The aerodynamic characteristics of wings with leading-edge vortex separation were predicted using a method based on a flow model with free vortex elements which are allowed to merge into a concentrated core. The calculated pressure distribution is more accurate than that predicted by methods with discrete vortex filaments alone. In addition, the computer time is reduced approximately by half

Lan, E.

Pao, J. L.

English

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(NASA-C$-162530) THE QUASI-VORTEX-LATTICE	 NOO-14052
METHOD FOR VINGS PITH EDGE VORTEX SEPARATION
Final report (Kansas Univ. Center for
Research, Inc.) 25 p HC A02/MF A01 CSCL 01A
	 Unclas
G3/02 46449
THE UNIVERSITY OF KANSAS CENTER FOR RESEARCH, INC.
2291 Irving Hill Drive—Campus West 	 s
Lawrence, Kansas 66045 	 s
THE QUASI-VORTEX-LATTICE MM FOR
WINGS WITH EDGE VORTEX SEPARATION
by
Jemw-Louh Pao and C. Edward Lan
CRINC-FRL-385-1
January, 1980
The Flight Research Laboratory
The University of Kansas Center for Research, Inc.
The University of Kansas
Lawrence, Kansas 66045
A Final Technical Report
Prepared under NASA Grant NSG 1537:
"Investigation of Optimum Wing Configurations with Edge Separation"
for
Langley Research Center
National Aeronautics and Space Administration
Uin 
this 
investigation, the quast-vortex-lattics, method is used to
predict the aerodynamic characteristics of vlt-'& with leading-edge vortex
separation. The method is based on a flow model with free vortex ele-
amts which are allowed to merge Into a concentrated core. The calculated
pressure distribution Is more accurate than that predicted by methods with
^'	
discrete vortex filaments alone. Tji addition, the computer time is reduced
approximately by half.
1
i
-	 g
Lff
hi
1. Introduction
Recently, lifting-surface methods have been applied to the calculation
of wing aerodynamics with edge vortex separation. Two popular methods are
of the doublet-panel type (Refs. 1 and 2) and the free vortices (Refs. 3 and
4). The panel method can predict accurate results. However, the computing
time involved is lengthy and it has not been extended to treat complex
geometries, such as wings with strakes and interacting surfaces. On the
other hand, it is easier to extend the method of free vortices to complex
geometries. However, the predicted pressure distribution is not accurate.
In addition, the leading-edge Kutta condition is not exactly satisfied
in the method of Ref. 3.
In Ref. 4, the quasi-vortex-lattice method was used, together with
discrete vortex filaments to model vortex flow effects. The leading-edge
Kutta condition is exactly satisfied and partial vortex separation is
allowed. But the predicted AC  distribution is more "diffused" than the
data show. This is because the effect of a concentrated vortex core in
the real flow cannot be accurately represented by a number of free vortices.
The objectives of this investigation are to improve the pressure predic-
tion of Ref. 4 by introducing a concentrated vortex core and to reduce the
computer time.
2. Description of Present Method
For convenience,Mehrotra's method (Ref. 4) will be called vortex-
filament model and the present method the core model. In the present
analysis, the basic assumptions are: (1) the wing is represented by a
bound vortex sheet, across which there exists a pressure difference;
r
(2) the separated flog along leading-edges is represented by forcrfree
leading-edge vortex elements which feed vorticity into a concentrated
core. During the iteration process, the following boundary conditions are
Imposed on both vortex-filament and core models: (1) The wing surface =at
be 3apesasahle; (2) gutta conditions are imposed along the leading- and
trailing-edges of the ping; (3) in the vortex-filament model, force-free
condition., is applied on the leading-edge vortex filaments and trailing
wake elements; (d) in the core model, force-free condition is applied on
the free vortex elements, concentrated core and trailing wake elements.
	
't.	 The computational procedures of the present method are as follows:
(1) Establish the leading-edge vortex filament_system.^rtth Iiehrotra's
method in two iterations.
i
(2) Find the centroid of the established leading-edge vortex filament system.
Note that to calculate the centroid of the leading-edge vortex filament
System, a series of cross flaw planes are considered, proceeding from
the wing apex toward the trailing-edge. The centroids of the vortex
f
filaments penetrating these planes are given by
4
Y	 ^,
.j
r^
where yyj, 
z  
are the centroid position in the jth plane,
*ij, 2 i are the intersection position of the i th vortex filament
with jth plane,
ri is the circulation around the ith vortex filament,
r j is the concentrated line vortex strength at the jth pie,
	
_.	 c
nj is the number of vortex filaments penetrating the j th plane.
2
r,
(3) Allow the isading-edge vortex elements to merge and feed vorticity
into the core through connecting segments.
(4) In order to have reasonable starting solution so that computer time
can be reduced, the initial a coordinates of concentrated core from
step 2 are modified to take the experimental values summarized by
t-	 Smith (Ref. S).
zv
 = 0. ts4 w t 0. 1
z V n 0. lass a + 0.1s8
a < 1.2
RZ G a c t. 2	 (3)
r
:i
2v s o. 13 a f 0.132
where,
= 4W" a l( A/c,) 
	 (4)
A is the aspect ratio, c  the root chord, and z  is the nondimensional
z-coordinate of the concentrated core, referring to the local semispan.
(S) During the iteration process, the new locations of the free vortex
elements, concentrated core and trailing wake are determined by satis-
fying the force free condition. Note that the force free condition
is not applied on both the feeding segments and the free sheet along
the very first vortex strip. The new locations of free vortex
elements and the core along the first vortex strip are obtained by
linear interpolation.
(6) From the starting solution of the core model, the core is forced to
move downward and outward twice, to obtain the force gradients. In each
case, relaxation paramters ( ay , A z) for determining new core coordinates
are assumed empirically as follows (See step 9):
Cts	 ^iK
second time
to
300
ms=s 
s — o.o2S'• .
 
	{T)
_
K
71	 (8)
^k
In sq. (S), Z% is the total force acting on the core in the s direction
and EF is the total force.	 In Sq. (6), CL is the initially calculated
lift coefficient and CL	is the lift coefficient by suction analogy.
-^
s
•
Note that A
	
and A z are assumed constant along the entire core.
y
-' (7)	 In the next iteration, A
y 
and Az of each segment of core are computed,
based on the difference of segment force in the proceeding two
Iterations.
-
" Cy; -	 —'Fa 1	 Cry (= o, r ^g	 (9)1 Fsz ^	 !^	 ^	 1	 Xs^
CX
_ : ^ FVJ
where the signs of Cyt and 
Czi 
are the same as those of F z2 and Fy2,
respectively.	 The physical idea is that the core should be moved in
the direction of reducing force magnitude.
U
(S) The new positions of free vortex elements in each iteration are
determined by the following equations,
^1 ^,
	
(11)C4X!
	 Vx
	
vs 	(12)
x
where Vx, Vy, VE are the total velocity components in the respective
coordinate directions. The new slopes of each segment of free vortex
elements are
(:tL / raw = S. (iz), + 4 t	 (13)4x oid
rfaz 1	 aZ 1^.^ o. ^ °Z-	 (14)
Since the length of each segment is conserved, therefore
as • eX ( t ^.AY 2. ^ 4z	 (15)
ox + x
S	 (16)
1+ rrAV 1s + 4Z
4;	 x W
It follows, that
ay	 (17)C°Y)new (4 X),
.,, X ^,aw
(4Z^*sw = ^4x^++sW ` Xft 	 (18)
S
6
Assume that
V	 VV ^tV3x
Yr
Yx	 V;
L Combining Eqs. (23). (25) and (26), (26) yields the following equation
for the ith segment:
f
F:[ 	 o	 ;r_{ 
X /ohl	 r(Ax (27)
F.
ta ' 	 es	 y ^t •x +x-1 	 +^ 	 riiVxl (28)
Then new slopes of ith segment of concentrated core are determined by:
^j	 rrr -1	 s	 + —	 y o	 f'4►x /rww	 r^l•x^ ^^	 Y  (29)
(	 r ^s 1 '^
*eM
(30)
Combining Egsi (27) through (30) results in
`
GY—x)%W
	 y;i
e; 	 eE	 Fyj	 (32)( 
J	
(
^
X ^y l •x eld 'f' A s r4^
,, Vxj
Since the segment length is conserved, it follows that
as s ax 1+ (It )16♦ 
x !	 (33)
7
,S
per	
I+ {t
son",
	 `
I
(. Y)%W in {Axe,, • (*).
Ax
aw 7x
Thus, the now positions of concentrated core are found as follows,
X4 wsw ! X,C f °X,O,	 (37)
Y4%awe
Ix
 yti t Y*ew
(10) The now positions fo trailing wake in each iteration are determined
in the sans manner as described in Ref. 4.
Note that the segment length is preserved along the free vortex
avant, concentrated core and trailing Wake. In addition, the
affect of the secondary vortices has not been included in the present
analysis.
30 Numerical Results
To show the improvement, those cases shows in Ref. 4 were re-calculated
and compared in Figures 1 through 12. The following observations can be
made:
(1) For the over-all aerodynamic characteristics (Figures 1 through S), the
results from both methods are similar, swept that the present results
are slightly more accurate at high angles of attack.
r
c
r:
(33)
(36)
Ev
9(2) Thera are significant improvements in the predicted AC p
 distribution,
In particular, at moderate to high angles of attack. Diffizoulty exists
at 1r,v angles of attack, because free vertex shvets from Mehrotra's
modal are relatively flat and the initial configurations in the present
model are more difficult to determine.
In most cases, final results can be obtained with two iterations in
Mehrotra's model and three iterations in the present model. The computer time is
approximately reduced by half as compared ddtth Ref. 4.
4. References
(1) Weber, J. A., ,rums, G. W.. Johnson, H. T., Lu, P. and Rubbert, P. ht.,
"Three-Dimensional Solution of Plows over Wisp with Leading-Up
Vortex Separation," AIM Journal, Vol. 14, April 1976, pp. 519-525.
(2) Johnson, F. T., Timoco, 1. 11., Lu, P. and Lpton, M. A., "Secant
Advances is the Solution of Three-Dimensional Finn over Wings with
Leading Up Vortex Separation," AIAA Paper 79-0252, 1979.
(3) Kandil, 0. A., Mock, D. T. and Nayfeh, A. No, "Nonlinear Prediction
of the Aerodynamic Loads on Lifting Surfaces," Journal of Aitcraft,
Vol. 13, Jas. 1976, pp. 22-28.
(4) Mahrotra, S. C. and Lan, C. L., "A Theoretical Investigation of the
Aerodynamics of Low-Aspect-Ratio Wings with Partial Leading-Up
Separation," NASA CR-145304, January 1978.
(5) Smith, J. N. 3. 0
 "Improved Calculation of Leading-Up Separation
from Slander. Thin. Delta Wings," Prot. Roy. Soc. A. 306, pp. 67-90,
1%8.
(6) link, P. T., "Wind-Tunnel Tests on a Slander Delta Wins at Sigh Inci-
dosce." Z. nugwissenschaften. Jahrg. 4, Heft 7, pp. 247-249, July 1956.
f'
10
t L(7) VM92 9 W. 5. 9 "Wfocts of lo@WlaCawsCm	 on Lm -Sp oed Characteristics
of Slender Delta Wep t " WASA CR-2002 j, 1972.
Lit	 (8) Bml*tto 0. L. and VIW O R. J., "spertuntal Investigation of
Influence of Zdge Us" on the Aerodymmmic Characteristics of low
Aspect-Ratio VImp at Low-Speeds," Jousal of Asromautietal sciescoo
VOL. 22, go. 6, pp. 317-533, Aug. 1955.
(9) Sisk, F. T. and Taylor, J., "Sam Low Speed Imporiviests, with 20 dog.
Delta, Vlaps" ARC R i M 3469, 1%7.
(10) Mars4es, D. J., Simpson, R. V. and Raimbird,, V. J., "Tbe now over
Delta Wimp at Low Speeds with L&Winrgdp Separation," The
College of Aeronautics, Cranfield, Sept. No. 114, Feb. 1958.
0.
-0.1
CM
-0.'
1.4
r-
1.0
0.8
^i
Cl
t 0.6
0.4
i
_ 0.2
0.
,
O	 Experiment (ref. 6)
--- — --- Present method
Mehrotra I s method (ref. 4}
11 , I	 1
,
,
,
,
It
,
itiiIt
,
lit
It
I ^
,
- 
►T - ^ 1	 1	 1	 1 f
1
1
U
0	 5	 10	 15	 20	 25	 M
oc (degree) ..
Figure 1. Variation of lift and pitching moment
(about 0.25 C) coefficients with angle of attack
for AR - 0.7053 delta wing
_	
z
12
0.
CH -0.1
-0. e
1.4
1.2
1.0
CL	0.8
0.6
0.4
0.2
0.
0
III
I
1
I
Q	 Experiment , (ref. 7 )
_—:Present method
Mlehrotra's method (ref. 4')
I
,.
1
I	 ;
^ I
••^T
1
1
_	
_ I
. i
1 1 1
1 ^
I
- i
1 1
I
i
1 I 1	 1
-
I
if
-
I1 ,
It it, t it till
_ i ^_1	 - I F• I i 1
i I I
r I
I
r I
I	 I.:
I 
5	 10	 15	 20	 25	 30	 3
oC (degree)
Figure 2. Variation of lift and pitching moment
(about 0.25 f) coefficients with angle of attack
for AR = 1.147 delta wing
13
1.!
1.4
1.2
1.0
CL
0.8
0.6
0.4
0.2
0.
Experiment (ref. 8)
-_^~__-•Present method
Mehrotra's method (ref.
T'
r
^	 1
i
t
•
1
i
i
1
1
^
1
i
.' 1
f
1
1
;
- 1 ,
t
;
0.	 5	 10	 15 20	 25	 An
cG (de,ree)
Figure 4. Variation of lift coefficient with angle
of attack for AR = 2.0 delta wing
14
-0.4
-0.!
0M1
-0.1
-1.0
-1.2
-1.4
-1.6
0.
-0. 2
1S
r
L 1	 1.+
0	 Experiment (ref. 8 )
---•-----	 Present method
--	 Mehrotra's method (ref.
s	 10	 15	 20	 25 30
oC (degree)
Figure S. Variation of pitching moment (about apex)
coefficient with angle of attack for AR - 2.0'
delta wing
16
TT-rE1]Er]=
r-1 I Ll
Experiment (ref. 9)
method
Mehrotra Is method (ref.4
1 R 1 R ...... ..........
x/%	 0.4583
t -r-'V I 1-1-r 11"T
I I LT I I I--
I	 i	 I	 I I	 I	 i I	 I I	 I f	 I	 I I-!-! 'T Lj-
1117!1111 A 44 i..OT--
1 1	 f
hEij
I	 I
I	 If
'1A
I	 T
I T	 f
X/CR 	 0-9166
+1	 1 1 1 T I
10:
U. 0.	 0.2	
1 
0.4 
2y/b(J- 
6	 0.8	 1.0
figure 6. Pressure distribution for
AR - 0.7053 delta wing at 10.0 degree
angle of attack
v
1.0
0.6
AC P
	
0.4
0.2
0.
1.0
0.8
0.6
AC P
0.4
0.2
3.
2.
l .1
0.
2.t
1.0
0.
0.	 0.2	 0.4	 0.6	 0.8	 1.0
2y/b(x)
Figure 7. Pressure distribution for
AR • 1.147 delta wing at 10.2 degree
angle of attack
17
t	 ACP
3.
2.
1.1
0•
2•a
1.0
0•
0.	 0.4	 00•0
AC 
O	 ExpeMment (ref. 7)
-------- Present matted
14throtra's method (ref.
t
i
i
x/CR = 0.5333
i
r;
i
i
i^
i
i
i
i
i
x/CR = 0.9333
i
e
s
18
2y/b(x)
Figure 8. Pressure distribution for
AR = 1.147 delta wing at 20.4 degree
angle of attack
AC 
P
1.0
2.0
ACP
1.0
0.
dd2-
0.
0.	 0.2	 0.4	 0.6	 0.8	 I.0
2.0
I'll	
A	 I3*0
.	 .	 . .	 . .	 . .	 .	 .
. .	 . . . .	 . .	 . .	 .	 . .	 .	 . .
L-L-W" ,I I L I
. .	 . .	 .	 . .	 . .	 .	 .
L .	 .	 . .	 . .	 .	 . .	 . . .
4	 1	 1 1	 1 W 444 1	 1	 1 11
...........
X/CR a 0.5333	 1 1	 f	 I I
I	 I	 I	 I	 I	 I	 I	 I	 .	 I	 I	 F	 T	 f	 F	 I I	 1	 1 1
I	 I	 i 	 I	 F	 T	 I	 I	 I	 I	 I	 j 	 I	 I	 I	 j 	 I	 I	 I I	 I	 I I
r- I	 I	 I	 I F FI I	 i	 i	 I I I I U-T f	 I
—	 ^	 ^	 I	 I .	
-	 .	 ,	 .	 I M F I	 I
,,fit ...	 ...........
Experiment (ref. 7
------ Present method
Mehrotra's method (ref.
I I	 I	 I	 T	 i0pr I	 I	 I	 I	 t	 a I	 I	 I	 a
0.9333
19
Zy/b(x)
Figure 9. Pressure distribution for
AR - 1.147 delta wing at 30.7 degree
angle of attack
20
0.
0.	 0.2	 0.4	 0.6	 0.8
	 1.0
2v/b(x)
ACP
1.0
0•
0"
2.0
3.0
7 T7-1
Experiment (ref. 10)
-------- Present method
 M ehrotra's method (ref. 4
i
+
t
r
^)	 nM CR	0. X67
a
+
X/CR • 0.8333
4-
.^
-
-.
law-
21
I
2.t
ACP
1.0
Figure 11. Pressure distribution for
AR - 1.4559 delta wing at 19.1 degree
angle of attack
0	 Experiment ( ref .1 A)
Present method
11-hrotra's method (ref.
r;
x/CR
 • 0.6467 `
LA	 ^._
x/CR • 0.6333;.
* -
+...
1.0
QUA ^
0.
22
0.	 0.2	 0.4 2y1b(%f	 0.8
	 1.0
Figure 12. Pressure distribution for
AR • 1.4559 delta wing at 23.9 degree
angle of attack


The quasi-vortex-lattice method for wings with edge vortex separation

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