The governing equations are the unsteady incompressible Navier-Stokes equation with cylindrical coordinates (r, θ, z) (Equation (1)) and the equation of continuity (Equation (2)).

$\frac{\partial u}{\partial t}+\left(u\cdot \nabla \right)u=-\nabla p+\frac{1}{\mathrm{Re}}{\nabla }^{2}u$(1)

$\nabla \cdot u=0$(2)

Here, u is a velocity vector with the components of (u, v, w), t is a dimensionless time based on a representative time, and p is the pressure. The discretization of the governing equations is based on the MAC method. For time integration, Euler’s explicit method is used. For space integration, the QUICK method is used for the convection terms, and the secondary central difference method is used for other terms. As an initial condition, the velocity is set to 0 in all regions. As boundary conditions, the no-slip condition at each cylinder wall is applied for the velocity, and the Neumann condition based on the equation of motion is applied for the pressure. Staggered grids are used as calculation grids and are assumed to have regular intervals in each direction. The number of the grid points in the radial direction is 41, and one in the axial direction is determined linearly according to the height of the cylinders. It was confirmed that the flow state does not change qualitatively even if the number of grids is doubled. The time step was also determined by verifying the Courant number and the diffusion number. The working fluid is assumed to be a glycerin aqueous solution.

To visualize the vortex structure in a flow, we use the Q-criterion to visualize the vortex. It is a way to define the vortex using the coincidence of the vortex region in the flow and the region where the second invariant of the velocity gradient tensor of the flow field is Q > 0. The velocity gradient tensor $D_{ij}$ can be written using the target component $S_{ij}$ and the non-target component ${\Omega }_{ij}$ as follows.

$\begin{array}{c}{D}_{ij}=\frac{\partial u_j}{\partial x_i}=S_{ij}+{\Omega }_{ij}\left(i,j=1,2,3\right)\end{array}$ (3)

The target component is called the strain rate tensor, and the non-target component is called the vorticity tensor. Each can be written as follows.

$\begin{array}{c}{S}_{ij}=\frac{1}{2}\left(D_{ij}+D_{ji}\right)\end{array}$ (4)

$\begin{array}{c}{\Omega }_{ij}=\frac{1}{2}\left(D_{ij}-D_{ji}\right)\end{array}$ (5)

The second invariant of the velocity gradient tensor, called the Q-criterion, can be written using $S_{ij}$ and ${\Omega }_{ij}$ as follows.

$\begin{array}{c}Q=\frac{1}{2}\left({\Omega }_{ij}{\Omega }_{ij}-S_{ij}{S}_{ij}\right)\end{array}$ (6)

Expanding the above equation in the cylindrical coordinate system (r, t, z) yields the following.

$Q=-\frac{1}{2}\left\{{\left(\frac{\partial u}{\partial r}\right)}^2+{\left(\frac{1}{r}\frac{\partial v}{\partial \theta }+\frac{u}{r}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2\right\}$

$\begin{array}{c}-\frac{\partial }{\partial r}\left(\frac{v}{r}\right)\frac{\partial u}{\partial \theta }-\frac{\partial w}{\partial r}\frac{\partial u}{\partial z}-\frac{1}{r}\frac{\partial w}{\partial \theta }\frac{\partial v}{\partial z}\end{array}$ (7)

From this equation, for Q > 0, we can define a region where the rotational motion is greater than the flow strain, which describes the vortex region.

There are four main representative vortex structures that appear in the top view. They are shown on Figure 2 . The left figure is Ring-like vortex. The figure on the right is Bead-like vortex consisting of a series of spherical or elliptical vortices in the shape of beads.

The top view shows a cross-section of 2 mm from the top inside the vessel as shown in Figure 3 . Previous studies have shown that this cross-section is the most likely to exhibit flow characteristics.

Figure 4 shows a visualization of the Q-value at 117 mm radius, Reynolds number 6000, and 100 s. For the top view, two layers of Ring-like vortices are formed. For the side view, two upper and lower target vortices are formed.

Figure 5 shows a visualization of the Q-value at 117 mm radius, 9000 Reynolds number, and 100 seconds. In the top view, long, stretched-looking Bead-like vortex can be observed, with elliptical vortices scattered irregularly on the outside. In the side view, the symmetry is broken, but a coherent vortex shape can be seen.

Figure 6 shows a visualization of the Q-value at 127 mm radius, Reynolds number 6000, and 100 seconds. A special vortex structure was observed under these conditions. In the top view, one stable Ring-like vortex was observed. In the side view, different numbers of vortices were observed in different areas. Four vortices were observed in reference section (a) and the surrounding area. The top two vortices are rounded and the bottom two are elliptical. The red region is biased toward the outside of the vessel. Three vortices were identified in and around the cross section (b) at 120˚ from the reference cross section (a). The top and middle vortices are identical to (a). In the section (c) at 264˚ from (a) and its vicinity, three vortices and one vortex that seems to have failed to form were observed. The lowest vortex has a collapsed shape. The top two vortices are almost the same shape as (a) and (b).

Figure 7 shows a visualization of the Q-value at 127 mm radius, 9000 Reynolds number, and 100 seconds. In the top view, Bead-like vortex with a series of tail-like vortices was observed on the outside, and a vortex transitioning from Ring-like vortex to Bead-like vortex was observed on the inside. In the side view, a collapsed vortex with a top and bottom target was observed.