On boundary-layer transition over a rotating broad cone

The route to turbulence in the boundary layer on a rotating broad cone is investigated using hot-wire anemometry measuring the azimuthal velocity. The stationary fundamental mode is triggered by 24 deterministic small roughness elements distributed evenly at a speciﬁc distance from the cone apex. The stationary vortices, having a wavenumber of 24, correspond to the fundamental mode and these are initially the dominant disturbance-energy carrying structures. This mode is found to saturate and is followed by rapid growth of the non-stationary primary mode as well as the stationary and non-stationary ﬁrst harmonics, leading to transition to turbulence. The amplitudes of these are plotted in a novel way to highlight the continued growth after saturation of the fundamental stationary mode.

; ψ = 90 • gives the rotating-disk case. When the cone (or disk) rotates, the fluid at the surface is forced to move circumferentially. Fluid in the boundary layer is also transported radially resulting in an inflectional radial velocity profile. On a broad cone, where the coneapex angle is large (ψ 50 • ), experiments show co-rotating vortices as for the disk case (ψ = 90 • ) [8]. On the other hand, for sharp cones (ψ 40 • ) centrifugal effects result in a Görtler type instability that dominates the flow [9]. Also for broad cones stability theory has shown that an absolute instability exists (see for instance Refs. [10,11]), similar to the one for the disk as suggested by Lingwood [4]. However since the stationary co-rotating vortices always exist in a physical experiment the role or importance of this absolute instability with respect to transition is not clear [12].
As shown in Fig. 1, an orthogonal coordinate system (x, θ, z) is defined on the cone surface with the origin located at the apex, where x, θ and z are the coordinates along the generating line of the cone (herein, the radial direction), the circumferential direction and the wall-normal direction. Lengths are normalized by a viscous length, δ * ν = ν * /(Ω * sin ψ), where ν * is the kinematic viscosity of the fluid and x = x * /δ * ν is the square root of the Reynolds number.
The experiments were conducted on a solid aluminium alloy cone, having a base diameter of 474 mm and an apex half-angle ψ = 60 • . The cone surface has a smooth finish (surface roughness of approx. 1 µm). It was mounted on an air bearing and rotated by a d.c.-motor at Ω * between 900 rpm and 1800 rpm around a vertical axis. The azimuthal velocity component was measured at a constant wall height z = 1.2 using a single hot-wire probe with its sensing element parallel to the x-direction at fixed points in the laboratory frame. The length and diameter of the wire were approximately 0.5 mm and 2.5 µm, resepectively. The signals from the anemometer and the tachometer attached to the spindle of the cone were simultaneously recorded for 1200 cone revolutions at a sampling rate of 720 data points per revolution. The velocity signal was post-processed using a high-pass filter (ω * /Ω * > 3.5), where ω * is the disturbance angular frequency (in the laboratory frame). Note that if the vortices are fixed with respect to the cone surface, then ω * /Ω * gives the azimuthal wavenumber. In the following, the measured velocity was normalized by the local wall velocity Ω * x * sin ψ. From the azimuthal velocity fluctuation v(t; θ, x), the stationary componentṽ(θ; x) was evaluated by phase averaging the fluctuation for every 5 revolutions. Further details of the setup as well as the mean flow can be found in Ref. [12].
The basic-flow characteristics of the cone flow are similar to that of the rotating disk (see Ref. [12]  mately 2 mm. The deterministically introduced disturbance gives a fixed fundamental wave number on the entire cone [12] and its initial development compares well with linear stability analysis. In the case without deterministic disturbances, e.g., a clean cone or with randomly distributed roughness elements, the wavenumber may vary but the disturbance development is still well predicted by linear theory [13].).
We present data from two different experiments: i) for a given rotational speed, varying the roughness height h * ; and ii) for a specific roughness height, varying the rotational speed.
In case i) the height is varied by layering elements, giving h * of approximately 4, 8, 13 and 17 µm. In case ii) h * ≈ 8 µm and Ω * was 900, 1200, 1500 or 1800 rpm. A case without roughness elements was also conducted, at the same four rotational speeds; the "clean" case. Figure 2 shows the transition position, x tr for all cases against the non-dimensional rough- Determining the transition position makes use of spectral information as described below. The clean case corresponds to h = 0 and x tr is seen to be rather insensitive to Ω * although increasing Ω * leads to slight decreases in x tr . As expected, with increasing h * (case i ), at a given rotational speed (here 900 rpm), the location of transition moves upstream (decreasing x tr ), whereas the behavior with an increasing rotational speed and fixed roughness position and height (case ii ), is non-monotonic; this will be explained below.
The development of the disturbances will first be discussed using the power-spectrum development with x as shown in  The roughness elements initially introduce disturbances at multiple harmonics (ω * /Ω * = 24, 48,...). In case i ), comparing Fig. 3(a) and (b), most of disturbances disappear below x ≈ 300 except the fundamental (ω * /Ω * = 24). Although the fundamental disturbance also decays, it begins to grow when entering the unstable region (beyond the neutral curve, i.e. x > 286). Further downstream (i.e. larger x) higher harmonics appear and at a distinct x the spectrum fills up indicating that transition has occurred. The comparison between Fig. 3(a) and (b) shows that increasing h * makes the harmonics appear at smaller x-locations and the transition shifts upstream.
Another way to study the transition scenario in more detail is to plot the development of the fundamental (ω * /Ω * = 24, dash-dotted lines) and first harmonic (ω * /Ω * = 48, solid lines) as in Fig. 4(a). As h * increases, the initial transient coupled to the disturbance increases.
After a short transient for x 295, the fundamental grows and the spatial growth rate shows good agreement with local linear stability analysis (LLSA) up to x ≈ 380 in Fig. 4(b).
In the linear region, the roughness height does not affect the growth rate but slightly affects the x-location where the growth rate begins to deviate from LLSA. The deviation occurs whenṽ rms of the fundamental reaches a certain magnitudeṽ rms, 24 ≈ 10 −2 (or the first has its maximum growth rate which is nearly double that of the fundamental, indicating a quadratic nonlinear process [14]. After nonlinear saturation, both the fundamental and first harmonic reach their maxima in the range 484 x 512, nearly corresponding to the transition locations marked by the arrows on the abscissa. Increasing the roughness height shifts the whole process upstream. We now focus on the power-spectrum density for case ii ) in Fig. 3 with a fixed roughness height (h * = 8 µm), but different Ω * : (a) 900 rpm and (c) 1800 rpm. When increasing the rotational speed the characteristic length scale δ * ν decreases, which brings two effects: i) increasing the amplitude of the initial disturbance h * /δ * ν ; and ii) shifting the location of the initial disturbance downstream (with respect to the normalized x-location). In Fig. 3(c), the roughness is within the unstable region and the fundamental begins to grow directly however, since the roughness element is at a larger x, the disturbance amplitude does not catch up with that introduced at smaller x despite the non-dimensional height of the roughness being larger. This leads somewhat counterintuitively to transition occurring around x = 497 compared to x = 487 at the lower rotational speed.
To get a better picture of the development of the amplitude of the fundamental and first harmonic of the stationary mode, we plot these as in Fig. 4. Figure 5 shows the development for the cases with different rotational speeds with fixed roughness height (h * = 8 µm). Here, it is clearly seen that for 900 rpm the fundamental decreases after the initial transient before it starts to amplify according to LLSA. However, for 1200 rpm (where the roughness element is within the unstable region), the fundamental does not decay but amplifies directly after the initial transient from a larger amplitude than that of the 900 rpm case at the same x and, therefore, its amplitude leads the 900 rpm case. For the two other cases (1500 and case. For 1800 rpm, the initial transient is slightly higher but the development is shifted downstream compared to the others. Just after the transient of each first harmonic, the initial decay rates follow LLSA (ω * /Ω * = 48) for all cases shown in Fig. 5(b). Then, the growth rates follow a similar pattern as for the fundamentals and their maximum growth rates are typically twice those of the fundamentals, except for 1800 rpm, which is slightly smaller. From what has been shown above, it is clear that the stationary vortices have a role in the transition scenario and initially they are the dominant disturbance-energy carrier.
In order to investigate the nonlinear interaction and transition process it is possible to plot harmonics as functions of this fundamental stationary mode (instead of plotting them against x). Some examples are shown in Fig. 6(a-c) where the fundamental non-stationary mode as well as the first harmonic of the stationary and non-stationary modes are shown.
The non-stationary component v was obtained by subtracting the stationary component from the total fluctuation signal. Here, we show data for four different Ω * and, after the initial transient, the disturbance amplitudes collapse for all three modes, albeit the highest rotational rate (red) has a slightly smaller slope (see also Fig. 5(b)). The growth rate of the first harmonic of the stationary disturbance is double that of the fundamental as indicated by the slope of 2 in the figure. A similar behavior can be seen for the non-stationary components. Figure 6(d-f) shows the ratio of the amplitude of each mode to that of the fundamental stationary mode v rms, 24 . As can be seen initially the non-stationary fundamental disturbance grows at the same rate as the stationary disturbance, i.e. the ratio of the non-stationary to the stationary fundamental disturbance remains a constant, nearly 10% shown in Fig. 6(d).
However, the first harmonic of the stationary disturbance as well as the non-stationary disturbances (both the fundamental and first harmonic) continue to grow when the fundamental mode has saturated ( v rms, 24 ≈ 8 × 10 −2 ). Eventually the non-stationary disturbances take over and dominates the disturbance energy; the amplitude ratio exceeds unity in Fig. 6(d) and (f), however at that point transition has already occurred and disturbance energy has spread over the whole spectrum.
This behavior can, of course, also be observed in Figs. 4(a) and 5(a) however, by plotting the harmonics against the amplitude of the fundamental disturbance, this development becomes clearly illuminated.
The results presented here seem to indicate that as the stationary mode saturates, a mean-flow modification with a three-dimensional base flow establishes. At this stage, a clear change in energy growth is seen, from growth of the primary stationary vortices to a growing unsteady primary mode as well as stationary and non-stationary harmonics. These become the dominant growing energy carriers during this stage of transition to turbulence.
Whether this is the same as the absolute secondary instability discussed in Ref. [7] or a different mechanism needs further investigation.

ACKNOWLEDGMENTS
We thank Dr Antonio Segalini for providing the local linear stability analysis results. This work was supported mainly by the Swedish Research Council (VR) through the ASTRID project, supporting the first author.