How to Design Bendway Weirs...Project Background • Physical Hydraulic Model Study – Determine...

How to Design Bendway Weirs

Project Background• U.S. Bureau of Reclamation: Middle

Rio Grande Channel Maintenance Program– 29-Mile Study Reach: Cochiti Dam to Bernalillo– Geomorphic Changes Due to Dam Construction– Meandering Threatening Critical Riverside

Facilities– Two Endangered Species and Degrading Habitat– Employ Native Material and Rock Weir

Techniques

Project Background

Project Background

• Physical Hydraulic Model Study

– Determine Design Criteria for Native

Material and Rock Weir Structures

• Bendway Weirs

• W-Weir, V-Weirs

• J-Hooks

• Root Wads

Study Objectives• Collect Empirical Data to Describe the Flow in

Bends

• Collect Empirical Data in an Effort to Quantify the Performance of Bendway Weirs While Varying Geometric Parameters

• Determine an Optimal Spacing for Bendway Weir Design

• Develop Design Criteria Applicable to Bends of Varying Geometry

Characteristic Prototype Model Scale FactorManning's' Roughness, n 0.027 0.018 LR

(1/6)

Design Flow (ft3/sec) 6000 12 LR(5/2)

Bend Type 1 Bottom Width (ft) 122 10.17 LR

Bend Type 1 Radius of Curvature (ft) 465 38.75 LR

Bend Type 3 Bottom Width (ft) 72 6 LR

Bend Type 3 Radius of Curvature (ft) 790 65.8 LR

Bed Slope (ft/ft) 0.000863 0.000863 1

Physical Model

Model Construction

Model Construction

Model Construction

Instrumentation• Mobile Instrumentation Cart w/ Standard Point

Gage• 122 Piezometer Taps and Stilling Wells• 3-D ADV Meter• Preston Tube

Baseline Data Collection• Model Flows 8, 12, 16, and 20 cfs • Measurements Collected Over Each Piezometer

Tap– WSE Using Piezometer Taps 60.001 ft– 3-D Velocity Profiles at 10% Depths– Preston Tube Shear

Baseline Data Analysis

• Water Surface Super Elevation– DZ = WSEX – WSED

• Velocity– Vector Mapping of 3-D Velocities

• Plan View• Cross Section: Helical Flow

• Shear Stress– Contour Mapping– Cross Section Distribution vs 1-D Model Output– Turbulence Stresses

Super Elevation 16cfsLeft Bank

-0.020-0.015-0.010-0.0050.0000.0050.0100.0150.020

0 2 4 6 8 10 12 14 16 18Cross Section

∆Ζ

(ft)

Piezo A Piezo B Piezo C

Upstream Bend

Downstream Bend

Super Elevation 16cfsRight Bank

-0.020-0.015-0.010-0.0050.0000.0050.0100.0150.020

0 2 4 6 8 10 12 14 16 18

Cross Section

∆Ζ

(ft)

Piezo E Piezo F Piezo G

Upstream Bend

Downstream Bend

0.911.11.21.31.41.51.61.71.81.922.12.22.32.4

Vel

ocity

(ft/s

)16cfs Downstream Bend

Flow

Cross Section 16

Cross Section 17

Cross Section 18

00.040.080.120.160.20.240.28

Lateral Velocity (ft/s)

16

17

18

Bou

ndar

y Sh

ear S

tress

(psf

)

Flow Direction

0

0.004

0.008

0.012

0.016

0.02

0.024

0.028

0.032

0.036

0.04

Boundary Shear Stress DistributionXSEC 6, 16cfs

0.000

0.004

0.008

0.012

0.016

0.020

0.024

0.028

0.032

0.036

0.0400 4 8 12 16 20

το

-0.100

-0.090

-0.080

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

0.000

Station (ft)

Outer Bank

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Flow Direction

Boundary Shear Stress DistributionXSEC 10, 16cfs

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.070 3 6 9 12 15

τ ο (p

sf)

- 0.1000

- 0.0900

- 0.0800

- 0.0700

- 0.0600

- 0.0500

- 0.0400

- 0.0300

- 0.0200

- 0.0100

0.0000

Station (ft)

Outer Bank

Bendway Weirs

• Height

• Length

• Orientation Angle

• Spacing Ratio

Literature Review• United Nations (1953)

• Indian Central Board of Irrigation and Power (1971)

• Richardson (1975)

• USACE (1980)

• Copeland (1983)

• Brown (1985)

• Maza Alvarez (1989)

• Derrick (1994 & 1998)

• Przedwojski (1995)

• Lagasse (1997)

• Lagrone (1998)

• Smith (1998)

• Heintz (2002)

Literature Review: Weir Spacing

Larc

Flow

Lproj,w

Weir Crest

Larc

Author Recommended Spacing Ratio

Type of Bank Remarks

1 Concave General Practice2-2.5 Convex General Practice4.29 Straight~5 Curves

Joglekar (1971) 2-2.5 Upstream GroynesUS Army (1984a) 2 Mississippi RiverMathes (1956) 1.5Strom (1962) 3-5

Acheson (1968) 3-4 Varies depending on curvature and stream slope

2-6 For bank protection

3-4 T-head groynes for navigation channels

Mamak (1956) 1.5-2 Deep channel for navigationBlench et al. (1976) 3.5Copeland (1983) >3 ConcaveKovacs el al. (1983) 1-2 Danube River

Mohan and Agraval (1979) 5 Submerged groynes of height one-third the depth

5.1-6.3 Straight

2.5-4 Curves

United Nations (1953)

Sloping crested weirs for bank protectionMaza Alvarez (1989)

Richardson et al. (1975)

Ahmad (1951)

Flow

Lproj,cw

Channel Top Width

Weir Crest

Literature Review: Weir LengthAuthor Suggested Length

United Nations (1953) "Start with a shorter length and extend the groynes after space between them has been silted up"

ICBIP (1971): No rules apply, build models to determine appropriated length

Richardson (1975) 50 feet or less

USACE (1980) Should be set at the desired constriction width of channel for navigation purposes

Brown (1985) Less Than 15% of bankfull channel width for impermeable structures

Maza Alvarez (1989) Less than 25% of bankfull channel width

Lagasse (1997) Less than 33% of bankfull channel width

Derrick (1998) Site-Specific Basis, engineering judgment

LaGrone (1998) 16.67%, not a design guideline but a site specific design

Literature Review: Orientation Angle

Flow

θ Weir Crest

Line Tangent to Bank

Author Range of Angles Suggested AngleBrown (1985) 30-150 150 decreasing to 90

Copeland (1983) 60-120 90Derrick (1994) 45-80 60

Indian Central Board of Irrigation and Power (1965): 60-80

Lagasse (1997) 50-85 60Mamak (1964): (Copeland

Literature Review) 70-80

Maza Alvarez (1989) 110 Richardson (1975) 60-150 70-80

Smith (1998) 60-75United Nations (1953) 60-80

USACE (1980) 100-105

Literature Review: Conclusions

• Design criteria are largely based upon engineering judgment and field experiences

• Typically, design criteria do not quantitatively explain changes in flow conditions due to bendway weir installations

• Cumulative effects of changing weir spacing, length, and angle are uncertain

Bendway Weirs: Design Review

• Weir Height– At or just below the bankfull or channel forming flow

depth

• Weir Length– 15 – 30% of the top width – Length perpendicular to the bank

• Weir Orientation Angle– Pointing upstream or perpendicular to the bank: 60 – 90

degree angle

• Spacing– Ratio between spacing and length, spacing ratio = 1 – 6.3

Test Matrix

Test Variable Number of Variations Variation Values

Discharge (cfs) 3 8, 12, 16

Spacing Ratio 4 3.4, 4.1, 5.9, 7.6

Weir Length 3 15%, 22%, 28%

Orientation Angle 2 90, 60

• 72 tests examining weir length, angle, and spacing

• 18 (and counting) supplemental tests, examining weir spacing

• Over 90 tests in all

Data Collection

Data Collection

Data Analysis

• MVR Regression Analysis

• Dimensional Analysis

• MVRout, MVRin, and MVRcenter Prediction Methods

CenterBase

aa MaxV

MaxVMVR =

Preliminary MVR AnalysisOuter Bank Maximum Total Velocity Ratio

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10

Spacing Ratio

MVR

out

16cfs

12cfs

8cfs Upstream

8cfs Downstream

CenterBase

aa MaxV

MaxVMVR =

Preliminary MVR AnalysisOuter Bank Maximum Velocity Ratio vs. Spacing Ratio,

12cfs

0

0.3

0.6

0.9

1.2

1.5

0 2 4 6 8 10

Spacing Ratio

MVR

out 28%

22%15%

• Distinct trends were observed for weirs having varying weir characteristics

Dimensional AnalysisSymbol Definition Dimensions

ρw Density of Water M/L3

νw Kinematic Viscosity of Water L2/Tυw Dynamic Viscosity of Water MT/L2

Symbol Definition DimensionsSo Bed Slope L/L

TWtestflow Top Width at Test Flow Lb Base Width LSs Side Slope L/Ln Mannings' Roughness T/L(1/3)

y Flow Depth Lr radius of curvature Lk Conveyance L3/TAc Area of the Channel at Test Flow L2

Symbol Definition DimensionsLproj,cw Projected Length of Weir Crest L

Lcw Weir Crest Length LLproj,w Projected Length of Weir L

Lw Weir Length Lhw Weir Height LLarc Arc Length Between Weirs Lθw Angle of weir with Respect to Perpendicular Transect L/L

Lperp Distance from Weir Tip Perpendicular to XS through Centerline LAw Area of weir projected on Perpendicular Transect L2

Symbol Definition DimensionsQ Discharge L3/Tg Gravity L/T2

Material Properties

Channel Properties

Weir Properties

External Properties

( )cwcwcwprojtestflowwwprojarc AALLTWhyLLfMVR ,,,,,,,, ,,=

Dimensional AnalysisBuckingham’s Pi theorem (Dimensional Analysis):

Identified the following dimensionless parameters

wproj

arc

LL

,1 =π

weirhy

=2πcwproj

testflow

LTW

,3 =π

cw

cwproj

LL ,

4 =πTotalAreaWeirArea

=5π

⎟⎟⎠

⎞⎜⎜⎝

⎛=

c

w

cw

cwproj

cwproj

testflow

wwproj

arc

AA

LL

LTW

hy

LL

fMVR ,,,, ,

,,

Data Analysis: Linear Regression

Necessary Analysis included:

• Multiple Linear Regression

• Natural Log Transformation of Intrinsically Linear data

• Best Subsets method to determine the most suitable regression model

• Analysis of Variance (ANOVA), contribution of independent variables, and determination of possible outliers

( )54321 ,,,, πππππfMVR =

Data Analysis: Multivariate Linear Regression (MVR Out)

Trial # Vars R-Sq Adj. R-Sq C-p s π1 π2 π3 π4 π51 1 38.6 37.4 62.9 0.413 X2 1 35.0 33.7 69.5 0.425 X3 1 28.5 27.1 81.4 0.446 X4 1 1.3 0.0 131.5 0.524 X5 1 1.0 0.0 132.0 0.524 X6 2 68.0 66.8 10.8 0.301 X X7 2 66.4 65.1 13.7 0.308 X X8 2 45.6 43.4 52.1 0.393 X X9 2 43.6 41.4 55.7 0.400 X X

10 2 39.0 36.6 64.3 0.416 X X11 3 73.3 71.7 3.2 0.278 X X X12 3 70.2 68.4 8.8 0.294 X X X13 3 68.6 66.8 11.7 0.301 X X X14 3 68.0 66.1 12.8 0.304 X X X15 3 66.4 64.4 15.7 0.311 X X X16 4 73.7 71.6 4.4 0.278 X X X X17 4 73.3 71.1 5.2 0.281 X X X X18 4 70.2 67.8 10.8 0.296 X X X X19 4 68.7 66.2 13.5 0.304 X X X X20 4 54.7 51.0 39.3 0.365 X X X X21 5 73.9 71.2 6.0 0.280 X X X X X

Weir Variables

• Weir Height

• Design Flow 12cfs

• Height of Weirs Equal to 12cfs Measured Depth

Orientation Angle

• Varying angle to bank

• Crest Width

• Set at 1ft

Weir Variables: Spacing

• Spacing Ratio Measurement

S = Larc/Lw

• Spacing Ratio: 3.4 - 8.4

Larc

Flow

Lproj,w

Weir Crest

Larc

Weir Variables: Orientation Angle

Flow

Lproj,cw

Channel Top Width

Weir Crest

Flow

θ Weir Crest

Line Tangent to Bank

Data Analysis: Multivariate Linear Regression

109.0

,

153.0700.0,

153.2

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=

wproj

arc

c

w

cw

cwproj

in

LL

AA

LL

MVR

859.0

35.2,

899.0

,019.0

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cw

cwproj

wproj

arc

out

AA

LL

LL

MVR

160.0567.0,

160.0

,

350.1 ⎟⎟⎠

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⎛=

c

w

cw

cwproj

wproj

arccenter A

AL

LLL

MVR

Data Analysis: Multivariate Linear Regression (MVR Out)

MVRout:: Observed vs. Predicted

0.0

0.4

0.8

1.2

0.0 0.4 0.8 1.2

Predicted MVRout

Obs

erve

d M

VRou

t

Test DataIdeal Fit

Error = Observed MVR – Predicted MVR

Average Error = 0.01

Average Absolute Error = 0.07

859.0

35.2,

899.0

,019.0

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=

c

w

cw

cwproj

wproj

arc

out

AA

LL

LL

MVR

Data Analysis: MultivariateLinear Regression (MVR Center)

MVRcenter: Observed vs. Predicted

1.0

1.2

1.4

1.6

1.8

1.0 1.2 1.4 1.6 1.8

Predicted MVRcenter

Obs

erve

d M

VRce

nter

Test DataIdeal Fit

Average Error = 0.00

Average Absolute Error = 0.07

160.0567.0,

160.0

,

350.1 ⎟⎟⎠

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⎛=

c

w

cw

cwproj

wproj

arccenter A

AL

LLL

MVR

Data Analysis: Multivariate Linear Regression (MVR In)

MVRin: Observed vs. Predicted

0.8

1.0

1.2

1.4

1.6

1.8

0.8 1.0 1.2 1.4 1.6 1.8

Predicted MVRin

Obs

erve

d M

VRin

Test DataIdeal Fit

Average Error = 0.00

Average Absolute Error = 0.05109.0

,

153.0700.0,

153.2

⎟⎟⎠

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=

wproj

arc

c

w

cw

cwproj

in

LL

AA

LL

MVR

Data Analysis: Multivariate Linear Regression (Summary)

wproj

arc

LL

,1 =π

cw

cwproj

LL ,

4 =πTotalAreaWeirArea

=5π

859.05

35.24

899.01019.0π

ππ•=outMVR

109.01

153.05

700.04153.2π

ππ=inMVR

160.05

567.04

160.01350.1 πππ=centerMVR

Design Example: Site Selection

0 2000’

Middle Rio Grande:

• 10 miles downstream of Cochiti Dam

Southwestern Willow Flycatcher

Design Example: Site Selection

• 2001 Aerial Photograph

• Bend Properties:

• rc = 578’

• Channel Top Width = 188.5’

• Channel Length = 1090’

• rc/TW = 3.07

0 2000’

rc

Bend Properties:

rc = 578’

Channel Top Width = 188.5’

Channel Length = 1090’

rc/TW = 3.07

Design Example: Site Selection

Top WidthRadius of Curvature

Relative Curvature Rc

(ft) (ft) dimensionless1 230.4 465 2.023 180 789.96 4.39

Bend Type

Design Example: Design ChannelDesign Channel Properties:

• Base Width = 80’

• Design Top Width = 134.2’

• Side Slope = 3:1 (H:V)

• Design Flow = 6000 cfs

• n = 0.027

• Bed Slope = 0.000863

VAQ =

21

32486.1 SAR

nQ = Flow depth = 9.24 ft

Velocity = 6.0 ft/sec

Design Example: Baseline Conditions

• Estimated Centerline Maximum baseline velocity = 6.0 ft/sec

• Estimated Outer Bank Maximum baseline velocity = (1.1)*(6.0 ft/s) = 6.62 ft/sec

• From Sediment Transport Analysis a 60 % reduction of baseline conditions is desired

• Desired Outer Bank Velocity = 2.65 ft/sec

40.00.665.2

===CenterBase

outout MaxV

MaxVMVR

Design Example: Preliminary Weir Design

• Weir Design depends upon design Top Width

• Start with 3 primary weir variables:

• Weir Length, Angle, and Spacing

• Keep two variables constant, change the third to achieve desired MVR results

Design Example: Preliminary Weir Design

• Weir Crest Length = 20%

• Orientation Angle = 75o

• Spacing = ?

• Calculate known weir variables

Projected Length of Weir Crest (ft) 26.84Length of Weir Crest (ft) 27.79Length of Weir (ft) 37.03Projected Length of Weir (ft) 35.77Area of Channel (ft) 995.33Projected area of Weir (ft) 161.19

wproj

arc

LL

,1 =π

cw

cwproj

LL ,

4 =πTotalAreaWeirArea

=5π

859.0

35.2,

899.0

,019.0

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=

c

w

cw

cwproj

wproj

arc

out

AA

LL

LL

MVR

Design Example: Preliminary Weir Design

• Solving for arc length yields a value of 203.8 ft

• A Spacing Ratio of 5.7 results (within tested limits)

Design Example: Preliminary Weir Design

0 5000 5000 500

CenterBase

InIn MaxV

MaxVMVR =

109.0

,

153.0700.0,

153.2

⎟⎟⎠

⎞⎜⎜⎝

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=

wproj

arc

c

w

cw

cwproj

in

LL

AA

LL

MVR

Design Example: Velocities along Other Axes

• Solving results in MVRin = 1.32

• Solving results in a Maximum predicted inner bank velocity of 7.89 ft/sec

• Is this acceptable?

160.0567.0,

160.0

,

350.1 ⎟⎟⎠

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c

w

cw

cwproj

wproj

arccenter A

AL

LLL

MVR

Design Example: Velocities along Other Axes

CenterBase

CenterCenter MaxV

MaxVMVR =

• Solving results in MVRcenter = 1.31

• Solving results in a Maximum predicted center channel velocity of 7.84 = ft/sec

• Is this acceptable?

Design Example: Examination of Sensitivity

Theta MaxVout

(degrees) (ft/sec)1 20 75 5.7 0.400 2.402 20 90 5.7 0.430 2.58

% Top WidthTrial # Spacing MVRout

Sensitivity of Spacing

Sensitivity of Angle

Sensitivity of Length

Trial #1: Lproj,w = 26.84’

Trial #4: Lproj,w = 33.55’

Theta MaxVout

(degrees) (ft/sec)1 20 75 5.7 0.400 2.403 20 75 4.6 0.329 1.97

Trial # Spacing MVRout% Top Width

Theta MaxVout

(degrees) (ft/sec)1 20 75 5.7 0.400 2.404 25 75 5.7 0.326 1.96

Spacing MVRout% Top WidthTrial #

Trial #1: Larc = 203.86’

Trial #3: Larc = 164.52’

Conclusions

• Regression analysis permits the prediction of MVR’s as a function of weir characteristics.

• Solutions may be optimized for specific project constraints/objectives.

• Design procedure is still evolving!

Questions?