NUMERICAL SIMULATION OF DETONATION PROCESS IN A TUBE

Transcription

1 Computational Fluid Dynamic JOURNAL vol. XX no. X XXXX 23 (pp.{) NUMERICAL SIMULATION OF DETONATION PROCESS IN A TUBE Hyungwon KIM, y Dale A. ANDERSON z Frank K. LU, z Donald R. WILSON z Abtract A two-dimenional time-accurate numerical model to imulate hydrogen{air detonation wave propagation and re ection wa developed. To contruct an e±cient numerical tool, while maintaining a reaonable accuracy, a two-tep global model wa elected and validated for a hydrogen{air mixture. The inherent ti ne in the chemical reaction model wa properly taken care of by a point-implicit treatment of ource term, together with the application of a \local ignition averaging model" to each meh where ignition tart. Calculation were performed to elect a cheme with adequate temporal and patial reolution for modeling the phyical proce for practical calculation. Calculation were compared againt Chapman{Jouguet theory a well a againt experiment. Key Word: Detonation, de agration-to-detonation tranition, Chapman{Jouguet, ignition 1 INTRODUCTION Depite the detructive nature of detonation, there i coniderable interet recently in exploiting detonation for certain application, uch a in propulion [1] and in high-enthalpy ground tet facilitie [2]. While there are many unreolved fundamental iue regarding initiation, tranition and propagation, for example, numerical modeling for obtaining engineering olution are ought for the above-mentioned application. Time-accurate computational uid dynamic (CFD) method can be ued to perform cycle analyi and performance optimization of pule detonation engine from the imulation of the correponding ow eld with variation in deign parameter. Thee method can alo be ued to upport the development of detonation driver. An unteady numerical imulation model for the above purpoe, emphaizing accuracy and e±ciency, i decribed. The objective i to contruct a two-dimenional time-accurate numerical imulation model, e±cient enough for deign parametric tudie Received on oo oo, oooo. y Agency for Defene Development, Taejon, Korea z Mechanical and Aeropace Engineering Department, Univerity of Texa at Arlington, Arlington, Texa, USA c Copyright: Japan Society of CFD/ CFD Journal 23 while maintaining a reaonable accuracy. The model i contructed to formulate the phyical phenomena a preciely a poible, including chemical and thermal non-equilibrium, and to numerically olve the reulting mathematical formulation a accurately a poible. Novel apect of thi work include a combination of point-implicit treatment and a \local ignition averaging model" (LIAM) applied to the global twotep reaction model for e±cient time-accurate olution of a propagating detonation wave. The partition of internal energy i baed on the two-temperature model, and the vibrational energy of each pecie i obtained by ubtracting out fully-excited tranlational and rotational energy from the total internal energy. Roe' ux-di erence plit cheme i combined with a Runge{Kutta integration cheme for an accurate capture of the hock wave in pace and in time. Extenive calculation are performed with numerical cheme of di erent order in pace and time, and with di erent meh ize to elect the proper cheme and meh ize to provide adequate reolution of the phyical proce. The preent model i validated by comparing the calculated data with thoe obtained from Chapman{Jouguet theory and experiment. 2 MATHEMATICAL FORMULATION 2.1 Governing Equation The time-dependent conervation equation governing an invicid, non-heat-conducting, reacting ga ow in which thermal nonequilibrium i modeled with a two-

2 2 temperature approximation are written in the conervation law form. Thi form i advantageou in numerical imulation ince it can correctly capture hock wave [3]. For a two-dimenional Carteian coordinate ytem, the ytem of equation take = S (1) where U i the vector of conerved variable, F and G are the convective ux vector, and S i the vector of ource term. Each vector i written a ½ ½ u ½u ½u 2 + p U = ½v ; F = ½uv ; ½e v 4 ½ue v ½E ½uE + pu ½ v w ½uv G = ½v 2 + p ; S = ½ve v 4w v ½vE + pv (2) The ubcript = 1; 2; 3;::: ; N where N i the number of pecie. The rt N row repreent pecie continuity, followed by the two momentum conervation equation for the mixture. The next row decribe the rate of change in the vibrational energy, and the - nal row i the total energy conervation equation. The term u and v are the velocitie in the x and y direction repectively, ½ i the mixture denity, p i the preure, e v i the vibrational energy, E i the total energy per unit ma of mixture, ½ i the -th pecie denity, w i the ma production rate of pecie per unit volume, and w v i the vibrational energy ource term which i de ned a w v = X Q v + X w e d; (3) The rt term on the RHS, Q v,repreentthevibrational energy exchange rate of pecie due to the relaxation proce with tranlational energy. The econd term on the RHS, w e d;, repreent the amount of vibrational energy gained or lot due to production or depletion of pecie from chemical reaction. 2.2 Thermodynamic Propertie In general, the internal energy of each pecie include a portion of the internal energy in thermodynamic equilibrium and the remaining portion in a nonequilibrium tate. The equilibrium portion of the internal energy i the contribution due to tranlation and internal mode that can be aumed to be in equilibrium at the tranlational temperature T.The remaining nonequilibrium portion i the contribution due to internal mode that are not in equilibrium at the tranlational temperature T, but may be aumed to atify a Boltzman ditribution at a di erent temperature. For the temperature range of interet, the rotational mode i aumed to be fully excited and in equilibrium with the tranlational temperature T, while the electronic excitation and free electron mode can be afely ignored. The vibrational mode remain a the only energy mode that could be in nonequilibrium with the tranlational temperature T. Thu, the pecie internal energy baed on a two-temperature model can be written a e = e eq; (T )+e v; (T v ) (4) where e eq; i the equilibrium portion of the internal energy and e v; i the vibrational energy which i not in thermodynamic equilibrium. The equilibrium portion of the energy can be further de ned a e eq; = Z T T ref (C v;t + C v;r) d + e ;o () where T ref i the reference temperature, e ;o i the energy of formation at the reference temperature, and C v;t and C v;r are the tranlational and rotational portion of the peci c heat at contant volume, repectively. Since the tranlational and rotational mode are aumed to be fully excited, C v;t and C v;r can be written a Cv;t =1: R=M ¹ (6) " Cv;r = ¹R=M ; diatomic molecule 1: R=M ¹ (7) ; polyatomic molecule where ¹ R i the univeral ga contant and M i the molecular weight of pecie. The energy of formation e ;o can be obtained from readily available heat of formation data a R e ;o = h ;o ¹ T M ref (8) Therefore, the equilibrium portion of energy can be written a R ¹ R e eq; (T )=K (T T M ref ) ¹ T M ref + h ;o (9) where K =1:, 2., 3. for monatomic, diatomic or linear polyatomic, and nonlinear polyatomic pecie repectively.

3 3 The heat capacity of the vibrational energy mode can be obtained from the fact that the tranlational and rotational heat capacitie are independent of temperature. Thi can be evaluated by utilizing a readily available curve t for total heat capacity evaluated at the temperature T v and ubtracting out the contant contribution from the tranlation and rotational heat capacitie, namely, where C v;v(t v )=C v(t v ) C v;t C v;r (1) C v (T v)=c p (T v) ¹ R M Cv and C p are peci c heat at contant volume and contant preure repectively. Curve t data for Cp(T ) can be found in the following form [4]: Cp (T )= ¹ R M X A k T k 1 (11) k=1 Therefore, Cv;v can be obtained a follow: 2 Ã X! ¹R A Cv;v (T M ktv k 1 7 ; diatomic v)= 2 Ãk=1 6 X! 4 ¹R A k T v k 1 4 ; polyatomic M k=1 (12) The pecie vibrational energy e v; can be obtained by integrating C v;v uch that e v; (T v )= Z Tv Cv;v( )d (13) T ref 2.3 Chemical Kinetic For accurate modeling of a detonation wave, epecially in the detonation front where rapid chemical reaction take place in the hock compreed region, pecie continuity equation baed on the chemical kinetic hould be olved together with the uid dynamic equation to account for poible chemical nonequilibrium. The ma production rate of pecie from chemical reaction can be written a [] XN r w = M ( ;r ;r )(R f;r R b;r ) (14) r=1 where M i the molecular weight of pecie, N r i the number of reaction, ;r and ;r are the toichiometric coe±cient for reactant and product, repectively, in the r reaction. The forward and backward reaction rate of the r reaction, R f;r and R b;r, repectively, are de ned by " # YN R f;r =1 K f;r (:1½ =M ) ;r " R b;r =1 =1 K b;r N # Y (:1½ =M ) ;r =1 () The factor 1, and.1 are required to convert from CGS to MKS unit, ince mot reaction rate data in the literature are found in CGS unit. The forward reaction rate coe±cient can be expreed by K f;r = A f;r T N f;r exp( E f;r = RT ¹ ) (16) where E f;r i the activation energy of the r-th forward reaction. The value of parameter A f;r, N f;r, E f;r are uually found tabulated according to the reaction involved. The backward reaction rate coe±cient i evaluated uing the equilibrium contant for the reaction uch that K b;r = K f;r =K c;r (17) 2.4 Vibrational Energy Relaxation The energy exchange between vibrational and tranlational mode due to intermolecular colliion i well decribed by the Landau{Teller formulation where it i aumed that the vibrational level of a molecule can change by only one quantum level at a time [6, 7]. The reulting energy exchange rate i given by e v;(t ) e v; Q v; = ½ (18) < > where e v; (T ) i the vibrational energy per unit ma of pecie evaluated at the local tranlational rotational temperature, and < > i the averaged Landau{Teller relaxation time of pecie given by [], XN X N < >= n j j n j (19) j=1 j=1 where j i the vibrational tranlational relaxation time of pecie caued by intermolecular colliion with pecie j, andn j i number denity of pecie j. The Landau{Teller interpecie relaxation time j can be obtained in econd uing the Millikan{White emi-empirical expreion [8] j = 1 p exp h A j ³ T 1=3 :¹ 1=4 j i 18:42 (2)

4 4 where p =preureinatm,a j =1: ¹ j µ j, ¹ j = M M j =(M + M j ) i the reduced ma, and µ j = characteritic vibrational temperature of a harmonic ocillator. The vibrational energy relaxation rate can be impli ed uing [] where X ½ e v; e v; < > ¼ X 1 v = ½ Cv;v T T v < > ¼ ½C v;v v (T T v) (21) X ½ =(M < >) X ½ =M Thi approximation reduce the number of pecie dependent parameter and impli e the evaluation of the vibrational relaxation a a ingle relaxation term multiplied by the di erence in the tranlational and vibrational temperature. When a point implicit formulation i ued on the ource term in the numerical algorithm, the above approximation greatly impli e an implicit treatment of the temperature di erence which drive the relaxation proce. 3 NUMERICAL FORMULATION 3.1 Finite-Volume Formulation A dicretized et of equation i derived from the governing partial di erential equation uing the nitevolume method. The advantage of thi method i it ue of the integral form of the equation, which enure conervation, and allow the correct treatment of dicontinuitie [3]. In the following derivation, the cell-centered approach will be decribed. For an arbitrary volume, encloed by a boundary ¾, the governing equation in integral form can be written a ZZZ I Ud + ~H ~nd¾ = Sd ¾ where ~H =(F; G) The unit vector ~n i normal to the in niteimal area d¾ and point outward. The rt tep to dicretize the above equation i to introduce volume averaged value of the conerved variable and the ource term a follow: <U>= 1 ZZZ Ud ; < S >= 1 ZZZ Sd (23) Theevolumeaveragedvariableareubtitutedinto the integral form of the governing equation to (<U> )+ (F; G) ~nd¾ =< S> (24) ¾ For a two-dimenional Carteian coordinate ytem where the computational cell i de ned by two contant line in both the x and y direction, the urface integral can be plit into four contribution, one from each bounding urface. When the index of the cell centered variable i (i; j), Eq. (24) can be written a I ZZ ZZ (F; G) ~nd¾ = Fd¾+ Fd¾ ¾ ¾ i+1=2 ¾ i 1=2 ZZ ZZ + Gd¾+ Gd¾ ¾ j+1=2 ¾ j 1=2 (2) Area-averaged value of uxe can be de ned uch that <F > i+1=2 = 1 ZZ Fd¾; ¾ i+1=2 ¾ i+1=2 ZZ 1 <G> j+1=2 = Gd¾ (26) ¾ j+1=2 ¾ j+1=2 where the bounding urface area ¾ i+1=2, ¾ j+1=2 actually repreent cell face length in the two-dimenional Carteian coordinate ytem. After ubtituting thee de nition of averaging into the Eq. (24) and dropping the bracket, the following dicrete form of the conervation equation written in a two-dimenional Carteian coordinate ytem can be obtained: i;j ¾ i+1=2 ¾ i 1=2 = F i+1=2 F i µ ¾ j+1=2 ¾ j 1=2 G j+1=2 G j 1=2 +S i;j (27) 3.2 Point Implicit Time Integration Nonequilibrium ow involving nite-rate chemitry and thermal energy relaxation often can be very dif- cult to olve numerically becaue of ti ne. The ti ne in term of the time cale can be de ned a the ratio of the larget to the mallet time cale uch that Sti ne = larget = mallet where can be any characteritic time in the ow eld. For reactive ow problem, there can be everal chemical time cale and relaxation time cale in addition to the uid dynamic time cale aociated with

5 convection. The ti ne parameter can be a high a O(1 6 ). The point implicit formulation evaluating the ource term at time level (n+1) ha been an e ective method ued to numerically integrate ti ytem [9]. The point implicit treatment i known to reduce the ti ne of the ytem by e ectively recaling all the characteritic time in the ow eld into the ame order of magnitude. Equation (27) i rewritten here with the ource term evaluated at the time level (n = µ Fi+1=2 n ¾ i+1=2 µ G n j+1=2 ¾ i 1=2 i 1=2 F n ¾ j+1=2 G n j 1=2 ¾ j 1=2 + S n+1 i;j (28) The ource vector i then linearized about the preent time level uch that µ S n+1 = S n + U When imple Euler time integration i ued, ubtituting thi linearization into the above equation and rearranging yield µ n t U = F n ¾ i+1=2 +F n i 1=2 ¾ i 1=2 ¾ j+1=2 G n j+1=2 + G n ¾ j 1=2 j 1=2 + Si;j n (3) Thee equation can be evaluated to get U entirely at the current time level at the expene of matrix inverion containing the ource term Jacobian. Temporal accuracy can be added uing Runge{ Kutta integration cheme intead of rt-order accurate Euler integration. The two-tep explicit Runge{ Kutta time integration cheme can be written a U n+ 1 2 = U n + 1 U n (31) U n+1 = U n U n ³2 U n where 1 =1., 2 =.,and³ 2 = -., and the three tep cheme i given by U n+ 1 3 = U n + 1 U n U n+ 2 3 = U n U n ³2 U n (32) U n+1 = U n U n ³3 U n+ 1 3 where 1 = 8=, 2 = =12, 3 = 3=4, and ³ 2 = 17=6, ³ 3 = = Flux-Di erence Split Algorithm The baic feature of the ux-di erence plit algorithm i the olution of a local Riemann problem at the cell interface to determine the cell-face ux. An approximate Riemann problem i ued with Roe' cheme, originally developed for a perfect ga [1] and later extended to a thermo-chemical non-equilibrium ga [11]. The ux-di erence cheme ued here i baed on the latter method. The approximate Rieman olver i implemented by computing the cell face ux a a ummation of the contribution from each wave component, F i+1=2 = 1 2 (F R + F L ) 1 2 NX +4 i=1 ^ i ^ i ^E i = 1 2 (F R + F L ) 1 2 ([[F ]] A +[[F ]] B +[[F ]] C ) (33) where ubcript R and L repreent the right and left tate repectively, i are eigenvalue, E i are eigenvector, i are correponding wave trength, and ^ indicate Roe-averaged quantitie. The [[F ]] A term correponding to the repeated eigenvalue i = u can be written a where 2 3 ^½ i µ [[F ]] A = [[½]] [[p]] ^u j^uj ^a 2 ^v ^e v ^H ^a 2 =(^ 1) 2 3 [[½ i =½]] +^½ j^uj [[v]] (34) [[e v ]] µ [[( )]] = ( ) R ( ) L XN µ =[[e v ]] ^ª i [[½ i =½]] + ^v[[v]] i=1 ª i = R it ~ i ~ 1 e eq;i + u2 + v 2 2 ~ = ~ C p ~C v = C p;t + C p;r C v;t + C v;r R = ¹ R=M

6 6 The [[F ]] B and [[F ]] C term which are contribution from the eigenvalue i = u a, arefoundtobe 2 3 ^½ i [[F ]] B;C = 1 ^u ^a ([[p]] ^½ ^a[[u]]) (^u ^a) 6 ^v 7 2^a 2 4 ^e v ^H ^u^a (3) For added patial accuracy, a higher-order approximation uing the MUSCL approach can be applied. When the MUSCL approach i employed, the primitive variable of the right and left tate at the cell interface are evaluated uing the following extrapolation formula [3] q L i+1=2 = q i q R i+1=2 = q i ¹± + q j 1= ¹± + q j+1=2 1 4 ¹± q j+1=2 ¹± q j+3=2 (36) where q i any primitive variable, the upercript L and R repreent left and right extrapolation repectively, and determine the type of extrapolation method uch that ( 1 2nd-order upwind cheme = 1=3 3rd-order upwind cheme (37) 1 2nd-order claic centered cheme In the above equation, the lope of the variable are limited to prevent nonphyical ocillation and to preerve the TVD (Total Variation Diminihing) property. The limited lope can be written uing the minmod limiter a follow: ¹± q i+1=2 =minmod(±q i+1=2 ;!±q i 1=2 ) (38) ¹± + q i+1=2 =minmod(±q i+1=2 ;!±q i+3=2 ) The minmod limiter i a function that elect the mallet number from a et when all have the ame ign but i zero if they have di erent ign uch that ( x if jxj < jyj and xy > min mod(x; y) = y if jxj > jyj if and xy > xy < (39) with the limit on! given a 1! (3 )=(1 ) (4) 3.4 Temperature Calculation The conerved variable at each cell center are updated uing Eq. (3) by a matrix inverion. From thee conerved variable, new value of the primitive variable, ½, u, v, e v, E, are eaily obtained. However, to cloe the problem, the temperature and vibrational temperature are determined at each iteration cycle. A Newton{Raphon method i ued to obtain the temperature in the following manner [, 12]: ½e X ³ ½ e T (k) ;T v (k) T (k+1) = T (k) + ½C v;tr ³ (k) (41) ½ e v; T T (k+1) v = T (k) v + ½e v X ½C v;v While the total internal energy e and vibrational energy e v are directly obtained from the updated conervative variable, the pecie internal energie e and the pecie vibrational energie e v; are calculated from the ga model uing the current value of both temperature. The iteration i carried out until converged value of both temperature are obtained. 4 VALIDATION STUDIES 4.1 H 2 {Air Reaction Model and Local Ignition Averaging The two-tep Roger{Chinitz reaction model i ued in thi tudy [13]. Thi model wa developed to repreent H 2 {air chemical kinetic with a few reaction tep a poible while till giving reaonably accurate global reult. Thi model conit of the following two tep: H 2 +O 2 Ã! 2OH 2OH + H 2 Ã! 2H 2 O (42) where the forward reaction rate contant are given by K f;r = A f;r (Á) T N f;r exp( E f;r = ¹ RT ) (43) and the pre-exponential A f;r (Á) i a function of the equivalence ratio Á, the fuel-to-air ratio divided by the toichiometric fuel-to-air ratio. Value of the parameter ued in thi model are " Af;1 (Á)=[8:917Á +(31:433=Á) 28:9] 1 47 N f;1 = 1 E f;1 = 4 86 cal/mole " Af;2 (Á)=[2: + (1:333=Á) :833Á]1 64 N f;2 = 13 E f;2 = 42 cal/mole (44) (4)

7 7 The backward reaction rate can be obtained from K b;r = K f;r =K c;r (46) where the equilibrium contant K c;r i given by.2 O 2 OH where K c;r = A c;r T N c;r exp( E c;r = ¹ RT ) (47) " Ac;1 =26: cm 3 =mole N c;1 = E c;1 = cal/mole (48) Specie Ma Fraction..1 " Ac;2 =2: cm 6 =mole 2 N c;2 =1 E c;2 = cal/mole (49) Thi model i valid for initial temperature of 1{ 2 K and equivalence ratio of.2{2. Since the chemitry model i not valid below 1 K, an ignition temperature mut be peci ed. Nitrogen i alo counted a a colliional partner in the thermodynamic model and the relaxation proce, but not included in the chemical reaction model ince the maximum temperature in the hydrogen{air reaction doe not reach the diociation temperature of nitrogen. Before an actual calculation uing the ow olver i made, the chemical kinetic model mut be examined to ee how each pecie concentration i changing, and on what time cale. Thi may provide ome inight into the ti ne of the ytem and ome clue for etablihing a ow olver time tep that permit pecie concentration to follow the correct kinetic. The ma production rate, Eq. (14), can be independently integrated uing the reaction data in Eq. (43){ (49) to yield the pecie ma fraction hitory. A typical reult obtained uing a Runge{Kutta integration method i in Fig. 1. The ma fraction of OH i een to rie very rapidly a oon a the ignition tart. The OH production reaction i intantaneou at it initial tage and goe to equilibrium very quickly in le than After that, the reaction eem to remain in equilibrium until the H 2 O production reaction begin around 1 9. It i intereting to note that all thee major change in pecie concentration take place within the rt 1 7, a time interval that i a typical uid dynamic time tep in the calculation. Moreover,theintegrationtimetephouldremainat or below 1 12 to enure table integration uing the Runge{Kutta cheme and to properly follow the chemical kinetic. Thi how the ti ne of the chemical reaction model. From the above obervation, it could be deduced that the integration time tep for the ow olver. H Time (ec) Fig.1: Specie ma fraction hitory from chemical kinetic. hould be le than 1 12 to properly include the chemical kinetic. Moreover, it hould be much le than thi order in region of OH production. However, it i practically impoible to ue thi mall time tep in the ow olver. When uing 1 12 aa ow olver time tep, 1 9 integration tep might be needed to olve a typical detonation wave propagation problem which ha a time cale of interet of 1 3. Thi would reult in 1 4 day of CPU time when 1 econd of CPU time per computation cycle, which i a proper etimate for thi code on a typical front-end worktation, i aumed. Fortunately, however, mot of thi ti ne problem can be taken care of by the point implicit treatment of ource term, through effectively recaling all the characteritic time cale involved. Thu, a typical uid dynamic time tep of the order of 1 7 can be afely ued throughout the calculation, ince the pecie production rate during thi time interval can be properly treated by the e ective recaling of the chemical reaction time cale. However,thevery rttimetepwhereallthedratic change take place within that hort period of time cannot be properly decribed by recaling time alone. Another pecial treatment for the igniting cell i needed to be able to ue a typical ow olver time tep. For thi purpoe, a \local ignition averaging model" (LIAM) i propoed. The baic idea for thi model come from the fact that the pecie ma fraction are changing dratically in a very hort period a oon a ignition tart and reache equilibrium oon afterward. LIAM eparate the cell in which the ignition condition i met and then integrate the chemical H 2 O

8 8 kinetic equation alone for that cell. A time tep le than 1 12 iuedintheintegrationwithinthe interval of the ow olver time tep. The average production rate of each pecie during thi time interval i then etimated uing w = ½ = t f () where ½ i the denity change of pecie, obtained from a eparate integration of the chemical kinetic equation during thi time interval, and t f i the ow olver time tep. The average value of the forward reaction rate for each reaction during thi interval can be etimated from R f;1 = W O2 =M O2 ; R f;2 = W OH =2M OH (1) Thee term mut be obtained for calculating the ource term Jacobian. Here, backward reaction rate during thi rt reaction time tep are aumed to be zero. LIAM turn out to work well together with the point implicit cheme to accurately decribe chemical kinetic in the ow olver uing a typical ow olver time tep of 1 7. Figure 2 how calculated reult of the pecie mole fraction hitory at a xed location inide the detonation chamber initially lled with hydrogen{ air mixture. Excellent agreement can be een with the equilibrium concentration data [14]. Thi aure that the chemical kinetic are properly modeled and coupled to the ow olver. Mole fraction O 2 OH H 2 O H 2 O 2, CEA H 2, CEA OH, CEA H 2 O, CEA Time (mec) Fig.2: Specie mole fraction hitory from the ow olver. 4.2 Spatial and Temporal Accuracy Generally peaking, a higher-order cheme may yield more accurate reult, in turn requiring more computing time. The preent computer model accommodate the option to chooe numerical cheme of both pace and time integration up to third-order accuracy. For temporal accuracy, Euler integration for rt-order, two-tep Runge{Kutta (RK) integration for econdorder, and three-tep RK for third-order are contained in the model. For patial accuracy, the MUSCL approach i ued for higher-order approximation. Thi ection examine the e ect of the order of the numerical cheme on the predicted detonation wave. Thi can then be ued to elect the order of the cheme neceary to provide adequate reolution of the phyical proce. Figure 3 i a comparion of detonation wave preure pro le along a detonation chamber when numerical cheme of di erent order of accuracy are implemented. Thi gure compare detonation wave pro- le, propagating into the quiecent hydrogen{air mixture initially at 1 atm. The higher-order calculation capture the higher peak preure a expected. However, it i intereting to note that the econd-order calculation i cloe to the third-order calculation, and that the overall hape of the two wave are almot the ame. Thi convergence trend can be een more clearly in Fig. 4 which how detonation wave velocitie a a function of ditance from the initiation point for each cheme. From thi obervation, a econd-order calculation i a reaonable choice when e±ciency and accuracy are conidered together. Thu, a econd-order accurate cheme in both pace and time i employed to be ued for further calculation, unle otherwie noted. 4.3 Meh Convergence A meh convergence tet i performed to determine the proper meh ize to inure accurate reolution of the phyical proce. The ame con guration and parameter are ued a before but with di erent meh ize. A econd-order accurate cheme in both pace and time i ued throughout the calculation. Figure how detonation wave pro le reulting from three di erent meh ize. Actual dimenional ize of., 2., and. mm are ued, repectively, in modeling a 2 m long tube with planar initiation. For calculation of both. and 2. mm meh ize, a time tep of 1 7 econd ha been ued uccefully to yield table olution. However, for the mallet meh ize of. mm, a maller time tep of 1 8 ha been ued, ince a time tep of 1 7 doenot yield a table olution for thi meh ize. The convergence trend can be een clearly from

9 9 2 2 Meh Convergence Tet ( Comparion of preure hitorie at t =.3 m ) It order 2nd order 3rd order x=.mm, t=1.e-7 x=2.mm, t=1.e-7 x=.mm, t=.e-8 Preure (atm.) Ditance from initiation point (m) Ditance from initiation point (m) Fig.3: Wave pro le from di erent order of accuracy. Fig.: Wave pro le from di erent meh ize. meh ize, a meh ize of 2. mm i reaonable. 2 2 Detonation Velocity (m/) 2 1t order in pace and time 2nd order in pace and time 3rd order in pace and time Detonation Velocity (m/) 2 x =. mm, t = 1.E-7 ec x = 2. mm, t = 1.E-7 ec x =. mm, t =.E-8 ec Ditance from initiation point (m) Fig.4: Detonation velocitie from di erent order of accuracy. Fig. 6 which depict detonation wave velocitie from the initiation point for each meh ize ued. The reult of the 2. mm meh how value almot converged to the. mm meh in both detonation velocity and overall wave hape. If we aume the computing time of 2. mm meh cae to be 1 CPU, then the correponding computing time of. mm and. mm cae will be about 1/4 CPU and CPU, repectively, for a two-dimenional calculation. When we take into account accuracy a well a e±ciency in chooing the Ditance from initiation point (m) Fig.6: Detonation velocitie from di erent meh ize. 4.4 Independence of the Geometry The propertie of fully-developed detonation wave hould be the ame regardle of the geometry involved whenever the initial condition and compoition of the fuel{air mixture are the ame. Four di erent calculation are performed here to con rm the reult to follow thi potulate. We take four cae with two kind of geometry and two initiation method, which are two-dimenional calculation with planar initia-

10 1 tion and with point initiation, and repeat the calculation for axiymmetric ow with the ame initiation method. The calculation domain i 68. cm 3.7 cm for both and lled with a toichiometric hydrogen{ air mixture at 1 atm. Initiation occur near the leftend wall. t = 1 µec t = 2 µec t = 3 µec Detonation Velocity (m/) 2 2 2D plane, Planar initiation 2D plane, Point initiation, on the axi 2D plane, Point initiation, on the wall Axi-ymmetric, Plane initiation Axi-ymmetric, Point initiation, on the axi Axi-ymmetric, Point initiation, on the wall Ditance from initiation point (m) t = 4 µec Fig.8: Detonation velocitie from di erent geometrie in ue. t = µec 2 t = 6 µec t = 7 µec t = 8 µec Detonation 1 2D plane, Planar initiation 2D plane, Point initiation, on the axi 2D plane, Point initiation, on the wall Axi-ymmetric, Plane initiation Axi-ymmetric, Point initiation, on the axi Axi-ymmetric, Point initiation, on the wall t = 9 µec Ditance from initiation point (m) Fig.9: Detonation preure from di erent geometrie in ue. Fig.7: Point initiation in axiymmetric geometry. Figure 7 how preure contour plot at peci- ed time for a point initiation cae in axi-ymmetric geometry. The formation of a planar detonation wave i clearly captured. A planar wave can be oberved to evolve from a pherical wave originated from a point initiation through the interaction of the re ecting wave. The calculated reult are ummarized in Fig. 8 and 9. Thee gure how detonation wave velocitie and the CJ plane preure, repectively, along two di erent line which are the lower boundary (actually, the centerline) and the upper wall. For point initiation cae, detonation velocitie and preure on the axi are oberved to remain lower than thoe of planar initiation cae but increaing, while the detonation velocitie and the preure on the wall are much higher than thoe of planar initiation cae and decreaing until the formation of the planar wave. The higher

11 Detonation Velocity (m/) 2 1 Thi work Theoretical CJ Detonation 1 Thi work Theoretical CJ Equivalence ratio Equivalence ratio Detonation Denity ( ρ / ρ 1 ) Thi work Theoretical CJ Detonation Temperature (K) Thi work Theoretical CJ Equivalence ratio Equivalence ratio Fig.1: Comparion with CJ theory. preure and velocity on the wall are due to the re ection of the detonation wave. The detonation velocitie and the preure for all four cae are oberved to converge to the ame value at a certain ditance from the initiation point where fully developed planar wave are formed, con rming the potulate tated above. 4. Comparion with Chapman{Jouguet Theory The calculated detonation wave propertie are compared with thoe obtained from CJ theory. Since the propertie of the fully-developed detonation wave converge to the ame value regardle of the geometry involved, a een in the previou ection, and ince we are intereted in the nal converged tate of the CJ condition, calculation can be properly performed on the one-dimenional planar geometry for the ake of e±ciency. The computational domain i thu compoed of a detonation tube of emi-in nite length lled with a toichiometric hydrogen{air mixture initially at 1 atm and 298. K. The detonation i initiated jut adjacent to the left-end wall, and the planar wave propagate to the right through the quiecent ga mixture. The detonation wave velocity, preure, denity, and temperature a a function of ditance from an initiation point are compared with the theoretical CJ value [14]. The converging trend of all the variable of the detonation wave i con rmed in each calculation of di erent equivalence ratio. Thee converged value can be compared to the theoretical CJ data. The reult are ummarized in Fig. 1, which depict the converged value of each detonation variable with varying equivalence ratio, and compare thee value with the theoretical CJ data. Excellent agreement between them i oberved. 4.6 Simulation of Shock-Induced Detonation Experiment were performed to validate the numerical imulation uing a detonation-driven hock tube. A chematic of the facility i hown in Fig. 11. The driver tube wa highly preurized initially with helium to 2 atm. The detonation tube wa lled with a toichiometric hydrogen{air mixture to 3 atm. Thee two tube were eparated by a double-diaphragm ection. When the double-diaphragm wa ruptured, a hock-induced detonation wa et up in the detonation tube. The preure hitorie were recorded at everal tation on the wall of the detonation tube

12 12 (PT2{PT). Figure 12 how the preure hitorie, recorded at tation 4 and correponding to cm and cm, repectively, from the location of the downtream diaphragm. The incident detonation wave and the re ected wave were clearly captured at each tation. The re ected wave traveled toward the left after re ecting from the diaphragm eparating the driven tube and detonation tube. It i intereting to note that the preure hitory after arrival of the re- ected wave i oberved to increae in tep. Driver Tube (2.4 mm ID) 3. m 2.74 m 3.16 m 3.7 m 3. m PT 1 PT 2 PT 3 PT 4 PT PT 6 PT 7 PT 8 PT 9 Double Diaphragm Thrut Stand Diaphragm Detonation Tube Driven Tube (2.4 mm ID) (41.2 mm ID) End Diaphragm Conductivity Channel (471 mm Length) Fig.11: Schematic of the UT Arlington detonationdriven hock tube Thi e ective preure wa aumed to act directly on the hydrogen{air mixture in the detonation tube a if the diaphragm wa e ectively removed at t =. Thee ectivepreurehouldbelowerthantheactual initial preure in the driver tube but higher than the preure inide the double diaphragm ection. A reaonable rt gue for the e ective preure in the driver tube wa atm. Another diaphragm eparated the right-end of the detonation tube and the driven tube. The inner diameter of the driven tube wa 4.11 cm, much maller than the.24 cm diameter detonation tube. The rightend of the detonation tube i modeled by a re ective boundary baed on the fact that the area opening to the driven tube, even after diaphragm rupture, correponded to only 7 percent of the cro-ectional area of the detonation tube. Figure 13 how the reulting preure hitorie at tation 4 and from the numerical imulation. The imulation nearly reproduce important feature of the experiment, in pite of the aumption and impli cation ued. The arrival time lag in the incident detonation wave between tation 4 and i oberved to be almot the ame a the experimental meaurement, which ugget excellent agreement in the detonation wave velocitie. The ame agreement i oberved in the re ected wave. The preure level of both incident and re ected wave are alo een to agree well, with the calculated preure hitory howed an increae at each tation after the arrival of the re ected wave. The tep-like preure rie the imulation almot exactly reproduce the experimental reult. 4 2 tation 4 tation Time (mec) tation 4 tation Fig.12: Preure meaurement at tation 4 and Numerical modeling of the experiment require that ome aumption and impli cation be made. For example, the double diaphragm ection meaured cm long and wa preurized initially to 1 atm, about a half of the preure in the driver tube. Therefore, complex wave interaction may exit inide the double diaphragm ection and driver tube until the econd diaphragm ruptured and hock-induced detonation wa initiated in the detonation tube adjacent to the diaphragm. The detail of the double diaphragm rupture were not modeled in the imulation but were replaced by an e ective preure inide the driver tube Time (mec) Fig.13: Calculated preure hitorie at tation 4 and.

13 13 16 t =. mec 3 16 t = 1. mec 3 16 t = 1. mec Preure Denity t = 2. mec 3 16 t = 2.3 mec 3 16 t = 2.4 mec t = 2. mec 3 16 t = 3. mec 3 16 t = 3.2 mec t = 3. mec 3 16 t = 4. mec Fig.14: Wave interation in hock-induced detonation tube.

14 14 Thi tep-like preure hitory i further examined. Figure 14 how the evolution of the preure and denity pro le inide the detonation tube. The incident detonation wave i clearly een to propagate to the right, and the driver material interface i alo clearly oberved from the denity pro le to follow the incident wave. The re ection of the incident wave from the right boundary can be een at 1. m. The re- ected wave propagate to the left until it meet the right-running material interface around 2.4 m. The left-running re ected wave re ect again from the material interface and, a a reult, a higher preure and denity are generated from thi re ection which propagate back to the right. The material interface travel back toward the left a can be een after 2.4 m. The right-running wave re ected from the material interface i oberved to re ect once again from the right boundary around 3.1 m and the reulting hockincreaed preure and denity are een to propagate back to the left. From thee obervation, the tep-like increaing pattern of the preure hitory turn out to be a reult of thee multiple wave re ection. The imulation reult uing the current numerical model i in excellent agreement with experimental data. From thi obervation, the aumption and the impli cation made to model the experiment are alo juti ed. The ue of an e ective driver preure appear to be a reaonable impli cation for modeling the double diaphragm ection. Thi imulation validate the current numerical model for calculating the unteady propagation of a detonation wave and it interaction with boundarie a well a other wave. CONCLUSIONS A numerical model to imulate hydrogen{air detonation wave propagation and re ection ha been preented. For the purpoe of contructing an e±cient numerical tool to be ued in parametric tudie, while maintaining a reaonable accuracy to be ued for analyi, a two-tep global model ha been elected and validated for the chemcal reaction of a hydrogen{ air mixture. The calculated reult from the preent model were compared with reult from Chapman{ Jouguet theory and experimental data. Excellent agreement wa oberved. The obervation validated the e±ciency and the accuracy of the preent model. Numerical cheme of di erent order were teted both in temporal and patial accuracy up to third-order. Meh convergence tet were performed for di erent meh ize. A econd-order accurate cheme in both pace and time integration applied to 2. mm meh ize eemed to be an appropriate choice from the trade-o of accuracy and e±ciency. Geometry independence of detonation wave propertie were alo con rmed a a validation proce of the preent numerical model. A hock-induced detonation experiment ha been imulated. The calculated reult almot exactly reproduced the experimental data, and thi provided a validation of the preent model in the unteady propagation of a detonation wave. REFERENCES [1] Kailaanath, K., \Review of propulion application of detonation wave," AIAA J., Vol. 38, No. 9, pp. 1798{178, 2. [2] Lu, F. K., Wilon, D. R., Bako, R. and Erdo, J. I., \Recent Advance in Detonation Technique for High-Enthalpy Facilitie," AIAA J., Vol. 38, No. 9, pp , 2. [3] Tannehill, J. C., Anderon, D. A. and Pletcher, R. H., Computational Fluid Mechanic and Heat Tranfer, 2nd ed., Taylor and Franci, [4] Gordon, S. and McBride, B. J., \Computer Program for Calculation of Complex Chemical Equilibrium Compoition, Rocket Performance, Incident and Re ected Shock, and Chapman{ Jouguet Detonation," NASA SP-273, [] Gno o, P. A., Gupta, R. N. and Shinn, J. L., \Conervation Equation and Phyical Model for Hyperonic Air Flow in Thermal and Chemical Nonequilibrium," NASA TP-2867, [6] Candler, G. V., \The Computation of Weakly Ionized Hyperonic Flow in Thermo-Chemical Nonequilibrium," Ph.D. Diertation, Stanford Univerity, June [7]Vincenti,W.G.andKruger,C.H.,Jr.,Introduction to Phyical Ga Dynamic, Krieger, Malabar, Florida, [8] Millikan, R. C. and White, D. R., \Sytematic of Vibrational Relaxation," J. of Chem. Phy., Vol. 39, No. 12, pp. 329{3213, [9] Buing, T. R. A. and Murman, E. M., \Finite- Volume Method for the Calculation of Compreible Chemically Reacting Flow," AIAA J., Vol. 26, No. 9, pp. 17{178, [1] Roe, P. L., \Approximate Riemann Solver, Parameter Vector, and Di erence Scheme," J. Comput. Phy., Vol. 43, No. 2, pp. 37{372, [11] Groman, B. and Cinnella P., \Flux-Split Algorithm for Flow with Non-equilibrium Chemitry and Vibrational Relaxation," J. of Comp. Phy., Vol. 88, No. 1, pp. 131{168, 199. [12] Munipalli, R., Kim, H., Anderon, D. A. and Wilon D. R., \Computation of Unteady Nonequilibrium Propulive Flow eld," AIAA Paper 97{ 274, [13] Roger, R. C. and Chinitz, W., \Uing a Global Hydrogen{Air Combution Model in Turbulent Reacting Flow Calculation," AIAA J., Vol. 21,

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