Bathtub vortex induced by instability

The driving mechanism and the swirl direction of the bathtub vortex are investigated by the linear stability analysis of the no-vortex flow as well as numerical simulations. We find that only systems having plane symmetries with respect to vertical planes deserve research for the swirl direction. The bathtub vortex appearing in a vessel with a rectangular cross section having a drain hole at the center of the bottom is proved to be induced by instability when the flow rate exceeds a threshold. The Coriolis force is capable of determining the swirl direction to be cyclonic.

Figure 1

Computational model and coordinates.

Figure 2

Flow fields for $\mathrm{Re}=50$ [(a)–(c)] and for $\mathrm{Re}=100$ [(d)–(f)], where $\mathrm{Ro}=\infty$. (a), (d) Paths of fluid particles initially aligned on lines at midheight of the inlets. (b), (e) Horizontal projection of paths. (c), (f) Isosurfaces of $m_z=0.12$ (counterclockwise, brown) and $m_z=-0.12$ (clockwise, blue).

Figure 3

Bifurcation diagram for $\mathrm{Ro}=\infty$. $v_1=v\left({\mathrm{P}}_1\right)$: Circumferential velocity at the representative point. Solid lines: Stable solutions; dashed line: unstable solution. Inset: Enlarged view of bifurcation diagram for both $\mathrm{Ro}=\infty$ (solid and dashed lines) and $\mathrm{Ro}=2.4\times {10}^5$ (dotted lines).

Figure 4

Eigenfunction of disturbance from the linear stability analysis (horizontal velocities on $z=1/2$ and paths of fluid particles). $\mathrm{Re}=65$; $\mathrm{Ro}=\infty$. Top view.

Figure 5

Time evolution of the total angular momentum $M_z$, angular momentum outflow $Q_2$, and torques due to pressure and viscous stress on the sidewalls and the bottom. $\mathrm{Re}=100$; $\mathrm{Ro}=\infty$. (a) Angular momentum $M_z$. (b) Inflow of the angular momentum $(-Q_2)$ from the outlet $(Q_2$: outflow); torques due to viscous stress $N_{\mathrm{v}x},N_{\mathrm{v}y}$, and $N_{\mathrm{v}z}$; torques due to pressure $N_{\mathrm{p}x}$ and $N_{\mathrm{p}y}$.

Figure 6

Pressure perturbation and torque density on the sidewalls. $\mathrm{Re}=65$; $\mathrm{Ro}=\infty$. (a), (c) Pressure (solid line) and torque density (dotted line) along the center line at midheight of the sidewalls. (b) Torque density on the sidewalls (the same on $\pm 3/2)$. (a) $y=3/2$, (c) $y=-3/2$.