Flow on a rotating cone with an apex angle of 15 degrees is studied through flake visualizations, both in still fluid and with axial flow. From recorded movies, the reflective light intensity at generating lines is extracted and its development is evaluated in space and time. The results show that increasing axial flow moves transition downstream, whereas increasing rotational speed has the opposite effect. We found that the transition process can be scaled by the von Kármán viscous length scale and a globally defined axial velocity.

The laminar similarity solution of the boundary layer on a rotating disk in still ambient fluid was presented by von Kármán more than 100 years ago. A cross-flow instability was discovered in 1955 through flow visualisation experiments, and that instability has since then been investigated through stability analysis, experiments and numerical simulations. Other studies have investigated if an absolute instability could explain the distinct transition Reynolds number seen in experiments. Later experiments and stability theory have taken on both broad and slender cones to elucidate the influence of the cone apex angle on stability and transition. If you are interested to contribute with research on rotating bodies the present paper can be seen as a guideline—quo vadis.

The boundary layer on rotating slender cones exhibits a centrifugal instability both in still fluid and at low axial velocity, in contrast to broad cones (and disks) that are mainly susceptible to a cross-flow instability. Here the influence of axial flow on the transition on a slender rotating cone with an apex angle of 15 degrees is studied in a towing tank using reflective flakes as tracers. A novel technique is used to evaluate the reflected light intensity from the flakes in terms of its probability density distribution (pdf) and root-mean-square (rms) to determine where transition occurs at different axial flow velocities. It is shown that axial flow stabilizes the boundary layer and our experiments together with others indicate that a Görtler number based on a length scale proportional to the boundary layer thickness is constant at transition for slender cones.

Rotating-disk flows were first considered by von Karman in a seminal paper in 1921, where boundary layers in general were discussed and, in two of the nine sections, results for the laminar and turbulent boundary layers over a rotating disk were presented. It was not until in 1955 that flow visualization discovered the existence of stationary cross-flow vortices on the disk prior to the transition to turbulence. The rotating disk can be seen as a special case of rotating cones, and recent research has shown that broad cones behave similarly to disks, whereas sharp cones are susceptible to a different type of instability. Here, we provide a review of the major developments since von Karman's work from 100 years ago, regarding instability, transition, and turbulence in the boundary layers, and we include some analysis not previously published.

The effect of polymer addition on transition to turbulence in a two-dimensional water-flow channel was experimen-tally investigated by flow visualization using reflective flakes. The flow entering the channel test section maintains a high disturbance level by expanding laterally after reaching a high Reynolds number upstream the test section. In order to obtain the intermittency factor (turbulence fraction), the visualized images were classified into non-turbulent and turbulent regions, and the streamwise scale of the streaks appearing in the non-turbulent region was estimated from the autocorrelation coefficient computed by shifting the images in the streamwise direction. The visualization results show that similar to the pure water case, intermittent flow with a patch-like distribution of turbulent and non-turbulent areas clustered by streamwise streaks is observed. The Reynolds number at which the intermittency increases shifts toward higher Reynolds numbers with increasing polymer concentration, indicating a delay of transition. The streaks appearing in the non-turbulent region elongate with increasing polymer con-centration. At high concentrations, straight elongated streaks penetrate through the turbulent regions, suggesting that the polymer addition affects the stability of the streaks. These changes of the streak behavior indicate that the polymer affects not only the transition Reynolds number but also the flow structure during the transition process.

It is now more than 100 years since the work of von Karman (1921) on the boundary-layer flow over a rotating disk was published in the first volume of Zeitschrift fufr Angewandte Mathematik und Mechanik (ZAMM, Vol. 1(4), pp 233-252). Recently, there has been a large amount of work undertaken addressing the instability and transition of the boundary-layer flows over rotating disks and cones using theoretical, numerical and experimental techniques. Here we will discuss some different methods to analyze experimental data that can give insight into the instability and transition to turbulence of boundary-layer flows over rotating slender and broad cones (including the disk). At first, we discuss the pdf-method (probability density function) that allows a simple way to determine regions of instability growth, transition and fully developed turbulence. Secondly, we look at various ways to use spectral information to investigate the boundary layers giving a deeper understanding of the transition process. Finally, a method to determine the most probable flow structure leading up to fully developed turbulence is discussed. We envisage that some of these methods can be useful in analyzing instability and transition also in other flow cases.

An experimental investigation of instabilities and transition in the boundary layer on a rotating broad (120 degrees apex angle) cone through hot-wire measurements combined with local linear stability analysis (LLSA) has been undertaken. The rotating-cone flow is susceptible to both cross-flow and centrifugal instabilities. For broad cones, the cross-flow instability dominates over the centrifugal instability, and vice versa for slender cones. Although stationary vortical disturbances from the cross-flow instability are dominant on the broad cone (in this case 24-26 vortices develop), we have identified an initially slowly growing nonstationary mode with a much smaller wavenumber, which close to transition increases its growth rate dramatically. We report on a detailed process to identify the wavenumber of the measured nonstationary disturbance, as well as quantitative comparisons between experimental results and LLSA.

The present study experimentally determines the transitional Reynolds number range for plane channel flow and characterizes its transitional state. The pressure along the channel is measured to determine the skin friction coefficient as function of Reynolds number from the laminar state, through the transitional region into the fully turbulent state. The flow structure was studied through flow visualisation which shows that as the Reynolds number increases from the laminar state the transitional region starts showing randomly occurring turbulent spots. With increasing Reynolds number the spots shift into oblique patches and bands of small scale turbulence that form across the channel width, together with large-scale streaky structures found in areas between the turbulent regions. An image analysing technique was used to determine the intermittency factor, i.e. the turbulence fraction in the flow, as function of Reynolds number. It is found that the skin friction coefficient reaches its turbulent value before the flow is fully turbulent (the intermittency factor is still below one). This suggests that the observed streaky structures in non-turbulent regions contribute to the enhancement of the wall-normal transfer of momentum. Also above the Reynolds numbers where the turbulent skin friction coefficient has been established large-scale features consisting of irregular streaky structures are found. They have an oblique shape similar to the non-turbulent and turbulent patches in the transitional flow indicating that the transition process is not fully complete even above the Reynolds number where the skin friction reaches its turbulent level.

Instability and transition in the boundary layer on a slender cone (600 apex angle) rotating in still fluid are investigated using hot-wire anemometry as well as through linear stability analysis. In contrast to broad cones (including the disk), where a cross-flow instability dominates the transition and different studies report similar transition Reynolds numbers, the reported transition Reynolds numbers on slender cones are scattered. The present experiments provide quantitative experimental datasets and the stability and transition are evaluated based on both the Reynolds number and a Girder number. The results consistently show that the instability development depends on the Gortler number rather than the Reynolds number and that transition starts at a well-defined Gortler number, whereas the transition Reynolds number depends on the rotational rate. The measured disturbance that first grows in the laminar region has a frequency approximately the same as or twice the rotational rate of the cone, which according to the stability analysis corresponds to the critical frequency of a slightly inclined vortex structure with respect to the cone axis or an axisymmetric vortex structure. These structures are similar to those observed in the flow visualisations of Kobayashi & Izumi (J. Fluid Mech., vol. 127, 1983, pp. 353-364) and considered as being due to a centrifugal instability.

We analyse the azimuthal velocity fluctuation in the boundary layer driven by a rotating slender cone with a half-cone apex angle of 30 ∘. The flow is dominated by a centrifugal instability, which develops into randomly occurring spiralling vortices travelling on the cone surface. Such non-stationary vortices are observed as an irregular wave packet-like fluctuation signal by a hot wire fixed in the lab frame of reference and the spectral map at different radial positions forms a smooth ridge, which is in contrast to the periodic time signal due to stationary crossflow vortices on broad cones, which gives rise to sharp spectral ridges. The present analysis decomposes the wave packet-like fluctuation using a short-time Fourier transform (STFT), revealing that the smooth spectral peak at a given radial position consists of waves with different frequencies. The most probable fundamental frequency follows the most unstable frequency according to linear stability theory. Also, we evaluate the amplitude of the harmonics of the most energetic mode around transition; quadratic nonlinear growth is observed until the amplitude of the fundamental mode saturates at transition. This behaviour is similar to that on broad cones although the primary instability and vortex structures are different.