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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