Full Text
Yu. P. Raiser
MOTION IN AN INHOMOGENEOUS ATMOSPHERE CAUSED BY A SHORT-DURATION PLANE IMPACT
(Presented by Academician Ya. B. Zel’dovich, 17 VI 1963)
HYDROMECHANICS
1. In the works of A. S. Kompaneets \((^{1,2})\), a strong point explosion in an inhomogeneous atmosphere is considered approximately. The solution \((^{1,2})\) loses validity when the shock wave moves upward from the point of explosion to very large distances and the pressure in the cavity that has formed falls to a very small value. The further propagation of the shock wave downward has much in common with the motion considered in the well-known problem of a short-duration impact on the surface of a gas bordering on a vacuum \((^{3-5})\).
Below the problem of a short-duration impact is solved for the case of an inhomogeneous atmosphere.
2. Let the initial density be distributed in space according to the barometric law
\[ \rho_0=\rho^{*}e^{x/\Delta},\qquad \Delta=\text{const}. \tag{1} \]
The initial pressure is equal to zero. At the moment \(t=0\), in a region of very small density at \(x\approx -\infty\), a plane impact or explosion is produced. A shock wave runs through the gas in the direction of increasing density, while the heated gas expands toward the vacuum. In the limiting motion there is a length scale \(\Delta\), but no scales of density or time; the coordinate \(x\) is determined only up to an additive constant\(^*\). Therefore the velocity of the shock-wave front is
\[ D=\dot X=\alpha\,\frac{\Delta}{t}, \tag{2} \]
where the coefficient \(\alpha\) depends only on the adiabatic exponent \(\gamma\). The coordinate of the front increases with time as
\[ X=\alpha\Delta\ln t+\text{const}. \tag{3} \]
The shock wave captures a mass of gas
\[ M=\int_{-\infty}^{X}\rho_0\,dx=\rho_0(X)\Delta, \tag{4} \]
which, by virtue of the condition \(\dot M=\rho_0(X)\dot X\), is equal to
\[ M=At^{\alpha}. \tag{5} \]
The integration constant \(A\), of dimension \(\text{g}\cdot\text{cm}^{-2}\cdot\text{s}^{-\alpha}\), is a parameter characterizing the intensity of the impact.
3. The dimensional features of the problem make it possible to represent the velocity, density, and pressure of the gas in the form
\[ u=\frac{2}{\gamma+1}\alpha\frac{\Delta}{t}\,v=u_{\phi}v,\qquad \rho=\frac{\gamma+1}{\gamma-1}\frac{At^{\alpha}}{\Delta}\,q=\rho_{\phi}q, \tag{6} \]
\[ p=\frac{2}{\gamma+1}\alpha^2\frac{\Delta A}{t^{2-\alpha}}\,f=p_{\phi}f, \]
\[ \overline{\phantom{aa}} \]
\(^*\) The quantity \(\rho^{*}\) is indeterminate because of arbitrariness in the choice of the origin for \(x\).
where \(u_{\phi}, \rho_{\phi}, p_{\phi}\) are the values at the shock-wave front, while the functions \(v, q, f\) depend only on the dimensionless distance \(\xi \equiv \dfrac{X-x}{\Delta}\), measured from the front, and on \(\gamma\). At the shock-wave front, for \(\xi=0\), \(v=q=f=1\).
The motion has a somewhat unusual self-similarity: the profiles of the gas-dynamic quantities move together with the shock-wave front, without stretching with time. However, in Lagrangian coordinates the motion is self-similar in the usual sense. Indeed, the mass Lagrangian coordinate is equal to
\[ m=\int_{-\infty}^{x}\rho(x)\,dx=\mathrm{const}\cdot M\int_{\xi}^{\infty}q(\xi)\,d\xi, \]
i.e., \(\xi\), and consequently \(v,q,f\), are functions of the self-similar variable \(\eta=m/M=m/At^{\alpha}\).
- Substituting expressions (6) into the equations of gas dynamics written in Lagrangian coordinates,
\[ \frac{\partial u}{\partial t}+\frac{\partial p}{\partial m}=0,\qquad \frac{\partial(1/\rho)}{\partial t}-\frac{\partial u}{\partial m}=0,\qquad p\rho^{-\gamma}=F(m). \tag{7} \]
we obtain equations for the functions—the representatives \(v(\eta), q(\eta), f(\eta)\):
\[ v+\alpha\eta v'=\alpha f';\qquad \frac{1}{q}+\eta\left(\frac{1}{q}\right)'=-\frac{2}{\gamma-1}v';\qquad fq^{-\gamma}\eta^{2/\alpha+\gamma-1}=1. \tag{8} \]
Integrating the second equation and eliminating \(q\) and \(v\) from the system, we find the basic equation of the problem:
\[ \frac{df}{d\eta} = \frac{\gamma+1}{2\alpha} \frac{ 1-\dfrac{\gamma-1}{\gamma+1}\left(1-\dfrac{2-\alpha}{\gamma}\right)f^{-1/\gamma}\eta^{-(2-\alpha)/\alpha\gamma} }{ 1-\dfrac{\gamma-1}{2\gamma}f^{-1/\gamma-1}\eta^{-(2-\alpha)/\alpha\gamma} }. \tag{9} \]
Since at \(x=-\infty\), or \(\eta=0\), the pressure is zero, the solution must pass through two points
\[ \eta=1,\quad f=1;\qquad \eta=0,\ f=0, \tag{10} \]
which determines the exponent \(\alpha\).
- In the case \(\gamma=2\), it is possible to find an exact analytic solution of the problem*. We have: \(\alpha=3/2\), \(M\sim t^{3/2}\), \(\dot X=3/2\,\Delta/t\),
\[ u_{\phi}\sim 1/t,\quad \rho_{\phi}\sim t^{3/2},\quad p_{\phi}\sim 1/t^{1/2}, \]
\[ f=\eta,\qquad q=\eta^{5/3},\qquad v={}^{3}/_{2}\left(1-{}^{1}/_{3}\eta^{-2/3}\right). \tag{11} \]
In Eulerian coordinates:
\[ f=(1+2\xi)^{-3/2},\quad q=(1+2\xi)^{-5/2},\quad v=1-\xi. \tag{12} \]
An analytic solution is also possible for \(\gamma=1\): \(\alpha=1\), \(f=\eta\), \(q=\eta^3\), \(v=1\). This case is of interest only from the point of view of bounding the exponent \(\alpha\), since it corresponds to infinite compression of the gas at the wave front, as a result of which, in Eulerian coordinates, \(v,q,f\) become \(\delta\)-functions.
Since the real values of \(\gamma\) for gases lie in the range \(1<\gamma<2\), it must be assumed that the corresponding values of \(\alpha\) lie in the interval \(1<\alpha<3/2\)**.
* This solution, written in Lagrangian coordinates, is completely analogous to the analytic solution of the usual problem of a short-time impulse, found for the special case \(\gamma=7.5\) (5).
** Consideration of the energy and momentum balances, analogous to (3), leads to the general restriction \(1<\alpha<2\).
Equation (9), as usual, has a singular point of saddle type, through which passes the integral curve corresponding to the solution. In the general case of an arbitrary \(\gamma\), the exponent \(\alpha\) and the solution are found by numerical integration by the trial method (see (4)). Figures 1 and 2 show the distributions, obtained in this way, of \(p\), \(\rho\), and \(u\) behind the shock wave for the case \(\gamma = 1.25\). In this case \(\alpha = 1.345\).
Fig. 1. Distribution of the pressure \(f\), density \(q\), and velocity \(v\) behind the shock wave in the mass coordinate
Fig. 2. Distribution of the pressure \(f\), density \(q\), and velocity \(v\) in space behind the shock wave
6. Let us return to the question of a point explosion in an inhomogeneous atmosphere. The motion of the shock wave downward acquires the character described here when the pressure in the cavity \(p_c\) becomes much smaller than the pressure at the shock front \(p_\phi\).* Suppose, for definiteness, that the transition to the new regime occurs at \(p_\phi/p_c = 10\). According to (2), this value corresponds to the time from the moment of explosion
\(t_1 \approx 19\tau\), where the time scale \(\tau\) is equal to
\(\tau = (\rho_{00}\Delta^5/E)^{1/2}\) (\(\rho_{00}\) is the air density at the point of explosion, \(E\) is the explosion energy). By the time \(t_1\), the shock wave has moved downward from the point of explosion by a distance \(z = 1.9\Delta\); the front velocity at this time is equal to
\(D_1 = 2.5 \cdot 10^{-2}\Delta/\tau\).
Let us extrapolate the limiting laws of propagation of the shock wave (3), (2) to the transition (“initial”) moment, and let us choose the coordinate and time in such a way that the initial condition \(D = D_1\) at \(X = 0\) is satisfied. We obtain, approximately with logarithmic accuracy,
\[ X = \alpha \Delta \ln \frac{D}{D_1} = \alpha \Delta \ln \frac{t}{\theta}, \]
where the parameters \(D_1\) and \(\theta\) are determined in terms of the explosion parameters by the expressions
\[ D_1 = 2.5 \cdot 10^{-2}\Delta/\tau = 2.5 \cdot 10^{-2}(E/\rho_{00}\Delta^3)^{1/2}, \qquad \theta = 40\alpha\tau^{**}. \]
In the same approximation,
\[ A = e^{1.9}\rho_{00}\Delta \theta^{-\alpha}. \]
An estimate using real numerical values of the parameters shows that, in the process of deceleration of the shock wave from the “transition” velocity \(D_1\) to a velocity of the order of \(1\) km/sec, several times greater than the speed of sound
* In a strong point explosion in a homogeneous atmosphere, \(p_\phi/p_c \approx 2.5\) (6).
** Let us note that the numerical values of \(D_1\) and \(\theta\) depend only weakly on the choice of the transition value \(p_\phi/p_c\). Thus, at the last moment calculated in (2), \(t \approx 23.4\tau\), close to the moment of “breakthrough” of the atmosphere, \(z = 2.0\Delta\), \(D_1 = 2.12 \cdot 10^{-2}\Delta/\tau\), \(p_\phi/p_c = 22\).
in cold air, the shock wave travels downward a distance of order \((2—3)\Delta\), which is added to the distance \(\simeq 2\Delta\) that follows from the theory \((^{1,2})\).
I express my gratitude to Ya. B. Zel’dovich and A. S. Kompaneets for their interest in the work, and to L. S. Nikol’skaya for the numerical calculations.
Received
7 VI 1963
REFERENCES
\(^{1}\) A. S. Kompaneets, DAN, 130, No. 5 (1960).
\(^{2}\) E. I. Andriankin, A. M. Kogan, A. S. Kompaneets, V. P. Krainov, Zhurn. prikl. mekh. i tekhn. fiz., No. 6, 3 (1962).
\(^{3}\) Ya. B. Zel’dovich, Akust. zhurn., 2, 1, 28 (1956).
\(^{4}\) V. B. Adamskii, Akust. zhurn., 2, No. 1, 3 (1956).
\(^{5}\) A. I. Zhukov, Ya. M. Kazhdan, Akust. zhurn., 2, No. 4, 352 (1956).
\(^{6}\) L. I. Sedov, Similarity and Dimensional Methods in Mechanics, 4th ed., Moscow, 1957.