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Numerical study is performed to investigate the swirling flow around a rotating disk in a cylindrical casing. The disk is supported by a thin driving shaft and it is settled at the center of the casing. The flow develops in the radial clearance between the disk tip and the side wall of the casing as well as in the axial clearance between the disk surfaces and the stationary circular end walls of the casing. Keeping the geometry of the casing and the size of the radial clearance constant, we compared the flows developing in the fields with small, medium and large axial clearances at the Reynolds number from 6000 to 30,000. When the rotation rate of the disk is small, steady Taylor vortices appear in the radial clearance. As the flow is accelerated, several tens of small vortices emerge around the disk tip. The axial position of these small vortices is near the end wall or the axial midplane of the casing. When the small vortices appear on one side of the end walls, the flow is not permanent but transitory, and a polygonal flow with larger several vortices appears. With further increase of the rotation rate, spiral structures emerge. The Reynolds number for the onset of the spiral structures is much smaller than that for the onset of the spiral rolls in rotor-stator disk flows with no radial clearance. The spiral structures in the present study are formed by the disturbances that are driven by a centrifugal instability in the radial clearance and they are penetrated radially inward along the circular end walls of the casing.

Flows around a rotating body and swirling flows around a stationary body give simple but important phenomena including Kármán flow, Bödewadt flow and Taylor vortex flow, and they have engaged historical interests in theoretical, experimental and numerical studies. The cylindrical rotor-stator cavity flow is the flow between a rotating disk and a stationary disk enclosed by an outer casing, and it presents one of the three-dimensional cross flow models [

Rotating flows can be found in fluid machinery and chemical reactors and they are examined to improve their performance [

In this study, we estimate the effect of the axial clearance (aspect ratio) and the rotation rate. The sizes of the outer casing and the radius of the disk are fixed, and three disks with different thicknesses are introduced in order to adjust the aspect ratio. The flow fields with these disks show systematic development of flow structures, as well as structures special to each field.

The schematic sketch of the flow field is shown in _{d}) and the thickness of the disk are 0.127 m and 0.03 m, respectively, and the inner radius and the height of the outer casing are 0.142 m and 0.04 m, respectively. The disk has a driving shaft of radius of 0.01 m at its center. The disk is placed at the center of the casing, and the axial clearance and the radial clearance are 0.005 m and 0.015 m, respectively. Two more disks with thicknesses of 0.032 m and 0.028 m are used. The angular velocity of the disk and the driving shaft is Ω. The reference velocity is the azimuthal velocity component at the tip of the rotating disk ΩR_{d} and the reference length is the radius of the rotating disk R_{d}. All physical variables are made dimensionless by these reference values. The coordinate system is a cylindrical system (r, θ, z), and its origin O is at the center of the lower end wall disk of

the casing. For the disks of thicknesses of 0.032 m, 0.030 m and 0.028 m, the aspect ratios Γ defined by the fraction of the dimensionless axial clearance h_{u} (=h_{l}) to the dimensionless disk radius r_{d} are 0.0315, 0.0394 and 0.0472, respectively. Schouveiler et al. [

The discretization method is based on the finite difference method and the details of the numerical method can be found in our previous paper [

where t is the time made dimensionless by 1/Ω, u is the velocity vector with its radial component u, azimuthal component v and axial component w, p is the pressure and Re is the Reynolds number defined by

The boundary conditions are given by the following equations:

On the inner stationary side of the casing

On the rotating shaft and the rotating disk

Initially, the flow is at rest and the disk is suddenly begins to rotate at a prescribed Reynolds number at t = 0.

The flow is observed in the axial section (i.e. (r, θ) plane), radial section (i.e. (θ, z) plane) and azimuthal section (i.e. (r, z) plane). The vortex structure of the flow is represented by the contour of the normalized helicity,

where ω is the vorticity vector.

In this section, we mainly show the result of the flow at Γ = 0.0394.

When the Reynolds number is low, Taylor vortices appear in the radial clearance between the rotating disk tip and the side wall of the casing.

While the flow at t = 60 fluctuates in the azimuthal direction, the flow at t = 300 is well formed.

After the sudden start of the disk rotation at t = 0, the flow at t = 60 is under development and the profile of the helicity exhibits its variation in the radial and azimuthal directions, and the flow pattern becomes axisymmetric by t = 300 (

When the Reynolds number is higher, a periodic pattern in the azimuthal direction appears in the fully developed flow.

of these small vortices is named as a bead-like flow. The bead-like flow at this Reynolds number is not stable but it is transitory. The small vortices around the disk merge with each other at t = 120 to 200 and another stable state appears. The flow at t = 230 has polygonal (heptagon) flow around the disk tip.

The flow in the (θ, z) plane is shown in

The counterclockwise rotating vortex causes radially outward flow near the bottom end wall. In Taylor- Couette system with finite cylinder length, the vortex with radially outward flow on the end wall is denoted as an anomalous vortex and the existence of the anomalous vortex explains one of the reasons why multiple flow modes appear [

small vortices appears around the disk tip. In

In the radial clearance, two main vortices appear as shown in

While the flow is radially outward in the Ekman layer on the rotating disk, the flow is radially inward in the Bödewadt layer on the stationary end wall of the casing.

As the Reynolds number increases further, new flow structures appear around the disk tip.

The two layered bead-like flow in

At much higher Reynolds number, the well-formed bead-like flow decays and the spiral structures extending toward the center of the disk are established. As an example of this flow,

When the disk is thicker and the aspect ratio Γ is 0.0315, one layered bead-like flow appears.

position where these small vortices emerge is around the midplane in the axial direction (around z = 0.156 in

The Reynolds number dependences of the flow patterns at Γ = 0.0315, 0.0394 and 0.0472 are shown in

The diagram at Γ = 0.0394 is shown in

The flow patterns at Γ = 0.0472 is depicted in

A systematic transition with the increase of the Reynolds number is found at all aspect ratios. First, Taylor vortices emerge in the radial clearance. Then, the bead-like flow appears, which has two layered structure consisting of small vortices near the end walls of the casing. At the higher Reynolds number, the flow has spiral structures around the disk tip. Some of the flow patterns such as the bead-like flow and the polygonal flow are also found in our experimental investigations [

The flow at Γ = 0.0394 contains the bead-like flow that has a vortex structure on one side of the end walls and transits to the polygonal flow. The bead-like and polygonal flow has an asymmetric structure in the axial direction and it appears at the relatively narrow range of the Reynolds number from 6500 to 7300. These flows are obtained from the numerical results of the governing equations. In experiment, these flows may be established, for example, by adjusting the acceleration ratio of the flow [

At the aspect ratios of 0.0315, 0.0394 and 0.0472, the numerically estimated Reynolds numbers for the onset of the two layered bead-like flow are 8300, 7400 and 6900, respectively, and the Reynolds number for the onset of the spiral structures are 15,000, 10,000 and 8600, respectively. That is, as the aspect ratio increases and the axial clearance widens, the flow tends to be unstable. This is supported by the fact that the upper limit of the Reynolds number for the spiral structures, above which the flow turns out to be turbulent, also becomes smaller with the increase of the aspect ratio.

In the enclosed rotor-stator flow with no radial clearance, the Reynolds numbers for the onset of the first unsteady instability of spiral rolls are 70,000, 50,000 and 40,000 for the aspect ratios Γ of 0.0315, 0.0394 and 0.0472, respectively [

The flow around a rotating disk with axial and radial clearances has been investigated by a numerical approach for three sizes of disk thickness. The effect of the aspect ratio and the Reynolds number on the flow patterns, and the development scenario have been examined. The range of the Reynolds number is from 6000 to 30,000. When the Reynolds number is small, Taylor vortices appear in the radial clearance between the rotating disk tip and the side wall of the outer stationary casing. Then, the bead-like flow consisting of a series of small vortices around the disk appears. The two-layered bead-like flow has vortex structures on the both end walls of the casing. The critical Reynolds number for the onset of the two layered bead-like flow increases as the axial clearance becomes narrower. Other than this bead-like flow, two kinds of the bead-like flows are found. The one is the flow that has a vortex structure only on one side of the end walls of the casing. This flow tends to transit to a polygonal flow. This axially asymmetric flow is found at the Reynolds number lower than that for the two- layered bead-like flow. The other is the flow that includes a vortex structure along the axial midplane. The axial position where the bead-like flow appears is near the boundary of the vortices in the azimuthal plane, where the flow is radially outward. As the Reynolds number further increases, spiral structures are formed around the disk tip. The Reynolds number for the appearance of the spiral structures is an order of magnitude smaller than the Reynolds number for the onset of the spiral rolls in the rotor-stator flow.

The authors would like to thank Mr. Shota Hara for his helpful contributions to this work. This work was partly supported by JSPS KAKENHI Grant Number 15K05791.

TakashiWatanabe,HiroyukiFurukawa,ShoheiFujisawa,SomaEndo, (2016) Effect of Axial Clearance on the Flow Structure around a Rotating Disk Enclosed in a Cylindrical Casing. Journal of Flow Control, Measurement & Visualization,04,1-12. doi: 10.4236/jfcmv.2016.41001

h_{c}: Casing thickness made dimensionless by R_{d} (-)

h_{d}: Disk thickness made dimensionless by R_{d} (-)

h_{l}, h_{u}: Dimensionless lower and upper axial clearances, h_{l} = h_{u} = (h_{c} − h_{d})/2 (-)

H_{n}: Normalized helicity (-)

p Pressure made dimensionless by ρ(Ω R_{d})^{2} (-)

r: Radial coordinate made dimensionless by R_{d} (-)

r_{c}: Inner radius of the casing made dimensionless by R_{d} (-)

R_{d}: Disk radius = 0.127 (m)

r_{d}: Radius of the disk made dimensionless by R_{d} = 1.0 (-)

r_{s}: Radius of the driving shaft made dimensionless by R_{d} (-)

t: Time made dimensionless by 1/Ω (-)

u: Velocity vector with components u, v and w made dimensionless by ΩR_{d} (-)

z: Axial coordinate made dimensionless by R_{d} (-)

Γ: Aspect ratio = h_{u}/r_{d} (-)

ν: Kinematic viscosity of the working fluid (m^{2}/s)

θ: Azimuthal angle (-)

ρ: Density of the working fluid (kg/m^{3})

Ω: Angular velocity of the disk and the driving shaft (1/s)

ω: Vorticity vector made dimensionless by Ω (-)