Nose-Tip Bluntness Effects on Transition at Hypersonic Speedsjjewell/documents/Jewell-2018-JSR-1...Nose-Tip Bluntness Effects on Transition at Hypersonic Speeds Pedro Paredes∗ National

Nose-Tip Bluntness Effects on Transition at Hypersonic Speeds

Pedro Paredes∗

National Institute of Aerospace, Hampton, Virginia 23666

Meelan M. Choudhari† and Fei Li‡

NASA Langley Research Center, Hampton, Virginia 23681

Joseph S. Jewell§ and Roger L. Kimmel¶

U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433

Eric C. Marineau**

Arnold Engineering Development Complex White Oak, Silver Spring, Maryland 20903

and

Guillaume Grossir††

von Kármán Institute for Fluid Dynamics, B-1640 Rhode-St-Genèse, Belgium

DOI: 10.2514/1.A34277

The existingdatabase of transitionmeasurements in hypersonic ground facilities has established that as thenose-tip

bluntness is increased the onset of boundary-layer transition over a circular cone at zero angle of attack shifts

downstream. However, this trend is reversed at sufficiently large values of the nose-tip Reynolds number, so the

transition onset location eventually moves upstream with a further increase in nose-tip bluntness. This transition-

reversal phenomenon was the focus of a collaborative investigation under the NATO Science and Technology

Organization (STO) Applied Vehicle Technology (AVT)-240 group. The current Paper provides an overview of that

effort, which included wind tunnel measurements and theoretical analysis related to modal and nonmodal

amplification of boundary-layer disturbances. Because modal amplification is too weak to initiate transition at large

bluntness values, transient growth has been investigated as the potential basis for a physics-based model for the

transition-reversal phenomenon. Results indicate that stationary disturbances that are initiated within the nose-tip

vicinity can undergo relatively significant nonmodal amplification that increases with the nose-tip bluntness. This

finding does not provide a definitive link between transient growth and the onset of transition but is qualitatively

consistent with the experimental observations that frustum transition during the reversal regimewas highly sensitive

to wall roughness and, furthermore, was dominated by disturbances originating near the nose tip. The present

analysis shows significant nonmodal growth of traveling disturbances that peak within the entropy layer and could

also play a role in the transition-reversal phenomenon if those disturbances are realizable in a natural disturbance

environment.

Nomenclature

E = total energy normF = disturbance frequencyG = energy gainht = total enthalpyhζ = spanwise metric factorhξ = streamwise metric factorJ = objective functionK = kinetic energy normk = peak-to-valley roughness heightL = reference length

LRN= length scale based on nose radius, m

M = Mach numberM = energy weight matrixm = azimuthal wave numberN = logarithmic amplification factor�q = vector of base flow variables~q = vector of perturbation variablesq = vector of amplitude variablesRekk = roughness-height Reynolds numberRN = nose radius, mReRN

= Reynolds number based on nose radius

ReξT = transition Reynolds number based on freestreamvelocity and transition location

Re∞ = freestream unit Reynolds numberrb = local radius of axisymmetric body at axial station of

interest, mT = temperature, KTw = wall temperature, KTw;ad = adiabatic wall temperature, K

(u, v, w) = streamwise, wall-normal, and spanwise velocitycomponents, m∕s

(x, y, z) = Cartesian coordinates, mα = streamwise wave number, m−1

ΔS = entropy increment, kg ⋅m2 ⋅ s−2 ⋅ K−1

Δξ = streamwise interval considered for optimal growthanalysis, m

δh = boundary-layer thickness, mδS = entropy-layer edge, mθ = cone half-angle, degκ = streamwise curvature, m−1

(ξ, η, ζ) = streamwise,wall-normal, and spanwisecoordinates,mρ = density, kg ⋅m−3

Presented as Paper 2018-0057 at the 2018 AIAA Aerospace SciencesMeeting, Kissimmee, FL, 8–12 January 2018; received 19 April 2018;revision received 23 August 2018; accepted for publication 25 August 2018;published online 26 November 2018. This material is declared a work of theU.S. Government and is not subject to copyright protection in the UnitedStates. All requests for copying and permission to reprint should be submittedto CCC at www.copyright.com; employ the ISSN 0022-4650 (print) or1533-6794 (online) to initiate your request. See also AIAA Rights andPermissionswww.aiaa.org/randp.

*Research Engineer, Computational AeroSciences Branch; also NASALangley Research Center. Senior Member AIAA.

†Research Scientist, Computational AeroSciences Branch. AssociateFellow AIAA.

‡Research Scientist, Computational AeroSciences Branch.§Research Scientist, (Spectral Energies, LLC), AFRL/RQHF. Senior

Member AIAA.¶Principal Aerospace Engineer, AFRL/RQHF. Associate Fellow AIAA.**Currently Program Officer, Hypersonics, Office of Naval Research.

Senior Member AIAA.††Postdoctoral Fellow, Aeronautics and Aerospace Department.

Article in Advance / 1

JOURNAL OF SPACECRAFT AND ROCKETS

http://dx.doi.org/10.2514/1.A34277

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ϕ = angular coordinate, radω = disturbance angular frequency, s−1

Subscripts

j = juncture locationT = transition location0 = initial position1 = final position∞ = freestream value

Superscripts

H = conjugate transpose� = dimensional value

I. Introduction

L AMINAR–TURBULENT transition of boundary-layer flowscan have a strong impact on the performance of hypersonic

vehicles because of its influence on the surface skin friction andaerodynamic heating. Therefore, the prediction and control oftransition onset and the associated variation in aerothermodynamicparameters in high-speed flows is a key issue for optimizing theperformance of the next-generation aerospace vehicles. Althoughmanypractical aerospacevehicles are blunt, themechanisms that leadto boundary-layer instability and transition on blunt geometries arenot well understood as of yet. A detailed review of boundary-layertransition over sharp and blunt cones in a hypersonic freestream isgiven by Schneider [1]. As described therein, both experimental andnumerical studies have shown that the modal growth of Mack-modeinstabilities (or, equivalently, the so-called second-mode waves) isresponsible for laminar–turbulent transition on sharp, axisymmetriccones at zero angle of attack. Studies have also shown that increasednose-tip bluntness has a stabilizing effect on the amplification ofMack-mode instabilities, which is consistent with the observationthat the onset of transition is displaced downstream as the nosebluntness is increased. However, while the boundary-layer flowcontinues to become more stable with increasing nose bluntness,experiments indicate that the downstream movement in transitionactually slows down and eventually reverses as the nose bluntnessexceeds a certain critical range of values. The observed reversal intransition onset at large values of nose bluntness is contrary to thepredictions of linear stability theory and therefore must be explainedusing a different paradigm. While no satisfactory explanation hasbeen proposed as of yet, one of the physical effects that have beensuspected to cause this transition reversal is the role of surfaceroughness.Earlier measurements related to the effect of nose bluntness on

frustum transition over hypersonic blunt cones have been thoroughlydocumented by Stetson [2]. He concluded that the details of the nose-tip flow played an important role in the transition-reversal process,even though the onset of transition occurred significantly fartherdownstream over the frustum of the cone. Stetson also observed thatthe measured transition locations within this regime were not easilyreproducible across different runs. At a fixed set of freestreamconditions, transition onset was found to vary across a wide range offrustum stations, and at times, the boundary-layer flow remainedlaminar over the entire cone. Nonaxisymmetric transition patternswere observed even at zero angle of attack, and the measured lengthof the transition zonewasmuch larger than that for coneswith smallervalues of nose bluntness. Finally, Stetson observed that frustumtransition in the transition-reversal regime was highly sensitive tosurface roughness in the nose-tip region. For smaller nose-tipbluntness before transition reversal, the surface finish on the nose tip(or the frustum) appeared to have no effect on frustum transition.Polishing the blunt nose tip before the wind tunnel run for the largebluntness cones resulted in either higher frustum transition Reynoldsnumbers or a completely laminar flow over the model. Primarily onthe basis of this last observation, Stetson speculated that frustumtransition for large bluntness cones was dominated by disturbances

originating near the nose tip. Therefore, roughness-induced transientgrowth appears to be a reasonable explanation for the transition-reversal phenomena in large bluntness cones at high speeds.Historically, the term bypass transition has been used to identify

transition paths that cannot be explained via modal amplification ofsmall-amplitude disturbances [3]. Examples of bypass transitioninclude the transition due to high levels of freestream disturbances,as, for example, in turbomachinery, or the subcritical transitionobserved in Poiseuille pipe flow experiments [4–6], transition due todistributed surface roughness on flat plates [7,8] or cones [9], andsubcritical transition observed on spherical forebodies [10–13].Because of the strongly favorable pressure gradient over blunt bodiessuch as hemispherical nose tips and spherical segment capsules, thelaminar boundary layer is highly stable, and hence the observed onsetof transition on such bodies has been known as the blunt-bodyparadox. In recent years, the phenomenon of transient, nonmodalamplification of disturbance energy has emerged as a possibleexplanation for many cases of bypass transition. Mathematically, thetransient, nonmodal growth is associated with the nonorthogonalityof the eigenvectors corresponding to the linear disturbance equations.Physically, the main growth mechanism corresponds to the lift-upeffect [14–16], which results from the conservation of horizontalmomentum in the course of spanwise-varying wall-normaldisplacement of the fluid particles. The actual growth in any givenscenario is determined by the details of the external disturbanceenvironment. However, an upper bound on the magnitude of energygain via transient growth can be predicted by using the so-calledoptimal growth theory, which is typically formulated tomaximize thedisturbance growth across a specified interval of streamwiselocations. Regardless of the flow Mach number [17,18], thedisturbances experiencing the highest magnitude of transient growthhave been found to be stationary streaks that arise from initialperturbations in the form of streamwise vortices. Schmid andHenningson [19,20] provide a thorough review of the transientgrowth theory and the earlier results from the literature.Reshotko and Tumin [21] were able to successfully correlate the

transition data from several wind tunnel experiments involvingsubcritical, nose-tip transition by using a semiempirical correlationderived from the optimal transient growth theory, which provides anupper bound on the magnitude of transient growth under suitableconstraints. They extended the ideas from Anderson et al. [22], whohad investigated subcritical transition in a flat plate boundary layerdue tomoderate to high levels of freestream turbulence (FST) and hadderived a correlation based on the optimal growth of boundary-layerdisturbances generated by the FST. Based on the hypothesis that asimilar disturbance growth could also be initiated by distributedsurface roughness over hypersonic blunt forebodies, Reshotko andTumin were able to develop a semiempirical transition correlation bylinking a critical disturbance amplitude required for the onset oftransition with the roughness height parameter via a gain functionbased on the optimal growth framework. Even though the relevanceof the linear, optimal disturbance growth concept to realistic, roughnose tips remains to be established, their work provides the firstphysics-based model toward a potential resolution of the blunt-bodyparadox. Unlike other established models based on empirical curvefits that are valid for a specific subclass of datasets, Reshotko andTumin’s optimal-growth-based transition criterion has been able toprovide a reasonable correlation with the measurements in variouswind tunnel and ballistic range facilities and for a broad range ofsurface temperature ratios. In follow-up work, Paredes et al. [23,24]revised the Reshotko and Tumin correlation by including the effectsof nonparallel basic state evolution, curvature terms, and thevariationof both inflow and outflow locations for the transient growth interval.Their results indicate that application of a more thoroughtheoretical framework reveals certain new features of optimalgrowth characteristics that were not indicated by the parallelframework used in the previous correlation. However, despitethese changes, the constants in the transition correlation remainclose to their original values.Notwithstanding the questions related to the physical relevance of

optimal growth theory, the successful correlation of much of the

2 Article in Advance / PAREDES ETAL.

available data by the Reshotko and Tumin correlation raises thepossibility that a similar framework could also correlate (or, perhaps,explain) the observations of transition reversal over blunt cones.That possibility was investigated during a collaborative effort underthe NATO STO group AVT-240 on hypersonic boundary-layertransition prediction focused on the problem of blunt cone transitionand the potential role of transient growth in the transition-reversalphenomenon. While the problems of blunt-body paradox andtransition reversal over blunt cones share a key similarity by way oftransition onset in the absence of (significant) modal instability, theyalso exhibit two major differences. First, the onset of transition in thelatter case is typically observed over the frustum of the cone, asopposed to the nose tip in the case of the blunt-body paradox. Assuch, the transient growth characteristics of blunt cone boundarylayers are also expected to be different and, in fact, more complexthan those over a blunt nose without any frustum. Besidesinvestigating the transient growth features over blunt cones, theNATO group’s effort also included wind tunnel measurements inboth the United States and Europe and a preliminary study related tothe effects of an azimuthally periodic array of roughness elementslocated near the sonic location over the nose tip. The current Paperprovides an overview of the collaborative work, including boththeoretical analysis and experimental measurements. The layout ofthe Paper is as follows. First, experimental transition measurementsover blunt cones at hypersonic freestream speeds from the literatureare summarized and compared in Sec. II. In Sec. III, the transientgrowth analysis to hypersonic blunt cones for selected flowconditions that match the experimental studies relevant to the NATOeffort are applied [2,25]. Those results were used to determine theazimuthal spacing between roughness elements for the experimentalmeasurements of roughness effects within the transition-reversalregime. Section IV outlines the preliminary findings from thatexperiment, while the summary and conclusions are presentedin Sec. V.

II. Overview of Transition Measurementsover Blunt Cones

The effect of nose-tip bluntness on boundary-layer transition isoften assessed by plotting the transition Reynolds number as afunction of the nose-tip radius Reynolds number in which bothReynolds numbers are based on the freestream conditions. Thetransition onset location ξT is determined by the change in slope ofthe axial distribution of the heat-transfer coefficient or the Stantonnumber. Jewell et al. [26] illustrate this procedure. The experimentaldata from Stetson [2] are plotted with the correlations from laminarand turbulent evolution of the Stanton number as a function of thelocal Reynolds number. The transition location was determined atthe location of the first experimental measurement that deviated fromthe laminar trend. The specific procedures to measure the transitiononset locations and the level of uncertainty are documented in thereferences corresponding to each dataset. Figure 1a presents theReynolds number at the start of transition ReξT as a function of

the nose-tip Reynolds numberReRNfor the experiments by Stetson in

the U.S. Air Force Research Laboratory (AFRL) Mach 6 high-Reynolds-number facility withRN � sharp (A1), 0.5 (A2), 1.0 (A3),1.5 (A4), 2.0 (A5), 2.5 (A6), 5.1 (A7), 7.6 (A8), 10.2 (A9), 12.7(A10), and 15.2 mm (A11) cones and by Aleksandrova et al. [27] inthe Central Aerohydrodynamic Institute UT-1M Ludwieg tube withRN � 0.5 (B1), 1 (B2), 2 (B3), 3 (B4), 4 (B5), 5 (B6), 6 (B7), 7 (B8),8 (B9), 10 (B10), 12 (B11), and 14mm (B12) cones. Both datasets areat a nominal freestream Mach number of 6 on straight 8 deg half-angle cones. The data from Stetson display two distinct regions,referred to as small bluntness, where the transition location movesdownstream with increased bluntness, and large bluntness, wherethe transition location rapidly moves upstream. The data fromAleksandrova et al. (indicated by star symbols) have a positive slope(small bluntness behavior) up to a criticalReRN

of 1.3 × 105. Beyondthis critical value, the transition appears to depend on uncontrolleddisturbances due to nose-tip roughness. In the critical region, groupsof points clustered by nose-tip radii of 3, 4, 5, 12, and 14mm exhibit adecrease in the transition Reynolds number with an increasing nose-tip Reynolds number, which is indicative of transition reversal. Withthe exception of the sharp and the 0.5 mm nose tips that consist ofsteel inserts, the cone model and nose-tip inserts were made of AG-4composite material. The roughness height of the AG-4 material wasnot specified byAleksandrova et al. but is expected to be rougher thanpolished steel. The shape of the transition front, whichwas quantifiedusing temperature-sensitive paint, revealed turbulent wedges atReynolds numbers above the critical value. The authors attribute theformation of such wedges to the presence of uncontrolled nose-tiproughness. The experiments from Aleksandrova et al. illustrate thatsurface roughness has a significant effect on the emergence of thetransition reversal. In Stetson’s Mach 6 experiments, the nominalmodel surface finish had a rms value of 15 μin: (0.38 μm), and bluntnose tips were polished before each run. The polished nose tips mostlikely explain why the small bluntness behavior was extended tonose-tip Reynolds numbers slightly above 9.0 × 105 in Stetson’sexperiments. The sensitivity of frustum transition to roughness at highnose-tip Reynolds numbers was investigated by Stetson by adding45–50 μin: (1.14–1.27 μm) rms roughness on the 0.6 in. (15.2 mm)nose tip. The added roughness caused early frustum transition.Figure 1b presents the Reynolds number at the start of transition as

a function of the nose-tip Reynolds number for three sets ofexperiments at nominal freestreamMach numbers between 9 and 10on straight slender cones. The plot includes the experiments ofStetson [2] in Tunnel F at Mach 9 on 7 deg half-angle cones withRN � sharp (C1), 1.5 (C2), 4.5 (C3), 7.5 (C4), 10.5 (C5), 14.0 (C6),22.5 (C7), and 55.4 (C8); ofMarineau et al. [25] in the U.S. Air ForceAEDCHypervelocityWind Tunnel Number 9 (Tunnel 9) at Mach 10on 7 deg half-angle cones with RN � 0.15 (D1), 5.1 (D2), 9.5 (D3),12.7 (D4), 25.4 (D5), and 50.1 mm (D6); and of Softley et al. [28,29]in the General Electric (G.E.) shock tunnel on 5 deg half-angle coneswith RN � 0.25 (E1), 0.51 (E2), 1.3 (E3), 12.7 (E4), 19.1 (E5), and25.4 mm (E6). As seen at Mach 6, the Mach 10 data also display thesmall bluntness and large bluntness regions. The boundary between

a) b)Fig. 1 TransitionReynoldsnumber basedon freestreamas a function of the noseReynoldsnumber at a)Mach 6 andb)Mach9 to 10,which illustrates theeffect of bluntness and the transition reversal.

Article in Advance / PAREDES ETAL. 3

the large bluntness and small bluntness occurs at varying Reynoldsnumbers of approximately 1.2 × 105, 3.9 × 105, and 9.0 × 105,respectively, for Softley et al., Marineau et al., and Stetson. Thereason for the variation in the critical Reynolds numbers among thethree datasets is not clear, since the respective experiments reportsimilar surface finishes of 30, 32, and 40 μin: (0.76, 0.81, and1.02 μm). Just like Stetson, Softley et al. and Marineau et al. alsopolished the nose tip before each run, but from the results, it appearsthat the polished surface finishmight have been smoother in Stetson’sexperiment.As noted by Stetson [30], the use of small bluntness and large

bluntness when discussing nose-tip bluntness effects can bemisleading, as the bluntness effect is related not only to the physicaldimensions of the nose tip but also to where transition occurs withrespect to the nose tip. As discussed by Muir and Trujillo [31], thenose-tip Reynolds number does not properly account for the variouseffects of the nose-tip radius and unit Reynolds numbers. Stetson andRushton [32] introduced the entropy-swallowing length as a parameterto relate the transition location to nose-tip bluntness effects. UsingRotta’s [33]correlation, the entropy-swallowing length is found to be afunction of �Re∞�1∕3 and �RN�4∕3. One disadvantage of the entropy-swallowing length as a correlation parameter is that it cannot be easilydefined for arbitrary geometries. Moreover, the blunt cone data areusually correlated by normalizing the transition locations (location orReynolds numbers) on blunt cones by the transition length on sharpcones in order to remove unit Reynolds number effects. Such anapproach cannot be extended to arbitrary geometries.Boundary-layer stability calculations have tried to explain the

transition behavior in hypersonic blunt cones. The effect of bluntnesson the second-mode instability was first investigated by Malik et al.[34] andHerbert and Esfahanian [35] andmore recently byMarineauet al. [25,36] and Jewell and Kimmel [37]. These recent studiesinclude parabolized stability equation (PSE) analysis of the historicalStetsonMach 6 andMach 9 blunt cone experiments using the STABLsoftware suite [38]. The stability analyses agree that the transitionreversal cannot be predicted by only considering Mack’s second-mode instability mechanism. This is because increased bluntnessstabilizes the second mode by moving the neutral point downstreamdue to local edge Mach number and local unit Reynolds numberreductions within the entropy layer. This implies that the transitionlocation based on Mack’s second-mode amplification keeps movingdownstream and eventually transition does not occur as the bluntnessincreases. In addition, the boundary-layer stability studies found thatthe first mode is also not destabilized by bluntness, so it cannot beresponsible for the transition reversal.Even if transition reversal cannot be predicted with linear stability

computations of the first and second modes, an approach combiningthe measurements and computations data will still be useful toevaluate where and when the transition process is no longerdominated by the second mode. Figure 2 presents a compilation ofsecond-mode transition N factors NT as a function of the nose-tipReynolds number ReRN

for the Mach 6 experiments of Stetson [2]from [37] (diamonds), Mach 9 experiments of Stetson [2] from [36](circles), andMach 10 experiments of Marineau et al. [25] (squares).For values ofReRN

below 1 × 104,NT increases withReRNas a result

of the increase in the unit Reynolds number. This behavior can belinked to an increase in the critical second-mode frequency fT,which implies a lower tunnel noise content near fT. This effect, firstdiscussed by Marineau et al. [25,36] has recently been modeled byBalakumar and Chou [39] with direct numerical simulations of theTunnel 9Mach 10 experiments. The approach includes the measuredfreestream noise spectrum and an empirical correlation (see [25]) todetermine the breakdown amplitude of the second mode. The sharpcone nose-tip radius was not specified in either the StetsonMach 6 orMach 9 experiments. To include these sharp cone data points inFig. 2, the sharp nose tipswere assumed to have the same radius as thesharp Tunnel 9 cone. A variation in the sharp cone nose-tip radiusdoes not change the trends, as it simply shifts the point left or right.For ReRN

between approximately 4 × 104 and 1 × 105, a steepdecrease inNT withReRN

is observed. Somewhere in this region, thetransition process appears to not be dominated by second-mode

amplification. Note that the departure from second-mode-dominatedtransition occurs before transition reversal. The decrease of HT

with bluntness was discussed by Marineau [36] and attributed to adecrease of the second-mode breakdown amplitudes and to anincrease in the initial amplitudes. The decrease in the breakdownamplitudes is linked to the lower edge Mach number, whereasthe increase in the initial amplitudes is related to the decrease in thecritical second-mode frequency fT as well as an increase in thereceptivity coefficient with ReRN

. For these conditions, the measuredand estimated transition N factors and the estimated receptivitycoefficients are shown in Figs. 3a and 3b, respectively.In contrast to Stetson’s blunt cone experiments, which measured

just the transition location, Marineau et al. [25] also measured theboundary-layer instabilities by using a large number of high-frequency response pressure sensors (PCB®-132). These measure-ments captured the evolution of the pressure fluctuations over thesurface of the cone. Figure 4 presents a map of the logarithm of thepressure power spectral density [log(PSD)] at Re∞ ≈ 17 × 106 m−1

forRN � 0.15, 5.1, 9.5, 12.7, 25.4, and 50.8 mm. ForRN ≥ 5.1 mm,the transition occurs before the entropy layer is swallowed. This leadsto a significant reduction of the edge Mach number compared to thesharp cone case. For the 5.1 mm nose tip in Fig. 4b, the edge Machnumber varies from 4.4 to 4.8 between the neutral point and the startof transition. The bluntness significantly delays the appearance of thesecond-mode waves and increases the distances over which growthand breakdown occur. As a result of these two factors, the transitionlocation is moved farther downstream on the blunt cone (from 0.25mon the sharp cone to 0.68 m on the 5.1 mm cone, based on heattransfer measurements). In addition, the second-mode frequenciesare significantly lower on the blunt cone due to the increasedboundary-layer thickness. As the nose-tip radius increases from 5.1to 9.5 mm, the transition location moves farther downstream, and theunstable second-mode frequencies are further decreased. Theincrease from 9.5 to 12.7 mm has a minor effect on the transitionlocation, which indicates that reversal is near. For the 12.7 mm nosetip, the start of transition occurs before significant growth of thesecond mode. This indicates that transition was not initiated bythe second-mode instability. However, farther downstreamwithin thetransitional region, the second-mode amplitudes keep increasing upto the downstream end of the cone. The 25.4 and 50.8 mm nose tipsare in the reversal regime as the transition location has movedupstream compared to those at the smaller radii. In addition,transition occurs before the appearance of second-mode waves. Thismakes sense, as the edgeMach number at the start of transition for the25.4 and 50.8mm radii are 3.3 and 3.2 respectively,which are too lowfor second-mode growth. The results for 25.4 and 50.8 mm nose tips

Fig. 2 Computed second-modeN factors at the experimental transitionlocation (start of transition) as a function of the nose-tip Reynoldsnumber. The same symbols and colors as in Fig. 1 are used to indicate thenose-tip radius.


also reveal that the mechanism responsible for the transition reversalhas a weak pressure signature, as no significant variation in thepressure PSD is found.

The use of fast-response heat flux sensors can help to characterizethe transition mechanisms on blunt cones. For instance, time-resolved heat transfer measurements performed at Mach 9 by

a) b)Fig. 3 Effect of bluntness on a) transitionN factors and b) receptivity for Stetson blunt cone experiments at Mach 6. The same symbols and colors as inFig. 1 are used to indicate the nose-tip radius.

a) b)

c) d)

e) f)Fig. 4 Contourmap of the logarithmof the pressure power spectral density log(PSD) for coneswith a)RN � 0.15 mm (ξT � 0.254 m), b)RN � 5.1 mm(ξT � 0.683 m), c) RN � 9.5 mm (ξT � 1.037 m), d) RN � 12.7 mm (ξT � 1.015 m), e) RN � 25.4 mm (ξT � 0.546 m), and f) RN � 50.8 mm(ξT � 0.504 m) at Re∞ ≈ 17 × 106 m−1 at Mach 10 in Tunnel 9. The vertical, black line denotes the measured transition location.


Zanchetta [40] in the Imperial College Gun Tunnel on a 5 deg half-angle cone revealed that in the reversal regime transitional events areformed in the near-nose region and convect downstream. Theformation frequency of the events was linked to the severity of theroughness environment. In certain cases, second-mode instabilitywaves and these transition events were occurring concurrentl, and theexperiments indicated that the second mode was responsible for thecompletion of transition. Recent laser-induced-fluorescence-based(LIF) schlieren measurements from Grossir et al. [41] on a blunt7 deg half-angle cone at Mach 11.9 in the von Kármán InstituteLongshot hypersonic wind-tunnel revealed disturbances with shapesquite different from the usual second-mode rope structures. Thedisturbances that extend above the edge of the boundary layer areseen in Fig. 5 for the 4.75 mm nose-tip radius. These disturbances,which were not present on schlieren images for sharper cones, couldbe a manifestation of the blunt cone transition mechanism leading tothe heat transfer fluctuation measured by Zanchetta.In summary, the experimental and numerical studies agree that

frustum transition in the reversal regime cannot be accounted for vialinear, modal instability analysis. Furthermore, the experimentalobservations agree that frustum transition on large bluntness cones ishighly sensitive to wall roughness and appears to be dominatedby disturbances that originate in the vicinity of the nose tip.Therefore, roughness-induced transient growth emerges as theprimary candidate for the experimentally observed trend in laminar–turbulent transition.

III. Transient Growth Analysis for HypersonicBlunt Cones

This section presents the transient growth analysis of blunt circularconeswith conditions selected tomatch a subset of the configurationsfrom the experiments conducted by Stetson [2] in the AFRLMach 6high-Reynolds-number facility and by Marineau et al. [25] in theArnold Engineering Development Complex (AEDC) Tunnel 9 atMach 10.Modal instability analysis for the AFRL configurations hasbeen already performed by Jewell and Kimmel [37], and Marineauet al. have described similar analysis for the AEDC configurations.They found that both first-mode and Mack-mode waves were eitherdamped or weakly unstable for the present configurations, andtherefore transition reversal cannot be predicted with the modalanalysis. Another modal instability mechanism that might play a rolein the transition reversal is the entropy-layer instability [42,43].However, although not shown here, our analysis did not revealed anysubstantially amplified entropy-layer modes for the configurations ofinterest. Therefore, the transient growth mechanism is investigatedas a potential cause for the transition reversal. First, the basic statesolutions are presented in Sec. III.A. Second, the transient growththeory is briefly introduced in Sec. III.B. Then, a detailed transientgrowth analysis of the selected blunt cones configurations ispresented in Sec. III.C.

A. Laminar Boundary Layer over Blunt Cones

The basic states used in the present analysis correspond to thelaminar boundary-layer flow over the selected blunt coneconfigurations. The laminar boundary-layer flows were computedby Jewell and Kimmel [37] and Marineau et al. [25] with reacting,axisymmetric Navier–Stokes equations on a structured grid. Thesolver was a version of the NASA data parallel-line relaxation(DPLR) code [44], which is included as part of the STABL-2D

software suite, as described by Johnson et al. [45,46]. This flowsolver employs a second-order-accurate finite-volume formulation.The inviscid fluxes are based on the modified Steger–Warming fluxvector splitting method with a monotonic upstream-center schemefor the conservation laws limiter. The time integration method is theimplicit, first-order DPLR method. The effects of chemistry andmolecular vibration are negligible at the present flow conditions.Thus, the high-enthalpy effects are omitted from the calculations. Theviscosity law used is Sutherland’s law, and the heat conductivity iscalculated using Eucken’s relation. Additional details about the basicstate solution and the grid convergence study are given by Jewell andKimmel for the AFRL configurations and by Marineau et al. for theAEDC configurations.

1. AFRL Configurations

The AFRL Mach 6 facility operates at stagnation pressures p0

from 700 to 2100 psi (4.83 to 14.5MPa). Theworking fluid is air andis treated as ideal gas because of the relatively low temperature andpressure. The blunt cones used in the experiments have a half-angle of8 deg and a base radius of 2.0 in. (0.0508 m). A total of 196experiments encompassing 108 unique conditions comprised theStetson [2] Mach 6 results. Table 1 shows the details of the fourconfigurations selected for the present analysis. The present analysisuses the 7 deg half-angle variable-bluntness cone that is currentlyused in the experiments in the AFRL Mach 6 facility. The thermalwall condition is isothermal with a constant wall temperature equalto �Tw � 300.0 K.The streamwise evolution of the boundary-layer thickness δh

and edge Mach number Me is plotted in Fig. 6. The boundary-layeredge, ηe � δh, is defined as the wall-normal position whereht∕ht;∞ � 0.995, with ht denoting the total enthalpy, i.e.,

ht � h� 0.5� �u2 � �v2 � �w2�, where h is the static enthalpy. Theevolution of boundary-layer thickness δh within the nose region andalong the entire geometry is shown in Figs. 6a and 6b, respectively.The boundary-layer thickness is nearly constant from the stagnationpoint up to ϕ ≈ 30 deg, which is characteristic of stagnationboundary-layer flow. From ϕ ≈ 45 deg, the boundary-layerthickness grows monotonically up to the end of the cone, as shownin Fig. 6b. Figures 6a and 6b also show that the boundary-layerthickness for the smaller nose radius (RN � 9.53 mm) cone issmaller than for the larger nose radius (RN � 15.24 mm) cone with

the same freestream conditions (Re � 91.4 × 106 m−1). Figures 6cand 6d show the evolution of the edge Mach number in the noseregion and in the entire cone, respectively. The edgeMach number isdetermined by inviscid theory to the leading order, and thereforethe evolution within the nose is nearly coincident for the fourconfigurations (Fig. 6c). However, the edge Mach number evolutionis clearly distinguishable from the smaller to the larger nose radiuscases when it is plotted against the streamwise coordinate for theentire geometry (Fig. 6d). The sonic location, which coincides withthe peak of the streamwisemass fluxwithin the nose [23,24], is foundat ϕ � 41.4 deg. The edge Mach number remains below Me � 3along the entire geometry for the large nose radius cones and onlybecomes slightly larger than Me � 3 at the end of the cone for thesmaller nose radius case.

2. AEDC Configurations

AEDC Tunnel 9 is a hypersonic, nitrogen gas, blowdown windtunnel with interchangeable nozzles that allow for testing at Mach

Fig. 5 LIF-based schlieren flowvisualization on4.75mmradius nose tip7 deg half-angle cone in the von Kármán Institute Longshot hypersonicwind-tunnel at M∞ � 11.9 and Re � 11.6 × 106 m−1. Fields of viewextend from x � 625 mm until the end of the cone at 806 mm.Disturbance extending past the boundary-layer thickness is visible.

Table 1 Details of the four AFRL configurations used inthe present study (wall temperature is �Tw � 300 K, andmeasured transition locations ξT are extracted from [26])

RN , mm Re∞ (×106 m−1) M∞ �T∞ �Tw∕ �Tw;ad ξT , m

5.080 91.4 5.9 76.74 0.57 0.16115.24 91.4 5.9 76.74 0.57 0.05115.24 60.9 5.9 76.74 0.57 0.22715.24 30.5 5.9 76.74 0.57 ——


numbers of 7, 8, 10, and 14 over a 0.177 × 106 m−1 to 158.8 ×106 m−1 unit Reynolds number range. A detailed description of thefacility can be found in [25]. The blunt cones used in the experimentsof Marineau et al. [25] had a base diameter of 15 in. (0.381 m) andinterchangeable nose tips with radii of 0.152–50.80 mm. The testmatrix for the 24 run test program is provided in [25]. The workingfluid is nitrogen. Table 2 shows the details of the four configurationsselected for the present analysis. The used thermal wall condition isisothermal with a constant wall temperature equal to �Tw � 300.0 K.The four configurations share a similar freestream unit Reynoldsnumber of Re∞ ≈ 17.5 × 106 m−1 and a freestreamMach number ofM∞ ≈ 9.78. The nose radius values vary from RN � 9.53 mmto RN � 50.8 mm.The boundary-layer thickness δh and edge Mach number Me for

the four configurations are plotted in Figs. 7a and 7b, respectively. Asobserved in the comparison of the AFRL configurations of Fig. 6b,despite the same freestream conditions, the boundary-layer thicknessin the frustum part of the cone is larger for larger nose radius values.Also, the edge Mach number is larger for the smaller nose radiuscases, although it remains smaller than Me � 4.5 for the fourconfigurations.

B. Transient Growth Theory

In this section, we outline the methodology used for transientgrowth analysis for the blunt cone. For the most part, this analysisfocuses on stationary disturbances, which experience the largestenergy growth in most flows. For the transient growth analysisinvolving the stationary disturbances, we use the linear PSEframework as explained in the literature [18,23,47–49]. Optimalgrowth analysis within the PSE framework also bears strong

similarities with the optimization approach based on the linearizedboundary-layer equations [17,22,50]. The advantage of the PSE-based formulation is that it is also applicable to more complex baseflows in which the flow evolves along the streamwise direction andthe boundary-layer approximation may not hold. We found that thePSE approach can also be extended to the unsteady disturbancesexamined in this Paper.Linear perturbations that are harmonic in time as well as in the

azimuthal direction, are written as,

~q�ξ; η; ζ; t� � q �ξ; η� exp�i�mζ − ωt�� c:c: (1)

where c.c. denotes a complex conjugate. The suitably non-dimensionalized, orthogonal, curvilinear coordinate system (ξ, η, ζ)denotes streamwise, wall-normal, and azimuthal coordinates, and(u, v, w) represent the corresponding velocity components. Densityand temperature are denoted by ρ and T. The streamwise andazimuthal wave numbers are α and m, respectively, and ω is theangular frequency of the perturbation. The Cartesian coordinates arerepresented by (x, y, z). The vector of perturbation fluid variables

is ~q�ξ; η; ζ; t� � � ~ρ; ~u; ~v; ~w; ~T�T , and the vector of disturbance

functions is q �ξ; η� � �ρ ; u ; v ; w ; T

�T . The vector of basic state

variables is �q�ξ; η� � ��ρ; �u; �v; �w; �T�T . The disturbance functions

q �ξ; η� satisfy the harmonic form of linearized Navier–Stokesequations (HLNSE),

LHLNSEq �ξ; η� �

�A� B

∂∂η

� C∂2

∂η2�D

1

hξ

∂∂ξ

�E1

hξ

∂2

∂ξ∂η

� F1

h2ξ

∂2

∂ξ2

�q �ξ; η� � 0 (2)

The linear operatorsA,B,C,D,E, andF are given by Pralits et al.[47] and Paredes [51], and hξ is the metric factor associated with the

streamwise curvature. The PSE derivation is based on isolatingthe rapid phase variations in the streamwise direction by introducingthe ansatz

q �ξ; η� � q�ξ; η� exp

�i

Zξ

ξ0

α�ξ 0� dξ 0�

(3)

Fig. 6 Streamwise evolution of a,b) boundary-layer thickness and c,d) edge Mach number of the laminar boundary-layer flows over the AFRLconfigurations. The legend refers toRN;S � 5.08 mm,RN;S � 15.24 mm,Re∞;L � 91.4 × 106 m−1,Re∞;M � 60.9 × 106 m−1,Re∞;S � 30.5 × 106 m−1.The transition locations from Table 1 are indicated in b with circles.

Table 2 Details of the fourAEDCconfigurations used in thepresent study (wall temperature is �Tw � 300, and measured

transition locations ξT are extracted from [25])

RN , mm Re∞, m−1 M∞ �T∞ �Tw∕ �Tw;ad ξT , m

9.530 17.6 × 106 9.797 51.01 0.340 1.03712.70 17.4 × 106 9.795 51.17 0.339 1.01525.40 17.3 × 106 9.791 51.13 0.340 0.54650.80 17.6 × 106 9.777 51.27 0.340 0.504


where the unknown wave number distribution α�ξ� is determined in

the course of the solution by imposing an additional constraint,

Zηq�

∂q∂ξ

hξhζ dη � 0 (4)

where the amplitude functions q�ξ; η� � �ρ; u; v; w; T�T vary slowlyin the streamwise direction in comparison with the phase term

exp�i∫ ξξ0α�ξ 0� dξ 0�. Substituting Eq. (3) into Eq. (2) and involving the

scale separation to neglect the viscous, streamwise derivative terms,

one obtains the PSE in the form

LPSEq�ξ; η� ��A� B

∂∂η

� C∂2

∂η2�D

1

h1

∂∂ξ

�q�ξ; η� � 0 (5)

Itmay be noted that Eq. (4) constrains the PSEdisturbances to have

a unique, local wave number at any streamwise location and therefore

the PSEs are likely to encounter difficulties when the perturbation

field includesmultiple disturbanceswith disparate variation along the

streamwise direction. Optimal growth perturbations are known to

include a superposition of multiple eigenmodes of the quasi-parallel

disturbances equations, and therefore additional simplificationsmust

accrue in applying the PSE for optimal growth predictions. This

simplification occurs when the streamwise disturbancewavelength is

much larger than the thickness of the mean boundary layer, so the

second-order, viscous derivative terms in ξ from the linearized

Navier–Stokes equations in Eq. (2) become uniformly smaller in

comparison with the viscous derivatives along the wall-normal

direction. Thus, after neglecting the second derivative terms in ξ, oneobtains the reduced form of the harmonic, linearized Navier–Stokes

equations (RHLNSE):

LRHLNSEq �ξ; η� �

�A� B

∂∂η

� C∂2

∂η2�D

1

hξ

∂∂ξ

�E1

hξ

∂2

∂ξ∂η

�q �ξ; η� � 0 (6)

These equations apply to all disturbances with a streamwise

scale that is sufficiently larger than the boundary-layer thickness.

Equations (5) and (6) retain some streamwise ellipticity via the

streamwise pressure gradient term ∂p∕∂ξ in the streamwise

momentum equation [52–55], and hence a marching solution would

not be feasible in all cases. However, our computations revealed that it

was feasible to obtain marching solutions to these equations for the

flow conditions of interest. For purely stationary disturbances (ω � 0and α � 0), this term can be dropped from the equations as justified in

[48,56]. Fornonstationaryperturbations (ω ≠ 0), a commonly adopted

solution is to replace ∂p∕∂ξ by ΩPNS∂p∕∂ξ, where ΩPNS is the

Vigneron parameter [57–59]. This parameter was originally

introduced for the integration of the parabolized Navier–Stokes

(PNS) equations and is determined by

ΩPNS � min�1;M2ξ� (7)

where Mξ is the local streamwise Mach number. The Vigneron

approximation ensures numerical stability of the marching scheme by

suppressing upstream influence within the solution. For locally

supersonic flow, the equations are not altered because ΩPNS � 1.For a disturbance field with α ≠ 0, a portion of the elliptic behavior isabsorbed in the wave part via the term iαp, and the residual upstreaminfluence can be suppressed by choosing a sufficiently large marching

step [54], without having to invoke the Vigneron approximation.

As described in Sec. III.C.3, the streamwise evolution of optimal

disturbances obtained using Eq. (7) was verified a posteriori using the

PSE (5) as well as the full set of linearized disturbance equations (2).The optimal initial disturbance ~q0 is defined as the initial (i.e.,

inflow) condition at ξ0 thatmaximizes the objective function J, whichis defined as ameasure of disturbance growth over a specified interval

[ξ0, ξ1]. The definitions used in the present study correspond to the

outlet energy gain J � Gout andmean energy gain J � Gmean and are

defined as

Gout � E�ξ1�E�ξ0�

(8)

Gmean � 1

ξ1 − ξ0

R ξ1ξ0E�ξ 0� dξ 0E�ξ0�

(9)

whereE denotes the energy norm of ~q. The energy norm is defined as

E�ξ� �Zηq�ξ; η�HMEq�ξ; η�h1h3 dη (10)

where h3 is the metric factor associated with the azimuthal curvature,

ME is the energy weight matrix, and the superscript H denotes the

conjugate transpose. The selection of J � Gout corresponds to the

outlet energy gain that is commonly used in studies of the optimal

perturbation problem [22,50]. The selection of J � Gmean defines the

mean energy gain and corresponds to the optimization of the mean

energy. This latter definition accounts for a possible overshoot in the

disturbance energy evolution that is not accounted for by the former

definition and is found to be present in hypersonic blunt forebodies

[23,24] aswell as in the nose tip of blunt cones as documented inwhat

follows.The choice of the energy norm is known to influence the optimal

initial perturbation as well as the magnitude of energy amplification

[17,49,60].Here,we use the positive-definite energynormderivedby

Mack [61] and used by Hanifi et al. [62] for transient growth

calculations, which is defined by

ME � diag

��T�ξ; η�

γ �ρ�ξ; η�M2; �ρ�ξ; η�; �ρ�ξ; η�; �ρ�ξ; η�; �ρ�ξ; η�

γ�γ − 1� �T�ξ; η�M2

�

(11)

a) b)Fig. 7 Streamwise evolution of a) boundary-layer thickness and b) edge Mach number of the laminar boundary-layer flows over the AEDCconfigurations. The transition locations from Table 2 are indicated in a with circles.


Additionally, the kinetic energy norm is also used for optimizationin this Paper. The kinetic energy of a perturbation is defined by

K�ξ� �Zηq�ξ; η�HMKq�ξ; η�hξhζ dη (12)

where

MK � diagh0; �ρ�ξ; η�; �ρ�ξ; η�; �ρ�ξ; η�; 0

i(13)

To differentiate when the total energy normE or the kinetic energynormK is used, a corresponding subscript is added to the energy gain,resulting in four options for the objective function:Gout

E ,GmeanE ,Gout

K ,andGmean

K . In the present Paper, the transient growth amplification isalso expressed in terms of the logarithmic amplification ratio, theso-calledN factor, based on the total energynorm,which is defined as

NE � 1

2ln �Gout

E � � −Z

ξ

ξ0

αi�ξ 0� dξ 0 � 1∕2 lnhE�ξ�∕E�ξ0�

i(14)

Furthermore, the present study uses the N factor based in otherdisturbance magnitude norms, as the kinetic energy NK, maximumtemperature NT , or maximum streamwise velocity Nu. Thesedefinitions are equivalent to that of Eq. (14), but with theirrespective norms.The variational formulation of the problem to determine the

maximum of the objective functional J leads to an optimality system[18], which is solved in an iterative manner, starting from a randomsolution at ξ0 that must satisfy the boundary conditions. The PSEs,L ~q � 0, are used to integrate ~q up to ξ1, where the final optimalitycondition is used to obtain the initial condition for the backwardadjoint PSE integration,L† ~q† � cmeanF� ~q�, where cmean � 0 for theoutlet energy gain optimization and cmean � 0 for the mean energygain optimization, and F� ~q� is a function of the direct solution [47].At ξ0, the adjoint solution is used to calculate the new initial conditionfor the forward PSE integration with the initial optimality condition.The iterative procedure finishes when the value of G has convergedup to a certain tolerance, which was set to 10−4 in the presentcomputations.Nonuniform stable high-order finite-difference schemes [63,64] of

sixth order are used for discretization of the PSE along the wall-normal coordinate. The discretized PSEs are integrated along thestreamwise coordinate by using second-order backward differ-entiation. The number of discretization points in both directions isvaried in selected cases to ensure convergence of the optimal gainpredictions. The wall-normal direction is discretized usingNy � 161, with the nodes being clustered toward the wall [64].No-slip, isothermal boundary conditions are used at the wall,i.e., u � v � w � T � 0. The amplitude functions are forced todecay at the far-field boundary by imposing the Dirichlet conditionsρ � u � w � T � 0, unless otherwise stated. The far-fieldboundary coordinate is set just below the shock layer. Verificationof the present optimal growth module against available transientgrowth results from the literature is shown in [18].In what follows, we study the axisymmetric boundary layer

over circular cones in hypersonic freestream flows. The freestreamconditions and geometries are selected to match a subset ofconfigurations used in the experiments conducted at theAFRL [2,37]and at AEDC [25]. For this problem, the computational coordinates(ξ, η, ζ) are defined as an orthogonal body-fitted coordinate system.The metric factors are defined as

hξ � 1� κη (15)

hζ � rb � η cos�θ� (16)

where κ denotes the streamwise curvature, rb is the local radius, and θis the local half-angle along the axisymmetric surface, i.e.,sin�θ� � drb∕dξ. For our study, the half-angle, θ, is 7 deg, and κ � 0

in the frustum region. The end of the nose and beginning of thefrustum is denoted as the juncture location that is defined asξj � RNπ∕2. The streamwise coordinatewithin the nose-tip region isrepresented by an angular coordinate defined as ϕ � ξ∕RN . Thenose-tip Reynolds number, ReRN

� �ρ∞ �u∞RN∕μ∞, is used to scalethe energy gain. The length scale LRN

� RN∕��ReRN

pis used to

normalize the spanwise disturbance wavelength defined asλ � 2πrb∕m.

C. Transient Growth Results

For a non-self-similar boundary layer such as the boundarylayer over blunt cones, both the initial and final locations must bevaried in order to obtain the overall picture of the optimal growthcharacteristics [23]. A special feature of the transient growth analysisfor the blunt cones of interest is that the results naturally split into twoparts, one that deals with transient growth intervals that are limited tothe nose region, where the results are expected to resemble those forthe hemisphere forebody reported by Paredes et al. [23], and a secondone that deals with transient growth intervals that extend into thefrustum region, where transition is observed in the experiments.Detailed transient growth results are first presented for the AFRLconfigurations in Secs. III.C.1 and III.C.2. Because stationarydisturbances usually yield the highest transient growth overall, amajority of the analysis in Sec. III is focused on the zero-frequencycase. However, motivated by the findings of Cook et al. [65], whoperformed a resolvent analysis of a similar blunt cone configurationat hypersonic conditions and reported significant nonmodal growthof planar, traveling waves inside the entropy layer, the transientgrowth analysis is extended to traveling disturbances in Sec. III.C.3.Finally, a brief summary of results is presented for the AEDCconfigurations in Sec. III.C.4.

1. Transient Growth Interval Within Nose Region

Herein, transient growth results with initial and final disturbancelocations within the nose region for the AFRL configurationsintroduced in Table 1 are investigated. The optimal mean total energygain Gmean

E and optimal mean kinetic energy gain GmeanK for the

RN � 15.24 mm and Re∞ � 91.4 × 106 m−1 case are plotted inFigs. 8a and 8b, respectively. Figure 8a shows that the highest totalenergy gain occurs for relatively short optimization intervals in thevicinity of the stagnation point, as indicated by the black line nearlyparallel to the lower boundary of the plot.However, the kinetic energybudget for these perturbations initiated near the stagnation point israther small. This fact is confirmed by the optimal mean kineticenergy gain Gmean

K plot of Fig. 8b. The optimal kinetic energy gainexhibits a maximum in the interior of the domain at ϕ0 � 42.4 degthat nearly coincides with the sonic location, ϕMe�1 � 41.4 deg.These results indicate the same features as the results reported byParedes et al. [23] for a hypersonic hemisphere forebody. Asexplained in [23], the selection of the kinetic energy norm alone ismotivated by the fact that a greater content of kinetic energy within aspecified total energy is likely to enhance the growth of secondaryinstabilities because the latter are driven by the strength of thestreamwise velocity shear and are less sensitive to the thermodynamicperturbations associated with the streaks.The optimal growth results for a specified inflow location ξ0 of the

flow are characterized in terms of the combination of azimuthal wavenumber m and outflow location ξ1 that lead to the maximum valueof the energy gain. Thus, the effect of nose-tip radius RN on themaximum value of the optimal energy gain, optimal wave number,and optimal growth interval, is analyzed next. Figure 9 shows theoptimal total and kinetic energy gains as a function of the inflowlocation (Figs. 9a and 9d) as well as the corresponding azimuthalwave number (Figs. 9b and 9e) and the optimal growth interval(Figs. 9c and 9f). Results are shown for the RN � 5.08 mm andRN � 15.24 mm cones at the same freestream unit Reynolds numberof Re∞ � 91.4 × 106 m−1. Figure 9a shows that both the total andkinetic energy gains are larger for the larger nose radius, although, asshown in Fig. 9d, the scaling is not perfectly linear because, asindicated by Paredes et al. [24], small deviations from the linear


scaling occur as a result of the differences in the ratio of boundary-layer thickness to the radius of the surface curvature. Figure 9b showsthat the optimal azimuthal wave number is nearly twice as large forthe larger nose radius cone in comparison with the case of the smallernose radius. The scaling of the initial disturbance wavelength withLRN

plotted in Fig. 9e shows a reasonable scaling with the boundary-layer thickness. On the other side, the optimal growth interval plottedin Fig. 9c does not scale with LRN

(or RN) as shown in Fig. 9f,presumably because the ratio of boundary-layer thickness to theradius of the surface curvature plays an important role for thisparameter.

2. Transient Growth Along Frustum Region

Next, transient growth across spatial intervals that extend into thefrustum region is studied in detail for the AFRL configurations. Inthis case, we find it more convenient to plot the transient growthamplification in terms of theN factor based on the total energy normdefined in Eq. (14). Results based on the kinetic energywere found tobe equivalent to those based on the total energy for the frustumregion. Figures 10a shows the N-factor contours for initial and finallocations on the frustum for the RN � 5.08 mm AFRL cone atRe∞ � 91.4 × 106 m−1. Similar results for the larger nose radius,RN � 15.24 mm, are shown in Fig. 10b. The N-factor values arelarger for the smaller nose radius cone except for initial and finallocations near the juncture of the cone, i.e., beginning of the frustum.The NE values are larger than 5.5 in the studied range of parameters

for the smaller nose radius case (Fig. 10a). For the larger nose radius

case, theNE values are larger than 4.5 in the range of ξ0 and ξ1 valuesstudied here, although the NE � 3.5 value is reached at smaller

values of ξ1 than for the smaller radius case when the initial

disturbance location is near the juncture location ξ0 ≈ ξj.Figures 11a and 11b show a magnified view of the N-factor

contours from Figs. 10a and 10b, respectively, for initial and final

locations in the vicinity of the juncture between the nose and the

frustum of the cone. Both figures show different behavior of the

transient growth amplification for disturbances initiated within

the nose region (ξ0 < ξj) and those that are initiated downstream ofthe juncture location (ξ0 ≥ ξj). The disturbances initiated in the nosehave a maximum for very short optimal growth intervals, as

previously shown in Fig. 9. For larger optimal growth intervals, the

transient growth amplification first decreases and then increases

again for ξ0 > ξj. On the other side, disturbances initiated

downstream of the nose (ξ0 ≥ ξj) experience a monotonic increase in

the energy gain factor as ξ1 is increased. Remarkably, disturbances

initiated in the vicinity of the juncture location (ξ0 ≈ ξj) experience aquite rapid growth for the more blunt case and short transient growth

intervals (Fig. 11b), resulting in relatively significant values of NE

just downstream of the juncture location. However, the analysis of

these two cases did not reveal a specific correlation between NE and

the onset of transition.The AFRL experiments investigated the effect of an azimuthally

periodic array of roughness elements mounted near the sonic point at

a) b)Fig. 8 Contours of a) optimalmean total energy gainGmean

E andb) optimalmeankinetic energy gainGmeanK within the nose region of theRN � 15.24 mm

and Re � 91.4 × 106 m−1 case. The solid line in the contour plot indicates the value of ϕ1 corresponding to maximum energy gain for a given ϕ0.

a) b) c)

d) e) f)Fig. 9 a,d) Optimal mean energy gain and corresponding b,e) azimuthal wave number and c,f) optimization interval within the nose region forRN;S � 5.08 mm and RN;L � 15.24 mm cones at same freestream unit Reynolds number, Re∞ � 91.4 × 106 m−1.


ϕ � 45 deg. To help gain some insight into the role of transientgrowth as a mechanism for roughness effects, we next examine thedetails of transient growth disturbances initiated in the nose region at

ϕ � ξ0∕RN � 45 deg, which coincides with the location of theroughness array. A description of the experimental findings isdeferred to Sec. IV. Figure 12 shows the mean total energy gain and

a) b)Fig. 10 Contours of N-factor values defined as NE � 1∕2 ln �Gout

E � in the frustum region of the a) RN � 5.08 mm and b) RN � 15.24 mm cones. Thejuncture location ξj is marked with a vertical and a horizontal dashed line. The freestream unit Reynolds number is Re � 91.4 × 106 m−1.

a) b)Fig. 11 Contours of N-factor values defined as NE � 1∕2 ln �Gout

E � in the vicinity of the nose region of the a) RN � 5.08 mm and b) RN � 15.24 mmcones. The juncture location ξj is marked with a vertical and a horizontal dashed line. The freestream unit Reynolds number is Re � 91.4 × 106 m−1.

a) b)

(c) d)Fig. 12 a,c) Optimalmean energy gain and b, d) corresponding azimuthal wave number with initial disturbance locationϕ0 � 45 deg (ξ0 � 0.04 m forthe RN � 5.08 mm cone and ξ0 � 0.012 m for the RN � 15.24 mm cone). The legend refers to RN;S � 5.08 mm, RN;L � 15.24 mm,Re∞;L � 91.4 × 106 m−1, Re∞;M � 60.9 × 106 m−1, and Re∞;S � 30.5 × 106 m−1.


corresponding azimuthal wave number as a function of the optimalgrowth interval, Δξ � ξ1 − ξ0, for the four AFRL configurationsshown in Table 1. Figure 12a shows that the trend previouslyobserved for ξ0 < ξj at Re � 91.4 × 106 m−1 (Fig. 11b) also appliesat other Reynolds numbers. Specifically, the optimal energy gain hasa maximum immediately downstream of the inflow station and thendecays up to a plateau zone before increasing again for longertransient growth intervals. For the same freestream unit Reynoldsnumber, the initial peak in optimal energy gain is larger for thelarger nose radius cone than for the smaller nose radius case, butthis situation is reversed for larger optimal growth intervals(Δξ > 0.02 m). Figure 12b shows that the three regions translate intoa discontinuous evolution of the corresponding azimuthal wavenumber. The scaling of the optimal mean total energy gain withReRN

(Fig. 12c) and of the corresponding initial azimuthal wavelengthwithLRN

(Fig. 12d) is reasonable for the cases with same nose radiusRN � 15.24 mm and different freestreamunit Reynolds numbers butnot for the smaller nose radius case (RN � 5.08 mm) and outside thenose region.Figure 13 shows further details of the transient growth

disturbances initiated in the nose at ϕ0 � 45 deg for the moreblunt cone at the highest Re∞ configuration (RN � 15.24 mm andRe∞ � 91.4 × 106 m−1). Three cases based on the trends observedin Fig. 12 are plotted: namely, a) Δξ � 0.0030 m and m � 420,b)Δξ � 0.023 m andm � 150, and c)Δξ � 0.091 m andm � 80.Case a corresponds to the first peak in mean energy gain that is largerfor the larger nose radius cone (Fig. 12a). The evolution of thedisturbance amplitude

��E∕E0

pfor case a shows a rapid rise to its peak

valuewithin a rather short distance from the initial disturbance locationand a slower subsequent decaywith values lower than1 for streamwiselocations in the frustum region. The initial optimal perturbation plottedin Fig. 13 is mostly contained within the boundary-layer thickness;hence, such initial disturbance profiles are better suited for excitationvia surface roughness than someother cases inwhich the initial profilesextend well outside of the boundary layer. The disturbance amplitudeevolution of case b has a smaller initial peak and then remains below 6along the streamwise domain plotted in Fig. 13. The initial optimalperturbation associatedwith this case is similar to that in case a, but thepeaks of the perturbation variables are located somewhat farther fromthe wall, which presumably makes this perturbation less likely to beexcited via surface roughness. The energy gain evolution for case c(Δξ � 0.091 m, m � 80) shows rather small disturbance amplifica-tion near the inflow location and then a monotonic amplification up toξ � 0.2 m. Figure 13 shows that the optimal initial perturbationshape in case c is more complex than that for cases a and b becausethe perturbation profiles for wall-normal and spanwise velocitycomponents have two and three peaks instead of one and two peaks,respectively. Also, the perturbation profiles have a larger wall-normalextension, and the peaks of these profiles are located farther from thewall. In summary, the results shown in Fig. 13 indicate that roughness-

induced perturbations at ϕ0 � 45 deg can experience transientgrowth in a short interval within the nose region. The transient growthstreaks can lead to the onset of nonstationary streak instabilities thattypically amplify rather rapidly and induce transition shortly after theironset; see [66–71] for details on secondary instability of streaks inhigh-speed boundary layers.Previously, disturbances initiated in the vicinity of the juncture

location, ξj � RNπ∕2, were found to experience a larger growth forthe larger nose radius cone and relatively short transient growthintervals. This finding is further investigated here. Figure 14 shows themean gain in total energy and corresponding azimuthal wave numberas a function of the optimal growth interval,Δξ � ξ1 − ξ0, for the fourAFRL configurations shown in Table 1. Figure 14a shows in detailsthat the trend observed in Fig. 11b for Re∞ � 91.4 × 106 m−1 is alsofound at other Reynolds numbers. Specifically, the energy gain has amonotonic increasing evolution asΔξ is increased, and the energy gainis larger for a larger nose radius for relatively short optimizationintervals (Δξ < 0.13 m). The difference between the energy gainvalues for the two nose radii (RN � 5.08 and 15.24 mm) and samefreestream unit Reynolds number (Re � 91.4 × 106 m−1) reachesabout a factor of 2 for Δξ � 0.05 m. The azimuthal wave numbersassociated with the optimal energy gain values plotted in Fig. 14a areshown in Fig. 14b. The azimuthal wave number quickly decreases andas the length of the transient growth interval increases, albeit at adecreasing rate. Compared to the large bluntness cases(RN � 15.24 mm), the smaller radius case (RN � 5.08 mm andRe � 91.4 × 106 m−1) shows a notable, different behavior of both theoptimal energy gain and the associated azimuthal wave numbers. Thisdifference is clearlyobserved in the scaledmeanenergygain and scaledinitial wavelength plotted in Figs. 14c and 14d, respectively. The largenose radius cases (RN � 15.24 mm) show a perfect scaling of bothproperties as the unit Reynolds number is varied, but the scaled valuescorresponding to the small radius case (RN � 5.08 mm) are clearlydifferent from the blunt nose cases. The reason for this poor scaling ofthe transient growth parameters with nose radius could be due to thelack of self-similarity of the boundary-layer profiles near the nose.Figure 15 shows further details of the transient growth disturbances

initiated at the juncture location for the RN � 15.24 mm and Re �91.4 × 106 m−1 configuration. The three cases shown in Fig. 15 forthe RN � 15.24 mm case correspond to a) Δξ � 0.0295 m andm � 200, b) Δξ � 0.0668 m and m � 150, and c) Δξ � 0.2066 mandm � 110. Figure 15 shows that the peak in disturbance amplitudemoves slightly downstream as the optimal growth interval becomeslonger from case a to case c. This peak is barely reached at the end ofthe cone length for case c. The optimal initial perturbations associatedwith the three cases are plotted in Fig. 15. The wall-normal extensionand the peaks of these profiles are located closer to the wall for shorteroptimal growth intervals, which makes them more closely relatedto roughness-induced perturbations. A wall-mounted device is notexpected to generate a perturbation with the wall-normal extension of

a) b) c)Fig. 13 Evolution of the disturbance amplitude

��E∕E0

pand corresponding initial optimal perturbations for the RN � 15.24 mm and Re∞ �

91.4 × 106 m−1 configuration. The selected optimization intervals and azimuthal wave numbers are a) Δξ � 0.0030 m and m � 420 (λ0∕δh � 0.89),b) Δξ � 0.023 m and m � 150 (λ0∕δh � 2.5), and c) Δξ � 0.091 m and m � 80 (λ0∕δh � 4.7). The initial disturbance location is set at the nose tip atϕ0 � 45 deg that corresponds to ξ � 0.012 m. The horizontal, dashed-double-dotted lines indicates the edge of the boundary layer based on totalenthalpy, η � δh.


the optimal perturbation corresponding to case c. Although not shownhere, equivalent results for the small radius case (RN � 5.08 mmand Re � 91.4 × 106 m−1) and an optimal growth interval of Δξ �0.0295 m show a peak disturbance amplitude equal to approxi-mately one-half of the peak value corresponding to the largeradius cone (RN � 15.24 mm and Re � 91.4 × 106 m−1) plotted inFig. 15. Similar to results for disturbances initiated near the soniclocation, these results with ξ0 � ξj indicate that roughness-inducedperturbations can experience greater transient growth for larger nose-tip bluntness at the same freestream conditions. Therefore, transitiononset could be driven by roughness-induced transient growth if thestreak amplitude required for streak instabilities is reached.

3. Transient Growth of Traveling Disturbances Along Frustum Region

Next,we present salient findings pertaining to the nonmodal growthof traveling disturbances over the frustum region of theAFRL cone forthe case of RN � 5.08 mm and Re � 91.4 × 106 m−1. A detailedparameter study of traveling-mode disturbances is deferred to thework

of Paredes et al. [72]. The procedure used for the transient growth

analysis of traveling disturbances is the same as that used earlier for

the stationary disturbances with α � 0, except with the addition of thestreamwise pressure gradient term that is approximated with Eq. (7)

based on the work by Vigneron et al. [57] for PNS equations. For the

results presented herein, we confine our attention to a transient growth

interval of �ξ0; ξ1� � �0.04; 0.161� m. The selected outflow location

corresponds to the transition location measured by Jewell et al. [26] as

indicated in Table 1, whereas the inflow location was chosen on the

basis of a parameter study, which showed that the maximum energy

gain occurs for ξ0 � 0.04 m for most combinations of frequency and

azimuthal wave number.Figure 16 shows the contours of the N factor based on the total

energy gainNE as a function of the frequency F and azimuthal wave

numberm. The outlet energy gain is selected as the objective functionfor the optimal growth analysis. One may observe that the maximum

energy gain of NE � 4.59 is achieved by a stationary, three-

dimensional perturbation with F � 0.0 kHz and m � 82. This

a) b)

c) d)Fig. 14 a,c) Optimal mean energy gain and b,d) corresponding azimuthal wave number with initial disturbance location set in the juncture atξ0 � ξj � RNπ∕2 (ξ0 � 0.008 m forRN � 5.08 mm and ξ0 � 0.024 m forRN � 15.24 mm). The legend refers toRN;S � 5.08 mm,RN;L � 15.24 mm,Re � 91.4 × 106 m−1, Re∞;M � 60.9 × 106 m−1, and Re∞;S � 30.5 × 106 m−1.

a) b) c)Fig. 15 Evolution of the disturbance amplitude

��E∕E0

pand corresponding initial optimal perturbations for the RN � 15.24 mm and Re∞ �

91.4 × 106 m−1 configuration. The selected optimization intervals and azimuthal wave numbers are a) Δξ � 0.0295 m andm � 200 (λ0∕δh � 0.82), b)Δξ � 0.0668 m and m � 150 (λ0∕δh � 1.1), and c) Δξ � 0.2066 m and m � 110 (λ0∕δh � 1.5). The initial disturbance location is set at the junctureξ0 � ξj that corresponds to ξ0 � 0.024 m. The horizontal, dashed-double-dotted lines indicates the edge of the boundary layer based on total enthalpy,η � δh.


stationary disturbance is the same as that studied previously inSec. III.C.2. Additionally, Fig. 16 shows a second, local maximum(NE � 3.27) in theN-factor contours for planar traveling waves withF � 340.0 kHz, m � 0. Additional results not presented herein forsharper cones (RN � 1.524 and 2.54 mm) have indicated that theenergy gain associated with this local maximum for travelingdisturbances increases with the nose bluntness. Those results alsoshowed that, unlike the case considered in Fig. 16, the peak N factorfor traveling disturbances is not always associatedwith planar waves.Figures 17 and 18 show further details concerning the streamwise

evolution of the optimal traveling disturbance [F � 340.0 kHz,m � 0, �ξ0; ξ1� � �0.04; 0.161� m, and J � Gout

E ]. Figure 17illustrates the axial growth in the disturbance amplitude in terms ofthe N-factors based on the total energy NE, kinetic energy NK,maximum temperature fluctuation NT , and maximum of thestreamwise velocity fluctuationNu. To check the grid convergence ofthe solution obtained on the basis of α � 0 and the PNSapproximation, results were obtained with two different streamwise

grids with nξ � 8192 and nξ � 16; 384, respectively. To confirm that

the Vigneron parameter does not have a significant influence on theN-factor evolution, additional results were obtained by marching thePSE with the same initial condition. The PSE solution is obtainedwithout invoking the Vigneron PNS approximation (i.e., with α ≠ 0and ΩPNS � 1). The number of streamwise points required to obtaingrid converged results with α ≠ 0 is approximately two orders ofmagnitude lower than the number of grid points required to resolve theentire waveform via α � 0. Additionally, the results based on theharmonic, linearized Navier–Stokes equation (2) were identical tothose shown here. The agreement between the various results isexcellent, indicating that the prediction accuracy is not compromisedby either physical or numerical approximation. An importantcharacteristic of the traveling-mode growth is the large differencebetween the growth factors for the temperature and streamwisevelocity perturbations. The maximum N factor based on thetemperature perturbation is NT � 6.82, whereas that based on thestreamwise velocity perturbation is substantially lower, Nu � 1.11.This difference is also reflected in the evolution of the total disturbanceenergy and the disturbance kinetic energy, indicating that the transientgrowth of the traveling disturbances is mainly reflected in thethermodynamic perturbations. The optimum initial disturbance at theinflow location is dominated by velocity fluctuations, which are anorder of magnitude larger than the temperature fluctuations. However,the rapid growth in temperature perturbation immediately downstreamof the inflow location allows the velocity and temperature perturbationto become comparable to each other. Indeed, Paredes et al. [72] havepointed out that, because of the sustained higher growth in temperatureperturbation over the remaining portion of the transient growth interval(albeit at a reduced pace in comparison with the inflow region), thetemperature fluctuations become the dominant component of theoutflow disturbance at ξ � ξ1. On the basis of initial value analysis oftransient nonmodal growth in compressible Poiseuille flows, Xie et al.[73] argued that the pressure-velocity coupling at higher Machnumbers played an important role in influencing the evolution ofdisturbances that are nearly aligned with the mean flow direction.However, the role of temperature fluctuations was not directlyinvestigated. Therefore, further work will be necessary to establishwhether or not the physical mechanisms discussed therein also play arole in the transient growth of planar disturbances observed inthis Paper.The streamwise evolution in the mode shape of the optimal

disturbance is plotted in Fig. 18. The normalized temperature~T∕max� ~T� and streamwise velocity ~u∕max� ~u� are mostlycontained between the boundary-layer edge δh and the entropy-layer edge δS, which is defined as the location where the localentropy increment with respect to the freestream value, i.e.,

ΔS � cp ln � �T∕ �T∞� − Rg ln � �p∕ �p∞�, is 0.25 times the entropy

increment at the wall [ΔS�ξ; η � δS� � 0.25ΔSwall]. The disturb-ance is initially tilted against the flow direction and increases inmagnitude while rotating downstream. This behavior stronglyresembles the Orr mechanism for the transient growth of planardisturbances [74,75], which has been widely studied forincompressible flows [20,76]. Nonmodal growth based on the Orrmechanism is attributed to an energy extraction from the mean shearby transporting momentum down the mean momentum gradientthrough the action of the Reynolds stress associated with theperturbation field. However, the energy amplification observed inthis compressible case is mainly attributed to the temperatureperturbation within the entropy layer. Similar to the LIF-basedschlieren images of Fig. 5, the peak disturbance magnitude at anyaxial location is observed outside of the mean boundary-layer edge.However, additional work is required to establish a more definitivelink between the experimental observations and the nonmodalgrowth mechanism reported herein. Furthermore, because thetraveling-mode structure in Fig. 18 has a relatively weak signaturewithin the boundary-layer region, its role in initiating boundary-layertransition also remains to be clarified. Similarly, nonlinearsimulations of these traveling instabilities, e.g., by means ofnonlinear PSEs, could provide some insight into the potential

Fig. 16 Contours of N-factor values defined as NE � 1∕2 ln �GoutE � in

the azimuthal wave number vs frequency plane in the frustum region ofthe RN � 5.08 mm cone with Re∞ � 91.4 × 106 m−1, ξ0 � 0.04 m, andξ1 � ξT � 0.161 m. The isocontours represents N-factor increments ofΔNE � 0.1. The solid black line denotes the azimuthal wave numbercorresponding tomaximumNE. The large black circles denote the (F,m)combination for maximumNE in both axes.

Fig. 17 N-factor evolution of the optimal disturbance withF � 340 kHz, m � 0, ξ0 � 0.04 m, and ξ1 � 0.161 m for the RN �5.08 mm cone with Re∞ � 91.4 × 106 m−1. The solid and dashed linesrepresent the solution with α � 0 and PNS approximation of Eq. (7) withnξ � n � 8192 points andnξ � 2n, respectively. The open circles denotethe solution with α ≠ 0 and no PNS approximation (ΩPNS � 1).


breakdown mechanism and the minimum initial amplitude of theoptimal disturbance that would likely result in the onset of transitionat the measured transition location.

4. Transient Growth on AEDC Configurations

Transient growth results for the AFRL cones showed thatdisturbances initiated in the vicinity of the juncture between the nosetip and the frustumof the cone can experience a rather strong transientamplification in a short streamwise distance. This amplification isstronger for the larger nose radius case at the same freestreamconditions. Based on that observation, results for the AEDCconfigurations are focused on disturbances initiated at the juncture ofthe cones to study the trend with this different set of flow andgeometry parameters. Figure 19a shows the evolution of optimal gainin mean energy gain as a function of the optimal growth interval fordisturbances initiated at the juncture location. Again, as the noseradius of the cone is increased, the optimal energy gain becomesnotably larger. Figure 19b indicates that the scaled energy gainvaluesGmean

E ∕ReRNare smaller for larger nose radius cases, indicating a

nonlinear increase in optimal energy gain with increasing nosebluntness. This trend is similar to the trend in Fig. 14c for the AFRLconfigurations.

IV. Roughness Effects on Laminar–TurbulentTransition on Blunt Cones

Between 1978 and 1982, Stetson performed a total of 196 sharp-and blunt-cone experiments on a thin-walled 8 deg half-angle, 4 in.(0.1016 m) diameter base cone at zero angle of attack in the AFRLMach 6 high-Reynolds-number facility. These measurements wereinitially reported in a 1983 paper [2], and recently Jewell andKimmel

[37] performed a computational analysis indicating that neither the

Mack second mode nor the first mode was likely to be the dominant

instabilitymechanism for nose tipswith a radius larger than 1mm, for

which transition was nevertheless consistently observed once

sufficient freestream Reynolds numbers were reached. Alternate

physical mechanisms may explain these results and include transient

growth or mean flow distortion from roughness.

More recently, experiments have been performed in the same

facility on a 7 deg half-angle variable-bluntness cone at zero angle of

Fig. 18 Contours of normalized temperature and streamwise velocity components of the optimal disturbance with F � 340 kHz,m � 0, ξ0 � 0.04 m,and ξ1 � 0.161 m for theRN � 5.08 mm cone withRe∞ � 91.4 × 106 m−1. The solid black line indicates the edge of the boundary layer δh. The dashedblack line indicates the edge of the entropy layer δS.

a) b)Fig. 19 Optimalmean energy gainwith initial disturbance location set at the juncture ξ0 � ξj � RNπ∕2 for the AEDC configurations (ξ0 � 0.015 m forRN � 9.53 mm, ξ0 � 0.020 m for RN � 12.7 mm, ξ0 � 0.040 m for RN � 25.4 mm, and ξ0 � 0.080 m for RN � 50.8 mm). The freestream unitReynolds number is Re∞ ≈ 17.5 × 106 m−1.

Fig. 20 Nose tip of RN � 5.08 mm with 210 equally spaced 15 μmcubical roughness elements thatwere lasermachined circumferentially at45 deg from the apex of the spherical tip.


attack with similar nose-tip bluntness values, which have also beenshown to transition from laminar to turbulent flowwithout significantpredicted amplification of first- or second-mode instability forsmooth nose tips with radius larger than 1 mm and sufficientfreestream Reynolds numbers [26]. A subset of the smooth cases,with RN � 5.08 mm and RN � 15.24 mm, have been repeated with15 μm laser-machined roughness elements of approximately cubicalshape located on the circle at 45 deg from the apex of the spherical tip,which is near the sonic line for zero angle of attack. The array ofroughness elements has been designed following the transient growthresults of Sec. III.C. The number of roughness elements has beenselected to match the disturbance azimuthal wave number that leadsto significant transient growth at the same conditions. Specifically,210 roughness elements were mounted on the small radius cone(RN � 5.08 mm), and 420 roughness elements were mounted on thelarge radius cone (RN � 15.24 mm). A picture of the small radiusnose tip with the array of roughness elements is shown in Fig. 20.The presence of roughness elements on the RN � 5.08 mm cone

had little discernible effect on the experimentally observed transitionlocation or heating levels across the operating conditions of theMach6 high-Reynolds-number facility. In contrast, roughness elements ofthe same shape and dimension on the RN � 15.24 mm cone had aprofound effect on transition above a critical Reynolds number. Thiseffect is illustrated in Figs. 21a and 21b, which present surface heatflux distributions measured by Medtherm coaxial thermocouplesfrom pairs of rough and smooth experiments, with roughness-heightRekk values matched to within 3% between the two sets. In Fig. 21a,the rough (Re∞ � 66.5 × 106 m−1, k∕δh � 0.381, andRekk � 294)and smooth (Re∞ � 66.5 × 106 m−1) surface heat transfer valuesfor the RN � 5.08 mm cone overlap. In Fig. 21b, the rough(Re∞ � 75.8 × 106 m−1, k∕δh � 0.225, and Rekk � 287) andsmooth (Re∞ � 76.9 × 106 m−1) cases for the RN � 15.24 mmcone are quite different, with the rough tip resulting in nearlyimmediate transition onset, while the equivalent smooth tip results inlaminar flow nearly to the end of the cone. This result indicates thatthe onset of transition in these data is associated with the combinedeffects of bluntness and roughness. Transient growth results shown inFig. 12a pertain to the growth of stationary disturbances initiatedwithin the nose-tip region at the same location where the roughnessarray was mounted in the AFRL experiments, i.e., θ0 � 45 deg. Thetransient growth results indicate that, for the same freestream unitReynolds number, the maximum energy gain for short optimizationintervals is about two times larger for the larger nose radius cone thanfor the smaller nose radius case. This finding could be related to theexperimental finding involving an increased effectiveness of theroughness array on a more blunt cone.

V. Conclusions

A collaborative effort by the NATO STO group AVT-240 onhypersonic boundary-layer transition prediction was devoted tobluntness effects on transition over circular cones at zero angle ofattack. This Paper presented an overview of that work, whichconsisted of transition measurements on blunt cone models with

varying degrees of nose-tip bluntness, measurements pertaining tothe combined effects of nose bluntness and controlled surfaceroughness on the nose tip, and theoretical analysis of optimaltransient growth on blunt cones.The following observations can be made on the basis of wind

tunnel measurements at freestream Mach numbers from 6 to 12:1) The recentmeasurements are consistent with the known trend of

an initially downstream movement in the transition onset location asthe nose-tip bluntness is increased. This trend confirms the stabilizingeffect of nose bluntness on the amplification of the Mack mode, i.e.,second-mode instability waves.2) The measurements also confirm the well-known observation

concerning a reversal in the previously mentioned trend atsufficiently large value of the nose-tip Reynolds number, and thetransition-reversal phenomenon cannot be explained via thehydrodynamic stability paradigm, due to the absence of anadequately significant modal amplification of first-mode, second-mode, and entropy-layer instabilities.3) Experimental measurements indicate that, in the transition-

reversal regime, boundary-layer transition over the frustum of thecone is highly sensitive to wall roughness. The onset of transitionappears to be dominated by disturbances that originate in the vicinityof the nose tip, suggesting roughness-induced transient growth as alikely cause for laminar–turbulent transition.

Optimal transient growth analysis has been conducted for thelaminar flow based on the solution of the Navier–Stokes equationsover a 7 deg half-angle variable-bluntness cone at zero angle of attackwith bluntness and freestreamconditions corresponding to a subset ofthe experiments of Stetson [2] at the AFRL Mach 6 high-Reynolds-number facility and Re∞ � 30.5–91.4 × 106 m−1 and of Marineauet al. [25] at the AEDCTunnel 9 facility with theMach 10 nozzle andRe∞ ≈ 17.5 × 106 m−1. Some interesting findings in regard to thetransient growth of both stationary and traveling disturbances havebeen uncovered in the course of this study. Estimates of themaximumpossible transient growth are obtained through a dual optimization[23,24] that varies both the initial and final locations of the transientgrowth interval. The following conclusions may be drawn on thebasis of the transient growth analysis:1) The optimal growth results can be naturally divided into two

parts. The first part deals with transient growth intervals limited to thenose region, and the second part applies to optimal growth intervalsthat extend to the frustum region that encompasses the range ofmeasured transition locations.2) Results for optimal growth intervals within the nose region are

similar to the findings by Paredes et al. [23] for a hemisphericalforebody. Specifically, the highest predicted growth in total energycorresponds to relatively short transient growth intervals in thevicinity of the stagnation point, but the kinetic energy growthassociated with these perturbations is rather small. The maximumgrowth in disturbance kinetic energy is found for perturbationsinitiated near the sonic point at an angle of 42.4 deg from the apex ofthe spherical tip. Furthermore, the wall-normal profiles of optimalinitial perturbations associated with the maximum energy gain areconcentrated within the boundary-layer thickness, i.e., close to the

a) b)Fig. 21 Heat flux distributions for a 7 deg half-angle cone atM∞ � 5.9with nose tips of a)RN � 5.08 mm and b)RN � 15.24 mm for a smooth tip and atip with a row of equally spaced 15 μm cubical roughness elements.


wall. More importantly, at a fixed set of freestream conditions,increased nose bluntness results in higher transient amplificationwithin the nose region.3) Based on the transient growth analysis for both AFRL and

AEDCconfigurations, onemay conclude that stationary disturbancesthat are initiated in the vicinity of the juncture between the nose tipand the frustum also exhibit a greater transient amplification forlarger values of nose-tip bluntness, but only for relatively shortoptimization intervals. Furthermore, for the large nose-tip bluntness,the wall-normal extension of the optimal initial perturbations issmall, indicating that wall roughness may be able to induce suchperturbations.4) The transient growth analysis of traveling disturbances showed

significant growth of planar waves that peak between the boundary-layer and entropy edges. The temperature component of thedisturbance can reach significantN factors at the measured transitionlocation,NT ≈ 7, while the streamwise velocity component is barelyamplified, Nu ≈ 1.We note that the actual generation (i.e., receptivity) of the transient

growth disturbances has not been addressed in this Paper.Disturbance profiles resulting from realistic external disturbancesusually result in a suboptimal transient growth. In that regard, thepresent results based on optimal initial disturbancesmay beviewed asproviding an upper bound on the nonmodal amplification due toeither spanwise periodic, roughness-induced, stationary disturbancesor freestream traveling waves.Finally, the roughness effects on transition reversal over a 7 deg

half-angle blunt cone at hypersonic flow are experimentallyinvestigated in the AFRL Mach 6 high-Reynolds-number facility.Transition measurements were made with an azimuthally periodicarray of 15 μm laser-machined roughness elements of approximatelycubical shape located at 45 deg from the apex of the spherical tip,which nearly coincides with the sonic location. The roughnessarray spacing (m � 210 for RN � 5.08 mm and m � 420 forRN � 15.24 mm) was designed to match the disturbance azimuthalwave number that is predicted to induce large transient growth underthe respective conditions. The following observations can bemade onthe basis of these measurements:1) The roughness array on the small nose-tip bluntness cone

showed no effect on the transition location at Re∞ ≈ 66 × 106 m−1

(i.e., k∕δh � 0.381, Rekk � 294).2) On the other hand, the experiments conducted with the large

nose-tip bluntness with Re∞ ≈ 76 × 106 m−1 (i.e., k∕δh � 0.225,Rekk � 287) showed a prominent effect of the roughness elements onthe transition onset, moving the transition front close to the nose tip,whereas the smooth tip baseline for the samemore blunt cone resultedin a laminar flow over the majority of the cone.3) The previously mentioned findings indicate that the transition

onset is associated with the combined effects of bluntness androughness.Finally, we note that wakes behind sufficiently large discrete

roughness elements can sustain large modal amplification of streakinstabilities as shown by Choudhari et al. [66,68], but the level ofamplification is sensitive to the roughness parameters and other flowdetails [67]. Because the measured onset of transition during theAFRL experiments was within a short distance behind the roughnessarray, wake instability may have played a role in the large upstreamshift in the transition location, and we are currently investigatingthat possibility via more detailed computations. However, a broaderset of measurements involving controlled surface roughness wouldbe useful in determining whether or not naturally occurring micro-roughness over the model surface may have been responsible for theobserved transition reversal in other blunt cone experiments. Effectsof the roughness location in the vicinity of the nose tip and thejuncture location would be an important aspect of such a parametricstudy. Flow visualizations accompanied by dynamic measurements,i.e., high-frequency response heat transfer or pressuremeasurements,in the region between the controlled surface roughness and the onsetof transition would go a long way in providing valuable cluesregarding the physical mechanisms at play. Finally, performingparallel measurements in both quiet and conventional wind tunnel

facilities, or in flight-test experiments, would provide furtherinformation regarding the role of both the freestream disturbancesand the nonmodal growth of traveling disturbances within theentropy layer during the transition-reversal phenomenon.

Acknowledgments

Part of this researchwas sponsored by theNASATransformationalTools and Technologies Project of the Transformative AeronauticsConcepts Program of the Aeronautics ResearchMission Directorate.A portion of the computational resources supporting this work wasprovided by the NASA High-End Computing Program through theNASA Advanced Supercomputing Division at Ames ResearchCenter. The authors would like to thank the Test ResourceManagement Center Test and Evaluation/Science and TechnologyProgram for its support. E. C. Marineau’s work was funded by Testand Evaluation/Science and Technology Program through the HighSpeed Systems Test area as part of the Center of Testing Excellence.The authors also acknowledge Benxin Wu and his group at PurdueUniversity for laser machining the nose-tip roughness elements. J. S.Jewell was supported by the National Research Council (NRC)Research Associateship for part of this work.

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