Boundary-Layer Stability Analysis for Stetson’s Mach 6Blunt-Cone Experiments

Joseph S. Jewell∗ and Roger L. Kimmel†

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

DOI: 10.2514/1.A33619

The results of a seminal experimental study of the effects of bluntness and swallowing length on transition on an

8 deg cone at zero angle of attack in Mach 6 high-Reynolds-number flow are analyzed with the STABL-2D

computational fluid dynamics code package. Mean flow solutions and parabolized stability analyses for a total of 11

different nose-tip bluntnesses, ranging from sharp to a 15.24 mm radius, are obtained. For the sharpest cases, theNfactor at transition is approximately seven; but, as bluntness increases and the calculated swallowing distance

lengthens, the computed N factor at the experimentally observed transition location drops below the level at which

Mack’s second mode would be expected to lead to transition. These results indicate that the dominant instability

mechanism for the bluntest cases is not the second mode, and oblique mode analysis also excludes these modes as the

dominant instability mechanism.

Nomenclature

h = enthalpy, KJ∕kgM = Mach numberN = amplification factorn = length tangential to surface, mp = pressure, kPaR = radius, mRe = Reynolds numberReunit = unit Reynolds number, 1∕mS = entropy, J∕�kg · K�St = Stanton numbers = surface length, mT = temperature, Ku = velocity, m∕sX = Stetson surface length, m−αi = disturbance growth rate, 1∕mβ = complex wave number, 1∕radδ = boundary-layer thickness, mρ = density, kg∕m3

ψ = phase angle, radΩ = most amplified frequency, kHzω = frequency, kHz

Subscripts

B = blunte = boundary-layer edge conditionsN = noseS = sharpTr = transition onsetw = wall conditions0 = stagnation conditions∞ = freestream conditions

I. Introduction

B OUNDARY-LAYER transition is a critical factor in the designof hypersonic vehicles, with profound impact on both heat

transfer and control characteristics. Although many practicalaerospace vehicles are blunt, the mechanisms that lead to boundary-layer instability and transition on sharp bodies are more thoroughlyunderstood at present. However, over the past half-century, a wealthof relevant wind-tunnel and flight data have been acquired.Analytical and computational techniques, as well as the rapiddevelopment of economical and powerful computer processors, havemade possible comprehensive computational analyses of existingexperimental datasets.Between 1978 and 1982, K. F. Stetson performed a total of 196

sharp- and blunt-cone experiments [1] on an 8 deg half-angle, 4 in.(10.16 cm) base diameter cone in the U.S. Air Force ResearchLaboratory’s (AFRL’s) Mach 6 high-Reynolds-number facility.These experiments were reported in a 1983 paper [1], along withresults from the Arnold Engineering Development Center’s(AEDC’s) Tunnel F with a larger cone at Mach 9. The resultsrecorded in Stetson’s laboratory notebook‡ provided further insightinto the dataset. Although a subset of the AEDC Mach 9 resultsreceived computational analysis [2], the AFRL Mach 6 results havenot (to date).The availability of well-tested mean flow and stability prediction

computer codes, along with multicore machines, now makes itpossible to analyze Stetson’s Mach 6 results with detail and fidelityunavailable in 1983 [1]. Marineau et al. [3] showed the utility ofapplying parabolized stability equation (PSE) stability predictions totransition experiments on blunt cones at Mach 10. Stetson’s Mach 6experiments [1] form a counterpoint to Marineau et al.’s Mach 10experiments [3]. Although the maximum length Reynolds numberswere comparable in the two experiments (47 × 106 forMarineau et al.[3], and 33 × 106 for Stetson [1]), the Stetson experiments werecarried out in a smaller wind tunnel at a higher freestream unitReynolds number. Since wind-tunnel noise scales on the unitReynolds number and wind-tunnel size, and roughness effectsdepend on the unit Reynolds number, the conditions for Stetson’sMach 6 experiments [1] were substantially different from Marineauet al.’s [3].

The AFRL Mach 6 facility operates at stagnation pressures p0

from 700 to 2100 psi (4.83 to 14.5 MPa). Details of the maximumand minimum Reynolds numbers conditions, along with a sampleintermediate case at themidpoint, are presented in Table 1. A total of196 experiments encompassing 108 unique conditions comprised

Presented as Paper 2016-0598 at the 54th AIAA Aerospace SciencesMeeting, San Diego, CA, 4–8 January 2016; received 1 April 2016; revisionreceived 30 July 2016; accepted for publication 1 August 2016; publishedonline 18 October 2016. This material is declared a work of the U.S.Government and is not subject to copyright protection in theUnited States. Allrequests for copying and permission to reprint should be submitted to CCC atwww.copyright.com; employ the ISSN 0022-4650 (print) or 1533-6794(online) to initiate your request. See also AIAARights and Permissions www.aiaa.org/randp.

*ResearchAerospace Engineer (NRCResearch Associate), AFRL/RQHF;jjewell@alumni.caltech.edu. Senior Member AIAA.

†Principal Aerospace Engineer, AFRL/RQHF. Associate Fellow AIAA.

‡Stetson, K. F., “Notes Related to Previous AIAA Papers on Blunt Cones,”PersonalCommunication to S. P. Schneider, PurdueUniv.,West Lafayette, IN,Dec. 2001.

Article in Advance / 1

JOURNAL OF SPACECRAFT AND ROCKETS

the Stetson [1]Mach 6 results (see Sec. III). Mean flow and stabilitycalculations for each unique condition were performed at acomputational cost of about 100 processor hours each.

II. Computational Methods

The mean flow over the cone is computed by the reacting,axisymmetric Navier–Stokes equations with a structured grid, usinga version of the NASA data parallel-line relaxation (DPLR) code [4],which is included as part of the STABL-2D software suite, asdescribed by Johnson [5] and Johnson et al. [6]. This flow solver isbased on the finite volume formulation. The use of an excludedvolume equation of state is not necessary because the static pressureover the cone is sufficiently low (typically 10–50 kPa) that the gas canbe treated as ideal. The mean flow is computed on a single-blockstructured grid with dimensions of 361 cells by 359 cells in thestreamwise and wall-normal directions, respectively. The inflow gascomposition in each case is air, with 0.233 O2 and 0.767 N2 massfractions. Although the computation includes chemistry, the impactof chemical reactions is negligible, as the local maximumtemperature does not exceed 611 K for any case.Grids for the 8 deg half-angle sharp cone and each of the 10

bluntness conditions (see Table 2) were generated using STABL-2Dbuilt-in grid generator, and mean flow solutions were examined toensure that at least 100 points normal to the cone surface were placedin the boundary layer for each stagnation pressure. For simplicity andto match the Stetson nomenclature, bluntness as a percentage of thebase radius of 2.0 in. (5.08 cm)was used to label the cases analyzed inthe present work. The boundary-layer profiles and edge propertieswere extracted from the mean flow solutions during postprocessing.The wall-normal extent of the grid increased down the length ofthe cone, from 0.25 mm at the tip to 50 mm at the base, allowing forthe shock to be fully containedwithin the grid for all cases tested. Thegridwas clustered at thewall, aswell as at the nose, in order to capturethe gradients in these locations. The nondimensionalwall distance y�for the grid, extracted from the DPLR solution for each case, waseverywhere less than one, where y� was used as a measure of localgrid quality at the wall in the wall-normal direction.A grid convergence study was performed on the RN � 0.508 mm

(1%) and RN � 5.08 mm (10%) bluntness cases at the study’smaximum unit Reynolds number (92.1 × 106∕m), which corre-sponded to the tunnel stagnation pressure p0 � 2100 psi(14.5 MPa). As the boundary layer was thinnest for the higheststagnation pressure, these represented the most challenging cases foreach grid to properly resolve. In addition to the normal 361streamwise and 359 wall-normal cell cases, grids of 301 × 299,

431 × 419, and 501 × 499 cells were computed for these two cases.The normal 361 × 359 cell case was found to be sufficientlyconverged for both bluntnesses examined. The stability results fromtheRN � 0.508 mm (1%) convergence study are presented in Fig. 1.Although there was a minimal discrepancy at the end of thecomputational geometry for the most coarse grid, convergencewas sufficiently achieved for the other three cases, includingthe 361 × 359 cell case, which was used for the rest of the study.

III. Mean Flow-Based Transition Correlations

Stetson [1] reported results by “normalizing” the transitionReynolds numbers for blunted cones by the transition Reynoldsnumbers for sharp cones at the same inflow conditions, which werecalculated as

XTrB

XTrS

� �ReX�TrB�ReX�TrS

�Reunit�eS�Reunit�eB

(1)

Stetson [1] found XTr values by examining heat transfer plots andselecting the first data point elevated above the laminar curve (seeFig. 2 for an illustration of this process) [7,8]. As the thermocoupleswere spaced at 0.5 in. (12.7 mm) intervals, this was the approximateuncertainty of the measurement. All of the transition locations in thepresent work were reported in Stetson’s notebook. The original heat

Table 1 Sample inflow conditions (minimum, maximum, andmidpoint Reynolds numbers) computed for each bluntness value

p0,psi

p0,MPa

Unit Re∞,×106∕m M∞

ρ∞,kg∕m3

P∞,kPa

T∞,K

U∞,m∕s Tw∕T0

700 4.83 30.7 5.9 0.154 3.40 76.7 1038 0.561400 9.65 61.4 5.9 0.308 6.80 76.7 1038 0.562100 14.5 92.1 5.9 0.461 10.2 76.7 1038 0.56

Table 2 Summary of gridsgenerated for the present study

RN, in. RN, mm Bluntness, %

0 0 00.02 0.508 10.04 1.016 20.06 1.524 30.08 2.032 40.10 2.540 50.20 5.080 100.30 7.620 150.40 10.16 200.50 12.70 250.60 15.24 30

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

s [m]

N fa

ctor

coarse (301x299)normal (361x359)fine (431x419)very fine (501x499)

Fig. 1 Stability results from the RN � 0.508 mm (1%), unit Reynolds

number 92.1 × 106∕m convergence study.

0.05 0.1 0.2 0.3x position [m]

106

107

10−4

10−3

10−2

Reynolds number

Sta

nto

n n

um

ber

laminar correlationlaminar (DPLR)Van Driest IIWhite and ChristophexperimentalTransition onset (Stetson [1])

Fig. 2 Stanton number vs Reynolds number for the sharpp0 � 700 psicondition, with laminar and turbulent correlations [7,8], as well aslaminar computation for comparison.

2 Article in Advance / JEWELL AND KIMMEL

transfer plots were not available, but one example from the sameseries of experiments (a sharp condition with p0 � 700 psi) wasavailable in Fig. 7 of [9]. The heat transfer results from this plot weredigitized, replotted in terms of the Stanton number and Reynoldsnumber, and are reproduced in Fig. 2 along with Stetson’s indicatedtransition location [1], as well as laminar and turbulent sharp-coneStanton number correlations [10] and laminar computations from thepresent work for comparison. The laminar computation andcorrelation both agreed well with the nondimensionalized heattransfer data, and the experimental points reached the turbulent valueafter transition onset.

The boundary-layer edge, throughout the present work, is definedas the point at which the derivative of the total enthalpy (see Fig. 3)along a line extending orthogonally from the surface of the conebecomes zero, as suggested by Bertin [11] for cases where thetotal enthalpy h near the edge of the boundary layer locally exceedsh∞. Stetson [1] used selected computations with two boundary-layer codes and used interpolations to find the unit Reynoldsnumber and Mach number at the boundary-layer edge andthroughout the entropy layer but noted that “it was not consideredpractical to make boundary-layer calculations for all of thegeometric and flow variations of the present investigation.” This is

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

Local Enthalpy h [KJ/kg]

n [m

m]

s = 0.1 ms = 0.2 ms = 0.3 ms = 0.4 mBoundary-layer edge, δ

a) Sharp tip

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

Local Enthalpy h [KJ/kg]

n [m

m]

s = 0.1 ms = 0.2 ms = 0.3 ms = 0.4 mBoundary-layer edge, δ

b) RN = 1.016 mm (2%)Fig. 3 Sharp and blunt total enthalpy profiles with boundary-layer edge indicated; p0 � 1400 psi (enthalpy is in kilojoules per kilogram).

x [m]

y [m

]

0 0.01 0.02 0.03 0.04 0.050

0.005

0.01

0.015

loca

l uni

t Re

[1/m

]

0

2

4

6

8x 10

7

a) Sharp tip

x [m]

y [m

]

0 0.01 0.02 0.03 0.04 0.050

0.005

0.01

0.015

loca

l uni

t Re

[1/m

]

0

2

4

6

8x 10

7

b) RN = 1.016 mm (2%)

Fig. 4 Sharp and blunt unit Reynolds number contours (detail): p0 � 1400 psi.

Article in Advance / JEWELL AND KIMMEL 3

now possible; in fact, the full flowfield including the shock wascalculated for each case in the present study, and two examples forsharp and blunt cases are presented in Fig. 4. Although thefreestream unit Reynolds number is 61.4 × 106∕m for both cases,the large “swallowing length” of the blunt case is evident as a largelocal region of low unit Reynolds number fluid next to the wall.The entropy layer swallowing length estimate of Rotta [12]

(XSW) was used by Stetson and Rushton [13] to correlate results,and it was applied to the current analysis. This correlation involvesthe edge unit Reynolds number, which can be difficult to define for ablunt cone. Because of this, it was considered worthwhile torecompute the edge unit Reynolds number using DPLR andcompare the derived swallowing length correlation to Stetson’soriginal results [1]. Stetson’s experimental data from Figure 9 of [1]are replotted in Figs. 5 and 6 with DPLR-derived values of the edgeunit Reynolds number to obtain the swallowing length from Rotta’scorrelation [12]. Figures 5 and 6 show no significant differences

from Stetson’s [1] Fig. 9 data, which are replotted from thecalculated values recorded in Stetson’s notebook as small dots. Themaximum difference from the data reported by Stetson was lessthan 2%.In addition to the Rotta estimate [12], DPLR solutionswere used to

visualize the entropy layer, as shown in Fig. 7. The Cantera [14]thermodynamics software package was used to calculate entropy foreach cell based on the computed gas composition, temperature, andpressure. The large entropy layer swallowing length of the blunt casewas evident as a large local region of high-entropy fluid behindthe curved shock and next to the wall, which was initially muchthicker than the boundary layer. The estimated swallowing lengthXSW calculated with the method of Rotta [12] as applied by Stetsonand Rushton [13] was 24.2 cm for this case, which was well beyondthe field of view in Fig. 7b.Following Marineau et al. [3] and Moraru [15], the Stetson [1]

results may also be presented, as in Fig. 8, as a function of the

10−3

10−2

10−1

100

101

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

XT

B

/XSW

XT

B

/XT

S

RN/R

B = 0.01

RN/R

B = 0.02

RN/R

B = 0.03

RN/R

B = 0.04

RN/R

B = 0.05

RN/R

B = 0.10

RN/R

B = 0.15

RN/R

B = 0.20

RN/R

B = 0.25

RN/R

B = 0.30

Stetson [1]

Fig. 5 Nose-tip bluntness normalized by swallowing length vs the transition location ratio.

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

XT

B

/XSW

(Re X

T

) B/(

Re X

T

) S

RN/R

B = 0.01

RN/R

B = 0.02

RN/R

B = 0.03

RN/R

B = 0.04

RN/R

B = 0.05

RN/R

B = 0.10

RN/R

B = 0.15

RN/R

B = 0.20

RN/R

B = 0.25

RN/R

B = 0.30

Stetson [1]

Fig. 6 Nose-tip bluntness normalized by swallowing length vs transition Reynolds number ratio.

4 Article in Advance / JEWELL AND KIMMEL

freestream Reynolds number calculated with the nose radius as therelevant length scale. In this approach, the freestream Reynoldsnumber at the observed transition location increases with the nose-tipReynolds number before dropping off sharply at about 9 × 105. Thispattern is consistent with that observed by Marineau et al. [3] andMoraru [15], although those authors found the dropoff point at thenose-tip Reynolds number of about 3 × 105.

IV. Stability Computations

The stability analyses are performed using the PSE-Chem solver,which is also part of the STABL-2D software suite. PSE-Chem [16]solves the reacting two-dimensional axisymmetric, linear parabolizedstability equations to predict the amplification of disturbances as they

interact with the boundary layer. The PSE-Chem solver includes

finite-rate chemistry and translational-vibrational energy exchange.The band of amplified frequencies within the boundary layer

predicted by linear stability theory (LST) is presented in a contour

plot in terms of disturbance growth rate −αi in Fig. 9. The most

amplified frequency predicted by a simple model based on edgevelocity and boundary-layer thickness is also plotted, and it shows

generally good agreement with the detailed computations.In the present work, the focus was on the two-dimensional (2-D)

secondmode, which should be dominant above approximatelyMach

4.5 [17] for conical geometrywith significant wall cooling.Marineau

et al. [3] also did not observe strong oblique or first-mode activity in

experimental results derived from a ray of surface-mounted pressure

transducers. Nevertheless, frequencies low enough to include the 2-D

x [m]

y [m

]

0 0.01 0.02 0.03 0.04 0.050

0.005

0.01

0.015

S [J

/(kg

.K)]

6500

7000

7500

a) Sharp tip

x [m]

y [m

]

0 0.01 0.02 0.03 0.04 0.050

0.005

0.01

0.015

S [J

/(kg

.K)]

6500

7000

7500

b) RN = 1.016 mm (2%)

Fig. 7 Sharp and blunt entropy contours (detail): p0 � 1400 psi.

104

105

106

107

106

107

108

ReR

N

Re X

TB

Fig. 8 Results as a function of nose-tip freestream Reynolds number, where ReXTB� Reunit;∞XTr and ReRN

� Reunit;∞RN.

Article in Advance / JEWELL AND KIMMEL 5

first mode were also examined comprehensively, and obliquemodes (with frequencies low enough to encompass the first mode)were included in the computation for a subset of the blunt cases.Specifically, an oblique testmatrix examined instabilities of spanwisecomplex wave numbers β from 1∕rad to 3000∕rad and frequenciesωfrom 20 to 800 kHz over the entire surface of the cone for each ofthe bluntnesses listed in Table 2. For each bluntness, the study’smaximum unit Reynolds number (92.1 × 106∕m), which corre-sponded to a stagnation pressure of p0 � 2100 psi (14.5 MPa), waschosen as the mean flow solution input for the oblique analysis. Theresults of this oblique mode study are tabulated in Table 3, whichrecords the oblique mode with the maximum N factor (where themaximumN factor of an obliquemode is greater than theN factor forthe most unstable 2-D mode at the same location) for each case.Significant amplification beyond that computed for the 2-D secondmode was not observed in the computational results for any of theoblique cases. No amplification greater than that of the 2-D secondmode was observed at any location for bluntnesses less thanRN � 10.16 mm, so these cases are omitted from Table 2.Computed second-mode N factors at the experimental transition

location are presented in Fig. 10. A strong trend with both nose-tipbluntness and swallowing length ratio is observed.Note that the sharpdata points (noted with an arrow) are located at infinity on the x axis,as the swallowing length approaches zero, and appear to be the

asymptotic value for the N factor at transition with decreasing nose-tip bluntness. An inset plot details the region of most rapid change inthe transitionN factor. Three regions are evident. ForXTr∕XSW < 0.3,the computedN factors at transition are less than one, indicating thatgrowth is minimal and implying that an alternate transitionmechanism, rather than the second mode, is important in this region,which is populated by the more blunt cases. This result is consistentwith past LST calculations on a smaller number of blunt hypersoniccone cases: for example, Lei and Zhong [18].Marineau et al. [3], whoobserved a similar effect experimentally, proposed that transientgrowth [19] or entropy-layer instability [20] might be plausiblecandidate mechanisms. From 0.3 < XTr∕XSW < 1.0, the computedN factor of transition rapidly increases. In this region, significantmodal growth occurs and may compete with other mechanisms toprovoke laminar-turbulent transition. For XTr∕XSW > 1.0, which

includes the sharpest cones in the dataset, a consistent computedtransitionN factor, fallingwithin a range of 6.9–7.7, is observed. Thisresult is consistent with Mack’s second mode [17] as the dominantinstability mechanism for the sharpest cases, and also is significantlyhigher than the typical value of transition onset of N ≈ 5–6, whichusually characterizes a “noisy” tunnel [21].Transition in conventional hypersonic wind tunnels on cones with

small bluntness has in some cases [22] been well correlated with Nfactors of about 5.5. Recent results [10,23,24] have found highertransitionN factors in some noisy hypersonic tunnels for cases wherethere is a mismatch between the strongest freestream noisefrequencies and the most unstable boundary-layer frequencies.Evidence supporting this mismatch hypothesis in previous studieshas included a dependency of the correlating N factor with the mostunstable frequency at transition, in an effort to account for decreasingtunnel noise amplitude at higher frequencies. In contrast, the presentdataset taken as a whole does not show any clear systematic variationin the computed most amplified frequency at the measured transitionlocation (see Fig. 11). In fact, within each nose bluntness subset, thetransition N factor decreases with increasing frequency, which is theopposite of the effect that would be expected for noise-dominatedtransition onset with the typical noise spectrum of a hypersonicnozzle, which has higher amplitudes (and therefore, it is inferred,lowerN factors of transition) at lower frequencies. The outliers above2000 kHz are likely the result of early bypass transition in theassociated experiments (caused by roughness, particulate, or anotherunknown factor) when the boundary layer is still relatively thin andthe calculated most unstable frequency is therefore large.The region inwhich the secondmode is significantmaybedefined by

the Mach number at the boundary-layer edge, which in Fig. 12a ispresented at the measured point of transition as a function ofXTr∕XSW.Figure 12b presents the variation of the computed N factor at themeasured transition location as a function of edge Mach number.The maximum computed second-mode N factors occur forMe > 4.5,whereas small second-mode N factors are computed for Me < 3.9,which is consistent with the predictions ofMack [17] for the variation inthe strength of the second-mode instability with Mach number. Sincebluntness and resultant entropy layer swallowing length effectivelymediate the edge Mach number, this may be the effect by which thesecond mode is emphasized or deemphasized in the transition process.However, it does not indicate the alternate instability mechanismresponsible for transition in themoreblunt caseswith a lower edgeMachnumber. The systematic obliquemode analysis described previously forthe highest-Reynolds-number case at each bluntness does not showsignificant amplification, which excludes oblique modes as thedominant instability mechanism for the blunt cases. As bluntnessincreases and the calculated swallowing distance lengthens, thecomputed N factor at the experimentally observed transition onset

Distance [mm]

Fre

quen

cy [k

Hz]

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500Most Amplified Frequency Correlation f(x) = 0.65U

e/(2δ

99)

−α i [1

/m]

−40

−20

0

20

40

60

80

Fig. 9 LST contours of −αi for RN � 1.016 mm (2%) for p0 � 1400 psi.

Table 3 Characteristics of the most unstablemodes for cases where oblique modes dominate

RN, in. RN, mm s, m ψ , deg ω, kHz Nmax

0.40 10.16 0.400 6.97 572.5 0.410.50 12.70 0.047 0.92 670.0 0.010.60 15.24 0.046 0.79 702.5 0.04

6 Article in Advance / JEWELL AND KIMMEL

10−3

10−2

10−1

100

101

0

1

2

3

4

5

6

7

8

XT

B

/XSW

N fa

ctor

at X

TB

sharp (XT

B

/XSW

= ∞)

RN/R

B = 0.01

RN/R

B = 0.02

RN/R

B = 0.03

RN/R

B = 0.04

RN/R

B = 0.05

RN/R

B = 0.10

RN/R

B = 0.15

RN/R

B = 0.20

RN/R

B = 0.25

RN/R

B = 0.30

0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

Fig. 10 ComputedN factor at experimentally measured transition location.

0 500 1000 1500 2000 2500 3000 35000

1

2

3

4

5

6

7

8

Most amplified frequency Ω at XT

B

[kHz]

N fa

ctor

at X

TB

sharpR

N/R

B = 0.01

RN/R

B = 0.02

RN/R

B = 0.03

RN/R

B = 0.04

RN/R

B = 0.05

RN/R

B = 0.10

RN/R

B = 0.15

RN/R

B = 0.20

RN/R

B = 0.25

RN/R

B = 0.30

Fig. 11 TransitionN factor vs computed most amplified frequency at transition location.

10−3

10−2

10−1

100

101

2.5

3

3.5

4

4.5

5

XT

B

/XSW

Med

ge a

t XT

B

sharp (XT

B

/XSW

= ∞)

RN/R

B = 0.01

RN/R

B = 0.02

RN/R

B = 0.03

RN/R

B = 0.04

RN/R

B = 0.05

RN/R

B = 0.10

RN/R

B = 0.15

RN/R

B = 0.20

RN/R

B = 0.25

RN/R

B = 0.30

a)

2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

Medge

at XT

B

N fa

ctor

at X

TB

sharp (XT

B

/XSW

= ∞)

RN/R

B = 0.01

RN/R

B = 0.02

RN/R

B = 0.03

RN/R

B = 0.04

RN/R

B = 0.05

RN/R

B = 0.10

RN/R

B = 0.15

RN/R

B = 0.20

RN/R

B = 0.25

RN/R

B = 0.30

b)

Fig. 12 Edge Mach number effects: a) Mach number vs XTr∕XSW , and b) N factor vs Mach number.

Article in Advance / JEWELL AND KIMMEL 7

location drops below the level at which either the first or the secondmode would be expected to lead to transition [25,26]. As the dominantinstabilitymechanism for the transition reversal is neither the first nor thesecond mode, it is not explained by LST.

V. Conclusions

A strong trend in transition onset N factor for both nose-tipbluntness and swallowing length ratio is observed in the resultscomputed (see Fig. 10) from the complete set of Stetson Mach 6conditions. As bluntness increases and the calculated swallowingdistance lengthens, the computed N factor at the experimentallyobserved transition location drops below the level at which Mack’ssecond mode would be expected to lead to transition [17]. Theseresults indicate that the dominant instability mechanism for themost blunt cases is likely not the second mode, which is consistentwith recent blunt-cone results at different conditions. A systematicexamination of oblique modes for a subset of the cases alsoindicates that these are not the dominant instability mechanisms, sothe transition reversal is not explained by linear stability theory.Alternate instability mechanisms include transient growth or meanflow distortion from roughness, as well as entropy-layer instability.Based upon the computed second-mode amplification factors eN,

transition onset in the air force research laboratory Mach 6 high-Reynolds-number facility is estimated to correspond toN ≈ 7 for thesharp and nearly sharp cases. These amplification values are high ascompared to themore typical value ofN ≈ 5–6 usually characterizinga noisy tunnel. One partial explanation may be a mismatch betweenthe strongest freestream noise frequencies and the most unstablesecond-mode boundary-layer frequencies; however, no systematicvariation in the computed most unstable frequency at the observedtransition location is seen in the present dataset. Measurements offreestream and boundary-layer instabilities in the Mach 6 tunnel, aswell as further analysis of the oblique first modes, will be essential forfurther investigation of this potential effect.

Acknowledgments

This research was performed while J. S. Jewell held a NationalResearch Council Research Associateship Award at the U.S. AirForce Research Laboratory. The authors thank Steve Schneiderof Purdue University for sharing his private communicationwith Ken Stetson, which provided additional clarification andtabulated data for the results reported in K. F. Stetson’s 1983 paperon Mach 6 blunt-cone experiments; Eric Marineau of the ArnoldEngineering Development Center for helpful discussion; andMatthew Tufts for his comments. J. S. Jewell thanks Ross Wagnildof Sandia National Laboratories for his patient advice on the use ofthe STABL code.

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A. DufreneAssociate Editor

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