Improved Solution for Potential Flow About Arbitrary A Xi Symmetric Bodies by the Use of a...

8/6/2019 Improved Solution for Potential Flow About Arbitrary A Xi Symmetric Bodies by the Use of a Higher-Order Surface S

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 5 (1975) 297-308.0 NORTH-HOLLAND PUBLISHING COMPANY

IMPROVED SOLUTION FOR POTENTIAL FLOW ABOUT ARBITRARYAXISYMMETRIC BODIES BY THE USE OF A HIGHER-ORDERSURFACE SOURCE METHOD *

John L. HESSSection Manager, Theoretical Aerodynamics, Douglas Aircraft Company, Long Beach, California, USA

Received 19 November 1974In recent years the surface-source method of calculating potential flow about arbitrary bodies has been developed

extensively and has proved to be a useful tool in a wide variety of low-speed design applications ranging from simpleshapes to complicated inlets with centerbodies, multi-element airfoils, and wing-fuse&e-pylon-nacelle combi-nations. Two-dimensional, axisymmetric, and three-dimensional methods have been developed. While the method isgenerally quite satisfactory, it is desirable to increase computational speed and accuracy for certain applications, par-ticularly interior flows and exterior flows about complicated multiple-body combinations. Such improvements can berealized by refining the formulation. In the basic method the profile curve of a two-dimensional or axisymmetric bodyis approximated by a large number of straight-line elements over each of which the source density is constant. Theso-called higher-order refinement consists of using curved surface elements and a source density that varies over anelement. This paper describes the analysis for the axisymmetric case and presents a number of test cases to show thelarge increases of speed and accuracy that can be obtained with the higher-order formulation.

1. Introduction

Recent years have been seen the development of very general surface singularity methods forthe calculation of potential flow about arbitrary configurations [ 1 . Moreover, these methodshave been applied successfully to a large number of practical design problems of low-speed flow[ 1,2]. The most common such method is the so-called surface-source method [ 21, which utilizesa source density distribution over the surface of the body about which flow is to be computed.Application of the boundary condition of zero (or prescribed) normal velocity on the body sur-face theoretically yields a Fredholm integral equation of the second kind for the source density.Once the source density is known, all other quantities, such as flow velocities and pressures, maybe obtained by integration. Separate procedures have been developed for calculating flow abouttwo-dimensional bodies, axisymmetric bodies, and three-dimensional bodies. For the case ofaxisymmetric bodies the flow itself does not necessarily have to be axisymmetric, but it may bea case of cross-flow for which the free-stream direction is perpendicular to the bodys symmetryaxis or a case of rotation of the body about an axis perpendicular to its symmetry axis [ 21.To implement this method for the computer, various approximations must be made. In particular, both the body shape and the source density distribution must be approximated in a formsuitable for machine computation. During most of the development of the surface-source methodthe profile curve defining an axisymmetric (or two-dimentional) body has been approximated by* The work was supported by NASA Lewis Lab., under Contract NAS3-18018.


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298 J.L. Hess, Impm ved solution for potential flow about arbitrary axisymm etric bodies

a large number N of small straight-line elements which form an inscribed polygon. Moreover, thesource density has been assumed to be constant over each straight-line element, although it variesfrom one element to another. This reduces the problem of determining the source density distri-bution to that of determining a finite number of values of the source density - one for each ele-ment. One point of each element, the midpoint for a straight-line element, is selected as the con-trol point where the normal-velocity boundary condition is to be applied. Formulas have beenderived that give the velocity at any point due to a unit value of source density on a straight-lineelement. From these formulas a matrix of vector velocities induced by the elements at the controlpoints can be obtained. Then the integral equation is replaced by a set of linear algebraic equationsfor the values of source density on the elements. The coefficient matrix of this set of equationsconsists of the set of normal velocities induced by the elements at each others control points,which is obtained by taking normal components of the basic induced velocity matrix. Finally,surface velocities at the control points are obtained by a matrix multiplication of the tangentialcomponents of the induced velocity matrix w?th the column of values of source density. The twomain parts of the computation are (I) the calculation of the N2 velocities that comprise the in-duced velocity matrix and (2) the solution of the linear equations for the values of source density.If a direct elimination solution is used, the computational magnitude of solving the linear equa-tions is proportional to N3.

The procedure described above, which uses flat surface elements and a piecewise-constantsource density distribution, is designated the base method. It has proved satisfactory in a widevariety of design applications [ 1,2] . However, it is evident that more elaborate procedures canbe formulated and that these should give a higher accuracy for a given element number N andthus an equal accuracy with a smaller element number. Moreover, if it is properly implemented,a more elaborate procedure would require only a slightly greater computing time than the basemethod for a given N. Because of the rapid variation of computing time with N, a reduction ofcomputing time for a given accuracy should be possible. This has been successfully accomplishedfor the case of two-dimensional bodies [ 1, 3351, in which case the application of main interestis a multi-element airfoil. Here the case of axisymmetric bodies is considered where the applicationof main interest is an inlet, possibly with centerbody and ring vanes. The technique employed isbasically the one used in two dimensions [ 31, which employs curved surface elements and a sourcdensity that varies over the element. Such an approach is designated a higher-order implementa-tion. For axisymmetric bodies these variations refer to the profile curve defining the body. Thecircumferential variations around the symmetry axis of element geometry and source density.are similar to those of the base method [ 21. Namely, a complete surface element is a portion ofa conoid (a cone frustum in the flat-element case), and the source density is independent ofcircumferential angle for axisymmetric flow and is proportional to the cosine of the circumferen-tial angle for cross-flow and for rotation.Use of higher-order implementations brings up the question of the consistency of the ordersof the approximation used for the element geometry and the approximation used for the sourcedensity. It is known [ 1,3] that consistency is obtained when the polynomial expressing theelement shape has a degree one higher than that defining the source density. Thus consistentformulations include: (1) flat-element ccnstant-source: (2) parabolic-element linear-source, and(3) cubic-element parabolic-source. In this paper parabolic elements are used together with apiecewise-parabolic source density. The extra inconsistent parabolic term in the source density


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J.L. Hess, Improved solution for potential flow about arbitrary axisym metric bodies 29

has been included to determine whether or not it gives increased accuracy in low-curvature regionIt turns out that, as the theory predicts, little or no gain in accuracy is obtained by including theparabolic source term.

2. Surface element geometryThe profile curve of the body about which flow is to be computed is specified as a table of

coordinates for N + 1 points (xi, yi), each of which is presumably exactly on the contour. By thismeans the contour is divided into N elementary arcs as shown in fig. 1. On each arc a controlpoint is selected by the following criterion: the normal projections of the endpoints of the elemen

Fig. 1. Division of a body contour into elementary arcs.

(xi, yi) and (xi+i, yi+i) on the line tangent to the arc at the control point are equidistant from thecontrol point. The slope of the tangent line at the control point is defined as the slope of theelement. If this tangent line is taken as the horizontal axis of a g, q coordinate system with thecontrol point as origin, the elementary arc is as shown in fig. 2. The equation of this arc may bewritten as a power series

q = ct2 + . . (1The arc length s along the arc measured from the control point is given by

d.s= [Jl +(~cE)~ + . ..I dt= [l + 2c2t2 + . .I d[.

Y. B

(2

Fig. 2. An elementary arc.


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J.L. Hess, Improved solution for potential flow about arbitrary axisym metric bodies

Thuss=t+;c2i3+... .

The basic reference coordinate system in which the body is defined has its x-axis as the symmetryaxis of the body. Let a point of the element have coordinates x = b, y = a in this system, then

~=~+(cos01)~---c(sincu)~2+... , a=y+(sincr)~+c(cosru)~2+... , (4where (11s the slope angle of the [-axis (tangent line) with respect to the x-axis, and X, v arereference coordinates of the control point.In principle, approximations of arbitrarjly high order could be generated by retaining sufficientterms in the above series. For present purposes the element is assume to be parabolic, and theabove series are terminated with the terms shown. For this approximation the control point lieson the pe~endicular bisector of the straight line between (xi, yi) and (x,, , yi+r ), and the slope othe element equals the slope of this straight line. Also, by inspection of (3) it can be seen that tothis order of approximation s may be substituted for t in (4). A circle is passed through the point(x~_~,JJ_~), (xi, vi), and (xi+%, i+r), and another circle is passed through the points (xi, yi),tx i+rp~i+r) and (xi+z,Y~+~). he half-curvature c of the parabolic element is set equal to half thegeometric mean of the curvatures of these two circles. Requiring the parabola to pass through theendpoints (xi, yi) and (xi+r JJ~+~ then uniquely defines the parabola and establishes the coordi- -nates x, y of the control point. For the first and last elements of a body one circle is not definedand c is set equal to half the curvature of the remaining circle.

3. Induced velocity matricesThe basic calculational task of the flow-calculation method is to compute the flow velocitiesinduced by the elements at each others control points. There are three types of induced velocitiecorresponding to three types of surface singularity: (1) a constant-strength ring source, which isappropriate for axisymmetric flow; (2) a ring source whose strength is proportional to the cosineof the circumferential angle, which is appropriate for cross flow and for body rotation; and (3)a constant-strength ring vortex, which is used to produce certain auxiliary solutions appropriateto axisymmetric flow about ring-airfoils and inlets. The velocities in the induced velocity matricesare obtained by integrating the ring source or vortex formulas over an element. Specifically, let aring singularity (source or vortex) have a radius a and lie in the plane x = b with its center on the

x-axis. Then the velocity at the point (x, y) due to this singularity isv= V[x, y, b, a] . (5

Specific formulas for the two types of ring source are contained in [ 21 and derived in [6] and [ 7Many of the functions appearing in the expressions are repeated, particularly the specific completelliptic integrals. The velocity components of a constant-strength ring vortex are related to thoseof a constant-strength ring source by [8] :


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J.L. Hess, Improved solution for potential flow about arbitrary axisymm etric bodies 30

V, (vortex) = i(t--b; V,(source) +z VY(source) ,(6x-bV,(vortex) =i VX(source) -a VY(source) .

Let the source density on the j-th element be denoted uj(s). The velocity induced by this sourcedensity at the control point of the i-th element (Xi, Ji) is obtained by integrating over the elementthat is

V; = J V[X, Vi>bj(S)>j(S)l oj(S)h > (7A s i

where Asj is the total arc length of the j-th element. Eq. (7) applies to both types of source singu-larity. This equation also applies to the vorticity singularity if uj(s) is replaced by the vorticitystrength pj(s). In evaluating the integral of (7), b and a are given by (4) with s replacing t.The source density may be written as a power series

u.(s) = u. + fJV)s + 0!2)s2 +J I I I . . . (8where uj, u(l) and uq2) are independent of s and represent, respectively, the value, first derivative,and half th: secondderivative of the source density at the control point of the j-th element. Inthe present analysis the source density is assumed to be at most parabolic, and (8) is terminatedwith the terms shown. Thus (7) becomes

vyj=uj J V[Xi, Yi, bj(s), Uj(S>I ds + Us S V[Xi, vi, bj(s), aj(s>I S dsA s j A sj

+ up J - VlXi> Vi> bj(S), aj(S)l s2 h 9 (9

orv; = vpTj + vpy +vpuy . (10

The vorticity distribution pj(i(s) is taken as constant so that only the first terms of (9) and (10) arpresent. In fact, the vorticity is handledexactly as in the base method [ 11 except that the integrais over a curved element rather than a flat one. The form of eq. (10) makes it easy to investigatethe effectiveness of retaining various terms in the source density expansion. The relative effective-ness of flat and curved elements can be investigated by setting c equal to either zero or nonzeroin eq. (4).

For j # i the integrals in (9) are evaluated by numerical integration using Simpsons rule with variable number of ordinates [ 21. It is evident from (9) that the three integrands are very similarand can be conveniently calculated together. For j = i the basic ring-source (or vortex) velocity


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302 J.L. Hess, Imp roved solut ion for potential flow about arbitrary axisymm etric bodies

V[x, .v, b, a] has a singularity of order 1 s at s = 0. Accordingly, it is necessary to make series ex-pansion in powers of s so that the singularity may be cancelled analytically. These expansions aresimilar to those used in the base method [ 31. Derivation of these expressions is straightforwardbut tedious and is not pursued here. Formulas for these expansions are contained in [ 91.

4. Organization of the calculationIn the most general case all three integrals of eq. (9) must be evaluated for both the axisymmetri

and the cross flow type of source density. Th;ls, including the vorticity, which requires only thefirst integral of (9), there are seven vector integrals in the higher-order analysis as opposed to threin the base method. In axisymmetric flow there are two velocity components, axial and radial,while cross flow has an additional circumferential component. Thus the total number of scalarintegrals can be seventeen rather than the seven of the base method. However, the final numberof scalar N X N induced velocity component matrices that must be stored and used is the same inboth the higher-order and the base methods, namely five.As described above, the normal velocity boundary condition is applied at the control point ofeach element to produce a number of linear equations equal to the number of elements. However,the variation of source density over an element is described by three parameters, ui, u,$, and ot2)IThe derivatives u(l) and o(2) are expressed in terms of values of u by differentiating the equationof the parabola &rough the three values ujPl , uj, and u~+~.That is

CJ!~) Djuj_lI + ~jUj + FjUj,, ) (11&) = Gjuj_, + Hjui + 1.0.I I 1+1 where the coefficients in (1 1) can be found from standard numerical differentiation formulas anddepend only on the lengths of the three elements [ 31. For the first (or last) element of a body thparabola whose equation is differentiated passes through the first (or last) three values of u, andthe formulas of (1 1) are modified accordingly. This last feature could introduce error if a smoothcontour is defined with the first and last points coincident, as for example, a torus. However, thiscase is too rare to be of great significance. The above procedure is appropriate for all inlets andducts and also for simply connected bodies, which are input from the upstream stagnation pointto the downstream stagnation point.

From (10) and (1 1) it is evident that the velocity induced at a point by an element is a linearcombination of three neighboring values of u. The matrix that is needed for subsequent calculationis the one giving the velocity at each control point due to each value of source density. As thecalculation proceeds to compute the velocity at a control point due to successive elements. thevelocity induced by the j-th element is not associated entirely with uj as in the base method, butcertain portions are associated with u~_~, uj, and u~+~in the obvious way. When all elements havebeen accounted for, the velocity at a control point due to each value of source density has beenformed as the sum of contributions from threp elements except for those due to the first and lastvalues, to which two elements contribute, and also those due to the third and third-to-last values,to which four elements contribute. The result of this phase of the computation is a matrix Vii ,


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J.L. Hess, Improved solution for potential flow about arbitrary axisym metric bodies 30

such that the velocities at the control points due to the body areN

Vi= c Viioi , i=l,2 ,..., N. (12j= lThere is a single vector matrix Vi i for the axisymmetric source densities, one for the cross-flowsource densities, and one for the axisymmetric vorticities, just as in the base method. Moreover,as in the base method, the vortex velocities are not saved individually. Instead, the velocities pro-duced by all elements at a control point are added together to give the velocity at that controlpoint due to a vortex sheet of constant unit strength. This auxiliary onset flow is used in certaininlet and ring-wing applications [ 91.

Thus the result of this phase of the calculation is a set of matrices Vi j and vorticity onset-flowsequivalent to those of the base method. In fact, all subsequent calculations [21 are identical forboth base and higher-order methods and do not depend on how the Vi i matrix was produced. Inparticular, the normal component of Vi i is the coefficient matrix of the linear equations for thevalues of source density. The right sides of these equations are the negatives of the normal com-ponents of the onset flows, either uniform or otherwise. Once solutions have been obtained, theflow velocity at each control point (or any other point) is calculated for each onset flow by addina sum of the form (12) to the onset-flow velocity at that point.

5 Comparison of calculated results with analytic solutionsTo determine the effectiveness of the higher-order technique and to evaluate the importance o

the various terms in the expansion, a considerable number of calculations have been performedfor a sphere and for a 8-to-l prolate spheroid, for which analytic solutions are available. The firstconclusion that can be drawn from these calculations is that the addition of the quadratic sourcedensity term CJ!*) never yields an appreciable increase in accuracy regardless of whether flat orcurved surfaceelements are used. In every case the solution that utilizes only (T. and u(l) is virtualindistinguishable from the one that also includes uj *) In what follows, the soldtion that utilizescurved surface elements and a linearly varying source density is denoted higher-order while theflat-element constant-source solution is denoted the base method. Solutions obtained with othercombinations of terms are labeled explicitly.

Calculations were performed for a sphere represented by 60 equal-length element of 3 sub-tended angle. Four solutions were obtained. In addition to the higher-order and the base method,two inconsistent solutions were obtained: that using curved-elements constant-sources and thatusing flat-elements linear-sources. The higher-order solution was also calculated for a 12-elementsphere whose elements each subtend a 15 angle. Results for a uniform onset flow parallel to thex-axis are presented in fig. 3, which shows differences between calculated and analytic surfacevelocities. The importance of mathematical consistency is evident. Accounting for either sourcevariation or element curvature separately produces no improvement on the base method. How-ever, when both are accounted for, as in the higher-order solution, the result is a large gain inaccuracy - about two orders of magnitude. Even the 12-element higher-order solution is an orderof magnitude more accurate than the 60-element base method and also requires an order of mag-nitude less computing time.


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304 J.L. Hess, Improved solution for potential flow about arbitrary axisym metric bodies

0 001

0.00

O,OOJ

0.001

o.ooiYMLVcdLCVW

0.001

c

-0,001

-0 002

-0.00~

-0.004

-0 005

BASE METHOD

HIGHER OD~zh _ _ _


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J.L. Hess, Improved solution for potential flow about arbitrary axisym metry bodies 305

-0003 - - 60 ELEMENTS------ 12 ELEMENTS

- SO ELEMENTS---- 12 ELEMENTS

_-BASE METHOD

Fig. 4b.

Fig. 4a.Fig. 4. Errors in surface velocity calculated by various methodfor a sphere in a uniform onset flow parallel to the y-axis.(a) Velocity in the xy-plane. (b) Velocity in the xz-plane.

~ BASE METHOD, 60 ELEMENTS---- HIGHER ORDER, 30 ELEMENTS

0.002 -

0 ! , / J---I _d--- ___oy6--------

______-_------- x-D.6 -0.4 -0.2 0

\LNF-0.002 L

Fig. 5. Errors in calculated surface velocity on an 8-to-1 prolate spheroid in a uniform onset flow parallel to the x-axis.


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306 J.L. Hess, improved solution for potential fl ow about arbitrary axisym metric bodies

004

0 03

tY

BASE METHOD, 60 ELEMENTS-_--- HIGHER ORDER, 30 ELEMENTS

Fig. 6a

-.----____ _____-__ -------- --,- -0 -& -0: b x/ -BASE METHOD, 60 ELEMENTS--- HIGHER ORDER, 30 ELEMENTS

Y

vb b b b b b b b b - m- b - b , ,

i Vi

Fig. 6b.Fig, 6. Errors in calculated surface velocity on an 8-to-1 prolate spheroid in a uniform onset flow parallel to the y-axis. (a) Velociin the ny-plane. (b) Velocity in the xz-plane.


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J.L. Hess, Improved solution for potential flow about arbitrary axisymmetric bodies 30one velocity component out of three does not change that basic conclusion that in general a con-sistent formulation is required to obtain a significant increase in accuracy.

The relative effectiveness of the two methods of solution for an 8-to-1 prolate spheroid is illus-trated by comparing results for the base method using 60 elements with results for the higher-order method using 30 elements. For these element numbers the higher-order method is approxi-mately four times faster than the base method. Somewhat different distributions of elementsare used for the two methods. The higher-order method concentrates elements in the high-curva-ture region to a greater degree than does the base method. Correspondingly, the higher-ordersolution uses very few elements in the low-curvature region, but it is able to maintain accuracyin this region bscause of the use of variable source density. Generally, each method is used withthe element distribution for which the best solution is obtained. (Similar experience is reportedin [ 31.) Differences between calculated and analytic surface velocities are shown in fig. 5 forthe axisymmetric case where the onset flow is parallel to the bodys symmetry axis (x-axis) andin fig. 6 for the cross-flow case where the onset flow is perpendicular to the bodys symmetryaxis (parallel to the y-axis). To put these results in perspective, the maximum value of surfacevelocity is about 1.0293 times freestream velocity for the axisymmetric case of fig. 5 and about1.9447 times free-stream velocity for the cross-flow case of fig. 6. In the xz-plane, fig. 6b, thevelocity is parallel to free-stream and has a constant magnitude of 1.9447. The curves of figs. 5and 6 emphatically show that in addition to beirg much faster the higher-order technique is alsomuch more accurate than the base method. The substantial gain in accuracy that can be achievedby use of the higher-order method for cases of smooth convex bodies is in marked contrast to thtwo-dimensional case [ 31.

6. Interior flow in ductsIn two dimensions [ 31 the greatest gains in accuracy from use of the higher-order method occu

for interior flow in ducts. To investigate the behavior of the present axisymmetric case, the firstgeometry selected is the analog of one previously considered in [ 31, namely the case of a uniformonset flow into the closed duct shown in fig. 7 (an interior hemisphere cylinder). The flow insidethe duct should be virtually stagnant, and the average axial velocity component at any axial loca-tion should be exactly zero. Thus, the average axial velocity is a measure of the calculationalerror. The base method with 55 elements gives a typical error of 15 percent of free-stream velocitUse of the higher-order technique reduces this error by a full two orders of magnitude.

A more practical case is that of a contracting duct of area ratio 16 as shown in fig. 8. The con-

Fig. 7. A closed duct showing control stations where average velocities are computed.


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308 J.L. Hess, Improved solution for potential flow about arbitrary axisym metry bodies

HIGHER ORDER I 0000 I.0125 0.9960 0.9938BASE METHOD 1.0000 0.9653 0.6794 0.7340

Fig. 8. Flow in a contracting duct of area ratio 16 with total velocity fluxes calculated at various locations by the higher-order andbase method.tracting section is a portion of a sine wave. By continuity the net flux of fluid at every axial statioshould be identical so that changes in this flux represent calculational error. Net fluxes have beencalculated at five stations as shown in fig. 8 and normalized with respect to the flux in the largeconstant-diameter section. It is evident that the use of the higher-order solution reduces the maximum error in flux by a factor of about 25.Acknowledgements

The enthusiastic assistance of Mrs. Sue Schimke was a highly significant factor in the successfulcompletion of this work. While her contributions were appreciated at all stages of the development,her running and analyzing of the various test cases was especially important. Another very sub-stantial contribution was made by Mr. Robert Martin, who developed the exceedingly lengthyexpansion formulas that are used to compute the self-effect of an element. The challenging taskof programming this method for the computer was accomplished with dispatch by Mr. DouglasFriedman.

Referencesilli21[31[ 4 1

1 5 1[ 6 1

[ 7 1[ 8 1[ 9 1

J.L. Hess, Review of integral-equation techniques for solving potential-flow problems with emphasis on the surface-sourcemethod, Comp. Meth. Appl. Mech. Eng. 5 (1975) 145-196.J.L. Hess and A.M.O. Smith, Calculation of potential flow about arbitrary bodies, Prog. in Aero. Sci. (Pergamon Press, NewYork, 1966).J.L. Hess, Higher-order numerical solution of the integral equation for the two-dimensional Neumann problem, Comp. Meth.Appl. Mech. Eng. 2 (1973) pp. 1-15.J.L. Hess, The use of higher-order surface singularity distributions to obtain improved potential flow solutions for two-dimensional lifting airfoils, Comp. Meth. Appl. Mech. Eng. 5 (1975) 11-36.J.H. Argyris, The impact of the digital computer on engineering sciences, Twelfth Lanchester Memorial Lecture, Aero. J.Royal Aero. Sot. 74, Nos. 709, 710 (1970).A.M.O. Smith and J. Pierce, Exact solution of the Neumann problem. Calculation of plane and axially symmetric flows aboutor within arbitrary boundaries, Douglas Aircraft Co. Report No. 26988 (Apr. 1958). (A brief summary is contained in theProceedings of the Third U.S. National Congress of Applied Mechanics, Brown University, 1958.)J.L. Hess, Calculation of potential flow about bodies of revolation having axes perpendicular to the free-stream direction. J.Aerospace Sciences 29 (162) p. 726.J.L. Hess, Extension of the Douglas Axisymmetric potential flow program to include the effects of ring vorticity with applica-tion to the problem of specified tangential velocity, Douglas Aircraft Co. Report No. DAC 33195 (June 1966).J.L. Hess and R.P. Martin, Improved solution for potential flow about arbitrary axisymmetric bodies by the use of a higher-order surface source method, NASA CR 134694 (1974).

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