A thin-layer reduced model for shallow viscoelastic flows Franois - PowerPoint PPT Presentation

A thin-layer reduced model for shallow viscoelastic flows François Bouchut 2 & Sébastien Boyaval 1 1 Univ. Paris Est , Laboratoire d’hydraulique Saint-Venant (ENPC – EDF R&D – CETMEF), Chatou, France & INRIA , MICMAC team 2 Univ. Paris Est , LAMA (Univ. Marne-la-Vallée) & CNRS July 2013, CEMRACS, Marseille

Outline Formal derivation of the mathematical model 1 Discretization of the new model 2 Numerical simulation & physical interpretation 3 F. Bouchut & S. Boyaval 2 / 36 Shallow viscoelastic fluids

Upper-Convected Maxwell (UCM) model Mass and momentum equations for incompressible fluid ( velocity u ; pressure p ; Cauchy stress − p I + τ ) with non-Newtonian rheology ( τ ≇ D ( u ) ≡ ( ∇ u + ∇ u T ) / 2): div u = 0 in D t , ∂ t u + ( u · ∇ ) u = − ∇ p + div τ + f in D t , � ∂ t τ + ( u · ∇ ) τ − ( ∇ u ) τ − τ ( ∇ u ) T � λ = η p D ( u ) − τ in D t , under gravity f ≡ − g e z in time-dependent domain D t ⊂ R 2 D t = { x = ( x , z ) , x ∈ ( 0 , L ) , 0 < z − b ( x ) < h ( t , x ) } F. Bouchut & S. Boyaval 3 / 36 Shallow viscoelastic fluids

Thin-layer geometry with non-folded interfaces z g n ( x ) h ( t , x ) b ( x ) x F. Bouchut & S. Boyaval 4 / 36 Shallow viscoelastic fluids

A free-surface boundary value problem We supply the UCM model with initial and boundary conditions u · n = 0 , for z = b ( x ) , x ∈ ( 0 , L ) , τ n = (( τ n ) · n ) n , for z = b ( x ) , x ∈ ( 0 , L ) , ∂ t h + u x ∂ x ( b + h ) = u z , for z = b ( x ) + h ( t , x ) , x ∈ ( 0 , L ) , ( p I − τ ) · ( − ∂ x ( b + h ) , 1 ) = 0 , for z = b ( x ) + h ( t , x ) , x ∈ ( 0 , L ) , where n is the unit normal vector at the bottom inward the fluid − ∂ x b 1 n x = � n z = � 1 + ( ∂ x b ) 2 . 1 + ( ∂ x b ) 2 F. Bouchut & S. Boyaval 5 / 36 Shallow viscoelastic fluids

∂ x u x + ∂ z u z = 0 , ∂ t u x + u x ∂ x u x + u z ∂ z u x = − ∂ x p + ∂ x τ xx + ∂ z τ xz , ∂ t u z + u x ∂ x u z + u z ∂ z u z = − ∂ z p + ∂ x τ xz + ∂ z τ zz − g , ∂ t τ xx + u x ∂ x τ xx + u z ∂ z τ xx = ( 2 ∂ x u x ) τ xx + ( 2 ∂ z u x ) τ xz + η p ∂ x u x − τ xx , λ ∂ t τ zz + u x ∂ x τ zz + u z ∂ z τ zz = ( 2 ∂ x u z ) τ xz + ( 2 ∂ z u z ) τ zz + η p ∂ z u z − τ zz , λ η p ( ∂ z u x + ∂ x u z ) − τ xz 2 ∂ t τ xz + u x ∂ x τ xz + u z ∂ z τ xz = ( ∂ x u z ) τ xx + ( ∂ z u x ) τ zz + λ u z = ( ∂ x b ) u x at z = b , � � − ( ∂ x b ) τ xx + τ xz = − ∂ x b − ( ∂ x b ) τ xz + τ zz at z = b , − ∂ x ( b + h )( p − τ xx ) − τ xz = 0 at z = b + h , ∂ x ( b + h ) τ xz + ( p − τ zz ) = 0 at z = b + h , ∂ t h + u x ∂ x ( b + h ) = u z at z = b + h . F. Bouchut & S. Boyaval 6 / 36 Shallow viscoelastic fluids

Long-wave asymptotic regime for shallow flows (H1) h ∼ ǫ as ǫ → 0 ∂ t = O ( 1 ) , ∂ x = O ( 1 ) , ∂ z = O ( 1 /ǫ ) � z (H2) ∂ x b = O ( ǫ ) ⇒ u z = ( ∂ x b ) u x | z = b − ∂ x u x = O ( ǫ ) b ⇒ ∂ x h = O ( ǫ ) i.e. long waves, since ∂ t h + u x ∂ x ( b + h ) = u z | z = b + h (H3) τ = O ( ǫ ) , hence also η p ∼ ǫ ( λ ∼ 1 ) , ⇒ ∂ z p = ∂ z τ zz − g + O ( ǫ ) (H4) motion by slice ∂ z u x = O ( 1 ) ⇒ ∂ z τ xz = D t u x + O ( ǫ ) compatible with τ xz | z = b , b + h if τ xz = O ( ǫ 2 ) , ⇒ ∂ z u x = O ( ǫ ) Momentum depth-average & u x ( t , x , z ) = u 0 x ( t , x ) + O ( ǫ 2 ) yields � b + h � b + h u x = ∂ t h 0 + ∂ x ( h 0 u 0 x )+ O ( ǫ 2 ) . . . 0 = ∂ x u x + ∂ z u z = ∂ t h + ∂ x b b F. Bouchut & S. Boyaval 7 / 36 Shallow viscoelastic fluids

Viscoelastic Saint-Venant equations Assuming ∂ z τ xx , ∂ z τ zz = O ( 1 ) , we get the closed system   ∂ t h + ∂ x ( hu x ) = 0 ,    � �  h ( u x ) 2 + g h 2    ∂ t ( hu x ) + ∂ x 2 + h ( τ zz − τ xx ) = − g ( ∂ x b ) h ,   ∂ t τ xx + u x ∂ x τ xx = 2 ( ∂ x u x ) τ xx + η p λ ∂ x u x − 1   λτ xx ,       ∂ t τ zz + u x ∂ x τ zz = − 2 ( ∂ x u x ) τ zz − η p λ ∂ x u x − 1   λτ zz . To explore the new reduced model: numerical simulations with a Finite-Volume scheme (conservativity). Anticipating stability of the FV scheme: mathematical entropy. F. Bouchut & S. Boyaval 8 / 36 Shallow viscoelastic fluids

UCM eqns naturally dissipate energy ! With σ = I + 2 λ η p τ , the UCM model rewrites � ∂ t σ + ( u · ∇ ) σ − ( ∇ u ) σ − σ ( ∇ u ) T � λ = I − σ in D t , and thermodynamics imposes σ s.p.d. and a free energy � 1 � � 2 | u | 2 + η p F ( u , σ ) = 4 λ I : ( σ − ln σ − I ) − f · x d x , (3) D t � dt F ( u , σ ) = − η p d I : ( σ + σ − 1 − 2 I ) d x . (4) 4 λ 2 D t F. Bouchut & S. Boyaval 9 / 36 Shallow viscoelastic fluids

Reformulation with energy dissipation  ∂ t h + ∂ x ( hu ) = 0 ,    � �   hu 2 + g h 2  2 + η p   ∂ t ( hu ) + ∂ x 2 λ h ( σ zz − σ xx ) = − gh ∂ x b ,   ∂ t σ xx + u ∂ x σ xx − 2 σ xx ∂ x u = 1 − σ xx   ,   λ     ∂ t σ zz + u ∂ x σ zz + 2 σ zz ∂ x u = 1 − σ zz   , λ � � hu 2 2 + g h 2 2 + gbh + η p ∂ t 4 λ h tr ( σ − ln σ − I ) � � u 2 � tr ( σ − ln σ − I ) ��� 2 + g ( h + b ) + η p + ∂ x hu + σ zz − σ xx 2 λ 2 = − η p 4 λ 2 h tr ( σ + [ σ ] − 1 − 2 I ) . F. Bouchut & S. Boyaval 10 / 36 Shallow viscoelastic fluids

Outline Formal derivation of the mathematical model 1 Discretization of the new model 2 Numerical simulation & physical interpretation 3 F. Bouchut & S. Boyaval 11 / 36 Shallow viscoelastic fluids

Flux (conservative) formulation ∂ t U + ∂ x F ( U ) = S   ∂ t h + ∂ x ( hu ) = 0 ,   � �   hu 2 + P ( h , s ) ∂ t ( hu ) + ∂ x = − gh ∂ x b , ( S )    ∂ t ( h s ) + ∂ x ( hu s ) = h S ( h , s )   , λ � � σ − 1 / 2 , σ 1 / 2 , P ( h , s ) = g h 2 2 + η p where s = xx zz 2 λ h ( σ zz − σ xx ) , h h � � − σ − 3 / 2 2 h ( 1 − σ xx ) , σ − 1 / 2 S ( h , s ) = xx zz 2 h ( 1 − σ zz ) is hyperbolic. � ∂ P � | s = gh + η p ∇ F : real eigenvalues ( 2 λ ( 3 σ zz + σ xx ) > 0) ∂ h � gh + η p λ 1 , 3 = u ± 2 λ ( 3 σ zz + σ xx ) g.n.l , λ 2 = u l.d. . Shall we discretize ( S ) by splitting: i) conservation ii) diffusion ? F. Bouchut & S. Boyaval 12 / 36 Shallow viscoelastic fluids

Problem: how to ensure stability There is a problem with discretizing ( S ) in conservative variables: the natural energy is not convex with respect to s ! E = hu 2 2 + g h 2 2 + gbh + η p � 4 λ h ( σ xx + σ zz − ln ( σ xx σ zz ) − 2 ) Now, convexity is essential to entropic stability of FV schemes (Jensen) and to preserve the invariant domain { h , σ xx , σ zz ≥ 0 } . F. Bouchut & S. Boyaval 13 / 36 Shallow viscoelastic fluids

A splitted Finite-Volume approach Free-energy-dissipating FV scheme: piecewise constants (anticipate: approximations of non-conservative variables) q ≡ ( q 1 , q 2 , q 3 , q 4 ) T := ( h , hu , h σ xx , h σ zz ) T on a mesh of R with cells ( x i − 1 / 2 , x i + 1 / 2 ) , i ∈ Z of x i − 1 / 2 + x i + 1 / 2 volumes ∆ x i = x i + 1 / 2 − x i − 1 / 2 at centers x i = 2 At each discrete time t n , variables updated by splitting: 1 Without source: Riemann problems (Godunov approach) 2 + topo h ∂ x b : preprocessing (hydrostatic reconstruction) 3 + dissipative sources in σ : implicit Main difficulties: free-energy dissipation + h , σ xx , σ zz ≥ 0 F. Bouchut & S. Boyaval 14 / 36 Shallow viscoelastic fluids

Finite Volume discretization q n i + 1 q n i x i x i + 1 x x i − 1 / 2 x i + 1 / 2 x i + 3 / 2 F. Bouchut & S. Boyaval 15 / 36 Shallow viscoelastic fluids

Step 1: Godunov approach � � x i + 1 / 2 n + 1 q n 1 1 x i − 1 / 2 q appr ( t n + 1 − 0 , · ) i ≈ ∆ x i q ( t n , · ) → q 2 = ∆ x i i ∆ x i � x − x i + 1 / 2 � q appr ( t , x ) = R , q n i , q n for x i < x < x i + 1 , i + 1 t − t n R ( x t , q l , q r ) is Riemann solver of the system without source, � t < − ∆ x i x 2 ∆ t ⇒ R ( x t , q i , q i + 1 ) = q i , + CFL for ∆ t = t n + 1 − t n . t > ∆ x i + 1 x 2 ∆ t ⇒ R ( x t , q i , q i + 1 ) = q i + 1 , �� 0 � � i + ∆ t n + 1 = q n R ( ξ, q n i , q n i + 1 ) − q n q 2 d ξ i i ∆ x i − ∆ x i / 2 � � ∆ x i / 2 � � R ( ξ, q n i − 1 , q n i ) − q n + d ξ i 0 F. Bouchut & S. Boyaval 16 / 36 Shallow viscoelastic fluids

Step 1: the free-energy flux condition � � � 0 1 n + 1 R ( ξ, q n i , q n E ( q 2 ) ≤ E i + 1 ) d ξ i ∆ x i − ∆ x i / 2 � � � ∆ x i / 2 1 R ( ξ, q n i − 1 , q n + E i ) d ξ ∆ x i 0 with Jensen inequality and the definitions (whatever G ) � �� � 0 � � � G l ( q l , q r ) = G ( q l ) − E R ( ξ, q l , q r ) − E q l d ξ, −∞ � �� � ∞ � � � G r ( q l , q r ) = G ( q r ) + E R ( ξ, q l , q r ) − E q r d ξ, 0 implies, provided G r ( q l , q r ) ≤ G ( q l , q r ) ≤ G l ( q l , q r ) � � ∆ t n + 1 E ( q n G ( q n i − 1 , q n i ) − G ( q n i , q n E ( q 2 ) ≤ i ) + i + 1 ) i ∆ x i F. Bouchut & S. Boyaval 17 / 36 Shallow viscoelastic fluids