Numerical Models
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Transcript of Numerical Models
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Numerical Models
Outline• Types of models• Discussion of three numerical models
(1D, 2D, 3D LES)
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Types of River Models (1)1. Conceptual
– Qualitative descriptions and predictions of landform and landscape evolution
– e.g., “cartoons” and +/- relationships2. 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
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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 recipes5. Cellular Automata
– Cells of a lattice that interact according to rules based on abstractions of physics
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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
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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
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Gradually-varied flow equation f
x SSgAxdgA
xVQ 0d
dd
d
Bed shear stress
sx kRuV 27.12log57.5
*
Conservation of fluid mass AVQ x
Bedload transport
cijkcijkkijbijk uuhPFi
**tan
Manning’s equation
342
22
RkVnS x
f
Conservation of suspended load
zzC
zzzCwV
xzCzV ijijijzijx )()()()(
Active layer thickness
50502
ca DT
Sediment continuity equation
0111
x
bix
bigt
bzp sijbij
j
bij
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
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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)
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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 load4. 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
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Degradation (Bennett and Bridge, 1995)
Equilibrium
Post-degradation
Equilibrium; steady, uniform flow
Post-degradation; steady, nonuniform flow
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(Bennett and Bridge, 1995)Aggradation
Eq.
Post-Agg.
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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
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0111
yVhc
xUhc
thc
Depth-integrated continuity equation
hfc
yD
xD
yhTc
xhTc
xzhcg
yUVhc
xUUhc
tUhc
dxbxxyxxxyxxs
)1()1()1()1(
111
hfc
yD
xD
yhTc
xhTc
yzhcg
yVVhc
xUVhc
tVhc
dybyyyyxyyyxs
)1()1()1(
)1(
111
Depth-integrated momentum equation
vvvdvvadd UDcCUC UUf
2
21
Drag force on vegetation
Uτ URgn
sb 3/1
2
Bed shear stress
Depth-averaged 2D numerical model(Wu et al., 2005)
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kxUT txx 3
22 ;
xV
yUTT tyxxy
; k
yVT tyy 3
22
vkbhk
t
k
t PPPyk
yxk
xykV
xkU
tk
k
cPPcPk
cyyxxy
Vx
Ut bh
tt2
231
Turbulence closures (+)
hIb
hIUmb
hUmm
D ss
ssxx
21212
2
12112
1111 3122
21
hIb
hIUmb
hUmm
DD ss
ssyxxy
22212
2
211222112
2111 31221
hIb
hIUmb
hUmm
D ss
ssyy
22222
2
22212
2121 3122
21
Dispersion terms in momentum equation attributed to helical flow (+)
Depth-averaged 2D numerical model(Wu et al., 2005)
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kksksyksxkk
sk
s
kkk
SScy
Dx
DyShc
yxShc
x
yVhSc
xUhSc
thSc
111
111
Conservation of suspended sediment (+)
011111
kbbkbkbybkbxbk qqc
Lyqc
xqc
tsc
Conservation of bedload (+)
kbbkkkskk
bm qq
LSS
tz
p
11 * Change in bed height
Depth-averaged 2D numerical model(Wu et al., 2005)
Two applications:1. Little Topashaw Creek, MS; channel adjustment to LWD
structures2. 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 sediment2. Bed is discretized using finite difference in time at cell centers3. Flow and sediment are decoupled
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(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
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Computational grid used in simulating LTC bend.
(Wu et al., 2005)
Little Topashaw Creek, MSComputational Grid
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Simulated flow field at LTC bend (Q=42.6 m3/s) 1 m/s
(Wu et al., 2005)
Little Topashaw Creek, MSFlow Vectors
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Simulated Flow, Little Topashaw Creek, MS
Without LWD With LWD(Wu et al., 2006)
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-2
-1.5
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
M easured Calcu lated
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, MSBed Adjustment
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(Bennett et al., accepted)
Physical Model
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(Bennett et al., accepted)
Physical Model
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0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0
D istan ce a lo n g flum e (m m )
0
2 0 0
4 0 0
6 0 0
Dis
tanc
e ac
ross
flum
e (m
m)
-8 5 -6 5 -4 5 -2 5 -5 5 2 5 4 5 6 5 8 5
0
2 0 0
4 0 0
6 0 0
D iffe ren ce in e lev a tio n (m m )
(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 ModelBed Adjustment
Observed
Predicted
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(Bennett et al., accepted)
20 0
40 0
60 0
Dis
tanc
e ac
ross
flum
e (m
m)
0 20 0 4 00 60 0 80 0 10 00 12 00 1 40 0
D istan ce a lo n g flu m e (m m )
2 0 0
4 0 0
6 0 0
(b )
(c )
20 0
40 0
60 0 (a )
m m /s
40 0
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 ModelFlow Vectors
Predicted
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(Bennett et al., accepted)
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 00 80 0 90 0 1 0 0 0 11 00 1 20 0 1 3 00 1 4 0 0
D is tan c e a lo ng flu m e (m m )
0
2 0 0
4 0 0
6 0 0
0 0 .3 0 .6 0 .9 1 .2 1 .5 1 .8 2 .1 2 .4 2 .7 3
0
2 0 0
4 0 0
6 0 0
Dis
tanc
e ac
ross
flum
e (m
m)
B ed sh ear s tre ss (P a )
(b )
(c )
0
2 0 0
4 0 0
6 0 0 (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 ModelBed Shear Stress
Predicted
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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
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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
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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)
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Mean velocity streamlines visualizing vortices inside the embayment region
(McCoy et al 2007)
Flow past Groynes
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Mean velocity streamlines visualizing vortex system in the downstream recirculation region
(McCoy et al 2007)
Flow past Groynes
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Instantaneous contours of contaminant concentration at groyne middepth (upper) and midwidth (lower)
(McCoy et al 2007)
Flow past Groynes
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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 tornado-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)
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Turbulent Flow over Fixed Dunes
(Bennett and Best, 1995)
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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 forquadrant two and four events
Three-dimensional view of instantaneous flow, where shadow area represents free surface, view of upper-half channel, and magnified view of free surface, wherethe labels U and D represent upwelling and downdraft.
(Yue et al., 2005b)(Yue et al., 2005a)
Flow over Dune: LES
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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)
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Fluvial ModelsConclusions• 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