# Unsteady Oblique Detonation Waves in a Tunnel Induced by Inflow Mach Number Variation

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## Abstract

**:**

_{0}) have been studied and the geometric model is a tunnel with an outward-deflection upper wall to mimic an engine outlet. Numerical results demonstrate that when M

_{0}deviates from the designed state, two typical wave structures arise, featuring a Mach stem of detonation or a post-corner recirculation zone. A sudden change in M

_{0}leads to the transition of these two structures, generating unsteady ODWs temporally with a multi-segment-complex wave surface caused by triple points. The wave structures near the corner have been analyzed in detail, revealing how the Mach stem and the recirculation zone evolve into each other. Furthermore, the effects of unsteady ODWs on hypersonic propulsion applications have been discussed, providing possible ways to suppress the Mach stem of detonation.

## 1. Introduction

## 2. Physical and Mathematical Model

_{0}are ignited via an ODW triggered by a solid wedge on the lower wall, as shown in Figure 1a. Different from most previous studies in free space, a channel with an outward deflection section or a turning corner is considered in this work, so the combustion product could expand and then provide greater thrust and impulse. In the ideal situation with a proper inflow, M

_{0}[20,21], the ODW surface reflects on the turning corner, reaching the best performance of ODE. If M

_{0}deviates slightly, the ODW could reflect before or behind the corner, as shown in Figure 1b. In this domain, H denotes the entrance height of the combustor; θ

_{1}and θ

_{2}are the wedge angle and the deflection angle of the upper wall, respectively; L and L

_{c}are the length of the computational domain and the upper wall before the corner.

_{I}and E

_{R}represent the activation energy of induction and heat release, respectively. Two constants, k

_{I}and k

_{R}, are used to control the chemical reaction rates in the induction and exothermic zones. In this study, k

_{R}is set to be 1.0, and k

_{I}= −u

_{vn}, for which u

_{vn}is the flow velocity behind the leading shock wave in the Zeldovich–von Neumann–Döring (ZND) detonation wave in a shock-fixed coordinate system. The variables, p, ρ, T, u, and v are the pressure, density, temperature, x-direction velocity, and y direction velocity, respectively. All of them are dimensionless with reference to the free flow state (in which, the tilde~denotes the original dimension quantity, and subscript 0 denotes the reference quantity of the free flow):

_{I}= 3.0T

_{S}, and E

_{R}= 1.0T

_{S}, where T

_{S}is the postshock temperature of Chapman–Jouguet (CJ) detonation and Q is the amount of chemical heat release. The induction region width ΔI of the one-dimensional ZND detonation wave is taken as unit length. The flow time is dimensionless with the sound velocity of the wavefront reactants, i.e., t = ΔI/c

_{0}, where c

_{0}is the initial sound velocity of the reactants. The selection of these parameters refers to the hydrogen–air mixture entering the ODE combustor while operating at a high altitude. Compared with previous studies [22,23,31], Q and E

_{I}decrease significantly to mimic the inflow with low density and high temperature, deriving from the high altitude and inlet compression, respectively.

_{1}= 150, L

_{c}= 185, L = 320, θ

_{1}= 20°, and θ

_{2}= 55°, among which L

_{c}is critical to obtain the typical flow-field structures. It is worth noting that the induction region width ΔI is the characteristic length scale of the ZND detonation wave and is also taken as the unit length. In this study, the length of the calculated domain is dimensionless by ΔI. As for the boundary conditions, the left entrance is modeled as a supersonic inflow boundary condition in which the flow parameters are constant; the right dashed lines correspond to the existence of the combustor and are set to be outflow boundaries, whose boundary parameters are extrapolated from the interior; the upper walls and lower wedge are characterized as slip boundaries.

## 3. Results and Discussion

#### 3.1. Basic Structures of Oblique Detonation and Resolution Study

_{c}is a critical parameter, which is chosen according to M

_{0}in this study. The basic simulations of ODWs without considering the upper wall at M

_{0}5.5 and 6.0 are performed to get the positions of ideal ODW surfaces. Finally, the L

_{c}is chosen as 185, which is between these two positions. It means that there is a certain M

_{0}between 5.5 and 6.0, at which the ODW reflects on the corner, and the ODE operates at the designed state.

_{0}are shown in Figure 2. It could be seen that the OSW induced by the lower wall turns into ODW near the position x = 40~60 gradually. The ODW surface extends downstream with the same smooth OSW–ODW transition, however, the ODW surface angle is affected by M

_{0}obviously. More importantly, the ODW surfaces reflect on different positions of the upper wall, before or behind the corner, resulting in different wave structures. Generally speaking, due to the different M

_{0}, the ODW could lead to two nonideal wave structures in a confined space whose geometric parameters are fixed.

_{0}5.5, the ODW surface reflects before the corner, leading to the Mach reflection and a Mach stem of detonation, as shown in Figure 3a. The Mach reflection changes the whole wave system, leading to a reflected shock wave, the second slip line, et al. On the other hand, the ODW surface reflects behind the corner with a high M

_{0}6.0, leading to an evidently different wave configuration. As shown in Figure 2b and Figure 3b, a recirculation zone forms behind the outward-turning corner, attributing the ODW decoupling influenced by the expansion wave. Despite different parameters of the chemical-reaction model, these wave structures have been observed in previous studies [24], though only steady structures with a fixed M

_{0}were investigated.

#### 3.2. Surface Evolution in the Structural Variation

_{0}based on the two steady structures. The structural variation of increasing M

_{0}is shown in Figure 5. Initially, an ODW with M

_{0}5.5 exists, whose structure features a Mach stem of detonation. Then, the inflow M

_{0}of the left boundary changes from 5.5 to 6.0, and the flow time is reset to be zero simultaneously. It is observed that a sudden increase of M

_{0}leads to the distortion of the upstream ODW surface, as shown in Figure 5a. Subsequently, triple points (TPs) form on the wave surface and propagate downstream, resulting in a multisegment complex wave surface, as shown in Figure 5b. TP1 is a left-running triple point facing downstream, and the other two belong to right-running triple points [41,42]. In fact, it could be seen that TP1 originated from the contact surface of velocity discontinuity, while TP2 and TP3 are from the self adjustment of the wave surface. Finally, these TPs move downstream and merge into the Mach stem. It should be noted that the contact surface of inflow velocity reaches the Mach stem before t = 31.97 (Figure 5c), however, TP2 and TP3 travel slowly and persist for a long time thereafter. This demonstrates that the ODW surface could self adjust to fit the increasing inflow M

_{0}to some extent, which is achieved by triple points behind the contact surface.

_{0}is shown in Figure 7. Initially, an ODW with M

_{0}6.0 exists, whose structure features a postcorner recirculation zone. Then, the inflow M

_{0}of the left boundary changes from 6.0 to 5.5, and the flow time is reset to zero simultaneously. Similarly, M

_{0}variation also leads to a distortional wave surface, generating two TPs. TP1 also originated from the interaction of the velocity contact and detonation front, however, the movement of the resulting TP1 is slightly slower than the contact surface. Notably, when the contact surface goes through the whole flow fields, TP2 still exists on the surface, which has a slow-moving process to adjust the wave surface. During this process, the recirculation zone becomes smaller but does not disappear until t = 70.09 (Figure 7f).

_{0}. To analyze the procedure of decreasing M

_{0}, the shock-front position along different lines is also recorded as a function of time, as shown in Figure 8. The wave-front position moves upstream, while TPs formed in this process are distinguished from Figure 6 in strength and function. Compared with Figure 6, the TP1 in Figure 8 leads to the first peak of these lines, causing the wave surface to suddenly move downstream, which prevented the whole wave surface from moving upstream. For these two processes, it is obvious that TPs formed on the wave surface both affect the movement of the detonation wave. The difference is that the action of the TPs is consistent with the direction of the wave surface movement for increasing M

_{0}while decreasing M

_{0}leads to an opposite result.

_{0}can both lead to structural variation, resulting in a multisegment-complex wave surface at the initial stage. For these two processes, the contact surface of inflow velocity could go through the whole flow fields quickly, while the detonation front would still be in a self-adjusting state and the original near-corner structures do not disappear immediately, demonstrating that the formation of the new structures mainly depends on the self-adjusting characteristic of the detonation wave.

#### 3.3. Discussion on the Near-Corner Structures

_{0}(increasing or decreasing) in steady cases involves triple points on ODWs structure. The triple points are critical to adjust the upstream and downstream of wave surfaces, but they only influence the development of the ODWs structure at the early stage. The other important phenomenon to be noted in this study is that unsteady ODWs are also concerned with the transition between the Mach stem of detonation and the postcorner recirculation zone which is named as the near-corner structures here.

_{0}. Figure 9 shows the evolution processes of local temperature fields of ODW near the corner, with the black curve denoting the sonic regions. When the M

_{0}increases from 5.5 to 6.0, the Mach stem moves downstream, finally inducing a recirculation zone. It is observed that the Mach stem becomes oblique and the subsonic zone shrinks, splitting into two small parts in Figure 9c. Ultimately, the lower subsonic zone disappears slowly due to the fact that the surface angle of ODW increases continuously, whereas the upper subsonic region is gradually building up and a relatively large recirculation zone arises here. The recirculation zone has been observed previously [23], whose formation is attributed to the reversed pressure gradient downstream. The low-temperature region could be observed in Figure 9c, enlarging together with the upper recirculation zone. In general, this work not only supports the formation mechanism of the recirculation zone proposed before but demonstrates that the original subsonic zone might evolve from a Mach stem of detonation and then grows from one extremely small point.

_{0}, respectively, whereas the red squares denote the state when the wave front reaches the turning point. It is observed that when M

_{0}increases from 5.5 to 6.0, the shock front quickly moves downstream and becomes steady within a short time. In contrast, the decreasing M

_{0}results in an extremely long transition time, which should be attributed to the slow-moving Mach stem. Even though the final near-corner structures in unsteady inflow, the M

_{0}remains the same with that steady flow. These processes exhibit that different wave dynamics could be involved with different spatial and temporal characteristics.

_{0}varies, which concerns with different subsonic zones. The subsonic zones are unavailable from the viewpoint of the ODE application since the ODE is designed to be supersonic in the whole flow tunnel. These results provide a physical description of ODW dynamics in a confined combustor and deepen our understanding of the ODW combustor design under the velocity variation conditions. When M

_{0}is increasing, the flow structures mainly include three major steps, i.e., from one subsonic zone (Mach stem) to two subsonic zones (recirculation zone and strong oblique detonation) and finally to one subsonic zone (large recirculation zone). Considering that the Mach stem is equivalent to a normal detonation wave with a high M

_{0}, the total pressure loss will be very high, so increasing M

_{0}could improve the impulse performance. On the contrary, the Mach stem forms accompanied by the decreasing M

_{0}, so the impulse performance becomes worse. Based on the present work, we found that the formation and moving of the Mach stem is a slow process, so it is possible to suppress it. For example, the Mach stem could be eliminated by adjusting the position of the upper-wall turning point, which could be achieved using a geometric variable device. For further consideration, the leaking slot located on the upper wall before the turning point could also be an option, which could weaken the Mach stem theoretically.

## 4. Conclusions

_{0}variation and ascertained the evolution processes of the ODW structure in a confined combustor, which is more in resemblance to the practical geometric model of the engine and the unsteady flow at different hypersonic flight conditions. The results demonstrate that when M

_{0}deviates from the designed state, two typical wave structures arise, featuring a Mach stem of detonation or a postcorner recirculation zone. Notably, decreasing or increasing M

_{0}can lead to the mutual transitions of these two typical structures, and a multisegment complex-wave surface caused by triple points appeared in both of the two processes. Meanwhile, the TP1 originated from the interaction of the velocity and detonation front going through the whole flow fields quickly, and the other TPs existing on the ODW surface move slowly to adjust the wave surface, demonstrating that the formation of new structures mainly depends on the self-adjusting characteristic of the detonation wave. The wave structures near the corner have been analyzed in detail, revealing that the formation and disappearance of the recirculation zone have their separate paths. The former originated from the splitting of the Mach stem and expands into a large one quickly, while the latter is squeezed by the ODW front and evolves into a Mach stem with a slow-moving process, demonstrating that different wave dynamics are involved in different spatial and temporal characteristics. Finally, this study discussed the effects of unsteady ODWs on propulsive application and illustrated that the Mach stem forming and moving in a slow process could be suppressed, which deepens our understanding of the ODW combustor design under velocity-variation conditions.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematics of the oblique detonation engine and (

**b**) the computational model of the combustor.

**Figure 2.**Temperature fields for ODWs in the case of (

**a**) M

_{0}= 5.5; (

**b**) M

_{0}= 6.0. Black curves represent streamlines.

**Figure 3.**Two typical wave structures: (

**a**) with a Mach stem of detonation, (

**b**) with a postcorner recirculation zone.

**Figure 4.**Temperature fields (

**a**) and pressure/temperature profiles (

**b**) along the line y = 100 in the case of M

_{0}= 5.5.

**Figure 5.**Mach number contours of ODW at different instants when M

_{0}increases from 5.5 to 6.0. (

**a**) t = 10.69, (

**b**) t = 26.65, (

**c**) t = 31.97, (

**d**) t = 47.93, (

**e**) t = 63.89, and (

**f**) t = 74.53.

**Figure 6.**Shock-front positions along the lines y = 60, 75, and 90 when M

_{0}increases from 5.5 to 6.0.

**Figure 7.**Velocity contours of ODW at different instants when M

_{0}decreases from 6.0 to 5.5. (

**a**) t = 8.94, (

**b**) t = 22.43, (

**c**) t = 31.97, (

**d**) t = 41.11, (

**e**) t = 50.81, (

**f**) t = 70.09.

**Figure 8.**Shock-front positions along the lines y = 60, 75, and 90 when M

_{0}decreases from 6.0 to 5.5.

**Figure 9.**Local temperature fields (with the black line denoting sonic curves) of ODWs when M

_{0}increases at different instants. (

**a**) t = 69.21, (

**b**) t = 74.53, (

**c**) t = 79.51, (

**d**) t = 97.81.

**Figure 10.**Local temperature fields (with the black line denoting sonic curves) of ODWs when M

_{0}decreases at different instants. (

**a**) t = 27.04, (

**b**) t = 50.81, (

**c**) t = 59.33, (

**d**) t = 67.78, (

**e**) t = 75.02, (

**f**) t = 80.80.

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**MDPI and ACS Style**

Niu, S.; Yang, P.; Wang, K.; Teng, H.
Unsteady Oblique Detonation Waves in a Tunnel Induced by Inflow Mach Number Variation. *Aerospace* **2023**, *10*, 330.
https://doi.org/10.3390/aerospace10040330

**AMA Style**

Niu S, Yang P, Wang K, Teng H.
Unsteady Oblique Detonation Waves in a Tunnel Induced by Inflow Mach Number Variation. *Aerospace*. 2023; 10(4):330.
https://doi.org/10.3390/aerospace10040330

**Chicago/Turabian Style**

Niu, Shuzhen, Pengfei Yang, Kuanliang Wang, and Honghui Teng.
2023. "Unsteady Oblique Detonation Waves in a Tunnel Induced by Inflow Mach Number Variation" *Aerospace* 10, no. 4: 330.
https://doi.org/10.3390/aerospace10040330