# Evolution of the Diffusion-Induced Flow over a Disk Submerged in a Stratified Viscous Fluid

### Abstract

The paper presents the results of the mathematically simulated evolution of a 3D diffusion-induced flow over a disk (with diameter *d* and thickness *H* = 0.76 *d*) immersed in a linearly density-stratified incompressible viscous fluid (described by a system of the Navier–Stokes equations in the Boussinesq approximation). The disk rests at the level of the neutral buoyancy (coinciding with its symmetry axis *z*) and disturbs the homogeneity of the background diffusion flux in the fluid forming a complex system of slow currents (internal gravitational 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. This work is the first to analyze in detail 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}. This mechanism is based on gravitational instability. The emergence of this instability is first detected at 0.473*T*_{b} at a height of 3.9 *d* above the center of the disk. The same mechanism is also implemented over the place where the body moves in the horizontal direction. The 3D vortex structure of the flow is visualized by constructing the isosurfaces of the imaginary part of the complex-conjugate eigenvalues of the velocity gradient tensor. In mathematical simulation, we employed a numerical method SMIF that has proved itself over three decades with an explicit hybrid finite-difference scheme for the approximation of the convective terms of the equations (second-order approximation, monotonicity).

## Keywords:

stratified viscous fluid diffusion internal waves crest trough convective cell disk visualization 3D vortex structure mathematical simulation## Notes

## REFERENCES

- 1.L. Prandtl and O. Tietjens,
*Hydro- und Aeromechanik*(Springer, Berlin, 1929, 1931) Vols. 1, 2.Google Scholar - 2.V. G. Baidulov and Yu. D. Chashechkin, “Boundary current induced by diffusion near a motionless horizontal cylinder in a continuously stratified fluid,” Izv., Atmos. Oceanic. Phys.
**32**, 751–756 (1996).Google Scholar - 3.V. G. Baydulov, P. V. Matyushin, and Yu. D. Chashechkin, “Structure of a diffusion-induced flow near a sphere in a continuously stratified fluid,” Dokl. Phys.
**50**, 195–199 (2005).CrossRefGoogle Scholar - 4.V. G. Baydulov, P. V. Matyushin, and Yu. D. Chashechkin, “Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid,” Fluid Dyn.
**42**, 255–267 (2007).MathSciNetCrossRefzbMATHGoogle Scholar - 5.M. R. Allshouse, M. F. Barad, and T. Peacock, “Propulsion generated by diffusion-driven flow,” Nat. Phys.
**6**, 516–519 (2010).CrossRefGoogle Scholar - 6.M. A. Page, “Fluid dynamics: Propelled by diffusion,” Nat. Phys.
**6**, 486–487 (2010).CrossRefGoogle Scholar - 7.M. J. Mercier, A. M. Ardekani, M. R. Allshouse, B. Doyle, and T. Peacock, “Self-propulsion of immersed objects via natural convection,” Phys. Rev. Lett.
**112**, 204501 (2014).CrossRefGoogle Scholar - 8.N. F. Dimitrieva and Yu. D. Chashechkin, “The structure of induced diffusion flows on a wedge with curved edges,” Phys. Oceanogr. No.
**3**, 70–78 (2016).Google Scholar - 9.O. M. Belotserkovskii, V. A. Gushchin, and V. N. Konshin, “Splitting method for studying stratified fluid flows with free surfaces,” USSR Comput. Math. Math. Phys.
**27**, 181–196 (1987).CrossRefGoogle Scholar - 10.P. V. Matyushin, “Numerical modeling of spatial separated flows of a homogeneous incompressible viscous fluid near a sphere,” Cand. Sci. (Phys. Math.) Dissertation (Moscow, 2003).Google Scholar
- 11.P. V. Matyushin, “The evolution of a stratified viscous fluid flow at the onset of body movement,” Protsessy Geosredakh, No.
**4**, 333-343 (2016).Google Scholar - 12.M. S. Chong, A. E. Perry, and B. J. Cantwell, “A general classification of three-dimentional flow field,” Phys. Fluids A
**2**, 765–777 (1990).MathSciNetCrossRefGoogle Scholar - 13.V. A. Gushchin and P. V. Matyushin, “Vortex formation mechanisms in the wake behind a sphere for 200 < Re < 380,” Fluid Dyn.
**41**, 795–809 (2006).CrossRefzbMATHGoogle Scholar - 14.V. A. Gushchin and P. V. Matyushin, “Numerical simulation and visualization of vortical structure transformation in the flow past a sphere at an increasing degree of stratification,” Comput. Math. Math. Phys.
**51**, 251–263 (2011).MathSciNetCrossRefzbMATHGoogle Scholar - 15.P. V. Matyushin, “Classification of flow regimes of stratified viscous fluid near the disk,” Protses. Geosredakh, No.
**4**, 678–687 (2017).Google Scholar - 16.V. A. Gushchin and P. V. Matyushin, “Simulation and study of stratified flows around finite bodies,” Comput. Math. Math. Phys.
**56**, 1034–1047 (2016).MathSciNetCrossRefzbMATHGoogle Scholar - 17.V. A. Gushchin, A. V. Kostomarov, and P. V. Matyushin, “3D Visualization of the Separated Fluid Flows,” J. Visual.
**7**, 143–150 (2004).CrossRefGoogle Scholar - 18.V. A. Gushchin and P. V. Matyushin, “Mathematical modelling of the 3D incompressible fluid flows,” Mat. Model.
**18**(5), 5–20 (2006).zbMATHGoogle Scholar - 19.V. A. Gushchin and P. V. Matyushin, “Classification of regimes of separated flows of fluid around a sphere for moderate Reynolds numbers,” in
*Mathematical Modeling: Problems and Results*(Nauka, Moscow, 2003), pp. 199–235 [in Russian].zbMATHGoogle Scholar - 20.P. V. Matyushin, “The vortex structures of the 3D separated stratified fluid flows around a sphere,” in
*Selected Papers of the International Conference on Fluxes and Structures in Fluids–2007, July 2–5,**2007, St. Petersburg, Russia,*Ed. by Yu. D. Chashechkin and V. G. Baydulov (IPMech RAS, Moscow, 2008), pp. 139–145.Google Scholar