### 1D Numerical Model

```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
• Primary purpose: To address erosion,
transport, and deposition of sediment
(sorting processes and bed adjustment)
• MIDAS: Model Investigating Density And
Size sorting
MIDAS
(van Niekerk et al., 1992a,b)
Q-flow discharge; S0-bed slope, Sf-friction
slope, R-hydraulic radius, ks-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
Bed shear stress
equation
Vx
 5.57log12.27R ks 
dQVx 
dd
 gA
 gAS0  S f  u*
dx
dx
Conservation of fluid mass
h
Q  Vx A

ibijk  Fij Pk
u*k  u*cij  k   cij 
tan 
Manning’s equation
Conservation of suspended load
Vx ( z )C ( z )ij   Vz  wij C ( z )ij    C ( z )ij 
n 2Vx2
Sf  2 4 3

 
k R
x
z
z 
z 

Active layer thickness
Ta  2 D50

 c 50

Sediment continuity equation
bz 
bi  bi 
1  p bij  1  1 bij  sij   0
t
 j    g x
x 
MIDAS
(van Niekerk et al., 1992a,b)
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-density distributions are adjusted
• If degradation 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)
Numerical Procedure (at every x, then t):
1. Gradually varied flow equation solved
using standard step-method, subject
to downstream boundary condition
and n
2. Shear stress (and bedload transport
(ib) determined
3. From ib, determine suspended load
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
Equilibrium;
Equilibrium
uniform flow
nonuniform
flow
(Bennett and Bridge, 1995)
Eq.
PostAgg.
(Bennett and Bridge, 1995)
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
Depth-averaged 2D numerical model
(Wu et al., 2005)
 1  c h  1  c Uh  1  c Vh 


0
t
x
y
 1  c Uh  1  c UUh  1  c UVh
t
  g (1  c)h

x


y
Depth-integrated continuity equation

 zs (1  c)hTxx   (1  c)hTxy  Dxx Dxy




 (1  c) bx  f dx h
x
x
y
x
y
Depthintegrated
momentum
equation
 1  c Vh   1  c UVh  1  c VVh 


t
x
y
 z  (1  c)hTyx  (1  c)hTyy  Dyx Dyy
  g (1  c)h s 



 (1  c) by  f dy h
y
x
y
x
y

 
1
2c
Cd  a U v U v  Cd  v
U v Uv
2
D
gn2
τ b  1 / 3 U U
Rs
fd 

Drag force on vegetation
Bed shear stress
Depth-averaged 2D numerical model
(Wu et al., 2005)

U 2
V 2
 V 
 U

;
Txx  2 t
 k ; Txy  Tyx  t 
T

2

 k


yy
t


x 3
y 3
 x 
  y


k
k
k    t
U
V
 
t
x
 y x   k


    t
U
V
 
t
x
y  x   

Turbulence
closures (+)
k     t k 
  
  Ph  Pkb  Pv  
x   y   k  y 
     t  

2
 
  c 1 Ph  c 3 P   Pb  c 2
 x   y     y 
k
k
2bs
bs2
1
2
Dxx  
1111U s h 
1112 IU s h  1212 I 2 h
mm  2
2m  1
3
bs
bs2
1
2
11 22  12 21 IUs h  12 22 I 2 h
Dxy  Dyx  
11 21U s h 
mm  2
2m  1
3
2bs
bs2
1
2
Dyy  
 21 21U s h 
 21 22 IU s h   22 22 I 2 h
mm  2
2m  1
3
Dispersion
terms in
momentum
equation
attributed to
helical flow
(+)
Depth-averaged 2D numerical model
(Wu et al., 2005)
1  c hSk  1  c UhSk  1  c VhSk 


t
x
y
Conservation
of suspended
sediment (+)
 
Sk   
Sk  Dsxk Dsyk

 s 1  c h

 s 1  c h


 sk 1  c Sk  Sk 
 x 
 x   y 
 y 
x
y
Conservation of
1  c sbk   bx 1  c qbk    by 1  c qbk
1


 1  c qbk  qbk   0
t
x
y
L
Change in bed height
1
 zb 

1  pm    sk S k  S*k   qbk  qbk 
L
 t  k


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
Two applications:
1. Little Topashaw Creek, MS; channel adjustment to LWD
structures
2. Physical model of alluvial adjustment to in-stream vegetation
LWD
(a) Map of study site, Little Topashaw Creek; (b) Photo facing
upstream. Shaded polygons are large wood structures
Little Topashaw Creek, MS
(Wu et al., 2005)
Little Topashaw Creek, MS
Computational Grid
Computational grid used
in simulating LTC bend.
(Wu et al., 2005)
Little Topashaw Creek, MS
Flow Vectors
1 m/s
Simulated flow field at
LTC bend (Q=42.6 m3/s)
(Wu et al., 2005)
Simulated Flow, Little Topashaw Creek, MS
Without LWD
(Wu et al., 2006)
With LWD
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.
Little Topashaw Creek, MS
1
deposition
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
erosion
-1
-1.5
Measured
Calculated
-2
(Wu et al., 2005)
0
20
40
60
80
Physical Model
(Bennett et al., accepted)
Physical Model
(Bennett et al., accepted)
(a)
600
Physical Model
Distance across flume (mm)
400
Observed
200
0
(b)
600
Predicted
400
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.
200
0
0
200
400
600
800
1000
1200
1400
Distance along flume (mm)
-85
-65
-45
-25
-5
5
25
45
65
Difference in elevation (mm)
85
(Bennett et al., accepted)
(a)
600
Physical Model
Flow Vectors
400
200
mm/s
Predicted
Distance across flume (mm)
400
(b)
600
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.
200
(c)
600
400
200
(Bennett et al., accepted)
0
200
400
600
800
1000
Distance along flume (mm)
1200
1400
(a)
600
Physical Model
Bed Shear Stress
400
200
Distance across flume (mm)
0
Predicted
(b)
600
400
200
0
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.
(c)
600
400
200
0
0
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
Distance along flume (mm)
(Bennett et al., accepted)
0
0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7
Bed shear stress (Pa)
3
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)
Flow past Groynes
(McCoy et al 2007)
Mean velocity streamlines visualizing vortices inside the embayment region
Flow past Groynes
(McCoy et al 2007)
Mean velocity streamlines visualizing vortex system in the downstream
recirculation region
Flow past Groynes
Instantaneous contours of contaminant concentration at groyne middepth
(upper) and midwidth (lower)
(McCoy et al 2007)
Flow past Plant Stem (cylinder)
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
the recirculation region on
the right side of the plant
stem using 3-D streamlines
(flat bed case).
(Neary et al., submitted)
Turbulent Flow over Fixed Dunes
(Bennett and Best, 1995)
Flow over Dune: LES
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
quadrant two and four events
(Yue et al., 2005a)
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)
Fluvial Models and River
Restoration
Future of stream restoration relies heavily
upon advancing current modeling capabilities
(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
• 1D models provide readily available
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
```