Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber

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1 Acta Mech Sin (212) 28(1):66 72 DOI 117/s y RESEARCH PAPER Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber Xu-Dong Zhang Bao-Chun Fan Ming-Yue Gui Zhen-Hua Pan Gang Dong Received: 2 January 21 / Revised: 26 December 21 / Accepted: 22 June 211 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 212 Abstract Gaseous detonation propagating in a toroidal chamber was numerically studied for hydrogen/oxygen/nitrogen mixtures The numerical method used is based on the three-dimensional Euler equations with detailed finiterate chemistry The results show that the calculated streak picture is in qualitative agreement with the picture recorded by a high speed streak camera from published literature The three-dimensional flow field induced by a continuously rotating detonation was visualized and distinctive features of the rotating detonations were clearly depicted Owing to the unconfined character of detonation wavelet, a deficit of detonation parameters was observed Due to the effects of wall geometries, the strength of the outside detonation front is stronger than that of the inside portion The detonation thus propagates with a constant circular velocity Numerical simulation also shows three-dimensional rotating detonation structures, which display specific feature of the detonationshock combined wave Discrete burning gas pockets are formed due to instability of the discontinuity It is believed that the present study could give an insight into the interesting properties of the continuously rotating detonation, and is thus beneficial to the design of continuous detonation propulsion systems 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 University of Science and Technology (KFJJ9-13) X-D Zhang B-C Fan ( ) M-Y Gui Z-H Pan G Dong Science and Technology on Transient Physics Laboratory, Nanjing University of Science and Technology, 2194 Nanjing, China bcfan@mailnjusteducn G Dong State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, 181 Beijing, China Keywords Continuously rotating detonation Threedimensional flow field structure Numerical study Detonation parameters deficit Effects of wall geometries 1 Introduction Due to very high energy release rate, less pollution and wider flying Mach number, which can be achieved in detonations, scientists and engineers concentrate on possible practical applications of this phenomenon for many years And it can be predicted that detonation-based fuel-burning devices will gradually provide more cost-effective and compact engines for sub-orbital spacecraft and other high-speed vehicles For the transient characteristic of detonation waves, the key point of the detonation engine is how to make the detonation waves stay in the combustor long enough Currently, there are three conceptions of detonation-based engines [1] The first one is pulse detonation engine (PDE), in which the thrust can be gained through periodic highfrequency pulse detonation The second one is oblique detonation wave engine (ODWE), in which the detonation wave is relatively static in the combustor, and the engine flight velocity should be higher than the reacting mixture detonation velocity However, there are some inherent difficulties in the above two kinds of engines, such as high-frequency ignition and the terrible noise in PDE, steady hypersonic incoming flow and high startup velocity in ODWE etc So far, both of them are immersed in their own troubles The third one is rotating detonation engine (RDE), in which a steady detonation wave is rotating azimuthally in an annular chamber consisting of two coaxial cylinders, meanwhile, a combustible gas is axially injected from the head end and the combustion products are thrown away from the detonation front due to the centrifugal force and rarefaction wave, as shown in Fig 1 The main advantages of RDE are its continuous operation and high combustion rate

Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber 67 Q t where + F ξ + G η + H ζ ρ 1 = S, (1) ρ 1 Ū Q = ρ Ns ρu, F = ρ Ns Ū ρūu + pξ x, ρv ρūv

2 Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber 67 Q t where + F ξ + G η + H ζ ρ 1 = S, (1) ρ 1 Ū Q = ρ Ns ρu, F = ρ Ns Ū ρūu + pξ x, ρv ρūv + pξ y ρw E ρūw + pξ z Ū(p + E) Fig 1 Schematic of the flow field in a rotating detonation engine ρ 1 V ρ 1 W Such rotating detonations have been experimentally observed by Voitsekhovskii [2] as early as 1959, and then Mikhailov [3] analyzed the structure of the detonation wave In 198s, Bykovskii s [4] experiments verified the possibility of controlling rotating detonation wave by using gaseous and liquid propellant fuel and injecting the fuel at subsonic speed in varying cross-section and constant sections of an annular combustor In recent years, investigations on the detonation engine have become popular, related researches have been done in France, America, Russia, Japan etc, and meaningful results have been obtained [5 1] Due to the three-dimensional structure of rotating detonation waves, experimental observations are limited to the visualization of detonations and in addition, a great number of previous numerical investigations are two-dimensional, therefore the flow pattern of the RDE is not yet fully understood [11 14] In this paper, three-dimensional Euler equations with 9 species 19 step reactions of the H 2 -AIR mixture are solved with high-resolution Roe scheme and MPI parallel computing The numerical method is validated first by a comparison between calculated and measured streak pictures Then, according to the numerical simulation results, the structure features of a detonation-shock combined wave propagating in an annular combustion chamber are described and the threedimensional flow patterns of the flow field induced by a rotating detonation are discussed 2 Numerical method and physical model 21 Governing equation With the assumption of negligible viscous effect, heat conductivity, and diffusion effects, the flow field is modeled using the three-dimensional chemical non-equilibrium Euler equations with chemical reaction in a non-dimensional and generalized body-fitted coordinate system G = S = ρ Ns V ρ Vu + pη x ρ Vv + pη y ρ Vw + pη z V(p + E) ω 1 ω Ns,, H = ρ Ns W ρ Wu + pζ x ρ Wv + pζ y ρ Ww + pζ z W(p + E) where x, y and z are the spatial variables of the orthogonal coordinate system, the components of the velocity in the computational coordination in ξ, η and ζ directions are, Ū = uξ x +vξ y +wξ z, V = uη x +vη y +wη z, W = uζ x +vζ y +wζ z, u, v and w are the components of the velocity vector in the orthogonal coordinates ρ is the density of the mixture, Ns Ns ρ k ρ = ρ k, ρ k = ρy k ; P is the pressure, P = RT; E is W k the total energy per volume, E = ρ T, c v dt ρ(u2 + v 2 + Ns w 2 ) + ρ k h k () ρ k, W k and Y k denote the density, molar mass and mass fraction of species k, respectively ω k is the mass production rate of species k, I ( Ns Ns ) ω k = W k (γ ki γ ki ) k fi [X k ] γ ki k bi [X k ] γ ki, (2) i=1

68 X-D Zhang, et al where [X k ] is the molar concentration of species k The indices i and k denote the number of species and number of reactions, respectively γ ki and γ ki denote the stoichiometric

3 68 X-D Zhang, et al where [X k ] is the molar concentration of species k The indices i and k denote the number of species and number of reactions, respectively γ ki and γ ki denote the stoichiometric coefficients for the reactants and products, respectively The forward rate constant k fi and backward k bi are obtained from Arrhenius law A detailed chemical kinetic mechanism for a stoichiometric H 2 -O 2 gas mixture diluted by N 2 (2H 2 + O N 2 ) is adopted to describe the chemical reactions including 9 species (O 2, O, H 2, H, OH, H 2 O, HO 2, H 2 O 2, N 2 ) and 19 reversible elementary reactions implemented by CHEMKIN-II code [16] The reference values are as follows: the pressure P = MPa, the temperature T = K, the characteristic length L = 1 m 22 Numerical method A second-order wave propagation algorithm based on Roe scheme is used for the numerical fluxes [15] In chemical reaction source terms, the LSODE package based on Gear scheme is used [17] To isolate the stiff chemical source term, in this work the time integration is performed using the second-order additive semi-implicit Runge-Kutta scheme [18] The schematic diagram of the physical model is shown in Fig 2 The annular combustion chamber is mainly consisted of two coaxial cylinders with a height of 1L, and its inner wall (B) and outer wall (C) are circular cylindrical surfaces with diameters of 12L and 16L, respectively Fig 2 A schematic diagram of the computational domain a Physical domain; b Computational domain Boundary and initial conditions are crucial in this study At the combustion entrance, the head wall boundary is set up as follows When the pressure of the head wall is higher than the reservoir pressure, the boundary is considered nonslip, adiabatic and non-catalytic, the fuel injection velocity is set to zero Otherwise, when the pressure is lower than the choking value, the injection velocity is given by the choking condition, and when the pressure is between the choking and reservoir ones, the injection velocity is calculated by the isentropic expansion based on the reservoir condition At the combustion exit, the outflow condition is employed The left and right boundaries of the computational domain in Fig 2b are subjected to period conditions The other boundaries are also considered as slip, adiabatic and non-catalytic solid conditions In order to get a specific initial flow field, the following procedure is adopted It is assumed first that a planar detonation propagates with CJ velocity along azimuthal direction in a two-dimensional flow field, which is initially filled with a quiescent hydrogen-air mixture at P = 25P and T = T ahead of the detonation wave, and is described by selfsimilar solutions behind the detonation The inlet and exit boundary conditions mentioned above are taken as the conditions at the upper and lower boundary, respectively Then a triangle layer of the fresh mixture is formed and grows as the flow field evolves after the computation software starts running When the triangle layer is large enough, the flow field is cut from the detonation wave to a location with a suitable height of the fresh mixture, the length of which is the same as the circumference of the three-dimensional chamber The cut flow field is finally taken as the initial flow field for the three-dimensional calculations The parallel implementation was carried out using MPICH2 with a zone decomposition method The grid number of the computational domain is ξ η ζ = Results and discussions 31 Numerical validations In our calculation, after several rotations, a stable rotating detonation is established As numerical validations, qualitative comparisons are executed between the measured results obtained by Bykovskii [5] and Wolanski [19] and our calculated results The streak pictures given in Fig 3 show a qualitative agreement, where the measured picture (Fig 3a) recorded by a high speed streak camera is taken from Ref [5]

Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber and the calculated picture (Fig 3b) is drawn based on the calculated temperature field of the

4 Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber and the calculated picture (Fig 3b) is drawn based on the calculated temperature field of the rotation detonation It can been seen from Fig 3 that the flow field can be divided into four marked areas by the detonation-shock combined wave, as called by Fujiwara and Tsuge in Refs [2, 21], and the two contact surfaces: an unburnt combustible gas area, a burnt gas area before the detonation-shock combined wave, a burnt gas area immediately behind the detonation wave and a compressed burnt gas area behind the shock wave 69 Figure 4 shows temporal variations of the pressure for the rotating detonation, where a and b represent measured [19] and calculated results, respectively The experimentally observed rotating detonation pressure is much lower than its CJ value which may arise from lots of experimental factors, and Fig 4 is used to give qualitative comparisons only The stable rotations can be verified from the very stable pressure peaks 3P 35P with a nearly repeatable peak-to-peak time interval 2328 µs shown in Fig 4 Fig 3 Streak pictures of the rotating detonation a Measured; b Calculated Fig 4 Pressure history at a location immediately behind the injection wall a Measured; b Calculated It can be deduced from the above description that a steadily rotating detonation with a planar front can be finally achieved in an annular combustion chamber, which implies that the propagating velocity of detonation wave near the outer wall is higher than that near the inner wall, and the rotating detonation should have a three dimensional structure 32 Three dimensional flow pattern The three dimensional structure of the specific unconfined detonation is depicted in Fig 5 Ahead of the detonation front, where the pressure is lower than the reservoir pressure, there exits an unburnt combustible gas layer on the head side of the combustor due to the gas injection from the reservoir, which sustains a continuously rotating detonation Although three sides of the detonation wavelet touch the inner, outer and head wall, another sides, ie underside, is in soft contact with the burnt gas, which generates transmitted shock waves combined with the detonation front The propagations of a continuously rotating detonation in an annular chamber consisting of two coaxial cylinders are shown in Fig 6, where the three-dimensional flow field is visualized and the distinctive features of flow patterns are Fig 5 Three-dimensional structure of the rotating detonation clearly depicted 33 Flow field of lateral cross-section A typical instantaneous flow field in the lateral cross-section at ξ = 16 is shown in Fig 7, which is divided into four distinguished areas by the detonation-shock combined wave and the two contact discontinuities as described in Sect 31

7 X-D Zhang, et al Fig 6 Evolutions of the rotating detonation in the annular chamber a t = µs; b t = 553 µs; c t = 124 µs; d t = 1467 µs; e t = 1925 µs; f t = 2489 µs; g t = 3148 µs; h t = 4145 µs

5 7 X-D Zhang, et al Fig 6 Evolutions of the rotating detonation in the annular chamber a t = µs; b t = 553 µs; c t = 124 µs; d t = 1467 µs; e t = 1925 µs; f t = 2489 µs; g t = 3148 µs; h t = 4145 µs Fig 7 Flow field structure in a lateral cross-section of the combustion chamber (during steady rotation) The fresh combustible mixture is injected into the chamber from the reservoir and forms an unburnt combustible gas layer near the head wall denoted as Zone I, where the rotating detonation wave propagates The burnt gas of the preceding cycle expelled by the fresh mixture of fuel is indicated as Zone II, where the unconfined detonation wave [22] turns into a transmitted shock wave due to the detonation interaction with the surface of Zone I and Zone II, which is called a detonation-shock combined wave by Fujiwara [2] Zone III represents the burnt gas immediately behind the present detonation front and Zone IV represents the compressed burnt gas behind the shock wave, both of which are separated by a contact discontinuity 34 Effects of wall geometries on flow fields The wall effects existing in the annular combustor can influence the flow field and strength of the detonation Both the lesser extent pressure effect due to geometrical compression and the overdriven due to Mach reflection lead to the fact that the strength of the outside detonation front is stronger than the inside portion affected by geometrical expansion The effects of the concave and convex walls thus permit a deto- nation to propagate with a constant circular velocity The distributions of H2 and OH mass fractions on the inner, middle and outer laterals at t = 8255 µs are shown in Fig 8a and Fig 8b, respectively, in term of which the chemical reaction zone can be described clearly There exists a thin area behind the detonation wave, where the fresh gas is burning dramatically Meanwhile, several burning gas pockets, generated by the instability of the interface and ignition by the compressed hot burnt gas behind the shock wave, can also be observed along the interface between Zones III and IV It is obvious that the reaction area near the inner convex wall is wider than that near the outer concave wall, and the reaction rate near the inner wall is lower than that near the outer wall due to the wall convergent/divergent effects The pressure distribution in the flow field on the inner, middle and outer laterals at t = 8255 µs is shown in Fig 9, which indicates that the rotating detonation has a character of unconfined detonation, ie the detonation wavelet entails an oblique shock wave which extends to the far downstream It is also shown that the strength of the outside detonation front is stronger than the inside portion, whereas the height of the detonation wavelet near the outer wall is less than that near the inner wall, also seen from Fig 8

Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber 71 Fig 8 Distributions of H 2 and OH on the inner, middle and outer laterals at t = 8255 µs a

velocities of the rotating detonation at the outer side and inner side are 1 8867 m/s and 1 46744 m/s, respectively, when the detonation rotates in a stable manner It is evident that the calculated

6 Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber 71 Fig 8 Distributions of H 2 and OH on the inner, middle and outer laterals at t = 8255 µs a H 2 distribution; b OH distribution Based on the Gorden-Mcbride code, the CJ detonation velocity in the mixture discussed in the paper is obtained as m/s However, the calculated propagation velocities of the rotating detonation at the outer side and inner side are m/s and m/s, respectively, when the detonation rotates in a stable manner It is evident that the calculated average propagation velocity of the rotating detonation ( m/s) is slightly lower than the CJ value, which can be attributed to the unconfined character of detonation wavelet, called as the detonation deficit Whereas the calculated average pressure just behind the detonation is also lower than the CJ value P CJ = 1518P, and the deficit is 248% Fig 9 Pressure distribution on the inner, middle and outer laterals at t = 8255 µs 4 Conclusions The velocity vectors of radial-azimuthal components on the head wall are shown in Fig 1 It can be seen that just behind the detonation wavelet, the outside fluid flows toward the outer wall whereas the inside fluid flows toward the inner wall But in the downstream flow field, more fluid tends to move toward the inner wall Fig 1 Velocity vector distribution on the head wall at t = 8255 µs Based on three-dimensional Euler equations associated with a detailed chemical reaction mechanism, a continuously rotating detonation in a hydrogen-air mixture has been modeled in an annular chamber consisting of two coaxial cylinders and investigated numerically, in which the wave propagation algorithm based on Roe scheme and the additive semiimplicit Runge-Kutta method are adopted to discretize the spatial derivatives and the time term, respectively A steadily propagating rotating detonation is achieved and its threedimensional flow field is visualized All typical features of a rotating detonation are captured and analyzed according to the calculated results The fresh combustible mixture injected into the chamber from the head wall, where the local pressure is lower than the reservoir pressure, forms a triangular layer of an unburnt combustible mixture ahead of the detonation, which provides a basic framework for a continuous propagation of the detonation Coupled with the wall convergent/divergent effects, a self-sustained rotating detonation with a planar front can be achieved in a toroidal area finally For a rotating detonation, the one side of the detonation wavelet is in soft contact with the burnt gas, which gener-

7 72 X-D Zhang, et al ates transmitted shock waves combined with the detonation front Before the detonation-shock combined wave, there are two distinguished zones separated by a contact discontinuity, ie the triangular fresh mixture layer and the burnt gas expelled by the injected fresh gas Meanwhile, two distinguished zones exist behind the combined wave, ie the burnt gas immediately behind the present detonation front and the compressed burnt gas behind the shock wave separated by a contact discontinuity emanating from the underside of the detonation, along which discrete burning gas pockets distribute due to the instability of the discontinuity The strength of the detonation along the outer wall is stronger than that along the inner wall, due to the convergence of the outer wall and the divergence of the inner wall Deficit of detonation parameters caused by the unconfined character of detonation wavelet can also be observed References 1 Daniau, E, Falempin, F, Zhdan, S: Pulsed and rotating detonation propulsion systems: First step toward operational engines AIAA (25) 2 Voitsekhovskii, BV: Stationary detonation Doklady Akademii Nauk UzSSR 129(6), (1959) 3 Mikhailov, VV, Topchiyan, ME: To the studies of continuous detonation in an annular channel Combustion, Explosion, and Shock Waves 1(4), (1965) 4 Bykovskii, FA, Mitrofanov, VV: Detonation combustion of a gas mixture in a cylindrical chamber Combustion, Explosion, and Shock Waves 16(5), (1981) 5 Bykovskii, FA, Vedernikov, EF: Continuous detonation combustion of an annular gas-mixture layer Combustion, Explosion, and Shock Waves 32(5), (1996) 6 Bykovskii, FA, Vedernikov, EF: Continuous detonation of a subsonic flow of a propellant Combustion, Explosion, and Shock Waves 39(3), (23) 7 Bykovskii, FA, Zhdan, SA, Vedernikov, EF: Continuous spin detonation Journal of Propulsion and Power, 22(6), (26) 8 Bykovskii, FA, Zhdan, SA, Vedernikov, EF: Continuous spin detonation of fuel-air mixtures Combustion, Explosion, and Shock Waves 42(4), (26) 9 Bykovskii, FA, Zhdan, SA, Vedernikov, EF: Continuous spin detonation of hydrogen-oxygen mixture 1 annular cylindrical combustors Combustion, Explosion, and Shock Waves 44(2), (28) 1 Lentsch, A, Bec, R, Serre, L, et al: Overview of current French activities on PDRE and continuous detonation wave rocket engines AIAA (25) 11 Daniau, E, Falempin, F, Getin, N, et al: Design of a continuous detonation wave engine for space application AIAA (26) 12 Zhdan, SA, Bykovskii, FA, Vedernikov, EF: Mathematical modeling of a rotating detonation wave in a hydrogenoxygen mixture Combustion, Explosion, and Shock Waves 43(4), (27) 13 Jiang XH, Fan BC, Dong G, et al: Numerical investigation on the flow field of rotating detonation wave Journal of Propulsion Technology, 28(4), (27) (in Chinese) 14 Hishida M, Fujiwara T, Wolanski P: Fundamentals of rotating detonations Shock Waves 19(1), 1 1 (29) 15 LeVeque, RJ: Finite Volume Methods for Hyperbolic Problems Cambridge University Press, Cambridge (ISBN ) (22) 16 Kee, RJ, Rupley, FM, Miller, JA: CHEMKIN II Sandia Report SAND 89-89, US DOE Sandia National Laboratories, Livermore, CA, (1989) 17 Radhakrishnan, K, Hindmarsh, AC: Description and use of LSODE, the livermore solver for ordinary differential equations LLNL Report UCRL-ID (1993) 18 Oran, ES, Young, TR, Boris, JP, et al: Weak and strong ignition I Numerical simulations of shock tube experiments Combustion and Flame 48, (1982) 19 Wolanski, P, Kindracki, J, Fujiwara, T: An experimental study of small rotating detonation engine In: Roy, G, Frolov, S, Sinibaldi, J eds Pulsed and Continuous Detonations Moscow, Torus Press, (26) 2 Fujiwara, T, Tsuge, S: Quasi-one-dimensional analysis of gaseous free detonations Journal of the Physical Society of Japan 33(1), (1972) 21 Tsuge, S, Fujiwara, T: On the propagation velocity of a detonation-shock combined wave Journal of Applied Mathematics and Mechanics 54(3), (1974) 22 Dabora, EK, Nicholls, JA, Morrison, RB: Tenth Symposium (International) on Combustion The Combustion Institute, Pittsburgh, (1965)

Fundamentals of Rotating Detonation. Toshi Fujiwara (Nagoya University)

Fundamentals of Rotating Detonation Toshi Fujiwara (Nagoya University) New experimental results Cylindical channel D=140/150mm Hydrogen air; p o =1.0bar Professor Piotr Wolanski P [bar] 10 9 8 7 6 5 4