Periodic oscillation and fine structure of wedge-induced oblique detonation waves

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1 Acta Mech. Sin. (211) 27(6): DOI 1.17/s y RESEARCH PAPER Periodic oscillation and fine structure of wedge-induced oblique detonation waves Ming-Yue Gui Bao-Chun Fan Gang Dong Received: 2 March 211 / Revised: 9 June 211 / Accepted: 12 July 211 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 211 Abstract An oblique detonation wave for a Mach 7 inlet flow over a long enough wedge of 3 turning angle is simulated numerically using Euler equation and one-step rection model. The fifth-order WENO scheme is adopted to capture the shock wave. The numerical results show that with the compression of the wedge wall the detonation wave front structure is divided into three sections: the ZND model-like strcuture, single-sided triple point structure and dual-headed triple point strucuture. The first structure is the smooth straight, and the second has the characteristic of the triple points propagating dowanstream only with the same velocity, while the dual-headed triple point structure is very complicated. The detonation waves facing upstream and downstream propagate with different velocities, in which the periodic collisions of the triple points cause the oscillation of the detonation wave front. This oscillation process has temporal and spatial periodicity. In addition, the triple point trace are recorded to obtain different cell structures in three sections. Keywords Oblique detonation wave Wedge Periodic oscillation Fine structure The project was supported by the National Natural Science Foundation of China (187296) and the Open Fund of State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology (KFJJ9-13). M.-Y. Gui B.-C. Fan ( ) G. Dong State Key Laboratory of Transient Physics, Nanjing University of Science and Technology, 2194 Nanjing, China bcfan@mail.njust.edu.cn G. Dong State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, 181 Beijing, China 1 Introduction Oblique detonation wave (ODW) stabilized over a body has been studied due to the ongoing development of high-speed propulsion systems, such as ODW engines and ram accelerators. Experimental observations by Lehr [1], Dabora et al. [2], Viguier et al. [3,4], Kamel et al. [5], and Morris et al. [6] show that at least two different kinds of overall flow configurations involving standing oblique detonation waves by the wedge are observed. The first is a direct initiation of a detonation wave at the wedge surface. The second possibility is an oblique shock wave/oblique detonation wave (OSW/ODW) transition which occurs at a certain distance from the apex of the wedge. The latter case provides the objectives of this paper. Numerical simulations of wedge-induced detonations conducted by many authors [7 9] numerically investigated the influence of the factors on the formation of a stable detonation. Recently, Choi et al. [1] numerically captured the unsteadiness ODW frontal structure induced by a wedge in a rectangular computational domain with a dimensionless size of They showed that the ODW instability has a strong dependence on the activation energy. However, only single-sided triple point was found. Later, Choi et al. [11] analyzed ODW structure with the flow turning angle varying from 35 to 38 and thought that flow turning angle was a key parameter for the onset condition of the dual-headed triple point structures. Wang et al. [12] numerically investigated the influence of different Mach numbers 6.8, 7. and 7.5 inlet flows on the oblique detonation wave induced by a wedge and obtained the transverse wave propagating both in one-way direction and two-way direction downstream behind the ODW. In this study, an oblique detonation wave induced by a long enough wedge is numerically analyzed by high reso-

2 Periodic oscillation and fine structure of wedge-induced oblique detonation waves 923 lution algorithm. With compression of the wedge wall, the front of an oblique detonation wave can be divided into three sections, and their flow field structures and oscillation features with temporal and spatial periodicity are significantly different, which are discussed in detail using numerical results. Simultaneously, the triple point traces are recorded to obtain different cell structures. where k, E a, and R are the pre-exponential factor, activation energy, and gas constant, respectively. The convective fluxes in Eq. (1) are integrated by the fifth-order weighted essentially non-oscillatory (WENO) scheme [13]. The discretized equations are integrated in time using a fourth-order accurate, four-stage Runge Kutta scheme. The schematic of oblique detonation wave induced by the wedge is shown in Fig. 1. Cartesian grid is aligned with the wedge surface. In the present study, a Mach 7 flow with a specific heat ratio of γ = 1.3 over a 3 wedge is treated. The heat release q and the activation energy E a are chosen as 1 and 3, respectively. These flow parameters are selected to weaken the restriction of grid resolution requirements from the chemical induction length scale analyses [1]. A rectangular computational domain is considered over the wedge with a dimensionless size of Ten additional cells are added upstream of the wedge surface to avoid the numerical reflection from the left boundary. Slip boundary conditions are specified at the wedge surface. Inflow conditions are fixed to the free values in both left and upper boundaries of the domain. Outflow conditions imposed to the right boundary specify zero gradients for all variables. 2 Physical model and numerical method The two-dimensional reactive Euler equations in the nondimensional form are given as ρ ρu ρv t ρu ρv ρe ρz = + x ρw ρu 2 + p ρuv (ρe + p)u ρzu + y ρuv ρv 2 + p (ρe + p)v ρzv, (1) where u and v are the velocity components in the physical (x, y) coordinates, t, p, ρ, and Z are the time, pressure, density, and mass fraction of product species, respectively. The total energy e is given by e = p/[ρ(γ 1)] + (u 2 + v 2 )/2 Zq, (2) where q represents the heat release. The chemical reaction is modeled by one-step irreversible reaction as follows w = (1 Z)k exp( E a /RT), (3) Fig. 1 Schematic of the computational domain attached to the wedge surface 3 Results and discussion 3.1 Numerical validation To test the mesh convergence, three different levels of grid resolution (i.e., 1 3, , and 2 6) are considered. Firstly, one-dimensional detonation is simulated. The results for mesh convergence for the peak pressure and detonation velocity are shown in Figs. 2a and 2b, respectively. It is shown that in the last two grid levels, the peak pressure and detonation velocity have little difference, which tend to the theoretical values. Figure 3 shows the temperature field of ODW with the three gird systems. The wave front associated with the 1 3 grid system is quite smooth. In the last two grid systems, the wave front instability is clearly observed, and both single-sided and dual-headed triple points occur. Therefore, the grid of is adopted in later study. 3.2 Fine structure of ODW The front of ODW induced by a wedge is composed of two different parts as shown in Fig. 4: the OSW, labeled AB, and the ODW, labeled BE, where the detonation onset takes place at the intersection between OSW and ODW. In addition, if the wedge is long enough, three regions in the flow field behind ODW can be defined following the x coordinate, i.e., section BC, CD and DE, according to the fine structures of oblique detonation as shown in Fig. 4, which is a plot of the pressure field of the oblique detonation structure for a Mach 7 inlet flow over a 3 wedge.

924 M.-Y. Gui, et al. Fig. 2 Mesh convergence for the simulation of one-dimensional detonation. a Peak pressure; b Detonation velocity a b c Fig.

4 Pressure field of the oblique detonation structure In section BC, the detonation has a Zeldovich von Neumann Doering (ZND) model-like structure with smooth straight front, which is supported and

Therefore, under actions of disturbance, collisions between shock wavelets occur in section CD, which leads to an abrupt transition from the incident shock to the oblique detonation wavelet denoted

3 924 M.-Y. Gui, et al. Fig. 2 Mesh convergence for the simulation of one-dimensional detonation. a Peak pressure; b Detonation velocity a b c Fig. 3 Temperature field of ODW with different grids: a 1 3; b ; c 2 6 Fig. 4 Pressure field of the oblique detonation structure In section BC, the detonation has a Zeldovich von Neumann Doering (ZND) model-like structure with smooth straight front, which is supported and stabilized by compression of the wedge wall. As the increase of distances from the wedge apex along the wall surface, the shock front becomes more sensitive to disturbance, due to extended flow space. Therefore, under actions of disturbance, collisions between shock wavelets occur in section CD, which leads to an abrupt transition from the incident shock to the oblique detonation wavelet denoted as D1 in Fig. 5b. Transverse waves emanate from collision points defined as the triple point. The close-up view of this structure in CD is shown in Fig. 5, where Fig. 5a is contours of reactive progress superimposed on the shadow pressure contour, and Fig. 5b is a schematic diagram. The combustible gases compressed by the shock S1 are ignited by the transverse wave and thus lead to the onset of the transverse detonation TS1. Both D1 and TS1 face upstream and move downstream due to supersonic incoming flow. As the leading shock front of D1 gradually curves downstream, its strength decreases. Finally, the decaying wave becomes a non-reactive shock wave decoupling from the reaction zone, and then will transit to a detonation again in the next cycle. All the transverse waves move downstream with almost the same velocities. In section DE, the further distortions of shock fronts appear, so that the abrupt transition from the incident shock to the oblique detonation wavelet can happen both upstream and downstream denoted as D2 and D1 in Fig. 6b, respectively. The close-up view of this structure of detonation is shown in Fig. 6. The combustible gases compressed by the shock S1 are ignited by both the transverse waves emanating from triple points TP1 and TP2, respectively, and the transverse detonations TS1 and TS2 are formed, where the incoming fresh flow penetrates into the transverse detonations TS1 and TS2 almost perpendicularly. It is noteworthy that

Periodic oscillation and fine structure of wedge-induced oblique detonation waves 925 Fig. 5 Close-up view of the wave front structure in section CD.

a Contours of reactive progress superimposed on the shadow pressure contour; b Schematic diagram D1 and TS1 face upstream and D2 and TS2 face downstream. Based on one-dimensional analysis by Yi et al.

4 Periodic oscillation and fine structure of wedge-induced oblique detonation waves 925 Fig. 5 Close-up view of the wave front structure in section CD. a Contours of reactive progress superimposed on the shadow pressure contour; b Schematic diagram Fig. 6 Close-up view of wave front structure in section DE. a Contours of reactive progress superimposed on the shadow pressure contour; b Schematic diagram D1 and TS1 face upstream and D2 and TS2 face downstream. Based on one-dimensional analysis by Yi et al. [14], in a supersonic flow, the propagating velocity of the detonation facing downstream is larger than that of the detonation facing upstream, which makes the triple points TP1 and TP2 move toward each other. During this process, the detonation waves D1 and D2 decay gradually and become the shock wave just before the collision of triple points TP1 and TP2. Subsequent to the collision, the transverse detonations TS1 and TS2 reflect and propagate away from each other, and new detonation wavelets form. Therefore, the oscillation process of the detonation wave front has temporal and spatial periodicity. 3.3 Periodic oscillation of ODW In order to describe the periodicity, different locations in one spatial cycle are chosen. Due to the only single-sided triple point in section CD, a fixed Point X in one spatial cycle ( X =.35) is chosen. Its oscillation feature of the detonation is demonstrated in Fig. 7. The curve of shock pressure is periodic and this temporal period is.9. At t =.42, a triple point occurs at X, the pressure is the highest and the shock front is coupled with the reaction. As the triple point moves downstream, the pressure decreases dramatically and then increases gradually. Meanwhile, the shock front is decoupled with the reaction and then coupled gradually, till the new triple point occurs. In section DE, the dual-headed triple point structure occurs, which causes the periodic complexity. Four fixed locations X 1 X 4 in one spatial cycle ( X =.55) are chosen, and the oscillations of detonation front at the different locations are also demonstrated in Fig. 8. Based on the calculations, the flow parameters on the detonation front in a cell are affected by four triple point collision events, in which the two collisions generate in the cell, denoted by subscripts i, and the other collisions happen in the adjacent upstream cell,

Whereas superscripts + and denote triple points TP1 and TP2 or the detonations facing upstream and downstream, respectively. At t =.

5 926 M.-Y. Gui, et al. Fig. 7 Oscillation feature of detonation in section CD. a Leading shock pressure history; b The reaction front superimposed on the Y-t diagram of shadow pressure distribution at a fixed X denoted by subscripts u. Whereas superscripts + and denote triple points TP1 and TP2 or the detonations facing upstream and downstream, respectively. At t =.42, a triple point collision 1 i just happens at the location X 1, which leads to the pressure increase as shown in Fig. 8a. After the collision, the detonations facing upstream and downstream denoted as 1 ± i are formed and propagate downstream with different velocities. Then, the second pressure peak 1 u occurs in Fig. 8a, corresponding with the triple point TP2 generated in the first collision in the adjacent cell. Subsequently, the third pressure peak 2 u occurs as the triple point TP2 generated in the second collision in the adjacent cell passes through X 1. Finally, a new temporal cycle begins, when the triple points collide again. Fig. 8 Oscillation feature of detonation wave at four locations in one cycle in section DE. a Location X 1 ; b Location X 2 ; c Location X 3 ; d Location X 4

Periodic oscillation and fine structure of wedge-induced oblique detonation waves 927 Fig. 8 Oscillation feature of detonation wave at four locations in one cycle in section DE.

The oscillations of the detonation are shown in Figs. 8b 8d. In Fig. 8b, pressure peaks 1 ± i occur due to the collision at X 1. In Fig. 8c, the second triple point collision 2 i occurs at the location X 3 at t =.

6 Periodic oscillation and fine structure of wedge-induced oblique detonation waves 927 Fig. 8 Oscillation feature of detonation wave at four locations in one cycle in section DE. a Location X 1 ; b Location X 2 ; c Location X 3 ; d Location X 4 (continued) For other fixed locations X 2 = X 1 + X/4, X 3 = X 1 + X/2 and X 4 = X X/4, X is width of the cell. The oscillations of the detonation are shown in Figs. 8b 8d. In Fig. 8b, pressure peaks 1 ± i occur due to the collision at X 1. In Fig. 8c, the second triple point collision 2 i occurs at the location X 3 at t =.426, and then pressure peaks 2 ± i occur in Fig. 8d due to the collision at X 3. Finally, a new spatial cycle will begin at X 5 = X 1 + X. It is obvious that the oscillation feature of the detonation at a fixed location is dependent on the features of the triple points passing through a temporal cycle. Therefore, the temporal oscillation patters of the detonation in one spatial cycle are different at different locations, but their temporal periods are the same being.13, which is longer than that in section CD. sets of triple points. The triple points facing upstream are the same as those in section CD, and the triple point facing downstream also moves downstream along the dashed lines with the velocity of 1.7, which causes the periodical collision and reflection of the two sets of triple points. Simultaneously, the incoming flow has a velocity component tangential to the oblique detonation wave. Finally the numerical smoke-foil in section DE is the inclining fish scale patterns, as shown in Fig Cell structure of ODW Figure 9 shows variations of the instantaneous shape of the ODW front with time. The triple point traces are also drawn in this image, in which solid lines correspond to the triple points facing upstream, and dashed lines, only appearing in DE, correspond to the triple points facing downstream. In section CD, all the triple points move downstream along the solid lines with the velocity of.67, which causes the formation of a set of parallel straight lines in the numerical smokefoil record (Fig. 1). While in section DE there are two Fig. 9 Variations of the instantaneous shape of the ODW front with time

7 928 M.-Y. Gui, et al. References Fig. 1 Numerical smoke-foil record of ODW 4 Conclusion Based on Euler equation and one-step irreversible reaction model, an oblique detonation wave for a Mach 7 inlet flow over a long wedge of 3 turning angle was investigated numerically. The fifth-order WENO scheme was adopted to capture the shock wave and a fourth-order accurate, fourstage Runge Kutta scheme was used to treat the time integration. Both calculated and measured images show a satisfactory qualitative agreement. The oblique detonation wave front is composed of two different parts: OSW and ODW. If the wedge is long enough, three regions in the flow field behind ODW can be defined, i.e., the ZND model-like strcuture, single-sided triple point structure and dual-headed triple point strucuture. The ZND model-like strcuture is the most simple and the smooth straight. In the single-sided triple point structure, the triple point propagates downstream only with the same velocity and oscillation of the detonation wave front occurs. While in the dual-headed triple point strucuture, the detonation waves facing upstream and downstream propagate with different velocities, in which oscillation of the detonation front from the collision of the triple points is very complex. This oscillation has temporal and spatial periodicity. The numerical smoke-foil record from the maximum pressure along the ODW front shows different cell structures in different sections. 1 Lehr, H.F.: Experients in shock-induced combustion. Acta Astronaut 17, (1972) 2 Dabora, E.K., Desbordes, D., Guerraud, C., et al.: Oblique detonation at hypersonic velocities. Progress in Astronautics and Aeronautics 133, (1991) 3 Viguier, C., Figueira da Silva, L., Desbordes, D., et al.: Onset of oblique detonation waves: comparison between experimental and numerical results for hydrogen-air mixtures. Proc. Combust. Inst. 26, (1996) 4 Viguier, C., Gourara, A., Desbordes, D.: H 2 -air and CH 4 - air detonations and combustions behind oblique shock waves. Proc. Combust. Inst. 27, (1998) 5 Kamel, M.R., Morris, C.I., Stouklov, I.G., et al.: PLIF imaging of hypersonic reactive flow around blunt bodies. Proc. Combust. Inst. 26, (1996) 6 Morris, C.I., Kamel, M.R., Hanson, R.K.: Shock-induced combustion in high-speed wedge flows. Proc. Combust. Inst. 27, (1998) 7 Li, C., Kailasanath, K., Oran, E.S.: Detonation structures behind oblique shocks. Physics of Fluids 4, (1994) 8 Papalexandris, M.V.: A numerical study of wedge-induced detonations. Combust. Flame 12, (2) 9 Figueira da Silva, L.F., Deshaies, B.: Stabilization of an oblique detonation wave by a wedge: a parametric numerical study. Combust. Flame 121, (2) 1 Choi, J.Y., Kim, D.W., Jeung, I.S., et al.: Cell-like structure of unstable oblique detonation wave from high-resolution numerical simulation. Proc. Combust. Inst. 31, (27) 11 Choi, J.Y., Shin, E.J.R., Cho, D.R., et al.: Onset condition of oblique detonation wave cell structures. AIAA (28) 12 Wang, A.F., Zhao, W., Jiang, Z.L.: Cellular structure of oblique detonation and propagating of transverse wave. Explosion and Shock Wave 3(4), (21) 13 Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hypersonic conservation laws. ICASE REPORT (1997) 14 Yi, T.H., Wilson, D.R., Lu, F.K.: Numerical study of unsteady detonation wave propagation in a supersonic combustion chamber. In: Proc. of 25th International Symposium on Shock Waves. Paper No. 141 (24), Bangalore, India, 7 12 July, 25, OrientLongman

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