A Gas-kinetic Scheme for Multimaterial Flows and Its Application in Chemical Reaction

Transcription

1 NASA/CR ICASE Report No A Gas-kinetic Scheme for Mutimateria Fows and Its Appication in Chemica Reaction Yongsheng Lian The Hong Kong University of Science & Technoogy Hong Kong Kun Xu The Hong Kong University of Science & Technoogy Hong Kong and ICASE Hampton Virginia Institute for Computer Appications in Science and Engineering NASA Langey Research Center Hampton VA Operated by Universities Space Research Association Nationa Aeronautics and Space Administration Langey Research Center Hampton Virginia Prepared for Langey Research Center under Contract NAS Juy 1999

2 A GAS-KINETIC SCHEME FOR MULTIMATERIAL FLOWS AND ITS APPLICATION IN CHEMICAL REACTION YONGSHENG LIAN AND KUN XU Abstract. This paper concerns the extension of the muticomponent gas-kinetic BGK-type scheme [6] to mutidimensiona chemica reactive fow cacuations. In the kinetic mode each component satisfies its individua gas-kinetic BGK equation and the equiibrium states of both components are couped in space and time due to the momentum and energy exchange in the course of partice coisions. At the same time according to the chemica reaction rue one component can be changed into another component with the reease of energy where the reactant and product coud have different γ. Many numerica test cases are incuded in this paper which show the robustness and accuracy of kinetic approach in the description of muticomponent reactive fows. Key words. gas-kinetic method muticomponent fow detonation wave Subject cassification. Appied Numerica Methods 1. Introduction. The deveopment of numerica methods for the mutimateria fows have attracted much attention in the past years [15 8 9]. One of the main appication for these methods is the chemica reactive fow cacuations [ 1 18]. Research of reactive fows especiay the detonation waves was pioneered by Zedovich von Neumann and Doering who deveoped a we known ZND mode. The ZND mode consists of a non-reactive shock foowed by a reaction zone. Ever since the mode was proposed ots of theoretica and numerica work on this probem have been done. Numerica cacuation of the ZND detonation was pioneered by Fickett and Wood [11]. They soved the one-dimensiona equations using the method of characteristics in conjunction with a shock fitting method. Longitudina instabiity waves were accuratey simuated. Later Taki and Fujiwara appied van Leer s upwind method to cacuate two-dimensiona traveing detonation waves [3 4]. They soved the Euer equations couped with two species equations. The chemica reaction was simuated by a two-step finite-rate mode and the transverse instabiities around shock front were ceary observed. It was pointed out by Coea et a in [5] that if the numerica resoution in the detonative shock front is not enough unphysica soution can be easiy generated such as the wrong shock speed. In order to avoid the unphysica soution Engquist and Sjögreen [6] obtained a high order TVD/ENO numerica method combined with Runge-Kutta time marching scheme to sove the combustion probem and designed a specia treatment in the shock region. Kaiasanath et. a [14] extended the Fux-Corrected Transport FCT agorithm for detonations. In the eary 9s Bourioux et.a. combined PPM scheme with conservative front tracking and adaptive mesh refinement in the study of detonative waves [ 3 4]. They showed the spatia-tempora structure of unstabe detonation in one and two spatia dimensions and found good agreement between the numerica simuation and the experimenta data. Quirk [1] addressed the particuar deficiency of the Godunov type upwind schemes in soving compex fow probems and suggested This research was supported by the Nationa Aeronautics and Space Administration under NASA Contract No. NAS whie the second author was in residence at the Institute for Computer Appications in Science and Engineering ICASE NASA Langey Research Center Hampton VA Additiona support was provided by Hong Kong Research Grant Counci through RGC97/98.HKUST6166/97P. Mathematics Department The Hong Kong University of Science & Technoogy Hong Kong. Mathematics Department The Hong Kong University of Science & Technoogy Hong Kong emai: makxu@uxmai.ust.hk; and ICASE Mai Stop 13C NASA Langey Research Center Hampton VA emai: kxu@icase.edu. 1

3 a hybrid scheme from which he successfuy simuated the gaoping in one and two-dimensiona detonations. Lindström [18] anayzed the poor convergence of inviscid Euer soutions in the study of detonative waves and suggested to sove the compressibe Navier-Stokes equations directy. Most recenty Hwang et a [13] pointed out that not ony the resoution of the reaction zone is important but aso the size of the computationa domain is critica in the capturing of correct detonative soutions. So far it is we recognized that a good scheme for the reactive fow must be abe to capture correct shock speed and resove wave structures in mutidimensiona case as we as present the correct period of the possibe unsteady osciation in the wave. Ever since the gas kinetic scheme was proposed for the compressibe fow simuations [5] due to its robustness and accuracy it has attracted more attentions in the CFD community. In this paper we are going to extend the muticomponent BGK sover [6] to high dimensions and deveop a new scheme with the incusion of reactive terms. The paper is organized as foows. Section introduces the governing equations for the chemica reactive fows in the D case and describes the numerica method. Section 3 is about the numerica experiments which incude non-reactive shock bubbe interaction and ZND wave cacuations in both 1D and D cases. We aso show a new exampe where the reactant and product coud have different γ. Different from previous approach [17] the current method foows the evoution of each species individuay and the scheme is more robust than the previous one.. Numerica Method. The focus of this section is to present a kinetic scheme to sove the foowing reacting compressibe Euer equations in the -D case ρ 1 ρ 1 U ρ 1 V KT ρ 1 ρ ρ U ρ V KT ρ 1 ρu + ρu.1 + P + ρuv = ρv ρuv ρv + P ρe t UρE + P x V ρe + P y KT Q ρ 1 where ρ 1 is the density of reactant ρ is the density of product ρ = ρ 1 + ρ is the tota density ρe is the tota energy which incude both kinetic and therma ones i.e. ρe = 1 ρu + V +P 1 /γ 1 1+P /γ 1. Here U V are the average fow veocities in the x and y directions respectivey. Each component has its specific heat ratios γ 1 and γ. P = P 1 + P is the tota pressure and Q is the amount of heat reeased per unit mass by reaction. The equation of state can be expressed as P 1 = ρ 1 RT and P = ρ RT. KT isthe chemica reactive rate which is a function of temperature. The specific form of KT wi be given in the numerica exampe section. The above reactive fow equations wi be soved in two steps. In the first step the nonreactive gas evoution parts are soved using the mutimateria gas-kinetic method. In the second step the source terms on the right hand side of Eq..1 are incuded in the update of fow variabes inside each ce..1. -D Muticomponent BGK Scheme Gas-kinetic Governing Equations. The focus of this subsection is to present the muticomponent BGK scheme in two dimensions. For two dimensiona probem the governing equation for the time evoution of each component is the BGK mode. f 1 t + uf x 1 + vf y 1 = g1 f 1 τ f t + uf x + vf y = g f τ

4 where f 1 and f are partice distribution functions for component 1 and gases and g 1 and g are the corresponding equiibrium states which f 1 and f approach. The reations between the distribution function and the macroscopic variabes are f 1 φ 1 α dξ1 + f φ α dξ = W.3 =ρ 1 ρ ρuρvρe T where dξ 1 = dudvdξ 1 dξ = dudvdξ φ 1 α =1 uv 1 u + v + ξ 1 T φ α = 1uv 1 u + v + ξ T are the moments for individua mass tota momentum and tota energy densities ξ1 = ξ 11 + ξ ξ1k 1 and ξ = ξ1 + ξ + + ξk. The integration eements are dξ 1 = dξ 11 dξ 1...dξ 1K1 and dξ = dξ 1 dξ...dξ K. K 1 and K are the degrees of the interna variabes ξ 1 and ξ which are reated to the specific heat ratios γ 1 and γ. For the two-dimensiona fow we have K 1 =5 3γ 1 /γ and K =5 3γ /γ Instead of individua mass momentum and energy conservation in a singe component fow for two component gas mixtures the compatibiity condition is.4 g 1 f 1 φ 1 α dξ 1 +g f φ α dξ = α = The equiibrium Maxweian distributions g 1 and g are generay defined as g 1 = ρ 1 λ 1 /π K 1 + e λ1u U1 +v V 1 +ξ 1 g = ρ λ /π K + e λu U +v V +ξ where λ 1 and λ are function of temperature. Due to the momentum and energy exchange in partice coisions in most cases the equiibrium states g 1 and g can be assumed to have the same veocity and temperature at any point in space and time. So based on given initia macroscopic variabes at any point in space and time W 1 = g 1 φ 1 α dξ1 =ρ 1 ρ 1 U 1 ρ 1 V 1 ρ 1 E 1 T.5 W = g φ α dξ =ρ ρ U ρ V ρ E T we can get the corresponding equiibrium vaues.6 W 1 = W = ρ 1 ρ 1 U ρ 1 V 1 ρ 1U + V + K 1 + T λ T ρ ρ U ρ V 1 ρ U + V + K + λ 3

5 where the common vaues of U V and λ can be obtained from the conservation requirements ρ = ρ 1 + ρ ρ 1 U 1 + ρ U = ρu ρ 1 V 1 + ρ V = ρv.7 ρ 1 E 1 + ρ E = ρu + V + K +ρ. 4λ With the definition of averaged vaue of interna degree of freedom K.8 K = ρ 1K 1 + ρ K ρ and the average γ.9 γ = K +4 K + the vaues U V and λ can be obtained from Eq..7 expicity.1 U = ρ 1U 1 + ρ U ρ.11 and.1 V = ρ 1V 1 + ρ V ρ λ = 1 K +ρ 4 ρ 1 E 1 + ρ E 1 ρu + V. As a resut the equiibrium states can be expressed as.13 g 1 = ρ 1 λ/π K 1 + e λu U +v V +ξ 1.14 g = ρ λ/π K + e λu U +v V +ξ. The governing equations. basicay correspond to viscous mutimateria governing equations and the scheme presented in the next section is actuay soving the Navier-Stokes fow equations where the dissipative coefficients are proportiona to the coision time τ [5]..1.. Muticomponent Gas-kinetic Scheme. Numericay the Botzmann equations. are soved using the spitting method. For exampe in the x direction we sove f 1 t + uf x 1 = g1 f 1 τ f t + uf x = g f τ 4

6 and in the y direction f 1 t + vf y 1 = g1 f 1 τ f t + vf y = g f. τ In each fractiona step the compatibiity condition.4 is sti satisfied. For the BGK mode in the x direction the equivaent integra soution of f at a ce interface x i+1/ and time t is.15 for component 1 and f 1 x i+1/ tuvξ 1 = 1 τ t g 1 x t uvξ 1 e t t /τ dt +e t/τ f 1 x i+1/ ut.16 f x i+1/ tuvξ = 1 τ t g x t uvξ e t t /τ dt +e t/τ f x i+1/ ut for component where x i+1/ is the ce interface and x = x i+1/ ut t the partice trajectory. There are four unknowns in Eq..15 and Eq..16. Two of them are initia gas distribution functions f 1 and f at the beginning of each time step t = and the others are g 1 and g in both space and time ocay around x i+1/ t=. Numericay at the beginning of each time step t = we have the macroscopic fow distributions inside each ce i W i =ρ 1 ρ ρuρvρe T i. From the discretized initia data we can appy the standard van Leer imiter L to interpoate the conservative variabes W i and get the reconstructed initia data.17 W i x =W i + Ls i+ s i x x i for x [x i 1/ x i+1/ ] and Wi x i 1/ W i x i+1/ are the reconstructed point-wise vaues at the ce interfaces x i 1/ and x i+1/. In order to simpify the notation in the foowing x i+1/ = is assumed. With the interpoated macroscopic fow distributions W i the initia distribution functions f 1 and f in Eq..15 and Eq..16 are constructed as 1+a 1 f 1 x g 1 x <.18 = 1+a 1 r x g r 1 x > for component 1 and.19 f = 1+a 1+a r x x g x < g r x > for component. The equiibrium states in Eq..15 and Eq..16 around x =t= are assumed to be. g 1 = 1+1 Hxā 1 x +Hxā 1 r x + Ā1 t g 1 5

7 and.1 g = 1+1 Hxā x +Hxā r x + Ā t g where Hx is the Heaviside function. g 1 and g are the initia equiibrium states ocated at the ce interface g 1 = ρ 1 λ /π K1+ e λu U +v V +ξ 1. g = ρ λ /π K+ e λu U +v V +ξ. The parameters a 1 r ā 1 r and Ā1 have the forms a 1 = a a 1 u + a 1 3 v + a 1 u + v + ξ1 4 a r 1 = a 1 r1 + a 1 r u + a1 r3 v + u + v + ξ a1 1 r4 ā 1 =ā 1 1 +ā 1 u +ā 1 3 v +ā 1 u + v + ξ1 4 ā 1 r =ā 1 r1 +ā 1 r u +ā1 r3 v +ā1 r4 u + v + ξ1 Ā 1 u + v + ξ = Ā1 1 + Ā1 u + Ā1 3 v + Ā A coefficients a 1 1 a 1... Ā 1 4 are oca constants. In order to determine a these unknowns the BGK scheme is summarized as foows. The equiibrium Maxweian distribution functions ocated on the eft side of the ce interface x i+1/ for component 1 and are and g 1 = ρ 1 λ /π K1+ e λ u U +v V +ξ 1.3 g = ρ λ /π K+ e λ u U +v V +ξ. At the ocation x = the reations.3 and.4 require W i x i+1/ ρ 1i ρ i ρu i ρv i ρe i x i+1/ = g 1 φ 1 α dξ1 + g φ α dξ = ρ 1 ρ ρu ρv ρe 6

8 and from which we have Simiary W i+1 x i+1/ ρ 1i+1 ρ i+1 ρu i+1 ρv i+1 ρe i+1 ρ 1r ρ r U r V r λ r ρ 1 ρ U V λ x i+1/ = = = g r 1 φ 1 α dξ 1 + g r φ α dξ = ρ 1i ρ i Ū i V i K 1+ ρ 1i+K + ρ i 4 ρe i 1 ρiū i + V ρ 1i+1 ρ i+1 Ū i+1 V i+1 i K 1+ ρ 1i+1+K + ρ i+1 4 ρe i+1 1 ρi+1ū i+1 + V i+1 x i+1/. x i+1/. ρ 1r ρ r ρu r ρv r ρe r Therefore g 1 g g r 1 and g r are totay determined. Since g 1 and g have the same temperature and veocity at any point in space and time as shown in Eq..6 the parameters a 1 1 a 1 a 1 3 a 1 4 are not totay independent. Since a 1 a 1 3 a 1 4 depend ony on derivatives of U V and λ common veocity and temperature in space and time require a a 1 = a a 3 a 1 3 = a 3 and a 4 a 1 4 = a 4. This is aso true among the parameters a 1 r a r...a1 r a r on the right hand side of a ce interface. So inside each ce i wehave ω 1 W i x i+1/ W i x i ω ω 3 x i+1/ x i.4 = ω 4 ω 5 a a u + v + ξ1 u + a 3 v + a 4 + a 1 + a u + v + ξ u + a 3 v + a 4 g 1 φ 1 α dξ 1 g φ α dξ. The above five equations uniquey determine the five unknowns a 1 1 a 1 a a 3 a 4 and the soutions is the foowing: Define Π 1 = ω 3 U ω 1 + ω Π = ω 4 V ω 1 + ω Π 3 = ω 5 U + V + K1+ λ U + V + K+ λ. ω 1 ω 7

9 The soutions of Eq..4 are a 4 = 8λ Π 3 U Π 1 V Π K 1 +ρ 1 +K +ρ λ a 3 = Π ρ 1 + ρ V a 4 ρ 1 + ρ λ λ a = Π 1 ρ 1 + ρ U a 4 ρ 1 + ρ λ a 1 = 1 ω ρ U a + V a 3 ρ U + V ρ a 1 1 = 1 ρ 1 ω 1 ρ 1 U a + V a 3 ρ 1 U + V + K + a 4 4λ + K 1 + 4λ a 4 With the same method a terms in a r 1 terms can be obtained. By taking the imits of t in Eq..15 and Eq..16 appying the compatibiity condition at x = x i+1/ t= and using Eq we get.5 ρ 1 ρ ρ U ρ V ρ E T = im e t/τ t = Hug 1 +1 Hug r 1 g 1 φ1 α dξ 1 + g φ α dξ f 1 x i+1/ utφ 1 dξ 1 + f x i+1/ utφ dξ φ 1 α dξ1 + Hug +1 Hug r. φ α dξ. The right hand side of the above equation can be evauated expicity using g 1 r in Eq..3. Therefore ρ 1 ρ λ U and V in Eq.. can be obtained from Eq..5. As a resut g 1 and g are totay determined. Then connecting the macroscopic variabes W =ρ 1 ρ ρ U ρ V ρ E T at the ce interface with the ce centered vaues in Eq..17 on both sides we obtain the sopes for the macroscopic variabes W W i x i x i+1/ x i and W i+1 x i+1 W x i+1 x i+1/ from which ā 1 and ā in Eq.. and ā 1 r and ā r in Eq..1 can be determined using the same techniques for soving Eq..4. At this point there are ony two unknowns Ā1 eft for the time evoution parts of the gas distribution functions in Eq.. and Eq..1. Substituting Eq..18 Eq..19 and Eq.. into the integra soutions Eq..15 and Eq..16 we get.6 f 1 x i+1/ tuvξ 1 =1 e t/τ g 1 + τt/τ 1+e t/τ Ā1 g 1 + τ 1+e t/τ +te t/τ ā 1 Hu+ā 1 r +e t/τ 1 uta 1 Hug 1 +1 uta 1 r 1 Hu 1 Hug1 r ug 1 and f x i+1/ tuvξ =1 e t/τ g + τt/τ 1+e t/τ Ā g 8

10 .7 + τ 1+e t/τ +te t/τ ā H[u]+ā r 1 Hu +e t/τ 1 uta Hug +1 uta r 1 Hug r ug. In order to evauate the unknowns Ā1 in the above two equations we can use the compatibiity condition at the ce interface x i+1/ on the whoe CFL time step t t g 1 f 1 φ 1 α dξ 1 dt +g f φ α dξ dt = from which we can get.8 g 1 Ā 1 φ 1 α dξ1 + g Ā φ α dξ = Ā Āu + Ā3v u + v + ξ + Ā4 1 g 1 φ1 α dξ 1 + Ā 1 + Āu + Ā3v u + v + ξ + Ā4 g φ α dξ = 1 [ γ 1 g 1 + γ u ā 1 Hu+ā 1 r 1 Hu g 1 γ +γ 3 Hug 1 +1 Hug r 1 ] +γ 4 u a 1 Hug 1 + a 1 r 1 Hug1 r φ 1 α dξ1 [ + γ 1 g + γ u ā Hu+ā r 1 Hu g +γ 3 Hug +1 Hug r ] +γ 4 u a Hug + a r 1 Hug r φ α dξ where γ = t τ1 e t/τ γ 1 = 1 e t/τ γ = t +τ1 e t/τ te t/τ γ 3 =1 e t/τ and γ 4 = τ1 e t/τ + te t/τ. The right hand side of the Eq..8 is known therefore a parameters in Ā 1 terms can be obtained expicity. 9

11 Finay the time-dependent numerica fuxes for component 1 and component gases across a ce interface can be obtained by taking the moments of the individua gas distribution functions f 1 and f in Eq..15 and Eq..16 separatey which are F ρ1 F ρ1u 1 = uφ 1 α f 1 x i+1/ tuvξ 1 dξ 1 F ρ1v 1 F ρ1e 1 i+1/ and F ρ F ρu F ρv = uφ α f x i+1/ tuvξ dξ. F ρe i+1/ Integrating the above time-dependent fux functions in a whoe time step t we can get the tota mass momentum and energy transports for each component from which the fow variabes in each ce can be updated... Reaction Step and Fow Update. After obtaining the fux functions across a ce interface we need to sove an ODE to account for the source term i.e. W t = S. More specificay inside each ce we need to sove ρ 1 t = KT ρ 1.9 ρ t = KT ρ 1 ρe t = KQ ρ 1. In the current study one step forward-euer method is used to sove the above equations. In summary the update of the fow variabes inside ce i j from step n to n+ 1 is through the foowing formuation Wij n+1 = Wij n 1 t t y F i 1/j F i+1/j dt + x G ij 1/ G ij+1/ dt + ts ij V where S ij is the corresponding source terms in ce i j F and G are numerica fuxes across ce interfaces by soving the muticomponent BGK equations and V is the area of the ce i j. 3. Numerica Exampes. In this section we are going to test the muticomponent BGK scheme for both nonreactive and reactive fow cacuations. For the viscous cacuations the coision time τ in the BGK scheme presented in the ast section is set to be τ = µ/p where µ is the dynamica viscosity coefficient and P is the tota oca pressure. For the Euer soutions the coision time in the cacuation is defined as τ =.5 t + P P r t P + P r 1

12 where t is the CFL time step and P and P r are the corresponding pressure terms in the states g and g r of the initia gas distribution function f. From the above expression we know that in the smooth region there are about coisions inside each time step in the current inviscid cacuations and the magnitude of corresponding numerica diffusion is about 1/1 of that in the Kinetic Fux Vector Spitting KFVS scheme [ 19 5]. Aso in comparison with the previous singe component kinetic method for the reactive fows [17] the current approach is more robust. The detai comparison is shown in [16] Nonreactive Mutimateria Fow Cacuations. In this subsection we are going to show two cases about the shock-bubbe interactions. The main difference between these two cases is about the initia density difference inside the bubbe which consequenty gives different fow pattern around materia interface. CASE1 A M s =1. shock wave in air hits a Heium cyindrica bubbe We examine the interaction of a M s =1. panar shock wave moving in the air with a cyindrica bubbe of Heium. Experimenta data can be found in [1] and numerica soutions using adaptive mesh refinement has been reported in []. Recenty a ghost fuid method has been appied to this case too [8]. A schematic description of computationa set-up is shown in Fig.4.1 where refection boundary conditions are used on the upper and ower boundaries. The initia fow distribution is determined from the standard shock reation with the given strength of the incident shock wave. The bubbe is assumed to be in both therma and mechanica equiibrium with the surrounding air. The non-dimensionaized initia conditions are W =ρ = 1U = V = P = 1γ = 1.4 pre-shock air W =ρ =1.3764U =.394V =P =1.5698γ =1.4 W =ρ =.1358U =V =P =1γ =1.67 Heium. post-shock air In the computation the nondimensiona ce size used is x = y =.5. In order to identify weak fow features which are often ost within contour pots we present a number of Schieren images. These pictures depict the magnitude of the gradient of the density fied ρ = ρ x + ρ 3.1 y and hence they may be viewed as ideaized images; the darker the image the arger the gradient. The density derivatives are computed using straightforward centra-differencing. The foowing noninear shading function φ is used to accentuate weak fow features [] 3. φ =exp k ρ ρ max where k is a constant which takes the vaue 1 for Heium and 6 for air. For R simuation in the next test case we use 1 for heavy fuid and 8 for air. Fig.4. shows snapshots of Schieren-type images at nondimensiona time t=. and t=15.. Before the shock hits the bubbe wigges usuay appear around the bubbe because the numerica scheme cannot precisey keep the sharp materia interface. The wigges spread in a directions. When they reach the soid wa they bounce back. But a these noise have a very sma magnitude. After the shock hits the bubbe the origina shock wave separates into a refected and a transmitted shock waves. A compex pattern of discontinuities has formed around the top and bottom of the bubbe. Since Heium has a ower 11

13 density in comparison with air any sma perturbation at the materia interface can easiy be ampified to form the instabiity. This instabiity at the materia interface is cosey reated to the Richtmyer-Meshkov instabiity. In comparison with the resut in [8] the current scheme coud capture the unstabe interface structure automaticay. The resut here is basicay consistent with both the experiment and that from the mesh-refinement study []. It is an interesting probem to further study shock-bubbe interaction case and understand the dynamics of any specia numerica treatment on the interface stabiity. In our cacuations we do not specificay pick up the ocation of interfaces. CASE A M s =1. shock wave hits a R cyindrica bubbe With the same scheme we investigate the interaction of a M s =1. panar shock wave moving in the air with a cyindrica bubbe of R. The main difference between this case and the previous one is that the density of the bubbe here is much arger than the density of air. The initia data is as foow W =ρ = 1U = V = P = 1γ = 1.4 pre-shock air W =ρ =1.3764U =.394V =P =1.5698γ =1.4 W =ρ =3.1538U =V =P =1γ =1.49 R. post-shock air In the numerica experiment we use x = y =.5. Fig.4.3 shows two snapshots of Schieren-type images at nondimensiona time t=. and t=15.. Due to the higher density in the bubbe region different from Case 1 the materia interface in this case is basicay stabe. This observation is aso consistent with the theoretica understanding and physica experiment. 3.. Reactive Fow Cacuations. The study of detonation wave has been undertaken theoreticay and computationay for over a century. The successfu theory of Ze dovich von Neumann and Doering has come to be a standard mode. The ZND soution for the reacting compressing Euer equations is described in [1] which consists of a non-reactive shook foowed by a reaction zone; both the shock and the reaction zone trave at a constant speed D. Given γ and heat reease Q there is a minimum shock speed the so-caed Chapman-Jougnet vaue D CJ above which the ZND soution can be constructed. The parameter which reates to the shock speed D of a given detonation wave to the CJ veocity D CJ is the overdrive factor f which is defined as 3.3 D f D CJ. The vaue of f determines the stabiity of the detonative front. In the foowing test cases we ony consider the reactive fow with two species i.e. the reactant and the product. The reactant is converted to the product by a one-step irreversibe reactive rue governed by Arrhenius kinetics. The factor KT which depends on the temperature is given by KT =K T α e E+ /T where K is a positive constant. In the current paper we assume that α = and the gas constant R is normaized to unity. Therefore the above temperature T is determined by T = P/ρ. One important parameter in the numerica cacuation of ZND soution is the haf-reaction ength L 1/ which is defined as the distance for haf-competion of the reactant starting from the shock front. Usuay the reaction prefactor K is seected such that the haf-reaction ength is unity. From the Arrhenius formua 1

14 the haf reaction ength is defined as 3.4 L 1/ = 1/ 1 D U K Zexp E+ T dz where D is the speed of the shock U is the post-shock fow speed. In the output of numerica resuts the mass fraction Z is defined as Z = ρ 1 ρ 1 + ρ. Case1: 1-D stabe ZND detonation: γ =1. Q = 5 E + =5. f = 1.8 This test case is from []. The pre-shock state is normaized to P = ρ =1andveocityU = V = the post-shock can be obtained using Chapman-Joguet condition. The prefactor K is chosen to be K = so that the ength of the haf-reaction zone L 1/ is unit. This case corresponds to the stabe ZND profie. The resuts with 1 and 4 points/l 1/ are shown in Fig.4.4 and 4.5. Case: 1-D unstabe detonation: γ =1.Q =5E + =5 f=1.6 In order to get a high quaity simuation resut for the unstabe overdriven detonation a high resoution soution is usuay required to resove the instabiity. At the same time the correct capturing of osciatory period requires a arge computationa domain. As pointed out in [13] for a particuar computation one can be tempted to keep ony a few points behind of the shock with the reasoning that the information behind the shock either never catches up with or does not affect the shock during the computation. However if too sma a computationa domain behind the shock is specified the points at the edge of and outside of the computationa domain cease to be updated after some time eading to a corruption of the data in that region. The U + c waves emanating from inappropriate boundary condition eventuay catch up with the shock itsef thus erroneousy aternate the shock properties. The anaysis in [13] shows that if one expects the numerica resuts at time t to be correct the computationa domain L and t must satisfy the foowing inequaity 3.5 t< L U + c D + L D where U is the speed of the post-shock fow and c is the sound speed. For the current test L shoud satisfy L 1.88t. This cassica unstabe detonation wave was first used by Fickett and Wood [11]. An important physica quaity for unstabe detonation is the pressure history at the precursor shock in the osciatory ZND wave as a function of time. For a stabe ZND wave this shock pressure history shoud exhibit sma fuctuations about the known precursor shock vaue and decay as time evoves. In the case of unstabe detonations the shock front pressure history makes arger excursions from the ZND vaue. For the case with γ =1. q = 5 E + = 5 and overdrive f = 1.6 according to Erpenbeck [7] this ZND profie is a reguar periodic pusating detonation with maximum shock pressure per period given by 11.1 ±. whie the unperturbed ZND shock pressure is In the current study the density and pressure are normaized to unit after shock. According to Q = 5γ =1. the CJ speed becomes D CJ = and the prefactor is chosen to be K = 3.75 so as to get unit haf-reaction ength. The post-shock state can be determined by Chapman-Jouguet condition with the 13

15 Tabe 3.1 Maxima and minima pressure vs. time for f=1.6 case and 8/L 1/. Time Maxima Time Minimum given shock speed. Due to the start-up numerica incompatibiity there is a arge initia shock pressure up to 114 at time t equa to 8 see Fig4.6. After t>15 the motion of the shock front becomes periodic. In this test we observe that at east points/l 1/ is needed for getting a correct unstabe ZND soution. In Fig.4.6 and 4.7 we show the numerica resuts with points/l 1/ and 4 points/l 1/ respectivey. At the same time the resut with 8 points/l 1/ is given as a reference. In Tabe3.1 the data of oca maximum and minimum pressure as a function of time are isted. Case3 Weak shock wave hitting the reactant In order to vaidate the muticomponent BGK scheme we design the foowing 1D case to simuate the chemica reaction in which the reactant and product have different γ. The initia condition is given beow W L =ρ L U L P L γ L = W M =ρ M U M P M γ M = W R =ρ R U R P R γ R = post-shock air pre-shock air Reactant. This case is about a weak shock wave with M =. hitting the reactant. We use the Arrhenius form for the reaction rate with E + = Q = 5 and K = 6.. The numerica ce size is x =1/. In Fig.4.8 we show the numerica resuts at time t =.. Since the shock is too weak to construct a ZND soution the fow motion ooks ony ike a two-component nonreactive gases. From Fig.4.8 we can see the ordinary incident shock moves faster than that of the transmitted shock and the weak refection wave moves backward. Case4 Strong shock wave hitting the reactant We increase the strength of the shock in Case3 up to M =8.. The initia condition is given beow W L =ρ L U L P L γ L = W M =ρ M U M P M γ M = W R =ρ R U R P R γ R = post-shock air pre-shock air Reactant. 14

16 In Fig.4.9 we show the numerica resuts at time t =.5. From the figure we observe that after the shock hits the reactant a ZND soution is obtained. Case5 Viscous Reactive Fow This case is from [18]. The initia data is a one-dimensiona ZND profie in the x-direction. The ZND wave connects the eft state ρ = U = V =ρ E = by a Chapman-Jouget detonation with the right state ρ r =1U r =V R =ρ r E r = 15. If no transverse gradient is present in the initia data the numerica scheme wi preserve the one-dimensiona ZND profie. So a periodic perturbation is imposed in the y-direction of the initia ZND profie where the initia data W x y is set to W ZND x + xnint.5 x cos4πy where NINTz is the nearest integer to z. The current test has Q = E + = 5 γ =1.. The reaction rate K is set to be 1 4. The coefficient of dynamica viscosity µ is set to 1.e-4. With the above choice of parameters the haf reaction ength L 1/ of the inviscid one-dimensiona Chapman-Jouget detonation wave is equa to.85. In our computation x = y = 1 8 is used. Therefore there are about 3 points/l 1/. Based on the anaysis in [13] in order to get an accurate soution it is sufficient to use a computationa domain x [ 1.]. At the eft and the right boundary we impose the eft and right states of the initia traveing wave soution. At the ower and upper boundaries periodic boundary conditions are used. Fig.4.1 shows a sequence of snapshots of the density distributions starting from the time t =.. Fig.4.11 is the snapshot of pressure at ater times when the ZND front has a reguar periodic osciating profie. The first picture is taken at t = 13 8 which is just after the coision of two tripe points. This figure ceary shows the formation of a Mach stem. In the next few snapshots the movement of tripe points aong the transverse shock front are ceary captured. A high pressure spot deveops at the ocation of tripe-point intersection. Fig.4.1 shows the snapshots of the temperature variations. More figures such as the mass fraction and vorticity are presented in [16]. This test case corresponds to the ceuar regime []. The hot spots in the shock front shoud dispay a reguar diamond propagating pattern such as observed in physica experiments. In Fig.4.13 we pot the numerica soot track of the ocation of shock front which is the successive geometric representation of the ZND front profie. Since ony the position of the shock front is recorded one dimensiona data is required at each output time. From the numerica soot track dispay we can easiy observe the formation of ceuar pattern. 4. Concusion. In this paper we have successfuy extended the BGK-type gas-kinetic scheme to mutidimensiona reactive fows. Since each component of the fow is captured individuay mass conservation is precisey preserved for each component in nonreactive mutimateria fow cacuations. For the reactive fow cacuations the mass exchange between different components has been impemented in the current kinetic method as we as the energy reease in the reaction process. Many numerica test cases vaidate the current approach and show the advantages of the kinetic scheme in the description of muticomponent fow cacuations. For exampe the unstabe and stabe materia interfaces are captured automaticay in the shock-bubbe interaction cases. 15

17 REFERENCES [1] P.L. Bhatnagar E.P. Gross and M. Krook A Mode for Coision Processes in Gases I: Sma Ampitude Processes in Charged and Neutra One-component Systems Phys.Rev pp [] A. Bourioux Numerica Study of Unsteady Detonations Ph.D. Thesis Princeton University [3] A. Bourioux J. Majda and V. Roytburd Theoretica and Numerica Structure for Unstabe One-dimensiona Detonations SIAM J. App. Math pp [4] A. Bourioux and J. Majda Theoretica and Numerica Structure for Unstabe Detonations Phi. Trans. R. Soc. Lond. A pp [5] P. Coea A. Majda and V. Roytburd Theoretica and Numerica Structure for Reacting Shock Waves SIAM J. Sci. and Stat. Comput pp [6] S. Engquist and B. Sjögreen Robust Difference Approximations of Stiff Inviscid Detonation Waves UCLA CAM Report [7] J.J. Erpenbeck Stabiity of ideaized one-reaction detonation Phys. Fuids pp [8] R.P. Fedkiw T. Asam B. Merriman and S. Osher A Non-osciatory Euerian Approach to Interfaces in Mutimateria Fows The Ghost Fuid Method J.Comput.Phys pp [9] R.P. Fedkiw X.D. Liu and S. Osher A Genera Technique for Eimination Spurious Osciations in Conservative Scheme for Muti-phase and Muti-species Euer Equations UCLA CAM Report [1] W. Fickett and W.C. Davis Detonation University of Caifornia Press Berkeey CA [11] W. Fickett and W.W. Wood Fow Cacuations for Pusating One-dimensiona Detonations Physics of Fuids 9 No pp [1] J.F. Haas and B. Sturtevant Interactions of weak shock waves with cyindrica and spherica gas inhomogeneities Journa of Fuid Mechanics pp [13] P. Hwang R.P. Fedkiw B. Merriman A.R. Karagozian and S.J. Osher Numerica Resoution of Pusating Detonation preprint 99-1 UCLA CAM reports [14] K. Kaiasanath E.S. Oran J.P. Boris and T.R. Young Determination of Detonation Ce Size and the Roe of Transverse Waves in Two-dimension Detonations Combustion and Fame pp [15] S. Karni Hybrid Mutifuid Agorithms SIAMJ. Sci. Comput pp [16] Y.S. Lian Two Component Gas-kinetic Scheme for Reactive Fows M.Phi Thesis Hong Kong University of Science and Technoogy [17] Y.S. Lian and K. Xu A Gas-kinetic Scheme for Reactive Fows ICASE Report No [18] D. Lindström Numerica Computation of Viscous Detonation Waves in Two Space Dimensions Report No Department of Computing Uppsaa University. [19] J.C. Manda and S.M. Deshpande Kinetic Fux Vector Spitting for Euer Equations Computers and Fuids pp [] D.I. Puin Direct Simuation Methods for Compressibe Inviscid Idea-gas Fow J. of Comput. Phys pp. 31. [1] J.J. Quirk Godunov-type Schemes Appied to Detonation Fows ICASE Report No [] J.J. Quirk and S. Karni In the dynamics of a shock bubbe interaction J. Fuid Mech

18 pp. 19. [3] S. Taki and T. Fujiwara Numerica Anaysis of Two Dimensiona Nonsteady Detonations AIAA Journa 16 No [4] B. van Leer Towards the Utimate Conservative Difference Scheme IV A New Approach to Numerica Convection J.Comput.Phys pp. 76. [5] K. Xu Gas-Kinetic Schemes for Unsteady Compressibe Fow Simuations von Karman Institute for Fuid Dynamics Lecture Series [6] K. Xu BGK-based Scheme for Muticomponent Fow Cacuations J. Comput. Phys pp

19 r=5 Pre-shock air Heium bubbe Post-shock air Fig Physica domain for shock-bubbe interaction Fig. 4.. Numerica Schieren images of the interaction of an M s=1. shock wave in the air moving from right to eft over an Heium cyindrica bubbe. The second image is the density distribution at time t = 15.. The third one is the density profie aong the centerine of the second figure. 18

20 Fig Numerica Schieren images of the interaction of an M s=1. shock wave moving from right to eft over an R cyindrica bubbe. The second image is the density distribution at time t = 15.. The thrid one shows the density profie aong the centerine of the second figure. 19

21 Front Pressure Front Pressure Front Pressure 1 9 Mesh=1 points/l 1/ Time 1 9 Mesh= points/l 1/ Time 1 9 Mesh=4 points/l 1/ Time Fig Mesh refinement study of the pressure history at the shock front for the stabe detonation wave where f=1.8 γ =1. Q = E + =5andL 1/ =1. CFL=.5.

22 Density 6 4 Pressure Veocity Mass Fraction Fig Numerica soutions soid ine of density ρ veocity U pressure P and mass fraction Z where f=1.8 γ =1. Q = E + =5 L 1/ =1. and1points/l 1/ CFL=.5. The dash ine is the exact soution. 1

23 Front Pressure Time Fig Loca maximum pressure variation as a function of time for the overdriven detonation where f=1.6 γ =1. Q = E + =5andL 1/ =1.. Soidine:8points/L 1/ and dash-dot ine: points/l 1/ CFL=.5.

24 Front Pressure Time Fig Loca maximum pressure variation as a function of time for the overdriven detonation where f=1.6 γ =1. Q = E + =5andL 1/ =1.. Soidine:8points/L 1/ dash-dot ine: 4 points/l 1/ CFL=.5. 3

25 Density Pressure Veocity Mass Fraction Density 1 1 Density Fig Weak shock wave M =. in the air γ =1.4 hitsthereactantgasγ =1.. x =1/. The reaction has E + = Q =5andK = 6. CFL=.5. 4

26 Density Pressure Veocity 1 5 Mass Fraction Density 1 4 Density Fig Strong shock wave M =8. in the air γ =1.4 hits the reactant gas γ =1.. x =1/. The reaction has E + = Q =5. andk = 6. CFL=.5. 5

27 T= T=1/16 T=/16 T=3/16 T=4/16 T=5/16 T=6/16 T=7/16 Fig Sequence of eight snapshots of density starting from time t =with the time increment of E + =5γ =1. x = y = 1 8 3points/L 1/. Shock moves from eft to right whereq = 6

28 A B C D E F G H I J Fig Sequence of ten snapshots of pressure starting from time t = with each time increment of 1 96 where Q = E + =5γ =1. x = y = 1 8 3points/L 1/. Shock moves from eft to right. 7

29 A B C D E F G H I J Fig Sequence of ten snapshots of temperature start from time t = with time increment of 1 96 whereq = E + = 5γ =1. x = y = 1 8 3points/L 1/. Shock moves from eft to right. 8

30 Fig Numerica soot track: successive geometric representation of the ZND shock front. Each ine represents the ocation of shock front. The vertica axis corresponding to the time. 9