Method for Rapid Modelling of Missile Aerodynamics

Antti Pankkonen*

Aalto University School of Engineering, Espoo, Finland

In previous work a high-order panel method was found to be a feasible tool for rapid aerodynamic modelling of missiles within preliminary design accuracy. The present paper addresses some issues identified within that work in an attempt to expand the aerodynamic modelling capability based on the panel method. A curve-fit scheme is developed to represent missile aerodynamics across the entire Mach number range. The results show that the aerodynamic coefficients can be modelled using a relatively simple analytic curve as a function of Mach number. The accuracy of the method appears to depend mostly on the original data points and to lesser extent on the curve-fit process.

Nomenclaturea,b,c,d,e,f = parameters of the aerodynamic coefficient fit curveC = unspecified aerodynamic coefficientCAF = axial force coefficientCDp = pressure drag coefficientCLα = lift coefficient slope w.r.t. Angle-of-attackCmα = pitching moment coefficient slope w.r.t. Angle-of-attackCmδ = pitching moment coefficient slope w.r.t. control deflectionCNF = normal force coefficientcp = pressure coefficientCPM = pitching moment coefficientM = Mach numberα = angle-of-attackφ = roll angle

Reference dimensions of the aerodynamic coefficients are maximum cross section area and maximum diameter of missile body.

I. Introductionn a previously published paper1 the panel method was found to be a feasible aerodynamic modelling tool within preliminary design accuracy. An application is, for example, model generation for performance-level trajectory

simulations where the details of the aerodynamics are not of importance. There are, however, several limitations to the application of the method. Most notably, linearisation of the potential equation underlying the method precludes realistic solutions at transonic and high supersonic Mach numbers.

IIn missile applications the panel method is most suitable for medium to long range missiles which have a sharp

nose and thus satisfy the inherent requirement that the slope of all solid surfaces be smaller than the Mach angle. This kind of missiles fly most of the time supersonically at low to moderate angle-of-attack, and operate only transiently at subsonic or transonic Mach numbers. Therefore the high Mach number limitation is considered more pressing and a method is sought to expand the computational results beyond the actual range of validity of the panel method.

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

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Copyright © 2011 by Antti Pankkonen. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

II. ConceptThe panel method alone can provide aerodynamic data at subsonic and supersonic Mach numbers. The range of

applicability is generally not sufficient for creation of a comprehensive aerodynamic model unless some interpolation and extrapolation of transonic and high supersonic regimes is performed. Interpolation of the transonic regime using a generic interpolation curve is not deemed critical in terms of accuracy for reasons outlined in the introduction. On the other hand, extrapolation of the supersonic regime may become prohibitively uncertain if data points are required at substantially higher Mach numbers than can be computationally obtained.

In order to improve the reliability of such extrapolations the Newtonian Impact Theory is applied to obtain aerodynamic coefficients in the hypersonic regime. More precisely, the results obtained by the theory are taken as limits as the Mach number approaches infinity. Given the available supersonic results from the panel method and the limit values from Newtonian theory, a fit curve can be introduced that approximates the supersonic data points and monotonically approaches the limit value at infinity. Thus, the data points beyond the range of validity of the panel method are obtained by interpolation instead of extrapolation, and therefore their reliability is substantially improved. It appears that for rapid modelling purposes a fit curve of the form

C M = aM bC∞ (1)

for any aerodynamic coefficient C as a function of Mach number M is sufficiently accurate in the supersonic regime, and it also satisfies the requirement for monotonous behaviour. In Equation 1 C∞ represents the aforementioned limit value at infinity, and a and b are parameters to be determined by fitting Eq. 1 to the supersonic data points.

A comprehensive representation of aerodynamic coefficients as a function of Mach number could be constructed piecewise by joining the subsonic and supersonic results with, for example, a third order polynomial that satisfies continuity of the value and slope of the coefficients across the transonic range, and extending the supersonic data using Equation 1. Here a more unified approach is taken by expanding the fit curve of Equation 1 in such a way as to represent also the subsonic and transonic regimes. Although the main aerodynamic prediction tool in this case is the panel method, the fit-based approach provides a convenient framework into which data can be added from any method and at any Mach number. Data from different sources can be overlapping and weight coefficients can be introduced to reflect the relative accuracy of the various sources. A further advantage is that the data points of the final aerodynamic model can be positioned along the Mach number scale independently of the data points originally computed.

The expanded fit curve is constructed by assuming a constant value for the coefficient in the incompressible regime, and devising a weighting function that blends the incompressible value and the supersonic value of Equation 1 in a way that realistically represents the transonic behaviour of the coefficient. Also, to remove the singularity of Eq. 1 at M=0, an arbitrary constant is added to the denominator of the supersonic part. Although this slightly alters the behaviour of the curve in the supersonic regime, the effect can be neglected since the exact form of Eq. 1 is arbitrary to begin with. The resulting fit curve is

C (M )=1+tanh[ d (M −e )]2⏟

weight

( aM b+1

+c⏟

supersonic

− f )+ f⏟

subsonic

(2)

where the physical significance of the various parameters is as follows:a: determines the supersonic peak value of the coefficientb: determines the rate of change of the coefficient in the supersonic regimec: limit value of the coefficient as Mach number approaches infinityd: determines the rate of change of the coefficient in (and the width of) the transonic regimee: determines the position of the transonic regimef: incompressible value of the coefficientThe parameter c is fixed to the limit value from the Newtonian theory, which leaves five free parameters to be

determined by the curve-fit algorithm. Theoretically five or more data points would be sufficient to fix all parameters, but in absence of transonic data points the parameters d and e are not positively determined because Equation 2 is fairly insensitive to these values in purely subsonic or supersonic regimes.

In case of zero-angle-of-attack axial force transonic data points can be obtained using a simplified computational method suitable for the rapid modelling concept, namely, application of the area-rule 2 with the notion that the wave-drag of a slender wing-body configuration in sonic flow is driven by the total cross-section area distribution in the stream-wise direction. If the actual geometry is replaced with an equivalent body-of-revolution while maintaining

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the area distribution, the flow solution reduces to two dimensions. Although transonic solutions must be based on non-linear flow equations such as the full potential equation or the Euler equations, the reduced dimensionality of the problem significantly simplifies grid generation and enables solutions to be obtained within minutes of computer time, which makes such solutions feasible within the present rapid modelling concept.

With the application of the area-rule principle as described above, all parameters of the fit curve can be positively determined in case of the axial pressure force coefficient. If an assumption is made that the transonic phenomena present in the flow-field affect all aerodynamic coefficients in a qualitatively similar fashion, the values of the parameters d and e obtained for the axial force coefficient fit-curve can be used also in the fit curves of other coefficients. This approach does not account for the effects of angle-of-attack which tend to widen the transonic regime. Although this tendency could be neglected in many cases given the relative unimportance of the transonic regime within the complete aerodynamic model, an approximate procedure has been considered to expand the present modelling method in this regard.

At non-zero angle-of-attack, or otherwise in absence of transonic data points, the parameters d and e can be determined by consideration of the bounds of the transonic regime, which can be identified using the panel method. Examination of surface Mach numbers of a flow solution will reveal if all panels are either subsonic or supersonic in accordance with the free-stream, or if there are both subsonic and supersonic panels present in which case the solution is by definition transonic. A simple iteration will solve for the subsonic and supersonic free-stream Mach numbers which bound the transonic regime. When the bounds are known, values of the parameters d and e can be selected such that the transonic “jump” represented by the weight function in Equation 2 is appropriately contained within the transonic regime.

III. Computational methods

A.Panel method softwareThe PANAIR program3 with certain modifications outlined in Reference 1 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. In all panel method computations the boundary conditions enforce velocity impermeability on solid walls, and surface pressure coefficients are computed from the isentropic formula. Wake modelling adheres to the principles propounded in Ref. 1.

In the previous panel method computations of Ref. 1 wings were modelled without thickness. Modelling is improved in the present work by constructing a finite-thickness panelling for the wings. However, the wings are not modelled as closed volumes with interior potential boundary conditions as is the case with the missile body. Instead, the zero-thickness doublet panel networks representing the average surface of the wings are overlaid with source panel networks that represent the actual wing surfaces. This type of modelling has several practical advantages over the closed-volume type. One is the simplicity of grid generation: there is no need to cap the root and tip of the wing, and the source panels need not abut the adjacent networks. Wake networks are attached to the average-plane doublet networks only. Another advantage is that, in contrast with the closed-volume modelling, local intersections of wake networks and wing panel networks do not cause numerical problems. The present type of wing modelling is also advantageous in terms of computational effort despite the fact that three layers of panel networks are required on the wing plan-form. This is because there is only one boundary condition per panel as opposed to two per panel in closed-volume modelling.

For robustness an approximate specification of thick wing boundary conditions has been found advantageous. Instead of explicitly specifying impermeability on all three layers of wing panelling, zero perturbation velocity is enforced on the average surface and impermeability of external surfaces is formulated as velocity difference being opposite to the onset flow in the surface normal direction. The underlying assumption is that the wing is sufficiently thin for the normal velocity on the internal surface to be considered equal to the average surface normal velocity.

B.Hypersonic computationThe Newtonian Impact Theory is applied to obtain aerodynamic coefficients at hypersonic Mach numbers.

Application of the Newtonian Impact Theory is straightforward as the local surface pressure coefficients are computed as a function of the local slope. Development of a computer program for this purpose is therefore a simple matter of combining the surface pressure computation with geometry-handling routines from an existing visualisation program for PANAIR solutions.

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C.Transonic pressure drag computationTwo-dimensional axi-symmetric, inviscid transonic solutions are obtained using the 2D-version of the Finflo-

flow solver (FINF2D) developed at Helsinki University of Technology. In the inviscid mode the program provides rapid numerical solutions to the Euler equations and computes distributed and integrated pressure forces. Grid generation for the solver is assisted by automatic computation of the wall contour of the equivalent body-of-revolution based on the surface grid.

IV. Results

A.Missile with strake-type wings and tailThe strake-wing missile configuration of Reference 4 was the subject of previous panel method computations1,

and the same configuration is revisited here in an attempt to expand the results. Figure 1 depicts the geometry and surface panelling of this missile. Panelling of the thick wings is shown in Figure 2. In Reference 1 longitudinal aerodynamic coefficients of this configuration were 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. The aerodynamic model thus obtained is expanded in the present work using the methodology set forth above.

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

Figure 2. Finite-thickness wing panel networks of the missile model.

At first the bounds of the transonic regime are estimated. Using the method described in Chapter II the curves depicted in Figure 3 are obtained. The percentages corresponding to the various pairs of curves indicate the fraction of surface panels that are required to be either subsonic or supersonic depending on the free-stream Mach number. A value less than 100% may give more realistic results if there are local disturbances in the solution which could have dominating effect on the results. In this case such are not evident, and the outer curves of Figure 3 are considered the bounds of the transonic regime. Mach numbers for the panel method computations are selected such that they remain outside the transonic regime at all angles-of-attack.

Fit curve parameters d and e which describe the transonic behaviour of aerodynamic coefficients are determined by the fit process at zero angle-of-attack, and their variation at higher angles is determined by requiring that the value of the weight function in Equation 2 remains constant along the bounding curves of Figure 3.

In Figure 4 are shown the computed axial force coefficients from the panel method, transonic Euler solutions, and the Newtonian impact theory, together with the fit curve approximating the data. Although local differences exist between the data and the fit curve, the overall fidelity of the fit appears sufficient.

The coincidence of PANAIR and FINF2D results at low supersonic Mach numbers evidences the validity of the area-rule concept. Further substantiation is presented in Figure 5, where the length-wise surface pressure distribution at Mach number 1.2 on the equivalent body-of-revolution, computed with FINF2D, is compared with the panel surface pressures of the actual geometry from PANAIR. Apart from the nose, where the peak pressure coefficient is not captured with the panel method, the results agree closely in regions where the actual geometry is axially symmetric or the cross section is invariant. There is some scatter in the panel method results near wing and tail leading-edges and trailing-edges as pressure coefficients are depicted from all wing and body panels. Despite the scatter, the essential trends of the pressure distributions are in reasonable agreement, and therefore the integrated axial force coefficients are closely matched.

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Figure 4. Zero angle-of-attack pressure drag coefficients from various methods and fit curve C(M).

Figure 5. Pressure coefficient distributions on equivalent body-of-revolution and actual geometry.

Figure 3. Bounds of the transonic regime defined with various criteria.

Axial forces at non-zero angles-of-attack are computed using the panel method and the Newtonian Impact Theory, as no rapid computational method is available for the transonic regime. The fit curve parameters describing the transonic behaviour of the aerodynamic coefficients are determined from the bounds of the transonic regime as described above. Using the fit curves obtained at several angles-of-attack, axial force curves as a function of angle-of-attack, shown in Figure 6, are constructed at all Mach numbers for which experimental data is presented in Ref. 4. Comparison with the experimental results of Ref. 4 exhibits good agreement of the trends, although absolute values differ due to lack of friction in the present computational results. At Mach numbers 2.0 and 2.36 the axial force appears nearly independent of angle-of-attack, with a decreasing trend at lower Mach numbers and an increasing trend at higher Mach numbers. This behaviour agrees with the experiments, although the cross-over of the supersonic axial force curves occurs at somewhat lower angle-of-attack. Of particular note are the results at Mach numbers 3.95 and 4.63 which lay well beyond the range of validity of the panel method, and thus support the use of Equation 1 for interpolating the high-supersonic results.

Normal force and pitching moment coefficients are treated similarly by applying Equation 2 to fit the subsonic and supersonic results from the panel method and limit values at M→∞ from the Newtonian impact theory. Also the transonic behaviour is modelled as per the axial force fit. The results computed from fit curves are shown in Figure 7, and exhibit fair agreement with the experimental values of Ref. 4.

A direct comparison between experimental normal force and pitching moment coefficients from Ref. 4, and computational results from the present method is shown in Figure 8. Here the coefficients are plotted as a function of the Mach number. Best agreement is evident at low angles-of-attack α≤10º although the pitching moment coefficient is systematically over-estimated. At high supersonic Mach numbers the computational values are obtained from the interpolation scheme based on Equation 1, which in this case seems to predict the high-supersonic trends only approximately, as the variation with Mach number of the experimental values appears to be more gradual than that of the computational results. However, the trends are established from the computed data points, which the fit curve approximates rather accurately, and therefore it appears that the differences arise mostly from the underlying panel method and other computational results.

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Figure 7. Normal force and pitching moment coefficients computed from fit curves.

Figure 6. Axial pressure force coefficients computed from fit curves.

B.Wing-control missileFor further testing the present

method is applied to estimate longitudinal stability and control derivatives of the Sparrow missile. Surface grid of this missile is shown in Figure 9. Experimental data in the supersonic regime are presented in Reference 5. Computational results for lift and pitching moment slopes and pitch control moment are shown in Figures 10-11 along with corresponding experimental data from Ref. 5. Slopes are computed as differences with 2º angle-of-attack variation and 5º control deflection. Agreement of the compu-tational and experimental results appears satisfactory, the local differences in pitching moment curves being equivalent to less than two percent difference in aerodynamic centre location along the missile length. Transonic computations are not performed in this case, and therefore the transonic fit parameters are

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Figure 8. Comparison of computational and experimental normal force and pitching moment coefficients.

Figure 10: Lift and pitch control moment slopes of the Sparrow missile.

Figure 9: Surface grid of the Sparrow missile.

estimated solely from the bounds of the transonic regime. Fitting of the lift coefficient slope and control moment slope is otherwise carried out as per the previous test case, but the behaviour of the pitching moment slope requires further attention.

Computational results for the pitching moment slope in the low supersonic regime exhibit variation with Mach number that is not attributable to transonic phenomena but rather results from variation of individual lift coefficient slopes of wing and tail surfaces together with wing-tail interference. Monotonous behaviour is evident at M>1.8 when the leading-edges of all surfaces are supersonic. The same phenomena are present in all results but their effect is significant only in pitching moment slope because the contributions of wing and tail surfaces have opposing signs and the net moment slope is small. In the curve-fit process the variation with Mach number is accounted for by allowing the “transonic” fit curve parameters d and e to vary instead of keeping them fixed with the values determined for the actual transonic regime. Whether this has an adverse effect on the transonic data cannot be quantitatively judged in absence of corresponding experimental data, but the overall transition from subsonic to supersonic regime appears reasonable and the low supersonic data is fitted with good fidelity.

V. ConclusionA curve-fit based modelling scheme was developed to extend the results from panel method to cover the entire

Mach number range. Additional rapid computational methods were utilised to obtain transonic and hypersonic data. The scheme is inherently suitable for combining aerodynamic data from any source, and the results from the panel method and other rapid modelling methods could readily be replaced or supplemented by, for example, CFD or experimental results if such are available.

The resulting accuracy of the fit curve is mostly dictated by the accuracy of the underlying data points, although the curve-fit process is by no means insignificant. The functional shape of the fit-curve was developed such that it represents what may be considered typical qualitative behaviour of aerodynamic coefficients, which seems to agree with the trends observed in the experimental data. Success of the fit process relies on the assumption that the trend of the highest supersonic data points is to monotonically approach the limit value at infinity, and data are available at sufficiently high Mach numbers for this trend to be evident.

Experience has shown that with automatic grid generation a comprehensive aerodynamic model can be built from scratch within one work-day. In a more general case where the grid must be manually created, another work-day is required for grid generation. Despite the simplifications made to the transonic pressure drag computations and relatively low number of data points, the computational effort required for the Euler solutions is similar to the bulk of the panel method solutions.

The present method appears usable as a rapid modelling tool and greatly expands the modelling capability based on the panel method. The remaining shortcomings are the same as identified in the previous work 1, that is, the lack of viscous drag and base pressure drag in axial force results, and preclusion of high angle-of-attack flow solutions. The lacking drag components can be readily estimated by semi-empirical methods which add no significant effort to the modelling work, but the angle-of-attack range can be extended only by extrapolation unless high angle-of-attack data are available from alternate methods.

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

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Figure 11: Pitching moment slope of the Sparrow missile.

References1 Pankkonen, A., "Some applications of a panel method to aerodynamic modelling of missiles," AIAA Paper 2010-7637, 2010.2 Whitcomb, R. T., "A Study of the Zero-Lift Drag-Rise Characteristics of Wing-Body Combinations Near the Speed of Sound," NACA RM L52H08, 1952.3 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 Monta, W. J., "Supersonic Aerodynamic Characteristics of a Sparrow III Type Missile Model with Wing Controls and Conparison with Existing Tail-control Results," NASA, TP-1078, 1977.

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