Görtler-number-based scaling of boundary-layer transition on rotating cones in axial inflow

Abstract This paper reports on the efficacy of the Görtler number in scaling the laminar-turbulent boundary-layer transition on rotating cones facing axial inflow. Depending on the half-cone angle $\psi$ and axial flow strength, the competing centrifugal and cross-flow instabilities dominate the transition. Traditionally, the flow is evaluated by using two parameters: the local meridional Reynolds number $Re_l$ comparing the inertial versus viscous effects and the local rotational speed ratio $S$ accounting for the boundary-layer skew. We focus on the centrifugal effects, and evaluate the flow fields and reported transition points using Görtler number based on the azimuthal momentum thickness of the similarity solution and local cone radius. The results show that Görtler number alone dominates the late stages of transition (maximum amplification and turbulence onset phases) for a wide range of investigated $S$ and half-cone angle ($15^{\circ } \leq \psi \leq 50^{\circ }$), although the early stage (critical phase) seems to be not determined by the Görtler number alone on the broader cones ($\psi =30^{\circ }$ and $50^{\circ }$) where the primary cross-flow instability dominates the flow. Overall, this indicates that the centrifugal effects play an important role in the boundary-layer transition on rotating cones in axial inflow.


Introduction
Rotating cones in axial inflow are one of the simplified models for probing the transition phenomena in three-dimensional boundary layers developed on aero-engine-nose-cones, launch vehicle tips, turbo-machinery rotors, etc.Generally, the rotation destabilises the boundary layer on a rotating cone such that the disturbances grow to form coherent vortex 2 structures close to the cone surface.In the presence of the meridional velocity component, the instability-induced vortices align in a spiral vortex pattern around the cone surface (Kobayashi & Izumi 1983).As the azimuthal wall velocity increases with the radius, the spiral vortices grow and eventually set on the turbulence (Kohama 1984a).In practice, this transition phenomena, i.e. the spiral vortex growth and the turbulence onset, affect the performance of an engineering system-by altering the momentum distribution near the wall, affecting the local skin friction and heat transfer.
Generally, a rotating-cone boundary layer undergoes transitions through three distinctly observable phases along the cone: 1) the critical phase where the instability-induced spiral vortices begin their growth, 2) the maximum amplification phase where the spiral vortex amplification peaks, where the vortices rapidly enhance mixing of the outer and inner flow, and 3) the turbulence onset phase where the velocity fluctuation spectra start resembling a general turbulence spectrum (Kobayashi et al. 1983;Kohama 1984b).The present article refers to these phases as transition points in geometric or flow-parameter spaces.In the transition region from the critical to turbulence onset, the instability-induced spiral vortices alter the thermal footprint and velocity distribution of the cone boundary layer.These effects are measurable and useful in identifying the phases of boundary-layer transition on rotating cones, as reported in the previous literature (Kobayashi et al. 1983(Kobayashi et al. , 1987;;Tambe et al. 2021;Kato et al. 2021).
On a cone/disk rotating in still fluid, transition has been evaluated by a single parameter such as rotational Reynolds number based on the local radius and wall velocity (Lingwood 1995(Lingwood , 1996;;Kobayashi & Izumi 1983;Garrett et al. 2009;Hussain et al. 2014).Recently, Kato et al. (2021) suggested another parameter, Görtler number  and experimentally showed that thickening of the boundary layer due to transition can be scaled by Görtler number rather than the Reynolds number on a  = 30 • cone.
When the cone is rotating in axial inflow, however, both cone rotation and axial inflow are two independent control parameters.Therefore, Kobayashi et al. (1983Kobayashi et al. ( , 1987) ) identified two flow parameters for scaling the boundary-layer transition on rotating cones in axial inflow: 1) the local rotational speed ratio  =  * Ω * / *  which accounts for the boundary-layer skew, and 2) the local Reynolds number   =  *  *  / * comparing the inertial versus viscous effects.Here, asterisk denotes dimensional variables; as schematically shown in figure 1,  * =  * sin  is the local cone radius,  * is the local meridional length from the cone apex, Ω * is the angular velocity of the cone,  *  is the meridional component of the boundary layer edge velocity,  * is the kinematic viscosity,  * ∞ is the free stream velocity,  * is the total meridional length of a cone, and  * ,  * are the meridional and azimuthal velocity components.
The present article reports the efficacy of Görtler-number-based scaling of the boundary layer transition on rotating cones in axial inflow, which reduces the two-parameter scaling (  − ) down to a single parameter .§2 describes the Görtler number and the basic flow formulation as functions of the local rotational speed ratio , which inversely represents the axial inflow strength.§3 describes the experiments, in which the flow data were obtained.§4 presents the measured flow data and reported transition points along conventional   −  scale as well as along Görtler number scale.§5 concludes the article.

Görtler number and the basic flow formulation
Görtler (1954) showed that due to the centrifugal effects, a boundary layer on a concave wall becomes unstable and its instability induces counter-rotating vortices-also known as Görtler vortices.Furthermore, the vortices form at a constant Görtler number-which is a product of two non-dimensional parameters: Reynolds number comparing the inertial versus viscous effects and a curvature term   accounting for the wall-normal extent of the viscous effects, e.g.boundary-layer thickness compared to the radius of wall curvature (Taylor 1923;Görtler 1954;Saric 1994).
On a rotating cone, the centrifugal effects exist because the viscous flow on the cone wall follows the curved motion of the rotating cone surface.The curved flow within the boundary layer affects its instability behaviour which can be evaluated by Görtler number  formulated with the azimuthal-momentum-based Reynolds number   * =  * Ω *  * / * and the curvature term   = √︁  * / * : Here, Ω * is the angular velocity and the azimuthal momentum thickness Here, V is the azimuthal velocity normalised by the local wall velocity  * Ω * ,  * is the wall-normal coordinate,  =  * / *  is the azimuthal momentum thickness normalised by the length scale  *  = √︁  * /Ω * ,  =  * / *  , and  =  * / *  .Since the strongest curvature appears at the rotating cone surface, the local cone radius  * is chosen as the radius of curvature in   ; and as the rotation adds momentum in the azimuthal direction, the azimuthal momentum thickness  * accounts for the wall-normal extent of viscous effects in   .
For a fixed non-dimensional radius , Görtler number only depends on the non-dimensional azimuthal momentum thickness  in equation (2.1).In the present work, we used the basic flow to compute  , assuming that the instability-induced distortion of the mean flow remains small until the maximum amplification phase, which will be validated through figures 4 and 5 in §4.In the rest of this section, we describe the basic flow formulation and the effect of axial flow (or ) on , which consequently affects the local Görtler number  for a given .
The basic flow is computed using the formulation given in section 2.3 of Hussain (2010), which is based on the formulation by Koh & Price (1967).Following the Mangler transformation, Hussain used a stream function-based similarity type transformation to obtain the governing equations in the non-dimensional form (Hussain 2010).The stream function , and  * =  * Ω * ((,  1 ) + 1).
(2.3) Here, is the scaled wall-normal coordinate  * / *  ,  is the exponent in the potential flow solution over a cone  *  =  *  *  , where  * is a constant; for  = 15 • , 30 • , and 50 • ,  = 0.0396, 0.117, and 0.3, respectively (Hussain 2010).The governing partial differential equations of the basic flow are as follows : Here, ′ denotes the / 1 .The boundary conditions are: (2.7) A commercial routine NAG D03PEF is used to obtain the basic flow solution.Hussain et al. (2016) have used this basic flow to successfully predict the trend of the critical Reynolds number for the instability onset on a rotating slender cone ( = 15 • ) in axial inflow-agreeing with the experimental results (Kobayashi et al. 1983;Tambe et al. 2021), also shown in figure 6(a).
Figure 2 shows the computed velocity profiles of the basic flow for three different halfcone angles  = 15 • , 30 • , and 50 • .When the axial inflow is dominant over the rotation (e.g.figure 2(a) at  = 0.32), the momentum is distributed in both azimuthal and meridional directions.However, when the rotation is dominant (e.g.figure 2(b) at  = 100), most of the momentum is distributed in the azimuthal direction.Furthermore, the meridional velocity takes the inflectional form, owing to the increased meridional pressure gradient caused by the strong rotation.
Owing to the transformation (equation (2.4)), the basic flow profiles for different half-cone angles  closely follow each other, especially in the azimuthal direction, see figures 2(a) and (b).Consequently, their azimuthal momentum thickness  1 (in  1 coordinates) follows a common trend with respect to the local rotational speed ratio , see figure 2(c).This shows that varying axial inflow influences the wall-normal distribution of the azimuthal momentum.When transformed back onto the physical coordinates , azimuthal momentum thickness  increases with  and decreases with the half-cone angle .Thus, at a fixed non-dimensional Focus on Fluids articles must not exceed this page length radius , increasing axial inflow strength or reducing  will lower the local Görtler number , weakening the centrifugal effects.This shows that the Görtler number formulation in equation (2.1) accounts for the axial flow strength (inverse of local rotational speed ratio ) through .

Methodology
The efficacy of Görtler number-based scaling for boundary layer transition on rotating cones is assessed by using surface temperature fluctuations and the velocity fields obtained as described in Tambe (2022) and by evaluating the Görtler numbers for the transition points reported along the two parameter   −  space in the literature.Parts of the raw data have been used to estimate the transition points reported by Tambe et al. (2021Tambe et al. ( , 2023)).
The experiments were performed at a low-speed open jet wind tunnel (named Wtunnel) at Aerospace Engineering, Delft University of Technology.The cones, made of polyoxymethylene (POM), were rotated in an axial inflow with the free stream velocity  * ∞ = 0.7 − 10.7 m/s and typical turbulence level below 0.01 * ∞ .Infrared thermography (IRT) is performed at a frequency of 200 Hz using an infrared camera FLIR (CEDIP) SC7300 Titanium to detect the thermal footprints of the instability-induced features, as described in Tambe et al. (2019Tambe et al. ( , 2021)).Moreover, the meridional velocity field is measured with two component particle image velocimetry (PIV) at a frequency of 2000 Hz, using a high-speed camera Photron Fastcam SA-1 and a high-speed laser Nd:YAG Quantronix Darwin Duo 527-80-M.The flow is seeded with smoke particles having a mean diameter of around 1m.Two-component velocity vector fields (with the vector pitch of around 2.6 × 10 −4 m) are obtained using a commercial software DaVis 8.4.0.

Results and discussions
During the boundary-layer transition on a rotating cone in axial inflow, the growing spiral vortices increase the surface temperature fluctuations, which is detected using infrared thermography (IRT) (Tambe et al. 2019(Tambe et al. , 2021)).Figure 3   / ′  , , which represents the rapid growth of the spiral vortices.Further downstream, the growth saturates at the  ′   / ′  , peak (the maximum amplification phase marked by the arrows), and subsequently, the flow becomes turbulent (the turbulence onset phase) (Tambe et al. 2021(Tambe et al. , 2023)).When the inflow Reynolds number   is increased at a fixed Ω * (consequently,   is decreased), the spiral vortex growth shifts downstream on the cone-showing that the scaled meridional length  * / *  does not scale the spiral vortex growth.In contrast, figure 3(b) shows that, on Görtler number scale, the temperature fluctuation profiles associated with the spiral vortex growth overlap with each other.This confirms that Görtler number is an appropriate parameter for scaling the spiral vortex growth region on a rotating cone in axial inflow.
The amplified spiral vortices begin to interact with the outer flow and enhance mixing; the enhanced mixing starts to increase the boundary layer thickness.For example, figure 4(a) and (b), show the mean meridional velocity fields over a rotating cone ( = 15 • ) on the local Reynolds number scale, at two different operating conditions.Here, the black dashed line marks the location of maximum amplification identified from the surface temperature fluctuations (e.g.figure 3).The solid and dotted black lines represent the boundary layer thicknesses  95,exp and  95,th obtained from the measured mean flow and computed basic flow, respectively;  95 is the wall-normal extent along  up to which the meridional velocity deficit (or excess, e.g. at high  figure 2(b)) is more than 5% of the outer irrotational flow velocity.Before the maximum amplification, the measured boundary layer thickness (solid line) follows that of the basic flow (dotted line) within 0.1 − 0.2 95,exp but starts to drastically deviate around the maximum amplification.The velocity fields in two different cases, as shown in the left columns of figures 4(a) and (b), do not align with each other on the local Reynolds number   scale-suggesting that   alone is not an appropriate scaling parameter for the rotating cones.However, on Görtler number scale, the near-wall velocity  For example, at a fixed local Reynolds number (  ), broader cones require a stronger rotation effect (higher ) to cause the boundary layer transition.Figure 5 shows the mean velocity fields for two rotating cones ( = 30 • and 50 • , respectively) on   and  scales.Due to the high rotation rates, typically  ≳ 5, the local meridional velocity is higher as compared to the boundary layer edge (e.g. as also seen in figure 2(b)).Similar to  = 15 • (figure 4), on broad cones  = 30 • and 50 • (figure 5), the measured boundary layer thickness (solid white line) is close to that of the basic flow (dotted white line) within 0.1 − 0.2 95,exp until the maximum amplification, beyond which it increases.For both these cones, the velocity fields align with each other on the Görtler number scale, unlike on the local Reynolds number scale.This shows that the Görtler number-based scaling is effective for a range of half cone angles: from slender ( = 15 • ) to broad ( = 50 • ) cones.The Görtler number related to the maximum amplification increases from around  ≈ 7.5 for a slender cone  = 15 • to around  ≈ 10 − 11 for the broader cones  = 30 • and 50 • .
To assess the generality of Görtler number-based scaling of rotating-cone boundary-layer transition, Görtler numbers are evaluated for the transition points reported by different studies in the literature in   −  parameter space, which used different wind tunnel facilities, different model sizes (base diameters  * =  * sin  = 0.047 − 0.1m), different measurement techniques (hot-wire anemometry, infrared thermography, etc.), transition criteria, freestream turbulence levels (0.05 − 1%), half-cone angles ( = 15 • , 30 • , and 50 • ) (Kobayashi et al. 1983;Kobayashi & Izumi 1983;Kobayashi et al. 1987;Tambe et al. 2021Tambe et al. , 2023)), and theoretical analysis (Hussain et al. 2016).The transition points (relating to critical, maximum amplification, and turbulence onset phases) are shown in   −  space directly as they are reported in the literature (figures 6(a)-(c)) and, after estimating their respective Görtler numbers, they are represented in  −  space (figures 6(d)-(f), transformed using equation (2.1), where  is obtained from the similarity solution as shown in figure 2(c) and  = √︁   /sin ).In   −  space (figures 6(a)-(c)), all transition points show a nearly log-linear behavior.For a  = 15 • cone (figure 6(a)), a good agreement between theory and different measurements is shown, confirming that   and  together are appropriate to represent the boundary-layer transition region on rotating cones.However, in  −  space (figure 6(d) for  = 15 • ), the transition points appear at respectively fixed Görtler numbersregardless of the local rotational speed ratio in the investigated range  ≳ 1.The Görtler numbers for the measured turbulence onset points by Kobayashi et al. (1987) in axial inflow  ≈ 2.5 − 4.5 agree with that of Kobayashi & Izumi (1983) in still fluid  ≈ ∞-both studies used the same measurement technique.Moreover, the critical points predicted by Hussain et al. (2016) also appear within a narrow range of Görtler number.Considering the small variation of the respective Görtler numbers relative to the uncertainty around ±0.1, we can conclude that the critical and the maximum amplification points as well as turbulence onset on the rotating 15 • cone are scaled by Görtler number regardless of the axial inflow for  ≳ 1.At low local rotational speed ratio, i.e.  << 1, the centrifugal effects are expected to be weak and different mechanisms dominate transition (Song et al. 2023).However, at  ≳ 1, the centrifugal instability is known to be dominant on the slender cone (Kobayashi et al. 1983;Kobayashi & Izumi 1983;Hussain et al. 2016;Kato et al. 2021), and the influence of  on Görtler number is reported here for the first time.Thus, when the transition is induced due to the strong rotation effect ( ≳ 1), Görtler number is an appropriate parameter to scale the boundary-layer transition on a rotating slender cone  = 15 • rather than using the two-parameter space   − .
For broader cones  = 30 • and 50 • , figures 6(e) and (f) show the transition points in  −  space, respectively.Unlike their near-log linear trends in the   −  space (figures 6(b) and (c)), the maximum amplification and turbulence onset points in  −  space appear from the similarity solution flow, and turbulent onset occur in a well-defined range of Görtler numbers for a wide range of investigated .However, the critical points vary with respect to the local rotational speed ratio  in  −  space at high rotational speed ratio  ≳ 5 (figure 6(e) and (f)), where, for broader cones  ⩾ 30 • , the primary crossflow instability dominates the flow rather than centrifugal instability (Kobayashi & Izumi 1983;Kobayashi et al. 1987;Garrett et al. 2010).
It is interesting that Görtler number dominates the maximum amplification and turbulence onset on broader cones  = 30 • and 50 • at higher  ≳ 5 (figures 5 and 6(e) and (f)), where the primary instability is the crossflow instability and forms co-rotating primary vortices.A possible explanation for this might be that the centrifugal effects cause a secondary instability; some measurements, e.g., figure 7 from Tambe et al. (2023) and also the top right of figure 8b from Kobayashi & Izumi (1983), show that new counter-rotating vortices emerge near the maximum amplification (after the primary co-rotating vortices have developed), which might be caused by centrifugal effects.Dominance of Görtler number in this region suggests that the centrifugal effects play an important role in the spiral vortex amplification and turbulence onset even on the broad cones where the crossflow primary instability dominates the flow initially.

Figure 1 :
Figure 1: Schematic of a rotating cone in axial inflow.

Figure 2 :
Figure 2: Three-dimensional boundary layer profiles on rotating cones ( = 15 • , 30 • , and 50 • ) with examples of (a) strong axial inflow  = 0.32 and (b) strong rotation  = 100.(c) Variation of the azimuthal momentum thickness with the local rotational speed ratio . 1 is the momentum thickness in the transformed wall-normal coordinate  1 . is the momentum thickness in the scaled wall normal coordinate  =  * / *  .
(a) shows the r.m.s. of surface temperature fluctuations along a cone with the half-cone angle  = 15 • at different operating conditions, i.e. different combinations of base rotational speed ratio   and inflow Reynolds number   .The temperature fluctuations are in terms of the normalised pixel intensity  ′   / ′  , .Critical and maximum amplification phases of the spiral vortex growth are identified for all profiles, examples are marked in figure 3(a).Here, the critical points are the intersection points (marked by squares) of the baseline noise level and the least-square linear fit through the rising  ′

Figure 3 :
Figure 3: Meridional profiles of r.m.s. of surface temperature fluctuations ( ′   / ′  , ), caused by the growth of instability-induced spiral vortices on a rotating cone ( = 15 • ), represented along (a) scaled meridional length  * / *  and (b) Görtler number scale.Squares and downward arrows represent the critical and maximum amplification points, respectively.

Figure 4 :
Figure 4: Mean meridional velocity field obtained from PIV over a rotating cone of  = 15 • on Reynolds number and Görtler number scales with : (a)   = 1.9 and   = 9.7 × 10 4 ; (b)   = 3.1 and   = 6.2 × 10 4 .Solid and dotted black lines represent the boundary layer thicknesses  95,exp and  95,th obtained from measured mean flow and computed basic flow, respectively.The black dashed line represents the maximum amplification identified from surface temperature fluctuations.