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^{1,a)}, P. Henrik Alfredsson

^{1,b)}and R. J. Lingwood

^{1,2,c)}

### Abstract

A new method of graphically representing the transition stages of a rotating-disk flow is presented. The probability density function contour map of the fluctuating azimuthal disturbance velocity is used to show the characteristics of the boundary-layer flow over the rotating disk as a function of Reynolds numbers. Compared with the variation of the disturbance amplitude (rms) or spectral distribution, this map more clearly shows the changing flow characteristics through the laminar, transitional, and turbulent regions. This method may also be useful to characterize the different stages in the transition process not only for the rotating-disk flow but also for other flows.

This research is supported by the Swedish Research Council (VR) and KTH. We also acknowledge the help from late Dr. Tim Nickels in arranging the loan of the experimental apparatus from the University of Cambridge Department of Engineering to KTH. We also thank the referees for useful comments and for pointing out Ref. 11 to us.

### Key Topics

- Flow instabilities
- 29.0
- Rotating flows
- 25.0
- Reynolds stress modeling
- 17.0
- Turbulent flows
- 12.0
- Transition to turbulence
- 5.0

## Figures

The experimental set-up of the rotating disk.

The experimental set-up of the rotating disk.

Mean azimuthal velocity profiles at *R* = 430 (○), 470 (*), 510 (×), 550 (□), 590 (◊), 630 (▽). Solid line is the laminar theory profile.

Mean azimuthal velocity profiles at *R* = 430 (○), 470 (*), 510 (×), 550 (□), 590 (◊), 630 (▽). Solid line is the laminar theory profile.

Profiles of *v* _{ rms }. The symbols are the same as in Fig 2.

Profiles of *v* _{ rms }. The symbols are the same as in Fig 2.

Fourier power spectra for ensemble-averaged time series measured at *z* = 1.3 at (a) *R* = 430, (b) *R* = 470, (c) *R* = 510, (d) *R* = 550, (e) *R* = 590, and (f) *R* = 630.

Fourier power spectra for ensemble-averaged time series measured at *z* = 1.3 at (a) *R* = 430, (b) *R* = 470, (c) *R* = 510, (d) *R* = 550, (e) *R* = 590, and (f) *R* = 630.

*v* _{ rms } variance measured at *z* = 1.3 and a constant rotational speed Ω* = 1400 rpm. Solid and dashed lines are exponential fittings for each instability region given as *v* _{ rms } ∼ exp (α*R*), where α is the growth rate. These coefficients for solid and dashed lines are α = 0.058 and 0.017, respectively. Circles denote unfiltered signal, triangles show band-pass filtered signal (17 < ω*/Ω* < 70) below *R* ⩽ 490 and high-passed filtered signal (17 < ω*/Ω*) for 495 ⩽ *R* ⩽ 525.

*v* _{ rms } variance measured at *z* = 1.3 and a constant rotational speed Ω* = 1400 rpm. Solid and dashed lines are exponential fittings for each instability region given as *v* _{ rms } ∼ exp (α*R*), where α is the growth rate. These coefficients for solid and dashed lines are α = 0.058 and 0.017, respectively. Circles denote unfiltered signal, triangles show band-pass filtered signal (17 < ω*/Ω* < 70) below *R* ⩽ 490 and high-passed filtered signal (17 < ω*/Ω*) for 495 ⩽ *R* ⩽ 525.

The PDF of the filtered instantaneous azimuthal fluctuation velocity *v* at *z* = 1.3 normalized by the wall speed. Filled contours indicate 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% of the local PDF value.

The PDF of the filtered instantaneous azimuthal fluctuation velocity *v* at *z* = 1.3 normalized by the wall speed. Filled contours indicate 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% of the local PDF value.

The PDF of the instantaneous azimuthal fluctuation velocity normalized by the wall speed to show the *z*-structure at (a) *R* = 470, (b) *R* = 490, (c) *R* = 510, (d) *R* = 530, (e) *R* = 550, (f) *R* = 570, (g) *R* = 590, (h) *R* = 610, and (i) *R* = 630, namely cases P02 to P09 in Table I. Filled contours indicate same as Fig. 6. The range of the abscissa is −0.5 to +0.5 for all *R*. Note that in the free stream far above the disk, the positive values of *v* emanates from the high velocity fluid near the disk giving a positive skewness, i.e., the picture here is opposite to the one that would be observed for the flow over a stationary plate where the skewness is negative close to the boundary-layer edge. The white + signs in (e) and (f) show the position of the double peaks.

The PDF of the instantaneous azimuthal fluctuation velocity normalized by the wall speed to show the *z*-structure at (a) *R* = 470, (b) *R* = 490, (c) *R* = 510, (d) *R* = 530, (e) *R* = 550, (f) *R* = 570, (g) *R* = 590, (h) *R* = 610, and (i) *R* = 630, namely cases P02 to P09 in Table I. Filled contours indicate same as Fig. 6. The range of the abscissa is −0.5 to +0.5 for all *R*. Note that in the free stream far above the disk, the positive values of *v* emanates from the high velocity fluid near the disk giving a positive skewness, i.e., the picture here is opposite to the one that would be observed for the flow over a stationary plate where the skewness is negative close to the boundary-layer edge. The white + signs in (e) and (f) show the position of the double peaks.

## Tables

Experimental conditions, where *r** and *z* represent the radial and axial positions of the probe, respectively.

Experimental conditions, where *r** and *z* represent the radial and axial positions of the probe, respectively.

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