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A unified second-order theory for airfoils of arbitrary shape in subsonic flow. The theory is based on recent advances in plane subsonic flow theory and includes second-order similarity rules for surface pressure and compressibility. The document also provides examples of the application of this theory to various airfoils.
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lilllllljjp~fl~yllll[l~~ : — TECHNICAL NOTE 3390
m
Milton D. Vau Dyke
Several recent advances in plane subsonic flow theory are combined into a unified second-order theory for airfoils of arbitrary shape. The solution.is reached in three steps: (^) The incompressible result is found by integration, it is converted into the corresponding subsonic com- pressible result by means of the second-order compressibility rule, and it is rendered uniformly valid near stagnation points by further simple rules. Solutions for a number of airfoils are given and are compared with the results of other theories and of experiment. (^) A straightforward computing scheme is outlined for calculating the pressures on any airfoil at any angle of attack.
Thin-airfoil theory provides a useful first approximation to the incompressible flow past two-dimensional airfoils, and the results can be immediately extended to subsonic compressible flow by the Prandtl- Glauert rule. It is natural to attempt to improve this simple theory by successive approximations so as to increase its accuracy for thicker air- foils and higher subsonic Mach numbers. (^) There results a series expansion of the flow quantities in powers (supplemented in some cases by logaritlxus) of the airfoil thickness ratio, camber ratio, and angle of attack.
For incompressible flow, the higher-order theory has been studied by various writers, in particular Riegels and Wittich (refs. 1 and 2) and Keune (ref. 3). A less straightforward series of approximations was developed by Goldstein (ref. 4). (^) Perhaps the most concise exposition of higher-order incompressible thin-airfoil (ref. 5).
For subsonic compressible flow, the undertaken by G&tler (ref. 6)... followed
theory is given by Lighthill
corresponding analysis was first by Hantzsche and Wendt (refs. 7 and 8), Schmieden and Kawalki (ref. 9), Kaplan (refs. 10 and 11); and Imai and Oyama (refs. 12 and 13).1 (^) These investigators treated only spe. cific simple shapes by rather laborious analysis. (^) Later, it was discovered ll?hesehistorical references are intended to be representative rather than exhaustive.
b solution for any airfoil. The reader interested only in calculating a specific case, without necessarily understanding the theory, can turn Q (^) directly to the section “PRACTICAL NLIMERZCAZCOMI?UTA!J?ION”on page 19.
FYom the preceding remarks it is clear that the solution is reached in tlu?eesteps. First, the formal second-order incompressible solution is found by intepyation. (^) Second, this is converted into the correspond- ing subsonic compressible solution by means of the second-order compress- ibility rule. Third, this is modified near stagnation points by the appropriate rules for round or sharp edges. These three steps will be considered successively.
Formal Incompressible Solution
Accordingly, consider an airfoil of moderate thickness and camber at a moderate engle of attack to a uniform subsonic stream (sketch (a)). It is essential that the x axis be chosen to pass through both the lead- ing snd trailing edges. Let the upper and lower surfaces of the airfoil be described by
Y(x) = C(x) t T(x)
(1) Ai- Y’ (^) Sketch (a)
where C(x) describes the mesn camber line and T(x) the thickness. The airfoil extends over the interval ASx~B, which is usually conveniently taken to be either -15x51 or O~x~l. All symbols are defined in Appendix A.
First-order solution.- In the first approximation of thin-airfoil theory, the condition of tangent flow at the airfpil surface is imposed e, on the two sides requires that
4
of the chord line y = O rather than at the surface, and
Y’(x) T=
=C’(x) *T’(x) (^) ( y=o
a The corresponding horizontal velocity disturbance on the chord line, which is required for calculating the surface pressure, consists of a term
associated with the airfoil thickness, and snother associated with its r csmber and angle of attack. (^) For the thickness —
Ult 1 f
‘T’(E) d~ T=~ (^) -E Ax. and for the csmber and angle of attack (^) .+
The latter result is due to Munk (ref. 25) and the former was apparently first given by Squire in a paper that is still not generally available. Cauchy principal values are indicated in each integral.
The surface speed is then given to a first approximation by .> Q Ult^ U1c
Second-order solution.- In the second ~pproximation, the tangency^ L condition is transferred from the airfoil surface to the.chord line by Taylor series expansion. The condition on the second-order increment in vertical velocity is thus found to be
V.Q
I
= C’2(X)+ T’2(x)
y=o
where
Ult CJX) =Tc+yT
T2(x) =~T+~C I
(We depart here from Lighthill’s notation ~n order to emphasize that the functions C2 and T2 are effectively the camber and thiclmess for some .— fictitious airfoil.) The problem is identical with that in first-order theory except for the condition at infinity, which is readily disposed of. Thus, corresponding to T2 is the increment in horizontal velocity^ -. (^) &J
the airfoil ordinates at”certain pivotal points multiplied by standard influence-coefficients. The details of this method, as adapted to thin- airfoil theory, sre given in Appendix C.
The numerical computing pro- ~ , J cedure is outlined.in the last section of tQis paper.
Second-Order Compressibility Rule
The secQnd-order counterpart of the Pr@idtl-Glauert compressibility rule is implicit.in an extension o.ftransonic similitude that was initi-
Imai soughtto improve the transonic similarity rule by retaining in its derivation all terms proportional to the square of the airfoil thiclmess except one appearing in the condition of taggent flow at the surface. The correlation of experimental data was uo~ appreciably improved, which led him to suggest that the neglected second-power term should also be included. This probably cannot be done. However, in attempting merely to reproduce Imai’s result as smnownced before publication, Hayes actually included that term in a second-order rule @r surface pressure.
Hayes’ result is that for two-dimensicmal subsonic or supersonic flow the ratio of the second-order to first-order pressure term on the surface is proportional to the parsmeter
(II -:21)3/2 (^) [ 7: 1~4 : 2(1 -Ma) (^) 1 (11)
where T is some measure of the thickness, camber, or angle of attack. Now at subsonic speeds the first-order pressure term is related to its value in incompressible flow by the Pradtl-Glauert rule. Combining
In incompressible flow the secgnd-order surface-pressure coefficient has the form
where the first-order term Cpl contains linear terms in thickness, cem- ber, and sngle of attack, end the second-order increment ACP2 contatis their squares and products. Then for the ssme airfoil in subsonic com- pressible flow, according to the compressibility rule, the pressure coef- ficient is
-.
..
. n-
e
.— .-.
—
.
C% =KICp= + K2(~p2) (1$3)). .
where
4 f J It has been pointed out that the formal thin-airfoil series requires modification near stagnation edges. The modification must in general be performed on the speed rather than the pressure. Hence the compressibil- ity rule for surface speed is required. It is readily found from the above rule for pressure by considering the small-disturbance series form of Bernoulli!s equation for compressible flow. Thus it is found that if the surface speed ratio in incompressible flow is
then at subsonic speeds
qM l.+KIA#+K*&+KZ - ()
u 2U 2 u
with
(lsa)
(lsb)
K= -1. M= (7 + l)M= + 4~ 2 8 j
This rule is seen to lack the fundamental simplicity sure.
Modification for Stagnation Edges
of the rule for pres-
Thin-airfoil theory is known to fail near leading and trailing edges if there is a stagnation point. The flow is actually brought to rest, but thin-airfoil theory predicts infinite speeds instead. If r is the dis- tance from the edge, the velocity contains powers of r-llz for a round edge and for any leading edge with.flow around it (associated with angle of attack), and powers of in r for a sharp edge. First-order theory contains first powers of these singularities, second-order theory their squsres, and so on, so that the formal thin-airfoil series ‘divergesin some neighbwhood of the edge. Not only are the velocities aud pressure incorrect near stagnation edges, but nonintegrable singularities appear in the higher-order expressions for aerodynamic forces.
k is at least a i Riegels found
first
approximation to the flow disturbances nesr the edge. a rule ~& first-order theory by considering the conform~l mapping, and Lighthill found a rule for second-order theory by considering a contraction of abscissas that shifts the thin-airfoil solution by half the radius of the edge.
In reference 20 corresponding rules have been developed for higher approximations, shsrp edges, and subsonic compressible flows. (^) The tech- nique used there is to consider the exact solution for some simple shape that approximates the airfoil In the vicinity of its edge. The ratio of the exact solution for the simple shape to its formal thin-airfoil series expansion serves as a multiplicative factor that corrects the series expansion for the actual airfoil. The result should then be simplified insofar as possible. The relevant rules will be summarized here; the details sre given in reference 20.
A round-nosed airfoil can be closely approximated by a parabola whose axis coincides with the initial camber line (sketch (d)). The exact solu- tion for incompressible flow past the parabola (resolved into stresming and circulatory components) leads to the following rule that converts the formal second-order solution “~^ for surface speed into a uniformly valid approxima- (^) A tion
g=
Here ~ is the abscissa measured from the edge into the airfoil, p is the edge radius, A is the initial angle of csm- (^) ~ her, and the *..signs refer, as usual, to the.upper and lower surfaces. Sketch (d)
This rule yields a uniform second approximation to the disturbances everywhere (except at the other edge, where additional modification may be required) if the rate of change of curvature of the profile is con- tinuous, which means that near the edge the thickness has the form
T(x) = bl&+b&031z+... (15)
However, airfoils of the NACA four- and five-digit series violate this requirement, their leading edges hawing the initial form
Hence the rule yields only a first approximation near the edge (while
leaving a second approximation elsewhere) snd can therefore be replaced’ by the simpler form — 6 ““
which is Ltghthill’s rule.
The rule for first-order theory is obtained by dropping the term p/4x. from equation (17). However, it is then advsmtageous to use the alternative form due to Riegels (ref. 2), which is correct to the same. order but a great deal more accurate (see r=. 20). It is simply
where q is the angle of the airfoil surface. Analogous rules can be found for the second-order theory, but their probably outweigh their slight advantages of accuracy and they will not be considered here.
alternative shortcomings A simplicity, so — .: The modification for a sharp edge is ~.md by considering inc?mpress-
y<
ible flow in an angle. If the edge is a trailing edge with Kutta condi- tion enforced, or a leading there is no flow around it,
edge at the id~al angle of attack, so that the second-ordE$ rule is ---
—.
where 5 is the semivertex angle, and “~ is the first-order solution. Otherwise, the circulatory part of the formal second-order solution, which ‘“ consists qf the terms singular like X.‘~12, must first he^ corrected 6ep- arately by the rule
Airfoils with two stagnation edges can be treated either by applying ‘ the appropriate correction separately at each edge, or by combining the rules. The combined rule for two round edges is given in equation (24)
of reference 20. Simtlarly, for a round edge at x = -1 and a sharp edge -
(with Kutta condition) at x = 1, the combined rule is G
angle of attack of the actual airfoil measured from the Ideal angle at which the stagnation point coincides with the vertex. It must be found as the coefficient of XO-112 in the first-order solution
w ‘C$yu..
The function Q^ is not known exactly, but a satisfactory approxima-^
tion is given by the Janzen-Rayleigh solution in powers of M2. For the
. corresponding to the ideal angle of attack
where a = O, the solution has been calculated to order M4 by Imai (ref. 23). (^) It is tabulated briefly in reference Xl for 7 = 7/5, where it is denotedby Q(x/p, M). For other angles of attack, the function (^). Q to order M2 can be extracted by a limiting process from Kaplan’s solution for an inclined ellipse (ref. 29), which gives> with ZO/P = Qj a/~ = *
‘(q’’M)‘ [**
(1- IF)J5$-$(2Q+V’)+
=@@ [( (^) ) (
-**+@ lnl-+l@-2q- 1-
The rule for shsrp noses in subsonic flow can be found by considering compressible flow in an angle. However, this basic solution is not yet available. For practical purposes the correction is probably negligible since it is appreciable over a much smaller neighborhood of a sharp edge than a round one. Moreover, sharp edges he usually trailing edges, in which case the details of the flow are altered by viscous effects.
AND OTHER TwlmRIEs
Incompressible Flow
It has been seen that the solution for subsonic flow depends on that for incompressible flow. It is therefore pertinent to test the second- order theory in the case of incompressible flow, where it can be checked against the exact results of conformal mapping.
.
w
Ellipse.- val -- 1 .
Consider am ellipse of thickness ratio T with the inter- as chord line. It is described by
Suppose that the Kutta condition is satisfied - the rear stagnation point coincides with the end of the major axis. Then the first-order solution for surface speed is found, from equations (3), (4), and (~), together with Appendix B, to be
Proceeding with
equations (6) to (9) gives the formal second-order result
r
1 -x (^) JT2_ X F
-x 1 a= ‘Tfa 1+X (^) 1- X2 ‘aTE-z (m)
This can be checked by expandin& the exact result, which
The formal second-order solution clearly breaks down near the ends of the ellipse. It is converted into a uniformly valid second approximation by applying equation (14) twice in succession, or using the combined rule of equation (24) of reference !20,which gives
These approximations are compared for an 18-percent-thick ellipse (which an NACA 0012 airfoil) at zero angle of
in figure 1 with the exact solution has nearly the same nose radius as attack. The precipitate descent of the formal second-order solut~on towsrd negative ‘fifin~tyis Just dis-
.
.
.
(30a)
approximation, the terms combined equally sharp
valid second the rule for
In deducing from this a uniformly independent of a are treated by edges that was described just after equation (21), with 5 = 2T. The terms 1. a are modified according to equation (~) tith X. = 1 + x. (Notice that no modification of these terms is required at the trailing edge.) The result is
2T
{
~=(1-xz)-1+;T12-(1+ x)~(l+x)-(1-x)~(l-x)] +
()[
2 ;T 3 -y (l-x2)- 3(l+x)ln (l+x) -3(1-x) ln(l-x)+
&l+3x)(l+x)lnw+x) +;(l-3x)(l-x)lnw- x)+.
;[2(l+x) ln(l+x)-(l+2x)ln (1-x)-k] }
( 30b)
These approximations we compared in figure 3 with the exact solution (ref. 31) for a circular-src airfoil 18 percent thick at zero angle of attack. Although the vertex angles are large in this example, the modifi- cation of the second-order solution is appreciable in such a small neigh- borhood of the edge that it would be invisible even on a much lsrger plot.
NACA 00XX airfoils.- Symmetrical (such as the NACA 0012) are naturally
. The airfoil of thickness ratio^ T^ is
airfoils of the NACA 0~ fsmily defined for the interval O ~ x < 1. described by (ref. 32)
b6xs + bsX4), o~.<1^ (31)
where
, (^) bl = 1.
b~ = -I. be = 1.421s b8 = -0.
With the aid OY Appendix B the first-order
T.
— . —
.
— solution is found to be
( 32a) (^) —
in agreement with the result given by Goldstein (ref. 26). Applying Riegels’ rule (eq. (18)) renders this a uniformly valid first approxima-
tion except very near the trailing edge.
The second-order terms in thickness, in addition to being very com- plicated, involve integrals that apparently cannot be evaluated in terms
of tabulated functions. Accordingly, the second-order terms have been calculated using the Germain-Watson-Thwaites-Webernumerical method dis- cussed in Appendix C, with N = 16. The accuracy is assured by the fact that cruder approximations results only slightly, as will be seen in a later
The formal second-order solution for surface form
of this approximation ‘
modify the numerical example.
speed therefore has the
+#=l+@++++& J ~~a-$a
where values of QT from equation (32a) and approximate numerical values
l x (^) QT QTT Q~~X
.05 1.836 .-3.35 (^) 5.55. .10 1.’714-1.00 3.25. .20 1.510 -.09C 1.65^. ;% ;.::9 X&C 1.00.
. 0 .58^.
o:% -0. -. .485 -. .238 -.
%a
0. . -. -. -. -.
First-order theory (or Prandtl- Glauert rule applied to exact incompressible value of 1.
K&rm&-Tsien rule
Second-order theory
Third-order theory
1.
1.
1.
Here the K&rn&-Tsien rule has been applied to the exact incompressible value of.the pressure coefficient, and-the speed ratio then calculated from Bernoulli’s equation. It is to be anticipated that second-order theory is more accurate than any of the couipressibilitycorrection for- mulas such as the K&m&n-Tsien rule, because it allows for a dependence on the particular airfoil shape and on the value of y. This is seen to be true for the ellipse.
In the seineway the second-order solutions are readily calculated. for the Joukowski and the biconvex airfoilsl and are found to agree tith the results that Hantzsche and Wendt obtained by laborious analysis. (^) .- —- NACA 0012 ai.rfoil.-The formal first- and second-order solutions for NACAOOXK airfoils in subsonic flow are eatiilyobtained from equations (13) and (32). The second-order solution can then be rendered uniformly valid near the nose using equation (22), although again the modification is sig- nificant in only a very small region of the nose.
For the NACA 0012 airfoil at zero angle of attack, Fmmons has calcu- lated the flow field at Mach numbers of O, 0.70, and 0.75 using the numer- ical relaxation method (ref. 35). The last of these Mach numbers is super- critical, so that the flow contains shock waves, and is beyond the scope of the present theory. The pressure distribution calculated by the relax- ation method for M = 0.70 is compared in figure 5 with the results of first- end second-order theory and various other approximations. The relaxation soluti.cmfor incompressible flow is also shown in comparison with Goldstein’s “exact” solution, and is seen to be inaccurate near the nose. (^) The solution for M = 0.70 probably contains similar inaccuracies, however, just as for the ellipse the pressure coefficients calculated by second-order theory may be slightly less negative thsn the true values near their minimum.
Experiments on NACA 0015 airfoil.- Experimental pressure distribu- tions in two-dimensional flow over the NACA 0015 airfoil at high subsonic speeds exe reported in reference 36. For zero ‘&gle of attack; the critl- ‘“ cal Mach number is approximately 0.70. The measurements at this Mach num- – ber are compared in figure 6 with the results of first- and second-order c
theory and of the two to the incompressible nately, the model was
19
common compressibility correction formulas applied flow values tabulated in reference 34. Unfortu- imperfectly constructed, and the ordinates were inaccurate nesr the nose and midchord. Otherwise, the measured pressures are in satisfactory accord with either second-order theory or the re”sults of the Karm&n-Tsien rule.
PRACTICAL -ICAL C@lPUTATION
The following computing procedure yields the formal second-order subsonic solution for the surface speed or pressure on any airfoil at my angle of attack. It requires a knowledge only of the airfoil ordinates at seven points along the chord. It is based on the foregoing theory together with the numerical method of Germain, Watson, Thwaites, snd Weber that is discussed in Appendix C.
Computing Procedure (N = 8) .
(1) Tabulate the ordinates Yu snd Yz of the upper snd lower sur-
. (^) faces at the seven pivotal points Xn listed in table I. (The x axis must pass through the leading and trailing edges.)
(2) Calculate the
T1=- 2
corresponding
(Yu - Yz),
values of
c S* (YIJ+Yz) (^) (35)
(3) Using the influence coefficients of tables 11 and 111, calculate
7 T! = z
ensT6 c!’= z
fnsCs s=1 S=l
T!” = I
gnsTs c“ =^ z
hnscs S=l S=l
(36)
