Self-Similar Flows with a Shock Wave Advancing toward the Center or Axis of Symmetry

Abstract

We study one-dimensional flows of an ideal (inviscid and non-heat-conducting) perfect gas with an adiabatic exponent γ behind a shock wave moving toward the center (ν = 3) or axis (ν = 2) of symmetry in a cold gas at rest. Flows with a reflected shock wave and flows terminating with simultaneous arrival of a shock wave and a piston, which has compressed the gas into a point or line, to the center of symmetry are admitted.

The first self-similar solution with a reflected shock wave was found by Guderley. In his solution the self-similarity exponent n = ${{n}_{*}}$(ν, γ) < 1 is determined from the condition that the flow under consideration does exist. At n = ${{n}_{*}}$(ν, γ) there can also exist self-similar solutions with a piston expanding from the center of symmetry after the arrival of a convergent shock wave to this point; these solutions have not been considered earlier. The attention to the flows with n ≠ ${{n}_{*}}$(ν, γ) was attracted in 2015–2017 by V. Kuropatenko with his colleagues. An analysis of these flows made below shows that the flows with a reflected shock wave constructed for n > ${{n}_{*}}$(ν, γ) are self-similar only in a curvilinear triangle in the coordinate-time plane. One of its sides is a segment of the trajectory of a self-similar shock wave that has not arrived at the center of symmetry. At n < ${{n}_{*}}$(ν, γ) these solutions become the completely self-similar flows of gas compression to a point constructed by Kuropatenko.

1 SELF-SIMILAR SOLUTIONS ON THE TIME-DEPENDENT SHOCK WAVE REFLECTION FROM AN AXIS OF CENTER OF SYMMETRY

In 1942 Guderley [1] (see also [2, 3]) solved the problem of the reflection of a shock wave (SW) from the axis or center of symmetry (below, from the “center of symmetry” (CS)). As a SW approaches the CS, its intensity increases without bound. It is possible that precisely for this reason the solutions searched in [1] were self-similar solutions with a strong SW IS moving throughout a cold gas at rest at a velocity D = D(t) < 0 with the time t reference point at the moment of SW IS arrival to the CS. For a perfect gas its radial velocity u, speed of sound а, density ρ, and pressure p behind a strong SW are as follows [2–5]:

$$u = \frac{2}{{\gamma + 1}}D,\quad a = - \frac{{\sqrt {2\gamma (\gamma - 1)} }}{{\gamma + 1}}D,\quad \rho = \frac{{\gamma + 1}}{{\gamma - 1}}{{\rho }_{0}},\quad p = \frac{2}{{\gamma + 1}}{{\rho }_{0}}{{D}^{2}}.$$

(1.1)

At a radial coordinate r the flow parameters can be represented in one of two equivalent forms with the self-similar variable ξ or τ

$$\begin{gathered} \xi = Cr \times \left\{ \begin{gathered} {{( - t)}^{{ - n}}},\quad t \leqslant 0, \hfill \\ {{t}^{{ - n}}},\quad t > 0; \hfill \\ \end{gathered} \right.\quad \tau = \frac{{Ct}}{{{{r}^{k}}}},\quad k = \frac{1}{n}, \\ u = n\frac{r}{t}U,\quad a = n\frac{r}{t}A,\quad \rho = {{\rho }_{0}}R,\quad p = {{\rho }_{0}}{{n}^{2}}\frac{{{{r}^{2}}}}{{{{t}^{2}}}}P,\quad P = \frac{R}{\gamma }{{A}^{2}}. \\ \end{gathered} $$

(1.2)

In Guderley’s solution the exponent n is to be determined, U, A, and R are functions of ξ or τ, and the constant С is arbitrary. Guderley worked with the coordinate ξ, which is finite on the SW IS, infinite at the r axis, and zero at the semi-axis t > 0. The variable τ, the same for all t, with C > 0 increases monotonically from a negative value on the SW IS to zero at the r axis and to infinity at the semi-axis t > 0.

The ordinary differential equations obtained by means of substituting the representations (1.2) with the variable τ into the equations governing the time-dependent flows of an ideal perfect gas having only one nonzero r-component of the velocity vector u reduce to the system of equations

$$\begin{gathered} \frac{{dA}}{{dU}} = \frac{{A{{f}_{2}}}}{{2(1 - U){{f}_{1}}}},\quad \frac{{d\tau }}{{dU}} = \gamma k\frac{{\tau f}}{{{{f}_{1}}}},\quad \frac{{d\ln R}}{{dU}} = \frac{{{{f}_{3}}}}{{(1 - U){{f}_{1}}}}, \\ f = {{(1 - U)}^{2}} - {{A}^{2}},\quad {{f}_{1}} = \gamma (1 - U)(k - U)U + (2k - 2 - \nu \gamma U){{A}^{2}}, \\ {{f}_{2}} = \gamma {\kern 1pt} \text{[}2(k - U) - \nu (\gamma - 1)U]f + (\gamma - 1){{f}_{1}},\quad {{f}_{3}} = {{f}_{1}} - \nu \gamma Uf. \\ \end{gathered} $$

(1.3)

The substitution of the same representations into relations (1.1), in which, due to these representations, D = r_IS/(kt) = nr_IS/t, yields

$${{U}_{{IS}}} \equiv U({{\tau }_{{IS}}}) = \frac{2}{{\gamma + 1}},\quad {{A}_{{IS}}} = - \frac{{\sqrt {2\gamma (\gamma - 1)} }}{{\gamma + 1}},\quad {{R}_{{IS}}} = \frac{{\gamma + 1}}{{\gamma - 1}}.$$

(1.4)

The constant С, which does not enter in Eqs. (1.3) and representations (1.4), can be chosen such that the quantity τ_IS is equal to any number, for example, −1. The representation of the SW IS as a power-law curve r ^{k > 1} = −Ct, that arrives to the CS at t = 0, does not ensure the existence of such a flow. An analysis of the integral curves and singular points of the first equation (1.3) can help to understand this fact. At the r axis and r > 0 the variable τ = 0, while u and a are finite. According to Eq. (1.2), this is possible only at U(0) = A(0) = 0. For this reason, for any k > 1 the integral curves of solutions containing the r axis or its finite interval originating at a point on IS with U and A from Eq. (1.4) must arrive at the origin A = U = 0 of the UA plane. However, the point on IS lies beneath the “sonic” line A = U − 1 presented by dashes in Fig. 1. At this line the function f from the second equation (1.3) is zero, while the integral curve proceeding from point IS must intersect it. For an arbitrary k the function  f₁ ≠ 0 at the point of intersection between the integral curve and the sonic line and the function τ has a local maximum τ = ${{\tau }_{{0*}}}$ at this point rather than increases monotonically; the occurrence of this maximum excludes the continuation of the solution toward the region, where τ > ${{\tau }_{{0*}}}$. In Fig. 1 and in what follows the arrows on the integral curves indicate the direction of the growth of the variable τ, while on the red curve the function f₁(U, A) is zero.

Fig. 1.

Integral curves, singular points, and “sonic” lines: f = 0 and SW RS (upper arrow) in Guderley’s solution for ν = 3, γ = 5/3, and ${{n}_{*}}$ = 0.68838.

Any point on the line A = U − 1 in the UA plane is associated with a curve with dr/dt = u − a in the rt plane, as in the case of the С⁻-characteristics. However, these curves are the envelopes of the С⁻-characteristics rather than the С⁻-characteristics themselves, since the compability conditions for С⁻-characteristics are not fulfilled on these curves at  f₁ ≠ 0. The envelope becomes a С⁻-characteristic, if two conditions are fulfilled at the intersection point, namely,  f = 0 and  f₁ = 0. Their consequence, that is, the equality f₂ = 0, makes the intersection point a singular point (saddle SP or node NP) of the first equation (1.3). The passage through the sonic line becomes possible along the integral curve coinciding with one of separatrices of the saddle or with the basic appendix of a node, while the choice n = ${{n}_{*}}$(ν,γ) makes it possible to join the singular point with the point IS by means of the integral curve. For ν = 3 and γ = 5/3 precisely this situation with the saddle SP and ${{n}_{*}}$ = 0.68838 is presented in Fig. 1. The origin A = U = 0 is a node of the first equation (1.3) with the bald integral curve that interested the sonic line and arrived at it. For ν = 2 and 3 and several values of γ the quantities ${{n}_{*}}$ = ${{n}_{*}}$(ν,γ) are presented in Table 1.

Table 1. Exponents n₀ = n₀(ν, γ) and ${{n}_{*}}$ = ${{n}_{*}}$(ν, γ)

At k = ${{k}_{*}}$ = 1/${{n}_{*}}$ > 1 the pressure р and the entropy function S = γр/ρ^γ behind the SW IS are infinite at the moment of its arrival at CS. With any increase in the reflected SW RS the function S remains infinite in the CS (at r = 0 and t > 0). However, at р = ∞ the gas expands instantaneously to a finite pressure and zero density, which, having conserved S = ∞, turn the speed of sound to infinity, a = (γp/ρ)^1/2 = ${{n}_{*}}$(r/t)A = ∞, at the semiaxis t > 0. Thus, А(τ = ∞) = ∞ and transition from А to the variable δ = 1/А lead to the equation

$$\begin{gathered} \frac{{d\delta }}{{dU}} = \frac{{\delta {{f}_{2}}}}{{2(U - 1){{f}_{1}}}},\quad \frac{{d\tau }}{{dU}} = \gamma {{k}_{*}}\frac{{\tau f}}{{{{f}_{1}}}}, \\ f = {{(1 - U)}^{2}}{{\delta }^{2}} - 1,\quad {{f}_{1}} = \gamma (1 - U)({{k}_{*}} - U)U{{\delta }^{2}} + 2({{k}_{*}} - 1) - \nu \gamma U, \\ {{f}_{2}} = (\gamma - 1){{f}_{1}} + \gamma [2({{k}_{*}} - U) - \nu (\gamma - 1)U]f. \\ \end{gathered} $$

(1.5)

The U axis, at which δ = 0, is the integral curve of the first equation (1.5). The other integral curves can arrive at this axis only through its singular points δ = 0, U = 1 and δ = 0, U = 2(${{k}_{*}}$ − 1)/(νγ). In Fig. 1b the former point is a node and the latter one is a saddle. In view of the second equation (1.5), the variable τ is finite at the node, whereas at the saddle τ = ∞. For this reason, a portion of the unknown integral curve is a segment of the saddle separatrix. The solution arrives jumpwise from point RS₋ at point RS₊ on the separatrix (upper arrow in Fig. 1a), and for U₊ and A₊ behind the reflected SW RS the following equations hold true [5]

$$\begin{gathered} {{U}_{ + }} = 1 - \frac{{2A_{ - }^{2} + (\gamma - 1){{{(1 - {{U}_{ - }})}}^{2}}}}{{(\gamma + 1)(1 - {{U}_{ - }})}}, \\ {{A}_{ + }} = \frac{{\sqrt {2(\gamma - 1)[\gamma {{{(1 - {{U}_{ - }})}}^{4}} - A_{ - }^{4}] + (6\gamma - {{\gamma }^{2}} - 1)A_{ - }^{2}{{{(1 - {{U}_{ - }})}}^{2}}} }}{{(\gamma + 1)(1 - {{U}_{ - }})}}. \\ \end{gathered} $$

(1.6)

When constructing the solution the point RS₋ with U₋ and A₋ on the integral curve that came from point IS is so chosen that, in accordance with Eqs. (1.6), the point RS₊ arrives at the separatrix of the saddle of the first equation (1.5) that comes from above in Fig. 1b.

The fourth quadrant of Fig. 1a and a portion of the rt-diagram corresponding to the bold segment of the integral curve connecting point IS with the origin U = A = 0 are presented in Fig. 2 on a larger scale. In Fig. 2b the left (right) bold curve that proceeds from point 1 is the SW IS (piston trajectory), while the segments connecting them are С⁻-characteristics. One of these segments that connects point 0⁻ with the origin is a С⁻-characteristic corresponding to the saddle SP on the sonic line in Fig. 2a. On the semiaxis t > 0, which is not presented on the rt-diagram, the gas velocity u = ${{n}_{*}}$rU/t = 0, as it must be in the CS.

Fig. 2.

Quadrant IV in the UA plane (а) and a part of the rt-diagram (b) of Guderley’s solution with SW reflection from the CS for ν = 3, γ = 5/3, and ${{n}_{*}}$ = 0.68838.

Although many researchers studied the Guderley solution, the most complete and important results have been obtained by the authors of this paper [5–8]. It is true that, starting from Guderley all the authors took no notice of the self-similar solutions with a piston that started its expansion from the CS at the moment of the arrival of the SW IS from the Guderley solution. We will fill this gap.

Letting A₋ = U₋ = 0 in Eq. (1.6) we arrive at the following values

$${{U}_{ + }} = {{U}_{{IS}}} = \frac{2}{{\gamma + 1}},\,\,\,\,{{A}_{ + }} = {{A}_{{IS}}} = \frac{{\sqrt {2\gamma (\gamma - 1)} }}{{\gamma + 1}}$$

(1.7)

behind the strong SW moving from the CS; in Fig. 1a it is shown by an arrow between the origin and the upper point IS. The SW with U_IS and А_IS from Eq. (1.7) is the most intense from the family of SWs, whose values of U₋ and A₋ correspond to the points on the segment of the bold integral curve from the second quadrant in Fig. 1a. The integral curves along which the values of U and А vary from the SW to the piston, are terminated at the node U = 1, δ = 0 in Fig. 1b. Figure 3 presents the rt-diagram for one of these self-similar flows with the former values of ν, γ, and ${{n}_{*}}$ but with U₋ ≈ –0.32 and A₋ = 0.50 and one more arrow in Fig. 1a. The left upper curve in Fig. 3 is the trajectory of a piston expanding from CS at t ≥ 0.

Fig. 3.

rt-Diagram of the self-similar flow with a piston that began to expand from the CS with arrival the SW IS to it for ν = 3, γ = 5/3, and n = ${{n}_{*}}$ = 0.68838.

2 KUROPATENKO’s SELF-SIMILAR SOLUTIONS WITH A SHOCK WAVE PROPAGATING TOWARD THE AXIS OR CENTER OF SYMMETRY

We will continue an investigation of flows with a SW propagating toward a CS in the case of the self-similarity exponent 1 < k ≠ ${{k}_{*}}$ = 1/${{n}_{*}}$. They were studied by Kuropatenko with his colleagues [9–13] who used, together with r, the Lagrangian mass variable m, where the self-similarity index is l = νn rather than n itself. Then it was accepted that the flows under consideration with a strong SW in motion in a cold gas at rest are generated by a piston traveling from point 1 with r = r₁, m = m₁, and t = t₁ < 0 toward the CS. In this case, the gas density ρ and velocity u, as well as the coordinate r, were sought using the separation of variables and invoking conditions (1.1) valid in the case of strong SWs, in the form:

$$\begin{gathered} \xi = \frac{m}{{{{m}_{1}}}}{{\left( {\frac{{{{t}_{1}}}}{t}} \right)}^{l}},\quad \rho = \frac{{\gamma + 1}}{{\gamma - 1}}{{\rho }_{0}}\delta (\xi ),\quad {{r}^{{\nu - 1}}}u = r_{1}^{{\nu - 1}}{{u}_{1}}{{\left( {\frac{t}{{{{t}_{1}}}}} \right)}^{{l - 1}}}Z(\xi ), \\ {{r}^{\nu }} = r_{1}^{\nu }{{\left( {\frac{t}{{{{t}_{1}}}}} \right)}^{l}}X(\xi ),\quad \frac{1}{{{{t}_{1}}}} = \nu \frac{{\gamma + 1}}{{2l{{r}_{1}}}}{{u}_{1}}, \\ \end{gathered} $$

(2.1)

$$\delta (1) = Z(1) = X(1) = 1.$$

(2.2)

Here, u₁ is the gas velocity behind the SW IS at point 1.

Substituting conditions (2.1) in the Euler equations governing the flows under consideration in the variables t and m leads to ordinary differential equations (the primes denote the differentiation with respect to ξ ≥ 1)

$$\begin{gathered} G\delta {\kern 1pt} ' = H,\quad GZ{\kern 1pt} ' = \frac{{\gamma - 1}}{{2{{\delta }^{2}}}}\xi H,\quad \xi X{\kern 1pt} ' = X - \frac{{2Z}}{{\gamma + 1}},\quad G = \frac{{\gamma {{\delta }^{{\gamma - 1}}}{{X}^{{2 - 2/\nu }}}}}{{{{\xi }^{{2/l - 2/\nu }}}}} - \frac{{\gamma - 1}}{{2{{\delta }^{2}}}}{{\xi }^{2}}, \\ H = 2\frac{{(\nu - l){{\delta }^{\gamma }}{{X}^{{2 - 2/\nu }}}}}{{\nu l{{\xi }^{{1 + 2/l - 2/\nu }}}}} - \frac{{l - 1}}{l}Z + \frac{{2(\nu - 1){{Z}^{2}}}}{{\nu (\gamma + 1)X}}. \\ \end{gathered} $$

(2.3)

In [9–13] the solution of the Cauchy problem for Eqs. (2.3) with the initial conditions (2.2) led to different results for n (or l = νn) greater or smaller than ${{n}_{*}}$ (or ${{l}_{*}}$ = ν${{n}_{*}}$). For n > ${{n}_{*}}$ the function ${{G}_{{0*}}}$ always turned to zero at a certain ${{\xi }_{{0*}}}$ > 1, the function ${{H}_{{0*}}}$ being in the general case different from zero. The assertion of the authors of [8–12] that in this case the curve m/m₁ = ${{\xi }_{{0*}}}$(t/t₁)^l arriving at the CS is a С^–-characteristic is erroneous. The equality ${{G}_{{0*}}}$ = 0 means that on this curve there holds the equality dr/dt = u – a, that is, the С^–-characteristics are only tangent to it. It becomes a С^–-characteristic only when the equality ${{H}_{{0*}}}$ = 0, equivalent to the compatibility condition for С^–characteristics, holds true on it, along with ${{G}_{{0*}}}$ = 0. At ${{H}_{{0*}}}$ ≠ 0 it is their envelope making impossible the construction of self-similar piston trajectories at ξ > ${{\xi }_{{0*}}}$ and t > ${{t}_{{0*}}}$, which indicates also the formation at point ${{0}_{*}}$ of a non-self-similar SW represented by the third segment of the boundary which does not reach the CS of the self-similar flow region, which is triangular in the rt plane.

The indices ${{n}_{*}}$(ν, γ) found in [9–13], at which the functions ${{G}_{{0*}}}$ and ${{H}_{{0*}}}$ turn simultaneously to zero for all ν and γ considered, are the same as those obtained in [1, 2, 5–8]. With account for this fact, which was not anyway mentioned in [9–13], and the greater simplicity of the customary self-similar approach we will continue an analysis of solutions for n ≠ ${{n}_{*}}$ within the framework of equations and conditions (1.2)–(1.4), assuming, following Kuropatenko, that the flows under study are produced by a piston that starts its motion toward the CS from point 1 in the rt plane, on the SW IS. In view of Eqs. (1.2) and (1.3), the following equalities hold on the piston trajectory dr/dt = u

$$\frac{{dt}}{t} = \frac{1}{{1 - U}}\frac{{d\tau }}{\tau },\quad \frac{{dr}}{r} = \frac{{U{\text{/}}k}}{{1 - U}}\frac{{d\tau }}{\tau }.$$

(2.4)

According to these equalities, the variations in time and the piston tralectory coordinate at 0 < U < 1 are analogous to the variation in the variable τ. Thus, if |τ₀| > 0 at the end of an integral curve, then |t₀| > 0 and r₀ > 0, that is, the piston does not reach the CS. If, on the contrary, τ₀ = 0, then t₀ = r₀ = 0 and these solutions describe the gas compression to a point.

The flows with |τ₀| > 0 are realized, if the integral curve of the first equation (1.3) started at point IS arrives at point ${{0}_{*}}$ on the line A = U − 1 in the UA plane with  f₁(${{U}_{{0*}}}$, ${{A}_{{0*}}}$) ≠ 0. The quantity ${{\tau }_{{0*}}}$ < 0 reaches a local minimum at this point; in view of this fact, the contibuation of the self-similar solution toward the values of τ greater than ${{\tau }_{{0*}}}$ turns out to be impossible. As in [9–13], this situation takes place always for 1 > n > ${{n}_{*}}$(ν, γ). The typical field of the integral curves and singular points of the first equations (1.3) in the fourth quadrant of the UA plane and an incomplete rt-diagram for this case are presented in Fig. 4.

Fig. 4.

UA plane (a) and incomplete rt-diagram (b) for a flow with a SW moving toward the CS for ν = 3, γ = 5/3, and n = 0.8 > ${{n}_{*}}$ = 0.68838.

As noted above, at f₁ ≠ 0 the point ${{0}_{*}}$ of the integral curve arriving onto the sonic line in the UA plane is associated with envelope Ct = r^k ${{\tau }_{{0*}}}$ of the С⁻-characteristics presented as a dashed line in the rt-diagram in Fig. 4b. This envelope, as the self-similar SW IS that has come from point 1 simulataneously with the piston along the trajectory with the equation Ct = −r^k, may at first glance arrive at CS. However, this will not happen, since the appearance of the envelope indicates the formation of a non-self-similar SW at the point ${{0}_{*}}$ of the piston trajectory. The SW that appeared at t < t_S outdistancing the С⁻-characteristic ${{0}_{*}}$–S and annealing it will overtake the self-similar SW IS before its arrival at CS. Therefore, at n > ${{n}_{*}}$ the self-similar solution can be realized only in a curvilinear triangle in the rt plane distant from the CS and confined by the piston trajectory 1–${{0}_{*}}$ and the segements of the self-similar SW IS and the non-self-similar SW that appeared at point ${{0}_{*}}$ of the piston trajectory and overtook it. In a much smaller vicinity of the CS the non-self-similar SW moving in the cold gas becomes the self-similar SW of Guderley’s solution with n = ${{n}_{*}}$(ν, γ).

More interesting self-similar solutions with the SW IS coming to the CS and a piston compressing the gas to a point are obtained in the case of the exponents n < ${{n}_{*}}$(ν, γ). At ${{n}_{*}}$ > n > n₀, where n₀ = n₀(ν, γ) is determined below (certain values of n₀ are given in Table 1), the solution is given by a segment of an integral curve of the first equation (1.3) with the origin at the point on IS with U and A from Eq. (1.4) and the end in a node beneath the sonic line from the fourth quadrant in the UA plane. The values of U and A at the node are determined by the equations f₁(U, A) = 0 and  f₂(U, A) = 0. In accordance with these equations, the following equalities and inequalities hold true for these n

$$\begin{gathered} {{n}_{*}}(\nu ,\gamma ) > n > {{n}_{0}} = \frac{2}{{2 + \nu (\gamma - 1)}},\quad U = \frac{{{{n}_{0}}}}{n} < 1, \\ {{A}^{2}} = B \equiv \frac{{\gamma (1 - U)(1 - nU)U}}{{(2 + \nu \gamma U)n - 2}} > 0,\quad A = - \sqrt B . \\ \end{gathered} $$

The quantity τ in the node turns out to be zero, thus leading, by virtue of Eqs. (2.4), to t = 0 and r = 0, that is, to a self-similar flow with a piston and a strong SW that compress the gas to a point. The above-said is illustrated in Fig. 5. The meaning of the lines and points presented in this figure is the same as in Figs. 1–4. Thus, in Fig. 5b the thin segments are the С^–-characteristics coming from the piston trajectory toward the SW IS.

Fig. 5.

UA plane (a) and rt-diagram (b) for a flow with a piston and a SW, which compress the gas to a point, for ν = 3, γ = 5/3, and n₀ = 0.5 < n = 0.6 < ${{n}_{*}}$ = 0.68838.

At even smaller values of the exponent n, 0 < n < n₀ < ${{n}_{*}}$, it is the degenerate singular point U = 1 and A = 0 that becomes the singular point, where the integral curve of the first equation (1.3) with the origin at the point on IS with U and A from Eq. (1.4) arrives. In accordance with [14] and the calculations made by us, the integral curves that gives the solution enters to this singular point being tangent to the vertical integral line U = 1 and realizing, as in the case presented in Fig. 5, the self-similar solution of the shock compression of the gas to a point (r = t = 0). In the vicinity of this singular point we obtain

$$U \approx 1 - K{{A}^{2}} < 1,\quad \tau \approx {{\tau }_{0}}\left[ {1 + \frac{{\gamma K{\text{/}}(1 - n)}}{{2 - \gamma (\gamma - 1)K}}{{A}^{2}}} \right],\quad 0 < K < \frac{2}{{\gamma (\gamma - 1)}}$$

with the integration constant K.

The results of the calculations performed for two such values of n, are presented in Figs. 6 and 7, which are similar with Fig. 5.

Fig. 6.

UA plane (a) and rt-diagram (b) for a flow with a piston and a SW which compress the gas to a point, for ν = 3, γ = 5/3, and n = 0.49 < n₀ = 0.5 < ${{n}_{*}}$ = 0.68838.

Fig. 7.

UA plane (a) and rt-diagram (b) for a flow with a piston and SW, which compress the gas to a point, for ν = 3, γ = 5/3, and n = 0.4 < n₀ = 0.5 < ${{n}_{*}}$ = 0.68838.

3 SUMMARY

The self-similar solutions with n = ${{n}_{*}}$ and a piston starting its expansion from the CS, where the SW IS comes, and the self-similar solutions obtained at 1 > n > ${{n}_{*}}$ in a curvilinear triangle in the rt plane with r > 0 present an unterest exceptionally for the theory. The self-similar solutions of shock compression of a gas to a point, which are obtained at ${{n}_{*}}$ > n > 0, present an interest also mainly for the theory, since in these cases the greater portion of the pistom work, and at n₀ > n even its entire work, is spent for an increase in the kinetic energy of the compressed gas.

In connection with the solutions for n ≠ ${{n}_{*}}$, the study made above, having specified certain details [9–13], demonstrates the applicability and a greater simplicity of the habitual self-similar approach with one equation (1.3) and the qualitative analysis of the integral curves in the UA plane, as compared with the separation of variables and the solution of the system of three equations (2.3). The main merit of Kuropatenko and his colleagues lies in the fact that they have found new self-similar solutions describing the compression of a cold gas at rest to a point.