Determination of the entrainment coefficient of a pure plume using the salt-bath technique

Abstract

The entrainment coefficient of the pure plume, the ratio of the radial velocity of the entraining fluid at the edge of the plume to the axial velocity, is characterized by a remarkable deviation from the actual value in references. To a turbulent flow with Gaussian profiles, the entrainment coefficient is accepted in a considerably wide range from 0.07 to 0.11 for pure plumes. In this study, the entrainment coefficient of a pure plume is determined by analyzing the stratified interface formed between fresh water and a buoyant fluid in a confined space. The interface position is determined using digital image processing while conducting salt-water experiments. The displacement of the virtual origin of the real plume is added to the original heights of the confined space and stratified interface because the plume in the laboratory is from a source of finite area rather than from a point. The two virtual source position determination methods are compared; the one where the ratio of the displacement of the virtual origin to the nozzle diameter is related only to the source parameter Γ is selected for the theoretical analysis. The experimental data of the source parameter Γ and corresponding entrainment coefficients indicate that the α value of the pure plume is 0.09251 when using the interpolation method. This value is validated through displacement experiments on a pure plume created by supplying salt water through the nozzle in a laboratory. Further, the entrainment coefficient of a turbulent jet is determined based on the stratified interface behavior in a confined space; the experimental results support the speculation that the entrainment coefficient of the pure jet should be 0.057. The experimental data of the entrainment coefficient α plotted against parameter Γ in the available literature are compared with those estimated from semi-empirical equations with two asymptotic values suggested in this paper.

Graphical abstract

Article Highlights

• Entrainment coefficient of a pure plume is determined by analyzing the interface between fresh water and buoyant fluid.

• The interface position is determined using digital image processing while conducting salt-water experiments.

• Two virtual source position determination methods are compared.

• The α value of the pure plume is determined to be 0.09251 when using the interpolation method.

• Experimental results support the speculation that the entrainment coefficient of the pure jet should be 0.057

Appendix. Verification of the entrainment coefficient of the pure jet

Based on Eqs. (12) and (13), the entrainment coefficient of a forced plume at a certain value of the parameter Γ depends on two asymptotic values: pure jet α_j and pure plume α_p. Fischer and List [40] and Carazzo et al. [23] presented comprehensive reviews of the entrainment coefficient of pure jets. The entrainment coefficient is in the range 0.045 < α_j < 0.067 for pure jets with turbulent velocity described by Gaussian profiles. The measurements made by Albertson et al. [42] for round jets suggest a spread rate of 0.114 (implying α_j = 0.057), and the general coefficient value has been adopted by Baines [43], List and Imberger [30], and Lee and Chu [44]. The experimental results obtained by Papanicolaou and List [45] show that the entrainment coefficient for a plume is close to 5/3 times the value for a jet, which can be verified from the measurement data obtained by Wang and Law [25] and the theoretical analysis results obtained by van Reeuwijk and Craske [41]. Once the entrainment coefficient of a pure plume is determined to be equal to 0.09251, the coefficient of a pure jet is 0.0555, which is very close to the value of 0.057. For the present purposes, α_j = 0.057 will be used throughout the data analyses hereinafter.

As discussed in Sect. 4.4, the rate of advance of the stratified interface of the buoyant fluid in a confined space can be employed to determine the value of α for an axisymmetric plume. If the jet fluid is marked by a dye, the accurate measurement of the position of the stratified interface may also provide an effective method for determining the value of α for a pure jet.

Based on the Eulerian integral model [44], the axial velocity and jet width of an axisymmetric jet can be written in terms of the nozzle diameter D and source velocity u₀ as

$$ \frac{{w_{m} }}{{w_{0} }} = \frac{1}{2\sqrt 2 }\left( \frac{z}{D} \right)^{ - 1} , $$

(19)

$$ b = 2\alpha z. $$

(20)

Note that M = M₀, and $M_{0} = \frac{\pi }{4}w_{0}^{2} D^{2}$ at the nozzle inlet.

Eqs. (3), (19), and (20) are combined at the stratification level in the confined space, and the velocity of the level dz₀/dt = U is also used. Measuring from the time when the stratification is formed, the relationship between the position z₀ and time t is integrated as

$$ \ln \frac{H}{{z_{0} }} = k\alpha t, $$

(21)

where

$$ k = \frac{{2\sqrt 2 M_{0}^{1/2} }}{{R^{2} \pi^{1/2} }}. $$

(22)

A smaller transparent tank (cross-sectional area of 17 cm × 18 cm and a depth of 33 cm) adopted in this group of experiments was similar to the plexiglass tank used in the experiment outlined in Sect. 4.2, and an opening with a diameter of 1.8 cm was also drilled on the side wall of the model, whose center is 30.5 cm above the bottom of the model box. The brine storage tank, source supply tank, and supply pipes for the brine solution were cleaned carefully before fresh water was mixed with a certain concentration of dye in the brine tank. In this group of experiments, the density of the fresh water solution containing the dye in the storage tank was 0.9958 g/cm³, which is the same as that of the fresh water of the environment, and the supply flow rate of the fresh water containing the dye was 60 ml/min. Two sheets of Perspex with a cross-sectional area of 16 cm × 16 cm were suspended horizontally in the plexiglass tank to prevent the bottom surface from creating disturbance in the tank, as shown in Fig. 

Fig. 12

Schematic of the setup for determining the entrainment coefficient of a pure jet

12, while the round jet fluid moves to the lower boundary and the mixing fluid begins to ascend. The two sheets were all evenly perforated with several openings of diameter 0.8 cm at the periphery of the sheets. One was fixed 6 cm from the bottom of the model, with a hole of diameter 12 cm at the center, and the other at 15 cm from the bottom with a 9 cm hole at the center. Four groups of repeated experiments were performed under the same conditions.

There are two main differences in the ascending of the visible interfaces in the confined space with respect to the two sources (buoyant plume and pure jet). For a turbulent plume, the heavy fluid reaches the model bottom and spreads out to the side walls to produce a discontinuous interface. The horizontal interface is then pushed upward, and the marked layer can be easily observed. However, for a pure jet, no discrepancy in the density exists between the entrainment fluid and the environment fluid. Therefore, there is no mixing of the entrainment fluid in the horizontal level because of the same density, which leads to a visible interface of unequal height. Moreover, because of the lack of density gradient in the horizontal level, a remarkable vertical circulation is induced, and the environment fluid at the four corners on the lower part of the Plexiglas tank is not entrained by the jet fluid during the experiment.

To employ the approximate position of the first front of the buoyant fluid with time to determine the entrainment coefficient of a pure jet, we take the average ratio of the non-occupied volume by the dye solution to the whole box volume as 6.82%. For a flow from a circular orifice of diameter 5 mm, the virtual origin position of the real jet source can be deduced from Eq. (20). With α_j = 0.057, the virtual origin correction for the jet position gives a constant value of z_j = 2.19 cm in these experiments. By fitting a straight line through the corrected data ln(H + z_j)/(z₀ + z_j) vs. kt, we determined the value of the jet entrainment coefficient, as shown in Fig. 

Fig. 13

Determining the entertainment coefficient of a jet

13.

In the figure, the experiment data shows a linear relationship between ln(H/z₀) and k·t, although the logarithm of ratio of height of the confined space to the stratification level measured in the middle of the experiment is a little higher than the data from fitting line. This discrepancy may be contributed to the stratification layer level harder to be observed in the space because of the same density between the entrainment fluid and the environment fluid. The slope of the fitted line yields α_j = 0.056, which is slightly lower than the accepted value of 0.057. The source conditions used in the experiments and the corresponding entrainment coefficient of the pure jet are listed in Table

Table 6 Experimental data correction of jet

6. For an axisymmetric jet, the measurement of the interface formed by the dye solution in the confined space supports the fact that the entrainment coefficient of the pure jet should be 0.057.