Modelos Hidrodinâmicos

Aula 4Equations for 3D and 2D Hydrodynamic Models.Parameters and Boundary and Initial Conditions

Mass conservation

)( PFnceluxDivergeDiffusiveFnceluxdivergeAdvectiveFmulationRateOfAccu

PFxP

xxPu

tP

jjj

j

If P is the volumic mass, that has no Sources or Sinkes and has no diffusion because the net movement of molecules is the velocity….

0

j

j

xu

t

0

j

j

xu

If incompressible:

Momentum Conservation

)( PFfusivoaDoFluxoDidivergêncivectivoaDoFluxoAddivergênciaçãoTaxaAcumul

PFxP

xxPu

tP

jjj

j

Se P for a quantidade de movimento por unidade de volume:

PFxu

xxuu

tu

j

i

jj

ij

i

Sources and sinks are Pressure forces (gravitational is zero because we are interested only on horizontal momentum)

j

i

jij

ij

i

xu

xxP

xuu

tu

Shallow Water Equations

• Hydrostatic Pressure (vertical aceleration negligeable).• If “z” is the vertical axis, pointing upwards:

ii

z

xg

xP

zgP

gdzP

gzP

)(

Equações das águas pouco profundas

)2,1(

i

xu

xxg

xuu

tu

j

i

jij

ij

i

0

j

j

xu )2,1(0

idzuxt ii

Using the Leibnitz rule these equations can be integrated on vertical to obtain

the equations of a 2D model.

xhu

xuudz

xdzxu

hz

z

hzz

z

hz

yhv

yvvdz

xdzyv

hz

z

hzz

z

hz

hzz

z

hz

wwdzzw

zw

yv

xu

xu

i

i

0

0

0;

yHV

xHU

t

vdzy

udzxt

th

yhv

xhu

th

dtdhw

yv

xu

tdtdw

hh

hzhzhz

zzz

The Finite Volume

The 2D case

xx hH xxxx hH

The Accumulation rate = flows in – flows out

yHv

xHu

t

HuHvHuHut

yxtvol

yyyxxx

1D CaseL

xAxxA

0

xQ

tL

AuAut

xLtAx

tvol

xxx

The Accumulation rate = flows in – flows out

Momentum: 1D Case

Lb: wet perimeter

Ls

A

bbss

bs

ixxx

xxx

LLx

gAxUQ

tQ

LLx

gAxUA

xxUQ

tQ

SSvolxUA

xUAAuUAUU

tAUx

tvolU

)((( 0

Horizontal diffusion is negligible compared to vertical diffusion

The 1D Spatial Grid

QiQi-1 Qi+1zizi-1

Discretization

0

:

0

2/2/

2/2/*

2/*

2/

*2/

*2/

*2/

*2/

*2/

*2/

xQQ

tL

LLx

gAxUQUQ

tQQ

ExplicitxQQ

tL

LLx

gAxUQUQ

tQQ

tx

tx

tx

ttxt

s

bbss

ttx

ttxxx

ttt

xxtx

ttxt

s

bbssxxxx

ttt

A staggered grid is convenient.Temporal discretization can be explicit, implicit ou Crank-Nicholson

Bottom shear stress

α

025.0

)(

.

23/4

22

n

mulaManningForURngUc

zutg

hfb

b

2D Case

fsj

i

jij

iji

j

i

fsj

i

jij

iji

HxUH

xHxg

xUHU

HtU

or

xHU

t

xUH

xxgH

xUHU

tHU

111

:

0

Stability

• Explicit (1D):

• Implicit: Incondicionally stable• Explicit 2D:

1

xtgH

1)2/(

xtgH

Boundary Conditions

02/2/

2/2/*

2/*

2/

xQQ

tL

LLx

gAxUQUQ

tQQ

tx

tx

tx

ttxt

s

bbss

ttx

ttxxx

ttt

Q2Q1 z2z1z0

• One can impose Free Surface levels and compute discharges or vice versa.• On sea side level is easier to know (tide) and on the land side river discharge use

to be easier.

Other boundary conditions

• Bathymetry!• Surface shear stress,• Diffusive fluxes,• Advective fluxes.

Initial conditions

• Discharges/velocities,• Levels.• The good thing is that dissipative systems have

low memory. Approximate initial conditions can be used. Usually zero velocity and horizontal free surface.

Parameters

• Friction coefficient,• Diffusion coefficient.• Surface friction coefficient if flux in not known

(e.g. from a meteorological model).