Triple configurations of pursuit shock waves in conditions of ambiguity of the solution

Introduction

Triple configurations of shock waves, present in stream and nozzle gas flows implemented in jet aviation and rocketry technologies, affect the performance of supersonic air intakes and other equipment based on jet flow technologies.

At the present time, developers keep on searching for effective solutions for ramjet, rotating, and pulse detonation engines, so the problem of analysing interaction between compression shocks, shock and explosive waves is especially relevant. To solve this problem, it is necessary to analyse all triple configurations that may form in given conditions, depending on device parameters. It is also important to analyse a variety of possible solutions to the problems regarding development of explosive-proof equipment, determination of damage effect caused by condensed substance explosion associated with irregular interaction of air shock waves and their Mach reflection.

This paper briefly reviews the properties of optimal triple configurations that correspond to the maximum variations in the parameters of flows after them and are inherent to the basic and alternative solutions within the framework of the local triple shock theory. In this respect, when searching for the optimal flow conditions in triple configurations, we must consider the ambiguity in solution to the constitutive system of equations.

General information on triple configurations

A triple configuration of compression shocks is a shock-wave structure consisting of three shock waves with a common triple point (point T in Fig. 1). Triple configurations of shock waves, being stationary within a selected coordinate system (compression shocks), are present in stream and nozzle gas flows implemented in jet aviation and rocketry technologies [1–3]. They affect the performance of supersonic air intakes and other equipment based on jet flow technologies [4, 5]. Triple configurations of moving (travelling) shock waves appear under their Mach reflection and irregular interaction [6–10], affecting the efficiency of the mechanical impact of an explosion, as well as the performance of explosion-proof devices intended to suppress the high-explosive effect [11–13], in particular, in case of multiple interaction of shock waves in confined volumes [14–16]. Gas flows having passed through different compression shock wave systems (sequence of shocks 1 and 2 or single shock 3) are separated by tangential discontinuity τ. The parameters of shocks are connected by the conditions of consistency on tangential discontinuity, written in the form [17–19]:

J₁ J₂ = J₃ ; (1)

β₁ + β₂ = β₃ . (2)

Here, J_i (i=1...3 ) – intensity of the i-th shock (ratio of static pressures after and before the shock);
β_i – flow turn angle on the surface of the i-th shock.

Angles β_i and Mach numbers M_i after the i-th shock are associated with shock intensity and Mach number M_k before the shock by the known [1] classic relationships.

Depending on the direction of flow turn on shocks 1–3, three types of triple configurations are distinguished. In configurations of the first type (TC-1, Fig. 1, a), flow turn on shock 1 occurs in a direction different from that on shocks 2 and 3. For example, at β₁ < 0 , angles β₂ > 0 , β₃ > 0. In configurations of the second type (TC-2, Fig. 1, b), the direction of turn on shock 2 is different from the others, and in configurations of the third type (TC-3, Fig. 1, c), flow turn occurs in the same direction on all the shocks. The stationary Mach configuration (SMC, Fig. 1, d) with direct main shock (β₃ = 0) and configuration TTC-2-3 (Fig. 1, e) with direct shock 2 (β₂ = 0) are transient.

Setting adiabatic index γ, Mach number M of flow before the triple point, and branching shock intensity J₁ does not always allow to explicitly define the properties of other shocks in the system of equations (1)–(2). The same parameters γ , M , and J₁ are matched by up to three physically based solutions with different values of β₂ and β₃ . The basic solution to the system of equations (1)–(2) is defined in the widest area of parameter space (γ, M, J₁ ), and two alternative solutions – only at subsets of the region of the basic solution definition. Triple configurations corresponding to the basic solution may belong to all three types, as well as to the transient configurations SMC and TTC-2-3. Alternative triple configurations (ATC) relate to the third type (see Fig. 1, c), and the flow after shock 2 is supersonic at that. They are formed as a result of interaction between pursuit shocks.

Many parameters of gas flows after triple configurations have substantial differences. Those of interest are the differences in stagnation pressures p₀, velocities V, flow rate functions q = ρV , flow strength d = ρV² , flow pulses j = ρ + ρV² after the triple point. The measure of difference here are their ratios on the tangential discontinuity sides. Triple configurations with extremal values of such ratios are called optimal configurations. Investigation of the optimal configurations may have practical importance in analysing the origin of self-oscillation regimes of flows in free and impact supersonic jets [20], when designing the equipment generating pulsating gas flows.

Further, we shall analyse the properties of optimal triple configurations corresponding to both the basic and alternative solutions. The numerical results are given for γ = 1, 4.

Optimal configurations corresponding to the basic solution

The properties of triple configurations of compression shock waves are analysed on the plane of parameters M and σ₁ (Fig. 2), where σ₁ – angle of shock 1 to the direction of flow before the shock. Angle σ₁ correlates with shock intensity J₁ as

J₁ = (1 + ε) M² sin² σ₁ - ε,

where ε = (γ − 1) / (γ + 1).

The range of angles σ₁ variation is limited from below by curve 1, which corresponds to shock transformation into a weak discontinuity (σ₁ = α(Μ) = arcsin (1/M), J₁ = 1). The values σ₁ are limited from above as well, at least by the requirement that shock 2 must exist in the supersonic flow after the branching shock. This requirement is conformed to by the region under curve 2, which is plotted proceeding from condition M₁ = 1 after shock 1.

For the existence of triple configuration, presence of a supersonic flow after shock 1 is insufficient. A solution to the system (1)–(2) exists only in the region between curves 1 и f₁ therefore curve f₁ is the exact upper boundary of the region under consideration and is defined by the equation common for curves f_i (i = 1, 2):

Curve f₁ starts in point F₁ on curve 1, where M_F1 = 1.245, M_F2 = 2.54.

Solutions to the system (1)–(2) in the region under consideration may correspond to configurations of different types. In the subregion between curves 1 and 3, configurations of TC-1 type are implemented; between curves 3 and 4 – TC-2, and between curves 4 and f₁ – TC-3. Curve 3 corresponds to the stationary Mach configuration and is plotted from solution to the equation

where a = (1− ε)(1+ ε J₁) ;

J_m – intensity of direct shock, formed in the flow with a given Mach number,

J_m =(1 + ε)Μ² -ε;

b = — [(1 + ε — ε² + ε³ )J₁² + ε(1 + ε) J₁ + (1 — ε)];

с = J₁ ((1 - ε²) J₁² - (1 + ε²) J₁ - 2ε).

Transient configurations TTC-2-3 (curve 4) are determined analytically as well [18, 19].

Intensities and other parameters of compression shock waves across the entire region of basic solution existence are changing continuously. The parameters of individual shocks take extremal and specific values (e.g., shocks 2 and 3 may correspond to the maximum deviation points, Crocco points, constant pressure points, and sonic point [19]).

The properties of flows after the triple configuration are determined from the system (1)–(2) and ratios on the compression shocks. For example, the ratios between total pressures p₀, velocities V, flow rate functions q, flow strengths d, flow pulses j at tangential discontinuity are as follows:

where – inverse ratio of densities on the shock.

The lower boundary of the region of solution existence (curve 1) corresponds to transformation of shock 1 and the upper boundary (curve f₁), of shock 2 – into a weak discontinuity. In these cases, all the considered parameter ratios after the triple point are equal to unity. At fixed Mach number M , the only extremum point of the considered functions in a range between the definition region boundaries is the maximum point. The configurations corresponding to these maxima are optimal at fixed Mach number.

The parameters of configurations optimal with respect to the target functions (4) are shown in Fig. 2 by curves 5–9, respectively. At low Mach numbers, these configurations belong to the third type. The intersections of curves 5–9 with curve 4 corresponds to the optimal transient configurations. In this case, the optimal ratios of parameters (I_po = 1,076; I_v = 1,085; I_q = 1,107; I_d = 1,201; I_j = 1,090) are low, and the ratios of Mach numbers (M = 1.596; M = 1.567; M = 1.571; M = 1.569; M = 1.584) are very close.

With an increase of Mach numbers, the optimal curves 5–9 come close and intersect in one point a, which corresponds to the stationary Mach configuration (SMC) with Mach number . The intensities of incident shock 1 and reflected shock 2 of the compression shock waves in such SMC are equal to:. It is proved [21, 22] that equality of the shock wave intensities leads to the total pressure maximum after the shock-wave system if the product of those intensities is a fixed value. It can be shown that in the SMC such a product ( J₃ intensity), even though it is not a fixed value, obeys the above theorem, therefore it is exactly the Mach configuration with equal shock intensities that is the optimal one. The parameter ratios after the optimal SMC:

At greater Mach numbers, configurations of the first type are optimal. The optimal values of target functions increase monotonically but in a limited way, while the optimal intensities of shocks 1 and 3 at M → ∞ tend to infinity. The configurations optimal in terms of I _p0 have the following finite limits:

The Mach number after shock 1 tends to the infinite limit (order √M) and after shocks 2 and 3 – to finite limits. Ratio I _p0 itself tends to the value

The limit values of other functions in configurations optimal in terms of I _p0 , are as follows:

and, as a rule, they are close to the optimal values reached on curves 6–9 (see Fig. 2): I_V → 5.261, I_d → 155.8, I_q → 30.41, I_j → 30.22; therefore, optimisation of configurations with respect to these parameters is sometimes substituted for optimisation as per I _p0 [18]. In configurations optimal with respect to these four parameters, intensity J₁ has order M₂ , and values M₁ and J₂ tend to high finite values. The angle of shock 1 tends to a low finite value at that, rather than to zero.

The optimal values (especially, the total pressure ratios) tend to their limits slowly: at M = 8 , optimal I _p0 = 19.36, and at M = 200 , I _p0 = 439.2. The optimisation of configurations leads to notable increase of the target functions. Thus, at M → ∞ , optimal I _p0 → 529.1, while I _p0 → 69.72 in the SMC and I _p0 → 1 in TTC-2-3.

Alternative triple configurations

Starting from certain Mach number (M = 2,542 in optimisation as per the ratio of total pressures), parameters ( M , σ₁ ) of the optimal basic configurations determine two more solutions, and at M > 2.61 – one solution that describes ATC of 2 6, 1 the third type corresponding to one of the alternative solutions, which exist along with the basic one at the same Mach numbers of the intensity flow of shock 1 (branching) and gas adiabat.

The alternative solutions to the system (1)–(2) appear on curve bc (see Fig. 2) as a result of decomposition of shock isomachs [13]. There are two different ATCs in curvilinear triangle F₂cb . One of the solutions in segment F₂c of curve f₂ corresponds to value J₁ < 1, due to which it stops being implemented. At the same time, a new and the only possible solution for ATC appears on curve f₂ , after point c . Curve f₂ and point F₂ are defined by relationship (3), and points b ( ) (M_b = 2.089) and c ( M_c = 3.117) – by high-degree (for point b – eighth) algebraic equations.

The maxima of equations (4) can be achieved in the ATCs corresponding to the solution which is continuous across the entire region beyond curves bc and f₂ (curves 10–14). At M → ∞ , the optimal value of I _p0 tends to the limit (5) and can be achieved at J₃/M² → C₁, J₁/M → √C₁, J₂/M → √C₁ . The flow turn angle on shock 3 in an optimal asymptotic ATC is opposite to its value in the “basic” configuration.

The limits of other parameter ratios in the optimal ATCs are at least comparable to the “basic” configurations: in the ATC, at M = 199.3, maximal I_V = 4.858, I_d = 133.1, I_q = 27.47, I_j = 28 and in the “basic” configurations, I_V = 5,257, I_d = 151, I_q = 29.23, I_j = 28.56. The relative position (from bottom to top) of the optimal curves 10–14 is opposite to the position of curves 5–9 at high Mach numbers.

With parameter γ increased, Mach numbers at which the ATCs are formed increase substantially and tend to infinity at γ → 5/3. At γ ≥ 5/3, the system of equations (1)–(2) has no more than one physically based solution.

Conclusion

The conducted calculation and parametric analysis of triple configurations forming under all theoretically feasible flow parameters before them serve for optimisation of systems and devices that employ the effects of interaction and reflection of compression shock waves, blast shock waves, and detonation waves.

The study demonstrates that triple configurations corresponding to different physically feasible solutions can be optimal: after such configurations, the maximum and quite high ratios of total pressures, velocities, flow strengths, and other flow parameters can be achieved on different sides of tangential discontinuity originating from the triple point. This statement holds true both for the basic (traditionally considered) and additional (alternative) solutions defining triple configurations, therefore, when searching for the optimal flow conditions in triple configurations, it is necessary to consider the ambiguity of solution to the defining system of equations.

The results obtained using theoretical and numerical methods can be used in various applications of gas dynamics. For instance, high total differential pressures in a supersonic gas jet initiate self-oscillation regimes when a jet interacts with obstacles, and lead to extreme acoustic and force loads when executing starting tasks. The different translational (transferred) impact of blast shock waves on bodies located above and below the triple point is achieved due to a considerable difference between flow strengths on the opposite sides of tangential discontinuity. This phenomenon can be used in design of explosion-proof devices and in an analysis of blast effect (especially in confined spaces with inevitable multiple reflection of shock waves and their irregular interaction). Moreover, high values of flow parameters after triple configurations hamper initiation of detonation in aircraft and rocket engines of appropriate type and shall be eliminated at the development phase of such devices.