Lecture Note
Lecture 1 Introduction An Unstructured Grid Finite-Volume
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University:
Boston University -
Course:
CAS PY 242 | Principles of General Physics 2 Academic year:
2024
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Views:
82
Pages:
52
Author:
vericepd4h
FVCOM-An unstructured grid Finite-Volume
Community Ocean Model
Chen-FVCOM-2013-01-Chile Critical Issues in Coastal Ocean Modeling
Irregular geometry
Mass Conservation ?
Steep topography
Intertidal wetlands Critical Issues for Global Ocean:
Basin-shelf
interaction
Basin scale
u Multi-scale dynamics:
Basin-shelf interaction,
convection via advection,
etc.
u Resolving irregular
coastal geometries
connected to the North
Atlantic Ocean and
Pacific Ocean
Coastal
process Possible solutions:
1. Finite-volume algorithms: Configuring the computational domains with
individual control volumes and calculating the variables by a net flux through
control volumes
An example:
∂ζ
∂(u D ) ∂(v D )
dxdy
=
−
[
∫∫ ∂t
∫∫ ∂x + ∂y ]dxdy = −∫S vn Dds
2. Unstructured grids: Configuring the computational domains with
unstructured grids to provide the accurate fitting of the irregular coastal geometry.
An example: FVCOM: Unstructured-grid, Finite-Volume Coastal Ocean Model
(Chen, C. R. H. Liu and R. C. Beardsley, JAOT, 2003)
F
• All variables are computed in the
integral form of the equations, which
provides a better representation of the
conservative laws of mass, momentum
and heat in the coastal region with
complex geometry.
• The numerical computational domain
consists of non-overlapping unstructured
cells.
• Combines the best from the finiteelement method for the geometric
flexibility and finite difference method
for the simplest discrete computation.
• Both current and tracer remain the
second-order accuracy.
F
u,v
u,v
u,v
F
F
F
u,v
u,v
u,v
F
F
F : " , T , S , ! , H , K m , K h , etc
yʹ′
u,v
u,v
u,v
u,v For example: The Continuity Equation:
: H, !, ", D, s, #, q2, q2l, Am, Kh
!
I
: u, v,
#$
#(u D ) #(v D )
!! #t dxdy = "!! [ #x + #y ]dxdy = "!S vn Dds FVCOM Wet/Dry Treatment Technology-Coastal Inundation
ζ
u, v Spherical Coordinate at the Arctic
In the Cartesian coordinates,
! dx = 0; ! dy = 0
s
A line integral is closed
s
In the spherical coordinates,
North Pole
" r cos$d# ! 0
A line integral is not always closed
s
' $$
(
&# 2
r cos "d!d"
r&"
?%
$(
s
)d! Spherical Coordinate FVCOM
u,vcosϕ
u,vcosϕ
F: Scalar variables such as ζ, T, S, Km, Kh,..and vertical velocity ω.
u,vcosϕ
•: The node of triangles where scalar variable or vertical velocity
is calculated
F
⊗: The centroid of a triangle where the horizontal velocity
is calculated.
u,vcosϕ
u,vcosϕ
Example: Continuity equation
u,vcosϕ
!$
1
!u D !v cos "D
+
(
+
)=0
!t
r cos " !#
!"
!&
"" !t r
#
T
2
cos $d%d$ + ""
#
1
!u D !v cos $D 2
(
+
)r cos $d%d$ = 0
r cos $ !%
!$
T
u,vcosϕ
u,vcosϕ
$'
r
= " [ ! ( Du ) d& " ! ( Dv cos & ) d% ]
$t
#
u,vcosϕ
T
T
T
u,vcosϕ
u,vcosϕ
u,vcosϕ
T
T
The gradient of the water temperature (or salinity) is
determined by the Greenʼs function through the
integration over the larger volume (with boundaries
linked to nodes). Treatment of North Pole
Node calculated using the polar stereographic projection
coordinate.
Centroid calculated using the polar stereographic projection
coordinate.
Node calculated directly in the spherical coordinate
system.
Centroid calculated directly in the spherical coordinate
system.
Conversion formulae between (up, vp) and (us, vs):
& u p # & ' sin (
$ ! = $$
$v !
% p " % cos (
' cos ( #& u s #
!!$$ !!
' sin ( "% vs "
dx
dt
dy
v p = hy
dt
u p = hx
1 + sin !
2
1 + sin !
hy =
2
hx = Nudging/OI
Assimilation
Ensemble/Reduced
Kalman Filters
North Pole Nested
System
Model Field
Sampling
Modules of FVCOM
Forcings:
Tides (Equilibrium+ O.B.)
Winds, Heat flux, Precipitation/
Evaporation
River discharges, Groundwater
O.B. fluxes
General Ocean
Turbulence Model
(GOTM)
3-D Sediment Model
FVCOM-Main Code
Cartesian/Spherical Coordinates
MPI Parallel
Surface Wave Model
Multiple Nesting
Non-hydrostatic
Existing Modules
Generalized Biological
Model
Water Quality Models
Adjoint Assimilation
Key:
3-D Wet/Dry Treatment
Multi OB Radiations
ViSiT Monitoring
NetCDF Output
Lagrangian-IBM
GUI Post-process Tools
Ice model
Under Development
Solver: Mode-split or semi-implicit; 2-D and 3-D,
Types: Research version (FVCOM-v2.7) and Forecast version (FVCOM-v3.1). ViSiT
Software developed by Lawrence Livermore National Laboratory
http://www.llnl.gov/visit/
• Open source; Parallel visualization; Interactive simulation support
• Multiple platform support (LINUX, UNIX, PC, MAC)
FVCOM Plug-in for VISIT
• FVCOM NETCDF files
• Visualization and animation of 3D
vector and scalar fields
• Database linking to NETCDF
formatted particle tracking output
Example of density isosurfaces from a
high resolution FVCOM GOM model FVCOM Users
38 countries(including HK/Taiwan)
>1000 users or institutions
Australia, Bangladesh, Brazil, Canada, Chile, China, Colombia, Denmark, Egypt, France, Germany, Hong Kong, Hungary
India, Indonesia, Iran, Israel, Italy, Jamaica, Japan, Korea, Kuwait, Malaysia, Mexico, Netherlands, New Zealand, Norway
Peru, Saudi Arabia, Singapore, Sweden, Taiwan, Thailand, Turkey, U.S., UK, Venezuela, Vietnam Hydrostatic FVCOM Validation Experiments
1.
Advection scheme;
2.
Wind-induced oscillation (POM, ECOM-si);
3.
Wind-induced waves over the slope bottom topography (POM, ECOM-si);
4.
Tidal Resonances in a semi-enclosed channel and a sector (POM, ECOM-si);
5.
Freshwater discharge plume (POM, ECOM-si);
6.
Bottom boundary layer over a step bottom slope (POM; ECOM-si)
7.
Equatorial Rossby soliton (ROMs)
8.
Wind-induced lateral boundary (ROMs)
9.
Super-critical current (ROMs)
Non-Hydrostatic FVCOM Validation Experiments
1.
2.
3.
Standing surface short gravity waves;
Lock exchanges;
Solitary waves in both homogeneous and layer fluids. 1. Advection Scheme
!F
!F
+C
=0
!x
!t
#
%
F(x,0) = $
%
&
5,
!2 " x " 2
0,
otherwise
and C = 1
Δt = 0.05, Δx=0.1
Upwind finitedifference stream
Central finitedifference stream
FVCOM finite-volume
flux scheme Δt = 0.005,
Δx=0.01
Upwind finitedifference stream
Central finitedifference stream
FVCOM finitevolume flux scheme Wind-induced oscillation
Wind is suddenly imposed at
initial
Linear, non-dimensional equations:
Wind
∂u
∂ζ
− v = −λ
∂t
∂r
ro=67.5 km
∂v
∂ζ
+ u = −λ
∂t
r∂θ
∂ζ λ ∂(ru) ∂v
+ [
+
]=0
∂t
∂r
∂θ
r
where
and
"=
d=75 m
gd
g&
& r cos '
; # = $ % $ˆ; $ˆ = o 4 ; & o = 3 4
ro f
ro f
"
u r =1 = 0; (u,v,") r =0 # finite; u t =0 = v t =0 = 0; " t =0 = $%ˆ (r,& )
!
Solution:
!
"
!o
! (r,! ,t) = 4 [ Ao (r) cos! + # ak Ak (r) cos(! ! " k t)]
!
k=1
"
!
A (r)
u(r,! ,t) = o3 [( o
!1)sin ! ! # bk Fk (r)sin(! ! " k t)]
r
!
k=1
"
! o dAo (r)
!1) cos! ! # bk Gk (r) cos(! ! " k t)]
v(r,! ,t) = 3 [(
dr
!
k=1
Reference:
Csanady ( 1968)
Birchfield (1969) Unstructured (FVCOM)
Structured (POM) Water elevation
Alongshore transport Radial mode: k=1, 2: gravity waves, k=3: topographic wave
Birchfield and Hickie (1977) JPO
Analytical
FVCOM (5 km)
POM (2.5 km)
1h
1d
Chen-FVCOM-2013-01-Chile 5 days
ζ: Elevation
V : Currents
FVCOM (5 km)
POM (2.5 km) Structured (POM)
Unstructured (FVCOM) Structured (POM)
Unstructured (FVCOM) Tidal Resonance in A Semi-closed Channel
Consider a 2-D linear, non-rotated initial
problem such as
{
∂Vr
∂η
+g
=0
∂t
∂r
∂Vθ
∂η
+g
=0
∂t
r∂θ
∂η ∂rVr H 0 ∂Vθ H 0
+
+
=0
∂t
r∂r
r∂θ
The solution:
η0 ( r, θ ) = [c1 J γ ( r
m
ω
gH 0
ω
) + c2Yγ m ( r
gH 0
)] ⋅ cos[
mπ (θ + α / 2)
α
where
c1 = A ⋅ Yγ'm ( L1
c2 = − A ⋅ J γ' m ( L1
γ m = mπ / α
ω
gH 0
) /[ J γ m ( L
ω
gH 0
) /[ J γ m ( L
ω
gH 0
)Yγ'm ( L1
ω
gH 0
)Yγ'm ( L1
ω
gH 0
) − J γ' m ( L1
ω
gH 0
) − J γ' m ( L1
ω
gH 0
)Yγ m ( L
ω
gH 0
)Yγ m ( L
ω
gH 0
)
ω
gH 0
)
] 1. Normal condition
(non-resonance)
2. Near-resonance
condition A Normal Case A Near-resonance Case Near-resonance, 2 km (FVCOM), 1 km (POM&ECOM-si) Near-resonance, 2km, Curvilinear 0.5-4 km
4 km
2 km Slope
topography
fitting Equatorial Rossby Soliton
L = 48
Grid A
C=0.
4
D=
24
Grid B
Periodic boundary
condition
1. Nonlinear shallow water equation in equatorial β-plane
2.
Inviscid flow
3.
Asymptotic solutions available to zero and first orders
(Boyd 1980,1985)
Grid C 0.05
0.125
0.25
Grid A
0.5
ROMS
(4th)
FVCOM
(2nd)
Δx
(ND)
SEOM
(7-9th)
hn/ht
Cn/Ct
hn/ht
Cn/Ct
hn/ht
Cn/Ct
0.5
0.472
0.917
0.884
1.088
0.923
0.98
0.25
0.846
0.984
0.926
0.993
0.929
0.99
0.125
0.92
0.984
0.923
0.986
0.937
0.989
0.05
0.935
0.983
0.936
0.983
0.915
0.98
hn: Computed peak of the sea surface elevation at 120 units
ht: Analytical peak of the sea surface elevation at 120 units
Cn: Computed average speed
Ct: Analytical averaged speed.
Grid B
Comments:
1. Analytical solution only represents the zero and
1st modes, while the numerical solution contains
a complete set of higher order modes. This is
not surprised to see numerical models can not
exactly reach the analytical solutions.
Grid C
2. FVCOM shows a fast convergence with
increase of horizontal resolution. Hydraulic Jump
No gradient condition
Analytical solution:
Maximum sea level: ! max = 0.5 m
u = 8.57 cm / s
v =0
! =0
(m)
! min = 0 m
Mean sea level:
! mean = 0.5 m
Mean velocity:
u = 7.956 m/s
Mean Froude #:
Fr =
Shock angle:
30o
φ = 8.95o
Minimum sea level:
(m)
u
gD
! = 30 0
Thickness:
! =0m
Mean deviation:
dy = 0 m
Characteristics:
•
Barotropic shallow water equations
•
No rotation considered, i.e. f =0, β = 0
•
Steady analytical solutions for u, ζ and the jump angle relative to the x axis.
= 2.075 The case with no horizontal diffusion: FVCOM quickly reaches steady status.
∆x = 0.5 m, ∆t = 0.002
Model
grids
∆x = 0.25 m, ∆t = 0.001
∆t
True
FVCOM
ROMs
∆x = 0.125 m, ∆t = 0.0005
ζmax
ζmin
ζmean
ū
Fr
α
δ
|dy|
0.5
0
0.5
7.956
2.075
30
0
0
80 X 60
0.002
0.688
-0.269
0.5
7.949
2.072
29.952
0.111
0.305
160 X 120
0.001
0.697
-0.268
0.499
7.951
2.073
30.030
0.063
0.151
320 X240
0.0005
0.696
-0.272
0.5
7.951
2.073
30.029
0.037
0.076
Reach an oscillatory solution without horizontal diffusion. Over shocking can be reduced by introducing a slope limiter method (Hubbard, J.
Comput. Phys., 1999).
Original code
Modified code with limiter 3-Dimensional Wind-Driven Flow in an Elongated, Rotating Basin
z
y
Governing equations:
x
2L
Length: 2L; width: 2B, and bathymetry:
h = h0 {0.08 + 0.92 " [ X ( x / L)(1 ! ( y / B) 2 )]}
where X(x) is a function in the form of
!
$& ) v + wz = 0
!
!
# ! !
% 2v
! fk ( v = ' g&* + K m %z 2
"
B.Cs:
r
's
$r
!v z =
&K m
#
!" vr = 0
w=0
at z = 0
w = 0 at z = %h
Steady status analytical solution for this linear
equation system is given as:
u + iv =
sinh[! ( z + h)] " x + i" y
cosh(!z )
#
[1 #
]
2
cosh(!h)
! cosh(!h)
!
1,
| x |< 1 & %x
$!
| x | &1 + %x 2
X ( x) = #
1&[
] , | x |' 1 & %x
!"
%x
w = " ! v y dz
∆x is a constant specified as 0.3% of the total
length of the basin.
where α2 = 2iδ-2 and δ = (2E)1/2 (E: Ekman number).
" L ! x ! L, " B ! y ! B Analytic
FVCOM ROMs
U
V
W
Analytic
Be aware that ROMs underestimates u and overestimates w (color bar scales
are different for analytical and ROMsʼ solutions). This figure is scanned from
Winantʼs working note. A non-hydrostatic problem: Lock-exchange flow
Light fluid
Heavy fluid
0.8m
Model setup:
• Initially g ' = gΔρ / ρ 0 = 0.01m / s 2
• Vertical 100 sigma layers
• Horizontal 400×5 nodes, dx = 0.002(m)
• no bottom friction and viscosity/diffusivity
0.1m Is the total energy conservative?
L/2 H
Potential Energy (PE ) =
! ! ! gz dx dz
" L/2 0
L/2 H
Kinetic Energy (K E )=
!!
" L/2 0
1
!V (u 2 + v 2 )dx dz
2
Under an inviscid condition, the total energy is conservative at an order of 10-4.
Blue line: PE
Red line: KE Comparison of FVCOM-NH with a high-order direct numerical simulation
(DNS) method, with constant horizontal and vertical eddy viscosity and tracer
diffusivity 1×10-6 m2/s:
FVCOM-NH
(DNS results from
Härtel et al., 2000)
ROMS’s results presented by Kanarska et al.
( 2007)
SUNTRAN’s results presented by Fringer et al.
(2006) with a first-order finite-volume scheme. Internal solitary waves breaking on a linear varying slope
ρ1
ρ2
h
H Photography
(Michallet and Ivey ,1999)
FVCOM-NH Photography
(Michallet and Ivey ,1999)
FVCOM-NH
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