TELEMAC-3D

Non hydrostatic 3D Navier-Stokes equations with a free surface The Telemac system

Why 3 dimensions?

Navier-Stokes equations

The 3 main difficulties

Building the mesh Survival in a moving mesh

Dynamic pressure

Other topics Dry zones Parallelism

The aspect ratio The 2-plane case Non linear waves

Applications

Jean-Michel Hervouet EDF Lab Laboratoire National d’Hydraulique et Environnement

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

Isaac Newton

Laboratoire National d’Hydraulique et Environnement

2

http://www.opentelemac.org

2500 active users

14000 messages on the forum

110 countries

130 persons in 2011 user club in Paris

12000 downloads of User Club proceedings in 2013

Groundwater flows

Sediment

Hydrodynamics

Water Quality

Waves

TELEMAC-2D TELEMAC-3D

SISYPHE Included in TELEMAC-2D or 3D

Coupling with Delwaq

TELEMAC-3D

ESTEL-2D ESTEL-3D

ARTEMIS

TOMAWAC

2 dimensions 3 dimensions

MASCARET

COURLIS - MOBILI

TRACER

1 dimension Mesh generators : Blue Kenue, Janet Pre-processing FUDAA-MASCARET

Post-processing FUDAA-MASCARET

FUDAA-PREPRO Blue KENUE (© CNRC-CHC),

TECPLOT / PARAVIEW

The Telemac-Mascaret hydroinformatic system

4

A consortium to provide manpower and steer developments

•  A first circle of industrial, engineering, institutional, academic partners willing to reinforce/enhance the development of the system, and ensuring the industrial quality and validation standards.

•  The group decides the integration process and development plan, promotion and assistance, … and engage resources/responsibility for developments, integration of new features, documentation, validation, promotion …). Minimum resources: 2 persons/year

•  A steering committee, advised by a scientific committee : –  Decides the technical, strategic and further developments

orientations, as well as the valorisation strategy –  Defines the priority in the common developments and way of doing

them (internal resources of the members, call to Open source community, ….)

–  Checks if the developments fulfil the desired quality standards (validation and documentation)

•  A specific internet site for the consortium:

–  To deliver and download the new reference versions –  To call Opensource community for new developments –  To receive new developments from the community –  To animate the community Forum of users

Where to get details?

Book: hydrodynamics of free surface flows Modelling with the finite element method (3D part mostly after Jean-Marc Janin) Telemac release notes: versions 5.7, 5.8, 5.9, 6.0, 6.1, 6.2, 6.3 On Telemac website www.opentelemac.org Astrid Decoene PhD: Hydrostatic model for 3D free surface flows and numerical schemes

Why 3 dimensions?

Because real life has 3 dimensions!

Courtesy Sébastien Bourban

Simplifying complexifies!

d

if d tends to infinity, the hydraulic radius tends to 0, the friction tends to infinity, the river stops flowing…

In 1D: a counter-example…

In 2D: a tracer depth-averaged concentration is not the conservative variable…

∫=s

f

Z

ZdzC

hc 1

Depth-averaged concentration conservative variable

∫=s

f

Z

ZdzCch

Differences in solutions…

H h

Saint-Venant theory, neglecting upstream velocity:

ghHlQ 2µ=

...3849.0332

==µ

After Bazin:

p

⎥⎦

⎤⎢⎣

⎡

++⎥⎦

⎤⎢⎣

⎡ += 2)(55.01003.0405.0pH

HH

µ

After Rehbock: µ between 0.4023 and 0.513

Discharge over a weir…

Simplifying complexifies!

Improvement of Saint-Venant equations by Boussinesq (1877)

Term added to left-hand side of Saint-Venant momentum equation:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

tUHdivgradH

tUdivgradH

!!

00

20

26

H0 reference depth. Propagation on dry bed unchanged, steady flow unchanged…

3D modelling of free surface flows, SimHydro 2012 Sophia-Antipolis 12-14 September 2012

Solitary wave in a canal with constant depth depth = 10 m wave height = 2 m

Mesh size: 0.1 m time step: 0.1 s C.F.L.: 10,84

Saint-Venant and Boussinesq solutions

20 s 40 s 60 s

What Navier-Stokes equations?

Telemac-3D

Reynolds Averaged Free surface 3D incompressible Navier-Stokes Equations

With or without hydrostatic pressure assumption

Advection and diffusion of tracers and/or sediment

Results : water depth, 3 components of velocity,

concentrations of tracers or sediment

What we would like to solve:

0)( =+∂

∂ Udivt

!ρ

ρ

FgdivUUtU !!!!!

++=⊗∇+∂

∂ )()()(σρ

ρ

Mass conservation with variable density:

Momentum with stress tensor , gravity and source terms:

Energy with thermal conduction and radiation:

rqUUete

+∇−∇=∇+∂

∂ )().()()( !!σρ

ρ

Assumptions and simplifications…

0)( =Udiv!

Newtonian fluid:

Incompressible fluid (constant density)?

But what about heat, salt, and sediments? Boussinesq approximation!

Dp µδσ 2+−= ( )UUD t!!

∇+∇=21

No specific equation for energy

Reynolds averaged:

ρµ

νν =ofinsteadt

The Boussinesq approximation (1/2)

0)(0)( ==>= UdivUdiv!!

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−∇−

00

111)(1ρρ

ρρρwrittenispin

Continuity equation:

Momentum equation:

...1 linearareandFEinsoρ

ρ

The Boussinesq approximation (2/2)

)(0

sZg ∇Δ

ρρ

∫Δ

+−+=sZ

zsatm dzgzZgpzp0

00 )()(ρρ

ρρHydrostatic pressure:

Compared to constant density, 2 extra terms in momentum: (second order terms neglected)

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ∇− ∫

sZ

zdzg

0ρρ

barotropic effect (depth independent)

baroclinic effect (depth dependent)

(This is hydrostatic pressure, only a part of the pressure…)

These terms apply to U and V, this is paradoxical…

The final equations: continuity and momentum

0)( =Udiv!

xt fUgraddivxpUgradU

tU

+ν+∂

∂

ρ−=+

∂

∂ ))((1)(.0

!

yt fVgraddivypVgradU

tV

+ν+∂

∂

ρ−=+

∂

∂ ))((1)(.0

!

zt fWgraddivzpWgradU

tW

+ν+∂

∂

ρ−=+

∂

∂ ))((1)(.0

!

Advection (non conservative form)

Turbulent diffusion

Source terms

(next slide…)

Pressure terms

Eulerian (fixed point…) derivative in time

General features of numerical schemes in Telemac-3D

Implicit schemes (linear systems solved with CG, GMRES, direct, etc.)

Matrices stored Element By Element (EBE) or Edge-Based

Depth, pressure, velocities colocalised

Advection schemes:

Method of characteristics: strong and weak form (ELLAM)

Streamline Upwind Petrov-Galerkin

N-Edge-based Residual Distributive (NERD, 2011)

Distributive schemes (N, PSI, prisms, enhanced predictor corrector)

The source terms in Telemac-3D (see subroutine trisou and wave_equation, version 7.0)

Buoyancy (heat, salt, sediment…) : in Fx, Fy (from Boussinesq approximation)

Coriolis force : in Fx, Fy , Fz not done…

Tide generating force : in Fx, Fy

Wave driven currents : in Fx, Fy , 3D term not done

Momentum from sources : in Fx, Fy , Fz not done…

Atmospheric pressure gradient : in Fx, Fy

Pressure gradients (other than buoyancy and non-hydrostatic pressure)

...1... +∂

∂−

∂

∂−=+

∂

∂

xZg

xp

tU s

atm

ρ

...1... +∂

∂−

∂

∂−=+

∂

∂

yZg

yp

tV s

atm

ρ

fs ZhZ +=

3D Navier-Stokes equations with a free surface

The 3 main numerical difficulties:

Computing the free surface and building the mesh

Writing equations in a moving mesh

Non hydrostatic pressure

Building the mesh

Computing the free surface and building the mesh

Goal: building a 3D mesh between the bottom and the free surface

Problem: how to get the free surface before the mesh?

Solution: a 2D mesh, prisms and the Saint-Venant continuity equation

Superimposing meshes of triangles

= 3D meshes of prisms

Every prism can be split into 3 tetrahedra…

But edges of 2 tetrahedra must not cross!

The Saint-Venant continuity equation

It is exact (no approximation in its derivation)

0)( =+∂∂ uhdivth !

01== ∫ dzU

hu s

f

Z

Z

!!with:

Idea: solving the Saint-Venant equations to get the depth

(the first versions of Telemac-3D called Telemac-2D)

Solving the Saint-Venant equations: a bad idea!

1)  Hidden hydrostatic approximation

2) An unexpected problem:

The sum over the vertical of discretised 3D continuity

equations is not the discretised Saint-Venant continuity equation

(sum over the vertical and discretisation do not commute)

Effect of neglecting the dynamic pressure to compute the free surface…

2006

2005

Sum over the vertical and discretisation do not commute…

Solution: first discretising the 3D equations, then summing them over the vertical

Computing the free surface: Solving a pseudo Saint-Venant continuity equation which is a discrete

sum over the vertical of 3D continuity equations:

0)( =+∂

∂∫ dzUdiv

th s

f

Z

Z

!

The 3D velocity is a predictor value taken from 3D momentum equations and computed in the 3D mesh of the previous time step

dzUsf

Z

Z∫!

is in fact a discrete sum over the vertical

(Importance of superimposing 2D meshes to get vertical lines!)

This predictor velocity would be explicit, but we can do better!

The idea of the pseudo-wave equation (1/2)

.........)(... ++∇−=+∂

∂fZhg

tU

Due to the hydrostatic part of pressure, there is a gradient of depth in the momentum equations:

The gradient of depth can remain implicit in the continuity equation…

Discretised, it is:

.........)(1

++∇Δ−=+

f

nnZhtgUU

The idea of the pseudo-wave equation (2/2)

RHShgradcdivth

=−∂

∂ ))(( 2

In the continuum:

D is a matrix defined on every vertical of the mesh, function of diffusion, friction, … General idea: the momentum equation is left partly unsolved when plugged in the continuity equation, but summing over the vertical suppresses this unsolved problem…

In discretisation, a bit more complex…

ghc =2

∫ −s

f

Z

Zhu dzDgbyreplacedc 12 θθ

See book page 158-164 and release notes 5.7 chapter 1.

Impossible d'afficher l'image. Votre ordinateur manque peut-être de mémoire pour ouvrir l'image ou l'image est endommagé

Impossible d'afficher l'image. Votre ordinateur manque peut-être de mémoire pour ouvrir l'image ou l'image est endommagée. Redémarrez l'ordinateur, puis ouvrez à nouveau le fichier. Si le x rouge est toujours affiché, vous devrez peut-être supprimer l'image avant de la réinsérer.

Bottom

Free surface

-6 -5 -4

-2

0

Impossible d'afficher l'image. Votre ordinateur manque peut-être de mémoire pour ouvrir l'image ou l'image est endommagé

Impossible d'afficher l'image. Votre ordinateur manque peut-être de mémoire pour ouvrir l'image ou l'image est endommagée. Redémarrez l'ordinateur, puis ouvrez à nouveau le fichier. Si le x rouge est toujours affiché, vous devrez peut-être supprimer l'image avant de la réinsérer.

Bottom

Free surface

-6 -5 -4

-2

0

Classical sigma transformation:

linear interpolation between bottom and free surface

Generalised sigma transformation:

Any function increasing from bottom to free surface

Now we can build the mesh between bottom and free surface

Survival in a moving mesh

What remains to do?

W computed to ensure

3D continuity * W* used for this

see wstarw.f

Advection-diffusion of W

Non hydrostatic pressure computed

New velocity field

Hydrostatic Non-hydrostatic

We have the depth, « some » velocities U and V

We need to have a divergence free velocity field (U,V,W)

We need to treat the tracers

Advection-diffusion of tracers, k, epsilon, sediment…

* We will see in what follows that W will not be used for advection, with hydrostatic assumption it is just built for visualisation

Now the big problem: 1+nif

nif

zyx

ni

ni

tfnotis

tff

,,

1

⎟⎠

⎞⎜⎝

⎛∂

∂

Δ

−+

Moving grid: the « relocalization » problem

What kind of do we need in a moving grid? tf∂

∂

The values of functions are given at mesh nodes, so we really want:

where i is a moving target in the real

world…

tff ni

ni

Δ

−+1

We need a coordinate system where the grid nodes will not move!

If we find a geometrical transformation (x,y,z) ! (x*,y*,z*)

So that x = x*, y = y*, and z* is constant for a mesh node The transformed mesh will not move!

and:

**,*, zyxtf⎟⎠

⎞⎜⎝

⎛∂

∂will be:

tff ni

ni

Δ

−+1

**,*, zyxtf⎟⎠

⎞⎜⎝

⎛∂

∂is an Arbitrary Lagrangian Eulerian derivative, see Astrid Decoene PhD

Sigma transformation

Z* between 0 and 1 hZz

ZZZz

z f

fs

f −=

−

−=* h

zz=

∂

∂

*

Generalised sigma transformation

zzz

zzzz

z ip

ipip

ip

Δ

−=

−

−=

+1

* zzz

Δ=∂

∂

*

a different sigma transformation for every layer

Zip+1

Zip

Δz

with this option the metrics in fixed planes will not depend on h

Advection and diffusion should be done in the transformed mesh…

0*

*****,*,

=∂

∂+

∂

∂+

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂=

zfW

yfV

xfU

tf

dtdf

zyx

It will thus do the relocalization

So far only advection is done in the transformed mesh

Diffusion is done in the fixed real mesh at time n+1 (see release notes 6.1 for investigating a diffusion matrix in the transformed mesh)

Problem: What is the vertical velocity in the transformed mesh?

0******,,

=∂

∂+

∂

∂+

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂==

zzW

yzV

xzU

tz

dtdzW

zyx

Hum…. we need to play with the sigma transformation… W* will be in fact given by the 3D continuity equation

in the transformed mesh…

When you look at the dark side, careful you

must be…

Basic rules and deadly traps:

xf

xfbutxx

∂

∂≠

∂

∂=

**

tzytzy xf

xfbecause

,,*,*,*⎟⎠

⎞⎜⎝

⎛∂

∂≠⎟

⎠

⎞⎜⎝

⎛∂

∂

**∫∫ ΩΩ

ΩΔ=Ω dfzdf

*)(*.*)(.*∫∫ ΩΩ

ΩΔ=Ω dfgradUzdfgradU!!

and

but…

****∫∫ ΩΩΩ

∂

∂Δ+

∂

∂Δ≠Ω

∂

∂+

∂

∂ dyfzV

xfUzd

yfV

xfU

Because some horizontal gradients in one space are vertical gradients in the other

tzytyxtzytzy xz

zf

xf

xf

,,,,*,,,,

**

⎟⎠

⎞⎜⎝

⎛∂

∂⎟⎠

⎞⎜⎝

⎛∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂=⎟

⎠

⎞⎜⎝

⎛∂

∂Basic rule

3D continuity in the transformed mesh

0zW

yV

xU

=∂∂

+∂∂

+∂∂

Becomes (with generalised sigma transformation):

0**)()()(

,,*,,*,,*,,

=⎟⎠

⎞⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂

tyxtzxtzyzyx zhW

yhV

xhU

th

0**)()()(

,,*,,*,,*,,

=⎟⎠

⎞⎜⎝

⎛∂

Δ∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

Δ∂+⎟

⎠

⎞⎜⎝

⎛∂

Δ∂+⎟

⎠

⎞⎜⎝

⎛∂

Δ∂

tyxtzxtzyzyx zzW

yzV

xzU

tz

The classical sigma transformation gives something close to Saint-Venant continuity:

∫∫ ∫ΓΩ Ω

+ ΓΨΔΔ−ΩΨΔΔ=ΩΨΔ−Δ*

*

* *

**1 *.**)(*(*)( dnUztdgradUztdzz iiinn !!!

is also in fact:

Splitting to get vertical and horizontal gradients:

∫∫ ∫∫ΓΩ Ω

+

Ω

ΓΨΔ+ΩΨΔ−ΩΨΔ−ΔΔ

=Ω∂

Ψ∂Δ

*

*

* *

*2

*1

*

*

*.**)(.(*)(1**

* dnUzdgradUzdzzt

dz

zW iiDinni !!!

0**)()()(

,,*,,*,,*,,

=⎟⎠

⎞⎜⎝

⎛∂

Δ∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

Δ∂+⎟

⎠

⎞⎜⎝

⎛∂

Δ∂+⎟

⎠

⎞⎜⎝

⎛∂

Δ∂

tyxtzxtzyzyx zzW

yzV

xzU

tz

0*)(**,,

=Δ+⎟⎠

⎞⎜⎝

⎛∂

Δ∂ Uzdivtz

zyx

!

After variational formulation and integration by parts:

These equations will give W* but…

∫∫ ∫∫ΓΩ Ω

+

Ω

ΓΨΔ+ΩΨΔ−ΩΨΔ−ΔΔ

=Ω∂

Ψ∂Δ

*

*

* *

*2

*1

*

*

*.*)(.(*)(1**

* dnUzdgradUzdzzt

dz

zW iiDinni !!!

The sum over the vertical of 3D continuity equations in the transformed mesh gives the Saint-Venant continuity equation solved for finding the depth:

3D continuity equations in the transformed mesh:

∑ ∫∫ ∑ ∫ΓΩ Ω

+ ΓΨΔ+ΩΨΔ−ΩΨ−Δ

=verticaloni

iverticaloni

iDDD

inn dnUzdgradUzdhh

t D *

*

*

**2

221 *.)(.()(102

!!!

0)( =∫ dzUdiv s

f

Z

Z

!Variational formulation of integrated by parts

The problem is well posed if the W* are located between planes:

( ) [ ] **)(*1**

1**1

*2/1 dzzzWzz

zWipz

ipzipip

jip ∫ −

−− Δ

−=Δ

(ip plane, j 2D point)

W*

ip

ip-1 j

Δz W* is a velocity in m/s

Because the sum of 3D continuity equations in the transformed mesh is imposed, we loose one degree of freedom per vertical…

W*

3

2

Δz W* is a flux in m/s

W* is computed in subroutine TRIDW2, from bottom to top

A simpler way to understand W* (1/2)

ΩΨΔ ∫Ω dzW Di2*   is a flux in m3/s

 flux that ensures the continuity of the point below,

1

A simpler way to understand W* (2/2)

ΩΨΔ ∫Ω dzW Di2*   is a flux in m3/s between 2 points on a vertical segment

This is a finite volume approach, and also exactly what distributive schemes do (finding fluxes along segments).

This will help for the tracer advection equation

Drawback: errors on the horizontal fluxes (Boussinesq approximation, finite element approximation, wrong forcing terms, etc.) are corrected by artificial vertical fluxes. Possible solution: see release notes 6.1, section 2.6, exact divergence-free fluxes given by the dynamic pressure computed in the transformed mesh (implemented in TRIDW3).

F.A.Q.: is ΔzW* equal to W-Wmesh ?

Answer: not exactly!

**,*, zyxmesh t

zW ⎟⎠

⎞⎜⎝

⎛∂

∂=

**

**)(

zfW

yfV

xfU

zfWW

yfV

xfU mesh ∂

∂+

∂

∂+

∂

∂=

∂

∂−+

∂

∂+

∂

∂(2.92 in book)

Take f = z:

*** **,**,

zWyzV

xzUWW

zxzymesh Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂=−

However Equation 2.92 shows that advection terms could be computed in the real mesh with a relative velocity (if compatibility with continuity strictly ensured)

Advection equation in the transformed mesh case of explicit scheme

0*)(.*(*)(

* *

**1

1 =ΩΨΔ+ΩΨΔ

−Δ∫ ∫

Ω Ω

++ dfgradUzd

tffz i

ni

nnn

!

Using the same W* as the continuity equation, it is compatible with it and ensures a perfect mass conservation.

∫∫ ∫ΓΩ Ω

+ ΓΨΔΔ−ΩΨΔΔ=ΩΨΔ−Δ*

*

* *

**1 *.**)(*(*)( dnUztdgradUztdzz iiinn !!!

new mesh old tracer

Principle of mass-conservation proof (1/2) case of explicit scheme

0*)(.*(*)(* *

**11 =ΩΨΔΔ+ΩΨ−Δ∫ ∫Ω Ω

++ dfgradUztdffz in

innn

!

∫∫ ∫ΓΩ Ω

+ ΓΨΔΔ−ΩΨΔΔ=ΩΨΔ−Δ*

*

* *

**1 *.**)(*(*)( dnUztdgradUztdzz iiinn !!!

Advection equation (1)

3D continuity equation (2)

∑=

+n

i

nif

1

)2()1(Forming gives:

∫∫ΓΩ

++ ΓΔΔ−=ΩΔ−Δ**

11 *.**)( dfnUztdfzfz nnnnn !!

i.e. final mass – initial mass = boundary fluxes during Δt

(if the advecting velocity field and the mesh is the same as in continuity equation…)

Principle of mass-conservation proof (2/2) semi-implicit scheme

0*)(.*(*)(* *

**1 =ΩΨΔΔ+ΩΨ−Δ∫ ∫Ω Ω

+ dfgradUztdffz iinn

!Advection equation (1)

∑=

+n

iif

1

)2()1(Forming gives:

∫∫ΓΩ

++ ΓΔΔ−=ΩΔ−Δ**

11 *.**)( dfnUztdfzfz nnnnn !!

Only if:

nh

nh zzz Δ−+Δ=Δ + )1(1 θθ n

fn

f fff )1(1 θθ −+= +

1=+ fh θθ

The dynamic pressure

Non hydrostatic continuity step

0)( ≠Udiv!

We have continuity in the transformed mesh, but still:

Why another continuity equation?

in the real mesh, and unknown pressure…

Solution: the Chorin algorithm

This part is like classical Navier-Stokes (and very bad!)

The Chorin algorithm (1/3)

)(1

0

1

d

hydnpgrad

tUU

ρ−=

Δ

−+!!

)())((0

hydd Udivpgradtdiv

!=

ρ

Δ

0)( 1 =+nUdiv!

Combining both gives the Poisson equation:

Last fractional step for the momentum:

We want:

We have done so far: gradientpressuredynamicexceptmomentumoftermsall

tUU nhyd

=Δ

−!!

)())((0

hydd Udivpgradtdiv

!=

ρ

Δ

Variational formulation and integration by parts:

ΩΨ−ΩΔ

Ψ=ΩΨΔ

∫∫∫ ΩΓΩdUdivdnptgraddgradptgrad i

hyddiid )().()().(

00

!

ρρ

Diffusion matrix, symmetric but

not so well conditioned

Boundary term, assumed 0

but this is wrong

Right-hand side

Boundary conditions: free surface pd=0, open boundaries: ???

The Chorin algorithm (2/3)

The Chorin algorithm (3/3)

)(0

1d

hydn ptgradUUρΔ

−=+!!

so:

Eventually:

∫

∫

Ω

Ω

ΩΨ

ΩΨΔ

≈Δ

d

dptgradptgrad

i

id

d

)()( 0

0

ρρ

But then compatibility lost: 0)( 1 ≠+nUdiv

!

but: linear linear piece-wise constant

In finite elements divergence(gradient(f)) is not Laplacian(f) (if done with integration by parts)

The many problems of Chorin algorithm

Laplacian not compatible Prescribed dynamic pressure = 0 at free surface (spoils divergence)

What dynamic pressure on open boundaries? What pressure gradient on solid boundaries?

Consequence:

But this velocity is not used for advection

More brainstorming on this:

Release notes 5.7 Chapter 2 (dealing with non linear waves) Release notes 6.1 Chapter 2 (on divergence free velocity fields)

0)( 1 ≠+nUdiv!

Other topics

Dry zones Parallelism

The aspect ratio The 2-plane case Non linear waves

Applications

The treatment of tidal flats and dry zones in Telemac

(How awful is the river without water!)

The presence of dry zones or tidal flats in 2D or 3D computational domains is a major difficulty

In environmental flows, nearly all computational domains have dry zones

Many solvers fail on dry zones

9 juin 2010 63

Problem 1: propagation on dry zone (wetting)

Problem 2: receding waters (drying)

Saint-Venant

Hydrostatic Navier-Stokes

Non-hydrostatic Navier-Stokes

Water at rest all equations

Problem 3: a seemingly easy but very difficult problem: a lake at rest

Checking the C-property

free surface = bottom elevation + water depth

free surface

bottom elevationmesh nodes

Linear interpolation of free surface leads to movements in the lake

Standard discretisation fails!

Problem 4: negative depths in numerical solutions

Continuity equation: 0)( =+∂

∂ uhdivth !

In finite elements: becomes: th∂

∂

thhMnn

Δ

−+1

In finite volumes: becomes: th∂

∂

thhSni

ni

i Δ

−+1

But: treated as: )( uhdiv ! [ ]))1(( 1 nnn uuhdiv !!θθ −++

∫∫∫ ΩΓΩΩΨ−Γ=ΩΨ dgraduhdnuhduhdiv nnn )(..)( !!!!

Boundary flux

Internal flux

Problem 5: Numerical pump

A bad solution: a minimum depth

If h < hmin then h=hmin

Problems: mass conservation, water flowing on banks

But how to find and treat dry zones if there is no threshold hmin ???

Other (not too good) solution: removing dry elements from the mesh

But how to find the shoreline, with hmin ???

shoreline

water

dry

Elements removed

Can we keep all elements and avoid using a threshold depth ?

Main problem of dry zones: their wrong free surface gradient

Bottom of a point in an element

Higher than the free surface of point in the same element

Comparing bottoms and free surfaces in an element (triangle) shows situations where the shoreline probably crosses the element. There is no need of threshold ! In such situations the gradient of the free surface must be

discarded or modified

Counter-example in an extreme case

The final solution involves a piece-wise linear definition of free surface with discontinuity on points

Additional problem in 3D

Elements without volume

A flood in a river: computation against observations (red line)

scan25 IGN (contrat marché n° M902C22360)

EDF R&D – LNHE – décembre 2005

scan25 IGN (contrat marché n° M902C22360)

EDF R&D – LNHE – décembre 2005

Parallelism in Telemac Domain decomposition without overlapping Single program Multiple data: (every processor runs Telemac on a part of the geometry) Distributed memory: each processor has its own memory and communicates with others via message transmission (pvm, mpi)

An example of domain decomposition

1 processor: scalar mode

More than 1 processor:

parallel mode

There is a number of free domain decomposers on Internet, we use Metis

http://glaros.dtc.umn.edu/gkhome/metis/metis/overview

Principle of parallel implementation in Telemac:

The final goal is to have parallel results strictly identical to scalar results (i.e. a difference is a bug)

All the numerical treatments must be the same

So far small differences may be observed, due to truncation errors

(remaining problems: assembly , dot products)

Typical differences are 1.D-6 on depth and velocity, if there are no bifurcations in the case

(von Karman eddies behind bridge piers are an example of bifurcations)

Adapting algorithms for parallelism is time consuming (but helps structuring and simplifying)

Debugging parallelism is painful

Is it possible to solve linear systems when points are scattered on several processors ?

Iterative solvers are series of matrix-vector products and dot products

Matrix-vector products need an assembly at interfaces: possible

(and easy with EBE or Edge-based matrices)

Dot-products require a sum over processors: possible

All processors will contribute to the same resolution:

They will treat their own points, will communicate for assembly of vectors at interfaces, they will communicate for computing the same dot

products

A single global resolution will thus be done collectively

The problem of characteristics and trajectories

In one time step a trajectory or a characteristic curve may cross an arbitrary number of

sub-domains. In 2008 Jacek Jankowski (Bundes Anstalt für Wasserbau, Germany) managed to write

an algorithm adapting the method of characteristics to domain-decomposition parallelism

In 2011: no difference between scalar and parallel!

Starting point Foot of the characteristic where the interpolation is done

The digit-to-digit equality between scalar and parallel runs

solved for trajectories

remaining problems: finite element assembly, dot products

finite element assembly: solved with integers

dot products: PhD at university of Perpignan (compensated

sums)

Parallelism: Malpasset test-case with Telemac version 6.2

Number of processors

Computer elapse time Speed-up

1 22 s 1 2 11 s 2 3 8 s 2.75 4 7 s 3.14 5 5 s 4.4 6 5 s 4.4 7 5 s 4.4 8 4 s 5.5

2D: small mesh

26000 elements, 1000 steps of 4 s

Number of processors

Computer elapse time Speed-up

1 545 s 1 2 264 s 2.06 3 181 s 3.01 4 140 s 3.89 5 122 s 4.46 6 102 s 5.34 7 96 s 5.67 8 85 s 6.4

2D: large mesh

104000 elements, 4000 steps of 1 s

8-core HP Z600 Linux Telemac-2D version 6.2

Number of processors

Computer elapse time Speed-up

1 112 s 1 2 61 s 1.83 3 44 s 2.54 4 33 s 3.39 5 28 s 4 6 24 s 4.66 7 22 s 5.09 8 20 s 5.6

3D: small mesh

26000 elements, 1000 steps of 4 s

Number of processors

Computer elapse time Speed-up

1 2285 s 1 2 1126 s 2.03 3 855 s 2.67 4 653 s 3.5 5 553 s 4.13 6 464 s 4.92 7 427 s 5.35 8 353 s 6.47

3D: large mesh

104000 elements, 4000 steps of 1 s

Parallelism: some results on IBM Blue Gene

Performances V5.9 (2008) : Calculs

BERRE sur Blue Gene P•Gros modèle 1 : test sur 57600 DT (16h de temps physique)

•12 083 160 prismes (31 plans horizontaux), DT=10s•256 à 8192 CPU

0

4

8

12

16

20

24

28

32

0 512 1024 1536 2048 2560 3072 3584 4096 4608 5120 5632 6144 6656 7168 7680 8192

Number of processors

Spee

d-up

(com

pare

d to

256

pro

cs)

TELEMAC3D speed-upIdeal speed-up

BERRE : 12 083160 mailles 3D

Large model: 12 083 160 prisms, with 31 horizontal planes, DT=10 s (2D mesh: 402772 triangles)

About 100 triangles per processor

Berre lagoon mesh seen from above

Side view of Berre lagoon mesh

The aspect ratio (open problem…)

A factor 1000 between horizontal and vertical scales!!!

Horizontal gradients are negligible compared to vertical gradients: solutions of diffusion-like equations tend to be 1-dimensional on the

vertical (this fact is even used as a preconditioning…). Turbulence models impacted

It is another problem for Chorin method: it will tend to correct the velocity field with artificial vertical velocities (like in the transformed mesh…)

)(0

1

dhydn

ptgradUUρΔ

−=+

Possible solution: an artificial factor on horizontal gradients (it works for Chorin but in this case what is pd ?)

The aspect ratio

The 2-plane case 2 planes suffice!

By the magic of finite element variational formulation, one layer of elements is

enough to solve the equations…

Essential boundary conditions (impermeability, friction, fluxes…) are plugged into equations, they do not replace them like Dirichlet conditions do.

Untrim code with 1 layer (1 velocity point per vertical) = Saint-Venant

Telemac-3D with 1 layer (2 velocity points per vertical) = something in between Saint-Venant and Navier-Stokes

The key point: including the dynamic pressure in the computation of the free surface…

Non linear waves

Solitary wave in a canal with constant depth depth = 10 m wave height = 2 m

Mesh size: 0.1 m time step: 0.1 s C.F.L.: 10,84

Navier-Stokes solution at 60 s with different numbers of

planes

Equations Computer time (HP Z600, 1 processor) Saint-Venant (with wave equation) 6 s

Saint-Venant (with primitive equations) 14 s Boussinesq (only works with primitive equations) 134 s Non hydrostatic Navier-Stokes 2 planes 28 s Non hydrostatic Navier-Stokes 3 planes 50 s Non hydrostatic Navier-Stokes 4 planes 78 s Non hydrostatic Navier-Stokes 5 planes 110 s Non hydrostatic Navier-Stokes 6 planes 157 s Non hydrostatic Navier-Stokes 7 planes 203 s Non hydrostatic Navier-Stokes 8 planes 242 s

Solitary wave in a canal with constant depth

Saint-Venant vs. Boussinesq vs. Navier-Stokes

Navier-Stokes are the right equations for non-linear waves!

10-plane mesh

Non linear waves

Navier-Stokes compared to Dingemans experiments

Applications

Thermal plumes Water quality

Sediment transport Tsunamis

Tides, storm surges Waves

Marine turbines Fish passes

3011703 elements 3240 steps of 5 s comparisons of Navier-Stokes, Saint-Venant and Boussinesq

cross sections after 8000 s

Lisbon tsunami (1st November 1755 09h40 am)

6023406 elements

93

3D simulation of tidal currents in Fundy Bay in Canada

(Canadian Hydraulic Center)

Evolution of salinity in the Berre lagoon

Influence of releases of Saint Chamas power plant salinity and discharges in the Caronte canal

Designing scenarii for releases

Salinity in Caronte canal between 17 November and 2 December 2005

EDF – R&D – Laboratoire National d’Hydraulique et Environnement

Salinity in the channel of Caronte in November 2005

Measured

Computed

Recent studies in the Berre lake (Nathalie Durand & Pierre Lang)

Sensitivity of mesh refinement (triangles split into 4)

1.2 million points, 96 processors on cluster Athos, 3.5 h CPU for 6 days

69 millions points, 1056 processors on Athos, 42 hours for 6 days

Telemac-3D: mixing of 2 fluids of different temperatures

With Kelvin-Helmholtz instabilities

Post-doc work of Lamia Abbas at LNHE

X 4 on the vertical

Telemac-3D: a boat in a lock

(study by HR-Wallingford for Panama locks)

What you have not seen…

Particles, oil spills, algae Advection schemes

Turbulence Thompson boundary conditions

Equations in Mercator projection Automatic boundary conditions for tides

Exchanges with atmosphere Sources, rain and evaporation

9 juin 2010

R&D Laboratoire National d'Hydraulique et Environnement

100

Thank you for your attention!

Manompana. Madagascar. 4 December 2007