On the Navier-Stokes- equations with the wall-eddy boundary - PowerPoint PPT Presentation

On the Navier-Stokes- αβ equations with the wall-eddy boundary conditions Gantumur Tsogtgerel McGill University BIRS Workshop on Regularized and LES Methods for Turbulence Banff Friday May 18, 2012

The problem The Navier-Stokes- αβ equations: ∂ t v − ∆ (1 − β 2 ∆ ) u + (grad v ) u + (grad u ) T v +∇ p = 0, v = (1 − α 2 ∆ ) u , ∇· u = 0, with α > β > 0 . Wall-eddy boundary conditions: � grad ω + γ (grad ω ) T � β 2 (1 − n ⊗ n ) n = ℓω , u = 0, with | γ | ≤ 1 and ℓ > 0 . [Fried&Gurtin’08] Study the spatial principal part: ∆ 2 u +∇ p = f , ∇· u = 0, & b.c.

Integration by parts Let G = grad ω + γ (grad ω ) T , with ω = curl u . Then � � � G : gradcurl φ = − div G · curl φ + Gn · curl φ Ω Ω ∂ Ω � � = − curldiv G · φ + Gn · curl φ + ( n × div G ) · φ ∂ Ω Ω Assume ∇ u = 0 and φ | ∂ Ω = 0 . Then we have curldiv G = − ∆ 2 u and g · curl φ = − ( n × g ) · ∂ n φ , hence � � � ∆ 2 u · φ − G : gradcurl φ = ( n × Gn ) · ∂ n φ . Ω Ω ∂ Ω The boundary condition is of the form − n × n × Gn = k ω , which implies ( k = ℓ / β 2 ). kn × ω = n × Gn If this is satisfied, and ∆ 2 u = 0 , then � � G : gradcurl φ + k ( n × ω ) · ∂ n φ = 0, ∀ φ : φ | ∂ Ω = 0. Ω ∂ Ω

Variational formulation Let V = { u ∈ D ( Ω ) : ∇· u = 0} , V = clos H 1 V , and V s = V ∩ H s ( Ω ) . Define the continuous bilinear form a : V 2 × V 2 → R by � � a ( u , φ ) = G : gradcurl φ + k ( n × ω ) · ∂ n φ , ∂ Ω Ω where k = ℓ / β 2 > 0 . This bilinear form is symmetric, since G : grad ψ = ω i , j ψ i , j + γω j , i ψ i , j = ω i , j ψ i , j + γω i , j ψ j , i , and ( n × ω ) · ∂ n φ = − ω · curl φ = − ( n × ∂ n u ) · ( n × ∂ n φ ) , the latter inequality true provided u | ∂ Ω = 0 . Let u ∈ V 4 satisfy a ( u , φ ) = ( f , φ ) L 2 for all φ ∈ V 2 , where f ∈ L 2 is a given function. Then ∆ 2 u +∇ p = f in Ω , u = n × n × Gn + k ω = 0 on ∂ Ω .

Coercivity: The volume term We want to show that a ( u , u ) ≥ c � u � 2 H 2 − C � u � 2 L 2 for u ∈ V 2 . Case γ = − 1 : � ( ω i , j − ω j , i ) ω i , j = 1 2 � curlcurl u � 2 L 2 = 1 2 � ∆ u � 2 L 2 ≥ c � u � 2 H 2 . Ω Case γ = 1 : Korn’s second inequality � ( ω i , j + ω j , i ) ω i , j ≥ c � ω � 2 H 1 . Ω Case | γ | < 1 : � � � ω i , j ω i , j ≤ ( ω i , j + γω j , i ) ω i , j +| γ | ω i , j ω i , j Ω Ω Ω To conclude the latter two cases, note that � u � H 2 ≤ C � ∆ u � L 2 = � curl ω � L 2 ≤ � ω � H 1 .

Coercivity: The boundary term We have established � | n × ∂ n u | 2 ≥ c � u � 2 a ( u , u ) ≥ c � u � 2 H 2 − kC � u � 2 H 2 − k H 3/2 . ∂ Ω In order for this to be positive, we need kC 2 P C < c , where C P is the constant of the Friedrichs inequality � u � H 3/2 ≤ C P � u � H 2 , that has the behaviour C 2 P ∼ diam( Ω ) . To conclude, we have a ( u , u ) ≥ c � u � 2 H 2 − C � u � 2 u ∈ V 2 , for L 2 and moreover there exists a constant δ > 0 such that ℓ δβ β < implies C = 0. diam( Ω )

Hilbert-Schmidt + elliptic regularity Define the operator A : V 2 → ( V 2 ) ′ by ( Au )( φ ) = a ( u , φ ) , and restrict its range to H = close L 2 V , i.e., consider A as an unbounded operator in H with the domain dom( A ) = { u ∈ V 2 : Au ∈ H } . Then A is self-adjoint and has countably many eigenvalues λ 1 ≤ λ 2 ≤ ... , with λ n → +∞ as n → ∞ . If ℓ > 0 is sufficiently small, then λ 1 > 0 . Moreover, the corresponding eigenfunctions form both an orthonormal basis in H , and a basis in V 2 , orthogonal with respect to a ( · , · ) + µ 〈· , ·〉 for some sufficiently large µ . Regularity results on the solutions of Au = f can be derived from the Agmon-Douglis-Nirenberg theory for elliptic systems. One also has a functional calculus, e.g., � g ( A ) u = g ( λ n ) 〈 u , v n 〉 v n n

Fixed-point formulation In H , and with f ∈ L 2 H , consider the initial value problem Λ = 1 − α 2 ∆ : V 2 → H . ∂ t Λ u + β 2 Au = f , where This is equivalent to ∂ t v + β 2 Λ − 1 2 A Λ − 1 v = Λ − 1 1 2 f , 2 u , with v = Λ 2 � �� � D implying that � t u ( t ) = Λ − 1 1 Λ − 1 2 e ( τ − t ) D Λ − 1 2 e − tD Λ 2 u (0) + 2 f ( τ )d τ . 0 Restricting attention to the time interval [0, T ] , let us write it as u = u 0 + Φ f . Let B ( v , u ) = P [(grad v ) u + (grad u ) T v ] , and let P : L 2 → H be the Leray projector. Then Navier-Stokes- αβ equations are ∂ t Λ u + β 2 Au − ∆ u + B ( Λ u , u ) = 0, or equivalently u = u 0 + Φ∆ u − Φ B ( Λ u , u ).

Local existence and blow-up criterion Recall the fixed-point formulation u = u 0 + Φ∆ u − Φ B ( Λ u , u ). Noting that “ B ( Λ u , u ) = ∂ ( Λ u · u ) ”, we can bound � B ( Λ u , u ) � H 1 � � u � 2 H 4 , and show that u �→ B ( Λ u , u ) is locally Lipschitz as a mapping V 4 → V 1 . Hence we can design a Banach fixed point iteration in V 4 , assuming that T > 0 is suitably small. This also gives the following blow-up criterion: If there is a finite time T ∗ < ∞ beyond which the solution cannot be continued, then it is necessary that � u ( t ) � H 4 → ∞ as t ր T ∗ . So global existence is proved if we show that � u ( t ) � H 4 is bounded by a finite constant depending on the time of assumed existence.

A priori estimates and global well-posedness Pairing ∂ t Λ u + β 2 Au − ∆ u + B ( Λ u , u ) = 0, ( ∗ ) with u , we get 1 d d t 〈 Λ u , u 〉+ β 2 〈 Au , u 〉+〈∇ u , ∇ u 〉 = 0, 2 which gives d d t � u � 2 H 1 + c � u � 2 H 2 ≤ C � u � 2 u ∈ L ∞ V ∩ L 2 V 2 . L 2 , implying If we act on ( ∗ ) by A before pairing with u , we get d d t � u � 2 H 3 + c � u � 2 H 4 ≤ C � u � 2 L 2 +|〈 AB ( Λ u , u ), u 〉| . Taking into account the bound |〈 B ( Λ u , u ), Au 〉| � � Λ u � H 1 � u � H 2 � Au � L 2 � ε � u � 2 H 4 + C ε � u � 2 H 2 � u � 2 H 3 , we get u ∈ L ∞ V 3 .

Similarly, if we act by A 2 before pairing with u , we get d d t � u � 2 H 5 + c � u � 2 H 6 ≤ C � u � 2 L 2 +|〈 A 2 B ( Λ u , u ), u 〉| . We have the bounds 1 3 2 B ( Λ u , u ), A 2 u 〉| � � B ( Λ u , u ) � H 2 � u � H 6 , |〈 A and � B ( Λ u , u ) � H 2 � � Λ u � H 3 � u � H 3 � � u � H 5 � u � H 3 , giving rise to d d t � u � 2 H 5 + c � u � 2 H 6 ≤ C � u � 2 L 2 + ε � u � 2 H 6 + C ε � u � 2 H 3 � u � 2 H 5 . Thus u ∈ L ∞ H 5 , and global existence follows.

Limit as α , β → 0 Let α n and β n be sequences satisfying 0 < α n ≤ c β n → 0 , and consider ∂ t Λ n u n + β 2 n Au n − ∆ u n + B ( Λ n u n , u n ) = 0, where Λ n has α n in it, and k = ℓ n / β 2 n is fixed, so that A does not change. Also, assume that the initial conditions are the same. Then we can show that u n ∈ L ∞ H ∩ L 2 V , β u n ∈ L 2 V 2 , α u n ∈ L ∞ V , with uniformly bounded norms. Hence there exists u ∈ L ∞ H ∩ L 2 V such that up to a subsequence L ∞ L 2 , L 2 H 1 . u n → u weak* in and u n → u weakly in Moreover, u is a weak solution of the Navier-Stokes equation. Note that the second order boundary condition will be lost under the limit.