UDC 532.1

HYDROMECHANICS

K. DZHUSUPOV, K. P. STANYUKOVICH

A REFLECTED ONE-DIMENSIONAL RAREFACTION WAVE IN A CONSTANT GRAVITATIONAL FIELD

(Presented by Academician L. I. Sedov on 2 IX 1969)

In the one-dimensional outflow of the products of an instantaneous explosion, situated in a constant gravitational field and obeying the equation of state \(P=\operatorname{const}\cdot \rho^k\), a rarefaction wave will travel to the left into the gas at rest; the solution of the equation for it was obtained for an arbitrary adiabat \(k\) in general form in work \((^1)\). The aim of the present article is to determine the motion that arises when the rarefaction wave is reflected from a rigid wall.

For the indicated flow the equations of gas dynamics, written in the form

\[ \begin{aligned} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} +\frac{2}{k-1}\,c\frac{\partial c}{\partial x}&=-a,\\ \frac{\partial c}{\partial t}+u\frac{\partial c}{\partial x} +\frac{k-1}{2}\,c\frac{\partial u}{\partial x}&=0, \end{aligned} \tag{1} \]

have the general solution \((^1)\)

\[ \psi=\frac{\partial^{\,n-1}}{\partial i^{\,n-1}} \frac{ F_1\!\left[\sqrt{2(2n+1)i+w}\right] + F_2\!\left[\sqrt{2(2n+1)i-w}\right] }{\sqrt{i}}, \]

\[ x=x_0+wt-at^2/2-\partial\psi/\partial w,\qquad t=\partial\psi/\partial i, \tag{2} \]

\[ x_0=-n(2n+1)c_{\mathrm{H}}^2/a,\qquad k=(2n+3)/(2n+1),\qquad n=0,1,2,\ldots \]

Here \(c\) is the speed of sound in the given medium; \(c_{\mathrm{H}}\) is the initial speed of sound in the section \(x=0\); \(i=c^2/(k-1)\) is the heat content of the gas; \(w=u+at\), where \(a\) is the acceleration of gravity and \(u\) is the flow velocity along the \(x\)-axis.

Let the rarefaction wave arise at the instant \(t=0\) in the section \(x=0\) and propagate in the negative direction of the \(x\)-axis. Since the compressed gas in the gravitational field before outflow was in a state of adiabatic equilibrium,

\[ c\frac{\partial c}{\partial x}=-\frac{k-1}{2}\,a,\qquad u=0 \]

or

\[ c^2=c_{\mathrm{H}}^2-(k-1)ax,\qquad u=0, \tag{3} \]

then the rarefaction wave, propagating according to the law \(dx/dt=-c\), will reach a rigid wall placed in the section \(x=-l\) after the interval

\[ t=t_l=\frac{2}{(k-1)a}\left[\sqrt{c_{\mathrm{H}}^2+(k-1)al}-c_{\mathrm{H}}\right] \tag{4} \]

and will be reflected from it. In this reflected wave the function \(F_2\) is retained:

\[ F_2=F_2[\omega-(w-\omega_{\mathrm{H}})] = \frac{1}{4a\,(n+1)!\,[2(2n+1)]^{\,n+1/2}} \times \]

\[ \times \frac{\partial^{\,n-1}}{\partial i^{\,n-1}} \frac{ \displaystyle\sum_{r=n+1}^{2n} A_r[\omega-(w+\omega_{\mathrm{H}})]^r }{\sqrt{i}}. \tag{5} \]

This is obvious, since the given function is determined by the condition

\[ w=\frac{2}{k-1}(c_{\mathrm{n}}-c) =\sqrt{2(2n+1)i_{\mathrm{n}}}-\sqrt{2(2n+1)i} =\omega_{\mathrm{n}}-\omega \]

on the right characteristic \(dx/dt=u+c\) and has a constant value on the left characteristic.

The function \(F_1=F_1[\sqrt{2(2n+1)i}+w]=F_1(\omega+w)\) must be found from the condition that \(u\equiv 0\) at \(x=-l\). Since \(F_1\) depends on arguments that are nothing other than the characteristic condition

\[ w+\frac{2}{k-1}c=\mathrm{const}, \]

where the constant at the time \(t=t_l\) is equal to

\[ \mathrm{const}=\frac{2}{k-1}\left[2\sqrt{c_{\mathrm{n}}^2+(k-1)al}-c_{\mathrm{n}}\right] =\frac{2}{k-1}c_{\mathrm{n}}^{*}, \]

the condition

\[ F_1=\mathrm{const}\quad \text{for}\quad w+\frac{2}{k-1}c=\frac{2}{k-1}c_{\mathrm{n}}^{*} \]

must be satisfied.

We seek the solution for \(\psi\) in the form

\[ \psi= -\frac{1}{4a(n+1)![2(2n+1)]^{n+1/2}} \left\{ \frac{\partial^{\,n-1}}{\partial i^{\,n-1}} \frac{\sum_{r=n+1}^{2n} A_r(\omega-w-\omega_{\mathrm{n}})^r}{\sqrt{i}} + \frac{\partial^{\,n-1}}{\partial i^{\,n-1}} \frac{F_1(\omega+w)}{\sqrt{i}} \right\}, \tag{6} \]

where the coefficients \(A_r\) are computed algebraically for each \(n\) (1).

In finding the explicit form of the function \(F_1\) for an arbitrary value of \(n\), computational difficulties arise. Below we consider the special case \(n=0\), for which it is easy to find the explicit form of the function \(F_1\), which will make it possible to construct the function \(F_1\) for other values of \(n\). For the indicated case \(F_2=0\), as is seen from (6), and \(\psi=F_1(\omega+w)\).

From (2)

\[ x=ut+at^2/2-F_1', \qquad ct=F_1'. \]

For \(x=-l\), \(u\equiv 0\), \(w=at\), \(w+c=at+c=z\), whence

\[ t=(z-c)/a,\qquad F_1'=c(z-c)/a \tag{7} \]

or

\[ aF_1'=(c-z+z)(z-c)=-(z-c)^2+z(z-c), \]

whence

\[ z-c=\frac{1}{2}\left(z\pm\sqrt{z^2-4aF_1'}\right). \]

Using the equality

\[ l=F_1'-a^2t^2/2a=F_1'-(z-c)^2/2a, \]

for \(F_1'\) we obtain the expression

\[ F_1'=\frac{3}{2}l+\frac{z^2}{9a}-\frac{z^2}{9a}\sqrt{z^2-6al} \]

or

\[ \frac{\partial \psi}{\partial w} = \frac{2}{3}l +\frac{(w+c)^2}{9a} -\frac{(w+c)}{9a}\sqrt{(w+c)^2-6al}. \tag{8} \]

Thus, knowledge of the derivative \(\partial\psi/\partial w\) determines the solution for the reflected wave.

It should be noted that this solution (for \(k=3\)) coincides with the result previously obtained by K. P. Stanyukovich by another method (the solution was “guessed”) (2):

\[ \sqrt{2ax+\alpha^2}+(at+\alpha)=2\sqrt{2ax+\alpha^2+2al}, \]

where \(\alpha=u+c\), which after an elementary transformation can be written as

\[ x=\frac{5}{9}(w+c)t-\frac{5}{18}at^2-\frac{2}{9a}(w+c)^2-\frac{4}{3}l+\frac{2(w+c)}{9a}\sqrt{(w+c)^2-6al}. \]

We obtain this same result if, in the expression \(x=wt-at^2/2-\partial\psi/\partial w\), the derivative \(\partial\psi/\partial w\) is replaced according to (8), which confirms the correctness of the approach in choosing the function \(F_1\).

For the case \(n=1\) \((k=5/3\)—a monatomic gas)

\[ i=3c^2/2,\qquad \omega=3c=\sqrt{6i}, \]

and solution (2) may be written in the form

\[ \psi=\frac{\omega_{\mathrm{H}}^2}{12a\omega}\,[\omega-(w+\omega_{\mathrm{H}})]^2+\frac{F_1(\omega+w)}{a\omega}, \tag{9} \]

\[ x=x_0+ut+\frac{at^2}{2}-\partial\psi/\partial w,\qquad t=\partial\psi/\partial i=\partial\psi/3c\,\partial c=3\,\partial\psi/\omega\,\partial\omega, \]

or, for \(x\) and \(t\), we have the expressions

\[ x=x_0+ut+\frac{at^2}{2}+\frac{\omega_{\mathrm{H}}^2}{6a\omega}\,[\omega-(w+\omega_{\mathrm{H}})]-\frac{F_1'}{a\omega}, \]

\[ t=-\frac{\omega_{\mathrm{H}}^2}{4a\omega^3}\,[\omega-(w+\omega_{\mathrm{H}})]^2 +\frac{\omega_{\mathrm{H}}^2}{2a\omega^2}\,[\omega-(w+\omega_{\mathrm{H}})] +\frac{3F_1'}{a\omega^2}-\frac{3F_1}{a\omega^3}. \]

Hence, under the condition that \(u=0,\ w=at\), at \(x=-l\) we have

\[ -l=x_0+\frac{at^2}{2}+\frac{\omega_{\mathrm{H}}^2}{6a\omega}\,[\omega-(at+\omega_{\mathrm{H}})]+\frac{F_1'}{u\omega}, \tag{10} \]

\[ t=-\frac{\omega_{\mathrm{H}}^2}{4a\omega^3}\,[\omega-(at+\omega_{\mathrm{H}})]^2 +\frac{\omega_{\mathrm{H}}^2}{2a\omega^2}\,[\omega-(at+\omega_{\mathrm{H}})] +\frac{3F_1'}{a\omega^2}-\frac{3F_1'}{a\omega^3}. \tag{11} \]

If we introduce a new variable

\[ z=w+\omega=at+\omega, \]

then

\[ F_1=F_1(w+\omega)=F_1(\omega+at)=F_1(z), \]

\[ at=at+\omega-\omega=z-\omega,\qquad \omega-(at+\omega_{\mathrm{H}})=2\omega-\omega_{\mathrm{H}}-z, \]

and relations (10) and (11) will take, respectively, the form

\[ -3(l+x_0)a=\frac{3(z-\omega)^2}{2} +\frac{\omega_{\mathrm{H}}^2}{2\omega}(2\omega-\omega_{\mathrm{H}}-z) +\frac{2F_1'}{\omega}, \]

\[ \omega(z-\omega)=-\frac{\omega_{\mathrm{H}}^2}{4\omega^2}(2\omega-\omega_{\mathrm{H}}-z)^2 +\frac{\omega_{\mathrm{H}}^2}{2\omega}(2\omega-\omega_{\mathrm{H}}-z) +\frac{3F_1'}{\omega}-\frac{3F_1'}{\omega^2}. \tag{12} \]

We transform the system of equations (12). The first equation gives

\[ F_1'=\frac{\omega_{\mathrm{H}}^2}{6}(\omega_{\mathrm{H}}+z) -\omega\left[\frac{\omega_{\mathrm{H}}^2}{3}+\frac{z^2}{2}+a(l+x_0)\right] +\frac{z\omega^2}{2}-\frac{\omega^3}{2}. \]

Hence, solving the cubic equation, we find \(\omega=\omega(F_1';z)\); substituting \(\omega=\omega(F_1';z)\) into the second equation, we arrive at the equation

\[ \Phi(F_1';F_1;z)=0, \tag{13} \]

solving which we formally find \(F_1 = F_1(z)\), which completely solves the posed problem.

It is obvious that already for \(k = 5/3\) \((n = 1)\) the problem is algebraically complicated, but can nevertheless be carried through to the end. For \(n = 2, 3, 4, \ldots\) this becomes impossible, since the equations contain \(\omega\) in a degree higher than the fourth. Therefore the problem must be solved approximately, approximating the isentrope \(P = A_0 \rho^k\) by the isentrope \(P = A \rho^3 - B\); in this case the accuracy will be sufficient, since in the reflected wave the pressure does not depend very strongly on the coordinate \(x\).

Moscow State University
named after M. V. Lomonosov

Received
7 V 1969

REFERENCES

1. K. P. Stanyukovich, K. Dzhusupov, DAN, 177, No. 4 (1967).
2. K. P. Stanyukovich, Unsteady Motions of a Continuous Medium, § 76, Moscow, 1955.