Investigation of the structures in the unstable rotating-cone boundary layer

This work reports on the unstable region and the transition process of the boundary-layer flow induced by a rotating cone with a half apex angle of 60 degrees using the probability density function (PDF) contour map of the azimuthal velocity fluctuation, which was first used by Imayama et al. [Phys. Fluids 24, 031701 (2012)] for the similar boundary-layer flow induced by a rotating disk. The PDF shows that the transition behavior of the rotating-cone flow is similar to that on the rotating disk. The effects of roughness elements on the cone surface have been examined. For the cone with roughnesses, we reconstructed the most probable vortex structure within the boundary layer from the hot-wire anemometry time signals. The results show that the PDF clearly describes the overturning process of the high-momentum upwelling of the spiral vortices, which due to vortex meandering cannot be detected in the phase-averaged velocity field reconstructed from the point measurements. At a late stage of the overturning process, our hot-wire measurements captured high-frequency oscillations, which may be related to secondary instability.

Figure 1

The coordinate system $(x,\theta ,z)$ for the rotating cone. We also define a radius $r=xsin\psi$. The velocity in the azimuthal direction is denoted as $V+v$ where $V(x,z)$ is the averaged velocity in $\theta$ (and time) and $v(t;x,z,\theta )$ is the deviation from $V$ at a given position $(x,z,\theta$) as function of time. The time and $\theta$ average of $v$ is zero, but the time average of $v$ is not in general zero at a fixed position in space.

Figure 2

The 60-degree cone with 24 artificial roughness elements that are uniformly mounted on the cone surface at $x^{*}=115.7\pm 0.5\mathrm{mm}$. At the top of the figure, the hot-wire probe can be seen. Cone rotation is anti-clockwise.

Figure 3

Profiles of azimuthal mean velocity $V$ (circles) in the laboratory frame (a) without and (b) with roughnesses at $x=267$. The solid curves show the similarity solution. The thick line at $z=2.81$ indicates the 90% boundary-layer thickness ${\delta }_{90}$ for the similarity solution, where $V$ becomes 0.10. The squares with the dashed lines show the measured 90% boundary-layer thickness.

Figure 4

Profiles of rms of azimuthal velocity $v_{\text{rms}}$ (circles) (a) without and (b) with roughnesses at $x=267$. The thick line at $z=2.81$ indicates the 90% boundary-layer thickness for the similarity solution, where $V$ becomes 0.10. The squares with the dashed lines show the measured 90% boundary-layer thickness. The dotted lines indicate the height $z=1.2$ where measurements for Figs. 5 and 5 were conducted.

Figure 5

The PDF of the azimuthal velocity fluctuation $v$: (a) clean disk at $z=1.3$, 1400 rpm from Ref. [20], (b) ${60}^{\circ }$ clean cone at $z=1.2$, 900 rpm, (c) ${60}^{\circ }$ cone with roughness elements at $z=1.2$, 900 rpm and (d) disk with 32 roughness elements at $z=1.3$, 1013 rpm from Ref. [15]. In (c) and (d), the roughness elements are located at $x=267$ and 287, respectively. Filled contours indicate 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% of the local maxima of PDF. Dashed lines (a) and (b) indicate transition locations reported in Ref. [22]. The alphabetical symbols indicate the sections shown in Fig. 7.

Figure 6

Fourier power spectra $E$ of azimuthal velocity component $v$ at $z=1.2$ for different $x$ locations on (a) clean cone and (b) cone with roughness elements: ${\omega }^{*}$ indicates the disturbance angular frequency in the laboratory frame. Thus, for vortices fixed with respect to the cone surface, ${\omega }^{*}/{\mathrm{\Omega }}^{*}$ indicates the azimuthal wave number.

Figure 7

The PDF of the azimuthal velocity fluctuation $v$ showing the $z$ structure at different $x$ locations: (A) $x=498$, (B) $x=516$, (C) $x=535$, (D) $x=554$, (E) $x=591$ on the clean cone [cf. Fig. 5], and (${\mathrm{A}}^{\prime }$) $x=461$, (${\mathrm{B}}^{\prime }$) $x=479$, (${\mathrm{C}}^{\prime }$) $x=498$, (${\mathrm{D}}^{\prime }$) $x=516$, (${\mathrm{E}}^{\prime }$) $x=535$ on the cone with roughness elements [cf. Fig. 5].

Figure 8

Velocity fluctuation $v$ for single revolutions (thin colored lines to the left; blue, green, and red lines indicate $v$ for revolution numbers 300, 301, and 302, respectively), phase-averaged velocity $\tilde{v}$ (thick black line to the left) and $v$ for the whole sample set (to the right) at $x=479$ (${\mathrm{B}}^{\prime }$ in Figs. 5 and 7). The line plots and contours (dark and light indicate negative and positive, respectively) are scaled by three times the local rms of the azimuthal velocity $v_{\text{rms}}$, corresponding to (a) $\pm 3v_{\text{rms}}=\pm 2.4\times {10}^{-1}$ at $z=0.6$; (b) $\pm 3v_{\text{rms}}=\pm 2.8\times {10}^{-1}$ at $z=1.2$; and (c) $\pm 3v_{\text{rms}}=\pm 2.0\times {10}^{-1}$ at $z=2.4$. The detected location of the vortex is marked by the red line on the right-hand plot (see ${10}^{\circ }\lesssim \theta \lesssim {30}^{\circ }$).

Figure 9

Velocity fluctuation $v$ for single revolutions (thin colored lines to the left; blue, green, and red lines indicate $v$ for revolution numbers 300, 301, and 302, respectively.), phase-averaged velocity $\tilde{v}$ (thick black line to the left) and $v$ for the whole sample set (to the right) at $x=498$ (${\mathrm{C}}^{\prime }$ in Figs. 5 and 7). The line plots and contours (dark and light indicate negative and positive, respectively) are scaled by three times the local rms of the azimuthal velocity $v_{\text{rms}}$, corresponding to (a) $\pm 3v_{\text{rms}}=\pm 3.7\times {10}^{-1}$ at $z=1.2$; (b) $\pm 3v_{\text{rms}}=\pm 2.6\times {10}^{-1}$ at $z=2.4$; and (c) $\pm 3v_{\text{rms}}=\pm 2.5\times {10}^{-1}$ at $z=3.2$. The detected location of the vortex is marked by the red line on the right-hand plot (see ${20}^{\circ }\lesssim \theta \lesssim {45}^{\circ }$).

Figure 10

Probability density function of the azimuthal location of the spiral vortex at (a) $x=461$, (b) $x=479$, and (c) $x=498$: the red cross shows the mean azimuthal locations of the whole population (approx. 900 revolutions) at each wall height. The most probable location of the vortex is determined by a linear fitting of the mean locations (shown by the straight lines). The most probable 100 samples were extracted around the most probable location of the vortex.

Figure 11

Reconstructed vortex structures based on the 100-sample population at $x=461$ (${\mathrm{A}}^{\prime }$ in Figs. 5 and 7): (a) phase-averaged azimuthal velocity $V+\langle v\rangle$; (b) $\langle v\rangle$; (c) $z$ gradient of the phase-averaged azimuthal velocity $\partial (V+\langle v\rangle )/\partial z$; (d) $\theta$ gradient of the of the phase-averaged azimuthal velocity $\partial \langle v\rangle /\left(r\partial \theta \right)$; and (e) rms of the 100 samples. The lines in (a), (c), and (d) are the same and show contours with a spacing of 0.1. The lines in (b) show contours with a spacing of 0.05. The green and red contours in (e) indicate isoline of $\pm 0.2$ in the $z$ and $\theta$ gradients of the mean flow. Negative contours are dashed.

Figure 12

Reconstructed vortex structures based on the 100-sample population at $x=479$ (${\mathrm{B}}^{\prime }$ in Figs. 5 and 7): (a) phase-averaged azimuthal velocity $V+\langle v\rangle$; (b) $\langle v\rangle$; (c) $z$-gradient of the phase-averaged azimuthal velocity $\partial (V+\langle v\rangle )/\partial z$; (d) $\theta$-gradient of the of the phase-averaged azimuthal velocity $\partial \langle v\rangle /(r\partial \theta$); and (e) rms of the 100 samples. The lines in (a), (c), and (d) are the same and show contours with a spacing of 0.1. The lines in (b) show contours with a spacing of 0.05. The green and red contours in (e) indicate isoline of $\pm 0.2$ in the $z$ and $\theta$ gradients of the mean flow. Negative contours are dashed.

Figure 13

Reconstructed vortex structures based on the 100-sample population at $x=498$ (${\mathrm{C}}^{\prime }$ in Figs. 5 and 7): (a) phase-averaged azimuthal velocity $V+\langle v\rangle$; (b) $\langle v\rangle$; (c) $z$ gradient of the phase-averaged azimuthal velocity $\partial (V+\langle v\rangle )/\partial z$; (d) $\theta$ gradient of the phase-averaged azimuthal velocity $\partial \langle v\rangle /(r\partial \theta$); and (e) rms of the 100 samples. The lines in (a), (c), and (d) are the same and show contours with a spacing of 0.1. The lines in (b) show contours with a spacing of 0.05. The green and red contours in (e) indicate isoline of $\pm 0.2$ in the $z$ and $\theta$ gradients of the mean flow. Negative contours are dashed.

Figure 14

Velocity fluctuation $v$ (gray contour) sorted by the detected vortex location (shown by the solid curve) at $x=498$ (${\mathrm{C}}^{\prime }$ in Figs. 5 and 7). The cross shows the mean location of the vortex at each wall height: (a) $z=1.2$, (b) $z=2.4$, and (c) $z=3.2$. The data between two squares were used for the reconstruction. The contours are scaled in the same way as in Fig. 9. The dashed lines indicate the data shown in Fig. 15.

Figure 15

Velocity fluctuation $v$ at $x=498$ (${\mathrm{C}}^{\prime }$ in Figs. 5 and 7) on the cone with roughnesses for different vortex locations during the vortex meandering: the curves from the bottom to top in each figure show the velocity fluctuation $v$ for the 20th, 150th, 300th, 600th, 750th, and 870th revolutions of the sorted data, corresponding to the data marked by the dashed lines in Fig. 14. The plots are scaled by twice the local rms of the azimuthal velocity $v_{\text{rms}}$, corresponding to (a) $\pm 2v_{\text{rms}}=\pm 2.5\times {10}^{-1}$ at $z=1.2$; (b) $\pm 2v_{\text{rms}}=\pm 1.7\times {10}^{-1}$ at $z=2.4$; and (c) $\pm 2v_{\text{rms}}=\pm 1.7\times {10}^{-1}$ at $z=3.2$.

Figure 16

Velocity fluctuation $v$ for typical single revolutions when the vortex is shifted (relative to the mean vortex position) upstream (bottom), intermediate (middle), and downstream (top). All data correspond to those marked by the dashed lines in Fig. 22 ($x=535$ on the clean cone, C in Figs. 5 and 7). Each plot is scaled by twice the local rms of the azimuthal velocity $v_{\text{rms}}$, corresponding to (a) $\pm 2v_{\text{rms}}=\pm 2.2\times {10}^{-1}$ at $z=1.2$; (a) $\pm 2v_{\text{rms}}=\pm 1.6\times {10}^{-1}$ at $z=2.4$; and (a) $\pm 2v_{\text{rms}}=\pm 1.4\times {10}^{-1}$ at $z=3.2$.

Figure 17

Phase-averaged velocity $\tilde{v}$ on the cone with roughnesses at $z=1.2$: The white dots are at $x=267$ and show the location of the roughnesses. The solid and dash-dotted lines mark the detected most probable vortex location and shifted location downstream based on the tracking of the local minimum of averaged $v$ for the 100-sample population. The dashed lines indicate the locations at $x=250$, 300, 350, 400, 450, 500, and 550. (b) Blow-up of (a) with the orthogonal curvilinear coordinate system ($x^{\prime }, y^{\prime }$) fixed on the vortex. The angle $\epsilon$ is the angle of the vortex ($y^{\prime }$ axis) with respect to the azimuthal direction.

Figure 18

Effect of the vortex meandering on the 90% boundary-layer thickness and spiral vortices as a function of $x$: (a) 90% boundary-layer thickness ${\delta }_{90}$, (b) angle of the vortex $\epsilon ,$ and (c) aspect ratio $AR$ of the vortex on the $x^{\prime }{z}^{\prime }$ plain. The black, red, and blue markers show quantities calculated by data for 100 revolutions, at which the vortex is located upstream of, intermediate to, and downstream of the mean, respectively. In (b), the dots correspond to the angle calculated based on the detected vortex axis (shown in Fig. 17) at $z=1.2$. The circles are interpolated angles at the location where the profile measurements were conducted. The squares in (a) show the time-averaged 90% boundary-layer thickness, shown in Fig. 3 as a reference. Sections at $x=461$, 479, 498 are denoted by ${\mathrm{A}}^{\prime }, {\mathrm{B}}^{\prime }, {\mathrm{C}}^{\prime }$, in Figs. 5 and 7.

Figure 19

Calibration voltages as a function of rotational speed for six different wall-normal positions. The figure also illustrates how, for a given $E_i^{*}$, we can determine the corresponding rotational speeds, $N_{i1}^{*}$ to $N_{i4}^{*}$.

Figure 20

Results of the optimization procedure to find the position relative to the wall.

Figure 21

Calibration curves including all data points in Fig. 19 where the optically determined wall distance is used for the points in (a), whereas the calibration curve using the optimized wall distance is shown in (b).

Figure 22

The PDF of the azimuthal velocity fluctuation $v$: (a) ${60}^{\circ }$ clean cone at $z=1.2$, 1800 rpm, (b) ${60}^{\circ }$ clean cone at $z=1.2$, 900 rpm. Filled contours indicate 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% of the local maxima of PDF.

Figure 23

Velocity fluctuation $v$ for the full sample population at $x=535$ on the clean cone (C in Figs. 5 and 7). The contours (dark and light indicate negative and positive, respectively) are scaled by three times the local rms of the azimuthal velocity $v_{\text{rms}}$, corresponding to (a) $\pm 3v_{\text{rms}}=\pm 3.3\times {10}^{-1}$ at $z=1.2$; (b) $\pm 3v_{\text{rms}}=\pm 2.4\times {10}^{-1}$ at $z=2.4$; and (c) $\pm 3v_{\text{rms}}=\pm 2.2\times {10}^{-1}$ at $z=3.2$. The three dashed lines from the left to right indicate where the vortex is shifted upstream, intermediate and downstream: the 415, 564, and 831 revolutions for (a), the 105, 359, and 524 revolutions for (b), and the 402, 444, and 523 revolutions for (c) are examined in Fig. 16.