Non-homogeneous incompressible Bingham flows with variable yield - PowerPoint PPT Presentation

Non-homogeneous incompressible Bingham flows with variable yield stress and application to volcanology. Jordane Mathé with Laurent Chupin (maths) and Karim Kelfoun (LMV) Laboratoire de Maths & Laboratoire Magmas et Volcans june 2015 Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 1 / 18

Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 2 / 18

Laboratory experiment Zoom into the granular column → O. Roche , LMV. Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 3 / 18

Laboratory experiment Zoom into the granular column → Fluidisation : injection of gas through a pore plate. O. Roche , LMV. Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 3 / 18

Figure: Non-fluidised → short runout distance Figure: Fluidised → long runout distance

Model 1 Numerical simulation 2 Perspectives 3 Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 5 / 18

Model 1 Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 6 / 18

Two phases for one fluid Consider one mixed fluid with variable density .

Two phases for one fluid Consider one mixed fluid with variable density . It satisfies the non-homogeneous incompressible Navier-Stokes equations:   div ( v ) = 0   ∂ t ρ + div ( ρ v ) = 0    ∂ t ( ρ v ) + div ( ρ v ⊗ v ) + ∇ p = ρ g + div ( S )

Two phases for one fluid Consider one mixed fluid with variable density . It satisfies the non-homogeneous incompressible Navier-Stokes equations:   div ( v ) = 0   ∂ t ρ + div ( ρ v ) = 0    ∂ t ( ρ v ) + div ( ρ v ⊗ v ) + ∇ p = ρ g + div ( S ) Unknown: v : velocity, p : total pressure, ρ : density.

Two phases for one fluid Consider one mixed fluid with variable density . It satisfies the non-homogeneous incompressible Navier-Stokes equations:   div ( v ) = 0   ∂ t ρ + div ( ρ v ) = 0    ∂ t ( ρ v ) + div ( ρ v ⊗ v ) + ∇ p = ρ g + div ( S ) Constant g : gravity.

Two phases for one fluid Consider one mixed fluid with variable density . It satisfies the non-homogeneous incompressible Navier-Stokes equations:   div ( v ) = 0   ∂ t ρ + div ( ρ v ) = 0    ∂ t ( ρ v ) + div ( ρ v ⊗ v ) + ∇ p = ρ g + div ( S ) Constant g : gravity. Rheology Let precise S .

Rheology In our case S = µ Dv + Dv = ˙ γ is the strain rate tensor, where µ is the effective viscosity

Rheology In our case S = µ Dv + Σ , Dv = ˙ γ is the strain rate tensor, Σ = q Dv where µ is the effective viscosity | Dv | ⇒ Bingham fluid with yield stress q

Rheology In our case S = µ Dv + Σ , Dv = ˙ γ is the strain rate tensor, Σ = q Dv where µ is the effective viscosity | Dv | ⇒ Bingham fluid with yield stress q Idea Let vary the yield stress q as a function of the interstitial gas pressure .

Variation of the yield stress Definition of the yield stress � atmospheric pressure: Coulomb friction = q high pressure: fluid Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 9 / 18

Variation of the yield stress Definition of the yield stress � atmospheric pressure: Coulomb friction = q high pressure: fluid � tan( δ )( ρ gh − gas pressure) if low gas pressure = q 0 if high gas pressure where δ is the internal friction angle. Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 9 / 18

Variation of the yield stress Definition of the yield stress � atmospheric pressure: Coulomb friction = q high pressure: fluid � tan( δ )( ρ gh − gas pressure) if low gas pressure = q 0 if high gas pressure tan( δ )( ρ gh − gas pressure) + = q where δ is the internal friction angle. Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 9 / 18

Equations of the model Noting p f the interstitial gas pressure, we obtain:  div ( v ) = 0     ∂ t ρ + div ( ρ v ) = 0       Dv  ( ρ gh − p f ) +  + ρ g  ∂ t ( ρ v ) + div ( ρ v ⊗ v ) − µ ∆ v + ∇ p = tan( δ ) div   � �� � | Dv |    yield q  

Equations of the model Noting p f the interstitial gas pressure, we obtain:  div ( v ) = 0     ∂ t ρ + div ( ρ v ) = 0       Dv  ( ρ gh − p f ) +  + ρ g  ∂ t ( ρ v ) + div ( ρ v ⊗ v ) − µ ∆ v + ∇ p = tan( δ ) div   � �� � | Dv |    yield q   ∂ t p f + v · ∇ p f − κ ∆ p f = 0 where κ is the diffusion coefficient.

Numerical simulation 2 Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 11 / 18

Dambreak: Mesh Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 12 / 18

Dambreak: how to treat the density? ∂ t ρ + v · ∇ ρ = 0 numerical method: RK3 TVD - WENO5 scheme. Jordane Mathé (Maths-LMV) Fluidized granular flows june 2015 12 / 18

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0 v n , Σ n , p n and p f n being calculated. n � 0

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0 v n , Σ n , p n and p f n being calculated. n � 0 v n +1 , Σ n +1 ) solution of We compute ( �  v n +1 − 4 v n + v n − 1 v n +1 ρ n +1 = div Σ n +1 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ � + ∇ p n    − e y ,  ρ n +1 ρ n +1  2 δ t Σ n +1 = P q n (Σ n +1 + r D � v n +1 + ε (Σ n − Σ n +1 )) ,   v n +1 �   �  � = 0 . � ∂ Ω

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0 v n , Σ n , p n and p f n being calculated. n � 0 v n +1 , Σ n +1 ) solution of We compute ( �  v n +1 − 4 v n + v n − 1 v n +1 ρ n +1 = div Σ n +1 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ � + ∇ p n    − e y ,  ρ n +1 ρ n +1  2 δ t Σ n +1 = P q n (Σ n +1 + r D � v n +1 + ε (Σ n − Σ n +1 )) ,   v n +1 �   �  � = 0 . � ∂ Ω We compute ( v n +1 , p n +1 ) thanks to the incompressibility constrain  v n +1 − � v n +1 2 3 ρ n +1 ∇ ( p n +1 − p n ) = 0 ,   + δ t �  �  div v n +1 = 0 , v n +1 · n ∂ Ω = 0 . �

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0 v n , Σ n , p n and p f n being calculated. n � 0 v n +1 , Σ n +1 ) solution of We compute ( �  v n +1 − 4 v n + v n − 1 v n +1 ρ n +1 = div Σ n +1 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ � + ∇ p n    − e y ,  ρ n +1 ρ n +1  2 δ t Σ n +1 = P q n (Σ n +1 + r D � v n +1 + ε (Σ n − Σ n +1 )) ,   v n +1 �   �  � = 0 . � ∂ Ω We compute ( v n +1 , p n +1 ) thanks to the incompressibility constrain  v n +1 − � v n +1 2 3 ρ n +1 ∇ ( p n +1 − p n ) = 0 ,   + δ t �  �  div v n +1 = 0 , v n +1 · n ∂ Ω = 0 . � We compute p f n +1 solution of 3 p f n +1 − 4 p f n + p f n − 1 + 2 v n +1 · ∇ p f n − v n +1 · ∇ p f n − 1 − ∆ p f n +1 = 0 . 2 δ t

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0 v n , Σ n , p n and p f n being calculated. n � 0 v n +1 , Σ n +1 ) solution of We compute ( �  v n +1 − 4 v n + v n − 1 v n +1 ρ n +1 = div Σ n +1 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ � + ∇ p n    − e y ,  ρ n +1 ρ n +1  2 δ t Σ n +1 = P q n (Σ n +1 + r D � v n +1 + ε (Σ n − Σ n +1 )) ,   v n +1 �   �  � = 0 . � ∂ Ω Σ n , 0 = Σ n , and ( v n , p n , p n ֒ → k = 0 f ) given. Σ n , k known, we first compute � v n , k solution of a → k � 0 ֒ Laplace-type problem then we project the stress tensor to obtain Σ n , k +1 :  v n , k − 4 v n + v n − 1 ρ n +1 + ∇ p n v n , k ρ n +1 = div Σ n , k 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ �   − e y ,   2 δ t ρ n +1  v n , k � � � = 0 , �    ∂ Ω   Σ n , k +1 = P q n (Σ n , k + r D � v n , k + ε (Σ n − Σ n , k )) .

Numerical scheme (without density) v 0 , Σ 0 , p 0 and p f 0 given. n = 0 v n , Σ n , p n and p f n being calculated. n � 0 v n +1 , Σ n +1 ) solution of We compute ( �  v n +1 − 4 v n + v n − 1 v n +1 ρ n +1 = div Σ n +1 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ � + ∇ p n    − e y ,  ρ n +1 ρ n +1  2 δ t Σ n +1 = P q n (Σ n +1 + r D � v n +1 + ε (Σ n − Σ n +1 )) ,   v n +1 �   �  � = 0 . � ∂ Ω Σ n , 0 = Σ n , and ( v n , p n , p n ֒ → k = 0 f ) given. Σ n , k known, we first compute � v n , k solution of a → k � 0 ֒ Laplace-type problem then we project the stress tensor to obtain Σ n , k +1 :  v n , k − 4 v n + v n − 1 ρ n +1 + ∇ p n v n , k ρ n +1 = div Σ n , k 3 � + 2 v n ·∇ v n − v n − 1 ·∇ v n − 1 − ∆ �   − e y ,   2 δ t ρ n +1  v n , k � � � = 0 , �    ∂ Ω   Σ n , k +1 = P q n (Σ n , k + r D � v n , k + ε (Σ n − Σ n , k )) .