Reattachment Line Research Articles

An experimental investigation on the flow control of the partially stepped NACA0012 airfoil at low Reynolds numbers

An experimental study of flow control around a NACA0012 airfoil modified with fully and partially step geometry was conducted at Reynolds numbers of 6 × 104 and 1.2 × 105. Partially step geometries, the location of the step on the airfoil, and Reynolds number are varied to show their effect on aerodynamic performance and flow structures on the airfoil. Experimental results show that fully and partially step geometries are effective for flow control around the airfoil and aerodynamic performance enhancement. The maximum increase in lift coefficient is approximately 46%, and the stall angle is shifted about 1° by the SM4 model at Re of 6x104 while the step geometry is on the pressure side. For a Reynolds number of 1.2 × 105, the highest increase in the lift-to-drag ratio of the airfoil is observed about 17.1 at an angle of attack of 6° by the SM2 model when step geometry is placed on the pressure side of the airfoil. In the event of partially or fully stepped at the pressure side of the model, step flow structures, including the reattachment line, recirculation zone, and corner eddy, are obtained. However, even though the step geometry is at the suction surface of the model, the formation of a laminar to turbulent transition is observed. The overall results suggest that the partially stepped geometry length, Reynolds number, and step location side on the airfoil have an important role in flow control to improve aerodynamic performance for various angles of attack.

Experimental investigation of the shock wave/boundary layer interaction at a compression corner and an incident shock wave generator

The separation of the boundary layer is one of the main failures in scramjets due to its influence on the amount of oxygen used in combustion, reducing thrust and specific impulse. The main objective of this paper was to perfom experimental investigations to study the shock wave/boundary layer interaction and boundary layer separation. The experimental investigation was done in the T2 shock tunnel using the throats/nozzles for theoretical Mach numbers of 6.5 and 8.3. Compression ramp (30° and 20°) models made of aluminum and printed on plastic filament in the 3D printer were designed and produced, as well as two shock generators models with 30°. The flow analysis was performed using pressure sensors and the schlieren image technique. In the cases of shock generators, the incident shock wave caused an increase in pressure and reached a plateau inside the separation bubble (from 81.84 kPa to 360.01 kPa). The highest pressure values were related to the reattachment line. Then, the expansion waves cause a reduction in pressure and adjacent phenomena cause increase in pressure. The ramps caused an increase in pressure and an effect close to a plateau. Then, the boundary layer is replaced and the pressure continues to increase until the end of the ramp, obtaining values from 15.9 kPa to 93.8 kPa. The highest pressure values were observed in the region of interaction of the incident shock wave with the reattachment shock. The pressure values between the aluminum and plastic filament models measured different values along the ramp (45.6 kPa and 28.8 kPa) in the last sensor. The images in schlieren were of excellent quality and the technique was highly efficient to visualize the phenomena. As results of these campaigns, it was possible to visualize the adverse pressure gradient, the flow separation structure, including the separation bubble and reattachment of the boundary layer.

Microramp wake impinging on canonical shock/boundary-layer interaction

We analyze the influence of microramp vortex generators (mVGs) on a canonical oblique shock wave/turbulent boundary-layer interaction (SBLI) in terms of mean flow field and unsteady dynamics. The flow configuration of our wall-resolved large-eddy simulations (LES) reproduces the experiment of Bo et al. [“Experimental investigation of the micro-ramp based shock wave and turbulent boundary-layer interaction control,” Phys. Fluids 24, 055110 (2012)]: a rake of microramps is inserted upstream of the SBLI, protruding by 0.476δ in a turbulent boundary layer (TBL) at free-stream Mach number M = 2.7 and corresponding to a Reynolds number based on the displacement thickness of Reθ=3600. The long integration time of 1672 Lsep/U∞ allows an accurate characterization of the low-frequency dynamics of the SBLI under the influence of the microramps. With respect to the reference SBLI without control devices, the mean flow field shows a new spatial organization of the recirculation bubble due to the mVGs' wake. The alternating high and low-speed zones in the near-wall region of the incoming TBL, induced by the counter-rotating streamwise vortices generated by the mVGs, trigger spanwise corrugations of the separation and reattachment lines and locally alter the reverse flow region. Tornado-like vortices are found in the vicinity of these zones, yielding a new fluid collection mechanism of the reverse flow region. These vortices redirect the fluid coming from regions outside of the wake in the incoming TBL to three key spanwise exit locations located in between the mVGs and at their centerline. Interestingly, power spectral densities of wall-pressure probes show a damping of the low-frequency dynamics of the reflected shock foot for spanwise stations aligned with the mVGs' wake, whereas this activity appears to be reinforced in the planes located in between the mVGs. However, we found no evidence of unsteady forcing linked to the high-frequency shedding of the coherent structures developing in the wake of the microramps. Dynamic mode decomposition highlights a significant change in the low-frequency dynamics, mostly affecting the mass budget of the recirculation bubble. The breathing of the recirculation zone that occurs at StL=0.1 for the SBLI without control devices (with StL=fLsep/U∞) appears to shift toward a lower frequency of StL=0.05. Remembering that the reflected shock foot motion is related to frequencies in the range StL=[0.03−0.05], the SBLI with upstream mVGs seems to highlight a synchronization of this motion with the breathing of the separation bubble.

An advanced aero-thermodynamic study of a heart-shaped dimpled pipe

The detailed flow field and local heat transfer enhancement in a pipe with heart-shaped dimples on the internal surface were experimentally studied. The flow structures were investigated at a flow regime of Re=20k in five different measurement planes around a dimple in the streamwise and spanwise directions using both planar PIV and stereo PIV. A ray tracing-based image correction method was utilized to overcome the important image distortions caused by the optically complex geometry of the dimple for the planar PIV measurements. Furthermore, the convective heat transfer coefficient on the wall of the dimple was investigated utilizing steady-state liquid crystal thermography (LCT) and infrared thermography within the flow regime range of Re=20k-60k. A comprehensive uncertainty analysis was conducted to account for the multiple measured quantities and corresponding error sources on such a complex geometry. The resulting heat transfer and flow fields revealed that maximum heat transfer enhancement is concentrated along the reattachment line downstream of the dimple where higher kinetic energy levels are observed. The enhancement factor, which is a measure of the heat transfer improvement relative to a smooth pipe, was evaluated on average at 1.7 across the dimpled surface, with local maxima reaching values up to 2.8. The dimpled pipe under examination had a relative skin friction coefficient of approximately 1.8 compared to a smooth pipe. This outperforms some of the roughness elements, such as ribs and spherical dimples, previously studied on similar circular channels in literature.

Vortex Structure Topology Analysis of the Transonic Rotor 37 Based on Large Eddy Simulation

Highly three–dimensional and complex flow structures are closely related to the aerodynamic losses occurring in the transonic axial–flow compressor. The large eddy simulation (LES) approach was adopted to study the aerodynamic performance of the NASA rotor 37 for the cases at the design, the near stall (NS), and the near choke (NC) flow rate. The internal flow vortex topology was analyzed by the Q–criterion method, the omega (Ω) vortex identification method, and the Liutex identification method. It was observed that the Q–criterion method was vulnerable to being influenced by the flow with high–shear deformation rate, especially near the end–wall regions. The Ω method was adopted to recognize the three–dimensional vortex structure with a higher precision than that of the Q–criterion method. Meanwhile, the Liutex vortex identification method showed a good performance in vortex identification, and the corresponding contribution of Liutex components in the vortex topology was analyzed. The results show that the high–vortex fields around the separation line and reattachment line had high vortex components in the x–axis, the tip clearance vortices presented a high–vortex component in the y–axis, and the suction side corner vortex possessed high–vortex components in the y– and z–axes.

Numerical study of transition process in different zones of a compressor cascade channel

Abstract The complex vortex structure compressor leads to the problem that the transition model is insufficient in predicting the flow instability of the compressor. In this paper, the rectangular cascade of compressor of different turning-angle conditions is taken as the object, and the transition characteristics on the end wall and the blade surface of the compressor cascade are in comparison by the method of large eddy simulation/LES. The effects of the horseshoe vortex and the separation bubble over the compressor cascade on the transition process are emphatically discussed. By analyzing characteristic parameters of the vortex structure, it is found that the separated transitional flow corresponds to multiple separations-and reattachments of the shedding vortex, and is affected by the cross-flow transition and the separate-transition. Finally, by discussing the instability of the separation line, reattachment line and the cross-flow inflection point of the separated transitional flow, it reveals that the transient disturbance caused by the vortex motion is an important reason affecting the prediction accuracy of the transition model.

Wind pressure distribution, non-Gaussian characteristics, and conical vortices of a large-span roof with variable arc angles

Understanding the characteristics and mechanism of wind pressure on long-span structures is important for structural design. In this study, the wind pressure distribution, non-Gaussian characteristics, and conical vortices of a typical large-span roof with variable arc angles were studied using wind tunnel tests on pressure measurements. The results indicate that the mean, the standard deviation(STD), and the negative extremum of the wind pressures on the corners of the roof are the most unfavorable when the corners are windward, and the values are related to arc angle. Additionally, the non-Gaussian characteristics of the roof are clear, the probability that the absolute value of skewness is greater than 0.25 exceeds 70%. The most unfavorable peak factors at the edge of the roof were larger, and they gradually decreased from the outside to the inside. Under the action of typical conical vortices, the mean, STD, skewness, and kurtosis values of wind pressures exhibit the distribution characteristics of “conical mountains,” and the wind pressures between the vortex axis, reattachment line, non-Gaussian rays, and along-wind axis varied between weak non-Gaussian (negative), Gaussian, stronger non-Gaussian (negative), and weak non-Gaussian (positive). The conical vortices exhibited stronger vortex shedding, and its energy was attenuated in the radial direction.

On the synchronisation of three-dimensional shock layer and laminar separation bubble instabilities in hypersonic flow over a double wedge

Linear three-dimensional instability is studied in the shock layer and the laminar separation bubble (LSB) induced by shock-wave/boundary-layer interactions in a Mach 7 flow of nitrogen over a double wedge with a$30^{\circ }\text {--}55^{\circ }$cross-sectional profile. At a free-stream unit Reynolds number$Re=5.2\times 10^{4}\,{\rm m}^{-1}$this flow exhibits rarefaction effects and has shock thicknesses comparable to the thickness of the boundary layer at separation. Flow features have been fully resolved using a high-fidelity massively parallel implementation of the direct simulation Monte Carlo method that captures the flow evolution from the inception of three-dimensionality, through linear growth of instabilities, to the early stages of nonlinear saturation. It is shown that the LSB sustains self-excited, small-amplitude perturbations that originate past the primary separation line and lead to spanwise-periodic wall striations inside the bubble and downstream of the primary reattachment line, as known from earlier experiments, simulations and instability analyses. A spanwise-periodic instability, synchronised with that in the separation zone, is identified herein for the first time, which exists in the internal structure of the separation and detached shock layers, and manifests itself as spanwise-periodic cats-eyes patterns in the global mode amplitude functions. The growth rate and the spanwise-periodicity length of linear disturbances in the shock layers and the LSB are found to be identical. Linear amplification of the most unstable three-dimensional flow perturbations leads to synchronised low-frequency unsteadiness of the triple point, with a Strouhal number of$St\approx 0.028$.

Skin-Friction-Based Identification of the Critical Lines in a Transonic, High Reynolds Number Flow via Temperature-Sensitive Paint.

In this contribution, three methodologies based on temperature-sensitive paint (TSP) data were further developed and applied for the optical determination of the critical locations of flow separation and reattachment in compressible, high Reynolds number flows. The methodologies rely on skin-friction extraction approaches developed for low-speed flows, which were adapted in this work to study flow separation and reattachment in the presence of shock-wave/boundary-layer interaction. In a first approach, skin-friction topological maps were obtained from time-averaged surface temperature distributions, thus enabling the identification of the critical lines as converging and diverging skin-friction lines. In the other two approaches, the critical lines were identified from the maps of the propagation celerity of temperature perturbations, which were determined from time-resolved TSP data. The experiments were conducted at a freestream Mach number of 0.72 and a chord Reynolds number of 9.7 million in the Transonic Wind Tunnel Göttingen on a VA-2 supercritical airfoil model, which was equipped with two exchangeable TSP modules specifically designed for transonic, high Reynolds number tests. The separation and reattachment lines identified via the three different TSP-based approaches were shown to be in mutual agreement, and were also found to be in agreement with reference experimental and numerical data.

Direct numerical simulation of a supersonic turbulent boundary layer over a compression–decompression corner

A direct numerical simulation of the interaction between a shock wave and the supersonic turbulent boundary layer in a compression–decompression corner with a fixed 24° deflection angle at Mach 2.9 is conducted. The characteristics of the shock interactions are investigated for two heights between the compression and decompression corners, corresponding to H/δref=4.25, 1.22, where δref denotes the reference turbulent boundary layer thickness. A classic shock wave/turbulent boundary layer interaction flow is reproduced in the higher case. For the lower case, the size of the separation region is significantly decreased, and the low-frequency unsteadiness is slightly suppressed in the interaction region, as assessed by analyzing the mean and fluctuating wall pressure. Flow patterns near the reattachment line show the existence of the Görtler vortices. By analyzing the curvature radius and Görtler number distribution, it was found that a strong centrifuge instability is reserved in the compression corner region and reversed in the decompression corner region due to the convex streamline curvature. The downstream flow of the decompression corner is relatively complex where the additional shocklet and new streamwise vortices are observed. A negative response mechanism is found regarding fluctuating wall-pressure signatures between the upstream and downstream of the decompression corner.

Analysis of the expanding process of turbulent separation bubble on an iced airfoil under stall conditions

The expanding process of turbulent separation bubble induced by a horn ice on the leading edge of an airfoil for business jet was investigated numerically with IDDES. Through comparing with the experiment results of critical stall condition, the fidelity of method for detailed simulation of ice-induced separation was verified firstly. Then the separation bubble under post-stall conditions was investigated in the aspects of both the features of averaged static flowfield and the behaviors of transient shear-layer vortices. The averaged results manifested that during the stall process, the bubble expanded with stretching and deformation for reattachment line and turbulence region were shifted towards trailing edge and spread continually. The corresponding extending and descending of pressure plateau with weak pressure gradient resulted in an early but mitigated stall pattern consequently. From the transient flowfield, it was observed that the entire evolution process of bubble was driven by the hairpin-type vortices generated from the shedding shear-layer with K-H instability. The propagation direction of these shear-layer vortices was affected by the entrainment of external flow. Under the post-stall condition, the motion of vortices towards the wall was delayed with the increasing of incidence and the interaction between vortices and wall was weakened correspondingly, which leaded the gradual decrease of reattachment effect and the continuous expanding of recirculation region.

Experimental and numerical study of laminar separation bubble formation on low Reynolds number airfoil with leading-edge tubercles

The present work reports the effect of leading-edge tubercles on aerodynamic performance and flow features of a cambered airfoil E216 at a Reynolds number of 100,000 and at various angles of attack in the pre-stall regime. Amplitude values of 2 mm, 4 mm and 8 mm and wavelength values of 15.5 mm, 31 mm and 62 mm are used for both experimental and simulation studies. The Transition-SST RANS model is used to simulate transition phenomenon (laminar separation bubble) and three-dimensional flow features over the airfoil. Wind tunnel experimental results are used for the performance analysis and the validation of the simulation methodology. The experimental values of $$C_\mathrm{l}$$ and $$C_\mathrm{d}$$ are 1.37 and 0.081, respectively, at a stall angle of $$12^{\circ }$$ for the plain airfoil. The experimental results show that the lift generated by tubercled airfoils is higher than that produced by the plain airfoil in the pre-stall region but lower at the stall angle. A maximum benefit of 4.51% in $$C_\mathrm{l}$$ is obtained for the tubercled airfoil with the highest amplitude (8 mm) and wavelength (64 mm) at $$6^{\circ }$$ angle of attack. A higher $$C_\mathrm{d}$$ is observed for all the tubercled airfoils than for the plain one. The simulation is mainly carried out to study the flow structure. Simulation results indicate the presence of laminar separation bubbles on the plain airfoil with a straight separation and reattachment line parallel to the trailing edge. The tubercles considerably altered the laminar separation bubble formation and the flow structure. A sinusoidal laminar separation bubble is formed on the tubercled airfoils with reduced bubble length. The laminar separation bubble along the trough is formed ahead of that at peak. Two pairs of counter-rotating vortices are formed on the airfoil surface along the trough at two different chord-wise locations which strongly alter the flow pattern over it. Prandtl’s secondary flow of the first kind is the key reason for the vortex formation.

Characteristics of the free-end mean pressure distribution for a surface-mounted finite-height cylinder

Wind tunnel experiments at a Reynolds number of Re = 6.5 × 104 were used to study the effect of aspect ratio and boundary layer thickness on the mean static pressure distribution on the free end of a surface-mounted finite-height cylinder. The cylinder's aspect ratio was changed in small increments from AR = 0.5 to AR = 11. Two different boundary layer thicknesses (relative to the cylinder diameter) were employed, δ/D = 0.6 and δ/D = 1.9. From analysis of the mean pressure contour plots, it was found that the sizes and locations of regions of lower pressure, adverse pressure gradient, and higher pressure, and the appearance of “eye-like spots”, are sensitive to both AR and δ/D. The adverse pressure gradient occurs just ahead of the mean reattachment line while the eye-like spots are related to termination points of the legs of the arch vortex within the free-end mean recirculation zone. The total normal force coefficient experienced by the cylinder is strongly influenced by the contribution of the wall shear stress on the sides of the cylinder, with a change in direction of the net vertical shear stress contribution occurring between AR = 7 and AR = 8 for both boundary layers.

Experimental and numerical study of wing boundary layer behavior in propeller flowfield

In this research, the effects of propeller slipstream on wing boundary layer and transition front were studied through wind tunnel tests and numerical analysis. In this respect, the flow around a NACA 6-series airfoil section was simulated in a computational fluid dynamics solver without and with propeller flowfield. The numerical study, presented in this paper, was concerned with the effect of propeller slipstream on both wing aerodynamics and boundary layer treatment. For experimental tests, oil flow visualization technique was used to determine laminar to turbulent transition location, laminar separation bubble with turbulent reattachment and turbulent separation line over the wing surfaces. Existence of propeller slipstream changed pressure and skin friction distribution over the wing surfaces, in both chordwise and spanwise directions and it hence affected on the wing loading distribution. Also upstream propeller affected on boundary layer characteristics including laminar/turbulent transition onset and separation over the wing. The results showed that the transition location moved toward the leading edge, and the separation bubble was washed out due to propeller slipstream.

Dynamic layer formation in the reattachment zone for a supersonic laminar separated flow

In this work, results of an experimental study of the flow structure in the region of reattachment of a supersonic laminar separated flow are reported. The separated flow is generated by a model shaped as a compression corner consisting of a horizontal flat plate followed by a ramp. The model width is equal to the plate length from the leading edge to the plate/ramp junction line. The ramp angle is varied in the interval φ = 20°–50°. The free-stream Mach number is M∞ = 6 or 8. The analysis of the Pitot pressure profiles near the ramp wall in the downstream direction behind the reattachment line confirms the existence of a high-pressure layer with the total pressure being appreciably greater than in the surrounding flow.

Numerical investigation of attached cavitating flow in thermo-sensitive fluid with special emphasis on thermal effect and shedding dynamics

This study aims to investigate the unsteady cavitation shedding dynamics flow around a NACA 0015 hydrofoil in thermo-sensitive fluid with thermal effect. The thermal effects are captured by a coupled solution of the continuity, momentum and energy equations, and the numerical results show a reasonable agreement with the available experiments. Time evolution process of the vortex structure is investigated. The ability and limitation of B-factor is analysed to evaluate the thermal effect, which can provide guidance for the further improvement of B-factor. The re-entrant jet and vortex structure show strong coherent relationship with the flow separation. The re-entrant jet promotes the formation of vortex structure, which in turn affects the separation flow of the flow field. Skin friction coefficient and boundary vorticity flux are applied to displace the flow separation on the hydrofoil surface. The results showed that the streamwise velocity decreases sharply in the vicinity of the collision between the re-entrant jet and the main stream, the skin friction streamline suddenly breaks off and separation or re-attachment line occurs. The intensity of the re-entrant jet inside the cavity becomes gradually weaker, the strength of the vortex is also weakened, which causes the skin friction coefficient in this region to be almost zero, and the phenomenon of flow separation and re-attachment is indistinct.

Flow effects in the reattachment region of supersonic laminar separated flow

The structure of a laminar separation flow around a compression corner at a free-stream Mach number M∞ = 6 is considered. The existence of a new element in the reattachment area—a high total-pressure layer—is shown. This layer is located downstream from the reattachment line above the boundary layer, and its total pressure is higher than that in either the boundary layer beneath it or the supersonic flow above it. Its appearance is due to the difference in the total pressure losses of the streamlines passing through a compression-wave fan and a shock wave in the reattachment zone. The vortex structure in the area of attachment is considered and the formation of streamwise vortices is shown. The origin of these vortices is similar to that of Taylor–Goertler vortices in subsonic flow. The pressure-pulsation distribution in the reattachment zone is also analyzed. It is shown that the spectrum of wall-pressure pulsations in the separation zone has two local maxima, the first caused by longitudinal fluctuations in the separation region and the second by pulsations of the vortices.

Large-Scale Separated Vortex Generated in a Wake Flow of Ahmed’s Body

Unsteady wake structures around a bluff body, which has a slanted rear surface, are investigated experimentally with measuring the velocity and the pressure fields. Velocity distribution obtained by Particle Image Velocimetry showed that vortices are intermittently generated and occasionally it grows to a large-scale vortex which separates the flow from the slanted surface. From the pressure distribution, which is simultaneously measured with the velocity field, it is clarified that the large-scale separated vortex has considerable extent in the spanwise direction. This fact suggests that the re-attachment line of the downwash is completely separated from the slanted surface when the large-scale vortex is generated. From the time-frequency analysis, which is applied to the velocity fluctuation measured by hot-wires, the flow observed at points downstream of the body exhibits the periodic fluctuation, whereas the flow near the slanted surface does not involve the periodic fluctuations. Instead, the intermittent fluctuation is observed for the case when the downwash of the flow structure is separated from and re-attaches to the slanted surface. The intermittent fluctuation on the slanted surface is observed regardless of the existence of upwash, whereas the periodic fluctuation at the point downstream vanished when the upwash is absent.

Robotic volumetric PIV of a full-scale cyclist

A novel approach to the measurement of large-scale complex aerodynamic flows is presented, based on the combination of coaxial volumetric velocimetry and robotics. Volumetric flow field measurements are obtained to determine the time-averaged properties of the velocity field developing around a three-dimensional full-scale reproduction of a professional cyclist. The working principles of robotic volumetric PIV are discussed on the basis of its main components: helium-filled soap bubbles as tracers; the compact coaxial volumetric velocimeter; a collaborative 6 degrees of freedom robot arm; particle image analysis based on Shake-the-Box algorithm and ensemble statistics to yield data on a Cartesian mesh in the physical domain. The spatial range covered by the robotic velocimeter and its aerodynamic invasiveness are characterised. The system has the potential to perform volumetric measurements in a domain of several cubic metres. The application to the very complex geometry of a full-scale cyclist in time-trial position is performed in a large aerodynamic wind tunnel at a flow speed of 14 m/s. The flow velocity in the near field of the cyclist body is gathered through 450 independent views encompassing a measurement volume of approximately 2 m3. The measurements include hidden regions between the arms and the legs, otherwise very difficult to access by conventional planar or tomographic PIV. The time-averaged velocity field depicts the main flow topology in terms of stagnation points and lines, separation and reattachment lines, trailing vortices and free shear layers. The wall boundary layers developing on the body surface hide below the level resolvable by the present measurements.

Experimental study of the mean structure and quasi-conical scaling of a swept-compression-ramp interaction at Mach 2

This study investigates the mean flow structure of two shock-wave boundary-layer interactions generated by moderately swept compression ramps in a Mach 2 flow. The ramps have a compression angle of either $19^{\circ }$ or $22.5^{\circ }$ and a sweep angle of $30^{\circ }$. The primary diagnostic methods used for this study are surface-streakline flow visualization and particle image velocimetry. The shock-wave boundary-layer interactions are shown to be quasi-conical, with the intermittent region, separation line and reattachment line all scaling in a self-similar manner outside of the inception region. This is one of the first studies to investigate the flow field of a swept ramp using particle image velocimetry, allowing more sensitive measurements of the velocity flow field than previously possible. It is observed that the streamwise velocity component outside of the separated flow reaches the quasi-conical state at the same time as the bulk surface flow features. However, the streamwise and cross-stream components within the separated flow take longer to recover to the quasi-conical state, which indicates that the inception region for these low-magnitude velocity components is actually larger than was previously assumed. Specific scaling laws reported previously in the literature are also investigated and the results of this study are shown to scale similarly to these related interactions. Certain limiting cases of the scaling laws are explored that have potential implications for the interpretation of cylindrical and quasi-conical scaling.