Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.Comment: submitted to Siam SIA

Nonlinear Acoustics in Underwater and Biomedical Applications: Array Performance Degradation and Time Reversal Invariance

This dissertation describes a model for acoustic propagation in inhomogeneous flu- ids, and explores the focusing by arrays onto targets under various conditions. The work explores the use of arrays, in particular the time reversal array, for underwater and biomedical applications. Aspects of propagation and phasing which can lead to reduced focusing effectiveness are described. An acoustic wave equation was derived for the propagation of finite-amplitude waves in lossy time-varying inhomogeneous fluid media. The equation was solved numerically in both Cartesian and cylindrical geometries using the finite-difference time-domain (FDTD) method. It was found that time reversal arrays are sensitive to several debilitating factors. Focusing ability was determined to be adequate in the presence of temporal jitter in the time reversed signal only up to about one-sixth of a period. Thermoviscous absorption also had a debilitating effect on focal pressure for both linear and nonlinear propagation. It was also found that nonlinearity leads to degradation of focal pressure through amplification of the received signal at the array, and enhanced absorption in the shocked waveforms. This dissertation also examined the heating effects of focused ultrasound in a tissue-like medium. The application considered is therapeutic heating for hyperther- mia. The acoustic model and a thermal model for tissue were coupled to solve for transient and steady temperature profiles in tissue-like media. The Pennes bioheat equation was solved using the FDTD method to calculate the temperature fields in tissue-like media from focused acoustic sources. It was found that the temperature-dependence of the medium's background prop- erties can play an important role in the temperature predictions. Finite-amplitude effects contributed excess heat when source conditions were provided for nonlinear ef- fects to manifest themselves. The effect of medium heterogeneity was also found to be important in redistributing the acoustic and temperature fields, creating regions with hotter and colder temperatures than the mean by local scattering and lensing action. These temperature excursions from the mean were found to increase monotonically with increasing contrast in the medium's properties.Office of Naval Research (Code 321-TS

Simulation 3D de la propagation d'ondes de choc acoustiques en atmosphère turbulente. Application au bang sonique

This thesis deals with the effects of atmospheric turbulence on the propagation of acoustical shock waves. These effects are of major interest for applications such as sonic boom, buzz saw noise or thunder. A numerical one-way method is developed to model and to simulate three-dimensional nonlinear propagation of acoustical shock waves in a moving heterogeneous medium. It relies on a split-step approach that permits to take into account efficiently the different involved physical mechanisms. To tackle realistic 3D problems (of order of one billion of degree of freedom), the implementation of the method is done using the parallel single program multiple data paradigm. Validity of this method is assessed using multiple test cases. The method is applied to investigate the effects of atmospheric turbulence on sonic boom propagation through the Planetary Boundary Layer. Hence, both under-track boom and boom in the shadow zone are studied for a hypersonic configuration developed in the European project ATLLAS II. Finally, the focusing of weak shock waves on a cusped caustic is simulated. It is the first study of the stability of a nonlinear caustic to flow perturbations to our knowledge.Cette thèse traite des effets de la turbulence atmosphérique sur la propagation d'ondes de choc acoustiques. Ces effets sont d'un grand intérêt pour des applications comme le bang sonique, le buzz saw noise ou le tonnerre. Une méthode numérique unidirectionnelle est développée pour modéliser et simuler la propagation tridimensionnelle d'ondes de choc acoustiques en milieu hétérogène en mouvement. Elle repose sur une approche à pas fractionnés qui permet de prendre en compte efficacement les différents mécanismes physiques présents. Pour s'attaquer à des problèmes 3D réalistes (de l'ordre du milliard de degré de liberté), l'implémentation de la méthode est réalisée en utilisant le paradigme de programmation parallèle " single program multiple data ". La validité de cette méthode est évaluée sur différents cas tests. La méthode est appliquée à l'étude des effets de la turbulence atmosphérique sur la propagation du bang sonique dans la couche limite planétaire. Ainsi, le bang sous trace et le bang dans la zone d'ombre sont calculés pour la configuration hypersonique développée dans le projet européen ATLLAS II. Enfin, la focalisation de chocs faibles sur une caustique cuspidée est simulée. Cela est, à notre connaissance, la première étude de la stabilité d'une caustique non linéaire à des perturbations dues à un écoulement

Adjoint-based optimization for inkjet printing

In this thesis the flow inside inkjet printhead microchannels is analysed using a two- parameter low Mach number expansion of the compressible Navier–Stokes equations and a reduced order model for the free surface flow inside the inkjet nozzle. The channel flow is separated into equations for an incompressible flow with no acoustic oscillations and equations for thermoviscous acoustic oscillations with no mean flow. This thesis concerns two types of optimal control problems. The optimal control problem of the first type is finding a velocity profile of the piezo-electric actuator that eliminates residual oscillations after a droplet is ejected. The cost function is the sum of the acoustic energy in the channel and the surface energy of the spherical cap of ink at the end of the nozzle at a given time. This problem is approached by obtaining the sensitivity of the total energy inside an inkjet microchannel with respect to boundary forcing using the adjoint method. Using gradient-based optimization algorithms, optimal waveforms are found that minimize the objective value at various final times and for geometries with increasing complexity. Physical interpretation to the optimal waveforms profiles is provided, and the exploited mechanisms are revealed. The optimal control problem of the second type is finding a shape of the inkjet printhead channel that maximises dissipation of the acoustic oscillations, without increasing the pressure drop required to drive the steady flow. Similarly, the adjoint approach is used to obtain the sensitivity of the acoustic flow eigenvalues with respect to boundary deformations in Hadamard form. Knowing the shape sensitivity of the incompressible flow viscous dissipation, the constrained optimization problem is solved to find a design that has the same viscous dissipation for the steady flow but a 40% larger decay rate for the oscillating flow. The final shape is not straightforward and would have been difficult to achieve through physical insight or trial and error. It could be improved further by adapting the parameters that describe the shape, but in this case the improvement would be small. The method is general and could be applied to many different applications in microfluidics. In summary, the methods in this thesis are promising techniques in the design and optimization of inkjet printheads. The discussed numerical techniques and the gained physical understanding can be used to automatically find the optimal design parameters, or, at a minimum, accelerate the experimental trial and error processes.This project has received funding from the European Unions Horizon 2020 research and innovation programme under Grant Agreement No. H2020-MSCA-ITN-2015

ACOUSTIC BESSEL AND VORTEX BEAMS: FORCES AND REFRACTION

Non-contact manipulation techniques or tweezers devices are invaluable for applications in physics, chemistry, biology, and engineering. Acoustic tweezers using either standing waves or focused beams have been investigated for more than a few decades with advantages of label-free operation, noninvasiveness, and biocompatibility when compared with the optical, magnetic, and electrical counterparts. Here, a new type of acoustic tweezers (i.e. acoustic tractors) is studied using acoustic Bessel and vortex beams that are able to pull objects against the beam\u27s propagation over centimeter ranges. Stable acoustic tractors require transversely stable trapping in addition to axially negative pulling. Hence, the transverse forces acting on a spherical particle centered on the axis of axisymmetric and vortex Bessel beams were first investigated by using both the Gorkov potential and the partial wave expansion with the trapping behaviors more flexible than trapping by standing waves and focused beams used in conventional acoustic tweezers. Then, the physical parameters desired for simultaneous trapping and pulling of particles of different sizes were examined. The results reveal the possibility of achieving a simultaneous pulling and trapping of a small particle using Bessel beams. In addition, the Born approximation method was used to analyze the transverse trapping force for spherical particles and particles of different shapes and orientations. Compared with the full solution from the partial wave expansion, the Born approximation can simplify the computation and can also provide insight into the transverse radiation force. In addition, a mathematical framework based on phase shifts adapted from quantum scattering theory was used to analyze the axial radiation force. This phase shift approach can allow one to engineer object and beam parameters to design experimentally achievable axially pulling forces. Furthermore, the effects of realistic factors such as gravity, buoyancy, and acoustic streaming were also evaluated. The work here is useful for the further study of acoustic radiation force and will lead to an experimental demonstration of stable acoustic tractor beams. The study will also guide particle manipulations with engineered objects

The 1995 NASA High-Speed Research Program Sonic Boom Workshop

The High-Speed Research Program and NASA Langley Research Center sponsored the NASA High-Speed Research Program Sonic Boom Workshop on September 12-13, 1995. The workshop was designed to bring together NASAs scientists and engineers and their counterparts in industry, other Government agencies, and academia working together in the sonic boom element of NASAs High-Speed Research Program. Specific objectives of this workshop were to (1) report the progress and status of research in sonic boom propagation, acceptability, and design; (2) promote and disseminate this technology within the appropriate technical communities; (3) help promote synergy among the scientists working in the Program; and (4) identify technology pacing the development of viable reduced-boom High-Speed Civil Transport concepts. The Workshop included these sessions: Session 1 - Sonic Boom Propagation (Theoretical); Session 2 - Sonic Boom Propagation (Experimental); and Session 3 - Acceptability Studies - Human and Animal

Numerical and experimental investigations of the acoustic standing wave resonator, pump, and micropump

The interactions of acoustic waves and thermoviscous fluids in closed cavities lead to some important physical phenomena such as, linear and nonlinear acoustic standing waves, and acoustic streaming which are very important in a wide range of engineering applications. The present dissertation is focused on the detailed investigation of standing wave dynamics in closed cavities. As a part of this research, novel numerical and experimental techniques are developed to analyze different phenomena caused by acoustic-fluid interaction. Using these techniques, the behavior of pressure, acoustic and streaming velocity fields inside the standing wave resonator, as well as the valveless acoustic pump and micropump are investigated. A new sixth-order accurate compact finite difference method for solving the Helmholtz equation with Neumann boundary conditions is developed. This scheme showed a better performance at higher wave numbers than the finite element method. A new fourth-order numerical scheme is also developed for solving highly nonlinear standing wave equations with no restriction on nonlinearity level and type of fluid. For highly nonlinear waves, the simulation results show the presence of a wavefront that travels along the resonator with very high pressure and velocity gradients. The slopes of the traveling velocity and pressure gradients, and the asymmetry in the pressure waveform are higher for CO 2 than those for air. The spatial and temporal variations of the nonlinear pressure and particle velocity fields inside a resonator are experimentally investigated at different frequencies and intensities. The effects of the excitation frequency and displacement on the streaming structure are also studied. It is found that, the classical streaming is not developed for Re s 1 50. Acoustic streaming patterns are also found to be significantly affected by transverse temperature gradient. A valveless acoustic standing wave pump is developed and the velocity fields inside this novel pump are analyzed. It is found that, the net flow rate of the pump increases with an increase in the pressure amplitude. The behavior of a novel acoustic micropump is also studied at a high frequency. The effect of the diffuser geometry on the pump performance is investigated. The results show that the maximum diffuser efficiency is achieved at the diffuser-nozzle element's half-angle of approximately 45

Symmetry analysis and hidden variational structure of Westervelt's equation in nonlinear acoustics

Westervelt's equation is a nonlinear wave equation that is widely used to model the propagation of sound waves in a compressible medium, with one important application being ultra-sound in human tissue. Two fundamental aspects of this equation -- symmetries and conservation laws -- are studied in the present work by modern methods. Numerous results are obtained: new conserved integrals; potential systems yielding hidden symmetries and nonlocal conservation laws; mapping of Westervelt's equation in the undamped case into a linear wave equation; exact solutions arising from the mapping; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and conservation laws.Comment: 23 pages; published versio