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Study on the effects of geometry on the initiation characteristics of the oblique detonation wave for hydrogen-air mixture
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Study on the effects of geometry on the initiation characteristics of the oblique detonation wave for hydrogen-air mixture Qiongyao Qin, Xiaobing Zhang* School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, PR China
article info
abstract
Article history:
The Oblique Detonation Wave Engine (ODWE) may act as a hypersonic propulsion system
Received 6 March 2019
operating at high Mach numbers, which is an important member in the family of
Received in revised form
Scramjet. Hydrogen is a promising fuel for Scramjet, which provides wider Mach number
22 April 2019
range and is environmentally friendly. The geometry of the engine greatly affects the
Accepted 25 April 2019
performance of the ODWE using hydrogen fuel. This investigation focuses on a novel
Available online 21 May 2019
wedge proposed recently, which may be utilized in scramjet engines. The wedge consists of two sub-wedges and a step. This research focuses on how the geometry of the wedge
Keywords:
affects the initiation characteristics of the oblique detonation. Simulations are conducted
Detonation
on basis of Euler equations and a 9-species and 19-reactions mechanism. It is found that
Geometry
a larger leading wedge angle leads to a shorter initiation length. A larger step angle in-
Initiation
duces a longer initiation length. Few effects are observed on the initiation characteristics
Combustion
for the current range of depth. The streamline surface at the rear of the step weakens the
Hydrogen
rear shock wave and induces a longer initiation length. The streamline surface at the tip of the step begins to take effect when the initiation position is away from the step. This research provides basis for understanding the performance of the oblique detonation wave under different geometries and provides theoretical basis for scramjet engine design. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction The Oblique Detonation Wave Engine (ODWE) may act as a hypersonic propulsion system operating at high Mach numbers [1]. Hydrogen is an ideal fuel for the Oblique Detonation Wave Engine. Hydrogen fuel provides a higher specific impulse than hydrocarbon fuels [2,3]. The higher cooling capacity of hydrogen and its faster reactions are required for higher Mach numbers [3]. It is envisioned that a hydrogen
economy will drive the propulsion system revolution towards the ultimate goal of silent aircrafts with zero harmful emissions [4]. Besides, Scramjet engines using hydrogen fuel and variable geometry do offer wider Mach number range capabilities [3]. Thus, it is of great importance to investigate the effects of geometry on the performance of the ODWE for hydrogen-air mixture. Although combustion control is widely used in traditional combustors [5], it is difficult to control the combustion in the ODWE. The reason is the high pressure, high temperature and hypersonic flow velocity ambient for
* Corresponding author. E-mail addresses: [email protected] (Q. Qin), [email protected] (X. Zhang). https://doi.org/10.1016/j.ijhydene.2019.04.248 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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Fig. 1 e Computational model.
the initiation of the oblique detonation wave. In our previous research [6], we designed a new wedge for controlling the detonation. The wedge contains two sub-wedges and a step, as shown in Fig. 1(a). The feasibility of the new wedge is verified in that exploratory research. The detonation is found to be hold behind the step. Since that the step and leading wedge play a vital role, the geometries of them are believed to affect the performance of the detonation. The current research is focusing on the effects of the leading wedge angle, step angle, step depth and streamline surface on the initiation characteristics of the oblique detonation wave for hydrogenair mixture. Nowadays, detonation phenomenon is being widely studied [7e11]. As a vital member in the detonation propulsion family [12], the oblique detonation has attracted more and more attentions. Lots of experimental researches are available about the initiation features [13e18]. The initiation of the detonation for hydrogen-air mixture and methane-air mixture was illustrated by Viguier et al. [13,14]. Many experiments about the oblique detonation waves generated by the hypersonic projectiles were performed [16e18]. In these experiments, the projectiles flew through the combustible gas at hypersonic speed and ignited the gas. The experimental results provided fundamental reference for the application of the ODWE. Besides, the bent tubes were utilized to generate detonations by Kasahara et al. [15]. This type of detonation may be put into application in hypersonic propulsion systems.
Table 1 e List of symbols. Nomenclature Q E F S u v p
solution vector Flux term Flux term source term velocity in x- direction velocity in y- direction pressure
T temperature E total energy per unit mass net generation rate of species u_ i r density Subscripts i species i N total of species
Lots of fundamental and valuable numerical investigations were conducted, which revealed the initiation features [19e31]. The impacts of the Mach number and pressure of incoming flow were analyzed by Teng et al. [25]. Different induction zone structures were observed under different incoming flow Mach numbers [27]. Besides, light was thrown on the instability features on the oblique detonation wave surface [32e34]. The previously mentioned detonation waves are mostly induced by semi-infinite wedges. The oblique detonation waves induced by different wedges were also investigated widely [1,6,35e39]. The oblique detonation waves generated by confined wedge were numerically studied by Lu
Table 2 e Reaction mechanism for hydrogen-air mixture.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)
Reaction
A
n
E
H2 þ O2 ¼ HO2 þ H H þ O2 ¼ OH þ O O þ H2 ¼ OH þ H OH þ H2 ¼ H þ H2O OH þ OH ¼ O þ H2O H þ OH þ M ¼ H2O þ M H þ H þ M ¼ H2 þ M H þ O þ M ¼ OH þ M H þ O2 þ M ¼ HO2 þ M O þ O þ M ¼ O2 þ M HO2 þ H ¼ OH þ OH HO2 þ H ¼ H2O þ O HO2 þ O ¼ O2 þ OH HO2 þ OH ¼ H2O þ O2 HO2 þ HO2 ¼ H2O2 þ O2 H þ H2O2 ¼ H2 þ HO2 O þ H2O2 ¼ OH þ HO2 OH þ H2O2 ¼ H2O þ HO2 H2O2 þ M ¼ OH þ OH þ M
1.00E14 2.60E14 1.80E10 2.20E13 6.30E12 2.20E22 6.40E17 6.00E16 2.10E15 6.00E13 1.40E14 1.00E13 1.50E13 8.00E12 2.00E12 1.40E12 1.40E13 6.10E12 1.20E17
0.0 0.0 1.0 0.0 0.0 2.0 1.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
56,000 16,800 8900 5150 1090 0.0 0.0 0.0 1000 1800 1080 1080 950 0.0 0.0 3600 6400 1430 45,500
The Arrhenius formula: k ¼ ATn eE=RT . Units are: seconds, moles, centimeters, calories and Kelvin. Third-body effects are: (6) H2O ¼ 6.0; (7) H20 ¼ 6.0, H2 ¼ 2.0; (8) H2O ¼ 5.0; (9) H2O ¼ 16.0, H2 ¼ 2.0; (19) H2O ¼ 15.0.
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Table 3 e Computational cases. Parameters Leading wedge Step depth Step angle Streamline surface
OA (mm)
Ma
h (mm)
q1
q2
q3
l (mm)
4 4 7 8
6.8 7 7 7
1 0.5, 1.5, 2 1.8 2.5
20 ~26 25 25 25
0 0 0 ~5 0
25 25 25 25
0, 1, 1.5, 2
Parametric variations are indicated in bold type.
Fig. 2 e (a) Pressure contours of the leading wedge angle of 26 and (b) temperature contours of the step depth of 2 mm.
Fig. 3 e Temperature profiles along the outlet boundary in y direction (a) leading wedge angle of 26 and (b) step depth of 2 mm.
et al. [35,36]. In these investigations, the wedges were placed in a channel. Interesting phenomenon was also observed on finite-length wedge [1,37e39]. Fang et al. numerically researched the influence of the expansion wave on the initiation over a finite-length wedge [1]. The hysteresis phenomenon of the oblique detonation wave on a finite-length wedge was observed by Liu et al. [40]. Qin & Zhang designed a new wedge for controlling the detonation and the effects of leading wedge length, Mach number and the rear wedge on the oblique detonation wave were analyzed [6]. However, the influence of the leading wedge angle, step depth, step angle and
streamline surface are not clear yet. The focus of this research is to investigate the influence of the geometry variation on the performance of the oblique detonation wave in hydrogen-air mixture. Parametric studies will be conducted to reveal the effects of these geometries. In current research, the initiation characteristics of the oblique detonation wave under different geometries will be evaluated based on the computational results. This research will discuss the mechanism of what makes the oblique detonation wave performs differently in hydrogen-air mixture. This study will reveal the effects of the
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Fig. 4 e Temperature contours of different leading wedge angles.
geometry on the initiation characteristics of the oblique detonation wave and provides basis for application of the novel wedge and scramjet engine design.
Numerical methods
3 3 3 3 2 2 2 r1 r1 u r1 v u_ 1 7 7 6… 7 6… 6… 6… 7 7 7 7 6 7 6 6 6 7 7 6 rN 7 6 rN u 6 rN v 6 u_ N 7 7 7 7 7 6 6 6 Q ¼6 7 E ¼ 6 2 F¼6 S¼6 7 7 6 0 7 7 7 6 ru 7 6 ru þ p 7 6 ruv 6 5 4 rv 5 4 ruv 4 rv2 þ p 5 40 5 rE uðrE þ pÞ 0 vðrE þ pÞ 2
(2)
Governing equations The total energy is given by: The governing equations in current research are Euler equations: vQ vE vF þ þ ¼S vt vx vy
(1)
p 1 E ¼ h þ u 2 þ v2 r 2
(3)
The symbols are listed in Table 1. The equation of state PN isp ¼ i¼1 Rðri =Mw; i ÞT. The spatial discretization is performed using the Finite Volume Method. The convective
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Fig. 5 e Initiation characteristics of different leading wedge angles.
Fig. 6 e Temperature contours of different step angles. fluxes are handled by the AUSM þ scheme, i.e., Advection Upstream Splitting Method. The reaction rates are computed from the Arrhenius equation. A hydrogen-air
reaction mechanism consisting of 9 species and 19 reactions (see Table 2) [41e43] is taken into consideration. This reaction mechanism has been validated in many
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Fig. 7 e Initiation characteristics of different step angles. investigations [6,39,43] and no more validation research will be conducted in current study.
valuable reference for understanding the features of detonation in ODWE.
Computational details
Results and discussions The previously proposed wedge is shown in Fig. 1(a). Different geometries will be simulated based on this model. The geometries that are going to be studies in current research are as follows: (1) Leading wedge angle: the angle indicated by q1 in Fig. 1(b). (2) Step angle: the angle indicated by q2 in Fig. 1(b), which is the angle between the step surface and the horizontal line. (3) Step depth: the vertical distance from the tip of the step (vertex A) to the rear wedge surface (BD), as shown in Fig. 1(b). (4) Streamline surface: the streamline surface is implemented by replacing the angles (OAB and ABD) with arcs, as shown in Fig. 1(c). The arc is part of a circle, which is tangent to the angle's sides. A parameter, l, is defined to specifying different types of streamline surfaces. All the parameters of different cases are listed in Table 3. OA is the leading wedge. Aircrafts equipped ODWE are supposed to operate on high altitude 25e35 km, and a wide range of Ma can be covered theoretically [28]. The free stream will go through a deflection before entering the combustor [28,44]. Besides, the realistic incoming flow involves complex fuel-air mixing. Although it agree better with the realistic flight by considering these aspects, it is not the focus of current research. The current research is aiming at revealing the initiation characteristics of oblique detonation wave under different geometries for hydrogen-air mixture. The atmospheric pressure and room temperature (100,000 Pa and 300 K) is chosen for the inlet conditions. The hydrogen and air are supposed to be well premixed and a stoichiometric hydrogen-air mixture (2H2 þ O2 þ 3.76 N2) is used as the free stream. The realistic incoming flow conditions will be considered in our future study. The oblique detonation wave formed under these initiation conditions, which have been adopted in many basic researches for ODWE [1,21,27,29], is believed to provide
Grid convergence test Fig. 2(a) gives the pressure contours of the case with a leading wedge angle of 26 . Fig. 2(b) gives the temperature contours of the case with a step depth of 2 mm. The grid resolution of the upper half is 25 m and the lower one is 50 mm. Slight difference is observed behind the triple points. The difference is the reflected shock waves generated by the shock wave/detonation wave interactions. The high resolution is better for capturing these shock waves, but this is not the aim of current research. Fig. 3 gives the temperature profiles at the outlet boundary. It shows that the computed detonation fronts are the same for three resolutions. The relative deviation of the peak pressure near the detonation front at the outlet of the case with a leading wedge angle of 26 is 3.73%. The relative deviation of the peak temperature near the detonation front at the outlet of the case with a step depth of 2 mm is 0.11%. Besides, the initiation lengths computed from three resolutions are the
Fig. 8 e Density contours with streamlines (q2 ¼ 5 ).
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Fig. 9 e Temperature contours of different step depths.
Fig. 10 e Initiation characteristics of different step depths.
same. The grid convergence test indicates that the low grid resolution (50 mm) is sufficient for the current research.
Effects of the leading wedge Before ignition of the detonation, a cold flow with chemical reactions suppressed is simulated. Two shock waves are formed on the leading wedge and rear wedge. The high temperature and high pressure behind the rear shock wave provides a pool for self-ignition of the oblique detonation wave. There are two typical transition types from the oblique shock wave to the oblique detonation wave [45], i.e., smooth one and abrupt one. In the smooth one, pressure waves caused by combustion strength the oblique shock wave gradually. While in an abrupt one, the pressure waves coalesce and form a combustion region of high pressure before interact with the shock wave. In current research, all the transitions are abrupt. The temperature contours computed under different leading wedge angles are shown in Fig. 4. The results of the case with a leading wedge angle of 25 is not shown currently because it can be found in our previous research [6]. The initiation position is approaching the step with increased leading wedge angle. The variation of the initiation length with the leading wedge angle is shown in Fig. 5. The initiation length is the direct initiation length defined in our previous work [6], which is the distance from the rear of the step to the initiation location of the detonation. It indicates that the initiation zone is shortening with increased leading wedge angle. A bigger leading wedge angle leads to larger flow deflection angle, which corresponds to a higher intensity of leading shock wave. Many researches have confirmed that higher temperature leads to a shorter initiation length [6,13,25]. This is
attributed to the nature of the combustion of the hydrogen-air mixture. The induction time of the hydrogen-air mixture strongly decreases with increase temperature [13]. Besides, the value of the initiation length can be estimated as the postshock velocity of the mixture multiplies the induction time. Thus, evaluating the post-shock temperature under different geometries can interpret the difference of the initiation characteristics of the oblique detonation wave. Fig. 5(b) shows curves of the temperature along the line withDy ¼ 0:75 mmover the step. The line is shown in Fig. 4(f). The temperature profiles cross two shock waves and an expansion wave between them. However, they do not cross the reaction zone. Twice increases and once decrease between them are observed. The first increase is caused by the leading shock wave and the second increase is induced by the rear shock wave. The decrease is attributed to the expansion wave induced by the tip of the step. The final temperature pointed out by a circle in Fig. 5(b) represents the status of the hydrogen-air mixture in the induction zone. It indicates that higher temperature in the induction zone is obtained with increased leading wedge angle. The source of the temperature difference is the leading shock wave, as shown in Fig. 5(b).
Effects of the step This section contains two parts: the effects of the step angle and the effects of the step depth. The results of different step angles are shown in Fig. 6. The oblique detonation wave is brought to the outlet with increased step angle. The corresponding initiation length is given in Fig. 7(a). A larger step angle induces a longer initiation length. The temperature curves along y ¼ 4.5 mm is plotted in Fig. 7(b). Unlike the temperature curves
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Fig. 11 e Temperature contours of different types of streamline surfaces.
Fig. 12 e Initiation characteristics of different types of streamline surfaces.
in Fig. 5(b), the curves coincide at the parts of the leading shock wave and the expansion wave. Although the temperature of 5 is higher than that of 0 at the end of the expansion wave, its final state after the rear shock is defeated. The density contours with streamlines of the case with a step angle of 5 is given in Fig. 8. It
is observed that the flow direction in the vicinity of the step is parallel to the surface, which means that the flow deflection angle is reduced to 20 . The decreased flow deflection angle weakens the intensity of the rear shock wave, leading to lower temperature after the rear shock.
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Fig. 13 e Temperature contours of different types of streamline surfaces.
Fig. 14 e Temperature profiles.
Fig. 9 shows the flow fields of different step depths. A visual resemblance is observed for the reaction zone. The difference exists in the expansion process over the step, which is attributed to the step depth. However, the difference of the expansion process produces few effects on the initiation characteristics. The detonation waves are triggered at the same position on the rear wedge. Fig. 10(a) exhibits the initiation lengths of different step depths. Only slight difference is observed with the variation of the depth. The temperature curves along y ¼ 3.0 mm interprets this phenomenon (Fig. 10(b)). The expansion processes end at different positions for different depths, but the peak temperatures after the rear shock wave are nearly the same. Besides, the case in Fig. 6(a) also proves that the step depth produces few effects on the initiation length. The case with a depth of 1 mm has been simulated in our previous work [6] and the value of the initiation length is very close to the value in current research.
Effects of the streamline surface A bigger value of l represents that more of the angle is replaced by arc. The behavior of the oblique detonation wave with the
variation of l is exhibited in Fig. 11. Both the tip and the rear of the step have streamline surfaces for these cases. The oblique detonation wave is observed moving towards the outlet as the value of l increases. The initiation length increases with l, as shown in Fig. 12(a). Fig. 12(b) shows the temperature curves along y ¼ 4.5 mm. The hydrogen-air mixture undergoes different types of expansion processes, but the temperature at the end of the expansion wave is the same. What makes a big difference is the rear shock wave. A larger value of l corresponds to weaker rear shock wave intensity, leading to lower temperature. By the way, the case with none streamline surface also acts as an evidence for the effects of the step depth. The step depth is 2.5 mm and the initiation is around 4.35 mm. The results are compared with two control groups. One is the group with only the tip of the step has streamline surface (named, ‘tip’ group) and the other is the group with only the rear of the step has streamline surface (named, ‘rear’ group). The flow fields are absolutely the same when only the tip has streamline surface (Fig. 13(a)e(c)). However, obvious difference is observed when only the rear has streamline surface (see Fig. 13(d)e(f)). The initiation characteristics are shown in Fig. 12(c) and (d). Slight decrease of the initiation length is found for the ‘tip’ group. The initiation length increases with the variation of l for the ‘rear’ group. The reason can be interpreted by the temperature. The temperature curves along y ¼ 4.5 mm coincide except the expansion period for the ‘tip’ group, as shown in Fig. 12(d). The temperature after the rear shock is the same, leading to a same initiation length. Situation is different for the ‘rear’ group. The rear shock wave is weakened by the streamline surface at the rear of the step, bringing the initiation close to the outlet. By comparing Fig. 12(a) and (c), it is noticed that the initiation length for the case with both ends have streamlines surface (l ¼ 2 mm) is 12.4 mm. The initiation length for the case with the rear has streamline surface (l ¼ 2 mm) is 8.8 mm. The addition of the streamline surface at the tip seems to affect the initiation greatly, which is against the conclusion above. The reason is obvious by analyzing the temperature curves along y ¼ 6.4 mm, where it is closer to the initiation location. The lines, y ¼ 6:4 mm andy ¼ 4:5 mm, are
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indicated by the white lines in Fig. 11(d). The streamline surface at the tip of the step weakens the rear parts of the two shock waves (see Fig. 14) However, this happens only when the initiation position is away from the step. More cases are necessary to find the critical point where the streamline surface at the tip begins to take effect.
Conclusion Numerical study is conducted to investigate the effects of the leading wedge angle, step angle, step depth and streamline surface on the initiation characteristics of the oblique detonation wave for hydrogen-air mixture. The oblique detonation wave formed in the hydrogen-air mixture performs differently with the geometry variation of the wedge. It shows that these geometries affect the initiation length through regulating the compression and expansion processes. A bigger leading wedge angle leads to higher leading shock wave intensity, inducing a shorter initiation length. The rear shock wave intensity reduces with increased step angle, bringing the initiation position close to the outlet. Step depth has few effects on the initiation length in current range. Streamline surface at the rear of the step dominates the initiation length when the detonation is close to the step. Streamline surface at the rear of the step weakens the rear shock wave and induces a longer initiation length. Streamline surface at the tip of the step begins to take effect when the initiation position is away from the step. This study reveals the effects of the geometry on the initiation characteristics for hydrogen-air mixture and provides basis for application of the novel wedge and scramjet engine design.
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[38] Liu Y, Han X, Yao S, Wang J. A numerical investigation of the prompt oblique detonation wave sustained by a finite-length wedge. Shock Waves 2016;26(6):729e39. [39] Choi J, Shin EJ, Jeung I. Unstable combustion induced by oblique shock waves at the non-attaching condition of the oblique detonation wave. Proc Combust Inst 2009;32:2387e96. [40] Liu Y, Wang L, Xiao B, Yan Z, Wang C. Hysteresis phenomenon of the oblique detonation wave. Combust Flame 2018;192:170e9. [41] Jachimowski CJ. An analytical study of the hydrogen-air reaction mechanism with application to scramjet combustion. NASA Tech Pap 1988:2791. [42] Yuan L, Tang T. Resolving the shock-induced combustion by an adaptive mesh redistribution method. J Comput Phys 2007;224:587e600. [43] Choi J, Jeung I, Yoon Y. Computational fluid dynamics algorithms for unsteady shock-induced combustion, part 1: Validation. AIAA J 2000;38:1179e87. [44] Zhang Y, Gong J, Wang T. Numerical study on initiation of oblique detonations in hydrogen-air mixtures with various equivalence ratios. Aero Sci Technol 2016;49:130e4. [45] Da Silva LFF, Deshaies B. Stabilization of an oblique detonation wave by a wedge: a parametric numerical study. Combust Flame 2000;121:152e66.
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Study on the effects of geometry on the initiation characteristics of the oblique detonation wave for hydrogen-air mixture Qiongyao Qin, Xiaobing Zhang* School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, PR China
article info
abstract
Article history:
The Oblique Detonation Wave Engine (ODWE) may act as a hypersonic propulsion system
Received 6 March 2019
operating at high Mach numbers, which is an important member in the family of
Received in revised form
Scramjet. Hydrogen is a promising fuel for Scramjet, which provides wider Mach number
22 April 2019
range and is environmentally friendly. The geometry of the engine greatly affects the
Accepted 25 April 2019
performance of the ODWE using hydrogen fuel. This investigation focuses on a novel
Available online 21 May 2019
wedge proposed recently, which may be utilized in scramjet engines. The wedge consists of two sub-wedges and a step. This research focuses on how the geometry of the wedge
Keywords:
affects the initiation characteristics of the oblique detonation. Simulations are conducted
Detonation
on basis of Euler equations and a 9-species and 19-reactions mechanism. It is found that
Geometry
a larger leading wedge angle leads to a shorter initiation length. A larger step angle in-
Initiation
duces a longer initiation length. Few effects are observed on the initiation characteristics
Combustion
for the current range of depth. The streamline surface at the rear of the step weakens the
Hydrogen
rear shock wave and induces a longer initiation length. The streamline surface at the tip of the step begins to take effect when the initiation position is away from the step. This research provides basis for understanding the performance of the oblique detonation wave under different geometries and provides theoretical basis for scramjet engine design. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction The Oblique Detonation Wave Engine (ODWE) may act as a hypersonic propulsion system operating at high Mach numbers [1]. Hydrogen is an ideal fuel for the Oblique Detonation Wave Engine. Hydrogen fuel provides a higher specific impulse than hydrocarbon fuels [2,3]. The higher cooling capacity of hydrogen and its faster reactions are required for higher Mach numbers [3]. It is envisioned that a hydrogen
economy will drive the propulsion system revolution towards the ultimate goal of silent aircrafts with zero harmful emissions [4]. Besides, Scramjet engines using hydrogen fuel and variable geometry do offer wider Mach number range capabilities [3]. Thus, it is of great importance to investigate the effects of geometry on the performance of the ODWE for hydrogen-air mixture. Although combustion control is widely used in traditional combustors [5], it is difficult to control the combustion in the ODWE. The reason is the high pressure, high temperature and hypersonic flow velocity ambient for
* Corresponding author. E-mail addresses: [email protected] (Q. Qin), [email protected] (X. Zhang). https://doi.org/10.1016/j.ijhydene.2019.04.248 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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Fig. 1 e Computational model.
the initiation of the oblique detonation wave. In our previous research [6], we designed a new wedge for controlling the detonation. The wedge contains two sub-wedges and a step, as shown in Fig. 1(a). The feasibility of the new wedge is verified in that exploratory research. The detonation is found to be hold behind the step. Since that the step and leading wedge play a vital role, the geometries of them are believed to affect the performance of the detonation. The current research is focusing on the effects of the leading wedge angle, step angle, step depth and streamline surface on the initiation characteristics of the oblique detonation wave for hydrogenair mixture. Nowadays, detonation phenomenon is being widely studied [7e11]. As a vital member in the detonation propulsion family [12], the oblique detonation has attracted more and more attentions. Lots of experimental researches are available about the initiation features [13e18]. The initiation of the detonation for hydrogen-air mixture and methane-air mixture was illustrated by Viguier et al. [13,14]. Many experiments about the oblique detonation waves generated by the hypersonic projectiles were performed [16e18]. In these experiments, the projectiles flew through the combustible gas at hypersonic speed and ignited the gas. The experimental results provided fundamental reference for the application of the ODWE. Besides, the bent tubes were utilized to generate detonations by Kasahara et al. [15]. This type of detonation may be put into application in hypersonic propulsion systems.
Table 1 e List of symbols. Nomenclature Q E F S u v p
solution vector Flux term Flux term source term velocity in x- direction velocity in y- direction pressure
T temperature E total energy per unit mass net generation rate of species u_ i r density Subscripts i species i N total of species
Lots of fundamental and valuable numerical investigations were conducted, which revealed the initiation features [19e31]. The impacts of the Mach number and pressure of incoming flow were analyzed by Teng et al. [25]. Different induction zone structures were observed under different incoming flow Mach numbers [27]. Besides, light was thrown on the instability features on the oblique detonation wave surface [32e34]. The previously mentioned detonation waves are mostly induced by semi-infinite wedges. The oblique detonation waves induced by different wedges were also investigated widely [1,6,35e39]. The oblique detonation waves generated by confined wedge were numerically studied by Lu
Table 2 e Reaction mechanism for hydrogen-air mixture.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)
Reaction
A
n
E
H2 þ O2 ¼ HO2 þ H H þ O2 ¼ OH þ O O þ H2 ¼ OH þ H OH þ H2 ¼ H þ H2O OH þ OH ¼ O þ H2O H þ OH þ M ¼ H2O þ M H þ H þ M ¼ H2 þ M H þ O þ M ¼ OH þ M H þ O2 þ M ¼ HO2 þ M O þ O þ M ¼ O2 þ M HO2 þ H ¼ OH þ OH HO2 þ H ¼ H2O þ O HO2 þ O ¼ O2 þ OH HO2 þ OH ¼ H2O þ O2 HO2 þ HO2 ¼ H2O2 þ O2 H þ H2O2 ¼ H2 þ HO2 O þ H2O2 ¼ OH þ HO2 OH þ H2O2 ¼ H2O þ HO2 H2O2 þ M ¼ OH þ OH þ M
1.00E14 2.60E14 1.80E10 2.20E13 6.30E12 2.20E22 6.40E17 6.00E16 2.10E15 6.00E13 1.40E14 1.00E13 1.50E13 8.00E12 2.00E12 1.40E12 1.40E13 6.10E12 1.20E17
0.0 0.0 1.0 0.0 0.0 2.0 1.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
56,000 16,800 8900 5150 1090 0.0 0.0 0.0 1000 1800 1080 1080 950 0.0 0.0 3600 6400 1430 45,500
The Arrhenius formula: k ¼ ATn eE=RT . Units are: seconds, moles, centimeters, calories and Kelvin. Third-body effects are: (6) H2O ¼ 6.0; (7) H20 ¼ 6.0, H2 ¼ 2.0; (8) H2O ¼ 5.0; (9) H2O ¼ 16.0, H2 ¼ 2.0; (19) H2O ¼ 15.0.
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Table 3 e Computational cases. Parameters Leading wedge Step depth Step angle Streamline surface
OA (mm)
Ma
h (mm)
q1
q2
q3
l (mm)
4 4 7 8
6.8 7 7 7
1 0.5, 1.5, 2 1.8 2.5
20 ~26 25 25 25
0 0 0 ~5 0
25 25 25 25
0, 1, 1.5, 2
Parametric variations are indicated in bold type.
Fig. 2 e (a) Pressure contours of the leading wedge angle of 26 and (b) temperature contours of the step depth of 2 mm.
Fig. 3 e Temperature profiles along the outlet boundary in y direction (a) leading wedge angle of 26 and (b) step depth of 2 mm.
et al. [35,36]. In these investigations, the wedges were placed in a channel. Interesting phenomenon was also observed on finite-length wedge [1,37e39]. Fang et al. numerically researched the influence of the expansion wave on the initiation over a finite-length wedge [1]. The hysteresis phenomenon of the oblique detonation wave on a finite-length wedge was observed by Liu et al. [40]. Qin & Zhang designed a new wedge for controlling the detonation and the effects of leading wedge length, Mach number and the rear wedge on the oblique detonation wave were analyzed [6]. However, the influence of the leading wedge angle, step depth, step angle and
streamline surface are not clear yet. The focus of this research is to investigate the influence of the geometry variation on the performance of the oblique detonation wave in hydrogen-air mixture. Parametric studies will be conducted to reveal the effects of these geometries. In current research, the initiation characteristics of the oblique detonation wave under different geometries will be evaluated based on the computational results. This research will discuss the mechanism of what makes the oblique detonation wave performs differently in hydrogen-air mixture. This study will reveal the effects of the
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Fig. 4 e Temperature contours of different leading wedge angles.
geometry on the initiation characteristics of the oblique detonation wave and provides basis for application of the novel wedge and scramjet engine design.
Numerical methods
3 3 3 3 2 2 2 r1 r1 u r1 v u_ 1 7 7 6… 7 6… 6… 6… 7 7 7 7 6 7 6 6 6 7 7 6 rN 7 6 rN u 6 rN v 6 u_ N 7 7 7 7 7 6 6 6 Q ¼6 7 E ¼ 6 2 F¼6 S¼6 7 7 6 0 7 7 7 6 ru 7 6 ru þ p 7 6 ruv 6 5 4 rv 5 4 ruv 4 rv2 þ p 5 40 5 rE uðrE þ pÞ 0 vðrE þ pÞ 2
(2)
Governing equations The total energy is given by: The governing equations in current research are Euler equations: vQ vE vF þ þ ¼S vt vx vy
(1)
p 1 E ¼ h þ u 2 þ v2 r 2
(3)
The symbols are listed in Table 1. The equation of state PN isp ¼ i¼1 Rðri =Mw; i ÞT. The spatial discretization is performed using the Finite Volume Method. The convective
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Fig. 5 e Initiation characteristics of different leading wedge angles.
Fig. 6 e Temperature contours of different step angles. fluxes are handled by the AUSM þ scheme, i.e., Advection Upstream Splitting Method. The reaction rates are computed from the Arrhenius equation. A hydrogen-air
reaction mechanism consisting of 9 species and 19 reactions (see Table 2) [41e43] is taken into consideration. This reaction mechanism has been validated in many
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Fig. 7 e Initiation characteristics of different step angles. investigations [6,39,43] and no more validation research will be conducted in current study.
valuable reference for understanding the features of detonation in ODWE.
Computational details
Results and discussions The previously proposed wedge is shown in Fig. 1(a). Different geometries will be simulated based on this model. The geometries that are going to be studies in current research are as follows: (1) Leading wedge angle: the angle indicated by q1 in Fig. 1(b). (2) Step angle: the angle indicated by q2 in Fig. 1(b), which is the angle between the step surface and the horizontal line. (3) Step depth: the vertical distance from the tip of the step (vertex A) to the rear wedge surface (BD), as shown in Fig. 1(b). (4) Streamline surface: the streamline surface is implemented by replacing the angles (OAB and ABD) with arcs, as shown in Fig. 1(c). The arc is part of a circle, which is tangent to the angle's sides. A parameter, l, is defined to specifying different types of streamline surfaces. All the parameters of different cases are listed in Table 3. OA is the leading wedge. Aircrafts equipped ODWE are supposed to operate on high altitude 25e35 km, and a wide range of Ma can be covered theoretically [28]. The free stream will go through a deflection before entering the combustor [28,44]. Besides, the realistic incoming flow involves complex fuel-air mixing. Although it agree better with the realistic flight by considering these aspects, it is not the focus of current research. The current research is aiming at revealing the initiation characteristics of oblique detonation wave under different geometries for hydrogen-air mixture. The atmospheric pressure and room temperature (100,000 Pa and 300 K) is chosen for the inlet conditions. The hydrogen and air are supposed to be well premixed and a stoichiometric hydrogen-air mixture (2H2 þ O2 þ 3.76 N2) is used as the free stream. The realistic incoming flow conditions will be considered in our future study. The oblique detonation wave formed under these initiation conditions, which have been adopted in many basic researches for ODWE [1,21,27,29], is believed to provide
Grid convergence test Fig. 2(a) gives the pressure contours of the case with a leading wedge angle of 26 . Fig. 2(b) gives the temperature contours of the case with a step depth of 2 mm. The grid resolution of the upper half is 25 m and the lower one is 50 mm. Slight difference is observed behind the triple points. The difference is the reflected shock waves generated by the shock wave/detonation wave interactions. The high resolution is better for capturing these shock waves, but this is not the aim of current research. Fig. 3 gives the temperature profiles at the outlet boundary. It shows that the computed detonation fronts are the same for three resolutions. The relative deviation of the peak pressure near the detonation front at the outlet of the case with a leading wedge angle of 26 is 3.73%. The relative deviation of the peak temperature near the detonation front at the outlet of the case with a step depth of 2 mm is 0.11%. Besides, the initiation lengths computed from three resolutions are the
Fig. 8 e Density contours with streamlines (q2 ¼ 5 ).
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Fig. 9 e Temperature contours of different step depths.
Fig. 10 e Initiation characteristics of different step depths.
same. The grid convergence test indicates that the low grid resolution (50 mm) is sufficient for the current research.
Effects of the leading wedge Before ignition of the detonation, a cold flow with chemical reactions suppressed is simulated. Two shock waves are formed on the leading wedge and rear wedge. The high temperature and high pressure behind the rear shock wave provides a pool for self-ignition of the oblique detonation wave. There are two typical transition types from the oblique shock wave to the oblique detonation wave [45], i.e., smooth one and abrupt one. In the smooth one, pressure waves caused by combustion strength the oblique shock wave gradually. While in an abrupt one, the pressure waves coalesce and form a combustion region of high pressure before interact with the shock wave. In current research, all the transitions are abrupt. The temperature contours computed under different leading wedge angles are shown in Fig. 4. The results of the case with a leading wedge angle of 25 is not shown currently because it can be found in our previous research [6]. The initiation position is approaching the step with increased leading wedge angle. The variation of the initiation length with the leading wedge angle is shown in Fig. 5. The initiation length is the direct initiation length defined in our previous work [6], which is the distance from the rear of the step to the initiation location of the detonation. It indicates that the initiation zone is shortening with increased leading wedge angle. A bigger leading wedge angle leads to larger flow deflection angle, which corresponds to a higher intensity of leading shock wave. Many researches have confirmed that higher temperature leads to a shorter initiation length [6,13,25]. This is
attributed to the nature of the combustion of the hydrogen-air mixture. The induction time of the hydrogen-air mixture strongly decreases with increase temperature [13]. Besides, the value of the initiation length can be estimated as the postshock velocity of the mixture multiplies the induction time. Thus, evaluating the post-shock temperature under different geometries can interpret the difference of the initiation characteristics of the oblique detonation wave. Fig. 5(b) shows curves of the temperature along the line withDy ¼ 0:75 mmover the step. The line is shown in Fig. 4(f). The temperature profiles cross two shock waves and an expansion wave between them. However, they do not cross the reaction zone. Twice increases and once decrease between them are observed. The first increase is caused by the leading shock wave and the second increase is induced by the rear shock wave. The decrease is attributed to the expansion wave induced by the tip of the step. The final temperature pointed out by a circle in Fig. 5(b) represents the status of the hydrogen-air mixture in the induction zone. It indicates that higher temperature in the induction zone is obtained with increased leading wedge angle. The source of the temperature difference is the leading shock wave, as shown in Fig. 5(b).
Effects of the step This section contains two parts: the effects of the step angle and the effects of the step depth. The results of different step angles are shown in Fig. 6. The oblique detonation wave is brought to the outlet with increased step angle. The corresponding initiation length is given in Fig. 7(a). A larger step angle induces a longer initiation length. The temperature curves along y ¼ 4.5 mm is plotted in Fig. 7(b). Unlike the temperature curves
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Fig. 11 e Temperature contours of different types of streamline surfaces.
Fig. 12 e Initiation characteristics of different types of streamline surfaces.
in Fig. 5(b), the curves coincide at the parts of the leading shock wave and the expansion wave. Although the temperature of 5 is higher than that of 0 at the end of the expansion wave, its final state after the rear shock is defeated. The density contours with streamlines of the case with a step angle of 5 is given in Fig. 8. It
is observed that the flow direction in the vicinity of the step is parallel to the surface, which means that the flow deflection angle is reduced to 20 . The decreased flow deflection angle weakens the intensity of the rear shock wave, leading to lower temperature after the rear shock.
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Fig. 13 e Temperature contours of different types of streamline surfaces.
Fig. 14 e Temperature profiles.
Fig. 9 shows the flow fields of different step depths. A visual resemblance is observed for the reaction zone. The difference exists in the expansion process over the step, which is attributed to the step depth. However, the difference of the expansion process produces few effects on the initiation characteristics. The detonation waves are triggered at the same position on the rear wedge. Fig. 10(a) exhibits the initiation lengths of different step depths. Only slight difference is observed with the variation of the depth. The temperature curves along y ¼ 3.0 mm interprets this phenomenon (Fig. 10(b)). The expansion processes end at different positions for different depths, but the peak temperatures after the rear shock wave are nearly the same. Besides, the case in Fig. 6(a) also proves that the step depth produces few effects on the initiation length. The case with a depth of 1 mm has been simulated in our previous work [6] and the value of the initiation length is very close to the value in current research.
Effects of the streamline surface A bigger value of l represents that more of the angle is replaced by arc. The behavior of the oblique detonation wave with the
variation of l is exhibited in Fig. 11. Both the tip and the rear of the step have streamline surfaces for these cases. The oblique detonation wave is observed moving towards the outlet as the value of l increases. The initiation length increases with l, as shown in Fig. 12(a). Fig. 12(b) shows the temperature curves along y ¼ 4.5 mm. The hydrogen-air mixture undergoes different types of expansion processes, but the temperature at the end of the expansion wave is the same. What makes a big difference is the rear shock wave. A larger value of l corresponds to weaker rear shock wave intensity, leading to lower temperature. By the way, the case with none streamline surface also acts as an evidence for the effects of the step depth. The step depth is 2.5 mm and the initiation is around 4.35 mm. The results are compared with two control groups. One is the group with only the tip of the step has streamline surface (named, ‘tip’ group) and the other is the group with only the rear of the step has streamline surface (named, ‘rear’ group). The flow fields are absolutely the same when only the tip has streamline surface (Fig. 13(a)e(c)). However, obvious difference is observed when only the rear has streamline surface (see Fig. 13(d)e(f)). The initiation characteristics are shown in Fig. 12(c) and (d). Slight decrease of the initiation length is found for the ‘tip’ group. The initiation length increases with the variation of l for the ‘rear’ group. The reason can be interpreted by the temperature. The temperature curves along y ¼ 4.5 mm coincide except the expansion period for the ‘tip’ group, as shown in Fig. 12(d). The temperature after the rear shock is the same, leading to a same initiation length. Situation is different for the ‘rear’ group. The rear shock wave is weakened by the streamline surface at the rear of the step, bringing the initiation close to the outlet. By comparing Fig. 12(a) and (c), it is noticed that the initiation length for the case with both ends have streamlines surface (l ¼ 2 mm) is 12.4 mm. The initiation length for the case with the rear has streamline surface (l ¼ 2 mm) is 8.8 mm. The addition of the streamline surface at the tip seems to affect the initiation greatly, which is against the conclusion above. The reason is obvious by analyzing the temperature curves along y ¼ 6.4 mm, where it is closer to the initiation location. The lines, y ¼ 6:4 mm andy ¼ 4:5 mm, are
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indicated by the white lines in Fig. 11(d). The streamline surface at the tip of the step weakens the rear parts of the two shock waves (see Fig. 14) However, this happens only when the initiation position is away from the step. More cases are necessary to find the critical point where the streamline surface at the tip begins to take effect.
Conclusion Numerical study is conducted to investigate the effects of the leading wedge angle, step angle, step depth and streamline surface on the initiation characteristics of the oblique detonation wave for hydrogen-air mixture. The oblique detonation wave formed in the hydrogen-air mixture performs differently with the geometry variation of the wedge. It shows that these geometries affect the initiation length through regulating the compression and expansion processes. A bigger leading wedge angle leads to higher leading shock wave intensity, inducing a shorter initiation length. The rear shock wave intensity reduces with increased step angle, bringing the initiation position close to the outlet. Step depth has few effects on the initiation length in current range. Streamline surface at the rear of the step dominates the initiation length when the detonation is close to the step. Streamline surface at the rear of the step weakens the rear shock wave and induces a longer initiation length. Streamline surface at the tip of the step begins to take effect when the initiation position is away from the step. This study reveals the effects of the geometry on the initiation characteristics for hydrogen-air mixture and provides basis for application of the novel wedge and scramjet engine design.
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