AIAA 2010-4676
28th AIAA Applied Aerodynamics Conference
28 June - 1 July 2010, Chicago, Illinois
Incompressible Flow about a Cone with Elliptical CrossSection
Mark Ishay1, Daniel Weihs2 and David Degani3
An analytical study of incompressible potential and boundary layer flowfields about a
forebody having the shape of a cone with an elliptical cross-section at zero angle of attack is
presented. We reduce the three-dimensional potential flow equations to generalized
axisymmetric equations by using an elliptical-conical coordinate system and, hence, allow
the use of a generalization of the Mangler transformation to obtain expressions for the
boundary layer velocity field. The solution is a function of the cone spatial vertex angle and
the cross-section ellipticity ratio. The friction and drag force on such cones are obtained. We
show that the drag on cones, in a range of ellipticities, is less than the drag on circular cones
of equal surface area. Several applications, including lifting body wave-rider configurations,
for symmetric and asymmetric elliptical cones, are presented.
Nomenclature
A1
a
b
C1
Cf
scaling constant for circular cone solution
long semi-axis of an ellipse
short semi-axis of an ellipse
scaling constant
skin friction coefficient
c
DEC
d
e
F
f
parameter in Elliptic-Conical coord. system
drag force
parameter in Elliptic-Conical coord. system
ellipticity ratio ( b / a )
auxiliary function
normalized Falkner-Skan function
î
Gn
l
Mm
m
n
Pn
q
Re
r0
unit vector
auxiliary function defining potential flow field in cylindrical coord. system
reference length
auxiliary function defining potential flow field in EC coord. system
parameter in the elliptic-conical coordinate system, defines the cone angle
circular cone angle parameter
Legendre functions
parameter of the potential flow solution
Reynolds number
reference value for r coordinate
r , , ˆ
Elliptic-Conical coordinates
1
PhD Student, Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel
Distinguished Professor, Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa,
Israel
3
Professor, Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel, Associate
Fellow AIAA.
2
Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
r , ,  Elliptic-Conical (EC) coordinates after transformation (2.3)
function of parameter r in the separable potential flow solution
R
auxiliary scaling function
S
s
perpendicular distance from the axis of symmetry to an arbitrary point on the surface
free stream velocity
U0
U
potential flow velocity along the surface
u
boundary layer velocity component in the x direction
velocity field
v
v
boundary layer velocity component in the y direction
X , Y , Z Cartesian coordinates
x
distance along cone’s generatrix
distance along a normal to the surface
y
Z0
cone length

0
i



0
0
local spatial half-vertex angle of a cone
maximum spatial half-vertex angle
parameters ( i  2,3 ) of the potential flow solution
Hartree parameter
polar angle parameter





w
0

polar angle
constant value of coordinate  , defines a specific cone
half vertex angle of the cone in the XY plane
function of parameter  in the separable potential flow solution
kinematic viscosity
dimensionless distance
density
parameter in the Elliptic-Conical coordinate system
wall shear stress
half vertex angle of the cone in the XZ plane
velocity potential
I. Introduction
Studies of the flow about a cone with circular and elliptic cross-sections have been of interest for many
investigators for several decades1-5. As Van Dyke1 pointed out back in the early fifties, the elliptic cone can become
a standard of comparison for supersonic flows past bodies without axial symmetry, just as the circular cone is used
as a standard for comparison for supersonic flows past bodies of revolution. The circular cone at angle of attack has
served as a model for many investigations of three-dimensional boundary layers4, 6. Most of the solutions are given
at the symmetry plane7-9, or as a three-dimensional approximate solution using perturbation methods10. Circular
cones at zero angle of attack are bodies of revolution and therefore have axisymmetric solutions3, 11.
There are many aeronautical applications in which their design requires that part of the configuration has, or
almost has, a conical shape with an elliptical cross-section, e.g., the Northrop F-5 or the NASA X-33 forebodies.
Other high-speed lifting bodies, like waveriders12, 13, may also be modeled as elliptic cones. Existing solutions for
flow about elliptic cones are for supersonic or hypersonic14, 15 flows as such aircraft are usually designed for
transonic or supersonic flights, i.e., compressible flows. However, the design should include the low speed,
incompressible flow regime.
In order to obtain a solution for incompressible flow about a semi-infinite cone with elliptical cross-section we
use an Elliptic-Conical coordinate system16. Use of this coordinate system allows us to mathematically treat the
three-dimensional flow field as a two-dimensional one in which the elliptic cone can be solved as a body of
revolution, using a generalization of the Mangler transformation for axisymmetric flow-fields17-19.
2
American Institute of Aeronautics and Astronautics
The Elliptic-Conical (EC) coordinate system is described in Part II. Using the EC coordinate system, we
examine the potential flow-field in Part III. The derivation of the boundary layer flow, wall-shear stress and drag
force along the elliptic cone are given in the Part IV. In Part V we evaluate the accuracy of the assumptions by
comparing the results with full Navier-Stokes numerical calculations. Solutions of different families of symmetric
and asymmetric elliptical cones and several applications, including wave-riders configurations, are presented in Part
VI.
II.
The Elliptic-Conical (EC) coordinate system
An elliptic cone can be represented in the elliptic-conical orthogonal coordinate system16  r ,  , ˆ  in terms of
Cartesian coordinates X , Y , Z as:
 2  rˆ  2

X  


 dc 



2
2
2
2
2
ˆ 
Y 2  r   d   d     ,
2


d  c2  d 2 


2
 2 r  c 2   2   c 2  ˆ 2  
Z



c2  c2  d 2 


(2.1)
with 0  ˆ 2  d 2   2  c 2 . The coordinate surface for   Const.   0 is, thus, a family of elliptic cones with their
axes coincident with the Z -axis (see Fig. 1(a)):
X2
Y2
Z2
(2.2)
F  X ,Y , Z   2  2
 2
0.
2
 0  0  d c   02
In order to reduce the number of constants, we introduce a transformation:


c
,
(2.3)
and the EC coordinate system equations (2.1) becomes
2 2
 2

2 
X  r

2



2
2
2
2 










Y 2  r 2
.
 2 1   2 




2
2
 Z 2  r 2 1    1    


1  2
(2.4)
The coordinate surface (2.2) is
X2

2
0

Y2
Z2

 0,
2
   1  02
(2.5)
2
0
where   ˆ / c,   d / c , so 0   2   2   2  1 , and   0  const is for a specific elliptic cone.
Fig. 1(b) describes a semi-infinite cone with an elliptical cross-section in the  ,  coordinate system. The local
spatial semi-vertex angle  is:
 1  02  1   2  
  cos 

 0
1   2  

1/ 2
1
.
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(2.6)
Z  ˆ  const r  const
X
Y (a)
(b)
Figure 1. (a) The Elliptic-Conical coordinates system, (adapted from Ref. 19), and a coordinate surface for
  const with their axes coincident with Z -axis. (b) Semi-Infinite cone with an elliptical cross-section in
the r , ,  coordinate system;  - local spatial half-vertex angle.
The coordinate surfaces (  ,   const - an elliptic cone, r  const - a sphere) are symmetrical about their respective
axes, so we have to solve the equations only for one quadrant, i.e. with all positive values for the coordinates. In
addition, there are singularities at the quadrant boundaries, which mean that in order to retrieve a circular cone the
equations need to be solved by a limiting process, or to make an approximation there.
Each cross section, projected to the plane normal to the longitudinal Z -axis (see Fig. 1) for a given cone, is an
ellipse. We define constants in Eq. (2.5) – the equation for a cross section – using an ellipticity ratio parameter
0  e  b / a  1 ( a and b are the long and the short semi-axes of an ellipse, respectively), where e  1 for a
circular cross-section, and e  0 for a flat plate. The limits e  1 and e  0 do not exist due to the definition of the
domain, 0   2   2   2  1 . Equation (2.5) can be written as:
X2
Y2
 2
1
2
2
2
 1  0  Z 0   1  02  Z 2
2
0
and
2
b
e2    
a

  2 1  02 
2  2
Z2
 0 2 1.
2
2
0
0 1  0 
2
Z
(2.7)
2
0
(2.8)
The constant  is defined:
 2  02 1  e 2 
and the spatial angle is a function of 0 and e :
cos 2  
1  02  1   2  .
1  02 1  e 2 
(2.9)
(2.10)
In order to locate edge points in the coordinate space for the extreme values of  ( 0   2   2 ), let's consider the
elliptic cross-section equation  X / a 2  Y / b 2  1 . Then, for the edge  values we get (see Fig. 1(b)):
  0 :  X , Y , Z    0, b, Z1 
.
(2.11)
   :  X , Y , Z    a, 0, Z 2 
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It should be noted that for a  b and constant r, according to Eq. (2.11), Z1  Z 2 . Hence, the line of r=const (Fig.
1(b)) for an elliptic cone, 0  const , is spatial. We have to be aware of this property while dealing with engineering
shapes, like elliptic cones, where their bases for a fixed length lie in a plane perpendicular to Z.
III.
Potential flow
A. Derivation of equations
In this section we consider the potential flow over a semi-infinite elliptic cone at zero angle of attack. Because of
conical similarity, the general form of the free stream velocity of flows over wedges20 or circular cones11 is
proportional to the distance from the vertex to power m , which is a function of the semi-vertex angle. For instance,
the equation for a circular cone relating n and the half-angle  0 , is Pn   cos  0   0 , where the prime denotes
differentiation and the Pn is the Legendre function21. For the generalized elliptical cone we solve the Laplace’
equation in the Elliptic-Conical coordinate system and obtain the relation between the parameter m  0  and the
spatial half-angle of the cone. We look for a solution of the potential flow of the form U  U1r  0  . Hence, we solve
the Laplace equation in the EC coordinate system along with an assumption that the cross-flow components are
negligible (we check the effect of this assumption analytically and numerically in Part V):
  ˆi  v  0 .
(3.1)

Substituting v   , we can have a separable solution of the form   R  r  M       . Because the boundary
m
conditions are independent of the coordinate r (as we show later), we can take   Const . With some tedious
algebraic manipulations (for details see Appendix A), this constant can be merged with the other constants of the
solution for R  r  . The equations for the velocity potential are:

2
R  22 R  0
r
r
2
2
   1   2  M     2 2   2  1 M    2 2   3  M  0 .
R 
(3.2)
 2   2  1   2     2   2  1     2  2   3    0
2
We assume  2  m  m  1 and  3  q  2  1 , but since   const as  2  2   3 , then q 
m  m  1 2
 , and the
 2 1
velocity potential is reduced to   R  r  M   , where the velocity components in the directions r and  ,
respectively, are
u

r
v
 2   2  1   2 
r  2   2 
1/ 2
1/ 2
1/ 2


.
(3.3)
Since no fluid penetrates the cone surface, we require the solutions of Eqs. (3.2) to satisfy the boundary condition
(3.4)
v   0   0 ,
where 0 defines a specific cone, (see Fig. 1(a)). A second condition is defined at the edge of the domain, i.e., for 
at the XY plane ( Z  0 ): Z 2  r 2 1   2  1   2  / 1   2   0 . As 0   2   2   2  1 this condition is satisfied
for   1 . On the symmetry plane the velocity in the r direction ( u ) should be zero, and the velocity in 
direction ( v ) can be defined in two ways: (1) the velocity in the  direction is a function of the coordinate r m 1 or,
(2) the derivative of the v velocity with respect to the distance r from the origin (or the cone vertex) is finite at this
distance, and, generally, is a function of r m  2 . Both options lead to the same boundary condition, as we shall see
later. From Eq. (3.2a) we obtain for the radial velocity component:
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American Institute of Aeronautics and Astronautics
R  r   C1r m  C2 r  ( m 1) .
(3.5)
From the symmetries of the Elliptic-Conical coordinate system, which means that for the same 0 there are two
elliptic cones, symmetric about XY plane (see Fig. 1(a)), the velocity component u    / r  vanishes as r tends
to zero. Therefore C2 must be zero and the power m satisfies the condition m  1 . Thus, the velocity potential can
be written as
(3.6)
  C1r m M m   .
Choosing a positive or negative value for C1 does not affect the analysis, but positive C1 gives a convenient
condition at the symmetry plane ( Z  0 ), where the velocity in  direction is in the Z -axis positive direction
(opposite to increasing  direction). The subscript m of the function M denotes the dependency of the solution on
parameter m , or on the spatial semi-vertex angle. The boundary conditions at the symmetry plane (conditions for
the u component and the two options for the v component mentioned earlier) are:
u   C1mr m 1 M m    0
 1
 2   2  1   2  M  
v   C1r
 1
.

 2   2 
 2   2 
M  
v
  C1 (m  1)r m  2
1   2 
1


2
2

r
   
1/ 2
(1)
1/ 2
m 1
m
1/ 2
(3.7)
1/ 2
(2)
Both lead to
1/ 2
m
1/ 2
M m   1  0
M m   1


 2   2 

  2   2  1   2 
1/ 2
1/ 2
1/ 2
 .


 1
(3.8)
From Eq. (3.2), by substituting  2 and  3 and the relations between them, we obtain
 2   2 1   2  M    2 2   2  1 M   m(m  1)  2   2  M  0 .
(3.9)
Equations (3.4), (3.8), and (3.9) define the mathematical problem completely. Usually, only two conditions are
necessary, but since 0 is not defined a priori, and the parameter m is a function of 0 , a third condition is needed.
We should keep in mind that the solutions depend on both  and m , although the function M m is a function of the
variable  only. Therefore, for specified values of 0 (which defines a specific cone and a value of parameter m )
and for a given  we get different values of M m . (It should be commented here that when the parameter m is a
real positive integer, the solutions are reduced to Lame functions22; for non-integer values of m , a generalized form
of the Lame functions can be written23. However, actual evaluation of these functions requires numerical
calculations anyway, so this direction was not followed).
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B. Solutions for non-integer m
In this section we discuss solutions
of Eq. (3.9) for non-integer values of
m as a function of two independent
variables  and  and as a function of
the ellipticity ratio, and show several
examples. For ellipticity ratio e  1 the
constant  tends to zero, according to
Eq. (2.9), and therefore   0 , which is
at
the
limit
of
the
domain
( 0   2   2   2  1 ).
Because
boundary conditions (3.7) depend on the
parameters  ,  ,  and do not compel
any external condition, the solution of
Eq. (3.9) is independent of the freestream velocity. Therefore, the constant
C1 (Eq. 3.6) is derived from the r component
of
the
velocity
equation vr  u  C1mr m 1 M m   ,
which depends only on 0 , by
equalizing the solution from Ref. 11 for
a circular cone and our solution for
e 1.
Equation (3.9) is now solved
numerically using a 4th order RungeKutta method. Figures 2 and 3 depict the
functions M m   and M m   for
several values of m . The value m  1.0
is the asymptotic value for a flat plate,
and is not in the solution domain. The
lowest numerically feasible m is about
1.02, which corresponds to a maximal
half-spatial angle of about 2°. As m
increases, finding a numerical solution is
more difficult: for a numerical accuracy
criterion of 105 , the highest value of m
which a solution can be obtained is about
3.1. Above this value of m the solution
does not converge (this value of m
represents a half-spatial angle of about
70°).
The dashed line in Fig. 2 denotes
  0 with intercepts for specified
values of m , marked by circles. This
line is actually a boundary for the
solution domain and defines specific
cones. The velocity component u (Eq.
(3.7)) for a given cone is proportional to
M m   , so the behavior of M m along
Figure 2. M m   as a function of  for selected values of m and
e  1 ; the dashed line denotes   0 with intercepts for specified
values of m marked by circles.
Figure 3. The gradient M m   as a function of  for selected values
of m and e  1 . The intercepts shown as circles along the abscissa
indicate values of 0 .
the curves of Fig. 2 is the same as for u , apart from the negative sign. The velocity component v (Eq. (3.7)) is
proportional to the gradient function M m . The boundary condition (3.7) compels the curves of function M m   to
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American Institute of Aeronautics and Astronautics
start at value zero at   1 , which is the symmetry plane ( Z  0 ), where the velocity is negative and has only an  component. At the cone surface (boundary condition (3.4)), the slope M m of an M m curve decreases to zero, so the
 -component of the velocity vanishes and the r -component is the one we are looking for, i.e., the potential flow
solution as a function of r m 1 , where m represents the local half-spatial angle  . The boundary condition at the
symmetry plane (3.8) causes all the curves to tend to infinity, M m   1   , as can be seen in Fig. 3. The circles
along the abscissa of Fig. 3 denote specific cones (   0 ), and have a zero value according to the boundary
condition (3.4).
To show the dependency on the ellipticity ratio ( e  b / a ), we take a particular elliptic cone with  0  100 , which
corresponds to 0  0.1736 , where tan  0 is the ratio of the ellipse long semi-axis at any cross-section to the
distance between this ellipse centroid and the cone apex. Figure 4(a) shows M m normalized by the value of M m for
 , as a function of 0   /   1 and for different values of the
circular cross-section ( e  1 ) and is denoted as M
m
ellipticity ratio e . From now on, all variables which are normalized this way (i.e., by the corresponding value for a
 as a function of
circular cross-section cone) are topped with “  ”. Figure 4(b) shows the normalized parameter m
e and  /  . We made the assumption of zero velocity in the azimuthal direction  , therefore the value of
u / C1  mr m 1 M m   for the same 0 and for r  const should be the same. Indeed, as the ellipticity ratio
decreases (and therefore m decreases), M m increases for the same values of 0 and  /  , hence, their product
(representing the potential velocity) remains the same, and an axisymmetric solution with respect to  is justified.
 as a function of  /  and of ellipticity ratio e , (b) m
 as a function of  /  and of
Figure 4. (a) M
m
ellipticity ratio e .
IV. The boundary-layer equations
We now solve the steady, viscous incompressible flow along a cone, after making the usual boundary layer
approximations and assuming no circumferential flow (Appendix B). The two-dimensional continuity equation is:


(4.1)
 su    sv   0 ,
x
y
where the coordinate x is the distance along the cone surface starting from the apex, and coordinate y is the
distance along a normal to the surface (Fig.1(b)). The distance s is the perpendicular distance from the axis of
8
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symmetry to an arbitrary point on the surface, which depends on the shape of the body's cross-section. For conical
surfaces with an elliptical cross-section, the function s  x  is:
s ( x)  x 02
1 2  2  2

 x F 0 ,  ,   .
1 2 1 2
(4.2)
We write s as a function of x because the values of the parameters 0 ,  , which define a family of elliptic cones,
and a value of  are given for each specific cross-section.
The momentum equation is:
u
u
dU
 2u
(4.3)
u
v
U
 2 ,
y
x
y
dx
where the velocity components u and v in the boundary layer are in the x - and y - directions, respectively, and U
denotes the value of u in the region of the potential flow just outside the boundary layer. The boundary conditions
are:
y  0, u  v  0
.
(4.4)
y  , u  U
The solution for the boundary layer equations under rotational symmetry was first given by Mangler17 in the form
described above. If l is the reference length, the transformation formulae read:
x
1
s  x
l2
sin   2 sin 
x  2  s 2  x  dx, y 
y,
l
l 0
s
,
(4.5)
l
1 ds 
u  u,
v  v 
yu  , U  U
s
s dx 
where a bar over independent and dependent variables relates to two-dimensional flows.
For flow at zero angle of attack the dimensionless distance is defined as:
  S  x 

y Ux
x 
,
1
(4.6)
2
where the function S  x  is given by
S  x 
s2 x
x
.
(4.7)
 s dx
2
0
For an elliptic cone S  x   3 , and  is related to m after application of the Mangler transformation, where U α
x  /  2    , as done by Evans11 for two-dimensional boundary layer.
The Mangler-type transformation given in Eq. (4.5) for the elliptic cone becomes:
x3
x  F2 2
3l .
x
yF y
l
(4.8)
The potential velocity U is obtained from the expression for u 0  of the solution for potential flow, by changing
the notation from r to the distance x :
U  C1mM m 0  x m 1 .
After transformation (4.8), this becomes:
9
American Institute of Aeronautics and Astronautics
(4.9)
 3l 2 
U  C1mM m 0   2 
F 
m 1
3
x
m 1
3
,
(4.10)
therefore the parameter  , known as the Hartree parameter, is related to m by
 2
m 1
.
m2
(4.11)
The velocity component in the x direction within the boundary layer is
written as
u  u  Uf    . It
depends on the distance from the
apex of the cone and on the distance
from the surface along a normal to the
surface, (Fig. 6). The equation to be
solved for f   is the Falkner-Skan
equation:
f   ff    1  f 2   0 ,
with boundary conditions:
f  0   f   0   0,
f   1
(4.12)
(4.13)
This equation, as in other boundary
layer formulations, is valid only from
a finite distance from the apex of the
cone. Our solution of (4.12) with
boundary conditions (4.13) depends
on the parameter m and on the
function M m , so the ellipticity ratio
and the azimuthal angle, expressed by
the variable  , are considered.
The numerical calculations are
done for a specific value of
0  0.3420 which represents a cone
with maximum half-spatial angle of
 0  20 . Figure 5 shows the values
of u / C1 as a function of x and
y

C1 / Z 0 cos  0

Figure 5. The normalized x -component velocity profiles, as a function
of the distance x from the apex and the normalized distance from the
surface along a normal to the surface. The dashed line denotes the
boundary layer thickness.
for an elliptic cone of ellipticity ratio e  0.8 ,  0  20 and for   0 . The boundary layer
thickness (the dashed line) is defined as 99% of the potential flow representing the free-stream velocity for the
boundary layer region.
After Eq. (4.12) is solved (numerically) we can calculate the skin friction coefficient from
m  2  12
(4.14)
C f   f 0
Re x ,
2
and the wall-shear stress
w 
1
U 02 C f .
2
By substituting the velocity profile (Eq.(4.9)), the skin friction takes the form
m

m2

C f   f 0
r 2,
2
C1mM m 0 
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(4.15)
(4.16)
where the bar denotes the appropriate two-dimensional solution, as defined in Eq. (4.5). The above equation for the
skin friction coefficient, and therefore the wall shear stress as well, depends only on the coordinate r for a given
elliptic cone. In order to calculate a friction drag force, we integrate the wall shear stress over the cone surface
r0 2
DEC     w  r  s  r  d  dr ,
(4.17)
0 0
where  is a polar angle for the specific cross-section. But since for a specific elliptic cone ( 0  const ), keeping
r  r0  const and varying the variable  for all the range ( 0   2   2 ) defines spatial lines outside the plane
normal to the longitudinal Z -axis (see Fig. 6), we have to adjust our solution for the friction drag for the elliptic
cones with constant length. Actually these lines ( 0  const and r  r0  const ) are the lines of equal potential
velocity, and therefore, equal wall-shear stress lines. It is analogous to circular cones where for a given distance
along the cone,  w  x  const  circular  const for polar angles 0    2 .
cone
Recalling that the solution obtained here is defined for each quadrant, because the  parameter varies between
0 and  , we see on Fig. 6 four generatrices: two for   0 (on the YZ plane) and two for    (on the XZ plane).
Choosing an elliptic cone of specific vertex angle (   0  const ) and length Z  Z 0 , will require to define r
coordinate as a function of the polar angle (  ). An elliptic cross section at Z  Z 0 can be defined by the parameter
:
X  a cos   Z 0
Y  b sin   eZ 0
0
1  02
0
1  02
cos 
.
sin 
The relationship between the polar
angle from the ellipse center  and
the
parameter
follows

Hence,
the
from tan   e tan  .
perpendicular distance from the axis
of symmetry to an arbitrary point on
the
ellipse
defined
as
s     a cos  1  e 2 tan 2  .
Using
Eq. (2.4), for a given elliptic cone
(   0 ), and using the parameter 
we have:
r 2  Z 2 Fr2 ,
(4.19)
where
1
1   2  sin 2   cos 2   .
1  02 
Now, as all the integrands for the
friction drag force (Eq. (4.17)) are
defined as functions of  and Z , we
can derive the drag for an elliptic
cone of a given length. In order to
simplify this expression, we recall
that the conic assumption made for
calculation of boundary layer flows.
Therefore, the ratio of the wall shear
stress for a given elliptic cone is:
Fr2 
Figure 6. Spatial lines for constant coordinates  and r .
11
American Institute of Aeronautics and Astronautics
(4.18)
 w r 
 w  rref

 r

r
 ref




m
2
.
(4.20)
Then, keeping  constant, for instance   0 , allows us to calculate the wall shear stress on   0 ray, as it is
shown in Fig. 6, using the same m and M m for the calculations. Hence, the drag force is
DEC 
 w  rref
where r0 is the radius on the ray   0 for a
given length Z 0 .
The drag force as a function of the
ellipticity ratio ( e  b / a ) is shown in Fig. 7.
Here, the drag force is normalized with the
one obtained for the case e  1 , which
represents the circular cross-section. We mark
 . The solution here
this normalized drag by D
is for  0  20 , or 0  0.3420 , and for a
cone length of Z  5.0 . The minimum drag,
for the same maximal spatial vertex angle and
the same length, obtained for an ellipticity
value near 0.53. Moreover, the elliptic cones
with ellipticity ratio values above 0.265 have
less friction drag than a circular cone. The
behavior of the drag force of the elliptic cone
differs from the behavior of the boundary
layer along an elliptic cylinder, where the
drag force increases as the ellipticity ratio
increases24.
rref m / 2

r0 2
r
m/ 2
s    d  dr ,
(4.21)
0 0
Figure 7. Drag force for an elliptic cone, normalized by the
drag force for e  1 , as a function of e .
V. Error estimation
We now estimate the error of the analytical solution of the previous two sections due to the assumptions made
(i.e., the velocity and its derivatives in the azimuthal direction  were neglected); we compare the theoretical
solutions with full three-dimensional numerical simulation for several elliptic and circular cones.
Using the Elliptic-Conical coordinate system enables us to transform the problem from the physical space into a
simpler mathematical one, in which we could assume that there is no flow in the azimuthal direction (Eq. (3.1)) and
therefore to obtain the expression for the potential flow of the form U  C1mM m   r m 1 . From the Bernoulli
equation, the pressure gradient is
1 dp
dU
U
 C12 m 2  m  1 M m2 r 2 m 3 ,
dr
 dr
(4.22)
and therefore
p   C12 m 2  m  1 M m2
r 2 m 1
.
2  m  1
(4.23)
As explained in Part IV, a fixed value of the coordinate r in the Elliptic-Conical coordinate system does not define
an elliptic cone with a constant given length (see Fig. 6), and we should use the transformation:
1 2
r 2  Z 02
,
(4.24)
1  02  1   2 
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American Institute of Aeronautics and Astronautics
where Z 0 is the cone length. Since we are interested only in the pressure gradient in the lateral direction (  ), the
pressure variation in this direction (using the conical assumption (4.20)) is:
p  Z 0 ,  
m 1
p 
 1  02 1  e 2  sin 2   cos 2   ,
(4.25)
p  Z 0 , ref 
where  is a polar angle parameter. In the same way,
the local velocity (U ) variation is U  p . Figure
p
U
1.0000
1.0000
8 shows the variation of the pressure and the local
velocity along the circumference on each cross section
for the elliptic cone 0  0.3420 , for a half-vertex
e  0.9
angel   20 , for ellipticity ratio e  0.3, 0.6, and
0.9. The variations are periodic since the solution is
symmetric about both axes. The left vertical axis
shows the pressure where the maximum difference is
about 1.5% for e  0.3 , the right vertical axis is for
local velocity variations with maximum difference of
e  0.6
0.85%. The local velocity, and therefore also the
pressure, has extremum points at k  2 , k  0, 1, 2 .
e  0.3
As ellipticity ratio decreases, the error increases. As
the maximum spatial vertex angle increases, the error
 /2
3 / 2 2 
0
increases as well; this is expected, since an increase in Figure 8. Normalized pressure and velocity U
the maximal vertex angle causes a larger deviation
variations (left and right abscissa, respectively) in
from the circular cone. However, even for the elliptic
cone with a maximal vertex angle of 80⁰ the pressure azimuthal direction  for the elliptic cone
variations are less than 12% and the local velocity 0  0.3420 for e  0.3, 0.6, 0.9 .
variations are less than 7%.
Thus, for practical engineering applications, the error is in the 3-5% range. In order to verify this, we performed
several CFD checks. The shape of the body chosen for the simulation is a long elliptic cone. The boundary
conditions are: uniform flow for the inlet condition and zero gradient for the pressure at the outlet (only a quarter of
the flow field was actually solved). Unlike the analytical assumptions, the cone length is finite for numerical tests, as
well as the distance from the body to all boundaries. The solution was performed using commercial package
Fluent®, version 6.3.26, for a steady, laminar,
-5
 w 2 x 10
incompressible flow. In order to compare theoretical and
Theoretical
numerical results for an elliptic cone, the scaling constant
1.8
Numerical
must be found by equalization of the solution about an
1.6
elliptic cone with e  1 and the circular one for the same
1.4
surface area. Thus, the scaling constant for a circular cone
1.2
has to be firstly obtained. In order to do it, numerical
simulations for an axisymmetric case of the cone were
1
performed and the scaling constant was derived from the
0.8
wall shear stress values. Equation (4.14) for the circular
11
0.6
cone, by substitution with the notations used by Evans is:
0.4
n

n2

(4.26)
C f   f 0
x 2,
0.2
2
 A1nGn  0 
0
where A1 , n, Gn , x are similar to C1 , m, M m , r and   cos  .
From the functional form of the shear stress
C f  Const  x  n / 2 matched with numerical results, we
derive the scaling constant A1 . For the elliptic cone, the
scaling constant is derived from the same equations:
0
1
2
3
4
5
Z Figure 9. Wall-shear stress as a function of a cone
length; theoretical and numerical results. The
results are for elliptic cone of ellipticity ratio
e  0.6 , maximum spatial half-vertex angle 20⁰,
Re Z  7300 , for the line of coordinate X  0 .
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American Institute of Aeronautics and Astronautics
2
 f  Ellipt  n
m  2 n Gn  0 Cone
 xref .
C1  A1
(4.27)
n  2 m M m  f 0Circ  rrefm
 Cone 
The reference values for circular and elliptic cones, where the wall-shear stress are equal, are related by the ratio of
square root of the ellipticity. Since the surface area is the parameter we want to preserve between circular and
elliptic cone (in order to compare between the solutions for e  1 ), which is represented by the square of the
reference length characteristics, this correlation is expected. Figure 9 presents the theoretical and numerical results
for wall shear stress as a function of cone length up to Z  5 , along the line of   0 which is for Cartesian
coordinates X  0 . The results are for an elliptic cone of ellipticity 0.6, for 0  0.3420 which represents
maximum spatial half -vertex angle of 20⁰, and Re Z  7300 . The differences between theoretical and numerical
results, both differential properties (like wall-shear stress) and integral properties (like drag force), for this case and
other cases checked, are within the range of 3-5%. There is a difference in the pressure values for polar angles of 0⁰ and 90° (Fig. 8), which causes a lateral flow in addition to the longitudinal. The numerical check for the lateral
velocity about an elliptic cone showed maximum values of 3-4% of the free-stream velocity, and fraction of percent
of the local longitudinal velocity. Due to the periodic behavior of the pressure and the velocity fields (Eq. (4.25) and
Fig. 8), the total lateral mass flux is zero, but the lateral velocities increase the total drag force by about 3-4%. The
errors resulting from all the above, including the approximations for the solution at quadrant edges (since the
Elliptic-Conical coordinate system doesn’t exist for limiting values of the coordinate  ), are, in all cases, less than
5%.
VI. Results
In this section we present results and discussion for different families of elliptic cones and several applications,
such as non-symmetric lifting bodies and wave-riders.
Normalized drag, D Figure 10 presents the drag force as
1.8
a function of ellipticity ratio,
 0 , Z  const ________________
normalized by D  e  1 for several
________________
Numerical
1.6
 0 , S   const cases. In order to define a cone, two
________________
geometrical properties have to be
________________
S Base , Z  const 1.4
defined. As mentioned earlier, the
discrepancy between numerical and
1.2
theoretical results for the drag is <5%.
The dash-dotted line is for a fixed
surface area and a maximum vertex
1
half-angle, so that the cone length
increases as ellipticity ratio decreases.
0.8
Dashed line is the results for the case
of a constant base area and a constant
0.6
cone length (and the spatial vertex
angle
varies
accordingly).
  e  changes linearly with e, having a
0.4
D
maximum at e  1 . This result is not a
0.53
0.263
0.2
surprise since the maximum surface
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
area is for e  1 and when ellipticity
ellipticity ratio, e
ratio decreases the surface area is
Figure 10. Normalized drag as a function of ellipticity ratio. Solid line
diminished. The solid line represents
for maximum spatial vertex half angle (  0 ) and the cone length
the drag force for cone bodies with
both vertex spatial half-angle and cone preserved, stars for numerical results, dash-dotted line for keeping
length kept constant,  0  20 and  0 and surface area constant, dashed line for preserving base area
Z  5.0 . Stars mark the numerical and cone length for all bodies.
results of the same case for ellipticity
ratio 0.4, 0.6 and 1.0.For this case, as long as the ellipticity ratio is higher than 0.263, the drag force on an elliptic
cone is lower than that for a circular cone, with a minimum for e =0.53. The friction-force difference between
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American Institute of Aeronautics and Astronautics
Normalized drag, Dmin circular and elliptic cones at this minimum point is about 6 percent for the specific elliptic cones family
( 0  0.3420 ). We checked other cases as well, including a case for constant volume of the cone (for which as an
ellipticity ratio decreases, the surface area of the cone decreases): there is no elliptic cone with friction force less
than for that for a circular cone.
In order to obtain the
ellipticity ratio of the elliptic cone with minimum drag force
for different families of elliptic
cones, we set to zero the
derivative of the drag force in
equation (4.21) with respect to e
D
(
 0 ). In Fig. 11 the left
e
emin vertical axis is the ellipticity
ratio, and the right axis is the
normalized drag force. The
horizontal axis presents different
families of elliptic cones, defined
eo by the maximum spatial halfvertex angle  0 . Each point on
line emin represents the ellipticity
ratio for a cone which its drag
force is the lowest for a given
cone family (i.e., same  0 ).
Maximum spatial half-vertex angle,  0 [⁰]
Each point on line eo represents
the ellipticity ratio for a cone Figure 11. Ellipticity ratio for elliptic cones with minimum and equal to
from a given cone family (i.e.,
circled cone drag force ( emin and eo respectively), as a function of a spatial
same  0 ) which its drag force is
half-vertex angle up to 40°. The left vertical axis is for ellipticity ratio, the
equal to the drag force of a
 min ).
right one is for normalized drag force ( D
circular cone with the same  0 .
For  0  40 there are no elliptic cones with a friction force lower than that for the circular cone with the same
vertex angle. As the vertex angle decreases, the ratio between the friction force on an elliptic cone with minimum
friction and a circular cone increases.
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American Institute of Aeronautics and Astronautics

D e=
2 0 e = 0.1 e = 0.2
e = 0.3
e = 0.4
e = 0.5
e = 0.6 Figure 12. Normalized drag force for different non-symmetric bodies with the same
maximum spatial half-vertex angle  0  20 and same length Z  5.0 .
Because of the nature of the Elliptic-Conical coordinate system (Eq. (2.4)) the solutions for the potential and
viscous flows were obtained for a quarter of a given elliptic cone. Using this property we can find solutions also for
non-symmetric bodies as shown in Fig. 12.
Here, friction forces for non-symmetric bodies,
(I)
(III) (II) normalized by the value for a symmetric body, are presented
for several cases. The symmetric case is for ellipticity ratio of
0.6 and  0  20 and a cone length Z 0  5.0 . The nonsymmetric cases are made of two halves of elliptic cones, Figure 13. Two main types of wave-rider’s
with the upper part having ellipticity e  0.6 and lower parts configuration: (I) negative dihedral, (II) positive
vary in the range of e  0.1  0.5 . As the lower part gets dihedral, (III) zero dihedral.
shallower, the drag force increases, with a minimum for
 L
e  0.5.
For lifting-body wave-rider configurations we follow the

D
Rasmussen13, where two main types of wave-rider
configurations were defined with cross-sections as showed in
Fig. 13. The lower surface is an elliptic cone and the upper
surface can have one of three possibilities: a negative
dihedral (case (I)), a positive dihedral (case (II)) and a zero
dihedral (case III). In order to calculate the friction force on
the upper surface we divide each side to strips with constant
width and, thus, changing the length in the flight direction,
and apply to each strip the Falkner-Skan20 solution with angle
of attack tan  h / Z 0 . h is the height of the upper
surface (for both cases (I) and (II)) from the zero dihedral
ellipticity ratio, e
line, and Z 0 is the length of the wave-rider. Since the
Figure 14. The differences of normalized drag
optimization of wave-rider configurations is not in the scope
and lift forces on wave-rider configurations as a
of this paper, rather, the influence of various lower surfaces
function of ellipticity ratio.
having elliptic cone shapes, we show in Fig. 14 only the
normalized drag and lift forces due to the lower surface. The case presented here is for the family of elliptic cones
0  0.3420 and for Z 0  5.0 . As we can see from Fig. 14, the lift force monotonically rises as the ellipticity ratio
increases and the drag force has a minimum at e  0.53 as already has been shown in Fig. 10.
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American Institute of Aeronautics and Astronautics
VII. Conclusion
An analytical solution which can estimate lift and drag forces for incompressible flows over elliptical-cone
configurations is conducted. The drag force for a range of ellipticities (ratio of small diameter to large diameter) is
shown to be less than that of the circular cone of equal surface area. The solution is reasonably accurate for elliptic
cones with half-vertex angles up to 20° and ellipticity ratio above 0.4. For half-vertex angles of 20° and ellipticity
ratio of 0.4, the error for the total drag is less than 5%.
Appendix A
The vector operators in the EC system, before applying the transformation (2.3) are given by Ref. 19 as
v    ˆi r

r

1
r   ˆ
2
2

1/ 2

ˆi  2  d 2   c 2   2    ˆi  2 ˆ 2 

ˆ d  
1/ 2
1/ 2
1/ 2

 c 2  ˆ 2 
1/ 2

ˆ

 2
 2
 
  d 2  c 2   2  2    2 2   d 2  c 2    



2
1



 

 2  2 



2
r r r 2  2  ˆ 2  
r


 
2
ˆ
2
2
2
2
ˆ
2
2
ˆ
ˆ


  d    c    2    2   d  c  
ˆ
ˆ 

(A.1)
2
Using the transformation (2.3) we have:

1
ˆ
2
2
v    ˆi r 
 i    
2
r r    2  
1/ 2
1/ 2
1   2 
1/ 2


 ˆi   2   2  1   2 
1/ 2
1/ 2
 

 
 2
 2
 

   2 1   2  2   2 2   2  1  



2
1



 



 2  2 
 2 2

2 
2
r r r     
r

 
2
2
2
2
2
     1     2   2    1  
2
(A.2)
By substituting v   , and assuming a separable solution of the form   R  r  M       :
RM  
 2   2 1   2  RM    2 2   2  1 RM   
2
1
R M   2 2

0
r
r    2    2   2  1   2  RM     2 2   2  1 RM  
where the primes denote differentiation. Then, dividing by R we have

 
 2
  d 2  c 2   2     2 2   d 2  c 2    
1
R 2 R


 

 2

0.
2
2

ˆ


R r R r     
  d 2  ˆ 2  c 2  ˆ 2 
 ˆ  2ˆ 2   d 2  c 2   


 
Because of the independence of the functions ( R, M ), the above equation can be rewritten as:
M 
M
 2
   2 1   2    2 2   2  1  

1


R
R
2

M
M
r2
 2

  ,
2 

R r R
      2   2   2     2 2   2  1   2
1 

  




while the right hand side can be rewritten in the same manner as
M 
M
 2   2 1   2    2 2   2  1   2 2 
M
M
,



2
2
2
2
2
2
      1   
  2    1   2    3




17
American Institute of Aeronautics and Astronautics
(A.3)
(A.4)
(A.5)
(A.6)
and the equations for the velocity potential are obtained as

2
R  R  22 R  0
r
r
 2   2 1   2  M    2 2   2  1 M    2 2   3  M  0 .
(A.7)
 2   2  1   2     2 2   2  1    2  2  3    0
Appendix B
Calculating the drag force requires a definition for area differential ( dA ) as a function of the distance from the apex
and along the axis of the cone ( Z ), and as a function of the azimuthal variable  . The differential displacement
along the cross-section curve at the given Z is defined as:
2
2
 dX   dY 
ds  
 
 d .
 d   d 
(B.1)
Substituting to the latter expressions for X and Y for constant 0 and at the distance r (or x for the local
coordinate system) from Eq. (4.19), leads to
ds 
r

02 
02   2  2
d .
1  2  2  2
(B.2)
The drag force is
Z s (   )
D  4

 w dsdZ
(B.3)
0 s (   0)
and substituting the expression for wall-shear stress from Eq. (4.15) we have:
Z  
D  4
0


0
 f 0
m  2 U 3 x 2 02   2  2
d dZ
0 
2
1 2  2  2
x 
(B.4)
Integral (B.4) cannot be determined analytically, because the parameters m,  , U , f 0 depend on the azimuthal
variable  , but it can be reduced to a single integral by substituting the expression for the potential velocity
U  C1mM m 0  r / l 
m 1
Z  
:
m2
D  4    f 0
2
0  0
C mM
1
m
0  x / l 
x

m 1 3
x

02 
02   2  2
d dZ
1 2  2  2
(B.5)
Also, from the geometry of the cone we can obtain the relation for the angle   cos 1  Z / r  , then
 
8 32 m
1  C1 0  mM m 0   1 2 02   2  2
m2
0 
Z  f 0 
d


3m
2
cos  0 
1  2  2  2
Lm 1
 0
 
3
D
(B.6)
where L is the reference length in the XYZ coordinate system, and cos  0  L / l .
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18
American Institute of Aeronautics and Astronautics
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