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Hysteresis phenomenon of the oblique detonation wave
Combustion and Flame 192 (2018) 170–179
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Hysteresis phenomenon of the oblique detonation wave Yu Liu∗, Lan Wang, Baoguo Xiao, Zhihui Yan, Chao Wang Science and Technology on Scramjet Laboratory of Hypervelocity Aerodynamics Institute, CARDC, Mianyang, Sichuan 621000, China
a r t i c l e
i n f o
Article history: Received 22 August 2017 Revised 24 October 2017 Accepted 12 February 2018
Keywords: Oblique detonation wave Hysteresis Strong shock solution Triple point CJ
a b s t r a c t Hysteresis phenomenon of the oblique detonation wave (ODW) is numerically studied. Two-dimensional unsteady reactive Euler equations are numerically solved as governing equations with a two-step reduced reaction mechanism. Wedge angle variation is realized by modifying inflow direction. It is found that hysteresis phenomenon does exist in ODW problem, i.e., the final state of the ODW is closely relevant to initial condition. Two types of hysteresis are discovered in this study: the hysteresis of upstreamdownstream triple point and the hysteresis of smooth-abrupt transition pattern. Detonation/Shock polar analysis on primary triple point structure of abrupt ODW demonstrates that the precursor shock of the ODW near primary triple point is actually a strong shock solution and therefore characterized by local detachment behavior which is responsible for primary triple point’s upstream moving. Similar to hysteresis of shock reflection, irreversibility is the mechanism for ODW’s hysteresis. It is found that hysteresis will disappear when the wedge angle is smaller than a certain value, which means that it may be impossible to obtain a standing Chapman–Jouguet (CJ) ODW without ignition delay or with short ignition delay at a CJ wedge angle via hysteresis. © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction The oblique detonation engine (ODE) [1,2] is one for highMach-number flight, using oblique detonation wave (ODW) at the combustor to produce thrusts and having advantages of high cycle efficiency and fast heat release and therefore short combustor length. In past decades, basic issues about the ODW involving initiation [3–7], standing stabilization [8–10] and cellular instability [11–13] have been concerned by many researchers. From practical perspective, initiation and standing stabilization of an ODW are of special importance because both of them are the key factors for an ODE to realize its performance advantages. Initiation of an ODW was often observed by researchers in experiments in which a high-velocity projectile passes through combustible gases. However, until 1990s, people paid more attention to shock-induced combustion (SIC) phenomenon than to ODW in high-velocity projectile experiments, like Lehr [14], McVey and Toong [15], Alpert and Toong [16], Matsuo and Fujiwara [17], and Matsuo and Fujii [18]. The oscillatory instability in SIC was comprehensively investigated in particular by these researchers. ODW’s initiation by a high-velocity projectile was first theoretically modeled by Vasiljev [3] and Lee [4], individually. Their ideas are to analogize this problem to cylindrical detonation initiation by a linear energy source and they pointed out that the energy ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Liu).
deposited per unit length by the projectile, which equals its drag force, should be no less than the critical energy per unit length required for initiation of a cylindrical detonation. After then, Ju et al. [19] argued that as an additional criterion to that proposed by Vasiljev [3] and Lee [4], the Damköhler number indicating the ratio of ignition delay time to flow characteristic time should be less than unity. Experiments by Verreault and Higgins [5] demonstrated that the energy criterion proposed by Vasiljev [3] and Lee [4] and the chemical kinetic criterion proposed by Ju et al. [19] do mutually control the ODW’s initiation by a high-velocity projectile. Besides, in experiments of Verreault and Higgins [5], four combustion regimes are found: prompt ODW regime (without leading shock or ignition delay), delayed ODW regime (with leading shock or ignition delay), instabilities regime and wave splitting regime. As for standing stabilization of an ODW, it is not as easy as that of an inert shock. Pratt et al. [20,21] made a detailed detonation polar analysis on standing ODW, pointing out that the ODW cannot be stabilized at the wedge that supports it unless the wedge angle lies between the minimum value called Chapman–Jouguet (CJ) wedge angle and the maximum value called detachment wedge angle. According to the detonation polar, they classified the ODW into three types: weak underdriven ODW, weak overdriven ODW and strong ODW, as shown in Fig. 1. Among these three types of ODW, the weak underdriven is unphysical since it violates the second law of thermodynamics. Ghorbanian and Sterling [22] supposed a double-wedge supersonic reacting flow and analyzed possible flow structures. They
https://doi.org/10.1016/j.combustflame.2018.02.010 0010-2180/© 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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this study, we will give another explanation for upstream moving of the triple point structure, via detonation/shock polar analysis. For ODW’s application in propulsion, a CJ ODW is always desirable because it has the minimum entropy increase. However, initiating and stabilizing a CJ ODW is not easy because of the small CJ wedge angle. One thought is that once the ODW is initiated, it is independent of downstream condition if post-detonation state is supersonic or sonic, as described in [29]. Thus, a CJ ODW with short ignition delay or even without ignition delay (prompt) is supposed to be possible if a highly overdriven ODW is first initiated by a large-angle wedge and then is decayed to a CJ ODW by decreasing the wedge angle. If this is true, or if hysteresis phenomenon exists and behaves as wished, it will be very helpful for ODW’s application in propulsion. Thus, to confirm hysteresis phenomenon of the ODW is the aim of this paper. As a matter of fact, hysteresis has already been discovered in shock reflection phenomenon (see [30] and [31]). In this study, the mechanism analysis of ODW’s hysteresis will also be conducted, by analogizing it to that of shock reflection. Since detonation simulation requires high grid resolution, adaptive mesh refinement technique is necessary to reduce total grid number. In this paper, such an open-source computational fluid dynamics (CFD) program called AMROC [32] is used to conduct numerical study. In next section, a brief introduction to the numerical method and model will be given. Fig. 1. Detonation polar drawn by Pratt et al. [21].
2. Numerical treatment 2.1. Governing equations The governing equations are two-dimensional unsteady reactive Euler equations given as: Continuity equation
∂ ρ /∂ t + ∂ (ρ u )/∂ x + ∂ (ρv )/∂ y = 0
(1)
where ρ , u, v, t, x and y denote the density, the x-direction velocity, the y-direction velocity, the time, the x coordinate and the y coordinate, respectively. Conservation equation of x-direction Momentum
∂ (ρ u )/∂ t + ∂ (ρ u2 + p)/∂ x + ∂ (ρ uv )/∂ y = 0
(2)
where p is the static pressure. Conservation equation of y-direction Momentum
∂ (ρv )/∂ t + ∂ (ρ uv )/∂ x + ∂ (ρv2 + p)/∂ y = 0
(3)
Conservation equation of energy Fig. 2. ODW at a double wedge assumed by Ghorbanian and Sterling [22].
∂ (ρ e )/∂ t + ∂ [u(ρ e + p)]/∂ x + ∂ [v(ρ e + p)]/∂ y = 0
(4)
where e is the specific total energy. For perfect gas, the equation of state is predicted a CJ ODW following the shock wave generated by the first wedge of a smaller angle and an overdriven ODW generated by the second wedge of a larger angle, as shown in Fig. 2 (Such a structure has been comfirmed by Liu et al. [23,24] in their numerical simulations with a reduced two-step reaction mechanism). Based on such a structure, they were the first to conduct detailed detonation/shock polar analysis on ODW structure. Kasahara et al. [25,26] and Maeda et al. [27,28] made a series of hypervelocity projectile experiments. They paid attention to the curvature effect and proposed the critical diameter criterion for a spherical projectile to stabilize an ODW. Liu et al. [23] paid particular attention to the triple point and the transverse shock of the ODW. They found the triple point structure can maintain stationary after initiation or propagate upstream to get closer to the wedge tip. They attributed the upstream propagation of the triple point to pressure variation due to inflow Mach number decrease or wedge angle increase. In
p = ρ RT
(5)
where R is the gas constant and T is the static temperature. Thus, the specific total energy can be expressed as
e = p/ρ (γ − 1 ) + (u2 + v2 )/2 − RT0 q˜
(6)
where γ , T0 and q˜ are the specific heat ratio, the static temperature of the inflow and the dimensionless local heat release using RT0 for normalization (q˜ = q/RT0 , where q is the dimensional local heat release), respectively. A two-step reduced reaction mechanism (see [33]) is employed, which helps mimic the feature of a chain-branching reaction and avoid huge computational expense of detailed reaction mechanism. The two-step reduced reaction mechanism consists of a thermally neutral induction step and a rapid heat release step that follows. The reaction rates equation for both steps are given as:
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∂ (ρξ )/∂ t + ∂ (ρ uξ )/∂ x + ∂ (ρvξ )/∂ y = H (1 − ξ )ρ kI exp [EI (1/RTS − 1/RT )]
(7)
∂ (ρλ )/∂ t + ∂ (ρ uλ )/∂ x + ∂ (ρvλ )/∂ y = [1 − H (1 − ξ )]ρ kR (1 − λ ) exp (−ER /RT )
(8)
where ξ , kI and EI are the reaction progress variable, the reaction rate constant and the activation energy of the induction step, respectively, and λ, kR and ER are that of the heat release step, respectively. Note that the reaction rate constant kI and kR control the induction length and the reaction length, respectively. TS is the temperature inside the induction region. H (1 − ξ ) is a step function written as:
H (1 − ξ ) =
= 1 if = 0 if
ξ <1 ξ ≥1
(9)
Clearly, at any instant the local heat release q˜ or q satisfies q˜ =
λQ˜ or q = λQ, where Q˜ and Q are the dimensionless and dimensional total heat release by the reactant (Q˜ = Q/RT0 ), respectively. 2.2. Numerical method and model As introduced above, AMROC is used to conduct numerical simulation in this study. It is an open-source CFD program based on finite volume method. Block-structured adaptive mesh refinement technique of AMROC allows it to simulate detonation with high resolution. MUSCL-Hancock scheme is employed for discretization with Riemann solver of hybrid Roe scheme. Van Albada slope limiter is used to ensure total variation diminishing (TVD) properties. For the source term treatment, Godunov splitting method is used to overcome the problem of stiff chemistry. AMROC has been validated to be very suitable for shock-induced combustion and ODW [32]. Thus, code validation is not conducted in this paper. All relevant details of AMROC can be found in [32]. Besides, grid convergence test will be given in next section. The computational domain is shown in Fig. 3, with a width of 200 mm and a height of 350 mm. The left and the top boundary are inflow boundaries, and the right boundary together with the bottom boundary in the range of −20 mm ≤ x < 0 mm is the outflow boundary. The wedge has a finite length of 120 mm, and the angle between the wedge surface and the horizontal direction is 30°. The reason for not using an infinite wedge, i.e., a truncated wedge, is mainly due to the treatment on outflow boundary condition. Since extrapolation is used for outflow boundary, a supersonic condition should be guaranteed. Thus, the finite-length wedge allows the flow to further expand and accelerate downstream so that it can maintain supersonic state at the right boundary. This is especially important for cases in which the ODW is close to detachment state and thus the post-ODW flow is subsonic. The angle between the inflow and the horizontal direction is denoted by θ . Thus, the real wedge angle θ satisfies θ = θ + 30◦ . In this study, θ can be varied while keeping the total velocity magnitude unchanged. Five values of θ from 0 to 4°, with an interval of 1°, are chosen so that five different wedge angles are tested: θ = 30◦ , θ = 31◦ , θ = 32◦ , θ = 33◦ and θ = 34◦ (detachment wedge angle). In order to investigate the hysteresis phenomenon of the ODW, a loop of wedge angle variation is simulated, i.e., 30°−31°−32°−33°−34°−33°−32°−31°−30°. We first start the simulation from θ = 30◦ case with reaction disabled, i.e., using the inert shock wave of θ = 30◦ case as initial condition. After then the reaction is turned on, so we can obtain the final state of θ = 30◦ case after sufficiently long CPU time. After we obtain the final state, we directly modify θ abruptly. In this way, we can vary the wedge angle simply by varying inflow direction, using current final state as initial condition for the next wedge angle. When the detachment wedge angle (θ = 34◦ ) is
Fig. 3. Computational domain and boundary conditions.
reached, we start to decrease the wedge angle along the path of 34°−33°−32°−31°−30°. In following sections, we will refer to the path of 30°−31°−32°−33°−34° as the positive path, and the path of 34°−33°−32°−31°−30° as the negative path. The simulated inflow is at a Mach number of 4.0 (2171.7 m/s), at a static temperature of 650 K and a static pressure of 20 kPa. The dimensionless total chemical energy release is set to be Q˜ = 8 which is moderate. The dimensionless activation energies of the induction step and the heat release step are EI /RTS = 4.8 and ER /RTS = 1.0, respectively. This approximately represents hydrogen/oxygen mixture according to [33]. The reaction rate constants of the induction step and the heat release step are set to certain values such that the induction length I and thermal pulse width R corresponding to a CJ detonation are 2.12 mm and 0.95 mm, respectively. The specific heat ratio is γ = 1.2 and the gas constant is R = 377.9 J/kg · K. It should be pointed out that in this paper, all the flow fields in different cases will be presented in the form of numerical schlieren, i.e., ∂ ρ /∂ y. 3. Results and discussions 3.1. Basic flow field of θ = 30◦ case In this section, we will start from the basic flow field of θ = 30◦ case, based on which grid convergence test and investigation of hysteresis phenomenon will be conducted. Figure 4 shows the basic flow field of θ = 30◦ case. It can be seen from Fig. 4a that the ODW consists of two sections, one of which is the stable Zeldovich–Neumann–Döring (ZND) ODW and the other is cellular ODW. According to Teng et al. [34], this is reasonable because a stable ODW tends to develop cellular instability
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Fig. 4. Flow field of θ = 30◦ case (a) the whole computational domain, (b) magnification of the ZND section.
in the far downstream. Figure 4b is the local magnification of the ZND section, from which it can be seen that a leading shock exists near the wedge tip with ignition delay behind it. A deflagration is triggered after the ignition delay and compression waves due to combustion are emanated which intersect with the leading shock and reflect to intersect with the deflagration again. During the interaction between the shock and the flame, the complex coupled structure consisting of the precursor shock and the flame front, i.e., the ODW, finally forms. It is very important to note that for θ = 30◦ case there is no transverse shock or typical primary triple point lying between the leading shock and the ODW. In another word, the transition pattern from the leading shock to the ODW is smooth and progressive. This is so-called smooth ODW according to Silva and Deshaies [8] and Teng and Jiang [35]. In comparison, an ODW with transverse shock and primary triple point structure is called abrupt ODW. In the following sections, the primary triple point structure as well as the transverse shock will be one of the objects that are focused on in this paper. In Fig. 4, the grid resolution is 30 pts/I, which will be justified to reach grid convergence in Section 3.2. 3.2. Grid convergence test Four different grid resolutions including 10 pts/I, 20 pts/I, 30 pts/I and 40 pts/I are tested. Figures 5 and 6 show the results of the whole computational domains and ZND sections under these different grid resolutions, respectively. It is seen that the global flow fields are similar to each other in a qualitative sense. However, 10 pts/I is too coarse to clearly distinguish the precursor shock and the flame front of the ODW. In comparison, 20 pts/I, 30 pts/I and 40 pts/I are all capable of capturing the fine structure. For ZND section, 30 pts/I has almost the same result with that of 40 pts/I. Focusing on the cellular instability, it is found that the cellular instability is triggered at a more upstream place for 20 pts/I, whereas 30 pts/I and 40 pts/I have almost the same cellular-instability-triggered position. Considering most
part of the cellular ODW is influenced or ‘contaminated’ by the expansion fan, it is not our main object of investigation. Thus, the results of grid convergence test confirm that 30 pts/I is enough to fulfill our requirement of grid resolution for current study. In order to highlight the hysteresis phenomenon, only the region where −20 mm ≤ x ≤ 120 mm and 0 mm ≤ y ≤ 130 mm, as shown in Fig. 6, will be displayed in Section 3.3. 3.3. Hysteresis phenomenon of the ODW: phenomenon description Based on the basic flow field of θ = 30◦ case, we start to vary the wedge angle in the way that has been introduced in Section 2.2. After a loop of wedge angle variation, the hysteresis process of the ODW is clearly shown in Fig. 7. Now detailed description will be given. Let us first focus on the positive path. It can be seen that when the wedge angle is changed from θ = 30◦ to θ = 31◦ , the transition pattern from the leading shock to the ODW is still smooth, with a reduced ignition delay due to the stronger leading shock. When the wedge angle is changed from θ = 31◦ to θ = 32◦ , the transition pattern is no longer smooth but abrupt, with an obvious primary triple point and a transverse shock (TS) existing. Note that the transverse shock for θ = 32◦ case in the positive path is in the downstream of the deflagration and thus the deflagration is complete for current case. However, when the wedge angle is further increased to θ = 33◦ , the triple point together with the transverse shock moves upstream and ‘swallows’ part of the deflagration, making it incomplete compared to the previous θ = 32◦ case. When the wedge angle is finally changed to detachment wedge angle, i.e., θ = 34◦ , a prompt detached ODW without ignition delay comes into being. It is important to note that the time needed for the triple point to move upstream and for the ODW to reach detachment state is significantly long (more than 1∼2 ms), which has also been demonstrated by Choi et al. [36]. After the detachment state for θ = 34◦ case is finally reached, the wedge angle is varied along the negative path. When the
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Fig. 5. The whole computational domain of grid convergence test for (a) 10 pts/I, (b) 20 pts/I, (c) 30 pts/I and (d) 40 pts/I.
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Fig. 6. ZND section of grid convergence test for (a) 10 pts/I, (b) 20 pts/I, (c) 30 pts/I and (d) 40 pts/I.
wedge angle is decreased to θ = 33◦ , the ODW recovers to what it looks like at the same wedge angle in the positive path. Thus, for θ = 33◦ case, no hysteresis is discovered. However, when the wedge angle is decreased from θ = 33◦ to θ = 32◦ , the situation is different. It is clear that for θ = 32◦ in the negative path the primary triple point together with the transverse shock still lies somewhere upstream and the deflagration remains to be partially ‘swallowed’. This is significantly different from the case at the same wedge angle in the positive path. Thus, hysteresis exists for θ = 32◦ case and we call such a type of hysteresis the hysteresis of upstream-downstream triple point. When the wedge angle is further decreased to θ = 31◦ , the primary triple point together with the transverse shock moves downstream, behind the deflagration. In another word, compared to the smooth ODW obtained for θ = 31◦ case in the positive path, abrupt ODW is obtained in the negative path for the same wedge angle. Such a type of hysteresis is called the hysteresis of smooth-abrupt transition pattern in this paper. When the wedge angle is finally decreased to θ = 30◦ , the transverse shock disappears and the ODW recovers to what it looks like in the positive path for the same wedge angle, i.e., hysteresis
ceases to exist. It is very interesting that although the ODW structure is closely relevant to initial condition from the perspective of the entire loop, the results are independent of initial condition along each of the two paths alone, positive or negative. That is to say, along the positive or negative path alone, we can obtain the same result for current wedge angle no matter which previous result is used as initial condition. In next section, mechanism for hysteresis phenomenon of the ODW will be discussed in detail. 3.4. Hysteresis phenomenon of the ODW: mechanism analysis In order to explain the mechanism for hysteresis phenomenon of the ODW, a detonation/shock polar analysis on the primary triple point of the abrupt ODW is conducted in this section. Such analysis is based on the so-called three-shock theory proposed by von Neumann for analyzing Mach reflection phenomenon of inert shocks (see [31]). To facilitate our analysis, the triple point structure of θ = 32◦ case in the positive path is particularly magnified in Fig. 8, and the corresponding schematic of the flow filed is shown in Fig. 9.
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Fig. 8. Magnification of the primary triple point of θ = 32◦ case in the positive path.
Fig. 7. Hysteresis process of the ODW.
Clearly, the flow field is divided into several regions by discontinuities and each region (or state) is denoted by a number in Fig. 9. The inflow has two paths to go across the triple point structure, one of which is from state 1 directly to state 5 (across the ODW) and the other is from state 1 to state 2 (across the leading shock) and thereafter to state 3 (across the compression waves) and finally to state 4 (across the transverse shock). State 4 and 5 should be matched with each other in the very neighborhood of the triple point along the slip line, i.e., having the same pressure and flow direction. Thus, such a match condition can be described by detonation/shock polar, as shown in Fig. 10a. It is seen that state 2 (denoted by A) can be found on the shock polar for state 1 (SP1) simply according to the wedge angle θ = 32◦ . Thereafter, the
Fig. 9. Schematic of the triple point structure of θ = 32◦ case in the positive path.
fluid particles are compressed by compression waves emanated by the deflagration and reach state 3 (denoted by B). Starting from B, the shock polar for state 3 (SP3) can thus be drawn. Note that SP3 has an opposite direction relative to SP1 and DP1 because the transverse shock is a right-going discontinuity whereas the leading shock and the ODW are both left-going. It is seen that SP3 has an intersection (denoted by C) with SP1 but has no intersection with DP1. Since state 4 and 5 in the very neighborhood of the triple point are both lying in the ODW’s induction region which is chemically frozen, the match condition should be indicated by C rather than D which indicates the post-ODW state in the far
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Fig. 11. Magnification of the primary triple point of θ = 33◦ case in the positive path.
Fig. 10. Detonation/shock polar in (a) P/P0 − θ and (b) β − θ , where P0 is the inflow static pressure, SP1 is the shock polar for state 1, DP1 is the detonation polar for state 1, SP3 is the shock polar for state 3, A denotes state 2, B denotes state 3, C denotes the match condition of state 4 and 5 in the very neighborhood of the triple point, and D denotes the post-ODW state in the far downstream.
downstream for current wedge angle. The dimensionless pressure of intersection C predicted by the detonation/shock polar is 16.75, in good agreement with CFD result which is 17.03. Clearly, intersection C is a strong shock solution. In Fig. 10b, each point corresponding to Fig. 10 a is given. It can be seen that the wave angle of intersection C is 79.2°. From the stream line in Fig. 8, it is also illustrated that near the triple point the wave angle of the ODW is quite large and it decreases gradually in the downstream. As we know, a strong shock solution does not exist for an attached oblique shock and it is only possible for a Mach stem or a detached shock. Thus, the ODW near the triple point can be considered as a Mach stem or locally detached. This is the basis of hysteresis phenomenon, as explained in the following texts.
Fig. 12. Schematic of the triple point structure of θ = 33◦ case in the positive path.
As the wedge angle is increased to θ = 33◦ , pressure of state 2 together with that of state 3 is also increased because of stronger leading shock. Consequently, SP3 moves rightward relative to that in Fig. 10a. According to von Neumann’s three-shock theory, this means an increased Mach stem height or increased detachment distance. Thus, the triple point moves upstream, getting closer to the wedge tip so that the Mach stem height or local detachment distance increases, as shown in Figs. 11 and 12. Once the triple point together with the transverse shock ‘swallows’ the deflagration, the fluid particles will no longer need to flow across the compression waves to reach state 3, and state 2 will play the role of state 3 in the detonation/shock polar. In this situation, SP3 will at first move rightward immediately after the wedge angle is changed from θ = 32◦ to θ = 33◦ , and thereafter move
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Fig. 13. Detonation/shock polar as the wedge angle changes from θ = 32◦ to θ = 33◦ , where SP3 is the shock polar for state 3 of θ = 32◦ case, SP3’ is the shock polar for state 3 during the initial period after the wedge angle is increased, and SP2 is the shock polar for state 2 of the final θ = 33◦ case.
leftward as the triple point moves upstream until it ‘swallows’ the deflagration, as shown in Fig. 13. Although SP2 is on the left of SP3’, it does not mean that the Mach stem height or detachment distance will decrease during the process because this process is kind of irreversible. Note that irreversibility is one of the main characteristics in hysteresis phenomenon of shock reflection. Thus, by analogy, the upstream movement of the triple point should also be irreversible to some extent. However, since SP3’ finally moves left after all, this is supposed to have some resistant effect against the triple point’s further upstream moving although the triple point cannot be brought back due to irreversibility. Perhaps this can explain why the upstream movement of the triple point takes quite a long time. In the negative path, as the wedge angle decreases, SP2 moves leftward and therefore the Mach stem height or local detachment distance also decreases. As a result, the triple point moves downstream. Again, irreversibility makes the flow structure fail to recover to what it is in the positive path for the same wedge angle. Thus, hysteresis shown in Fig. 7 comes into being. It should be pointed out that for θ = 33◦ , the triple point is so close to the wedge tip both in positive and negative paths that hysteresis can hardly be distinguished. The hysteresis phenomenon ceases to exist when the wedge angle is decreased to θ = 30◦ . It is also interesting to note that for θ = 31◦ the hysteresis of smooth-abrupt transition pattern exists. Since criteria of ODW’s transition pattern have already been proposed (see [8] and [35]), they seem not appropriate for explaining current hysteresis of smooth-abrupt transition pattern. 3.5. Further discussions The results in this study are quite disappointing for propulsion application of the ODW. As described in Section 1, in earlier studies, ODWs are supposed to be self-sustained once initiated and be independent of downstream condition if the wedge angle is equal to or smaller than CJ value because in such situations sonic condition is reached behind the ODW. Therefore, the ODW is thought to be able to stand on the wedge arbitrarily even if the wedge angle is smaller than CJ value. For oblique detonation
engine, CJ ODW is always desirable because it has the minimum entropy increase. However, CJ ODW is corresponding to small wedge angle which tends to hinder detonation initiation and standing stabilization due to lower energy deposition and longer induction length. Thus, before this study, we wished to obtain CJ ODW on a small-angle wedge by first initiating ODW on a largeangle wedge and afterward lowering the wedge angle to CJ value. Unfortunately, it has been demonstrated in current study that the hysteresis phenomenon does not behave as wished. If the wedge angle decreases along the negative path, ignition delay will continuously increase. Furthermore, it seems that hysteresis cannot be maintained when the wedge angle is decreased to a certain value, which means the flow field is independent of initial condition for small wedge angles. Thus, it is reasonable to expect that a standing CJ ODW is actually not possible when the wedge angle is small enough even if it has already been somehow initiated, because the ignition delay distance will be far beyond the wedge length. Maybe some artificial measures are needed to interfere in the hysteresis process so that the phenomenon can behave as we wish, i.e., obtaining a CJ ODW without ignition delay or with short ignition delay when the wedge angle is decreased to CJ value. Another important thing is the physical essence deep inside the hysteresis phenomenon of the ODW. Although we have explained the mechanism of hysteresis by analogy with that of shock reflection, there is still a remarkable difference between the two. For shock reflection, hysteresis phenomenon is corresponding to a dual-solution domain according to von Neumann’s three-shock theory. Thus, hysteresis phenomenon of shock reflection has a reasonable explanation in mathematics. In contrast, there is no dual-solution domain for hysteresis of the ODW, as shown in Figs. 10 and 13. Therefore, in future work, it is necessary to deeply understand the physical essence of ODW’s hysteresis, which is still unclear. 4. Conclusions 1. Hysteresis phenomenon of the ODW is confirmed. Two types of hysteresis are found: the hysteresis of upstreamdownstream triple point and the hysteresis of smoothabrupt transition pattern. 2. Along each of the two paths alone, positive or negative, the final state of the ODW is independent of the initial condition. 3. For an abrupt ODW, detonation/shock polar analysis shows that the precursor shock of the ODW near the primary triple point is actually a strong shock solution and therefore is a Mach stem or locally detached. 4. The mechanism of hysteresis phenomenon of the ODW is also explained by detonation/shock polar analysis. Similar to hysteresis phenomenon of shock reflection, irreversibility should be responsible for ODW’s hysteresis. Once the wedge angle is smaller than a certain value, hysteresis phenomenon will disappear. This means the desire of obtaining a standing CJ ODW without ignition delay or with short ignition delay at a CJ wedge angle via hysteresis is probably impossible. Acknowledgments The research is supported by The National Natural Science Foundation of China NSFC Nos. 91641126 and 11702316. References [1] J.P. Sislian, H. Schirmer, R. Dudebout, J. Schumacher, Propulsive performance of hypersonic oblique detonation wave and shock-induced combustion ramjets, J. Propul. Power 17 (2001) 599–604.
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Hysteresis phenomenon of the oblique detonation wave Yu Liu∗, Lan Wang, Baoguo Xiao, Zhihui Yan, Chao Wang Science and Technology on Scramjet Laboratory of Hypervelocity Aerodynamics Institute, CARDC, Mianyang, Sichuan 621000, China
a r t i c l e
i n f o
Article history: Received 22 August 2017 Revised 24 October 2017 Accepted 12 February 2018
Keywords: Oblique detonation wave Hysteresis Strong shock solution Triple point CJ
a b s t r a c t Hysteresis phenomenon of the oblique detonation wave (ODW) is numerically studied. Two-dimensional unsteady reactive Euler equations are numerically solved as governing equations with a two-step reduced reaction mechanism. Wedge angle variation is realized by modifying inflow direction. It is found that hysteresis phenomenon does exist in ODW problem, i.e., the final state of the ODW is closely relevant to initial condition. Two types of hysteresis are discovered in this study: the hysteresis of upstreamdownstream triple point and the hysteresis of smooth-abrupt transition pattern. Detonation/Shock polar analysis on primary triple point structure of abrupt ODW demonstrates that the precursor shock of the ODW near primary triple point is actually a strong shock solution and therefore characterized by local detachment behavior which is responsible for primary triple point’s upstream moving. Similar to hysteresis of shock reflection, irreversibility is the mechanism for ODW’s hysteresis. It is found that hysteresis will disappear when the wedge angle is smaller than a certain value, which means that it may be impossible to obtain a standing Chapman–Jouguet (CJ) ODW without ignition delay or with short ignition delay at a CJ wedge angle via hysteresis. © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction The oblique detonation engine (ODE) [1,2] is one for highMach-number flight, using oblique detonation wave (ODW) at the combustor to produce thrusts and having advantages of high cycle efficiency and fast heat release and therefore short combustor length. In past decades, basic issues about the ODW involving initiation [3–7], standing stabilization [8–10] and cellular instability [11–13] have been concerned by many researchers. From practical perspective, initiation and standing stabilization of an ODW are of special importance because both of them are the key factors for an ODE to realize its performance advantages. Initiation of an ODW was often observed by researchers in experiments in which a high-velocity projectile passes through combustible gases. However, until 1990s, people paid more attention to shock-induced combustion (SIC) phenomenon than to ODW in high-velocity projectile experiments, like Lehr [14], McVey and Toong [15], Alpert and Toong [16], Matsuo and Fujiwara [17], and Matsuo and Fujii [18]. The oscillatory instability in SIC was comprehensively investigated in particular by these researchers. ODW’s initiation by a high-velocity projectile was first theoretically modeled by Vasiljev [3] and Lee [4], individually. Their ideas are to analogize this problem to cylindrical detonation initiation by a linear energy source and they pointed out that the energy ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Liu).
deposited per unit length by the projectile, which equals its drag force, should be no less than the critical energy per unit length required for initiation of a cylindrical detonation. After then, Ju et al. [19] argued that as an additional criterion to that proposed by Vasiljev [3] and Lee [4], the Damköhler number indicating the ratio of ignition delay time to flow characteristic time should be less than unity. Experiments by Verreault and Higgins [5] demonstrated that the energy criterion proposed by Vasiljev [3] and Lee [4] and the chemical kinetic criterion proposed by Ju et al. [19] do mutually control the ODW’s initiation by a high-velocity projectile. Besides, in experiments of Verreault and Higgins [5], four combustion regimes are found: prompt ODW regime (without leading shock or ignition delay), delayed ODW regime (with leading shock or ignition delay), instabilities regime and wave splitting regime. As for standing stabilization of an ODW, it is not as easy as that of an inert shock. Pratt et al. [20,21] made a detailed detonation polar analysis on standing ODW, pointing out that the ODW cannot be stabilized at the wedge that supports it unless the wedge angle lies between the minimum value called Chapman–Jouguet (CJ) wedge angle and the maximum value called detachment wedge angle. According to the detonation polar, they classified the ODW into three types: weak underdriven ODW, weak overdriven ODW and strong ODW, as shown in Fig. 1. Among these three types of ODW, the weak underdriven is unphysical since it violates the second law of thermodynamics. Ghorbanian and Sterling [22] supposed a double-wedge supersonic reacting flow and analyzed possible flow structures. They
https://doi.org/10.1016/j.combustflame.2018.02.010 0010-2180/© 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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this study, we will give another explanation for upstream moving of the triple point structure, via detonation/shock polar analysis. For ODW’s application in propulsion, a CJ ODW is always desirable because it has the minimum entropy increase. However, initiating and stabilizing a CJ ODW is not easy because of the small CJ wedge angle. One thought is that once the ODW is initiated, it is independent of downstream condition if post-detonation state is supersonic or sonic, as described in [29]. Thus, a CJ ODW with short ignition delay or even without ignition delay (prompt) is supposed to be possible if a highly overdriven ODW is first initiated by a large-angle wedge and then is decayed to a CJ ODW by decreasing the wedge angle. If this is true, or if hysteresis phenomenon exists and behaves as wished, it will be very helpful for ODW’s application in propulsion. Thus, to confirm hysteresis phenomenon of the ODW is the aim of this paper. As a matter of fact, hysteresis has already been discovered in shock reflection phenomenon (see [30] and [31]). In this study, the mechanism analysis of ODW’s hysteresis will also be conducted, by analogizing it to that of shock reflection. Since detonation simulation requires high grid resolution, adaptive mesh refinement technique is necessary to reduce total grid number. In this paper, such an open-source computational fluid dynamics (CFD) program called AMROC [32] is used to conduct numerical study. In next section, a brief introduction to the numerical method and model will be given. Fig. 1. Detonation polar drawn by Pratt et al. [21].
2. Numerical treatment 2.1. Governing equations The governing equations are two-dimensional unsteady reactive Euler equations given as: Continuity equation
∂ ρ /∂ t + ∂ (ρ u )/∂ x + ∂ (ρv )/∂ y = 0
(1)
where ρ , u, v, t, x and y denote the density, the x-direction velocity, the y-direction velocity, the time, the x coordinate and the y coordinate, respectively. Conservation equation of x-direction Momentum
∂ (ρ u )/∂ t + ∂ (ρ u2 + p)/∂ x + ∂ (ρ uv )/∂ y = 0
(2)
where p is the static pressure. Conservation equation of y-direction Momentum
∂ (ρv )/∂ t + ∂ (ρ uv )/∂ x + ∂ (ρv2 + p)/∂ y = 0
(3)
Conservation equation of energy Fig. 2. ODW at a double wedge assumed by Ghorbanian and Sterling [22].
∂ (ρ e )/∂ t + ∂ [u(ρ e + p)]/∂ x + ∂ [v(ρ e + p)]/∂ y = 0
(4)
where e is the specific total energy. For perfect gas, the equation of state is predicted a CJ ODW following the shock wave generated by the first wedge of a smaller angle and an overdriven ODW generated by the second wedge of a larger angle, as shown in Fig. 2 (Such a structure has been comfirmed by Liu et al. [23,24] in their numerical simulations with a reduced two-step reaction mechanism). Based on such a structure, they were the first to conduct detailed detonation/shock polar analysis on ODW structure. Kasahara et al. [25,26] and Maeda et al. [27,28] made a series of hypervelocity projectile experiments. They paid attention to the curvature effect and proposed the critical diameter criterion for a spherical projectile to stabilize an ODW. Liu et al. [23] paid particular attention to the triple point and the transverse shock of the ODW. They found the triple point structure can maintain stationary after initiation or propagate upstream to get closer to the wedge tip. They attributed the upstream propagation of the triple point to pressure variation due to inflow Mach number decrease or wedge angle increase. In
p = ρ RT
(5)
where R is the gas constant and T is the static temperature. Thus, the specific total energy can be expressed as
e = p/ρ (γ − 1 ) + (u2 + v2 )/2 − RT0 q˜
(6)
where γ , T0 and q˜ are the specific heat ratio, the static temperature of the inflow and the dimensionless local heat release using RT0 for normalization (q˜ = q/RT0 , where q is the dimensional local heat release), respectively. A two-step reduced reaction mechanism (see [33]) is employed, which helps mimic the feature of a chain-branching reaction and avoid huge computational expense of detailed reaction mechanism. The two-step reduced reaction mechanism consists of a thermally neutral induction step and a rapid heat release step that follows. The reaction rates equation for both steps are given as:
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∂ (ρξ )/∂ t + ∂ (ρ uξ )/∂ x + ∂ (ρvξ )/∂ y = H (1 − ξ )ρ kI exp [EI (1/RTS − 1/RT )]
(7)
∂ (ρλ )/∂ t + ∂ (ρ uλ )/∂ x + ∂ (ρvλ )/∂ y = [1 − H (1 − ξ )]ρ kR (1 − λ ) exp (−ER /RT )
(8)
where ξ , kI and EI are the reaction progress variable, the reaction rate constant and the activation energy of the induction step, respectively, and λ, kR and ER are that of the heat release step, respectively. Note that the reaction rate constant kI and kR control the induction length and the reaction length, respectively. TS is the temperature inside the induction region. H (1 − ξ ) is a step function written as:
H (1 − ξ ) =
= 1 if = 0 if
ξ <1 ξ ≥1
(9)
Clearly, at any instant the local heat release q˜ or q satisfies q˜ =
λQ˜ or q = λQ, where Q˜ and Q are the dimensionless and dimensional total heat release by the reactant (Q˜ = Q/RT0 ), respectively. 2.2. Numerical method and model As introduced above, AMROC is used to conduct numerical simulation in this study. It is an open-source CFD program based on finite volume method. Block-structured adaptive mesh refinement technique of AMROC allows it to simulate detonation with high resolution. MUSCL-Hancock scheme is employed for discretization with Riemann solver of hybrid Roe scheme. Van Albada slope limiter is used to ensure total variation diminishing (TVD) properties. For the source term treatment, Godunov splitting method is used to overcome the problem of stiff chemistry. AMROC has been validated to be very suitable for shock-induced combustion and ODW [32]. Thus, code validation is not conducted in this paper. All relevant details of AMROC can be found in [32]. Besides, grid convergence test will be given in next section. The computational domain is shown in Fig. 3, with a width of 200 mm and a height of 350 mm. The left and the top boundary are inflow boundaries, and the right boundary together with the bottom boundary in the range of −20 mm ≤ x < 0 mm is the outflow boundary. The wedge has a finite length of 120 mm, and the angle between the wedge surface and the horizontal direction is 30°. The reason for not using an infinite wedge, i.e., a truncated wedge, is mainly due to the treatment on outflow boundary condition. Since extrapolation is used for outflow boundary, a supersonic condition should be guaranteed. Thus, the finite-length wedge allows the flow to further expand and accelerate downstream so that it can maintain supersonic state at the right boundary. This is especially important for cases in which the ODW is close to detachment state and thus the post-ODW flow is subsonic. The angle between the inflow and the horizontal direction is denoted by θ . Thus, the real wedge angle θ satisfies θ = θ + 30◦ . In this study, θ can be varied while keeping the total velocity magnitude unchanged. Five values of θ from 0 to 4°, with an interval of 1°, are chosen so that five different wedge angles are tested: θ = 30◦ , θ = 31◦ , θ = 32◦ , θ = 33◦ and θ = 34◦ (detachment wedge angle). In order to investigate the hysteresis phenomenon of the ODW, a loop of wedge angle variation is simulated, i.e., 30°−31°−32°−33°−34°−33°−32°−31°−30°. We first start the simulation from θ = 30◦ case with reaction disabled, i.e., using the inert shock wave of θ = 30◦ case as initial condition. After then the reaction is turned on, so we can obtain the final state of θ = 30◦ case after sufficiently long CPU time. After we obtain the final state, we directly modify θ abruptly. In this way, we can vary the wedge angle simply by varying inflow direction, using current final state as initial condition for the next wedge angle. When the detachment wedge angle (θ = 34◦ ) is
Fig. 3. Computational domain and boundary conditions.
reached, we start to decrease the wedge angle along the path of 34°−33°−32°−31°−30°. In following sections, we will refer to the path of 30°−31°−32°−33°−34° as the positive path, and the path of 34°−33°−32°−31°−30° as the negative path. The simulated inflow is at a Mach number of 4.0 (2171.7 m/s), at a static temperature of 650 K and a static pressure of 20 kPa. The dimensionless total chemical energy release is set to be Q˜ = 8 which is moderate. The dimensionless activation energies of the induction step and the heat release step are EI /RTS = 4.8 and ER /RTS = 1.0, respectively. This approximately represents hydrogen/oxygen mixture according to [33]. The reaction rate constants of the induction step and the heat release step are set to certain values such that the induction length I and thermal pulse width R corresponding to a CJ detonation are 2.12 mm and 0.95 mm, respectively. The specific heat ratio is γ = 1.2 and the gas constant is R = 377.9 J/kg · K. It should be pointed out that in this paper, all the flow fields in different cases will be presented in the form of numerical schlieren, i.e., ∂ ρ /∂ y. 3. Results and discussions 3.1. Basic flow field of θ = 30◦ case In this section, we will start from the basic flow field of θ = 30◦ case, based on which grid convergence test and investigation of hysteresis phenomenon will be conducted. Figure 4 shows the basic flow field of θ = 30◦ case. It can be seen from Fig. 4a that the ODW consists of two sections, one of which is the stable Zeldovich–Neumann–Döring (ZND) ODW and the other is cellular ODW. According to Teng et al. [34], this is reasonable because a stable ODW tends to develop cellular instability
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Fig. 4. Flow field of θ = 30◦ case (a) the whole computational domain, (b) magnification of the ZND section.
in the far downstream. Figure 4b is the local magnification of the ZND section, from which it can be seen that a leading shock exists near the wedge tip with ignition delay behind it. A deflagration is triggered after the ignition delay and compression waves due to combustion are emanated which intersect with the leading shock and reflect to intersect with the deflagration again. During the interaction between the shock and the flame, the complex coupled structure consisting of the precursor shock and the flame front, i.e., the ODW, finally forms. It is very important to note that for θ = 30◦ case there is no transverse shock or typical primary triple point lying between the leading shock and the ODW. In another word, the transition pattern from the leading shock to the ODW is smooth and progressive. This is so-called smooth ODW according to Silva and Deshaies [8] and Teng and Jiang [35]. In comparison, an ODW with transverse shock and primary triple point structure is called abrupt ODW. In the following sections, the primary triple point structure as well as the transverse shock will be one of the objects that are focused on in this paper. In Fig. 4, the grid resolution is 30 pts/I, which will be justified to reach grid convergence in Section 3.2. 3.2. Grid convergence test Four different grid resolutions including 10 pts/I, 20 pts/I, 30 pts/I and 40 pts/I are tested. Figures 5 and 6 show the results of the whole computational domains and ZND sections under these different grid resolutions, respectively. It is seen that the global flow fields are similar to each other in a qualitative sense. However, 10 pts/I is too coarse to clearly distinguish the precursor shock and the flame front of the ODW. In comparison, 20 pts/I, 30 pts/I and 40 pts/I are all capable of capturing the fine structure. For ZND section, 30 pts/I has almost the same result with that of 40 pts/I. Focusing on the cellular instability, it is found that the cellular instability is triggered at a more upstream place for 20 pts/I, whereas 30 pts/I and 40 pts/I have almost the same cellular-instability-triggered position. Considering most
part of the cellular ODW is influenced or ‘contaminated’ by the expansion fan, it is not our main object of investigation. Thus, the results of grid convergence test confirm that 30 pts/I is enough to fulfill our requirement of grid resolution for current study. In order to highlight the hysteresis phenomenon, only the region where −20 mm ≤ x ≤ 120 mm and 0 mm ≤ y ≤ 130 mm, as shown in Fig. 6, will be displayed in Section 3.3. 3.3. Hysteresis phenomenon of the ODW: phenomenon description Based on the basic flow field of θ = 30◦ case, we start to vary the wedge angle in the way that has been introduced in Section 2.2. After a loop of wedge angle variation, the hysteresis process of the ODW is clearly shown in Fig. 7. Now detailed description will be given. Let us first focus on the positive path. It can be seen that when the wedge angle is changed from θ = 30◦ to θ = 31◦ , the transition pattern from the leading shock to the ODW is still smooth, with a reduced ignition delay due to the stronger leading shock. When the wedge angle is changed from θ = 31◦ to θ = 32◦ , the transition pattern is no longer smooth but abrupt, with an obvious primary triple point and a transverse shock (TS) existing. Note that the transverse shock for θ = 32◦ case in the positive path is in the downstream of the deflagration and thus the deflagration is complete for current case. However, when the wedge angle is further increased to θ = 33◦ , the triple point together with the transverse shock moves upstream and ‘swallows’ part of the deflagration, making it incomplete compared to the previous θ = 32◦ case. When the wedge angle is finally changed to detachment wedge angle, i.e., θ = 34◦ , a prompt detached ODW without ignition delay comes into being. It is important to note that the time needed for the triple point to move upstream and for the ODW to reach detachment state is significantly long (more than 1∼2 ms), which has also been demonstrated by Choi et al. [36]. After the detachment state for θ = 34◦ case is finally reached, the wedge angle is varied along the negative path. When the
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Fig. 5. The whole computational domain of grid convergence test for (a) 10 pts/I, (b) 20 pts/I, (c) 30 pts/I and (d) 40 pts/I.
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Fig. 6. ZND section of grid convergence test for (a) 10 pts/I, (b) 20 pts/I, (c) 30 pts/I and (d) 40 pts/I.
wedge angle is decreased to θ = 33◦ , the ODW recovers to what it looks like at the same wedge angle in the positive path. Thus, for θ = 33◦ case, no hysteresis is discovered. However, when the wedge angle is decreased from θ = 33◦ to θ = 32◦ , the situation is different. It is clear that for θ = 32◦ in the negative path the primary triple point together with the transverse shock still lies somewhere upstream and the deflagration remains to be partially ‘swallowed’. This is significantly different from the case at the same wedge angle in the positive path. Thus, hysteresis exists for θ = 32◦ case and we call such a type of hysteresis the hysteresis of upstream-downstream triple point. When the wedge angle is further decreased to θ = 31◦ , the primary triple point together with the transverse shock moves downstream, behind the deflagration. In another word, compared to the smooth ODW obtained for θ = 31◦ case in the positive path, abrupt ODW is obtained in the negative path for the same wedge angle. Such a type of hysteresis is called the hysteresis of smooth-abrupt transition pattern in this paper. When the wedge angle is finally decreased to θ = 30◦ , the transverse shock disappears and the ODW recovers to what it looks like in the positive path for the same wedge angle, i.e., hysteresis
ceases to exist. It is very interesting that although the ODW structure is closely relevant to initial condition from the perspective of the entire loop, the results are independent of initial condition along each of the two paths alone, positive or negative. That is to say, along the positive or negative path alone, we can obtain the same result for current wedge angle no matter which previous result is used as initial condition. In next section, mechanism for hysteresis phenomenon of the ODW will be discussed in detail. 3.4. Hysteresis phenomenon of the ODW: mechanism analysis In order to explain the mechanism for hysteresis phenomenon of the ODW, a detonation/shock polar analysis on the primary triple point of the abrupt ODW is conducted in this section. Such analysis is based on the so-called three-shock theory proposed by von Neumann for analyzing Mach reflection phenomenon of inert shocks (see [31]). To facilitate our analysis, the triple point structure of θ = 32◦ case in the positive path is particularly magnified in Fig. 8, and the corresponding schematic of the flow filed is shown in Fig. 9.
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Fig. 8. Magnification of the primary triple point of θ = 32◦ case in the positive path.
Fig. 7. Hysteresis process of the ODW.
Clearly, the flow field is divided into several regions by discontinuities and each region (or state) is denoted by a number in Fig. 9. The inflow has two paths to go across the triple point structure, one of which is from state 1 directly to state 5 (across the ODW) and the other is from state 1 to state 2 (across the leading shock) and thereafter to state 3 (across the compression waves) and finally to state 4 (across the transverse shock). State 4 and 5 should be matched with each other in the very neighborhood of the triple point along the slip line, i.e., having the same pressure and flow direction. Thus, such a match condition can be described by detonation/shock polar, as shown in Fig. 10a. It is seen that state 2 (denoted by A) can be found on the shock polar for state 1 (SP1) simply according to the wedge angle θ = 32◦ . Thereafter, the
Fig. 9. Schematic of the triple point structure of θ = 32◦ case in the positive path.
fluid particles are compressed by compression waves emanated by the deflagration and reach state 3 (denoted by B). Starting from B, the shock polar for state 3 (SP3) can thus be drawn. Note that SP3 has an opposite direction relative to SP1 and DP1 because the transverse shock is a right-going discontinuity whereas the leading shock and the ODW are both left-going. It is seen that SP3 has an intersection (denoted by C) with SP1 but has no intersection with DP1. Since state 4 and 5 in the very neighborhood of the triple point are both lying in the ODW’s induction region which is chemically frozen, the match condition should be indicated by C rather than D which indicates the post-ODW state in the far
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Fig. 11. Magnification of the primary triple point of θ = 33◦ case in the positive path.
Fig. 10. Detonation/shock polar in (a) P/P0 − θ and (b) β − θ , where P0 is the inflow static pressure, SP1 is the shock polar for state 1, DP1 is the detonation polar for state 1, SP3 is the shock polar for state 3, A denotes state 2, B denotes state 3, C denotes the match condition of state 4 and 5 in the very neighborhood of the triple point, and D denotes the post-ODW state in the far downstream.
downstream for current wedge angle. The dimensionless pressure of intersection C predicted by the detonation/shock polar is 16.75, in good agreement with CFD result which is 17.03. Clearly, intersection C is a strong shock solution. In Fig. 10b, each point corresponding to Fig. 10 a is given. It can be seen that the wave angle of intersection C is 79.2°. From the stream line in Fig. 8, it is also illustrated that near the triple point the wave angle of the ODW is quite large and it decreases gradually in the downstream. As we know, a strong shock solution does not exist for an attached oblique shock and it is only possible for a Mach stem or a detached shock. Thus, the ODW near the triple point can be considered as a Mach stem or locally detached. This is the basis of hysteresis phenomenon, as explained in the following texts.
Fig. 12. Schematic of the triple point structure of θ = 33◦ case in the positive path.
As the wedge angle is increased to θ = 33◦ , pressure of state 2 together with that of state 3 is also increased because of stronger leading shock. Consequently, SP3 moves rightward relative to that in Fig. 10a. According to von Neumann’s three-shock theory, this means an increased Mach stem height or increased detachment distance. Thus, the triple point moves upstream, getting closer to the wedge tip so that the Mach stem height or local detachment distance increases, as shown in Figs. 11 and 12. Once the triple point together with the transverse shock ‘swallows’ the deflagration, the fluid particles will no longer need to flow across the compression waves to reach state 3, and state 2 will play the role of state 3 in the detonation/shock polar. In this situation, SP3 will at first move rightward immediately after the wedge angle is changed from θ = 32◦ to θ = 33◦ , and thereafter move
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Fig. 13. Detonation/shock polar as the wedge angle changes from θ = 32◦ to θ = 33◦ , where SP3 is the shock polar for state 3 of θ = 32◦ case, SP3’ is the shock polar for state 3 during the initial period after the wedge angle is increased, and SP2 is the shock polar for state 2 of the final θ = 33◦ case.
leftward as the triple point moves upstream until it ‘swallows’ the deflagration, as shown in Fig. 13. Although SP2 is on the left of SP3’, it does not mean that the Mach stem height or detachment distance will decrease during the process because this process is kind of irreversible. Note that irreversibility is one of the main characteristics in hysteresis phenomenon of shock reflection. Thus, by analogy, the upstream movement of the triple point should also be irreversible to some extent. However, since SP3’ finally moves left after all, this is supposed to have some resistant effect against the triple point’s further upstream moving although the triple point cannot be brought back due to irreversibility. Perhaps this can explain why the upstream movement of the triple point takes quite a long time. In the negative path, as the wedge angle decreases, SP2 moves leftward and therefore the Mach stem height or local detachment distance also decreases. As a result, the triple point moves downstream. Again, irreversibility makes the flow structure fail to recover to what it is in the positive path for the same wedge angle. Thus, hysteresis shown in Fig. 7 comes into being. It should be pointed out that for θ = 33◦ , the triple point is so close to the wedge tip both in positive and negative paths that hysteresis can hardly be distinguished. The hysteresis phenomenon ceases to exist when the wedge angle is decreased to θ = 30◦ . It is also interesting to note that for θ = 31◦ the hysteresis of smooth-abrupt transition pattern exists. Since criteria of ODW’s transition pattern have already been proposed (see [8] and [35]), they seem not appropriate for explaining current hysteresis of smooth-abrupt transition pattern. 3.5. Further discussions The results in this study are quite disappointing for propulsion application of the ODW. As described in Section 1, in earlier studies, ODWs are supposed to be self-sustained once initiated and be independent of downstream condition if the wedge angle is equal to or smaller than CJ value because in such situations sonic condition is reached behind the ODW. Therefore, the ODW is thought to be able to stand on the wedge arbitrarily even if the wedge angle is smaller than CJ value. For oblique detonation
engine, CJ ODW is always desirable because it has the minimum entropy increase. However, CJ ODW is corresponding to small wedge angle which tends to hinder detonation initiation and standing stabilization due to lower energy deposition and longer induction length. Thus, before this study, we wished to obtain CJ ODW on a small-angle wedge by first initiating ODW on a largeangle wedge and afterward lowering the wedge angle to CJ value. Unfortunately, it has been demonstrated in current study that the hysteresis phenomenon does not behave as wished. If the wedge angle decreases along the negative path, ignition delay will continuously increase. Furthermore, it seems that hysteresis cannot be maintained when the wedge angle is decreased to a certain value, which means the flow field is independent of initial condition for small wedge angles. Thus, it is reasonable to expect that a standing CJ ODW is actually not possible when the wedge angle is small enough even if it has already been somehow initiated, because the ignition delay distance will be far beyond the wedge length. Maybe some artificial measures are needed to interfere in the hysteresis process so that the phenomenon can behave as we wish, i.e., obtaining a CJ ODW without ignition delay or with short ignition delay when the wedge angle is decreased to CJ value. Another important thing is the physical essence deep inside the hysteresis phenomenon of the ODW. Although we have explained the mechanism of hysteresis by analogy with that of shock reflection, there is still a remarkable difference between the two. For shock reflection, hysteresis phenomenon is corresponding to a dual-solution domain according to von Neumann’s three-shock theory. Thus, hysteresis phenomenon of shock reflection has a reasonable explanation in mathematics. In contrast, there is no dual-solution domain for hysteresis of the ODW, as shown in Figs. 10 and 13. Therefore, in future work, it is necessary to deeply understand the physical essence of ODW’s hysteresis, which is still unclear. 4. Conclusions 1. Hysteresis phenomenon of the ODW is confirmed. Two types of hysteresis are found: the hysteresis of upstreamdownstream triple point and the hysteresis of smoothabrupt transition pattern. 2. Along each of the two paths alone, positive or negative, the final state of the ODW is independent of the initial condition. 3. For an abrupt ODW, detonation/shock polar analysis shows that the precursor shock of the ODW near the primary triple point is actually a strong shock solution and therefore is a Mach stem or locally detached. 4. The mechanism of hysteresis phenomenon of the ODW is also explained by detonation/shock polar analysis. Similar to hysteresis phenomenon of shock reflection, irreversibility should be responsible for ODW’s hysteresis. Once the wedge angle is smaller than a certain value, hysteresis phenomenon will disappear. This means the desire of obtaining a standing CJ ODW without ignition delay or with short ignition delay at a CJ wedge angle via hysteresis is probably impossible. Acknowledgments The research is supported by The National Natural Science Foundation of China NSFC Nos. 91641126 and 11702316. References [1] J.P. Sislian, H. Schirmer, R. Dudebout, J. Schumacher, Propulsive performance of hypersonic oblique detonation wave and shock-induced combustion ramjets, J. Propul. Power 17 (2001) 599–604.
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