Simulations of Boundary-Layer Response to Laser-GeneratedDisturbances at Mach 6

Dirk Heitmann∗ and Rolf Radespiel†

Technische Universität Braunschweig, 38106 Braunschweig, Germany

DOI: 10.2514/1.A32282

A finite volume Navier–Stokes code was used to simulate the interaction of a laser-induced disturbance with a

conical boundary layer at Mach 5.85. The simulations start 0.2 μs after an optically induced breakdown that

generates an initial disturbance. The initial disturbance was determined from Taylor’s intense explosion similarity

law and this disturbance is imposed into the flowfield of a 7 deg half-angle cone. The initial disturbance parameters,

such as energy, were obtained from comparisons to experimental data. The disturbance was imposed some distance

above the model and the boundary layer, so that mainly the propagating shock wave interacts with the boundary

layer. This interaction results in the formation of a second-modewave packet, and its generation and development are

described. Also, comparisons to experimental data and to results of the linear stability theory are given. Furthermore,

some disturbance parameters were varied, such as released energy or the number of laser pulses.

Nomenclature

A = (wave) amplitudea = speed of sound, m∕sE = laser energy, Jf = frequency, Hzf = similarity variableL = length, mM = Mach numberPr = Prandtl number, 0.72p = pressure, PaR = radius of shock wave, mR = specific gas constant, 284 J�kgK�−1Re = Reynolds numberr = radius, mSu = Sutherland constant, 110.4 KT = temperature, Kt = time, sU = velocity, m∕sX, Y, Z = coordinates, mX�, Y�, Z� = coordinates of unrolled cone, m−αi = spatial growth rate, 1∕mγ = ratio of heat capacities, 1.4δ = boundary-layer thickness, mη = nondimensional radiusΘ = angle, degλ = wavelength, m? = heat conductivity, J�msK�−1μ = dynamic viscosity, Nsm−2

ρ = density, kg∕m3

Φ, Ψ = similarity variableΨ = wave propagation direction, deg

Subscripts

e = boundary-layer edge0 = undisturbed

2 = at shock front∞ = freestream

I. Introduction

T HE transition from laminar to turbulent flow is still poorlyunderstood, especially for high-speed flows. The transition

scenario consists of three steps: receptivity, boundary-layerdisturbance growth, and the nonlinear breakdown to turbulence.Although plenty of subsonic data and quite sophisticated models areavailable, so far it has not been possible to reliably predict thetransition location in high-speed flows. The largest uncertaintiesresult from the receptivity and nonlinear breakdown stages, whereasthe linear stage of the disturbance growth (for simple geometries likecones) are understood quite well. In a two-dimensional (2-D)compressible boundary layer at high Mach number, there are twotypes of instabilities: first mode (or Tollmien–Schlichting wave) andthe secondmode orMack [1]mode. Because the amplification rate ofthe second mode is much higher, it is typically assumed that thisinstability drives the transition.There are three possible approaches for the investigation of the

transition scenario, the usage of analytic models, the evaluation ofexperimental data, and numerical simulations. Each of them has theirown assets and drawbacks and a combined approach appearsreasonable. Here, the focus is laid on numerical simulations toreproduce an experiment reported in Heitmann et al. [2]. A similarexperiment with slightly different flow and disturbance parameterswas also reported in Heitmann et al. [3].For design purposes, themost common approach is the analytic eN

method, which attempts to correlate transition location solely withthe linear growth of instability waves. This approach lacks physicalmodeling of the nonlinear breakdown and the receptivity. The linearstep of the transition can be described with the linear stability theory(LST) (see, e.g., Mack [1]). By the use of nonlinear parabolizedstability equations, the initial phase of the third step can be describedas well. However, there is no validated model for the first stepavailable in super- and hypersonic flows and hence transitionprediction is questionable. Another problem is that usually thedisturbance environment is not unknown.Hence, even if a receptivitymodel was available, this step could nevertheless not be predicted.In principle, the numerical approach [i.e., mainly direct numerical

simulation (DNS). Thatmeans omit computations] could result in themost reliable data, but due to its large demand of computing power itis limited to simple geometries. Furthermore, DNS computationsoften face the problem that initial disturbances of the flowfield arenot well known. Often, it is used to gain additional insights intoexperiments (see, e.g., the DNS computations of Sivasubramanian

Presented as Paper 2011-3875 at the 41st AIAA Fluid DynamicsConference and Exhibit, Honolulu, Hawaii, 27–30 June 2011; received 22November 2011; revision received 21 June 2012; accepted for publication 22June 2012; published online 25 January 2013. Copyright © 2012 by theAmerican Institute of Aeronautics and Astronautics, Inc. All rights reserved.Copies of this paper may be made for personal or internal use, on conditionthat the copier pay the $10.00 per-copy fee to the Copyright Clearance Center,Inc., 222 RosewoodDrive, Danvers,MA 01923; include the code 1533-6794/13 and $10.00 in correspondence with the CCC.

*Research Assistant, Institute of Fluid Mechanics, Bienroder Weg 3.†Professor, Institute of FluidMechanics, BienroderWeg 3. SeniorMember

AIAA.

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et al. [4], in which flow conditions and wave excitation are similar tothe experiment of Bountin et al. [5]).The experimental investigation also includes several problems

ranging from the flow quality of conventional facilities to arguableabilities of the measurement equipment to resolve the instabilitieswith their low amplitudes and high frequencies. A further difficultyconsists of introducing controlled disturbances. For experiments insubsonic flows, remarkable progress was obtained when artificialdisturbances were introduced to generate tailored waves. Therefore,there is a demand for similar purturbators at higher speeds. Especiallybecause of the need for high-frequency excitation, the design of suchpurturbators is difficult. Only two kinds of purturbators are reportedin the literature. In some references, artificial disturbances aregenerated inside the boundary layer using an electric glow discharge(see, e.g., [6–8]). In other references, a high-power laser is used forthe disturbance generation in the freestream or on the model. Forexample, Glumac et al. [9] and Salyer et al. [10] described laser-generated hot spots and their evolution, and Schmisseur et al. [11]analyzed the response of a boundary layer of an elliptic cone to asmall volume of heated gas that remains from a laser-generatedplasma after some microseconds. Besides, laser disturbances wereused for various kinds of flow control (see, e.g., [12–14]).Although electric purturbators are reported more frequently in the

literature on boundary-layer stability experiments, an optical systemmight offer some advantages. Glow dischargers might disturb theflowmechanically evenwhen they are switched off and their locationcannot be altered once they are installed. With an optical purturbator,the disturbance shape can be changed from apoint to a line, or even anextended area, simply by beam-shaping optics, and the location canbe continuously adjusted without modifying the model. In addition,the amplitude can be altered over a large range, and the frequency isadjustable by simply setting the time delay between consecutivepulses. Furthermore, the laser disturbance can be introducedsomewhere in the flowfield (i.e., off the surface, so that the receptivitystep can also be treated).The purturbator in the present paper consists mainly of an

impulsive pressure disturbance. A single laser-induced sphericallypropagating disturbance above the model is thereby simulated. Thedisturbance consists of a shock wave and a heated gas volume in thecenter. Because the disturbance is introduced some distance abovethe model, mainly the shock wave interacts with the boundary layer.This stimulation generates a disturbed packet of waves, whichspreads out as it travels downstream. The disturbance excites allpossible modes and the wave is formed by selective amplification ofthe most unstable waves. Such a broadband stimulation possiblyresembles a natural transition scenario better than a harmonicdisturbance generator. Most computations described in the presentpaper were done for single-pulse excitation. Also, some disturbanceparameters were varied, such as pulse energy and the number of laserpulses. This means that, in some computations, the boundary layerwas perturbed with two consecutive pulses with different temporalspacing to impose some frequency content at the inverse pulsespacing.

II. Numerical Procedure

The calculations consist of three steps:1) At first, the steady flowfield around a 7 deg half-angle cone is

computed. The wall is set to isothermal conditions (Twall � 300 K).This should approximately match the experimental conditionsbecause, in the experiment, themodel heating during thewind-tunnelrun time is small. Results for the base flow are given in Sec. III.A.2) This steady solution is modified by implementing a disturbance

into the flowfield. In the experiments, a high-power laser pulse wasused to generate a spherically propagating pressure disturbancesimilar to a very small explosion. The disturbance modeling isdescribed in Sec. II.C.3) An unsteady simulation follows to capture the disturbance

development and the generation and evolution of a second-mode-likewave train.

A. Flow Solver

The triangular adaptive upwind (TAU) code of the GermanAerospace Center (DLR) was used for all simulations. This code isdescribed in Gerhold [15] and has been validated for hypersonicflows in Mack and Hannemann [16]. The code discretizes theReynolds-averaged Navier–Stokes (RANS) equations by a finitevolume technique on hybrid grids. However, for the presentcomputations, the code was run in laminar mode (i.e., the Navier–Stokes equationswere solved). For discretizing the convective fluxes,different upwind schemes are available (Van Leer, AUSMDV, Roe,MAPS�, EFM, AUSMP, and AUSMPWP). In Wolf et al. [17],different flux schemes were applied to three test cases (stationarynormal shock, shock reflection on a wedge, and Sod’s shock-tubeproblem) and the pressure weighted advection upstream splittingmethod (AUSMPWP) resolved the flow phenomena best because it isleast dissipative and diffusive. Therefore, this second-order-accuratescheme is used here. To close the system of differential equations,further relations are required. The applied code uses the ideal gas lawp � ρRT and the viscosity is modeled according to Sutherland:

μ�T� � μ0

�T

T0

�32 T0 � SuT � Su

with Su � 110.4 K, reference temperature T0 � 273 K, andreference viscosity μ0 � 1.716 × 10−5 Nsm−2. The heat conduc-tivity is computed from the viscosity by the relation

λ � γ

γ − 1R

μ

Pr

with the ratio of heat conductivities γ � 1.4, the specific gas constantR � 287 J kgK−1, and the Prandtl number Pr � 0.72.Unsteady simulations are carried out using an implicit dual-time

stepping approach, which calculates for each physical time step a“pseudosteady” solution of the RANS or Navier–Stokes equations.This time integration is second-order accurate. At the beginning ofeach simulation, the physical time step was set to 1E-9s. This wasnecessary due to the large velocities (>1000 m∕s) in the excitationregion and the small cell size. After some time steps, when thevelocities had decreased, the time step size was increasedsuccessively to 2E-8s.The code uses the NetCDF format for grid and solution files,

allowing the user to easily modify variables. This was necessarywhen the initial disturbance was implemented into the flowfield.Furthermore, parallel computing is possible via domain splitting andMessage Passing Interface (MPI) communication.

B. Grid

A hybrid grid consisting of approximately 32 million grid pointswas generated to mesh a 90 deg segment of the 7 deg half-anglecone (length 0.5m). The nominally sharp cone tipwas roundedwith aradius of 4 μm, because a sharp tip cannot be generated usinghexahedrons. The radius was chosen to be small enough to preventany influence on the boundary layerwithin the region of interest. Thisis at positions downstream of the initial disturbance positionX � 0.2825 m. Near the tip region, cell clustering was necessary toimprove calculation stability. The hybrid mesh around the tip isdepicted in Fig. 1.Because of the requirement of high resolution in the boundary

layer, very small cells in that region were used. The clustering nearthe cone wall is shown in Fig. 2. An unstructured layer separates thefine grid in the near wall region (>3.5 boundary-layer thicknesses)from the remaining coarser grid part. Thiswas necessary to reduce thenumber of cells and the associated computational costs. Besides, anunstructured layer perpendicular to the flow direction (at a positionupstream of the disturbance) was included to obtain higher resolutionin the circumferential direction and to further reduce the cell size. Thecell size in the region, where second-mode waves are expected, is inthe flow direction 0.14 mm (35 points per wavelength), in the cone-circumferential direction 0.51 mm, and in the wall-normal direction

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0.001–0.11 mm (50 points inside the boundary layer). The lateralboundaries of the computational domain are defined as symmetryplanes. Note that the disturbances remain small enough to not reachthe lateral grid boundaries.

C. Laser-Generated Disturbance

In experiments reported earlier (Heitmann et al. [2]), a high-powerlaser was used to introduce artificial disturbances. If a certain powerdensity is exceeded, air is no longer transparent. Instead, the laserpower is absorbed and a plasma is ignited. The gas in the focal volumeis strongly heated and a shock wave propagates away from thedisturbance position.Taylor’s [18] similarity law for intense explosions was used as the

initial condition for simulating such a disturbance. The distribution ofpressure, density, and radial velocity can be obtained from this lawand it is summarized here for completeness. The similarityassumptions for an expanding blast wave of constant total energy are

p∕p0 � R−3f�η� ρ∕ρ0 � Ψ�η� u � R−3∕2Φ�η�

for pressure, density, and radial velocity, respectively, where R is theradius of the shock, p0 and ρ0 are pressure and density of theundisturbed gas, η � r∕R is the nondimensional radius, andf,Ψ, andΦ are functions of η. These functions are tabulated in Taylor [18] forγ � 1.4. The blast wave radius can be obtained from

R � S�γ�t25�E

ρ0

�15

where S is a function of the specific heats of the gas, and E is thereleased energy.The value for the released energy is difficult to determine,

therefore, a variation of this parameter is presented in Sec. III.B. In theexperiments, the amount of released energy was not assessable,because only a fraction ϵ · Elaser of the laser energy (Elaser � 250 mJ)is converted into the disturbance. The value of ϵ depends on manyfactors, ranging from beam quality in space and time through gasproperties, such as gas species or density. Therefore, one can findvarious values for ϵ in the literature. Because of the low-density flowtreated here, especially the density ρ is important. Mori et al. [19]investigated the influence of reducing the pressure and associateddensity. Reducing the density results in a reduced ϵ. At a pressureof p0 � 20 kPa, they determined ϵ � 0.2, whereas at ambientconditions, ϵ ≈ 0.5 is possible. For the much lower pressures treatedhere (p0 ≈ 600 Pa in most of the simulations), no information isgiven. Therefore, by comparing the measured shock wave prop-agation velocity with Euler computations, the energy conversionwas determined to be ϵ < 0.001. This seems to be in agreement withexperiments in which the laser power was measured before andbehind the breakdown region and where no difference could befound.The disturbance has to be specified at a time when the explosion is

still strong. This is necessary because, in the model, the Rankine–Hugoniot relations are simplified and only valid as long as p∕p0 islarge. In Korobeinikov [20], the necessary pressure ratio (shockpressure to undisturbed pressure) for air is given as p∕p0 > 20, sothat differences between the disturbance behavior and Taylorapproximation are small. The pressure ratio can be adjusted by settingthe time. In the present study, this time was set to Δt � 0.2 μs afterthe optical breakdown, and this always resulted in ratios larger than50.

r = 4 µm

Fig. 1 Rounded cone tip.

Fig. 2 Grid refinement near cone surface.

η = r/R

p/p 2,

ρ/ρ 2,

U/U

2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

pressuredensityvelocity

Fig. 3 Taylor similarity law for determining the distribution ofpressure, density, and radial velocity of initial disturbance.

+ + + + + + + ++

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+ + + + + + + + + + + + + +

+ ++ +

++ + + + +

++++

++ ++

+

+

+

+

+

+

++ + + + + + + + + + +

+ + + + + + + + + + + + ++

+++

++

+

+

+

+

+ + + + + + + + ++ + + + + + + + + + + + + + + + + + + +

+

++ + + +

X, m

p,kP

a

0 0.0005 0.001 0.0015 0.002

0

20

40

60

80

100

+

∆t = 0.2 µsinitial disturbance

∆t = 0.3 µs

0.5 µs

1.0 µs

Taylor similarity solutionEuler calculationNavier – Stokes calculation

Fig. 4 Comparison of disturbance development between Taylorapproximation, Euler, and Navier–Stokes calculations.

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Also, more complex models are reported in the literature that takethe counterpressure into account and for which this restriction doesnot apply (see, e.g., Sedov [21] or Bach and Lee [22], and referencestherein). Such an approach had allowed the prescription of the initialdisturbance at a later time instance. However, instead, the Taylorsimilarity solution was applied. Because the interaction of thedisturbance with the boundary layer takes place at aboutΔt � 10 μsand, in total, 200 μs were computed, the savings in computationalcosts did not justify the application of more complex models.Results from the Taylor law in still air are given in Fig. 3. Pressure,

density, and velocity are scaled using their peak value at the shockfront. In Fig. 4, the pressure distribution of one of the used initialdisturbances and its development in the first microsecond are shown.Therefore, Euler and Navier–Stokes computations were done. Inthe beginning, this development should be similar to the Taylorapproximation, otherwise the influence of the counterpressurewouldalready be too strong and the starting solution would not havebeen appropriate. Therefore, for Δt � 0.3 μs, a similarity solutionaccording to Taylor is given and compared with Euler and Navier–Stokes computations. The Euler computation agrees well with thesimilarity law. This indicates that, at Δt � 0.2 μs, the time to whichthe starting point of the simulation was set, the flow phenomenaare well described by the similarity law. The Navier–Stokescomputation shows some oscillations in the beginning (Δt � 0.3 μsat X � 0.0002 m), but these oscillations disappear later, so that, atΔt � 0.5 μs, no difference to the Euler computation is visible. Theseoscillations are not critical because the interaction with the boundarylayer does not take place before Δt > 10 μs.

III. Results

A. Base Flow

At first, some results of the base flow calculation are shown. Theflow conditions were chosen similar to experiments reported inHeitmann et al. [2], and they are summarized in Table 1 alongwith thegeometry parameters. In Fig. 5, the pressure distribution around thecone is depicted. The disturbance position and the measurementpositions in the experiments are also indicated. The shock and thepressure increase toward the model, according to the Taylor–Macollequation (see, e.g., Sims [23]), are clearly visible. In addition, aMachwave is emanating from the surface at X � 0.3 m, but this wave isbarely visible. It originates from a 2-D roughness strip (3 mm widthand 0.1 mm height), which was present to exactly resemble theexperiments. The flow around this roughness is shown in Fig. 6, inwhich velocity contours along with streamlines are given. In theexperiment, it was confirmed that the roughness only locally affectedthe flow and did not cause transition. The roughness Reynoldsnumber is about Rek ≈ 1.6 and transition should not be affected aslong as Rek < 10 · · · 25 (see review by Schneider [24]). Because ofthe rather small roughness height, it was difficult to grid this part

accurately. Within the height of 0.1 mm, 10 cells are placed and theroughness element was generated by cutting it out of the existingstructured grid. The grid in the vicinity of the roughness element isdepicted in Fig. 7.The stability behavior strongly depends on the boundary-layer

shape. To assure that the boundary layer is accurately resolved, wall-normal profiles were extracted and these profiles are compared withanalytical boundary-layer solutions in Fig. 8. For the analyticalsolution, a flat plate boundary layer was computed according to vanDriest [25]. The input variables, such asMach number and freestreamtemperature, were calculated from the Taylor–Macoll equation [23].Then, the solution was Mangler transformed [26] to obtain a conicalboundary layer. The agreement for velocity and temperature dis-tributions at all three positions is good. The derivatives of theboundary-layer profile represent a more sensitive measure forcomparing the computed boundary layer to the analytical solution.These derivatives are furthermore important, because they appear inthe stability equations (see, for example, Schrauf [27]). The first twoderivatives for both, the velocity and also the temperature, aredepicted in Fig. 9 for the X � 0.2 m position. Also, for thederivatives, the agreement between the numerical and analyticalsolution is reasonable, and this is considered a validity check of thebase flow calculation.

B. Disturbed Flow

At first, results of single-pulse stimulation are given and comparedwith experimental data reported earlier (see Heitmann et al.[2]). Afterward, some parameter variations are presented. Severalcomputations were performed, some to assess the effect of para-meters that could not reliably be determined in the experiment, such

Table 1 Flow and geometry parameter

Geometry data Flow conditions

Lcone, m 0.5 p∞, Pa 311Θcone, deg 7.0 T∞, K 54rnose, μm 4.0 M∞, – 5.85— — Re∞, m

−1 4.91 × 106

— — Ree, m−1 6.25 × 106

Fig. 6 Velocity field around roughness element.

X, m

Z,m

0.299 0.3 0.301 0.302 0.303

0.037

0.038

0.039

7°-half-angle cone

Fig. 7 Grid in the vicinity of the roughness element.

Fig. 5 Distribution of static pressure in the flowfield around the cone.

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as the impact of the energy deposited in the disturbance and theinfluence of the roughness strip. Furthermore, simulations with twosequential pulses were conducted in an attempt to impose somefrequency content. Results obtained on a locally refined grid are alsoreported, to assess the effects of numerical discretization on thecomputed flow data.

1. Single-Pulse Stimulation

The disturbance was introduced at an axial distance from the tip of0.2825 m and 5 mm above the surface in the center of the 90 degsegment of the computational domain. The simulation starts Δt �0.2 μs after the optical breakdown, and the initial disturbance has aradius of about 1 mm, depending on the released energy. The firstinteraction with the boundary layer takes place about 10 μs later. Atthat time, a spherically propagating shock wave and a heated gasvolume (hot spot) in the center remain from the initial disturbance.The temperature in the disturbed region is very high (T > 1E5 K).This is high enough to imply high-temperature and nonideal gaseffects such as dissociation. As the hot spot spreads out, thetemperature very quickly decreases, for example, after 5 μs, the

temperature has dropped to approximately 2400 K and after 10 μs,when the interaction with the boundary layer takes place, to about1400 K. Furthermore, the heated gas volume stays throughout theentire computation above the boundary layer. Hence, althoughnonideal gas effects might be present in the beginning, thecomputation of the interaction with the boundary layer is notcorrupted. Because the disturbance was introduced 5 mm above thesurface, mainly the pressure disturbance interacts with the boundarylayer. Figure 10 shows the wall-normal temperature distribution atΔt � 52 μs through the disturbance center. For comparison, theundisturbed boundary layer is also given. The boundary layer isslightly distorted, but the hot spot is still 5 mm above the surface (i.e.,some distance away from the boundary layer).In Fig. 11, the density gradient of the flowfield around the

disturbance is shown at different times to visualize the disturbanceshock. In the upper part, the shock wave from the cone tip is visible.Also, the edge of the boundary layer and the weak disturbance fromthe roughness strip can be seen. In the upper figure, the unstructuredlayer shown in Fig. 2 is indicated. The shock wave gets blurred whenit propagates into this region. At 10 μs, the disturbance hits themodel

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xxxxx

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xxxxxx

x

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xxxxxxxxxxxxxxxxxx

x

U/Ue

Z*,

m

0 0.2 0.4 0.6 0.8 10

0.001

0.002

0.003

x

X = 100 mm

X = 200 mm

X = 400 mm

Similarity ProfileTAU

a) Streamwise velocity

xxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xx

xxxxxx

xx

xx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

x

x

T/Te

Z*,

m

1 2 3 4 50

0.001

0.002

0.003

x

X = 100 mm

X = 200 mm

X = 400 mm

Similarity ProfileTAU

b) Temperature

Fig. 8 Comparison of boundary-layer profiles computed by the TAU finite volume code and the Mangler-transformed flat plate similarity profiles.

dU/dZ*, 105 s-1

Z*,

mm

0 2 4 60

0.5

1

1.5

2

2.5

3

3.5

4

similarity solutionTAU

d²U/dZ*², 108 m-1 s-1

Z*,

mm

-15 -10 -5 0 50

0.5

1

1.5

2

2.5

3

3.5

4

dT/dZ*, 105 K m-1

Z*,

mm

-3 -2 -1 0 10

0.5

1

1.5

2

2.5

3

3.5

4

d²T/dZ*², 108 K m-2

Z*,

mm

-5 0 5 100

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 9 Comparison of boundary-layer derivatives computed by the TAU finite volume code and theMangler-transformed flat plate similarity solution,X � 0.2 m.

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for the first time and the shock reflection is visible. The other figuresshow the disturbance of the boundary layer, the reflection of theshock, and the distortion of the reflected shockwhen it passes throughthe hot spot.In Fig. 12, the surface pressure fluctuations at different times after

the laser pulse are shown. The downstream-traveling part of thespherical disturbance appears in all figures, whereas the upstreamtraveling part of the wave quickly disappears. Already, at Δt �50 μs, some kind of two-dimensional structures have developed

behind the shock wave. In the following, these structures spread outand form a wave train of several oscillations at Δt � 200 μs. Thiswave is a second-mode instability wave and its properties, such aswavelength and frequency content, are in agreement with data fromthe literature. The eigenfunction was extracted from the simulationby computing themean and fluctuations from several boundary-layerprofiles at a fixed time over a narrow spatial range. This eigenfunctionis shown in Fig. 13. The mean curve is similar to the temperatureprofiles already shown in Sec. III.A. The fluctuations show one largemaximum near the boundary-layer edge and a smaller one near thewall. At the wall, the fluctuations fall to zero due to the isothermalboundary condition. In experimental visualizations of the densitygradient (schlieren images), large second-modewaves appear as ropewaves (see, e.g., Demetriades [28]). Suchwaves are also visible in thecomputed schlieren image (see Fig. 14). The boundary layer is alsoindicated and the rope-like structures are mainly near the boundary-layer (BL) edge, where the fluctuations are maximal.In the experiment, surface pressure fluctuations were measured at

six positions. The gauges were located at axial distances from the tipbetween 270 and 395 mm with a spacing of 25 mm. At thesepositions, time traces from the numerical simulation were extractedand they are depicted in Fig. 15 alongwith the experimental data. Thelines on the bottom represent the most upstream sensor and the timetraces for the following sensor positions are shifted upward. Theinitial disturbance positionwas behind the firstmeasurement positionand hence, at X � 270 mm, no disturbance is visible in thesimulation. The experimental curve shows a very small peak att � 15 μs, which possibly represents the upstream propagating part

T, K

Z*,

m

100 150 200 250 3000

0.005

0.01

0.015X = 0.3265 m∆t = 52.0 µs

hot spot(disturbance center)

temperatureboundary layer

Fig. 10 Temperature distribution in the wall-normal direction throughthe hot spot (E � 0.0625 mJ).

Fig. 11 Density gradient of the initial disturbance development (E � 0.25 mJ).

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of the disturbance. At X � 295 mm, both the experimental and thenumerical curve show a pressure peak of about 500 Pa at t � 15 μs.Such a sharp pressure rise excites all possible modes, stable andunstable ones, and in the beginning, transient growth seems to occur.A rather large amplitude wave appears behind the shock atX � 320 mm. Its amplitude slightly decreases until X � 345 mm,presumably due to the damping of intrinsically stable modes, whichwere also excited by the impulsive pressure disturbance. In thefollowing, the wave is amplified and it gets separated from the stillvisible shock due to its smaller propagation velocity. The convectionvelocity is approximately 85% of the boundary-layer edge velocityUe, instead ofUe � a (a being the local speed of sound). Of course,the measurement includes a certain amount of noise. The rearmostsensor especially shows a low-frequency oscillation of about 20 kHzover the entire period shown in Fig. 15. The amplitude of this

oscillation is about the same as that of the second-mode packet.However, this oscillation is not related to the perturbation. Instead, itis considered to be related to the instrumentation because only thissensor shows this low-frequency oscillation. In different experi-ments, a few sensors always showed similar low-frequency waves,and this seems to occur when the gauges are not mounted completelystress free. Such stress, for example, is generated when the differentborings of the mounting hole are not exactly aligned. The agreementbetween experiment and computation is nevertheless quite good.Both the amplitude and the temporal evolution of the wave areproperly reproduced by the simulation.At the rear positions, some oscillations are visible between the

shock wave and the wave packet, for example, for the rearmostposition between 120 and 150 μs. These are interpreted as fastacoustic waves and their propagation velocity supports thisassumption. The waves are also visible in the visualization of thesurface pressure fluctuations in Fig. 12. There they appear in the formof spherical waves that precede the second-mode wave packet. FromFig. 12, it is visible that thesewaves develop from the initially sphericshock wave. This wave spreads out, that is, the number of cyclesincreases to three at Δt � 200 μs, whereas the amplitude decreases.In the following, time traces like in Fig. 15 were used to calculate

disturbance spectra. These are depicted in Fig. 16 for three positions.Therefore, short time traces of 100 μs in length with the wave in thecenter were used. The spectra show a growing peak between 100 and200 kHz related to the second mode. A slight decrease of the mostunstable frequency due to the tuning of the instability to theboundary-layer thickness is visible. Furthermore, a harmonic starts toevolve at a frequency of about 300 kHz. These spectra were used tocompute the growth rate according to the relation

−αi �ln�A2∕A1�

ΔX�

given by Mack [1], with A1 and A2 being the amplitudes atconsecutive positions, and ΔX� being the surface distance between

Fig. 12 Surface pressure fluctuations at different times after thedisturbance (E � 0.0625 mJ).

T/Te, -

T’/Te, -

Z*/

δ,-

1 2 3 4 5

0 0.1 0.2 0.3 0.4

0

0.5

1

1.5

2Laser:E = 0.0625 mJ

BL profiles extracted between:0.390 < X [m] < 0.435

at ∆t = 200.0 µs

meanfluctuations

Fig. 13 Fluctuations of the temperature in the area of the stimulatedwave (E � 0.0625 mJ).

Fig. 14 Computed schlieren image in the area of the stimulated wave(E � 0.25 mJ).

t, µs0 50 100 150 200

ExperimentTAU simulation

500

0

p, P

a

X = 395 mm

X = 270 mm

X = 295 mm

X = 320 mm

X = 345 mm

X = 370 mm

Fig. 15 Surface pressure fluctuations at different positions on the conesurface to compare TAU simulation (E � 0.0625 mJ) and experiment.

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them. This growth rate in the unstable frequency range is also shownin Fig. 16. Besides, an amplification rate obtained from LST is givenin the figure. Therefore, the stability code LILO described in Schrauf[27]was used. The code allows for the computation of local aswell asnonlocal stability behavior in compressible media. Amplificationrates can be computed for a single station, but also an integration overseveral stations is possible to determine N factors. The boundary-layer profile was extracted from the steady simulation at X � 0.4 m.For the present computation, frequency and propagation directionwere specified to close the equation system. In the frequency range ofthe second mode, only marginal deviations appear for both theamplification and the unstable frequency range. The LST curveshows damping for frequencies below 120 kHz. In the range of thefirst mode (about 60 kHz), the growth rate increases so that this modeis approximately neutral. In this figure, only 2-D waves wereconsidered. This is reasonable for the second mode. However, thefirst mode exhibits larger growth for oblique waves than for 2-Dwaves. This is depicted in Fig. 17, in which the amplification rate isgiven in dependency of frequency and propagation direction Ψ. Forthe first mode, the propagation direction is related to the wavefrequency. It is most unstable for f ≈ 40 kHz and Ψ ≈ 60 deg. Thesecond mode is most unstable forΨ ≈ 0 deg. With increasingΨ, thegrowth gets smaller until it approaches zero at about 40 deg.The spectra in Fig. 16 showed not only growing amplitudes for

frequencies in the second-mode range but also at about 20 kHz. Asevident fromFig. 17, this is close to the oblique first-mode instability.This instability exhibits a maximal spatial amplification of−αi ≈ 6 m−1. Because the impulsive stimulation excites all possiblemodes, the first mode might also be present at X � 400 mm.Furthermore, the time traces used for the spectra computation includethe fast acoustic waves that precede the second-mode wave packet.Although their amplitudes do not grow, they spread out, as can be

seen in Fig. 12. Both the first-mode instability and the fast acousticwaves contribute to the growing low-frequency peak in Fig. 16.

2. Effect of Disturbance Energy

The excitation energy is very small and could not be measuredreliably in the experiment. Hence, a variation of this parameter wasperformed in the numerical work. For different excitation energies,the fluctuations of the surface pressure are quite similar to the dataalready shown. For stimulation with moderately higher energy, thewave’s amplitude increases. This is depicted in Fig. 18, in whichdisturbance spectra computed at theX � 395 mm location are givenfor three excitation energies. For the E � 0.125 mJ case, theamplitude related to the second mode has increased compared withthe stimulation with smaller energy. Also, the amplitude of theharmonic at f � 300 kHz increases. In the case of the highestexcitation energy considered here, the second-mode amplitudedecreases and the peak is slightly shifted to lower frequencies. Itseems, that, in this case, a critical amplitude was attained, wherefromno further amplification in the frequency range of the second modetakes place.

3. Multipulse Stimulation

In experiments, a stimulationwith two consecutive pulseswas alsotested (see Heitmann et al. [3]). For the corresponding computations,the pulse energy was set to 0.0625 mJ and different time distancesbetween the pulses were applied. With such double pulses, at leastsome frequency content can be imposed at the inverse of the pulsespacing. Although such a stimulation still excites all modes, thosemodes at the desired frequency are predominantly stimulated. As aconsequence, the phase of transient growth in the beginning isreduced and the wave attains larger amplitudes. Besides, for certainpulse spacings, the fast acoustic waves in front of thewave packet aredetuned and disappear more quickly. This is depicted in Fig. 19, inwhich a comparison of the surface pressure fluctuations betweenstimulation with a single pulse and with two pulses is shown. In thisimage, the cone surface is mapped onto planar X�, Y� coordinates.Despite the stimulation with adjusted frequency, all developing

wave packets exhibit the same frequency content. This is aconsequence of the strong frequency selectivity of the boundarylayer. The spectra in Fig. 20 do not show extraordinarily largeamplitudes at the set frequencies. However, because the doublepulses influence the broadband stimulation, the initial amplitudes ofthe unstable modes are also changed and, therefore, the wave’samplitude near the end of the cone changes. This is in agreement withthe corresponding measurements (see Heitmann et al. [3]).

4. Effect of Roughness Element

Asmentioned in Sec. III.A, inmost of the simulations, a roughnesselement was present to exactly resemble the conditions of theexperiment. In the experiment, it seemed that thewave attained larger

f, kHz

p’,P

a/H

z

- αi,

m-1

0 100 200 300 400 500

0

10

20

0

25

X = 395 mmX = 400 mmX = 405 mm

TAULST

amplification rate -αi:

spectra (TAU):

Fig. 16 From TAU simulations computed disturbance spectra andcomparison of growth rates to linear stability computations, forLSTonly2-D waves were considered.

Fig. 17 Propagation direction of Mack [1] first and second modes,X � 400 mm.

f, kHz

p’,P

a/H

z

0 100 200 300 400 500

0

10

20

30

40

E = 0.2500 mJE = 0.1250 mJE = 0.0625 mJ

stimulation withsingle pulse of energy:

Fig. 18 Disturbance spectra atX � 395 mm for wave stimulation withdifferent pulse energies.

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amplitudes if such a roughness strip was present, but a quantificationof this effect was not possible. Therefore, such a variation was alsoconducted numerically. For these simulations, a pulse energy ofE � 0.0625 mJ was chosen and the time traces of the pressurefluctuations at X � 395 mm are depicted in Fig. 21. The shock andthe fast acoustic waves in front of the wave packet are similar in bothcases. At t ≈ 150 μs, the stimulated wave reaches the position and, inthe presence of the roughness, thewave’s amplitude is approximately25% larger. Figure 22 shows the corresponding disturbance spectraand they also show the amplitude increase when the roughness ispresent. Although the effect seemed to be larger in the experiments,the simulations show the same trend. Other parameters, such asfrequency content or spatial dimensions (not shown) are very similar.To show the effect of the roughness strip images of the density

gradients in the vicinity of it are given in Fig. 23 for two timeinstances. The roughness strip (3.0 × 0.1 mm2) is positioned atX � 0.3 m. In the left image (Δt � 15 μs), the spherical shock wavehas already reached the wall. It is slightly distorted from passingthrough the boundary layer and the reflection from thewall is visible.At this time, it has not yet reached the roughness element. Somemicroseconds later, an additional spherical wave with the center onthe wall is generated when the forward moving part of the blast wavepasses the roughness. Because it is hardly visible, the region ismarked in the right image. This wave is still rather small and it islocated between 0.299 < X,m < 0.303. Thus, roughly speaking, theflow behaves in a similar way as if the flow had been stimulated withtwo pulses. As mentioned before, the effect seems to be smaller thanin the experiment.However, if themechanism is considered similar toa double-pulse stimulation, it can be assumed that the sensitivity isvery large to several parameters, such as distance of initialperturbation position (X � 282.5 mm) to roughness strip and initialbehavior of the blast wave. It is probable that small differences inthese parameters caused the discrepancies between experimental andnumerical results.

C. Computation on Locally Refined Grid

The applied flow solver is second-order accurate in space and time.Because of this rather low order, the computations were checkedwithrespect to the cell size and to the time step size. Only results from thegrid check are shown here and results concerning the time step sizeare only mentioned.At three times during the computations (Δt � 50; 100; 150 μs),

the time step sizewas reduced by a factor of 10 and the pressure at thewall and wall-normal profiles through the disturbance center werecompared with results, where the standard time step size was used.No differences could be found, and so the time step size wassufficiently small. Note that the maximum time step size was 2E-8s.Hence, the used time stepswere two orders ofmagnitude smaller thanthe time constant of the relevant flow phenomena, which is in therange of microseconds.A refinement of the entire grid in all three dimensions was not

possible due to limited computational resources. Instead, only a smallregion of the already rather fine block near the wall was refined bydoubling the amount of cells in all three directions. Hence, within thisregion, the number of cells was increased by a factor of 8. The totalnumber of grid pointswas then about 40million instead of 32million.This region covered 30 deg in the circumferential direction and about3.5 cm in the axial direction. In Fig. 24, partial views of thisrefinement are shown. This refined block is located at the rear of thecone. Thus, this study allows statements about the behavior of thewave packet, but not necessarily about its generation, which takesplace farther upstream.The simulation with pulse energy E � 0.0625 mJ was chosen for

this grid refinement test. Figure 25 shows the surface pressurefluctuations at Δt � 200 μs for both grids on the unrolled cone. Atthis time, the wave has already passed through the refined region.At the tail of the wave, the lateral dimensions are slightly larger forthe refined grid, but the differences are very small. In Fig. 26, the

Fig. 19 Surface pressure fluctuations at Δt � 200 μs for stimulationwith one and two pulses (ΔtLaser � 6.8 μs).

f, kHz

p’,P

a/H

z

0 100 200 300 400 500

0

5

10

15

20

25

30

35

twopulses

⎫⎬⎭

single pulsef = 102 kHzf = 147 kHzf = 208 kHz

Fig. 20 Comparison of disturbance spectra atX � 395 mm for double-pulse stimulation with different pulse spacings, E � 0.0625 mJ.

t, µs80 100 120 140 160 180 200

with roughness

w/o roughness

50

0

-50

p,P

a

50

0

-50

p,P

a

Fig. 21 Time traces of stimulated waves (E � 0.0625 mJ) with andwithout roughness element.

f, kHz

p’,P

a/H

z

0 100 200 300 400 500

0

5

10

15

20

w/o roughnesswith roughness(3.0 × 0.1 mm2)

X = 395 mm

Fig. 22 Disturbance spectra of stimulated waves (E � 0.0625 mJ) withand without roughness element.

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extracted time traces for X � 370 and 395 mm are shown, so thatquantitative information is also available. TheX � 370 mm positionis at the beginning of the refined block and hence there are nodifferences. The other position is near the end of the block and thereare also only marginal differences between the computations. Themaximumdifference found in terms of the surface pressurewas about0.8%.This study showed that, once the wave is generated, its behavior is

properly resolved by the computation. This is further confirmed bythe agreement with calculations, according to the linear stabilitytheory. However, no statements are possible about the grid resolutionin the region where receptivity takes place (i.e., further upstream).The agreement to experimental data at different positions on the conemight also be considered as indication for a sufficient grid resolutionin this cone region. Of course, this indication is not an undoubtedproof but, due to the computational costs, a refinement of the entiregrid was not possible. We also note that the grid contained a cellclustering in the receptivity region (at X ≈ 0.3 m). There, the

minimum cell dimension in the flow direction was about 0.01 mminstead of 0.14 mm farther downstream. In the downstream region,this spacing was chosen to accommodate 35 cells per second-modewavelength. However, in the receptivity region, other wavelengthsmight be relevant and, therefore, the spacing was chosen to besmaller.In conclusion, it appears that the computational grid used in the

present study was chosen to be fine enough to capture the relevantflowphysics. For the linear phase ofwavegrowth, this was confirmedby a grid refinement study. For the receptivity phase farther upstream,we conclude this from the agreement to experimental data.

IV. Conclusions

The German Aerospace Center (DLR) triangular adaptive upwindcodewas used to perform time-resolved simulations of the interactionof a laser-generated disturbance with a conical hypersonic boundarylayer. The simulations were performed to supplement anotherexperiment, in which only pressure fluctuations at a few discretepositions could be determined. Some additional information could be

Fig. 23 Visualization of the density gradient in the vicinity of the roughness element and its effect on the blast wave development.

X, m

Z,m

0.36 0.365 0.37 0.375 0.38

0.045

0.05

0.055

0.06

a) Refinement in wall normal direction b) Refinement in circumferential directionFig. 24 Views of local grid refinement.

Fig. 25 Comparison of surface pressure fluctuations atΔt � 200 μs onstandard and locally refined grid.

t, µs100 150 200

100

0

-100

p,P

a

standard gridlocally refined grid

X = 370 mm

X = 395 mm

Fig. 26 Comparison of surface pressure fluctuations of two positions onstandard and locally refined grid.

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gained from the simulations that could not be extracted from theexperiments. This includes, for example, the eigenfunction of thestimulated instability wave or the influence of the roughness element.Also, parameters such as the energy that was released in thedisturbance were varied. This variation was conducted to assess theuncertainty in the corresponding experiment.Each simulation consists of three steps: 1) base flow computation,

2) implementation of an explosion at a very early stage, and 3) anunsteady computation. The base flow computation was checked bycomparing computed boundary-layer profiles at several axiallocations with analytical solutions. Also, for their derivatives,reasonable agreement was found. The boundary-layer profiles werefurthermore used as input for stability analysis.During the unsteady simulation, the prescribed blast wave spreads

out and represents an impulsive pressure disturbance that interactswith the boundary layer, whereas the hot gas in the disturbance centerstays above the conical boundary layer. This stimulation results in theformation of two differentwave trains. The upstream-traveling part ofthe spherical disturbance is rapidly attenuated and vanishes, but thedownstream-traveling part remains visible until the end of thesimulations. The initially large propagation velocity decreases andthe shock waveweakens to an acoustic wave. Although its amplitudedecreases, the wave spreads out and consists of three to four cycles atthe end of the simulations. Behind these fast acoustic waves, asecond-mode-like wave train evolves. Because of the differentpropagation velocities of these two waves, they get separated.However, the computational domain is too small for the waves tocompletely separate from each other. Hence, the view on the second-mode wave packet remains limited due to the remnants of theexpanding disturbance. However, these remnants have their origin inthe applied perturbation method and were also present in theexperiments.Several parameters of the second-mode wave packet could be

extracted. For example, the frequency content and the amplificationrate agreed with stability computations. The computed spectra alsoshowed growing amplitudes in a broad frequency bandwith a peak at20 kHz. This is in the range of the oblique first mode. These waveswere also excited by the impulsive broadband stimulation. Besides,the fast acoustic waves contribute to this low-frequency signal. Otherparameters of the second-mode wave packet agreed with data fromthe literature, such as the wavelength, which is about two boundary-layer thicknesses, and the rope-like structure in computed schlierenimages. Furthermore, the results were compared with experimentaldata and the agreement was reasonable.With these simulations and the corresponding experiments, it was

shown that a second-mode wave packet can be generated fromdisturbances off the surface. For a start, a perturbation approach waschosen that could be generated comparatively easily in theexperiment. Furthermore, this perturbation approach simplified thenumerical simulations because high-temperature effects in thebreakdown region could be neglected. This was possible because thehot gas in the disturbance center does not enter the boundary layer.Besides this perturbation approach, the laser purturbator offersvarious other possibilities for inserting either temperature or pressuredisturbances or a combination of both. Future investigations willfocus on disturbances in front of the bow shock to resemble a naturaltransition scenario better.

Acknowledgments

This research was supported by the German Research Foundation(DFG) within the project RA 595/15-1. The authors gratefullyacknowledge theNorth-German SupercomputingAlliance, where allsimulationswere performed. Furthermore, the authors are grateful forthe kind assistance Stefan Hein of DLR Göttingen provided for theanalytic boundary-layer solutions.

References

[1] Mack, L.M., “BoundaryLayer Linear Stability Theory,” SpecialCourseon Stability and Transition of Laminar Flow, AGARD Rept. 709,March 1984, pp. 379–409.

[2] Heitmann, D., Radespiel, R., and Kähler, C., “Investigation of theResponse of a Hypersonic 2D Boundary Layer to Controlled AcousticDisturbances,” 48th AIAA Aerospace Sciences Meeting Including the

New Horizons Forum and Aerospace Exposition, AIAA Paper 2010-536, Jan. 2010.

[3] Heitmann, D., Radespiel, R., and Knauss, H., “Experimental Study ofMach 6 Boundary Layer Response to Laser Generated Disturbances,”41st AIAA Fluid Dynamics Conference and Exhibit, AIAAPaper 2011-3876, 2011.

[4] Sivasubramanian, J., Mayer, C., Laible, A., and Fasel, H., “NumericalInvestigation of Wavepackets in a Hypersonic Cone Boundary Layer atMach 6,” 39th AIAA Fluid Dynamics Conference, AIAA Paper 2009-3560, 2009.

[5] Bountin, D., Shiplyuk, A., and Maslov, A., “Evolution of NonlinearProcesses in aHypersonicBoundaryLayer on a SharpCone,” Journal ofFluid Mechanics, Vol. 611, Sept. 2008, pp. 427–442.doi:10.1017/S0022112008003030

[6] Kosinov, A. D., Maslov, A. A., and Shevelkov, S. G., “Experiments onthe Stability of Supersonic Boundary Layers,” Journal of Fluid

Mechanics, Vol. 219, Oct. 1990, pp. 621–633.doi:10.1017/S0022112090003111

[7] Ladoon, D. W., and Schneider, S. P., “Measurements of ControlledWave Packets at Mach 4 on a Cone at Angle of Attack,” 36th AerospaceSciences Meeting and Exhibit, AIAA Paper 1998-0436, 1998.

[8] Kendall, J. M., “Supersonic Boundary Layer Stability Experiments,”Boundary Layer Transition Study Group Meeting, Vol 2, BSD Rept.TR-67-213, 1967.

[9] Glumac, N., Elliott, G., and Boguszko, M., “Temporal and SpatialEvolution of a Laser Spark in Air,” AIAA Journal, Vol. 43, No. 9, 2005,pp. 1984–1994.doi:10.2514/1.14886

[10] Salyer, T. R., Collicott, S.H., andSchneider, S. P., “Characterizing LaserGenerated Hot Spots for Receptivity Studies,” AIAA Journal, Vol. 44,No. 12, 2006, pp. 2871–2878.doi:10.2514/1.13023

[11] Schmisseur, J. D., Schneider, S. P., and Collicott, S. H., “SupersonicBoundary Layer Response to Optically Generated Freestream Dis-turbances,” Experiments in Fluids, Vol. 33, No. 2, 2002, pp. 225–232.doi:10.1007/s00348-001-0392-5

[12] Elliott, G. S., Crawford, J., and Mosedale, A., “Enhancement of LargeScale Structures in Supersonic Axisymmetric Jets Using LaserExcitation,” 36th Aerospace Sciences Meeting and Exhibit, AIAAPaper 1998-0331, 1998.

[13] Yan, H., Adelgren, R., Elliott, G., Knight, D., Beutner, T., and Ivanov,M., “Laser Energy Deposition in Intersecting Shocks,” First Flow

Control Conference, AIAA Paper 2002-2729, 2002.[14] Adelgren, R. G., Yan, H., Elliott, G. S., Knight, D. D., Beutner, J. T.,

Zheltovodov, A. A., Ivanov, M., and Khotyanovsky, D., “LocalizedFlow Control by Laser Energy Deposition Applied to Edney IV ShockImpingement and Intersecting Shocks,” 41st Aerospace Sciences

Meeting and Exhibit, AIAA Paper 2003-0031, 2003.[15] Gerhold, T., “Overview of the Hybrid RANS Code Tau, in

MEGAFLOW—Numerical Flow Simulation for Aircraft Design,”Notes on Numerical Fluid Mechanics and Multidisciplinary Design,Vol. 89, Springer–Verlag, Berlin, 2005, pp. 81–92.

[16] Mack, A., and Hannemann, V., “Validation of the Unstructured DLR-TAU-Code for Hypersonic Flows,” 32nd Fluid Dynamics Conference

and Exhibit, AIAA Paper 2002-3111, 2002.[17] Wolf, T., Estorf, M., and Radespiel, R., “Simulation of the Time-

Dependent Flow Field in the Hypersonic Ludwieg Tube Braunschweig,”4th Atmospheric Reentry Vehicles and Systems, Arcachon, France, 2005.

[18] Taylor, G. I., “The Formation of a Blast Wave by a Very IntenseExplosion. I. Theoretical Discussion,” Proceedings of the Royal Societyof London, Series A, Mathematical and Physical Sciences, Vol. 201,No. 1065, 1950, pp. 159–174.doi:10.1098/rspa.1950.0049

[19] Mori, K., Komurasaki, K., and Arakawa, Y., “Energy Transferfrom a Laser Pulse to a Blast Wave in Reduced-Pressure AirAtmospheres,” Journal of Applied Physics, Vol. 95, No. 11, 2004,pp. 5979–5983.doi:10.1063/1.1710721

[20] Korobeinikov, V. P., “Gas Dynamics of Explosions,” Annual Review of

Fluid Mechanics, Vol. 3, Jan. 1971, pp. 317–346.doi:10.1146/annurev.fl.03.010171.001533

[21] Sedov, L. I., Similarity and Dimensional Methods in Mechanics, CRCPress, 10th ed., Boca Raton, FL, 1993, pp. 296–311.

[22] Bach, G. G., and Lee, J. H. S., “Analytical Solution for Blast Waves,”AIAA Journal, Vol. 8, No. 2, 1970, pp. 271–275.doi:10.2514/3.5655

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[23] Sims, J. L., “Tables For Supersonic Flow Around Right Circular Conesat Zero Angle of Attack,” NASA SP-3004, 1964.

[24] Schneider, S. P., “Effects of Roughness on Hypersonic Boundary-LayerTransition,” Journal of Spacecraft and Rockets, Vol. 45, No. 2, 2008,pp. 193–209.doi:10.2514/1.29713

[25] Van Driest, E. R., “Investigation of Laminar Boundary Layer inCompressible Fluids Using the Crocco Method,” NACA TN-2597,1952.

[26] Mangler, W., “Zusammenhang Zwischen Ebenen und Rotationssymme-trischen Grenzschichten in Kompressiblen Flüssigkeiten,” Zeitschrift für

AngewandteMathematikundMechanik,Vol. 28,No.4,1948, pp. 97–103.doi:10.1002/zamm.19480280401

[27] Schrauf, G., “LILO 2.0—User’s Guide and Tutorial,” GSSC TR-6,2004.

[28] Demetriades, A., “Hypersonic Viscous Flow over a Slender ConePart III: Laminar Instability and Transition,” AIAA Seventh Fluid and

Plasma Dynamics Conference, AIAA Paper 1974-0535, June 1974.

R. KimmelAssociate Editor

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