Critical review on different types of plumes and jets

Reviewing Cipollina (2005) and Pantokratoras (2003)

Literature Review , 2015 , 11 Pages , Grade: 3

Environmental Sciences

Two papers are critically reviewed in this study: “Bench-Scale Investigation of Inclined Dense Jets” by Cipolla and others (2005) and “Vertical Penetration of Double-Diffusive Water Plumes Discharged Vertically Downward” by Pantokratoras (2003).

The main goal is to provide new experimental information on the behavior of dense jets issuing from nozzles variously inclined on the horizontal plane. Also, to develop suitable Froude number equation for double diffusive plumes, in which discharge temperature is greater than that of the ambient water whereas the discharge salinity is smaller than that of the ambient sea water.

Another objective is to prove the Turner`s Equation (Pantokratoras 2003) is unsuitable for double diffusive plumes and to develop this equation.
Most of the investigations on dense jet diffusion dealt with jets issuing vertically in the environment, despite the fact that it has been long known that inclined dense jets result in higher values of dilution.

The water temperature in the tank (1.5 m long, 0.45 m wide, filled with water up to 0.60 m above the bottom) was in the range 18–20° C, ν ~10-6 m2/s. The solution employed to feed the jets was colored and prepared by dissolving NaCl in tap water in a suitable lab-scale rectangular batch storage tank. Three different jet densities were tested: 1,055, 1,098, and 1,179 kg/m3. Brass nozzles have four different diameters (d0) (1.12, 2, 3, and 4 mm). The initial discharge Reynolds numbers ranged from 2,500 to 10,400, depending on nozzle size and jet features, and therefore flow at the nozzle outlet was always turbulent.

2

The Distance of the Jet Impact Point from the Origin (X

i

in meters): It is an important

design parameter for an underwater diffuser as when designing the diffuser it has often to be

decided whether and where sea floor protection against erosion is to be provided.

(1)

F is densimetric Froude number, (--),

a

= density of ambient water (kg/m3),

0

=

density of jet at its source (kg/m3), u

0

= velocity of issuing jet=Q/(×d

02

/4) (m/s), g =gravity

constant (m/s

2

) and Q = jet volumetric flow rate (m3/s) in (Eq. (1)); Froude number has a simple

proportionality with X

i

; (X

i

/d

0

=k

i

×F). k

i

value was obtained 2.25 for 60° inclined jets.

ki ,kz ,kx ,ky = coefficients of proportionality between normalized X

i

, Z, X, Y and F(-)

Maximum Rise Level (Z in meters): Simple proportionality is found again between

Maximum Rise Level (Z in meters) and Froude number (Z/d

0

=k

z

× F), with proportionality

coefficients k

z

of 1.08, 1.61, and 2.32 for jet initial inclinations of 30, 45, and 60°, respectively.

Centerline Maximum Coordinates (X and Y in meters): X and Y are proportional to

the Froude number over the whole investigated range (X/d

0

=k

x

× F, Y /d

0

=k

y

× F) with k

x

values

of 1.95, 1.80, 1.42, and k

y

values of 0.79, 1.17, 1.77 for

0

=30, 45, and 60°, respectively.

Viscosity effect: Jets characterized by two modified values of viscosity (1.77 and

2.97×10

-3

Pa s). In Fig. 2, the Z values obtained with the modified viscosity jets are compared

with each other as well as with data obtained with unmodified viscosity jets (characterized by

viscosities ranging from 10

-3

to 1.5 × 10

-3

Pa s, depending on salt concentration, i.e., on jet

density). Observation of Fig. 2 shows that there is no significant effect of viscosity on the

behavior of dense jets. Notably, this is consistent with the assumption of results independent of

Reynolds number (Re = (u

o

× d

0

) ÷ ).

---

3

Figure 1: Jet geometrical parameters.

Figure 2: Effect of jet viscosity on max. rise level.

This research is on negatively buoyant jets, where the jet fluid has higher density than

the ambient fluid and sinks to the floor. We told mostly about horizontal or vertical positively

buoyant jets in the lecture instead of inclined dense jets. Hence, this paper provides a useful

contribution to literature.

---

4

Recommendation: A larger tank would be used for less constraint on the range

of parameters such as the nozzle diameter, inclination angle and flow rate.

Critical review of "Vertical Penetration of Double-Diffusive Water Plumes

Discharged Vertically Downward": This paper has been written by Pantokratoras (2003). The

title of this paper is (see Fig. 3). A discharge with no buoyancy is referred to as a nonbuoyant

jet or pure jet. A release of buoyancy only (no initial momentum) is called a pure plume.

A discharge with both momentum and buoyancy is call a buoyant jet or forced plume.

Positively buoyant flows are defined where the buoyancy force acts vertically upwards against

the gravity force; negative buoyancy is defined as acting downwards in the direction of the

gravity force (Otranto 2004). In this study buoyant jet in other words forced plume is dealt with.

The main goal: To develop suitable Froude number equation for double diffusive

plumes, in which discharge temperature is greater than that of the ambient water whereas the

discharge salinity is smaller than that of the ambient sea water. Another objective is to prove

the Turner`s Equation (Pantokratoras 2003) is unsuitable for double diffusive plumes and to

develop this equation.

Turner`s Equation: Turner developed the following equation for vertical penetration:

(2))

Where Y

m

=terminal rise height; D=jet exit diameter; c=constant; and F

0

=initial

densimetric Froude number.

Initial Densimetric Froude Number:

(3)

In this Froude number u

0

and

0

=jet exit velocity and density and

a

=ambient density.

The constant c takes the value of 1.74 according to Turner's measurements.

Pantokratoras (1999) showed that this densimetric Froude number is unsuitable for

heated water jets and must be replaced by the effective Froude number.

Effective Froude number for heated water jets:

---

5

(4)

Where T

o

and T

a

are initial (jet exit) and ambient temperatures (°C) and (T

a

) =thermal

expansion coefficient of water taken at ambient temperature. This finding concerns pure water

jets. However, in most discharges from power, manufacturing, and wastewater treatment plants

into the sea, there is both a temperature and salinity difference between the discharge and the

ambient water. In this case, we have thermosolutal or double-diffusive plumes with two

buoyancy components (heat and salt) which both oppose the downward flow.

Eq. (2) including the effective Froude number is not valid in the temperature region 4

T

a

10°C where the water density­temperature relationship becomes strongly nonlinear.

Maximum densities appear also in saline water.

Eight equations are written for an inclined jet (Pantokratoras 2003).

Figure 3: Schematic Diagram of Inclined Buoyant Jet Discharged Downwards.

(5)

---

6

(6)

(7)

(8)

(9)

Geometric Properties

(10)

(11)

(12)

Practical salinity (at atmospheric pressure) (Tomczak 2000):

S = 0.0080 - 0.1692 K

1/2

+ 25.3853 K + 14.0941 K

3/2

- 7.0261 K

2

+ 2.7081 K

5/2

(13)

Centerline density (s) (according to International Equation of State for Seawater):

Figure 4: Temperature-Density-Salinity (TDS) Diagram (Huntley 2005).

---

7

The equation of state for water: It's given by the following equation

(Pantokratoras 1999):

= 1+3.3 × 10

-8

T

3

­ 7.4 × 10

-6

T

2

+ 5.1 ×

10

-5

T ­ 9.5 × 10

-5

(14)

The entrainment coefficient

e

is given by the following equation (Pantokratoras 2003):

e

= 0.0535-(0.0535-0.0833)(R/R

p

)

2

(15)

Where R

p

= 0.557 (plume Richardson number) and R is the local Richardson number

given by

(16)

In Eq. (16), F

L

= the local Froude defined as

(17)

The parameter , which is the spreading ratio between the temperature and velocity

profile, takes the value of 1.16 according to Fan and Brooks (1969). The entrainment coefficient:

e

= 0.0535+0.5524/F

L 2

(18)

There are eight unknowns: (1) centerline velocity u(s); (2) centerline temperature T(s)

(°C); (3) centerline salinity S(s); (4) 1/e jet half-width of velocity profile b; (5) angle of

inclination ; (6) and (7) the x and y coordinates of the jet axis; and (8) centerline density (s).

The flow establishment zone is taken 6.2D long (xo = 6.2D cos

o

and yo= 6.2D sin

o

)

according to experimental data by Albertson et al. (1950) (Pantokratoras 2003). Boundary

conditions:

u(s) = u(0)

b = D/2

=

o

The ratio (T(s)-T

a

)/ (T

o

-T

a

) takes the value of 0.70 and (S(s)-S

a

)/ (S

o

-S

a

) =0.70.

Effective Froude number for Double-diffusive / thermosolutal plumes:

(19)

---

8

Where T

o

and T

a

=initial and ambient temperatures (°C); S

o

and S

a

are initial and ambient

salinities; and (S

a

)=saline expansion coefficient of water taken at ambient salinity. The thermal

and saline expansion coefficients are defined as follows (Pantokratoras 2003)

(20)

(21)

e.g.: for temperature 10 °C, salinity 30 ppt, 100 dbar pressure, alpha (°C

-1

)=1.586×10

-4

,

beta (ppt

-1

)= 7.609×10

-4

(Gill, A. E. 1982)

.

The nondimensional distance Y

m

/D has been calculated experimentally for a water jet

discharged vertically downward with the following conditions: u

o

=80 cm/s, D=1 cm, T

o

=40°C,

T

a

=10°C, and S

0

=0 ppt. The ambient salinity took the values 0, 5, 10, 20, 30, and 40 ppt. From

Table 1 it is seen that the values of the constant c which correspond to the effective Froude

number (Table 1-last col.) are close to 1.74. Hence, Turner's equation is valid also for double-

diffusive plumes if effective Froude number is used, instead of the densimetric Froude number.

Vertical penetration for Double-diffusive / thermosolutal plumes:

(22)

Table 1. Values of Coefficient c for a Double-Diffusive Plume for Different Ambient

Salinities

In the temperature range of 4 T

a

10°C distance Ym /D has been calculated for a wide

range of effective Froude numbers (10­500). From Fig. 5, it is seen that, at low ambient

temperatures, the coefficient c is a function of both the ambient temperature and effective

Froude number.

---

9

Figure 5: Variation of constant c with effective Froude number and ambient

temperature for a vertical fresh water jet discharged downward. The horizontal dashed line

corresponds to c=1.74.

In this paper both experimental and numerical methods are used simultaneously

to achieve the objective. The research is on positively buoyant plumes, where plume density is

lower. In the lecture we told about ordinary densimetric Froude number equation. However

according to results of this paper, for the extreme cases (such as thermosolutal plumes),

densimetric Froude number equation is not suitable and has to be improved in accordance with

the flow types and conditions. This paper provides a useful contribution to literature by

revealing the unsuitability of densimetric Froude number for double diffusive plumes.

Recommendations for future study

Future study can be implemented on multiport diffusers which are a linear structure

consisting of many more or less closely spaced ports or nozzles which inject a series of turbulent

jets at high velocity into the ambient receiving water body. Additionally boundary interaction

can be investigated which separates near-field (where there is a mixing zone presents, which is

the region in which the initial dilution of a discharge occurs) from far-field mixing processes.

End of the buoyant jet phase and transition to passive diffusion can be showed. Maybe a new

study combines these two papers generates better results.

Future study may be done on more complicated flow pattern such as buoyant jets, which

combines jets and plumes to form unique flow characteristics. For instance if the distance from

---

10

the source (z-defined as "s" by Pantokratoras 2003) is much greater than l

M

(z l

M

), then the

flow will resemble a plume, properties are dominated by buoyancy flux. If z l

M

, the flow will

be largely influenced by momentum and initial conditions, thus being jet-like (Otranto 2004).

Momentum length scale is

l

M

=M

3/4

/B

1/2

(23)

or,

z/l

M

=B

1/2

×z / M

3/4

(24)

If B then l

M

(Plume like)

If M then l

M

(Jet like)

Conclusions

New experimental information on the behavior of dense jets issuing from nozzles

variously inclined on the horizontal plane was investigated. Additionally, a new effective

Froude number is introduced for double-diffusive/thermosolutal plumes. Both papers are quite

successful and available for improvement for future studies.

---

11

References

Cipollina A., Brucato A., Grisafi F., and Nicosia S. (2005). ASCE Journal of Hydraulic

Engineering, Vol. 131, No. 11, pp. 1017-1022.

Fan, L. N., and Brooks, N. H. (1969). ``Numerical solutions of turbulent buoyant jet

problems.'' Rep. No. KH-R-18, W. M. Keck Laboratory of Hydraulics and Water Resources,

California Institute of Technology, Pasadena, Calif.

Gill, A. E., (1982). Properties of water (and seawater), using the UNESCO equation of

state. Atmosphere-ocean dynamics. Academic Press, New York, USA.

Huntley, Anthony C. (2005). Homepage, Saddleback University, California, USA.

http://www.saddleback.edu/faculty/thuntley/ms20labs/SEAWATER%20DENSITY.pdf

Otranto, Fabio (2004). Laboratory Modeling of Sea-brine discharges. Thesis Submitted

in partial fulfillment of the requirements for the degree of Bachelor of Engineering

(Environmental) with honors, The University of Western Australia, Australia.

Pantokratoras, A. (1999). ``Vertical penetration of inclined heated water jets discharged

downward.'' J. Environ. Eng., 125(4), 389­393.

Pantokratoras (2003). Vertical penetration of double-diffussive water plumes

discharged vertically downwards, ASCE Journal of Hydraulic Engineering, Vol. 129, No. 7.,

pp. 541-545.

Tomczak, M. (2000). Homepage, Flinders University, Adelaide, Australia.

http://www.es.flinders.edu.au/~mattom/IntroOc/notes/lecture03.html

Frequently asked questions

What is the distance of the jet impact point from the origin (Xi in meters)?

It is a design parameter for an underwater diffuser used to decide whether and where sea floor protection against erosion is to be provided.

What is the formula for the densimetric Froude number?

F a = density of ambient water (kg/m3), 0 = density of jet at its source (kg/m3), u0 = velocity of issuing jet=Q/( ×d02/4) (m/s), g =gravity constant (m/s2) and Q = jet volumetric flow rate (m3/s).

How are the maximum rise level (Z), centerline maximum coordinates (X and Y) related to the Froude number?

There is a simple proportionality between the Maximum Rise Level (Z) and Froude number (Z/d0=kz × F). X and Y are also proportional to the Froude number (X/d0=kx × F, Y /d0 =ky × F).

What is the impact of viscosity on dense jets?

There is no significant effect of viscosity on the behavior of dense jets, consistent with results independent of Reynolds number (Re = (uo × d0) ÷ ).

What is the research about?

This research is on negatively buoyant jets, where the jet fluid has higher density than the ambient fluid and sinks to the floor, opposed to horizontal or vertical positively buoyant jets.

What is a recommendation of the research?

A larger tank would be used for less constraint on the range of parameters such as the nozzle diameter, inclination angle and flow rate.

What is a nonbuoyant jet or pure jet?

A discharge with no buoyancy.

What is a pure plume?

A release of buoyancy only (no initial momentum).

What is a buoyant jet or forced plume?

A discharge with both momentum and buoyancy.

What is the main goal of the Double-Diffusive Water Plumes research?

To develop a suitable Froude number equation for double diffusive plumes and to improve the Turner’s Equation.

What is Turner's Equation?

An equation for vertical penetration (Ym =c×D×F0).

What are the 8 equations written for an inclined jet based on?

Pantokratoras 2003.

How is salinity defined in the equations?

Practical salinity (at atmospheric pressure) is a complex polynomial function of K.

What is the equation of state for water used in the research?

= 1+3.3 × 10-8T 3 ­ 7.4 × 10-6 T 2 + 5.1 × 10-5T ­ 9.5 × 10-5

What is the entrainment coefficient equation?

e = 0.0535-(0.0535-0.0833)(R/Rp)2

What are the eight unknowns in this study?

(1) centerline velocity u(s); (2) centerline temperature T(s) (°C); (3) centerline salinity S(s); (4) 1/e jet half-width of velocity profile b; (5) angle of inclination ; (6) and (7) the x and y coordinates of the jet axis; and (8) centerline density (s).

What's the constant value to determine the flow establishment zone and source locations

according to experimental data by Albertson et al. (1950) (Pantokratoras 2003), Boundary conditions = 6.2D

What are the formulas for an Effective Froude number for Double-diffusive / thermosolutal plumes?

(See Pantokratoras 2003)

What are the formulas for thermal and saline expansion coefficients?

(See Pantokratoras 2003)

Is Turner's equation valid for double-diffusive plumes?

Yes, if the effective Froude number is used instead of the densimetric Froude number.

What can be implemented on multiport diffusers?

Future study can be implemented on multiport diffusers which are a linear structure consisting of many more or less closely spaced ports or nozzles which inject a series of turbulent jets at high velocity into the ambient receiving water body.

What is a momentum length scale

lM=M3/4/B1/2

What are the main conclusions of the research?

New experimental information on the behavior of dense jets issuing from nozzles variously inclined on the horizontal plane was investigated. Additionally, a new effective Froude number is introduced for double-diffusive/thermosolutal plumes. Both papers are quite successful and available for improvement for future studies.

• College: Akdeniz University (Vocational School of Technical Sciences)
• Grade: 3
• Author: Muhammed Ernur Akiner (Author)
• Publication Year: 2015
• Pages: 11
• Catalog Number: V306805
• ISBN (eBook): 9783668077102
• ISBN (Book): 9783668077119
• Language: English
• Tags: Dilution, Froude number, inclined jets, nozzle, plumes, turbulent