Control of Shock-Wave / Boundary-Layer Interaction Using Volumetric Energy Deposition

Transcription

1 46 th AIAA Aerospace Sciences Meeting, 7 1 Jan. 28, Reno NV Control of Shock-Wave / Boundary-Layer Interaction Using Volumetric Energy Deposition Jonathan Poggie Air Force Research Laboratory, Wright-Patterson AFB, OH USA A numerical study of three Mach 14 compression ramp flows was carried out in order to determine whether a moderate power input through an electromagnetic actuator located upstream of separation would lead to beneficial structural changes in the flow. First, high resolution calculations were carried out for a flat-plate boundary layer flow, and volumetric energy addition was found to introduce streamwise vorticity and three-dimensionality into the flow. Then the 15, 18, and 24 compression ramp configurations of Holden and Moselle (ARL Report 7-2) were examined, and reasonable agreement was obtained between calculations of the baseline flow-fields and experimental measurements on the wall centerline. 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, and heat transfer directly downstream of the actuator, with a corresponding increase at, and outboard of, the actuator. Volumetric energy deposition is judged to have useful flow control applications, but careful consideration of possible penalties must be made in any design application. I. Introduction Interest in electromagnetic control of high-speed flows dates to the mid-195s, when the problem of hypersonic atmospheric entry was first being explored. Given the high temperatures in the shock layer around a reentry vehicle, and the concomitant ionization and electrical conductivity, it was natural to consider exploiting electromagnetic effects for flow control. Interest in large-scale plasma-aerodynamics has waxed and waned several times in the intervening decades, with the initial enthusiasm eventually damped each time by the realities of the weight and complexity of high-strength magnets and power conditioning equipment. Because of their favorable weight and power consumption properties, small-scale actuators based on glow and arc discharges have become increasingly popular in recent years. The Air Force Research Laboratory Computational Sciences Branch has been involved in the numerical modeling of electromagnetic flow control devices for several years. 1 4 This high-fidelity modeling, employing a magnetohydrodynamic model or a drift-diffusion-poisson model, is computationally expensive, however. Many questions of engineering design interest can be answered using a reduced-order model, in which the action of the actuator is represented by specified force and energy source terms in the fluid conservation equations. This was the approach taken in recent work on the control of a Mach 14 compression ramp flow. 5, 6 Twoand three-dimensional calculations were carried out for a series of compression ramp flows originally studied by Holden and Moselle. 7 Validation calculations were first carried out on the baseline flow, and reasonable agreement was obtained with pressure, skin friction, and heat flux data. The effects of steady body forces, volumetric heating, and surface heating were then explored for the 24 ramp configuration. Both forms of heating, as well as upstream- and upward-directed body forces, were found to have a beneficial effect on the flow. Best results were obtained with surface heating and with upward-directed body forces. Actuation was found to cause the shear layer to reattach on the ramp at a slightly shallower angle, leading to reduced velocity and temperature gradients at reattachment, and thus a reduction in the heat transfer rate. Senior Aerospace Engineer, AFRL/VAAC, Bldg. 146 Rm. 225, 221 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. 1 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

2 Finally, periodic, unsteady actuation with heating and and outward/upward force were explored. The unsteady actuator was found to introduce a region of hot, slow fluid in the boundary layer flow that convected downstream through reattachment. As it passed through the reattachment region, it led to reduced heating near the model centerline, but increased heating outboard. After the structure passed, the flow returned to its previous state. The present paper revisits this problem in more detail. Well-resolved calculations have been carried out, first for the flat-plate boundary layer alone, and then for the full ramp configuration. An attempt has been made to resolve the vortices introduced into the boundary layer flow by the actuators. Flow control studies have been carried out for all three ramp configurations (15, 18, and 24 ), rather than just the fully-separated 24 case studied earlier. II. Methods An implicit, central-difference scheme was employed to solve the fluid conservation laws with a model source term for volumetric energy deposition. Calculations were carried out with a code, PS3D, developed by 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: E t ρ + (ρu) = (1) t (ρu)+ (ρuu Σ) = (2) t + (ue Σ u + Q) = S (3) where ρ is the gas density, u is its velocity, Σ is the total stress tensor, E = ρ(ɛ + u 2 /2) is the total fluid energy, ɛ is the internal energy, and Q is the heat flux. An energy source term S is included on the right hand side of the energy equation. The total stress tensor Σ is given by the usual constitutive equation for a Newtonian fluid and the heat flux Q follows Fourier s heat conduction law: ( ui Σ ij = pδ ij + µ + u ) j 2 x j x i 3 µ u k δ ij (4) x k Q i = k T (5) x i where p is the pressure, µ is the viscosity, and k is the thermal conductivity. The transport coefficients were evaluated using the correlations given in Ref. 8. The working fluid (air) was assumed to be a calorically and thermally perfect gas: ɛ = c v T and p = ρrt, where T is the temperature, c v is the specific heat, and R is the ideal gas constant. A phenomenological heating model was considered, and its effects on the flow were evaluated. The volumetric heating model consisted of an exponential decay over an ellipsoidal region: S = Q π 3/2 abc exp [ ( ) 2 ˆx a (ŷ ) 2 (ẑ ) ] 2 b c and was added to the total energy equation. The following rotation and translation of the coordinates allowed the position and orientation of the energy deposition region to be adjusted: (6) ˆx =(x x ) cos θ (z z ) sin θ ŷ =(y y ) ẑ =(x x ) sin θ +(z z ) cos θ (7a) (7b) (7c) The angle θ was taken to be the angle between the major axis of the elliptical energy deposition region and the 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 those developed by Beam and Warming 9 and Pulliam. 1 All calculations were carried out using double-precision 2 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

3 arithmetic. Applying the standard transformation from physical coordinates (x, y, z) to grid coordinates (ξ, η, ζ), the conservation equations (1) (3) can be written in the form: U t + E ξ + F η + G ζ = E v ξ + F v η + G v ζ + S (8) where the usual notation 11 is used. For example, U =[ρ, ρu, E] T is the the vector of dependent variables, E is a flux, U = U/J, E =(ξ x E + ξ y F + ξ z G)/J, and J is the Jacobian of the grid transformation. Writing Eq. (8) as U/ t = R, and discretizing in time, we have: (1 + θ)u n+1 (1 + 2θ)U n + θu n 1 = tr n+1 (9) where θ = for an implicit Euler scheme and θ =1/2 for a three point backward scheme. We introduce subiterations such that U n+1 U p+1, with U = U p+1 U p. The right hand side R n+1 is linearized in the standard thin layer manner. Collecting the implicit terms on the left hand side, and introducing approximate factoring and a subiteration time step ˆt gives: [ I ˆt ] 1+θ (B + δ ξa 1 + δ ξ R 1 δ ξ + D iξ ) [ I ˆt ] 1+θ (δ ηa 2 + δ η R 2 δ η + D iη ) [ I ˆt ] (1) 1+θ (δ ζa 3 + δ ζ R 3 δ ζ + D iζ ) U = ˆt 1+θ {(1 + θ)u p (1 + 2θ)U n + θu n 1 } R p D e U p t where B is the source Jacobian, and A 1 3 and R 1 3 are flux Jacobians. The spatial derivatives are evaluated using second order central differences. The symbols D i and D e are, respectively, the implicit and explicit damping operators described by Pulliam. 1 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 commands. 14 Further, the code is set up to run in either a time-accurate mode, or with local time stepping to accelerate convergence. It was found to be efficient in many cases to compute an initial solution using a low-storage fourth-order Runge-Kutta time-integration method (e.g., see Sec of Ref. 11) and local time-stepping, and then to compute 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 originally studied by Holden and Moselle. 7 These flows have been used as a benchmark case in a number of previous computational studies The ramp configuration consisted of an initial flat plate, of length 439 mm and width 61 mm, mounted parallel to the freestream, followed by a second plate, inclined to the freestream by an angle of 15, 18, or 24. The freestream Mach number was 14.1, the Reynolds number based on the length of the initial flat plate was , and the ratio of wall temperature to freestream temperature was 4.1. As a preliminary step, a set of calculations was carried out to determine the effect of volumetric energy deposition on the initial flat plate boundary layer flow (Fig. 1a). This allowed careful resolution of the perturbed boundary layer that, in the control cases, forms the inflow boundary condition of the separated zone. Then the full flat plate and ramp configuration was computed for each of the three ramp angles, and compared to the Holden-Moselle experiments (Fig. 1b). Finally, the effect of energy deposition on the three flows was assessed. 3 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

4 A. Boundary Layer Flow A set of calculations was carried out to determine the effect of volumetric energy deposition on the initial flat plate boundary layer flow. The configuration was a sharp leading edge flat plate, 439 mm long and 3 mm wide. (Note that the plate is narrower than the ramp considered in the next section.) The freestream conditions were identical to those of the Holden-Moselle experiments: M = 14.1, Re L = , and T w /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 x = 22 mm, y = 8 mm, z = 15 mm. The characteristic dimensions of the ellipsoid were a = 2 mm, b = 5 mm, c = 2 mm. The inclination angle was θ =, and the total power deposited in the flow was Q = 1 W. A grid resolution study was carried out for this flow, with grids of points, points, and points. Profiles of the wall centerline properties are shown in Fig. 2. The figure also shows corresponding experimental data for the three ramp flows, in the relevant region upstream of the shock interaction. The pressure coefficient, skin friction coefficient, and heat transfer coefficient (Stanton number) are defined respectively as: = 2(p w p )/(ρ U ), 2 =2τ w /(ρ U ), 2 and = q w /[ρ U (H H w )]. 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 coefficient is shown in Fig. 2a. The solution is seen to be grid-converged for both the baseline cases (solid lines) and the heating cases (dash-dot lines). The numerical results fall somewhat above the experimental data. Similar results are seen for the skin friction coefficient (Fig. 2b) and heat transfer coefficient (Fig. 2c), although finer grids are required to resolve these quantities for the cases with energy deposition. Figure 3 illustrates the effect 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 shown in Fig. 3a. For the baseline solution, the flow properties are uniform along the z-direction. With the presence of volumetric heat release, the flow diverges around a virtual bump in the boundary layer. There is a corresponding introduction of a w-component of velocity into the boundary flow. This is illustrated in Fig. 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 at the shock.) Density contours and stream ribbons are shown in a three-dimensional view of the computational domain in Fig. 3c. A density perturbation is evident at the wall, just downstream of the heat release zone. A region of reduced density is seen where the wake of the heating zone intersects the outlet plane, and intersection of the 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 ribbons are initially perpendicular to the flat plate, but as they near the heating region, the lower portion of the boundary layer is diverted outward. This velocity gradient w/ y corresponds to a streamwise component of vorticity ω x. Contours and iso-surfaces of streamwise vorticity are shown in Fig. 3d. The effect is seen to be similar to flow over a small bump in the boundary layer, with a vortex system wrapped around the disturbance. B. Compression Ramp Calculations were carried out for the full compression ramp configurations using grids of and points. Again, each ramp configuration consisted of an initial flat plate of length L = 439 mm and width W = 61 mm, mounted parallel to the freestream, followed by a second plate, inclined to the freestream by an angle of θ = 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 layer calculations were employed: x = 22 mm, y = 8 mm, z = 35 mm, a = 2 mm, b = 5 mm, c = 2 mm, θ =, and Q = 1 W. The present calculations differ from the calculations described in previous work. 5, 6 The energy deposition is less localized; the parameters a = b = c = 5 mm were used in the earlier papers. Further, the grid resolution has 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 computational results are fairly close to the experimental values, and the calculations on the two grids produce similar 4 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

5 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 agreement with 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 reflecting the 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 a penalty of additional heat transfer near the energy source. Temperature contours in the symmetry plane x = 35 mm (side view) are shown in Fig. 6. The same general flow structure is present for each of the baseline cases (Figs. 6a,c,e). A weak shock is seen emanating from the plate leading edge, generated by the hypersonic boundary layer displacement effect. A strong oblique shock, due to the turning angle θ 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 edge shock and ramp shock in a complex manner near reattachment. (Hung and MacCormack 15 have identified this as an Edney 21 Type VI interaction.) There is a striking thinning of the boundary layer there. This region is often called the boundary layer neck. 15 Farther downstream, the boundary layer is altered by the presence 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 appears in the boundary layer, just downstream of the energy addition. There is a weak shock associated with this perturbation. Downstream, the flow structure is altered. In particular, a region of hot gas appears above the ramp corner and near the boundary layer neck. Maximum temperatures for each case are shown in Table 1. Since T max < 17 K for all cases, the perfect gas model should be a reasonable assumption for all cases. Case Baseline Heating 15 ramp 115 K 1657 K 18 ramp 1248 K 1657 K 24 ramp 1526 K 166 K Table 1. Maximum static temperature for each ramp flow calculation, grid. Figure 7 shows the wall heat flux distribution for each case. The view is downward onto the plate and ramp, with the corner located at = 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 at =.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 offsetting the advantage of control. The extent of this penalty region diminishes with increasing ramp angle and interaction strength. IV. Summary and Conclusions A numerical study of three Mach 14 compression ramp flows was carried out in order to determine whether a moderate power input through an electromagnetic actuator located upstream of separation could lead to beneficial structural changes in the flow. The effect of the actuator was represented by a reducedorder model, in which a specified heat source term was introduced in the energy conservation equation. The work of two recent papers 5, 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 energy addition 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 experimental measurements on the wall centerline. Grid resolution studies showed the importance of using adequate 5 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

6 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 volumetric heating with a 5 mm characteristic dimension), a penalty in off-centerline heating was not observed. For the unsteady actuation examined in Ref. 6, the penalty in off-centerline heating was observed for certain stages in the actuation cycle, but the total energy deposition for control was greater (2 kw vs..1 kw). The effects seen in the present paper are not affected qualitatively by grid resolution: the coarse grid cases display the same heat transfer pattern as the fine grid solutions shown in Fig. 7, but with lower heating rates overall. A similar result is obtained when the total energy deposited in the flow is reduced from 1 W to 2 W. (For brevity, the latter results are not presented in this paper.) Thus, the off-centerline heat transfer penalty seems to be related to the larger spanwise length scale used in the present work as compared to Refs. 5, 6 (2 mm vs. 5 mm). These results merit further investigation. Overall, heat addition was seen to have a strong effect on the flowfield. There appears to be some potential for flow control applications, but careful consideration of possible penalties must be made in order to obtain a useful application of this technique. Acknowledgments This project was sponsored in part by the Air Force Office of Scientific Research under grants monitored by J. Schmisseur and F. Fahroo, and by grants of High Performance Computing time from the following Department of Defense Major Shared Resource Centers: the Army High Performance Computing Research Center (AHPCRC), the Aeronautical Systems Center (ASC), and the Naval Oceanographic Office (NAVO). References 1 Poggie, J. and Gaitonde, D. V., Magnetic Control of Flow Past a Blunt Body: Numerical Validation and Exploration, Physics of Fluids, Vol. 14, No. 5, 22, pp Gaitonde, D. V. and Poggie, J., Implicit Technique for Three-Dimensional Turbulent Magnetoaerodynamics, AIAA Journal, Vol. 41, No. 11, 23, pp Poggie, J. and Sternberg, N., Transition from the Constant Ion Mobility Regime to the Ion-Atom Charge-Exchange Regime for Bounded Collisional Plasmas, Physics of Plasmas, Vol. 12, No. 2, Feb Poggie, J., Discharge Modeling for Flow Control Applications, AIAA Paper , American Institute of Aeronautics and Astronautics, Reston VA, January Poggie, J., Plasma-Based Control of Shock-Wave / Boundary-Layer Interaction, AIAA Paper 26-17, American Institute of Aeronautics and Astronautics, Reston VA, January Poggie, J., Plasma-Based Hypersonic Flow Control, AIAA Paper , American Institute of Aeronautics and Astronautics, Reston VA, June Holden, M. S. and Moselle, J. R., Theoretical and Experimental Studies of the Shock Wave - Boundary Layer Interaction on Compression Surfaces in Hypersonic Flow, Tech. Rep. ARL 7-2, Aerospace Research Laboratories, January White, F. M., Viscous Fluid Flow, McGraw-Hill, New York, 2nd ed., Beam, R. and Warming, R., An Implicit Factored Scheme for the Compressible Navier-Stokes Equations, AIAA Journal, Vol. 16, No. 4, 1978, pp Pulliam, T. H., Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow, AIAA Journal, Vol. 18, No. 2, 198, pp Hoffmann, K. A. and Chiang, S. T., Computational Fluid Dynamics, Engineering Educational System, Wichita KS, 4th ed., 2, 2 vols. 12 Jameson, 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 , American Institute of Aeronautics and Astronautics, Reston, VA, Chandra, R., Dagum, L., Kohr, D., Maydan, D., McDonald, J., and Menon, R., Parallel Programming in OpenMP, Academic Press, San Diego, Gropp, W., Lusk, E., and Skjellum, A., Using MPI: Portable Parallel Programming with the Message-Passing Interface, The MIT Press, Cambridge, MA, 2nd ed., Hung, C. M. and MacCormack, R. W., Numerical Solutions of Supersonic and Hypersonic Laminar Compression Ramp Flows, AIAA Journal, Vol. 14, No. 4, 1976, pp Power, G. D. and Barber, T. J., Analysis of Complex Hypersonic Flows with Strong Viscous/Inviscid Interaction, AIAA Journal, Vol. 26, No. 7, 1988, pp Rizzetta, D. and Mach, K., Comparative Numerical Study of Hypersonic Compression Ramp Flows, AIAA Paper , American Institute of Aeronautics and Astronautics, Reston, VA, June of 12 American Institute of Aeronautics and Astronautics Paper 28-19

7 18 Rudy, D. H., Thomas, J. L., Kumar, A., Gnoffo, P. A., and Chakravarthy, S. R., Computation of Laminar Hypersonic Compression-Corner Flows, AIAA Journal, Vol. 29, No. 7, 1991, pp Gaitonde, D. and Shang, J. S., The Performance of Flux-Split Algorithms in High-Speed Viscous Flows, AIAA Paper , American Institute of Aeronautics and Astronautics, Reston, VA, January Updike, G. A., Shang, J. S., and Gaitonde, D. V., Hypersonic Separated Flow Control Using Magneto-Aerodynamic Interaction, AIAA Paper , American Institute of Aeronautics and Astronautics, Reston, VA, January Edney, B., Anomalous Heat-Transfer and Pressure Distributions on Blunt Bodies at Hypersonic Speeds in the Presence of an Impinging Shock, FFA Report 116, Aeronautical Research Institute of Sweden, Stockholm, Sweden, February (a) Flat plate. (b) Compression ramp. Figure 1. Configuration for flat plate and ramp computations. Approximate location of energy deposition indicated with colored circles. Side and rear boundaries included to illustrate extent of computational domain experiment, 15 deg ramp experiment, 18 deg ramp experiment, 24 deg ramp computation, baseline, 5x5x5 grid computation, baseline, 1x1x1 grid computation, baseline, 2x2x2 grid computation, heating, 5x5x5 grid computation, heating, 1x1x1 grid computation, heating, 2x2x2 grid (a) Pressure coefficient (b) Skin friction coefficient (c) Heat transfer coefficient. Figure 2. Grid resolution study: wall centerline profiles for the flat plate boundary layer flow. 7 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

8 (a) Wall skin friction, contour interval (b) Sectional streamlines and w-velocity in outlet plane, contour interval 5 m/s. (c) Density contours and stream ribbons, contour interval kg/m 3. (d) Contours and iso-surfaces of streamwise vorticity (ω x), contour interval.5 s 1. Figure 3. Effect of volumetric heating on flat plate boundary layer flow, grid. 8 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

9 experiment, 15 deg ramp 215x6x6 grid, 15 deg ramp 415x12x12 grid, 15 deg ramp (a) Pressure coefficient, 15 -ramp. (b) Skin friction coefficient, 15 -ramp. (c) Heat transfer coefficient, 15 -ramp experiment, 18 deg ramp 215x6x6 grid, 18 deg ramp 415x12x12 grid, 18 deg ramp (d) Pressure coefficient, 18 -ramp. (e) Skin friction coefficient, 18 -ramp. (f) Heat transfer coefficient, 18 -ramp experiment, 24 deg ramp 215x6x6 grid, 24 deg ramp 415x12x12 grid, 24 deg ramp (g) Pressure coefficient, 24 -ramp (h) Skin friction coefficient, 24 -ramp (i) Heat transfer coefficient, 24 -ramp. Figure 4. Comparison of computation and experiment for baseline compression ramp flow. 9 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

10 heating baseline (a) Pressure coefficient, 15 -ramp (b) Skin friction coefficient, 15 -ramp (c) Heat transfer coefficient, 15 -ramp heating baseline (d) Pressure coefficient, 18 -ramp (e) Skin friction coefficient, 18 -ramp (f) Heat transfer coefficient, 18 -ramp heating baseline (g) Pressure coefficient, 24 -ramp. (h) Skin friction coefficient, 24 -ramp. (i) Heat transfer coefficient, 24 -ramp. Figure 5. Effect of volumetric heating on centerline properties of compression ramp flow, grid. 1 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

11 (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, 5 K contour, grid. 11 of 12 American Institute of Aeronautics and Astronautics Paper 28-19

12 (a) Baseline case, 15 -ramp, contour interval (b) Heating case, 15 -ramp, contour interval (c) Baseline case, 18 -ramp, contour interval (d) Heating case, 18 -ramp, contour interval (e) Baseline case, 24 -ramp, contour interval (f) Heating case, 24 -ramp, contour interval Figure 7. Wall heat flux distribution for each case, grid. 12 of 12 American Institute of Aeronautics and Astronautics Paper 28-19