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NASA Contractor Report 3062
NASA CR 3062 C.
A Higher Order Panel Method
for Linearized Supersonic Flow
F. Edward Ehlers, Michael A. Epton,
Forrester T. Johnson, Alfred E. Magnus,
and Paul E. Rubbert
CONTRACT NAS2-
MAY 1979
FOR EARLY DOMESTIC DISSEMINATION Because of its significant early commercial potential, this information, which has been developed under a U.S. Gov- ernment program, is being disseminated within the United States in advance of general publication. This information may be duplicated and used by the recipient with the ex- press limitation that it not be published. Release of this information to other domestic parties by the recipient shall be made subject to these limitations. Foreign release may be made only with prior NASA ap- proval and appropriate export licenses_~;his+e+erd shall be marked on any reproduction of th]s.iTfor,matr,on rn;w,hole or in part. ,,Q,“. Date for general release. Decem!$1+9F
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FED0 DOCUMENT
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CONTENTS
Page
1.0 SUMMARY............................ 1
2.0 INTRODUCTiON......................... 2 2.1 Historical Development of Panel Methods.............. 2 2.2 Technical Approach and Development of the Present Supersonic Panel Method......................... 4
3.0 THEORY OF LINEARIZED COMPRESSIBLE FLOW........... 6
Derivation of Basic Equations ................... 6 Uniqueness of the Mass Flux Boundary Conditions in Subsonic Flow .... 8 Euler Momentum Theorem for Mass Flux Boundary Conditions in Linearized Theory ...................... 11 Integral Equations for Supersonic Linearized Flow ........... 14 Integral Equation for Super-inclinedPanels .............. 18 Uniqueness of the Solutions with Potential and Linear Mass Flux Boundary Conditions on Super-inclinedSurfaces ............ 20 Integral Equation for Subinclined Panels ............... 23 Uniqueness of Solution on Subinclined Surfaces ............ 26 The Velocity Components Neglecting Vortices at Panel Edges ...... 30 Boundary Conditions on Wings and Bodies .............. 3 1
GENERAL DESCRIPTION OF THE SUPERSONIC PANEL METHOD..
4.1 Earlier Six Parameter Spline Method............. 4.2 Improved Nine Parameter Doublet Spline and Improved Paneling To Obtain Continuous Doublet Strength and Geometry...... 4.3 Combined Source and Doublet Panels............. 4.4 Efficient Computations of the Influence Coefficients.......
AERODYNAMIC INFLUENCE COEFFICIENTS FOR SUBINCLINED DOUBLET PANELS....................
PANELEDGE SINGULARITIES FOR DOUBLET PANELS......
INFLUENCE COEFFICIENTS FOR SUBINCLINED DOUBLET PANELS NEGLECTING THE EDGE VORTEX..............
AERODYNAMIC INFLUENCE COEFFICIENTS FOR SUBINCLINED SOURCE PANELS......................
AERODYNAMIC INFLUENCE COEFFICIENTS FOR SUPERINCLINED PANELS.......................... 9.1 Velocity Potential for Source and Doublet Panels........ 9.2 Velocity Components for the Doublet Distribution Without Edge Vortices......................
... ill
CONTENTS (concluded)
Page
APPENDIX E. RELATIONSHIP^ OF SUPERSONIC PANEL IN INFLUENCE COEFFICIENTS TO SUBSONIC PANEL INFLUENCE COEFFICIENTS..................... 148
REFERENCES............................ 153
FIGURES..............................
NOTATION AND SYMBOLS
PCI
P
i
B
P
C
CP
F
Hk (0)
I (^11)
m
ml
M
?I
n,
nxcjnyc3nzc
P
pn
QI
Qm
Matrix of transformation from compressible to panel coordinate system
Matrix of transformation from reference to compressible coordinate system
ith row of Matrix Ar
=pT-
Matrix relating normal to conormal
velocity of sound
=(p - p0) /% pOU02 Pressurecoefficient
Force on surface
=J x2 - rk (e)2 Hyperbolic distance from kth panel edge r = rk (0)
=(y2 - y1) I( x2 - x1)
Parameter in oblique transformation
Free stream Mach number
=n ny, n, Unit normal vector
= (ry M2) n,, ny, nz Conormal vector
Components of unit normal in compressible coordinate system
Pressure,Newtons/square meter
= zJsnR ds/r
ReplaceszQmO and defined by equation (A 19)
= XmQO+ Ql /m
vi
xm A xm
Zi
XC’Yc, zc
xp Yp zr
Ye, ze
ym
$rn
‘rn
zr
ZR
%
PC
=s,-s/m
=mx,form<l
xjcij Farfield panel coordinates
Compressible coordinate system
Reference coordinate system of configuration
Panel edge coordinates for superinclined panels
= s - s,/m Oblique coordinate to x,
= mYm =ms-s,,^ for^ hn^ Ii^1
=xm2+z 2 /m^ 2 A,z,^ =m2z m =cm2+z
‘(YmlR2-Ym2’1) ~[Ym1Ym2+(*-~2)R~R~]
= R/( s, - ms )
GREEK SYMBOLS
Angle of attack of free stream velocity vector with respect to reference coordinate system
Angle of yaw of free stream with respect to reference coordinate system
=&
=JIG
Polar coordinate angle
Angles of rotation in transformation to panel coordinate system
Source strength
= l/m
Set of singularity values associated with the panel system
Doublet strength
Density, kilograms/meter
... Vlll
= l/R,
Perturbation velocity potential in meters
Equivalent incompressible velocity potential in meters
Variables of integration over planar subinclined panel
Variables of integration over planar subinclined panel
=aa-- a
ax, ay, az
Vector differential operator
NOTATION
Subscripts 1 and 2 used to designate quantities evaluated at points 1 and 2 of panel edge
a/an, Conormal derivative
- Denotes upper and lower sides of surface when used as superscripts
I
Denotes finite part of integral
-1 As superscript on matrix denotes inverse
T As superscript on matrix denotes transpose
V = (a/ax, alay, a/az)
v = (-a/at,^ a/an, ajay)
V’ = (a/at,^ a/a7j,^ a/a0^ or sometimes^ (a/aq, a/at)
Proportional to
ix
2.0 INTRODUCTION
2.1 HISTORICAL DEVELOPMENT OF PANEL METHODS
Methods based on linearized theory with singularity distributions over panels represen- ting aircraft surfaces have been found especially useful tools for analyzing the aerodynamic forces on aircraft. These methods can treat configurations of general shape which are not tractible by direct mathematical analysis. Boundary conditions are applied at discrete points associatedwith each panel of the surface. The required integrals are usually evaluated in closed form, and a set of linear equations results to be solved for the required parameters. These methods have been especially successful for subsonic or incompressible flow and since a good summary is given in Ashley and Rodden [ 1I and Rubbert and Saaris [ 21, they will not be discussedhere. The most recent method is that of Johnson and Rubbert [3]. Much of the technique for supersonic flow presented here is derived from their method.
In recent years, there are three panel methods based on linearized supersonic flow which are noteworthy. Woodward 141 used constant pressure and constant source panels and applied tangential flow boundary conditions on wings and bodies. His method was later improved by utilizing constant line sources and vortices which vary linearly in the streamwise direction on each panel (reference 15I 1.
To obtain differentiability of the computed pressure, Mercer, Weber, and Lesferd [ 61 used singularity splines providing continuity of the vorticity. Difficulties arose at discon- tinuities in the plan form for the vorticity method which were alleviated by rounding such corners. Piecewise linear pressuredistributions were obtained for planar wings with linear boundary conditions instead of the stepwise distribution from the constant singularity panel methods.
Morino, Chen, and Suciu [ 71 describe a method using Green’s formula for the potential. The boundary conditions of tangential flow are not applied directly to the potential in the useful way but are inserted into the source term in Green’s theorem. Requiring continuity of tangential flow near the surface insures that the boundary conditions are satisfied. The configuration is divided into quadrilaterals defined by an array of grid points on the surface. The surface is approximated for each panel by fitting a hyperboloidal surface through the four points which maintains geometric continuity with adjacent panels. The quantities, such as values of doublet and source strengths, were assumedconstant on each panel leading to an influence coefficient method with a set of linear equations to be solved for the values of the unknown doublet strengths.
In the appendix to reference [ 61, Mercer, Weber, and Lesferd suggesta doublet Mach line panel method using a fourth degreepolynomial in characteristic coordinates for the doublet distribution on each panel to analyze planar wings. This type of paneling has a num- ber of positive features:
.
- Discontinuities in pressureoccur across Mach lines from planform corners and can be easily taken into account with Special Mach line paneling.
- The aerodynamic influence coefficients in the characteristic coordinates are very sim- ple, closed form expressions.
- Becauseof the domain of influence of the characteristic strips, the matrix of coefficients for the parameters can be made triangular by proper ordering, and hence the solution is exceedingly fast.
- A continuous pressuredistribution^ is obtained.
This method, along with a similar source panel technique, was derived and tested as part of the present contract. The basic theory and results are presented in a separatecontractors report (reference [ 81). The source panel method was stable and accurate for both analysis and design boundary conditions. The doublet method was also stable for the region down- stream of the Mach line from the corner formed by the supersonic leading edge but was unstable for the region downstream of the Mach line from the comer formed by the supersonic.and subsonic portions of the leading edge. The causeof this instability was not pursued, principally because.themethod was confined to planar wings with linearized boundary conditions and hence has limited application, but also becausethe subsonic method of Johnson and Rubbert [ 3 ] appeared to be a more promising approach.
The most advanced panel method for subsonic flow is that developed by Johnson and Rubbert [ 31. The technique usesa quadratic approximation for curved panels with a quad- ratic distribution of doublet strength and a linear distribution of source strength applied to each panel. The vanishing of the normal component of the massflux is applied as boundary conditions to solid surfaces instead of the vanishing of the normal component of velocity as in most of the earlier methods. This is discussedin section 3. The aerodynamic influence coefficients are correct to the first order in relative panel curvature. The method has the following properties:
- It is insensitive to how the configuration is paneled; and, hence, allows the use^ of auto- matic paneling programs. It does not require special experience in applying the program to practical engineering problems.
- It is economical in computation cost. For a given accuracy, the method requires fewer panels than the constant singularity panel techniques.
- It offers a wide variety in the choice of modeling techniques, including both design and analysis boundary conditions.
The numerical method is stable, accurate, flexible, and efficient, essential properties for any numerical method to be useful and practical for engineering design and analysis. It was felt that the approach was ideal for supersonic flow as well. The techniques for treating the geometry and the mode of paneling in the subsonic method were utilized with very little changein the initial supersonic method, along with the quadratic doublet and linear source distribution on each panel. In subsonic flow, disturbances due to discontinuities of the doub- let strength at panel edgesor gaps in panel geometry decreasein intensity with distance from their origin. In supersonic flow, similar discontinuities produce disturbances which do not decay with distance but introduce infinite singularities along Mach cones emanating from panel
n Z aijXi = bj,^ j^ =^ 1,2,..n i= 1
where aij is the matrix of aerodynamic influence coefficients, Xi are the singularity strengths. Once the Xi are found the velocity and pressuremay be computed anwhere in the flow field.
Johnson and Rubbert [33 also included a correction to the aerodynamic influence coef- ficients due to panel curvature, by expanding the potential for a curved panel in terms of the small quantities defining the panel shapeand retaining only the first order terms. The same approach failed for supersonic flow becauseof the singularity on the Mach cone which was of higher order for the curvature terms and, hence, the pressureswere non-integrable. With flat panels, the method produced excellent results for a wide range of configurations, and these are reported in section 12.
Hess [ 91 has shown that a discontinuity in doublet strength induces the same velocity field as a line vortex. (The relationship between doublet sheets,vortex sheetsand line vor- tices is explained in Appendix A of reference [ 91). Even when the doublet is continuous from one panel edge to the adjoining panel edge, gaps between panels will produce infinite singu- larities on the Mach cones from the panel corners becausethe gap prevents the cancelling of the two line vortices. These singularities in supersonic flow do not decay with distance from the edge as in subsonic flow but propagate along Mach cones nearly unattenuated (see section A8.1 in Appendix A). For some paneling configurations, these disturbances impinge upon control points and causelarge oscillations in pressure. It was felt that since the discontinui- ties in doublet strength were small, the problem could be eliminated by discarding the line integrals of the doublet strength along panel edgeswhich produce the line vortices. (For example, the second integral of the equation following equation (7.1)). When this was done the method proved to be inaccurate even for small discontinuities of doublet strength or geo- metry, although this approach was successful for subsonic flow.
It became apparent that, for supersonic flow, the gaps at panel edgesmust be eliminated by a panel system that maintains continuity of geometry and by a spline system which ensures continuity of doublet strength. This was achieved by dividing the basic non-planar panel formed by the four points on the curved surface into 8 triangular panels in such a way that no gaps occur in the geometry (see figure 1). A quadratic distribution of doublet strength is pres- cribed over each triangular subpanel in such a way that doublet strength is continuous over the entire surface. This leads to a 9 parameter spline instead of the former 6 parameter spline; and with the combined source and doublet paneling yields satisfactory solutions for the super- sonic flow over complicated wing body configurations.
Section 3 presents a derivation of the basic integral equations of linearized theory with appropriate boundary conditions, while section 4 presents a more detailed description of the panel method. Derivation of specific formulas for the method are given in Section 5 through
- For those interested only in an overall view of the panel method, section 4 and also sec- tions 12 and 13 giving results and conclusion may be read without the necessity of having read the complete theory.
3.0 THEORY OF LINEARIZED COMPRESSIBLE FLOW
3.1 DERIVATION OF BASIC EQUATIONS
To derive the basic equation we follow the analysis of Ward [ 101 and consider Euler’s equations for steady, inviscid flow in the form
vx&=o (^) (3.1)
v.p~,=o
PVC l vvc+vp=o
(3.2)
where vc = u + v, u is the free stream velocity vector, and v the perturbation velocity vec- tor. The quantities p and p are the pressureand density, respectively. With the subscript 0 denoting free stream values of the quantities, linearizing of equations (3.1) to (3.3) yields
vxv=o
pov. v+v. v(p-po) = 0
POU- vv+ “(p-PO) = 0
Since for isentropic flow dp = c2dp zc02dp, where c is the velocity of sound, ehillating p and p between equations (3.5) and (3.6) gives a single differential equation for the velocity, namely,
v. [V - a(~*v)/c()q = 0
(3.6)
or v.w=o (3.7)
we have from equation (3.10)
PVC - p($^ = po [ Tm(Eq/c()2^ +... ]^ = pow^ +...^ (3.11)
Hence equation (3.7) is shown to be the conservation equation for the perturbation mass flux vector.
The conditions of irrotationality, equation (3.1), is satisfied by introducing a velocity poten- tial. Let v= uovqb (3.12)
then the conservation equation (3.7) becomes, with u in the x direction,
(l-M2bxx + @yy+ @zz= 0 (3.13)
and is easily recognized as the classic Prandtl-Glauert equation for linearized compressible flow. From equation (3.10) the pressure coefficient becomes
cP= - 1
P-PO
pouo
=- 24, - (LGx2+dJy2+9z2) (3.14)
where
fl2=l-M
When p2~ 2 is dropped this is the slender body approximation tothe pressure. For flat wings, the quadrttic terms are often neglected. However, we often compute pressureby the slender body approximation or by the complete isentropic relation.
3.2 UNIQUENESS OF THE MASS FLUX BOUNDARY CONDITIONS
IN SUBSONIC FLOW
To demonstrate uniqueness of the solution of equation (3.7) or (3.13) under certain boundary conditions for subsonic flow, we apply the divergence theorem to @Wfor a volume V enclosed by a surface S. We obtain, usingv. W = 0,
‘- W.v$dv= 2 3u- +^ v2^ +w9^ dv^ (3.15)
where fi= ( n^ X’ “y’ “z >.is the outward normal to the surface S. Let S be divided into two parts with $ prescribed on Sl and 6’ l fi prescribed on S2. Since the differential equation is linear, the difference of two solutions is also a solution and equation (3.15) holds. Let 4 and W denote.the difference of two solutions which satisfy the sameprescribed boundary conditions. Then the surface integral on the left hand side of the equation (3.15) vanishes. The volume integral on the right side is identically equal to zero. Since the terms of the inte- grand are always positive, the integral can be zero only if the terms of the integrand are zero everywhere, and the two solutions are identical. Hence, the solution of V- W = 0 will be unique with C$or W l fi prescribed as boundary conditions on the surface S. This can be shown for infinite regions when @goes to zero like l/r as r goes to infmity. In the light of the uni- quenesstheorem it is appropriate in our panel methods for analyzing the flow over bodies to use linearized mass flux boundary conditions. On solid boundaries this takes the form
(~iW).::=O (3.16)
or for the velocity potential in equation (3.12) and a free stream velocity in the x direction
(1 +82~x)nx+Oyny+~znz=0 (3.17)
This equation can be written in a different form by introducing the conormal vector
EC= P^2 n,, ny, nz
We obtain for equation (3.17);
nx + a4/an, = 0
The conorrnal derivative of the potential then can be interpreted as the normal component of the perturbation massflux to the surface.
The differential equation (3.13), the pressurerelation in equation (3.14) and boundary conditions in (3.17) comprise the fundamental boundary value problem we are solving for analyzing aircraft configurations by the methods described here. When the flow is subsonic, the equations can be converted to the incompressible flow by introducing the incompressible velocity potential
@= @i/P (3.18)
