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 ...


I. INTRODUCTION
Flows induced by rotating cones and disks have been investigated as generic models of three-dimensional boundary layers, e.g., flow over a swept wing [1], flow on turbomachinery blades [2] and flows related to some chemical applications such as desalination [3] or condensation [4]. The rotating-cone flow is the general case ( Fig. 1) where the rotating-disk flow is a special case with a cone angle of ψ = 90 • . When the cone angle is large (50 • ψ ≤ 90 • ), the laminar boundary layer developing from the apex (or center in the case of a disk) is susceptible to an inviscid cross-flow instability, whereas for small cone angles (ψ 30 • ), the flow is susceptible to centrifugal instability (ψ = 0 • corresponding to the flow induced by a rotating cylinder) [5]. Thus, the rotating-cone flow is a fundamental and interesting flow manifesting different instabilities depending on the cone angle.
Here, we consider a cone rotating at a rotational rate Ω * in still fluid ( * denotes a dimensional quantity). An orthogonal coordinate system (x, θ, z) is defined as shown in Fig. 1, where x and z are respectively the coordinates along the generating line of the cone and the wall-normal direction with the origin located at the apex, and are non-dimensionalized by a viscous length, δ * ν = ν * /(Ω * sin ψ), where ν * is the kinematic viscosity of the surrounding fluid. θ is the azimuthal coordinate in a frame rotating with the cone surface. When the cone is rotating a boundary layer forms where the fluid is forced to move with the cone in the azimuthal direction due to the no-slip condition at the surface, but it is also transported in the x-direction, hence the cone acts as a centrifugal pump. To compensate for the flow in the x-direction, fluid is drawn towards the cone surface.
The flow on a rotating disk is known as the von Kármán disk flow. In his seminal paper [6], von Kármán showed that there exists a similarity solution for the laminar boundary layer on the rotating disk. A similarity solution can also be obtained for the rotating-cone case as shown by Ref. [7], which (using the viscous length δ * ν as a scaling parameter) gives the same similarity equations as those derived by von Kármán for the rotating disk. The length along the cone surface is non-dimensionalized as x = x * /δ * ν and the square of this value can be seen as a Reynolds number, Re = Ω * x * 2 sin ψ ν * .
On the wider cone (50 • ψ ≤ 90 • ), the flow is susceptible to inviscid cross-flow instability [8,9] at sufficiently high Reynolds numbers. This instability gives rise to slightly inclined disturbances with respect to the azimuthal direction and grow with increasing x to become co-rotating spiral vortices. The disturbances most often and easily observed are stationary with respect to the rotating surface (i.e. fixed in θ for a specified x) because, unless a particular time-dependent disturbance is artificially introduced, the disturbances are primarily triggered continuously by unavoidable surface roughness. For increasing x (or Re) the disturbance amplitude saturates and the stationary vortices "collapse" leading to transition from a laminar to a turbulent boundary layer. These vortices are quite similar to those observed in swept-wing boundary layer [10,11]. In the recent work by Groot et al. [11] a comprehensive collection of references to swept-wing boundary-layer stability and transition can be found.
The stability of the boundary layer on the rotating disk was analyzed by Lingwood [12,13] who found an absolute instability, which was proposed to be responsible for the particular transition Reynolds number found in different experimental studies at that time on smooth disks, although on rough disks transition will occur at a lower Reynolds numbers [14].
This has led to further studies both through experiments (e.g. [15]) and direct numerical simulations [16] of the disk flow (an extensive list of the relevant literature for the rotating disk can be found in [15,16] and also in a recent review [17]).
The recent simulations by Appelquist et al. [16] were able to shed some further light on the transition scenarios on the disk. In the simulations the disturbances were triggered by a stationary forcing at the wall (imitating discrete surface roughness) and two different transition scenarios were found, depending on the height of the surface roughnesses or rather the amplitude of the stationary forcing. With low-amplitude forcing, the onset of secondary instability was observed before the amplitude of the cross-flow vortices reached the saturation level but the flow first became turbulent after the primary disturbance amplitude reached the saturation level. In the case of high-amplitude forcing, the primary disturbance amplitude reached the saturated level before the onset of the secondary absolute instability, and the flow underwent transition at the critical Reynolds number for the secondary global instability.
In the latter case, compared with the former, the secondary instability was observed over a wider range of Reynolds numbers, especially where the spatial gradients of the mean velocity are large.
These cases with low/high-amplitude forcing correspond to experiments without/with artificial roughness elements [15,18], typically with a height of some micrometers, i.e. about two orders of magnitude smaller than δ * ν . However, the secondary instability has not been directly observed in experiments although some flow visualization photographs indicate the appearance of small wave-length disturbances on top of the stationary vortices (e.g., Fig. 8 in Ref. 19) and some kinks in measured time signals are reported (e.g., Fig. 11 in Ref. 18).
For the transition on the clean disk, an interesting method describing the transition stages was proposed by Imayama et al [20]. The method graphically shows different flow characteristics of the transition using the normalized probability density function (PDF) contour map of the azimuthal velocity fluctuation. The PDF method clearly illuminates changes in the wall-normal structure of the flow. They inferred that a topological change in the PDF around 2.0 ≤ z ≤ 2.8 ( Fig. 7 in Ref. 20) may represent a secondary instability.
However, as yet, it has not been shown conclusively whether this change corresponds to a structural change in the spiral vortices or to a secondary instability. This paper is the first experimental work to report on the detailed changes the cross-flow vortices on the cone undergo as they develop. Through measurements of the azimuthal velocity on a cone with ψ = 60 • , for which the inviscid cross-flow instability is expected to be the primary instability (similar to the disk flow), this work aims to address the following issues: (i) effects of the micrometer-sized artificial (regularly spaced in the azimuthal direction) roughness elements on the PDF; (ii) interpretation of the PDF, especially the topological change of the PDF with respect to the vortex structure; (iii) reconstruction of the vortex structure during its development in x; and (iv) exploration of the experimental data to find signs of the secondary instability. In addition, the paper provides a comparison of the cone case with the disk case and comparisons between the PDF and other common measures describing the transition, e.g., root mean square (rms) and Fourier power spectra of the velocity fluctuations.
The paper is organized as follows. Section II briefly describes the experimental setup and methodology and section III describes and discusses the experimental results. It is divided into four subsections: III A shows the mean flow and rms profiles on the cone, whereas III B shows the PDF plots for two different cone cases and makes a comparison with two previously measured disk cases. In III C, a method that allows the vortex structures to be reconstructed from single point data are presented and the resulting vortex structures are discussed in detail. III D shows results where the secondary instability is detected in the some velocity signals and how it is related to vortex meandering. Finally, section IV gives the conclusion of the work. Appendix A describes in some detail a new calibration procedure developed for the hot wires, Appendix B provides a comparison of two different rotational speeds to verify that Reynolds number similarity is valid and Appendix C gives some examples of merging and splitting of vortices on the clean cone.

II. EXPERIMENTAL SET-UP
The rotating-cone facility consists of a precision-made solid aluminium alloy cone mounted on an air bearing and driven by a d.c.-motor. The cone has a base diameter of 474 mm. The cone surface was smoothly finished (resulting in a surface roughness of approx. 1 µm). The cone was accurately aligned and rotated on the vertical axis (the rotational imbalance was approx. 10 µm at the edge). At the edge of the cone, a fixed, horizontal wooden annular plate was positioned flush with the cone surface. The cone was spun at a rotational speed of 900 rpm. This rotational speed was chosen so that transition occurs far enough away from the edge of the cone (x = 629) while the vortex structures can still be sufficiently resolved spatially within the boundary layer.
A single hot-wire probe with its sensing element parallel to the x-direction was used to roughness element Cone rotation is anti-clockwise. measure the azimuthal velocity component. The hot wire has a sensing length of approx. 0.5 mm and a diameter of 2.5 µm. The probe was moved by means of a two-axes traverse system in the x-and z-directions. The velocity was measured at fixed points in the laboratory frame. The signals from a Constant Temperature Anemometer (CTA) and the tachometer of the driving system were recorded simultaneously during approx. 60 s at a sampling rate of 720 data points per cone revolution. In the following the measured velocity was normalized by the local wall velocity V * wall = Ω * x * sin ψ. From the simultaneous recordings, it is possible to reconstruct stationary flow structures. For this kind of analysis the velocity signal was post-processed using a high-pass filter (ω * /Ω * > 15) for x < 510 on the clean cone and for x < 430 on the cone with roughnesses, where ω * is the disturbance angular frequency (in the lab frame). Note that if the vortices are fixed with respect to the cone surface, then ω * /Ω * = ω gives the azimuthal wavenumber. For accurate calibration and positioning of the hot wire in the boundary layer, a new calibration methodology was developed. In the work on the rotating disk by Imayama et al. [20] the calibration of the hot wire was completed using the laminar boundary layer profile by first optically determining the position of the probe accurately with respect to the disk surface. However, for the cone case the optical determination of distance from the wall was less accurate due to the curved surface; instead the calibration curve and the distance from the cone surface to the hot wire was simultaneously obtained based on a least-square method from measurements in the laminar boundary layer. The details can be found in Appendix. A.
In order to introduce reproducible stationary disturbance, artificial roughness elements were mounted on the cone surface. As shown in Fig 13045) was used to create the roughness elements, which was also the case for Ref. 15. Each element has a circular shape with a diameter and height of approximately 2 mm and 4 µm, which correspond to about 5δ * ν and 0.01δ * ν , respectively.  thickness δ * 90 , where V becomes 10% of V * wall , also agrees with that based on the similarity solution, i.e. z = 2.81. Below x = 498 and x = 461 without/with roughnesses, the difference between the measured mean velocity and the similarity solution does not exceed 5% of the local wall velocity. The artificial roughnesses do not seem to affect the time-averaged laminar profiles although the roughnesses promote transition. Beyond the laminar regime, the boundary-layer thickness increases significantly and the profile approaches the typical turbulent profile.
A corresponding behavior is also seen in the rms of the azimuthal velocity v rms in Fig. 4.
In the laminar regime, one peak of v rms is observed, which is caused by the spiral cross-flow vortices. During the transition process the boundary layer thickens, and the peak of v rms becomes broader and extends further into the outer layer. In order to further investigate the process of transition to turbulence, we use the PDF approach in the same manner as Ref. 20 in section III B, where we compare the transition process on the cone with that on the disk using the PDF map of v around the local maximum of v rms (as shown by the dotted line in  of the outer contour of the PDF changes suddenly and PDF begins to be skewed.   As another characterization, Fourier power spectra from time signals measured on the cone at z = 1.2 are shown in Fig. 6. Here, we calculated the spectrum with a frequency resolution of ∆ω * /Ω * = 0.2 based on Welch's method with a segment length of 5 cone revolutions and 0% overlap (The time signal was split into segments and the spectrum was calculated for each segment. Finally, these spectra were averaged.). The power spectral density E of the fluctuation v (without high-pass filtering) is shown for different x-locations on the clean cone (a) and the cone with roughness elements (b). On the clean cone, a broad peak for 12 ω * /Ω * 36 is observed at x = 328 for which the amplitude increases with x. This broad peak contains spikes at integer values of ω * /Ω * , indicating the stationary disturbances with respect to the cone surface, namely, the stationary spiral vortices. Similar spikes were also observed on the clean disk in Ref. [18,23]. Initially, the dominant wavenumber in the peak changes within a range of 20 to 26 in repeated measurements with the same setup The data for the three revolutions indicate that the branched maximum in z 1.2 at x = 498 in Fig. 7(C ) is caused by a plateau in the v-signal seen for the three revolutions in Fig. 9(b) and (c). As z increases, the plateau is observed at higher values of v. Therefore, the branched maximum in Figure 7(C ) shifts toward higher v. A similar plateau or small kink is also shown in Fig. 8(b) although it is not as clear. According to Fig. 7, this feature seems to be common both in the cases with and without roughnesses.
The gray contours on the right of Figs. 8 and 9 show that the spiral vortices actually move in the azimuthal direction during the cone revolutions although they are usually denoted as "stationary" vortices. Interestingly, all vortices shift in the same way simultaneously and the number of vortices remains the same as the number of roughnesses, which is consistent with the sharp peak in the spectra shown in Fig. 6(b). The red line on each contour plot of  shown (corresponding to sections A , B and C in Fig. 5 and Fig. 7). Plot (a) in each figure shows the phase average of the azimuthal velocity field V + v of the sample population, where V (z; x) is the mean velocity shown in Fig. 3 A similar inclined structure is also observed in DNS for the disk case ( Fig. 11 in Ref. 24).
In Fig. 12(b), the high-momentum upwelling region begins to be distorted and a spacing appears between high-and low-momentum regions in 1.  Fig. 7(B ).
Further away from the wall, the branched maximum in the PDF indicates the top part of the high-momentum upwelling. In Fig. 13(b), the distortion and spacing become more significant and the branched maximum extends toward the outer layer in Fig. 7(C ). Thus, the branched maximum in Fig. 7(B ) and (C ) indicates the distortion of the high-momentum area, or overturning process of the high-momentum upwelling.
The detailed overturning process discussed above is not observable in the phase-averaged velocity fieldṽ. As observed in Fig. 8 and Fig. 9, the vortex meandering causes phase-shifting of the waveforms and each plateau is smoothed out in the phase-averaged velocity. However, the PDF extracts only the value of v without the phase and is, therefore, unaffected by the phase-shifts. Thus, the PDF detects the overturning process from the point measurement without any complicated vortex-location detection. The PDF can probably also detect the overturning process (in a time-averaged sense) on the clean cone (see the similar branched maxima in Fig. 7(C) and in Fig. 7(C )), even when vortex splitting and merging occur and the reconstruction of the snapshot becomes more difficult. This is clearly a distinct advantage of the PDF method.

D. Secondary instability and vortex meandering
To investigate the secondary instability, we evaluate the mean-flow distortion. In Figs Another point of interest is the relation between the secondary instability and vortex meandering. Figure 14 shows the "sorted" azimuthal velocity fluctuation v at x = 498 for three different z; the same data as Fig. 9 are shown but all the revolutions are sorted according to the detected vortex location, which is marked by the solid white curve. As Taking a close look at Fig. 9, one can see the irregular oscillations, e.g., around the 700th revolution in Fig. 9(a) but not in Fig. 8.   For a clearer visualization of this trend, the velocity fluctuation v at some single revolutions marked by the dashed lines in Fig. 14 is plotted as curves in Fig. 15; the sorted v for the 20th, 150th, 300th, 600th, 750th, 870th revolutions corresponds to the curves from the bottom to the top of each sub-plot in Fig. 15. Note that the most probable 100 samples that were used for the reconstruction of the "snapshot" in the previous section correspond to data between two squares in Fig. 14 and are located in the range between the 300th and 600th revolutions. Figure 15 clearly indicates that v begins to contain high-frequency oscillations at higher values of the sorted cone revolution, corresponding to that the vortex is shifted downstream (larger θ in Fig. 14), at all wall heights. Especially, data for the 750th and 870th revolutions contain waves, with a wavelength much shorter than the one of the primary vortex (15 • in θ). It should be mentioned that the waveforms for the 750th and 870th revolutions are similar to the ones shown in figures 18(b) and (c) in Ref. [16].
A similar evaluation was also made for the clean cone data and the velocity fluctuation  To characterize the effects of the vortex meandering on the spiral vortex structure, we also consider the aspect ratio of the vortex cross section. Figure 17(a) shows the phase-averaged velocity fieldṽ at z = 1.2 on the cone with artificial roughnesses in a gray contour scale.
The solid and dash-dotted lines indicate the mean vortex locations detected by searching for the minimum of v for the 100 samples around the mean vortex location and for the 100 samples when the vortex are located farthest downstream (the last 100 revolutions in Fig. 14(a), for example. For each case, no fitting was applied for this case). Here, we define a new orthogonal curvilinear coordinate system (x , y , z ) on the detected vortex locations as shown in Fig. 17(b). The x -and y -axes are normal and parallel to the vortex axes. The z -axis is normal to the cone surface (same as z). The angle is defined as the angle of the y -axis with respect to the azimuthal direction. We estimate the aspect ratio of the spiral vortex in the x z -plane, AR = δx /δ 90 , where the width of the vortex is defined as δx = (2πx sin ψ sin )/24. The vortex height is estimated as the 90% boundary-layer thickness δ 90 . It should be noted that δ 90 is a rough measure of the vortex height and tends to underestimate the vortex height in the transition region (see Fig. 13: the high-momentum upwell is located clearly beyond the 90% boundary-layer thickness, δ 90 = 4.6.).
The effects of the vortex meandering on the vortex aspect ratio as a function of x are shown in Fig. 18. In this figure we divide the vortices into three categories (with 100 revolutions each) depending on their azimuthal location relative to the mean vortex position at each x-and z-position: those located upstream (having the vortex axis at smaller θ, i.e., the first 100 revolutions in Fig. 14, for example) are marked by black symbols; those located downstream (having the vortex axis at larger θ, i.e., the last 100 revolutions in Fig. 14) are marked by blue symbols; and those located around the mean are here called "intermediate" (100 revolutions around the average) marked by red symbols. Figure 18(a) shows the 90% boundary-layer thickness δ 90 for the three cases as well as the time-averaged thickness, whereas Fig. 18(b) shows the angle of the vortex with respect to the azimuthal direction as shown in Fig. 17(b). In Fig. 18(c) the aspect ratio AR of the vortex in the x z -plane is

IV. CONCLUSIONS
The present work reports on the unstable region and the transition process of the boundary-layer flow on a rotating cone with a half angle of 60 degrees using the PDF method introduced by Imayama et al. [20]. The PDF of the azimuthal velocity fluctuation v shows the transition process in a consistent manner both on the rotating disk and on the rotating cone, and in both cases with and without artificial roughness elements (Fig. 5).
For the early development stage of the spiral cross-flow vortices, the PDF shows the difference due to the uniformity of the initial disturbances; in the case with artificial roughness elements mounted uniformly in the azimuthal direction, the PDF has two maxima corresponding to positive and negative values of v. On the clean cone/disk, the PDF has only one maximum around v = 0 because of the amplitude modulation of the v-signal. In the transition stage, however, the PDFs show a similar pattern for both the disk and cone, with and without the roughness elements.
In addition to the PDF, the velocity fluctuations were carefully examined and the most probable vortex structure was reconstructed from the time signals measured by a single point hot wire (Figs. 11, 12 and 13). The results show that the PDF detects the overturning process of the high-momentum upwelling of the spiral vortices (Fig. 7). This overturning process cannot be detected in the phase-averaged velocity because of the meandering of the spiral vortices. Thus, the PDF method has a distinct advantage when assessing the structural nature and development of the cross-flow vortices from single-point measurements. The PDF method easily detects the overturning process without any simultaneous plane measurement such as PIV or complicated vortex reconstruction (which was undertaken in this work in order to verify the method).
Also, our measurement captured the high-frequency oscillations, which may be related  rotating cone is known (see for instance Ref. [9]).
The calibration procedure is based on the velocity distribution within the boundary layer being known as function of the wall-normal coordinate and the wall speed. If the wall distance were known accurately, it would then be easy to find the calibration curve but, in the present case, the absolute distance to the wall was not known accurately enough.
However, by taking calibration points at different rotational speeds at several fixed wall distances, where the increments between the various wall distances were accurately known, it was possible to estimate the position of the wall relative to the hot-wire sensor through an error minimization procedure. Such a data set of calibration points is shown in Fig. 19 for five different rotational speeds and six different wall-normal positions.
The velocity at the position of the hot wire is given as where z * m is the indicated wall distance of the traversing system (adjusted using the optical camera technique) and z * 0 is the unknown error in the wall distance. Here, in order to regard z * 0 as a constant, we aligned the traverse system with the generating line of the cone using a laser distance meter; the laser distance meter was mounted on the traversing system in the same way as the hot-wire probe and it ensured the distance from the cone surface remained in the range of ±10 micrometers (less than 0.1 in the non-dimensional z) at different x-locations.
By choosing a specific voltage (say E * i ) and using Fig. 19 and interpolating to get the corresponding rotational speed for the different heights (Ω * ij = 2πN ij /60), we know that for all heights we should have the same velocity. We illustrate this in Fig. 19 by . For each voltage E * i we calculate the variance of the fitting to Eq. A1 by varying z * 0 in 1 µm steps. We do that for all j values and then calculate the total rms error by taking the square root of the sum of the variances. For the data in Fig. 19 we use voltages in the range E * i = 4.1 to 5.0 Volt, with 0.1 Volt increments, i.e. ten different voltages. Each voltage can be obtained from different heights z * ij and corresponding different rotational rates Ω * ij that are interpolated from the data as shown in Fig. 19. Given E * i is a function of velocity independent of position and angular velocity, it is now possible to obtain an estimate of the real distance from the wall. For a given E * i we can estimate the velocity at each of the points j (1 ≤ j ≤ n i ) for a given displacement z * 0 from Eq. A1 such that V * ij = Ω * ij x * sin ψf (z * ij + z * 0 )/δ * ν .
However, since we do not know the value of z * 0 we sweep z * 0 over a certain range with small steps (1 µm) and calculate V * ij for each z * 0 . By calculating the average velocity for each value of z * 0 , we obtain an estimate of the velocity, V * i (z * 0 ) at that value of E * i , and a corresponding measure of the deviation for each value of z * 0 as We do this operation for all values of E * i that we have chosen (1 ≤ i ≤ M ) and calculate Finally, we find the value of z * 0 that gives the smallest total deviation (see Fig. 20 where we have used M = 10) and assume that this displacement gives us the best estimate of the distance to the wall as z * m + z * 0 . The result of the optimization procedure is found in Fig. 20, which shows a distinct error minimum. The preliminary optical determination of the wall position is seen to be off by approximately 96 µm. As can be seen, the rms error in velocity is less than 0.1% of the wall velocity.
The resulting calibration curves are shown in Fig. 21, both with the approximate, optically determined, wall position (left) and with the wall position determined from the optimization procedure (right). The line is obtained from a least-square fit of the modified King's law [25] given by where E * is the anemometer voltage at the velocity V * , E * 0 is the voltage at zero velocity taken when the cone is at rest, and k * 1 , k * 2 and n are calibration constants. According to the original King's law n = 0.50 for an infinitely long cylinder, here we find a value of 0.48.
This kind of method to calibrate hot wires against laminar boundary profiles can be used for different flows where the exact wall position is unknown. The methodology does not only give a calibration curve but also gives a good estimate of the wall position.  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).

Appendix B: Effect of the rotational speed
In this paper we present results from the 60 degree rotating cone with a rotation speed of 900 rpm. The instability and further development of the vortex structure should be solely dependent on the Reynolds number and not on the rotation rate per se. However, the rotation speed affects the viscous length scale of the flow ( ν * /(Ω * sin ψ)) so both the length x * along the generatrix as well as the height of any roughness, which is inevitable on the cone surface, in terms of the viscous length scale vary with the rotational speed. In   Fig. 22, the PDF plots of the disturbance as function of x for both the 1800 rpm (a) and 900 rpm (b) clean cone cases are shown; the latter shows the same data as in Fig. 5(b). It is clear from the figures that there is almost no difference between the two cases showing that the Reynolds number similarity holds as expected. It also shows that for these experiments the different non-dimensional lengths from the apex to the rim of the cone do not influence the transition process.
Appendix C: Vortex splitting and merging on the clean cone In this section, we report the splitting and merging of the spiral vortices showing the velocity fluctuation v in a similar way to that in Figs. 8 and 9. Figure 23 shows v for the whole sample set at x = 535 on the clean cone (to show the splitting and merging, only half of the cone is shown.). In the area occupied by the vortices with lower amplitude (θ 75 • ), some vortices merge into one vortex and one vortex splits into two vortices. In Fig. 23(a), for example, a vortex at θ = 90 • splits into two around the 142nd revolution and two vortices merge into one at θ = 105 • around the 624th revolution. Because of the merging and splitting, the number of vortices changes through the revolutions. In contrast, vortices with larger amplitude (θ 75 • ) do not merge nor split although the vortex meandering occurs in a similar manner to the case with the artificial roughness elements (Figs. 8 and 9).