# Numerical Models-2011

Τεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 4 χρόνια και 5 μήνες)

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Numerical Models

Outline

Types of models

Discussion of three numerical models
(1D, 2D, 3D LES)

Types of River Models (1)

1.
Conceptual

Qualitative descriptions and predictions of landform
and landscape evolution

e.g., “cartoons” and +/
-

relationships

2.
Empirical

Functional relationships based on data

May include statistical relationships

e.g., hydraulic geometry

3.
Analytical

Derive new functional relationship based on
physical processes and conservation principles
(mass, energy, momentum); deterministic

e.g., sediment transport equations

Types of River Models (2)

4.
Numerical

Represents all relevant physical processes in a set
of governing equations

Conservation of fluid mass, energy, momentum
(fluid and sediment)

1D, 2D, or 3D; dynamic; coupled or decoupled,
science of numerical recipes

5.
Cellular Automata

Cells of a lattice that interact according to rules
based on abstractions of physics

Discussion of Three Models

1D numerical model routes flow and
sediment along a single channel

2D numerical model that routes flow and
sediment within curved channels

3D Large Eddy Simulation (LES) model
that routes flow in complex channels

1D Numerical Model

Route flow and sediment (decoupled) in a
straight, non
-
bifurcating alluvial channel

1D: width
-

and depth
-
averaged

transport, and deposition of sediment

MIDAS:
M
odel
I
nvestigating
D
ensity
A
nd
S
ize sorting

-
varied flow
equation

f
x
S
S
gA
x
d
gA
x
V
Q

0
d
d
d
d

Bed shear stress

s
x
k
R
u
V
27
.
12
log
57
.
5
*

Conservation of fluid mass

A
V
Q
x

cij
k
cij
k
k
ij
bijk
u
u
h
P
F
i

*
*
tan

Manning’s equation

3
4
2
2
2
R
k
V
n
S
x
f

z
z
C
z
z
z
C
w
V
x
z
C
z
V
ij
ij
ij
z
ij
x
)
(
)
(
)
(
)
(

Active layer thickness

50
50
2
c
a
D
T

Sediment continuity equation

0
1
1
1

x
bi
x
bi
g
t
bz
p
sij
bij
j
bij

MIDAS

(van Niekerk et al., 1992a,b)

Q
-
flow discharge; S
0
-
bed slope, S
f
-
friction
slope, R
-
s
-
roughness height,
w
-
settling velocity, ijk
-
coefficients for gain size
and density, and bed shear stress, F
-
proportion
of ij, P
-
proportion of shear stress k

Treatment of bed
:

Active layer: what is available for transport in a given time
-

and space
-
step

Particle exchange between active layer and moving bed occurs during
each time
-
step; grain size
-

If

occurs: active layer is replenished from below

If
deposition

occurs: active layer moves upward

Assume fluid flow and sediment transport are over time
-
step

MIDAS

(van Niekerk et al., 1992a,b)

MIDAS

(van Niekerk et al., 1992a,b)

Numerical Procedure (at every
x
, then
t
)
:

1.
using standard step
-
method, subject
to downstream boundary condition
and n

2.
(
i
b
) determined

3.
From
i
b

4.
Bed continuity equation solved at
each node using a modified
Preissmann

scheme (nearly a central
[finite] difference scheme)

5.
New grain size
-
density distributions,
as modified by erosion or deposition

(Bennett and Bridge, 1995)

Equilibrium

Post
-

Equilibrium;
uniform flow

Post
-

nonuniform
flow

(Bennett and Bridge, 1995)

Eq.

Post
-

Agg.

2D Numerical Model

Route flow and sediment (decoupled) in a
straight to mildly sinuous, non
-
bifurcating alluvial
channel with vegetation

2D: depth
-
averaged

Depth
-
integrating the time
-

and space averaged
3D Navier
-
Stokes equations

Considers the dispersion terms associated with
helical flow

Explicitly addresses the effects of vegetation in
stream corridor

0
1
1
1

y
Vh
c
x
Uh
c
t
h
c

Depth
-
integrated continuity equation

h
f
c
y
D
x
D
y
hT
c
x
hT
c
x
z
h
c
g
y
UVh
c
x
UUh
c
t
Uh
c
dx
bx
xy
xx
xy
xx
s

)
1
(
)
1
(
)
1
(
)
1
(
1
1
1

h
f
c
y
D
x
D
y
hT
c
x
hT
c
y
z
h
c
g
y
VVh
c
x
UVh
c
t
Vh
c
dy
by
yy
yx
yy
yx
s

)
1
(
)
1
(
)
1
(
)
1
(
1
1
1

Depth
-
integrated
momentum
equation

v
v
v
d
v
v
a
d
d
U
D
c
C
U
C
U
U
f



2
2
1

Drag force on vegetation

U
τ
U
R
gn
s
b
3
/
1
2

Bed shear stress

Depth
-
averaged 2D numerical model

(Wu et al., 2005)

k
x
U
T
t
xx
3
2
2


;

x
V
y
U
T
T
t
yx
xy


;
k
y
V
T
t
yy
3
2
2



v
kb
h
k
t
k
t
P
P
P
y
k
y
x
k
x
y
k
V
x
k
U
t
k

k
c
P
P
c
P
k
c
y
y
x
x
y
V
x
U
t
b
h
t
t
2
2
3
1

Turbulence
closures (+)

h
I
b
h
IU
m
b
h
U
m
m
D
s
s
s
s
xx
2
12
12
2
12
11
2
11
11
3
1
2
2
2
1

h
I
b
h
IU
m
b
h
U
m
m
D
D
s
s
s
s
yx
xy
2
22
12
2
21
12
22
11
2
21
11
3
1
2
2
1

h
I
b
h
IU
m
b
h
U
m
m
D
s
s
s
s
yy
2
22
22
2
22
21
2
21
21
3
1
2
2
2
1

Dispersion
terms in
momentum
equation
attributed to
helical flow
(+)

Depth
-
averaged 2D numerical model

(Wu et al., 2005)

k
k
sk
syk
sxk
k
s
k
s
k
k
k
S
S
c
y
D
x
D
y
S
h
c
y
x
S
h
c
x
y
VhS
c
x
UhS
c
t
hS
c

1
1
1
1
1
1


Conservation
of suspended
sediment
(+)

0
1
1
1
1
1

k
b
bk
bk
by
bk
bx
bk
q
q
c
L
y
q
c
x
q
c
t
s
c

Conservation of

k
b
bk
k
k
sk
k
b
m
q
q
L
S
S
t
z
p

1
1
*


Change in bed height

Depth
-
averaged 2D numerical model

(Wu et al., 2005)

Two applications:

1.
Little Topashaw Creek, MS; channel adjustment to LWD
structures

2.
Physical model of alluvial adjustment to in
-
stream vegetation

Numerical Procedure
:

1.
Governing equations are
discretized

using a finite volume method on a curvilinear, non
-
orthogonal grid
for flow and sediment

2.
Bed is
discretized

using finite difference in time at cell centers

3.
Flow and sediment are decoupled

(a) Map of study site, Little Topashaw Creek; (b) Photo facing
upstream. Shaded polygons are large wood structures

(Wu et al., 2005)

LWD

Little Topashaw Creek, MS

Computational grid used
in simulating LTC bend.

(Wu et al., 2005)

Little Topashaw Creek, MS

Computational Grid

Simulated flow field at
LTC bend (Q=42.6 m
3
/s)

1
m
/
s
(Wu et al., 2005)

Little Topashaw Creek, MS

Flow Vectors

Simulated Flow, Little Topashaw Creek, MS

Without LWD

With LWD

(Wu et al., 2006)

-2
-1.5
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Measured
Calculated
0
20
40
60
80
Measured and simulated bed changes between June
2000 and August 2001. Units of bed change and
scale are m, and the contour interval is 0.25 m.

erosion

deposition

(Wu et al., 2005)

Little Topashaw Creek, MS

(Bennett et al., accepted)

Physical Model

(Bennett et al., accepted)

Physical Model

0
200
400
600
800
1000
1200
1400
Distance along flume (mm)
0
200
400
600
D
i
s
t
a
n
c
e

a
c
r
o
s
s

f
l
u
m
e

(
m
m
)
-85
-65
-45
-25
-5
5
25
45
65
85
0
200
400
600
Difference in elevation (mm)
(a)
(b)
Contour plots of changes in
bed surface topography in
response to the rectangular
vegetation zone with VD =
2.94 m
-
1

as (a) observed in the
experiment and as (b)
predicted using the numerical
model. Flow is left to right.

(Bennett et al., accepted)

Physical Model

Observed

Predicted

(Bennett et al., accepted)

200
400
600
D
i
s
t
a
n
c
e

a
c
r
o
s
s

f
l
u
m
e

(
m
m
)
0
200
400
600
800
1000
1200
1400
Distance along flume (mm)
200
400
600
(b)
(c)
200
400
600
(a)
mm/s
400
Simulated depth
-
averaged flow vectors for
the trapezoidal channel with (a) no vegetation
present, and in response to the rectangular
vegetation zone (shown here as a lined box)
at a density of 2.94 m
-
1

at (b) the beginning
and (c) conclusion of the experiment.

Physical Model

Flow Vectors

Predicted

(Bennett et al., accepted)

0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
Distance along flume (mm)
0
200
400
600
0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3
0
200
400
600
D
i
s
t
a
n
c
e

a
c
r
o
s
s

f
l
u
m
e

(
m
m
)
Bed shear stress (Pa)
(b)
(c)
0
200
400
600
(a)
Contour plots of simulated distributions
of bed shear stress for the trapezoidal
channel with (a) no vegetation present,
and in response to the rectangular
vegetation zone (shown here as a
lined box) at a density of 2.94 m
-
1

at
(b) the beginning and (c) conclusion of
the experiment. Flow is left to right.

Physical Model

Bed Shear Stress

Predicted

3D Numerical Models

Classic in 3D modeling is to close the Navier

Stokes
equations by:

Reynolds decomposition of the velocity components into mean
and fluctuating components

Employ a Boussinesq approximation to link the resulting
Reynolds stresses to properties of the time
-
averaged flow
(Reynolds
-
averaged Navier

Stokes (RANS) approach)

Mixing
-
length model is one such closure scheme where the characteristic
length and timescales of the turbulence are prescribed a priori

So
-
called
k

model still the most popular way of determining these length
and timescales from properties of the flow

RANS focused on the accurate representation of the
mean flow field

3D Numerical Models

Large Eddy Simulation
(LES) resolves the turbulence
above a particular filter scale, rather than resolving
variations greater than the integral timescale as occurs
in RANS

Can yield accurate results in situations where turbulent
structures of importance to the modeler are generated at a
variety of scales

LES calculates the properties of all eddies larger than
the filter size and models those smaller than this scale
by a subgrid
-
scale (SGS) turbulence transport model

3D LES Model

LES

equations are derived by applying a filter to the
Navier

Stokes equations

RANS

approaches to modeling the Navier

Stokes
equations decompose the velocity in to mean and
fluctuating components, whereas
LES

is based upon a
length scale for a filter, often taken to be equal to the grid
size employed

Important differences of
LES

vs.
RANS

LES

equations retain a time derivative (why
LES

can be
employed to give time
-
transient solutions)

Additional stress term contains more components than the
Reynolds stresses in
RANS

(Smagorinsky SGS model is most
commonly used for subgrid
-
scale solutions)

Mean velocity streamlines visualizing vortices inside the embayment region

(McCoy et al 2007)

Flow past Groynes

Mean velocity streamlines visualizing vortex system in the downstream
recirculation region

(McCoy et al 2007)

Flow past Groynes

Instantaneous contours of contaminant concentration at groyne middepth
(upper) and midwidth (lower)

(McCoy et al 2007)

Flow past Groynes

Visualization of horseshoe vortex system in the mean flow and associated upwelling motions
downstream of the plant stem a) flat bed b) deformed bed

Visualization of the
-
like vortex inside
the recirculation region on
the right side of the plant
stem using 3
-
D streamlines
(flat bed case).

(Neary et al., submitted)

Flow past Plant Stem (cylinder)

Turbulent Flow over Fixed Dunes

(Bennett and Best, 1995)

Instantaneous velocity fluctuation fields of u
and w in the middle plane of the channel.
Dashed lines represent the instantaneous
free
-
surface positions. Q2 and Q4 stand for

Three
-
dimensional view of instantaneous flow, where
shadow area represents free surface, view of upper
-
half
channel, and magnified view of free surface, where

the labels U and D represent upwelling and downdraft.

(Yue et al., 2005b)

(Yue et al., 2005a)

Flow over Dune: LES

Fluvial
Models and River
Restoration

Future of stream restoration relies heavily
(tools)

Use models to verify field and laboratory data

Use models to assess various restoration
strategies (rapidly, cheaply, and without harm to
the environment)

Fluvial Models

Conclusions

quantitative information of erosion, transport
and deposition within river corridors in the
downstream direction, but not laterally

2D and 3D models provide the highest fidelity
of turbulent flow in downstream and lateral
directions (as well as vertical directions with
3D codes), but require

Much expertise in fluid mechanics and numerical
techniques

Much computer capability