Evolution of the diffusion-induced flow over a disk, submerged in a stratified viscous fluid

P. V. Matyushin

Matematičeskoe modelirovanie, Tome 30 (2018) no. 11, pp. 44-58.

Résumé

In the present paper the results of mathematical modeling of the evolution of the $\mathrm{3D}$ diffusion-induced flow over a disk (with diameter $d$ and thickness $H = 0.76\cdot d$), immersed in a linearly density stratified incompressible viscous fluid (described by the Navier–Stokes equations in the Boussinesq approximation), are shown. The disk rests at the level of the neutral buoyancy (which coincides with its axis of symmetry $z$) and disturbs the homogeneity of the background diffusion flux in the fluid, forming a complex system of the slow currents (gravitational internal waves). Over time, two thin horizontal convection cells are formed at the upper and lower parts of the disk, stretching parallel to the $z$ axis and adjacent to the base cell with thickness $d/2$. For the first time the fundamental mechanism for the formation of each new half-wave near the vertical axis $x$ (passing through the center of the disk) during half the buoyancy period of the fluid $T_b$ is analyzed in detail. This mechanism is based on gravitational instability. The beginning of this instability was fixed at $0.473\cdot T_b$ at a height of $3.9\cdot d$ above the center of the disk. The same mechanism is also realized over the place where the body moves in the horizontal direction. The $\mathrm{3D}$ vortex structure of the flow is visualized by the isosurfaces of the imaginary part of the conjugate eigenvalues of the velocity gradient tensor. The method SMIF with an explicit hybrid finite difference scheme for the approximation of the convective terms of the equations (second-order approximation, monotonicity), which has proved itself over the past $30$ years, is used for the mathematical modeling.

Détail

Keywords: stratified viscous fluid, internal waves, crest, trough, convective cell, disk, visualization, mathematical modeling.
Mots-clés : diffusion, $\mathrm{3D}$ vortex structure

Bibliographie

[1] L. Prandtl, O. Tietjens, Hydro- und Aeromechanik, v. 1, Springer, Berlin, 1929, 335 pp. ; 1931, 245 pp. | MR

[2] V.G. Baidulov, Iu.D. Chashechkin, “Pogranichnye techeniia, indutsirovannye diffuziei okolo nepodvizhnogo gorizontalnogo tsilindra v nepreryvno stratifitsirovannoi zhidkosti”, Izvestiia AN. Fizika atmosfery i okeana, 32:6 (1996), 818–823

[3] V.G. Baydulov, P.V. Matyushin, Yu.D. Chashechkin, “Structure of a diffusion-induced flow near a sphere in a continuously stratified fluid”, Doklady Physics, 50:4 (2005), 195–199 | DOI | MR

[4] V.G. Baydulov, P.V. Matyushin, Yu.D. Chashechkin, “Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid”, Fluid Dynamics, 42:2 (2007), 255–267 | DOI | MR

[5] M.R. Allshouse, M.F. Barad, T. Peacock, “Propulsion generated by diffusion-driven flow”, Nature Physics, 6 (2010), 516–519 | DOI

[6] M.A. Page, “Fluid dynamics: Propelled by diffusion”, Nature Physics, 6 (2010), 486–487 | DOI

[7] M.J. Mercier, A.M. Ardekani, M.R. Allshouse, B. Doyle, T. Peacock, “Self-propulsion of immersed objects via natural convection”, Phys. Rev. Lett., 112 (2014), 204501, 5 pp. | DOI

[8] N.F. Dimitrieva, Yu.D. Chashechkin, “The structure of induced diffusion flows on a wedge with curved edges”, Physical oceanography, 2016, no. 3, 70–78

[9] O.M. Belotserkovskii, V.A. Gushchin, V.N. Konshin, “Splitting method for studying stratified fluid flows with free surfaces”, USSR Computational Mathematics and Mathematical Physics, 27:2 (1987), 181–196 | DOI | Zbl

[10] P.V. Matiushin, Chislennoe modelirovanie prostransvennykh otryvnykh techenii odnorodnoi neszhimaemoi viazkoi zhidkosti okolo sfery, Diss. ... kand. f.-m. n., M., 2003, 194 pp.

[11] P.V. Matiushin, “Evoliutsiia techeniia stratifitsirovannoi viazkoi zhidkosti pri nachale dvizheniia tela”, Nauchnyi zhurnal «Protsessy v geosredakh», 2016, no. 4(9), 333–343

[12] M.S. Chong, A.E. Perry, B.J. Cantwell, “A general classification of three-dimentional flow field”, Phys. Fluids A, 2 (1990), 765–777 | DOI | MR

[13] V.A. Gushchin, P.V. Matyushin, “Vortex formation mechanisms in the wake behind a sphere for 200 Re 380”, Fluid Dynamics, 41:5 (2006), 795–809 | DOI | Zbl

[14] V.A. Gushchin, P.V. Matyushin, “Numerical Simulation and Visualization of Vortical Structure Transformation in the Flow past a Sphere at an Increasing Degree of Stratification”, Comp. Math. and Math. Physics, 51:2 (2011), 251–263 | DOI | MR | Zbl

[15] P.V. Matiushin, “Klassifikatsiia rezhimov techenii stratifitsirovannoi viazkoi zhidkosti okolo diska”, Nauchnyi zhurnal «Protsessy v geosredakh», 2017, no. 4(13), 678–687 | Zbl

[16] V.A. Gushchin, P.V. Matyushin, “Simulation and study of stratified flows around finite bodies”, Comp. Math. and Math. Physics, 56:6 (2016), 1034–1047 | DOI | MR | Zbl

[17] V.A. Gushchin, A.V. Kostomarov, P.V. Matyushin, “3D Visualization of the Separated Fluid Flows”, J. of Visualization, 7:2 (2004), 143–150 | DOI

[18] V.A. Gushchin, P.V. Matiushin, “Matematicheskoe modelirovanie prostranstvennykh techenii neszhimaemoi zhidkosti”, Matemat. modelirovanie, 18:5 (2006), 5–20

[19] V.A. Gushchin, P.V. Matiushin, “Klassifikatsiia rezhimov otryvnykh techenii zhidkosti okolo sfery pri umerennykh chislakh Reinoldsa”, Matematicheskoe modelirovanie: problemy i rezultaty, Nauka, M., 2003, 199–235

[20] P.V. Matyushin, “The vortex structures of the 3D separated stratified fluid flows around a sphere”, Selected Papers of the International conference «Fluxes and Structures in Fluids-2007» (July 2–5, 2007, St. Petersburg, Russia), eds. Yu.D. Chashechkin, V.G. Baydulov, IPMech RAS, M., 2008, 139–145