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TOWARD AN INVESTIGATION OF THE STABILITY OF DISCONTINUOUS SOLUTIONS OF THE GAS-DYNAMIC EQUATIONS
S. K. ASLANOV
In [1] the problem of stability was considered for a discontinuous solution of the gas-dynamic equations describing a plane shock wave that occupies the position \(y,z\). A perturbation \(\psi(y,z,t)\) imposed on the discontinuity \(x=0\) will, according to the laws of shock waves, propagate into the region \(x>0\). Linearization about the unperturbed state of the partial differential equations for this region and of the matching conditions at the discontinuity gives a homogeneous boundary-value problem for small perturbations. Erpenbeck [1] transforms the latter into a boundary-value problem for an ordinary differential equation (in the independent variable \(x\)) by means of a Fourier transform in the coordinates \(y,z\) and a Laplace transform in time \(t\). As a result, the determination of the stability of the discontinuous solution under consideration is reduced to determining the nature of the roots of a certain function (depending on this solution and on the equation of state), which stands in the denominator of \(\xi(\tau)\), the Fourier–Laplace transform of the perturbation of the position of the discontinuity \(\psi(y,z,t)\).
The positivity of the real part of at least one of the indicated roots (poles of \(\xi(\tau)\)) then entails exponential growth in time of small perturbations and, consequently, serves as a criterion for instability of the discontinuous solution under study, which is written in the form [1]
\[ F_s = l - 1 + \frac{u}{c} < 0,\qquad l = 2 - \frac{u^2}{c^2}(V_1 - V)\left(\frac{\partial p}{\partial S}\right)_V \frac{1}{T}, \tag{1} \]
where \(u\) is velocity; \(p\) is pressure; \(T\) is temperature; \(c\) is the speed of sound; \(V\) is specific volume; \(S\) is entropy; the unit subscript denotes gas-dynamic parameters ahead of the discontinuity \((x<0)\), while no subscript denotes those behind the discontinuity \((x>0)\);
\[ p = p(V,S) \tag{2} \]
is the equation of state of the medium. In the opposite case \((F_s>0)\) there is no exponential growth in time \(t\) of arbitrarily small perturbations, and Erpenbeck [1] regards \(F_s>0\) as the condition for stability of the shock wave in the linear approximation, thereby defining \(F_s=0\) as the boundary separating the regions of stability and instability of the given discontinuous solution. However, such an approach is mathematically incorrect, since for the region \(F_s>0\) one can assert only the impossibility of roots with a positive real part, but not the complete absence of roots with zero real part (corresponding to conservation of the perturbation amplitude in time).
To obtain a more definite picture of the perturbations in time \(t\) in the region \(F_s>0\), one must find the inverse Laplace transform of \(\xi(\tau)\), i.e., return to the Fourier transform of the perturbation of the position of the discontinuity \(\psi(y,z,t)\). Since \(\xi(\tau)\) has a cut along the imaginary axis [1] (p. 1185), this operation is apparently very difficult and was not carried out by Erpenbeck.
As will be shown below on the basis of our previous work [2], the real parts of the region \(F_s>0\) correspond to imaginary eigenvalues of the problem. And this part of the region \(F_s>0\) represents the so-called doubtful case of the linear theory of stability and requires nonlinear analysis. This circumstance is aggravated by the fact that, as we make clear [2], precisely in the indicated part of the region \(F_s>0\) lies the case most important for gas dynamics: a shock wave in a perfect gas, with respect to which Erpenbeck concludes stability.
First of all, let us establish the relation between \(\left(\dfrac{\partial p}{\partial S}\right)_V\) and \(\left(\dfrac{\partial V}{\partial p}\right)_H\), where the subscript \(H\) denotes differentiation along the shock adiabat, whose equation is written in the form
\[ W_1-W+\frac{1}{2}(V_1+V)(p-p_1)=0. \tag{3} \]
Differentiating the latter and taking into account the thermodynamic identity \(dW=T\,dS+V\,dp\) and the mechanical conservation laws at the shock [3]
\[ p-p_1=-\frac{u^2}{V^2}(V-V_1), \tag{4} \]
we shall have
\[ 2T\,dS=(p-p_1)\,dV-(V-V_1)\,dp. \tag{5} \]
Therefore the equation of state (2) along the shock adiabat (5) is transformed into \(p=p(V)\), i.e., (5) is written in the following form:
\[ 2T\,dS=\left[p-p_1-(V-V_1)\left(\frac{\partial p}{\partial V}\right)_H\right]dV. \tag{6} \]
On the other hand, by differentiating (2), equation (5) can be represented in the form
\[ -\left[T+\frac{1}{2}(V-V_1)\left(\frac{\partial p}{\partial S}\right)_V\right]dS +\frac{1}{2}\left[p-p_1-(V-V_1)\left(\frac{\partial p}{\partial V}\right)_S\right]dV=0. \]
Substitution of the latter into (6), taking into account (4) and the obvious identity [3] \(\left(\dfrac{\partial p}{\partial V}\right)_S=-\dfrac{c^2}{V^2}\), finally gives
\[ \left(\frac{\partial V}{\partial p}\right)_H =-\frac{V^2}{u^2}\left[1-\frac{2}{l}\left(1-\frac{u^2}{c^2}\right)\right]. \tag{7} \]
We next pass to our notation [2]. Then the state before the shock \((x<0)\) corresponds to the zero subscript, and the state behind the shock \((x>0)\) to the unit subscript. The velocity \(u\) will now be denoted by \(v\), and
\[ m=\frac{v_0^2}{V_0^2}\left(\frac{\partial V_1}{\partial p_1}\right)_H =\frac{v^2}{V^2}\left(\frac{\partial V_1}{\partial p_1}\right)_H . \]
In this notation \(F_s\) from (1), using (7), assumes the following form:
\[ F_s=1+M_1-2\,\frac{m+M_1^2}{m+1},\qquad M_1=\frac{v_1}{c_1}<1. \tag{8} \]
Hence one obtains the condition for instability of the shock wave \((F_s<0)\)
\[ m>1+2M_1,\qquad m<-1, \tag{9} \]
which, as was to be expected, coincides with our results [2] and with the work [4], corresponding to exponential growth of perturbations in \(t\). As for the region \(F_s>0\), i.e., in our notation \(-1<m<1+2M_1\), in it, in turn, along with the region of exponential decrease of stability perturbations, contrary to Erpenbeck’s assertion [1], there is a region of the doubtful case of the linear theory of stability, for which the perturbation amplitude remains constant in \(t\). This follows from [2], where the boundaries of the “doubtful” region of interest to us were found:
\[ M_1>\frac{\delta}{2-\delta},\qquad 2M_1+1>m>m_+=-1+\frac{\delta\eta^2}{1-\eta\sqrt{1-\delta}}; \]
\[ M_1<\frac{\delta}{2-\delta},\qquad 1-2M_1<m<1+2M_1; \tag{10} \]
\[ \eta^2=1-M_1^2,\qquad \delta=\frac{v_1}{v_0}=\frac{V_1}{V_0}. \]
Moreover, in the case of a shock wave in a perfect gas, which is the most important and interesting one for gas dynamics, we have succeeded in explicitly expressing the dependence of the perturbations on time [2], which falls within the interval of the doubtful case and requires a nonlinear approach to the problem, or, as was done by us [2], allowance for the dissipative effect of viscosity, which stabilizes the plane shock wave.
References
- Erpenbeck J. J. The Physics of Fluids, 5, No. 10, 1181—1187, 1962.
- Aslanov S. K. Differential Equations, 2, No. 8, 1115—1131, 1966.
- Landau L. D.; Lifshitz E. M. Mechanics of Continuous Media. Moscow, GTTI, 1953.
- D’yakov S. P. ZhETF, 27, 3 (9), 287—295, 1954.
Received by the editors
December 10, 1965
Odessa State University
named after I. I. Mechnikov