Abstract Generated abstract
This paper studies one-dimensional gas motion behind a piston moving at constant velocity in a medium with a weak power-law density inhomogeneity. Using dimensional variables, the solution is written as a perturbation of Sedov’s self-similar homogeneous-medium solution, leading after linearization to ordinary differential equations for corrections to velocity, pressure, and density. The analysis develops series solutions near the piston in terms of the homogeneous velocity variable, identifies logarithmic singular behavior from the characteristic equation, and applies strong-shock boundary conditions to determine the remaining constants. Explicit approximate correction formulas are given for both noninteger and integer values of the inhomogeneity exponent divided by the wave geometry parameter, with computed examples for plane, cylindrical, and spherical cases when the exponent is one.
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HYDROMECHANICS
M. P. MIKHAILOVA
MOTION OF A PISTON WITH CONSTANT VELOCITY IN AN INHOMOGENEOUS MEDIUM
(Presented by Academician L. I. Sedov, 10 VII 1961)
Consider the motion of a gas behind a piston which expands with constant velocity \(u\) in a medium whose density \(\rho_n\) varies according to the law
\[ \rho_n=\rho_1[1-\varepsilon r^\chi], \tag{1} \]
where \(r\) is the linear coordinate; \(\varepsilon\) is a small parameter; \(\chi, \rho_1\) are constants. The unknown functions are the pressure \(p\), the density \(\rho\), and the velocity \(v\), which depend on the variables \(t, r\) and on the parameters \(p_1, \rho_1, \varepsilon, \chi, \gamma\), where \(\gamma=c_p/c_v\). From these quantities one can form only two dimensionless variables
\[ \lambda=\frac{\gamma p_1 t^2}{\rho_1 r^2},\qquad \mu=\varepsilon r^\chi . \]
Thus the sought dimensional functions can be represented through dimensionless functions depending on the dimensionless variables:
\[ v=\frac{r}{t}\,\overline V(\lambda,\mu),\qquad p=\rho_1\left(\frac{r}{t}\right)^2\overline P(\lambda,\mu),\qquad \rho=\rho_1\overline R(\lambda,\mu). \tag{2} \]
The problem of the motion of a gas behind a piston moving with constant velocity in a homogeneous medium was solved by L. I. Sedov \({}^{(1)}\). Let \(V_0(\lambda)\), \(P_0(\lambda)\), \(R'(\lambda)\) be the solutions of this problem. Then we represent the linearized sought solutions in the form
\[ \overline V=V_0(\lambda)+\mu V(\lambda),\qquad \overline P=P_0(\lambda)+\mu P(\lambda),\qquad \overline R=R_0(\lambda)+\mu R(\lambda). \tag{3} \]
The basic equations of one-dimensional unsteady motion have the form
\[ \frac{\partial v}{\partial t} +v\frac{\partial v}{\partial r} +\frac{1}{\rho}\frac{\partial p}{\partial r}=0, \]
\[ \frac{\partial \rho}{\partial t} +\frac{\partial \rho v}{\partial r} +(\nu-1)\frac{\rho v}{r}=0, \tag{4} \]
\[ \frac{\partial}{\partial t}\left(\frac{p}{\rho^\gamma}\right) +v\frac{\partial}{\partial r}\left(\frac{p}{\rho^\gamma}\right)=0. \]
Here \(\nu=1,2,3\), respectively, for plane, cylindrical, and spherical waves. After passing to dimensionless variables and varying with respect to \(\mu\), we obtain a system of ordinary differential equations for \(V, P, R\). From the independent variable \(\lambda\) we pass to the independent variable
variable \(V_0\), and we shall seek solutions in the form
\[ V=(1-V_0)^S \sum_{n=0}^{\infty} a_{ni}(1-V_0)^n, \]
\[ P=(1-V_0)^S \sum_{n=0}^{\infty} b_{ni}(1-V_0)^n, \tag{5} \]
\[ R=(1-V_0)^S \sum_{n=0}^{\infty} c_{ni}(1-V_0)^n. \]
As is known, in the case of self-similar motion the solutions for \(V_0\), \(P_0\), \(R_0\) are not expressed in analytic form. After passing to the independent variable \(V_0\), approximate solutions can be found by expanding the desired functions in powers of \((1-V_0)\):
\[ \lambda=\lambda_{\mathrm{p}}\left[1-\frac{2}{\nu}(1-V_0)+\ldots\right], \]
\[ P_0=P_{0\mathrm{p}}\left[1-\frac{2}{\nu}(1-V_0)+\ldots\right], \tag{6} \]
\[ R_0=R_{0\mathrm{p}}\left[1+\chi(1-V_0)^2+\ldots\right], \]
where \(\lambda_{\mathrm{p}}, P_{0\mathrm{p}}, R_{0\mathrm{p}}\) are the values of the functions at the piston.
The characteristic equation of the system, if the equalities (6) are taken into account, will be
\[ S^2\left(S-\frac{\chi}{\nu}\right)=0. \tag{7} \]
The roots \(S_1=S_2=0\) correspond to a solution with a logarithmic singularity \((^2)\).
Let us now consider the boundary conditions. At the piston \(\vartheta=u\), and since \(V_0=1\), then \(V(1)=0\). Ahead of the piston, at some distance, a shock wave is moving. We shall assume that the piston velocity is large, and take the conditions at the shock wave in the form
\[ \vartheta_2=\frac{2}{\gamma+1}c,\qquad \rho_2=\frac{\gamma+1}{\gamma-1}\rho_H,\qquad p_2=\frac{2}{\gamma+1}\rho_H c^2, \tag{8} \]
where \(c\) is the velocity of the shock wave.
The conditions at the shock wave, after passage to dimensionless variables and variation with respect to \(\mu\), have the form
\[ V_2=\left[\frac{2(1+\chi)(1+\lambda^*)}{\gamma+1}-V_0(\lambda^*)+2\lambda^*\left(\frac{dV_0}{d\lambda}\right)_{\lambda=\lambda^*}\right]a-\frac{2\lambda^*}{\gamma+1}, \]
\[ P_2=\left[2\lambda^*\left(\frac{dP_0}{d\lambda}\right)_{\lambda=\lambda^*}-2P_0(\lambda^*)+\frac{4(1+\chi)}{\gamma+1}\right]a-\frac{2}{\gamma+1}, \tag{9} \]
\[ R_2=\left[2\lambda^*\left(\frac{dR_0}{d\lambda}\right)_{\lambda=\lambda^*}+\frac{4(\gamma+1)(1+\chi)\lambda^*}{\gamma-1}\right]a-\frac{\gamma+1}{\gamma-1}, \]
where \(\lambda^*\) is the value of \(\lambda\) at the shock wave, and \(a\) is a constant.
The radius vector \(r_2\) at the shock wave is represented in the form
\[ r_2=r_{20}[1+a\mu^*]. \tag{10} \]
The solutions for \(V, P, R\) take the following form if \((1-V_0)^2\) is neglected and \(\chi/\nu\) is not an integer:
\[ V=\frac{\chi}{\nu}\frac{b_{01}}{\gamma P_{0\mathrm{p}}}(1-V_0)\ln(1-V_0) +\frac{2\chi c_{03}}{R_{0\mathrm{p}}(\chi+\nu)}(1-V_0)^{\chi/\nu+1}. \]
\[ P=b_{01}\left\{\left[1-\frac{\chi+2}{\nu}(1-V_0)\right]\ln(1-V_0)+1+\frac{4}{\nu}(1-V_0)\right\}, \tag{11} \]
\[ R=\frac{R_{0\mathrm{p}}}{\gamma P_{0\mathrm{p}}}b_{01} \left\{\left[1-\frac{\chi}{\nu}(1-V_0)\right]\ln(1-V_0)+1+\frac{\nu}{\chi}+\frac{4}{\nu-\chi}(1-V_0)\right\} +c_{03}(1-V_0)^{\chi/\nu}. \]
In these formulas the condition at the piston has already been taken into account. The unknown constants \(b_{01}, c_{03}, a\) are determined from the conditions at the shock wave.
If \(\chi/\nu\) is an integer, then the solutions will have the form
\[ V=\frac{\chi}{\nu}\frac{b_{01}}{\gamma P_{0\text{p}}}(1-V_0)\ln(1-V_0), \]
\[ P=b_{01}\left\{\left[1-\frac{\chi+2}{\nu}(1-V_0)\right]\ln(1-V_0)+1+\frac{4}{\nu}(1-V_0)\right\}, \tag{12} \]
\[ R=\left[\frac{R_{0\text{p}}}{\gamma P_{0\text{p}}}b_{01}+c_{11}(1-V_0)\right]\ln(1-V_0) +\left(1+\frac{\nu}{\chi}\right)\frac{R_{0\text{p}}}{\gamma P_{0\text{p}}}b_{01} +c_{11}(1-V_0) \]
Fig. 1
Fig. 2
Fig. 3
The values of \(V, P, R\) computed by us as functions of \(V_0\) for \(\chi=1\) in the cases \(\nu=1,2,3\) are presented in the form of graphs (Figs. 1, 2, 3).
Received
30 VI 1961
CITED LITERATURE
- L. I. Sedov, Methods of Similarity and Dimensionality in Mechanics, Moscow, 1957.
- G. Piaggio, Integration of Differential Equations, 1933.