International Journal of Energetica (IJECA)  

https://www.ijeca.info 

ISSN: 2543-3717 Volume 5. Issue 2. 2020                                                                                                           Page 42-46 

 

 IJECA-ISSN: 2543-3717. December 2020                                                                                                Page 42 

  

 

Numerical simulation of the flow rate effect on the evolution of a 

negatively buoyant water jet 

 
Oumaima Eleuch

1
, Noureddine Latrache

2
, Sobhi Frikha

1
, Abderrahmane Khechekhouche

3,1
, Zied Driss

*1
 

 
1
Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax, TUNISIA 

2
University of Brest, FRE CNRS 3744 IRDL, 29238 Brest, FRANCE 

3
Technology faculty, University of El-Oued, ALGERIA 

 

Email
*
: zied.driss@enis.tn 

 

Abstract – A numerical simulation study was done on a penetrating pure water jet injected into 

another surrounding salt water miscible with negative buoyancy conditions. For the 

incompressible filtered Navier-Stokes equations and the sum of the fluid model volume, we used a 

transient Computational Fluid Dynamics (CFD) solver (VOF model). A finite volume 

discretization method using Open Source code given in Open Foam 2.3.0. was used to solve these 

equations. The flow has a significant impact in the laminar system on the evolution of the jet in 

terms of subsequent permanent phase as well as transient regime. 

 

Keywords: Pure water jet, saltwater, transient penetration, stationary profile. 

 

Received: 21/11/2020 – Accepted: 25/12/2020 

 

I. Introduction 
 

The mixing of fluids at a difference in temperature, 

concentration or density has been the subject of several 

scientific studies in the world. When a fluid is pumped 

into another fluid of different density, fountains or 

negatively buoyant plumes arise where the buoyancy 

force opposes the momentum flux. Prior to collapsing 

back around itself, the injected fluid penetrates a distance 

into the environment. In a number of configurations, 

including round fountains, planar fountains, fountains 

affecting a solid surface and fountains penetrating an 

interface, there have been various studies examining 

fountain behavior. Depending on the ratio of buoyancy 

and momentum flux, and also as laminar or turbulent, 

fountain flow can be defined as weak or forced. Using 

numerical simulation over the parametric range 

0.25<Fr<10, 50<Re<150, Pr=7, 300 and 700 [1], the 

findings show that the behavior of plane laminar 

fountains with parabolic velocity inlet profile. The flow 

behavior is most strongly influenced by the number of 

Froude and, to a lesser extent, by the number of 

Reynolds, especially for poor fountains at low numbers 

of Reynolds. In order to examine their results, a group of  

 

 

 

researchers examined the upward discharge of a floating 

plume in a two-layer stratified ambient medium and used 

the phenomenon of the stratification which is based on 

the difference in the density parameter in the flow [2]. 

The turbulent fountains were the subject of most early 

studies [3-5]. In comparison, only a few studies have 

been devoted to the case of laminar jets. In these cases, 

the weak laminar plane fountains exhibit both a uniform 

and parabolic profile of the discharge velocity at the 

source. In addition, the original, temporary and final 

characteristics of the fountain heights and the times for 

the front of the fountain to hit these heights were 

calculated and scaled [6]. 

The effect of flow rate on the penetration of a laminar 

pure water jet within miscible saltwater is investigated 

numerically in this paper. The increase in the flow rate 

allows the penetration depth to increase and the steady-

state phase to be steadily reached. 

II. Numerical method 

The 3D numerical method was adopted to simulate the 

flow of the buoyant pure water jet in miscible saltwater. 



Oumaima Eleuch et al. 

IJECA-ISSN: 2543-3717. December 2020                                                                                                                Page 43 
 

The laminar flow is governed by the equations of Navier-

Stokes and the volume of the fluid model with the 

Boussinesq approximation, which were solved by the 

finite volume method treated by Open Foam [7]. The 

further dependence test of the mesh is performed to a 

grid system of 77x150x77. Figure 1 presents the 

computational domain and boundary conditions. The 

inlet boundary of the jet is defined as velocity-inlet. A 

value of p = 101325 Pa is known for the outlet pressure, 

which means that the fluid will exit the model to an 

environment of static atmospheric pressure at the 

opening of the tank, where the nozzle is. 

 
 

Figure 1. Boundary conditions 

III. Results and Discussion 

 

III.1. Volume fraction 

The jet evolution penetrates in the salt water through the 

nozzle with a diameter D = 1.37 mm and for different 

values of the flow rate equal to Q = 0.047 cm
3
.s

-1
, Q = 

0.065 cm
3
.s

-1
, Q = 0.086 cm

3
.s

-1 
and Q = 0.132 cm

3
.s

-1
. 

The results obtained are shown in the figures for different 

instances equal to t = 0.08 s in Figure 2, t = 0.25 s in 

Figure 3, t = 0.5 s in Figure 4, t = 1 s in Figure 5 and t = 

1.5 s in Figure 6.  

Based on these results, it has been observed an increase 

in the depth and the width of the jet in each instance 

when the flow rate increases. This fact is due to the 

increase of the initial velocity of the flow with a constant 

section of the nozzle that allows the increase of the 

Reynolds number corresponding to a laminar regime. In 

these conditions, it has been observed that there is more 

mixing layer in the boundary between the jet of lighter 

water and the denser water when the flow rate rises. 

Indeed, the head of the jet contains more pure water 

(α=1) for the small values of the flow rate Q. 

At t=0.08 s 

   
 (a) 

Q=0.047 cm
3
.s

-1
 

(b) 

Q=0.065 cm
3
.s

-1
 

   
 (c) 

Q=0.086 cm
3
.s

-1
 

(d) 

Q=0.132 cm
3
.s

-1
 

 

Figure 2. Distribution of the volume fraction of the jet in the plane z=0 

mm at t=0.08 s 

 

 

   
 (a) 

Q=0.047 cm
3
.s

-1
 

(b) 

Q=0.065 cm
3
.s

-1
 



Oumaima Eleuch et al. 

IJECA-ISSN: 2543-3717. December 2020                                                                                                                Page 44 
 

   
 (c) 

Q=0.086 cm
3
.s

-1
 

(d) 

Q=0.132 cm
3
.s

-1
 

 

Figure 3. Distribution of the volume fraction of the jet in the plane z=0 

mm at t=0.25 s 

  

 

   
 (a) 

Q=0.047 cm
3
.s

-1
 

(b) 

Q=0.065 cm
3
.s

-1
 

   
 (c) 

Q=0.086 cm
3
.s

-1
 

(d) 

Q=0.132 cm
3
.s

-1
 

Figure 4. Distribution of the volume fraction of the jet in the plane z=0 

mm at t=0.5 s 

   

 (a) 
Q=0.047 cm

3
.s

-1
 

 

(b) 

Q=0.065 cm
3
.s

-1
 

   
 (c) 

Q=0.086 cm
3
.s

-1
 

(d) 

Q=0.132 cm
3
.s

-1
 

 

Figure 5. Distribution of the volume fraction of the jet in the plane z=0 

mm at t=1 s 

 

III.2. Transient penetration 

The beginning of penetration of the liquid jet is made by 

the initial velocity. This velocity decreases up to zero 

because of the inverse buoyancy force and the viscous 

drag in the whole tank despite that the jet continues to 

penetrate in the outer liquid by gravity effect. In this 

transient regime, we have performed for a few values of 

the flow rate Q the temporal evolution of the depth as 

shown in Figure 6. According to these data, it has been 

reported that the depth of the penetration increases with 

the flow rate Q. Furthermore, the jet takes more time in 

the transient penetration for the high flow rate. However, 

when the flow rate decreases, the maximum depth and 

the steady-state system are easily reached by the jet. In 

Table 1, the maximal penetration depth of the jet and the 

transient time needed to achieve this maximal depth are 

summarized for four flow rates in the same conditions. 



Oumaima Eleuch et al. 

IJECA-ISSN: 2543-3717. December 2020                                                                                                                Page 45 
 

 
Figure 6. Penetration of the jet head 

 

Table 1. Variation of the maximal penetration depth for four different 

flow rates 

Flow rate 

Q (cm
3
.s

-1
) 

Time 

t (s) 

Maximal penetration depth  

H (m) 

0.047 0.75 0.0064 

0.065 1 0.0096 

0.086 1.5 0.014 

0.132 2 0.02595 

 

III.3. Stationary profile of the jet 

 

The variance of the penetration depth H of the jet is 

presented in Figure 7 as a function of it’s a dimensionless 

width at four different injection flow rates. These results 

confirm that the profile of the jet in depth and width at 

the same time just after the steady-state is reached under 

the same condition. For each value of Q, the curve has a 

horizontal asymptote in H=Hs. Indeed, it has been 

observed that once the jet penetrates the tank, the depth 

increases with the width in the transient phase with the 

presence of miscibility between the outer and the inner 

liquids. When the jet reaches the steady-state regime and 

the depth rest constant with H=Hs in few seconds before 

rising back, the head of the jet continues to expel 

radially.  

To compare between the considered flow rates, it has 

been observed that for the big value of flow rate Q=0.132 

cm
3
.s

-1
, the jet is wider and spreads easily despite it needs 

more time to reaches the steady-state regime. Thus, the 

decreasing of the flow rate allows decreasing of the 

width of the jet with low penetration depth and fast 

steady-state phase. 

 
 

Figure 7. Profile of the jet for different flow rates 

IV. Conclusion 

In this paper, negatively buoyant jet flow, induced by 

vertical injection of pure water downward into saltwater, 

has been affected by the flow rate of the jet. The 

numerical results show that the penetration depth of the 

jet inside the denser water increases with the increase of 

the flow rate. Whereas, the jet reaches slowly the 

maximal depth and the stationary regime. 

 

Nomenclature 
 

D 

Fr 

H 

P  

Pr  

Q 

Re  

t 

V 

α 

: Diameter (mm) 

: Froude number 

: Penetration depth  (m) 

: Pressure (Pa) 

: Prandtl number 

: Flow rate (cm
3
.s

-1
)  

: Reynolds number 

: Time (s) 

: Velocity (m/s) 

: Jet contains, α=1 for pure water 

 

References 

[1] N. Srinarayana,  S.W. Armfield, L. Wenxian,  “Behavior 

of laminar plane fountains with a parabolic inlet velocity 

profile in a homogeneous fluid”, International Journal of 

Thermal Sciences, Vol 67, 2013, pp. 87-95.  

[2] G. Noutsopoulos, K. Nanou,  “The round jet in a two-layer 

stratified ambient”, Proc. Intl Symp. on Buoyant Flows: 

Athens-Greece 1–5, 1986, pp. 165–183.  

[3] T. Mizushina, F. Ogino, H. Takeuchi, H. Ikawa, “An 

experimental study of vertical turbulent jet with negative 

buoyancy”, Warme and Stoffubertragung (Thermo and 



Oumaima Eleuch et al. 

IJECA-ISSN: 2543-3717. December 2020                                                                                                                Page 46 
 

Fluid Dynamics), Vol 16, 1982, pp. 15–21.  

[4] I. H. Campbell,  J. S. Turner, “Fountains in magma 

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[6] W. Lin, S. W. Armfield, “Direct simulation of weak 

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