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Buoyant Jets and Plumes:Theory and Models

Environmental Hydraulics

Definitions

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

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Positive and Negative Buoyancy

Positive Negative

Negative/positive

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⎜ ⎟∂ ∂⎝ ⎠

Model of Buoyant Jet

Assumptions:

• hydrostatic pressure

• small variations in density

• symmetric jet

• self-similar properties

• no influence from boundaries

• density linearly related to concentration

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Geometrical conditions:

sindzds

= θ

cosdxds

= θ

Continuity (water):

max0

( 2 ) 2 sd u rdr u rds

∞

ρ π = α ρ π∫

Continuity (pollutant):

0 00

2cu rdr c Q∞

π =∫

Momentum equation (x-direction):

20 0 0

0

cos 2 cosu rdr Q u∞

θ ρ π = θ ρ∫

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Momentum equation (z-direction):

20 0 0

0 0 0

sin 2 sin ( 2 )s

u rdr Q u g rdr ds∞ ∞

θ ρ π = θ ρ + Δρ π∫ ∫ ∫

Self-similarity:

2

max

2

max

exp( ( ) )

exp( ( ) )

u rku s

c rkc s

= − ⋅

= −μ ⋅ ⋅

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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Typical values on self-similarity coefficients:

Fr0 k μ

0 96 0.74∞ 77 0.80

Homogenization before Horizontal Spreading

max 1.4after

cc

≈

Regard temperature as a ”pollution” with:

0

* r

r

T TTT T−

=−

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Example of Discharge of Wastewater

Wastewater, at a flow rate of 0.1 m3/s and with a density of 1005 kg/m3, is discharged horizontally through a circular pipe at a depth of 20 m below the sea surface. The density of the seawater is assumed to be constant, ρr = 1025 kg/m3. What is the dilution at the jet axis immediately below the water surfaceand how far from the discharge point will the jet reach the water surface? Make calculations for an initial jet diameter of D0 = 0.25 m and D0 = 0.20 m.

01 2 20

0.1 0.1 2.04 m/s/ 4 0.25 / 4

uD

= = =π π⋅

Initial velocity:

02 20.1 3.18 m/s

0.20 / 4u = =

π⋅

Density difference: ρr – ρ0 = 1025 – 1005 = 20 kg/m3

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Densimetric Froude number:

012.04 9.2320 0.25

1005

Fg

= =⋅ ⋅

023.18 16.120 0.20

1005

Fg

= =⋅ ⋅

z/Do + x/Do and Fo => solution through Figure 40

Water surface at: z/Do=20/0.25=80 + Fo=9.23

⇒ x/Do=45 x=11 m

Sm=50

Water surface at: z/Do=20/0.20=100 + Fo=16.1

⇒ x/Do=70 x=14 m

Sm=57

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Effects of Receiving Waters (Ambient)

Density-stratified shear flow

Jet in cross flow (calculated streamlines)

• buoyancy

• stratification

• currents