Some applications of a panel method to aerodynamic modelling of missiles

Antti Pankkonen*

Aalto University School of Science and Technology(Helsinki University of Technology), Espoo, Finland

A high-order panel method is applied to compute longitudinal aerodynamic characteristics of several missile configurations for which wind-tunnel test data are available. Feasibility and limitations of the panel method in missile aerodynamic modelling are then assessed based on comparison of computational and experimental results. An effort is made to extend the range of applicability of the panel method to moderate angles of attack by modelling separated vortices. Results show that non-linear aerodynamic characteristics of conventional missile configurations can be predicted within preliminary design accuracy at low to moderate angles of attack.

NomenclatureCm = pitching moment coefficient (Cm = CMY)CN = normal force coefficient (CN = -CZ)CX = force coefficient in the x direction (forward)CY = force coefficient in the y direction (right)CZ = force coefficient in the z direction (down)CMX = moment coefficient around the x axisCMY = moment coefficient around the y axisCMZ = moment coefficient around the z axisD = maximum body diameterM = Mach numberXcp = location of centre-of-pressureα = angle-of-attackδ = pitch control deflectionφ = roll orientation

Reference dimensions of the aerodynamic coefficients are maximum cross section area and maximum diameter of missile body. Delta Δ preceding a symbol indicates difference.

I. Introductionlthough it has been decades since the panel methods were on the cutting-edge of computational aerodynamics, these methods still find use as rapid modelling tools. Short set-up and execution time on a modern computer

and ability to handle arbitrary geometries make panel methods an attractive alternative as a midway between state-of-the-art CFD and handbook methods. Coupled with automatic grid generation, panel methods can rapidly produce aerodynamic data for flight simulation and other applications.

ASeveral limitations and approximations must be accepted when utilising panel methods. This is especially true in

the supersonic regime, where the geometry must be sharp-nosed and relatively slender. In supersonic cases the maximum local slope of solid walls is constrained by the Mach angle, which yields an upper limit for Mach number on a given geometry. The panel methods being based on the linearised potential equation means that the transonic regime must be considered off-limits, although computation is possible very near Mach number 1. The same is true

* Research Engineer, Department of Applied Mechanics, Flight Mechanics Research Group. AIAA Member.

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Copyright © 2010 by Antti Pankkonen. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

for flow cases that are dominated by viscous effects. For more comprehensive overview and theoretical background of panel methods the reader is referred to Erickson's paper1. Previous applications of a panel method to missile aerodynamic modelling have been presented by Fornasier and d'Espiney2.

II. Computational methods

A.Panel method softwareThe PANAIR program3 is utilised to solve the potential flow around missile configurations. The program is

capable of solving both subsonic and supersonic flow around an arbitrary geometry subject to the underlying limitations of panel methods in general. The program has been modified to improve computational performance by replacing several temporary files with memory storage. Also handling of wake network abutments has been made more robust so that partially coinciding wake networks may exist in the grid without the program trying to create unwanted abutments.

The geometry input to PANAIR is composed of topologically rectangular panel networks. A variety of different surface singularity types and boundary condition specifications are available to formulate the problem. Impermeable surfaces are generally modelled using panels with quadratically varying doublet strength and/or linearly varying source strength. Continuity of source and doublet strength is enforced at panel network abutments, and user-specified boundary conditions are enforced on panel centre control points.

The program computes surface pressure coefficients from perturbation velocities using linear, slender body, second order, or isentropic pressure formula according to the analyst's choice. The isentropic formula is used throughout this study.

B.Batch-run utilityBuilding an aerodynamic data set generally requires dozens or hundreds of computations in varying flight

conditions. In order to utilise the panel method efficiently and conveniently, a batch-run utility has been developed to invoke grid generation and solution of the cases, and collection and post-processing of the results. Aerodynamic coefficients can be tabulated for plotting or formatted as input for flight simulation.

C.Grid generationAn automatic grid generator has been developed to rapidly produce surface and wake grids for conventional

missile configurations. Wake geometry depends on angle-of-attack, roll orientation, and Mach number as described in section II.D. Hence, a single grid is not sufficient to produce aerodynamic data in varying flight conditions, and automatic grid generation is necessary to reduce workload of the analyst. The grid generator produces a complete PANAIR input file that contains the flow case specification and program control parameters in addition to the geometry and boundary conditions.

Missile configuration is built up of an axially symmetric body and up to five wing groups. Wings are modelled as zero-thickness doublet networks, whereas body surface is represented by composite source-doublet networks. Body surface is divided into panel networks at wing leading-edges and trailing-edges and surface slope discontinuity points in the lengthwise direction, and along wing root chord lines in the transverse direction. Boundary conditions are set to enforce velocity impermeability at panel centre control points, and body surface panels also enforce zero perturbation potential inside the body.

D.Wake modellingA prescribed wake is used to model the trailing vorticity of the lifting configuration. As a basic rule, the grid

lines of wake panel networks are aligned with the free-stream whenever this is possible without intersecting body surface networks. Special treatment is required for wake networks that emanate from wings on the free-stream side of the body. Near the body these wakes are deflected outwards from the centre-line to keep them outside the body.

Wing trailing-edge wakes are extended as constant-strength networks to the sideline of the body behind the wing in order to avoid the singularity that would otherwise present itself at the junction of wing trailing edge and body. In case of a deflected wing the root chord becomes separated from the body, and the wake is therefore extended along the root chord to support non-zero doublet strength along the edge. Otherwise a high doublicity gradient, or vorticity, would concentrate near the root chord causing unrealistic local velocities.

Wake networks are also attached to wing tips and leading edges when necessary to represent the side-edge or leading-edge vortices of low-aspect-ratio wings and delta-wings. The separated vortices are not discrete, indeed it is impossible to model a concentrated vortex with a continuous doublet network, and no effort is made to exactly

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represent the shape of separated leading-edge vortices. Instead, a set of generic rules is used to locate maximum vorticity above and slightly behind the leading-edge in order to model the gross effect of a separated vortex. Leading-edge vortices are included only in cases where the leading-edge is subsonic and the inflow angle in a plane perpendicular to the leading-edge exceeds 10º.

III. Results

A.Missile with strake-type wings and tailIn Reference 4, comprehensive wind tunnel measurements were carried out on a missile model with strake-type

wings and tail. Figure 1 depicts the geometry and surface panelling of this missile. Composition of the surface and wake panel networks at α=10º and φ=45º is shown in Figure 2.

Longitudinal aerodynamic coefficients of this configuration are computed at four Mach numbers (0.6, 1.18, 1.7, and 2.86), two roll orientations (0º and 45º), and six angles-of-attack from 0 to 18 degrees. A half-grid is sufficient for the computations because only symmetric cases are considered. The half-grid consists of 932 surface panels at 0º roll orientation as the wings coinciding with the plane of symmetry are omitted, and 1084 surface panels at 45º roll orientation. A sequential single-process batch-run of the 48 cases takes about 10 minutes of computer time on a commonplace workstation. Results at various Mach numbers and roll orientations are plotted in Figures 3-5. Figure 6 illustrates the distribution of surface pressure coefficients in one of the cases.

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Figure 1. Surface panelling of missile model with strake-type wings and tail.

Figure 2. Surface and wake panel networks of the missile model.

Direct comparison of computational axial force results of Fig. 3 with wind tunnel measurements is obviously pointless since the panel method solution represents inviscid flow. Even so, it can be seen that supersonic pressure drag increment and effects of angle-of-attack are predicted within reasonable accuracy despite the zero-thickness wings.

On the other hand, normal force and pitching moment coefficients of Figures 4 and 5 are comparable to wind tunnel results, and show fair agreement with the experiments of Ref. 4. The reference results are not reproduced here.

Non-linear behaviour of normal force and pitching moment coefficients is evident in Figures 4 and 5, and results from the side-edge vortices of the strake-type wings and variation of wing-tail interference due to wake geometry.

Figure 4. Normal force (CN=-CZ) and pitching moment (CMY) coefficients at various Mach numbers.

Figure 5. Normal force (CN=-CZ) and pitching moment (CMY) coefficients at various roll angles.

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Figure 3. Axial force coefficients at various Mach numbers.

A quantitative comparison of the results at various roll orientations (Fig. 5) with experimental results of Ref. 4 is presented in Figure 7. Normal force coefficients are seen to agree within ±5%. The disagreement of centre-of-pressure locations seems rather severe at first glance, but the maximum difference, 0.67 diameters, is actually about 5% of the length of the missile and occurs at low angle-of-attack where the net aerodynamic force is small.

0 2 4 6 8 10 12 14 16 18 20-0.06

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Figure 7. Relative difference of experimental and computational normal forces and centre-of-pressure locations at various roll orientations.

To illustrate the effect of wake modelling on the aerodynamic coefficients, the computations at various Mach numbers are repeated with a fixed wake aligned with the body centre-line. Normal force and pitching moment results are plotted in Figure 8. As expected the coefficient slopes at zero angle-of-attack are predicted reasonably well with the fixed wake, but at increasing angle-of-attack the coefficients deviate wildly from the test data of Ref. 4 and the results of Figure 4 obtained with more comprehensive wake model. The non-linearity of aerodynamic coefficients in Figure 8 arises mostly from the isentropic formula used in surface pressure computation.

Figure 9 illustrates the surface pressure coefficient in one case with fixed wake model. Steep pressure gradients on the tail are due to the body-aligned wake that coincides with the tail and causes strong interference effects. Comparison with Figure 6 indicates that aligning the wake with the free-stream reduces interference strength as the wake is deflected away from the tail. The increased downwash from the side-edge vortices of the strake-wings causes reduction in tail normal force, but this is more than compensated by marked increase in wing and rear body normal force due to the vortices.

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Figure 6. Top-side surface pressure coefficient (difference on wings) at α=10º, φ=0º, M=1.7.

Figure 8. Normal force and pitching moment coefficients with fixed wake model.

B.Double-canard missile noseSupersonic wind tunnel measurements have been

previously carried out at Helsinki University of Technology using a missile nose model with two different nose shapes and several wing configurations5. One of these configurations, namely a sharp-nosed double-canard configuration shown in Figure 10, is chosen for comparison with panel method results. The forward wings are fixed and the rearward wings are deflected for pitch control. Computations were made at Mach number 1.6 with various control deflections. At this Mach number the leading-edges of the wings are supersonic, and separated vortices are therefore not included in the wake model. Again a half-grid is used due to symmetry, the number of surface panels being 524. The total of 21 cases is completed in just over one minute of computer time.

Computational results are shown in Figure 11, and corresponding experimental results in Figure 12. Agreement of the results is fair at small control deflections, but the effect of large control deflection is underestimated at low angle-of-attack and, although the relative differences remain small as seen below, becomes qualitatively incorrect at large angle-of-attack.

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Figure 10: Missile nose model panelling.

Figure 9. Top-side surface pressure coefficient (difference on wings) at α=10º, φ=0º, M=1.7 with fixed wake model.

Figure 11. Computational normal force and pitching moment coefficients coefficients at various control deflections.

Figure 12. Experimental normal force and pitching moment coefficients at various control deflections.5

Quantitative assessment of the above results is presented in Figure 13, which shows the relative difference of experimental and computational normal forces and difference of centre-of-pressure locations. Differences in zero- control-deflection normal force coefficients are similar in magnitude to the previous case, and larger local differences are evident with increasing control deflections. Good agreement of centre-of-pressure locations is in part due to the short overall length of the model, which results in short moment arms of the components and small pitching moment coefficients.

0 5 10 15 20 25 30-0.2

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Figure 13. Relative difference of experimental and computational normal forces and centre-of-pressure locations at various control deflections.

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C.Missile with delta-wing and tailIn Reference 2, a delta-wing missile model was used for comparisons between the HISSS panel code and

experimental results at Mach numbers 0.8 and 2.0. Incidentally, the HISSS program shares the theoretical background and methodology of PANAIR. Full configuration test cases are repeated here using PANAIR and a more comprehensive wake model as described in section II.D. Surface geometry and panelling of this model is depicted in Figure 14. Sweep angle of the delta-wing is 70º and thus the leading-edge remains subsonic also in the supersonic case. Wake networks, whose geometry at α=15º is shown in Figure 15, are therefore attached to leading-edges to represent separated vortices. The half-grid used in the computations consists of 930 surface panels, and the solution of the 16 cases requires about 2.5 minutes of computer time.

Comparison of the results shown in Figure 16 with Reference 2 indicates that the effect of angle-of-attack on wake geometry and inclusion of prescribed leading-edge vortices, even if their exact shape is not realistically modelled, brings forth the non-linear character of aerodynamic coefficients. The result is a remarkably better approximation of the experimental data than is attainable with simpler fixed wake geometry.

Despite the improvement due to vortex modelling, a systematic under-prediction of normal force coefficients is evident in Figure 17. Differences in centre-of-pressure locations appear reasonable, the maximum being about 3% of the missile length.

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Figure 14. Surface panelling of missile model with delta-wing and tail.

Figure 15. Detail of wake networks of delta-wing missile.

Figure 16. Normal force coefficients and centre-of-pressure locations at various Mach numbers.

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Figure 17. Relative difference of experimental and computational normal forces and centre-of-pressure locations at various Mach numbers.

IV. ConclusionA panel method was applied to compute longitudinal aerodynamic characteristics of various missile

configurations. Comparison of the computational results with experimental results shows that aerodynamic coefficients can be predicted within preliminary design accuracy up to moderate angle-of-attack and small control deflection. The highest angle-of-attack computed herein was 24º and the results were satisfactory up to 10º positive control deflection. The panel method is therefore considered a useful tool in aerodynamic modelling. It is possible to compose a comprehensive simulation model of a missile using mostly panel method results, provided that exact behaviour at transonic speeds or high angles-of-attack is not significant. The only piece of data required from other sources is the zero angle-of-attack drag curve, which can be combined with the panel method results to create a comprehensive axial force representation as a function of Mach number, angle-of-attack, and control deflection.

Successful computation of aerodynamic coefficients at moderate angles-of-attack requires some a-priori knowledge of the essential flow phenomena in the case at hand. Separated vortices may have major effect on the aerodynamics and therefore their existence and approximate geometry must be known, and wake-type networks must be present in the computational grid in order to model the effects of these vortices. The present results suggest that the exact shape of the separated vortices is not critical as far as the total aerodynamic coefficients are concerned, although prediction improvement might be realisable with more accurate vortex modelling especially in the delta-wing case.

The panel method is non-iterative when a prescribed wake model is used, and thus free from instability and convergence issues inherent in iterative numerical schemes. Therefore, coupled with automatic grid generation it is suitable for batch-runs which robustly provide comprehensive aerodynamic data sets without intervention from the analyst. Of course, review of the results is always necessary to detect any anomalous solutions that may arise, for example, due to locally invalid panel geometry.

Future applications of the panel method will include more general, non-cruciform configurations and asymmetric

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flight conditions, where examination of the lateral-directional aerodynamic characteristics is of interest. Also, investigation of dynamic stability derivatives is possible using the panel method.

AcknowledgementsThe present study was conducted as a part of a research project funded by the Finnish Air Force. The author

wishes to thank the FiAF for continuing support of research activities at Helsinki University of Technology.

References1 Erickson, L. L., "Panel Methods - An Introduction," NASA, TP 2995, 1990.2 Fornasier, L., d' Espiney, P., "Prediction of Missile Stability Using the HISSS Panel Code," Recherche Aérospatiale 1989-4, 1989.3 Saaris, G. R., "A502 User's Manual - PAN AIR Technology Program for Solving Problems of Potential Flow about Arbitrary Configurations," Boeing, 1992.4 Allen, J. M., "Aerodynamics of an Axisymmetric Missile Concept Having Cruciform Strakes and In-Line Tail Fins From Mach 0.6 to 4.63," NASA, TM-2005-213541, 2005.5 Siiropää, V., "Ohjusten siipien aerodynamiikka suurilla kohtauskulmilla," Helsinki University of Technology, Laboratory of Aerodynamics, Report T-179 (Unpublished, In Finnish), 2002.

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