Numerical modeling of shallow hydrodinamic flows by …€¦ · Numerical modeling of Ocean...
Transcript of Numerical modeling of shallow hydrodinamic flows by …€¦ · Numerical modeling of Ocean...
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Numerical modeling of shallow hydrodinamic flows bymixed finite element methods
Tomas Chacon RebolloBCAM
Workshop ”Environmental Mathematics Day”February 26, 2013
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Sketch of the talk
1 Some remarks on numerical modeling of Ocean dynamics.
2 Primitive Equations of the Ocean: Reduced formulation
3 Mixed approximations.
4 Stabilized Method
5 Numerical results
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Numerical modeling of Ocean dynamics
Ocean dynamics largely determine the local climate of several areas,so as the global climate.The top 5m-depth oceanic layer contains as much heat as theatmosphere.
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Numerical modeling of Ocean dynamics
Oceanic upwellings generate 60% of world fisheries.
Numerical modeling of ocean flow is less developed than atmosphericflow.
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Numerical modeling of Ocean dynamics
Numerical models of oceanic flow separately deal with:
The surface dynamics (much faster)The surface mixing layer andThe inner flow dynamics.
The surface dynamics are modeled by 2D Shallow Water models.
We here focus on the other two.
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The Primitive Equations of the Ocean
Primitive Equations: Geometry
Ω = (x, z) ∈ Rd , x ∈ ω, −D(x) < z < 0, ω ⊂ Rd−1
∂Ω = Γs ∪ Γb : Γs ≡ ω × 0 Sea surface , Γb Bottom and sidewalls
Ω
Γs
Γb
D
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The Primitive Equations of the Ocean
Problem statement: Primitive Equations of the Ocean
Obtain U = (u, u3) : Ω× (0,T ) 7→ Rd Velocityand P : Ω× (0,T ) 7→ R Pressure
such that
∂tu + U · ∇u− µ∆u + α k× u +∇HP = f in Ω× (0,T )
∂vP = −ρg in Ω× (0,T )
∇ ·U = 0 in Ω× (0,T )
−µ∂u∂n|Γs = τw , u3|Γs = 0 in (0,T )
u|Γb= 0, u3 · n3|Γb
= 0 in (0,T )
u(0) = u0 in Ω.
f : Source term.
τw : Surface wind tension
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The Primitive Equations of the Ocean
Problem Statement: Rigid-lid assumption
Free-surface equation:
u3 = ∂tη + u · ∇Hη at x3 = η(x, t).
It comes from∂
∂tof the free-surface equation,
x3(t) = η(x(t), t).
A particular solution is
η ≡ 0, u3 = 0 at x3 = 0.
This is the Rigid-lid assumption.
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The Primitive Equations of the Ocean
Problem Statement: Reduced Formulation
Obtain u : Ω× [0,T ] 7→ Rd−1 Horizontal velocityand p : ω × (0,T ) 7→ R Surface pressure, s. th.
∂tu + U · ∇u− µ∆u + α k× u +∇Hp = f in Ω× (0,T )
∇H · < u >= 0 in ω × (0,T ),
−µ∂u∂n|Γs = τw , u|Γb
= 0 in (0,T ),
u(0) = u0 in Ω,
where U = (u, u3), with u3(x , z) =
∫ 0
z∇H · u(x , s) ds;
< u > (x) =
∫ 0
−D(x)u(x , s)ds
The 3d pressure is recovered by P(x , z) = p(x) + ρ gz .
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The Primitive Equations of the Ocean
Mathematical analysis
Existence of weak solutions (H1 regularity in space): Lions, Temam &Wang (1992), Lewandowski (1997), Chacon & Guillen (2000).
Difficulty: Low regularity of convection term (u3, ∂zu3 only L2
regularity in space).
Existence and uniqueness of strong solutions (H2 regularity in space):Kobelkov (2006), Cao and Titi (2007), Kukavica and Ziane (2007).
Main point: Additional regularity of surface pressure, coming from its2D nature.
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Mixed FE discretization
Prismatic FE spaces
Approximate Ω by polyhedric domains Ωh by:
dh
z = - D (x)h
Construct a prismatic grid Th of Ωh.
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Mixed FE discretization
Mixed FE spaces: Example of prismatic grid
Swimming pool
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Mixed FE discretization
Prismatic FE spaces
We consider prismatic FE spaces by
V kh =
φh ∈ C(Ωh) ∩H1
b(Ωh) | ∀K ∈ Th, φh|K ∈ (Pk(x)⊗ Pk(z))2
Mkh =
qh ∈ C (ωh) | ∀T ∈ Ch, qh|T ∈ Pk(x) and
∫Ω qh dxdz = 0
,
k ≥ 1;
where H1b(Ωh) =
φ ∈ [H1(Ωh)]2 |φ|Γb
= 0.
Key point: The 3D interpolate of a 2D discrete pressure remains 2D(Prismatic structure of the grid).
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Mixed FE discretization
Prismatic FE spaces
´ Nodos de presion (en superficie)
Nodos de velocidad horizontal
Location of dofs for (P2 − P1) Mixed FE.
´ Nodos de presion (en superficie)
Nodos de velocidad horizontalNodos de velocidad horizontal
Nodos de velocidad vertical
´ Nodos de presion
Comparison of dofs for Mixed FE for Non-reduced vs Reducedformulations.
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Mixed FE discretization
Oceanic inf-sup condition
Key for the stability of the surface pressure discretization:
β ‖qh‖L20(Ωh) ≤ sup
vh∈Vh
(∇H · 〈vh〉, qh)ωh
|∇vh|L20(Ωh)
∀qh ∈ Mh,
Sets compatibility conditions between the velocity and the pressurespaces Vh and Mh
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Mixed FE discretization
How to design pairs of spaces satisfying the oceanic inf-supcondition?
Recipe:
Start from a pair of 3D velocity-pressure spaces Wh and Qh satisfyingthe standard inf-sup condition:
β ‖qh‖L20(Ωh) ≤ sup
wh∈Wh
(∇ ·wh, qh)Ωh
|∇wh|L20(Ωh)
∀qh ∈ Qh,
Set
Vh: Horizontal components of the velocities of Wh
Mh: Pressures of Qh that do not depend on z .
Then the pair (Vh,Mh) satisfies the oceanic inf-sup condition
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Mixed FE discretization
Discrete Problem
Given (unh, pnh), find (un+1
h , pn+1h ) ∈ Vh ×Mh such that
∀ (vh, qh) ∈ Vh ×Mh,
(un+1h − unh
k, vh) +
((Un
h · ∇)un+1h , vh
)+ (∇νun+1
h ,∇νvh)+
+ f (k× un+1h , vh)− (pn+1
h ,∇H · 〈vh〉)ω+= 〈ln+1, vh〉;
(∇H · 〈un+1h 〉, qh)ω = 0,
(1)
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Mixed FE discretization
Stabilized Method
Purpose: Avoid to use purely stabilizing degrees of freedom.
Given (unh, pnh), find (un+1
h , pn+1h ) ∈ Vk ×Mk such that
∀ (vh, qh) ∈ Vh ×Mh,
(un+1h − unh
k, vh) +
((Un
h · ∇)un+1h , vh
)+ (∇νun+1
h ,∇νvh)+
+ f (k× un+1h , vh)− (pn+1
h ,∇H · 〈vh〉)ω+
+∑K∈Th
τK
∫K
(Unh · ∇un+1
h )(Unh · ∇vh)Dh dx dz
= 〈ln+1, vh〉;(∇H · 〈un+1
h 〉, qh)ω +∑T∈Ch
τT Dh (∇Hpn+1h ,∇Hqh)T = 0,
(2)where τK and τT are stabilization coefficients.
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Mixed FE discretization
Reduced inf-sup condition
Lemma
There exists a constant β > 0 independent of h such that ∀qh ∈ Mkh ,
β ‖qh‖L20(Ωh) ≤ sup
vh∈Vkh−0
(∇H · 〈vh〉, qh)ωh
|∇vh|L20(Ωh)
+
∑K∈Th
h2K‖∇qh‖2
Lα(K)
1/2
. (3)
Stabilized methods introduce the additional term in their formulation.
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Mixed FE discretization
Stabilization coefficients
τT = C h2T .
τK = τK (ReK ) = γhK
UnK
min(ReK ,P); with ReK =Un
KhK
ν.
τK takes into account the local balance between convection anddiffusion.
τK = O(h2K ), τT = O(hT )2.
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Mixed FE discretization Numerical Analysis
Key points for numerical analysis
Stabilizing terms represented by bubble FE spaces (Chacon 1998).
Numerical scheme cast as a mixed method for an augmentedvariational formulation.
Underlying inf-sup condition (Velocity+Bubble, Pressure) thatensures stability.
Standard tools of functional analysis used to perform the numericalanalysis.
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Simulation of interaction wind friction ↔ Coriolis force.
Horizontal dimensions (m): ω = [0, 104]× [0, 5× 103];
Minimum depth: 50m; Maximum depth: 100m.
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Imposed wind: (7.5, 0, 0) m/s. Latitude: 45 N.
Asymptotic analysis (As Ek → +∞, free space):Surface velocity points 45 to the right of the wind.
Surface velocity
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Surface pressure
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Projection of velocity on a stream-wise vertical plane cut
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
3D velocity on a stream-wise vertical plane cut
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Projection of velocity on a span-wise vertical plane cut
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Mixed FE discretization Numerical experiments
Wind-induced Upwelling and Downwelling flow in Genevalake
Surface wind: v = 7.5 (cos 45, sin 45) m/s. Latitude: 45 N.
Horizontal dimensions (m): 65 Km long, 13 Km large.
Maximum depth: 300m.
Isobath lines every 50m.
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Vertical velocity on horizontal cut plane z = −50 after 12h.
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Mixed FE discretization Numerical experiments
Wind-induced flow in a basin
Vertical velocity on horizontal cut plane z = −50 after 24h.
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Orthogonal Sub-Scales method
Derivation of OSS method for linearized problem
Consider the linearized steady PE: Given aH ∈ H1(Ω)d ,
Obtain y : Ω× [0,T ] 7→ Rd−1 Horizontal velocityand p : ω × (0,T ) 7→ R Surface pressure, s. th.
a · ∇y − µ∆y + k× αy +∇Hp = f in Ω
∇H · < y >= 0 in ω,
−µ∂y
∂n|Γs = τw , u|Γb
= 0 ,
where a = (aH , a3), with a3(x , z) =
∫ 0
z∇H · aH(x , s) ds;
< y > (x) =
∫ 0
−D(x)y(x , s)ds
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Orthogonal Sub-Scales method
Derivation of OSS method for linearized problem
The OSS method is a Variational Multi-Scale method.
It starts from the variational formulation:Obtain (y , p) ∈ H1
b(Ω)d−1 × L3/20 (Ω, ∂3) such that
B(a; y , p; w , q) = F (v), ∀(w , q) ∈W 1,4b (Ω)d−1 × L2
0(Ω, ∂3)
with
B(a; y , p; w , q) = −(a · ∇w , y) + µ(∇y ,∇w) + (αk× y ,w)
− (∇H · w , p) + (∇H · y , q);
F (v) = (f , v) + (τw ,w)Γs ;
H1b(Ω) = y ∈ H1(Ω), y |Γb
= 0
W 1,4b (Ω) = w ∈W 1,4(Ω), w |Γb
= 0
Lr0(Ω, ∂3) = q ∈ Lr (Ω), ∂3q = 0,
∫Ω q = 0
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Orthogonal Sub-Scales method
Variational Multiscale Method
Condensed notation: u = (y , p), v = (w , q)
(P)
Obtain u = uh + u ∈ U = Uh
⊕U s. t.
B(uh + u, v) = L(v), ∀v ∈ U
v = vh, (Ph)
Obtain uh ∈ Uh s. t.B(uh, vh) + B(u, vh) = L(vh), ∀vh ∈ Uh
v = v , (P)
Obtain u ∈ U s. t.
B(uh, v) + B(u, v) = L(v), ∀v ∈ U
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Orthogonal Sub-Scales method
Modeling of sub-grid scales
So, u satisfies
B(u, v) = L(v)− B(uh, v) = 〈Rh, v〉, ∀v ∈ U,
where Rh is the residual associated to uh.
Then, u = M(Rh)
Modeling of sub-scales: u = M(Rh) ' Mh(Rh)
Modeled equation for large scales:
(Ph)
Obtain uh ∈ Uh
B(uh, vh) + B(Mh(Rh), vh) = L(vh), ∀vh ∈ Uh
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Orthogonal Sub-Scales method
Modeling of sub-grid scales
Modeling of subscales (Codina 2002, Chacon, Gomez & Sanchez2010)
u ' Mh(Rh) = −τK (I − Πτ )(Rh) on K ,
where
Πτ is the orthogonal projection on Uh with respect to
(vh,wh)τ =∑K∈Th
(τK vh,wh)
τK is the stabilization matrix,
τK =
[τ1K 0
0 τ2K
].
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Orthogonal Sub-Scales method
Model for large scales
Obtain (yh, ph) ∈ Uh such that
B(yh, ph; vh, qh)−∑K∈Th
τ1K (F∗(vh, qh), (I − Πτ1)(F(yh, ph)))K
+∑K∈Th
τ2K (∇H · vh, (I − Πτ2)(∇H · yh))K
= F (vh) +∑K∈Th
τ1K (F∗(vh, qh), (I − Πτ1)(f ))K , ∀(vh, qh) ∈ Uh
where
F(yh, ph) = a · ∇yh − ν∆yh + α k× yh +∇Hph,
F∗(vh, qh) : Adjoint of F .
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Orthogonal Sub-Scales method
OSS model for non-linear PE
OSS method for non-linear PE: Replace a by yh = (yh, y3h).
Obtain (yh, ph) ∈ Uh = V kh ×Mk
h such that
B(yh; yh, ph; vh, qh)−∑K∈Fh
τ1K (F∗(vh, qh), (I − Πτ1)(F(yh, ph)))K
+∑K∈Fh
τ2K (∇H · vh, (I − Πτ2)(∇H · yh))K
= F (vh) +∑K∈Th
τ1K (F∗(vh, qh), (I − Πτ1)(f ))K , ∀(vh, qh) ∈ Uh
whereF(yh, ph) = yh · ∇yh − ν∆yh + α k× yh +∇Hph,
Interactions large-small scales due to convection are neglected.
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Orthogonal Sub-Scales method
Advantages of OSS method
No need of nested grids.
Stabilization of only the sub-scales not represented in large-scalesspace.
Reduced numerical diffusion.
No need of further modeling to simulate turbulent flows (onceinteractions large-small scales due to convection taken into account)(?).
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Orthogonal Sub-Scales method Analysis of OSS method
Convergence
Theorem
Assume that the triangulations Thh>0 are regular, τ1K = O(h2K ) and
τ2K = O(1), then
1 The OSS discretization of the non-linear PE admits a unique solution(yh, ph) ∈ Uh = V k
h ×Mkh which is bounded in
H1b(Ω)d−1 × L
3/20 (Ω, ∂3).
2 The sequence (yh, ph)h>0 contains a subsequence which is weakly
convergent in H1b(Ω)d−1 × L
3/20 (Ω, ∂3) to a solution of the non-linear
PE.
3 If the weak solution of PE belongs to W 1,3(Ω)d−1, then theconvergence is strong.
The same analysis technique as for Penalty method applies
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Orthogonal Sub-Scales method Analysis of OSS method
Stability: Discrete inf-sup condition
Lemma
Assume that the triangulations Thh>0 are regular, and thatτK = O(h2
K ). Then there exists a constant C > 0 such that ∀ph ∈ Mkh ,
C‖ph‖Lα(Ω) ≤ supvh∈V k
h
(∇H · vh, ph)
‖vh‖W 1,α′b
+
∑K∈Th
hαK‖Πτ1(∇qh)‖αLα(K)
1/α′
+
[R∑i=1
(sup
wh∈Yh(θi )
(∇ · wh, qh)
|wh|W 1,αb (θi )
)α]1/α′
,
where Yh(θi ) =[V kh |θi∩ H1
0 (θi )]d
, and θi is the support of the P1 basis
function located at node i .If the grids are uniformly regular, the red term is not needed.
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Orthogonal Sub-Scales method Analysis of OSS method
Error estimates
Theorem
Under the hypothesis of the convergence theorem, assume also that thecontinuous solution of the PE satisfies (u, p) ∈ Hk+1(Ω)× Hk(ω), forsome k , l ≥ 1 and that the data are small enough. Then, the followingerror estimates hold,
|uh − u|1,Ω + ‖p − ph‖L3/2(Ω) ≤ C hk .
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Orthogonal Sub-Scales method Numerical Results
Numerical Results: 2D Flows
Solution of discrete non-linear problem through evolution approach
1∆t (yn+1 − yn) + yn · ∇yn+1 − µ∆yn+1 + ∂xpn+1 = f in Ω ⊂ R2
∂x < yn+1 >= 0 in ω ⊂ R
y 0 = y0
yn+1|Γb = 0, µ∂yn+1
∂n |Γs = τw
Implementation with FreeFem++
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments:Testing the convergence order
Smooth solution on a square
Horizontal velocity
h P1b-P1 OSS Order in H1-norm
0,072 0.0586436 0.00369733
0.036 0.0283502 0.00101718 1.81623
0.018 0.013938 0.00314393 1.67336
0.014 0.011539 0.000211092 1.5564
Table 1: Estimated convergence orders for horizontal velocity.
Pressure
h P1b-P1 OSS Order in L2-norm
0.072 0.000932524 0.00045671
0.036 0.000327411 0.00123518 1.84027
0.018 0.000115275 3.7988e-5 1.68045
0.014 7.65232e-5 2.51929e-5 1.60469
Table 2: Estimated convergence orders for surface pressure.
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Testing the rigid-lid condition
Rigid-lid condition (yz = 0 at the surface z = 0) imposed as
∇H · < y >= 0.
.
Discretization by OSS method:
(∇H · < yh >, qh)ω+∑K∈Th
τ1K (∇Hqh, (I − Πτ1)(F(yh, ph)− fh))K = 0.
We check
nh =‖yzh‖L2(ω)
‖∇yh‖L2(Ω):
h nh conv. order
0.122193 0.00897865
0.0625004 0.00343854
0.0359585 0.00143265 0.91218872
Table 3: Convergence of nh
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Convex and non-convexgeometries
Convex and non-convex geometries f = 0, τw = 1, µ = 0.5
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Convex and non-convexgeometries
Convex and non-convex geometries f = 0, τw = 1, µ = 0.5
Horizontal Velocity
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Convex and non-convexgeometries
Convex and non-convex geometries f = 0, τw = 1, µ = 0.5
Vertical Velocity
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Convex and non-convexgeometries
Convex and non-convex geometries f = 0, τw = 1, µ = 0.5
Velocity Field
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Convex and non-convexgeometries
Convex and non-convex geometries f = 0, τw = 1, µ = 0.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4
−3
−2
−1
0
1
2
3
4Presión
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5
−4
−3
−2
−1
0
1
2
3
4
5Presión
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Testing stabilization properties
Stabilization properties:
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Testing stabilization properties
Stabilization properties: Stabilization of the pressure
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
300
350
400
EstableP1−burbuja/P1
Pressures. Re=100
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Orthogonal Sub-Scales method Numerical Results
Numerical experiments: Testing stabilization properties
Stabilization properties: Stabilization of the convection
Velocity Field. Re=400. P1-Bubble/P1
Velocity Field. Re=400. OSS
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Future work
Present and future work
High-order penalty-stabilized methods: Replace (for instance)
(∇H · 〈un+1h 〉, qh)ω +
∑T∈Ch
τT Dh (∇Hpn+1h ,∇Hqh)T = 0, ∀qh ∈ Mk
h ,
by
(∇H · 〈un+1h 〉, qh)ω +
∑T∈Ch
τT Dh ((I − Πh)∇Hpn+1h ,∇Hqh)T = 0,
where Πh is an interpolation or projection operator on the velocityspace V k
h .
Turbulence modeling: Include large-small scale interaction terms inOSS method.
A posteriori error estimates + Grid adaptation: Use the modeled u′ aserror indicator.