Air entrainment and associated energy dissipation in steady and unsteady plunging jets at free surface

Applied Ocean Research 30 (2008) 37–45

Contents lists available at ScienceDirect

Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Air entrainment and associated energy dissipation in steady and unsteady plunging jets at free surface Ashabul Hoque a,∗ , Shin-ichi Aoki b a Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh b Department of Architecture and Civil Engineering, Toyohashi University of Technology, Toyohashi 441-8580, Japan

article

info

Article history: Received 27 May 2007 Received in revised form 11 March 2008 Accepted 18 March 2008 Available online 5 May 2008 Keywords: Air bubble entrainment Plunging water jet Plunging wave breaking Potential energy Thermodynamical work Energy dissipation rate

a b s t r a c t A theoretical formulation that describes energy transformation and dissipation caused by air bubble entrainment in water is proposed, in which the potential energy increase, induced by air bubble displacement and thermodynamical work done on air bubbles, are considered. Laboratory experiments were carried out with measurements of void fraction distributions under vertical steady plunging jets and plunging breakers of water waves. The energy dissipation rate has been estimated from measured data of void fractions and compared with the total energy dissipation rate. The ratios of energy dissipation rate caused by bubble entrainment to total dissipation rate have been found to be 4%–8% for vertical circular plunging jets and 36%–40% for plunging breakers. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Free-surface aerated flows are frequently encountered, for example, at waterfalls, spillways and breaking waves (Chanson [5]). In the surf zone, a plunging jet due to wave breaking is often a primary cause of air entrainment. During stormy weather, the sea surface in the surf zone becomes almost white as the result of entrained air bubbles and splashed droplets. The importance of air entrainment caused by breaking waves in wave energy dissipation in the surf zone is readily understood because most of the wave energy is dissipated in such an aerated surf zone. Several researchers (e.g. Bin [1]; Bonetto and Lahey [2]; Cumming and Chanson [10–12]; Brattberg and Chanson [3]; Chanson and Brattberg [4]; Hoque [15]) studied air entrainment characteristics at plunging jets and revealed some specific characteristics of the air–water flows induced by the jet. However, energy aspects such as energy transformation and energy dissipation mechanisms through air bubble entrainment were not quantified well (Hoque and Aoki [17]). Similarly, in coastal and ocean hydrodynamics, properties of the air–water field in the surf zone and the influence of entrained air on the wave transformation were not investigated thoroughly (Hwung et al. [19]; Cox and Shin [9];

∗ Corresponding author. Tel.: +880 721 750183. E-mail addresses: [email protected] (A. Hoque), [email protected] (S.-i. Aoki). 0141-1187/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2008.03.004

Chanson and Lee [7]; Lamarre and Melville [20]), although some researchers pointed out that the energy dissipation was related to the turbulent air–water flows (e.g. Horikawa and Kuo [18]). The situation derives from the complexity of the phenomena involved and the difficulty of measurements in the air–water mixture. In addition, since air bubbles are not properly scaled by the Froude similitude in small-scale models (Chanson et al. [6]), the effects of air bubbles may have been overlooked in small size experiments. Führböter [14] proposed a simple model of wave height decay in the surf zone assuming that entrained air bubbles first increase the potential energy of water by displacement of water mass upward and then the potential energy is dissipated during air bubble detrainment. In this development, the void fraction is explicitly incorporated, but Führböter [14] did not consider the rate of energy dissipation that is introduced in the energy balance equation in most of wave decay models proposed (e.g. Dally et al. [13]) and he neglected thermodynamical work done on air bubbles. Cox and Shin [9] showed experimental data of the void fraction and turbulent intensity in the surf zone and discussed some relation between them. More recently, Chanson et al. [8] showed a comparison of air entrainment and also bubble size among freshwater, seawater and saltwater. In seawater, the air entrainment rate in terms of void fractions volume flow rates was found significantly less than in freshwater whereas in saltwater it was intermediate between the seawater and freshwater results. The present paper extends the development of Führböter [14] by incorporating a formulation of the energy dissipation rate

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

38

and by quantifying the contribution of air bubbles to total energy dissipation rate in steady vertical circular plunging jets and unsteady plunging breakers of water waves. Although the proposed theory explains only a part of the total energy dissipating in the system, it provides some insights on the contribution of air bubbles. 2. Theory In the following, we discuss energy aspects in air bubble entrainment–detrainment process: i.e., work done on the air–water system to displace air into water and keep a steady void fraction. First of all, we consider the work to displace air into water in the form of air bubbles. The work W includes two constituents: work done on water, Wwater and on air, Wair . W = Wwater + Wair .

(1)

The work done on water Wwater may be further divided into two components: an increasing potential and kinetic energy of water which includes turbulence generated. Wair is the thermodynamical work to compress air into water. Potential energy increase caused by air entrainment into water. The increase in potential energy of water is caused by the entrainment of a volume of air and the resulting upward displacement of water. It may be estimated from the distributions of void fraction. For the simplicity, it is assumed that air bubbles are entrained in water uniformly in the horizontal plane. Hence a water column with unit area is considered as shown in Fig. 1. The water level rise 1h above the still water depth h caused by entrained air bubbles is given by Z h+1h c(z)dz (2) 1h = 0

where c(z) represents the vertical distribution of void fraction and z is the vertical elevation positive upwards with z = 0 at the bottom. 1h equals the volume of air per unit surface area in the column. The increase in potential energy of water, 1Ew , induced by the entrained air bubbles is defined as a difference between potential energies with and without air bubbles:

1Ew = E(with bubbles) − E0 (without bubbles) Z

h+1h

= 0

= −ρg

Z

ρ(1 − c(z))gzdz − h+1h

Z 0

h

1h



1h 2   Z h+1h Z h+1h 1h = −ρg c(z)zdz + ρg h + c(z)dz 2 0 0   Z h+1h 1h − z dz . = ρg c(z) h + 2 0 0

We define a characteristic length la as R h+1h c(z)(h + 12h − z)dz 0 la = R h+1h c(z)dz 0 R h+1h c(z)(h + 12h − z)dz 0

=

1h

.

(3)

4γ

(7)

D(z)

h+1h

Z z

ρg(1 − c(z))dz.

(8)

Let us consider an air bubble of volume va at atmospheric pressure pa . The volume v of the same bubble at an elevation z becomes (assuming an isothermal compression): pa va = pv.

(9)

The work Wb required to compress the air bubble from va to v is Z v Z v pa v a va Wb = − pdv = − dv = pa va ln va

(4)

(5)

This equation can be extended to the case of an uneven surface rise where the air bubble distribution is not horizontally uniform, if we use the air volume in water, Va , instead of 1h.

1Ea = ρgla Va .

p(z) = pa + pw (z) +

pw (z) =

This length corresponds to the distance between the barycenter of total entrained air and z = h + 12h . Thus, the increase in potential energy simply becomes

1Ew = ρgla 1h.

Note that the unit of 1Ew in the above equation is different from Eq. (5), since Eq. (5) represents an increase in potential energy per unit area. Work done on air bubbles. We here estimate the work done on air bubbles displaced into water. Assuming spherical bubbles, the static pressure inside an air bubble of diameter D at an elevation z is

where pa , γ and pw represent the atmospheric pressure, surface tension and relative pressure of ambient water, respectively. The pressure of ambient water is defined as

ρgzdz

 c(z)zdz + ρg h +

Fig. 1. Sketch of (a) water level rises by entrained air and (b) schematic diagram and definitions of variables.

(6)

va

= p(z)v(z) ln

p(z) pa

v

v

.

(10)

The total work on all entrained air bubbles per unit area, WT , is deduced from Eq. (10), assuming v(z) = c(z)dz: Z WT =

0

h+1h

c(z)p(z) ln

p(z) pa

dz .

(11)

An approximate expression of the integrand is derived assuming that the pressure increment 1p = pw (z) + D4(γz) in Eq. (7) is

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

39

In the same way, we estimate the rate of work on air to maintain air bubbles at a depth z in water. Taking the time derivative of WT (Eq. (13)), " Z # Z h+1h ∂ h+1h 4γ ∂ WT = c(z) ρg (1 − c(z))dz + dz

∂t

∂t

0

D(z)

0

∂ ∂t 0 Z 4γ ∂ h+1h dz + c(z) ∂t 0 D(z)

= ρg

Z

h+1h

 Z c(z) h − z +

= −wr ρg 1h −

Fig. 2. Sketch of water pool.

sufficiently small compared to the atmospheric pressure pa , i.e., the air entrainment is restricted to the region near water surface. Taylor expansion of the integrand gives c(z)p(z) ln

p(z) pa

4γ



≈ c(z) pw (z) +

D(z)



.

(12)

0

= 0

D(z)

z

h+1h



c(z)

ρg(h − z +

Z 0

z

c(z)dz) +

4γ D(z)



dz.

(13)

Energy dissipation rate. Next, let us estimate the power or rate of work to maintain the entrained air bubbles in water. The estimate is based upon the idea that the necessary rate of work is the same as the energy lost per unit time as air bubbles rise and escape at the free surface. We assume that all the air bubbles rise at the same rise velocity, wr . The time variations of void fraction c(z, t) are expressed as c(z, t) = c(z − wr t)

(14)

and

∂c ∂c = −wr . ∂t ∂z

(15)

The time derivative of the increase in potential energy of water 1Ew (Eq. (3)) provides the decreasing rate of the potential energy as the result of air bubbles rise and escape:

∂ ∂E 1Ew = = ∂t ∂t

Z

h+1h

0

ρ(1 − c(z))gzdz.

(16)

Combining Leibniz’s formula and Eq. (15), and assuming c(z = 0, t) = 0, the following expression is derived after some algebra.

∂ 1Ew = −ρgwr 1h. ∂t

(17)

c(z)dz dz

!

{c(z)}2 dz − wr

c0

4γ

1 − c0 D

(19)

where c0 denotes the void fraction just below the water surface. In the derivation, we assume a small vertical variation in bubble diameter D and let D be constant. To extend Eq. (19) for the case of uneven water surface elevation, we use Va instead of 1h and introduce the aerated surface areaR A in the second in Eq. R h+term 1h h+1h c(z)dz  0 {c(z)}2 dz, (19). Furthermore, assuming 1h = 0 which is valid for a small void fraction, Eq. (19) yields   ∂ c0 4γ A WT = −wr ρgVa + . (20) ∂t 1 − c0 D

For example, if we consider c0 = 0.20 and h = 0.50 (m) then we can calculate 1h = c0 /(1 − c0 )∗ h (m3 /m2 ) and A = (h + 1h) (m2 /m). Moreover we know, ρ = 1000 (kg/m3 ), γ = 0.073 (N/m) and D = 0.003 (m). Using these values we can show that the second term in the above equation is small, so the rate of work yields simply R = −2wr ρgVa .

(22)

If the energy is supplied from water motion, such as plunging jets or plunging breakers, the same amount of energy must be continuously dissipated and the dissipation rate can be estimated using Eq. (21) or Eq. (22). In practical applications, we have to give a value to the rise velocity wr . According to visual observations, air bubble rise in plunging jets and plunging breaking waves is neither uniform nor steady. However, we should consider that wr is an averaged rate of replacement of bubbles. Thus we here use a value wr = 25 cm/s, which is the typical averaged rise velocity of bubbles with diameter between 0.2 to 20 mm (Wood [22]; Chanson [5]). In our experiments, the diameters of the air bubbles were observed to lie in this range. Furthermore, the equations cannot be applied strictly to unsteady air entrainment, such as in waves, since the theory assume semi-steady air entrainment. Nevertheless, we will use mean void fraction data averaged over one wave period in an energy balance equation in which wave energy is treated as an averaged quantity. 3. Experiments 3.1. Vertical circular plunging jet (steady)

For uneven water surface rise, Eq. (17) can be extended as

∂ 1Ew = −ρgwr Va ∂t

0

h+1h

0



The total rate of work R that is necessary to maintain a steady void fraction against the air detrainment (i.e. bubble rise) is   ∂ ∂ c0 4γ A R= 1Ew + WT = −wr 2ρgVa + . (21) ∂t ∂t 1 − c0 D

Substituting Eqs. (8) and (12) into Eq. (11), we obtain " Z # Z h+1h h+1h 4γ c(z) ρg (1 − c(z))dz + WT = dz Z

Z

z

(18)

where Va is the air volume in water. The final result is a logical outcome if we consider one air bubble with a volume of Va rising with the velocity wr .

Fig. 2 shows the experimental setup and a definition sketch of vertical plunging jets. Experiments were carried out in a 2.0 m long 0.10 m wide and 0.74 m deep glass wall pool. A straight cylindrical nozzle (PVC) was used with inside diameter of 1.25 cm and a height of 1.0 m to generate a circular vertical plunging jet. The recirculating water was supplied by pumps. The flow rate was

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

40

Table 1 Experimental investigations of vertical circular plunging jet flows Run No.

x1 (cm)

Qw (cm3 /s)

V0 (cm/s)

V1 (cm/s)

r1 (cm)

Va (cm3 )

R (J/s)

fR (%)

Cir-1 Cir-2 Cir-3 Cir-4 Cir-5

10 5 2.5 5 5

270 270 270 350 370

208.5 208.5 208.5 272.0 286.0

251.0 231.0 220.0 289.0 310.0

0.585 0.611 0.626 0.623 0.625

13.4 9.49 5.80 16.7 23.5

0.066 0.046 0.028 0.080 0.114

7.80 6.40 4.28 5.48 6.42

measured with a volume per time technique with errors being less than 2%. Fig. 3 shows a photograph of typical bubble cloud observed in the experiments. The experimental cases (Cir-1 to Cir-5) are summarized in Table 1, where x1 is the distance between the nozzle and the water surface, Qw the flow rate of a jet, Va the velocity at the nozzle, and r1 the radius of a jet at the impingement point. Visual observations showed predominantly entrained bubble sizes are between 0.5 and 5 mm. A KanomaxTM system 7931 single-tip L-shape conductivity probe (diameter of inner electrode: 0.1 mm) was used to measure the void fractions (Fig. 4). The measurement principle is based upon the difference in electrical resistivity between air and water when an air bubble excites the tip of the probe. The resistance of water is one thousand times lower than that of air. When the probe tip is in contact with water, current will flow between the tip and the supporting metal; when it is in contact with air no current will flow. A typical response of the probe is illustrated in Fig. 4. The local void fraction is defined as a ratio of time that the probe tip is in air to the total measuring time: c=

X 1ti i∈tm

(23)

tm

where i corresponds to an individual bubble, 1t is the period that probe tip is in the air bubble and tm is the total measuring time. 3.2. Two dimensional wave breaking (unsteady) Instrumentation and experimental cases. The experiments were performed in a 20 m long, 0.8 m wide and 0.6 m deep wave

Fig. 3. A photograph of vertical circular plunging jet.

flume filled with tap water to a depth of 0.4 m as shown in Fig. 5. A sloping bottom (1:9.5) was installed at 9.65 m from a wave generator. The effect of air bubble on the wave gauge was tested in separate experiments and revealed that the wave gauges recorded reasonably accurately the rise in water level in air–water mixture

Fig. 4. Sketch of (a) conductivity probe and (b) output signal from a conductivity probe.

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

41

Fig. 5. Sketch of the wave flume.

Fig. 7. Sketch of (a) output signal from a probe during the passage of a bubble and (b) response of the wave gauge and probe. Fig. 6. Air entrainment by plunging breaker: (a) photograph and (b) sketch.

with an error of the same order of magnitude as the bubble foam thickness (Hoque and Aoki [16]). The L-shape conductivity probe tip was set up in the opposite direction of wave propagation and void fraction was measured with a grid spacing of 5–25 cm increment along the channel and 2 cm increment in the depth for 2-dimensional wave breaking. In the measurements at a location close to the free surface, the conductivity probe was not always immersed during the wave period. Measurements were carried out along the centerline of the wave flume and all the data were recorded with the scan rate of 1 kHz per channel for several hundreds of waves. A high-speed digital video camera was also used to visualize the breaking events. Experimental wave conditions are summarized in Table 2 for three tests in plunging breakers. To assess the breaker type, values of the surf similarity parameter I0 are indicated in the table. The visualized air entrainment at the wave breaking is shown in Fig. 6. Air bubbles were entrained at the plunging points, rather than at the breaking point. Thus in the transition zone between the breaking and plunging point, little air bubbles were observed in water. Analysis of data from conductivity probe. A schematic diagram of responses of the conductivity probe and wave gauge are illustrated

in Fig. 7. When an air bubble is pierced by the probe tip, the output yields a square pulse. Fig. 8 shows a typical time series of surface elevation and air bubbles pulsed obtained respectively from the wave gauge and the conductivity probe located at 2 cm, 0 cm and –4 cm above the still water level. In the figure, vertical positions of the conductivity probe are indicated by dashed lines and outputs from the probe are plotted as square waves taking values of 1 (in air) or 0 (in water) alternately. The conductivity probe comes out of the water when the free surface elevation drops below the probe position. For the probe position close to the free surface, the period 1t during which the probe is immersed (see Fig. 7) was relatively short compared to the wave period but many pulses or air bubbles were observed in the period. The calculation technique of void fraction is as follows. In the period 1t, the total time of air bubble encounter, 1τ , is given as X 1τ = 1τi (24) i∈1t

where i corresponds an individual air bubble detected during 1t and 1τi is the period that the air bubble takes when it passes the probe tip. The air concentration or void fraction c averaged over the period 1t is obtained from the following definition: c=

1τ . 1t

(25)

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

42

Table 2 Experimental investigations of plunging wave breaking flows Run No.

H0 (m)

PL-1 PL-2 PL-3

0.125 0.145 0.166

T (s)

H0 /L0

Hb (m)

Hb /hb

I0

R (J/s m)

fR (%)

1.80

0.024 0.028 0.032

0.180 0.198 0.207

1.01 1.05 1.07

0.677 0.627 0.586

12.40 14.00 15.50

39.70 37.24 35.60

Fig. 8. Examples of records of wave gauges and conductivity probe at different elevation: z = 2; 0; −4 cm (H0/L0 = 0.024).

Fig. 9 presents variation of void fractions as functions of number of waves in one series of experiments, where the void fractions were averaged over one wave period. Approximately 350–420 waves were extracted from a 10 to 12 min data recording for each depth. It was observed that the duration of breaking event, 1t, was approximately 1/8, 1/4 and 1/2 of the wave period (T = 1.8 s) for the Fig. 9(a), (b) and (c), respectively. Averaging c and 1t over all the individual waves gave a mean void fraction and mean breaking duration at several locations inside the surf zone. 4. Results and discussion 4.1. Vertical circular plunging jet (steady) Void fraction distributions and volume of entrained air. Fig. 10 shows typical void fraction distributions at several locations below the impingement point for one set of flow conditions. The void fraction data are plotted as functions of the radial distance normal to the jet centerline. The two data sets for x − x1 = 0.01 m and 0.015 m show mostly zero concentration implying that these data were located inside the developing flow region. The maximum void

fraction reached 33% at x − x1 = 0.01 m and the peak value of void fraction was found to decrease with increasing distance x. Below the developing flow region, the distribution of void fraction became flat with peak values at the centerline (r = 0). These tendencies were about the same for all other experimental cases. The volume of entrained air per unit depth, Va∗ , was estimated from the measured void fraction distributions c(r) as  Z 2π Z rb Va∗ = rc(r)dr dθ (26) 0

0

where rb is the distance from the jet’s centerline to the outer edge of the void fraction distribution. The value of rb was taken large enough not to influence the air volume estimated. The measurements were performed up to the deepest position where air bubbles were not found. Fig. 11 shows vertical distributions of air volume per unit depth, Va∗ , which indicates that the volume of entrained air per unit depth increased with increasing jet impact velocity. Similar results were obtained by Bonetto and Lahey [2]. In the figure, the volume of air per unit depth increased in the deeper position, where air bubbles were redistributed or trapped due to vortices (e.g. Wood [22], Chanson [5]). The total air volume Va is obtained by integrating Va∗ over the depth of bubble cloud.

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

43

Fig. 9. Variations in void fractions averaged over one wave period at different elevation: z = 2; 0; −4 cm (H0/L0 = 0.024).

Fig. 10. Local averaged void fraction distributions: (Cir-1, x1 = 10 cm, V1 = 251 cm/s).

Energy dissipation rate. From the estimated air volume, the rate of energy dissipation R was obtained from Eq. (22) corresponding to each run and listed in Table 1. As the impact velocity V1 increases, energy dissipation rate R also increases. As recognized from Eq. (22), the plot of energy dissipation rate will take the same curve as the air volume Va (Fig. 11), because energy dissipation rate is proportional to the entrained air volume assuming constant rise velocity wr . The energy transport rate of vertical circular plunging jet, Ej , is defined as Ej =

1 2

ρw Qw V12 .

(27)

Fig. 11. Vertical distribution of entrained air volume per unit depth.

In the steady state, the whole of this energy is lost within the water pool. It is defined the ratio of Ed to R as fR : fR (%) =

R Ej

∗ 100.

(28)

fR is the ratio of energy dissipation rate estimated from entrained air bubbles to the total energy dissipation rate. The values of fR for

each experimental case is shown in Table 1. The energy dissipation rate estimated from entrained air is only 4%–8% of total energy

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

44

Fig. 12. Vertical distribution of void fractions (H0 /L0 = 0.024).

Fig. 13. Void fraction distribution as a function of dimensionless breaking duration time (H0 /L0 = 0.024).

dissipation rate. Although there may be errors in estimating Va and wr , the results show that most of the energy seems to be transformed directly to kinetic energy or turbulence in vertical steady plunging jets. 4.2. Two dimensional wave breaking (unsteady) In Fig. 12, the mean void fractions during breaking event are presented at several locations, where x − xb denotes the distance measured from the breaking point (Fig. 5). Fig. 12 shows that void fraction has its maximum at x − xb = 0.60 m and in fact, this position is close to the plunging point. Moreover, the data clearly show the significant void fraction at the second plunging point (x − xb = 0.80 m). The plunging points were also confirmed by the high-speed video visualization. The data of void fraction consistently decays exponentially with the depth (Hoque and Aoki [16]). Wu [23], Stanton et al. [21] and Hwung et al. [19] found similar trends of void fraction distribution for the large-scale experiments and field measurement. Furthermore, experimental results indicate that the maximum void fraction reach 25% near the still water level. Similar results were measured by Hwung et al. [19] for plunging breakers. The relations between the void fraction c and the duration of breaking event 1t/T at different elevations are illustrated in Fig. 13 which show that void fraction decays almost linearly as the duration of breaking event increases. Volume and potential energy of entrained air. The time averaged volume of entrained air per unit length and width, Va∗ , were estimated from the void fraction distributions (Fig. 12) using the following equation: Z η+ 1t(z) Va∗ = c(z) dz (29) T

−h

where η+ is the instantaneous water surface elevation. Fig. 14 shows the horizontal variation of the air volume Va∗ . In the transition zone (Fig. 6), the void fraction is almost zero indicating no contribution to total air volume in this domain. The maximum air volume was found near the plunging point. From the end of transition zone to the point of maximum penetration, Va∗ rapidly increases and subsequently decreases. Energy dissipation rate. For the steady vertical plunging jets, the energy dissipation rate R was estimated by using Eq. (22) where the air volume Va was obtained by integrating the vertically integrated void fraction Va∗ : Z x2 R = 2ρgwr Va∗ dx (30) x1

Fig. 14. Horizontal distribution of air volumes per unit length.

where x1 and x2 represent the offshore and onshore limits of aerated zone. On the other hand, the energy flux for waves of amplitude a at the water depth h is given as 12 ρga2 (gh)1/2 . Thus the rate of energy dissipation in the aerated zone will be estimated by 1

ρg[{a2 (gh)1/2 }1 − {a2 (gh)1/2 }2 ] (31) 2 where subscripts 1 and 2 represent the position of x1 and x2 , respectively (see Fig. 6). Finally, the ratio of energy dissipation rate estimated from air bubbles to the total energy dissipation rate can be expressed as Ew =

fR (%) =

R Ew

∗ 100.

(32)

The estimated values of fR are shown in Table 2. The contribution of the air bubbles was found about 36%–40% with small difference in wave conditions. The values of fR is found to be considerably large compared with the steady plunging jets. The fact simply means that the air volume entrained by unit power is larger in plunging breakers than in plunging jets, in other words, plunging breakers displace air into water more efficiently than plunging jets. Although one of the reasons for the difference may come from the difference between steady and unsteady phenomena, the results in the present study indicate that the air bubble entrainment–detrainment process itself contributes greatly to wave energy dissipation in the surf zone.

A. Hoque, S.-i. Aoki / Applied Ocean Research 30 (2008) 37–45

5. Conclusions In this paper, a theoretical formulation that describes energy transformation and dissipation caused by air bubble entrainment in water is proposed, in which the potential energy increases by air bubble displacement and thermo-dynamical work done on air bubbles are considered. Laboratory experiments were carried out with measurements of void fraction distributions under vertical steady plunging jets and plunging breakers of water waves. The energy dissipation rate was estimated from measured void fractions and compared with the total energy dissipation rate. The ratios of energy dissipation rate caused by bubble entrainment to total dissipation rate are estimated to be 4%–8% for vertical circular plunging jets and 36%–40% for plunging breakers. The results show there is an inconsistency for plunging jets and plunging breakers, possibly because of shallow water depth at breaking waves. The high ratio in the plunging breakers indicates that the air bubble entrainment–detrainment process itself contributes greatly to wave energy dissipation in the surf zone. Acknowledgement The authors are indebted to Dr. Hubert Chanson for many valuable suggestions and encouragements on the work. References [1] Bin AK. Gas entrainment by plunging liquid jets. Chem Eng Sci 1993;48: 3585–630. [2] Bonetto F, Lahey RT. An experimental study on air carry under due to a plunging liquid jet. Intl J Multiphase flow 1993;19:281–94. [3] Brattberg T, Chanson H. Air entrainment and air bubble dispersion at twodimensional plunging water jets. Chem Eng Sci 1998;53:4113–27.

45

[4] Chanson H, Brattberg T. Experimental study of the air–water shear flow in a hydraulic jump. Intl J Multiphase Flow 2000;26:583–607. [5] Chanson H. Air bubble entrainment in free-surface turbulent shear flow. Academic Press; 1997. p. 401. [6] Chanson H, Aoki S, Hoque A. Physical modeling and similitude of air bubble entrainment at vertical circular plunging jets. Chem Eng Sci 2004;59:747–58. [7] Chanson H, Lee JF. Plunging jet characteristics of plunging breakers. Coastal Eng 1997;31:125–41. [8] Chanson H, Aoki S, Hoque A. Bubble entrainment and dispersion in plunging jet flows: Freshwater versus seawater. J Coastal Research 2006;22:664–77. [9] Cox DT, Shin S. Laboratory measurements of void fraction and turbulence in the bore region of surf zone waves. J Eng Mech 2003;1197–205. [10] Cumming PD, Chanson H. An experimental study of individual air bubble entrainment at a planar plunging jet. Trans IchemE 1999;77:159–64. [11] Cumming PD, Chanson H. Air entrainment in the developing flow region of plunging jets-part 1: Theoretical development. J Fluid Eng, Trans, ASME 1997; 119:597–602. [12] Cumming PD, Chanson H. Air entrainment in the developing flow region of plunging jets-part 2: Theoretical development. J Fluid Eng, Trans, ASME 1997; 119:603–8. [13] Dally WR, Dean RG, Dalrymple RA. Wave height variation across beaches of arbitrary profile. J Geophys Res 1985;90:11917–27. [14] Führböter A. Air entrainment and energy dissipation in breakers. Proc. of ICCE 1970;391–8. [15] Hoque A. Air bubble entrainment by breaking waves and associated energy dissipation. Ph.D. thesis. TUT, Japan; 2002. p. 151. [16] Hoque A, Aoki S. Distribution of void fraction under breaking waves in the surf zone. Ocean Eng 2005;32:1829–40. [17] Hoque A, Aoki S. Energy transformation due to air bubble entrainment by vertical plunging jet and hydraulic jump. Proc. of APACE 2001;1:136–44. [18] Horikawa K, Kuo CT. A study of wave transformation inside the surf zone. In: Proc of 10th ICCE, ASCE. 1966. p. 217–33. [19] Hwung HH, Chyan JM, Chung YC. Energy dissipation and air bubbles mixing inside surf zone. In: Proc 23rd ICCE, ASCE. 1992. p. 308–21. [20] Lamarre L, Melville WK. Air entrainment and dissipation in breaking waves. Nature 1991;351:469–72. [21] Stanton TP, Thornton EB. Profiles of void fraction and turbulent dissipation under breaking waves in the surf zone. In: Proc 27th ICCE. 2002. vol. 1, pp. 70–9. [22] Wood IR. Air entrainment in free-surface flows. Balkema; 1991. p. 142. [23] Wu J. Bubbles in the near-surface ocean: A general description. J Geophys Res 1988;93:587–90.