³) the basic relations accompanying the interference of shock waves were established; diffraction of waves by a cylinder, a sphere, and a cone was considered in article (⁴); work (⁵) is devoted to unsteady flow around a wing. A survey of foreign investigations may be found in (⁶). Experimental investigations carried out recently have made it possible to establish certain criteria determining the formation of the bow shock in supersonic flow around a sphere and a cylinder (⁷, ⁸). However, the interrelation between the position of the unsteady bow wave reflected from the body and the corresponding pressure on the surface has not yet been clarified, and at present there are no data of any kind concerning the time required for the pressure on the surface of bodies to become established, a matter of considerable importance in applied aerodynamics.

In order to fill the indicated gaps the present work was undertaken. The unsteady flow around cylinders with flat and spherical bluntness was investigated, as well as around a cylinder with elongation \(L/2R = 3.13\), whose axis of symmetry is directed perpendicular to the velocity of the incident flow. The experiments were carried out in a shock tube assembled according to a single-diaphragm scheme. The measurement of unsteady pressures and shock-wave velocities was carried out by methods analogous to those in works (⁹, ¹⁰). The Mach numbers \(M\) of the incident wave were varied within the range from 1.5 to 6.0, which corresponded to variation of the Mach numbers \(M_\infty\) of the flow behind the wave from 0.6 to 2.1.

The physical picture of the interaction of a shock wave with a blunted body is as follows. In the case of a cylinder having a flat bluntness, at the initial instant after reflection of the incident wave from the end face the pressure is constant over the entire surface of the end face and is equal to the pressure obtained when a shock wave is reflected from a flat infinite wall. With time the reflected shock wave moves away from the end face, forming into a stationary bow compression shock. Disturbances arising at the edge of the bluntness propagate toward the center of the end face in the form of an axisymmetric rarefaction wave moving with the speed of sound \(a\) behind the reflected shock wave. After the rarefaction wave reaches the center of the end face, its reflection occurs. In the direction from the center toward the periphery of the end face, the reflected rarefaction wave begins to move. If one introduces into the consideration the dimensionless time \(\tau_w = wt/\Delta_0\) and the relative unsteady wave standoff \(\delta = \delta_0/\Delta_0\), where \(w\) is the velocity of the reflected wave at the initial instant of time, \(\Delta_0\) is the magnitude of the maximum (steady) standoff of the reflected wave from the plane of the end face, \(\delta_0\) is the unsteady standoff of the reflected wave from the end face, and \(t\) is the current time, then the dependence of \(\delta\) on \(\tau_w\) is found to be linear in that

over the time interval where the rarefaction wave has not yet caught up with the reflected wave (Fig. 1a). The dimensionless time \(\tau_1\) at which the rarefaction wave catches up with the reflected wave on the axis of symmetry of the cylinder can be expressed as

\[ \tau_1 = R/\Delta_0 \sqrt{(a/w)^2 - 1}, \]

where \(R\) is the radius of the cylinder. On the interval \(\tau_w \leq \tau_1\), the velocity of the reflected shock wave on the cylinder axis is constant and equal to \(w\). As \(M\) increases,

Fig. 1. Dependence of \(\delta\) on \(\tau_w\): \(a\)—cylinder with a flat bluntness; \(b\)—cylinder with a spherical bluntness; \(c\)—cylinder exposed to flow transverse to the axis of symmetry. \(1\)—\(M = 2.5\) \((M_\infty = 1.2)\); \(2\)—\(M = 3.2\) \((M_\infty = 1.4)\); \(3\)—\(M = 3.9\) \((M_\infty = 1.6)\); \(4\)—\(M = 5.6\) \((M_\infty = 2.0)\)

\(\tau_1\) approaches unity, which is caused mainly by a decrease in \(\Delta_0\). Consequently, the end of the formation of the steady bow shock for large values of \(M\) occurs at smaller \(\tau_w\) (Fig. 1a).

In the case of a cylinder with spherical bluntness and a cylinder exposed to flow transverse to the axis of symmetry, the velocity of the reflected shock wave on the

Fig. 2. Dependence of \(\Delta p\) on \(\tau_a\) at the critical point. \(1\)—cylinder with a flat bluntness; \(2\)—cylinder with a spherical bluntness; \(3\)—cylinder exposed to flow transverse to the axis of symmetry. \(a\)—\(M = 1.5\) \((M_\infty = 0.6)\); \(b\)—\(M = 2.5\) \((M_\infty = 1.2)\); \(c\)—\(M = 3.9\) \((M_\infty = 1.6)\); \(d\)—\(M = 5.6\) \((M_\infty = 2.0)\)

axis of the cylinder begins to decrease much earlier than for flat bluntness (Fig. 1b, c). This is associated with the earlier arrival of disturbances from the curvilinear portions of the surface. Therefore the linear segment of the dependence of \(\delta\) on \(\tau_w\) is insignificant. The tangent of the angle of inclination of the curves \(\delta = f(\tau_w)\) at the point \(\tau_w = 0\) does not depend on \(M\) and is equal to unity (Fig. 1).

It is convenient to consider the dimensionless unsteady pressure \(\Delta p = (p - p_c)/(p_0 - p_c)\) as a function of the dimensionless time \(\tau_a = at/2R\), where \(p\) is the unsteady pressure on the surface; \(p_0\) is the maximum value of the pressure corresponding to plane reflection of the wave from an infinite wall; \(p_c\) is the value of the steady pressure on the surface.

Numerous experiments carried out with models of different diameter showed that throughout the investigated range of Mach numbers \(M\) the form of the function \(\Delta p = \varphi(\tau_a)\) is practically independent of \(M\) (Fig. 2). This is true both for supersonic and for subsonic values of \(M_\infty\) of the flow moving behind the shock wave, where the reflected shock wave recedes from the cylinder to an unlimited distance with time.

The dimensionless time \(\tau_a^0\), corresponding to the arrival of the expansion wave at the point \(\rho\) of the flat bluntness, can be determined from the formula \(\tau_a^0 =\)

Fig. 3. Dependence of \(\Delta p\) on \(\tau_a\) for different points of the flat bluntness.
\(1\) — \(\rho = 0\); \(2\) — \(0.35\); \(3\) — \(0.525\); \(4\) — \(0.685\)

Fig. 4. Position of the reflected shock wave \(\delta_1\), corresponding to the realization of the steady pressure at the critical point.
\(1\) — cylinder with flat bluntness, \(2\) — cylinder with spherical bluntness, \(3\) — cylinder in crossflow, normal to the axis of symmetry

\[ = 0.5(1-\rho). \]

Before the arrival of the expansion wave \((\tau_a < \tau_a^0)\), the quantity \(\Delta p\) remains constant, equal to unity (Fig. 2, 1 and Fig. 3). At subsequent times \((\tau_a > \tau_a^0)\) the pressure falls, approaching the value \(\Delta p = 0\), corresponding to steady flow. In the case \(\rho = 0\) (the critical point), in the region \(\tau_a \geq \tau_a^0\) the dependence of \(\Delta p\) on \(\tau_a\) is approximated by the curve

\[ \Delta p = 0.2440/(\tau_a - 0.3080) - 0.2745. \]

It is important to note that, in practice, the realization of the steady pressure occurs simultaneously over the entire surface of the face and, independently of the number \(M\), corresponds to \(\tau_a \approx 1.0—1.2\) (Fig. 3).

In the case of a cylinder with spherical bluntness and a cylinder in crossflow normal to the axis, the pressure at the critical point begins to fall practically immediately after reflection of the incident wave (Fig. 2, 2, 3).

The dependence of \(\Delta p\) on \(\tau_a\) for the critical point of a cylinder with spherical bluntness is approximated by the curve

\[ \Delta p = 0.1552/(\tau_a - 0.1333) - 0.1667, \]

and in the case of a cylinder in crossflow normal to the axis, by the curve

\[ \Delta p = 0.7030/(\tau_a - 0.5870) - 0.1957. \]

The realization of the steady pressure at the critical point occurs: for a cylinder in crossflow normal to the axis, at \(\tau_a \approx 2—3.5\), and for a cylinder with spherical bluntness, at \(\tau_a \approx 0.8\) (Fig. 2, 2, 3).

The relationship between the instant at which the steady pressure is realized at the critical point of the bluntness and the value of the corresponding relative standoff \(\delta_1\) can be established by using the relation
\[ \tau_w=\frac{2wR}{a\Delta_0}\tau_0 . \]
Substituting here the values of \(\tau_a\) corresponding to the instant at which the steady pressure is realized, we obtain the dependence of \(\delta_1\) on \(M\) (Fig. 4), on the basis of which one can draw an interesting conclusion: the steady pressure at the critical point is established faster than the steady bow wave is formed. The delay in the formation of the bow wave is especially pronounced for small Mach numbers; for example, in the case of a cylinder with a flat bluntness, at \(M \approx 3.0\) the steady pressure on the surface of the flat bluntness is realized when the shock wave has moved away to a distance amounting to about 60% of the distance corresponding to the steady bow wave (Fig. 4). At larger Mach numbers (of the order of 5.5), establishment of the pressure on the flat bluntness occurs at \(\delta_1 \approx 0.9\). It may be expected that for \(M > 6\) the completion of bow-wave formation will occur practically simultaneously with the establishment of the steady pressure on the surface.

The author thanks S. S. Semenov for useful discussions of the article, Kh. A. Rakhmatulin for his attention to the work, and also V. A. Braslavets, V. S. Lelekov, V. P. Fedotov, A. I. Kharitonov, and Yu. A. Tsvetaev for their participation in developing the measurement technique and carrying out the experiments.

CITED LITERATURE

1. V. V. Rusanov, Zhurn. vychislit. matem. i matem. fiz., 1, 2, 267 (1961).
2. G. B. Whitham, J. Fluid Mech., 2, 2 (1957).
3. G. B. Whitham, J. Fluid Mech., 5, 3, 369 (1959).
4. A. E. Bryson, R. W. F. Gross, J. Fluid Mech., 10, 1, 1 (1961).
5. A. I. Golubinskii, Inzh. zhurn., 3, 3, 442 (1963).
6. D. G. Pack, J. Fluid Mech., 18, 4, 549 (1964).
7. F. V. Shugaev, Zhurn. prikl. mekh. i tekhn. fiz., No. 6, 104 (1963).
8. M. P. Syshchikova, M. K. Berezkina, A. N. Semenov, ZhETF, 34, 11, 2015 (1964).
9. Collection: Shock Tubes, IL, 1962.
10. G. D. Salamandra, T. V. Bazhenova et al., Some Methods for Investigating Rapid Processes and Their Application to the Study of Detonation-Wave Formation, Publishing House of the Academy of Sciences of the USSR, 1960.