DSpace at VNU: On a two-fluid model of two-phase compressible flows and its numerical approximation

Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns On a two-fluid model of two-phase compressible flows and its numerical approximation Mai Duc Thanh Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam a r t i c l e i n f o Article history: Received 22 September 2010 Accepted May 2011 Available online 13 May 2011 Keywords: Two-fluid Two-phase flow Conservation law Source term Lax–Friedrichs Numerical scheme a b s t r a c t We consider a two-fluid model of two-phase compressible flows First, we derive several forms of the model and of the equations of state The governing equations in all the forms contain source terms representing the exchanges of momentum and energy between the two phases These source terms cause unstability for standard numerical schemes Using the above forms of equations of state, we construct a stable numerical approximation for this two-fluid model That only the source terms cause the oscillations suggests us to minimize the effects of source terms by reducing their amount By an algebraic operator, we transform the system to a new one which contains only one source term Then, we discretize the source term by making use of stationary solutions We also present many numerical tests to show that while standard numerical schemes give oscillations, our scheme is stable and numerically convergent Ó 2011 Elsevier B.V All rights reserved Introduction We consider in the present paper the stratified flow model for the two-fluid model in one-dimensional space variable The model with gravity consisting of governing equations is given by (see Staedtke et al [13] and García-Cascales and Paillốre [6]): @ t ag qg ị ỵ @ x ag qg ug ị ẳ 0; @ t ag qg ug ị ỵ @ x ag qg u2g ỵ pịị ẳ p@ x ag ỵ ag qg g; @ t ag qg Eg ị ỵ @ x ag qg Eg ỵ pịug ị ẳ p@ t ag ỵ ag qg ug g; @ t al ql ị ỵ @ x al ql ul ị ẳ 0; 1:1ị @ t al ql ul ị ỵ @ x al ql u2l ỵ pịị ẳ p@ x al ỵ al ql g; @ t al ql El ị ỵ @ x al ql El ỵ pịul ị ẳ p@ t al ỵ al ql ul g; where is the volume fraction, qi is the density, ui is the velocity, ei is the internal energy and Ei ẳ ei ỵ u2i ; is the total energy, g is the gravity constant in the model with gravity, and g = in the model without gravity, and the subscript ‘‘i’’ can be ‘‘g’’ or ‘‘l’’, representing the gas or liquid phase of fluids respectively The volume fractions of the fluid satisfy ag ỵ al ẳ 1: E-mail addresses: mdthanh@hcmiu.edu.vn, hatothanh@yahoo.com 1007-5704/$ - see front matter Ó 2011 Elsevier B.V All rights reserved doi:10.1016/j.cnsns.2011.05.010 196 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 The source terms in non-conservative form on the right-hand side (1.1) are the interphase interaction terms, indicating the momentum and energy exchanged between phases The system of equations is closed by supplementing the equations of state of gas and liquid phases In this paper, we assume that the gas phase has the equation of state of the perfect gas for the air p ¼ ðcg À 1Þqg eg ; ð1:2Þ and we use the stiffened gas equation of state for the liquid phase p ¼ clcÀ1 ql C pl T l À p1 l el ¼ C pl cl T l ỵ pq1 1:3ị l p ẳ cl 1ịql el ỵ cl p1 : For the tests, we will take (see Chang and Liou [2], for example) kJ cg ¼ 1:4; C gp ¼ 1:012 kg:K ; kJ cl ¼ 1:9276; C lp ¼ 8:07673 kg:K ; p1 ẳ 11:5968 103 at: 1:4ị Compressible multi-fluid flow models such as (1.1) has been widely used to model multi-phase flows, see for example Ishii [7], Stewart and Wendroff [14], and Chang and Liou [2] However, there are concerns for this modeling First, multi-fluid models are often not to be hyperbolic, and this would lead to an ill-posed initial-valued problem Moreover, analytical form of the characteristic fields are not available Hence, it is hard to use the conventional Roe or Godunov type of schemes to calculate the numerical fluxes Therefore, additional equations or correction terms must be included to make it well-posed and analytical form of characteristic fields can be derived Various types of corrected source terms of models of two-phase flows were presented by Garcià-Cascales and Paillère [4] Second, the system (1.1) is not in conservative form and can be understood in the sense of nonconservative product, which was introduced by Dal Maso et al [3] For related works, but rather for simpler models, were done by Marchesin and Paes-Leme [12], LeFloch and Thanh [10,11], and Thanh [16], where the Riemann problem is solved Third, but may not the last, source terms cause lots of inconveniences in approximating physical solutions of the system as the errors may become larger as the meshes are refined Therefore, constructing a stable scheme will be important for study and applications Numerical treatments for systems of balance law with source terms that not rely on the analytic forms of the characteristic fields have been attracted many authors In the case of a single conservation law, well-balance schemes which are stable were constructed by Greenberg and Leroux [5], and Botchorishvili et al [1] In the system case, a well-balance and stable scheme for the model of fluid flows in a nozzle with variable cross-section was constructed and its properties were established by Kröner and Thanh [9], Kröner et al [8] A well-balance scheme for shallow water equations was constructed by Thanh et al [15] Recently, a well-balance scheme for a one-pressure model of two-phase flows, where one phase is compressible, the other phase is incompressible, was established by Thanh and Ismail [17], where the sources are reduced to one phase In this paper we present several forms of the system (1.1) and the equations of state, and then using these forms of equations of state we construct a numerical scheme that is numerically stable for the system (1.1) We observe that only the nonconservative terms cause the oscillations for standard schemes This suggests us to minimize the effects of source terms by reducing the number of source terms involving in the system By algebraic addition, we reduce the system to the new one containing exactly one source term The conservative equations can be dealt with using a convenient standard numerical scheme Motivated by our earlier works, we discretize the source term in the non-conservative equation using stationary waves in the gas phase The paper is organized as follows In Section we provide results which describes basic properties of the two-fluid model We solve present basic facts that are useful for the computations of the next sections In Section we will adapt a standard scheme and construct a stable scheme for (1.1) In Section we present several tests which show that the adaptive scheme is numerically stable Finally, in Section we provide some discussions and conclusions Several forms of the model and properties 2.1 Other forms of the two-fluid model First, we derive several equivalent forms for the two-fluid model (1.1) for smooth solutions Each of the equivalent form differs from each other by the equation of balance of energy There are three equivalent models where the equation for the balance of energy in each phase is written in terms of the total energy as in (1.1), in terms of the internal energy, or in terms of the specific entropy 2.1.1 Model involving equation for internal energy First, we check that the system (1.1) is equivalent to the following system @ t qi ị ỵ @ x qi ui ị ẳ 0; @ t qi ui ị ỵ @ x qi u2i ỵ pịị ẳ p@ x ỵ qi g; @ t qi ei ị ỵ @ x qi ei ui ị ẳ p@ t ỵ @ x ui ịị; 2:1ị i ẳ g; l: 197 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Indeed, let us rewrite the equations of balance of momentum in each phase as ui @ t ðai qi Þ þ qi @ t ui þ qi ui @ x ui ỵ ui @ x qi ui ị ỵ @ x p ẳ qi g; for i = g, l Multiplying both sides of the last equation by ui, i = g, l, and then applying the chain rule, we obtain u2i @ t ðai qi ị ỵ qi @ t  2  2 ui u ỵ qi ui @ x i þ u2i @ x ðai qi ui Þ þ ui @ x p ¼ qi ui g; 2 or     u2i u2 u2 ð@ t qi ị ỵ @ x qi ui ịị ỵ @ t qi i ỵ @ x qi ui i ỵ ui @ x p ¼ qi ui g; 2 i ¼ g; l: Due to the conservation of mass, this yields     u2 u2 @ t qi i ỵ @ x qi ui i ỵ ui @ x p ¼ qi ui g; 2 i ẳ g; l: 2:2ị Adding Eq (2.2) to the equation for internal energy in (2.1), we get        u2 u2 ỵ @ x qi ei ỵ ui ỵ p@ t þ ðp@ x ðai ui Þ þ ui @ x pị ẳ qi ui g; @ t qi ei ỵ 2 i ẳ g; l; or        u2 u2 @ t qi ei ỵ ỵ @ x qi ei ỵ ui ỵ p@ t ỵ @ x ui pị ẳ qi ui g; 2 i ¼ g; l: The last equation gives          u2 u2 @ t qi ei ỵ ỵ @ x qi ei ỵ ỵ p ui ỵ p@ t ẳ qi ui g; 2 i ¼ g; l; or @ t qi Ei ị ỵ @ x qi Ei ỵ pịui ị ỵ p@ t ¼ qi ui g; i ¼ g; l; ð2:3Þ which gives the equations of balance of energy in (1.1) Thus, the system (2.1) and the system (1.1) are equivalent 2.1.2 Model involving equation for the specific energy Furthermore, the equation for internal energy in (2.1) can be written as @ t qi ịei ỵ qi ị@ t ei ỵ ei @ x qi ui ị ỵ qi ui ị@ x ei ỵ p@ t ỵ @ x ui ịị ẳ 0; i ẳ g; l; or ei @ t qi ị ỵ @ x qi ui ịị ỵ qi @ t ei ỵ ui @ x ei ị ỵ p@ t ỵ @ x ui ịị ẳ 0; i ¼ g; l: Therefore, equations for conservation of mass imply that  qi @ t ei ỵ ui @ x ei ị ỵ p @ t ỵ @ x ðai ui Þ À qi  ð@ t qi ị ỵ @ x qi ui ÞÞ ¼ 0; i ¼ g; l; or  qi @ t ei ỵ ui @ x ei ị þ p @ t þ @ x ðai ui ị qi  @ t qi ỵ qi @ t ỵ qi @ x ui ị ỵ ui @ x qi ị ẳ 0: Cancel the terms to get qi ð@ t ei þ ui @ x ei Þ À pai qi ð@ t qi ỵ ui @ x qi ị ẳ 0; or  qi @ t ei ỵ ui @ x ei  @ q ỵ u @ q Þ ¼ 0: t i i x i p qi Re-arranging terms, we get qi   p p ¼ 0; @ t ei À @ t qi ị ỵ ui @ x ei @ x qi qi qi 198 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 or qi       1 ỵ ui @ x ei ỵ p@ x ẳ 0; @ t ei þ p@ t qi qi i ¼ g; l: Using thermodynamical identity T i dSi ẳ dei ỵ pd qi ; i ¼ g; l; we can rewrite the above equation as qi ðT i @ t Si ỵ ui T i @ x Si ị ẳ 0; or qi @ t Si ỵ ui @ x Si ị ẳ 0; i ẳ g; l: 2:4ị Adding the multiple of equations for conservation of mass to Eq (2.4): qi @ t Si ỵ ui @ x Si ị ỵ Si @ t qi ị ỵ @ x qi ui ịị ẳ 0; we deduce the equations for entropy: @ t ðai qi Si Þ ỵ @ x qi Si ui ị ẳ 0; i ¼ g; l: The above calculation shows that the system (2.1) can be written as @ t ðai qi ị ỵ @ x qi ui ị ẳ 0; @ t qi ui ị ỵ @ x qi u2i ỵ pịị ẳ p@ x ỵ qi g; @ t qi Si ị ỵ @ x qi Si ui ị ẳ 0; 2:5ị i ¼ g; l: 2.2 Equations of states Each fluid is characterized by its equation of state In this section, we will establish the equations of state in terms of the pressure and the entropy as the thermodynamical independent variables For the gas phase the polytropic ideal gas has the equation of state of the form p ¼ qg Rg T g ẳ cg 1ịqg eg ; R T eg ¼ C gv T ¼ c gÀ1g ; ð2:6Þ hg ¼ C pg T g g where Rg is the specific gas constant; Rg ¼ gg R, where g is the mole-mass fraction, and R is the universal gas constant Let C gv ¼ c Rg g À1 R c be the specific heat at constant volume and C gp ¼ c gÀ1g be the specific heat at constant pressure, so that Cgp = cgCgv g Using thermodynamical identity, we have dqg deg dSg ¼ C gv À ðcg À 1Þ eg qg ! This gives eg Sg À S ¼ C gv log gà ! c À1 qgg or c À1 eg ¼ qgg exp   Sg À Sg à : C gv Therefore, c p ¼ ðcg À 1Þqgg exp   Sg À Sg à : C gv 2:7ị Thus, qg ẳ qg p; Sg ị ¼ p cg À !1=cg exp   Sg à À Sg : C gp ð2:8Þ M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 199 Furthermore, by denition, the enthalpy is p hg ẳ eg ỵ qg ẳ e ỵ cg 1ịe ẳ cg e: So that hg ¼ cg exp   Sg À Sgà cg À1 qg ; C gv or, for short, c hg ẳ hg qg ; Sg ị ẳ GSịqgg where GSị ẳ cg exp ;   Sg À Sgà : C gv ð2:9Þ Substituting qg from (2.8) into (2.9), and re-arranging terms, we get hg ¼ hg p; Sg ị ẳ cg cg 1ị cg 1ị=cg pðcg À1Þ=cg exp   Sg À Sg à : C gp ð2:10Þ Next, we consider the liquid phase where the equations of state are given by p ¼ ðcl À 1Þql C lv T l À p1 ; el ẳ C lv T l ỵ p1 ql 2:11ị ; cl l where C lv ¼ c RÀ1 be the specific heat at constant volume and C lp ¼ cRlÀ1 be the specific heat at constant pressure, so that l l Clp = clClv, where Rl is the specific gas constant Substituting el from (2.11) into the thermodynamical identity del = TldSl À pdvl, vl = 1/ql, we obtain dðC v l T l ỵ p1 v l ị ẳ T l dSl À pdv l : This yields dSl ẳ C lv dT l p ỵ p1 ỵ dv l Tl Tl or dT l dv l dSl ẳ ỵ cl 1ị : C lv Tl vl Hence, Sl À Sl à c À1 ¼ logðT l v l l Þ C lv so that c À1 T l ¼ ql l exp   Sl À Slà : C lv ð2:12Þ Substituting Tl from (2.12) into (2.11), we obtain p ẳ cl 1ịC lv exp   Sl À Slà cl ql À p1 : C lv This implies ql ẳ ql p; Sl ị ẳ  1=cl   p ỵ p1 S À Sl : exp l Rl C lp ð2:13Þ Moreover, the enthalpy in the liquid phase is given by hl ẳ el ỵ p ql ẳ C lv T l þ p1 ql þ ðcl À 1ÞC lv T l ¼ cl C lv T l : Substituting Tl from (2.12) into the last equation gives c À1 hl ¼ C lp exp ðSl À Slà C lv Þql l or c hl ẳ hl ql ; Sl ị ¼ LðSÞql l ; where LðSÞ ¼ C lp exp   Sl À Sl à : C lv ð2:14Þ 200 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Substituting ql from (2.13) into (2.14), we obtain hl ẳ hl p; Sl ị ẳ C lp  c 1ị=cl   p ỵ p1 l Sl Sl à : exp Rl C lp ð2:15Þ 2.3 Non-hyperbolic system of balance law Let us choose the pressure p and the entropies Sg, Sl as the thermodynamical independent variables Then, the unknown function, or the state variable, can be found under the form V ¼ ðag ; p; ug ; Ss ; ul ; Sl ÞT : ð2:16Þ To find the Jacobian matrix of (1.1) or (2.5) for this variable V, we need to re-write (2.5) in the form @ t V ỵ AVị@ x V ẳ 0; 2:17ị for which A(V) is the Jacobian First, it is derived from (2.5) that the equations for entropy can be written as @ t Si ỵ ui @ x Si ẳ 0; i ẳ g; l: 2:18ị Since qi = qi(p, Si), i = g, l, the chain rule implies that the equations for mass can be written as ð@ p qi ị@ t p ỵ @ Si qi ị@ t Si ị ỵ qi @ t ỵ ui @ p qi ị@ x p ỵ @ Si qi ị@ x Si ị ỵ qi @ x ui ị ẳ 0; i ẳ g; l or @ p qi ị@ t p ỵ qi @ t ỵ @ Si qi ị@ t Si ỵ ui @ x Si ị ỵ ui @ p qi ị@ x p ỵ qi @ x ui Þ ¼ 0; i ¼ g; l: Using (2.18), we get from the last equation @ p ðqi Þ@ t p ỵ qi @ t ỵ ui @ p qi ị@ x p ỵ qi @ x ui ị ẳ 0; i ẳ g; l: The last two equations (i = g, l) yield an equation for the volume fraction al @ p ql ịqg ỵ ag @ p qg ịql ị@ t at ỵ ap @ p ðqg Þal @ p ðql Þðug À ul ị@ x p ỵ qg al @ p ql ị@ x ðag ug Þ À ql ag @ p ðqg Þ@ x ðal ul Þ ¼ 0; ð2:19Þ and an equation for the pressure ql ag @ p qg ỵ qg al @ p ql ị@ t p ỵ ql ag ug @ p qg ỵ qg al ul @ p ql ị@ x p ỵ ql qg @ x ag ug ỵ al ul ị ẳ 0: 2:20ị Next, the equations for mass can be written as qi @ t ui ỵ ui @ t qi ị þ ui @ x ðai qi ui Þ þ qi ui @ x ui ỵ @ x p þ p@ x @ x ¼ p@ x ; for i = g, l Canceling the terms and using the equations of mass, we obtain @ t ui þ ui @ x ui þ qi @ x p ẳ 0; i ẳ g; l: 2:21ị Thus, we obtain a system from (2.18, 2.21): @ t at ỵ a @ p ðq Þq l l  g þag @ p ðqg Þql  @ t p þ q ag @ p q g þqg al @ p ql l  ap @ p ðqg Þal @ p ql ịug ul ị@ x p ỵ qg al @ p ðql Þ@ x ðag ug Þ À ql ag @ p ðqg Þ@ x ðal ul Þ ẳ 0;  ql ag ug @ p qg ỵ qg al ul @ p ql ị@ x p ỵ ql qg @ x ag ug ỵ al ul ị ẳ 0; @ t ug ỵ ug @ x ug ỵ q1 @ x p ẳ 0; 2:22ị g @ t Sg ỵ ug @ x Sg ẳ 0; @ t Sl ỵ ul @ x Sl ẳ 0; @ t ul ỵ ul @ x ul ỵ q1 @ x p ¼ 0: l Setting a1 ¼ qg ug al @ p ql ỵql ul ag @ p qg al @ p ql ịqg ỵag @ p qg ịql ; a4 ẳ a @ p ql ịqg q q u u ị ; b2 ẳ ; b4 ẳ b1 ẳ a @ p q lịqg ỵgag @ pl q l l g g ịql qq a b3 ẳ a @ p q ịq l ỵgagg@ p q l ag @ p ðqg Þal @ p ðql Þðug Àul Þ al @ p ql ịqg ỵag @ p qg ịql ; q a @ p ðql Þag ; l g ỵag @ p qg ịql a3 ẳ a @ p qgịql l a2 ẳ l g g ịql l q a @ p qg ịal ; l g ỵag @ p qg ịql ql ag ug @ p qg ỵqg al ul @ p ql al @ p ðql Þqg þag @ p ðqg Þql ; ql qg al al @ p ql ịqg ỵag @ p qg ịql ; 201 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 we can write the Jacobian matrix of the system (2.22) as a1 Bb B B B0 B AVị ẳ B B0 B B0 @ a3 a4 b2 b3 b4 u 0 0 u 0 ul 0C C C 0C C C: 0C C 0C A 0 0 ul qg ql a2 ð2:23Þ A straightforward calculation shows that the characteristic polynomial det (A(V) À kI) for the matrix (2.23) is given by Pkị ẳ ðug À kÞðul À kÞQ ðkÞ ð2:24Þ where Q ðkÞ ¼ ða1 À kÞðb2 À kÞðug À kÞðul À kÞ a2 b1 ug kịul kị ỵ a4 b1 ql b3 qg ul kị ỵ b4 ql ! ug kị a1 kị ỵ a3 b1 qg ðul À kÞ ðug À kÞ: The polynomial P(k), however, may or may not have a complete set of real zeros This can be seen as the polynomial Q(k) may have or may not have four real zeros, as illustrated by the Fig Consequently, the system is hyperbolic in certain regions of the phase domain and is not hyperbolic in other regions Construction of the stable numerical scheme Since the gravity terms agqgg, alqlg are regular terms and not play any significant role, in the following we restrict our consideration to the case without gravity to simplify our argument This can be formally done by letting g = Let Dt and Dx be the given uniform time step and the spacial mesh size, respectively Set xj ¼ jDx; Denote by U nj k¼ Dt : Dx j Z; tn ¼ nDt; n N: the approximate value of the value U(xj, tn) of the exact solution U at the point (x, t) = (xj, tn) We also set Adding up the equations of balance law of momentum and re-arranging the equations, we obtain the following equivalent system @ t ag qg ị ỵ @ x ag qg ug ị ẳ 0; @ t ag qg Sg ị ỵ @ x ag qg Sg ug ị ẳ 0; @ t al ql ị ỵ @ x al ql ul ị ẳ 0; @ t al ql Sl ị ỵ @ x al ql Sl ul ị ẳ 0; @ t ag qg ug ỵ al ql ul ị ỵ @ x ag qg u2g ỵ pị ỵ al ql u2l ỵ pịị ẳ 0; @ t ag qg ug ị ỵ @ x ag qg u2g ỵ pịị ẳ p@ x ag : Fig Graphs of Q (k) defined by (2.25) show two cases: (a) there are four zeros (left); (b) there are only two zeros ð3:1Þ 202 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 The first five equations are conservative Thus, we can apply an appropriate standard numerical scheme for the first five conservative equations of (3.1) The last equation of (3.1) is nonconservative Motivated by our earlier works, [9,17], we discretize the source term of the last equation using the stationary waves in the gas phase Precisely, we denote 1 ag qg ug ag qg C C B B ag qg Sg ug ag qg Sg C C B B C C B B C; C; f Uị ẳ B a q u a q U¼B l l l l l C C B B C C B B al ql Sl ul al ql Sl A A @ @ 2 ag qg ug ỵ al ql ul ỵ p ag qg ug ỵ al ql ul V ẳ ag qg ug ị; gVị ẳ ag qg u2g ỵ pịị: 3:2ị We now dene a composite scheme   U nỵ1 ẳ U nj k FU nj ; U njỵ1 ị FðU njÀ1 ; U nj Þ ; j   V nỵ1 ẳ V nj k GV nj ; V njỵ1; ị GV nj1;ỵ ; V nj ị ; j ð3:3Þ where F and G are convenient standard numerical uxes The vectors V njỵ1;ặ in (3.3) will be determined as follows, relying on the previous result that the entropy Sg is conserved across stationary waves We thus compute qng;jỵ1; ; ung;jỵ1; from the equations ang;j qng;jỵ1; ung;jỵ1; ẳ ang;jỵ1 qng;jỵ1 ung;jỵ1 ; ung;jỵ1; ị2 ỵ hg qng;jỵ1; ; Sng;jỵ1 ị ẳ ung;jỵ1 ị2 3:4ị ỵ hg qng;jỵ1 ; Sng;jỵ1 ị; and compute qng;j1;ỵ ; ung;j1;ỵ from the equations ang;j qng;j1;ỵ ung;j1;ỵ ẳ ang;j1 qng;j1 ung;j1 ; ung;j1;ỵ ị2 ỵ hg qng;j1;ỵ ; Sng;j1 ị ẳ ung;j1 ị2 3:5ị ỵ hg qng;j1 ; Sng;j1 ị: For simplicity we omit the indices for the grids (j, n), j Z, n N in (3.4) and (3.5) In both cases we are led to resolve for (qg, ug) from the simultaneous nonlinear equations ag qg ug ¼ ag;0 qg;0 ug;0 :¼ mg ; ðug Þ2 ỵ hg qg ; Sg;0 ị ẳ ug;0 ị2 ỵ hg qg;0 ; Sg;0 ị :ẳ qg ; 3:6ị i ¼ g; l; Substituting ug = mg/agqg from the first equation of (3.6) to the second equation, after re-arranging terms, we obtain the following nonlinear equation for the density equation F g qg ị :ẳ u2g;0 ỵ 2hg qg;0 ; Sg;0 ÞÞq2g À 2q2g hg ðqg ; Sg;0 Þ À  ag;0 ug;0 qg;0 ag 2 ¼ 2q2g ðqg À hg ðqg ; Sg;0 ÞÞ À  2 mg ag ẳ 0: 3:7ị Let us investigate properties of the function Fg It is derived from (3.7) that F g qg ị ẳ 2u2g;0 ỵ 2hg qg;0 ; Sg;0 ịịqg À 4qg hg ðq; Sg;0 Þ À 2q2g hg;qg ðqg ; Sg;0 ị dqg ẳ 2u2g;0 ỵ 2hg qg;0 ; Sg;0 ÞÞqg À 4qg hg ðqg ; Sg;0 Þ À 2qg pq qg ; Sg;0 ị ẳ 2qg ẳ 2qg u2g;0 ỵ Z qg;0 pqg s; Sg;0 ị s qg ! u2g;0 ỵ2 Z qg;0 qg ! hqg s; Sg;0 Þds À pqg ðqg ; Sg;0 Þ ds À pqg ðqg ; Sg;0 Þ ; which has the same sign as Gg qg ị :ẳ U0 qg ị 2qg ẳ u2g;0 ỵ Z qg;0 pqg s; Sg;0 ị qg s ds À pqg ðqg ; Sg;0 Þ: Moreover, it is not difficult to check that 2pqg ðqg ; Sg;0 ị ỵ qg pqg qg qg ; Sg;0 ị > 0; i ẳ g; l; 3:8ị M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 203 This implies that  Gg qg ị  :ẳ 2hqg qg ; Sg;0 ị ỵ pqg qg qg ; Sg;0 ịị ẳ 2pqg qg ; Sg;0 ị ỵ qg pqg qg ðqg ; Sg;0 Þ < qg : dqg qg ð3:9Þ Moreover, it is easy to check that Gg qg ị > as qg ! 0ỵ; Gg ðqg Þ < for large qg ; i ¼ g; l: Hence, there exists exactly one value qg = qg,max such that Gg(qg,max) = 0, Gg(qg) > iff qg < qg,max, and Gg(qg) < iff qg > qg,max for i = g, l Moreover, a straightforward calculation shows that qg;max ¼ ql;max ¼  2qg cg cg ỵ1ị  2ql C lp exp exp   Sg à ÀSg;0 C gv Slà ÀSl;0 C lv c 1À1 c 1À1 l g ; ð3:10Þ : Consequently, F g ðqg Þ dqg > 0; qg < qg;max ; F g ðqg Þ dqg < 0; qg > qg;max ; F g qg ị dqg ẳ 0; qg ẳ qg;max ; i ẳ g; l: 3:11ị Observe that F g qg ẳ 0ị < 0; F g qg ị ! as qg ! ỵ1: 3:12ị Thus, (3.7) has a solution iff F g ðqg;max Þ P   ag;0 ug;0 qg;0 ; ag ð3:13Þ or, equivalently, ag;0 jug;0 jqg;0 ag P amin U g;0 ị :ẳ q : F g ðqg;max Þ ð3:14Þ In this case, we can easily see that (3.7), and, therefore, system (3.6), has two roots, denoted by ng,1(Ug,0, ag) ng,2(Ug,0, ag), which coincide iff ag = ag,min(Ug,0) The fact that ag = ag,0 still gives us a solution of (3.6) and, therefore, of (3.7), ag,0 has to satisfy ag;min ðU g;0 Þ ag;0 : Fig Test 1: usual discretization of the right-hand side does not give a satisfactory result 204 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 To make sure that the scheme always works, we practically assign the new value for ag by anew ¼ maxfag ; amin ðU g;0 Þg: g ð3:15Þ Test cases In this section, we will present several numerical tests For simplicity, we use the standard Lax–Friedrichs scheme for the two numerical uxes F and G in (3.3): U nỵ1 ẳ 12 U njỵ1 ỵ U nj1 ị 2k f U njỵ1 ị f U nj1 ịị; j V nỵ1 ẳ 12 V njỵ1; ỵ V nj1;ỵ ị 2k gV njỵ1; ị gV nj1;ỵ ịị: j ð4:1Þ For comparison purposes, we take the classical Lax–Friedrichs scheme with a usual discretization of the right-hand side, says, the central difference formula The solution will be computed on the interval [À1, 1] of the x-space for 300 mesh points and at the time t = 0.1 All the tests show that the classical scheme is unstable, but our adaptive scheme is stable In fact, let us consider the Riemann problem for the system (1.1) with the Riemann data ðag;0 ; p0 ; ug;0 ; Sg;0 ; ul;0 ; Sl;0 ÞðxÞ ¼  if x 0; UL ; where UL, UR are given for each test Fig Test 1: adaptive scheme (3.3) can give a stable solution Fig Test 2: usual discretization of the right-hand side does not give a satisfactory result M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 205 Test We can take Sgà ¼ 0:2; Slà ¼ 0; and U L ¼ ð0:7; 1; 1; 1; 1; 1ị; U R ẳ 0:8; 1:1; 1:1; 1:1; 1:2; 1:1Þ: ð4:2Þ Fig shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result.Fig indicates that our adaptive scheme (3.3) gives stable solutions Test We may choose Sgà ¼ 0; Slà ¼ 0:1; and U L ¼ ð0:6; 1:1; 1; 0:5; 0:5; 0:5ị; U R ẳ 0:5; 1; 1:2; 0:6; 0:8; 0:8Þ: Fig Test 2: adaptive scheme (3.3) can give a stable solution Fig Test 3: usual discretization of the right-hand side does not give a satisfactory result ð4:3Þ 206 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Fig shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result Fig indicates that our adaptive scheme (3.3) gives stable solutions Test We choose Sgà ¼ 0; Slà ¼ 0; and U L ¼ ð0:2; 0:9; 1; 0:3; 0:8; 0:5Þ; U R ¼ ð0:3; 1; 0:3; 0:2; 1; 0:4Þ: ð4:4Þ Fig shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result Fig indicates that our adaptive scheme (3.3) gives stable solutions Test We can take Sgà ¼ 2; Slà ¼ 1; Fig Test 3: adaptive scheme (3.3) can give a stable solution Fig Test 4: usual discretization of the right-hand side does not give a satisfactory result M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 207 and U L ẳ 0:5; 1; 1; 0:5; 0:1; 0:5ị; U R ẳ 0:4; 1; 0:9; 0:6; 0:2; 0:4ị: 4:5ị Fig shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result Fig indicates that our adaptive scheme (3.3) gives stable solutions Test We can take Sgà ¼ 0:2; Slà ¼ À0:1; and U L ¼ ð0:5; 2; 1; 0:5; 1; 0:5ị; U R ẳ 0:6; 2:1; 1:1; 0:6; 1:2; 0:4Þ: ð4:6Þ Fig 10 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result Fig 11 indicates that our adaptive scheme (3.3) gives stable solutions Fig Test 4: adaptive scheme (3.3) can give a stable solution Fig 10 Test 5: usual discretization of the right-hand side does not give a satisfactory result 208 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Test We can take Sgà ¼ 0:2; Slà ¼ À0:1; and U L ¼ 0:5; 2; 1; 0:5; 1; 0:5ị; U R ẳ 0:4; 2:2; 0:9; 0:6; 1:1; 0:4Þ: ð4:7Þ Fig 12 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result Fig 13 indicates that our adaptive scheme (3.3) gives stable solutions Test In this test, let us take Sgà ¼ À0:1; Slà ¼ 0:1; and U L ¼ ð0:5; 1:2; 1; 0:5; 0:5; 0:5ị; U R ẳ 0:4; 1; 1:1; 0:6; 0:8; 0:8Þ: ð4:8Þ We compute the solution at t = 0.1 and x [À1, 1] with different meshes: 300, 1000, and 2000 mesh points The test shows that the approximate solutions are numerically stable, see Figs 14–17 Fig 11 Test 5: adaptive scheme (3.3) can give a stable solution Fig 12 Test 6: usual discretization of the right-hand side does not give a satisfactory result M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Fig 13 Test 6: adaptive scheme (3.3) can give a stable solution Fig 14 Test 7: the volume fraction computed at t = 0.1 and x [À1, 1] with 300, 1000, and 2000 mesh points Fig 15 Test 7: the pressure computed at t = 0.1 and x [À1, 1] with 300, 1000, and 2000 mesh points 209 210 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Fig 16 Test 7: the gas velocity computed at t = 0.1 and x [À1, 1] with 300, 1000, and 2000 mesh points Fig 17 Test 7: the liquid velocity computed at t = 0.1 and x [À1, 1] with 300, 1000, and 2000 mesh points Thus, we have seen by the above tests that our numerical scheme is stable It generates approximate solutions which are numerically convergent Conclusions Two-fluid models of two-phase flows possess complicated properties and raise challenging problems in both theoretical study of the analytic solutions as well as in numerical treatments This is mainly because the system is not hyperbolic and the analytic form of characteristic fields is not available The Riemann problem cannot be solved by the standard way and the conventional Godunov-type schemes cannot be applied In this paper, we derive several equivalent forms of the two-fluid model, and we build an adaptive scheme for the two-fluid model of two-phase compressible flows Many numerical tests show that our scheme is stable and numerically convergent This gain gives a convenient way to deal with numerical approximations of two-fluid model of two-phase flows, since standard numerical schemes cannot efficiently work for these models References [1] Botchorishvili R, Perthame B, Vasseur A Equilibrium schemes for scalar conservation laws with stiff sources Math Comput 2003;72:131–57 [2] Chang CH, Liou MS A conservative compressible multifluid model for multiphase flow: shock-interface interaction problems In: 17th AIAA computational fluid dynamics conference, June 6-9, 2005, Toronto, Ontario, Canada, American Institute of Aeronautics and Astronautics Paper, 20055344, 2005 [3] Dal Maso G, LeFloch PG, Murat F Definition and weak stability of nonconservative products J Math Pures Appl 1995;74:483–548 M.D Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 211 [4] Garcià-Cascales JR, Paillère H Application of AUSM schemes to multi-dimensional compressible two-phase flow problems Nucl Eng Design 2006;236:1225–39 [5] Greenberg JM, Leroux AY A well-balanced scheme for the numerical processing of source terms in hyperbolic equations SIAM J Numer Anal 1996;33:1–16 [6] García-Cascales JR, Paillère H Application of ausm schemes to multi-dimensional compressible two-phase flow problems Nucl Eng Design 2006;236(12):1225–39 [7] Ishii M Thermo-fluid dynamic theory of two-phase flow Paris: Eyrolles; 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Toronto, Ontario, Canada, American Institute of Aeronautics and Astronautics Paper, 20055344, 2005 [3] Dal Maso G, LeFloch PG, Murat F Definition and weak stability of nonconservative products J Math... show that our scheme is stable and numerically convergent This gain gives a convenient way to deal with numerical approximations of two-fluid model of two-phase flows, since standard numerical schemes... Equations of states Each fluid is characterized by its equation of state In this section, we will establish the equations of state in terms of the pressure and the entropy as the thermodynamical