Quiescent Water Research Articles

Electro-hydro-elastic modeling of Structure-Borne Traveling Waves and their application to aquatic swimming motions

Structure-Borne Traveling Waves (SBTWs) can replicate the combined undulatory and oscillatory motions utilized by body and/or caudal fin (BCF) aquatic swimmers. However, when SBTWs are generated in a dense fluid media, such as water, there is poor understanding of the Fluid–Structure Interactions (FSI). There is also limited understanding of how to achieve the full undulatory motions. This study demonstrates the effect of FSI on SBTWs and investigates the relationship between SBTW quality (undulatory vs. oscillatory motion) and forcing input. SBTWs function by leveraging a surface’s structural properties (mode shapes and natural frequencies) to deform a surface into a dynamic waveform. SBTWs are a combination of traveling and standing waves, and can range in quality from pure traveling wave (undulatory) to pure standing wave (oscillatory). SBTWs require a small number of actuators, can be generated on any thin-walled surface (flat or curved), and have customizable parameters (direction, frequency, wavelength). However, SBTWs are sensitive to FSI due to their dependence on the structural properties. In this study, an electro-hydro-elastic (EHE) model is developed to capture the effect of FSI on SBTWs and investigate how the forcing input affects the resultant waveform (undulatory vs. oscillatory). The EHE model consists of a clamped-free beam with discrete, piezoelectric actuators in a quiescent fluid. Slender beam theory and the hydrodynamic function are used to capture the added fluid mass and viscous damping, assuming small displacement amplitudes. The model accurately replicates experimental SBTWs in quiescent water across a range of parameters and is able to predict the SBTW quality. It is shown that the quality is maximized (i.e. pure traveling wave/undulatory motion) when the actuators provide equal forcing to the structure. This can be achieved through judicious actuator placement and control of the relative voltage inputs. This study demonstrates that SBTWs can be generated in quiescent water and they can replicate the combined undulatory and oscillatory motions seen in BCF swimming. This represents a critical step towards implementing SBTWs for underwater propulsion, and future studies should evaluate the propulsive capabilities (i.e. thrust).

Experiments and modeling of bubbles colliding head‐on in water

AbstractExperiments and modeling of head‐on collision of a pair of bubbles driven by buoyancy in quiescent water are presented. The experiments showed that two bubbles after contacting for a few milliseconds may coalesce or bounce back depending on the collision velocity. As theoretical comparisons, the influence of velocity boundary conditions on the liquid film drainage and the characteristics of two film drainage models with the assumptions of “partially” and fully mobile interfaces under variable‐velocity approach condition are studied. The “partially” mobile model is found to be easier to predict rebound case and longer drainage time, and could not obtain the drainage process where the film is thinned to the thickness below 10−7 m at both low and relatively high Re. As a whole, the fully mobile model with variable‐velocity approach shows more reasonable prediction results for the coalescence and rebound of bubbles in the ultrapure water.

Rhamonolipids produced by Pseudomonas aeruginosa promotes methane hydrates formation in fixed bed silica gel medium

Natural gas hydrates are seen as an alternative future energy source. They have also been valued for their carbon dioxide capturing capability, gas separation, desalination, natural gas storage and transportation. Developing economical and viable gas hydrate based technology is one of the most promising research areas of present decade. Successful commercialization of gas hydrate based technology is often curtailed due to slow formation rate. The present study evaluates biosurfactant as a kinetic promoter of methane hydrates formation in a fixed bed C type silica gel medium. Biosurfactant was produced by growing Pseudomonas aeruginosa strain A11 in glycerol supplemented mineral salt medium. Biosurfactant characterization with FTIR, NMR and MALDI-TOF spectroscopy reveled it to be a glycolipids type biosurfactant namely rhamnolipids. Saturating C type silica gel with of 100 ppm rhamnolipids solution enhanced the rate of methane hydrates formation by reducing the induction time. Mole of methane consumed and percentage of water to hydrate conversion was observed to be more in 1000 ppm rhamnolipids saturated C type silica gel as compared to quiescent water system and water saturated silica gel system. Overall results suggest that rhamonolipids produced by strain A11 in combination with silica gel can be utilized as environmentally safe kinetic promoter for methane hydrate formation.

Wake of transversely rotating and translating sphere in quiescent water at low Reynolds number

The unsteady wake of a rotating and translating sphere is experimentally investigated at Reynolds numbers (Re) ranging from 115 to 550 for two rotational speeds. Non-dimensional rotational speed, $$\Omega^{ * }$$ , is defined as the ratio of the maximum azimuthal speed of the sphere and the speed of translation. Motion of the sphere is generated using a specially designed mechanism in a channel, and its wake has been visualized using fluorescent dye. Unlike the steady flow past a stationary sphere where the limiting Reynolds number is 270, the flow becomes unsteady when sufficient rotation rate is introduced at much lower Reynolds number ( $${\text{Re}} = 115,\Omega^{ * } = 0.375$$ ). The transition from one-sided to double-sided hairpin vortex occurs at one critical Re for a given rotational speed, and it switches back to one-sided hairpin vortex shedding at another critical Re. Particle image velocimetry results demonstrate the vorticity distribution in the wake and the variation of the wake width with axial distance. At very low Re, velocity profiles in the wake are nearly self-similar. The strength of the vortex rings weakens in the far wake. The decay of centerline velocity in axial direction confirms the dissipation of the vorticity in the far wake. In this unsteady flow, the time-averaged value of the separation angle is higher than that of the steady flow on the advancing side due to enhanced adverse pressure gradient. The Strouhal number calculated from the dye visualization in this range of $$\Omega^{ * }$$ and Re varies from 0.175 to 0.25, a clear increasing trend both with increasing Re and $$\Omega^{ * }$$ . The coefficient of drag evaluated at the center plane remains constant at 0.94 for the present investigation up to $${\text{Re}} = 550$$ . The wavelength of the unstable hairpin vortices is found to scale inversely with the translational velocity of the sphere.

Temperature effects on transient behaviour of a free-falling liquid gallium drop

The present investigation presents the results of an experimental study of a free-falling liquid gallium drop in a quiescent water. Experiments were performed with different drops size in the range of 2.67 mm to 5.56 mm. All experiments were carried out under isothermal conditions, where the liquid Gallium temperature equals the temperature of the quiescent water inside the column. The temperature of both phases was investigated in the range of 30 °C to 70 °C. The study focuses on the transient behaviour of the falling drops and their aspect ratio as well as the effect of size, viscosity ratio and temperature. The captured high speed images are studied to quantify the evolution of the aspect ratio of the falling Gallium drops in time. Transient oscillations of the aspect ratio are found to exhibit a decaying trend in time. Amplitude and frequency of these oscillation were analysed. The frequency of aspect ratio oscillations were found to decrease with Eötvös number following the fD-3/2 law according to Rayleigh theory The free fall velocity of the Gallium drop is found to be highly correlated to the aspect ratio oscillations. Furthermore, whilst the oscillations amplitude was found to be sensitive to viscosity ratio, the frequency of oscillations was not affected in any noticeable way. The trends reported are limited to the range of experimental conditions investigated in this study.

An experimental study on the heat transfer by a single bubble wake rising near a vertical heated wall

We experimentally investigate the effect of a single bubble wake on the convective heat transfer (cooling) on a vertical heated wall in a quiescent water (room temperature) tank. The wall temperature is measured with an infrared camera, and gas-liquid velocity fields are also measured simultaneously using a two-phase particle image velocimetry. We consider two conditions of bubble that rises along two-dimensional zigzagging and straight paths, while varying the bubble-wall distance. When it is sufficiently close to the wall for the zigzagging bubble to impact on the wall, the local heat transfer increases up to eight times compared to that by natural convection. As the bubble-wall distance increases, the amount of heat transfer enhancement decreases, while there is a time delay between the instants of maximum Nusselt number and closest bubble-wall distance. This indicates that the bubble-induced liquid flow is the main source of heat transfer, rather than the physical collision to the wall, confirmed by the strongest correlation between the wall-normal liquid velocity and Nusselt number variation. Although the instantaneous maximum rise is smaller, the total augmentation of heat transfer from the wall is maximized at an intermediate distance (about 2.5 times of bubble size), which is determined by the effective time span and area under the effect of bubble wake. When the bubble rises in a straight path, however, the bubble is not effective for heat transfer. Even for the smallest bubble-wall distance, at which the bubble is attracted (and collides) toward the wall, the variation of Nusselt number is quite below 10%. We find that the linearly rising bubble wake exists inside the thermal-boundary layer, while the zigzagging bubble wake significantly disturbs it. Therefore, the main mechanism of the surface cooling is the disruption of a thermal-boundary layer with the entrained bulk liquid by the bubble wake.

Numerical study on evolution of weakly nonlinear waves into an acoustic soliton in bubbly water

Finite but small amplitude (i.e., weakly nonlinear) propagation of pressure (or acoustic) waves in an initially quiescent water uniformly containing many spherical microbubbles is numerically studied. Especially, an effective equation for the case of low frequency long wave is the Korteweg–de Vries–Burgers (KdVB) equation, which consists of the nonlinear, dissipation, and dispersion terms. Two major solutions of KdVB equation are a shock wave (i.e., dissipative case) and a soliton (i.e., dispersive case). The purpose of this study is the elucidation of spatio-temporal evolution of these three terms (not coefficients) and that of resultant waveforms. The results are summarized as follows: (a) a solitary pulse waves were formed by the nonlinear and dispersion terms and well-known nonlinear interaction was observed; (b) the number of formed solitons was dependent on the initial waveform such as Gaussian, shock, and rectangular waveforms; (c) the number of formed solitons was strongly dependent on the initial void fraction.

Experiments on fall of open tori

Fall of open tori horizontally dropped in quiescent water is experimentally investigated. The fall is controlled by the Galileo number, the torus aspect ratio and the solid to fluid density ratio. The dependence of the descent velocity and of the drag coefficient on the control parameters is first analyzed. As the control parameters are varied, vertical translation, zigzagging and spiraling descent are observed. These different descent modes are characterized and a phase diagram of the falling tori are presented as function of the three control parameters.

Angular momentum tearing mechanism investigation through intermolecular at the bubble interface

Two-phase flow with gas-liquid component is commonly applied in industries, specifically in the refinery process of liquid products. Oil products with bubbles contents are undesirable in a production process. This paper describes an investigation of a process mechanism regarding the bubble breakup of the two-phase injection into quiescent water. The analytical model was developed based on the force mechanism of water flow at the bubble interface. The inertia force of water flow continually pushes the bubble while the drag force resists it. The bubble gets shapes change that affects the hydrodynamic flow around the bubble. Vortices with high energy density impact and make the stress interface over its strength so that the interface gets tear. The experiment was carried out by observing in the middle part of the injected flow. It was found that the forming process of bubble breakup can be explained as the following steps: 1) sweep model is a bubble pushed by the inertial force of water flow. The viscous force of water shears the surface of the bubble. The effect of both forces, the bubble changes its shape. Then trailing vortex starts to appear in near bubble tail. The second flow of water is in around of the bubble to strengthen the vortex energy density that causes fragments to detach from the parent bubble; 2) stretching model, the apparent bubble has high momentum force infiltrated in stagnant water depth and bubble ends are stretched out by the inertial force of the bubble and viscous force of water. The bubble surface has experienced stretching and tearing become splitting away. Based on the finding, the breakup process is highly dependent on the momentum of water flow, which triggers the secondary flow as the initial process of vortex flow, and it causes the tear of the bubble surface due to angular momentum

Hydrodynamics of an asymmetric heave plate for a point absorber wave energy converter

In oscillatory flow, drag and inertial forces experienced by flat plates and cylinders depend on the Keulegan–Carpenter number. Here we demonstrate that this extends to an asymmetric hexagonal conic heave plate suitable for a point absorber wave energy converter. The forces on three geometric scales of this plate were measured throughout forced oscillation experiments in quiescent water. From these measurements, phase-invariant and phase-dependent coefficients of drag and added mass are calculated using the Morison equation. For low amplitude oscillations, the total force experienced by the plate is well-described by phase-invariant coefficients that scale with the Keulegan–Carpenter number. However, for larger oscillations, maximum forces are better described by phase-dependent coefficients. Flow visualization is used to interpret the phase variations.

Velocity Field around a Rigid Flapping Wing with a Winglet in Quiescent Water

This study investigated the effect of a winglet on the velocity field around a rigid flapping wing. Two-dimensional particle image velocimetry was used to capture the velocity field of asymmetric one-degree-of-freedom flapping motion. A comparison was conducted between wings with and without a winglet at two flapping frequencies, namely 1.5 and 2.0 Hz. The effect of the winglet on the velocity field was determined by systematically comparing the velocity fields for several wing phase angles during the downstroke and upstroke. The presence of a winglet considerably affected the flow field around the wingtip, residual flow, and added mass interaction. The added mass was lower and residual flow was weaker for the wings with a winglet than for the wings without a winglet. The added mass and velocity magnitudes of the flow field increased proportionally with the flapping frequency.

Dynamic behavior of a deformable bubble rising near a vertical wire-mesh in the quiescent water

Because wire-meshes have a great potential application for designing high-efficient heat and mass transfer systems, a deformable bubble of Re ~ 800 rising in still tap-water near a vertical mesh of different porosity is studied using a high-speed shadowgraphy apparatus including two mutually perpendicular projections. We focus on the effect of the boundary conditions with three types of meshes (Mesh-L, Mesh-M, and Mesh-S) and initial wall distance s* on the dynamic behavior of the bubbles rising by the presence of the meshes. The dynamic parameters, shape oscillation, and energy components are discussed, compared to the cases of the free-rising bubbles and that near impermeable walls. Results show that the near-mesh bubble always travels within a three-dimensional spiral trajectory at different initial wall distances, similar to the corresponding free-spiraling bubble. The mesh-bubble collisions only happen on the Mesh-L and Mesh-M, at the smallest initial distance s* = 0.11, due to their lower pressure drop before and after the mesh as a liquid through it and causing the stronger added mass effect on the bubbles. Compared to the no-slip wall, the mesh, as a slippery boundary condition of non-zero wall-normal velocity, has insignificant interference on the bubble wake, suppressing the symmetric vortex shedding and then the transition of the bubble path from spiraling into zigzagging. The interference of the mesh can lead to the transition and obvious fluctuation in the characteristic frequencies of the lateral migration, velocity, and symmetric shape-oscillation. In terms of energy, the horizontal momentum components of a near-mesh rising bubble are transferred to each other to maintain the spiral motion. The absence of bubble-mesh collisions may explain why just the energy dissipation ED generated by Mesh-S changes slightly and maintains at a lower level at s* < 9.05, while ED caused by other walls or meshes cannot achieve this low level until s* > 4.58. It is expected that our findings will be helpful to predict or modify the near-mesh gas void-fraction distribution and to improve mass, heat transfer in related engineered systems.

Path instability of a compressible air bubble rising in quiescent water with consideration of variable thermophysical properties

A numerical study of a compressible air bubble rising in water with variable thermophysical properties is conducted. The coupled level set and volume of fluid method is adopted to track the interface. Empirical correlations in terms of temperature for thermophysical properties and equation of state for air are considered in the simulations. We simulate a compressible 3D static air bubble in water for the first time. The volume change of the bubble and the pressure difference between the inside and outside of the static bubble agree well with the theoretical solutions. We study the zigzag motion of a compressible air bubble considering the temperature difference with respect to the surrounding water. The mechanism of the zigzag motion is clarified by analyzing the shape evolution and the wake structure. The periodic shedding of hairpin vortex with 2R mode is observed during the zigzag motion. There are two periods of the bubble shape oscillation within one zigzag cycle. We obtain the relation between the oscillation frequency of the aspect ratio (fE), the zigzag path (fp) and vorticity component in X and Y direction accumulated on the bubble surface (fv) as fE=2fp=2fv.

On the Unsteady Wake of a Rigid Plate Under Constant Acceleration and Deceleration

AbstractLaboratory experiments are performed to explore distinct features of the unsteady wake induced by a rigid plate undergoing a series of accelerated and decelerated phases in a quiescent water tank. Each plate motion pattern is defined by a constant acceleration from the rest followed by the same magnitude deceleration until rest; i.e., both of them lasting the same time interval. Flow statistics are obtained for various plate acceleration/deceleration motions in a plane normal to the plate span using particle image velocimetry. Results revealed that the relative near-wake bulk flow underwent acceleration during 75% of the plate cycle; the associated maximum velocity was reached approximately at the middle of the deceleration phase in all the cases. Transversal velocity profiles in the near wake were significantly modulated by the magnitude and sign of the plate acceleration. The dynamics of the large-scale coherent motions shed by the plate was also dependent on plate acceleration pattern. However, the near-wake nondimensional circulation strength reached a plateau roughly at the end of the acceleration phase and was approximately conserved despite the modulation of the deceleration phase. In the early stage of the plate motion, the Lagrangian flow acceleration was dominant; however, the local acceleration dominated at the end of the cycle.

Impact of bubble size on the integral characteristics of bubble plumes in quiescent and unstratified water

We present an experimental study to investigate the impact of bubble size on behavior of bubble plumes from the integral point of view. Two distinct types of bubble plume were generated under the same volumetric gas flow rate: many small bubbles using an air-stone diffuser and a few large bubbles using a single-orifice diffuser. Using Particle Image Velocimetry, we measured the velocity field in the plumes, and derived the integral characteristics for comparison, including the evolution of plume width, centerline velocity, volume flux, momentum flux, momentum amplification factor, and the entrainment coefficient. The measurement data show that the evolution of all the parameters in the plumes are influenced strongly by the different bubble sizes and population. The plumes with many small bubbles behave closer to the single-phase coherent plumes, showing a linear growth of plume width and a decreasing centerline velocity. These characteristics are modified in the plumes with fewer but larger bubbles. With the same total initial buoyancy flux from the source, fewer but larger bubbles give rise to weaker mean flow characteristics, i.e., smaller plume velocity and weaker fluxes of volume and momentum. This modification is attributed to the different mechanisms in fundamental transport of energy and momentum between mean and fluctuating components. In the single-orifice cases, the plumes are influenced by stronger but less frequent bubble wakes, compared to those more coherent bubble plumes in the air-stone cases. Higher momentum amplification factors were found in bubble plumes with fewer but larger bubbles, indicating a stronger ratio of turbulence to mean in these plumes. Despite the modification, all bubble plumes support a typical shear entrainment process within 40 cm height-of-rise. Hence, the integral model and the universal scaling using the plume length scale D and the bubble slip velocity Ws were found to describe the integral behavior of bubble plumes generally well in the range of our experimental parameters, regardless of the different bubble sizes and population.

Nonlinear vibrations of a nuclear fuel rod supported by spacer grids

The internal components of Pressurized Water Reactors (PWRs), particularly nuclear fuel assemblies, must be able to withstand Flow Induced Vibrations (FIVs) during operating conditions and during extreme accident conditions, such as earthquakes. Nuclear fuel assemblies are composed of long slender tubes, filled with uranium pellets that are bundled together by periodic support provided by spacer grids. Spacer grids are square structures used to increase thermal mixing in the core and provide support to the fuel rods and guide tubes allowing for the installation and removal of nuclear fuel rods. Nevertheless, spacer grids constitute a nonlinear flexible boundary condition experiencing friction forces and impacts complicating the dynamics of the fuel rod-spacer grid system. In order to improve safety margins in the design of nuclear fuel assemblies, it is of great interest to understand the nonlinear behavior at the interface of the spacer grids with the fuel rods, investigating the complexity due to the nonlinear evolution of the system stiffness and damping properties. In particular, the effect of the constant evolution of the vibration amplitude as a function of the change of the excitation forces on the dynamics of the fuel rod response is still undetermined. Experiments were carried out in quiescent water and in air to understand the nonlinear vibration response of a single zirconium fuel rod supported by spacer grids. The vibration response under a step-sine harmonic excitation at different force amplitude levels in the frequency neighborhood of the fundamental mode was measured. The response of the rod displayed nonlinear phenomena such as the shift of the resonant frequencies, multiple solutions with some instabilities (jumps) and hysteresis, and a weak one-to-one internal resonance. Tests were performed on an empty rod and on a rod filled with tungsten pellets representative of nuclear fuel. The pellets were let free to move and were subsequently blocked axially to reproduce the effect of the beginning-of-life constraint in operational nuclear plants. The experimental data were processed by means of a simplified identification procedure to extract the damping parameters of the vibrating system. The equivalent viscous damping is found to increase and to be a function of the level of excitation and of the peak vibration amplitude.

Experimental study on a yawed square cylinder in oscillatory flows

The vortical structures around a yawed square cylinder oscillating in quiescent water are investigated using the particle image velocimetry technique. Following a previous study on the hydrodynamics (Lou et al., 2017), the present experiments are performed at different yaw angles (α) and Keulegan-Carpenter (KC) numbers to correlate the independence principle (IP) to vortical flow structures. At KC = 6, the vortex pair shows no shedding. Similar vortex patterns at different yaw angles result in similar hydrodynamic behaviors, which validates the IP at small KC numbers. The single and double pairs of vortex shedding regimes are observed for α= 0° at KC = 11 and 19, respectively, and the vortex shedding process is determined by the movement of the cylinder as well as the interaction between vortices and shear layers. As α increases to 45°, the shear layers are stretched and show attachment to the upper and lower sides of the cylinder for most of the time within one oscillating cycle. The shedding is only observed at the end of each half cycle and the vortices are found to reattach to the cylinder body. The subsequent drag force behavior of the yawed cylinder displays significant differences from that at α= 0° and hence the IP is no longer applicable at KC = 11 and 19. When KC increases to 25, a three pairs of vortex shedding regime can be observed at both α= 0° and 45°. A similar flow feature, characterized by the shear layer attachment when the cylinder is at the neutral position and the vortex shedding at the end of each half cycle, has been found for both α= 0° and 45°. This result indicates that the IP becomes valid for the yawed square cylinder when the KC is sufficiently large until it is analogous to the steady flow.

Effects of Baffle Width on Heat Transfer to an Immersed Coil Heat Exchanger: Experimental Optimization

Abstract In this work, we investigate the effects of the width of an annular baffle region on natural convection heat transfer to an immersed, coiled heat exchanger in an otherwise quiescent sensible hot water storage tank. In the experiments, the coiled heat exchanger sits in an annular region created by the tank wall and a straight, cylindrical baffle. The width of this baffle region is 1.5, 2, 3, or 4 times the heat exchanger diameter. These experiments are compared to each other and to corresponding control experiments with no baffle. In general, all baffles create considerable benefits over their respective control experiments, consistent with past studies. The considered metrics of heat transfer rate, fraction of energy discharged from the tank, heat exchanger outlet temperature, and heat exchanger effectiveness show that heat transfer is improved slightly by narrowing the baffle region. For example, relative to their respective controls, the energy extracted from the tank after 30 min of discharge in the 1.5D, 2D, 3D, and 4D experiments is 23.3%, 20.8%, 18.1%, and 14.6% higher, respectively. This improvement in natural convection heat transfer as the baffle region narrows is attributed to the increasing thermal stratification observed with increasingly narrow baffle regions.

Bubble formation and evolution behavior from vertical wall orifice

Bubble formation is an integral part of the two-phase flow science. Through numerical simulation and experiments using different air flow rates and orifice diameters, the present study aims at investigating the behavior of bubble formation and evolution from vertical wall orifice in quiescent pure water. For the experiments, the images of the bubble formation process under different working conditions were recorded using a high-speed camera and analyzed the entire process. The bubble formation process can be divided into three stages, namely nucleation, stable growth, and necking. According to the obtained results, bubble forms only when the air-phase pressure exceeds the threshold pressure at wall orifice. Due to the influence of the threshold pressure and buoyancy, the bubble volume decreases with an increase in the wall orifice diameter for the same flow rate. Moreover, the volume of fluid method is applied to simulate bubble formation in a three-dimensional model and the “buffer volume” is considered in the simulation model. The simulation results matched well with the experimental data, which proves the existence of threshold pressure and the periodic pressure fluctuation at the wall orifice.

Derivation of an amplitude equation for pressure wave propagation in polydisperse bubbly liquids

A weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent water uniformly containing small gas bubbles is theoretically investigated. In the present study, the bubbles do not coalesce, break up, appear, and disappear. The bubbles are spherical, and these oscillations are spherically symmetric. In addition, the viscosity of gas inside the bubbles and the thermal conductivities of the both phases are neglected. Although abovementioned assumptions were used in our previous studies [e.g., Kanagawa et al., J. Fluid Sci. Technol.(2010); Kanagawa, J. Acoust. Soc. Am. (2015)] and classical studies [e.g., van Wijngaarden, J. Fluid Mech. (1968)], we here incorporate polydispersity of bubbly liquids. From the method of multiple scales, an amplitude (or a nonlinear wave) equation describing a long-range propagation of waves in bubbly liquids is derived from the basic equations in a two-fluid model. By comparing the present result with the previous results assuming monodispersity, we qualitatively and quantitatively discuss an effect of polydispersity on the wave propagation process in bubbly liquids.