Brazilian Journal
of Chemical ISSN 0104-6632 Printed in Brazil Engineering www.abeq.org.br/bjche
Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
A COMPARISON OF HYPERBOLIC SOLVERS II: AUSM-TYPE AND HYBRID LAX-WENDROFF- LAX-FRIEDRICHS METHODS FOR TWO-PHASE FLOWS
R. M. L. Coelho1*, P. L. C. Lage2 and A. Silva Telles1
1Curso de Pós-Graduação em Tecnologia de Processos Químicos e Bioquímicos (TPQB), Departamento de Engenharia Química, Escola de Química, Bloco E, Centro de Tecnologia, Universidade Federal do Rio de Janeiro, CEP: 21949-900, Rio de Janeiro - RJ, Brazil. 2Programa de Engenharia Química, COPPE, Universidade Federal do Rio de Janeiro, P.O. Box 68502, CEP: 21945-970, Rio de Janeiro - RJ, Brazil. *Current address: Research and Development Center (PETROBRAS /CENPES/EB-E&P/EPEP), Phone: + (55) (21) 3865-7406, Fax: + (55) (21) 3865-6793, Cidade Universitária, Q.7, Ilha do Fundão, CEP: 21949-900, Rio de Janeiro - RJ, Brazil. E-mail: [email protected]
(Submitted: October 23, 2008 ; Revised: October 30, 2009 ; Accepted: January 4, 2010)
Abstract - Riemann-solver based schemes are difficult and sometimes impossible to be applied for complex flows due to the required average state. Other methods that do not use Riemann-solvers are best suited for such cases. Among them, AUSM+, AUSMDV and the recently proposed Hybrid Lax-Friedrichs-Lax-Wendroff (HLFW) have been extended to two-phase flows. The eigenstructure of the two-fluid model is complex due to the phase interactions, leading to numerous numerical difficulties. One of them is the well-posedness of the equation system because it may lose hyperbolicity. Therefore, the methods that are not based on the wave structure and that are not TVNI could lead to strong oscillations. The common strategy to handle this problem is the adoption of a pressure correction due to interfacial effects. In this work, this procedure was applied to HLFW and AUSM-type methods and their results analyzed. The AUSM+ and AUSMDV were extended to achieve second-order using the MUSCL strategy for which a conservative and a non-conservative formulation were tested. Additionally, several AUSMDV weighting functions were compared. The first and second-order AUSM-type and HLFW methods were compared for the solution of the water faucet and the shock tube benchmark problems. The pressure correction strategy was efficient to ensure hyperbolicity, but numerical diffusion increased. The MUSCL AUSMDV and HLFW methods with pressure correction strategy were, on average, the best of the analyzed methods for these test problems. The HLFW was more stable than the other methods when the pressure correction was considered. Keywords: Hyperbolic conservation laws; AUSM+; AUMSDV; Hybrid schemes; Two-phase flow; Two-fluid model.
INTRODUCTION macroscopic relations, such as an equation of state (EOS) and interfacial flux equations, which represent The simulation of two-phase flows is important in the behavior of the associated microscopic many industrial applications and it is governed by phenomena. These relations strongly influence the the conservation of mass, momentum and energy and structure and dynamics of the waves in the flow. the associated constitutive equations. The later are Molecular dynamic models have been locally applied
*To whom correspondence should be addressed
154 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles to model the interfaces in two-phase flows. However, are several successful methods that account for the it is not feasible to apply it to the whole flow domain. flow wave structure through a superposition of Therefore, average Navier-Stokes models have been Riemann solutions. These are local solutions of the used in which the singular interface with its appropriate hyperbolic equations which are composed of jump conditions are averaged. This approach has been elementary waves (Menikoff and Plhor, 1989). These proposed in Ishii (1975) in the classical two-fluid methods are based on characteristic upwind model. The two-fluid model is based on field averaging differencing and have been applied from nuclear and and accounts for both mechanical and thermal non- oil industries to solid combustion. The emergence of equilibrium, being able to deal with a wide range of TVNI (total variation non increasing, also popularly flow patterns. The two-fluid model includes the known as TVD, total variation diminishing) methods equations for mass, momentum and energy (Harten, 1983) made the accurate capture of conservation for each phase. It is possible to sum the discontinuities possible. The pioneer method was phase momentum equations to obtain a single equation that of Godunov (Godunov, 1959) which is based on for the whole mixture which requires additional the exact solution of the Riemann problem at the relations for the phase slip velocity. This is the idea interfaces. Its advantage is the assurance of the behind the drift-flux model. It is possible to further density, pressure and volume fraction positivity, but simplify the model by eliminating the acceleration its drawbacks are its large computer cost and the lack terms (no pressure wave model Masella et al., 1998), in of generality due to its requirement of an analytical which the flow is driven by the mass transport. integration of the Riemann invariants. Frequently, the validation limits of the EOS’s Flux difference splitting (FDS) schemes are used are violated at the interfaces and the variables modifications of classical upwind schemes. Such evaluated from them are inaccurate or physically schemes are accurate for all speed flows and two- unrealistic (negative pressures, densities and volume phase flows but time consuming due to the repeated fractions). Besides, in the case of multicomponent evaluation of the Jacobian matrix of the flux with mixtures, the notion of partial pressure is valid only respect to the conservative variables and its with the local thermodynamic equilibrium eigenstructure and they do not guarantee positivity of assumption, which is acceptable when all density and pressure. These schemes are components are perfectly mixed and the number of approximate Riemann solvers and the most popular molecular collisions is so large that temperature is one is due to Roe (Roe, 1981). The generalization of rapidly homogenized. When two fluids are not the FDS scheme is difficult because it is not always perfectly mixed, there is an interface which is the possible to analytically obtain the necessary locus of the collisions between their molecules. expressions for the average state in which the Therefore, at an interface the local thermodynamic Jacobian matrix must be evaluated. This class of equilibrium does not hold anymore. It is expected methods was adopted by several authors that the interface propagates with an intermediate (Sainsaulieu, 1995; Toumi, 1996, Toumi and velocity and has an intermediate pressure which shall Kumbaro 1996; Tiselj and Petelin, 1997; Fjelde and be modeled. This implies the presence of non- Karlsen, 2002; Cortes et al., 1998; Romate, 1998; conservative terms in the two-fluid model which Faille, 1999; Saurel and Abgrall, 1999). must be carefully modeled to achieve a stable The flux vector splitting schemes (FVS) are hyperbolic model. Otherwise, the two-fluid model simplifications of the FDS schemes and are based on becomes ill-posed, resulting in numerical instabilities scalar rather than matrix calculations. Therefore, during its solution. these methods are less time consuming but are more The conservation equations for advection- diffusive. Recently, great effort has been spent on dominated two-phase flow problems can be written developing hybrid schemes in order to achieve FVS as a set of conservation equations whose solution for efficiency with FDS accuracy. These methods are the flow variables may present large gradients or well-known as Advection Upstream Splitting even discontinuities. Most of the existing Methods (AUSM). The AUSM scheme (Liou and commercial codes, such as OLGA (Bendiksen et al., Steffen, 1993) uses the idea of splitting the flux into 1991), are based on very diffusive simple upwind a convective part upwinded in the flow direction and pressure-based solvers which may lead to unrealistic a pressure part upwinded according to the acoustic solutions due to their failure in solving the flow properties of the flow. The flow direction is wave field. These methods have limited accuracy determined by the Mach number sign considering compared to density-based solvers when strong information for the left and right state for a given cell compressibility is present. On the other hand, there face. This approach allows the capture of stationary
Brazilian Journal of Chemical Engineering
A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 155 contact discontinuity with reduced numerical diffusion in the whole cell instead of the common approach even with first-order discretization. However, for where the dissipation is evaluated at the interface. oblique shocks (not aligned with the mesh), the AUSM Schemes like hybrid Lax-Friedrichs Lax-Wendroff method admits highly non-monotone solutions for (De Vuyst , 2004), AUSM (Liou, and Steffen, 1993), pressure and density, leading to difficulties in AUSM+ (Liou, 1996) and AUSMDV (Wada and convergence. Thereafter, Wada and Liou (1997) Liou, 1997, Liou, 200) are not based on the proposed a method called AUSMDV, which mitigates characteristic analysis of the flow and do not require the occurrence of non-monotonic solutions allowing the model to be hyperbolic. However, as shall be robust resolution of shock contact (steady and moving) shown in this work, their accuracy is improved if the waves but introduced the carbuncle effect. Recently, model is hyperbolic. Liou (1996) proposed a modified scheme called Another class of methods not studied here are the AUSM+, which is capable of solving contact relaxation schemes (Evje and Fjelde, 2002), which discontinuities exactly without non-monotonic are capable of dealing with strong non-linearities. problems. Besides, the method preserves the positivity However, they require a case-by-case analysis of the of pressure, density and volume fraction. sub-characteristic stability based on entropy The application of the density-based solvers to the compatibility or Chapman-Enskog-like expansion two-fluid model is not simple due to the non-hyperbolic (De Vuyst, 2004). nature of the two-fluid model, the presence of non- Many other authors adopted different methods in conservative terms, stiff source terms, complex order to solve two-phase flows. Recently, Lorentzen interfacial relations, phase appearance and and Fjelde (2005) applied a slope limiter to the finite disappearance and phase change. These methods also element predictor-corrector shooting technique for the experience loss of accuracy for low Mach number drift-flux model. Wackers and Koren (2005) applied a flows. In such cases, AUSM+ behaves more like a HLL Riemann Solver to a new model with five central difference discretization scheme. This may lead equations. Cascales and Salvador (2006) applied a high to odd-even decoupling and it is then necessary to resolution method and a slope limiter to the two-fluid couple velocity and pressure (Liou, 1999). The larger model. Guinot (2001a, b) applied the Godunov method the difference between the speed of sound and the to solve a simplified two-phase flow model. Castro and velocity, the stronger the odd-even decoupling effect is. Toro (2006) compared a Riemann Solver and a second- Some strategies have been proposed for dealing with order extension using MUSCL and ADER strategies to one and two-phase low Mach number flows in the the Saurel and Abgrall (1999) five equation model. AUSM+ context (Liou, 1999 and Paillère et al., 2003). Bedjaoui and Sainsaulieu (1997) examined the effects AUSM+ has been used in many works for two-phase of diffusion and relaxation parameters on the stability drift flux (Liou, 1999, Paillère et al., 1999, Fjelde, of the solution of a non-hyperbolic two-fluid model, 2002, Evje and Fjelde, 2003) and two-fluid (Niu, 2001, proving its stability depending upon the initial Paillère et al., 2003, Evje and flatten, 2003, De Vuyst, conditions. 2004) models. As the ideal EOS is not capable of representing It is well-known that first-order schemes tend to non-classical waves, many numerical methods were present numerical diffusion, while the second-order recently extended to real materials. In our previous schemes tend to present oscillations near high work (Coelho at al., 2006), a comparison was made gradient regions. De Vuyst (2004) presented a novel regarding the accuracy and CPU time consumption hybrid method and applied it to some one-phase flow of second-order extensions of Roe’s Riemann solver, problems and to the water faucet two-phase flow VFRoe, AUSM+ and Hybrid Lax-Friedrichs Lax- problem. The idea of this method (HLFW) is to Wendroff for ideal and real gases modeled with evaluate the flux with a weighted combination of cubic equations of state, including non-classical second-order Lax-Wendroff and first-order Lax- wave phenomena. Friedrichs schemes. The weighting parameter is the In this work, the AUSM-type and Hybrid Lax- key to the success of this type of methods. The Wendroff-Lax-Friedrichs methods were compared for author proposed a concept of numerical dissipation two-phase flow simulations of benchmark problems. A by convexity in which the entropy and its flux are stiffened gas EOS was adopted in order to make not directly used for the dissipation evaluation. This possible comparison with some results presented in the allows the dissipation to be expressed as a simple literature. This paper is organized as follows: section 2 convex function, but also allows some entropy is a summary of the two-fluid model equations. In violation. The proposed dissipation term is evaluated section 3, a summary of the AUSM+, AUSMDV and
Brazilian Journal of Chemical Engineering Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
156 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles
Hybrid Lax-Friedrichs-Lax-Wendroff (HLFW) where ρ is the density, v is the velocity, pi is the methods and the MUSCL (Monotone Upstream- k Centered Scheme for Conservation Laws) strategy due interface pressure, which represents the effects of to Van Leer (1977 and 1979) are presented. In section surface tension and low wavelength dynamics, α is D 4, the discretization schemes of additional terms in the the volume fraction, Fk represents the interfacial two-fluid model are described. In section 5, the results and wall drag forces, gx is the gravity acceleration of the application of these methods to two-phase flows component along the x-coordinate and subscripts g are discussed. The methods were compared in terms of and l refer to gas and liquid phases, respectively. accuracy for different mesh sizes. The usage of the Defining Δpi = p - pi the following pressure correction term that enforces hyperbolicity and k k k of different expressions for the weighting function of transformation could be applied: the AUSMDV scheme was also analyzed. Additionally, ∂(p)α∂α∂ ∂ pi MUSCL strategies based on primitive and conservative kk−=αΔ+αppii k⎡⎤ k (4) variables were compared. ∂xxxkkkk∂∂⎣⎦ ∂ x
leading to the following form: THE TWO-FLUID MODEL t The set of hyperbolic one-dimensional conservation u[=αρgg , αρ ggg v, αραρ ll , lll v], laws is generically written as: 2i f(u)=αρ [ggg v , α g ( ρ gg v +Δ p g ), ∂∂uf(u) +=g(u) 2it ∂∂tx αρlllv, α l ( ρ ll v +Δ p)] l (5) (1) u(x,0)= u (x) 0 (u)=−α∂∂++αρ [0, piD / x F g ,0, g gg g ggx where f(u) is the physical flux and g(u) represents iD a source/sink term, like the friction loss. The −αll ∂p/ ∂ x + α llxl ρ g + F ] linearized form of Eq. (1) is given by: In the present work, the pressures and the interfacial ∂∂uu ∂ f(u) pressures were assumed to be equal for both phases, +=A(u) g(u), A(u) = (2) ii i ∂∂tx ∂ u that is, pp= gl= p and pp==gl p. The authors in the references (Toumi, 1996, Cortes et al., 1998, where A is the Jacobian matrix. The Riemann Evje and Fjelde, 2003) used a perturbation method in solvers require the calculation of the eigenvalues and order to obtain the approximate eigenvalues of the eigenvectors of the Jacobian matrix, which is system. It is important to highlight that, if Δp0= , evaluated with an average state. This average state is the model becomes non-hyperbolic due to the difficult and sometimes impossible to be obtained. existence of complex eigenvalues. The system of equations of the two-fluid model t (Ishii, 1975) applied to two-phase flow and with no Let w,v,v,p=α⎣⎡ ggl⎦⎤ be the vector of non- source/sink terms could be written in the form of Eq. conservative variables. It is necessary to obtain the (1) as follows: transformation of u into w and vice-versa. The following equation is a relation between u and the t u[=αρgg , αρ ggg v, αραρ ll , lll v] , phase velocities:
2 f(u)=αρ [ggg v , α g ( ρ gg v + p g ), vg21l43= u / u , v= u / u (6)
2t αρlllv, α l ( ρ ll v + p)] l (3) Besides, iD g(u)=∂α∂++αρ [0,pggggx / x F g ,0, g αρgg(p) = u 1 (7) p/xgF]iD∂α ∂ + α ρ + l lllxl αρll(p)=−α (1 g )ρ l (p)= u 3
Brazilian Journal of Chemical Engineering
A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 157
These equations lead to the following non-linear the AUSM method. relation, which can be solved to give the pressure The AUSM+ is as accurate as the flux splitting and, thus, the densities: methods due to Roe (1981) without the wave decomposition, leading to a large reduction of CPU ⎡⎤ time consumption and making the formulation u1 ⎢⎥1(p)u0− ρl3−= (8) simple and quite general. The AUSM+ is able to ⎣⎦⎢⎥ρg (p) solve the contact discontinuities exactly, making it suitable for viscous flow calculations. The transformation of w into u is trivial. The The AUSM+ flux for the system given by eqs. (1) complete description of the system is accomplished and (3) is given by the following expression, where by specifying an EOS. For an isentropic evolution, the pressure and mass flux parts are separated: the following relation was assumed: AUSM 2t F(u)[v;(vp)]k,j+ 1/2 = αρkkk α k ρ kk + k = γk pAkk=ρ−Πk k (9) j1/2+ j1/2++ j1/2 mPkkΨ+ k where γ is the ratio of the specific heats and Ak j1/2++j1/2t+ j1/2 Ψ=kkk[1; v ] ; P = (11) and Πk are constants depending upon the fluid. This EOS reproduces the stiffened gas EOS, j1/2j1/2t++ [0;αkk p ] p(/c)(/c)=ργγkk −ρ0 , if A(1/c)= γk and kk k k AUSM AUSM AUSM t 0 γk F(u)[F;F]j+++ 1/2= g,j 1/2 l,j 1/2 Π=ρk (/c). k where the subscript k indicates the kth phase (gas, g, or liquid, l) and the following upwind formula based AUSM+, AUSMDV AND THE HYBRID LAX- j1/2+ j1/2+ FRIEDRICHS-LAX-WENDROFF METHODS on the sign of mk is used to compute Ψk (Paillère et al., 2003):
We are interested in solving the Finite Volume m j1/2+ m(u)(u) j1/2+ Ψ j1/2+ =Ψ+Ψ+k ⎡ L R ⎤ discretization of Eq. (1) with the following form: kk 2 ⎣ kkkk⎦ (12) tt+Δ t t m j1/2+ ujj=−ε u [Fj+− 1/2 (u) − F j 1/2 (u)] +Δ t g(u j ) (10) k ⎡⎤LR Ψ−Ψkk(ukk ) (u ) 2 ⎣⎦ where ε=Δt/Δx, F j1/2+ and F j1/2− are the numerical using the m j1/2+ value obtained by the following flux calculated at the interface j-1/2 and j+1/2 of the k volume j. For the Euler explicit the numerical flux is steps: evaluated at the t time level and for the Euler implicit it is evaluated at the t+Δt time level. (i) Evaluation of left and right Mach numbers calculated by using a mean sound speed at the AUSM Type Methods interface:
Liou and Steffen (1993) proposed a scheme based j1/2++ L R L,R L,R j1/2 ccc,Mv/ckkkk==kk (13) on the separation of a convective flux and a pressure flux, presenting some drawbacks, especially in the (ii) Then, the interface pressure is calculated by case of strong shocks. Wada and Liou (1997) averaging the left and right pressures: presented an improved scheme called AUSMDV, which is a blend of the FDS and FVS methods. j1/2j1/2++ + L L However, AUSMDV does not have the property of αkkp(M)(p)=℘kk α + (14) capturing a stationary shock. Thereafter, Liou (1996) − RR presented a novel scheme called AUSM+, which is ℘α(Mkk )( p) capable of capturing stationary shock and contact ± discontinuity waves, preserves the positivity of where ℘ are consistent, differentiable and symmetric pressure and density and improves the accuracy of polynomial functions.
Brazilian Journal of Chemical Engineering Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
158 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles
(iii) the interface Mach number is calculated as a These polynomials were developed in order to polynomial function of the Mach numbers given in satisfy some properties (see Liou 1996, for details). the first step: They are responsible for the more general characteristic of the AUSM+ scheme. Liou (1996) j1/2++ L − R ± M(M)(M)k =μkk +μ (15) adopted the criterion in which μ (M) has an inflection point at M0= and ℘± (M) has two where the superscripts “+” and “–” are associated inflection points for M= ± 1, leading to A = 3/16 with the right and left running waves, respectively. and B = 1/8. These parameter values could also be changed in order to improve AUSM+ results, but (iv) finally, the mass flux is given as follows: they were left fixed as suggested in Liou (1996). The AUSM+ suffers of odd-even decoupling for ⎡⎤()(MαρLj1/2j1/2++ + |M|) + low Mach numbers, losing accuracy. This is j++ 1/2 j 1/2 ⎢⎥k kk m0.5c kk= (16) particularly important for liquid flows where the sound ⎢⎥R j++ 1/2 j 1/2 ⎣⎦()(Mαρk kk − |M|) speed is usually three orders of magnitude higher than the velocity. As cited in the introduction, density-based The common speed of sound defined in Eq. (12) solvers exhibit a stiffness problem for low Mach is the key to the AUSM+ accuracy instead of the number where it is necessary to couple pressure and density. For this purpose, a cut-off procedure was simple upwind ( cc,forj1/2++++=≥= j m0, j1/2 cc j1/2 j1, kkk kk adopted by Paillère et al. (2003) in order to limit the otherwise). Liou (1996) presented a correct sound speed and consequently the Mach number. This expression for the interface sound speed based on the procedure was applied to the liquid phase in the present isoenergetic condition. This involves an EOS work and is explained in the following. dependent expression. In this paper this was avoid by using Eq. (12), as suggested by the author, which 1) Modify Eq. (16) to add a pressure diffusion term does not guarantee the exact capture of stationary for the liquid phase given by: shocks. Besides, the first RHS term of Eq. (11) is a weighted Mach number averaged flux. The j1/2++ j1/2 mmm ll=+p (21) j1/2+ coefficient mk is merely a scalar, reducing CPU demand for the flux evaluation and keeping the 2) Redefine the interface numerical speed of sound, algorithm free of jacobian evaluations. The original Eq. (13), to: AUSM+ polynomials are: ⎛⎞LR j1/2++ j1/21u⎡⎤ll u L R ± cf =+⎜⎟⎢⎥ cc,ll M(M)(M1 =± |M|)/2 (17) l ⎜⎟LR ⎝⎠2 ⎣⎦ccll
± ⎪⎧M (M), if | M |≥ 1 Mv/c L,R= L,R j+ 1/2 (22) M(M)± = 1 (18) lll 2 ⎨ 2 ⎩⎪±±(M 1) / 4, otherwise (1−+ M22 ) M 2 4M 2 j1/2+ 00 f(M)= 2 ± 1M+ 0 ⎧M1 (M), ⎪if | M |≥ 1 ⎪ where M is a cut-off Mach number, say 10-4. μ=± (M) (19) 0 ⎨ ⎪M(M)116BM(M),± ⎡⎤∓ ∓ ⎪ 22⎣⎦ 3) compute ⎩otherwise j1/2++ L + L − R + R M(M)M(M)(M)M(M)=μl1l − −μ l1l + . ± ⎧M1 (M) / M, ⎪if |M |≥ 1 4) evaluate pressure diffusion term as: ± ⎪ ℘ (M) = ⎨ (20) ± ∓ ⎛⎞ LR ⎪±−M(M)2⎡⎤∓ M 16AMM(M), 11j1/2++ j1/2 α−αlL pp lR 22⎣⎦mc =−⎜⎟ 1M (23) ⎪ p 2 ll⎜⎟2j1/22 + ⎩otherwise ⎝⎠M(c)0 l
Brazilian Journal of Chemical Engineering
A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 159
This last term adds the necessary dissipation relaxation parameter χ for each phase k can be scaled for low Mach numbers. written as: Before the AUSM+ formulation, the authors in Wada and Liou (1997) presented an improvement to LW L R F(u,u,)0.5ffχ =+− AUSM creating the AUSMDV scheme. This method k,j+ 1/2k,j k,j+ 1 ( k k ) presented some problems with the carbuncle effect (27) RL which were solved in AUSM+. However, for one- ⎡f(u,u)φ + χ (ff)−−⎤ ε ⎢ ()k,j k,j+ 1 kk⎥ dimensional problems, we believe it remains − ⎢ ⎥ desirable and the results confirmed this expectation. 2χ ⎢f(u,u)()φ k,j k,j+ 1 ⎥ The scheme is obtained by the substitution of Eq. ⎣ ⎦ (16) with the following expression: L R where ff(u)k = k,j , ff(u)k = k,j+ 1 and φ is a ⎡⎤L + ()Mαρkk k,j1/2+ + consistent mean state function, that is, it has the mc j1/2++= j1/2⎢⎥ (24) kk⎢⎥ property that φ(u,u)= u . In the present work, a ()M)αρ R − ⎣⎦⎢⎥kk k,j1/2+ -2 constant value of χ = -10 and φ(u,v)=+ 0.5(u v) where: were used as suggested by the author. The modified Lax-Friedrichs numerical flux is given by: L,R ± 2Y w1/2 = LR (25) MLF L R YY+ F(u,u)0.5ffk,j+ 1/2k,j k,j+ 1 = ( k+− k ) (28) Additionally: 1 uu− 4ε ()k,j+ 1 k,j ±±±L,R Mw(M)k,j1/2++=μ j1/2 k + (26) Then, the weighted flux at tn is given by: ±±L,R 1w− j1/2+ )M(M) 1 k nnn F(u,u)k,j+ 1/2 k,j k,j+ 1 = Wada and Liou (1997) adopted Y = p/ρ. n MLF n n Thereafter, Liou (2000) suggested Y = 1/ρ in order θ+k,j+ 1/2 F(u,u)k,j+ 1/2 k,j k,j+ 1 (29) to eliminate shock instability (carbuncle n LW n n phenomenon) and Evje and Flatten (2003) adopted (1− θk,j+ 1/2 )Fk,j+ 1/2 (uk,j ,u k,j+ 1 ,χ ) Y=ρ/α. One should bear in mind that the shock instability is not important in the one-dimensional nnn where the weighting factor, θ=θθmax( , ) , problems analyzed in this paper. In the analysis of k,j++ 1/2 k,j k,j 1 the results, the difference between these options will depends upon the local dissipation function, η, be further discussed. which is defined as:
n1+ n n1+ n Hybrid Lax-Wendroff-Lax Friedrichs Method ηθ=k,j( k,j ) S(uk,j ) − S(u k,j ) +
De Vuyst (2004) proposed a novel approach for ⎧G(u,u)fFnnnnn⎡⎤−+⎫ numerical solution of hyperbolic systems using a ⎪ k,j−− 1/2 k,j 1 k,j ⎣⎦k,j k,j− 1/2 ⎪ +ε hybrid method that combines first and second order ⎨ ⎬ ⎪G(u,u)Ffnnnn⎡⎤− n⎪ numerical fluxes. The author presented the basis for ⎩⎭k,j++ 1/2 k,j k,j 1 ⎣⎦k,j+ 1/2 k,j generalizing the calculation of the optimal weighting (30) nn2 nnn of such fluxes. The weight is calculated from a S(uk,j )= || u k,j || /2, G k,j++ 1/2 (u k,j ,u k,j 1 ) = proposed concept of local dissipation by a convexity function which differs from the traditional entropy ∇+S(unn u )/2(uu)/2=+nn dissipation functions because it does not need an u ()k,j k,j+ 1 k,j+ 1 k,j entropy flux expression and allows some relaxation of the entropy inequality (e.g., entropy violation). which, in its turn, depends on the numerical flux The author used the Lax-Wendroff and the Lax- given by Eq. (28) and on the final updated field, n1+ Friedrichs fluxes. The Lax-Wendroff flux with a uk,j . The method was implemented by the
Brazilian Journal of Chemical Engineering Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
160 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles algorithm suggested in De Vuyst (2004) to achieve η For an approximate Riemann solver, the last step n is the solution of the Riemann problem at the "j+1/2" ≤ 0, which calculates θk,j as the smallest values that R L n1+ n face with the values U j e U j1+ as left and right make ηk,j negative. Let Δθk,j > 0 be a small n,q values, respectively. For the AUSM type methods increment and build the sequences {θk,j } as follows: R L the values of U e U substitute the values of Uj j j1+ n,.0 and Uj+1 in the flux evaluation. The Δ j values are let q = 0 and θ=k,j 0 and obtained as follows: n1+ n,q n,q while ηθ>ξk,j ( k,j ) and θk,j ≤ 1 do ⎧max[0,min(ΓΔj−+ 1/2 , Δ j 1/2 ), ⎪ n,q+ 1 n,q n ⎪ θ=θ+Δθk,j k,j k,j min(Δ j−+ 1/2 ,ΓΔ j 1/2 )] → Δ j + 1/2 > 0 ⎪ Δ= qq1=+ j ⎨min[0,max(ΓΔ , Δ ), (33) ⎪ j−+ 1/2 j 1/2 nn,q ⎪ set θ=θk,j k,j and calculate ⎪max(Δ j−+ 1/2 ,ΓΔ j 1/2 )] → Δ j + 1/2 < 0 ⎩⎪ nnn Δ=−uu, Δ=− u u θ=θθk,j++ 1/2max( k,j , k,j 1 ) j−+ 1/2jj1−+ j 1/2 j1j evaluate the numerical flux, Eq. (29) where Γ = 1 and 2 leads to the MINMOD (MINBEE) and SUPERBEE schemes, respectively. calculate the updated field, Eq. (10) The MUSCL data reconstruction can be applied evaluate ηθn1+ ( n,q ) to primitive variables instead of conservative k,j k,j variables, but not when using the previously shown algorithm, because Eq. (32) requires that the end while. dimension of the reconstructed data vector be equal where ξ is a small positive number. A Newton- to the flux vector dimension which, in turn, is equal Raphson like algorithm could reduce CPU demand. to the number of conservative variables. However, in In Eq. (29), S is the convexity function and G its some tests, the MUSCL-Hancock applied to AUSM+ derivative. presented too much pressure oscillation. Therefore, a primitive variable based data reconstruction Extension of AUSM-Type Methods to Second-Order algorithm was also tested. A primitive-variable version of the MUSCL-Hancock algorithm can be The extension of the present methods to second- found in Toro (1999), but it requires the Jacobian order follows Van Leer (1977, 1979) and the matrix evaluation. Therefore, a slope limiter scheme MUSCL method modified later by Van Leer (1985) was adopted. It consists of Eq. (31) with W (the (MUSCL-Hancock, see Toro, 1999, for details). The vector of primitive variables) in substitution to U. first step is called data reconstruction and consists of Finally, the following expression for Δ instead of changing the u value as follows: j Eq. (33): LL Uujj=−Δj /2 (31) Δ=Θjj(r) Δ RR Uujj=+Δj /2 Δ=j(0.5 + 0.5 ϖΔ ) j−+ 1/2 + (0.5 − 0.5 ϖΔ ) j 1/2 (34)
The second step is the evolution step given by: r/=Δj−+ 1/2 Δ j 1/2
L LLRε UU=+⎡⎤ F(U)F(U) − where Δ j1/2− and Δ j1/2+ have the same definition as jj2 ⎣⎦ j j (32) in Eq. (33), Θ (r) is a slope limiter and ϖ ∈[-1,1]. R ε UU=+RLR⎡⎤ F(U)F(U) − The MINMOD (MINBEE) reconstruction is jj2 ⎣⎦ j j obtained if the slope limiter is taken as:
Brazilian Journal of Chemical Engineering
A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 161
⎧0 , r< 0 where CD is a positive drag coefficient that depends ⎪ upon the phase and interfacial properties and the ⎪ Θ=(r) r , 0≤≤ r 1 (35) Dw ⎨ flow pattern, a is the specific area and Fg is the ⎪ ⎧⎫2 min⎨⎬ 1, , r> 1 wall drag force. As for all interphase change fluxes, ⎪ ⎩⎭1rr−ϖ+ + ϖ ⎩ we have:
Di Di DISCRETIZATION OF THE REMAINING FFlg=− (38) TERMS Phase Appearance and Disappearance Interfacial Transfer Terms As the formation of vacuum is a challenge to The two-fluid model is non-hyperbolic for single phase flow simulation, the phase appearance i ppkk= . In order to regularize this term, some and disappearance are important numerical interfacial pressure models were created. A summary difficulties in multiphase flow simulation. First, the of these models can be found in (Cortes et al., 1998). numerical dissipation of the scheme must be The following form for the interfacial pressure was evaluated for conditions where the volume fraction adopted in (Bestion, 1990, Cortes et al., 1998, Evje tends to zero or one. The AUSM+ flux, Eq. (11), is and Flatten, 2003, Paillère et al., 2003) (with still non-singular since the Mach number is bounded. ppii== p iand pp== p): The corresponding velocity bounds imply that the lg lg velocity of the disappearing phase must tend to the remaining phase velocity as it disappears. For αραρ gg ll 2 example, if the gas volume fraction tends to 0, the ppigl−=σ (vv) − (36) ()αρ+αρgl lg following expression is used: where σ is a positive constant. For σ ≥ 1 the two-fluid v()u/u[1-()]u/ugg21g43= α+α (39) model becomes hyperbolic, while for σ < 1, the eigenvalues of the flux jacobian matrix flux become where ()α is a positive function equal to 1 if complexes. One should note that the value of σ = 1 α>αmin, with αmin very close to zero and tending (used in CATHARE code) is a limit based on an smoothly to zero if α<αmin. approximate eigenvalues analysis and values of σ ≥ 1.2 are recommended to achieve hyperbolicity. The effect Source Terms Treatment of this parameter on the contact wave is important and will be presented in the results. The authors in Paillère Following Paillère et al. (2003), the source terms et al. (2003) presented results for σ=2 and σ=5, but the were discretized with a simple central formula as adoption of these values is questionable. As equation follows: (36) is more a mathematical artifact to keep the model hyperbolicity than a physical model, simulation results ∂αk αk,j+ 1−α k,j− 1 with \sigma=0 are also included to show how much its (piji,j )= p (40) ∂Δx2x usage, which implies in additional numerical diffusion, affects the results. Besides, it is not clear if the methods The remaining gravity and drag terms were treated here could keep useful results for reasonable evaluated at cell j. One could observe that all refined meshes even if the hyperbolicity is lost since they do not make directly use of the hyperbolicity. discretizations were based on the two-fluid model The interfacial drag force depends upon the flow form given in Eq. (3). Some authors adopted the pattern, which is, in turn, dependent upon the form given in Eq. (5) where the flux presents Δ p superficial velocities of the phases. One possible instead of p and the non-conservative term includes form for this term is: the pressure gradient instead of the volume fraction gradient. Evje and Flatten (2003) used this form but DDiDw FFFgg=+ g = treated the non-conservative term as part of the (37) AUSM flux instead of considering it a source term. CD Dw Cascales and Salvador (2006) adopted both central −αρ−gla(vv)|vv|F g l g −+ l g 8 and backward discretization for this term.
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162 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles
For the Hybrid Lax-Wendroff-Lax Friedrichs frequently adopted. Therefore, the MUSCL-Hancock method, the treatment of the non-conservative terms strategy was substituted by a primitive (non- is included in the flux. The modified Lax-Wendroff conservative) variable algorithm described previously. flux given by Eq. (27) is substituted by: However, as shown in Figure 1 (a) and (b), the results are slightly worse than the ones obtained using the LW L R conservative variable approach. Similar results were F(u,u,)0.5ffk,j+ 1/2k,j k,j+ 1 χ=( k + k ) − observed for the water faucet problem. Therefore, only RL results for the conservative variable MUSCL scheme ε ⎡⎤f(u,u)φ+χ−− (ff) − ⎢⎥()k,j k,j+ 1 kk (41) are presented in the following. Figure 2 (a) and (b) 2χ f(u,u)φ ⎣⎦⎢⎥()k,j k,j+ 1 show the results for different choices of the weighting function and σ for AUSMDV scheme. It can be seen ⎡⎤n that the use of σ=1.2 led to better results since it ε f(u,u)()φ k,j k,j++ 1+ χ Bj1/2+ (u,u) k,j 1−− k,j − ⎢⎥produces less oscillation. Figure 2 (c) and (d) show the 2χ f(u,u)φ ⎣⎦⎢⎥()k,j k,j+ 1 results for different choices of the weighting function and σ=1.2 for the AUSMDV and MUSCL-AUSMDV and the variables update is also changed to: for 1000 and 10000 cell meshes. Liou (2000) suggested Y = 1/ρ and one can see that this option n1+ n n n UU(FF)jj=−ε−j+− 1/2 j 1/2 − gives similar results when compared to Y = p/ρ. For (42) refined meshes, only the Evje and Flatten (2003) ε ⎡⎤nnnnnn suggestion, Y=ρ/α, could represent the two separate B(UU)B(UU)j−+ 1/2jj1−+−+ j 1/2 j1j − 2 ⎣⎦waves pattern. However, the Y = ρ/α option applied to AUSMDV produced some oscillations. This effect is where B is the non-conservative term even worse for MUSCL-AUSMDV. The options Y = i i ([0, p ∂α/x ∂ ,0, p ∂α/x ∂ ) and Bj+1/2=B[(Uj+Uj+1)/2]. g g l l p/ρ and Y = 1/ρ produce too much diffusion for AUSMDV. Since this is a shock tube problem, the less diffusive option is usually considered to be more NUMERICAL RESULTS suitable. Therefore, for this example, the option Y=ρ/α seems to be more adequate for AUSMDV, despite In our comparison study, the accuracy of the some oscillatory behavior. For the MUSCL-AUSMDV methods was compared for different mesh sizes. The scheme, Y=ρ/α produces unacceptably large shock tube and the water faucet test problems were oscillations. Therefore, although Y=1/ρ produces too employed. Particularly, the effects of the pressure much numerical diffusion, it was adopted as the option correction term and of the choice of the AUSMDV for MUSCL-AUSMDV, which was capable of weighting term were examined. Some complex cases capturing the two separate waves only for the 10000 such as phase separation were not included in this cell mesh. analysis because a special treatment for the model Figures 3 to 6 show results for localized regions that singularity at each limit of the phase fraction value have been magnified to highlight the differences among would have to be included and analyzed. the schemes. Comparing these results, the following remarks can be made: Shock Tube Problem (i) The AUSMDV scheme was more diffusive than the corresponding AUSM+ scheme and the We consider a shock tube problem with large MUSCL AUSMDV is more diffusive than MUSCL relative velocity proposed in Cortes et al. (1998), which AUSM+ with or without pressure correction. For σ = is a Riemann problem with the left and right initial 0, all methods presented unstable solutions and no states given by wL=[0.71, 65, 1, 265000] and wR=[0.7, solution for a 10000 cell mesh was possible. 50, 1, 265000], respectively. The tube length is 100. (ii) In the results presented by Evje and Flatten Since there is no analytical solution available, the (2003) with a Roe Riemann solver, two separate void results from the HLFW method using a 10000-cell fraction waves can be seen. In our results, this mesh were adopted as reference values. The MUSCL scenario is observed for meshes up to 1000 cells for AUSM+ was adopted in Fjelde (2002) for the drift flux all σ. However, the two separate waves are model, but their MUSCL reconstruction used non- accompanied by overshoots for all AUSM type conservative instead of conservative variables, as methods with some advantage for AUSMDV. For
Brazilian Journal of Chemical Engineering
A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 163 the HLFW methods the overshoot was observed for in which MUSCL AUSMDV was slightly better than σ = 0, but not for σ up to 1.2. Therefore, the convex HLFW. The MUSCL AUSM+ presented results with function proposed in De Vuyst (2004) was efficient strong oscillatory behavior for all mesh sizes and it is for this case. not recommended. (iii) Although the usage of σ = 1.2 reduced the (vi) The authors in Paillère et al. (2003) observed numerical oscillations for both pressure and void that the value of σ strongly affects a contact fraction, the amplitude of the two separate waves discontinuity. In Figure 2 (a), it is possible to remains slightly higher than the Riemann solver observe the effect of σ=2. For all methods and solution used as reference by Evje and Flatten (2003). weighting parameter functions, the contact (iv) The MUSCL-Hancock strategy presented the discontinuity wave showed the expected positive expected behavior, favoring numerical oscillations for pressure gradient at x=0.5 for σ=0 while the results meshes up to 200 cells. Therefore, the extension to for σ=2 showed a negative pressure slope at this second-order of such methods by MUSCL was suitable point. Therefore, values of σ higher than 1.2 should only for small meshes for the adopted Y expression. be considered with care. (v) From the results, the most suitable methods It should be pointed out that the AUSMDV less were the MUSCL AUSMDV and HLFW schemes diffusive weighting function (Y = 1/ρ) was adopted to with the pressure correction (σ= 1.2 ). The accuracy generate these results, aiming to represent the two of HLFW was superior to that of MUSCL AUSMDV separate void fraction waves. However, it actually for all mesh sizes. The MUSCL AUSMDV presented added considerable numerical oscillation. The other small oscillations only for the 10000-cell mesh. This weighting functions were not tried because they are result is particularly interesting because this behavior expected to produce more numerical diffusion and the is different from the results for one-phase flows weighting function used already gave results with larger obtained in our previous work (Coelho et al., 2006), numerical diffusion than those obtained with HLFW.
0.302 100 cell mesh 0.300 primitive conservative 0.298 reference CFL = 0.1 σ=0 g
α 0.296
0.294
0.292
0.290
44 46 48 50 52 54 x (a)
271000
270000
269000
p CFL = 0.1 268000 100 cell mesh σ = 0 primitive 267000 conservative reference 266000
265000 10 20 30 40 50 60 70 80 90 x (b) Figure 1: Comparison for the shock tube problem results of the AUSM+ with MUSCL strategy for primitive and conservative variables for (a) volume fraction and (b) pressure (at t = 0.1 s). Reference values from HLFW 10000 cell mesh.
Brazilian Journal of Chemical Engineering Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
164 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles
271000 0.302 100 cell mesh 270000 0.300 CFL = 0.1 σ=0 AUSMDV 100 cell mesh Y=ρ/α 269000 σ=0 0.298 Y=p/ρ Y=ρ/α Y=1/ρ CFL = 0.1 p Y=p/ρ 0.296 AUSMDV 268000 g σ=1.2
Y=1/ρ α Y=ρ/α σ=1.2 0.294 Y=p/ρ 267000 Y=ρ/α Y=1/ρ Y=p/ρ 0.292 reference 266000 Y=1/ρ reference 0.290 265000 0.288 0 102030405060708090100 0 102030405060708090100 x x
(a) (b) 0.304 0.304 1000 cell mesh 10000 cell mesh 0.302 Y=ρ/α 0.302 Y=ρ/α Y=p/ρ Y=p/ρ 0.300 0.300 Y=1/ρ Y=1/ρ 0.298 0.298 MUSCL MUSCL CFL = 0.1 Y=ρ/α 0.296 Y=ρ/α
0.296 g g σ =1.2 α α Y=p/ρ Y=p/ρ 0.294 0.294 Y=1/ρ Y=1/ρ CFL = 0.1 reference 0.292 reference 0.292 σ =1.2 0.290 0.290 0.288 0.288 0.286 43 44 45 46 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 x x (c) (d) Figure 2: Comparison for the shock tube problem results for the AUSMDV with different weighting functions and σ values for a 100 cell mesh: (a) pressure and (b) volume fraction, and for AUSMDV and MUSCL-AUSMDV for σ = 1.2 on different meshes sizes (at t = 0.1 s): (c) 1000 cells and (b) 10000 cells. Reference values from HLFW 10000 cell mesh.
0.304 0.302 100 cell mesh 200 cell mesh 0.302 AUSM+ 0.300 AUSM+ AUSMDV 0.300 AUSMDV 0.298 HLFW HLFW MUSCL 0.298 MUSCL AUSM+ 0.296 0.296 AUSM+ g g
CFL = 0.1 α AUSMDV CFL = 0.1 α AUSMDV σ = 0 0.294 reference σ = 0 0.294 reference
0.292 0.292 0.290 0.290 0.288 0.288 44 45 46 47 48 49 50 51 52 53 54 44 45 46 47 48 49 50 51 52 53 54 x x
(a) (b)
0.306 0.304 1000 cell mesh 0.302 AUSM+ AUSMDV 0.300 HLFW 0.298 MUSCL
0.296 AUSM+ g
α 0.294 AUSMDV reference CFL = 0.1 0.292 σ = 0 0.290 0.288 0.286 46 47 48 49 50 51 52 53 x (c) Figure 3: Comparisons of void fraction for the shock tube problem. Results of the AUSM+, AUSMDV, AUSM+ and AUSMDV MUSCL schemes and HLFW for 100 (a), 200 (b), 1000 (c) cell meshes without interfacial pressure correction (at t = 0.1 s). Reference values from HLFW 10000 cell mesh.
Brazilian Journal of Chemical Engineering
A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 165
0.304 0.304 100 cell mesh 0.302 0.302 200 cell mesh AUSM+ AUSM+ 0.300 AUSMDV 0.300 AUSMDV HLFW HLFW 0.298 0.298 MUSCL MUSCL AUSM+ CFL = 0.1 0.296 AUSM+ CFL = 0.1 0.296
g g AUSMDV σ = 1.2 AUSMDV σ = 1.2 α α 0.294 reference 0.294 reference 0.292 0.292 0.290 0.290 0.288 0.288 44 45 46 47 48 49 50 51 52 53 54 44 45 46 47 48 49 50 51 52 53 54 x x
(a) (b)
0.302 0.302 1000 cell mesh 10000 cell mesh 0.300 0.300 AUSM+ AUSM+ 0.298 AUSMDV 0.298 AUSMDV HLFW HLFW 0.296 MUSCL CFL = 0.1 0.296 MUSCL g g
σ = 1.2 α α AUSM+ AUSM+ 0.294 0.294 AUSMDV AUSMDV reference CFL = 0.1 0.292 0.292 σ = 1.2 0.290 0.290 0.288 0.288 46 47 48 49 50 51 52 53 48 49 50 51 52 x x (c) (d) Figure 4: Comparisons of void fraction for the shock tube problem. Results of the AUSM+, AUSMDV, AUSM+ and AUSMDV MUSCL schemes and HLFW for 100 (a), 200 (b), 1000 (c) and 10000 (d) cell meshes with interfacial pressure correction (σ = 1.2, t = 0.1 s). Reference values from HLFW 10000 cell mesh. 271000
271000 CFL = 0.1 σ = 0 270000 CFL = 0.1 270000 σ = 0 269000 200 cell mesh 269000
100 cell mesh p AUSM+ AUSM+ 268000 AUSMDV
p 268000 AUSMDV HLFW HLFW MUSCL 267000 267000 MUSCL AUSM+ AUSM+ AUSMDV 266000 266000 AUSMDV reference reference 265000 265000 0 102030405060708090100 0 102030405060708090100 x x (a) (b)
271000
270000 CFL = 0.1 σ = 0 269000 1000 cell mesh AUSM+
268000 AUSMDV
p HLFW MUSCL 267000 AUSM+ AUSMDV 266000 reference
265000 0 102030405060708090100 x (c) Figure 5: Comparisons of pressure profiles for the shock tube problem. Results of the AUSM+, AUSMDV, AUSM+ and AUSMDV MUSCL schemes and HLFW for 100 (a), 200 (b), 1000 (c) cell meshes without interfacial pressure correction (at t = 0.1 s). Reference values from HLFW 10000 cell mesh.
Brazilian Journal of Chemical Engineering Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
166 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles
271000 271000 CFL = 0.1 CFL = 0.1 σ = 1.2 270000 σ = 1.2 270000
269000 269000 200 cell mesh
100 cell mesh p p AUSM+ AUSM+ 268000
268000 AUSMDV AUSMDV HLFW HLFW 267000 267000 MUSCL MUSCL AUSM+ 266000 AUSM+ 266000 AUSMDV AUSMDV reference 265000 reference 265000 0 20406080100 0 102030405060708090100 x x (a) (b)
271000 271000
CFL = 0.1 270000 σ = 1.2 270000 CFL = 0.1 269000 1000 cell mesh 269000 σ = 1.2 10000 cell mesh AUSM+ AUSM+ p p AUSMDV 268000 268000 AUSMDV HLFW HLFW MUSCL MUSCL 267000 267000 AUSM+ AUSM+ AUSMDV AUSMDV 266000 reference 266000
265000 265000 0 102030405060708090100 20 30 40 50 60 70 80 90 x x (c) (d)
Figure 6: Comparisons of pressure profiles for the shock tube problem. Results of the AUSM+, AUSMDV, AUSM+ and AUSMDV MUSCL schemes and HLFW for 100 (a), 200 (b), 1000 (c) and 10000 (d) cell meshes with interfacial pressure correction (σ = 1.2, t = 0.1 s). Reference values from HLFW 10000 cell mesh.
Water Faucet
The water faucet problem (Ransom, 1987) became a benchmark and has been extensively used by many authors in order to test numerical schemes in tracking volume fraction fronts (Toumi, 1996, Coquel et al., 1997, Paillère et al., 2003, Evje and Flatten, 2003, De Vuyst, 2004). This problem is a study of the gravity effects on a vertical downward water jet flow. Consider a vertical pipe of length 12 5 m initially at a uniform state with p = 10 Pa, αl = 0.80, vg = 0 and vl = 10 m/s. The pressure at the outlet is kept constant at the initial value and, at the inlet, the values of the remaining variables are kept constant and equal to their initial values. A scheme of the problem is shown in Figure 7. This problem has the following analytical solution:
⎧ αl0v l0 2 ⎪1−<+ , if x gt / 2 vl0 t α= 2 (43) g ⎨ v2gxl0 + ⎪ ⎩αg0, otherwise Figure 7: Scheme of the water faucet problem.
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A Comparison of Hyperbolic Solvers II: AUSM-Type and Hybrid Lax-Wendroff-Lax-Friedrichs Methods for Two-Phase Flows 167
The problem was also solved for different choices Figures 9 (a) to (c) shows the comparison of the weighing function, Y, in the AUSMDV between the solutions obtained for the water faucet scheme. The results are presented in Figure 8. The problem for 320, 640 and 1280-cell meshes, using adoption of Y=α/ρ instead of Y=p/ρ or Y=1/ρ AUSM+, AUSMDV (Y=p/ρ), these two methods produced no improvement in AUSMDV or in with MUSCL-Hancock data reconstruction (Y=1/ρ) MUSCL-AUSMDV when replacing Y= 1/ρ. Liou and the HLFW method without interfacial pressure (2000) has observed that Y= 1/ρ eliminates the correction, that is, Δ p= 0 . Figure 10 (a) to (c) shows carbuncle phenomenon, but this is not important for the same solution but with σ = 1.2 in Eq. (36) for one-dimensional problems. In our tests, no Δp calculation. The following behavior can be appreciable differences were observed between Y= observed from Figure 9: 1/ρ and Y=p/ρ for AUSMDV for the water faucet (i) The AUSMDV scheme was more diffusive problem. However, Y=p/ρ was unstable for than the corresponding AUSM+ scheme and the MUSCL-AUSMDV and the usage of Y=1/ρ MUSCL AUSMDV is more diffusive than MUSCL produced much better results then the other options. AUSM+ with or without pressure correction. Particularly, for refined meshes, the usage of Y=1/ρ (ii) Since the methods are not TVD, for σ =0 the showed results without the strong oscillations and oscillations tended to grow when the mesh was with an accuracy equivalent of the HLFW method. refined or when the order is extended and this was This is due to the more diffusive characteristic of the confirmed for both MUSCL and HLFW schemes. scheme imparted by Y=1/ρ, which was prohibitive AUSM+ presented some oscillations for the 320 cell- for the shock tube problem. Therefore, Y= p/ρ and mesh, which increased when the mesh was refined Y= 1/ρ were considered to be the better choices for and AUSMDV presented strong numerical diffusion. the water faucet problem for the AUSMDV and The MUSCL second-order extended schemes MUSCL-AUSMDV methods, respectively. presented strong oscillations for the 320-cell mesh,
0.50 while the HLFW scheme presented some oscillation for this mesh and both could not converge for the 0.45 1280 cell mesh CFL = 0.3 reference 1280-cell mesh. However, HLFW presented an σ = 1.2 AUSMDV 0.40 Y=ρ/α oscillatory solution for the 640-cell mesh, while the Y=p/ρ MUSCL-AUSM+ schemes could not be solved for 0.35 MUSCL-AUSMDV Y=ρ/α both 640 and 1280-cell meshes. MUSCL-AUSMDV
g α Y=1/ρ 0.30 also presented strong oscillations for the 640-cell mesh. 0.25 From Figure 10, it can be seen that:
0.20 (iii) The pressure correction term increased the
0123456789101112 accuracy of all schemes. It made the solution with x MUSCL schemes possible for all mesh sizes. Figure 8: Comparison for the water faucet problem (iv) As observed in the shock-tube problem, the results for AUSMDV with different weighting accuracy of HLFW was the best among the methods functions (at t = 0.5 s). analyzed.
0.50
0.45 Exact 0.45 Exact 320 cell mesh 640 cell mesh 0.40 AUSM+ 0.40 AUSM+ AUSMDV AUSMDV 0.35 HLFW 0.35 HLFW g
α MUSCL MUSCL g
0.30 AUSM+ α AUSMDV 0.30 CFL = 0.3 AUSMDV 0.25 σ = 0 CFL = 0.3 0.25 σ = 0 0.20 0.20 0.15 0.15 012345678910 012345678910 x x (a) (b)
Brazilian Journal of Chemical Engineering Vol. 27, No. 01, pp. 153 - 171, January - March, 2010
168 R. M. L. Coelho, P. L. C. Lage and A. Silva Telles
0.50 CFL = 0.3 0.45 σ = 0
0.40
0.35 Exact g
α 1280 cell mesh 0.30 AUSM+ AUSMDV 0.25 320 cell mesh HLFW 0.20
0.15 012345678 x (c) Figure 9: Comparisons of void fraction profiles for the water faucet problem. Results of the AUSM+, AUSMDV, AUSM+ and AUSMDV MUSCL schemes and HLFW for 320 (a), 640 (b) and 1280 (c) cell meshes without pressure correction (at t = 0.5 s).
0.45 0.45 Exact CFL = 0.3 Exact 320 cell mesh σ = 1.2 640 cell mesh 0.40 CFL = 0.3 AUSM+ 0.40 σ = 1.2 AUSM+ AUSMDV AUSMDV HLFW 0.35 0.35 HLFW MUSCL MUSCL g
AUSM+ g
AUSM+ α α 0.30 AUSMDV 0.30 AUSMDV
0.25 0.25
0.20 0.20
012345678910 012345678910 x x (a) (b)
0.45 CFL = 0.3 σ = 1.2 0.40
0.35
g Exact α 1280 cell mesh 0.30 AUSM+ AUSMDV HLFW 0.25 MUSCL AUSM+ 0.20 AUSMDV
0123456789 x (c) Figure 10: Comparisons of void fraction profiles for the water faucet problem. Results of the AUSM+, AUSMDV, AUSM+ and AUSMDV MUSCL schemes and HLFW for 320 (a), 640 (b) and 1280 (c) cell meshes with pressure correction (σ=1.2 and at t = 0.5 s).
CONCLUSIONS to be not only problem dependent, but also order dependent, requiring different expressions for the The AUSM+, AUSMDV, Hybrid Lax-Friedrichs- AUSMDV and the extended second-order MUSCL- Lax-Wendroff, and MUSCL-Hancock second-order AUSMDV schemes. Primitive and conservative extensions of AUSM+ and AUSMDV methods were MUSCL-Hancock extensions were compared and the compared in terms of accuracy using two benchmark conservative variable option was slightly better. All problems. Three AUSMDV weighting functions schemes were compared with and without the were compared and the most suitable one was found interfacial pressure correction term.
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For the water faucet test problem, the pressure this method has an iterative optimization of a correction term improved considerably the solution. weighting parameter, which increases accuracy but The results without the pressure correction term also increases the CPU time consumption. However, showed that the HLFW method was stable only for unlike the one-phase flow analysis, where the coarse meshes. The same observation is valid for the accuracy of the Hybrid Lax-Friedrichs-Lax- MUSCL second-order extensions of the AUSM-type Wendroff method was comparable to AUSM type methods. AUSMDV was stable for large meshes, but methods but with larger CPU time consumption showed more numerical diffusion than the results (Coelho et al., 2006), this method is an alternative using the HLFW method for a mesh four times for Riemann solvers and AUSM type solvers with coarser. The AUSM+ method presented high reasonable accuracy and CPU time for two-phase overshoots for all mesh sizes. The results with the flows. As it does not require the jacobian matrix pressure correction term showed much less evaluation, it can be easily generalized for complex oscillation. This term stabilized the solution for flow closure relations. refined meshes for all methods including the MUSCL second-order extensions. However, only the MUSCL- AUSMDV and HLFW methods gave oscillation-free ACKNOWLEDGEMENTS results. The MUSCL second-order extensions were unstable without the pressure correction term. In this The authors acknowledge the financial support case, an oscillation-free solution was obtained only for provided by PRH-ANP/MME/MCT – Agência the AUSMDV method. Nacional do Petróleo – Brazil and CNPq (grant no. In the large relative velocity shock tube test 301548/2005-6). problem, the two separated void fraction waves could be simulated with AUSM type methods only for refined meshes. However, the Hybrid Lax-Friedrichs- NOMENCLATURE Lax-Wendroff method was capable of handling this two separate void fraction scenario with coarser a specific area meshes. As in the case of the water faucet, it was not A Jacobian matrix possible to obtain stable solutions for refined meshes Ak EOS parameter for all methods without the pressure correction term. B vector of non-conservative The results with the pressure correction term also terms showed more numerical diffusion, but with acceptable c speed of sound results for coarse meshes. However, in such meshes C drag coefficient the two separate waves were not captured. The results d Cp specific heat at constant for refined meshes showed that only the MUSCL- pressure AUSMDV and HLFW methods were capable of Cv specific heat at constant eliminating the numerical oscillations with high volume accuracy results. Different from the water faucet CFL Courant-Friedrichs-Lewy problem, the better choices for the AUSMDV number, CFL = Δt weighting function were different for the first-order (c+max(v))/Δx and second-order schemes. In the water faucet flux vector problem, the requirements on numerical diffusion f could be relaxed, allowing MUSCL-AUSMDV to F numeric flux vector D drag force due to interface give better results with the more diffusive choice for Fk the weighting function. and wall on phase k Di interfacial drag force on The results show that it is not possible to Fk generalize which AUSMDV weighting function is phase k Dw wall drag force on phase k the best. This choice is case and order dependent. Fk For all cases, AUSMDV produced additional gx component of the gravity diffusion and more stable results than the AUSM+ acceleration vector in the x method. For both test problems, the best results were direction obtained from the MUSCL-Hancock second-order g non-homogeneous vector in extension of AUSMDV and the Hybrid Lax- conservation equations Friedrichs-Lax-Wendroff methods, with some G derivative of the convex advantage for the latter. One should bear in mind that function
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