numerical modeling of shallow hydrodinamic flows by …€¦ · numerical modeling of ocean...

Numerical modeling of shallow hydrodinamic flows bymixed finite element methods

Tomas Chacon RebolloBCAM

Workshop ”Environmental Mathematics Day”February 26, 2013

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

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.


Oceanic upwellings generate 60% of world fisheries.

Numerical modeling of ocean flow is less developed than atmosphericflow.


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.

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


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


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.


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 .


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.

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.


Mixed FE spaces: Example of prismatic grid

Swimming pool


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).


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.


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


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:


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


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)


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.


Reduced inf-sup condition

Lemma

There exists a constant β > 0 independent of h such that ∀qh ∈ Mkh ,


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.


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.

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.

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.



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



Surface pressure



Projection of velocity on a stream-wise vertical plane cut



3D velocity on a stream-wise vertical plane cut



Projection of velocity on a span-wise vertical plane cut


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.



Vertical velocity on horizontal cut plane z = −50 after 12h.



Vertical velocity on horizontal cut plane z = −50 after 24h.

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


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


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


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


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

].


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 .


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.


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)(?).

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


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.


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 .

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++


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.


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


Numerical experiments: Convex and non-convexgeometries

Convex and non-convex geometries f = 0, τw = 1, µ = 0.5




Horizontal Velocity




Vertical Velocity




Velocity Field




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


Numerical experiments: Testing stabilization properties

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



Stabilization properties: Stabilization of the convection

Velocity Field. Re=400. P1-Bubble/P1

Velocity Field. Re=400. OSS

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.

numerical modeling of shallow hydrodinamic flows by …€¦ · numerical modeling of ocean...

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