UDC 533.6.011.72

AERODYNAMICS

M. P. MIKHAILOVA

MOTION OF A PISTON IN A HEAT-CONDUCTING AND VISCOUS MEDIUM

(Presented by Academician L. I. Sedov on 22 X 1965)

Consider the one-dimensional motion of a gas ahead of a piston that moves with constant velocity \(U\) in a heat-conducting and viscous medium. The basic equations of one-dimensional unsteady motion of a viscous and heat-conducting medium have the form

\[ \frac{\partial \rho}{\partial t} + \frac{\partial(\rho u)}{\partial r} + (\nu-1)\frac{\rho u}{r}=0, \]

\[ \frac{\partial\left(r^{\nu-1}\rho u\right)}{\partial r} + \frac{\partial}{\partial r} \left[ r^{\nu-1}\left(\rho u^2-p_{rr}\right) \right] + (\nu-1)r^{\nu-2}p_{nn}=0, \]

\[ \frac{\partial}{\partial t} \left[ \rho r^{\nu-1}\left(\frac{u^2}{2}+\varepsilon\right) \right] + \frac{\partial}{\partial r}r^{\nu-1} \left[ \rho u\left(\frac{u^2}{2}+\varepsilon\right)-p_{rr}u-\chi\frac{\partial T}{\partial r} \right]=0, \tag{1} \]

\[ p_{rr}=-p+\lambda \left[ \frac{\partial u}{\partial r}+(\nu-1)\frac{u}{r} \right] +2\mu\frac{\partial u}{\partial r}, \]

\[ p_{nn}=-p+\lambda \left[ \frac{\partial u}{\partial r}+(\nu-1)\frac{u}{r} \right] +2\mu\frac{u}{r}, \]

where \(t\) is time; \(r\) is the coordinate; \(\rho\) is density; \(u\) is the velocity component; \(\varepsilon\) is internal energy; \(T\) is temperature; \(\chi\) is the coefficient of heat conductivity; \(p\) is pressure; \(p_{rr}, p_{nn}\) are stress coordinates; \(\lambda, \mu\) are viscosity coefficients; \(\nu=1,2,3\), respectively, for plane, cylindrical, and spherical pistons. Suppose that \(p=R\rho T\) and \(\varepsilon=C_vT\), where \(R\) is the gas constant and \(C_v\) is the specific heat conductivity.

Instead of the dimensional independent variables \(t\) and \(r\), take the dimensionless independent variables
\[ \xi=\frac{\gamma p_1}{\rho_1}\frac{t^2}{r^2} \quad\text{and}\quad \eta=\frac{\chi t}{\rho_1 C_v r^2}, \]
where \(\rho_1, p_1\) are the density and pressure ahead of the shock wave. We also introduce the dimensionless constants \(q_1=2\mu C_v/\chi\) and \(q_2=\lambda C_v/\chi\). The sought dimensional functions are represented in terms of dimensionless functions depending on the dimensionless variables by

\[ u=\frac{r}{t}V_1(\xi,\eta),\qquad p=\rho_1\frac{r^2}{t^2}P_1(\xi,\eta),\qquad \rho=\rho_1R_1(\xi,\eta). \tag{2} \]

Let \(V_0(\xi), P_0(\xi), R_0(\xi)\) be the solutions of the problem for self-similar motion obtained by L. I. Sedov \((^1)\). Then the required linearized solutions can be represented in the form

\[ V_1=V_0(\xi)+\eta V(\xi),\qquad P_1=P_0(\xi)+\eta P(\xi),\qquad R_1=R_0(\xi)+\eta R(\xi). \tag{3} \]

After passing to dimensionless variables and varying with respect to \(\eta\), we obtain a system of ordinary differential equations for \(V, P, R\). Assuming that the velocity of the piston is large and that \(\xi^2\) may be neglected, the system takes the form

\[ 2\xi(1-V_0)\,dR/d\xi - 2\xi R_0\,dV/d\xi + \left[(\nu-2)R_0-2\xi\,dR_0/d\xi\right]V + \left[(\nu-2)V_0+1-2\xi\,dV_0/d\xi\right]R =0, \]

\[ \begin{aligned} &2\xi(1-2V_0)R_0\,dV/d\xi + 2\xi V_0(1-V_0)\,dR/d\xi - 2\xi\,dP/d\xi \\ &\quad + 2\left\{\xi(1-2V_0)\,dR_0/d\xi + \left[(\nu-1)V_0-2\xi\,dV_0/d\xi\right]R_0\right\}V \\ &\quad + \left[2\xi(1-2V_0)\,dV_0/d\xi+\nu V_0^2\right]R + 2\xi(\nu-2)(q_1+2q_2)\,dV_0/d\xi =0, \end{aligned} \tag{4} \]

\[ \begin{aligned} &\left[V_0 R_0 (2-3V_0)-\frac{2\gamma P_0}{\gamma-1}\right]\xi \frac{dV}{d\xi} +2\xi \frac{1-\gamma V_0}{\gamma-1}\frac{dP}{d\xi} +\xi V_0^2(1-V_0)\frac{dR}{d\xi}+\\ &+\left[2\xi V_0\left(1-\frac{3}{2}V_0\right)\frac{dR_0}{d\xi} +2\xi R_0(1-3V_0)\frac{dV_0}{d\xi} -V_0R_0+\frac{3}{2}\nu V_0^2R_0+\right.\\ &\left.\qquad+\frac{\nu\gamma}{\gamma-1}P_0-\frac{2\xi\gamma}{\gamma-1}\frac{dP_0}{d\xi}\right]V +\frac{1}{\gamma-1}\left(\nu\gamma V_0-1-2\xi\gamma\frac{dV_0}{d\xi}\right)P+\\ &+\left[\frac{V_0^2}{2}(\nu V_0-1) +2\xi V_0\left(1-\frac{3}{2}V_0\right)\frac{dV_0}{d\xi}\right]R -\nu(q_1+\nu q_2)V_0^2+\\ &+(\nu q_1+(3\nu-2)q_2) \left[2\xi V_0\frac{dV_0}{d\xi} -\frac{2\gamma}{\gamma-1}\left(\frac{P_0}{R_0} -\frac{\xi}{R_0}\frac{dP_0}{d\xi} +\xi\frac{P_0}{R_0^2}\frac{dR_0}{d\xi}\right)\right]=0. \end{aligned} \]

First we solve the system of homogeneous differential equations.

Fig. 1

Fig. 2

We denote these solutions by \(\overline V, \overline P, \overline R\). From the independent variable \(\xi\) we pass to the independent variable \(V_0\) and seek solutions in the form

\[ \overline V=(1-V_0)^s\sum_{n=0}^{\infty}a_{ni}(1-V_0)^n,\qquad \overline P=(1-V_0)^s\sum_{n=0}^{\infty}b_{ni}(1-V_0)^n, \]

\[ \overline R=(1-V_0)^s\sum_{n=0}^{\infty}c_{ni}(1-V_0)^n,\qquad i=1,2,3,4. \tag{5} \]

We also represent \(\xi, P_0, R_0\) as series in powers of \((1-V_0)\)

\[ \xi=\xi_n\left[1-\frac{2}{\nu}(1-V_0)+\cdots\right],\qquad P_0=P_{0n}\left[1-\frac{2}{\nu}(1-V_0)+\cdots\right], \]

\[ R_0=R_{0n}\left[1-\frac{\nu-1}{2\nu}\frac{R_{0n}}{\gamma P_{0n}}(1-V_0)^2+\cdots\right]; \tag{6} \]

\(\xi_n, P_{0n}, R_{0n}\) are the values of the functions at the piston.

The characteristic equation of the system is

\[ s^2\left[\frac{\gamma}{\gamma-1}\frac{P_{0n}}{R_{0n}} \left(s+\frac{1}{\nu}\right)-\frac{1}{\nu}\right]=0. \tag{7} \]

The roots \(s_1=s_2=0\) correspond to solutions with a logarithmic singularity \((^2)\). For \(\gamma=1.4\), the characteristic root \(s_3=13/35,\; 5/14,\; 29/105\), respectively for \(\nu=1,2,3\).

Particular solutions of the system of differential equations with the right-hand side are likewise sought in the form of series in powers of \((1-V_0)\).

The required solutions must satisfy the boundary conditions at the piston and at the shock wave. At the piston \(u=U\), and since \(V_0=1\), then

$V(1)=0$. Ahead of the shock wave the gas is at rest, the density and pressure are constant, and therefore the conditions at the shock wave can be written in the form

\[ \rho_1 c=\rho_2(c-u_2), \qquad p_{rr_1}=\rho_1 c u_2+p_{rr_2}, \tag{8} \]

\[ \frac{c p_1}{\gamma-1} = \rho_1 c\left(\frac{u_2^2}{2}+\frac{p_2}{(\gamma-1)\rho_2}\right) +p_{rr_2}u_2 +\frac{\kappa}{C_v(\gamma-1)}\,\frac{\partial}{\partial r}\frac{p_2}{\rho_2}, \]

where $c$ is the velocity of the shock wave. The conditions at the shock wave after passing to dimensionless variables and varying with respect to $\eta$ have the form

\[ \left( R_2\left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*} -2\xi^* a\left.\frac{dR_0}{dV_0}\right|_{V_0=V_0^*} \right)(1-V_0^*)- \]

\[ - R_0\left[ (aV_0^*+V_2)\left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*} +2a\xi^* \right]=0 \]

\[ (aP_0^*+P_2)\left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*} -2a\xi^*\left.\frac{dP_0}{dV_0}\right|_{V_0=V_0^*} -(\nu q_2+q_1)V_0^*\left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*} + \]

\[ +(a+q_1+q_2)2\xi^* = V_2\left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*}, \tag{9} \]

\[ 2\frac{1}{\gamma-1} \left( \frac{P_0^*}{R_0}\left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*} -\xi^*\left.\frac{d}{dV_0}\frac{P_0}{R_0}\right|_{V_0=V_0^*} \right) +aV_0^*\left(\left.V_0^*\right|_{V_0=V_0^*}+2\xi^*\right) = \]

\[ = \left( V_0^*V_2-\frac{P_2}{R_0}+\frac{P_0^*}{R_0^2}R_2 \right) \left.\frac{d\xi}{dV_0}\right|_{V_0=V_0^*}, \]

where $V_0^*$ is the value of $V_0$ at the shock wave, and $a$ is a constant.

The radius vector of the shock wave is represented in the form

\[ r_2=r_{20}(1+a\eta+\ldots), \tag{10} \]

where $r_{20}$ is the radius vector of the shock wave for self-similar motion.

The solutions for $V$, $P$, and $R$ have the following form, if one retains only $1-V_0$ to the first degree:

Fig. 3

\[ V=(1-V_0)\left\{ [1+\ln(1-V_0)]a_{11} +\left.\frac{\partial a_{11}}{\partial s}\right|_{s=0} +a_1 \right\}+\ldots, \]

\[ P=[1+\ln(1-V_0)][b_{01}+b_{11}(1-V_0)] +\left( \left.\frac{\partial b_{11}}{\partial s}\right|_{s=0} +b_1 \right)(1-V_0)+\ldots, \]

\[ R=[1+\ln(1-V_0)][c_{01}+c_{11}(1-V_0)] +\left.\frac{\partial c_{01}}{\partial s}\right|_{s=0} +\left.\frac{\partial c_{11}}{\partial s}\right|_{s=0}(1-V_0)+ \]

\[ +c_{03}(1-V_0)^{s_3}+c_0+c_1(1-V_0)+\ldots \tag{11} \]

In these formulas the condition at the piston has already been taken into account. The constants $b_{01}$, $c_{03}$, and $a$ are found from the conditions at the shock wave, while the constants $a_1$, $b_1$, $c_0$, $c_1$ are known coefficients of particular solutions of the inhomogeneous system.

Computations of $V$, $P$, and $R$ were carried out for Prandtl number $\Pr=1$ for $\nu=1, 2, 3$; they are presented in the form of graphs (see Figs. 1, 2, and 3).

Moscow Institute
of Railway Transport Engineers

Received
5 X 1965

References

1. L. I. Sedov, Similarity and Dimensional Methods in Mechanics, Moscow, 1961.
2. G. T. Padé, Integration of Differential Equations, Moscow–Leningrad, 1933.