Formal derivation of Saint-Venant-Exner-like model: Vertically - PowerPoint PPT Presentation

Formal derivation of Saint-Venant-Exner-like model: Vertically averaged Vlasov-Navier-Stokes equations Mehmet Ersoy 1 Timack Ngom 1 , 2 1 LAMA, UMR 5127 CNRS, Université de Savoie, 2 LANI, Université Gaston Berger de Saint-Louis (Sénégal) J. DYNAMO Rennes, 17-19 March, 2010.

Outline of the talk Introduction Formal derivation of the “mixed” CNSEs Formal derivation of the MENT model The non-dimensional “mixed” system System vertically averaged Examples Example 1: a viscous sedimentation model Example 2: the Grass sedimentation model Perspective

Outline Introduction Formal derivation of the “mixed” CNSEs Formal derivation of the MENT model The non-dimensional “mixed” system System vertically averaged Examples Example 1: a viscous sedimentation model Example 2: the Grass sedimentation model Perspective

The Saint-Venant-Exner model Saint-Venant equations for the hydrodynamic part:  ∂ t h + div ( q ) = 0 ,  � q ⊗ q � � � g h 2 (1)  ∂ t q + div + ∇ = − gh ∇ b h 2 + a bedload transport equation for the morphodynamic part: ∂ t b + ξ div ( q b ( h , q )) = 0 (2)

The Saint-Venant-Exner model Saint-Venant equations for the hydrodynamic part:  ∂ t h + div ( q ) = 0 ,  � q ⊗ q � � � g h 2 (1)  ∂ t q + div + ∇ = − gh ∇ b h 2 + a bedload transport equation for the morphodynamic part: ∂ t b + ξ div ( q b ( h , q )) = 0 (2) with ◮ h the water height from the surface z = b ( t , x ) , ◮ q = hu the water discharge, ◮ q b the sediment discharge (given by an empirical law: Grass equation [G81], The Meyer-Peter and M˝ uller equation [MPM48]), ◮ and ξ = 1 / ( 1 − ψ ) the porosity of the sediment layer. [MPM48] E. Meyer-Peter and R. Müller , Formula for bed-load transport, Rep. 2nd Meet. Int. Assoc. Hydraul. Struct. Res., 39–64 , 1948. [G81] A.J. Grass , Sediment transport by waves and currents, SERC London Cent. Mar. Technol. Report No. FL29 , 1981.

Outline Introduction Formal derivation of the “mixed” CNSEs Formal derivation of the MENT model The non-dimensional “mixed” system System vertically averaged Examples Example 1: a viscous sedimentation model Example 2: the Grass sedimentation model Perspective

Vlasov equation for sediments ∂ t f + div x ( vf ) + div v (( F + � g ) f ) = r ∆ v f (3) where ◮ f the density of particles, ◮ � g is the gravity vector ( 0 , 0 , − g ) t , and ◮ F is the Stokes drag force: F = 6 πµ a ( u − v ) with a = cte the radius , M 4 3 π a 3 the mass, M = ρ p (4) ρ p the mass density of sediments, and µ a characteristic viscosity, u velocity of the fluid ◮ r ∆ v f is the Brownian motion of the particles with r > 0 is the velocity of the diffusivity given by the Einstein formula: r = kT 6 πµ a = kT 9 µ (5) M M M 2 a 2 ρ p in which k is the Boltzmann constant, T > 0 is the temperature of the suspension, assumed constant.

Compressible Navier-Stokes equations   ∂ t ρ w + div ( ρ w u ) = 0 ,     ∂ t ( ρ w u ) + div x ( ρ w u ⊗ u ) + ∂ x 3 ( ρ w u v ) + ∇ x p ( ρ )  � �     = div x ( µ 1 ( ρ ) D x ( u )) + ∂ x 3 µ 2 ( ρ )( ∂ x 3 u + ∇ x u 3 ) + ∇ x ( λ ( ρ ) div ( u ))    + F , ∂ t ( ρ w u 3 ) + div x ( ρ w u u 3 ) + ∂ x 3 ( ρ w u 2  3 ) + ∂ x 3 p ( ρ )  � �     = div x µ 2 ( ρ )( ∂ x 3 u + ∇ x u 3 ) + ∂ x 3 ( µ 3 ( ρ ) ∂ x 3 u 3 ) + ∂ x 3 ( λ ( ρ ) div ( u ))     p = p ( t , x ) = g h ( t , x )   ρ 2 ( t , x )  4 ρ f (6) with u = ( u , u 3 ) , x = ( x , x 3 ) and µ i � = µ j . ◮ ρ w the density of the fluid, ρ s the macroscopic density of � sediments, ρ = ρ w + ρ s with ρ s = R 3 f dv , ◮ F the coupling of fluid-sediment interaction, including the gravity source term: � R 3 Ffdv + ρ w � F = − g (7) [BM10] D. Bresch and G. Métivier , Anelastic Limits for Euler Type Systems, In preparation , 2010.

Fluid sediment coupling  �� 6 πµ a � � 9 µ = kT   ( u − v ) + � ∂ t f + div x ( vf ) + div v g f ∆ v f ,   2 a 2 ρ p  M M        ∂ t ρ w + div ( ρ w u ) = 0 ,     ∂ t ( ρ w u ) + div x ( ρ w u ⊗ u ) + ∂ x 3 ( ρ w u v ) + ∇ x p ( ρ )   � �    = div x ( µ 1 ( ρ ) D x ( u )) + ∂ x 3 µ 2 ( ρ )( ∂ x 3 u + ∇ x u 3 ) + ∇ x ( λ ( ρ ) div ( u ))     + F ,         ∂ t ( ρ w u 3 ) + div x ( ρ w u u 3 ) + ∂ x 3 ( ρ w u 2 3 ) + ∂ x 3 p ( ρ )   � �    = div x µ 2 ( ρ )( ∂ x 3 u + ∇ x u 3 ) + ∂ x 3 ( µ 3 ( ρ ) ∂ x 3 u 3 )     + ∂ x 3 ( λ ( ρ ) div ( u )) (8)

With boundary conditions: free surface: a normal stress continuity. movable bed: a general wall-law condition and continuity of the velocity at the interface x 3 = b ( t , x ) . kinematic: ??? � replaced with the equation: � 1 + |∇ x b | 2 u | x 3 = b · n b S = ∂ t b + (9) � 1 + |∇ x b | 2 u | x 3 = b · n b may plays the role of incoming and and S − outgoing particles. [MSR03] N. Masmoudi and L. Saint-Raymond , From the Boltzmann Equation to the Stokes-Fourier System in a Bounded Domain, Communications on pure and applied mathematics, 53(9):1263–1293 ,2003.

Dimensionless number and asymptotic ordering Let ◮ √ θ be the fluctuation of kinetic velocity, ◮ U be a characteristic vertical velocity of the fluid, ◮ T be a characteristic time, ◮ τ be a relaxation time, ◮ L be a characteristic vertical height, and √ � a � 2 ρ p θ C = T F = g T E = 2 B = U , τ , √ , C (10) 9 ρ f θ L with the following asymptotic regime: ρ p C = 1 = O ( 1 ) , B = O ( 1 ) , ε , F = O ( 1 ) , E = O ( 1 ) . (11) ρ f [GJV] T. Goudon and P-E. Jabin and A. Vasseur , Hydrodynamic limit for the Vlasov-Navier-Stokes Equations. I. Light particles regime, Indiana Univ. Math. J., 53(6):1495–1515 ,2004.

Approximation at main order with respect to ε Asymptotic expansion of f , u , p and ρ as: f = f 0 + ε f 1 + O ( ε 2 ) , . . . Then at order 1 /ε  div v (( u 0 − v ) f 0 − ∇ v f 0 ) = 0 ,  (12) �  R 3 ( v − u 0 ) f 0 dv = 0 . Let ρ s and V be the macroscopic density and the macroscopic speed V of particles: � � � � � ρ s 1 = f dv (13) ρ s V v R 3 Then Equations (12) provide: 1 2 � u 0 − v � 2 and V 0 = u 0 . f 0 = s e − 1 ( 2 π ) 3 / 2 ρ 0 (14)

Approximation at main order with respect to ε Then at order 1: Integrating Vlasov equation against 1 and v :  ∂ t ρ 0 s + B div ( ρ 0 s u 0 ) = 0      s u 0 ⊗ u 0 ) + B ∇ x ( ρ 0 ∂ t ( ρ 0 s u 0 ) + B div x ( ρ 0 (12) s ) � �    R 3 ( u 0 − v ) f 1 dv + R 3 u 1 f 0 dv − F ρ 0 s �  = k  On the other hand, the dimensionless CNSEs ( Σ being the anisotropic viscous tensor):  ∂ t ρ 0 w + B div ( ρ 0 w u 0 ) = 0 ,   �   w u 0 ⊗ u 0 ) + B ∇ x p 0 = 2 E div (Σ 0 : D ( u 0 )) ∂ t ( ρ 0 w u 0 ) + B div x ( ρ 0 � � �   R 3 ( v − u 0 ) f 1 dv − R 3 u 1 f 0 dv − F ρ 0 w �  + ∇ ( λ div ( u 0 )) + k .  (13)

Adding two system and returning to physical variables, we obtain the “mixed” model:  ∂ t ρ + div ( ρ u ) = 0 ,     ∂ t ( ρ u ) + div x ( ρ u ⊗ u ) + ∂ x 3 ( ρ u v ) + ∇ x P   � �     = div x ( µ 1 ( ρ ) D x ( u )) + ∂ x 3 µ 2 ( ρ )( ∂ x 3 u + ∇ x u 3 )    + ∇ x ( λ ( ρ ) div ( u )) (14)     ∂ t ( ρ u 3 ) + div x ( ρ u u 3 ) + ∂ x 3 ( ρ u 2  3 ) + ∂ x 3 P  � �     = div x µ 2 ( ρ )( ∂ x 3 u + ∇ x u 3 ) + ∂ x 3 ( µ 3 ( ρ ) ∂ x 3 u 3 )    + ∂ x 3 ( λ ( ρ ) div ( u )) where P = p + θρ s .

Outline Introduction Formal derivation of the “mixed” CNSEs Formal derivation of the MENT model The non-dimensional “mixed” system System vertically averaged Examples Example 1: a viscous sedimentation model Example 2: the Grass sedimentation model Perspective

Asymptotic analysis: “thin layer” ◮ Vertical movements are assumed small with respect to horizontal one, ◮ Vertical length is assumed small with respect to horizontal one, i.e. we compare: ◮ L and L (the characteristic length of the domain), ◮ U and U (the characteristic horizontal velocity of the fluid) .

Outline Introduction Formal derivation of the “mixed” CNSEs Formal derivation of the MENT model The non-dimensional “mixed” system System vertically averaged Examples Example 1: a viscous sedimentation model Example 2: the Grass sedimentation model Perspective

We introduce a small parameter such as: ε ≈ L L ≈ U U . and ◮ T such as T = L / U , ◮ ρ such as P = ρ U 2 , t = t x = x y = y u = u v = v � � � � � T , L , L , U , U , p = P ρ = ρ ρ s = ρ s H = H b = b � � � p , � ρ, � ρ , L , L , ¯ ¯ ¯ λ = λ µ j = µ j � λ, � , j = 1 , 2 , 3 . ¯ µ j ¯

Asymptotic ordering With µ i ( ρ ) = ε i − 1 ν i ( ρ ) , i = 1 , 2 , 3 and λ ( ρ ) = ε 2 γ ( ρ ) . (15) Re i Re λ where U Re i = ρ UL Re λ = ρ UL F r = √ g L , , λ . (16) µ i is the Froude number F r , the Reynolds number associated to the viscosity µ i (i=1,2,3), Re i and the Reynolds number associated to the viscosity λ , Re λ . We also set S = ε U .