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Turbulent Jets:Theory and Models

Environmental Hydraulics

Definitions

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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Examples of Jets and Plumes

Momentum jet Thermal plume

Jet in air

Buoyant jet

Momentum:

Buoyancy: ( )ρ

ρ ρs

u A

gV−

2

Densimetric Froude number:

'

ρ ρ'

ρ

d

d

s

uFr

g l

g g

=

−= ld : a length scale

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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

Circular Jet

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

Zone of established flow (fully developed jet)

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Zone of Flow Establishment (ZFE)

Extends from the discharge point until water entrained at the edges of the jet affects the centerline velocity.

Velocity profile has a top-hat distribution initially, but attains a Gaussian shape at the end of the ZFE.

Flow in ZFE:

. . . oo o o

Q x xx D

Q D D

⎛ ⎞⎟⎜= + + ≤⎟⎜ ⎟⎟⎜⎝ ⎠

2

1 0 083 0 0128 6 2

(Albertson et al. 1948)

Zone of Established Flow (ZEF)

Extends from where the water entrained at the edges of the jet affects the centerline velocity and to infinity.

Velocity profile has a Gaussian shape (concentration also).

Flow in ZEF:

. . oo o

Q xx D

Q D= >0 32 6 2

(Albertson et al. 1948)

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Flow Development in a Jet

Characteristic length scale: oQ o

o

Ql A

M= =

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

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Increase of flow occurs through entrainment of ambient water.

π s i

dQr v

dx=2

Δ π Δs iQ r v x=2

Mass balance equation for water:

max

..

πis

v Q uD r

= ≅00

0 32 10 05

2

Using previous expression for flow:

Entrainment velocity is about 5% of velocity at jet axis

Plane Jet

Rectangular slot with large width in relation to height.

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Zone of Flow Establishment (ZFE)

Extends from the discharge point until water entrained at the edges of the jet affects the centerline velocity.

Velocity profile has a top hat distribution initially, but attains a Gaussian shape at the end of the ZFE.

Flow in ZFE:

. .o

q xx d

q d= + ≤1 0 080 5 2

(Albertson et al. 1948)

Zone of Established Flow (ZEF)

Extends from where the water entrained at the edges of the jet affects the centerline velocity and to infinity.

Velocity profile has a Gaussian shape (concentration also).

Flow in ZEF:

. .o

q xx d

q d= >0 62 5 2

(Albertson et al. 1948)

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Velocity and concentration in the plane jet:

Centerline velocity and concentration:

max .u du x=

0

6 2

max .c dc x=

0

2 0

max

expu y

u x

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

250

max

expc y

c x

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

225

Wastewater with a pollutant concentration of 20 mg/l is to be discharged into the sea. Calculate the velocity and pollutant concentration at the central axis 20 meters from where the water was discharged if:

• discharge occurs via a circular jet and the pipe opening is 0.1 m2 with a wastewater flow rate of 200 l/s.

• discharge occurs via a plane jet and the height of the opening is 0.1 m with a wastewater flow rate of 200 l/s per meter.

Sample Problem

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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

Top-hat velocity distribution:

π πor

T TQ rudr r u= =∫ 2

0

2

ρ π ρπor

T TM ru dr r u= =∫ 2 2 2

0

2

Taylor’s entrainment hypothesis:

α εe c Tu u u= = π αT Tq r u=2

uT

rT

x

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Equations to solve:

( )π π αT T T T

dr u r u

dx=2 2

( )ρπ T T

dr u

dx=2 2 0 constantT T o or u r u= =2 2 2 2

o oT

T

r ur

u=

2 22

2

αo o

To

r uu

r x=+2

Solution:

αT or r x= +2

Flow rate:

απ α πT T o o

o

xQ r u x r u

r

⎛ ⎞⎟⎜= + = + ⎟⎜ ⎟⎟⎜⎝ ⎠2 2 2

2 1

αo o

Q xQ r= +1 2

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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

Virtual Origin

-4.00 0.00 4.00 8.00 12.00x/D

0.00

0.40

0.80

1.20

1.60

2.00

b/D

1800 rpm Coflow

1800 rpm Counter flow

1200 rpm Counter flow

Jets typically exhibit linear spreading:

Mr mx=

m is about 0.11

Virtual source

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Width Parameters for Turbulent Round Jets

α . .

α . .j

p

= ±

= ±

0 0535 0 0025

0 0833 0 0042

(jet)

(plume)

Gaussian velocity distribution:

π M MQ r u= 2

max

expM

u ru r

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

2

2 self-similar velocity profile

ρπ M MM r u= 2 212

Compare with top-hat distribution:T Mr r= 2