Matthieu Guédra

On the Use of a Complex Frequency for the Description of Thermoacoustic Engines

In this paper, a formulation is proposed to describe the process of thermoacoustic amplification in thermoacoustic engines. This formulation is based on the introduction of a complex frequency which is calculated from the transfer matrices of the thermoacoustic core and its surrounding components. The real part of this complex frequency represents the frequency of self-sustained acoustic oscillations, while its imaginary part characterizes the amplification/attenuation of the wave due to the thermoacoustic process. This formalism can be applied to any type of thermoacoustic engine including stack-based or regenerator-based systems as well as straight, closed loop or coaxial duct geometries. It can be applied to the calculation of the threshold of thermoacoustic instability, but it is also well-suited for the description of the transient regime of wave amplitude growth and saturation due to non linear processes. All of the above mentioned aspects are described in this paper.

Theoretical prediction of the onset of thermoacoustic instability from the experimental transfer matrix of a thermoacoustic core.

The aim of this paper is to propose a method to predict the onset conditions of the thermoacoustic instability for various thermoacoustic engines. As an accurate modeling of the heat exchangers and the stack submitted to a temperature gradient is a difficult task, an experimental approach for the characterization of the amplifying properties of the thermoacoustic core is proposed. An experimental apparatus is presented which allows to measure the transfer matrix of a thermoacoustic core under various heating conditions by means of a four-microphone method. An analytical model for the prediction of the onset conditions from this measured transfer matrix is developed. The experimental data are introduced in the model and theoretical predictions of the onset conditions are compared with those actually observed in standing-wave and traveling-wave engines. The results show good agreement between predictions from the model and experiments.

Experimental and theoretical study of the dynamics of self-sustained oscillations in a standing wave thermoacoustic engine

A model for the description of the transient regime leading to steady-state sound in a quarter-wavelength thermoacoustic prime mover is proposed, which is based on the description of the unsteady heat transfer in the system, coupled with an ordinary differential equation describing wave amplitude growth/attenuation. The equations are derived by considering a cross-sectional averaged temperature distribution along the resonator, and by assuming that both the characteristic time associated with heat diffusion through the stack and that associated with the thermoacoustic amplification are much larger than the acoustic period. Attention is here focused on the only mechanism of saturation due to heat transport by sound within the stack. The numerical solving of the governing equations leads to the prediction of the transient regime, which is compared with experimental results for several values of the heat power supplied to the system and for several positions of the stack in the resonator. The model reproduce...

Influence of shell compressibility on the ultrasonic properties of polydispersed suspensions of nanometric encapsulated droplets

Liquid droplets of nanometric size encapsulated by a polymer shell are envisioned for targeted drug delivery in therapeutic applications. Unlike standard micrometric gas-filled contrast agents used for medical imaging, these particles present a thick shell and a weakly compressible core. Hence, their dynamical behavior may be out of the range of validity of the models available for the description of encapsulated bubbles. In the present paper, a model for the ultrasound dispersion and absorption in a suspension of nanodroplets is proposed, accounting for both dilatational and translational motions of the particle. The radial motion is modeled by a generalized Rayleigh-Plesset-like equation which takes into account the compressibility of the viscoelastic shell, as well as the one of the core. The effect of the polydispersity of particles in size and shell thickness is introduced in the coupled balance equations which govern the motion of the particles in the surrounding fluid. Both effects of shell compressibility and polydispersity are quantified through the dispersion and absorption curves obtained on a wide ultrasonic frequency range. Finally, some results for larger gas-filled particles are also provided, revealing the limit of the role of the shell compressibility.

A model for acoustic vaporization of encapsulated droplets

The use of encapsulated liquid nanoparticles is currently largely investigated for medical applications, mainly because their reduced size allows them to enter targeted areas which cannot be reached by large microbubbles (contrast agents). Low-boiling point perfluorocarbon droplets can be vaporized on-site under the action of the ultrasonic field, in order to turn them into echogeneous-eventually cavitating-microbubbles. This paper presents a theoretical model describing this phenomenon, paying particular attention to the finite size of the droplet and its encapsulation by a thin viscoelastic layer. Numerical simulations are done for droplets of radii 1 and 10 μm and for frequencies of 1-5 MHz. Results reveal that droplet surface tension and shell rigidity are responsible for an increase of the acoustic droplet vaporization threshold. Furthermore, this threshold does not vary monotonically with frequency, and an optimal frequency can be found to minimize it. Finally, the role of some physical properties on the dynamics of the particle is analyzed, such as the contrast of inner and outer liquids densities and the mechanical properties of the shell.

Account of heat convection by Rayleigh streaming in the description of wave amplitude growth and stabilization in a standing wave thermoacoustic prime-mover.

This study focuses on the transient regime of wave amplitude growth and stabilization occuring into a standing wave thermoacoustic engine. Experiments are performed on a standing wave thermoacoustic oscillator. They show that the transient regime leading to steady state sound exhibits complicated dynamics, like the systematic overshoot of wave amplitude before its final stabilization, and the spontaneous and periodic switch on/off of the thermoacoustic instability at constant heat power supply. A simplified model is presented which describes wave amplitude growth from the coupled equations governing thermoacoustic amplification and unsteady heat transfer. In this model, the assumption of a one-dimensional temperature profile is retained and the equations describing heat transfer through the thermoacoustic core are coupled to that describing wave amplitude growth. These equations include the simplified description of two processes saturating wave amplitude growth, i.e. thermoacoustic heat pumping by acoust...

A model for acoustic vaporization dynamics of a bubble/droplet system encapsulated within a hyperelastic shell

Nanodroplets have great, promising medical applications such as contrast imaging, embolotherapy, or targeted drug delivery. Their functions can be mechanically activated by means of focused ultrasound inducing a phase change of the inner liquid known as the acoustic droplet vaporization (ADV) process. In this context, a four-phases (vapor + liquid + shell + surrounding environment) model of ADV is proposed. Attention is especially devoted to the mechanical properties of the encapsulating shell, incorporating the well-known strain-softening behavior of Mooney-Rivlin material adapted to very large deformations of soft, nearly incompressible materials. Various responses to ultrasound excitation are illustrated, depending on linear and nonlinear mechanical shell properties and acoustical excitation parameters. Different classes of ADV outcomes are exhibited, and a relevant threshold ensuring complete vaporization of the inner liquid layer is defined. The dependence of this threshold with acoustical, geometrical, and mechanical parameters is also provided.

A derivation of the stable cavitation threshold accounting for bubble-bubble interactions

The subharmonic emission of sound coming from the nonlinear response of a bubble population is the most used indicator for stable cavitation. When driven at twice their resonance frequency, bubbles can exhibit subharmonic spherical oscillations if the acoustic pressure amplitude exceeds a threshold value. Although various theoretical derivations exist for the subharmonic emission by free or coated bubbles, they all rest on the single bubble model. In this paper, we propose an analytical expression of the subharmonic threshold for interacting bubbles in a homogeneous, monodisperse cloud. This theory predicts a shift of the subharmonic resonance frequency and a decrease of the corresponding pressure threshold due to the interactions. For a given sonication frequency, these results show that an optimal value of the interaction strength (i.e. the number density of bubbles) can be found for which the subharmonic threshold is minimum, which is consistent with recently published experiments conducted on ultrasound contrast agents.

Numerical investigations of single bubble oscillations generated by a dual frequency excitation

The oscillations of a single bubble excited with a dual frequency acoustic field are numerically investigated. Computations are made for an air bubble in water exposed to an acoustic field with a linearly varying amplitude. The bubble response to an excitation containing two frequencies f1 = 500kHz and f2 = 400kHz at the same amplitude is compared to the monofrequency case where only f1 is present. Time-frequency representations show a sharp transition in the bifrequency case, for which the low frequency component f2 becomes resonant while the high frequency component f1 is strongly attenuated. The temporal evolution of the power spectra reveals that the resonance of the low frequency component is correlated with the time varying mean radius of the bubble. It is also observed that the total power of the bubble response in the bifrequency case can reach almost twice the power obtained in the monofrequency case, which indicates a strong enhancement of the cavitating behavior of the bubble for this specific frequency combination.

Triggering of surface modes by bubble coalescence at high pressure amplitudes

In order to study surface instabilities of small bubbles, a single spherical bubble is usually trapped in an ultrasound field and submitted to increasing acoustic pressure until reaching the necessary threshold. In the current work, coalescence between two bubbles is used as a trigger for non-spherical oscillations. Experiments are conducted in water with air bubbles of radii ranking from of 10 to 80 μm, at a driving frequency of 30 kHz and captured at 67 kHz. While most literature deals with bubble coalescence at relatively low pressure amplitudes implying spherical bubbles, coalescence at high pressure amplitudes (in the present case up to 30 kPa) leads to surface instabilities during and after the coalescence. During the impact and the immediately following oscillations, transitory surface deformations appear. After this transitory period, the bubbles stabilize either being in a purely spherical oscillation mode or exhibiting a stable surface mode. We analyze under which conditions either of the two ca...