1

Course Summary – Environmental Hydraulics

Environmental Hydraulics

Environmental hydraulics:

Hydrodynamic aspects of water quality management in natural bodies of water (Fischer et al. 1979).

2

Different Types of Pollution Discharge

Municipal wastewater

treated wastewater

combined sewer overflow (CSO)

storm water

Industrial wastewater

Cooling water

Water Quality Criteria

Related to the use of the receiving waters and present conditions:

• water supply (household, industry)

• fishing

• recreation

• irrigation

• transportation

• natural values

3

Receiving Water Types

SNV: Swedish EPA

• flowing water

• shallow lakes (depth < 12 – 15 m)

• deep lakes (depth > 12 – 15 m)

• estuaries

• open coastal areas

Near-Field and Far-Field Zone

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Balance Equations for Water and Pollutants

Box models: based on the conservation of mass (water, pollutant)

Define a suitable control volume (a fixed volume in space through whose boundary mass can be transported). The control volume can be of arbitrary size and shape, but the control surface must be closed.

Nominal Retention Time

VT

q=

The time (T) it takes to replenish the water in the receiving water through the flow rate q assuming no mixing (”piston flow”).

Example of retention times:Lake Vättern 10 yr

The Baltic Sea 20-40 yr

Lake Vänern 150 yr

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Complete and Instantaneous Mixing

( )d

cV q cqdt

= ⋅ −0

Mass balance equation (box model):

Assumptions: steady-state flow, no pollutant in inflow, complete and instantaneous mixing

Estuarine Water Exchange

E

Qf

Qf, cin

V, c

csea

Mass balance:

( ) ( )f in sea f

dcV Q c Ec E Q c kVc

dt= + − + −

E: water exchange form tide, wind etc

6

Mechanisms for Mixing and Transport

• diffusion (molecular, turbulent)

• dispersion

• advection

The General Transport Equation

Advection-Diffusion (AD) Equation

( ) ( ) ( )x y z

c c c c c c cu v w D D D

t x y z x x y y z z∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

( )x

c c cu D

t x x x∂ ∂ ∂ ∂+ =∂ ∂ ∂ ∂

3-D:

1-D:

Change in concentration with time at x,y,z

Change due to advection Change due to

diffusion

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Fick’s Law

Valid for molecular diffusion:

mol

dcq D

dx=− ⋅

Dmol = molecular diffusion coefficient (for specific tracer)

Transport of Tracer in a Pipe

( )m m md

c c cU E

t x x x∂ ∂ ∂∂+ =∂ ∂ ∂ ∂

1D approach:

U = Q/A,Q = flow rateA = cross sectional area

advectiondiffusion (dispersion)

change in concentration with time

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Solution:

( )( , ) exp

ρ πm

dd

M x Utc x t

E tA E t

⎛ ⎞− ⎟⎜= ⋅ − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

44

Gaussian (normal) distribution

Estimate D based on field measurements with tracer:

rhodamine

9

( )exp

πM x Ut y

cDt Dt

⎛ ⎞− + ⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2 20

4 4

Spreading of injected tracer cloud (2D in space):

c c c cU D D

t x x y∂ ∂ ∂ ∂+ = +∂ ∂ ∂ ∂

2 2

2 2

Solution:

Definitions: Jets and Plumes

Jet = boundary layer flow originating from a source of momentum

Plume = boundary layer flow originating from a source of buoyancy

Buoyant jet (forced plume) = boundary layer flow originating from a source of momentum and buoyancy

Boundary layer: high rate of change across some direction(s)

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Circular Jet

Zone of flow establishment (jet development; 6-10Do)

Zone of established flow (fully developed jet)

Jet behavior depends on:

• jet parameters

diameter (Do), velocity (Uo)

• environmental parameters (receiving water)

ambient velocity (Ua)

• geometrical factors

water depth (h), orientation of discharge

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Velocity and concentration in the circular jet:

Centerline velocity and concentration:

max .u Du x= 0

0

6 2

max .c Dc x= 0

0

5 6

max

expu r

u x

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

277

max

expc r

c x

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

262

Self-Similarity

Velocity (and concentration) profiles look the same everywhere properly scaled.

max

ΨM

u ru r

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

Scaling parameters:

• maximum (centerline) velocity

• jet width

Example, Gaussian profile:max

expM

u ru r

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

2

0.00 0.10 0.20 0.30 0.40ξ = r/(x+a)

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

U_ /U

m

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Model of Circular Jet

Δx

Q Q+ΔQ

q

Volume conservation:

dQq

dx=

Momentum conservation:

dMdx=0

πor

Q rudr= ∫0

2

ρ πor

M ru dr= ∫ 2

0

2

Buoyant Jet Evolution

Zone of jet evolution:

1. jet development (ZFE)2. fully developed jet (ZEF)3. final vertical elevation of jet4. horizontal spreading

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Densimetric Froude Number

00

00

0

r

uFg D

=ρ −ρρ

00

0

0

0

'

' r

uFg D

g

=

ρ −ρ=

ρ

Related is the Richardson number:

21

o

RiF

=

(in oceanography: )2

/ uRi gz z∂ρ ∂⎛ ⎞= − ρ⎜ ⎟∂ ∂⎝ ⎠

Buoyant Jet Trajectories

00

00

0

0

,

r

mm

o o

uFg D

cSc

x zD D

=ρ −ρρ

=

Governing parameters:

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Homogenization before Horizontal Spreading

max 1.4after

cc

≈

Regard temperature as a ”pollution” with:

0

* r

r

T TTT T−

=−

Outfall and Diffuser System

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Stratified Receiving Waters

Causes of density differences:

• salinity (halocline)

• temperature (thermocline)

• suspended solids (lutocline)

4 8 12 16 20 24Water Temperature (degrees)

-25

-20

-15

-10

-5

0

Wat

er D

epth

(m)

Month

FEB

APR

JUN

AUG

OCT

DEC Temperature distribution Nainital Lake

Deep Lakes

Lake BaikalMax. depth 1637 m

Dead SeaMax. depth 330 m

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Estuary

Definition: a semi-enclosed coastal body of water having free connection to the open sea and within which sea water is measurably diluted with fresh water deriving from land drainage

(UNESCO)

Estuary Classification

According to circulation and salinity distribution:

1. salt wedge estuary

2. highly stratified estuary

3. slightly stratified estuary

4. vertically mixed estuary

5. inverse estuary

6. intermittent estuary

Types 1-4 involves advection of freshwater from a river + introduction of sea water through turbulent mixing (typically tidal motion).

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Heat Exchange

Important for circulation in a receiving water.

Determines the rate at which artificially added heat is transferred to the atmosphere

Examples:

• annual temperature variation and stratification in a lake

• evaporation

• discharge of cooling water from fossil and nuclear power plants

Heat Exchange Mechanisms

Heat exchange at a water surface:

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Equilibrium Temperature

Equilibrium temperature (TE): for given meteorological conditions, the temperature that corresponds to a neat heat flow of zero at the water surface

⇒ The temperature that the water surface will approach under constant meteorological conditions

TS < TE : heating up

TS > TE : cooling down

For small deviations between TS and TE:

Φn = K (TE – TS)

K: heat exchange coefficient

Approximate expression for K:

223.7 (0.0613 )(70 3.5 )K Wβ= + + +

20.0454 0.00192 0.000156T Tβ ββ = + +

2dTs TTβ

+=

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Discharge of Cooling Water to River

2

2( ) ( )

xd T d T KBu E T

dx dx cAΔ Δ

= − Δρ

Dissolved Oxygen in Water

Necessary for (aerobic) life in water.

Air is dissolved at the water surface and then transported into the water mass by turbulence and/or currents.

Simple oxygen balance for well-mixed conditions (uniform conditions over a cross section):

1 ( )mdCVol k A C Cdt

= −

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Inflow of oxygen proportional to the deficit in the water volume under study:

( )mdC r C Cdt

= −

r : re-aeration coefficient (depends on flow conditions and exposed surface area to water volume

r = 10-5 – 10-4 s-1

Oxygen Consuming Substances

Example: municipal and industrial discharge of organic and inorganic matter

Characterized through BOD(t) or COD(t)

(biological and chemical oxygen demand, respectively, expressed in mg O2/liter H2O)

Model of degradation (first-order reaction):

( ) ( )d BOD t K BOD tdt

= − ⋅

K: degradation coefficient

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Oxygen balance:

( ) ( )mdC r C C K BOD tdt

= − − ⋅

where:

( )0( ) expBOD t BOD Kt= ⋅ −

(solution to a first-order reaction)

Mechanisms for Water Exchange

• wind

• waves

• tide

• seiching

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Wind

*

0

*

( ) ln

o

air

v zw zz

v

=κ

τ=

ρ

Velocity profile:

(shear velocity)

Surface Shear Stress

20 10D airC wτ = ρ air flow over water/land

2s DC uτ = ρ water flow over bottom

Force balance:

10 100.035airs o u w wρτ = τ => = =

ρ

Water surface velocity 3-4% of wind speed

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Waves - Induced Flows

• oscillatory flows

• mean flows (longshore current, undertow, rip current)

• turbulence (bottom boundary, breaking)

Oscillatory flow: not net flow (advection), little mixing

Wave-Induced Mean Flows

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Tide-Induced Water Level Variations

Types of Tide

semi-diurnal

diurnal

mixed

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Seiching

Modes in Closed and Open-Ended Basins

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Inclination of Water Surface due to Wind

Force balance:

0 b

w

dhdx gh

τ + τ=

ρ

Inclination of Density Interface

Surface inclination is small but a compensating tilt in the density interface becomes large.

Force balance (neglecting water movement and associated shear stresses): 2 1 1

2 1

dh dhdx dx

ρ= −

ρ −ρ

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Saltwater Wedge

Equation for interface shape:

1 1

1 2 2

211

2

1

1

ydy ySdx F

ρ+ρ

= −ρ

− −ρ

(from energy equation and momentum equation)

Saltwater penetration occurs if:

1/ 2

2 1

2oF

⎛ ⎞ρ −ρ< ⎜ ⎟ρ⎝ ⎠

Internal Waves

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Receiving Water Study

• establish knowledge on existing water quality

• forecast changes in water quality

• estimate receiving water capacity

• establish monitoring program

Objectives:

Parameters to Characterize Receiving Waters

• salinity

• temperature

• currents

• water level variations

• mixing characteristics

• wind

• topography

Capacity to receive waste water is determined by:

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Sediment Transport

• properties of sediment

• current boundary layers

• threshold of motion

• bed features

• suspended load transport

• bed load transport

• total load transport

Properties of Sediment

• grain size (diameter)

• density

• porosity

• concentration (by volume or mass)

• angle of repose

• permeability (fluidization)

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Current Boundary Layers

Velocity profile over a loose boundary (bed).

Vertical gradients are much larger than horizontal ones.

Logarithmic velocity profiles:

*

*

( ) lnκ

τρ

o

o

u zU z

z

u

=

=

Mean velocity: ( )h

U U z dzh= ∫

0

1

κ .=0 4

Von Karman’s constant

Bed Roughness Length

*

*

*

*

νν

ν

ν

so

so

s so

u kz

u

kz

u

k u kz

= <

= +

= >

59

30 9

7030

Smooth flow

Transitional flow

Rough flow

Nikuradse roughness:

.sk d= 502 5

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Current Skin Friction Shear Stress (Flat Bed)

Shear stress: τ ρo DC U= 2

Drag coefficient:( )κln /D

o

Cz h

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜ +⎝ ⎠

2

1

Relationship with other friction coefficients:

/D

f g gnC

C h= = =

2

2 1 38

( Darcy-Weisbach / Chezy / Manning )

Threshold of Motion

Conditions for initiation of motion.

Shields diagram

*τρ ( ) ν

cr u df

g s d

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠−50

501(Shields 1936)

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Bed Features

A variety of features appears on a loose bed exposed to flowing water:

• ripples

• dunes

• antidunes

Sediment Transport Modes

• bed load

along the bottom; particles in contact; bottom shear stress important

• suspended load

in the water column; particles sustained by turbulence; concentration profiles develop

bed load suspended load sheet flow

Increasing Shields number

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Suspended Load

Settling velocity less than upward turbulent component of velocity (for grains to remain in suspension).

Important parameter: ws/u*

( ) ( )a

h

ssz

q c z u z dz= ∫

Suspended Sediment Concentration Profiles

Exponential (constant diffusivity):

( ) exp sR

o

wC z C z

K

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

if ws/Ko> 4: weak suspension

if ws/Ko < 0.5: strong suspension

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Settling Velocity

Depends on:

• particle diameter

• particle density

• particle concentration

• particle shape

• viscosity of water (temperature)

• turbulence

/

*

( )ν

g sD d

⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

1 3

502

1

Dimensionless grain size for characterization of settling velocity:

Suspended Load Transport

expo sss c R

s o

K w hq U c

w K

⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥= − − ⎟⎜ ⎟⎢ ⎥⎟⎜⎝ ⎠⎣ ⎦1

Integrate product between concentration and velocity over the vertical.

For the exponential concentration profile and constant velocity:

θθexp .

θcr

R cRc A⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠4 5

( )*. exp .cRA D−= ⋅ −33 5 10 0 3

Reference concentration (Camenen and Larson 2007):

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Bed Load Transport Formulas

( )Φ θ θ/

cr= − 3 28

Meyer-Peter and Müller (1948):

( )Φ θ θ θ/cr= −1 212

Nielsen (1992):

/ θΦ θ exp .

θcr

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠3 212 4 5

Camenen and Larson (2006):

Structures and Flows in Nature

• forces on structures

• local scour

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Forces on Structures due to Water Flow

ρD DF C U A= 212

Drag force:

Lift force:

ρL LF C U A= 212

(+ inertia force for oscillatory flows)

CD depends on Re + shape (friction + form)

Local Scour

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Scour around Detached Bodies

Example: bridge piers, piles

Downflow in front, accelerated flow at the side, wake vortices on the back

⇒ Erosion

Once a hole is formed, recirculation occurs

Scour around a Cylinder

Design formula:

α tanhs os u

h hK K K

D D

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠2