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Introduction Thermoacoustic instabilities in liquid rocket engines are a big challenge to overcome in the development of recent rockets design. Perturbation of liquid jet submitted to transverse high amplitude acoustic wave is here studied as one of the contributions to these instabilities. At large scale two phenomena has been experimentally evidenced for liquid jets, flattening and deviation. A semi-analytical model is presented here to compute the local radiation pressure as the main reason for liquid jet or droplet deformation. Radiation pressure distribution model for circular cylinder was already studied in order to see how the deformation of liquid jets and droplets starts. For a good interpretation of this deformation, acoustic circular model was modified to take into account the real shape of liquid elements modified by acoustics. Deformation of liquid jets and droplet are also encountered in acoustic levitation and ultrasonic standing wave. Acoustic model • Exploit the radiation pressure on a curved surface in order to understand how acoustic can deform liquid jets or droplets. Liquid jet and droplets deformation by non-uniform radiation pressure distribution R. HERRERA, J.-B. BLAISOT, C. RICHARD & F. BAILLOT Conclusions Perspectives • Extend model for the extreme aspect ratios ( اͳ and بͳሻ • Couple the acoustic model to the behavior law at the liquid interface to take into account the surface tension action. • Compute stable static shape of liquid jets and droplets submitted to transverse acoustic waves as function of acoustic intensity. • Compute dynamical deformation for instable jets or droplets. • For deformable objects o Suction and compressions Lflattening ^ acoustic axis o Aspect ratio () decrease L suction amplified L flattening enhanced • Radiation pressure acts as a unbalanced effect conducing always to increase the deformation as the object is deformed. CNES JC2, Toulouse 13-15/11/2019 ߩ ǣ density of surrounding fluid ǣ speed of sound in surrounding fluid ȉ : time average on a period ܪ ሺ ݎఏ ሻ : cylindrical Hankel function ݎఏ : spherical Hankel function ሺߠݏሻ : Legendre polynomial For elliptic cylinders or ellipsoids : • On the acoustic axis ( ߠൌ Ͳι , ߠൌ ͳͺͲι and ߠൌ ͵Ͳι ), the maximum compression (+) is maintained at the same value regardless the aspect ratio. • In the plane perpendicular to the wave ( ߠൌ ͻͲι and ߠൌ ʹͲι ), the maximum suction effect (-) increases as the aspect ratio decreases. ++ : Acoustic axis -- : Perpendicular plane to the wave Objective Interpretation Results SuctionL object deformation object deformation La/b decreasesL Suction increases ++ : Acoustic axis -- : Perpendicular plane to the wave Polar representation of radiation pressure distribution for three typical elliptic cross-sections or ellipsoids. LFlattening enhanced + + : Acoustic axis • Whatever and excluding near pressure antinodes o Maximum compression effect occurs in the vicinity of ߠൌ Ͳι , ߠൌ ͳͺͲι and ߠൌ ͵Ͳι (acoustic axis) o Maximum suction effect occurs in the vicinity of ߠൌ ͻͲι and ߠൌ ʹͲι (^ acoustic axis) + + + - - + + + - - Radiation pressure distribution on an elliptic cylinder in the acoustic velocity antinode for different aspect ratios ൌ Radiation pressure distribution on an ellipsoid in the acoustic velocity antinode for different aspect ratios ൌ References F. Baillot, J. B. Blaisot, G. Boisdron, C. Dumouchel, « Behavior of an air-assisted jet submitted to a transverse high frequency acoustic field ». J. Fluid Mech., 640, 307-344, 2009. A. Ficuciello, J.B. Blaisot, C. Richard, F. Baillot, « Investigation of air-assisted sprays submitted to high frequency transverse acoustic fields: Droplet clustering ». Physics of Fluids, 29, 067103, 2017. F. G. Mitri, « Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves », J. Appl. Phys., 118, 214903, 2015. L. V. King, « On the acoustic radiation pressure on spheres ». Proc. R. Soc. Lond. A, 147,212-240, 1934. A. Zhuk, « Radiation force acting on a cylindrical particle in a sound field ». Soviet Applied Mechanics, Kluwer Academic Publishers-Plenum Publishers, 22, 689-693, 1986. W. Wei, D. B. Thiessen and P. L. Marston, « Acoustic radiation force on a compressible cylinder in a standing wave ». J. Acoust. Soc. Am. 116 201, 2004. Liquid jets are consider as infinite cylinders of elliptic cross section Incident acoustic field (standing wave ) is expressed using potential velocity as: ߶ ൌ ܣ ఠ௧ ௫ା ௫ା Scattered acoustic field is modeled using second solution of Helmholtz equation in cylindrical coordinates: ߶ ௦ ൌ ܣ ఠ௧ ୀ ஶ ܥ ܪ ሺ ݎఏ ሻ ݏ ߠScattered coefficient is calculated using the boundary condition of the object at rest ቚ ߶ߘ௦ ڄ ை ൌ ቚ െ ߶ߘ ڄ ை Resulting acoustic field is computed as : ߶ൌ߶ ߶ ௦ Expression of the radiation pressure on the object ௗ ሺߠǡ ݎൌ ݎ ሻൌ െ ߩ ʹ ߶ߘଶ ߩ ʹ ଶ ߶ ݐଶ For axissymmetric ellipsoids only the model for the scattered field is changed: ൌ ࣓ ୀ ஶ ሺ ሻ Elliptic Cross section a y x O b
