dns of hypersonic boundary layer

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Theoret. Comput. Fluid Dynamics (1998) 11: 4967

Theoretical and Computational

Springer-Verlag 1998

Fluid Dynamics

Direct Numerical Simulation of Hypersonic Boundary-LayerFlow on a Flared Cone1

C. David Pruett

Department of Mathematics, James Madison University,

Harrisonburg, VA 22807, U.S.A.

Chau-Lyan Chang

High Technology Corporation, Hampton, VA 23666, U.S.A.

Communicated by M. Y. Hussaini

Received 20 March 1997 and accepted 21 May 1997

Abstract. The forced transition of the boundary layer on an axisymmetric flared cone in Mach 6 flow

is simulated by the method of spatial direct numerical simulation (DNS). The full effects of the flared

afterbody are incorporated into the governing equations and boundary conditions; these effects include

nonzero streamwise surface curvature, adverse streamwise pressure gradient, and decreasing boundary-layer

edge Mach number. Transition is precipitated by periodic forcing at the computational inflow boundarywith perturbations derived from parabolized stability equation (PSE) methodology and based, in part, on

frequency spectra available from physical experiments. Significant qualitative differences are shown to exist

between the present results and those obtained previously for a cone without afterbody flare. In both cases,

the primary instability is of second-mode type; however, frequencies are much higher for the flared cone

because of the decrease in boundary-layer thickness in the flared region. Moreover, Goertler modes, which

are linearly stable for the straight cone, are unstable in regions of concave body flare. Reynolds stresses,

which peak near the critical layer for the straight cone, exhibit peaks close to the wall for the flared cone.

The cumulative effect appears to be that transition onset is shifted upstream for the flared cone. However, the

length of the transition zone may possibly be greater because of the seemingly more gradual nature of the

transition process on the flared cone.

1. Introduction

The computation documented in the recent papers of Pruett et al. (1995), Pruett and Chang (1995), and Pruett

and Zang (1995), represents a first attempt to simulate numerically laminarturbulent transition in a high-

speed boundary-layer flow for Reynolds numbers and configurations of engineering interest. Specifically,

their work examined the laminar breakdown of a Mach 8 boundary-layer flow on an axisymmetric sharp cone

in the absence of a streamwise pressure gradient. The computation, an approximate analog of the sharp-cone

experiment of Stetson et al. (1983), was motivated by the hope that, when experiment and computation

are both brought to bear on the same physical problem, the results will be more enlightening than when ei-

ther approach is used alone. In this spirit, in this paper we take a further step in the direction of configuration

1 The first author gratefully acknowledges the financial support of the Air Force Office of Scientific Research through Grant

F49620-95-1-0146 (monitored by Dr. Leoridas Sakell).

49


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50 C.D. Pruett and C.-L. Chang

DNS by simulating transitional hypersonic flow on a flared cone; again the choice of test case is motivated

by previous experimental investigations, which we discuss briefly below.

It is now well established both experimentally and computationally that the high-frequency second-

mode disturbances predicted by Mack (1984) of the basis of linear stability theory (LST) provide the

primary instability mechanism for transition to turbulence in high-speed boundary-layer flows. Specifically,

second-mode instability waves have been observed experimentally in the hypersonic boundary layers of

right circular cones by several experimentalists, among them Stetson and Kimmel (1992), who summarize a

body of work in their 1992 paper. That second-mode disturbances can lead to transition on a cone has been

verified computationally by the present authors (Pruett and Chang, 1995). Because of a generally stabilizing

trend as Mach number is increased (Mack, 1984), in high-speed boundary layers, the transition zone (i.e.,

the region between transition onset and fully turbulent flow) can be quite long. It is natural to wonder how

the results for straight cones translate to generalized axisymmetric forebodies. In particular, how are the

primary instability, the transition mechanism, the location of transition onset, and the length of the transition

zone altered by variations in parameters such as surface curvature?

Recently, several experiments have been conducted on cones in hypersonic wind tunnels to assess the

effects on boundary-layer stability and transition of streamwise surface curvature(Kimmel, 1993; Lachowicz

et al., 1996a,b), of wall cooling (Blanchard and Selby, 1996), and of angle of attack (Doggett et al., 1997). Of

these influences, surface curvature is of primary interest here. In general, the effects of streamwise surfacecurvature are both overt and subtle. The dominant effects of body curvature on transition are the induced

changes in the mean flow. Concave curvature, for example, produces a positive (adverse) streamwise pressure

gradient, a decrease in mean edge Mach number, an increase in mean wall temperature, and a decrease in

boundary-layer thickness. More subtle effects include Goertler instabilities, possible competition between

second-mode and Goertler instabilities, and the presence of additional curvature terms in the equations of

motion that must be considered in any theoretical or computational context.

Of the experiments mentioned above, Kimmels (1993) was strictly a transition experiment, in which the

primary goal was determination of the effects of parameter variation on the location of transition onset and

the length of the transition region. Specifically, Kimmel examined the boundary-layer flow on five different

conical configurations at several values of freestream unit Reynolds number. For each configuration, an

ogive or flared afterbody was appended to a (right circular) conical forebody, which remained the same

for all configurations. By varying the geometry of the afterbody, a range of nearly linear favorable (forogival afterbodies) and adverse (for flared afterbodies) pressure gradients was obtained. Transition locations

were determined solely on the basis of surface heat transfer. In general, Kimmel found that although

favorable pressure gradients delay transition, and adverse pressure gradients promote transition, favorable

pressure gradients caused a shorter transition zone. For high-speed boundary-layer flows, the transition

zone can be quite long, and the effect on the length of the transition zone of an adverse pressure gradient

was inconclusive. Kimmels transition experiments were conducted in the Mach 8 wind tunnel at the Arnold

Engineering Development Center, a conventional facility with relatively high levels of freestream turbulence.

In contrast, the stability and (natural) transition experiments of Lachowicz et al. (1996a,b), Doggett et

al. (1997), and Blanchard and Selby (1996) are unique in that they were conducted in a quiet tunnel,

the Mach 6 pilot facility at the NASA Langley Research Center, which was designed specifically to reduce

freestream turbulenc significantly. The characteristics of the facility itself are documented in Blanchard et al.

(1997). The experiment of Lachowicz et al. (1996a,b), the first conducted after the facility study, examined

boundary-layer stability on an axisymmetric flared cone with a nominally adiabatic wall. In this experimentthe goal was to identify the primary mechanism of transition as well as the transition location and the

extent of the transition region. Transition onset was estimated from surface temperature measurements, for

which the model was instrumented with thermocouples. Run times were sufficiently long so that the model

surface was considered to be in thermal equilibrium. Remarkably different surface temperature distributions

were obtained dependent upon whether or not the tunnel was operating in its quiet mode (with tunnel-wall

boundary-layer bleed valves open). (See Figure 2 of Lachowicz et al. (1996b).) Boundary-layer total-

temperature and mass-flux profiles and disturbance spectra were also obtained by hot-wire anemometry. In

addition, a few images were obtained by schlieren interferometry. The experiment involved a single flared-

cone configuration, for which the flared afterbody produced a nearly linear adverse pressure gradient, the

effects of which were of primary concern. The body flare was also essential to the experiment in another

aspectwithout it transition could not have been achieved within the quiet core of the tunnel. The later


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Direct Numerical Simulation of Hypersonic Boundary-Layer Flow on a Flared Cone 51

(and complementary) experiments of Doggett et al. (1997) and Blanchard and Selby (1996) expanded on

the initial study of Lachowicz et al. (1996a,b) by examining the effects on stability of angle-of-attack and

wall cooling, respectively. All three experiments involved flared-cone models whose boundary layers were

characterized by adverse pressure gradients. To complement this experimental work, theoretical studies were

conducted by Balakumar and Malik (1994), who used LST to predict transition locations on the basis of the

integrated growth rates (N-factors) of selected disturbances. For the flared-cone configuration of Lachowiczet al. (1996a,b), Balakumar and Malik (1994) predicted that second-mode disturbances with frequencies in

the neighborhood of 230 kHz were most likely to lead to transition. The interested reader is referred to the

recent paper of Wilkinson (1997) that reviews the NASA Langley quiet-tunnel experiments.

For our present work, a computational test case has been designed based on the configurationa flared

cone in nominal thermal equilibriumand flow conditions of the experimental investigation of Lachowicz

et al. (1996a,b). This choice of test case affords a rare opportunity to examine a single high-speed transi-

tional boundary-layer flow by theoretical (Balakumar and Malik, 1994), experimental (Lachowicz et al.,

1996a,b), and computational means. We confess at the outset that such comparisons are never exact, and

the reader should be cautioned as to the limitations of each viewpoint. The N-factor analysis attempts to

predict transition, which is a fundamentally nonlinear phenomenon, on the basis of linear theory. The lim-

itations of the experiment include a focus only on the primary instability mechanism (i.e., no information

is available regarding three dimensionality of the flow) and a lack of specificity in regard to the disturbanceenvironment. Moreover, as the first of the three quiet-tunnel experiments, the experiment of Lachowicz et al.

(1996a,b) experienced some problems with the signal-to-noise ratio of the hot-wire anemometer, and there

was difficulty in calibrating data collected very near the model surface. Whereas the physical experiment

examined natural transition, for reasons of computational efficiency, the numerical experiment examines

forced transition, for which the boundary layer is seeded with disturbances of specific rather than random fre-

quencies. Unfortunately, the resolution requirements for the simulation of natural transition rendered such a

computation impractical at the time due to the computational resources available. Moreover, also because of

practical limitations in computer resources, we simulate only the early stages of laminarturbulent transition

on the flared cone; the simulation of end-stage transition is not attempted. Consequently, the reader should

consider these theoretical, experimental, and computational results as different perspectives on similar, but

not identical, objects. Despite the limitations and inconsistencies, our hope is that the composite picture of

transition on a flared cone is clearer than that of each of the individual views.To summarize, our motivations are to advance the state-of-the-art in configuration DNS and to initiate a

computational study of the effects of surface curvature on transitional high-speed flows. To these ends, we

adapt the approach and algorithm of Pruett et al. (1995). The effects of streamwise surface curvature, mean

streamwise pressure gradient, and varying boundary-layer edge conditions are incorporated fully into the

governing equations and boundary conditions. The particular test case affords an opportunity to compare

the present results not only with theoretical and experimental results but also with computational results of

transitional flow on a straight cone (Pruett and Chang, 1995; Pruett and Zang, 1995).

In the next section we briefly discuss the governing equations for flow along axisymmetric bodies with

streamwise curvature. The computational test case is summarized in Section 3. Section 4 addresses the

numerical methodology, in general, and the changes to the approach of Pruett et al. (1995) necessitated

by the flared-cone geometry, in particular. In Section 5 we exploit nonlinear parabolized stability equation

(PSE) methodology to develop a viable transition mechanism based on second-mode primary instability.

Results are presented in Section 6, and, where appropriate, comparisons are made with previous theoretical,experimental, or computational results. Conclusions are offered in Section 7.

2. Governing Equations

We consider the compressible flow of air, for which p, , T, , and denote its pressure, density, temper-ature, viscosity, and thermal conductivity, respectively. We assume that the flow is exactly governed by the

compressible NavierStokes equations (CNSE), which can be found in many references on fluid mechanics.

If we restrict attention to axisymmetric bodies and define x as the surface arc length, as the azimuthalangle, and z as the coordinate normal to the wall, then in body-fitted coordinates, the CNSE assume the


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form given in Pruett et al. (1995), where u, v, and w denote the velocity components in the streamwise,azimuthal, and wall-normal directions, respectively. We note that the equations given in Pruett et al. include

both transverse and streamwise curvature terms and incorporate the notation R(x) and (x) to denote thebody radius and the angle of the surface tangent relative to the axis, respectively, as functions of arc length.

(See Figure 2 of Pruett et al. (1995).) Four dimensionless parameters arise from nondimensionalization

of the governing equations: Mach number M, Reynolds number Re, Prandtl number Pr, and the ratio ofspecific heats, . Throughout this work, Pr and are assumed to be constants with values of 0.72 and 1.4,respectively. Finally, the dependency of viscosity on temperature is modeled by Sutherlands law, and = /Pr.

3. Test Case

The experiment of Lachowicz et al. (1996a,b), on which the numerical computation is based, examines

Mach 6 flow on a 20-in-long flared-cone model at a (nominal) zero angle of attack. The model consists

of a 5 half-angle axisymmetric sharp cone to which a flared afterbody is appended. The afterbody, which

begins 10 axial inches from the apex of the cone, is defined by a circular-arc flare with a constant radius of

curvature of 93.07 in; a curvature discontinuity exists at the flare interface. The nominal preshock freestream

conditions, which we use for the numerical experiment, are

M = 6.0,T

= 98.78R(Tt = 810.0R),p

= 11.86 psf(pt = 18,720 psf),

Re1 = 2,728,087 ft1,

(1)

where Re1 is the freestream unit Reynolds number. Throughout this work, asterisks denote dimensionalquantities, and the subscripts , t, and e denote freestream, total, and boundary-layer edge conditions,respectively. The actual freestream conditions as reported by Lachowicz et al. (1996a,b), which differ

slightly from the nominal values above, are M = 5.93 and Re

1 = 2.82 106

per foot. To facilitatelater comparisons with the experimental results of Lachowicz et al. (1996a,b) and the theoretical results of

Balakumar and Malik (1994), we define as follows the root Reynolds number based on freestream values

of density, velocity, and viscosity:

ReL =Re1x

=

ux

. (2)

A precise thermal boundary condition at the model surface cannot be ascertained on the basis of information

reported about the experiment. Run times were long, and it is believed that the wall behaved approximately

adiabatically under equilibrium conditions. However, the model was thin-skinned and uninsulated, and

mounting hardware prevented the attainment of thermal equilibrium near the models base. Given the uncer-

tainty as to the experimental thermal condition and the fact that we consider only the early stages of transition

for the DNS, for which the mean flow has laminar characteristics, we have adopted the boundary conditionof Pruett and Chang (1995). The use of this thermal condition is advantageous in that it allows comparison of

simulated flared-cone and straight-cone results for the same boundary conditions. The boundary conditions

are specified precisely later.

4. Methodology

Roughly speaking, spatial DNS is a four-step process. In the first step a laminar and steady base state is

defined, whose stability is to be investigated. The base state is then subject to time-dependent disturbances,

which are imposed at or near the computational inflow boundary. The fluctuations are usually obtained from

linear or nonlinear stability theory; specification of the disturbances comprises the second step. The third


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step involves computing the evolution of the disturbances by numerically solving the three-dimensional and

unsteady CNSE. The temporally and spatially evolving numerical solution is sampled and stored periodically

in time for the subsequent postprocessing and analysis that constitute the final step. In this section we address

steps one and three; steps two and four are addressed in the next two sections, respectively.

To obtain the laminar base state for their stability calculations, Balakumar and Malik (1994) computed

the flow over the flared-cone configuration by both Euler and NavierStokes techniques. The NavierStokes

solution was sensitive to the choice of limiter (dissipation) and required very high resolution within the

wall layer to provide accurate first and second wall-normal derivatives, which were needed for accurate

linear-stability analysis. They found that the pressure distributions computed by the two methods agreed

very well except near the tip of the cone. Moreover, N-factors computed by the two methods agreed to

within approximately 10%. Because the difference in N-factors was small, for computational efficiency,

they recommended an Euler/boundary-layer approach over the solution of the full CNSE.

For the present work, we also favor a boundary-layer equation approach. Specifically, we exploit the

spectrally accurate boundary-layer code of Pruett and Streett (1991) with the modification for varying

edge conditions proposed by Pruett (1993). The postshock edge pressure distribution pe (x) needed by the

boundary-layer code is obtained from the NavierStokes solution of Balakumar and Malik (1994). In the

vicinity of the tip (a region excluded in the DNS calculation), the pressure distribution is modified by lin-

early extrapolating upstream from downstream values. For self-consistency, all other boundary-layer edgeconditions are inferred from the pressure, based on the assumption of isentropic postshock flow. Although

this assumption is not strictly valid in the flared region because of the gradually increasing shock angle, the

shock is highly oblique and weak, and errors committed by assuming isentropic postshock flow are small, as

will be confirmed shortly. Figure 1 presents the radius R of the cone as a function of surface arc length x.The edge Mach number, edge temperature, and edge pressure distributions are also provided as functions of

x. Note that the edge Mach number decreases almost linearly from nearly 5.6 at the beginning of the flareto 4.91 at the end of the model. The close agreement between our present end-of-model Mach number and

that reported by Balakumar and Malik (1994) (who observe a Mach number of 4.93 at the end of the flare)

serves to validate the approximation of isentropic postshock flow. In contrast to the edge Mach number,

edge pressure and temperature both increase significantly and nearly linearly in the flare region. The stream-

wise evolution of the laminar boundary-layer thickness is shown in Figure 2. Although for experimentalists

the boundary-layer thickness

is the conventional measure of boundary-layer growth, for computational

Figure 1. Radius and boundary-layer edge conditions versus surface arc length x for axisymmetric flared-cone model in Mach 6 flow.

Units for temperature and pressure are degrees Rankine and psf, respectively.


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Figure 2. Computationally (dashed line) and experimentally

(symbols) derived values of boundary-layer thickness versus

surface arc length for flared-cone model. For reference, computa-

tionally derived displacement thickness 1

(solid line) is also pro-

vided. Boundary-layer growth reverses just downstream of body

flare, which begins at thelocation indicatedby vertical dashedline.

purposes the displacement thickness 1 , an integral quantity given for axisymmetric bodies in White (1974)(or in Equation (13) of Pruett et al. (1995)), is preferred. Both quantities are presented in Figure 2. For the

Mach-number range of the flare region, is approximately 1.2 1 . For comparison, measured values of taken from the essentially laminar-flow region of the experiment of Lachowicz et al. (1996a) are alsopresented. The experimental and computational values agree remarkably well; both manifest thinning of

the boundary layer in the flare region. Dimensional temperature and streamwise velocity profiles for an

adiabatic wall are shown in Figure 3. The profiles, which correspond to axial station x = 1.083 ft (13.0in) in the flare region, are virtually indistinguishable from the adiabatic-wall results shown in Figure 13b of

Balakumar and Malik (1994), obtained at the same station. It is also instructive to view the same profiles in

normalized units, as shown in Figure 4. For this figure, lengths have been normalized by the local boundary-

layer displacement thickness 1 and temperature and velocity have been normalized by their respectiveedge values. Note that the velocity distribution is nearly linear across the boundary layer, a characteristic of

high-speed boundary layers. The shoulder of the boundary layer lies approximately 1.25 1 from the wall.If we bear in mind that second-mode disturbances typically travel at 9095% of the boundary-layer edge

velocity, it can be inferred from Figure 4 that the critical layer (where u and disturbance phase velocity areequal) lies approximately one boundary-layer thickness from the wall. Also note that, because of frictionalheating, the wall temperature for this high-speed flow is nearly six times that at the boundary layer edge.

The boundary-layer solution is interpolated onto a DNS grid using a spectrally accurate interpolation

code, which extrapolates outside the boundary-layer region by solving the continuity equation exactly in the

asymptotic limit as z . The asymptotic outer solution is matched to the inner solution at the boundary-layer edge. Special attention is paid to computing the wall-normal velocity accurately, for which we use

the method of Pruett (1993). A typical wall-normal velocity distribution versus z is shown in Figure 5. Foraxisymmetric cones, the wall-normal velocity is negative, which implies that the far-field computational

boundary should be treated as an inflow boundary.

Figure 3. Dimensional temperature and streamwise velocity distributions versus wall-normal coordinate z at axial station x = 13 in.

Profiles are virtually indistinguishable from those shown in Figure 13 of Balakumar and Malik (1994).


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Figure 4. Normalized temperature and streamwise velocity distributions versus locally normalized wall-normal coordinate z at axial

station x = 13 in.

The special care used to match inner andouter solutions consistentlyis necessaryto prevent discontinuities

in higher derivatives of the basic state, which would contaminate the subsequent stability analysis. As a

consistency check, we typically compute the residuals of the time-independent and dimensionless CNSE byusing the interpolated boundary-layer solution as an initial state. Residuals that are quite small, say of order

104, as shown in Figure 6, indicate that the boundary-layer solution closely approximates the exact solution.

Because the residuals reflect the sum of approximation errors, interpolation errors, numerical errors, and

blunders, small residual errors provide a stringent test of self-consistency among the various numerical tools

exploited by DNS. Indeed, on several occasions we have uncovered and corrected algorithmic bugs simply

by examining the steady-state residuals.

For the DNS calculation, we adapt the techniques of Pruett et al. (1995). Their fully explicit collo-

cation scheme exploits the third-order low-storage RungeKutta scheme of Williamson (1980) for time

advancement and both spectral and central compact-difference methods for spatial discretization. Specifi-

cally, spectral-collocation methods (Canuto et al., 1988) are used in the azimuthal direction, in which the

flow is periodic, whereas the sixth-order compact-difference methods of Lele (1992) are exploited in the

aperiodic streamwise and wall-normal directions. Modest filtering is used to correct the innate tendencyof central-difference approximations toward oddeven decoupling of the solution in the far field whenever

resolution is marginal. Filtering also significantly reduces numerical reflections at the outflow boundary.

A discussion of the necessity for and implementation of filtering can be found in Pruett et al. (1995). With re-

Figure 5. Normalized wall-normal velocity versus locally nor-

malized wall-normal coordinate z at axial station x = 13 in.

Figure6. Residuals of governingequationsalong inflow boundary

of computational domain of DNS.


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gard to the spectral-collocation scheme used in the azimuthal direction, we implement dealiasing according

to the 12

-rule to suppress high-frequency oscillations that could lead to numerical instability. As in Pruett et

al. (1995), we formulate the energy equation in terms of the pressure, and, for computational efficiency, we

enforce symmetry with respect to the plane = 0.Minor modifications to the scheme of Pruett et al. are necessitated by particulars of the current flared-

cone problem. Because boundary-layer edge conditions vary for the current problem, flow quantities are

nondimensionalized by the preshock freestream temperature, velocity, and density, etc., given or implied

in (1). Previously, flow quantities were scaled by appropriate postshock values at the boundary-layer edge,

which were presumed constant along the axisymmetric cone without flare. For both the work of Pruett et

al. (1995) and the current work, lengths are scaled by the boundary-layer displacement thickness 1 at thecomputational inflow boundary xin. Previously, R(x) was computed internally and (x) was constant; forthe present problem, both quantities are computed externally by the boundary-layer code and read as input

to the DNS algorithm. As was done in Pruett et al., grid spacing is uniform in the x direction, and, in thewall-normal direction, grid points are tightly clustered near the wall and near the critical layer. For the present

problem, the DNS computation includes only a region of flow aft of the flare, where the boundary-layer

thickness diminishes. (Refer to Figure 2.) This is in contrast to the growing boundary layer along the cone

without flare. In either case, to account for the variation in boundary-layer thickness, we exploit a mapping

of the formz = f(x). (3)

However, for the present problem, the mapping function is linear and is given by

f(x) = 1 0.05x xoutxin xout

, (4)

where xout is the coordinate of the computational outflow boundary. For all practical purposes, = 1corresponds to a distance one (local) boundary-layer displacement thickness from the surface.

The boundary conditions for the present problem are taken without modification from Pruett et al. (1995).

In brief, all flow variables are specified along the inflow boundary, a buffer domain approach (Streett and

Macaraeg, 1989/90) is exploited at the outflow boundary, and Thompson (1987) conditions are adapted

along the far-field boundary = max. At the wall, all velocity components vanish; the thermal boundary

condition may be either adiabatic or isothermal. As mentioned previously, to facilitate comparison with thestraight-cone results of Pruett and Chang (1995), we adopt their hybrid thermal boundary condition whereby

wall temperature is held fixed at its laminar adiabatic-wall value. Density at the wall is computed directly

from the governing equations.

5. Primary Instability and Transition Mechanism

As mentioned previously, an advantage in considering the present flared-cone configuration is that both

theoretical and experimental data regarding the primary instability are available. As observed by Balakumar

and Malik (1994), for the flared cone, the gradual reduction in boundary-layer thickness in the flare region

allows unstable disturbances of fixed frequency to experience amplification over longer regions than for

the straight cone, whose boundary layer continuously thickens. Their linear stability analyses predicted thattransition was likely to result from a primary second-mode disturbance of frequency 230 kHz, which attained

the largest integrated growth rate (N-factor). Remarkably, the frequency data obtained by Lachowicz et al.

(1996a,b) using hot-wire anemometry are in very close agreement with the predictions of Balakumar and

Malik (1994). Specifically, the experimental results show that the most unstable frequency F in the flareregion lies in the range of 218228 kHz. Unfortunately, the signal-to-noise levels of the data of Lachowicz

et al. (1996a,b) are too low in the region ahead of the flare to provide useful information there.

As mentioned in the Introduction, we believe that well-resolved DNS of natural transition is currently

impractical for existing supercomputers; consequently, we limit our consideration to forced transition with

monochromatic forcing. Ultimately, transition is a three-dimensional and nonlinear process, and additional

information beyond the frequency content of the primary disturbance is needed to identify a viable transition

mechanism. Specifically, information regarding disturbance three dimensionality is essential. To date, we


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know of no hypersonic stability or transition experiment in which the disturbance environment is well

defined, with the sole exception of the stability experiment of Cavalieri (1995) in which a glow-discharge

device was used to seed the boundary layer of a cone with linear two- and three-dimensional disturbances of

specified frequencies. Unfortunately, the results of this pioneering experiment were not available at the time

of our computation. Neither is such information provided by the experiment of Lachowicz et al. (1996a,b)

or the theoretical study of Balakumar and Malik (1994), on which the present numerical experiment is

based. Therefore, to construct a viable transition scenario, we must turn to numerical experimentation, for

which we exploit parabolized stability equation (PSE) methodology as adapted to compressible flows by

Chang et al. (1991). Unlike LST, nonlinear PSE methodology treats both meanflow nonparallelism and

moderately nonlinear wave interactions. In the PSE approach, disturbances are Fourier-transformed in the

periodic temporal and azimuthal dimensions. Each harmonic is associated with an integer pair (l, m), wherethe integers l and m identify the harmonic with respect to the fundamental values of temporal frequency and azimuthal wave number , respectively. Harmonics (l, m) are axisymmetric ifm = 0, oblique (helical)ifm = 0, and stationary if l = 0. The Fourier component (0,0) represents the nonlinear distortion of thebase state. Because of the azimuthal periodicity of the flow on a cone, the wavenumber of helical modes

must satisfy R = n, where n is an integer. For conical bodies, this implies that necessarily varieswith x. In the flared region, modes of zero temporal frequency are referred to as Goertler modes and are

associated with concave streamwise surface curvature. We mention in passing that, because computationsare performed in Fourier space in PSE methodology, subharmonics of fundamental disturbances arise only

if explicitly introduced. In contrast, in physical experiments or DNS, subharmonic frequencies may emerge

spontaneously from low-level physical or numerical noise.

Parameter studies based on nonlinear PSE methodology confirmed, as expected, that a fundamental (1,

0) second-mode disturbance of circular frequency = 2F was highly unstable for frequency F = 230kHz. Moreover, helical modes of the same frequency were also unstable for a wide range of values ofn. In particular, n = R = 15 was shown to lead to highly unstable disturbances. On the basis of thisstudy, a transition scenario was constructed based on the nonlinear interactions of a triad of finite-amplitude

Fourier harmonics (1, 0), (1, 1), and (1, -1) (and their complex conjugates), where all three components

have the same rms amplitude of 1% in their maximum temperature fluctuation (when averaged in time

and in the azimuthal dimension). The effect of summing the equal and opposite finite-amplitude helical

components (1, 1) and (1, -1) is to produce a spectrum of three-dimensional disturbances, including the(0, 2) harmonic, which is known to promote laminar breakdown. It should be noted that the axisymmetric

second-mode (1, 0) component alone is insufficient to trigger transition, an inherently three-dimensional

phenomenon. The Fourier amplitudes (not rms) of selected harmonics from the PSE calculation are shown

in Figure 7. The PSE computation spanned the region x = 0.58 ft (ReL = 1258), ahead of the flare, tox = 1.4 ft (ReL = 1954) in the flare zone. At the inflow boundary, the calculation was initialized withthe disturbance triad defined above. The initial amplitudes of the axisymmetric and helical components

were chosen arbitrarily to establish transition on the flared afterbody. All other harmonics shown in the

figure emerged spontaneously from the nonlinear interactions among the fundamental triad. Figure 7 shows

saturation of the fundamental components of the disturbance near x = 1.1 ft occurring simultaneously with

Figure 7. Streamwise evolution of temperature maxima of se-

lectedharmonicsof primary disturbancesobtainedby PSE method.

First andsecond indicesidentify temporal andspanwise harmonics

with respect to fundamental temporal frequency and spanwise

wave number , respectively, where R = 15.


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Figure 8. Growth rates of Goertler modes versus fundamental az-

imuthal wavenumber n = R at ReL = 1630 (x = 0.974 ft)

obtained by linear PSEmethodology. (Growth rates arenondimen-

sionalized according to Mack using local boundary-layer length

scale.)

significant distortion of the mean flow (0, 0) and rapid growth of higher harmonics to significant amplitudes.

These conditions indicate that the flow is transitioning from a laminar to a turbulent state.

Of particular interest here are the stationary (Goertler) modes (0, m). For the flared cone with an adverse

pressure gradient, the PSE method predicts linearly unstable Goertler modes for a broad range of valuesm. Specifically, as shown in Figure 8, Goertler modes are highly unstable for fundamental azimuthalwavenumbers n = R ranging from less than 10 to more than 80. In contrast, Goertler modes are linearlystable for the cone without flare. Whereas, stationary harmonics are indeed observed for both the straight

cone and the flared cone in the region ahead of the flare, they are not eigenfunctions of the flow and thus owe

their existence to other instability mechanisms. Specifically, such harmonics emerge initially due to linear

spatial transient effects, but their growth is sustained only through nonlinear interactions (as observed for

the straight cone by Pruett and Chang (1995)). In contrast, true Goertler modes are able to extract energy

directly from the mean flow. Given (potentially) different energy transfer mechanisms, significant qualitative

differences in the respective transition processes of straight and flared cones should not be surprising.

6. Results

The computational domain of the DNS encompassed a section along the body downstream of the beginning

of the flare. Specifically, the domain spanned the region

xin = 0.9 x 1.2 = xoutft,

0 2

n(n = 15),

0 7.5.

(5)

In terms ofReL, the computational domain of the DNS spanned 1567 ReL 1809. To facilitatecomparisons of the DNS results with those of Lachowicz et al. (1996a,b), Figure 9 shows the dependence

ofReL on x. The length scale (xin) = 0.00439 in.

The time-periodic inflow condition at xin = 0.9 was derived from the PSE results at the correspondingstation. At this station, the fundamental components ((1, 0), (1, 1), and (1, -1)) of the disturbance had attained

relatively large amplitudes, and numerous harmonics of the disturbance showed significant amplitudes. All

significant harmonics were incorporated into the DNS inflow condition. Relative to the wavelength of

the fundamental components of the disturbance, the computational domain encompassed approximately

26 wavelengths in streamwise extent. The resolution of the computational grid was 2304 18 128 inthe streamwise, azimuthal, and wall-normal directions, respectively. At this streamwise grid density, each

fundamental wavelength was resolved by approximately 90 grid points, sufficient, according to Pruett et

al. (1995), to resolve at least the first six streamwise harmonics. The azimuthal resolution, significantly

less than that of Pruett et al. (1995), was sufficient only for the initial stages of laminar breakdown, and

was dictated largely by available computational resources. Consequently, the simulation was halted prior to

endstage transition.


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Figure 9. Root Reynolds numberversussurfacearc lengthfor flow

conditions of test case. DNS computation spans 1567 ReL

1809, as indicated by vertical dashed lines.

Figure 10. Comparison of DNS and PSE results for selected har-

monics.

The DNS computation was performed on the Waterways Experiment Station (WES) Cray C90 at Vicks-

burg, Mississippi. Approximately 250 hours of single processor time and 115 megawords of main memory

were required for the computation. The code is optimized to run in parallel, and during the production

runs, we typically requested three CPUs. In physical terms, the computation proceeded for 30 periods of

oscillation at the fundamental frequency. The dominant instability waves were observed to propagate down-

stream at approximately 90% of the boundary-layer edge velocity; consequently, the physical computation

time was approximately that necessary for the fundamental instability wave to complete one traverse of the

computational domain.

During the computation, the numerical solution was periodically written to output and archived at a

rate of 64 samples per period of the fundamental frequency. These data were later postprocessed to extract

quantitative and qualitative information about the evolution of the flow. Most of the results to follow were

obtained by Fourier time-series analysis of the final period (2930) of the computation.

Figure 10, obtained by Fourier transforms in the periodic temporal and azimuthal dimensions, displays

the streamwise evolution of the amplitudes of the temperature fluctuation of selected harmonics of the DNS

calculation. The corresponding PSE results are also provided for comparison. The DNS and PSE results

agree well for approximately the first third of the computational domain of the DNS. Downstream of the

first third, the DNS and PSE results gradually diverge, but the methods continue to indicate agreement in

the qualitative behavior of the selected harmonics. Downstream ofx = 1.15 ft, some transient effects ofthe leading wavefront remain; here the DNS results cannot be considered fully developed and should be

disregarded.

Figures 11 and 12 display the structure (amplitude envelope) of the velocity and temperature components,

respectively, of selected harmonicsatx = 1.05 ft, the midpoint of the computational domain of the DNS. For

comparison, PSE results are also presented. In general, the largest fluctuation amplitude in the temperatureis experienced at the critical layer, whereas, for the dominant harmonics, velocity fluctuations are strongest

nearer to thewall.The agreementbetween DNSand PSEmethodscan be considered good only in a qualitative

sense. The magnitude of the differences comes somewhat as a surprise, given the closer agreement between

PSE and DNS obtained by Pruett and Chang (1995) for the cone without flare. We have considered a number

of possible explanations, none of them satisfactory. Rather than to speculate, we prefer to admit simply that

the origin of the disagreement remains unknown. Whatever the explanation, it should be kept in mind that

the DNS is initiated well into the transition process, in a region of moderately nonlinear interactions not far

upstream of saturation of the fundamental. That PSE and DNS agree even moderately well at this relatively

late stage is testament to the potential of PSE as a transition predictor.

Figure 13 displays the streamwise evolution of various mean quantities computed by averaging both

temporally and spatially (over the azimuthal coordinate). Here, where used, overbars denote averaged


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Figure 11. Amplitudes of velocity fluctuations of selected harmonics versus scaled wall-normal coordinate at x = 1.05 ft.

Figure 12. Amplitudes of temperature fluctuations of selected harmonics versus scaled wall-normal coordinate at x = 1.05 ft.


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Figure 13. Streamwise evolution of selected mean quantities.

quantities. For comparison, the corresponding quantities are shown also for the undisturbed initial laminar

state. Clear indications of transitioning flow are evident: principally a dramatic drop in the shape factor

H= 1/

2 below its laminar value (where

2 is the boundary-layer momentum thickness). We note that the

laminar shape factor, which is nearly constant for the cone without flare (Pruett and Zang, 1995), diminishes

with x in the flared region for the present configuration because of the decrease in Mach number. At thisstage of the transition process, the decrease inHis due almost entirely to an increase in momentum thickness2 rather than to a decrease in displacement thickness

1 . Near the end of the computational domain, the

skin-friction coefficient Cf has increased approximately 25% over its laminar value, where

Cf =2

walleu

2e,

wall = u

z|z=0.

(6)

However, at the same station, the coefficient Ch of thermal stress at the wall has doubled. Taken together,the plots for Cf and Ch call attention to an ambiguity in defining the location x

tr of transition onset for

high-speed flows. On the basis of minimum wall shear,xtr 0.95 ft. On the basis of minimum heat transfer,transition onset apparently occurs upstream of the computational inflow boundary of the DNS.

Figure 14 presents the streamwise evolution of the principal components of the Reynolds stress tensor,

namely uu, vv, and ww, where primes denote fluctuations about the mean state. The streamwiseand wall-normal stresses are observed to be of nearly equal magnitudes and peaked quite close to the wall.

Similarly, turbulent Mach number Mt and (density-weighted) turbulent kineetic energy K(Figure 15) also


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Figure 14. Streamwise evolution of principal components of Reynolds stress.

Figure 15. Streamwise evolution of turbulent Mach number, turbulent kinetic energy, and rms density fluctuation, each averaged

temporally and azimuthally.

exhibit peaks close to the wall. (Turbulent kinetic energy is definedas one-half thetrace of theReynolds stress

tensor.) The same figure presents evolution of the rms fluctuation in density. Large rms density fluctuations,

as high as 30% of the reference density, are observed at the critical layer and also at the wall.

6.1. Comparisons with the Experiment of Lachowicz et al.

Because the paths to transition differ fundamentally for the computational and physical experiments (i.e.,

forced versus natural transition), caution must be exercised in drawing comparisons and contrasts. However,

a few quantitative and qualitative comparisons can be attempted. First, as discussed previously (Figure 2),

the experimentally and computationally derived valued of boundary-layer thickness compare very well in


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the essentially laminar region. Second, in Figure 9 of Lachowicz et al. (1996a) rms mass-flux and total-

temperature fluctuation maxima are presented for a station that lies in the transition zone. Mass-flux spectra

through the boundary layer are also shown in Figure 12 of that paper. For comparison, we present Figure 16,

which shows the streamwise evolution of both the rawand normalized rms mass-flux fluctuations. The shapes

of the computationally and experimentally derived mass-flux profiles are remarkably similar, although the

computational results attain higher amplitudes (perhaps because energy in the physical flow is distributed

over a range of frequencies). Both show peaks at the critical layer, one boundary-layer thickness from the

surface. Because of hot-wire calibration difficulties near the wall, the experimental results are given only

for 23 z. (The reader is cautioned that the variable has a different usage in this paper from its usage in

Lachowicz et al. (1996a), for which 6.) Third, in Figure 9 of Lachowicz et al. (1996b), disturbancesof frequencies in the 200 kHz range are shown to grow linearly over the region 1880 ReL 2050,in close agreement with LST, and to saturate downstream of ReL = 2050. In particular, the dimensionalmass-flux growth rate of the disturbance of 234 kHz is calculated to be 12.9 per foot, compared with a rate

of 13.9 predicted by LST. For comparison, Figure 17 presents the streamwise evolution of the (1, 0) com-

ponent of the fundamental disturbance obtained by PSE methodology. The growth is approximately linear

in the region immediately downstream of the body flare; however, due to a relatively large initial amplitude,

saturation begins considerably further upstream than in the experiment, at approximately ReL = 1700. The

average growth rate at the beginning of the flare region is approximately 9.2 per foot, somewhat lower thaneither the experimental and theoretical values, presumably because nonlinear effects are already significant

at the higher amplitude. Fourth, Figure 24 of Lachowicz et al. (1996a) shows significant energy in the first

harmonic (430 kHz) of the fundamental disturbance (230 kHz) at a station near the end of the model. At this

location the amplitude of the first harmonic is roughly one order of magnitude below that of the fundamental.

Similarly, Figure 10 of the present paper shows considerable energy in the first axisymmetric harmonic (2, 0).

Figure 16. Streamwise evolution of raw and normalized rms mass-flux fluctuations. For legend, see Figure 14.

Figure 17. Maximum amplitude versus ReL of temperature of

fundamental (1, 0) component of disturbance as obtained by PSE

(dashed line) and DNS (solid line). Location corresponding to be-

ginning of flared afterbody identified by the vertical dashed line.


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Figure 18. Computational flow image of instantaneous wall-normal density gradient showing rope-like appearance of dominant

instability waves.

The ratio of the amplitudes of the fundamental to the first harmonic at the location of saturation of the

fundamental is about 5:1. (However, as cautioned by Wilkinson (personal communication), the frequency

of the first harmonic lies considerably outside the bandwidth (350 kHz) of the hot-wire anemometer in the

experiment; thus, amplitude data at this frequency cannot be considered accurate.)Finally, we call attention to Figure 3 of Lachowicz et al. (1996a), a schlieren photograph that shows

a wave-like appearance at the boundary-layer edge in the flared-cone experiment. Similar structures have

been observed in hypersonic boundary-layer flows by several other experimentalists and have been referred

to previously as rope-like waves. See, for example, the excellent review by Smith (1994). Computational

flow imaging (CFI) of the present numerical experiment also clearly reveals instability waves of rope-like

appearance (Figure 18). To approximate crudely the physics involved in schlieren imagery, the instantaneous

wall-normal density gradient has been averaged over the azimuthal coordinate and projected as a two-

dimensional image in greyscale. The numerically derived image is qualitatively similar in appearance to the

actual schlieren image obtained by Lachowicz et al. (1996a). In particular, the wavelength of the instability

is very nearly twice the boundary-layer thickness, in keeping with the nature of second-mode disturbances

and the observations of Lachowicz et al. (1996a).

6.2. Comparisons with the Computations of Pruett and Coworkers

It is well known from experience with incompressible and low-speed boundary-layer flows that an adverse

pressure gradient destabilizes Goertler modes, and Goertler instability was anticipated for the present flared-

cone test case. This suspicion was verified by PSE methodology; Goertler modes were indeed found to be

linearly unstable for a broad range ofn = R (Figure 8). In contrast, Goertler modes are linearly stable forthe cone without flare and grow only through nonlinear interactions. As shown in Figure 11, the amplitude

of the streamwise velocity fluctuation of the principal Goertler harmonic (0, 1) is strongly peaked close to

the wall. Surprisingly, the fundamental (1, 0) and (1, 1) components also exhibit their strongest streamwise

velocity fluctuations very close to the wall. In contrast, Pruett and Chang (1995) observe the (1, 0) harmonic

to have approximately equal peaks in velocity amplitude near the wall and near the critical layer.

Perhaps the mostprofound difference between the present results and previouscomputationalexperiments

on the straight cone concerns the rms density fluctuation. For the flared cone, density fluctuations as largeas 30% of the reference density are observed at the wall (Figure 15), whereas in Figure 7 of Pruett and Zang

(1995), duplicated here for completeness as Figure 19, density fluctuations are nonzero but small at the wall.

Both configurations show rms density fluctuation peaks near the critical layer. We speculate that the large

density fluctuation near the wall is driven by the streamwise gradient in mean wall temperature in the flare

region (Figure 20).

The differences in velocity and density fluctuations between the straight-cone and flared-cone cases

apparently combine to result in a striking contrast in the behavior of the Reynolds stresses relative to the

observations of Pruett and Zang (1995). As observed in Figures 8 and 9 of Pruett and Zang, the dominant

peaks in Reynolds stress occur initially about a displacement thickness from the wall and gradually migrate

toward the wall as nonlinear effects become stronger. In distinct contrast, as shown in Figures 14 and 15 of

the present paper, the turbulent kinetic energy develops and remains close to the wall along the flared cone.


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Figure 19. Streamwise evolution of temporally and azimuthally

averaged rms density fluctuation for cone without flare adapted

from Figure 7 of Pruett and Zang (1995). Note: Reynolds numbers

here are based on local (not freestream) reference values and donot correspond directly with (2).

Figure 20. Laminar adiabatic-wall temperature versus arc length.

We also observe that, for the cone without flare, the Reynolds stresses are dominated by their streamwise

component, whereas along the flared cone the streamwise and wall-normal stresses are of nearly equal

magnitudes.

It is also instructive to compare the present results with the earlier computational studies of Pruett and

Zang (1995), who examined the laminar breakdown of hypersonic boundary-layer flow on axisymmetric

bodies by means of temporal, rather than spatial, DNS. In particular, they considered external flows of

Mach 4.5 along a cylinder and Mach 6.8 along a cone. Initial disturbances in the form of primary and

subharmonic secondary instability waves were used to trigger transitional states. Despite the fact that the

transition mechanism was fundamentally different from that later used by Pruett and Zang (1995) for spatial

DNS, the qualitative nature of the transition processes for both the Mach 4.5 and Mach 6.8 test cases wassimilar to that described above for the cone without flare. These results (and others) strongly suggest that

the qualitative nature of a transitioning hypersonic boundary-layer flow is not terribly sensitive to changes

in either Mach number or moderate transverse curvature (Pruett et al., 1991). Moreover, once triggered, the

transition process seems to be relatively insensitive to the triggering mechanism. In contrast, the present

results suggest that nonzero streamwise curvature has a marked effect on the nature of transition.

In summary, transition is qualitatively different for the straight and flared cones. Because most of the

turbulent activity is confined initially to the critical layer in the boundary layer of the straight cone, transition

onset, as manifest by surface changes to the mean state, is delayed but relatively abrupt. As shown in Figures

6 and 11 of Pruett and Zang (1995), significant changes in skin friction and shape factor are evident only

considerably downstream of the station where the fundamental harmonics experience saturation. In contrast,

for the flared cone, significant deviations from laminar values are observed to occur in shape factor, skin

friction, and thermal stress at stations upstream of saturation of the fundamental. Qualitatively, it appears

that transition on the flared cone is, for lack of a better term, mushy. Transition onset occurs earlier, but

the transition process is gradual.

6.3. Comparisons with the Experiment of Kimmel

Althoughthe results of Kimmel (1993) are admittedly inconclusive, and the present computational results are

admittedly incomplete, we attempt to draw a parallel. As mentioned previously, because different researchers

use different measures of transition onset, it is difficult to draw definitive comparisons between computation

and experiment, and in our judgement, the transition community would benefit by consensus in the definition

of transition onset for high-speed boundary-layer flows. However, our overall impression that, relative to the

straight cone, transition onset occurs earlier on the flared-cone, is consistent with Kimmels observation that


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an adverse pressure gradient moves the onset of transition forward. Moreover, the impression that transition

occurs more gradually on the flared cone is not inconsistent with his observations that favorable pressure

gradients cause a shorter transition zone and that adverse pressure gradients may possibly lengthen the

transition zone.

7. Conclusions

The laminarturbulent transition of the boundary layer on an axisymmetric flared cone in Mach 6 flow

has been simulated by nonlinear PSE and spatial DNS techniques. The full effects of streamwise surface

curvature have been taken into account in the governing equations and boundary conditions. These effects

include adverse streamwise pressure gradient, decreasing edge Mach number, decreasing boundary-layer

thickness, and increasing wall temperature. Wherever possible, results from the present computation have

been compared with those obtained from previous physical and numerical experiments.

Physical experiments (Lachowicz et al., 1996a,b), N-factor studies based on compressible LST (Balaku-

mar and Malik, 1994), nonlinear PSE methodology, and DNS all confirm that disturbances of very high

frequency (on the order of 230 kHz), provide the primary instability mechanism that precipitates transition.

Disturbances in this frequency range are classified as second-mode disturbances in the nomenclature of

Mack (1984). A further indication that the primary disturbances are of second-mode type is the appearance

of so-called rope-like waves in schlieren images from the physical experiment or in computational flow

imagery from the numerical experiment. These waves manifest a wavelength that is approximately twice the

boundary-layer thickness, in keeping with the predictions of LST for second-mode disturbances. Because an

adverse pressure gradient results in diminishing boundary-layer thickness in the flare region, the disturbance

frequencies observed on the flared cone are higher than for a cone without flare.

The results of the present numerical flared-cone experiment differ fundamentally from those of Pruett

and Chang (1995), who simulated laminar breakdown on a right circular cone without a flared afterbody. In

particular, for the present configuration, the dominant Reynolds stress components, turbulent kinetic energy,

and turbulent Mach number all manifest maxima close to the wall, rather than near the critical layer as

observed previously. In addition to a large fluctuation amplitude in density at the critical layer, as observed

also for the straight cone, large density fluctuations are observed at the wall for the flared-cone configuration.Moreover, for the flared-cone, Goertler modes of instability (i.e., harmonics of zero frequency) are shown

to be linearly unstable, whereas they are stable for the cone without flare.

In general, these differences result in qualitatively different transition processes on straight and flared

cones. Relative to the straight cone, transition on the flared cone appears to begin earlier and to proceed more

gradually. The present computational results would seem to be consistent with the experimental results of

Kimmel (1993).

Acknowledgement. The first author is indebted to Dr. Jason Lachowicz, for helpful discussions and data from the experiment

of Lachowicz et al. (1996a,b), and to Dr. Steve Wilkinson of NASA Langley Research Center, for many clarifying comments

regarding the Langley quiet-tunnel experiments.

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