Numerical simulation of high-speed turbulent water jets in air

Abstract

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

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

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) studied the pressure distribution on a target plate placed at a given axial distance from the nozzle. They demonstrated that the 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 = 2 mm and nozzle exit (subscript “0”) velocity of around V ₀ = 155 m/s. They found that the centreline jet velocity remains constant and equal to V ₀ for more than 100D and then linearly decays to 0.25V ₀ at about 2500D. Surprisingly, severe air entrainment causes the water (subscript “w”) volume fraction α _w 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 α _w 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 R _i varies as

Outside of this region is the droplet flow region, of which the radial width R _o varies as

where k ₁ and C are spread coefficients related as

and k ₂ 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

where is the mass flux in the axial direction of water droplets given by

and x and r are the axial and radial coordinates of a point in the jet. Further, ρ _w is the density of water; α _w (x,r), the volume fraction; V _w (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

where is the nozzle radius and , the mass flux of water droplets at nozzle exit. Using Eqs. (4) and (6), a relation between the centreline mass flux and the nozzle exit mass flux is obtained as

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

Let

where α _w0 is the volume fraction and V _w0 the axial velocity of water droplets at the nozzle exit. α _w0 is assumed to be 100%.

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

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).

Figure 2 Computational domain, boundary conditions and meshing

FLUENT was applied as the flow solver. The Eulerian multiphase model and the standard k–ϵ 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,

The term is the mass transfer from the water phase to the air (subscript “a”) phase. In the physical problem, the surrounding air is entrained into the jet and the mass of air in the jet increases. To implement this process numerically, both and are mass source terms for the water phase, were set to zero, leaving 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

The source term due to the momentum transfer ( ) in Eq. (12) is automatically handled by FLUENT once the mass transfer is specified, namely by

The term in Eq. (12) represents the inter-phase interaction force and is the inter-phase momentum exchange coefficient. The incorporation of Eq. (13) in the continuity equation is accomplished, using user defined functions in FLUENT.

The k–ϵ mixture turbulence model was used for turbulence modelling. The transport equations are:

where ρ _m is the mixture density and , the mixture velocity. The turbulent viscosity μ _t,m and the production of turbulent kinetic energy G _k,m are calculated as

The model constants are the standard values C _1ϵ  = 1.44, C _2ϵ . =  1.92, C_μ  = 0.09, σ _k  = 1.0, σ_ϵ  = 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 10⁻⁶ and the time step size was 10⁻⁵ 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 published test data of Rajaratnam et al. (1994, 1998) and Leach et al. (1966). Rajaratnam et al. found that the jet centreline velocity V ₀ remains constant for more than 100D and then decays linearly to about 0.25V ₀ at about 2500D. Severe air entrainment causes the water volume fraction α _w to fall drastically from 20% at 20D to 5% at 100D. Figures 3 and 4 confirm that the simulation accurately predicts the centreline characteristics.

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

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

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

Figure 5 shows the velocity profiles for x/D = 100, 200 and 300. In comparison to Rajaratnam et al. (1994), the velocity distribution 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 applications; 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 = 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 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.

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

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

The radial distribution of the volume fraction and the water-phase velocity within x/D = 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 = 30. Figure 9 shows that the volume fraction of water drops from 0.43 at x/D = 10 to 0.21 at x/D = 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.

Figure 8 Water-phase velocity at x/D = 10, 20 and 30

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

From an application point of view, the pressure distribution on a target (subscript “T”) plate P _T placed perpendicularly to the jet flow field is of prime concern. Since the jet loses a sufficient amount of centreline pressure P _T(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.

Figure 10 Normalized pressure distribution on a target plate placed at 76D and comparison with Leach et al. (1966)

The numerical simulation matches well near the centreline but deviates slightly toward the edge. Leach et al. (1966) used a third-order polynomial curve fit for their test data to represent 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.

Figure 11 Experimental normalized target pressure along radial direction and comparison with Leach et al. (1966) (from Guha 2008)

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

5 Conclusions

Numerical simulations were performed to capture the entrainment 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	=	diameter of nozzle
F	=	momentum source term
G	=	production of turbulent kinetic energy
k 1, C	=	spread coefficients
	=	axial mass flux of water droplets
	=	mass transfer
P	=	pressure
r	=	radial distance
R	=	radial width of jet droplet zone
S	=	mass source term
x	=	axial distance

Greek symbols

ϵ	=	turbulent dissipation rate
μ	=	viscosity
ρ	=	density

Subscripts

a	=	air
i	=	inner
m	=	mixture
o	=	outer
t	=	turbulent
w	=	water
0	=	nozzle outlet

Additional information

Notes on contributors

Anirban Guha

Department of Civil Engineering, University of British Columbia, Vancouver, BC, Canada V6T1Z4