Shape Optimization of an AxisymmetricBlunt Body in Hypersonic Flow forReducing Drag and Heat Transfer

Christopher Seager* and Ramesh K. Agarwal**

Washington University in St. Louis, St. Louis, MO. 63130

A large design concern for high-speed vehicles such asnext generation launch vehicles or reusable spacecraft isthe drag and heat transfer experienced at hypersonicvelocities. In this paper, the optimized shapes for minimumdrag and heat transfer for an axisymmetric blunt body aredeveloped using computational fluid dynamics (CFD) softwarein conjunction with a genetic algorithm (GA). For flow fieldcalculations, the commercial flow solver ANSYS FLUENT isemployed to solve the unsteady compressible ReynoldsAveraged Navier-Stokes equations (RANS) including the SST k-ω turbulence model. The hypersonic body is optimized using amulti-objective genetic algorithm (MOGA) to minimize bothdrag and heat transfer. The MOGA creates a Pareto-optimalfront containing the optimized shapes for various relativeweights of drag and heat transfer The results show asignificant decrease in both drag and heat transfer, andexhibit the expected body profile changes. This allows formore complex shapes to be simulated and optimized in thefuture.

I. Introduction

Hypersonic vehicles experience large drag forces and highrates of heat transfer during flight, as is the case duringatmospheric re-entry. Due to the engineering challengesassociated with handling these high forces and fluxes, changingthe vehicle shape to reduce drag and heat transfer is a primaryengineering concern. Testing the design in hypersonic flow isdifficult, however. The cost and technical issues associated withhypersonic wind tunnel testing make routine testing impractical,and experimental data from actual flight vehicles at high Machnumbers is challenging to obtain. This can be seen by the high

cost and timeline of NASA’s X series [7]. Because of theselimitations for physical testing, CFD technology is an attractivealternative to aid in the design of future hypersonic vehicles.

In this paper, the commercial CFD solver ANSYS FLUENT solvesthe RANS equations in conjunction with the SST k-ω turbulencemodel. The implementation of a turbulence model, as opposed tolaminar or inviscid flow, moderately affects the drag coefficientbut is extremely important in estimating wall heat flux [10]. Asthe body shape is optimized for both minimum drag and heattransfer, error in either design variable leads to error in theoptimized shape. Therefore both the turbulence model and meshindependence are vital to the simulation’s validity.

Drag and heat transfer on the blunt body are minimizedthrough a genetic algorithm. A genetic algorithm (GA) is anoptimization tool based on biological evolution. It iterates onindividual body shapes over generations, discarding the worstshapes and saving and altering the best in terms of the givendesign variables. It should be noted that the GA should beexercised very carefully in determining the optimal shape assmall shape changes can lead to large differences in drag andheat transfer values. Thus, a well-written GA not only providesthe optimal shape, but also is efficient in decreasing the totalcomputational time needed for convergence.

II. Brief Review of Hypersonic Flow Characteristics

Hypersonic flow involves many complex physical phenomenathat make aerodynamic design of hypersonic vehicles difficult.These complex phenomena also complicate flow modeling andsimulation. Strong shockwaves close to the body wall,shock/boundary layer interactions, transitional and turbulentflow, and molecular dissociation of air all affect the solution,and are computationally expensive to simulate. The primary focusof this paper is the drag and heat transfer associated withhypersonic flow.

Aerodynamic drag can be divided into two components:pressure drag and friction drag [1]. In supersonic cases, wavedrag, or the increased pressure behind a shockwave, dominatespressure drag. Wave drag occurs because the front of the vehicleexperiences large pressures due to the shockwaves, and the aft ofthe vehicle experiences smaller pressures due to expansion waves.

Summing these pressures over the area of the body gives a netforce opposite the direction of motion. Inside the shockwave,pressure drag is influenced by flow separation and shock/boundarylayer interaction, which also sum to a net force opposite thedirection of motion. Friction drag is due to the velocitygradient in the boundary layer present at the wall surface. Aturbulent boundary layer has a larger velocity gradient than alaminar one, and therefore imparts a higher skin friction drag onthe body. Combining wave drag with pressure drag and skinfriction, the drag on a hypersonic vehicle with a turbulentboundary layer is large. The total drag of the vehicle can beexpressed as:

D=12CDρ∞ V∞

2S (1)

where CD is the total drag coefficient, ρ∞ is the free-streamdensity, V∞ is the free-stream velocity, and S is a referencearea (in this paper, the cross sectional area of the body facingthe free-stream). Therefore, for a given cross sectional shapetraveling at a given altitude and speed, CD is the only parameterthat can be changed to reduce drag. Reducing CD can beaccomplished by changing the shape of the body.

Heat transfer at hypersonic speeds occurs due to aerodynamicheating. The free stream contains high amounts of kinetic energythat is converted to thermal energy when the molecules passthrough the shockwave. Air temperatures at the stagnation pointfor hypersonic vehicles can be thousands of degrees. These hightemperatures, combined with a turbulent boundary layer anddecreased shock standoff distance, cause large amounts of heattransfer into the body [2,14]. Re-entry vehicles generally have ablunt leading edge to separate the shockwave from the body andincrease shock standoff distance, in addition to a heat resistantthermal protection system (TPS). This poses a challenge forhypersonic vehicles that want to minimize drag and heat transfer,as the blunt leading edge lowers maximum heat transfer butresults in higher drag. Thus, there is a balance between findingminimum drag while keeping the heat transfer below a criticalvalue.

III. Brief Description of Genetic Algorithm and ShapeParameterization

Genetic algorithms are a class of stochastic optimizationalgorithms inspired by biological evolution [13]. In a GA, a setor generation of input vectors, called individuals, is iterated over,successively combining traits of the best individuals untilconvergence is achieved [3]. In general, a GA employs thefollowing steps [6,12].

1. Initialization: Randomly generate N individuals.2. Evaluation: Evaluate the fitness of each individual.3. Natural Selection: Remove a subset of the individuals (often the

individuals that have the lowest fitness).4. Reproduction: Create a new generation of individuals.

a. Parent Selection: Select pairs of individuals to producean offspring.

b. Crossover: Exchange information between the parentindividuals.

c. Mutation: Randomly alter some small percentage of thepopulation.

5. Check for Convergence: If the solution has converged, return thefittest individual. If the solution has not yet converged,repeat the cycle starting at step 2 with the new generation.

Multi-objective genetic algorithms (MOGAs) differ from single-objective genetic algorithms (SOGAs) in that the fitness ofindividuals is not determined by a single design variable, but bytwo or more optimization objectives [15]. These objectives areoften conflicting [9]. This is the case in this paper, as dragcoefficient and maximum heat flux tend to be inversely related.Thus, a unique solution depends on the relative weights of eachoptimization objective. A rigorous MOGA gives a set of optimalsolutions, known as a Pareto-optimal front of solutions. A givenPareto-optimal solution contains the best solutions for any givenweights of the design objectives. Each solution is the best forits own set of weights, but not necessarily the best for allcases.

IV. The Shape Optimization Process

This section of the paper is split into four parts thatexplain each individual process of optimization. Figure 2 showsthe generalized map for the optimization process. The processesare: 1) Shape Generation, 2) Meshing, 3) Simulating, and 4)Optimizing.

1) Shape GenerationBezier curve polynomials are used to create the body profile. A

Bezier curve is a parametric curve that isdefined by control points, where the curve itself traces a smoothpath between the control points. They have favorablecharacteristics for use in a GA in that the initial and finalpoints and slopes can be fixed, control point information betweenindividuals is easily swapped, and changing a control pointresults in a similar change to the curve itself [16]. Figure 2shows a set, or generation, of three random curves. The greenstars represent the control points, whereas the red linerepresents the resulting curve. Note that in all cases, the firsttwo and last two control points are fixed to ensure a bluntleading edge and appropriate size constraints on the shape. Theshape is axisymmetric about the “x” axis, traveling to the leftat hypersonic speeds.

Shape Generation Meshing

SimulatingOptimizing

Body Shape

Objective

Values

Information for new

generation of curves

Figure 1: General schematic of information flow in optimization process.

Figure 2: Three randomly generated curves and their Bezier Points

2) MeshingWhile the control points (green stars in Figure 2) are saved

for future use of the shape, the curve itself (red line in Figure2) is parameterized and loaded into ICEM Meshing Software to bemeshed. ICEM is called with a pre-written meshing script thatcreates the same mesh profile for different curve shapes: the farfield boundaries remain the same, as does the boundary spacing,but the wall profile adjusts to the newly called curve. The meshfor each of our randomly generated curves is shown in Figure 3.They are axisymmetric about the horizontal axis, and theboundaries lie just far enough away from the bodies to captureany shockwaves. The mesh is segmented, with high cell densitynear the wall for accurate turbulence and heat transfer modeling,and increased coarseness beyond the shock to increase computationspeed. This particular mesh gives a compromise between accuracyand computation cost, as will be described later in the Results andDiscussion section.

3) SimulatingThe mesh generated in step 2 is loaded into commercial CFD

software ANSYS FLUENT. FLUENT finds surface quantities of themesh, including skin friction, wall pressure coefficient, andtotal surface heat flux, by solving the RANS equations and theSST k-ω turbulence model of the flow field. Information for the

Figure 3: Meshed random curves

simulations is given in Table 1 [5]. This simulation is alsoautomated by using a script. The script loads the mesh, runs thesimulations, and outputs the data of interest: drag coefficientand maximum wall heat flux. Figure 4 shows Mach number contoursof our randomly generated curves after they were simulated.

4) MOGAThe above steps repeat until information for all individuals

in the generation are recorded. Then the drag and heat transfervalues are loaded into the MOGA, which runs evaluation, naturalselection, and reproduction functions on the data. Throughout theentirety of the optimization process, the objective informationfor each body shape is always retained to the same body shape.

Figure 4: Mach number contours of randomly generated curves

Curve 1High DragHigh Heat Transfer

Curve 2Low Drag

Medium Heat Transfer

Curve 3High Drag

Low Heat Transfer

Table 1: FLUENT Simulation Parameters

Boundary Condition Parameter Value

Pressure Far-Field

Mach Number 7.1Gauge Pressure (Pa) 663Temperature (K) 293

Wall Constant Temperature(K)

1000

Pressure Outlet Operating Pressure(Pa)

0

Axis -- --

4a) Evaluation and Natural SelectionThe objective values for each curve are analyzed and

compared to each other. In our case, minimizing the valuesis desired. Each objective value is sorted from best toworst (where the lowest value is the best, and highest isthe worst) and then combined to form an overall fitness ofthe curve. Changing the weight of one objective over theother (e.g., dividing the fitness associated with heat fluxin half while keeping drag the same) allows for creation ofthe Pareto-Optimal front by defining which curves are morelikely to be fit than others.

Simply removing the least fit half of individuals fromfurther use is natural selection. Thus, only the fittesthalf of data “survives” and has a chance to reproduce andinfluence the next generation.

Looking at the randomly generated curves in Figure 4,“Curve 1” has poor drag and heat transfer characteristics,and would likely get discarded during natural selection. Ifdrag were the more important design objective, “Curve 2”would likely have a better fitness, whereas if heat transferwere the more important design objective, “Curve 3” wouldlikely have the better fitness. If both design objectiveswere weighted the same or very close to one another, then,depending on the values of drag and heat transfer, either“Curve 2” or “Curve 3” could have the better fitness.

4b) ReproductionReproduction involves parent selection, crossover, and

mutation. From the remaining data, pairs of parent shapesare selected using a skewed random draw; the more fit afunction is, the higher its probability of being selected.The paired curves then undergo crossover, where they exchangeBezier point information. The exchange can be a completeswap of certain points or a weighted average of thecharacteristics of certain points. Both the type ofcrossover and the type of exchange are selected at random.This process repeats until the desired amount of daughtershapes are created, and a new generation is formed.

A final step is mutation. Randomly selected Bezierpoints of randomly selected curves are randomly altered,adding more diversity into the generation to increase rangeof simulated shapes.

4c) Check for ConvergenceFinally, the MOGA checks if the curves satisfy

convergence criteria. The criteria involve likeness of thegeneration, the differences in objective values to theprevious generations, and total number of generations run.If the system has converged, MOGA gives the fittestindividual. If the system has not yet converged, the newgeneration is loaded into the shape generating function andthe process repeats.

Figure 5 shows a more detailed information schematic of theoptimization process.

V. Results and Discussion

A. Mesh ValidationThe accuracy of any CFD simulation is largely dependent on

the mesh that is used. In this optimization of hypersonic flow,very small wall spacing is required to accurately model heattransfer, but the distance can be increased outside the boundarylayer, and especially outside the region of shock [11]. Thispaper carried out a mesh validation study by creating anextremely dense uniform mesh on both a spherical and a randomlygenerated shape. These individual simulations were run with 3rd

Begin Optimization

Program

Generate N random shapes for initial generation

Mesh Shapes

Simulate

Shapes

MOGA

Evaluation

Natural Selectio

nReproductio

n

Converged?

Output fittest individua

l

YES

NOGenerate shapes of new generation

Figure 5: Detailed schematic of information flow in optimization process

order scheme accuracy, and obtained a more or less “exact”solution to the simulation (the 631x1250 size mesh in Table 2).

These meshes were extremely computationally expensive,though, and as the GA requires simulating a moderate number ofindividuals over some dozens of generations, they would not bepractical given the available computation power. Thus, this meshvalidation study focused on finding a much faster mesh whoseresulting objective values differed only slightly from theprevious dense computations. Table 2 details a variety of themeshes used and their resulting objective values (only the valuesfor the spherical simulations are shown). Mesh types includeuniform, exponential, and segmented construction, along withvaried wall and far field refinements. The study found a meshthat gave low error in drag and heat transfer values, but with aruntime suitable for optimization (the 35-105-10x400 segmented meshin Table 2). The details of this mesh are given in Figure 6.

Table 2: Mesh Validation Study

Type ofMesh

Spherical GeometryMesh Size

DragCoefficient

CD error Max Heat Trasfer(W/m2)

MaxHTerror

Max y+

Uniform80x200 .95574 .1 3080000 29.3120x200 .95534 .1 2820000 18.6160x400 .95511 .1 2600000 9.3200x400 .95503 0 2590000 8.8

631x1250 .95463 ---------- 2380000 -----------

Exponential

100x200 (1e-5m walldistance)

.95503 0 2850000 20

100x200 (5e-6m walldistance)

.95495 0 2900000 22

100x200 (1e-6m walldistance)

.95545 .1 2900000 22

Segmented

390-10x400 .95455 0 2470000 485-115x400 .9547 0 2580000 8

44-146-10x400 .95503 0 2550000 725-75-10x400 .95464 0 2580000 835-105-10x400 .95471 0 2560000 735-105-10x250 .95487 0 2690000 13

B: IndividualCaseResults

Figure 6 shows theresults obtained for equallyweighted fitness (1 to 1).

Figure 7 shows results for heattransfer weighted twice the drag (.5 to 1)

Figure 5: Mesh used for computations – 35-105-

10x400

Pressure OutletPressure Far-Field

Wall(400

Uniformly Spaced Cells)

Axis35

uniformly spaced cells

105 exponentially spaced cells

10 cells outside of

shock region

Figure 8 shows results for drag weighted 75% that of heat

transfer(.75 to 1).

Figure 6: Equal-weighted optimization results - Mach contours, wall skin friction and pressure coefficient, wall heat flux, and convergence graphs.

Drag Coefficient: 0.745 (22% reduction)

Maximum Heat Transfer: 2230000 W/m2 (13% reduction)

Figure 7: Results for heat transfer favored 2 to 1 – Mach contours, wall skin friction and pressure coefficient, and wall heat flux plots.

Drag Coefficient: .960 (1% increase)

Maximum Heat Transfer: 1900000 W/m2 (26% reduction)

Figure 8: Results for heat transfer favored by 4 to 3 – Mach contours, wall skin and pressure coefficients, and wall heat transfer.

Drag Coefficient: 0.825 (13% reduction)

Maximum Heat Transfer: 2070000 W/m2 (19% reduction)

C: Creation of a Pareto-Optimal FrontCombining the results from part “B” creates a Pareto-optimal

front, shown in Figure 9.

Figure 9: Pareto-optimal front with design parameter comparisons and shape comparisons

D: DiscussionThe results from each individual simulation agree with

predictions and are plausible. The shockwaves lie wellwithin the coarser, outer-boundary of the mesh, and the skinfriction, pressure coefficient, and total surface heattransfer plots correspond well to the body shapes. Thus, themesh was sufficiently far from the wall to not influence thesimulations, and the computed parameters from thesimulations vary along the wall shape as they should. Addingto the plausibility, each simulation ran until all residualswere well below 1x10-3 and the drag and various heattransfer values had converged.

The shape of the lowest equal weighted case matcheswell with the results from [4] and [5]. The former showsexperimentally that a blunted cone shape has lower drag thana standard cone at supersonic speeds, and the optimizedshapes appear to be approaching a blunted cone shape with abluntness ratio of around 0.4 as drag importance isincreased. The optimized shape found in [5] focused only ondrag, and as our drag importance increases, the optimizedshape from this paper appears to be approaching somethingsimilar to their shape as well. Heat transfer values arehigher, but of a similar magnitude of those given in [8].This is because turbulent flow leads to greater heattransfer than laminar conditions.

The combined results also agree as expected. Heattransfer decreased more than the drag coefficient as theweights skewed towards heat flux. The geometry followed theexpected paths too, with the optimized shapes havingincreasingly larger leading edge radii of curvature as heattransfer begins to dominate the optimization variables. Evenso, the curves maintain somewhat of a streamlined shape tokeep drag from becoming too large. Plotting the Pareto-optimal front shows a well-fit curve connecting the threecases, and is extrapolated backwards for drag-favoredoptimization. Previous simulations support the approximatevalues of the variables, as prior drag-only optimizationsconverged to a drag coefficient value of about 0.63. Furthersimulations are working on finding the shapes of drag-favored optimization through using the MOGA, with theexpectation that they lie near to the Pareto-optimal front.

VI: Future Work This paper shows that a MOGA is a practical way ofoptimizing hypersonic shapes for reduced drag and heat transfer.From this initial research, many options are available forimmediate analysis. First and foremost are simulations of drag-favored optimization: shifting the design objective weights sothat drag is more important than heat transfer. Following this,using a larger number of Bezier control points or implementingdifferent turbulence models would show how much the results relyon given modeling parameters. Different shapes can be optimizedas well, as this only requires a change in the generatingfunction, mesh, and simulation. More complex axisymmetric shapescan be optimized, along with the eventual optimization ofaxisymmetric shapes at angles of attack or optimization of fullythree-dimensional bodies. Further flow complexities such asmolecular dissociation and rarified effects can be introduced,observing the effect they have on the shape.

VII. ConclusionThis study focused on using a multi-objective genetic

algorithm to minimize the drag coefficient and heat transfer foran axisymmetric blunt body in hypersonic flow. Bezier polynomialswere used to create shape profiles that were meshed and solved bythe commercial CFD solver ANSYS FLUENT. FLUENT solved for theflow field of each shape by solving the Reynolds-averaged Navier-Stokes equations and the k-ω SST turbulence model. Acomprehensive mesh validation study found a mesh of reasonablecomputation cost that experienced less than 10% error for bothdrag and heat transfer calculations to that of an extremely densemesh. Individual results were analyzed before being combined intoa Pareto-optimal front. The front displayed expectedcharacteristics, such as the leading edge radius of curvatureincreasing the higher heat transfer fitness was weighted, andvice versa. Future work includes simulating the current bluntbody using dissociation and/or rarefied models, changingturbulence models and degrees of freedom of the system, andoptimizing other axisymmetric and non-axisymmetric shapes.

VIII. References

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