Solutions for a hyperbolic model of multiphase flow and related numerical issues Debora Amadori University of L’Aquila (Italy) AMIS2012, June 20 2012
Outline Outline Part I: A multiphase flow model 1 Description Basics on the homogeneous system Comparison with other models Solutions for the homogeneous system The algorithm The system with a reaction term Part II: Error estimates for scalar laws with source 2 The setting Fractional Step/ Well Balanced at a glance L 1 stability result
Part I: A multiphase flow model Description A 1–D model for multiphase reactive flow v t − u x = 0 u t + p ( v, λ ) x = 0 = α λ t τ ( p − p e ) λ ( λ − 1) v > 0 : specific volume, u : velocity, λ : mass density fraction of vapor in the fluid 0 ≤ λ ≤ 1; λ = 0 : liquid ; λ = 1 : vapor ;
Part I: A multiphase flow model Description A 1–D model for multiphase reactive flow v t − u x = 0 u t + p ( v, λ ) x = 0 = α λ t τ ( p − p e ) λ ( λ − 1) v > 0 : specific volume, u : velocity, λ : mass density fraction of vapor in the fluid 0 ≤ λ ≤ 1; λ = 0 : liquid ; λ = 1 : vapor ; α > 0 , p e a fixed equilibrium pressure, τ a reaction time p : pressure p = p ( v, λ ) = A ( λ ) A ′ ( λ ) > 0 , A ( λ ) > 0 , v γ [Fan, SIAM J. Appl. Math. (2000)]
Part I: A multiphase flow model Description λ = 0 λ = 1 p stable liquid P ✏ ✻ metastable vapor q P ✏ ✮ ✏ p e . . ✲ . . ✾ ✘ ✘ metastable liquid stable vapor . . . . . . . . . . ✲ . . v v m v M The system shows all major wave patterns observed in shock tube experiments on retrograde fluids. [Thompson, Carofano, Kim, 1986; Thompson, Chaves, Meier, Kim and Speckmann, 1987] . The model may take into account viscosity terms v t − u x = 0 u t + p x = εu xx = α λ t τ ( p − p e ) λ ( λ − 1) + bελ xx Riemann problem for the 0 -viscosity and 0 -relaxation limit: [Corli and Fan (2005)] .
Part I: A multiphase flow model Description Aim of this work For a fixed relaxation time τ > 0 , look for global (in time) solutions of the Cauchy problem with large BV data � � ( v, u, λ )(0 , x ) = v o ( x ) , u o ( x ) , λ o ( x ) , In the spirit of the fractional step method [Dafermos& Hsiao, 1982] , the analysis will consider the homogeneous system: the 3 × 3 system of conservation laws 8 = 0 v t − u x < (H) u t + p ( v, λ ) x = 0 λ t = 0 : the complete system, where the source term is added by time discretization.
Part I: A multiphase flow model Description Aim of this work For a fixed relaxation time τ > 0 , look for global (in time) solutions of the Cauchy problem with large BV data � � ( v, u, λ )(0 , x ) = v o ( x ) , u o ( x ) , λ o ( x ) , In the spirit of the fractional step method [Dafermos& Hsiao, 1982] , the analysis will consider the homogeneous system: the 3 × 3 system of conservation laws 8 = 0 v t − u x < (H) u t + p ( v, λ ) x = 0 λ t = 0 : the complete system, where the source term is added by time discretization. The relaxation limit τ → 0 . [Joint work with: Andrea Corli ( University of Ferrara )]
Part I: A multiphase flow model Basics on the homogeneous system The homogeneous system Assume γ = 1 . The 3 × 3 system v t − u x = 0 (H) u t + ( A ( λ ) /v ) x = 0 λ t = 0 � e 1 , 3 = ±√− p v = ± is strictly hyperbolic: A ( λ ) /v, e 2 = 0 . Two fields are GNL, one is LD.
Part I: A multiphase flow model Basics on the homogeneous system The homogeneous system Assume γ = 1 . The 3 × 3 system v t − u x = 0 (H) u t + ( A ( λ ) /v ) x = 0 λ t = 0 � e 1 , 3 = ±√− p v = ± is strictly hyperbolic: A ( λ ) /v, e 2 = 0 . Two fields are GNL, one is LD. • From the third equation: λ = λ o ( x ) ⇒ 2 × 2 system with non-homogeneous flux, possibly discontinuous.
Part I: A multiphase flow model Basics on the homogeneous system The homogeneous system Assume γ = 1 . The 3 × 3 system v t − u x = 0 (H) u t + ( A ( λ ) /v ) x = 0 λ t = 0 � e 1 , 3 = ±√− p v = ± is strictly hyperbolic: A ( λ ) /v, e 2 = 0 . Two fields are GNL, one is LD. • From the third equation: λ = λ o ( x ) ⇒ 2 × 2 system with non-homogeneous flux, possibly discontinuous. � λ ℓ x < 0 • Special case: λ o ( x ) = λ r x > 0
Part I: A multiphase flow model Basics on the homogeneous system The homogeneous system Assume γ = 1 . The 3 × 3 system v t − u x = 0 (H) u t + ( A ( λ ) /v ) x = 0 λ t = 0 � e 1 , 3 = ±√− p v = ± is strictly hyperbolic: A ( λ ) /v, e 2 = 0 . Two fields are GNL, one is LD. • From the third equation: λ = λ o ( x ) ⇒ 2 × 2 system with non-homogeneous flux, possibly discontinuous. � λ ℓ x < 0 • Special case: λ o ( x ) = λ r x > 0 Look for weak BV solutions, globally defined in time, with possibly large data
Part I: A multiphase flow model Comparison with other models Comparison with other models In Eulerian coordinates, for p = A ( λ ) ρ and ρ = 1 /v , the system (H) rewrites as ρ t + ( ρu ) x = 0 , � � ρu 2 + p ( ρ, λ ) ( ρu ) t + = 0 , x ( ρλ ) t + ( ρλu ) x = 0 . In [Benzoni-Gavage, 1991] many models for diphasic flows are proposed, for instance ( ρ l R l ) t + ( ρ l R l u l ) x = 0 ( ρ g R g ) t + ( ρ g R g u g ) x = 0 ( ρ l R l u l + ρ g R g u g ) t + ( ρ l R l u 2 l + ρ g R g u 2 g + p ) x = 0 . Here l and g stand for liquid and gas ; ρ l , R l , u l are the liquid density, phase fraction, velocity, and analogously for the gas, R l + R g = 1 , p = Aρ g .
Part I: A multiphase flow model Comparison with other models Comparison with other models If u l = u g , ρ l = 1 and R l > 0 , define the concentration c = ρ g R g then [Peng, 1994] ρ l R l ( R l ) t + ( R l u ) x = 0 ( R l c ) t + ( R l cu ) x = 0 (P) � � R l (1 + c ) u 2 + p ( R l (1 + c ) u ) t + = 0 x R l with p = Ac . 1 − R l System (P) is strictly hyperbolic for c > 0 ; the eigenvalues coincide with u at c = 0 . If c ≡ 0 then (P) reduces to the pressureless gasdynamics system.
Part I: A multiphase flow model Comparison with other models Comparison with other models If u l = u g , ρ l = 1 and R l > 0 , define the concentration c = ρ g R g then [Peng, 1994] ρ l R l ( R l ) t + ( R l u ) x = 0 ( R l c ) t + ( R l cu ) x = 0 (P) � � R l (1 + c ) u 2 + p ( R l (1 + c ) u ) t + = 0 x R l with p = Ac . 1 − R l System (P) is strictly hyperbolic for c > 0 ; the eigenvalues coincide with u at c = 0 . If c ≡ 0 then (P) reduces to the pressureless gasdynamics system. c System (P) is analogous to (H) , with λ = 1 + c , but the pressure laws differ when λ , c ∼ 0 .
Part I: A multiphase flow model Comparison with other models The Riemann problem The Cauchy problem for v t − u x = 0 � ( v ℓ , u ℓ , λ ℓ ) � � x < 0 A ( λ ) u t + = 0 ( v, u, λ )(0 , x ) = v ( v r , u r , λ r ) x > 0 x λ t = 0 can be solved for any pair of initial data (with v ℓ , v r > 0 and λ ℓ , λ r ∈ [0 , 1] ). ε 1 ε 2 ε 3 ❅ ✁✁ ❅ ❅ ✁ Phase waves are stationary: p , u are conserved across them (“kinetic condition”). p ℓ = p r u ℓ = u r
Part I: A multiphase flow model Comparison with other models Known results For small BV data: well-posedness of the Cauchy problem. [Glimm 1965; Bressan, Hyperbolic systems... , 2000.] If λ is constant : the Cauchy problem for v t − u x = 0 , u t + ( A/v ) x = 0 has a global solution if only Tot . Var . ( v o , u o ) < ∞ [Nishida (1968)] If p ( v ) = A v γ , with γ > 1 global existence if ( γ − 1)Tot . Var . ( u o , v o ) is small [Nishida-Smoller, DiPerna 1973] . If λ is not constant, [Peng (1994)] ; results by compensated compactness: [B´ ereux, Bonnetier, & LeFloch (1997); Gosse (2001); Lu (2003)] , but for different pressure laws. For the full Euler system: [Liu (1977), Temple (1981)] .
Part I: A multiphase flow model Solutions for the homogeneous system Solutions for the homogeneous system √ Assume: v o ( x ) ≥ v > 0 , a = A , λ o ( x ) ∈ [0 , 1] and define � n | a ( λ ( x j )) − a ( λ ( x j − 1 )) | A o = 2 sup a ( λ ( x j )) + a ( λ ( x j − 1 )) . j =1 where the supremum is taken over all n ≥ 1 , x o < x 1 < . . . < x n .
Part I: A multiphase flow model Solutions for the homogeneous system Solutions for the homogeneous system √ Assume: v o ( x ) ≥ v > 0 , a = A , λ o ( x ) ∈ [0 , 1] and define � n | a ( λ ( x j )) − a ( λ ( x j − 1 )) | A o = 2 sup a ( λ ( x j )) + a ( λ ( x j − 1 )) . j =1 where the supremum is taken over all n ≥ 1 , x o < x 1 < . . . < x n . � � �� Observe that A o ∼ Tot . Var . log a ( λ o ) .
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