numerical models

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 +/- 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

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

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

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

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



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

Degradation (Bennett and Bridge, 1995)

Equilibrium

Post-degradation

Equilibrium; steady, uniform flow

Post-degradation; steady, nonuniform flow

(Bennett and Bridge, 1995)Aggradation

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

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)

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 (+)


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


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

(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, MSComputational Grid

Simulated flow field at LTC bend (Q=42.6 m3/s) 1 m/s

(Wu et al., 2005)

Little Topashaw Creek, MSFlow 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

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

(Bennett et al., accepted)

Physical Model

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.


Physical ModelBed Adjustment

Observed

Predicted


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


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

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 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)

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 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

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 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

numerical models

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