46th AIAA Aerospace Sciences Meeting, 7–10 Jan. 2008, Reno NV

Control of Shock-Wave / Boundary-Layer InteractionUsing Volumetric Energy Deposition

Jonathan Poggie!

Air Force Research Laboratory, Wright-Patterson AFB, OH 45433-7512 USA

A numerical study of three Mach 14 compression ramp flows was carried out in order todetermine whether a moderate power input through an electromagnetic actuator locatedupstream of separation would lead to beneficial structural changes in the flow. First, highresolution calculations were carried out for a flat-plate boundary layer flow, and volumetricenergy addition was found to introduce streamwise vorticity and three-dimensionality intothe flow. Then the 15!, 18!, and 24! compression ramp configurations of Holden andMoselle (ARL Report 70-0002) were examined, and reasonable agreement was obtainedbetween calculations of the baseline flow-fields and experimental measurements on the wallcenterline. For all three configurations, volumetric heat addition in the incoming boundary,at a station half-way down the flat plate, lead to decreased pressure, skin friction, andheat transfer directly downstream of the actuator, with a corresponding increase at, andoutboard of, the actuator. Volumetric energy deposition is judged to have useful flowcontrol applications, but careful consideration of possible penalties must be made in anydesign application.

I. Introduction

Interest in electromagnetic control of high-speed flows dates to the mid-1950s, when the problem ofhypersonic atmospheric entry was first being explored. Given the high temperatures in the shock layeraround a reentry vehicle, and the concomitant ionization and electrical conductivity, it was natural toconsider exploiting electromagnetic e!ects for flow control. Interest in large-scale plasma-aerodynamics haswaxed and waned several times in the intervening decades, with the initial enthusiasm eventually dampedeach time by the realities of the weight and complexity of high-strength magnets and power conditioningequipment. Because of their favorable weight and power consumption properties, small-scale actuators basedon glow and arc discharges have become increasingly popular in recent years.

The Air Force Research Laboratory Computational Sciences Branch has been involved in the numericalmodeling of electromagnetic flow control devices for several years.1–4 This high-fidelity modeling, employinga magnetohydrodynamic model or a drift-di!usion-Poisson model, is computationally expensive, however.Many questions of engineering design interest can be answered using a reduced-order model, in which theaction of the actuator is represented by specified force and energy source terms in the fluid conservationequations.

This was the approach taken in recent work on the control of a Mach 14 compression ramp flow.5,6 Two-and three-dimensional calculations were carried out for a series of compression ramp flows originally studiedby Holden and Moselle.7 Validation calculations were first carried out on the baseline flow, and reasonableagreement was obtained with pressure, skin friction, and heat flux data. The e!ects of steady body forces,volumetric heating, and surface heating were then explored for the 24! ramp configuration. Both forms ofheating, as well as upstream- and upward-directed body forces, were found to have a beneficial e!ect onthe flow. Best results were obtained with surface heating and with upward-directed body forces. Actuationwas found to cause the shear layer to reattach on the ramp at a slightly shallower angle, leading to reducedvelocity and temperature gradients at reattachment, and thus a reduction in the heat transfer rate.

"Senior Aerospace Engineer, AFRL/VAAC, Bldg. 146 Rm. 225, 2210 Eighth St. Associate Fellow AIAA.This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

Finally, periodic, unsteady actuation with heating and and outward/upward force were explored. Theunsteady actuator was found to introduce a region of hot, slow fluid in the boundary layer flow that convecteddownstream through reattachment. As it passed through the reattachment region, it led to reduced heatingnear the model centerline, but increased heating outboard. After the structure passed, the flow returned toits previous state.

The present paper revisits this problem in more detail. Well-resolved calculations have been carriedout, first for the flat-plate boundary layer alone, and then for the full ramp configuration. An attempt hasbeen made to resolve the vortices introduced into the boundary layer flow by the actuators. Flow controlstudies have been carried out for all three ramp configurations (15!, 18!, and 24!), rather than just thefully-separated 24! case studied earlier.

II. Methods

An implicit, central-di!erence scheme was employed to solve the fluid conservation laws with a modelsource term for volumetric energy deposition. Calculations were carried out with a code, PS3D, developedby the author. The physical model and numerical procedure are described in this section.

The conservation of mass, momentum, and energy for the overall gas is expressed as:

!"

!t+! · ("u) = 0 (1)

!

!t("u) +! · ("uu"!) = 0 (2)

!E!t

+! · (uE "! · u + Q) = S (3)

where " is the gas density, u is its velocity, ! is the total stress tensor, E = "(# + u2/2) is the total fluidenergy, # is the internal energy, and Q is the heat flux. An energy source term S is included on the righthand side of the energy equation.

The total stress tensor ! is given by the usual constitutive equation for a Newtonian fluid and the heatflux Q follows Fourier’s heat conduction law:

"ij = "p$ij + µ

!!ui

!xj+

!uj

!xi

"" 2

3µ

!uk

!xk$ij (4)

Qi = "k!T

!xi(5)

where p is the pressure, µ is the viscosity, and k is the thermal conductivity. The transport coe#cients wereevaluated using the correlations given in Ref. 8. The working fluid (air) was assumed to be a calorically andthermally perfect gas: # = cvT and p = "RT , where T is the temperature, cv is the specific heat, and R isthe ideal gas constant.

A phenomenological heating model was considered, and its e!ects on the flow were evaluated. Thevolumetric heating model consisted of an exponential decay over an ellipsoidal region:

S =Q

%3/2abcexp

#"

!x

a

"2

"!

y

b

"2

"!

z

c

"2$

(6)

and was added to the total energy equation. The following rotation and translation of the coordinatesallowed the position and orientation of the energy deposition region to be adjusted:

x = (x" x0) cos & " (z " z0) sin & (7a)y = (y " y0) (7b)z = (x" x0) sin & + (z " z0) cos & (7c)

The angle & was taken to be the angle between the major axis of the elliptical energy deposition region andthe x-axis. Note that

%%%"#" S(x, y, z) dx dy dz = Q.

The conservation laws were solved using an approximately-factored, implicit scheme, related to thosedeveloped by Beam and Warming9 and Pulliam.10 All calculations were carried out using double-precision

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American Institute of Aeronautics and Astronautics Paper 2008-1090

arithmetic. Applying the standard transformation from physical coordinates (x, y, z) to grid coordinates(', (, )), the conservation equations (1)–(3) can be written in the form:

!U

!t+

!E

!'+

!F

!(+

!G

!)=

!Ev

!'+

!F v

!(+

!Gv

!)+ S (8)

where the usual notation11 is used. For example, U = [", "u, E ]T is the the vector of dependent variables, Eis a flux, U = U/J , E = ('xE + 'yF + 'zG)/J , and J is the Jacobian of the grid transformation.

Writing Eq. (8) as !U/!t = R, and discretizing in time, we have:

(1 + &)Un+1 " (1 + 2&)Un + &Un#1 = $tRn+1 (9)

where & = 0 for an implicit Euler scheme and & = 1/2 for a three point backward scheme. We introducesubiterations such that U

n+1 # Up+1, with $U = U

p+1 " Up. The right hand side Rn+1 is linearized

in the standard ‘thin layer’ manner. Collecting the implicit terms on the left hand side, and introducingapproximate factoring and a subiteration time step $t gives:

&I " $t

1 + &(B + $!A1 + $!R1$! + Di!)

'$

&I " $t

1 + &($"A2 + $"R2$" + Di")

'$

&I " $t

1 + &($#A3 + $#R3$# + Di#)

'$U =

" $t

1 + &

((1 + &)Up " (1 + 2&)Un + &U

n#1

$t"Rp "DeU

p

)

(10)

where B is the source Jacobian, and A1#3 and R1#3 are flux Jacobians. The spatial derivatives are evaluatedusing second order central di!erences.

The symbols Di and De are, respectively, the implicit and explicit damping operators described byPulliam.10 The explicit damping operator uses a nonlinear blend of second- and fourth-order damping.12

In the implementation of the computer code, multi-level parallelism is exploited by using vectorization,multi-threading with OpenMP commands,13 and multi-block decomposition implemented through MPI com-mands.14 Further, the code is set up to run in either a time-accurate mode, or with local time stepping toaccelerate convergence.

It was found to be e#cient in many cases to compute an initial solution using a low-storage fourth-orderRunge-Kutta time-integration method (e.g., see Sec. 6.6.8 of Ref. 11) and local time-stepping, and then tocompute the final solution using the implicit method with a global time-step, as described above.

III. Results

Three-dimensional calculations were carried out for a series of Mach 14 compression ramp flows originallystudied by Holden and Moselle.7 These flows have been used as a benchmark case in a number of previouscomputational studies.15–20 The ramp configuration consisted of an initial flat plate, of length 439 mm andwidth 610 mm, mounted parallel to the freestream, followed by a second plate, inclined to the freestreamby an angle of 15!, 18!, or 24!. The freestream Mach number was 14.1, the Reynolds number based on thelength of the initial flat plate was 1.04 $ 105, and the ratio of wall temperature to freestream temperaturewas 4.1.

As a preliminary step, a set of calculations was carried out to determine the e!ect of volumetric energydeposition on the initial flat plate boundary layer flow (Fig. 1a). This allowed careful resolution of theperturbed boundary layer that, in the control cases, forms the inflow boundary condition of the separatedzone. Then the full flat plate and ramp configuration was computed for each of the three ramp angles, andcompared to the Holden-Moselle experiments (Fig. 1b). Finally, the e!ect of energy deposition on the threeflows was assessed.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

A. Boundary Layer Flow

A set of calculations was carried out to determine the e!ect of volumetric energy deposition on the initial flatplate boundary layer flow. The configuration was a sharp leading edge flat plate, 439 mm long and 300 mmwide. (Note that the plate is narrower than the ramp considered in the next section.) The freestreamconditions were identical to those of the Holden-Moselle experiments: M = 14.1, ReL = 1.04 $ 105, andTw/T" = 4.1. Periodic boundary conditions were applied in the z-direction.

For the energy deposition cases, an ellipsoidal energy source (Eqs. 6-7) was situated at x0 = 220 mm,y0 = 8 mm, z0 = 150 mm. The characteristic dimensions of the ellipsoid were a = 20 mm, b = 5 mm,c = 20 mm. The inclination angle was & = 0, and the total power deposited in the flow was Q = 100 W.

A grid resolution study was carried out for this flow, with grids of 50$50$50 points, 100$100$100 points,and 200 $ 200 $ 200 points. Profiles of the wall centerline properties are shown in Fig. 2. The figure alsoshows corresponding experimental data for the three ramp flows, in the relevant region upstream of the shockinteraction. The pressure coe#cient, skin friction coe#cient, and heat transfer coe#cient (Stanton number)are defined respectively as: Cp = 2(pw"p")/(""U2

"), Cf = 2*w/(""U2"), and Ch = qw/[""U"(H""Hw)].

Here H is the total enthalpy and the subscripts% and w indicate that the quantity is evaluated, respectively,in the freestream or at the wall.

The pressure coe#cient is shown in Fig. 2a. The solution is seen to be grid-converged for both thebaseline cases (solid lines) and the heating cases (dash-dot lines). The numerical results fall somewhat abovethe experimental data. Similar results are seen for the skin friction coe#cient (Fig. 2b) and heat transfercoe#cient (Fig. 2c), although finer grids are required to resolve these quantities for the cases with energydeposition.

Figure 3 illustrates the e!ect of energy deposition on the boundary layer flow for the fine-grid solution.The magnitude of the skin friction, along with selected trajectories of the shear stress vector, are shownin Fig. 3a. For the baseline solution, the flow properties are uniform along the z-direction. With thepresence of volumetric heat release, the flow diverges around a virtual bump in the boundary layer. Thereis a corresponding introduction of a w-component of velocity into the boundary flow. This is illustrated inFig. 3b, which shows the magnitude of the spanwise velocity in the outlet plane of the computational domain(x = 439 mm), along with selected sectional streamlines. (Note that, in the figure, these streamlines end atthe shock.)

Density contours and stream ribbons are shown in a three-dimensional view of the computational domainin Fig. 3c. A density perturbation is evident at the wall, just downstream of the heat release zone. A regionof reduced density is seen where the wake of the heating zone intersects the outlet plane, and intersection ofthe weak shock introduced by the thermal bump with the outlet plane appears as a red ‘frown.’

Two stream ribbons are shown in the figure to indicate the distortion of the boundary layer. The ribbonsare initially perpendicular to the flat plate, but as they near the heating region, the lower portion of theboundary layer is diverted outward. This velocity gradient !w/!y corresponds to a streamwise componentof vorticity +x. Contours and iso-surfaces of streamwise vorticity are shown in Fig. 3d. The e!ect is seento be similar to flow over a small bump in the boundary layer, with a vortex system wrapped around thedisturbance.

B. Compression Ramp

Calculations were carried out for the full compression ramp configurations using grids of 215$ 60$ 60 and415$120$120 points. Again, each ramp configuration consisted of an initial flat plate of length L = 439 mmand width W = 610 mm, mounted parallel to the freestream, followed by a second plate, inclined to thefreestream by an angle of & = 15!, 18!, or 24!. Periodic boundary conditions were applied in the z-direction.For the energy addition (Eqs. 6-7) cases, parameters similar to those used in the flat plate boundary layercalculations were employed: x0 = 220 mm, y0 = 8 mm, z0 = 305 mm, a = 20 mm, b = 5 mm, c = 20 mm,& = 0, and Q = 100 W.

The present calculations di!er from the calculations described in previous work.5,6 The energy depositionis less localized; the parameters a = b = c = 5 mm were used in the earlier papers. Further, the grid resolutionhas been increased in the present work, both with increased mesh size and increased grid clustering.

Figure 4 compares computation and experiment for the wall centerline properties of the baseline flow.Both the coarse- and fine-grid numerical solutions are shown. For the 15! case, Figs. 4a-c, the computationalresults are fairly close to the experimental values, and the calculations on the two grids produce similar

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American Institute of Aeronautics and Astronautics Paper 2008-1090

results. The peak heat transfer rate is somewhat over-predicted. Similar results are obtained for the 18!ramp, Figs. 4d-f. For the fully separated 24! ramp case, only the fine grid solution has acceptable agreementwith the experimental data.

Figure 5 shows centerline profiles for the 15!, 18!, and 24! ramps, with and without energy addition.Only modest changes are seen in the pressure distribution (Figs. 5a,d,g) with heating, primarily reflectingthe perturbation near the heat source and a downstream shift of the reattachment location. In each case,heating is seen to reduce the skin friction (Figs. 5b,e,h) and heat transfer (Figs. 5c,f,i) on the ramp, for apenalty of additional heat transfer near the energy source.

Temperature contours in the symmetry plane x = 305 mm (side view) are shown in Fig. 6. The samegeneral flow structure is present for each of the baseline cases (Figs. 6a,c,e). A weak shock is seen emanatingfrom the plate leading edge, generated by the hypersonic boundary layer displacement e!ect. A strongoblique shock, due to the turning angle & of the ramp, is seen farther downstream. In the 24! case, a shock/ compression-wave system appears with flow separation, and these waves interact with the leading edgeshock and ramp shock in a complex manner near reattachment. (Hung and MacCormack15 have identifiedthis as an Edney21 Type VI interaction.) There is a striking thinning of the boundary layer there. Thisregion is often called the boundary layer ‘neck.’15 Farther downstream, the boundary layer is altered by thepresence of an embedded shear layer generated by the shock intersection near the neck.

With the presence of the volumetric heat addition (Figs. 6b,d,f), a zone of increased temperature appearsin the boundary layer, just downstream of the energy addition. There is a weak shock associated with thisperturbation. Downstream, the flow structure is altered. In particular, a region of hot gas appears abovethe ramp corner and near the boundary layer neck.

Maximum temperatures for each case are shown in Table 1. Since Tmax < 1700 K for all cases, the perfectgas model should be a reasonable assumption for all cases.

Case Baseline Heating15! ramp 1105 K 1657 K18! ramp 1248 K 1657 K24! ramp 1526 K 1660 K

Table 1. Maximum static temperature for each ramp flow calculation, 415"120"120 grid.

Figure 7 shows the wall heat flux distribution for each case. The view is downward onto the plateand ramp, with the corner located at x/L = 1 in each case, and flow from left to right. The left column(Figs. 7a,c,e) shows the baseline wall heat flux, which is uniform in the z-direction. The right column(Figs. 7b,d,f) shows the cases with volumetric energy addition.

In each control case, a spot with a higher heating rate is seen just downstream of the energy addition atx/L = 0.5. On the ramp, in a strip aligned with the actuator, the heat transfer rate is significantly reduced,as was seen in Fig. 5. Outboard of this strip, an increased heat transfer rate is seen, possibly o!setting theadvantage of control. The extent of this penalty region diminishes with increasing ramp angle and interactionstrength.

IV. Summary and Conclusions

A numerical study of three Mach 14 compression ramp flows was carried out in order to determinewhether a moderate power input through an electromagnetic actuator located upstream of separation couldlead to beneficial structural changes in the flow. The e!ect of the actuator was represented by a reduced-order model, in which a specified heat source term was introduced in the energy conservation equation. Thework of two recent papers5,6 was revisited in more detail, addressing three ramp configurations (15!, 18!,and 24!) studied experimentally by Holden and Moselle.7

First, high resolution calculations were carried out for a flat-plate boundary layer flow. Volumetric energyaddition was found to introduce streamwise vorticity and three-dimensionality into the flow.

Next, the 15!, 18!, and 24! compression ramp configurations of Holden and Moselle were examined,and reasonable agreement was obtained between calculations of the baseline flow-fields and experimentalmeasurements on the wall centerline. Grid resolution studies showed the importance of using adequate

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American Institute of Aeronautics and Astronautics Paper 2008-1090

spatial resolution in computations of this class of flows.For all three ramp angles, volumetric heat addition in the incoming boundary, at a station half-way down

the flat plate, lead to decreased pressure, skin friction, and heat transfer directly downstream of the actuator.A corresponding increase was observed at and outboard of the actuator.

In Ref. 5, which employed coarser grid resolution and more localized heating (surface heating, and volu-metric heating with a 5 mm characteristic dimension), a penalty in o!-centerline heating was not observed.For the unsteady actuation examined in Ref. 6, the penalty in o!-centerline heating was observed for certainstages in the actuation cycle, but the total energy deposition for control was greater (2 kW vs. 0.1 kW).

The e!ects seen in the present paper are not a!ected qualitatively by grid resolution: the coarse grid casesdisplay the same heat transfer pattern as the fine grid solutions shown in Fig. 7, but with lower heating ratesoverall. A similar result is obtained when the total energy deposited in the flow is reduced from 100 W to20 W. (For brevity, the latter results are not presented in this paper.) Thus, the o!-centerline heat transferpenalty seems to be related to the larger spanwise length scale used in the present work as compared toRefs. 5, 6 (20 mm vs. 5 mm). These results merit further investigation.

Overall, heat addition was seen to have a strong e!ect on the flowfield. There appears to be somepotential for flow control applications, but careful consideration of possible penalties must be made in orderto obtain a useful application of this technique.

Acknowledgments

This project was sponsored in part by the Air Force O#ce of Scientific Research under grants monitoredby J. Schmisseur and F. Fahroo, and by grants of High Performance Computing time from the followingDepartment of Defense Major Shared Resource Centers: the Army High Performance Computing ResearchCenter (AHPCRC), the Aeronautical Systems Center (ASC), and the Naval Oceanographic O#ce (NAVO).

References

1Poggie, J. and Gaitonde, D. V., “Magnetic Control of Flow Past a Blunt Body: Numerical Validation and Exploration,”Physics of Fluids, Vol. 14, No. 5, 2002, pp. 1720–1731.

2Gaitonde, D. V. and Poggie, J., “Implicit Technique for Three-Dimensional Turbulent Magnetoaerodynamics,” AIAAJournal , Vol. 41, No. 11, 2003, pp. 2179–2191.

3Poggie, J. and Sternberg, N., “Transition from the Constant Ion Mobility Regime to the Ion-Atom Charge-ExchangeRegime for Bounded Collisional Plasmas,” Physics of Plasmas, Vol. 12, No. 2, Feb. 2005.

4Poggie, J., “Discharge Modeling for Flow Control Applications,” AIAA Paper 2008-1357, American Institute of Aero-nautics and Astronautics, Reston VA, January 2008.

5Poggie, J., “Plasma-Based Control of Shock-Wave / Boundary-Layer Interaction,” AIAA Paper 2006-1007, AmericanInstitute of Aeronautics and Astronautics, Reston VA, January 2006.

6Poggie, J., “Plasma-Based Hypersonic Flow Control,” AIAA Paper 2006-3567, American Institute of Aeronautics andAstronautics, Reston VA, June 2006.

7Holden, M. S. and Moselle, J. R., “Theoretical and Experimental Studies of the Shock Wave - Boundary Layer Interactionon Compression Surfaces in Hypersonic Flow,” Tech. Rep. ARL 70-0002, Aerospace Research Laboratories, January 1970.

8White, F. M., Viscous Fluid Flow , McGraw-Hill, New York, 2nd ed., 1991.9Beam, R. and Warming, R., “An Implicit Factored Scheme for the Compressible Navier-Stokes Equations,” AIAA

Journal , Vol. 16, No. 4, 1978, pp. 393–402.10Pulliam, T. H., “Implicit Finite-Di!erence Simulations of Three-Dimensional Compressible Flow,” AIAA Journal , Vol. 18,

No. 2, 1980, pp. 159–167.11Ho!mann, K. A. and Chiang, S. T., Computational Fluid Dynamics, Engineering Educational System, Wichita KS, 4th

ed., 2000, 2 vols.12Jameson, A., Schmidt, W., and Turkel, E., “Numerical Solutions of the Euler Equations by a Finite Volume Method Using

Runge-Kutta Time Stepping Schemes,” AIAA Paper 81-1259, American Institute of Aeronautics and Astronautics, Reston, VA,1981.

13Chandra, R., Dagum, L., Kohr, D., Maydan, D., McDonald, J., and Menon, R., Parallel Programming in OpenMP ,Academic Press, San Diego, 2001.

14Gropp, W., Lusk, E., and Skjellum, A., Using MPI: Portable Parallel Programming with the Message-Passing Interface,The MIT Press, Cambridge, MA, 2nd ed., 1999.

15Hung, C. M. and MacCormack, R. W., “Numerical Solutions of Supersonic and Hypersonic Laminar Compression RampFlows,” AIAA Journal , Vol. 14, No. 4, 1976, pp. 475–481.

16Power, G. D. and Barber, T. J., “Analysis of Complex Hypersonic Flows with Strong Viscous/Inviscid Interaction,”AIAA Journal , Vol. 26, No. 7, 1988, pp. 832–840.

17Rizzetta, D. and Mach, K., “Comparative Numerical Study of Hypersonic Compression Ramp Flows,” AIAA Paper89-1877, American Institute of Aeronautics and Astronautics, Reston, VA, June 1989.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

18Rudy, D. H., Thomas, J. L., Kumar, A., Gno!o, P. A., and Chakravarthy, S. R., “Computation of Laminar HypersonicCompression-Corner Flows,” AIAA Journal , Vol. 29, No. 7, 1991, pp. 1108–1113.

19Gaitonde, D. and Shang, J. S., “The Performance of Flux-Split Algorithms in High-Speed Viscous Flows,” AIAA Paper92-0186, American Institute of Aeronautics and Astronautics, Reston, VA, January 1992.

20Updike, G. A., Shang, J. S., and Gaitonde, D. V., “Hypersonic Separated Flow Control Using Magneto-AerodynamicInteraction,” AIAA Paper 2005-0164, American Institute of Aeronautics and Astronautics, Reston, VA, January 2005.

21Edney, B., “Anomalous Heat-Transfer and Pressure Distributions on Blunt Bodies at Hypersonic Speeds in the Presenceof an Impinging Shock,” FFA Report 116, Aeronautical Research Institute of Sweden, Stockholm, Sweden, February 1968.

(a) Flat plate. (b) Compression ramp.

Figure 1. Configuration for flat plate and ramp computations. Approximate location of energy depositionindicated with colored circles. Side and rear boundaries included to illustrate extent of computational domain.

x/L

Cp

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

experiment, 15 deg rampexperiment, 18 deg rampexperiment, 24 deg rampcomputation, baseline, 50x50x50 gridcomputation, baseline, 100x100x100 gridcomputation, baseline, 200x200x200 gridcomputation, heating, 50x50x50 gridcomputation, heating, 100x100x100 gridcomputation, heating, 200x200x200 grid

(a) Pressure coe"cient.

x/L

Cf

0 0.2 0.4 0.6 0.8 10

0.01

0.02

(b) Skin friction coe"cient.

x/L

Ch

0 0.2 0.4 0.6 0.8 10

0.01

0.02

(c) Heat transfer coe"cient.

Figure 2. Grid resolution study: wall centerline profiles for the flat plate boundary layer flow.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

(a) Wall skin friction, contour interval 1" 10#3. (b) Sectional streamlines and w-velocity in outlet plane, con-tour interval 5 m/s.

(c) Density contours and stream ribbons, contour interval 1"10#4 kg/m3.

(d) Contours and iso-surfaces of streamwise vorticity (!x),contour interval 0.5 s#1.

Figure 3. E!ect of volumetric heating on flat plate boundary layer flow, 200"200"200 grid.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

x/L

Cp

0 0.5 1 1.5 20

0.1

0.2

0.3experiment, 15 deg ramp215x60x60 grid, 15 deg ramp415x120x120 grid, 15 deg ramp

(a) Pressure coe"cient, 15!-ramp.

x/L

Cf

0 0.5 1 1.5 2

0

0.005

0.01

0.015

(b) Skin friction coe"cient, 15!-ramp.

x/L

Ch

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

(c) Heat transfer coe"cient, 15!-ramp.

x/L

Cp

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4experiment, 18 deg ramp215x60x60 grid, 18 deg ramp415x120x120 grid, 18 deg ramp

(d) Pressure coe"cient, 18!-ramp.

x/L

Cf

0 0.5 1 1.5 2

0

0.005

0.01

0.015

0.02

(e) Skin friction coe"cient, 18!-ramp.

x/L

Ch

0 0.5 1 1.5 20

0.005

0.01

0.015

(f) Heat transfer coe"cient, 18!-ramp.

x/L

Cp

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1experiment, 24 deg ramp215x60x60 grid, 24 deg ramp415x120x120 grid, 24 deg ramp

(g) Pressure coe"cient, 24!-ramp.

x/L

Cf

0 0.5 1 1.5 2-0.01

0

0.01

0.02

0.03

0.04

(h) Skin friction coe"cient, 24!-ramp.

x/L

Ch

0 0.5 1 1.5 20

0.01

0.02

0.03

(i) Heat transfer coe"cient, 24!-ramp.

Figure 4. Comparison of computation and experiment for baseline compression ramp flow.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

x/L

Cp

0 0.5 1 1.5 20

0.1

0.2

0.3heatingbaseline

(a) Pressure coe"cient, 15!-ramp.

x/L

Cf

0 0.5 1 1.5 2

0

0.005

0.01

0.015

(b) Skin friction coe"cient, 15!-ramp.

x/L

Ch

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

(c) Heat transfer coe"cient, 15!-ramp.

x/L

Cp

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4heatingbaseline

(d) Pressure coe"cient, 18!-ramp.

x/L

Cf

0 0.5 1 1.5 2

0

0.01

0.02

(e) Skin friction coe"cient, 18!-ramp.

x/L

Ch

0 0.5 1 1.5 20

0.005

0.01

0.015

(f) Heat transfer coe"cient, 18!-ramp.

x/L

Cp

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8heatingbaseline

(g) Pressure coe"cient, 24!-ramp.

x/L

Cf

0 0.5 1 1.5 2-0.01

0

0.01

0.02

0.03

0.04

(h) Skin friction coe"cient, 24!-ramp.

x/L

Ch

0 0.5 1 1.5 20

0.01

0.02

0.03

(i) Heat transfer coe"cient, 24!-ramp.

Figure 5. E!ect of volumetric heating on centerline properties of compression ramp flow, 415"120"120 grid.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

(a) Baseline case, 15!-ramp. (b) Heating case, 15!-ramp.

(c) Baseline case, 18!-ramp. (d) Heating case, 18!-ramp.

(e) Baseline case, 24!-ramp. (f) Heating case, 24!-ramp.

Figure 6. Temperature distribution for each case, 50 K contour, 415"120"120 grid.

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American Institute of Aeronautics and Astronautics Paper 2008-1090

(a) Baseline case, 15!-ramp, contour interval 5" 10#4. (b) Heating case, 15!-ramp, contour interval 5" 10#4.

(c) Baseline case, 18!-ramp, contour interval 1" 10#3. (d) Heating case, 18!-ramp, contour interval 1" 10#3.

(e) Baseline case, 24!-ramp, contour interval 2" 10#3. (f) Heating case, 24!-ramp, contour interval 2" 10#3.

Figure 7. Wall heat flux distribution for each case, 415"120"120 grid.

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American Institute of Aeronautics and Astronautics Paper 2008-1090