Hydromechanics

V. A. Smirnov

Interaction of a Simple Wave with a Contact Discontinuity

(Presented by Academician A. A. Dorodnitsyn, 7 III 1960)

The problem considered is an element of many practical problems of one-dimensional gas flow. Numerical computation is possible by the method of characteristics or by the method of finite differences (¹).

Below an exact solution of the problem is given for a discrete series of values of the adiabatic exponent \(\kappa\), defined in the form

\[ \kappa=\frac{2n+3}{2n+1},\quad n\ \text{integer}. \tag{1} \]

The scheme of the flow regions when a simple wave passes through a contact discontinuity is given in Fig. 1.

In general, 4 cases are possible (a combination of a rarefaction or compression wave and a discontinuity in which the speed of sound on the right is greater \(T_>\) or less \(T_<\)). Shown here is a rarefaction wave passing through the discontinuity \(T_>\).

In regions \(I\), \(III\), and \(IV\) there are simple waves, respectively incident, reflected, and transmitted. In region \(II\) there is a general unsteady flow, representing the interaction of the incident simple wave with the wave reflected from the contact discontinuity.

For the discrete values of \(\kappa\) (or \(n\)) under consideration, the flow in region \(II\) is determined by a function \(\chi(c,v)\) (²) of the form

\[ \chi(c,v)= \left(\frac{\partial}{c\,\partial c}\right)^{n-1} \left[ \frac{1}{c}\,\varphi\left(c+\frac{\kappa-1}{2}v\right) + \frac{1}{c}\,\psi\left(c-\frac{\kappa-1}{2}v\right) \right], \tag{2} \]

where \(v\) is the velocity of the gas particles; \(c\) is the speed of sound; \(\varphi\) and \(\psi\) are arbitrary functions.

The coordinates of the flow plane are expressed through \(\chi(c,v)\) in the form

\[ t=\frac{\kappa-1}{2c}\frac{\partial\chi}{\partial c}, \quad x=vt-\frac{\partial\chi}{\partial v}. \tag{3} \]

The functions \(\varphi\) and \(\psi\) in the problem under consideration are determined by the boundary conditions on the characteristic \(AB\) and the segment \(AC\) of the contact discontinuity. The condition on \(AB\) will be (²)

\[ \chi(c,v)=-\int f(v)\,dv \quad \text{for}\quad c-\frac{\kappa-1}{2}v=c_0. \tag{4} \]

Here \(f(v)\) is a function characterizing the incident simple wave:

\[ x=(v+c)t+f(v); \]

\(c_0\) is the speed of sound in the undisturbed gas to the left of the contact discontinuity.

This condition gives an ordinary linear differential equation of order \((n-1)\) with a right-hand side for the function \(\varphi\). Condition (4) has been studied in solved problems on the interaction of simple waves (³), reflection of a simple wave from a wall, and from a free boundary. It has been shown

(3), that the general solution of the homogeneous equation corresponding to (4) may be set equal to zero, since it represents \(\varphi(u)\) in the form of a polynomial in \(u\) with coefficients proportional to \(b_1, b_2, \ldots, b_{n-1}\), \(\psi(c_0), \psi'(c_0), \ldots, \psi^{(n-1)}(c_0)\) (where \(b_1, b_2, \ldots, b_{n-1}\) are constants of the general solution), and all these constants are immaterial, since they disappear in the expression \(\chi(c,v)\).

Thus, condition (4) determines the function \(\varphi(u)\) as a particular solution of the nonhomogeneous linear equation following from (4), if one sets

\[ \psi(c_0)=\psi'(c_0)=\ldots=\psi^{(n-1)}(c_0)=0. \tag{5} \]

The condition at the contact discontinuity is obtained from the following considerations:

1. The discontinuity does not move relative to the gas particles, i.e., on it

\[ \frac{dx}{dt}=v. \tag{6} \]

Fig. 1

1. The flow to the right of the discontinuity is a simple wave, in which

\[ c_2-\frac{\varkappa-1}{2}v=c_{20} \tag{7} \]

(\(c_2\) is the speed of sound to the right of the discontinuity, \(c_{20}\) is the speed of sound in the undisturbed gas on the right).

1. The ratio of the densities \(\rho_1/\rho_2\) on both sides of the discontinuity does not change when the simple wave passes. This follows from the isentropic nature of the flow on each side of the discontinuity,

\[ \frac{\rho_1}{\rho'_1}=\left(\frac{p_1}{p'_1}\right)^{\varkappa}, \qquad \frac{\rho_1}{\rho'_1}=\left(\frac{\rho_2}{\rho'_2}\right)^{\varkappa}, \]

whence

\[ \frac{\rho_1}{\rho_2}=\frac{\rho'_1}{\rho'_2}=\mathrm{const}. \]

1. The pressure is continuous in passing through the contact discontinuity. It follows from this that

\[ \frac{c}{c_2}= \sqrt{\frac{\varkappa p}{\rho_1}}\Big/ \sqrt{\frac{\varkappa p}{\rho_2}} = \sqrt{\frac{\rho_2}{\rho_1}}, \]

and relation (7) may be written on the contact discontinuity for the gas on the left in the form

\[ c\sqrt{\frac{\rho_1}{\rho_2}}-\frac{\varkappa-1}{2}v = c_0\sqrt{\frac{\rho_1}{\rho_2}}. \tag{7a} \]

Thus, it must be that

\[ \frac{dx}{dt}=v \quad\text{along the line on which}\quad c=c_0+\frac{\varkappa-1}{2}\sqrt{\frac{\rho_2}{\rho_1}}\,v. \]

Using (3), after simple calculations we obtain the condition at the contact discontinuity for the flow in region \(II\):

\[ \frac{\partial^2\chi}{\partial v^2} + \frac{\varkappa-1}{2}\sqrt{\frac{\rho_2}{\rho_1}} \frac{\partial^2\chi}{\partial v\,\partial c} - \frac{\varkappa-1}{2c} \frac{\partial\chi}{\partial c} =0 \quad \text{for } c=c_0+\frac{\varkappa-1}{2}\sqrt{\frac{\rho_2}{\rho_1}}\,v. \tag{8} \]

This condition gives an ordinary differential Euler equation with a right-hand side for determining the function \(\psi\).

For example, for \(\chi = 1.4\) \((n = 2)\) and for the case of a centered rarefaction wave \((f(v) = 0)\), from condition (4) the function \(\varphi(u) \equiv 0\). Then condition (8) for determining \(\psi(u)\) gives the equation

\[ a^{3}c^{3}\xi''' + 3(2-a)a^{2}c^{2}\xi'' + 3(5-a)(1-a)ac\xi' + 15(1-a)^{2}\xi = 0, \]

where \(a = \sqrt{\rho_{2}/\rho_{1}}\), \(\xi = \xi(c)\), \(\psi(u) = \xi[(au-c_{0})/(1-a)]\).

The constants entering into the solution of the equation following from (8) are determined by conditions (5) and by the condition that the coordinates of point \(A\) of the contact discontinuity, determined from (3) (for \(c = c_{0}\), \(v = 0\)), coincide with the values determined directly from the given incident wave and the initial position of the contact discontinuity (one condition).

Institute of Mechanics
Academy of Sciences of the USSR

Received
2 III 1960

REFERENCES

¹ R. Courant, K. Friedrichs, Supersonic Flow and Shock Waves, IL, 1950.
² L. D. Landau, E. M. Lifshitz, Mechanics of Continuous Media, Moscow, 1954.
³ K. P. Stanyukovich, Theory of Unsteady Gas Motions, Moscow, 1948.