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Crossflow Instability on a Yawed Cone at Mach 6
Available online at www.sciencedirect.com
ScienceDirect Procedia IUTAM 14 (2015) 15 – 25
IUTAM ABCM Symposium on Laminar Turbulent Transition
Crossflow instability on a yawed cone at Mach 6 Stuart A. Craig∗, William S. Saric Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Abstract Boundary-layer measurements were performed in a Mach 6, low-disturbance wind tunnel on a 7◦ cone at 5.6◦ angle of incidence such that the model was primarily subject to crossflow instability. Constant-temperature hot-wire anemometry was used to measure the streamwise mass flux at a series of planes normal to the cone axis. A dominant stationary wave is observed to achieve nonlinear saturation at approximately 23% with maximum rms fluctuations of approximately 8%. Additionally, traveling crossflow waves were observed in a broad frequency band centered at f = 40 kHz and secondary instabilities were observed in a broad band centered at f = 100 kHz. Measurements show excellent agreement with previously published computational results and are qualitatively similar to studies performed in subsonic flows. Transition to turbulence was not observed to occur under these conditions. c 2015 2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) Keywords: Laminar-turbulent transition; boundary layer; hypersonic; crossflow instability
1. Introduction The order-of-magnitude increase in surface heat transfer rate and viscous drag upon transition to turbulence has important ramifications in the design of hypersonic vehicles. However, the lack of reliable transition prediction methods has historically led to the overdesign of these systems and inefficient vehicles as designers are forced to rely on simplified correlation methods that are often non-physical and geometry-specific. The development of a physics-based approach to transition prediction therefore remains an important challenge for the design of high-speed vehicles. Boundary-layer instabilities take a variety of forms depending on the flow geometry and conditions. On many geometries common in high-speed flight—for example swept surfaces and cones at nonzero angle of incidence—the crossflow instability dominates the transition process. Crossflow arises in regions with a pressure gradient and curved streamlines on these bodies 1 . The inviscid streamlines are curved as a result of the combined influence of sweep (or angle of attack) and the pressure gradient. The pressure gradient, being nonparallel to the inviscid streamlines, has a component normal to the streamlines which provides the centripetal force associated with their curvature. While the pressure gradient is constant in the wall-normal direction through the boundary layer, the velocity is reduced near the wall. The pressure gradient therefore no longer balances with the required centripetal force and results in a nonzero ∗
Corresponding author. E-mail address: [email protected]
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.019
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yt wt
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stream-normal velocity component known as crossflow. The resulting three-dimensional boundary layer (Fig. 1) is then unstable to waves oriented in the crossflow direction. These boundary layers are typically unstable to both stationary and traveling waves. While the traveling waves often grow more rapidly, in low-speed flows the origin of the two varieties is different, and as such, which dominates transition is determined by several factors. In subsonic flows, stationary waves have been shown to originate from surface roughness features and their initial amplitudes and wavenumber spectra depend on the character and amplitude of the surface roughness. Traveling waves, by contrast, are extraordinarily sensitive to free-stream turbulence 2 . Early evidence from the present study (discussed below) suggests that stationary waves in the high-speed case follow a similar rule. To date, no such study has been undertaken establishing a similar link for traveling waves. The stationary wave on a cone manifests in the boundary layer initially as an azimuthal modulation to the mean flow, eventually growing into a series of co-rotating vortices aligned nearly with the inviscid streamlines 1 (see Fig. 4). These vortices adhere initially to linear stability theory 3 , but quickly become nonlinear as they modify the mean flow by drawing O(1) momentum fluid down into the boundary layer. The resulting modified boundary layer often contains several new inflection points, leading to secondary instabilities that rapidly break down to turbulence 4,5,6,7 . Given the ubiquity of the swept wing on modern aircraft, crossflow has been the subject of intense study over the past several decades 1 . The bulk of these studies have been in subsonic flows, however, while hypersonic crossflow has remained relatively untouched, comprised only by a handful of computational and experimental studies. Several direct numerical simulations (DNS) were performed probing the effects of periodic, discrete roughness 8 and distributed roughness 9 on a right-circular cone with 7◦ half angle at 6◦ , a geometry which has been adopted as a standardized geometry. An additional computational tool was developed on this geometry utilizing the more lightweight nonlinear parabolized stability equations (NPSE) 10 . Experimental data are more sparse, including an early experiment by Stetson 11 and a series of more recent experiments closely related to the computational works of Balakumar and Owens 8 and Gronvall et al. 9 in a Mach 6 quiet Ludwieg tube 12 . The existing experimental data include temperature and heat flux data as well as flow visualization, but lack boundary layer measurements. The present study utilizes the newly recommissioned Mach 6 Quiet Tunnel at Texas A&M University to provide such data.
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Type-K thermocouples (×10) 0.106 m
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2. Experimental setup and method 2.1. Wind tunnel facility The present experiments were performed in the Mach 6 Quiet Tunnel at Texas A&M University (M6QT). The facility began life at the NASA Langley Research Center and was transferred to Texas A&M in 2005 where the facility underwent an extensive modification and integration effort at the National Aerothermochemistry Laboratory 13 followed by a series of second-mode experiments 14,15,16 . M6QT is a low-disturbance, hypersonic wind tunnel with free-stream pressure fluctuations of pt2,rms /pt2 < 0.1% (0.1% represents the noise floor for the relevant instrumentation), making the facility suitable for boundary-layer stability experiments. The facility operates in a pressure-vacuum blowdown mode with an open free-jet test section and is capable of approximately 40 seconds of on-condition run time. The tunnel is heated to a total temperature of T t1 = 430 K in order to prevent liquefaction through the expansion. The settling chamber total pressure range is 400 kPa ≤ pt1 ≤ 950 kPa under quiet conditions corresponding to a free-stream unit Reynolds number range of 4.6 × 106 m−1 ≤ Re/L ≤ 10.6 × 106 m−1 . For the present experiments, pt1 ≈ 896 kPa and Re/L ≈ 10 × 106 m−1 . 2.2. Cone model The model in the present study (Fig. 2) is a 7◦ right circular cone placed at 5.6◦ angle of attack. It is constructed of 17-4 PH heat-treated stainless steel and features thick walls to minimize temperature variation during each run. It is 0.430 m in length with a 0.155 m removable and interchangeable tip. The tip and the mounting shaft are each keyed to ensure the repeatability of the mounting orientation. The tip installed in the present experiments is a smooth, sharp tip of radius rtip ≤ 50 μm and is considered aerodynamically sharp. The rms surface roughness is approximately Rq = 2.15 μm. Along one ray of the frustum is an array of ten surface-mounted type-K thermocouples used for monitoring the wall temperature during testing. The hollow frustum is fitted with a PID-controlled internal heating system and the model is heated to just below T aw prior to each run. During the preheat, the wall temperature continues to rise until the cone reaches T w /T aw = 1 for the duration of the run. During a typical run, a single thermocouple varies in temperature by ΔT < 1 K. The thermocouples nearest the tip and base are typically at a constant T w /T aw = 0.988 and T w /T aw = 0.963 as a result of their proximity to the large thermal masses in the tip and mount respectively. The aftmost measurement location is at the condition T w /T aw = 0.988. 2.3. Instrumentation The present study utilizes a modified A. A. Lab Systems AN-1003 constant-temperature hot-wire anemometer for point measurements of mass flux. Wires are operated at a high temperature loading factor (τ ≈ 0.8), and as such the wire is primarily sensitive to mass flux 17 . Hot-wire probes are designed and manufactured in-house using 90%/10% platinum/rhodium Wollaston wire. The active element is 5 μm in diameter and 650 μm in length (L/D ≈ 130). In the present study, the wire voltage is uncalibrated. The anemometer has a constant frequency response up to a corner frequency of fc ≈ 180 kHz.
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Fig. 3: Cylindrical-coordinate probe traversing mechanism.
Hot-wire probes are positioned using a three-axis probe traversing system operating in a cylindrical coordinate system. The system allows approximately 200◦ of travel about the azimuth of the cone at a rate of 10◦ /s, 120 mm of travel along the axis of the cone, and 40 mm of travel radially. The traverse position (rt , θt , zt ) is calibrated such that θt = 0◦ and θt = 180◦ correspond to the windward and leeward rays of the cone respectively, zt = 0 corresponds to the tip of the cone, and rt = 0 is set locally to the wall location. The entire system is operated through a custom LabVIEW motion control and data acquisition program. 2.4. Method A typical tunnel run involved starting with the probe at θ = 0◦ and the maximum r-position in order to remove it from the flow, protecting it from the starting shock and minimizing tunnel blockage. The traverse then moved to the measurement location (θ ≈ 120◦ ) conducted an azimuthal sweep of measurement points, and then moved into position behind the cone for protection from the unstart shock. Each data point was sampled at 1 MHz for 100 ms. A typical run consisted of between 30 and 40 data points. Contours were generated by combining a series of constant-height azimuthal runs into a two-dimensional grid at three locations of constant z: 360 mm, 370 mm, and 380 mm. Hot-wire voltages were split into mean and fluctuating components, V(r, θ, z, t) = V(r, θ, z) + v (r, θ, z, t), by averaging and AC coupling of the signal respectively. 3. Results 3.1. Mean flow and the stationary wave Mean flow contours were created by averaging the hot-wire signal to obtain V(r, θ, z). The resulting voltages were then normalized against V w and V fs , the hot-wire voltages in the near-wall and free-stream regions of the flow respectively. In the most upstream location (z = 360 mm), the mean flow reflects the early development of the stationary crossflow wave, including the convection of high-momentum fluid from the free-stream region toward the wall (Fig. 4, top). By z = 370 mm (Fig. 4, center), the vortices have developed further and begun the rollover characteristic of crossflow-dominated boundary layers. At this point it appears to have reached a fully-developed state,
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and by z = 380 mm (Fig. 4, bottom), the structure has begun to degrade, with the low- and high-speed protrusions beginning to recede slightly. These data may also be analyzed by separating the grids into a series of vertical profiles (Fig. 5). Early in the development of the stationary wave, the series of mean flow profiles (plots of normalized V, Fig. 5a,d,g) may be averaged over the azimuthal range (V mean , black lines) and the shape of the undisturbed boundary layer recovered (Fig. 5a). As the stationary wave grows, the mean profiles develop an additional bend and a pair of additional inflection points (Fig. 5d,g). Additionally, the average profile may be subtracted from each individual mean flow profile to produce a series of mean defect profiles (V − V mean , Fig. 5c,f,i) showing the departure from the mean as a function of the boundary-layer height. From these plots, it is clear that the early modification to the mean flow is concentrated high in the boundary layer (Fig. 5c) and works its way toward the wall as the development of the stationary wave continues (Fig. 5f,i). Stationary wave mode shapes can be calculated by calculating the rms of the stationary wave at each height in the boundary layer. This information is already contained within the series of defect profiles. The series of points obtained by considering a single height on each of these profiles represents an azimuthal wave describing the modification to the mean flow. Taking the rms of this wave and plotting against the boundary layer height results in the stationary wave mode shapes shown in Fig. 6.
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Immediately evident in plots of the mode shapes is the fact that saturation is reached rapidly at approximately 23%. At the upstream location, the stationary wave has nearly reached saturation already and all subsequent growth is in the secondary peak on the underside of the curve, which matches the growth pattern observed in the series of defect profiles. In low-speed flows, the development of this secondary lobe was indicative of the onset of nonlinearity in the mean flow modification 18 , though it was observed above the primary peak on the curve. The saturation and subsequent slow attenuation also quantifies the qualitative observation from the contours that the high- and lowmomentum intrusions are beginning to recede. 3.2. Boundary-layer fluctuations The hot-wire signals were also AC-coupled in order to isolate the fluctuating component of the voltage, V . Con are plotted at each location in Fig. 7. These contours, in agreement with the Vrms profiles in Fig. 5b,e,h, tours of Vrms show a slow growth in overall fluctuation levels from 7% at z = 360 mm up to 8% at z = 380 mm. Fluctuations are concentrated in the regions of maximum mass flux gradient, particularly along the leeward side of and in the high-speed troughs between each vortex. At z = 380 mm, the low-disturbance region inside each vortex has begun to exhibit higher fluctuation levels. Power spectra were calculated for each data point in order to explore the frequency content of the fluctuations. The data was arranged as a series of contours at each frequency bin in the discrete Fourier transform and the spatial evolution of the frequency content in each bin was observed. Of particular note were two frequency bins: a lowfrequency bin, 15 kHz ≤ f ≤ 60 kHz; and a high-frequency bin, 80 kHz ≤ f ≤ 130 kHz. The power spectra for each point were integrated over each bin to calculate the total signal energy in the bin at a given sampling point. The results were then normalized across each contour with respect to each bin at a given axial location. Contours of the signal energy are shown in Figs. 8 and 9 for the low- and high-frequency bins respectively. The contour series for the low-frequency band (Fig. 8) shows a band of energy around the windward side of each vortex that shifts further into the troughs between vortices as the flow develops. Comparing the location of this contours, it is evident that while these frequencies account for little of the initial frequency band with the overall Vrms
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fluctuation energy, a large portion of the downstream growth occurs in this band. This frequency band coincides with that of the most amplified traveling crossflow waves according to LPSE calculations 10 and is very similar in location to traveling crossflow in similar low-speed studies 5,6,7 . The high-frequency band (Fig. 9) exists along the leeward side edge of the vortices. This band is localized in a location comparable to that of the onset of secondary instability in low-speed flows 5,6,7 and the observed frequencies compare well with such low-speed cases based on convective scales in the flow and takes the form f ≈ 0.4Ue /δ. It is therefore believed that this frequency band represents to secondary instability. However, contrary to the low-speed case, explosive growth is not observed immediately following the onset of secondary instability and transition to turbulence is never observed. 3.3. A note on surface roughness and stationary crossflow During the course of this study, one data contour was taken and then the roughness on the model was modified slightly in character but not in amplitude. Another contour was taken at the same location and while the amplitude of the vortices were identical to those measured prior to the change, the azimuthal location shifted slightly. In all other cases where roughness was not changed, all runs were repeatable. Therefore, it is likely that, as in the low-speed
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case, the formation of stationary crossflow waves originates at surface imperfections near the neutral point, and a slight change in the character of the roughness in that region affects the eventual development of the stationary wave downstream.
4. Conclusions Experiments were performed in the low-disturbance M6QT at Texas A&M University on the crossflow instability on a yawed, 7◦ cone at 5.6◦ yaw in a Mach 6 flow. Hot-wire measurements were made at several axial locations on the cone and the development of both the steady and unsteady components of the flow were investigated. The mean flow was dominated by the effect of a stationary crossflow wave that was observed to quickly saturate at an amplitude of approximately 23%. Voltage fluctuations grew only slightly from 7% to 8% over the measure-
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ment range. The spectral content of the measurements indicates the presence of a traveling crossflow wave growing concurrently with the stationary mode as well as the existence of secondary instability. These results compare well with computational studies on the same geometry and with analogous studies, both computational and experimental, in low-speed flows. Stationary-wave saturation occurs at levels comparable to lowspeed flows and its amplitude decreases slightly after saturation. The traveling wave is comparably located to its lowspeed counterpart and its frequency shows good agreement with LPSE calculations. Finally, secondary instability was observed to be located in a region nearly identical to that of the low-speed case and its frequency showed a relationship with convective flow scales similar to that of the low-speed case of the form f ≈ 0.4Ue /δ. Transition to turbulence is not observed in the present experiments. Since secondary instabilities do not show the expected explosive growth common at low-speeds and stationary-wave saturation occurs far downstream (z/L = 0.86), it is believed that the receptivity process on the smooth tip does not provide sufficient initial amplitude for transition to occur naturally on the model.
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Acknowledgements Support from the AFOSR/NASA National Center for Hypersonic Laminar-Turbulent Transition Research through Grant FA9550-09-1-0341 is gratefully acknowledged. References 1. Saric, W.S., Reed, H.L., White, E.B.. Stability and Transition of Three-Dimensional Boundary Layers. Annu Rev Fluid Mech 2003; 35(1):413–440. doi:10.1146/annurev.fluid.35.101101.161045. 2. Deyhle, H., Bippes, H.. Disturbance growth in an unstable three-dimensional boundary layer and its dependence on environmental conditions. J Fluid Mech 1996;316:73–113. doi:10.1017/S0022112096000456. 3. Mack, L.M.. Boundary-Layer Linear Stability Theory. AGARD Report No 709 1984;. 4. Kohama, Y., Saric, W.S., Hoos, J.A.. A high-frequency, secondary instability of crossflow vortices that leads to transition. In: Boundary layer transition and control; Proceedings of the Conference, Univ. of Cambridge. Cambridge, UK: Royal Aeronautical Society; 1991, p. 4.1–4.13. 5. Wassermann, P., Kloker, M.. Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer. Journal of Fluid Mechanics 2002;456:49–84. doi:10.1017/S0022112001007418. 6. White, E.B., Saric, W.S.. Secondary instability of crossflow vortices. J Fluid Mech 2005;525:275–308. doi:10.1017/ S002211200400268X. 7. Bonfigli, G., Kloker, M.. Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation. J Fluid Mech 2007;583:229. doi:10.1017/S0022112007006179. 8. Balakumar, P., Owens, L.R.. Stability of Hypersonic Boundary Layers on a Cone at an Angle of Attack. In: 40th AIAA Fluid Dynamics Conference. Chicago, IL: AIAA 2010-4718; 2010, . 9. Gronvall, J.E., Johnson, H.B., Candler, G.V.. Hypersonic Three-Dimensional Boundary Layer Transition on a Cone at Angle of Attack. In: 42nd AIAA Fluid Dynamic Conference. New Orleans, LA: AIAA 2012-2822; 2012, . 10. Kuehl, J.J., Perez, E., Reed, H.L.. JoKHeR: NPSE Simulations of Hypersonic Crossflow Instability. In: 50th AIAA Aerospace Sciences Meeting. Nashville, TN: AIAA 2012-0921; 2012, . 11. Stetson, K.F.. Mach 6 Experiments of Transition on a Cone at Angle of Attack. Journal of Spacecraft and Rockets 1982;19(5):397–403. doi:10.2514/3.62276. 12. Swanson, E.O., Schneider, S.P.. Boundary-Layer Transition on Cones at Angle of Attack in a Mach-6 Quiet Tunnel. In: 49th AIAA Aerospace Sciences Meeting. Orlando, FL: AIAA 2010-1062; 2010, . 13. Hofferth, J.W., Bowersox, R.D.W., Saric, W.S.. The Mach 6 Quiet Tunnel at Texas A&M: Quiet Flow Performance. In: 27th AIAA Aerodynamic Measurement Technology and Ground Testing Conference. Chicago, IL: AIAA 2010-4794; 2010, . 14. Hofferth, J.W., Saric, W.S.. Boundary-Layer Transition on a Flared Cone in the Texas A&M Mach 6 Quiet Tunnel. In: 50th AIAA Aerospace Sciences Meeting. Nashville, TN: AIAA 2012-0923; 2012, . 15. Hofferth, J.W., Saric, W.S., Kuehl, J.J., Perez, E., Kocian, T.S., Reed, H.L.. Boundary-Layer Instability & Transition on a Flared Cone in a Mach 6 Quiet Wind Tunnel. In: RTO-AVT-200-10. 2012, . 16. Hofferth, J.W., Humble, R.A., Floryan, D.C.. High-Bandwidth Optical Measurements of the Second-Mode Instability in a Mach 6 Quiet Tunnel. In: 51st AIAA Aerospace Sciences Meeting. Grapevine, TX: AIAA 2013-0378; 2013, . 17. Smits, A.J., Hayakawa, K., Muck, K.C.. Constant Temperature Hot-wire Anemometer Practice In Supersonic Flows - Part 1: The Normal Wire. Exp Fluids 1983;1(2):83–92. doi:10.1007/BF00266260. 18. Reibert, M.S., Saric, W.S., Carrillo, R.B.J., Chapman, K.L.. Experiments in nonlinear saturation of stationary crossflow vortices in a swept-wing boundary layer. In: 34th AIAA Aerospace Sciences Meeting. Reno, NV: AIAA 96-0184; 1996, .
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IUTAM ABCM Symposium on Laminar Turbulent Transition
Crossflow instability on a yawed cone at Mach 6 Stuart A. Craig∗, William S. Saric Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Abstract Boundary-layer measurements were performed in a Mach 6, low-disturbance wind tunnel on a 7◦ cone at 5.6◦ angle of incidence such that the model was primarily subject to crossflow instability. Constant-temperature hot-wire anemometry was used to measure the streamwise mass flux at a series of planes normal to the cone axis. A dominant stationary wave is observed to achieve nonlinear saturation at approximately 23% with maximum rms fluctuations of approximately 8%. Additionally, traveling crossflow waves were observed in a broad frequency band centered at f = 40 kHz and secondary instabilities were observed in a broad band centered at f = 100 kHz. Measurements show excellent agreement with previously published computational results and are qualitatively similar to studies performed in subsonic flows. Transition to turbulence was not observed to occur under these conditions. c 2015 2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) Keywords: Laminar-turbulent transition; boundary layer; hypersonic; crossflow instability
1. Introduction The order-of-magnitude increase in surface heat transfer rate and viscous drag upon transition to turbulence has important ramifications in the design of hypersonic vehicles. However, the lack of reliable transition prediction methods has historically led to the overdesign of these systems and inefficient vehicles as designers are forced to rely on simplified correlation methods that are often non-physical and geometry-specific. The development of a physics-based approach to transition prediction therefore remains an important challenge for the design of high-speed vehicles. Boundary-layer instabilities take a variety of forms depending on the flow geometry and conditions. On many geometries common in high-speed flight—for example swept surfaces and cones at nonzero angle of incidence—the crossflow instability dominates the transition process. Crossflow arises in regions with a pressure gradient and curved streamlines on these bodies 1 . The inviscid streamlines are curved as a result of the combined influence of sweep (or angle of attack) and the pressure gradient. The pressure gradient, being nonparallel to the inviscid streamlines, has a component normal to the streamlines which provides the centripetal force associated with their curvature. While the pressure gradient is constant in the wall-normal direction through the boundary layer, the velocity is reduced near the wall. The pressure gradient therefore no longer balances with the required centripetal force and results in a nonzero ∗
Corresponding author. E-mail address: [email protected]
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.019
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stream-normal velocity component known as crossflow. The resulting three-dimensional boundary layer (Fig. 1) is then unstable to waves oriented in the crossflow direction. These boundary layers are typically unstable to both stationary and traveling waves. While the traveling waves often grow more rapidly, in low-speed flows the origin of the two varieties is different, and as such, which dominates transition is determined by several factors. In subsonic flows, stationary waves have been shown to originate from surface roughness features and their initial amplitudes and wavenumber spectra depend on the character and amplitude of the surface roughness. Traveling waves, by contrast, are extraordinarily sensitive to free-stream turbulence 2 . Early evidence from the present study (discussed below) suggests that stationary waves in the high-speed case follow a similar rule. To date, no such study has been undertaken establishing a similar link for traveling waves. The stationary wave on a cone manifests in the boundary layer initially as an azimuthal modulation to the mean flow, eventually growing into a series of co-rotating vortices aligned nearly with the inviscid streamlines 1 (see Fig. 4). These vortices adhere initially to linear stability theory 3 , but quickly become nonlinear as they modify the mean flow by drawing O(1) momentum fluid down into the boundary layer. The resulting modified boundary layer often contains several new inflection points, leading to secondary instabilities that rapidly break down to turbulence 4,5,6,7 . Given the ubiquity of the swept wing on modern aircraft, crossflow has been the subject of intense study over the past several decades 1 . The bulk of these studies have been in subsonic flows, however, while hypersonic crossflow has remained relatively untouched, comprised only by a handful of computational and experimental studies. Several direct numerical simulations (DNS) were performed probing the effects of periodic, discrete roughness 8 and distributed roughness 9 on a right-circular cone with 7◦ half angle at 6◦ , a geometry which has been adopted as a standardized geometry. An additional computational tool was developed on this geometry utilizing the more lightweight nonlinear parabolized stability equations (NPSE) 10 . Experimental data are more sparse, including an early experiment by Stetson 11 and a series of more recent experiments closely related to the computational works of Balakumar and Owens 8 and Gronvall et al. 9 in a Mach 6 quiet Ludwieg tube 12 . The existing experimental data include temperature and heat flux data as well as flow visualization, but lack boundary layer measurements. The present study utilizes the newly recommissioned Mach 6 Quiet Tunnel at Texas A&M University to provide such data.
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Type-K thermocouples (×10) 0.106 m
R50 μm
0.155 m 0.430 m Fig. 2: Schematic of the 7◦ cone model.
2. Experimental setup and method 2.1. Wind tunnel facility The present experiments were performed in the Mach 6 Quiet Tunnel at Texas A&M University (M6QT). The facility began life at the NASA Langley Research Center and was transferred to Texas A&M in 2005 where the facility underwent an extensive modification and integration effort at the National Aerothermochemistry Laboratory 13 followed by a series of second-mode experiments 14,15,16 . M6QT is a low-disturbance, hypersonic wind tunnel with free-stream pressure fluctuations of pt2,rms /pt2 < 0.1% (0.1% represents the noise floor for the relevant instrumentation), making the facility suitable for boundary-layer stability experiments. The facility operates in a pressure-vacuum blowdown mode with an open free-jet test section and is capable of approximately 40 seconds of on-condition run time. The tunnel is heated to a total temperature of T t1 = 430 K in order to prevent liquefaction through the expansion. The settling chamber total pressure range is 400 kPa ≤ pt1 ≤ 950 kPa under quiet conditions corresponding to a free-stream unit Reynolds number range of 4.6 × 106 m−1 ≤ Re/L ≤ 10.6 × 106 m−1 . For the present experiments, pt1 ≈ 896 kPa and Re/L ≈ 10 × 106 m−1 . 2.2. Cone model The model in the present study (Fig. 2) is a 7◦ right circular cone placed at 5.6◦ angle of attack. It is constructed of 17-4 PH heat-treated stainless steel and features thick walls to minimize temperature variation during each run. It is 0.430 m in length with a 0.155 m removable and interchangeable tip. The tip and the mounting shaft are each keyed to ensure the repeatability of the mounting orientation. The tip installed in the present experiments is a smooth, sharp tip of radius rtip ≤ 50 μm and is considered aerodynamically sharp. The rms surface roughness is approximately Rq = 2.15 μm. Along one ray of the frustum is an array of ten surface-mounted type-K thermocouples used for monitoring the wall temperature during testing. The hollow frustum is fitted with a PID-controlled internal heating system and the model is heated to just below T aw prior to each run. During the preheat, the wall temperature continues to rise until the cone reaches T w /T aw = 1 for the duration of the run. During a typical run, a single thermocouple varies in temperature by ΔT < 1 K. The thermocouples nearest the tip and base are typically at a constant T w /T aw = 0.988 and T w /T aw = 0.963 as a result of their proximity to the large thermal masses in the tip and mount respectively. The aftmost measurement location is at the condition T w /T aw = 0.988. 2.3. Instrumentation The present study utilizes a modified A. A. Lab Systems AN-1003 constant-temperature hot-wire anemometer for point measurements of mass flux. Wires are operated at a high temperature loading factor (τ ≈ 0.8), and as such the wire is primarily sensitive to mass flux 17 . Hot-wire probes are designed and manufactured in-house using 90%/10% platinum/rhodium Wollaston wire. The active element is 5 μm in diameter and 650 μm in length (L/D ≈ 130). In the present study, the wire voltage is uncalibrated. The anemometer has a constant frequency response up to a corner frequency of fc ≈ 180 kHz.
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Fig. 3: Cylindrical-coordinate probe traversing mechanism.
Hot-wire probes are positioned using a three-axis probe traversing system operating in a cylindrical coordinate system. The system allows approximately 200◦ of travel about the azimuth of the cone at a rate of 10◦ /s, 120 mm of travel along the axis of the cone, and 40 mm of travel radially. The traverse position (rt , θt , zt ) is calibrated such that θt = 0◦ and θt = 180◦ correspond to the windward and leeward rays of the cone respectively, zt = 0 corresponds to the tip of the cone, and rt = 0 is set locally to the wall location. The entire system is operated through a custom LabVIEW motion control and data acquisition program. 2.4. Method A typical tunnel run involved starting with the probe at θ = 0◦ and the maximum r-position in order to remove it from the flow, protecting it from the starting shock and minimizing tunnel blockage. The traverse then moved to the measurement location (θ ≈ 120◦ ) conducted an azimuthal sweep of measurement points, and then moved into position behind the cone for protection from the unstart shock. Each data point was sampled at 1 MHz for 100 ms. A typical run consisted of between 30 and 40 data points. Contours were generated by combining a series of constant-height azimuthal runs into a two-dimensional grid at three locations of constant z: 360 mm, 370 mm, and 380 mm. Hot-wire voltages were split into mean and fluctuating components, V(r, θ, z, t) = V(r, θ, z) + v (r, θ, z, t), by averaging and AC coupling of the signal respectively. 3. Results 3.1. Mean flow and the stationary wave Mean flow contours were created by averaging the hot-wire signal to obtain V(r, θ, z). The resulting voltages were then normalized against V w and V fs , the hot-wire voltages in the near-wall and free-stream regions of the flow respectively. In the most upstream location (z = 360 mm), the mean flow reflects the early development of the stationary crossflow wave, including the convection of high-momentum fluid from the free-stream region toward the wall (Fig. 4, top). By z = 370 mm (Fig. 4, center), the vortices have developed further and begun the rollover characteristic of crossflow-dominated boundary layers. At this point it appears to have reached a fully-developed state,
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1 0.8
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and by z = 380 mm (Fig. 4, bottom), the structure has begun to degrade, with the low- and high-speed protrusions beginning to recede slightly. These data may also be analyzed by separating the grids into a series of vertical profiles (Fig. 5). Early in the development of the stationary wave, the series of mean flow profiles (plots of normalized V, Fig. 5a,d,g) may be averaged over the azimuthal range (V mean , black lines) and the shape of the undisturbed boundary layer recovered (Fig. 5a). As the stationary wave grows, the mean profiles develop an additional bend and a pair of additional inflection points (Fig. 5d,g). Additionally, the average profile may be subtracted from each individual mean flow profile to produce a series of mean defect profiles (V − V mean , Fig. 5c,f,i) showing the departure from the mean as a function of the boundary-layer height. From these plots, it is clear that the early modification to the mean flow is concentrated high in the boundary layer (Fig. 5c) and works its way toward the wall as the development of the stationary wave continues (Fig. 5f,i). Stationary wave mode shapes can be calculated by calculating the rms of the stationary wave at each height in the boundary layer. This information is already contained within the series of defect profiles. The series of points obtained by considering a single height on each of these profiles represents an azimuthal wave describing the modification to the mean flow. Taking the rms of this wave and plotting against the boundary layer height results in the stationary wave mode shapes shown in Fig. 6.
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Immediately evident in plots of the mode shapes is the fact that saturation is reached rapidly at approximately 23%. At the upstream location, the stationary wave has nearly reached saturation already and all subsequent growth is in the secondary peak on the underside of the curve, which matches the growth pattern observed in the series of defect profiles. In low-speed flows, the development of this secondary lobe was indicative of the onset of nonlinearity in the mean flow modification 18 , though it was observed above the primary peak on the curve. The saturation and subsequent slow attenuation also quantifies the qualitative observation from the contours that the high- and lowmomentum intrusions are beginning to recede. 3.2. Boundary-layer fluctuations The hot-wire signals were also AC-coupled in order to isolate the fluctuating component of the voltage, V . Con are plotted at each location in Fig. 7. These contours, in agreement with the Vrms profiles in Fig. 5b,e,h, tours of Vrms show a slow growth in overall fluctuation levels from 7% at z = 360 mm up to 8% at z = 380 mm. Fluctuations are concentrated in the regions of maximum mass flux gradient, particularly along the leeward side of and in the high-speed troughs between each vortex. At z = 380 mm, the low-disturbance region inside each vortex has begun to exhibit higher fluctuation levels. Power spectra were calculated for each data point in order to explore the frequency content of the fluctuations. The data was arranged as a series of contours at each frequency bin in the discrete Fourier transform and the spatial evolution of the frequency content in each bin was observed. Of particular note were two frequency bins: a lowfrequency bin, 15 kHz ≤ f ≤ 60 kHz; and a high-frequency bin, 80 kHz ≤ f ≤ 130 kHz. The power spectra for each point were integrated over each bin to calculate the total signal energy in the bin at a given sampling point. The results were then normalized across each contour with respect to each bin at a given axial location. Contours of the signal energy are shown in Figs. 8 and 9 for the low- and high-frequency bins respectively. The contour series for the low-frequency band (Fig. 8) shows a band of energy around the windward side of each vortex that shifts further into the troughs between vortices as the flow develops. Comparing the location of this contours, it is evident that while these frequencies account for little of the initial frequency band with the overall Vrms
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fluctuation energy, a large portion of the downstream growth occurs in this band. This frequency band coincides with that of the most amplified traveling crossflow waves according to LPSE calculations 10 and is very similar in location to traveling crossflow in similar low-speed studies 5,6,7 . The high-frequency band (Fig. 9) exists along the leeward side edge of the vortices. This band is localized in a location comparable to that of the onset of secondary instability in low-speed flows 5,6,7 and the observed frequencies compare well with such low-speed cases based on convective scales in the flow and takes the form f ≈ 0.4Ue /δ. It is therefore believed that this frequency band represents to secondary instability. However, contrary to the low-speed case, explosive growth is not observed immediately following the onset of secondary instability and transition to turbulence is never observed. 3.3. A note on surface roughness and stationary crossflow During the course of this study, one data contour was taken and then the roughness on the model was modified slightly in character but not in amplitude. Another contour was taken at the same location and while the amplitude of the vortices were identical to those measured prior to the change, the azimuthal location shifted slightly. In all other cases where roughness was not changed, all runs were repeatable. Therefore, it is likely that, as in the low-speed
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case, the formation of stationary crossflow waves originates at surface imperfections near the neutral point, and a slight change in the character of the roughness in that region affects the eventual development of the stationary wave downstream.
4. Conclusions Experiments were performed in the low-disturbance M6QT at Texas A&M University on the crossflow instability on a yawed, 7◦ cone at 5.6◦ yaw in a Mach 6 flow. Hot-wire measurements were made at several axial locations on the cone and the development of both the steady and unsteady components of the flow were investigated. The mean flow was dominated by the effect of a stationary crossflow wave that was observed to quickly saturate at an amplitude of approximately 23%. Voltage fluctuations grew only slightly from 7% to 8% over the measure-
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ment range. The spectral content of the measurements indicates the presence of a traveling crossflow wave growing concurrently with the stationary mode as well as the existence of secondary instability. These results compare well with computational studies on the same geometry and with analogous studies, both computational and experimental, in low-speed flows. Stationary-wave saturation occurs at levels comparable to lowspeed flows and its amplitude decreases slightly after saturation. The traveling wave is comparably located to its lowspeed counterpart and its frequency shows good agreement with LPSE calculations. Finally, secondary instability was observed to be located in a region nearly identical to that of the low-speed case and its frequency showed a relationship with convective flow scales similar to that of the low-speed case of the form f ≈ 0.4Ue /δ. Transition to turbulence is not observed in the present experiments. Since secondary instabilities do not show the expected explosive growth common at low-speeds and stationary-wave saturation occurs far downstream (z/L = 0.86), it is believed that the receptivity process on the smooth tip does not provide sufficient initial amplitude for transition to occur naturally on the model.
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Acknowledgements Support from the AFOSR/NASA National Center for Hypersonic Laminar-Turbulent Transition Research through Grant FA9550-09-1-0341 is gratefully acknowledged. References 1. Saric, W.S., Reed, H.L., White, E.B.. Stability and Transition of Three-Dimensional Boundary Layers. Annu Rev Fluid Mech 2003; 35(1):413–440. doi:10.1146/annurev.fluid.35.101101.161045. 2. Deyhle, H., Bippes, H.. Disturbance growth in an unstable three-dimensional boundary layer and its dependence on environmental conditions. J Fluid Mech 1996;316:73–113. doi:10.1017/S0022112096000456. 3. Mack, L.M.. Boundary-Layer Linear Stability Theory. AGARD Report No 709 1984;. 4. Kohama, Y., Saric, W.S., Hoos, J.A.. A high-frequency, secondary instability of crossflow vortices that leads to transition. In: Boundary layer transition and control; Proceedings of the Conference, Univ. of Cambridge. Cambridge, UK: Royal Aeronautical Society; 1991, p. 4.1–4.13. 5. Wassermann, P., Kloker, M.. Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer. Journal of Fluid Mechanics 2002;456:49–84. doi:10.1017/S0022112001007418. 6. White, E.B., Saric, W.S.. Secondary instability of crossflow vortices. J Fluid Mech 2005;525:275–308. doi:10.1017/ S002211200400268X. 7. Bonfigli, G., Kloker, M.. Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation. J Fluid Mech 2007;583:229. doi:10.1017/S0022112007006179. 8. Balakumar, P., Owens, L.R.. Stability of Hypersonic Boundary Layers on a Cone at an Angle of Attack. In: 40th AIAA Fluid Dynamics Conference. Chicago, IL: AIAA 2010-4718; 2010, . 9. Gronvall, J.E., Johnson, H.B., Candler, G.V.. Hypersonic Three-Dimensional Boundary Layer Transition on a Cone at Angle of Attack. In: 42nd AIAA Fluid Dynamic Conference. New Orleans, LA: AIAA 2012-2822; 2012, . 10. Kuehl, J.J., Perez, E., Reed, H.L.. JoKHeR: NPSE Simulations of Hypersonic Crossflow Instability. In: 50th AIAA Aerospace Sciences Meeting. Nashville, TN: AIAA 2012-0921; 2012, . 11. Stetson, K.F.. Mach 6 Experiments of Transition on a Cone at Angle of Attack. Journal of Spacecraft and Rockets 1982;19(5):397–403. doi:10.2514/3.62276. 12. Swanson, E.O., Schneider, S.P.. Boundary-Layer Transition on Cones at Angle of Attack in a Mach-6 Quiet Tunnel. In: 49th AIAA Aerospace Sciences Meeting. Orlando, FL: AIAA 2010-1062; 2010, . 13. Hofferth, J.W., Bowersox, R.D.W., Saric, W.S.. The Mach 6 Quiet Tunnel at Texas A&M: Quiet Flow Performance. In: 27th AIAA Aerodynamic Measurement Technology and Ground Testing Conference. Chicago, IL: AIAA 2010-4794; 2010, . 14. Hofferth, J.W., Saric, W.S.. Boundary-Layer Transition on a Flared Cone in the Texas A&M Mach 6 Quiet Tunnel. In: 50th AIAA Aerospace Sciences Meeting. Nashville, TN: AIAA 2012-0923; 2012, . 15. Hofferth, J.W., Saric, W.S., Kuehl, J.J., Perez, E., Kocian, T.S., Reed, H.L.. Boundary-Layer Instability & Transition on a Flared Cone in a Mach 6 Quiet Wind Tunnel. In: RTO-AVT-200-10. 2012, . 16. Hofferth, J.W., Humble, R.A., Floryan, D.C.. High-Bandwidth Optical Measurements of the Second-Mode Instability in a Mach 6 Quiet Tunnel. In: 51st AIAA Aerospace Sciences Meeting. Grapevine, TX: AIAA 2013-0378; 2013, . 17. Smits, A.J., Hayakawa, K., Muck, K.C.. Constant Temperature Hot-wire Anemometer Practice In Supersonic Flows - Part 1: The Normal Wire. Exp Fluids 1983;1(2):83–92. doi:10.1007/BF00266260. 18. Reibert, M.S., Saric, W.S., Carrillo, R.B.J., Chapman, K.L.. Experiments in nonlinear saturation of stationary crossflow vortices in a swept-wing boundary layer. In: 34th AIAA Aerospace Sciences Meeting. Reno, NV: AIAA 96-0184; 1996, .
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