Application to Coupled Flow Problems Daniel Arndt - PowerPoint PPT Presentation

Application to Coupled Flow Problems Daniel Arndt Georg-August-Universit¨ at G¨ ottingen Institute for Numerical and Applied Mathematics 5th deal.II-Workshop August 3-7, 2015

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Table of Contents Introduction 1 Rotating Frame of Reference 2 Nodal-based MHD 3 Non-Isothermal Flow 4 Daniel Arndt Application to Coupled Flow Problems 2

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Table of Contents Introduction 1 Rotating Frame of Reference 2 Nodal-based MHD 3 Non-Isothermal Flow 4 Daniel Arndt Application to Coupled Flow Problems 3

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary The Full Set of Equations Velocity and Pressure ∂ t u − ν ∆ u + ( u · ∇ ) u + ∇ p + 2 ω × u = f u − βθ g + ( ∇ × b ) × b , ∇ · u = 0 Magnetic Field ∂ t b + λ ∇ × ( ∇ × b ) − ∇ × ( u × b ) = f b , ∇ · b = 0 Temperature ∂ t θ − α ∆ θ + ( u · ∇ ) θ = f θ Daniel Arndt Application to Coupled Flow Problems 4

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Local Projection Stabilization Idea Separate discrete function spaces into small and large scales Add stabilization terms only on small scales. Notations and prerequisites Family of shape-regular macro decompositions {M h } Let D M ⊂ [ L ∞ ( M )] d denote a FE space on M ∈ M h . For each M ∈ M h , let π M : [ L 2 ( M )] d → D M be the orthogonal L 2 -projection. κ M = Id − π M fluctuation operator Averaged streamline direction u M ∈ R d : | u M | ≤ C � u � L ∞ ( M ) , � u − u M � L ∞ ( M ) ≤ Ch M | u | W 1 , ∞ ( M ) Daniel Arndt Application to Coupled Flow Problems 5

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Assumptions Assumption - Approximation It holds for all w ∈ W l , 2 ( M ), M ∈ M h and l ≤ s ≤ k � κ M w � L 2 ( M ) ≤ Ch l M � w � W l , 2 ( M ) Assumption - Inf-Sup Stability Consider FE spaces ( V h , Q h ) satisfying a discrete inf-sup-condition: ( ∇ · v , q ) inf sup ≥ β > 0 �∇ v � L 2 (Ω) � q � L 2 (Ω) q ∈ Q h \{ 0 } v ∈ V h \{ 0 } V ❤ div := { v h ∈ V h | ( ∇ · v h , q h ) = 0 ∀ q h ∈ Q h } � = { 0 } ⇒ Daniel Arndt Application to Coupled Flow Problems 6

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Table of Contents Introduction 1 Rotating Frame of Reference 2 Analytical Results Numerical Results Nodal-based MHD 3 Non-Isothermal Flow 4 Daniel Arndt Application to Coupled Flow Problems 7

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Frames of Reference Navier Stokes Equations in an Inertial Frame of Reference ∂ u ∂ t + ( u · ∇ ) u − ν ∆ u + ∇ p = f in Ω × (0 , T ) ∇ · u = 0 in Ω × (0 , T ) Ω ⊂ R d bounded polyhedral domain Navier Stokes Equations in a Rotating Frame of Reference ∂ v ∂ t + ( v · ∇ ) v − ν ∆ v + 2 ω × v + ∇ � p = f in Ω × (0 , T ) ∇ · v = 0 in Ω × (0 , T ) ω × ( ω × r ) = − 1 p = p − 1 2 ∇ ( ω × r ) 2 2( ω × r ) 2 � Daniel Arndt Application to Coupled Flow Problems 8

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Weak Formulation Find U h = ( u h , p h ) : (0 , T ) → V h × Q h , such that ( ∂ t u h , v h ) + A G ( u h , U h , V h ) + (2 ω × u h , v h ) = ( f , v h ) for all V ❤ = ( v h , q h ) ∈ V h × Q h where A G ( w ; U , V ) := a G ( U , V ) + c ( w ; u , v ) a G ( U , V ) := ν ( ∇ u , ∇ v ) − ( p , ∇ · v ) + ( q , ∇ · u ) c ( w , u , v ) := (( w · ∇ ) u , v ) − (( w · ∇ ) v , u ) 2 Daniel Arndt Application to Coupled Flow Problems 9

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Stabilization Terms LPS Streamline upwind Petrov-Galerkin (SUPG) � s u ( w h ; u h , v h ) := τ M ( w M )( κ M (( w M · ∇ ) u h ) , κ M (( w M · ∇ ) v h )) M M ∈M h grad-div � t h ( w h ; u h , v h ) := γ M ( w M )( ∇ · u h , ∇ · v h ) M M ∈M h LPS Coriolis stabilization � a h ( w h ; u h , v h ) := α M ( w M )( κ M ( ω M × u h ) , κ M ( ω M × v h )) M M ∈M h Daniel Arndt Application to Coupled Flow Problems 10

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Convergence Result Theorem For a sufficiently smooth solution we obtain for e h = u ❤ − j u u : � t � e h � 2 ||| e h ( τ ) ||| 2 LPS d τ ≤ C exp( C G t ) h 2 k L ∞ (0 , t ;[ L 2 (Ω)] d ) + 0 with a Gronwall constant C G ( u ) = 1 + C | u | L ∞ (0 , T ; W 1 , ∞ (Ω)) + Ch � u � 2 L ∞ (0 , T ; W 1 , ∞ (Ω)) The parameters have to satisfy (1 ≤ s ≤ k ): √ ν h 2( k − s ) M h M ≤ C τ M ≤ τ 0 | u M | 2 � u h � L ∞ ( M ) h 2( k − s − 1) M γ M = γ 0 α M ≤ α 0 � ω � 2 L ∞ ( M ) Daniel Arndt Application to Coupled Flow Problems 11

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Numerical Results, Rotating Poiseuille Flow Ω = [ − 2 , 2] × [ − 1 , 1] � (1 − y 2 , 0) T , x = − 2 u ( x , y ) = | y | = 1 , ( ∇ u · n )( x = 2 , y ) = 0 (0 , 0) T , ν = 10 − 3 u 0 = 0 , p 0 = 0 , f = 0 ω = (0 , 0 , 100) , Flow for the parameters ω = (0 , 0 , 1) , ν = 10 − 1 Daniel Arndt Application to Coupled Flow Problems 12

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Poiseuille Flow, grad-div Daniel Arndt Application to Coupled Flow Problems 13

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Poiseuille Flow, grad-div Daniel Arndt Application to Coupled Flow Problems 14

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Poiseuille Flow, SUPG Coriolis Daniel Arndt Application to Coupled Flow Problems 15

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Poiseuille Flow, SUPG Coriolis Daniel Arndt Application to Coupled Flow Problems 16

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Poiseuille Flow, SUPG Coriolis Adaptive Daniel Arndt Application to Coupled Flow Problems 17

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Rotating Poiseuille Flow, SUPG Coriolis Adaptive Daniel Arndt Application to Coupled Flow Problems 18

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Taylor-Proudman, Stewartson Layer Navier Stokes Equations in a Rotating Frame of Reference ∂ u ∂ t + Ro ( u · ∇ ) u + 2ˆ e z × u = Ek ∆ u − ∇ p ∇ · u = 0 u = r sin θ ˆ e φ at r = r i u = 0 at r = r o r i = 1 / 2 r o = 3 / 2 ν Ro := ∆Ω / Ω Ek := Ω( r o − r i ) 2 Daniel Arndt Application to Coupled Flow Problems 19

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Taylor-Proudman, Stewartson Layer Fluid structure between two rotating spheres Figure: Ek = 10 − 6 Daniel Arndt Application to Coupled Flow Problems 20

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Taylor-Proudman, Stewartson Layer Fluid structure between two rotating spheres Figure: Ek = 10 − 4 Ro = − . 5 Daniel Arndt Application to Coupled Flow Problems 21

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Table of Contents Introduction 1 Rotating Frame of Reference 2 Nodal-based MHD 3 Analytical Results Numerical Results Non-Isothermal Flow 4 Daniel Arndt Application to Coupled Flow Problems 22

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Incompressible Nodal-based MHD - Joint work with Benjamin Wacker Stationary Linearized Incompressible Nodal-based MHD model − ν ∆ u + ( a · ∇ ) u + ∇ p − ( ∇ × b ) × d = f u , ∇ · u = 0 , λ ∇ × ( ∇ × b ) + ∇ r − ∇ × ( u × d ) = f b , ∇ · b = 0 u velocity field, p kinematic pressure a extrapolation for u b induced magnetic field, r magnetic pseudo pressure d extrapolation for b ν kinematic viscosity, λ magnetic diffusivity Daniel Arndt Application to Coupled Flow Problems 23

Introduction Rotating Frame of Reference Nodal-based MHD Non-Isothermal Flow Summary Weak Formulation Find U h := ( u h , b h , p h , r h ) ∈ V h × C h × Q h × S h such that A G , u ( U h , V h ) + A G , b ( U h , V h ) = � f u , v � + � f b , c � , for all V h := ( v h , c h , q h , s h ) ∈ V h × C h × Q h × S h . A G , u ( U , V ) = ν ( ∇ u , ∇ v ) + � a · ∇ u , v � − � ( ∇ × b ) × d , v � − ( p , ∇ · v ) + ( ∇ · u , q ) A G , b ( U , V ) = λ ( ∇ × b , ∇ × c ) − �∇ × ( u × d ) , c � + ( ∇ r , c ) − ( b , ∇ s ) Daniel Arndt Application to Coupled Flow Problems 24