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)
1
Examples of Jets and Plumes
Thermal plume
Momentum jet
Jet in
air
Buoyant jet
Momentum: ρu2 A
Buoyancy:
(ρ s − ρ) gV
Densimetric Froude number:
Frd =
g' =
u
g ' ld
ρs −ρ
g
ρ
ld : a length scale
2
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)
3
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:
2
⎛x⎞
Q
x
= 1 + 0.083 + 0.0128 ⎜⎜ ⎟⎟⎟
⎜⎝ Do ⎠
Qo
Do
x ≤ 6.2 Do
(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:
Q
x
= 0.32
Qo
Do
x > 6.2 Do
(Albertson et al. 1948)
4
Flow Development in a Jet
Characteristic length scale: lQ =
Qo
= Ao
Mo
Centerline velocity and concentration:
umax
D
= 6.2 0
u0
x
cmax
D
= 5.6 0
c0
x
Velocity and concentration in the circular jet:
u
umax
c
cmax
⎛
r2 ⎞
= exp ⎜⎜−77 2 ⎟⎟⎟
⎜⎝
x ⎠
⎛
r2 ⎞
= exp ⎜⎜−62 2 ⎟⎟⎟
⎜⎝
x ⎠
5
Increase of flow occurs through entrainment of ambient water.
Mass balance equation for water:
ΔQ = 2πrs vi Δx
dQ
= 2πrs vi
dx
Using previous expression for flow:
vi = Q0
0.32 1
≅ 0.05umax
D0 2πrs
Entrainment velocity is about 5% of velocity at jet axis
Plane Jet
Rectangular slot with large width in relation to height.
6
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:
q
x
= 1 + 0.080
qo
d
x ≤ 5.2d
(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:
q
x
= 0.62
qo
d
x > 5.2d
(Albertson et al. 1948)
7
Centerline velocity and concentration:
umax
d
= 6 .2
u0
x
cmax
d
= 2.0
c0
x
Velocity and concentration in the plane jet:
⎛
y2 ⎞
= exp ⎜⎜−50 2 ⎟⎟⎟
⎜⎝
x ⎠
u
umax
c
cmax
⎛
y2 ⎞⎟
⎜
= exp ⎜−25 2 ⎟⎟
⎜⎝
x ⎠
Sample Problem
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.
8
Model of Circular Jet
q
Volume conservation:
Q
Q+ΔQ
dQ
=q
dx
Δx
Momentum conservation:
dM
=0
dx
ro
Q = ∫ 2πrudr
0
ro
M = ∫ ρ2πru2 dr
0
Top-hat velocity distribution:
rT
ro
Q = ∫ 2πrudr = πrT2 uT
x
0
uT
ro
M = ∫ ρ2πru2 dr = ρπrT2 uT2
0
Taylor’s entrainment hypothesis:
ue = αuc = εuT
q = 2πrT αuT
9
Equations to solve:
d
πrT2 uT ) = 2πrT αuT
(
dx
d
(ρπrT2uT2 ) = 0
dx
rT2 uT2 = ro2 uo2 = constant
rT2 =
ro2 uo2
uT2
Solution:
uT =
ro uo
ro + 2αx
rT = ro + 2αx
Flow rate:
⎛
2αx ⎞⎟
Q = πrT2uT + 2αx = πro2 uo ⎜⎜1 +
⎟
⎜⎝
ro ⎠⎟
Q
x
= 1 + 2α
Qo
ro
10
Self-Similarity
Velocity (and concentration) profiles look the same
everywhere properly scaled.
1.00
umax
⎛r⎞
= Ψ ⎜⎜ ⎟⎟⎟
⎜⎝ rM ⎠
0.80
0.60
_
U /Um
u
0.40
0.20
0.00
Scaling parameters:
-0.20
0.00
• maximum (centerline) velocity
0.10
0.20
ξ = r/(x+a)
0.30
0.40
• jet width
Example, Gaussian profile:
u
umax
⎛ r2 ⎞
= exp ⎜⎜− 2 ⎟⎟⎟
⎜⎝ rM ⎠
Virtual Origin
Jets typically exhibit linear
spreading:
2.00
1800 rpm Coflow
1800 rpm Counter flow
1.60
rM = mx
1200 rpm Counter flow
b/D
1.20
m is about 0.11
0.80
0.40
0.00
-4.00
0.00
4.00
8.00
12.00
x/D
Virtual
source
11
Width Parameters for Turbulent Round Jets
α j = 0.0535 ± 0.0025
(jet)
α p = 0.0833 ± 0.0042
(plume)
Gaussian velocity distribution:
u
umax
⎛ r2 ⎞
⎜⎜− ⎟⎟
exp
=
⎜⎝ rM2 ⎠⎟
self-similar velocity profile
Q = πrM2 u M
1
M = ρπrM2 u2M
2
Compare with top-hat distribution:
rT = 2rM
12