particle size distributions in turbulent air-water …

PARTICLE SIZE DISTRIBUTIONS IN TURBULENT AIR-WATER FLOWS

MATTHIAS KRAMER

UNSW Canberra, School of Engineering and Information Technology, Campbell, ACT, Australia

e-mail [email protected]

The University of Queensland, School of Civil Engineering, Brisbane, QLD 4072, Australia

ABSTRACT Turbulent multiphase gas-liquid flows occur in a large variety of natural environments and play an important role in many industrial processes. A common characteristic of these flows is the presence of a dispersed phase, for example air bubbles or water droplets, that are embedded in a continuous phase. In highly-aerated flows, bubbles and droplets influence the energy dissipation and make a substantial contribution in terms of air-water mass transfer. The chord-sizes and the size distribution of dispersed liquid particles are traditionally measured with phase-detection intrusive probes. Despite the long-standing use of this measurement technique, the interpretation of measurement results in turbulent flows is still challenging. The present study investigated the effects of turbulent motion on measured chord-lengths by means of stochastic modelling. A virtual dual-tip phase-detection probe was placed within a pattern of synthetic particles, impacting the tips of the probe. A newly developed adaptive window cross-correlation (AWCC) technique was shown to outperform conventional approaches, demonstrating that instantaneous velocities must be considered when calculating chord length distributions to avoid erroneous results. The approach was further applied to real two-phase flow signals recorded in a laboratory stepped spillway, providing an unbiased assessment of chord-length distributions in turbulent air-water flows.

Keywords: Intrusive phase-detection probes, chord-length distribution, turbulence, signal processing, adaptive window cross-correlation technique (AWCC)

1 INTRODUCTION

The sizes and the size distribution of dispersed liquid particles (bubbles, droplets) play an important role in

gas-liquid flows that occur in natural or human-made environments such as hydraulic structures, chemical

reactors or river cascades. To understand the dynamics of these flows, it is important to accurately measure

gas-liquid flow properties. Since the early works of Neal and Bankoff (1964); Jones and Delhaye (1976) and

Herringe and Davis (1976), intrusive phase-detection probes have been widely used to characterize bubbly

flows with low void fractions (Kataoka et al. 1986; Revankar and Ishii 1993) and highly aerated flows (Cartellier

and Achard 1991; Chanson and Toombes 2002; Felder and Chanson 2015). Phase-detection probes with

varying numbers of tips (single-tip, dual-tip or four-tip) are designed to pierce bubbles or droplets and to

instantaneously infer the phase that surrounds the tips. Commonly used instrumentation comprises conductivity

and fiber-optical probes, which work upon the principles of changing conductivity or light reflection respectively

(Valero and Bung 2016, Felder and Pfister 2017). Signal processing allows to evaluate basic two-phase flow

parameters, including mean (time-averaged) void fraction (𝐶), mean bubble count rate (𝐹), chord-lengths (𝑙𝑐ℎ),

mean interfacial area (𝐴) and mean interfacial velocities (𝑈). Event-based techniques are used for the

computation of interfacial velocities in flows with low void fractions (𝐶 < 0.1) (Revankar and Ishii 1993), whereas

cross-correlation techniques are indispensable for highly aerated flows (0.1 < 𝐶 < 0.9). Recently, the use of

stochastic signals helped to gain new insights into measurement accuracy in highly-aerated flows as

demonstrated in Bung and Valero (2016). Kramer et al. (2019a) developed an adaptive window cross-correlation

(AWCC) technique that enables the estimation of pseudo-instantaneous velocities (𝑢), one-dimensional energy

spectra and turbulence intensities 𝑇𝑢 = 𝑢′𝑟𝑚𝑠/𝑈 [1]

in highly aerated flows, where 𝑢′𝑟𝑚𝑠 is the root-mean-square of velocity fluctuations 𝑢′ = 𝑢 − 𝑈. Consequently, the focus of this study is to improve the computation of chord-lengths in turbulent flow regions by taking the instantaneous velocity estimation of the AWCC technique into account.

1.1 Particle shapes, chord-times and chord-lengths in highly-aerated flows A key feature of natural and human-made air-water flows is the existence of a wide range of void fractions,

typically ranging from nearly zero next to a solid boundary to almost unity closest to the free-surface, where the latter is highly dynamic (Bung 2013; Kramer et al. 2019b) and associated with the growth of free-surface disturbances (Valero and Bung 2018; Valero 2018). The bubbly flow is commonly defined for 𝐶 < 0.3 and air bubbles are dispersed in a continuous water phase. High-speed videos suggest a homogeneous mixture of individual air bubbles of irregular shape, ranging from spherical to ellipsoidal (Halbronn et al. 1953; Straub and Lamb 1953). The intermediate region with void fractions between 0.3 < 𝐶 < 0.7 is characterized by an air–water

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

doi:10.3850/38WC092019-0680

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mailto:[email protected]


mixture with a balanced ratio between air and water entities, by collisions, deformations, coalescence and reformations of ’bubbles’ and ’droplets’ (Felder and Chanson 2016). At relatively high void fractions, ejected droplets are dispersed in air, defining the so-called upper region (𝐶 > 0.7). In contrast to bubbles and droplets in free rise or fall (Clift et al. 1978), only little information on the shape of dispersed liquid particles in flow regions with higher velocities and void fractions exceeding 𝐶 > 0.3 is available. Intrusive phase-detection probes intersect dispersed particles at chords that most often do not correspond to their centerline size. More general, any one-dimensional air- or water-entity bounded on each side is referred to as an ‘air-bubble’ or respectively ‘water-droplet’ with measured chord-time 𝑡𝑐ℎ (Toombes 2002). It must be noted that the intrusive measurement of chord times assumes that the piercing process of the dispersed phase is ideal and not affected by the tip size of the phase-detection probe or wetting and drying processes. Once the chord-times are obtained, the chord-lengths can readily be computed as

𝑙𝑐ℎ = 𝑡𝑐ℎ𝑢 [2]

with 𝑢 the instantaneous interfacial velocity. The first time-resolved velocity measurements in highly-aerated flows were introduced by Kramer et al. (2019a) and all previous studies assumed that the particle velocity was equal to the mean air-water interfacial velocity, replacing the velocity term in eq. [2] with the mean interfacial velocity 𝑈. Chord-length distributions in highly-aerated flows have been frequently reported and table 1 gives an overview of selected key studies addressing open channels, stepped spillways and hydraulic jumps. More research on the air-water flow structure was undertaken in bubble columns and fluidized beds, typically at lower void fractions and turbulence levels, and the most important contributions relate chord-length distributions to particle size distributions (section 1.2).

Table 1. Selected studies on flow structure in highly-aerated flows; note that Felder and Chanson (2016) presented chord-time distributions only.

Reference Application Velocity term in

eq. [2] Fitted distribution

Chanson (1997) Open channel 𝑈 (mean) lognormal Chanson and Toombes (2002) Stepped spillway 𝑈 (mean) - Toombes (2002) Single step, stepped spillway 𝑈 (mean) lognormal Wang (2014) Hydraulic jump 𝑈 (mean) - Felder and Chanson (2016) Stepped spillway - - Present study Synthetic signals, stepped spillway 𝑢 (instantaneous) Gamma, lognormal

1.2 Relation between chord-length distributions and particle size distributions The tips of intrusive phase-detection probes intersect dispersed phase particles not only at its centerline

but also at chord lengths other than the major axis. As the centerline size (diameter or major axis) is of primary interest, a geometrical relation between the chord-length 𝑙𝑐ℎ and the major axis (or the semi-major axis 𝑟) of the pierced particles was subject to previous research (i.e. Herringe and Davis 1976; Clark and Turton 1988; Liu and Clark 1995; Santana et al. 2006; Sobrino et al. 2009; Ruedisuli et al. 2012). In general, the probability density function of measured chord lengths for all sizes of particles can be described by integrating the joint distribution function of chord lengths and semi-major axes (Herringe and Davis 1976)

𝑝(𝑙𝑐ℎ) = ∫ 𝑝(𝑙𝑐ℎ , 𝑟) 𝑑𝑟∞

0 [3]

where 𝑝(𝑙𝑐ℎ) is the probability density function of the chord lengths and 𝑝(𝑙𝑐ℎ , 𝑟) is the joint probability distribution of chord lengths and semi-major axes. Based on the conditional probability theorem, the joint probability distribution 𝑝(𝑙𝑐ℎ , 𝑟) of particles with semi-axes 𝑟, intersected with chord lengths 𝑙𝑐ℎ, is given by

𝑝(𝑙𝑐ℎ , 𝑟) = 𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) 𝑝(𝑟) [4]

with 𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) the conditional probability and 𝑝(𝑟) the probability density function of semi-major axes of piercedparticles. Inserting eq. [4] into [3] yields a relation between chord lengths of pierced particles and their centerline size

𝑝(𝑙𝑐ℎ) = ∫ 𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) 𝑝(𝑟) 𝑑𝑟∞

0[5]

The conditional chord length distributions 𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) were derived by Herringe and Davis (1976) for horizontallyintersected spheres

𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) =𝑙𝑐ℎ2𝑟2

[6]

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and by Clark and Turton (1988) for various shapes of vertically intersected particles, including spheres,

ellipsoids and capped ellipsoids. As 𝑝(𝑙𝑐ℎ) and 𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) are known from measurements and geometrical relationships, eq. [5] can be solved in terms of 𝑝(𝑟) by using one of the following approaches, including numerical backward transformation (Clark and Turton 1988; Turton and Clark 1989), analytical backward transformation (Liu and Clark 1995; Clark et al. 1996), non-parametrical backward transformation (Liu et al. 1996) and maximum entropy methods (Santana et al. 2006; Sobrino et al. 2009).

All mentioned approaches have their intrinsic advantages and disadvantages and there are several underlying assumptions, as summarized in Ruedisuli et al. (2012). The shape of a probability density function can further be characterized in terms of statistical moments and it was found that the relation between mean chord length 𝐿𝑐ℎ and mean major-axis 𝐷 can be expressed as 𝐿𝑐ℎ = 2/3 𝐷 (Glicksman et al. 1987; Kalkach-Navarro 1993; Liu and Bankoff 1993; Ruedisuli et al. 2012). Overall, most of the previous research was undertaken for vertical bubbly flows in fluidized beds or bubble columns. While the developed theory (equations [3] to [6]) is applicable to highly-aerated flows, it is known that vertical bubbly flows are fundamentally differentfrom aerated open-channel flows in terms of turbulence levels and particle sizes.

2 SYNTHETIC SIGNALS Synthetic signals were generated to demonstrate discrepancies in computed chord-lengths and chord-

length distributions that result from different velocity terms in eq. [2]. A pattern of spherical, non-overlapping particles was generated and randomly distributed within a control volume that had dimensions of 40 mm height, 40 mm width and 3*104 mm length, the latter corresponding to a mean velocity of 3 m/s and a sampling duration of 10 s (figure 1, left, only a small part of the control volume is shown). A virtual dual-tip phase-detection probe was centered at the control volume’s outlet boundary. The probe had ideally thin tips that were separated by 4 mm in the streamwise direction (𝑥), having no separation in the normal (𝑦) and transverse (𝑧) direction.

Figure 1. Characterization of synthetic signals (left) generated particles and location of the virtual probe tips, indicated as red points; only a small part of the control volume is shown; flow direction in positive 𝑥-direction (middle) simulated time series (𝑈 = 3 m/s and 𝑇𝑢 = 0.2 ) and time series obtained with the AWCC technique (right) binary output signal (𝑆) of the leading and trailing tips of the virtual phase-detection probe; only 0.02 s are shown for clarity.

The distribution of particle diameters was following a Gamma distribution, being computed as

𝑓(𝑑; 𝑘, 𝜆) = 𝑑𝑘−1𝑒

−𝑑𝜆

𝜆𝑘𝛤(𝑑) for 𝑑 > 0 and 𝑘, 𝜆 > 0 [7]

where 𝑑 is the diameter, 𝛤 is the Gamma function and the parameters 𝑘 and 𝜆 are the shape and the scale parameter, respectively. The particles were transported with fluctuating velocities (figure 1, middle), generated by means of the Langevin (1908) equation, similar to Bung and Valero (2017) and Kramer et al. (2019a). The equation was originally developed to simulate Brownian movement as a stochastic process 𝑢∗ with zero mean, and can be written as (Pope 2000)

𝑢∗(𝑡 + 𝛿𝑡) = 𝑢∗(𝑡) (1 − 𝛿𝑡

𝑇𝑥) + 𝑢𝑟

∗𝑚𝑠 (

2𝛿𝑡

𝑇𝑥)12𝜉(𝑡) [8]

with 𝑇𝑥 the integral time-scale, 𝜉(𝑡) a standardized Gaussian variable and 𝛿𝑡 the time step fulfilling 𝛿𝑡 ≪ 𝑇𝑥. The modelled stochastic variable 𝑢∗ corresponds to a velocity fluctuation 𝑢′ (𝑢∗ ∼ 𝑢′) and can be added to a mean velocity 𝑈, allowing to generate velocity time series with defined integral time-scales 𝑇𝑥 and turbulence intensities 𝑇𝑢. The modelled flow of the present investigation was one-dimensional (no motion in normal or transverse direction) and all particles were transported with the same streamwise velocity, corresponding to a flow field with homogeneous, anisotropic turbulence. It was further assumed that the piercing process was ideal, and no particle-tip interactions, surface deformations or break-up processes were considered. During the simulation, the binary output of a virtual probe tip (figure 1, right) was evaluated at each sampling time-step by taking into consideration whether the probe tip was surrounded by the dispersed phase


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𝑆(𝑡) =

{

1 𝑖𝑓 (𝐶𝑃𝑥 − 𝑥𝑖(𝑡)

𝑎𝑖)

2

+ (𝐶𝑃𝑦 − 𝑦𝑖(𝑡)

𝑏𝑖)

2

+ (𝐶𝑃𝑧 − 𝑧𝑖(𝑡)

𝑐𝑖)

2

≤ 1

0 𝑖𝑓 (𝐶𝑃𝑥 − 𝑥𝑖(𝑡)

𝑎𝑖)

2

+ (𝐶𝑃𝑦 − 𝑦𝑖(𝑡)

𝑏𝑖)

2

+ (𝐶𝑃𝑧 − 𝑧𝑖(𝑡)

𝑐𝑖)

2

> 1

[9]

where 𝑆(𝑡) is the output signal of a probe tip at the time 𝑡, 𝐶𝑃𝑥,𝐶𝑃𝑦,𝐶𝑃𝑧 are the coordinates of the tip (leading or

trailing), 𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 are the center coordinates of the 𝑖𝑡ℎ particle and 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 are the corresponding semi-axes. Thesignals of the virtual probe were processed with the AWCC technique (figure 1, middle), as detailed in Kramer et al. (2019a). The size of the correlation windows was adjusted to include 5 dispersed phase particles (𝑁𝑝 = 5)

and filtering criteria were set to 𝑅𝑥𝑦,𝑚𝑎𝑥 = 0.5 and 𝑆𝑃𝑅 = 0.6, where 𝑅𝑥𝑦,𝑚𝑎𝑥 is the required maximum cross-

correlation coefficient and 𝑆𝑃𝑅 is the secondary peak ratio, defined as the ratio of the second highest peak to the highest peak of a cross-correlation function.

Table 2 gives an overview of the performed simulations, including flow characteristics, particle characteristics and measured mean chord lengths. Altogether, 5 simulations were conducted (table 2). Turbulence intensities were varied between 𝑇𝑢 = 0.1 to 0.5 and the chord lenghts were calculated from the output signal of the virtual probe by means of eq. [2], substituting the velocity term by (1) the instantaneous interfacial velocity and (2) the mean (time-averaged) interfacial velocity. Simulations results for different turbulence levels are presented in figure 2. The left column shows the diameter distribution of the randomly distributed spherical particles (simulation input). Resulting chord length distributions are shown in the middle column (instantaneous velocity case) and the right column (mean velocity assumption).

Table 2. Synthetic signals: flow description, particle characteristics, measured mean chord lengths 𝐿𝑐ℎ and deviations; 𝑈: mean velocity; 𝑇𝑢: turbulence intensity; 𝐶: void fraction; 𝐷: diameter of dispersed particles (mean value); 𝑘: shape parameter; 𝜆: scale parameter; 𝑢: instantaneous velocity.

Flow characteristics Particle characteristics 𝐿𝑐ℎ 𝐿𝑐ℎ Deviation 𝑈 𝑇𝑢 𝐶 𝐷 Shape 𝑘 𝜆 𝑢 (eq. [2]) 𝑈 (eq. [2]) 𝑒

(m/s) (-) (-) (mm) (-) (-) (-) (mm) (mm) (%)

3.0 0.1 0.8 1.5 spherical 1.5 1.0 2.36 2.37 0.4

3.0 0.2 0.8 1.5 spherical 1.5 1.0 2.35 2.42 3.0

3.0 0.3 0.8 1.5 spherical 1.5 1.0 2.41 2.59 7.1

3.0 0.4 0.8 1.5 spherical 1.5 1.0 2.42 2.84 17.9

3.0 0.5 0.8 1.5 spherical 1.5 1.0 2.35 3.07 27.2

2.1 Chord lengths calculated with instantaneous velocities It was observed that the chord length distributions had a greater variance than the particle diameter

distributions, which was because most of the small particles did not hit the probe tips during the simulation. Due

to this effect, the mean chord length was larger than the mean diameter for the Gamma distributed particle

diameters (table 2) and could be approximanted with 𝐿𝑐ℎ ≈ 1.6𝐷. This relation also holds for a void fraction of

1 − 𝐶 = 0.2, given that simulated particles may represent any dispersed phase in a two-phase gas-liquid flow.

In contrast, the relationship 𝐿𝑐ℎ = 2/3𝐷 (section 1.2) has been extensively used for estimating the interfacial

area in aerated open-channel flows, for example Toombes (2002) and Chanson (2002). Current results suggest

that previous methods must be revisited because the ratio of mean chord length and mean diameter varies with

regard to particle shapes and size distributions. The results were further independent of the turbulence intensity

and the chord length distribution could be fitted with a Gamma distribution, yielding 𝑘 = 2.2 and 𝜆 = 1.

2.2 Chord lengths calculated with mean velocities Chord lengths computed with the mean velocity (eq. [2]) are shown for different turbulence intensities in the

right column of figure 2. The mean chord lengths were dependent on the turbulence intensity, for example larger mean chord lengths were observed at higher turbulence levels. This was due to the fact that chord times, recorded at other velocities than the mean, were wrongly mapped onto the chord length histogram. To give an example, assume a flow with mean velocity of 𝑈 = 3 m/s and an uniform particle size of 𝐷 = 1 mm. The turbulence intensity is 𝑇𝑢 = 0.5, implying that 𝑢′𝑟𝑚𝑠 = 1.5 m/s. The chord times of a particle that is pierced at its centreline and at velocities of 𝑢 = 1.5 m/s and 4.5 m/s gives 𝑡𝑐ℎ = 𝑙𝑐ℎ/𝑢 = 6.67 ∗ 10

−4 s and 2.22 ∗ 10−4 s,respectively. Computing the chord size with the mean velocity would result in chord lengths of 𝑙𝑐ℎ = 2 mm and 0.67 mm (instead of the correct value 𝑙𝑐ℎ = 1 mm), showing that particles hitting the probe at low velocities results in a larger error than particles hitting the probe at high velocities, explaining the increase in mean chord length. From these results and the drawn example, it becomes clear that previous assumptions of similar particle

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and mean velocities (table 1) introduced errors into the estimated chord lengths. These errors, with respect to the mean chord length and considering the characteristics of the present simulations, were in the order of 7% for 𝑇𝑢 = 0.3 and 27% for 𝑇𝑢 = 0.5 (table 2, last column). The synthetic signals show that instantaneous velocity estimates must be used for an unbiased calculation of chord lenghts.

Figure 2. Simulated particle-size distributions and resulting chord length distributions, measured with a virtual phase detection probe for different turbulence levels; turbulence intensities (𝑇𝑢) and shape (𝑘) and scale

parameters (𝜆) of fitted Gamma distribution are indicated; the bin width of the histograms is set to 0.25 mm (left column) diameter distribution of dispersed particles (simulation input) (middle column) measured chord length distributions using the instantaneous interfacial velocity estimation (right column) measured chord length distribution using the mean interfacial velocity estimation.

3 STEPPED SPILLWAY APPLICATION Chord sizes and size distributions were evaluated for the highly turbulent air-water flow down a laboratory

stepped spillway (figure 3). The data were originally published in Kramer et. al (2019) and are re-analysed herein. The stepped chute had a slope of 𝜃 = 45° and consisted of 12 steps with a length of 𝑙 = 0.1 m and a vertical height of ℎ = 0.1 m. Further description of the facility can be found in Zhang (2017); Kramer and Chanson (2018); Kramer and Chanson (2019); Kramer et al. (2019b). The chute was operated under skimming flow conditions, corresponding to a specific discharge of 𝑞 = 0.11 m2/s, a dimensionless discharge of 𝑑𝑐/ℎ =1.1 and a Reynolds number of 𝑅𝑒 = 4𝑞/𝜈 = 4.4 ∗ 105; with the critical depth 𝑑𝑐 = (𝑞

2/𝑔)1/3, 𝑔 the gravitational

acceleration and 𝜈 the kinematic viscosity of water. A dual-tip conductivity phase-detection probe (figure 3, right; inner diameter: 0.25 mm; outer diameter: 0.8 mm; longitudinal tip separation 𝛥𝑥 = 4.7 mm) was mounted at the centreline of the channel and a full vertical profile was measured at the 8th step edge. The sampling rate and sampling duration were 20 kHz and 90 s, respectively.

In consistency with section 2, instantaneous velocity estimations were used to calculate chord lengths on the basis of eq. [2]. The instantaneous velocities were derived with the AWCC technique and similar processing parameters as those for the synthetic signals were adopted (𝑁𝑝 = 10, 𝑅𝑥𝑦,𝑚𝑎𝑥 = 0.5, 𝑆𝑃𝑅 = 0.6). Chord lengths

were only computed if the filtering criteria were fulfilled. Figure 4 shows basic parameters of the recorded vertical profile, where the void fraction exhibited a typical S-shaped profile and the bubble count rate was maximum at an elevation where the void fraction was 𝐶 ≈ 0.4 (figure 4, left). The interfacial velocity followed a power law profile and the shape of the turbulence intensity profile was like turbulence intensity distributions in monophasic open channel flows (Nezu and Nakagawa 1993, Kramer et al. 2019a), with almost constant values of 𝑇𝑢 ≈0.2 above the characteristic elevation where 𝐶 = 0.5 (𝑌50) (figure 4, middle). The mean chord lengths of air increased with increasing elevation from the invert and the mean chord lengths of water showed an opposite behavior, both curves intersecting each other at 𝑌50 (figure 4, right). In this context, it is understood that a chord length may refer to any bounded, one-dimensional air- or water-entity.


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Figure 3. Experimental setup and instrumentation (left) physical model of the stepped spillway; flow direction from top left to bottom right (middle) top view of the stepped chute including inlet section (right) simplified sketch of the deployed dual-tip phase-detection conductivity probe; air and water chord lengths are indicated; longitudinal tip separation 𝛥𝑥 = 4.7 mm.

Figure 4. Vertical profile of basic flow parameters of the highly-aerated flow down the stepped spillway, evaluated at the 8th step edge (left) mean void fraction and mean bubble count rate (middle) mean interfacial velocity and turbulence intensity (right) mean chord lengths and integral auto-correlation length scales.

Figure 5. Chord length distributions (subscript a: air; subscript w: water) of the stepped spillway flow (8th step edge) in the bubbly flow region, intermediate flow region and upper flow region; parameters of the fitted Gamma and lognormal distributions are indicated; the bin width of histograms is set to 0.25 mm (left column) bubbly flow region at 𝑦/𝑌50 = 0.5 with 𝐶 = 0.2 (middle column) intermediate flow region at 𝑦/𝑌50 = 1.0 with 𝐶 = 0.5 (right column) upper flow region at 𝑦/𝑌50 = 1.5 with 𝐶 = 0.8.

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For completeness, integral auto-correlation length scales 𝐿𝑥𝑥 (Chanson and Carosi 2007) were calculated as follows

𝐿𝑥𝑥 = 𝑈∫ 𝑅𝑥𝑥(𝜏)𝑑𝜏∞

𝜏=0[10]

where 𝑅𝑥𝑥 is the auto-correlation coefficient and 𝜏 is the time-lag of the auto-correlation function of the binarized probe output 𝑆. Integral length scales agreed favourably with 𝐿𝑥𝑥 ≈ min (𝐿𝑐ℎ,𝑎; 𝐿𝑐ℎ,𝑤), showing that 𝐿𝑥𝑥 is not a characteristic of the longitudinal size of large turbulent eddies (Chanson and Carosi 2007), but rather an expression of the mean chord length in combination with the spatial distribution of particles.

Figure 5 shows measured chord lengths of both phases for elevations of 𝑦/𝑌50 = 0.5 (left column; 𝐶 = 0.2) 𝑦/𝑌50 = 1.0 (middle column; 𝐶 = 0.5) and 𝑦/𝑌50 = 1.5 (right column; 𝐶 = 0.8). The selected elevations are representative for the bubbly flow region (𝐶 < 0.3), the intermediate region (0.3 < 𝐶 < 0.7) and the upper spray region (𝐶 > 0.7). The air-water flow structure of the bubbly flow region consisted of individual air bubbles that were advected with the continuous water phase, represented by high probabilities of small air chord-lengths between 0 and 0.5 mm (figure 5, left column). The chord-length distribution at 𝑦/𝑌50 = 1.0 was quite similar for both phases, with a slightly larger amount of small air chord-lengths between 0 and 0.25 mm (figure 5, middle column). This similarity is also confirmed in terms of mean chord lengths, which were 𝐿𝑐ℎ,𝑤 = 1.02 mm for the water phase and 𝐿𝑐ℎ,𝑎 = 0.98 mm for the air phase at 𝑦/𝑌50 = 1.0 (also shown in figure 4, right). In the upper region at 𝑦/𝑌50 = 1.5, the flow was characterized by ejected droplets, splashing and free-surface instabilities, evolving as ligaments from the bulk flow. The chord-lengths of the dispersed phase were slightly larger in the upper flow region than in the bubbly flow region (figure 5, left and right column). For example, water chord-lengths had highest probabilities between 0 and 0.75 mm (spray region), while air chord-lengths had highest probabilities between 0 and 0.5 mm (bubbly flow).

Gamma and lognormal distributions (equations [7] and [11]) were fitted to the computed chord-lengths (air and water) at each measurement point of the vertical profile and, with the lognormal distribution computed as

𝑓(𝑙𝑐ℎ; 𝜇, 𝜎) =1

𝑙𝑐ℎ 𝜎√2𝜋𝑒𝑥𝑝 (−

(ln 𝑙𝑐ℎ−𝜇)2

2𝜎2) for 𝑑 > 0 [11]

where 𝜇 and 𝜎 are the distribution parameters. The vertical evolution of the distribution parameters 𝑘, 𝜆, 𝜇 and 𝜎 showed that the air chord distributions became wider with increasing distance from the step-edge and vice versa for the water chord distributions (figure 6, left and middle column). The goodness of the fitted distributions was evaluated by means of the normalized root mean square error (NRSME), being an estimator of the overall deviations between best-fit distributions and measured values. The NRMSE value varies between −∞ to unity, where the latter represents a perfect fit. It is seen that both distributions are fitting the measured chord lengths, with the Gamma distribution being a better fit for the water chord-lengths and the lognormal distribution being slightly more suitable for the air-chord lengths (figure 6, right column). Overall, the current methodology in measuring and characterizing chord length distributions presents a systematic approach that can be applied to other turbulent air-water flows such as hydraulic jumps, plunging jets, breaking waves, etc.

Figure 6. Descriptive statistics of chord length distributions (subscript a: air; subscript w: water) in a fully aerated stepped spillway flow for a vertical profile, measured at the 8th step edge (left column) shape (𝑘) and

scale parameters (𝜆) of fitted Gamma distributions (middle column) parameters 𝜇 and 𝜎 of fitted lognormal distributions (right column) normalized root mean square errors (NRSME).


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4 CONCLUSIONS The present work investigated chord-lengths and chord-length distributions of dispersed liquid particles in

turbulent air-water flows, which are traditionally measured with intrusive phase-detection probes. Based on synthetic signals of a virtual probe, it was demonstrated that the computation of chord-lengths with mean velocities introduced errors in the order of up to 30% (for 𝑇𝑢 = 0.5 and in terms of the mean chord length). Therefore, it is necessary to adopt instantaneous velocity estimates, which can be obtained with a novel adaptive window cross-correlation technique. Assuming an underlying Gamma distribution of particle diameters, the synthetic signals showed that the mean chord length was larger than the mean diameter, contrasting previous assumptions. Although a wider range of underlying distributions needs to be tested, the present findings suggest that the computation of the interfacial area in highly-aerated flows must be revisited. Subsequently, the methodology was applied to the air-water flow down a laboratory stepped spillway. For the first time, unbiased chord-length distributions in turbulent air-water flows were measured and characterized with Gamma and lognormal distributions. Both distributions fitted the measured chord lengths, with the Gamma distribution being a better fit for the water chord-lengths and the lognormal distribution being slightly more suitable for the air-chord lengths. Altogether, the present investigation aims to improve the interpretation of phase-detection probe measurements in turbulent air-water flows by using instantaneous velocity estimates. Future research should focus on inferring underlying particle size distributions and improving the computation of the interfacial area in highly-aerated flows.

ACKNOWLEDGEMENTS The author thanks Jason Van Der Gevel and Stewart Matthews for the technical assistance. Discussions

with Hubert Chanson are acknowledged. Matthias Kramer was supported by DFG grant no. KR 4872/2-1.

NOMENCLATURE

𝐴 interfacial area (mean) (m2) 𝑎 major semi-axis of an ellipsoidal particle (mm)

𝑏 minor semi-axis of an ellipsoidal particle (mm)

𝐶 void fraction (mean) (-)

𝐶𝑃𝑥 streamwise coordinate of a probe tip (mm) 𝐶𝑃𝑦 vertical coordinate of a probe tip (mm)

𝐶𝑃𝑧 transverse coordinate of a probe tip (mm)

𝑐 minor semi-axis of ellipsoidal particle (mm)

𝐷 major axis of an ellipsoidal particle (mean) (mm)

𝑑 diameter of spherical particle or major axis of an ellipsoidal particle (mm)

𝑑𝑐 critical depth (m)

𝑒 relative deviation of mean chord lenghts (%) 𝐹 bubble/droplet count rate (mean) (1/s) 𝑔 gravitational acceleration (m/s2)

ℎ vertical step height (m)

𝑖 counter for particles (-)

𝑘 Shape parameter of the Gamma function (-)

𝐿𝑐ℎ chord length (mean) (mm)

𝐿𝑥𝑥 integral length scale (mm) 𝑙 horizontal step length (m) 𝑙𝑐ℎ chord length (mm) 𝑁𝑝 number of dispersed phase particles for AWCC (-)

𝑝(𝑙𝑐ℎ) probability density function of measured chord lengths (-)

𝑝(𝑙𝑐ℎ , 𝑟) joint probability of measured chord lengths and particles with semi-major axes 𝑟 (-) 𝑝( 𝑙𝑐ℎ ∣∣ 𝑟 ) conditional probability of chord lengths and particles with semi-major axes 𝑟 (-) 𝑝(𝑟) probability density function of semi-major axes of pierced particles (-)

𝑞 specific discharge (m2/s)

𝑟 radius of spherical particle or semi-major axis of ellipsoid (mm)

𝑅𝑒 Reynolds number (-)

𝑅𝑥𝑥 coefficient of the auto-correlation function (-) 𝑅𝑥𝑦,𝑚𝑎𝑥 maximum cross-correlation coefficient (-)

𝑆 binary output of a virtual probe tip (-)

𝑆𝑃𝑅 secondary peak ration (-)

𝑡 time (s)

𝑡𝑐ℎ chord time (s)

𝑇𝑢 streamwise turbulence intensity (-)

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𝑇𝑥 velocity integral-time scale (s) 𝑈 velocity (mean) (m/s)

𝑢 instantaneous velocity (m/s)

𝑢′ fluctuating velocity component (m/s)

𝑢∗ modelled stochastic variable (m/s)

𝑢′𝑟𝑚𝑠 root-mean-square of velocity fluctuations (m/s)

𝑥𝑖 streamwise center coordinate of the 𝑖𝑡ℎ particle (mm)

𝑌50 elevation where 𝐶 = 0.5 (mm)

𝑦𝑖 vertical center coordinate of the 𝑖𝑡ℎ particle (mm)

𝑧𝑖 transverse center coordinate of the 𝑖𝑡ℎ particle (mm)

𝛤 Gamma function 𝛥𝑥 streamwise tip separation (mm) 𝜉 standardized Gaussian variable (-)

𝜆 scale parameter of the Gamma function (-)

𝜇 parameter of lognormal distribution (-)

𝜈 kinematic viscosity of water (m2/s)

𝜎 parameter of lognormal distribution (-)

𝜏 time-lag of the auto-correlation function (s)

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