On the Structure Stability of a Neutrally Stable Shock Wave in a Gas and on Spontaneous Emission of Perturbations

Abstract

We analyze the stability of the structure of a neutrally stable shock wave, which is also referred to as a spontaneously emitting shock wave. We have obtained solutions describing linear sinusoidal perturbations emerging in the structure and downflow which depend on the coordinate along the emitting structure. It is shown that these perturbations decay with time. We have also considered the limit transition for the dimensionless width of the structure, which tends to zero. The results obtained in the limit coincide with classical results when the shock wave is treated as the discontinuity surface.

1 INTRODUCTION

Analysis of the linear stability of shock waves in gases [1–5] has shown that depending on the equation of state of the gas and the shock wave intensity, a plane shock wave can be stable asymptotically (shock waves in a perfect gas are stable), unstable (when perturbations of the shock wave surface and of hydrodynamic quantities in the vicinity of the wave increase exponentially with time), and stable neutrally. In the latter case, there exist sinusoidal perturbations of the shock wave surface and the flow behind it, which do not grow and do not decay with time. The conditions in which a shock wave is neutrally stable were obtained in aforementioned publications and have form

$$\frac{{1 - {{{({{{v}}_{2}}{\text{/}}{{c}_{2}})}}^{2}} - {{{v}}_{1}}{{{v}}_{2}}{\text{/}}c_{2}^{2}}}{{1 - {{{({{{v}}_{2}}{\text{/}}{{c}_{2}})}}^{2}} + {{{v}}_{1}}{{{v}}_{2}}{\text{/}}c_{2}^{2}}} < {{j}^{2}}{{\left( {\frac{{d{{V}_{2}}}}{{d{{p}_{2}}}}} \right)}_{H}} < 1 + 2\frac{{{{{v}}_{2}}}}{{{{c}_{2}}}}.$$

(1)

Here, ${v}$ is the velocity component normal to the shock wave, c is the velocity of sound, V = 1/ρ is the specific volume, j² = (p₂ – p₁)/(V₁ – V₂) is the square of the mass flux, subscripts “1” and “2” correspond to quantities in front of and behind the shock wave, and subscript “H” on the derivative indicates that it is taken along the shock adiabat.

In analysis of 2D perturbations, we assume that the x axis is directed along the normal to the shock wave and the unperturbed velocity is directed along the x axis. We assume that the y axis lies on the unperturbed surface of the shock wave. Then the aforementioned sinusoidal perturbation of the shock wave surface can be written in form

$$\begin{gathered} x = \xi (y,t),\quad \xi (y,t) = \operatorname{Re} A\exp i(ky - \omega t), \\ A = {\text{const}}{\text{.}} \\ \end{gathered} $$

(2)

The dispersion equation, which is a consequence of the linearized boundary conditions on the shock wave, defines the relation

$$\frac{\omega }{k} = \pm W.$$

(3)

For neutrally stable waves, W is real-valued and independent of k. Velocity W is “supersonic” in the sense that

$$W > \sqrt {c_{2}^{2} - {v}_{2}^{2}} .$$

(4)

The expression on the right-hand side of this equality is the velocity of sound propagation along the ray parallel to the shock wave.

If a perturbation of form (2) propagates over the surface of a neutrally stable shock wave and conditions (3) and (4) are satisfied, acoustic, entropy, and vortex perturbations propagate into the region behind the shock wave and do not decay with increasing distance from it; therefore, neutrally stable shock waves are also referred to as spontaneously emitting shock waves (SESW).

Since only ratio ω/k is defined in expression (2) for neutral perturbations, and the value k is arbitrary, we can compose from the solutions satisfying condition (3) perturbation ξ(y, t) of an arbitrary form (including a perturbation with the leading front), which propagates over the shock wave with velocity W. These perturbations as a superposition of solutions of form (2) correspond to the linear approximation. In the nonlinear approximation the solution was obtained, in which after the point moving along SESW the shock wave splits into two shocks or into a shock and a Prandtl–Mayer wave. If we place the origin of coordinates at this point, the flow will depend only on the polar angle [6].

In review [7] devoted to stability of shock waves, the following remark was made: since velocity W of propagation of perturbations of the shock wave surface is supersonic in the sense that inequality (4) holds, no perturbations from the flow behind the shock wave can arrive at the point moving with velocity W over the shock wave if these perturbations have been concentrated at the initial instant in the vicinity of this point and have not arrived from infinity. For this reason, it was declared in [7] that perturbations of form (2), (3) do not exist. Further, it was noted in [7] that the real-valueness of quantity ω/k indicates that the reflection coefficient for acoustic perturbation catching up with the shock wave in the case of neutrally stable waves turns to infinity for a certain angle of incidence. The reflection coefficient of weakly nonlinear waves calculated in [7, 8] turned out to be finite, but for certain conditions tends to infinity if the amplitude of incident wave tends to zero.

Later, in connection with possible applications, there appeared a number of publications [9–14], in which the flows of high-density gases with neutrally stable shock waves were studied.

Publications [9–11] were devoted to numerical experiments with neutrally stable shock waves and their interaction with perturbations. The conclusions drawn from these numerical experiments concerning spontaneously emitting shock waves lead us to the fact that spontaneous (not induced by external actions) generation of acoustic waves was not observed in numerical experiments. In our opinion, the results reported in [9–11] give no grounds for the final conclusion because of the imperfection of any method of numerical analysis of gas flows with shock waves and the nonlinearity of calculated examples. It was noted in [9–11] that the results obtained in [7, 8] are hypothetical and contradictory to a certain extent.

It should be noted that in the numerical calculations performed in [15], which is connected ideologically and methodically with [9–11], undamped multidimensional oscillations of a shock wave with the formation of a cellular structure of the front were obtained in numerical calculations. The existence of such a cellular structure was observed in [15] in the numerical calculation in the region of ambiguous representation of the discontinuity and, according to the author of [15], is a consequence of this indeterminacy (i.e., a nonlinear effect). It is also stated in [15] that one can speak of neutrally stable shock waves only assuming that induced perturbations of their surface decay much more slowly than in the case of an absolutely stable shock wave, and the spontaneous emission of acoustic waves by the front of a neutrally stable shock wave, which was predicted in the linear theory, is not observed at all.

It can be noted that in publications [7, 8], the origin of processes associated with evolution of perturbations with a neutrally stable shock wave has not been determined. No indications were given either concerning the construction of the solution for the linearized problem of the behavior of perturbations of neutrally stable shock waves with account for the concepts formulated in these articles. However, the formal construction of the solution to the problem with initial data [3] leads to the emergence of perturbations emitted by a shock wave into the region behind it (the source of these perturbations in the form of perturbations of the shape of the shock wave does not decay with time).

In this connection, the following questions arise: what are the mechanisms of perturbation transfer along the shock wave and how can a correct description be obtained for linear perturbations of neutrally stable shock waves?

In [16], it was proposed that the SESW structure be analyzed for determining the possibility of perturbation transfer over the shock wave and then the limit transition in which the width of the structure tends to zero be considered. A brief description of the results that will be described below was given in [17]. Here, we consider the simplest model of SESW structure. We assume that a conventional stable shock wave (the same as in a perfect gas) moves at front, and behind this wave, there is a flow of an inviscid non-heat-conducting gas, the relaxation processes in which lead to a decrease in adiabatic exponent γ. A transition to the flow in front of the shock wave to the flow at a large distance behind the structure corresponds to a shock transition in the SESW.

In Section 2, we choose parameters of a certain SESW, the structure of which will be analyzed further. In Section 3, we formulate the assumptions adopted in the description of the relaxation process and obtain the solution to the problem of the SESW structure. In Section 4, we consider the equations describing linear perturbations (depending harmonically on y) in the SESW structure and downflow. In Section 5, we formulate the conditions ensuring the absence of perturbations arriving from infinity and construct the solutions for perturbations consisting of waves propagating downflow from the shock wave. We describe the method for numerical calculation of eigenfrequencies and represent the results of these calculations in the form of a dependence of the eigenfrequency on k; we also consider the limit transition for k → 0 (k is the wavenumber in the y direction) or, which is the same, for the dimensionless width of the structure (divided by the perturbation wavelength in the y direction) tending to zero. In Section 6, we formulate the conclusion concerning the SESW, which are obtained for this limit transition, and discuss the results.

2 CHOICE OF PARAMETERS FOR SPONTANEOUSLY EMITTING SHOCK WAVES

In this section, the shock wave is treated as a surface of discontinuity separating two gas flows. In accordance with inequality (1), we will mark by subscripts “1” and “2” the flow parameters in front of the shock wave and behind it, respectively. We choose the parameters of the flow and of the shock wave so that conditions (1) are satisfied. We assume that intrinsic (chemical or relaxation) processes, the rates of which will be considered as limited in next sections, can occur in the gas. In analysis of the shock wave as a discontinuity process in this section, we assume that these processes occur instantaneously, and the unperturbed flow behind the shock wave and in front of it is a homogeneous flow of an inviscid non-heat-conducting gas.

For specifying the properties of the gas, we must define the internal energy of unit mass,

$$\varepsilon = \varepsilon (p,V).$$

(5)

The relations on the shock wave have form [4]

$$\varepsilon ({{p}_{2}},{{V}_{2}}) - \varepsilon ({{p}_{1}},{{V}_{1}}) - \frac{1}{2}({{V}_{1}} - {{V}_{2}})({{p}_{1}} + {{p}_{2}}) = 0,$$

(6)

$${{j}^{2}} = \frac{{{{p}_{2}} - {{p}_{1}}}}{{{{V}_{1}} - {{V}_{2}}}}.$$

(7)

Equation (6) is known as the shock adiabat or the Hugoniot adiabat (H). Relation (7) expresses mass flux j = ρ₁${{{v}}_{1}}$ = ρ₂${{{v}}_{2}}$ through the unit area of the shock wave. This equality is a consequence of the continuity of the mass fluxes and of the normal component of the momentum on the shock wave.

In inequalities (1) expressing the condition of spontaneous emission of the shock wave, we must determine quantities c₂ and (dV₂/dp₂)_H. Quantity c₂ is the equilibrium velocity of sound for the state behind the relaxation zone.

Let us determine the value of c₂ as the velocity of an infinitely weak shock wave in the vicinity of state 2. We will use Eq. (6) of the shock adiabat, assuming that state 1 is close to state 2 so that

$$d\varepsilon = d{{\varepsilon }_{2}} - d{{\varepsilon }_{1}},\quad {{V}_{1}} - {{V}_{2}} = - dV.$$

In the absence of viscosity and heat conduction, we then obtain the following relations that hold for a quasi-equilibrium process in the vicinity of state 2:

$$\begin{gathered} d{{\varepsilon }_{2}} + {{p}_{2}}d{{V}_{2}} = 0,\quad ({{\varepsilon }_{p}})_{2}^{'}d{{p}_{2}} + [({{\varepsilon }_{V}})_{2}^{'} + {{p}_{2}}]d{{V}_{2}} = 0, \\ c_{2}^{2} = \frac{{d{{p}_{2}}}}{{d{{\rho }_{2}}}} = \frac{{V_{2}^{2}[{{{(\varepsilon _{V}^{'})}}_{2}} + {{p}_{2}}]}}{{{{{(\varepsilon _{p}^{'})}}_{2}}}}, \\ \varepsilon _{V}^{'} = {{\left( {\frac{{\partial \varepsilon }}{{\partial V}}} \right)}_{p}},\quad \varepsilon _{p}^{'} = {{\left( {\frac{{\partial \varepsilon }}{{\partial p}}} \right)}_{V}}. \\ \end{gathered} $$

(8)

Let us now find the value of (dV₂/dp₂)_H. For this, we take the differential of Eq. (6), assuming that state 1 is given:

$$\begin{gathered} {{(\varepsilon _{p}^{'})}_{2}}d{{p}_{2}} + {{(\varepsilon _{V}^{'})}_{2}}d{{V}_{2}} + \frac{1}{2}({{p}_{2}} + {{p}_{1}})d{{V}_{2}} \\ - \frac{1}{2}({{V}_{1}} - {{V}_{2}})d{{p}_{2}} = 0, \\ \end{gathered} $$

(9)

whence

$${{\left( {\frac{{d{{V}_{2}}}}{{d{{p}_{2}}}}} \right)}_{H}} = - \frac{{{{{(\varepsilon _{p}^{'})}}_{2}} - ({{V}_{1}} - {{V}_{2}}){\text{/}}2}}{{{{{(\varepsilon _{V}^{'})}}_{2}} + ({{p}_{2}} + {{p}_{1}}){\text{/}}2}}.$$

(10)

Velocity ${v}$ can be found from equality

$${v} = jV,$$

(11)

where j is defined in accordance with relation (7).

Using equalities (11), (7), and (8), we transform the combination of terms in the numerator on the right-hand side of inequalities (1),

$$\frac{{{{{v}}_{2}}({{{v}}_{1}} + {{{v}}_{2}})}}{{c_{2}^{2}}} = \left( {\frac{{{{V}_{1}}}}{{{{V}_{2}}}} + 1} \right)\frac{{{{p}_{2}} - {{p}_{1}}}}{{{{V}_{1}} - {{V}_{2}}}}\frac{{{{{(\varepsilon _{p}^{'})}}_{2}}}}{{{{{(\varepsilon _{V}^{'})}}_{2}} + {{p}_{2}}}}.$$

(12)

We denote

$$\frac{{{{{(\varepsilon _{p}^{'})}}_{2}}}}{{{{V}_{1}} - {{V}_{2}}}} = X > 0,\quad \frac{{{{{(\varepsilon _{V}^{'})}}_{2}} + {{p}_{2}}}}{{{{p}_{2}} - {{p}_{1}}}} = Y > 0.$$

(13)

In accordance with relation (8), inequality Y > 0 follows from inequalities $\varepsilon _{{p2}}^{'}$ > 0 and $c_{2}^{2}$ > 0. Relation (12) takes form

$$\frac{{{{{v}}_{2}}({{{v}}_{1}} + {{{v}}_{2}})}}{{c_{2}^{2}}} = \left( {\frac{{{{V}_{1}}}}{{{{V}_{2}}}} + 1} \right)\frac{X}{Y}.$$

(14)

Let us transform analogously the denominator of the fraction in inequalities (1). This gives the following expression for the left-hand side of inequalities (1):

$$\frac{{1 - [{{{({{{v}}_{2}}{\text{/}}{{c}_{2}})}}^{2}} + {{{v}}_{1}}{{{v}}_{2}}{\text{/}}c_{2}^{2}]}}{{1 - [{{{({{{v}}_{2}}{\text{/}}{{c}_{2}})}}^{2}} - {{{v}}_{1}}{{{v}}_{2}}{\text{/}}c_{2}^{2}]}} = \frac{{Y - ({{V}_{1}}{\text{/}}{{V}_{2}} + 1)X}}{{Y + ({{V}_{1}}{\text{/}}{{V}_{2}} - 1)X}}.$$

The middle expression in inequalities (1) with account for relation (10) can be reduced to

$$ - \frac{{{{p}_{2}} - {{p}_{1}}}}{{{{V}_{1}} - {{V}_{2}}}}\frac{{{{{(\varepsilon _{p}^{'})}}_{2}} - ({{V}_{1}} - {{V}_{2}}){\text{/}}2}}{{{{{(\varepsilon _{V}^{'})}}_{2}} + ({{V}_{1}} + {{V}_{2}}){\text{/}}2}} = - \frac{{X - 1{\text{/}}2}}{{Y - 1{\text{/}}2}}.$$

(15)

Therefore, the left inequality in (1) takes form

$$\frac{{({{V}_{1}}{\text{/}}{{V}_{2}} + 1)X - Y}}{{({{V}_{1}}{\text{/}}{{V}_{2}} - 1)X + Y}} > \frac{{X - 1{\text{/}}2}}{{Y - 1{\text{/}}2}}.$$

(16)

For choosing the shock wave with the perturbation emitted by it, we must take the values of X and Y such that inequalities (1) be satisfied. Without considering the solution to inequality (16) in the whole, we confine our analysis to the following range of their values:

$$X > \frac{1}{2},\quad Y > \frac{1}{2}.$$

(17)

In this case, middle expression (15) in inequalities (1) turns out to be negative (i.e., the slope of shock adiabat (dV₂/dp₂)_H in state 2 is negative). In this case, the right inequality in (1) holds automatically because 1 + 2${{{v}}_{2}}$/c₂ > 0. The fulfillment of the left inequality was reduced to the fulfillment of inequality (16). Under limitations (17), inequality (16) can be reduced to

$$(Y - X)\left[ {Y - 1 - \left( {\frac{{{{V}_{1}}}}{{{{V}_{2}}}} - 1} \right)X} \right] < 0.$$

Thus, for the fulfillment of the left inequality in (1), it is sufficient to choose the values of X and Y so that they satisfy inequalities (17) as well as inequalities

$$X < Y < 1 + \left( {\frac{{{{V}_{1}}}}{{{{V}_{2}}}} - 1} \right)X.$$

(18)

For a perfect gas, the internal energy has form

$$\varepsilon = \frac{{pV}}{{\gamma - 1}},\quad {\text{or}}\quad p = (\gamma - 1)\rho \varepsilon ,\quad \gamma = {\text{const}}{\text{.}}$$

(19)

Factor (γ – 1) for a perfect gas is known to depend on the number of degrees of freedom of gas molecules.

If a relaxation process occurs, the fraction of energy corresponding to translational degrees of freedom of all particles (molecules, ions, and electrons) also depends on the energy spent (stored) during the dissociation and ionization. Therefore, we can write

$$p = \rho \varepsilon f,\quad {\text{or}}\quad \varepsilon = pV{\text{/}}f.$$

(20)

Here, f is the fraction of energy corresponding to translational degrees of freedom and ε is the total energy of the unit mass of the gas in all degrees of freedom, which also includes the chemical energy, dissociation energy, and ionization energy.

Comparison of relations (19) and (20) shows that f and (γ – 1) play the same role in the equation of state; therefore, we will henceforth not distinguish between f and (γ – 1). It is well known that dissociation and ionization in hypersonic aerodynamics are taken into account in many cases by a change in the value of γ [18]. In analysis of the structure and possible nonstationary processes, we will use equalities (20), but f will be considered a variable quantity for which a differential equation will be postulated. The equilibrium value of f corresponding to given values of V and p will be marked by subscript “e”.

Let us choose the values ensuring the fulfillment of inequalities (1):

$$\begin{gathered} {{f}_{{e1}}} = 0.4,\quad {{f}_{{e2}}} = 0.2,\quad {{V}_{2}}{\text{/}}{{V}_{1}} = 1{\text{/}}5, \\ {{p}_{2}}{\text{/}}{{p}_{1}} \approx 4.8,\quad X = 2.5,\quad Y = 9. \\ \end{gathered} $$

(21)

Ratio p₂/p₁ was obtained using¹Eq. (6).

Using inequalities (20) and (13), we can obtain the derivatives of f_e = pV/ε(p, V) in state 2:

$${{\left( {\frac{{\partial {{f}_{e}}}}{{\partial p}}} \right)}_{2}} = \frac{{({{f}_{e}})_{2}^{2}}}{{{{p}_{2}}}}\left[ {f_{2}^{{ - 1}} - X\left( {\frac{{{{V}_{1}}}}{{{{V}_{2}}}} - 1} \right)} \right] \approx - \frac{{0.200}}{{{{p}_{2}}}},$$

(22)

$${{\left( {\frac{{\partial {{f}_{e}}}}{{\partial V}}} \right)}_{2}} = \frac{{({{f}_{e}})_{2}^{2}}}{{{{V}_{2}}}}[f_{2}^{{ - 1}} + 1 - Y(1 - {{p}_{1}}p_{2}^{{ - 1}})] \approx - \frac{{0.045}}{{{{V}_{2}}}}.$$

(23)

The derivative along the Rayleigh–Michelson straight line in state 2 has form

$$\begin{gathered} {{\left( {\frac{{d{{f}_{e}}}}{{dV}}} \right)}_{{M2}}} = {{\left( {\frac{{\partial {{f}_{e}}}}{{\partial V}}} \right)}_{2}} + {{\left( {\frac{{\partial {{f}_{e}}}}{{\partial p}}} \right)}_{2}}{{\left( {\frac{{\partial p}}{{\partial V}}} \right)}_{{M2}}} \\ = - \frac{{0.045}}{{{{V}_{2}}}} + \frac{{0.2({{p}_{2}} - {{p}_{1}})}}{{{{p}_{2}}({{V}_{1}} - {{V}_{2}})}} \approx - \frac{{0.005}}{{{{V}_{2}}}}. \\ \end{gathered} $$

(24)

3 STRUCTURE OF SPONTANEOUSLY EMITTING SHOCK WAVES, ASSOCIATED WITH THE RELAXATION PROCESS

As noted above, equilibrium states of a gas are determined by equalities (20), in which

$$f = {{f}_{e}}(p,V),$$

(25)

where f_e is considered as a known function that will be defined below. We assume that behind the leading gasdynamic shock wave, there is a flow of an inviscid non-heat-conducting gas, which is determined by the relaxation process in which function f tends to its equilibrium value f_e.

The relaxation process in a gas particle must be described by a differential equation that we write in the simplest form:

$$\frac{{df}}{{dt}} = - \lambda (f - {{f}_{e}}(p,V)),\quad \lambda > 0.$$

(26)

Quantity λ may depend on the current state of the gas and will be chosen constant for convenience.

We further introduce the following notation. Quantities without subscripts are running values in the structure. Quantities with subscript “1” correspond to the flow in front of the leading shock wave; subscript “0” marks the quantities immediately behind the leading shock wave, while quantities with subscript “2” correspond to the flow behind the relaxation zone. We assume that f₀ ≡ γ₁ – 1 = f₁ immediately behind the leading shock wave; i.e., we assume that γ does not change in this shock wave (γ₀ = γ₁). This equality is based on the idea that the leading shock wave is narrow (on the order of a few mean free paths), and the value of f does not change in it because f varies with a finite rate.

Let us consider the flow in which all values depend on variable x and function f = γ – 1 varies in accordance with Eq. (26) from f₀ = f₁until the equality f = f_e(p, V) is satisfied. If the value of function f is known for a certain x, the flow parameters can be calculated from the conservation laws that are observed in the case of a 1D steady-state flow.

The fulfillment of equality (7) does not depend on the relaxation process. In a 1D steady-state flow, j = const. Then equality (7) defines on the (V, p) plane the Rayleigh–Michelson straight line. The values of V(x) and p(x) for a steady-state flow lie on this straight line.

For γ₁ = γ₀, the shock adiabat is a hyperbole with asymptotes [4]

$$\frac{V}{{{{V}_{1}}}} = \frac{{{{\gamma }_{1}} - 1}}{{{{\gamma }_{2}} + 1}},\quad \frac{p}{{{{p}_{1}}}} = - \frac{{{{\gamma }_{1}} - 1}}{{{{\gamma }_{2}} + 1}},$$

where V₁, p₁ is the state in front of the shock wave.

This hyperbole crosses straight line (7) at the initial point and at point V = V₀, p = p₀ corresponding to the state immediately behind the leading shock wave. Further, we observe the 1D flow upon a decrease in γ, for which the conservation laws hold. In this case, the values of all quantities are determined by the value of γ like in the shock wave (i.e., in the discontinuity at which the value of γ varies from γ₁ to the running value of this quantity).

Equation (6) for the shock adiabat in the case of variation of the value of γ in the shock wave has form [4]

$$\begin{gathered} H({{\gamma }_{1}},\gamma ,{{V}_{1}},{{p}_{1}},V,p) = 0,\quad H({{\gamma }_{1}},\gamma ,{{V}_{1}},{{p}_{1}},V,p) \\ \equiv \frac{1}{{\gamma - 1}}pV - \frac{1}{{{{\gamma }_{1}} - 1}}{{p}_{1}}{{V}_{1}} - \frac{1}{2}({{V}_{1}} - V)({{p}_{1}} + p). \\ \end{gathered} $$

(27)

In the case considered here, γ (γ < γ₁) is the current value in the structure; we assume that γ → γ₂ for x → ∞. Relation (27) leads to

$$P = \frac{{({{\gamma }_{1}} + 1){\text{/}}({{\gamma }_{1}} - 1) - U}}{{(\gamma + 1){\text{/}}(\gamma - 1)U - 1}},\quad P = \frac{p}{{{{p}_{1}}}},\quad U = \frac{V}{{{{V}_{1}}}}.$$

(28)

For γ < γ₁, hyperbola (28) passes through point U = 1, P = (γ – 1)/(γ₁ – 1) located in the (V, p) plane below initial point U₁ = 1, P₁ = 1.

For different values of γ, the shock adiabats defined by Eq. (27) do not intersect with one another, and shock adiabats with a smaller value of γ lie below the shock adiabats with a larger value of γ.

On Rayleigh–Michelson straight line (7), there exists point V₀, p₀ at γ = γ₁, which represents the values of quantities V and p immediately behind the leading shock wave. The state corresponding to the current value of γ is represented by the point of intersection of shock adiabat (27) with Rayleigh–Michelson straight line (7). Upon a decrease of γ, this point moves along the Rayleigh–Michelson straight line from point V₀, p₀ towards lower values of V.

Using Eq. (27), we obtain f as a function of the points on Rayleigh–Michelson straight line (7):

$$\begin{gathered} f \equiv \gamma - 1 = \frac{{pV}}{{{{p}_{1}}{{V}_{1}}{\text{/}}{{f}_{1}} + ({{V}_{1}} - V)({{p}_{1}} + p){\text{/}}2}} \\ = \frac{{U[1 + {{J}^{2}}(1 - U)]}}{{1{\text{/(}}{{\gamma }_{1}} - 1) + (1 - U)[2 + {{J}^{2}}(1 - U){\text{/}}2]}}, \\ \end{gathered} $$

(29)

$${{J}^{2}} \equiv \frac{{{{P}_{2}} - 1}}{{1 - {{U}_{2}}}} = {{j}^{2}}\frac{{{{V}_{1}}}}{{{{p}_{1}}}}.$$

(30)

Solving these relations, we obtain explicit expressions for the parameters of the background steady-state flow in the form of functions depending on f and, in accordance with relation (29), on functions U.

Let us now specify the value of function f_e and it derivative on the Rayleigh–Michelson straight line. Function f_e(U) must satisfy conditions

$${{f}_{e}}(1) = 0.4,\quad {{f}_{e}}(0.2) = 0.2.$$

(31)

In analysis of the structure, we must know the value of function f_e on the Rayleigh–Michelson straight line only on segment [U₂, U₀], where U₀ is the value of U behind the leading shock wave (U₀ ≈ 0.412, U₂ = 0.2). We will henceforth represent function f_e(U) by a linear function of U on segment [U₂, U₀] (Fig. 1) with the coefficient given by equality (24):

$${{f}_{e}} = 0.2 + {{\left( {\frac{{d{{f}_{e}}}}{{dU}}} \right)}_{{M2}}}(U - 0.2).$$

(32)

Fig. 1.

Plots of functions f_e(U) (lower line) and f(u) on the Rayleigh–Michelson straight line in the interval (U₂, U₀), where U₀ is the value of U behind the bow shock wave, U₀ ≈ 0.412.

The representation of function f_e along the Rayleigh–Michelson straight line in the form of a linear function with the coefficient the same as at final point U = U₂ = 0.2 of the structure makes it possible to define the partial derivatives of function f_e(V, p) on a segment of the Rayleigh–Michelson straight line by constants equal to their values (22), (23) in state 2:

$$\frac{{\partial {{f}_{e}}}}{{\partial p}} = {{\left( {\frac{{\partial {{f}_{e}}}}{{\partial p}}} \right)}_{2}},$$

(33)

$$\frac{{\partial {{f}_{e}}}}{{\partial V}} = {{\left( {\frac{{\partial {{f}_{e}}}}{{\partial V}}} \right)}_{2}}.$$

(34)

Thus, we have determined the dependence of all quantities in the structure on U. To find U(x) on the Rayleigh–Michelson straight line, we write Eq. (26) as

$$\frac{{df}}{{dx}} = - \alpha \frac{{f - {{f}_{e}}}}{U},\quad \alpha = \frac{\lambda }{{j{{V}_{1}}}}.$$

(35)

Equation (35) can be wrtten as

$$\frac{{dx}}{{dU}} = - \frac{{Uf(U){\kern 1pt} '}}{{\alpha [f(U) - {{f}_{e}}(U)]}}.$$

(36)

We have used the fact that f and f_e on the Rayleigh–Michelson straight line are expressed, in accordance with relations (30) and (32), in terms of U. In the structure, U decreases, f – f_e → 0, and x → ∞ for U → U₂ (see Fig. 1).

From expression (36) for α = const, we obtain

$$\alpha x = - \int\limits_{{{U}_{0}}}^U {\frac{{sf(s){\kern 1pt} '{\kern 1pt} ds}}{{[f(s) - {{f}_{e}}(s)]}},} $$

(37)

because immediately behind the leading shock wave for x = 0, we have U|_{x = +0} = U₀. Solving Eq. (37), we find that the variation of the specific volume in the structure of a shock wave with relaxation is described by function U = U₀(x), which is represented graphically in Fig. 2. Here and below, superscript “0” marks the functions characterizing the structure of the discontinuity for x > 0.

Fig. 2.

Shape of the shock wave structure with account for relaxation U = U₀(x) for α = 1.

It can be seen from Fig. 2 that for α = 1, we can set the effective width of the structure in the given case δ = 1. The expressions for the velocity and pressure in the shock wave are determined from equation (30) of the Rayleigh–Michelson straight line under the assumption that the mass flux is preset and ${{{v}}_{x}}$ = jV.

In Fig. 3, the shock adiabat of the leading shock wave is represented by curve 1, and the Rayleigh–Michelson straight line is shown by 4. Curve 2 is the shock adiabat for a gas in which p₂V₂ = f₂ε, f₂ = f_e2 = 0.2, Curve 3 is the shock adiabat for the case when quantity f_e(p, ρ) is given by relations (32), (33), and (34). Adiabats 2 and 3 have the same value of f = 0.2 at the final point. The difference between these curves is due to the fact that f_e = const in the former case, while in the latter case, f_e is a variable quantity that is defined in the vicinity of the Rayleigh–Michelson straight line by the aforementioned equalities. It should be noted that the assumptions concerning the f_e(p, ρ) dependence as if render the medium softer, which is manifested in Fig. 3 by the smaller slope of curve 3 as compared to curve 2. Figure 3 corresponds to the adopted parameters of the shock wave, the structure of which will be tested for stability.

Fig. 3.

Shock adiabats and the Rayleigh–Michelson straight line: (1) γ = γ₁ = 1.4; (2) γ = 1.2, γ₁ = 1.4; (3) γ = 1 + f_e(p, ρ), γ₁ = 1.4.

4 PERTURBATIONS OF THE STRUCTURE OF SPONTANEOUSLY EMITTING SHOCK WAVES

System of equations in variables ρ, ${{{v}}_{x}}$, ${{{v}}_{y}}$, and ε, which describes 2D flows of an inviscid non-heat-conducting gas in the relaxation zone, has the standard form expressing the conservations laws of mass, two momentum components, and energy:

$$\begin{gathered} \frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial x}}(\rho {{{v}}_{x}}) + \frac{\partial }{{\partial y}}(\rho {{{v}}_{y}}) = 0, \\ \frac{\partial }{{\partial t}}{{{v}}_{x}} + {{{v}}_{x}}\frac{\partial }{{\partial x}}{{{v}}_{x}} + {{{v}}_{y}}\frac{\partial }{{\partial y}}{{{v}}_{x}} + \frac{1}{\rho }\frac{\partial }{{\partial x}}p = 0, \\ \frac{\partial }{{\partial t}}{{{v}}_{y}} + {{{v}}_{x}}\frac{\partial }{{\partial x}}{{{v}}_{y}} + {{{v}}_{y}}\frac{\partial }{{\partial y}}{{{v}}_{y}} + \frac{1}{\rho }\frac{\partial }{{\partial y}}p = 0, \\ \frac{{\partial \varepsilon }}{{\partial t}} + {{{v}}_{x}}\frac{{\partial \varepsilon }}{{\partial x}} + {{{v}}_{y}}\frac{{\partial \varepsilon }}{{\partial y}} + \frac{p}{\rho }\left( {\frac{{\partial {{{v}}_{x}}}}{{\partial x}} + \frac{{\partial {{{v}}_{y}}}}{{\partial y}}} \right) = 0. \\ \end{gathered} $$

(38)

The additional equation for f and the equation of state have form

$$\frac{{\partial f}}{{\partial t}} + {{{v}}_{j}}\frac{{\partial f}}{{\partial {{x}_{j}}}} = - \lambda [f - {{f}_{e}}(p,\rho )],\quad p = \rho \varepsilon f.$$

(39)

Let us consider a steady-state flow of a relaxing gas with a leading shock wave in plane x = 0. We analyze small linear perturbations of the flow. Passing to dimensionless dependent and independent variables, we obtain

$$\begin{gathered} t \to \frac{L}{{{{{v}}_{2}}}}t,\quad x \to Lx,\quad y \to Ly, \\ \rho \to {{\rho }_{2}}\rho ,\quad V \to {{V}_{2}}V, \\ \end{gathered} $$

$$p \to {{p}_{2}}p,\quad \varepsilon \to \frac{{{{p}_{2}}}}{{{{\rho }_{2}}}}\varepsilon ,\quad {{{v}}_{{x,y}}} \to {{{v}}_{2}}{{{v}}_{{x,y}}},\quad \tilde {\zeta } \to L\tilde {\zeta },$$

where L is the characteristic width of the discontinuity structure, which can be written as L = 1/α and x = $\tilde {\zeta }$(y, t) is the expression for the perturbed front of the leading shock wave. We set

$$\left( \begin{gathered} \rho \\ {{{v}}_{x}} \\ {{{v}}_{y}} \\ \varepsilon \\ f \\ \end{gathered} \right) = \left( \begin{gathered} {{\rho }^{0}}(x) \\ {{u}^{0}}(x) \\ 0 \\ {{\varepsilon }^{0}}(x) \\ {{f}^{0}}(x) \\ \end{gathered} \right) + \left( \begin{gathered} \delta \rho \\ \delta {{{v}}_{x}} \\ \delta {{{v}}_{y}} \\ \delta \varepsilon \\ \delta f \\ \end{gathered} \right),$$

$$\left( \begin{gathered} \delta \rho \\ \delta {{{v}}_{x}} \\ \delta {{{v}}_{y}} \\ \delta \varepsilon \\ \delta f \\ \end{gathered} \right) = \left( \begin{gathered} \hat {\rho }(x) \\ {{{{\hat {v}}}}_{x}}(x) \\ {{{{\hat {v}}}}_{y}}(x) \\ \hat {\varepsilon }(x) \\ \hat {f}(x) \\ \end{gathered} \right){{e}^{{iky}}}{{e}^{{ - i\omega t}}},$$

where k = k_yL, k_y being the wavenumber of the monochromatic wave in the direction along the discontinuity. Henceforth, k is assumed to be real-valued everywhere, while ω is complex-valued in the general case.

Linearizing expressions (38) and (39), we obtain the system of equations for variables $\hat {\rho }$(x), ${{{\hat {v}}}_{x}}$(x), ${{{\hat {v}}}_{y}}$(x), $\hat {\varepsilon }$(x), and $\hat {f}$(x):

$${\mathbf{By}}{\kern 1pt} ' = {\mathbf{Ay}},$$

(40)

where prime indicates the differentiation with respect to x; y = {$\hat {\rho }$, ${{{\hat {v}}}_{x}}$, ${{{\hat {v}}}_{y}}$, $\hat {\varepsilon }$, $\hat {f}{{\} }^{ \top }}$, and

$${\mathbf{A}} = \left( {\begin{array}{*{20}{c}} {\frac{{d{{u}^{0}}}}{{dx}} - i\omega }&{\frac{{d{{\rho }^{0}}}}{{dx}}}&{i{{\rho }^{0}}k}&0&0 \\ { - \frac{{Q{{f}^{0}}{{\varepsilon }^{0}}}}{{{{\rho }^{{02}}}}}\frac{{d{{\rho }^{0}}}}{{dx}}}&{\frac{{d{{u}^{0}}}}{{dx}} - i\omega }&0&{\frac{Q}{{{{\rho }^{0}}}}\frac{{d{{f}^{0}}{{\rho }^{0}}}}{{dx}}}&{\frac{Q}{{{{\rho }^{0}}}}d\frac{{d{{\rho }^{0}}{{\varepsilon }^{0}}}}{{dx}}} \\ {i\frac{{{{\varepsilon }^{0}}{{f}^{0}}kQ}}{{{{\rho }^{0}}}}}&0&{ - i\omega }&{ik{{f}^{0}}Q}&{ik{{\varepsilon }^{0}}Q} \\ 0&{\frac{{d{{\varepsilon }^{0}}}}{{dx}}}&{i{{\varepsilon }^{0}}{{f}^{0}}k}&{\frac{{d{{u}^{0}}}}{{dx}}{{f}^{0}} - i\omega }&{{{\varepsilon }^{0}}\frac{{d{{u}^{0}}}}{{dx}}} \\ { - \frac{{\partial {{f}_{e}}}}{{\partial p}}{{f}^{0}}{{\varepsilon }^{0}} - \frac{{\partial {{f}_{e}}}}{{\partial \rho }}}&{\frac{{d{{f}^{0}}}}{{dx}}}&0&{ - \frac{{\partial {{f}_{e}}}}{{\partial p}}{{f}^{0}}{{\rho }^{0}}}&{ - {{\varepsilon }^{0}}{{\rho }^{0}}\frac{{\partial {{f}_{e}}}}{{\partial p}} - i\omega + 1} \end{array}} \right),$$

$${\mathbf{B}} = \left( {\begin{array}{*{20}{c}} { - {{u}_{0}}}&{ - {{\rho }_{0}}}&0&0&0 \\ { - \frac{{{{\varepsilon }_{0}}{{f}_{0}}Q}}{{{{\rho }_{0}}}}}&{ - {{u}_{0}}}&0&{ - {{f}_{0}}Q}&{ - {{\varepsilon }_{0}}Q} \\ 0&0&{ - {{u}_{0}}}&0&0 \\ 0&{ - {{\varepsilon }_{0}}{{f}_{0}}}&0&{ - {{u}_{0}}}&0 \\ 0&0&0&0&{ - {{u}_{0}}} \end{array}} \right),$$

where

$$Q = \frac{{{{V}_{1}} - 1}}{{1 - {{p}_{1}}}}.$$

Ultimately, Eqs. (40) take the form of dynamic system

$${\mathbf{y}}{\kern 1pt} ' = {\mathbf{Cy}},\quad {\mathbf{C}} = {{{\mathbf{B}}}^{{ - 1}}}{\mathbf{A}}$$

(41)

with coefficients depending on x.

For linear perturbations, we have

$$n = {{(1, - ik\tilde {\zeta })}^{ \top }},\quad {\boldsymbol{\tau }} = {{(ik\tilde {\zeta },1)}^{ \top }},$$

$$\tilde {\zeta }(y,t) = \zeta {{e}^{{iky - i\omega t}}},\quad {{D}_{x}} = - i\omega \tilde {\zeta },$$

is the x component of the velocity of the discontinuity, and n and τ are the vectors of the normal and tangent to the front, respectively.

Let us set

$${\mathbf{v}} = {{({{{v}}_{x}},{{{v}}_{y}})}^{ \top }}.$$

The boundary conditions at the leading shock wave x = $\tilde {\zeta }$(y, t) are as follows [4]: equality of the tangential components of velocity

$$({{{\mathbf{v}}}_{1}} - {{{\mathbf{v}}}_{0}}) \cdot {\boldsymbol{\tau }} = 0;$$

(42)

and jump of normal velocities

$$({{{\mathbf{v}}}_{1}} - {{{\mathbf{v}}}_{0}}) \cdot {\mathbf{n}} = \sqrt {({{p}_{0}} - {{p}_{1}})({{V}_{1}} - {{V}_{0}})} .$$

(43)

Denoting by D = Dn the velocity of the discontinuity, we obtain the relation for the velocity of the discontinuity as a function of p and V,

$${{[({{{\mathbf{v}}}_{1}} - {\mathbf{D}}) \cdot {\mathbf{n}}]}^{2}} = V_{1}^{2}\frac{{{{p}_{0}} - {{p}_{1}}}}{{{{V}_{1}} - {{V}_{0}}}};$$

(44)

the relation expressing the energy conservation at the discontinuity,

$${{\varepsilon }_{0}} - {{\varepsilon }_{1}} - \frac{1}{2}({{V}_{1}} - {{V}_{0}})({{p}_{1}} + {{p}_{0}}) = 0;$$

(45)

and the continuity of f at the discontinuity,

$${{f}_{1}} = {{f}_{0}}.$$

(46)

Relations (42)–(46) and expressions p = p(ρ, ε, f) in (38) imply that for perturbations {$\hat {\rho }$, ${{{\hat {v}}}_{x}}$, ${{{\hat {v}}}_{y}}$, $\hat {\varepsilon }$, $\hat {f}{{\} }^{ \top }}$ for x = 0, eliminating ζ from the boundary conditions, we obtain

$$\begin{gathered} \frac{{{{V}_{1}}}}{2}\left[ {{{V}_{1}}{{\beta }^{2}} + \frac{{{{\varepsilon }_{0}}(1 - \beta ){{f}_{0}}}}{{{{p}_{0}} - {{p}_{1}}}}} \right]\hat {\rho } + \frac{{(1 - \beta ){{f}_{0}}}}{{2\beta ({{p}_{0}} - {{p}_{1}})}}\hat {\varepsilon } + {{{{\hat {v}}}}_{x}} \\ - \frac{i}{{2k}}\left[ {\left( {\frac{{{{f}_{0}}{{\varepsilon }_{0}}}}{{{{p}_{0}} - {{p}_{1}}}} + \frac{{{{V}_{1}}{{\beta }^{2}}}}{{1 - \beta }}} \right)\frac{{d{{\rho }^{0}}}}{{dx}} + \frac{{{{f}_{0}}}}{{({{p}_{0}} - {{p}_{1}})\beta {{V}_{1}}}}} \right. \\ \left. { \times \frac{{d{{\varepsilon }^{0}}}}{{dx}} + \frac{2}{{(1 - \beta ){{V}_{1}}}}\frac{{d{{{v}}^{0}}}}{{dx}}} \right]{{{{\hat {v}}}}_{y}} = 0, \\ \end{gathered} $$

$$\begin{gathered} \left( { - \frac{{{{f}_{0}}{{\varepsilon }_{0}}}}{{{{p}_{0}} - {{p}_{1}}}} + \frac{{{{V}_{1}}{{\beta }^{2}}}}{{1 - \beta }}} \right)\hat {\rho } - \frac{{{{f}_{0}}}}{{{{V}_{1}}({{p}_{0}} - {{p}_{1}})\beta }}\hat {\varepsilon } \\ + \frac{i}{{k(1 - \beta )}}\left[ {\left( {\frac{{{{f}_{0}}{{\varepsilon }_{0}}}}{{{{V}_{1}}({{p}_{0}} - {{p}_{1}})}} - \frac{{{{\beta }^{2}}}}{{(1 - \beta )}}} \right)\frac{{d{{\rho }^{0}}}}{{dx}}} \right. \\ \left. { + \frac{{{{f}_{0}}}}{{V_{1}^{2}({{p}_{0}} - {{p}_{1}})\beta }}\frac{{d{{\varepsilon }^{0}}}}{{dx}} + \frac{{2\eta }}{{V_{1}^{2}}}} \right]{{{{\hat {v}}}}_{y}} = 0, \\ \end{gathered} $$

(47)

$$\begin{gathered} \text{[}{{\beta }^{2}}{{V}_{1}}({{p}_{0}} + {{p}_{1}}) + {{f}_{0}}{{\varepsilon }_{0}}(1 - \beta )]{{V}_{1}}\hat {\rho } \\ + \left( {\frac{{{{f}_{0}}}}{\beta } - {{f}_{0}} - 2} \right)\hat {\varepsilon } + \frac{i}{{(1 - \beta )k}}\left[ {({{f}_{0}}{{\varepsilon }_{0}}{{{( - 1 + \beta )}}_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}} \right. \\ \, - {{\beta }^{2}}{{V}_{1}}({{p}_{0}} + {{p}_{1}}))\frac{{d{{\rho }^{0}}}}{{dx}}\left. { + \frac{{(( - 1 + \beta ){{f}_{0}} + 2\beta )}}{{{{V}_{1}}\beta }}\frac{{d{{\varepsilon }^{0}}}}{{dx}}} \right]{{{{\hat {v}}}}_{y}} = 0, \\ - \frac{i}{{k{{V}_{1}}(1 - \beta )}}\frac{{d{{f}^{0}}}}{{dx}}{{{{\hat {v}}}}_{y}} + \hat {f} = 0, \\ \end{gathered} $$

where

$$\beta = {{U}_{0}} = 0.412.$$

It should be recalled that dimensionless quantities with subscript “0” characterize the state immediately behind the leading shock wave.

Equations (41) with boundary conditions (47) serve for determining the behavior of perturbations in region x > 0. Pay attention to the fact that relations (47) is the system of four linear homogeneous equations connecting five aforementioned quantities; therefore, one more boundary condition is required for the full description of perturbations. Physically, this is due to the fact that five quantities δρ, δ${{{v}}_{x}}$, δ${{{v}}_{y}}$, δε, and δf in the region behind the leading shock wave can be represented in the general case as five waves, four of which propagate from the shock wave, and one wave comes from infinity from behind. The boundary conditions are the conditions of reflection of this wave, which determine the four departing waves. This aspect will be considered below.

Analogously to [19–21], we introduce the dynamic system conjugate to (41):

$${\mathbf{z}}{\kern 1pt} ' = - {\mathbf{zC}}.$$

(48)

The solutions to systems of equations (41) and (48) are known to possess the following property:

$$({\mathbf{y}},{\mathbf{z}}){\kern 1pt} ' = 0,$$

(49)

this is established by direct verification. Here, (y, z) is the scalar product in ℝ₅.

Using the solutions to Eqs. (41) and (49), we obtain the condition for the absence of acoustic perturbations arriving from x = ∞.

Asymptotic system (41) for x → ∞ has form

$${\mathbf{\tilde {y}}}{\kern 1pt} ' = {{{\mathbf{C}}}_{\infty }}{\mathbf{\tilde {y}}},$$

(50)

where

$${{{\mathbf{C}}}_{\infty }} = \mathop {\lim }\limits_{x \to \infty } {\mathbf{C}}$$

is a matrix with constant coefficients of k and ω:

$${{{\mathbf{C}}}_{\infty }} = \left( {\begin{array}{*{20}{c}} {0.758 + 0.00198i\omega }&{0.198i\omega }&{0.198ik}&{0.200 - 0.200i\omega }&{9.98 - 4.99i\omega } \\ { - 0.758 + 0.998i\omega }&{ - 0.198i\omega }&{ - 1.20ik}&{ - 0.200 + 0.200i\omega }&{ - 9.98 + 4.99i\omega } \\ { - 5.04ik}&0&{i\omega }&{ - 1.01ik}&{ - 25.2ik} \\ {0.758 - 0.998i\omega }&{0.198i\omega }&{0.198ik}&{0.200 + 0.800i\omega }&{9.98 - 4.99i\omega } \\ { - 0.152}&0&0&{ - 0.04}&{ - 2.0 + i\omega } \end{array}} \right).$$

We have assumed that ρ₂, p₂, ${{{v}}_{{x2}}}$ = 1, ε₂ = $f_{2}^{{ - 1}}$ = 5, V₁ = 5, p₁ = 1/4.83, ∂f_e/∂ρ = 0.045, and ∂f_e/∂p = –0.2.

For given k and ω, Eq. (50) has solutions

$${{{\mathbf{\tilde {y}}}}_{i}} = {{c}_{i}}{{{\mathbf{r}}}_{i}}\exp {{\mu }_{i}}x,\quad i = 1,...,5,$$

where μ_i are the eigenvalues of matrix C_∞, r_i are the right eigenvectors corresponding to them, and c_i are arbitrary constants. Let us suppose that the wave arriving from infinity from behind corresponds to subscript “5,” the eigenvalue of limiting matrix is μ₅, the right eigenvector is r₅, and the left eigenvector (row vector) is l₅. The condition of the absence of the arriving wave is c₅ = 0.

System (48) for x → ∞ has solutions of form

$${{{\mathbf{\tilde {z}}}}_{i}} = {{c}_{{zi}}}{{{\mathbf{l}}}_{i}}\exp ( - {{\mu }_{i}}x),\quad i = 1,...,5,$$

where l_i are the left eigenvectors (row vectors) of matrix C_∞, and c_zi are arbitrary constants.

We write the solution to system (41) in form

$${\mathbf{y}} = \sum\limits_{i = 1}^5 {{{c}_{i}}{{{\mathbf{F}}}_{{\mathbf{i}}}}(x),} $$

(51)

where

$${{{\mathbf{F}}}_{{\mathbf{i}}}}(x) \to {{{\mathbf{r}}}_{i}}\exp {{\mu }_{i}}x$$

(52)

for x → ∞.

Let us consider the solution to system (48),

$${\mathbf{z}} = {{{\mathbf{G}}}_{{\mathbf{5}}}}(x),$$

such that

$${{{\mathbf{G}}}_{{\mathbf{5}}}}(x) \to {{{\mathbf{l}}}_{5}}\exp ( - {{\mu }_{5}}x)$$

(53)

for x → ∞.

Since left eigenvector l₅ is orthogonal to right eigenvectors r_i, i = 1, …, 4, we have

$$\begin{gathered} \left( {\mathop {\lim }\limits_{x \to \infty } {{{\mathbf{G}}}_{{\mathbf{5}}}}(x),\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{\mathbf{i}} = {\mathbf{1}}}^{\mathbf{4}} {{{{\mathbf{c}}}_{{\mathbf{i}}}}{{{\mathbf{F}}}_{{\mathbf{i}}}}({\mathbf{x}})} } \right) \\ = \left( {{{{\mathbf{l}}}_{5}}\exp ( - {{\mu }_{5}}x),\sum\limits_{{\mathbf{i}} = {\mathbf{1}}}^{\mathbf{4}} {{{{\mathbf{c}}}_{{\mathbf{i}}}}{{{\mathbf{r}}}_{{\mathbf{i}}}}\exp ({{\mu }_{{\mathbf{i}}}}{\mathbf{x}})} } \right) \\ = \sum\limits_{{\mathbf{i}} = {\mathbf{1}}}^{\mathbf{4}} {{{{\mathbf{c}}}_{{\mathbf{i}}}}} ({{{\mathbf{l}}}_{{\mathbf{5}}}},{{{\mathbf{r}}}_{{\mathbf{i}}}})\exp (({{\mu }_{{\mathbf{i}}}} - {{\mu }_{{\mathbf{5}}}}){\mathbf{x}}) = {\mathbf{0}}. \\ \end{gathered} $$

(54)

Then relations (49), (51), (53), and (54) give (see, for example, [22])

$$\begin{gathered} ({{{\mathbf{G}}}_{{\mathbf{5}}}}(x),{\mathbf{y}}({\mathbf{x}})) = \left( {\mathop {\lim }\limits_{x \to \infty } {{{\mathbf{G}}}_{{\mathbf{5}}}}(x),\mathop {\lim }\limits_{x \to \infty } {\mathbf{y}}({\mathbf{x}})} \right) \\ = \left( {\mathop {\lim }\limits_{x \to \infty } {{{\mathbf{G}}}_{{\mathbf{5}}}}(x),\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{\mathbf{i}} = {\mathbf{1}}}^{\mathbf{5}} {{{{\mathbf{c}}}_{{\mathbf{i}}}}{{{\mathbf{F}}}_{{\mathbf{i}}}}({\mathbf{x}})} } \right) \\ = \left( {\mathop {\lim }\limits_{x \to \infty } {{{\mathbf{G}}}_{{\mathbf{5}}}}(x),\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{\mathbf{i}} = {\mathbf{1}}}^{\mathbf{4}} {{{{\mathbf{c}}}_{{\mathbf{i}}}}{{{\mathbf{F}}}_{{\mathbf{i}}}}({\mathbf{x}}) + \mathop {\lim }\limits_{x \to \infty } {{c}_{5}}{{{\mathbf{F}}}_{{\mathbf{5}}}}(x)} } \right) \\ = ({{{\mathbf{l}}}_{5}}\exp ( - {{\mu }_{5}}x),{{c}_{5}}{{{\mathbf{r}}}_{5}}\exp ({{\mu }_{5}}x)) = {{c}_{5}}({{{\mathbf{l}}}_{5}},{{{\mathbf{r}}}_{5}}). \\ \end{gathered} $$

(55)

The scalar product of the left and right eigenvectors corresponding to the same eigenvalue differs from zero; therefore, the condition

$${{c}_{5}} = 0$$

(56)

of absence of arriving perturbations can be written as

$$({{{\mathbf{G}}}_{{\mathbf{5}}}}(x),{\mathbf{y}}({\mathbf{x}})) = 0.$$

(57)

In this case, by virtue of relation (49), equality (57) holds for all x ≥ 0. Let us write Eq. (57) for x = 0:

$$({{{\mathbf{G}}}_{{\mathbf{5}}}}({\mathbf{0}}),{\mathbf{y}}({\mathbf{0}})) = 0.$$

(58)

This equation is the missing boundary condition mentioned above for x = 0. The fulfillment of this condition is the necessary and sufficient condition of the absence of an acoustic perturbation arriving at the structure from behind.

Boundary conditions (47) together with Eq. (58) forms a homogeneous system of linear equations with matrix further denoted as H in five quantities $\hat {\rho }$, ${{{v}}_{x}}$, ${{{v}}_{y}}$, $\hat {\varepsilon }$, and $\hat {f}$ for x = 0. Coefficients of matrix H are equal to the coefficients of these unknowns in Eqs. (47) and Eq. (58). This homogeneous system has a nontrivial solution if its determinant equals zero:

$$\det {\mathbf{H}} = 0.$$

(59)

This condition ensures the fulfillment of the boundary conditions on the leading shock wave and the absence of perturbations arriving from infinity. It is an equation in ω for a preset k.

The goal of testing of the basis solution for stability is to determine such η = –iω and k, for which detH = 0 and Reη > 0 or to demonstrate that such values of η and k do not exist.

In other words, the search for an unstable discrete spectrum of operator (41) (for eigenvalues located in the right half-plane Ω⁺ of spectral parameter η) is reduced to determining zeros of detH(η, k) lying in half-plane Ω⁺ for real-valued k. For a preset k, the number of zeros of detH(η, k) lying within a closed contour on the η plane can be calculated using the principle of argument and is determined by the total number of turns of the image of detH(η, k) of the chosen close contour around zero.

5 ANALYSIS OF STABILITY OF THE SHOCK WAVE STRUCTURE

To determine function detH(η, k), we must find the values of function G₅(0). To this end, the Cauchy problem for system of equations (48) is solved numerically with given values of η and k with following condition:

$${{{\mathbf{G}}}_{{\mathbf{5}}}}({{x}_{r}}) = {{{\mathbf{l}}}_{{\mathbf{5}}}}.$$

The value of x_r is chosen quite large so that the parameters of the main flow change insignificantly for x > x_r , and the components of column G₅(0) can be calculated with a sufficiently high accuracy. The accuracy of calculation for quantity x_r was verified by comparing the results obtained when the boundary condition was specified for x = x_r and x = 2x_r.

The numerical solution of system of equations (48) is performed by backward integration from x = x_r to x = 0 by the second-order Runge–Kutta method with a constant integration step. The accuracy of the solution was verified by comparing the results obtained with integration steps differing by two times.

For the arriving acoustic wave for values of η lying in the right half-plane or on the imaginary axis, the value of μ₅ has a positive real part, and solution G₅(x) at the initial stage of the backward integration increases exponentially in accordance with relation (53). To reduce computational difficulties associated with a rapid increase of the solution, it was proposed [19] that the solution be sought in form

$${{{\mathbf{G}}}_{{\mathbf{5}}}}(x) = {{{\mathbf{\bar {G}}}}_{5}}(x)\exp ( - {{\mu }_{5}}x),$$

where function ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(x) satisfies the condition

$${{{\mathbf{\bar {G}}}}_{5}}({{x}_{r}}) = {{{\mathbf{l}}}_{{\mathbf{5}}}}$$

(60)

and is a solution to system

$$\frac{{d{{{{\mathbf{\bar {G}}}}}_{5}}(x)}}{{dx}} = - {{{\mathbf{\bar {G}}}}_{5}}(x)({\mathbf{C}} - {{\mu }_{{\mathbf{5}}}}{\mathbf{I}}).$$

(61)

Here, I is the unit matrix.

Matrix (C – μ₅I) in the limit for x → ∞ has left eigenvector l₅ that corresponds to eigenvalue 0; for this reason, function ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(x) varies much more slowly than G₅(x).

Analogously, we introduce new function ${\mathbf{\bar {y}}}$(x) such that y(x) = ${\mathbf{\bar {y}}}$(x)exp(μ₅x). This function coincides with y(x) for x = 0 and satisfies equation

$$\frac{{d{\mathbf{\bar {y}}}(x)}}{{dx}} = ({\mathbf{C}} - {{\mu }_{{\mathbf{5}}}}{\mathbf{I}}){\mathbf{\bar {y}}}(x).$$

(62)

In this case, because of the fulfillment of relations

$$\begin{gathered} ({{{\mathbf{G}}}_{{\mathbf{5}}}}(0),{\mathbf{y}}(0)) = ({{{{\mathbf{\bar {G}}}}}_{5}}(0),{\mathbf{\bar {y}}}(0)), \\ {{{\mathbf{G}}}_{{\mathbf{5}}}}(0) = {{{{\mathbf{\bar {G}}}}}_{5}}(0),\quad {\mathbf{y}}({\mathbf{0}}) = {\mathbf{\bar {y}}}(0) \\ \end{gathered} $$

(63)

coefficients of matrix H remain unchanged.

Left eigenvector l₅ that is used in the formulation of condition (60) is defined to within a complex-valued constant. The components of row vector ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0) depend continuously on η if it is chosen so that one of components of row vector ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0) remains unchanged. In the calculations given below, the fourth component of this row vector was chosen constant and equal to 1 (the choice of component is indifferent). For row vector ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0) for which the fourth component equals unity, we introduce notation ${{{\mathbf{\hat {G}}}}_{5}}$(0). Obviously, we have

$${{{\mathbf{\hat {G}}}}_{5}}(0) = {{{\mathbf{\bar {G}}}}_{5}}(0){\text{/}}\bar {G}_{5}^{4}(0).$$

The algorithm of calculation of detH(η, k) includes the following steps.

For given values of k and η, we determine μ₅ (eigenvalue of matrix C_∞) corresponding to the arriving acoustic wave and the left eigenvector l₅ corresponding to this eigenvalue and having the fourth component equal to 1.

Let us solve the Cauchy problem for a preset value of ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(x_r) = l₅ for system of differential equations (61) and determine the components of vector ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0).

Let us find components of vector ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0).

For given values of k, η, and for determined ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0), we calculate matrix H(η, k) and find its determinant.

We introduce notation

$${{D}_{h}}(\eta ,k) = \det {\mathbf{H}}(\eta ,k),$$

if the determinant has been obtained using vector ${{{\mathbf{\hat {G}}}}_{{\mathbf{5}}}}$(0).

Figure 4 shows closed contour ABC on the (Re(ω), Im(ω)) plane, which is the boundary of a sector of radius 100. The sector symmetric to that plotted relative to the Im(ω) axis can be disregarded in view of the obvious symmetry of the problem relative to the substitution of –y for y.

Fig. 4.

Contour on the (Re(ω), Im(ω)) plane.

The value of |D_h| varies strongly for real-valued ω in the limits 0 < ω < 100. For this reason, details of the image of ABC contour differ significantly in scales. To verify that the image of the ABC contour does not embrace the origin of the coordinates, we must either depict the details of this contour on different graphs on different scales or use the method proposed in [19]. In this method, the graph is plotted not for function D_h, but for function D_h/|D_h|^a, where exponent a is chosen depending on the form of variation of function D_h on the contour. Obviously, the image of the ABC contour obtained with the help of function D_h/|D_h|^a embraces the origin of coordinate if and only if this is valid for the image obtained using function D_h.

The results of calculations are shown in Figs. 5, 6, and 7 for values of k = 0.0512, 1.54, and 5.12, respectively. The last of the three values of k is 100 times larger than the first one. It can be seen from all three figures that the image of the ABC contour does not embrace the origin; therefore, function D_h(ω) does not vanish for the values of ω corresponding to points belonging to sector ABC. It should be noted that Fig. 7 shows the graph of function D_h(ω) itself.

Fig. 5.

Image D_h(ω, k) of the boundary of the sector shown in Fig. 4. Calculation was made for λ = 1 and k = 0.0512.

Fig. 6.

Image D_h(ω, k) of the boundary of the sector shown in Fig. 4. Calculation was made for λ = 1 and k = 1.54.

Fig. 7.

Image D_h(ω, k) of the boundary of the sector shown in Fig. 4. Calculation was made for λ = 1 and k = 5.12.

The contours in Figs. 5, 6, and 7 have a segment of the zigzag shape. The zigzag between points D and E in Fig. 6 corresponds to the variation of the real value of ω from 1.5 to 6. The curves in Fig. 8 represent functions Re(k_x) and Im(k_x) depending on the real value of ω for Im(ω) = 0. Here, k_x = –iμ₅ is the wavenumber corresponding to the arriving acoustic wave for x → ∞. It can be seen from Fig. 8 that Im(k_x) experiences the strongest variation between points D and E and approaches its asymptotic value at point E. In addition, on the left of point D and on the right of point E, the slope of function Re(k_x) is close to constant, but different values. Obviously, together with k_x, the value of l₅ varies equally strongly for 1.5 < ω < 6, which leads to a change in the components of column vector ${{{\mathbf{\bar {G}}}}_{{\mathbf{5}}}}$(0) and D_h.

Fig. 8.

Re(k_x) and Im(k_x) for real values of ω. Calculation was made for λ = 1 and k = 1.54.

As shown above, function D_h(ω, k) does not vanish for |ω| ≤ 100, Re(ω) ≥ 0, and Im(ω) ≥ 0. Let us now show that function D_h does not vanish for |ω| > 100, Re(ω) ≥ 0, and Im(ω) ≥ 0 (i.e., in the first quadrant outside of sector ABC in Fig. 4). We introduce auxiliary function

$${{D}_{M}}(\omega ,k) = \frac{{i{{D}_{h}}(\omega ,k)}}{\omega }.$$

Since parameter ω does not vanish outside of sector ABC either, zeros of function D_M(ω, k) and D_h(ω, k) coincide.

Calculations show that function D_M(ω, k) has limit iD_∞(k) for |ω| → ∞. This limit depends on k, and D_∞(k) is a positive real-valued function. Figure 9 shows the contours with arcs B₁C₁, B₂C₂, and B₃C₃ of radius |ω| equal to 104, 524, and 1000, respectively. The images of these curves plotted with the help of function D_M(ω, k) are shown in Fig. 10. It can be seen that for |ω| → ∞, the image of the arc is contracted to point C_∞; in this case, contour B₁C₁C_∞ does not embrace the origin.

Fig. 9.

Contours on the (Re(ω), Im(ω)) plane.

Fig. 10.

Images of the boundaries of the contours shown in Fig. 9. Calculation was made for λ = 1 and k = 5.12.

The calculations performed in a wide range of variation of ω and k revealed the absence of zeroth of function D_h(ω, k) for Im(ω) ≥ 0. However, these zeros exist for Im(ω) < 0 for all values of k considered here.

The approximate value of ω for which function D_h(ω) has zero value was determined by the half division method. In the third quadrant, a rectangle with sides parallel to the coordinate axes was constructed. The rectangle was large enough, and its image obtained using function D_h(ω) embraced the origin. The first stage of the further procedure of refinement the position of the root involved the construction of rectangles embedded into one another, such that the image of each rectangle contained the origin. When each side of a rectangle became quite small, a contour in the shape of a circle was used. The image of a small-radius circle was also close to the circle, and the coordinates of two sequential positions of the centers of the circle and of its image were used for obtaining a new position of the center of the circle on the plane corresponding to complex values of ω.

Figures 11 and 12 illustrate the example of approximate determination of the value of ω, for which function D_h(ω) has zero value. Figure 11 shows a circle on the (Re(ω), In(ω)) plane. Figure 12 shows the image of this circle in which the origin is located. The calculations were performed with an integration step decreasing by half from curve 1 to curve 4. Curves 3 and 4 are spaced closely on the graph; therefore, the accuracy of calculation with the step corresponding to curve 3 can be considered sufficient. In plotting curve 3, we used 4800 integration steps.

Fig. 11.

Contour on the (Re(ω), Im(ω)) plane; k = 0.0254.

Fig. 12.

Image of the circle shown in Fig. 11; k = 0.0254, x_r = 15.6: (1) 1200, (2) 2400, (3) 4800, and (4) 9600 steps.

As can be seen from Fig. 11, the radius of the contour was chosen so that to ensure the determination of the real and imaginary parts of ω to the third significant digit.

As noted above, the accuracy of calculation depends on the value of parameter x_r. The backward integration of Eq. (61) begins from this value of the spatial coordinate. We assume that asymptotic form (50) ensures the required accuracy of calculation for this value of the coordinate. Figure 13 shows the results of calculation for values of x_r differing by two times for a constant integration step the same as for curve 3 in Fig. 12. Curves 2 and 3 are in good agreement with the results obtained for x_r = 15.6 and 31.2.

Fig. 13.

Image of the circle shown in Fig. 11; k = 0.0254, integration step Δx = 0.003255: x_r = (1) 7.81, (2) 15.62, (3) 31.24.

Such a procedure of estimating the accuracy of calculation from the integration step as well as from the value of x_r was performed in all calculations. The value of x_r = 15.6 ensured a sufficiently high accuracy for all values of ω used here, and the required number of steps increased with increasing |ω| and amounted to 372 000 for |ω| = 1000.

Table 1 contains the determined values of ω, for which D_h(ω) = 0.

Table 1

k	Re(ω)/k	Im(ω)/k2
0.001	1.65	–0.800
0.003	1.65	–0.800
0.01	1.65	–0.800
0.0254	1.65	–0.805
0.1	1.65	–0.833
0.15	1.65	–0.838

The data given in Table 1 show that Re(ω)/k and Im(ω)/k² weakly depend on k in the interval considered here (see Figs. 14 and 15). In the range of k values in which the values of Re(ω)/k = W and Im(ω/k²) = ‒b can be considered constant, the following equality holds:

$$\omega = Wk - ib{{k}^{2}}.$$

(64)

Fig. 14.

Dependence of Re(ω)/k on k provided that D_h(ω) = 0.

Fig. 15.

Dependence of Im(ω)/k² on k provided that D_h(ω) = 0.

This equality corresponds to differential equation

$$\frac{{\partial \xi }}{{\partial t}} + W\frac{{\partial \xi }}{{\partial y}} = b\frac{{{{\partial }^{2}}\xi }}{{\partial {{y}^{2}}}}.$$

(65)

6 CONCLUSIONS

We have constructed the solution to the problem of the structure of a spontaneously emitting shock wave (SESW) in a gas and demonstrated the stability of this structure in the linear approximation. The results correspond to a specific shock wave with a specific model of its structure. However, these results show that the argument refuting the existence of SESW put forth in [7] and doubts about the absence of spontaneous emission that can issue from numerical experiments [9–11] are unjustified. We believe that there are no grounds for discrediting the conclusions drawn in the initial classical works [1–3] concerning SESW and for doubting the existence of nonlinear solutions given in [6]. It is also important to note the asymptotic stability of the SESW structure in the concrete case considered here. Naturally, the conclusion about the stability of the structure refers to the specific case and depends on the model of the structure; however, there are certain grounds to believe that if a structure is associated with dissipative processes, it turns out to be stable.

Let us comment on dependence (64) obtained from dispersion relation (59). Ratio ω/(${{{v}}_{1}}$k_y) depends on the value of quantity k = k_yδ (δ is the effective width of the structure). As follows from Table 1, for k ≪ 1 (i.e., l/δ ≫ 1, where l is the length of the perturbation wave in the y direction), disregarding terms k² as well as the terms with higher powers of k_y, we obtain a linear dependence of ω on k_y (the same as in analysis of the discontinuity disregarding its structure). This result is expected because in this approximation, the flow in the vicinity of each point of the perturbed structure occurs as in a stationary structure with a 2D shock wave with the fulfillment of the conservation laws. The next term in the expansion of ω in k takes into account the curvature of the structure and nonstationary form of the flow in it. As shown in this study, the existence of the discontinuity structure leads to the emergence of a dissipative term proportional to k² in the dispersion equation; therefore, the perturbation of the shape of a quite long wave propagating over the SESW in the positive y direction (or in the negative direction upon the sign reversal of W) is described by Eq. (65).

Finally, let us consider how perturbations are transmitted along an SESW. We analyze the stability of perturbations of a shock wave with the front perpendicular to the velocity of the incoming flow. Figure 16 shows a stationary SESW located on the y axis above point A. Let us suppose that the perturbation specified by the initial conditions is such that a part of the shock wave, remaining planar, has slightly (by a small angle θ) changed its direction (straight line AB in Fig. 16). This leads to a change in its velocity. The point of intersection of the perturbed and unperturbed shock waves moves over the unperturbed shock wave with a certain velocity W. If condition (4) is satisfied (i.e., W > $\sqrt {{{c}^{2}} - {{{v}}^{2}}}$, c and ${v}$ being the velocity of sound and the normal velocity of the gas behind the shock wave), this means that the shock wave is a spontaneously emitting wave. Obviously, the velocity of the point of intersection of two shock waves propagating in close directions is determined by derivative ∂D/∂θ, where D is the shock wave velocity. The fulfillment of aforementioned condition (4) requires that derivative ∂D/∂θ be quite large. In the linear approximation, velocity W is independent of θ; therefore, linear perturbations of any form propagate over the shock wave without distortions. If we take into account the term with the second derivative on the right-hand side of Eq. (65), the obtuse angle between the perturbed and unperturbed shock waves will be smoothed as shown by dashed line in Fig. 16. Representing the solution to Eq. (65) in the form of the Fourier integral with respect to k_y and bearing in mind that each value of k_y corresponds to a certain system of waves emitted into the region behind the shock wave, we can obtain (in explicit form) a wave of perturbations wave shape, which propagates over the shock wave with an arbitrary initial dependence on y and the corresponding system of small perturbations propagating from the shock wave. The shock wave itself in this case serves as a conductor of perturbations (waveguide) which receives energy from the gas flowing through it.

Fig. 16.

Propagations of perturbations over a shock wave.