Boundary-layer instability and transition on a flared

Int. J. Engineering Systems Modelling and Simulation, Vol. 5, Nos. 1/2/3, 2013

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Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel Jerrod Hofferth*, William Saric, Joseph Kuehl, Eduardo Perez, Travis Kocian and Helen Reed Aerospace Engineering, Texas A&M University, 701 H.R. Bright Building, 3141 TAMU, College Station, TX 77843-3141, USA E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: Measurements of boundary-layer transition location and boundary layer profiles on a sharp-tipped 5º-half-angle flared cone were made in a low-disturbance Mach 6 wind tunnel. Uncalibrated boundary-layer profiles of mean and fluctuating voltage representative of mass flux are obtained using constant temperature hot-wire anemometry at several axial locations, and are notionally compared with preliminary simulations. Spectral energy content is observed between 250 and 310 kHz – the first measurements of frequencies typical of the second mode instability at Texas A&M. Growth of this high-frequency content is compared with N-factor results from linear parabolised stability equation (LPSE) computations. Possible sources of disagreement between the experimental and computed frequencies for second-mode growth are discussed, as are future improvements to the hotwire anemometry technique. Nevertheless, the successful measurement of high-frequency content highlighted here constitutes an important step toward acquisition of calibrated measurements of hypersonic boundary-layer instabilities to be used as code validation. Keywords: hypersonics; quiet tunnels; low disturbance; boundary layer stability; laminar turbulent transition; second-mode instabilities; hotwire anemometry; linear stability theory; LST; parabolised stability equations; PSEs. Reference to this paper should be made as follows: Hofferth, J., Saric, W., Kuehl, J., Perez, E., Kocian, T. and Reed, H. (2013) ‘Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel’, Int. J. Engineering Systems Modelling and Simulation, Vol. 5, Nos. 1/2/3, pp.109–124. Biographical notes: Jerrod Hofferth is a PhD candidate studying under the advisement of Dr. William Saric. He has been primarily responsible for the reconstruction and characterisation of the NASA Langley Mach 6 Quiet Tunnel at Texas A&M’s National Aerothermochemistry Laboratory, and is currently conducting the initial series of boundary-layer stability experiments in the new installation. He has held an ASEE/NASA Aeronautics Fellowship (2009–2011), during which time he completed an internship with NASA Langley’s quiet-tunnel group in the flow physics and control branch. William Saric is a University Distinguished Professor and George Eppright ‘26 Chair in Engineering at Texas A&M University. He is the Director of the AFOSR/NASA National Centre for Hypersonic Transition Research. He received his PhD from the Illinois Institute of Technology in 1968 and has held appointments at Sandia National Laboratories, Virginia Tech., and Arizona State University. He is a member of the National Academy of Engineering and received the AIAA Fluid Dynamics Award (2003), the SES G.I. Taylor Medal (1993), and the AGARD (NATO) Scientific Achievement Award (1996). He is a fellow of AIAA, APS, and ASME. Joseph Kuehl is an Assistant Research Scientist with a split position between the National Centre for Hypersonic Transition Research and the Geochemical and Environmental Research Group at Texas A&M University. He received his PhDs in Mechanical Engineering and Physical Oceanography from the University of Rhode Island and Graduate School of Oceanography (2009). He developed the JoKHeR NPSE code in collaboration with Dr. Helen Reed for

Copyright © 2013 Inderscience Enterprises Ltd.

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J. Hofferth et al. hypersonic boundary layer stability applications. His other research interests include geophysical fluid dynamics (gap-leaping boundary currents and ocean observing systems) and non-linear vibrations (time series and finite time invariant manifold analysis). Eduardo Perez is currently pursuing his Master of Science degree in Aerospace Engineering at Texas A&M University under the advisement of Dr. Helen Reed. He received his Bachelor of Science degree in the 2009 from Texas A&M. As a graduate research assistant, he uses CFD and PSE/NPSE to study the laminar-to-turbulent transition of three-dimensional boundary layers in hypersonic flow. Travis Kocian is a graduate from Texas A&M University in 2012 with a Bachelor of Science degree in Aerospace Engineering and is currently pursuing his Master of Science degree under the advisement of Dr. Helen Reed. His research interests include CFD and PSE/NPSE boundary-layer stability analyses on various representative hypersonic geometries. Helen Reed is a Professor of Aerospace Engineering at Texas A&M University. She received her PhD from Virginia Tech in 1981 and has held appointments at NASA Langley Research Centre, Stanford University, Sandia National Laboratories, Tohoku University, and Arizona State University. She is a member of the National Research Council’s Aeronautics and Space Engineering Board (ASEB), she received the AIAA/ASEE Atwood Award (2007), and she was elected to the Academy of Engineering Excellence (2008) and the Committee of 100 (2010) at Virginia Tech. She is a Fellow of AIAA, APS, and ASME. This paper is a revised and expanded version of a paper entitled ‘Comparison of experimental and computational boundary-layer profiles and instability growth on a flared cone in a Mach 6 quiet flow’ presented at the 47th Applied Aerodynamics Symposium of the French Aeronautics and Astronautics Society, Paris, 26–28 March 2012.

1

Introduction

Understanding the process of laminar-to-turbulent boundary layer transition in the hypersonic regime is crucial to the design of a safe and efficient air vehicle, yet it poses formidable challenges to theoreticians and experimentalists alike. For the theoretician, the underlying physics portray a multifold process that can evolve in many different ways depending upon a number of parameters (Fedorov, 2011). For the experimentalist, flight tests are prohibitively expensive and flight-relevant wind tunnel experiments are limited to a few unique facilities, as discussed in an extensive review of efforts at NASA and Purdue University given by Schneider (2007). Despite the tremendous progress made in direct numerical simulations (DNSs), high-fidelity vehicle-scale simulations remain presently out of reach, and recourse to simplifying assumptions must still be made. Historically, the resulting uncertainties in our predictive capabilities have ultimately led to overdesign of thermal protection systems and suboptimal vehicle efficiencies. For these reasons, the Mach 6 Nozzle Test Chamber (M6NTC) was originally designed, built, and operated at NASA Langley as part of a programme spanning the 1970s, 80s, and 90s for the study of supersonic and hypersonic boundary-layer stability and transition in a quiet freestream environment (Wilkinson, 1997). Through the use of a bleed slot upstream of the throat and a highly-polished nozzle wall, transition on the nozzle wall is delayed and a test region with very low freestream disturbance levels (P′t2,RMS/Pt2 < 0.1%) is achieved (Chen et al., 1993). This enables sensitive measurements of boundary-layer instabilities largely free from the adverse effects of

freestream noise common to conventional supersonic wind tunnel facilities but absent in flight. The Langley M6NTC was decommissioned in the late 1990s and has recently been reestablished at Texas A&M University (Hofferth et al., 2010; Hofferth and Saric, 2012) as the Mach 6 Quiet Tunnel (M6QT). Currently, a combined theoretical, computational, and experimental effort is underway at Texas A&M and other universities collaborating under the NASA/AFOSR National Centre for Hypersonic Laminar-Turbulent Transition Research. A review of the many current activities of the centre is provided in Saric (2012). A key goal of the centre is to experimentally validate in the M6QT new and evolving computational tools based on linear stability theory (LST), parabolised stability equations (PSE/NPSE), and DNS. This work will follow the example of Lyttle et al. (2005), who, through careful consideration of the full range of experimental conditions, found excellent quantitative agreement between the second-mode growth predicted by their LST study and observed on blunted cones at Mach 6 and 8. In order to guide diagnostic capability development in the newly-reestablished M6QT, as well as to provide a solid foundation for new stability and transition research conducted at Texas A&M, activity has thus far centred around: 1

reevaluating the performance of the tunnel relative to historical NASA Langley data (Hofferth et al., 2010)

2

reproducing and building upon the stability experiments of Lachowicz et al. (1996) and Doggett et al. (1997) using the same flared cone model.

Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel The work presented below describes in detail the latest progress toward the latter of these two goals, with comparison to modern computations of the basic state and stability analyses.

2

Experimental setup

2.1 Facility and test environment Experiments were conducted in the Texas A&M University M6QT. In its present installation, the facility operates in a pressure-vacuum blowdown infrastructure with a typical on-condition test time of 30 seconds. Up to six of these full-duration runs can be executed in a single day (Hofferth et al., 2010). Freestream flow quality in the tunnel’s current installation has been observed to be very similar to that observed at NASA Langley. At Langley, a sharp rise in hot-wire fluctuation was observed at the nozzle exit plane on the centreline at Re = 10.0 × 106 m–1 (Blanchard et al., 1997), and at Texas A&M, the same sharp rise is observed in Kulite fast-response Pitot pressure data at the same physical location beginning at Re = 10.6 × 106 m–1 (Hofferth and Saric, 2012). Under quiet conditions at Re = 10.0 × 106 m–1, the Mach number in the test area was determined to be uniform at M = 5.91 ± 1.4% at NASA Langley, and the character of this variation was verified to be nearly identical recently at Texas A&M (Hofferth and Saric, 2012). For the boundary-layer measurements presented below, the tunnel operated at a nominal unit Reynolds number, Figure 1

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Re = 10 × 106 m–1 (stagnation pressure Pt1 = 906 kPa, stagnation temperature Tt1 = 430 K). Figure 1 shows contours of freestream root-mean-square (RMS) Pitot pressure disturbance levels as measured by a Kulite high-frequency pressure transducer for unit Reynolds numbers of 8 and 10 × 106 m–1, superimposed with a depiction of the installed location of the flared cone model. Note that the axes are normalised by the nozzle exit radius, rexit, and nozzle length, L. Clearly visible in the freestream data is the onset of nozzle-wall transition and the resulting impingement of pressure disturbances onto the flared cone model. At Re = 10 × 106 m–1, it can be seen that a substantial portion of the upstream part of the cone lies within a quiet – or low-disturbance – portion of the freestream environment, whereas the portion of the cone model downstream of Xc ≈ 0.305 m (12″) lies within a region of higher disturbance levels. This is not significantly dissimilar from the amount of noise impingement present for the previous experiments at NASA Langley; Lachowicz (1995) noted noise impingement for Xc > 0.362 m (14.25″) for a slightly lower Re = 9.3 × 106 m–1. Although the authors consider the freestream environments of the historical and present experiments to be very similar, future work will explicitly address the receptivity of the boundary layer to this noise impingement downstream and the resulting impact on stability and transition.

M6QT freestream noise contours, with cone model placement (see online version for colours)

Note: Presented with further discussion as Figure 3 in Hofferth and Saric (2012).

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2.2 Test model The model used in the present study is the Langley 93-10 flared cone, shown in Figure 2. Its geometry consists of a 5º half-angle right-circular conical profile for the first 0.25 m (10″) of axial distance, followed by a tangent flare of radius 2.36 m (93.071″) until the base of the cone at the 0.51 m (20″) axial station. The adverse pressure gradient imposed by the flare serves to destabilise the second-mode instability, promoting more substantial growth within the small test environment of the M6QT. The base diameter is 0.12 m. The model is of a thin-wall stainless steel construction with nominal wall thickness of 1.8 mm. The first 38 mm of length can be removed and replaced with an arbitrary nose-tip shape. Available nose pieces with sharp (2.5 µm nominal radius) tips are of primary interest for these initial experiments, but several tips of larger radii are also available. The model is instrumented along two axial rays separated by 180º azimuthally. Along one ray is a total of 29 pressure ports (8 in the straight-wall section, and 21 along the flare) 1 mm in diameter, whereas the opposite ray contains a total of 51 Type K thermocouples (8 in the straight-wall section, and 43 along the flare) welded to the interior wall. The wall thickness beneath the array of thermocouples is reduced to 0.8 mm in order to reduce the thermal inertia, improving measurement response time. Coordinate tolerances on the model are quoted as being within 25 µm on the radius in the straight section, and 50 µm on the flare. Coordinate measurements at Langley Figure 2

revealed a maximum RMS radius error less than 2.8% of the model boundary-layer thickness (Lachowicz et al., 1996). The model is polished to a surface finish of 50–100 nm RMS for the first 0.18 m of length, and 150–200 nm thereafter. Surface waviness is within 0.2 µm/mm. For the present study, the cone was installed at nominally zero angle of attack. Using the confocal-laser displacement device as described in Hofferth and Saric (2012), the geometric misalignment was determined to be 0.08º in pitch (nose up, bottom side windward) and 0.14º in yaw (nose left). As shown in the schematic in Figure 3, this misalignment puts the cone ray along which measurements are taken – via both the embedded thermocouples and boundary-layer hot-wire scans – in a slightly windward, 3D condition. The data collected by Lachowicz et al. (1996) were taken under 0.2º windward misalignment with pitch misalignment under 0.1º, so the conditions for the two studies are similar.

2.3 Hot-wire anemometry A constant-temperature anemometer (CTA) system is used in the present study. An A.A. Lab Systems AN-1003 anemometer unit was used to operate hot-wire sensors at a single, high temperature loading factor (τ ~ 1), such that the wire was primarily sensitive to mass flux (Smits et al., 1983).

Cross-sectional schematic of the NASA Langley 93-10 model and embedded instrumentation

Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel Figure 3

Cone alignment diagram (looking upstream through cone) (see online version for colours)

Figure 4

Hot-wire probe, 2.5 μm in diameter, with relatively short (L/D ≈ 100) active element (see online version for colours)

Note: Scale in (mm) (see online version for colours)

For protection against wire breakage during the tunnel start and unstart processes, the hot-wire probe begins parked just behind the base of the cone model, emerges once flow is established, and returns to a location behind base of the cone prior to tunnel shutdown. Operating in this manner, there is sufficient time in a single run to acquire data at 50 to 60 boundary-layer data points at a single axial location, or perhaps 20–25 points in boundary layers at each of two axial locations. With this limited run time, a multi-run procedure must be employed in order to fully tune a wire and prepare it for proper data collection. After a wire has been constructed and annealed, it is installed into the tunnel test section and mounted to the traversing mechanism. Next, a motion programme is manually configured which sets an array of points between approximately 250 μm and 4 mm from the model surface at the desired axial location. The tunnel is then preheated subsonically, during which time a conservative tuning balance is set on the anemometer to ensure stable operation during the full hypersonic run. When the run begins, a full boundary layer profile is

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acquired normally in order to determine the height in the motion programme that corresponds to the location of maximum RMS fluctuation. For the second run, the wire is placed directly in this maximum RMS location for the full duration of the run, so that bridge tuning can be specifically optimised using square wave injection at test conditions. For subsequent runs, these tuning, gain, and offset settings are fixed and the wire can be operated at other axial locations immediately. At the completion of a measurement campaign (or when a wire breaks), the wall is found in the traverse coordinate system at each axial location by carefully seeking electrical continuity between the prong tips and the model surface. It is only then that actual wall-normal distances are known. To automate this process in the future, new probe bodies may employ a fouling wire offset a known distance from the prong tips, with wall-finding circuitry integrated into the traverse controller, as was originally done by Lachowicz et al. (1996). Hot-wire data are sampled at 1 MHz, with a 250 ms residence time at each location within the boundary layer.

2.4 Run conditions Relevant run conditions for the hot-wire measurements presented here are listed in Table 1. Average cone temperature is provided but is to be considered only notional, as it is non-uniform both spatially along the cone axis and temporally as it cools per the data of §3.1. Furthermore, because the thermocouple array is beneath the ray of hot-wire motion, there is a noticeable rise in measured wall temperature behind the probe as it descends into the boundary layer. The transition-location data of §3.1 were taken prior to the hot-wire runs listed in Table 1, in order to avoid the influence of the hot-wire presence on the measurements.

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Table 1

Run conditions for hot-wire measurements

Run number

Cone axial station Xc (m) (in)

Pt1 [kPa (psia)]

Tt1 (K)

Average cone temp Tw (K)

1266

0.41 (16.0)

905.9 (131.4)

443.4

1261

0.46 (18.0)

908.7 (131.8)

429.2

1263

0.48 (19.0)

905.9 (131.4)

1264

0.50 (19.5)

905.3 (131.3)

Figure 5

Unit Re (106m–1)

R = sqrt (ReS)

410.1

9.7

1991

399.5

10.3

2174

427.1

398.4

10.3

2240

428.2

397.4

10.3

2264

Average cone wall temperature rates, in quiet mode (bleed valves open) and noisy mode (bleed valves closed) at Re = 9.9 × 106 m–1

0.2

0.1

Quiet flow (bleed valves open) Noisy flow (bleed valves closed)

ΔTw/Δt [K/s]

0

-0.1

-0.2

Transition under noisy flow

-0.3

-0.4 500

1000

Transition under quiet flow

1500 R = Re1/2 S

2000

2500

Note: Error bars indicate typical drift in rates over run duration.

3

Results and discussion

3.1 Surface temperature measurements As the simplest available diagnostic for the initial flaredcone experiments, transition on the model was monitored using the embedded array of 51 thermocouples. In previous experiments conducted in the tunnel at NASA Langley, where available tunnel run times exceeded 30 minutes at constant condition, this thermocouple array was used to determine adiabatic wall recovery temperatures, identifying regions of transitional or turbulent flow by the higher steady-state temperatures present due to a higher recovery factor. However, in the tunnel’s present installation, where run time does not exceed 40 seconds, the model surface does not have sufficient time to reach adiabatic conditions. Furthermore, because 10 to 15 minutes of subsonic convective preheating of the tunnel facility is required to establish the proper stagnation temperature prior to a full tunnel run, the cone model begins the run with wall temperatures higher than the adiabatic condition by approximately 10%. Therefore, for the duration of a given tunnel run, the cone wall temperature varies, and transition can only be observed in the differing rates of temperature change; the wall is cooled relatively slowly in regions of laminar flow, is cooled quickly under a turbulent

boundary layer, and is relatively heated in regions of transitional flow. Figure 5 presents typical profiles of the rate of temperature change along the axis of the cone for both quiet and noisy flow at Re = 9.9 × 106 m–1. The noisy flow condition was obtained with the nozzle-throat boundary-layer extraction valves closed; this is only ever done only for demonstration purposes, and does not represent a relevant flow condition. Cone axial stations have been cast in terms of cone Reynolds number, defined using freestream conditions and the distance along the cone arc surface, S: R = ReS =

ρ∞U ∞ S , μ∞

(1)

where ρ∞ and U∞ are the freestream density and velocity, respectively, and μ∞ is the freestream absolute viscosity evaluated using Sutherland’s Law. Error bars in Figure 5 represent the slight slowing of the rate of temperature change over the duration of the run as the adiabatic condition is approached (but never reached); magnitudes of the drift are typically 0.05 to 0.1 K/s for a typical run. This drift does not substantially change the character of the curve and thus does not significantly affect the interpretation of transition location.

Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel

onset moves aft to R = 2,100 (Xc = 0.44 m), and fully turbulent flow is not observed on the cone. Table 2 compares the transition-onset Reynolds numbers observed in the current campaign with those observed by Lachowicz et al. (1996), here cast in terms of ReS for easier comparison.

Under noisy conditions (square markers), transition appears to begin near R = 1,700 (Xc = 0.3 m), with fully-turbulent flow for R > 2,000. The slower transient response aft of R = 2,200 is an artefact of the thermal inertia of the 13 mm-thick mounting structure at the cone base. With the bleed valves opened (circular markers), a low-disturbance test environment is created, and transition Table 2

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Comparison of cone transition Reynolds numbers Freestream unit Re (m–1)

Reynolds number at transition onset, ReS,tr (m–1)

Lachowicz et al. (1996)

9.3 × 106

4.2 × 106

Adiabatic wall; ReS,tr calculated from temperatures; noise impingement for Xc > 14.25″

Hofferth and Saric (2012)

9.9 × 106

4.4 × 106

Tw = f(Xc, t); ReS,tr calculated from temperature rates; noise impingement for Xc > 12″

Dataset

Figure 6

Notes

Uncalibrated CTA voltage profiles (high overheat ratio; approx. mass flux) 3 Mean RMS

2.5

Mean RMS

Z [mm]

2

1.5

1

Xc = 0.41 m [16.0″] 6 -1 Re = 9.7 × 10 m R = Re1/2 = 1991 S

0.5

Xc = 0.46 m [18.0″] 6 -1 Re = 10.3 × 10 m R = Re1/2 = 2174 S

0

(a)

(b)

3 Mean RMS

2.5

Mean RMS

Z [mm]

2

1.5

1

Xc = 0.48 m [19.0″] Re = 10.3 × 106 m-1 R = Re1/2 = 2240 S

0.5

0

0

0.2

0.4 0.6 Normalised Voltage Range

(c)

0.8

Xc = 0.50 m [19.5″] Re = 10.3× 106 m-1 R = Re1/2 = 2264 S 1

0

0.2

0.4 0.6 Normalised Voltage Range

(d)

0.8

1

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Although the agreement here (≈5% in ReS,tr for similar freestream Re) is quite encouraging, the authors are reluctant to draw any strong conclusions from the direct comparison, as the differences in the experiments are numerous; these include any differences in facility flow quality, uncertainty in model angles of attack, and, most importantly, the necessarily and substantially different thermal conditions on the model wall and the difference in thermocouple interpretation that this requires. Thus, the most important conclusion to be drawn from this comparison is a general one: after years of storage, relocation and reactivation of the M6QT facility and its models, transition onset on the Langley 93-10 model remains very near its base at the tunnel’s high-Reynolds number quiet flow condition, as it was before. Current practice is to monitor the thermocouple array in the above manner during all tunnel runs to provide a convenient independent verification of the state of the boundary layer during hot-wire measurements.

3.2 Boundary-layer mean and RMS profiles Subsequent to the work presented in Hofferth and Saric (2012), hotwires of 2.5 μm core diameter rather than 5 μm were put into service and tuning of the anemometer circuit was further optimised during a tunnel run at the location of maximum RMS voltage intensity in the boundary layer. With these improvements, an impulse-response time constant of approximately 2.5 μs was achieved in the boundary layer, indicating a –3 dB rolloff point in the frequency response near 333 kHz. Uncalibrated mean and RMS anemometer output voltage are plotted normalised in Figure 6 for each axial location surveyed. Profiles of mean CTA voltage are displayed as solid black lines, normalised by the maximum and minimum CTA voltages encountered in the scan (an approximately 3-volt spread, prenormalisation). Profiles of RMS CTA voltage (calculated from the broadband signal) are in dashed lines, normalised by the maximum RMS voltage in each boundary layer. The shape of the mean voltage profiles is largely well behaved and consistent with expectation, although a mild overshoot in the mean voltage is present near the boundarylayer edge. Kendall (1957) observed an overshoot in impact pressure profiles on a flat plate at Mach 5.8 acquired when using only the largest of three Pitot probes. It is believed that the overshoot observed in the present dataset is likely due to a similar probe-interference effect, whereby a static pressure rise due to the probe lifts the boundary layer and modifies the effect of the probe shock to produce the overshoot. Future studies will employ smaller, morestreamlined probe bodies to test this hypothesis. Note also in the mean flow profiles that data at locations nearest the wall may include additional effects not present elsewhere in the boundary layer, chief among them the modified response of the hot-wire in transonic/subsonic flow. At each axial location considered, the height at which the maximum RMS fluctuation is observed corresponds to 80%–90% of the boundary-layer thickness, in agreement

with theoretical expectation based on the location of the critical layer, with the present computations for mode shapes of ρU disturbances, and with the observations of Lachowicz et al. (1996). Figure 7 presents – for observation of general trends only – the experimental mean boundary-layer profiles together with computed mass flux profiles, each nondimensionalised by their local edge values. In contrast to Figure 6, the experimental data here have been normalised between 0.1 and 1 to casually reflect that the measurements do not actually reach the wall, and that sensitivity of the hotwire becomes complicated as the flow becomes subsonic [e.g., the ‘kink’ in the data of Figure 7(b) near the amplitude of 0.1]. Due to this and to the fact that hot-wire voltage does not vary linearly with mass flux, a direct comparison between experiment and computation cannot occur until calibrated hot-wire measurements are obtained. One additional difficulty in making such comparisons – one that would be present even if the hotwire were calibrated – is that the cone wall temperature is slowly varying as the boundary layer is being scanned. That is, the shape and thickness of the boundary layer is being physically distorted while it is being measured. Specifically, the boundary layer thins (due to the cooling of the wall) as the measurement point is traversed top-to-bottom. Thus, a different experimental profile is obtained when the boundary layer is scanned bottom-to-top instead. This may be accounted for in future simulations by performing several ‘quasi-steady’ computations at different wall temperatures, according to the wall temperatures at the time each point was acquired in the experimental profile. For the present computational simulations, a structured two-dimensional grid was generated in the Pointwise meshing software. The grid size encompasses 599 grid points along the cone surface and 1,499 in the wall-normal direction, with 912 in the shock layer and 330 in the boundary layer. Emphasis was placed on clustering cells in the boundary-layer and shock regions to capture high gradients. The flow over the cone is solved as steady-state laminar flow using general aerodynamic simulation programme (GASP) computational software, Version 5.1. The code solves the unsteady Navier-Stokes equations using a cell-centred finite volume scheme. The flow simulation was performed in GASP as an axisymmetric case at zero angle of attack (i.e., the slight model misalignment is not accounted for in the simulation). The computations are performed using the Roe-Harten implementation for the inviscid fluxes, with a van Albada limiter equal to 0.33 and a 3rd-order upwind biased spatial accuracy. A no-slip wall boundary condition was applied at the cone surface with a wall temperature of Tw = 398 K. Stability computations were performed on solutions from several grids of varying mesh density, and the present mesh was determined to be grid-converged by comparison of the linear parabolised stability equation (LPSE) N-factors, which are sensitive to very small changes in primary flow quantities (Perez et al., 2012).


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Normalised uncalibrated CTA voltage profiles (high overheat ratio; approx. mass flux) vs. normalised mass flux profiles from calculation 3 Computation Experiment

2.5

Computation Experiment

Z [mm]

2

1.5

1

Xc = 0.41 m [16.0″] 6 -1 Re = 9.7 1/2 × 10 m R = ReS = 1991

0.5

Xc = 0.46 m [18.0″] 6 -1 Re = 10.3 1/2 × 10 m R = ReS = 2174

0

(a)

(b)

3 Computation Experiment

2.5


Z [mm]

2

1.5

1

Xc = 0.48 m [19.0″] 6 -1 Re = 10.3 × 10 m R = Re1/2 = 2240 S

0.5

0

0

0.2 0.4 0.6 0.8 Normalised Voltage / Mass Flux Range

(c)

Without attempting to draw unreasonably significant conclusions from Figures 7(a) to 7(d) given the aforementioned caveats, one can verify that indeed the boundary-layer thicknesses and general shape of the laminar boundary layer are at least very similar. The differences in boundary-layer thickness, δ99 (see also Figure 8), calculated at 0.99 times the edge mass flux, may be due to the spatial and temporal variation in the cone wall temperature (modelled computationally as a constant 398 K). The slightly windward misalignment of the experimental measurement ray relative to the axisymmetric computation is likely not a dominant factor in the differences in δ99, as this would manifest in the opposite direction, with the experimental δ99 less than the computed value.

Xc = 0.50 m [19.5″] 6 -1 Re = 10.3× 10 m R = Re1/2 = 2264 S 1

0

0.2 0.4 0.6 0.8 Normalised Voltage / Mass Flux Range

1

(d)

3.3 Boundary layer spectra and early observations of the second-mode instability Figure 9 presents spectrograms of the unsteady hot-wire signal for each of the four axial cone stations sampled. Power spectra were computed from the anemometer voltage output using Welch’s method (4,096 points per Hamming window, 3,500 point overlap) at each height in the boundary layer. Spectra were then smoothed for additional clarity using frequency averaging over bins 1 kHz wide. The smoothed spectra were then plotted as contours to show the spatial variation of the measured frequency content across the boundary layer. Greyscale levels in the spectrogram are mapped to the log of the amplitude of the power spectra, and the four images use the same scale.

118 Figure 8

J. Hofferth et al. Boundary-layer thickness distribution, with comparison to computation 3 Computation (0° AoA) Experiment

2.5

δ99, ρU [mm]

2

1.5

1

0.5

Re ≈10 × 106 m-1 0

0

0.1

0.2

0.3

0.4

Xc [m]

Figure 9

Spectra across the boundary layer at each axial station

(b)

(a)

(a)

(c) Note: Colourmap ranges are consistent between axial stations.

(d)

0.5

Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel The most notable feature in the spectrograms is the prominent development of energy content between 250 and 310 kHz, first appearing at Xc = 0.46 m (18 in), larger in intensity at Xc = 0.48 m (19 in), and present but without additional growth at Xc = 0.50 m (19.5 in). This represents the first measurement of energy content within the expected regime for second-mode instabilities in the M6QT at its Texas A&M installation. Earlier studies (Hofferth and Saric, 2012) had used 5-μm wires with insufficient tuning to resolve fluctuating content above ~150 kHz. Though the tuning for the present study is much improved and is sufficient to observe the presence of content at these frequencies, suspicion remains regarding their amplitudes, as the nature of a –3 dB (50%!) rolloff near 333 kHz implies some attenuation of frequencies lower than this. Further characterisation of the frequency response in the rolloff region is required in order to obtain confidence in amplitudes of second-mode measurements. Further optimisation of bridge tuning may also be required. Also evident in the spectrograms is the presence of at least two additional sharp peaks throughout the dataset near 190 and 210–225 kHz. The peak at 190 kHz remains for all runs and does not shift as a function of height in the boundary layer. The second peak occurs at slightly different frequencies in each run, and for all axial stations the frequency slightly increases (to 225–240 kHz) as distance from the wall is reduced. Both peaks appear excited by the presence of high flow fluctuation levels, with higher amplitudes observed over much of the boundary layer relative to outside it. Neither of these sharp peaks is present in a ‘flow-off’ dataset, taken with the wire operating at the same temperature as during a run with all tunnel electronics

prone to interference (e.g., traversing mechanism, tunnel heaters) active, but in a quiescent test section. However, it is still expected that these peaks may be related to probe vibration and/or strain-gaging effects rather than actual flow phenomena. These probe effects may indeed manifest in the signal only under the presence of flow and have their frequencies modified by either the changing velocities or effective overheat ratio in the boundary layer. This will be investigated in detail in future work. As mentioned in §2.3, the first step in determining the cause of these sharp peaks will be to follow established guidelines (Kovasznay, 1953; Smits et al., 1983; Smits and Dussauge, 1989) by providing slack in the wire and/or dampening on the mounting joints to minimise the detrimental effects of strain-gaging and vibration. Spectra at the locations of maximum RMS voltage at each axial location are extracted from Figure 9 and plotted together in Figure 10. This view offers more quantitative insight into the development of frequencies and amplitudes along the cone axis. Significant growth of the second-mode energy content is observed between Xc = 0.41 m (16 in) to 0.48 m (19 in), but amplitudes between Xc = 0.48 m (19 in) and 0.50 m (19.5 in) remain largely unchanged. Additionally, one can observe the centre of the energy band indeed shifting slightly toward higher frequencies at downstream stations [e.g., between Xc = 0.41 m (16 in) to 0.48 m (19 in)], as expected due to the slight thinning of the boundary layer in accordance with the pressure gradient imposed by the flare. A comparison between these spectra and computed N-factors from LPSE computations is presented next in §3.4.

Spectra from the maximum-RMS heights at the four axial stations surveyed Possible wire strain-gaging effects

CTA Power Spectra [V2/Hz]

Figure 10

119

Exp, Xc = 0.41 m Exp, Xc = 0.46 m Exp, Xc = 0.48 m Exp, Xc = 0.50 m

Second-mode energy: 250−310 kHz

10-11

10-12

Re ≈ 10 × 106 m-1 100

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3.4 Stability computations and comparison with experiment Stability computations were conducted for comparison with experiment using an internally developed NPSE code, JoKHeR, described in detail by Kuehl et al. (2012) and Reed et al. (2012). Currently, JoKHeR is capable of running quasi-3D NPSE. That is, it calculates first- and second-mode and crossflow stability in a 3D boundary layer but is restricted to assume that the flow is uniform perpendicular to the chosen arbitrary marching path. The code is written in a general orthogonal curvilinear coordinate system and uses a primitive variable formulation. Both LPSE and LST analysis are considered in this work. •

•

LST. LST has been the most widely used approximate method for stability analysis in the aerospace community. In this approach, the total flow is separated into a steady basic state and an unsteady disturbance of the form φ(y)ei(αx+βz–ωt). The basic state is, itself, a solution to the full Navier-Stokes equations and represents the flow that exists in the absence of any environmental disturbances or forcing. The basic state is assumed locally parallel so that the wall-normal velocity is set to zero and the flow quantities are functions of the wall-normal direction only. The disturbances are assumed to be small enough, allowing the non-linear terms to be neglected. With these approximations and homogeneous boundary conditions, the disturbance equations feature coefficients which depend on the wall-normal coordinate only, and thus the separation of variables into normal modes is possible. This results in a local eigenvalue problem. The inclusion of surface and trajectory curvature terms in this formulation is not standard, as they tend to balance the neglected non-parallel boundary layer effect. However, in the case of cones and other highly curved bodies, these curvature terms play a significant role and are therefore retained in the present LST analysis. Non-linear parabolised stability equations (NPSE). NPSE lies between the extremes of LST and DNS and accounts for curvature, non-parallel, and non-linear effects. Validation of the NPSE with experiments has been proven by the community; with appropriate modelling of the operating and disturbance input conditions, the agreement among theory, computations, and experiments has shown to be remarkable. As with LST, the flow is decomposed into a basic state plus a disturbance, and one assumes the basic state to be a solution to the original equations of motion. A scaling analysis based on the slow growth of the boundary layer in a direction parallel to the surface (for example, the local inviscid streamline direction) allows second derivatives in that direction to be neglected, which nearly parabolises the disturbance

equations and permits a marching solution. Unless otherwise stated, we will refer to the local marching direction, the wall normal direction, and the mutually perpendicular direction to the prior two as x, y, and z, respectively. For quasi 3-D disturbances, the variables φ = [u, v, w, ρ, T]T take the form

φ ( x, y, z , t ) = φ ( x , y ) + φ ′( x, y, z, t ),

(2)

and Fourier (normal mode) decomposition of the disturbances leads to

φ′ =

K

K

∑∑ ⎡⎣φ( x , y) A( x)e

i ( k β z − nω t ) ⎤

⎦

−K −K

(3)

where x = x / Re and A( x) = e ∫

i α ( x ) dx

with complex streamwise wave number α ( x ). A normalisation condition is applied to the shape function, φ ( x , y ),

in order to maintain the assumption that the shape function is slowly varying in the marching direction. Thus, exponential growth is accounted for by the amplitude function, A(x), leaving the shape function amplitude order one. Harmonic balancing is used to identify non-linear modal interactions, though in the present work LPSE is applied in which non-linear effects are neglected and only a single monochromatic wave is considered. Using the basic state results selectively shown in Figure 7, local growth rates from LST were computed and are presented below in Figure 11. As expected, the frequencies for significant local growth rates for the second mode instability are highly tuned to the boundary-layer thickness. Although the LST calculation of Figure 11 is both readily obtained and instructive in demonstrating the high degree of sensitivity of the unstable second-mode passband to boundary layer thickness, the LPSE technique was used for subsequent analyses due to its inclusion of curvature and non-parallel effects. First, Figure 12 presents normalised, LPSE-computed mode shapes of mass flux perturbations for the locally most-amplified frequency, with comparison to experimental profiles of fluctuating voltage at each axial station on the cone. In contrast to the broadband RMS profiles presented in Figure 6, the RMS profiles here are generated from data filtered to include only the 230–330 kHz passband where the second mode is observed in order to more-directly compare to the single-frequency LPSE profiles. The agreement in instability mode shapes here is excellent, with locations of maximum RMS only diverging from the computation slightly at the last two axial stations in an amount commensurate with the divergence in observed δ99 seen in Figure 8, likely attributed to the early stages of transition onset.


121

LST-computed local growth rates along the cone axis −6

5

x 10

0.5

X = 483 mm c

0 −6 x 10 5

0.45

Xc = 406 mm

0.35

X = 330 mm

−αi/Re

c

0.3

0 −6 x 10 5

X = 254 mm

0.25

c

0 −6 x 10 5

0.2

Xc = 178 mm

0.15

0 −6 x 10 5

0.1

Xc = 102 mm 0 50

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1

Frequency (kHz) Figure 12

Axial Distance (m)

0.4

0 −6 x 10 5

δ

99

1.5

2

0.05

(mm)

LPSE-computed mode shapes of fluctuating mass flux for locally-most amplified frequencies, compared with experimental fluctuating voltage profiles, filtered to the second-mode passband, 230–330 kHz 3

2.5



Xc = 0.41 m [16.0″] 6 -1 Re/L = 9.7 × 10 m R = Re1/2 = 1991 S

Xc = 0.46 m [18.0″] 6 -1 Re/L = 10.3 × 10 m R = Re1/2 = 2174 S

Z [mm]

2

1.5

1

0.5

0

(a)

(b)

3

2.5



Xc = 0.48 m [19.0″] Re/L = 10.3 × 106 m-1 R = Re1/2 S = 2240

Xc = 0.50 m [19.5″] Re/L = 10.3 × 106 m-1 R = Re1/2 S = 2264

Z [mm]

2

1.5

1

0.5

0

0

0.2 0.4 0.6 0.8 Normalised ρURMS Fluctuation Mode Shapes

(c)

1

0

0.2 0.4 0.6 0.8 Normalised ρURMS Fluctuation Mode Shapes

(d)

1

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To determine integrated growth for comparison to experiment, N-factors were computed from amplitudes of mass flux disturbances calculated using LPSE. These are shown in Figure 13 for frequencies between 180 kHz and 312 kHz. At Xc = 0.44 m (17 in), where transition onset was observed in the surface temperature data of §2.1, the N-factor is approximately 13.5. Figure 14 provides a comparison of the experimental spectra of Figure 10 with the LPSE N-factor results of Figure 13 at each of the surveyed axial stations on the cone. Because the hot-wire data are currently uncalibrated, proper scaling of the experimental spectra relative to the LPSE N-factor results is not possible, and meaningful comparisons of relative amplitude growth cannot be made. It remains instructive, however, to notionally plot the two datasets together to identify frequencies for peak growth and how they evolve downstream. Most evident in Figure 14 is the key discrepancy between experiment and computations: the most unstable frequency increases from 226 kHz at Xc = 0.4 m to 236 kHz at Xc = 0.5 m as computed by the LPSE method, whereas the experimental data show the centre of energy near 280 kHz. This seems to contradict the comparison between computed and experimental boundary layer thickness presented in Figure 8, which would have suggested slightly lower experimental second-mode frequencies due to the relatively higher observed δ99. However, it is important to again highlight the many complications with the current notional comparison of computed and experimental δ99, from lack of calibration to cone wall temperature variation. Considering these complications in comparing δ99, the agreement in the unstable frequency band is the more appropriate indicator of agreement between computation and experiment. Two key remaining issues in the computation may affect this agreement: Figure 13

1

the slight windward misalignment of the model not accounted for in the 2-D simulation

2

the non-uniform experimental cone wall temperature, currently modelled in the simulation as a constant.

It remains possible that implementing the slightly windward misalignment of the cone in the computation will thin the computed boundary layer and increase the computed frequencies for second-mode growth to more closely match experiment. Similarly, implementing the non-uniform average cone wall temperature will better model the evolution of the boundary-layer thickness, affecting local growth rates and frequencies for peak integrated growth. All of these effects are being quantified in ongoing computational studies.

4

Conclusions and future work

The work presented above represents the latest developments in experimental and computational capabilities at Texas A&M for hypersonic stability and transition on the Langley 93-10 flared cone. Second-mode instability waves are observed in uncalibrated hot-wire measurements at several axial locations, over which significant growth is observed. This successful measurement of the second-mode instability and comparison with new in-house LST and LPSE simulations represents a significant advancement over the earlier hot-wire data recently presented without computational comparison by Hofferth and Saric (2012).

LPSE N-factors (a) shown for various frequencies with streamwise development along the cone axis, and (b) shown versus frequency for the four axial stations surveyed experimentally

16 187 kHz 200 kHz 212 kHz 225 kHz 237 kHz 250 kHz 262 kHz 275 kHz 287 kHz 300 kHz 312 kHz

N-Factor, LPSE, (ρU)max

14 12 10 8 6 4

Xc = 0.41 m Xc = 0.46 m Xc = 0.48 m Xc = 0.50 m

2 0

0

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(a)

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240 260 f [kHz]

(b)

280

300

320

Boundary-layer instability and transition on a flared cone in a Mach 6 quiet wind tunnel Experimental spectra at maximum RMS locations compared with LPSE N-factor results

Xc = 0.41 m [16.0″] N-factor, LPSE Xc = 0.46 m [18.0″] N-factor, LPSE Xc = 0.48 m [19.0″] N-factor, LPSE Xc = 0.50 m [19.5″] N-factor, LPSE

2

CTA Power Spectra [V /Hz]

Offset in unstable passband frequencies

10

-11

18

16

14

12

10

-12

100

N-Factor, LPSE, (ρU)max

Figure 14

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250

Current efforts are focused on identifying and eliminating the sources of the above disagreement between computed and observed second-mode most-amplified frequencies. Recent high-bandwidth optical measurements using focused schlieren deflectometry have demonstrated substantially stronger agreement with the LPSE-predicted peak frequencies (only a ≈10–15 kHz discrepancy) and illustrated a strong sensitivity to very small misalignments of the cone (8 kHz per 0.1º). Additional work is being done computationally and experimentally. On the computational side, the slowly-varying, non-uniform cone model wall temperature will be simulated, and three-dimensional simulations will be performed to account for the slight experimental misalignment. On the experimental side, the focus will be on providing calibrated hot-wire data; multiple overheat ratios will be used to separate mass flux and total temperature contributions, signals will be compensated for freestream total temperature drift during a run, hot-wire frequency response will be further optimised, and the probe design will be streamlined to minimise adverse influence. At each stage, experimental results will continue to be compared with evolving computations of both the mean flow and instability development on the cone. Once diagnostic capabilities have adequately matured and agreement with computation is achieved, the techniques will be applied to a variety of new projects: azimuthal measurements of the structure of non-linear interactions near transition onset on the Langley 93-10 flared cone, measurements of transient growth on a new straight cone with a laser-sintered blunt tip with prescribed distributed roughness, and study of crossflow on a 7º sharp-tipped straight cone at 6º angle of attack.

300 f [kHz]

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Acknowledgements This work was conducted as part of the AFOSR/NASA National Centre for Hypersonic Research in LaminarTurbulent Transition through Grant FA9550-09-1-0341 whose support is gratefully acknowledged. This grant also supported much of the reestablishment and characterisation of the Texas A&M M6QT. The authors are grateful to Pointwise for the gridding software and to AeroSoft for their support with GASP. The authors thank the Texas Advanced Computing Centre (TACC), a high-performance computing resource centre at the University of Texas at Austin. Finally, the authors would like to thank Alex Craig, Raymond Humble, Michael Semper, and Nicole Sharp for their support throughout this work.

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Hofferth, J.W. and Saric, W.S. (2012) ‘Boundary-layer transition on a flared cone in the Texas A&M Mach 6 quiet tunnel’, AIAA Paper 2012-0923. Hofferth, J.W., Bowersox, R.D.W. and Saric, W.S. (2010) ‘The Mach 6 quiet tunnel at Texas A&M: quiet flow performance’, AIAA Paper 2010-4794. Kendall, J.M. (1957) ‘An experimental investigation of leading edge shock-wave – boundary-layer interaction at Mach 5.8’, J. Aero. Sci., Vol. 24, No. 1, pp.47–56. Kovasznay, L.S.G. (1953) ‘Turbulence in supersonic flow’, J. Aero. Sci., Vol. 20, No. 10, pp.657–674. Kuehl, J.J., Perez, E. and Reed, H.L. (2012) ‘JoKHeR: NPSE Simulations of hypersonic crossflow instability’, AIAA Paper 2012-0921. Lachowicz, J.T. (1995) Hypersonic Boundary Layer Stability Experiments in a Quiet Wind Tunnel with Bluntness Effects, PhD dissertation, Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, NC, November. Lachowicz, J.T., Chokani, N.D. and Wilkinson, S.P. (1996) ‘Boundary Layer stability measurements in a hypersonic quiet tunnel’, AIAA J., Vol. 34, No. 12, pp.2496–2500. Lyttle, I.J., Reed, H.L., Shiplyuk, A.N., Maslov, A.A., Buntin, D.A. and Schneider, S.P. (2005) ‘Numerical-experimental comparisons of second-mode behavior for blunted cones’, AIAA J., Vol. 43, No. 8, pp.1734–1743.

Perez, E., Kocian, T., Kuehl, J.J. and Reed, H.L. (2012) ‘Stability of hypersonic compression cones’, AIAA Paper 2012-2962. Reed, H.L., Kuehl, J.J., Perez, E., Kocian, T., Hofferth, J.W. and Saric, W.S. (2012) ‘Nonlinear parabolized stability equation simulations in hypersonic flows’, Presented at RTO AVT-200 Symposium, April, Paper 7. Saric, W.S. (2012) ‘AFOSR/NASA national science center for hypersonic laminar-turbulent transition’, Presented at RTO AVT-200 Symposium, April, Paper 5. Schneider, S. (2007) ‘The development of hypersonic quiet tunnels’, AIAA Paper 2007-4486. Smits, A.J. and Dussauge, J-P. (1989) ‘Hot-wire anemometry in supersonic flow’, AGARD No. 315, Chapter 5. Smits, A.J., Hawakawa, K. and Muck, C.K. (1983) ‘Constant-temperature hot-wire anemometer practice in supersonic flows – part 1: the normal wire’, Exps. Fluids, Vol. 1, No. 2, pp.83–92. Wilkinson, S.P. (1997) ‘A review of hypersonic boundary layer stability experiments in a quiet mach 6 wind tunnel’, AIAA Paper 1997-1819.