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Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid

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Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid acoustics Article Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid Jitendra A. Kewalramani 1,* , Zhenting Zou 2 , Richard W. Marsh 3, Bruce G. Bukiet 4 and Jay N. Meegoda 1 1 Department of Civil & Environmental Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA; [email protected] 2 Dynamic Engineering Consultants, Chester, NJ 07102, USA; [email protected] 3 Department of Chemical & Materials Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA; [email protected] 4 Department of Mathematical Science, New Jersey Institute of Technology, Newark, NJ 07102, USA; [email protected] * Correspondence: [email protected] Received: 2 February 2020; Accepted: 29 February 2020; Published: 3 March 2020 Abstract: In this paper, nonlinearity associated with intense ultrasound is studied by using the one-dimensional motion of nonlinear shock wave in an ideal fluid. In nonlinear acoustics, the wave speed of different segments of a waveform is different, which causes distortion in the waveform and can result in the formation of a shock (discontinuity). Acoustic pressure of high-intensity waves causes particles in the ideal fluid to vibrate forward and backward, and this disturbance is of relatively large magnitude due to high-intensities, which leads to nonlinearity in the waveform. In this research, this vibration of fluid due to the intense ultrasonic wave is modeled as a fluid pushed by one complete cycle of piston. In a piston cycle, as it moves forward, it causes fluid particles to compress, which may lead to the formation of a shock (discontinuity). Then as the piston retracts, a forward-moving rarefaction, a smooth fan zone of continuously changing pressure, density, and velocity is generated. When the piston stops at the end of the cycle, another shock is sent forward into the medium. The variation in wave speed over the entire waveform is calculated by solving a Riemann problem. This study examined the interaction of shocks with a rarefaction. The flow field resulting from these interactions shows that the shock waves are attenuated to a Mach wave, and the pressure distribution within the flow field shows the initial wave is dissipated. The developed theory is applied to waves generated by 20 KHz, 500 KHz, and 2 MHz transducers with 50, 150, 500, and 1500 W power levels to explore the effect of frequency and power on the generation and decay of shock waves. This work enhances the understanding of the interactions of high-intensity ultrasonic waves with fluids. Keywords: power ultrasound; nonlinear wave propagation; shock; rarefaction 1. Introduction Ultrasound is a type of sound wave with frequencies ranging from 20 KHz up to several gigahertz. For practical applications, ultrasound is mostly generated by ultrasonic transducers and propagates into the subject material. Ultrasonics is a branch of acoustics dealing with the generation and use of inaudible sound waves [1]. Applications of ultrasonics are rigidly classified as being of either low intensity (popularly known as non-destructive applications) or high-intensity (also known as power ultrasonics) [1]. A low-intensity acoustic wave in a homogeneous medium propagates at a certain speed (relative to the medium), and deformations produced by the wave in the medium are purely elastic. Ultrasonic non-destructive testing and imaging used as means of exploration, detection, Acoustics 2020, 2, 147–163; doi:10.3390/acoustics2010011 www.mdpi.com/journal/acoustics Acoustics 2019, 2 FOR PEER REVIEW 4 Acoustics 2020, 2 148 detection, and information (e.g., the location of a crack or determination of material properties), are some of the promising low-intensity applications. and informationHigh-intensity (e.g., acoustic the location wave of behaves a crack ordifferen determinationtly than oflow-intensity: material properties), they can are permanently some of the changepromising the low-intensityphysical, chemical, applications. or biological properties or, if intense enough, even destroy the medium to whichHigh-intensity it is applied acoustic [1]. At high wave intensity, behaves di thefferently amplitudes than low-intensity: of vibration used they are can sufficiently permanently high change that nonlinearthe physical, propagation chemical, or occurs. biological The properties uses of or,high if-power intense enough,ultrasonics even include destroy thecleaning, medium enhancing to which chemicalit is applied reactions, [1]. At highemulsification, intensity, the dispersion, amplitudes welding of vibration of metals used and are polymers, sufficiently machining high that and nonlinear metal formingpropagation in solids occurs. and The fluids, uses offood high-power processing, ultrasonics ultrasonic include agglomeration, cleaning, enhancing etc. The nonlinear chemical nature reactions, of intenseemulsification, acoustic dispersion, waves underlies welding many of metals applications and polymers, of high-intensity machiningand acoustic metal waves forming [2,3]. in solids The branchand fluids, of acoustics food processing, dealing with ultrasonic high-intensity agglomeration, acoustic etc.is called The nonlinearnonlinear natureacoustics. of intense acoustic wavesIn underliesordinary elastic many applicationsmedia, finite-amplitude of high-intensity sound acoustic propagation waves has [2 ,been3]. The intensively branch of studied acoustics to understanddealing with many high-intensity nonlinear acoustic acoustic is effects called nonlinearsuch as cumulative acoustics. wave distortion with propagation distanceIn ordinary as well as elastic the concept media, of finite-amplitude the parametric ar soundray [4]. propagation Nonlinear acoustics has been is intensively based on a studiednonlinear to theoryunderstand of elasticity many nonlinear [5]. Once acoustic the original effects sinusoidal such as cumulative finite-amplitude wave distortionwave propagates with propagation from an ultrasonicdistance as source well as into the concepta fluid medium, of the parametric the wave array will [be4]. exposed Nonlinear to acousticstwo effects is basedinfluencing on a nonlinear its time course:theory of(1) elasticitythe dissipation [5]. Oncearising the from original viscosity, sinusoidal heat conductivity, finite-amplitude and relaxation wave propagates processes from and (2) an theultrasonic nonlinearity source leading into a fluidto the medium, formation the of wave higher will harmonics be exposed to the totwo fundamental effects influencing frequency its of time the wavecourse: [6]. (1) Understanding the dissipation the arising basic frommechanisms viscosity, of heat the conductivity,nonlinear effects and of relaxation intense acoustic processes waves and (2)is criticalthe nonlinearity to studying leading the distribution to the formation of high-intensi of higherty harmonics acoustic waves to the in fundamental a system [7]. frequency The purpose of the of thiswave paper [6]. Understandingis to understand the and basic model mechanisms the nonlinea of ther behavior nonlinear of high-intensi effects of intensety acoustic acoustic waves. waves The is nonlinearitycritical to studying associated the with distribution intense ofultrasound high-intensity is studied acoustic by using waves the in one-dimensional a system [7]. The motion purpose of nonlinearof this paper shock is towave understand in an ideal and fluid. model the nonlinear behavior of high-intensity acoustic waves. The nonlinearity associated with intense ultrasound is studied by using the one-dimensional motion of 2.nonlinear Nonlinear shock Wave wave Propagation in an ideal and fluid. Shock Formation 2. NonlinearIn linearWave wave Propagation motion, disturbances and Shock Formationin a medium can be defined as the result of three independent modes: the acoustic, entropy, and vorticity modes [2,8]. If wave intensity increases, the In linear wave motion, disturbances in a medium can be defined as the result of three independent disturbances in a medium becomes large enough such that nonlinear wave propagation starts [3,7]. modes: the acoustic, entropy, and vorticity modes [2,8]. If wave intensity increases, the disturbances in The nonlinear elastic behavior of the materials becomes progressively more important at high- a medium becomes large enough such that nonlinear wave propagation starts [3,7]. The nonlinear amplitude wave excitation [3,9]. As the speed of the disturbance (or particle velocity) () increases, elastic behavior of the materials becomes progressively more important at high-amplitude wave linear wave motion ceases, and therefore the superposition principle is no longer valid. Thus, the (u) disturbanceexcitation [3 ,cannot9]. As thebe speeddescribed of the in disturbancethe form of (orthe particle three independent velocity) modesincreases, [8]. linearThese wave modes motion start interactingceases, and with therefore one another. the superposition As shown principle in Figure is 1, no the longer following valid. three Thus, interactions the disturbance between cannot these be modesdescribed are inactivated the form due of theto high-intensities: three independent the modessound-sound, [8]. These sound-vorticity modes start interactingand sound-entropy with one interactionsanother. As shown[2,8]. inDue Figure to1 , thethe followingsound-sound three interactionsinteractions, between harmonic these modesgenerations are activated and self- due demodulationsto high-intensities: occur the within sound-sound, the acoustic sound-vorticity mode [2,8]. andThen, sound-entropy the acoustic interactionsmode
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