Technical note

Numerical simulation of high-speed turbulent water jets in air

ANIRBAN GUHA, Department of Mechanical, Automotive and Materials Engineering, University of Windsor,Windsor, ON, Canada N9B3P4. Present address: Department of Civil Engineering, University of British Columbia,Vancouver, BC, Canada V6T1Z4.Email: aguha@interchange.ubc.ca (author for correspondence)

RONALDM. BARRON,Department of Mechanical, Automotive and Materials Engineering, University of Windsor,

Windsor, ON, Canada N9B3P4.Email: az3@uwindsor.ca

RAM BALACHANDAR (IAHR Member), Department of Civil and Environmental Engineering, Universityof Windsor, Windsor, ON, Canada N9B3P4.Email: rambala@uwindsor.ca

ABSTRACTNumerical simulation of high-speed turbulent water jets in air and its validation with experimental data has not been reported in the literature. It istherefore aimed to simulate the physics of these high-speed water jets and compare the results with the existing experimental works. High-speedwater jets diffuse in the surrounding atmosphere by the processes of mass and momentum transfer. Air is entrained into the jet stream and theentire process contributes to jet spreading and subsequent pressure decay. Hence the physical problem is in the category of multiphase flows, forwhich mass and momentum transfer is to be determined to simulate the problem. Using the Eulerian multiphase and the k–e turbulence models,plus a novel numerical model for mass and momentum transfer, the simulation was achieved. The results reasonably predict the flow physics ofhigh-speed water jets in air.

Keywords: CFD, jet cleaning, multiphase flow, numerical modeling, turbulence, water jet

1 Introduction

High-speed turbulent water jets having velocity of 80–200 m/s

in air are extensively used in industrial cleaning operations. They

exhibit a high velocity coherent core surrounded by an annular

cloud of water droplets moving in an entrained air stream.

Leu et al. (1998) discussed the anatomy of these high-speed

jets (Fig. 1). Much like Rajaratnam et al. (1994, 1998), theydivided the jet into three distinct regions:

(1) Potential core regions close to the nozzle exit, instabilities

cause eddies resulting in a transfer of mass and momentum

between air and water with air entrainment breaking up

the continuous water into droplets. There remains a

wedge-shaped potential core surrounded by a mixing layer

in which the velocity is equal to the nozzle exit velocity.

(2) Main regionwhere air dynamics and continuous interaction of

water with surrounding air results in the break-up of the water

jet stream into droplets. There is a high degree of air entrain-

ment and the size of water droplets decreases with the increase

of radial distance from the axis. Due to momentum transfer

to the surrounding air, the mean velocity of the water jet

decreases and the jet expands. The jet region close to the

jet-axis is called the water droplet zone. Between the latter

and the surrounding air, there is a water mist zone in which

drops are very small and the velocity is almost negligible.

(3) Diffused droplet region, where extremely small droplets

of negligible velocity are produced by complete jet

disintegration.

Although the characteristics of submerged high-speed water

jets were thoroughly studied (Long et al. 1991, or Wu et al.1995), few experimental studies on high-speed water jets in air

have been reported in the literature. Leach et al. (1966) studiedthe pressure distribution on a target plate placed at a given

axial distance from the nozzle. They demonstrated that the

Journal of Hydraulic Research Vol. 48, No. 1 (2010), pp. 124–129

doi:10.1080/00221680903568667

# 2010 International Association for Hydro-Environment Engineering and Research

Revision received 27 August 2009/Open for discussion until 31 August 2010.

ISSN 0022-1686 print/ISSN 1814-2079 onlinehttp://www.informaworld.com

124

normalized pressure distribution along the centreline of a jet

depends on the nozzle geometry while it is independent in the

radial direction. The normalized pressure becomes equal to

the ambient pressure at a distance of around 1.3 times the

nozzle exit diameter D from the centreline. Outside this

region, the shear stress is too small to clean the target surface.

They also found that the normalized pressure distribution was

similar for both various inlet pressure conditions and nozzle

geometries.

Yanaida and Ohashi (1980) did similar work and developed

a mathematical expression for the centreline pressure.

Unfortunately, their curve did not provide satisfactory results

for the relevant axial distances in cleaning operations.

Rajaratnam et al. (1994, 1998) used a converging-straight

nozzle of D ¼ 2 mm and nozzle exit (subscript “0”) velocity

of around V0 ¼ 155 m/s. They found that the centreline jet

velocity remains constant and equal to V0 for more than 100Dand then linearly decays to 0.25V0 at about 2500D. Surprisingly,

severe air entrainment causes the water (subscript “w”) volume

fraction aw is the ratio of volume of a particular phase to

sum of all phases present in the mixture to fall drastically.

Measurements along the centreline indicate that aw at 20D is

20%, at 100D is 5% and at 200D is just 2%.

To the best of our knowledge, numerical simulation of this

problem has been reported in the literature in only one instance

yet the results were not validated against test results (Liu et al.2004). Also their results do not simulate the actual physics

as experimentally observed by Rajaratnam et al. (1994). Thus

the flow physics of high-speed turbulent water jets in air are

simulated. The next step will be to validate the results with the

available test data.

2 Novel mass-flux model

Due to Leu et al. (1998), the potential core and the water droplet

zones (Fig. 1) are of prime importance for industrial cleaning,

since these zones have a significant momentum to clean a

surface. Yanaida and Ohashi (1980) analysed the problem by

dividing the jet flow according to radial distance from the

centreline (Fig. 1). The inner region corresponds to a continuous

flow region, of which the radial width Ri varies as

Ri ¼ k1ffiffiffix

pþ k2 (1)

Outside of this region is the droplet flow region, of which the

radial width Ro varies as

Ro ¼ Cxþ k2 (2)

where k1 and C are spread coefficients related as

k1 ¼ 1:9C (3)

and k2 is the parameter depending on nozzle radius. Subscripts

“i” and “o” relate to inner and outer, respectively.

According to Erastov’s experiment (Abramovich 1963), the

mass flow rate of these water jets follow

_M ðx; rÞ_M ðx; 0Þ

¼ 1�r

R

� �1:5� �3

(4)

where _M ðx; rÞ is the mass flux in the axial direction of water

droplets given by

_M ðx; rÞ ¼ awðx; rÞ � rw � Vwðx; rÞ (5)

and x and r are the axial and radial coordinates of a point in the

jet. Further, rw is the density of water; aw(x,r), the volume

fraction; Vw(x,r), the axial velocity of water droplets,

respectively. According to the mass conservation principle, the

mass flow rate at any cross-section of the jet is equal to the

mass flow rate at the nozzle exit. If the droplet flow is assumed

to be a continuum, then this principle can be represented as

_M 0p R0ð Þ2¼ 2p

ðR0

_M ðx; rÞrd rð Þ (6)

where R0 is the nozzle radius and _M0, the mass flux of water dro-

plets at nozzle exit. Using Eqs. (4) and (6), a relation between the

centreline mass flux and the nozzle exit mass flux is obtained as

_M ðx; 0Þ ¼5:62 _M0R2

0

R2(7)

The mass flux of water droplets at any point in the jet can be

expressed in terms of the nozzle exit mass flux by substituting

Eq. (4) in Eq. (7), resulting in

_M ðx; rÞ ¼5:62 _M 0R2

N

R21�

r

R

� �1:5� �3

(8)

Let

_M 0 ¼ rw0 � aw0 � Vw0 (9)

Figure 1 Anatomy of high-speed water jets in air (Leu et al. 1998)

Journal of Hydraulic Research Vol. 48, No. 1 (2010) Numerical simulation of high-speed turbulent water jets in air 125

where aw0 is the volume fraction and Vw0 the axial velocity of

water droplets at the nozzle exit. aw0 is assumed to be 100%.

Substituting Eq. (9) into Eq. (8) gives

_M ðx; rÞ ¼5:62� rw � aw0 � Vw0 � R2

0

R21�

r

R

� �1:5� �3

(10)

Equation (10) is the polynomial function based on an empirical

mass-flux model. If the nozzle exit velocity is properly known,

this model can be used to estimate the flow characteristics of

high-speed water jets in air.

3 Numerical simulation

The objective is to perform numerical simulations of high-speed

turbulent water jets in air and to compare the results with

published test data of Rajaratnam et al. (1994, 1998) and

Leach et al. (1966). Equation (10) needs therefore to be

coupled with the continuity and momentum equations of

turbulent multiphase flows.

The computational domain (Fig. 2) and a structured grid

system were created in the commercial mesh generation

package GAMBIT. Since this problem involves circular jets,

only half of the domain was simulated in a two-dimensional

axis-symmetric space. The computational space was 1000 mm �

500 mm, and a tightly clustered grid was ensured in the regions

where larger flow gradients are expected. The radial extent of

the domain was large enough to ensure that the pressure outlet

boundary condition (set to atmospheric pressure) and the wall

boundary conditions can be accurately applied, i.e. without

adversely affecting the flow field. The radial width of the velocity

inlet boundary (set at 155 m/s) was 1 mm as per the test conditions

of Rajaratnam et al. (1994, 1998).FLUENT was applied as the flow solver. The Eulerian

multiphase model and the standard k–1 turbulence model with

standard wall functions were used to capture the flow physics.

Water was treated as the secondary phase. The drag coefficient

between the phases was determined by the Schiller–Naumann

equation (Schiller and Naumann 1935). The continuity and

momentum equations for the water phase in the Eulerian

model for multiphase flows are, respectively,

@ðawrwÞ

@tþ r � awrw~vw

� �

¼Xi¼w;a

ð _ma!w � _mw!aÞ þ Sw(11)

@ðawrw~vwÞ

@tþ r � awrw~vw~vw

� �¼ �awrpþ r � ��tw

þ awrw gQ

þ

Xi¼w;a

Kwað~vw � ~vaÞ þ _ma!w~va!w � _mw!a~vw!a

� þ ~Fw

(12)

The term _mw!a is the mass transfer from the water phase to the

air (subscript “a”) phase. In the physical problem, the surround-

ing air is entrained into the jet and the mass of air in the jet

increases. To implement this process numerically, both _ma!w

and Sw are mass source terms for the water phase, were set to

zero, leaving _mw!a as the only mass source term at the right

hand side of Eq. (11). Note that physically there is no mass

transfer between air and water; it is used because of the ease in

numerical implementation in FLUENT. Since the mass flux of

the water phase at all points in the domain is known from the

empirical mass flux model by Eq. (10), it was incorporated

into the continuity equation (11) as

_mw!a ¼ r � ð _M ; 0Þ (13)

The source term due to the momentum transfer ( _mw!a~vw!a) in

Eq. (12) is automatically handled by FLUENT once the mass

transfer is specified, namely by

~vw!a ¼ ~va if _mw!a . 0

~vw!a ¼ ~vw if _mw!a < 0 (14)

The term Kwað~vw � ~vaÞ in Eq. (12) represents the inter-phase

interaction force and Kwais the inter-phase momentum exchange

coefficient. The incorporationofEq. (13) in the continuity equation

is accomplished, using user defined functions in FLUENT.

The k–1 mixture turbulence model was used for turbulence

modelling. The transport equations are:

@ðrmkÞ

@tþ r � rmk~vm

� �¼ r �

t;m

skrk

�þ Gk;m � rm1 (15)

@ðrm1Þ

@tþ r � rm1~vm

� �

¼ r �t;m

s1

r1

�þ1

kðC11Gk;m � C21rm1Þ;

(16)

Figure 2 Computational domain, boundary conditions and meshing

126 A. Guha et al. Journal of Hydraulic Research Vol. 48, No. 1 (2010)

where rm is the mixture density and ~vm, the mixture velocity. The

turbulent viscosity mt,m and the production of turbulent kinetic

energy Gk,m are calculated as

mt;m ¼ rmCm

k2

1(17)

Gk;m ¼ mt;m r~vm þ r~vm� �T� �

(18)

The model constants are the standard values C11 ¼ 1.44, C21. ¼

1.92, Cm ¼ 0.09, sk ¼ 1.0, s1 ¼ 1.3. Standard wall functions

were used to model near wall flows. For brevity, the description

of standard wall functions is not discussed. Interested readers

refer to the FLUENT 6.3.26 user manual for details.

Pressure–velocity coupling was achieved using the phase-

coupled SIMPLE algorithm. All the residuals tolerances were

set to 1026 and the time step size was 1025 s. The program

was run for a time long enough to attain quasi-steady state.

The default under-relaxation parameters of FLUENT were

used in the computation. The discretization schemes used in

the simulation are listed in Table 1.

4 Results

Figures 3–5 compare the simulation results with that of the pub-

lished test data of Rajaratnam et al. (1994, 1998) and Leach et al.(1966). Rajaratnam et al. found that the jet centreline velocity V0

remains constant for more than 100D and then decays linearly to

about 0.25V0 at about 2500D. Severe air entrainment causes the

water volume fraction aw to fall drastically from 20% at 20Dto 5% at 100D. Figures 3 and 4 confirm that the simulation

accurately predicts the centreline characteristics.

Figure 5 shows the velocity profiles for x/D ¼ 100, 200 and

300. In comparison to Rajaratnam et al. (1994), the velocity distri-bution gives good results within a radial width of 5D. Outside

this region, the water mist zone is more prominent. Since the mist

zone is formed of sparse droplet flows, the continuum hypothesis

as a basic assumption of Eulerian model becomes invalid; hence,

the model is no longer suitable to capture the flow physics.

Note that the mist zone has little effect in cleaning appli-

cations; hence its modelling is not a major concern. Thus, we

can conclude that the simulation results match reasonably well

with the test data of Rajaratnam et al. (1994, 1998).Figures 6 and 7 show the velocity and volume fraction

contours of the water-phase up to x/D ¼ 10. These figures are

drawn to the same geometric scale, giving a quantitative

comparison between the two contours. The volume fraction

contour shows that the water-phase volume fraction decays

sharply with increased radial distance while the velocity

contour indicates that the velocity magnitude remains almost

constant for considerable radial distance. The velocity contour

is much wider than the volume fraction contour. This observation

is in agreement with Rajaratnam and Albers (1998) yet they did

not provide the results of volume fraction distribution in the

radial direction. Thus, it can be concluded that a considerable

Figure 5 Velocity distribution at x/D ¼ 100, 200, 300 and comparisonwith experimental results of Rajaratnam et al. (1994)

Table 1 Discretization schemes for jet flow

Variable Discretization scheme

Time First-order implicit

Momentum QUICK

Volume fraction QUICK

Turbulent kinetic energy Second-order upwind

Turbulent dissipation rate Second-order upwind

Figure 4 Numerical simulations of normalized centreline water-phasevelocity and comparison with experimental results of Rajaratnam et al.(1994)

Figure 3 Numerical simulation of decay of centreline water-phasevolume fraction and comparison with experimental results ofRajaratnam et al. (1998)

Journal of Hydraulic Research Vol. 48, No. 1 (2010) Numerical simulation of high-speed turbulent water jets in air 127

amount of air is entrained within the jet. Near the outer jet region,

the co-flowing air carries the water droplets (of negligible

volume fraction) and has considerably high velocity. Near the

centreline, the entrained air has a relatively high volume fraction

increasing radially, and moving with identical velocity as the

water phase.

The radial distribution of the volume fraction and the water-

phase velocity within x/D ¼ 30 is of major importance in

cleaning and cutting applications. Figures 8 and 9 respectively

show the water-phase velocity and volume fraction distributions

at various axial locations. From Fig. 8, it is obvious that the

potential core still exists at x/D ¼ 30. Figure 9 shows that the

volume fraction of water drops from 0.43 at x/D ¼ 10 to 0.21

at x/D ¼ 30, indicating the amount of air entrainment along

the centreline. The distribution of water-phase volume fraction

is expected to be Gaussian (Rajaratnam and Albers 1998), but

the simulation results show a distribution close to Gaussian

with a bulge at the jet–air interface. Since it is impossible to

predict the mist region with an Eulerian approach, the volume

fraction of water actually lost as mist numerically accumulates

near the jet–air interface and produces the erroneous bulging

effect. The bulging effect flattens out with increased axial

distance. The entrained air flows with the same velocity as the

water-phase, but owing to low air density in comparison to

water (�1:815); the momentum delivered to cutting or cleaning

surface is significantly reduced.

From an application point of view, the pressure distribution on

a target (subscript “T”) plate PT placed perpendicularly to the jet

flow field is of prime concern. Since the jet loses a sufficient

amount of centreline pressure PT(x,0) as it travels, the target

plate should be kept near the nozzle exit to ensure efficient

cutting or cleaning. It is essential for the simulation to predict

the pressure distribution at the target plate accurately, hence

the test conditions with a jet velocity of 350 m/s and nozzle

radius of 0.5 mm of Leach et al. (1966) were numerically

implemented. Figure 10 compares the simulation results with

the experiment.

The numerical simulation matches well near the centreline

but deviates slightly toward the edge. Leach et al. (1966) useda third-order polynomial curve fit for their test data to represent

Figure 7 Contour of water-phase velocity in jet (within x/D ¼ 10)

Figure 6 Contour of water-phase volume fraction in jet (within x/D ¼ 10)

Figure 9 Water-phase volume fraction at x/D ¼ 10, 20 and 30

Figure 8 Water-phase velocity at x/D ¼ 10, 20 and 30Figure 10 Normalized pressure distribution on a target plate placed at76D and comparison with Leach et al. (1966)

128 A. Guha et al. Journal of Hydraulic Research Vol. 48, No. 1 (2010)

the radial pressure distribution. According to Guha (2008),

the test results for different nozzle exit velocities indicate

that the Gaussian fit is more appropriate (Fig. 11). Since the

present simulation results resemble the Gaussian distribution,

the flow physics are more accurately predicted than by the

experiments.

5 Conclusions

Numerical simulations were performed to capture the entrain-

ment of surrounding air into high-speed water jets. The

simulation reasonably predicts velocity, pressure and volume

fraction distributions of high-speed water jets in air. The results

accurately describe the centreline characteristics, but under-

predict the velocity and over-predict the volume fraction

distribution near the jet edge. Since the near-edge region is

predominantly a sparse droplet flow region, the Eulerian models

fail to accurately capture the physics. The proposed simulation

methodology is helpful for predicting the flow behaviour of jets

used in industrial cleaning applications since these focus on the

near-field jet region.

Notation

D ¼ diameter of nozzle

F ¼ momentum source term

G ¼ production of turbulent kinetic energy

k1, C ¼ spread coefficients_M ðx; rÞ ¼ axial mass flux of water droplets

_m ¼ mass transfer

P ¼ pressure

r ¼ radial distance

R ¼ radial width of jet droplet zone

S ¼ mass source term

x ¼ axial distance

Greek symbols

1 ¼ turbulent dissipation rate

m ¼ viscosity

r ¼ density

Subscripts

a air

i inner

m mixture

o outer

t turbulent

w water

0 nozzle outlet

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Journal of Hydraulic Research Vol. 48, No. 1 (2010) Numerical simulation of high-speed turbulent water jets in air 129