A short tutorial in the application of topological ideas to the intepretation of oil flow patterns is presented. Topological concepts such as critical points, phase portraits, topological stability, and indexing are discussed. These concepts are used in an ordered procedure to construct phase portraits of skin friction lines with oil flow patterns for a wing-body combination and two angles of attack. The relationship between the skin friction phase portrait and planar cuts of the velocity field is also discussed

Fearn, Richard L.

Yates, Leslie A.

English

NASA Technical Reports Server

3 1176013444261 
CR 4168 19880016959 NASA- -
NASA Contractor Report 4168 
The Determination of the 
Topological Structure of Skin 
Friction Lines on a Rectangular 
Wing-Body ~Combination 
Leslie A. Yates and Richard L. Fearn 
GRANT NAGl-368 
AUGUST 1988 
NI\S/\ \ \11\\\'\ '\11 \\\\ 111'\ \\11\ 1'1\\ 1111' \1'1 \11\ NF01802 
https://ntrs.nasa.gov/search.jsp?R=19880016959 2020-03-20T06:48:32+00:00Z
NASA Contracto.r Report 4168 
The Determination of the 
TopologicallStructure of Skin 
Friction Line~s on a Rectangular 
Wing-Body <:ombination 
Leslie A. Yates and Richard L. Fearn 
University of Florida 
Gainesville, Florida 
Prepared for 
Langley Research Center 
under Grant NAG 1-368 
NI\SI\ 
National Aeronautics 
and Space Administration 
Scientific and Technical 
Information Division 
1988 
Abstract 
A short tutorial in the application of topological ideas to the interpretation of oil flow patterns is presented. 
Topological concepts such as critical points, phase portraits, topological stability, and indexing are discussed. These 
concepts are used in an ordered procedure to construct phase portraits of skin friction lines from oil flow patterns 
for a wing-body combination at two angles-of-attack. The relationship between the skin friction phase portrait and 
planar cuts of the velocity field is also discussed . 
Introduction 
The determination of the topology of skin friction or limiting flow lines can facilitate the interpretation and 
discussion of separated flows. The report "Topology of Two-Dimensional and Three-Dimensional Separated Flows" 
by Tobak and Peake provides an extensive treatment of the practical applications of topology to separated flows 
(Reference 1). This report augments the Tobak and Peake report and ptovides a step by step procedure for 
determining skin friction lines from oil flow photographs. Futhermore, a relationship between the skin friction 
lines and two-dimensional calculations is discussed. 
Rules for Construction of the Phase Portrait 
The skin friction, (Tan 1'y), defines a continuous vector field upon a body's surface. Curves which are everywhere 
tangent to the skin friction vectors are called skin friction lines and satisfy the equation 
At any point (x,y) on the surface where the magnitude I(j)f the skin friction is nonzero, the slope of the skin friction 
line is uniquely determined, and only one line passes through (x, y). If 1're = 1'y = 0 at a point (xo, Yo), the slope 
of the skin friction line is undefined at (xo, yo), and more than one skin friction line may pass through (xo, Yo). A 
point at which 1're = 1'y = 0 is called a critical point; a curve upon which 1're = 1'y = 0 is a critical line. Lighthill 
(Reference 2) provides a clear and extensive treatment of continuous vector fields and critical points. 
In determining the topological structure of skin friction lines on any three-dimensional body, care must be taken 
to insure the continuity of the vector field constructed from the skin friction lines and to insure the structural stability 
of the phase portrait (pattern of skin friction lines). For the vector field to be continuous, structures such as the 
following 
------------ C 
-c 
t t t t t 
must be ruled out unless C is a critical line. 
A phase portrait is structurally unstable if small perturbations in the flow parameters produce structural changes 
in the phase portrait. A structural change occurs when the initial pattern of skin friction lines cannot be continuously 
mapped onto the pattern of skin friction lines for the perturbed state. In other words, if the initial skin friction lines 
are represented by rubber bands, these bands cannot be stretched or deformed in any manner to give the pattern 
of the perturbed state. The perturbed state can only be obtained from the initial state by cutting and re-gluing 
some of the bands. For example, consider the nonsimple critical point illustrated below. Small perturbations in flow 
parameters can change this pattern of skin friction lines to a pattern in which no critical points occur or to a pattern 
which has two nodes and two saddles. 
nonsimple critical point stable patterns 
Phase portraits with only simple critical points (nodes, spiral nodes, and saddle points) and no saddle-saddle 
connections are structurally stable. The types of simple critical points are illustrated below. 
node spiral node 
(focus) 
saddle point 
Once the continuous vector field has been constructed from the skin friction lines and the critical points have 
been identified, the topological rules for three-dimensional bodies can be used to verify the results. For a simply 
connected body, the rule is 
"DN - "DS = 2, 
where "DN is the number of nodes and "DS is the number of saddles. This rule must be applied to the entire surface 
of the body. Since only a portion of the vector field is usually available from flow visualization studies, it is often 
not possible to use this rule. 
A second method which can be used for verification of the pattern of skin friction lines is indexing. Let the region 
2 
y 
S bounded by the curve 1 be the area of interest. As 1 is traversed in the counterclockwise direction, the angle 
which the vector field makes with respect to a fixed vector varies. The index I is given by 
I == /:)./21'1, 
where b. is the total change in the angle. The topological rule for the region 5 becomes 
lJN -lJS == I. 
If there are regions 51, ... , Sn contained in 5 for which the vector field cannot be determined because the flow is 
weak or time varying, it is possible to remove these regions as follows. A curve Ci is drawn such that Si is entirely 
contained within Ci and the vector field is defined at every point on Ci' An index Ii can be determined for the 
region enclosed by Ci. 
As an example of the indexing procedure, consider the following representation of the crosshatched areas in 
Figure 1£. 
At each point on the closed curve drawn about the crosshatched area, the direction of the skin friction lines is known. 
As the curve is traversed in a counterclockwise direction from A to B, the angle which the skin friction line makes 
with respect to the vector at A changes continuously from 0 to 1'1. From B to A, this angle changes continuously 
from 1'1 to 21'1. The total change is 21'1, and the index for this region is 1. 
Once the index I and indices Ii have been determined for S and the Si, the topological rule governing the number 
of nodes and saddles in the region 5' which excludes the Si is given by 
This rule can then be used to check the topological interpretation of the skin friction lines when only a portion of 
the skin friction field is available. 
3 
Step by Step Procedure for the Construction of a Phase Portrait 
In this section, the procedure for constructing phase portraits of skin friction lines on a rectangular wing-body 
combination is demonstrated. The oil flow photographs which define the skin friction lines were obtained from a 
series of experiments conducted by S. Kjelgaard and W. L. Sellers at NASA Langley Research Center. In their tests, 
Kjelgaard and Sellers used a wing with an aspect ratio of 7.0. The wind tunnel dynamic pressures ranged from 10.5 
and 21.5 psf, and the angle of attack ranged from 6° to 22°. In Figure 1, the step-by-step construction of the skin 
friction phase portrait from the oil flow pattern for the wing-body combination is shown. Figure la presents the oil 
flow pattern for the wing-body configuration at an angle-of-attack of 20°. 
Step 1 (Figure Ib). We begin with a region which the direction of the flow is known. For this example, we are 
confident that the flow at the trailing edge in the region shaded with the fine dot pattern is in the reverse direction. 
The skin friction lines are shown in this region with their direction defined by the direction of the flow near the wing 
surface. 
Step 2. The vector fields for the wing tip regions are determined in Figure Ie. Since our aim is to construct a 
continuous vector field with no structural instabilities and hence, no critical lines, the directions of the skin friction 
lines in the regions shaded by the medium dot pattern are automatically defined by the adjacent fine dot patterned 
region. Note that in the wing tip region, there appear to be lines of attachment which feed the wing tip vortices. 
Step 3. The vector field for the center of the wing is determined in Figure ld. Again, the directions of the skin 
friction lines in the region shaded by the coarse dot pattern are automatically defined by those in the adjacent fine 
dot pattern regions. Even though two separate fine dot pattern regions are connected through the coarse dot pattern 
region, no discontinuities or critical lines occurred in this construction. If they had, it would have been necessary to 
reconsider the assumptions of step 1. 
Step 4. The vector fields for the regions near the leading edge are determined in Figure Ie. These regions are shaded 
with slash pattern. At the junction of the fine dot patterned regions and slash patterned regions on the left wing, 
the skin friction lines behave as follows: 
111/ 
If the skin friction lines in the fine dot pattern region connect with those in the slash patterned region at a finite 
angle, there is either a discontinuity in the vector field or a structural instability. Presumably, the skin friction lines 
in the fine dot patterned regions asymptotically approach a separation line which is parallel to the skin friction lines 
in the slash patterned region: 
Separation line 
The resolution of the oil is not fine enough to observe this behavior. 
In the forward section of the wing, there are areas in which the skin friction lines are not well defined by the oil. 
These will be indexed out. 
4 
Step 6 (Figure 1£). In the final step the nodes, spiral nodes, and saddle points are identified. Nodes and spiral nodes 
are points where many skin friction lines converge. Saddle points are points which are avoided by the skin friction 
lines and demonstrate the behavior illustrated previously. Both crosshatched areas have an index of 1. These areas 
behave as nodes of attachment and have been identified as such. 
To check the above results, a curve, "Y, is drawn just inside the wing edge. This curve avoids all critical points 
since the direction of the vector field is not defined at a critical point. The index of the region interior to the curve 
"Y is found as follows. 
y 
B c 
In the above representation of the curve "Y. and the skin friction vectors along "Y, points A, B, C, and D on "Y were 
chosen so that the skin friction vectors at th~~se points would be aligned. From A to B; the change in the angle which 
the skin friction makes with respect to the vector at A is 21r. From B to C, the skin friction at first goes through 
an angle change of -1r and then change of 11". Hence, the total change in angle is O. From C to D and D to A, the 
change in the angle is 21r. Therefore, traversing "Y in a counterclockwise direction gives a total change of 61r and an 
index of 3. The topological rule for the region interior to "Y is 
EN - ES = 3. 
In Figure 1f, the number of nodes inside "Y is 6, the number of saddle points is 3, and the topological rule is satisfied. 
A topological evaluation of the same wing-body combination at a = 16° is included in Figure 2. In this case, 
either the reversed flow region (shaded with fine dots) on the left hand side or the attached flow region on the right 
hand side (shaded with a slash pattern) could have been used for the beginning step. The reversed flow region was 
chosen so that the step by step procedure outlined above could be followed. 
Comparison to Two Dimensional Calculations 
If the three-dimensional flow field about the rectangular wing-body combination is known, the phase portrait of 
the velocity components in any plane cutting through the wing can be constructed in a manner similar to that used 
for skin friction lines. Since this plane contains a wing cross section and only in-plane velocity components, half 
nodes,N', and half saddles,S', may appear at the wing boundary, and saddle point to saddle point connections are 
allowed. The topological rule which must be satisified by this portrait is 
1, 1, EN + - EN - ES - - ES = -1. 
2 2 
Although the full three-dimensional flow field is not available from oil flow photographs, the direction of the flow 
near the wing surface can be determined from the skin friction lines. In a plane cutting through the wing at line D 
in Figure 1, the skin friction lines indicate that near the wing the in-plane velocity components are in the upstream 
direction at the leading and trailing edges. There is a small region in which the velocity components are in the 
5 
0 
c 
N 
" tS 
a 
.j...l 0 
ttl M 
C:::~ 
c: 
~ .j...l 
t.. 
ttl ~ 
r-- til :;:, 0.. ~ 
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ttl 
LC') 
.j...l r--
U N QJ 
t.. 
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0 
0-
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0 
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0 
or-
Lt... 
6 
,----
I 
I 
Attachment line 
ID 
Figure 1 b 
Figure 1 c 
Figure 1 f 
7 
-, 
I 
I 
I 
-- -~ 
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Attachment line 
0 
~ 
...... 
(j 
~ 
CO 
O"l 
....... 
0'1 
~ I.,C") 
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,...... 
::l II 
O"l 
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8 
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N S Figure 2f SN 
9 
...... 
o 
Figure 3. Phase portrait for the velocity fielrl calculated by a two 
dimensional thin layer Navier-Stokes corle. 
~ I=-_~--'------------
downstream direction. 
In two-dimensional calculations by Anderson, Thomas, and Rumsey (Reference 3) using thin layer Navier-Stokes 
equations for a NACA 0012 wing at a = 18°, the flow has a large recirculation region. Unpublished data from 
Rumsey of NASA-Langley Research Center indicate a small bubble at x = 0.3c within this recirculation region. 
Figure 3 shows the phase portrait for the two-dimensional solution which used the maximum number of iterations 
allowed in Reference 3. This phase portrait satisfies the topological rule stated above for a plane cutting through a 
three-dimensional body. 
N ear the airfoil surface, the two-dimensional flow at the leading and trailing edges is in the upstream direction. 
In the bubble, the flow is in the downstrea.m direction. This agrees qualitatively with the behavior observed along 
the line D in Figure 1. 
It should be noted that the skin friction lines indicate that there may be a large component of the velocity which 
is parallel to the leading edge. This component and its effects are not included in the two dimensional model. As 
a result, planes which cut through other sections of the wing need not give flows near the surface similar to that of 
the two dimensional calculation. 
CONCLUSIONS 
The use of topological ideas aids in the interpretation of the flow about three-dimensional bodies. For the 
rectangular wing-body combination studied, this approach readily identified a reverse flow region at the leading 
edge and the locations of flow separation and atta.chment. Comparison of the flow direction at the wing's surface 
determined from the skin friction lines and a two dimensional thin-layer Navier-Stokes code showed favorable 
agreement for certain regions of the wing. 
REFERENCES 
1. Tobak, M. and Peake, D. J.: Topology of Two-Dimensional and Three-Dimensional Separated Flows, AIAA Paper 
79-1480,1979. 
2. Light hill , M. J.: Attachment and Separation in Three-Dimensional Flow, Ed. by L. Rosenhead, Oxford Univ. 
Press, 1963, pp.72-82. 
3. Anderson, W. K., Thomas, J. L., Rumsey, C. L.: Application of Thin Layer Navier-Stokes Equations near 
Maximum Lift, AIAA Paper 84-0049, 1984. 
11 
I\U\SI\ 
Nallonal AerCYlau!CS arc! 
Scace J\Qmlr'usrrall()n 
1 Report No 
NASA CR-4168 
Report Documentation Page 
2 Government Accession No 3 RecIpient's Catalog No 
4 Title and Subtitle 5 Report Date 
The Determinatlon of the Topological Structure of Skin August 1988 
Frlction Lines on a Rectangular Wlng-Body Comblnation 
6 Performing Organization Code 
7 Author(s) 
Leslle A. Yates and Richard L. Fearn 
9 Performing Organization Name and Address 
Unlversity of Florlda 
Dept. of Englneerlng SClence, Mechanics, and Aerospace 
Engineenng 
8 Performing Organization Report No 
10 Work Unit No 
505-60-11-03 
11 Contract or Grant No 
NAGl-368 
Gal nesvlll e, FL 32612 13 Type of Report and Period Covered ~----------~----~~--------------------------~ 12 Sponsoring Agency Name and Address 
National Aeronautlcs and Space Administration 
Langley Research Center 
Hampton, VA 23665 
15 Supplementary Notes 
Langley Technlcal Monltor: Scott O. KJelgaard 
16 Abstract 
Contractor Report 
14 Sponsoring Agency Code 
A short tutorlal ln the applicatlon of topologlcal ideas to the lnterpretatlon of 
011 flow patterns 15 presented. Topological concepts such as critical pOlnts, 
phase portralts, topologlcal stablllty, and indexing are discussed. These 
concepts are used in an ordered procedure to construct phase portraits of skln 
frlctlon lines from oil flow patterns for a wing-body comblnation at two angles-
of-attack. The relatlonship between the skin friction phase portralt and planar 
cuts of the veloclty field is also dlscussed. 
17 Key Words (Suggested by Author(s)) 18 Distribution Statement 
Topology; Separated Flows; Oil Flows; 
Surface Flows 
19 Security Classlf (of this report) 20 Security Classlf (of this page) 
Unclasslfled Unclasslfled 
Unclassified - Unlimited 
Subject Category 02 
21 No of pages 22 Price 
16 A01 
NASA FORM 1626 OCT 86 For sale by the NatIOnal Techmcal InformatIOn ServIce, Sprmgfield, Vlrgmla 22161-2171 '1ASA-Langley, 1988 
End of Document 


The determination of the topological structure of skin friction lines on a rectangular wing-body combination

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