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A study of longitudinal processes and interactions in compressible viscous flows. (English) Zbl 07193511
Summary: Fluid motion has two well-known fundamental processes: the vector transverse process characterized by vorticity, and the scalar longitudinal process consisting of a sound mode and an entropy mode, characterized by dilatation and thermodynamic variables. The existing theories for the sound mode involve the multi-variable issue and its associated difficulty of source identification. In this paper, we define the source of sound inside the fluid by the objective causality inherent in dynamic equations relevant to a longitudinal process, which naturally favours the material time-rate operator $$D/Dt$$ rather than the local time-rate operator $$\partial/\partial t$$, and describes the sound mode by inhomogeneous advective wave equations. The sources of sound physical production inside the fluid are then examined at two levels. For the conventional formulation in terms of thermodynamic variables at the first level, we show that the universal kinematic source can be condensed to a scalar invariant of the surface deformation tensor. Further, in the formulation in terms of dilatation at the second level, we find that the sound mode in viscous and heat-conducting flow has sources from rich nonlinear couplings of vorticity, entropy and surface deformation, which cannot be disclosed at the first level. Preliminary numerical demonstration of the theoretical findings is made for two typical compressible flows, i.e. the interaction of two corotating Gaussian vortices and the unsteady type IV shock/shock interaction. The results obtained in this study provide a new theoretical basis for, and physical insight into, understanding various nonlinear longitudinal processes and the interactions therein.
##### MSC:
 76 Fluid mechanics
##### Keywords:
compressible flows; mathematical foundations
Full Text:
##### References:
 [1] Blokhintzev, D.1946The propagation of sound in an inhomogeneous and moving medium I. J. Acoust. Soc. Am.18 (2), 322-328. [2] Chu, B. T. & Kovásznay, L. S.1958Non-linear interactions in a viscous heat-conducting compressible gas. J. Fluid Mech.3 (5), 494-514. [3] Chu, Y. B. & Lu, X. Y.2012Characteristics of unsteady type IV shock/shock interaction. Shock Waves22 (3), 225-235. [4] Chu, Y. B. & Lu, X. Y.2013Topological evolution in compressible turbulent boundary layers. J. Fluid Mech.733, 414-438. · Zbl 1294.76175 [5] Colonius, T., Lele, S. K. & Moin, P.1994The scattering of sound waves by a vortex: numerical simulations and analytical solutions. J. Fluid Mech.260, 271-298. [6] Courant, R. & Friedrichs, K. O.1948Supersonic Flow and Shock Waves, . Interscience. · Zbl 0041.11302 [7] Eldredge, J. D., Colonius, T. & Leonard, A.2002A vortex particle method for two-dimensional compressible flow. J. Comput. Phys.179 (2), 371-399. · Zbl 1130.76393 [8] Emmons, H. W.(Ed.) 1958Fundamentals of Gas Dynamics, . Princeton University Press. [9] Enflo, B. O. & Hedberg, C. M.2002Theory of Nonlinear Acoustics in Fluids, 1st edn. Springer. · Zbl 1049.76001 [10] Eyink, G. L. & Drivas, T. D.2018Cascades and dissipative anomalies in compressible fluid turbulence. Phys. Rev. X8, 011022. · Zbl 1419.76286 [11] Ffowcs Williams, J. E.1977Aeroacoustics. Annu. Rev. Fluid Mech.9 (1), 447-468. [12] Goldstein, M. E.1976Aeroacoustics. McGraw-Hill. [13] González, D. R., Speth, R. L., Gaitonde, D. V. & Lewis, M. J.2016Finite-time Lyapunov exponent-based analysis for compressible flows. Chaos26 (8), 083112. [14] Haller, G.2001Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D149 (4), 248-277. · Zbl 1015.76077 [15] Han, S., Luo, Y. & Zhang, S.2019The relation between finite-time Lyapunov exponent and acoustic wave. AIAA J.57 (12), 5114-5125. [16] Howe, M. S.1975Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech.71, 625-673. · Zbl 0325.76117 [17] Howe, M. S.1998Acoustics of Fluid-Structure Interactions, . Cambridge University Press. · Zbl 0921.76002 [18] Jiang, G. S. & Shu, C. W.1996Efficient implementation of weighted ENO schemes. J. Comput. Phys.126 (1), 202-228. · Zbl 0877.65065 [19] Jordan, P. & Gervais, Y.2008Subsonic jet aeroacoustics: associating experiment, modelling and simulation. Exp. Fluids44 (1), 1-21. [20] Kovásznay, L. S.1953Turbulence in supersonic flow. J. Aeronaut. Sci.20 (10), 657-674. · Zbl 0051.42201 [21] Lagerstrom, P. A.1964Laminar flow theory. In The Theory of Laminar Flows (ed. Moore, F. K.), . Princeton University Press. [22] Lagerstrom, P. A., Cole, J. D. & Trilling, L.1949 Problems in the theory of viscous compressible fluids. Tech. Rep. 6. GALCIT. [23] Liepmann, H. W. & Roshko, A.1957Elements of Gasdynamics. Dover. · Zbl 0078.39901 [24] Lighthill, M. J.1952On sound generated aerodynamically I. General theory. Proc. R. Soc. Lond. A211 (1107), 564-587. · Zbl 0049.25905 [25] Lighthill, M. J.1956Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanics (ed. Batchelor, G. K. & Davies, R. M.), pp. 250-351. Cambridge University Press. [26] Lighthill, M. J.1978Waves in Fluids, Cambridge University Press. · Zbl 0375.76001 [27] Lilley, G. M.1974On the noise from air jets. In Noise Mechanisms, AGARD-CP-131, pp. 13.1-13.12. [28] Liu, L. Q.2018Unified Theoretical Foundations of Lift and Drag in Viscous and Compressible External Flows. Springer. [29] Mao, F., Shi, Y. P., Xuan, L. J., Su, W. D. & Wu, J. Z.2011On the governing equations for the compressing process and its coupling with other processes. Sci. China Phys., Mech. Astronomy54 (6), 1154-1168. [30] Mao, F., Shi, Y. P. & Wu, J. Z.2010On a general theory for compressing process and aeroacoustics: linear analysis. Acta Mechanica Sin.26 (3), 355-364. · Zbl 1269.76108 [31] Meneveau, C.2011Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech.43 (1), 219-245. · Zbl 1299.76088 [32] Mitchell, B. E., Lele, S. K. & Moin, P.1995Direct computation of the sound from a compressible co-rotating vortex pair. J. Fluid Mech.285, 181-202. · Zbl 0848.76085 [33] Ostashev, V.1997Acoustics in Moving Inhomogeneous Media. CRC Press. · Zbl 0896.76085 [34] Phillips, O. M.1960On the generation of sound by supersonic turbulent shear layers. J. Fluid Mech.9 (1), 1-28. · Zbl 0097.41502 [35] Pierce, A. D.1981Acoustics: An Introduction to Its Physical Principles and Applications. Springer. [36] Serrin, J. B.1959Mathematical principles of classical fluid mechanics. In Fluid Dynamics I/Strömungsmechanik I (ed. Flugge, S. & Truesdell, C.), , pp. 125-263. Springer. [37] Tam, C. K., Webb, J. C. & Dong, Z.1993A study of the short wave components in computational acoustics. J. Comput. Acoust.1 (1), 1-30. · Zbl 1360.76303 [38] Truesdell, C.1954The Kinematics of Vorticity. Indiana University Press. · Zbl 0056.18606 [39] Wang, M., Freund, J. B. & Lele, S. K.2006Computational prediction of flow-generated sound. Annu. Rev. Fluid Mech.38 (1), 483-512. · Zbl 1100.76058 [40] Wang, L. & Lu, X. Y.2012Flow topology in compressible turbulent boundary layer. J. Fluid Mech.703, 255-278. · Zbl 1248.76085 [41] Whitham, G. B.1974Linear and Nonlinear Waves, John Wiley & Sons. [42] Wilczek, M. & Meneveau, C.2014Pressure Hessian and viscous contributions to velocity gradient statistics based on gaussian random fields. J. Fluid Mech.756, 191-225. · Zbl 1327.76078 [43] Wu, T. Y.1956Small perturbations in the unsteady flow of a compressible viscous and heat-conducting fluid. J. Math. Phys.35 (1-4), 13-27. · Zbl 0072.20202 [44] Wu, J. Z., Liu, L. Q. & Liu, T. S.2018Fundamental theories of aerodynamic force in viscous and compressible complex flows. Prog. Aerosp. Sci.99, 27-63. [45] Wu, J. Z., Ma, H. Y. & Zhou, M. D.2015Vortical Flows. Springer. [46] Wu, J. Z., Zhou, Y. & Fan, M.1999A note on kinetic energy, dissipation and enstrophy. Phys. Fluids11 (2), 503-505. · Zbl 1147.76537 [47] Zhang, S. H., Li, H., Liu, X. L., Zhang, H. X. & Shu, C. W.2013Classification and sound generation of two-dimensional interaction of two Taylor vortices. Phys. Fluids25 (5), 056103. [48] Zhu, Y. D., Chen, X., Wu, J. Z., Chen, S. Y., Lee, C. B. & Gad-El Hak, M.2018Aerodynamic heating in transitional hypersonic boundary layers: role of second-mode instability. Phys. Fluids30 (1), 011701.
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