ON APPLYING PISTON THEORY IN INTERFERENCE FLOWS

Marius-Corne Meijer∗ , Laurent Dala∗,∗∗ , Peter G. Karkle∗∗∗∗University of Pretoria, ∗∗Council for Scientific and Industrial Research, ∗∗∗Central

Aerohydrodynamic Institute

Keywords: wing-body interference, piston theory, approximate modelling, slender body

Abstract

The theoretical basis for applying pistontheory to a slender wing-body combination isconsidered and a numerical example is givenwith comparison to experimental data and otherprediction methods.

1 Introduction

Piston theory [1] has long been used to predictsurface pressures on wings and on panels insupersonic flows. Classical piston theory (CPT)is defined by modelling perturbations relativeto the freestream flow. It has been extendedto account for upstream influence in the caseof airfoils and flat panels [2], and to accountfor curvature in the case of shells with nocrossflow [3]. CPT has also been used toadd nonlinear thickness effects to otherwiselinear panel methods [4] and to estimate theeffective airfoil shape in hypersonic flows due tothe boundary layer displacement thickness [5].Local piston theory [6] (LPT) is defined bymodelling perturbations relative to an existingmean steady-state solution. LPT applied relativeto Euler solutions has seen increased use dueto its associated reduction of computationalcost relative to full unsteady Euler solution,and has recently seen increased application tovehicles [7] rather than to isolated surfaces. Theaccuracy of the Euler-based LPT results hasseen to be lower [7] when applied to vehicleswith aerodynamically interfering componentsthan when applied to isolated surfaces. This,along with the recent application of LPT relativeto Navier-Stokes solutions [8], has motivated

the present investigation into the basis for theapplication of piston theory to interference flows.

2 Theoretical basis

Piston theory is a special case of the unsteadyanalogy, which relates the steady (or unsteady)flow in N dimensions to the unsteady flow inN − 1 dimensions – in particular, at hypersonicspeeds, this is known as the hypersonicequivalence principle. Various milestones inthe reduction of the order of the equationsfor slender bodies have been achieved, withnotable contributions due to Ilyushin [9] at lowincidences and due to Sychev [10] at highincidences. A summary of the various similitudesand their relation to each other is given in [11].It has been found [12] that in certain cases,the similitudes (such as Sychev’s [10]) correlatewell with experimental data well outside of thelimiting assumptions imposed in their theoreticalderivation.

Piston theory may be derived from the Eulerequations when the ratios of certain lengthsor gradients and velocities in the flow becomesmall parameters and the associated terms arediscarded. In particular, piston theory requiresthat gradients in two orthogonal directions beneglected, as illustrated in Fig. 1, where asthe unsteady analogy and other related methodstypically only require gradients along the bodyaxis to be neglected. A mathematical treatmentof these considerations and the conditions underwhich they break down is given in [13]; here,the ideas will be illustrated by considering onlyone component of the momentum equation andconsidering typical crossflow-plane flowfields.

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M.-C MEIJER, L. DALA, P.G. KARKLE

slender body /slender wing theory

strip theory

Fig. 1 : Planes normal to which gradients areneglected in piston theory.

Consider the third component ofthe momentum equation, with termsnondimensionalized with respect to referencevalues such that the terms and their derivativesare O(1). The velocities ui are referenced tovalues Ui, coordinates ξi are nondimensionalizedwith respect to lengths Li, and the dimensionlesspressure p and density ρ have reference valuespR and ρR respectively:(

L3U1

L1U3

)u1

∂u3

∂ξ1+

(L3U2

L2U3

)u2

∂u3

∂ξ2+

+u3∂u3

∂ξ3=−

(U2

1

U23

pR

ρRU21

)1ρ

∂p∂ξ3

(1)

The bracketed terms are dimensionlessparameters of the local flow problem; forgiven magnitudes of these parameters, the orderof magnitude of perturbations to ui may beconsidered and a truncation up to a certainorder of smallness may be made, potentiallyremoving a variable ui from the equation. Thevarious methods that result from the unsteadyanalogy differ in the particulars of the order ofthe various dimensionless parameters and theorder of the perturbations. Piston theory, as apoint-function relation between pressure andvelocity, requires the gradients in the ξ1 andξ2-directions to be neglected. This effectivelyprecludes its application in regions where the

crossflow velocity or acceleration tangential tothe surface is of the order of the piston velocity –this may occur in various regions in the flowfieldas illustrated in Fig. 2, typically where significantsurface curvature exists: the wing-body junction,the wing-tips, and potentially on the body surfacein crossflow.

M∞ sinα

Nose shock

Regions of largeflow gradients due towing-body geometry

Wing vortex

Fig. 2 : Typical flow in the crossflow plane atsmall incidence.

3 Application

The application of classical piston theory to abody with a monoplane wing is illustrated in thissection. The geometry considered is describedin Fig. 3, corresponding to the model geometryof a series of experimental work by Fellows &Carter [14]. The results presented in [14] arelimited to the normal-force coefficient for thevarious model components across the spectrumof angle-of-attack (α) and freestream Machnumber (M∞); the pressure distribution at a givencombination of M∞ and α for all the componentswas not given. A systematic comparison toexperiment of the pressure distribution predictedby piston theory was thus not possible. However,for the portion of the wing-body combination aft

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PISTON THEORY IN INTERFERENCE FLOWS

of the wing root, a comparison of the normal-coefficients could be made for the combination(CN), the interference of the wing on thebody (CNB(W )

), and for the wing including theinterference from the body (CNB(W )

). The valuesare related by:

CN =CNB +CNB(W )+CNW (B) (2)

where CNB is the normal-force coefficient of theisolated body. The additional terms are relatedto the normal-force coefficient of the isolatedwing (CNW ) through carry-over factors KB(W ) andKW (B):

CNB(W )= KB(W )CNW , CNW (B) = KW (B)CNW (3)

It is important to note that in the followingsections, CN and CNB(W )

refer to the portion of thebody aft of the root of the wing, and not to theentire body. The results from applying classicalpiston theory to the geometry are compared tothose obtained from a public-domain version ofthe NASA-AMES WingBody panel code, as wellas to experiment and approximate methods tomodel interference.

LN

LLW

D

sW

LcΛLE

Wing 1

Wing 8

R

Wing 1: ΛLE = 78.2◦, sW = 3”

( RD) = 0.30984( x

D)−0.03989( RD)

2−0.00261( RD)

3

Common: LW = 14.384”, Lc = 0.75”, t = 0.202”L = 19.5”, D = 1.5”, LN = 3D

Wing 8: ΛLE = 85.5◦, sW = 1.125”

Fig. 3 : Geometry definition.

3.1 Reference Data and Methods

The experimental values of the normal-forcecoefficient were obtained in [14] by integrationof the pressure distribution over the surfaces ofthe various model components – this allowed forthe experimental CNW (B) to be obtained directly;the value of CNB(W )

was deduced from themeasured value of CNB +CNB(W )

and from testsof the isolated body (CNB). These results, alongwith values from various prediction methods,are shown in Figs. 5–10. The normal-forcecoefficient of the isolated wing (CNW ) was alsodetermined by experiment.

Fellows & Carter [14] applied the carry-overfactors KB(W ) and KW (B) to the experimentally-measured CNW to obtain predictions for thenormal-force coefficient of the portion of thewing-body combination subject to interference.The values used for the carry-over factorswere taken from slender body theory [15](SBT), upwash theory [15] (UT), and the P-N-K method [15] (PNK), and are summarized inTable 1 and Table 2.

Table 1: Wing-on-body carry-over factor, KB(W )

Method M∞ = 2.0 M∞ = 2.8Wing 1 SBT 0.278 0.278

PNK 0.246 0.232Wing 8 SBT 0.611 0.611

PNK 0.581 0.565

Table 2: Body-on-wing carry-over factor, KW (B)

Method Wing 1 Wing 8SBT 1.162 1.349UT 1.300 1.522

3.2 Piston Theory

Classical piston theory was used to obtainpredictions for the isolated body (CNB) and theisolated wing (CNW ); no attempt was made tomodel the interference, i.e., CNB(W )

= 0 andCNW (B) = CNW . For reference, the normal-forcecoefficient of the nose alone was also computedfor comparison to experiment; the results are

3

M.-C MEIJER, L. DALA, P.G. KARKLE

shown in Fig. 5 and Fig. 6. The coordinatesystem used in defining the downwash equationsis shown in Fig. 4. The pressure coefficient as

x

r

θ

V∞α

θ = 0

Fig. 4 : Coordinate system.

modelled by classical piston theory is given by:

CP =2

M∞

(c1w+ c2M∞w2 + c3M2

∞w3) (4)

in which w is the downwash nondimensionalizedwith respect to the freestream velocity, andfor which the coefficients c1, c2, and c3 aredependent on the pressure equation used. Inthe case of expansion, the limiting value of CPcorresponds to that at vacuum. The generalexpressions for the coefficients are given in [16];for the present case the coefficients are listed inTable 3, and are seen to be close to those due toLighthill [17].

Table 3: Piston theory pressure coefficients

M∞ = 2.0 M∞ = 2.8c1 1.155 1.071c2 0.733 0.642c3, w > 0 0.254 0.181c3, w≤ 0 0.234 0.185

The dimensionless downwash w is defined as:

w≡−V∞ · n (5)

where V∞ is the unit vector of the freestreamvelocity and n is the unit normal vector ofthe body or wing surface. The dimensionlessdownwash may be split into a component (wa)associated with surface gradients relative to thecomponent of V∞ directed along the axis of thebody (x-direction) and into a component (wc)associated with surface gradients relative to thecomponent of V∞ lying in the crossflow plane:

w = wa + wc (6)

The unit normal vector of the surface for the bodyis given as:

n =−sinφ x+ cosφsinφ r+ cosφcosθθθθ (7)

In this formulation, the following relations hold:

Body: wa = cosαsinφ (8)wc =−sinαcosφcosθ (9)

Wing: wa ≈ 0 (10)

wc ≈{−sinα, upper+sinα, lower (11)

The normal-force coefficient is then obtained byintegrating the pressure over the surface:

CNB =− 1SW

2π∫0

L∫L−LW

CPRcosφcosθ dθdx (12)

CNW =− 1SW

∫∫wing

CPR dS (13)

In each case, the normal-force coefficient isreferenced to the wing area (SW ) of the wingconsidered. The results obtained using 1st-order and 3rd-order piston theory with nointerference modelling are shown in Figs. 5–10. It is noted that classical piston theorywith the downwash equation as defined hereproduces a lifting pressure-distribution aroundthe cylindrical portion of the body for all α 6= 0.

4 Discussion

For a discussion on the results of slender bodytheory (SBT), the P-N-K method (PNK), and of

4

PISTON THEORY IN INTERFERENCE FLOWS

upwash theory (UT) and their correlation to theexperimental results, the reader is referred to thereport of Fellows & Carter [14]; here, it is onlyremarked that the departure of these results fromthe experimental data highlights the fact that theinterference is not linear with α. The correlationof the WingBody panel code and of piston theoryto the experimental results is discussed here, withreference to Figs. 5–10.

4.1 WingBody Code

Firstly, it is noted that all the results from theWingBody code are linear with α, as is expectedfor the linearized potential flow code. Fairlygood correlation with experiment is obtained forα < 5◦ for all the coefficients in the case ofWing 1; thereafter, the nonlinear force of thebody leads to significant differences, as seen inFig. 5. In the case of the smaller-span Wing8, very poor agreement is obtained for the wingnormal-force at both Mach numbers, as seen inFig. 8, with a significant over prediction in thenormal-force slope at α = 0◦. Considering thegood agreement of the SBT prediction of CNW (B)

at low α, which is obtained from the experimentalCNW , and considering the significantly betteragreement of the WingBody results for Wing1, it is suggested that the large difference maybe attributed to either numerical issues in thecode arising from poorly conditioned panels overthe majority of the highly swept (85.5◦) wing.Similar overprediction in the body normal-forceis obtained, as seen in Fig. 6. The results fromthe WingBody panel code suggest that a simplelinear potential-flow method may be suitable topredict the interference and overall loads for α <5◦ in the Mach range considered, provided thatthe leading-edge sweep is not extreme (ΛLE <80◦). Improved prediction is expected fromcodes in which the vortex sheet from the wingleading-edge is modelled. The WingBody codeprovides a significantly better prediction of thebody normal-force slope at α = 0◦ compared topiston theory in both cases and Mach numbers.This is not surprising, as the downwash equation,Eq. (5), is a poor representation of the flow for thelow-subsonic crossflow Mach number M∞ sinα.

4.2 Piston Theory

The aforementioned issue of the downwashequation for the body in piston theory at low-subsonic crossflow Mach numbers is evident inthe results for the body normal-force slope atα = 0◦ in Fig. 5 and Fig. 6. Eq. (5) resultsin a non-zero lifting pressure on the cylindricalportion of the body for all α; this is inconsistentwith the concept of established flow over acylinder in the crossflow plane at low α, whichsuggests that only the nose contributes to thenormal-force. Isolating the normal-force fromthe nose alone in Fig. 5 and Fig. 6 shows thatgood correlation with the nose normal-force fromexperiment is obtained by 1st-order piston theoryfor the available data range of 0◦ ≤ α ≤ 25◦,with improved agreement at the higher Machnumber. Comparison of the slope of the nose-alone normal-force in Fig. 6 from piston theoryto the experimental slope for the body in thepresence of the wing suggests that for α < 5◦ thenose-alone prediction serves as an approximationto the overall lift on the body for Wing 8.

Investigation of the results in Fig. 5 and Fig. 6at high α shows that the impact-like downwashequation, Eq. (9), becomes more appropriateat higher crossflow Mach numbers, as M∞ sinα

approaches and exceeds unity and the Sychev-regime is approached. The sudden change inslope of the body normal-force noted for 1st-order piston theory arises due to the leeside flowreaching vacuum in the 1st-order model; it isnoted that this is delayed when the 3rd-ordermodel is used. It is noted that the normal-force slope from 3rd-order piston theory andfrom experiment both appear to asymptote toa near-constant value for α > 15◦, with betteragreement at M∞ = 2. From Eq. (4) and Eq. (9)it is seen that for the cylindrical portion of thebody, CNB ∝ c3M∞ sin3

α, which differs from theSychev similarity parameter of M∞ sinα; thus,the correlation in slopes over the range of α

shown at M∞ = 2, as noted for both wing-bodycombinations, may be a happy coincidence.

5

M.-C MEIJER, L. DALA, P.G. KARKLE

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

α [◦]

CN

B+

CN

B(W

)

ΛLE = 78.2◦, M∞ = 2.0

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rd

Nose alone

(a)

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

α [◦]

CN

B+

CN

B(W

)

ΛLE = 78.2◦, M∞ = 2.8

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rd

Nose alone

(b)

Fig. 5 : Body normal-force coefficient in the presence of Wing-1: (a) M∞ = 2.0, (b) M∞ = 2.8.

0 10 20 30 400

0.5

1

1.5

2

α [◦]

CN

B+

CN

B(W

)

ΛLE = 85.5◦, M∞ = 2.0

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rd

Nose alone

(a)

0 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

α [◦]

CN

B+

CN

B(W

)

ΛLE = 85.5◦, M∞ = 2.8

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rd

Nose alone

(b)

Fig. 6 : Body normal-force coefficient in the presence of Wing-8: (a) M∞ = 2.0, (b) M∞ = 2.8.

6

PISTON THEORY IN INTERFERENCE FLOWS

0 10 20 30 400

0.5

1

1.5

2

α [◦]

CN

W(B)

ΛLE = 78.2◦, M∞ = 2.0

Experiment [14]WingBody codeUT [14]SBT [14]PT 1stPT 3rd

(a)

0 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

α [◦]

CN

W(B)

ΛLE = 78.2◦, M∞ = 2.8

Experiment [14]WingBody codeUT [14]SBT [14]PT 1stPT 3rd

(b)

Fig. 7 : Wing-1 normal-force coefficient in the presence of the body: (a) M∞ = 2.0, (b) M∞ = 2.8.

0 10 20 30 400

0.5

1

1.5

2

2.5

3

α [◦]

CN

W(B)

ΛLE = 85.5◦, M∞ = 2.0

Experiment [14]WingBody codeUT [14]SBT [14]PT 1stPT 3rd

(a)

0 10 20 30 400

0.5

1

1.5

2

2.5

α [◦]

CN

W(B)

ΛLE = 85.5◦, M∞ = 2.8

Experiment [14]WingBody codeUT [14]SBT [14]PT 1stPT 3rd

(b)

Fig. 8 : Wing-8 normal-force coefficient in the presence of the body: (a) M∞ = 2.0, (b) M∞ = 2.8.

7

M.-C MEIJER, L. DALA, P.G. KARKLE

0 10 20 30 400

0.5

1

1.5

2

2.5

α [◦]

CN

ΛLE = 78.2◦, M∞ = 2.0

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rdPT 3rd*

* Wing withnose alone

(a)

0 10 20 30 400

0.5

1

1.5

2

α [◦]

CN

ΛLE = 78.2◦, M∞ = 2.8

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rdPT 3rd*

* Wing withnose alone

(b)

Fig. 9 : Overall normal-force coefficient for the Wing-1-body configuration : (a) M∞ = 2.0, (b) M∞ = 2.8.

0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

α [◦]

CN

ΛLE = 85.5◦, M∞ = 2.8

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rdPT 3rd*

* Wing withnose alone

(a)

0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

α [◦]

CN

ΛLE = 85.5◦, M∞ = 2.8

Experiment [14]WingBody codePNK [14]SBT [14]PT 1stPT 3rdPT 3rd*

* Wing withnose alone

(b)

Fig. 10 : Overall normal-force coefficient for the Wing-8-body configuration : (a) M∞ = 2.0, (b) M∞ =2.8.

8

PISTON THEORY IN INTERFERENCE FLOWS

Consideration of Figs. 9–10 shows thatpiston theory consistently over-predicts thewing normal-force slope, with degradation inperformance as α increases. The best agreementis obtained at low incidence for the less-sweptWing 1 at the higher Mach number of M∞ = 2.8:this is expected, as this represents the closestcombination of parameters to the domain forwhich piston theory is theoretically valid. Onceagain, the rapid change in the normal-force slopefor 1st-order piston theory is due to vacuum beingreached in the 1st-order model. In consideringthe results for CNW as predicted by piston theory,it is noted that for the given geometries, thicknesseffects are negligible and nonlinearities are dueto the proportionality CNW ∝ c3M∞ sin3

α. Theworse prediction of dCNW (B)/dα at α = 0◦ notedfor Wing 8 relative to Wing 1 is consistent withthe smaller portions of the flowfield in which bothspanwise and axial gradients may be neglected,as assumed in piston theory.

The preceding comments regarding CNB +CNB(W )

and CNW as predicted by piston theory arereflected in the results for the overall normal-force (CN) of the wing-body section, as shown inFig. 5 and Fig. 6. The prediction by 3rd-orderpiston theory for the wing-alone and the nose-alone is included to provide an estimate of theimprovement to dCN/dα at α = 0◦ that mightbe achieved through more appropriate modellingof the pressure on the cylindrical body at low-subsonic crossflow Mach numbers.

5 Conclusions

The theoretical basis for applying piston theoryrequires that velocity gradients in two orthogonaldirections (one being the axis for the unsteadyanalogy) be negligible relative to gradients in athird direction (the orientation of the cylinder).Theoretical considerations for the validity ofthese assumptions suggest that classical pistontheory may be applied for α ≈ 0◦ in regionssufficiently far from the wing-body junctionand from the wing-tips and at sufficiently highMach numbers. Outside of these parameters,the flowfield is 2D unsteady in the cross-flow plane and interference must be accounted

for. Comparison of piston theory predictions toexperimental results for highly swept wing-bodycombinations suggest that classical piston theorymay not be applied without modification; resultsfor the body at high-subsonic to supersoniccrossflow Mach numbers suggest preliminaryprediction of dCN/dα may be possible usingclassical piston theory.

Acknowledgements

This paper is dedicated to Prof. P. G. Karkle, whopassed away before the completion of the work.

References

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[2] Dowell EH and Bliss DB. New look at unsteadysupersonic potential flow aerodynamics andpiston theory. AIAA Journal, Vol. 51, No. 9,pp 2278-2281, 2013.

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[7] Shi X, Tang G, Yang B, Li H. Supersonic flutteranalysis of vehicles at incidence based on localpiston theory. Journal of Aircraft, Vol. 49, No. 1,pp 333–337, 2012.

[8] Han H, Zhang C, Wang F. An approximatemodel of unsteady aerodynamics for hypersonicproblems at high altitude. Chinese Journal ofTheoretical and Applied Mechanics, Vol. 45,No. 5, pp 690-698, 2013. (Abstract in English)

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M.-C MEIJER, L. DALA, P.G. KARKLE

[9] Ilyushin AA. The law of plane sections inthe aerodynamics of high supersonic speeds.Journal of Applied Mathematics and Mechanics,Vol. 20, No. 6, 1956.

[10] Sychev VV. Three-dimensional hypersonic gasflow past slender bodies at high angles of attack.Journal of Applied Mathematics and Mechanics,Vol. 24, No. 2, pp 296-306, 1960.

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[12] Voevodenko NV and Panteleev IM. Numericalmodeling of supersonic flow over wings withvarying aspect ratio on a broad range of anglesof attack using the law of plane sections. FluidDynamics, Vol. 27, No. 2, pp 239-244, 1992.

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6 Contact Author Email Address

mariuscmeijer@gmail.com

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