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CHAPTER 5 REVIEW OF 1D OPEN CHANNEL HYDRAULICS

Dam at Hiram Falls on the Saco River near Hiram,

Maine, USA

TOPICS REVIEWED

- This e-book is not intended to include a full

treatment of open channel flow. It is assumed

that the reader has had a course in open channel

flow, or has access to texts that cover the

field. Nearly all undergraduate texts in fluid

mechanics for civil engineers have sections on

open channel flow (e.g. Crowe et al., 2001).

Three texts that specifically focus on open

channel flow are those by Henderson (1966),

Chaudhry (1993) and Jain (2000). - Topics treated here include
- Relations for boundary resistance
- Normal (steady, uniform) flow
- St. Venant shallow water equations
- Gradually varied flow
- Froude number subcritical, critical and

supercritical flow - Classification of backwater curves
- Numerical calculation of backwater curves

SIMPLIFICATION OF CHANNEL CROSS-SECTIONAL SHAPE

River channel cross sections have complicated

shapes. In a 1D analysis, it is appropriate to

approximate the shape as a rectangle, so that B

denotes channel width and H denotes channel depth

(reflecting the cross-sectionally averaged depth

of the actual cross-section). As was seen in

Chapter 3, natural channels are generally wide in

the sense that Hbf/Bbf ltlt 1, where the subscript

bf denotes bankfull. As a result the

hydraulic radius Rh is usually approximated

accurately by the average depth. In terms of a

rectangular channel,

THE SHIELDS NUMBER A KEY DIMENSIONLESS PARAMETER

QUANTIFYING SEDIMENT MOBILITY

?b boundary shear stress at the bed ( bed drag

force acting on the flow per unit bed area)

M/L/T2 ?c Coulomb coefficient of resistance

of a granule on a granular bed 1 Recalling

that R (?s/?) 1, the Shields Number ? is

defined as

It can be interpreted as a ratio scaling the

ratio impelling force of flow drag acting on a

particle to the Coulomb force resisting motion

acting on the same particle, so that

The characterization of bed mobility thus

requires a quantification of boundary shear

stress at the bed.

QUANTIFICATION OF BOUNDARY SHEAR STRESS AT THE BED

U cross-sectionally averaged flow velocity (?

depth-averaged flow velocity in the wide

channels studied here) L/T u shear

velocity L/T Cf dimensionless bed

resistance coefficient 1 Cz dimensionless

Chezy resistance coefficient 1

RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW

Keulegan (1938) formulation

where ? 0.4 denotes the dimensionless Karman

constant and ks a roughness height

characterizing the bumpiness of the bed L.

Manning-Strickler formulation

where ?r is a dimensionless constant between 8

and 9. Parker (1991) suggested a value of ?r of

8.1 for gravel-bed streams.

Roughness height over a flat bed (no bedforms)

where Ds90 denotes the surface sediment size such

that 90 percent of the surface material is finer,

and nk is a dimensionless number between 1.5 and

3. For example, Kamphuis (1974) evaluated nk as

equal to 2.

COMPARISION OF KEULEGAN AND MANNING-STRICKLER

RELATIONS ?r 8.1

Note that Cz does not vary strongly with depth.

It is often approximated as a constant in

broad-brush calculations.

BED RESISTANCE RELATION FOR MOBILE-BED FLUME

EXPERIMENTS

Sediment transport relations for rivers have

traditionally been determined using a simplified

analog a straight, rectangular flume with

smooth, vertical sidewalls. Meyer-Peter and

Müller (1948) used two famous early data sets of

flume data on sediment transport to determine

their famous sediment transport relation

(introduced later). These are a) a subset of the

data of Gilbert (1914) collected at Berkeley,

California (D50 3.17 mm, 4.94 mm and 7.01 mm)

and the set due to Meyer-Peter et al. (1934)

collected at E.T.H., Zurich, Switzerland (D50

5.21 mm and 28.65 mm).

Bedforms such as dunes were present in many of

the experiments in these two sets. In the case

of 116 experiments of Gilbert and 52 experiments

of Meyer-Peter et al., it was reported that no

bedforms were present and that sediment was

transported under flat-bed conditions. Wong

(2003) used this data set to study bed resistance

over a mobile bed without bedforms.

Flume at Tsukuba University, Japan (flow turned

off). Image courtesy H. Ikeda. Note that

bedforms known as linguoid bars cover the bed.

BED RESISTANCE RELATION FOR MOBILE-BED FLUME

EXPERIMENTS contd.

Most laboratory flumes are not wide enough to

prevent sidewall effects. Vanoni (1975),

however, reports a method by which sidewall

effects can be removed from the data. As a

result, depth H is replaced by the hydraulic

radius of the bed region Rb. (Not to worry, Rb ?

H as H/B ? 0). Wong (2003) used this procedure

to remove sidewall effects from the

previously-mentioned data of Gilbert (1914) and

Meyer-Peter et al. (1934). The material used in

all the experiments in question was quite

well-sorted. Wong (2003) estimated a value of

D90 from the experiments using the given values

of median size D50 and geometric standard

deviation ?g, and the following relation for a

log-normal grain size distribution Wong then

estimated ks as equal to 2D90 in accordance with

the result of Kamphuis (1974), and ?s in the

Manning-Strickler resistance relation as 8.1 in

accordance with Parker (1991). The excellent

agreement with the data is shown on the next page.

TEST OF RESISTANCE RELATION AGAINST MOBILE-BED

DATA WITHOUT BEDFORMS FROM LABORATORY FLUMES

NORMAL FLOW Normal flow is an equilibrium state

defined by a perfect balance between the

downstream gravitational impelling force and

resistive bed force. The resulting flow is

constant in time and in the downstream, or x

direction.

- Parameters
- x downstream coordinate L
- H flow depth L
- U flow velocity L/T
- qw water discharge per unit width L2T-1
- B width L
- Qw qwB water discharge L3/T
- g acceleration of gravity L/T2
- bed angle 1
- tb bed boundary shear stress M/L/T2
- S tan? streamwise bed slope 1
- (cos ? ? 1 sin ? ? tan ? ? S)
- water density M/L3

As can be seen from Chapter 3, the bed slope

angle ? of the great majority of alluvial rivers

is sufficiently small to allow the approximations

NORMAL FLOW contd.

Conservation of water mass ( conservation of

water volume as water can be treated as

incompressible)

Conservation of downstream momentum Impelling

force (downstream component of weight of water)

resistive force

Reduce to obtain depth-slope product rule for

normal flow

ESTIMATED CHEZY RESISTANCE COEFFICIENTS FOR

BANKFULL FLOW BASED ON NORMAL FLOW ASSUMPTION FOR

u The plot below is from Chapter 3

RELATION BETWEEN qw, S and H AT NORMAL EQUILIBRIUM

Reduce the relation for momentum conservation ?b

?gHS with the resistance form ?b ?CfU2

Generalized Chezy velocity relation

or

Further eliminating U with the relation for water

mass conservation qw UH and solving for flow

depth

Relation for Shields stress ?? at normal

equilibrium (for sediment mobility calculations)

ESTIMATED SHIELDS NUMBERS FOR BANKFULL FLOW BASED

ON NORMAL FLOW ASSUMPTION FOR ?b The plot

below is from Chapter 3

RELATIONS AT NORMAL EQUILIBRIUM WITH

MANNING-STRICKLER RESISTANCE FORMULATION

Solve for H to find

Solve for U to find

Manning-Strickler velocity relation (n

Mannings n)

Relation for Shields stress ?? at normal

equilibrium (for sediment mobility calculations)

BUT NOT ALL OPEN-CHANNEL FLOWS ARE AT OR CLOSE TO

EQUILIBRIUM!

And therefore the calculation of bed shear stress

as ?b ?gHS is not always accurate. In such

cases it is necessary to compute the

disquilibrium (e.g. gradually varied) flow and

calculate the bed shear stress from the relation

Flow over a free overfall (waterfall) usually

takes the form of an M2 curve.

Flow into standing water (lake or reservoir)

usually takes the form of an M1 curve.

A key dimensionless parameter describing the way

in which open-channel flow can deviate from

normal equilibrium is the Froude number Fr

NON-STEADY, NON-UNIFORM 1D OPEN CHANNEL

FLOWS St. Venant Shallow Water Equations

- x boundary (bed) attached nearly horizontal

coordinate L - y upward normal coordinate L
- bed elevation L
- S tan? ? - ??/?x 1
- H normal (nearly vertical) flow depth L
- Here normal means perpendicular to the bed

and has nothing to do with normal flow in the

sense of equilibrium.

Bed and water surface slopes exaggerated below

for clarity.

Relation for water mass conservation (continuity)

Relation for momentum conservation

DERIVATION EQUATION OF CONSERVATION OF OF WATER

MASS

Q UHB volume water discharge L3/T ?Q

Mass water discharge ?UHB M/T ?/?t(Mass in

control volume) Net mass inflow rate

Reducing under assumption of constant B

STREAMWISE MOMENTUM DISCHARGE Momentum flows!

Qm ?U2HB streamwise discharge of streamwise

momentum ML/T2. The derivation follows below.

Momentum crossing left face in time Dt

(?HBU2Dt) mass x velocity Qm momentum

crossing per unit time, (Momentum crossing in

Dt)/ Dt ?U2HB

Note that the streamwise momentum discharge has

the same units as force, and is often referred to

as the streamwise inertial force.

STREAMWISE PRESSURE FORCE

The flow is assumed to be gradually varying, i.e.

the spatial scale Lx of variation in the

streamwise direction satisfies the condition H/Lx

ltlt 1. Under this assumption the pressure p can

be approximated as hydrostatic. Where z an

upward normal coordinate from the bed,

p pressure (normal stress) M/L/T2

Integrate and evaluate the constant of

integration under the condition of zero (gage)

pressure at the water surface, where y H, to

get

Integrate the above relation over the

cross-sectional area to find the streamwise

pressure force Fp

Fp pressure force ML/T2

DERIVATION EQUATION OF CONSERVATION OF

STREAMWISE MOMENTUM

?/?t(Momentum in control volume) net momentum

inflow rate sum of forces Sum of forces

downstream gravitational force resistive force

pressure force at x pressure force at x ?x

or reducing,

CASE OF STEADY, GRADUALLY VARIED FLOW

Reduce equation of water mass conservation and

integrate

constant

Thus

Reduce equation of streamwise momentum

conservation

But with water conservation

So that momentum conservation reduces to

THE BACKWATER EQUATION

Reduce

with

to get the backwater equation

where

Here Fr denotes the Froude number of the flow and

Sf denotes the friction slope. For steady flow

over a fixed bed, bed slope S (which can be a

function of x) and constant water discharge per

unit width qw are specified, so that the

backwater equation specified a first-order

differential equation in H, requiring a specified

value of H at some point as a boundary condition.

NORMAL AND CRITICAL DEPTH

Consider the case of constant bed slope S.

Setting the numerator of the right-hand side

backwater equation zero, so that S Sf

(friction slope equals bed slope) recovers the

condition of normal equilibrium, at which normal

depth Hn prevails

Setting the denominator of the right-hand side of

the backwater equation zero yields the

condition of Froude-critical flow, at which Fr

1 and depth the critical value Hc

At any given point in a gradually varied flow the

depth H may differ from both Hn and Hc. If Fr

qw/(gH3)1/2 lt 1 the flow slow and deep and is

termed subcritical if on the other hand Fr gt 1

the flow is swift and shallow and is termed

supercritical. The great majority of flows in

alluvial rivers are subcritical, but

supercritical flows do occur. Supercritical

flows are common during floods in steep bedrock

rivers.

COMPUTATION OF BACKWATER CURVES

The case of constant bed slope S is considered as

an example. Let water discharge qw and bed slope

S be given. In the case of constant bed friction

coefficient Cf, let Cf be given. In the case of

Cf specified by the Manning-Strickler relation,

let ?r and ks be given. Compute Hc Compute

Hn If Hn gt Hc then (Fr)n lt 1 normal flow is

subcritical, defining a mild bed slope. If Hn lt

Hc then (Fr)n gt 1 normal flow is supercritical,

defining a steep bed slope.

or

where x1 is a starting point. Integrate upstream

if the flow at the starting point is

subcritical, and integrate downstream if it is

supercritical.

Requires 1 b.c. for unique solution

COMPUTATION OF BACKWATER CURVES contd.

Flow at a point relative to critical flow note

that

It follows that 1 Fr2(H) lt 0 if H lt Hc, and 1

Fr2 gt 0 if H gt Hc.

Flow at a point relative to normal flow note

that for the case of constant Cf

and for the case of the Manning-Strickler

relation

It follows in either case that S Sf(H) lt 0 if H

lt Hn, and S Sf(H) gt 0 if H gt Hn.

MILD BACKWATER CURVES M1, M2 AND M3

Again the case of constant bed slope S is

considered. Recall that

A bed slope is considered mild if Hn gt Hc. This

is the most common case in alluvial rivers.

There are three possible cases.

Depth increases downstream, decreases upstream

M1 H1 gt Hn gt Hc

Depth decreases downstream, increases upstream

M2 Hn gt H1 gt Hc

Depth increases downstream, decreases upstream

M3 Hn gt Hc gt H1

M1 CURVE

M1 H1 gt Hn gt Hc

Water surface elevation ? ? H (remember H is

measured normal to the bed, but is nearly

vertical as long as S ltlt 1). Note that Fr lt 1 at

x1 integrate upstream. Starting and normal

(equilibrium) flows are subcritical. As H

increases downstream, both Sf and Fr decrease

toward 0. Far downstream, dH/dx S ? d?/dx

d/dx(H ?) constant ponded water As H

decreases upstream, Sf approaches S while Fr

remains lt 1. Far upstream, normal flow is

approached.

The M1 curve describes subcritical flow into

ponded water.

Bed slope has been exaggerated for clarity.

M2 CURVE

M1 Hn gt H1 gt Hc

Note that Fr lt 1 at x1 integrate upstream.

Starting and normal (equilibrium) flows are

subcritical. As H decreases downstream, both Sf

and Fr increase, and Fr increases toward 1. At

some point downstream, Fr 1 and dH/dx - ?

free overfall (waterfall). As H increases

upstream, Sf approaches S while Fr remains lt

1. Far upstream, normal flow is approached.

The M2 curve describes subcritical flow over a

free overfall.

Bed slope has been exaggerated for clarity.

M3 CURVE

M1 Hn gt Hc gt H1

Note that Fr gt 1 at x1 integrate downstream.

The starting flow is supercritical, but the

equilibrium (normal) flow is subcritical,

requiring an intervening hydraulic jump. As H

increases downstream, both Sf and Fr decrease,

and Fr decreases toward 1. At the point where Fr

would equal 1, dH/dx would equal ?. Before this

state is reached, however, the flow must undergo

a hydraulic jump to subcritical flow.

Subcritical flow can make the transition to

supercritical flow without a hydraulic jump

supercritical flow cannot make the transition to

subcritical flow without one. Hydraulic jumps

are discussed in more detail in Chapter 23.

The M3 curve describes supercritical flow from a

sluice gate.

Bed slope has been exaggerated for clarity.

HYDRAULIC JUMP

In addition to M1, M2, and M3 curves, there is

also the family of steep S1, S2 and S3 curves

corresponding to the case for which Hc gt Hn

(normal flow is supercritical). These curves

tend to be very short, and are not covered in

detail here.

CALCULATION OF BACKWATER CURVES

Here the case of subcritical flow is considered,

so that the direction of integration is upstream.

Let x1 be the starting point where H1 is given,

and let ?x denote the step length, so that xn1

xn - ?x. (Note that xn1 is upstream of xn.)

Furthermore, denote the function S-Sf(H)/(1

Fr2(H) as F(H). In an Euler step scheme,

or thus

A better scheme is a predictor-corrector scheme,

according to which

A predictor-corrector scheme is used in the

spreadsheet RTe-bookBackwater.xls. This

spreadsheet is used in the calculations of

the next few slides.

BACKWATER MEDIATES THE UPSTREAM EFFECT OF BASE

LEVEL (ELEVATION OF STANDING WATER)

A WORKED EXAMPLE (constant Cz) S 0.00025 Cz

22 qw 5.7 m2/s D 0.6 mm R 1.65 H1 30 m H1

gt Hn gt Hc so M1 curve

Example calculate the variation in H and tb

?CfU2 in x upstream of x1 (here set equal to 0)

until H is within 1 percent of Hn

RESULTS OF CALCULATION PROFILES OF DEPTH H, BED

SHEAR STRESS ?b AND FLOW VELOCITY U

H

tb

U

RESULTS OF CALCULATION PROFILES OF BED ELEVATION

h AND WATER SURFACE ELEVATION x

x

h

REFERENCES FOR CHAPTER 5

Chaudhry, M. H., 1993, Open-Channel Flow,

Prentice-Hall, Englewood Cliffs, 483 p. Crowe, C.

T., Elger, D. F. and Robertson, J. A., 2001,

Engineering Fluid Mechanics, John Wiley and sons,

New York, 7th Edition, 714 p. Gilbert, G.K.,

1914, Transportation of Debris by Running Water,

Professional Paper 86, U.S. Geological

Survey. Jain, S. C., 2000, Open-Channel Flow,

John Wiley and Sons, New York, 344 p. Kamphuis,

J. W., 1974, Determination of sand roughness for

fixed beds, Journal of Hydraulic Research, 12(2)

193-202. Keulegan, G. H., 1938, Laws of turbulent

flow in open channels, National Bureau of

Standards Research Paper RP 1151, USA. Henderson,

F. M., 1966, Open Channel Flow, Macmillan, New

York, 522 p. Meyer-Peter, E., Favre, H. and

Einstein, H.A., 1934, Neuere Versuchsresultate

über den Geschiebetrieb, Schweizerische

Bauzeitung, E.T.H., 103(13), Zurich,

Switzerland. Meyer-Peter, E. and Müller, R.,

1948, Formulas for Bed-Load Transport,

Proceedings, 2nd Congress, International

Association of Hydraulic Research, Stockholm

39-64. Parker, G., 1991, Selective sorting and

abrasion of river gravel. II Applications,

Journal of Hydraulic Engineering, 117(2)

150-171. Vanoni, V.A., 1975, Sedimentation

Engineering, ASCE Manuals and Reports on

Engineering Practice No. 54, American Society of

Civil Engineers (ASCE), New York. Wong, M.,

2003, Does the bedload equation of Meyer-Peter

and Müller fit its own data?, Proceedings, 30th

Congress, International Association of Hydraulic

Research, Thessaloniki, J.F.K. Competition

Volume 73-80.